CHAPTER 1: INTRODUCTION

Transcription

1 CHAPTER : INTRODUCTION. Backgroun to the Thesis A etaile esign, manufacture an testing of an etening boom was mae within a teaching company scheme between Brunel an Niftylift in 999 []. The close, pentagonal steel tube was selecte to provie the require strength an stiffness with internal capacity for the hyraulic piping an cables. Rees an Joyce [, ] selecte a close, pentagonal steel tube section to resist local buckling whilst limiting the en eflection to a value pre-etermine from the weight of two men an their baggage containe within the cage at the en of a fully etene boom. Typically, from a trailer platform vehicle with stowe height of. m, the etene telescopic beam can reach a vertical height 7. m an provie a horiontal outreach of 8.7 m when subjecte to an en loa of 00 kg. The self-propelle version offers slightly improve telescoping length imensions: vertical lift: m, horiontal reach m, couple with the ae avantage of a moveable base []. A recent British Stanar has recognise that strength, stiffness an stability are issues concerning safety in these elevate platforms []. An initial analytical moel of the eflection profile uner the etene outreach conition in these vehicles was evelope from applying a strain energy metho (i.e. to a steppe, tubular, cantilever beam []). Deflection eperiments were conucte upon a full-scale, three-boom cantilever in which the en fiings attempte to match those arising within the knuckle-en, the pivot pin an ram fiing attachment positions for the vehicle. Aitional strains were measure with gauges bone to outer surfaces at positions along each boom where local concentrations arose from cut-outs for cabling an in areas of contact between ajacent lengths. The testing showe that whilst the analytical moel provie for the global eflections with goo accuracy, further work woul be require to account for the effect of the local eformation behaviour upon the esign. Moreover, the self-weight of steel booms ae to eflection an loa carrying capacity to an etent that the performance of alternative, lighter materials was eeme worthy of investigation. The present proposal is to eamine these aspects in more etail especially in contet of an optimum esign. It is epecte that commercial software packages such as ABAQUS/CAE will be employe for this purpose. The reliance place upon these packages emans that they be accompanie by valiating analytical solutions where the latter are available. The

2 early work has showe that energy methos incluing Castigliano s theorem an the principle of virtual work may be use for checking isplacements but only at specific points along the booms. More recently a continuous isplacement function has been sought from applying Macaulay s metho to a simplifie two-boom, tubular square-section laboratory cantilever uner en loa []. The principle of successive integration from curvature to give slope an from slope to give isplacement was applie to the scale-own laboratory moel. Normally, Macaulay s metho works only with a uniform cross-section but here an allowance for the changing section area is require. The piecewise scheme evelope here was first applie to a three section telescoping cantilever beam assembly which amits three ifferent secon moments of area: those within the two boom regions plus one overlap region, when the compatibility between rotations an isplacements at their interfaces is ensure. This technique was then applie to a two section cantilever assembly, which is the caniate assembly eamine throughout the course of this thesis. Surface stresses an strains may be calculate from classical mechanics, allowing for the changing section an bening moment with length. Displacements an longituinal surface strains were measure at various length positions with minimum an maimum overlap. This provie the isplacement an strain profiles eperimentally so enabling a comparison with the theory. While the eflecte shape is continuous, the strain profile is interrupte at the overlap as we might epect from its strengthening influence. The theory reprouces the eperimental behaviour well enough for it to be use as the valiating tool escribe above. Where the theoretical preictions are overlai to reveal error there may be a nee for further refinements to be mae as require. Buckling is a relatively new problem that has been ientifie in the two-boom section cantilever. There is an urgent nee to evelop a theory an marry it with eperimental preictions to account for these phenomena that is common amongst such sections. As oppose to the eflection an stress behaviour which are alreay well ocumente in stuent projects unertaken at Niftylift [, 6, 7, 8], buckling poses a ifficulty in that, consierable resources woul nee to be evote in terms of the creation of a test rig an suitable apparatus to moel real life working conitions. Both global an local buckling theories, especially those propagate using the energy methos are of vital importance, an these theories must be aapte to the assembly. A means of verifying the same must also be evise.

3 . Statement of the Problem Given that telescopic sections are fining increasing applications in those instances where space is at a premium an reuction in weight an material cost is a necessity, it is vital that a esign theory an subsequent methoology is generate wherein the behaviour of the telescoping assembly is accounte for, in terms of its possible application areas. Despite numerous avances over the ages in the fiel of mechanics, not to mention increasing use of telescoping sections all aroun us from simple househol appliances to comple retractable staium roofing, the lack of literature in the public omain with regars to telescoping sections is a eterrent to the further use of this simple yet highly effective structure. This thesis aims to she light on the uniqueness of the telescoping assembly an its etaile analysis an bring it forth into the public omain as there is an absence of material pertaining to telescoping structures in the wier engineering contet as compare to the information that is available in the private sector omain.. Aim an Objectives Aim: To provie a esign methoology to estimate stiffness relate properties in telescopic cantilever beams as use for lifting evices. Stiffness refers to the loa - isplacement behaviour for which the relate properties are to inclue beam geometry, material ensity an strength. Thus, for eample, when minimising weight of the telescopic beam structure use as a lift an reach evice, its stiffness an loa carrying capacity are to be preserve. Objectives:. To stuy known eflection, buckling an stress analyses methos an aapt them to thin-walle, telescopic tubular sections using analytical an numerical techniques. The techniques to be eamine inclue Macaulay s fleure equation, Mohr s moment area metho, energy methos incluing Castigliano s theorem, the principal of virtual work, the Rayleigh-Rit metho an moelling with finite elements,. To buil a specific analytical moel most suitable for the preiction of eflections stress istribution an buckling loa characteristics in this structure,. To valiate the moel propose through the conuct of eperiments on scale beam structures,. To apply the moel to case stuies incluing crane an elevating platform structures,

4 5. To isseminate preicte an eperimental results through publication. Analyses an selective laboratory eperiments are to be conucte to meet the esign specification. The theoretical analyses are to envelop both analytical an numerical approaches. The eperiments will employ scale moels for laboratory testing to appraise each theory. Applications will then be mae to commercial evices that employ the telescopic beam structure.. Summary Finings The iterature review on Classical Mechanics ientifie the three major areas of investigation into the telescoping arrangement as follows. i. Design of the telescoping sections for their many applications often requires estimates of eflections at various length positions. The evelopment of analytical methos for estimating eflection an stress for beams in bening were evelope in the 8 th century by Euler an Bernouli an are escribe in many tetbooks [5-8]. The beam eflection is foun by four common methos: (i) irect integration [5-8], (ii) Macaulay s step function [8, 9], (iii) Mohr s theorems [-] an (iv) strain energy [-]. Despite uniform section beams being well-serve by the classical theory they are less often use for eflection analyses of variable section beams incluing tapere, steppe an telescopic esigns [9, 0]. Application of classical theory an its aaptation to the telescoping arrangement is to be eamine. ii. The nature in which buckling occurs within the telescoping cantilever assembly is to be establishe, as: (a) a local buckling prouce within the iniviual rectangular hollow sections an (b) a global buckling wherein the structure in its entirety unergoes buckling. In the former, transverse shear an torsion in iniviual rectangular hollow section members is given special consieration because of their ability to sustain a constant shear flow irrespective of wall thickness (unlike an open section). However pure fleure is only possible when shear forces act at the shear centre of a member. A combine loaing scenario is eamine whereupon the telescoping arrangement is subjecte to bening, torsion an shear. Consiering the latter, principles of application of energy methos are scrutinise an the means by which they can be aapte to the telescoping arrangement are stuie in etail. The

5 telescoping assembly in its entirety is eamine as a cantilever-column in that one en is fie an the other is free. In orer to arrive at the critical buckling loa of the assembly it was imperative to unerstan the concepts of bening an potential energy systems. Base on the finings from the literature, a three-pronge methoology was evelope to match the propose theory with the results obtaine from eperimental work conucte as well as from Finite Element Analyses..5 Structure of the Thesis Chapter outlines the literature that was ientifie to have a vital role in achieving the summary finings presente above in.. Chapter proposes the Tip Reaction Moel as applie to the two section telescopic cantilever beam assembly an provies the etaile tip eflection analysis of the same by the four common methos namely; Direct Integration, Macaulay s Step Function, Mohr s Theorem an Strain Energy Principles. Deflection preiction techniques have been aapte to the two section telescopic cantilever beam assembly. A C-program evelope using a irect integration technique, has been generate an is currently in use in inustry. Chapter provies the buckling methoology wherein, by working from first principles, a better unerstaning of the compleity of the problem was attaine. This involve a etaile unerstaning an application of energy methos to the case at han. What was referre to by Timoshenko as the energy metho is use as a base an applie to the cantilever column as etaile in [5]. This approach in turn is verifie by valiating the result obtaine against the criteria for the Euler critical buckling loa for a column having one en fie an the other free as its bounary conitions. A general form for preicting the critical buckling loa eactly is thus erive an is applie to suit each case iniviually taking into account the ifferent cross sectional secon moments an lengths. Chapter 5 eamines the phenomenon of shear, torsion an a combination of both in the iniviual rectangular hollow sections that comprise the overall telescoping assembly. The concept of constant shear flow in the walls of the sections as well as the importance of the shear centre is ientifie an applie to the structure. Importance is lai on the esign 5

6 optimisation of the hollow rectangular constituent beam sections of the assembly, wherein the longer limbs of the cross section, can be ajuste to raise the stress to a preetermine esign stress value, thereby saving weight, allowing for greater efficiency in loa bearing an simultaneous buckling at the loa limit. Chapter 6 escribes the bening an shear stress analysis of the two section telescoping assembly when it is subjecte to both in-line an offset loaing. Thus combine loaing is eamine whereupon the telescoping arrangement is subjecte to bening, torsion an shear. Stresses generate are plotte graphically in which the accompanying analyses performe here originate from the secon of two publishe papers, emanating from this work [9]. As such the results obtaine are verifie. Chapter 7 provies in ample etail the FE analysis that was applie to the two section telescoping arrangement in orer to obtain the tip eflections for ifferent loaings, the stresses generate as a result of subjecting the telescoping arrangement to both inline an offset loaing an finally the etermination of the critical buckling loa. The results thus obtaine were use to verify the theoretically etermine eflection, stress an buckling moels. Chapter 8 etails the eperimental work that was conucte on a two section telescoping cantilever beam an the results obtaine for eflection an stress analysis as well as buckling loa etermination. The eperimental section eamines the variations in eflections an stresses for varying governing parameters, especially the effect the change in overlap length has on the overall strength of the structure. Also of importance is the stress analysis wherein the strain gauges reaings are converte to their equivalent principal stress magnitues an compare with their theoretically erive counterparts. The test beam itself was subjecte to both in-line an offset loaing in orer to mimic the possible loaing configurations of the telescopic assembly in its working environment. The test rig evelope is the closest approimation possible to an encastre fiing. Chapter 9 compares an iscusses the results obtaine from the three-pronge approach as applie to the two section telescoping cantilever beam assembly an raws conclusions, whilst also making recommenations for future work. 6

7 Appeni A etails the Tip Reaction Moel as applie to the two section telescopic cantilever beam assembly. Appeni B outlines the C program that uses the Macaulay s theorem erive eflection equations to calculate an preict the tip eflections of the two section telescopic cantilever beam assembly, for given configurations. A similar program written an evelope for the three section telescopic cantilever beam assembly is in commercial application. Appeni C outlines the in-line loaing inuce stress analysis of the caniate assembly. Appeni D analyses the offset loaing inuce stress analysis of the two section telescoping arrangement. Appeni E is the first paper publishe as a by-prouct of this thesis. It is entitle The Telescopic Cantilever Beam Assembly: Part Deflection Analysis. Appeni F represents the secon paper publishe, entitle The Telescopic Cantilever Beam Assembly: Part Stress Analysis. Taken together, parts an provie an analytical theory for bening of a iscontinuous beam that i not eist heretofore, thereby obviating the nee for a numerical solution. Appeni G outlines in ehaustive etail the FEA methoology an proceure for eflection, stress an buckling etermination, for the given assembly. Appeni H etails the strain gauging principles, an application techniques that were use to conuct the stress analysis of the telescoping assembly. 7

8 CHAPTER : ITERATURE REVIEW. Introuction This thesis has its origins in Abraham s issertation entitle Establishing a Methoology for the Finite Element Analysis of a Comple Boom Assembly for Estimating Stress Hot Spots an Deflections, [8] which focusse entirely on the Finite Element Analysis (FEA) of the HR5N telescopic boom assembly manufacture by Niftylift, a renowne manufacturer of access platforms, both trailer-mounte an self-propelle. FEA was performe on the aforementione telescopic boom assembly uner the action of concentrate an istribute loaing in orer to observe, ientify an interpret the results of combine loaing. At the time of unertaking of the project very little was unerstoo about the egree of compleity an the nature of moelling the boom assembly in FEA. The software moule use was ABAQUS/CAE an although Niftylift being manufacturers were aept at analysing the assembly the results generate were not entirely unerstoo. The project starte out initially as a means of verification of eperimental results using FEA of the boom assembly but ue to the ifficulty of full-scale testing, the former was limite an eventually the project became entirely software riven. Although the compleity of the boom assembly analyse in this project an those past [6-8] is much greater than the two section telescoping cantilever assembly great pains has been taken in this instance to introuce an apply well-known an age ol concepts to a new structure an therein lies the novelty or the contribution to knowlege. The theories reviewe here are by no means novel or unique but have been establishe through the ages. However in ealing with a new structure, a starting point was neee an the sections that follow aim to follow on from this objective. At the outset, the author wishes to epress that iniviual symbols an coorinates for each of the theories eamine through the course of this chapter, have been aopte from their original sources. 8

9 . Curvature Bening Moment Relationship The Curvature-Bening Moment Relationship etaile below is the basis an start of many a tet on Soli Mechanics [5 8]. The objective of this section is to allow the reaer to see behin the thought process an reasoning behin the chapters that follow. A X O rig in a l S h a p e B y Y O θ+θ θ RR ϕ A B C s D D e fle cte S h a p e Figure.: Beam in Bening (Aapte from [0]) The beam AB, lying along the -ais, is loae with positive bening moment M. The bening moment causes the beam to eform into an arc of raius R. Accoring to the sign convention, the centre of curvature O will then be above the -ais. A short element of beam CD of length s subtens angle about O, where s R.. The change in angle at the centre of curvature, O, must equal the change in graient of the tangent, hence. In moving from point C to D in the irection of positive, an taking into account the sign convention: s R. R., or. R s Sagging curvature is taken to be positive an is efine by a raius of curvature R with centre at O. Since the eflection of the beam is small compare to the raius of curvature it can be approimate that s an hence. R It can be assume that for small eflections the slope Y tan such that: 9

10 Y R But M I y E R (.) where M is the bening moment about the neutral ais R is the raius of curvature of the beam σ is the stress inuce as a result of the bening moment M y is the perpenicular istance to the neutral ais Y is the eflection inuce E is the Young s Moulus of Elasticity of the beam material I is the secon moment of area about the neutral ais M Equation (.) yiels the relation R EI. Rearranging the terms yiels: M EI Y which requires two successive integrations with constants introuce to match the bounary conitions in orer to epress the eflection Y in terms of the un-eflecte position.. Macaulay s Step Function Metho With successive integration of the fleure equation (.) iscontinuities will arise in the moment epression when passes points of concentrate forces, moments an abrupt changes in istribute loaing. If Equation (.) is applie to regions in where M() is continuous, the greater becomes the number of ifferential equations an integration constants to be solve. Fortunately the amount of work involve may be lessene when M() in Equation (.) is replace with a step function M(-a) as outline in [0] M Y (.) a EI 0

11 in which a efines positions at which iscontinuities arise. Equation (.) permits successive integration of a single step function as is applie to the three section telescoping cantilever beam as shown in []. The following rules are to be kept in min. (a) Take the origin for at the left-han en. (b) Where necessary eten an counterbalance uniformly istribute loaing to the right-han en. (c) et concentrate moments lie at the position [-a] 0. () Establish the function M[-a] in the furthest right portion of the beam. Sagging moments are positive. (e) Integrate such terms as [-a] in the form a. These terms are to be ignore when, in substituting values for the value of the bracket [] becomes negative. (f) Apply the known slope an eflection values to fin the constants of integration.. Mohr s Moment Area Theorems The moment area metho is convenient in case of beams acte upon by point loas in which case the bening moment area consists of triangles an rectangles. In the case of istribute loa the etermination of the position of the M-iagram s centroi itself involves integration an as such it no longer remains simpler than Macaulay s metho. However, this metho may be use in certain stanar cases of istribute loa where the position of the centroi of the bening moment area is known. This section shows the reaer the way in which the author unerstoo the moment area theorem. The manner in which it has been applie to the same to the two section telescoping cantilever beam assembly is etaile in.5. an.5.. Consier a beam AB as shown in Figure. (a) carrying such a loa that it has a bening iagram as shown in Figure. (b). et the beam ben into AC D B as shown in Figure. (c). Now consier an element of small length CD of the beam at a istance from B as shown in Figures. (a) an. (b). et M = Bening moment between C an D δ = ength of CD R = Raius efine by the eflecte beam

12 δθ = Angle inclue between the tangent at C an D facing the atum or the change of slope over the elementary portion δ A = Area of bening moment iagram over the entire span = Horiontal istance of centre of gravity G of the entire bening moment iagram from the atum θ = The angle in raians, inclue between the tangents rawn at the etremities of the beam From the geometry of the eflecte beam it can be seen that C D =Rδθ or δ = Rδθ (.) R A C δ D B (a) l A G. C M B (b) O D δθ A R B (c) C D δθ P Q θ Datum Figure.: Beam in Bening (Aapte from [9])

13 But M I y E R an hence R M EI. As oppose to Equation (.), the negative sign is not consiere whilst ealing with areas. Substituting this value of R in Equation (.) reveals: M (.5) EI The total change of slope from A to B may be foun out by integrating Equation (.5) between the limits 0 to l as: l M EI 0 EI 0 M But 0 M = Area of the Bening Moment Diagram This leas us to the first of the two Mohr s theorems as follows: A (.6) EI Drawing tangents at C an D an then making them met at P an Q on the atum line through B as shown in Figure. (c). From the geometry of the figure it can also be seen that the tangents at C an D also subten an angle of δθ ' M M. PQ ' (.7) EI EI The total intercept may be foun by integrating Equation (.7) above between the limits 0 to l as: Y l M. ' EI 0 EI l 0 M.. ' But M.. = Moment of area of the Bening Moment Diagram over portion δ about the atum

14 0 M.. = Moment of Area of the Bening Moment Diagram over the entire span δ about the atum = A. An finally the last of the two Mohr s theorems can be epresse as: Y A' (.8) EI A point to remember is that in Equations (.6) an (.8) the I value may be constant or may be variable. Equations (.6) an (.8) are Mohr s first an secon theorems more commonly referre to as the moment-area equations. When these equations (.6) an (.8) are employe to fin the slope an eflection at a given point in a beam respectively, their application epens upon the manner in which the beam is supporte..5 Deflection Theorems in Brief Design of cantilever beams for their many applications often requires estimates of eflections at various length positions. The evelopment of analytical methos for estimating eflection are escribe in many tetbooks [5-8]. The beam eflection y is foun by four common methos: (i) irect integration [5-8], (ii) Macaulay s step function [5-8,, ], (iii) Mohr s theorems [5 8, 9] an (iv) strain energy [6,, ]. Both (i) an (ii) are base on the fleure equation erive above Y M EI (.) The prouct EI is the fleural rigiity which is constant in a uniform cross-section. Sagging an hogging moments are taken to be positive an negative, respectively. The moment function, M() in Equation (.), is the bening moment epresse in term of the length position. The irect integration metho (i) aopts successive integrations of Equation (.) leaing to the slope Y/ an then the isplacement Y. Metho (i) is restricte to relatively simple loaing, incluing that consiere here, which oes not lea to iscontinuous M() epressions. Macaulay s step-function technique (ii) is use where moment iscontinuities

15 o arise at span positions where aitional concentrate loa are applie an also, for a uniform loaing that oes not eten to the full length. Mohr place a geometrical interpretation upon the bening-moment iagram when integrating Equation (.) for slope an eflection. When A an B are separate points on the moment iagram (M() vs. ), for which B is a point of ero slope an the eflection at A is require, then Mohr s two theorems (iii) state: Slope at A EI Area of the M-iagram between A an B Deflection of A relative to B EI First moment of area the M-iagram between B an A about A. When strain energy methos (iv) are use to estimate beam eflection the energy store through an internal stress an strain is equate to the work one by eternal forces an moments. Two useful interpretations of this approach, aopte for FE analyses, lie in the theorems of Castigliano an the principle of virtual work, which are ealt with in the sections that follow [6]. Despite uniform section beams being well-serve by the classical theory it is less often use for eflection analyses of variable section beams incluing tapere, steppe an telescopic esigns [9, 0]. Here, it is more likely that FE is aopte to ensure that a given eflection allowance is not eceee. The sections that follow etail the methos of eflection etermination..6 Principle of Superposition This principle, as state in [0], states that the total elastic isplacement at a point in a structure uner a given combination of eternally applie loaing may be obtaine by summing the isplacements at that point when the loas are applie to their position inepenently in any sequence. This principle is amply etaile an applie to the two section telescoping cantilever beam assembly, in an.5.. These etail the Mohr s moment area metho when applie to the tip loae an uniformly istribute loae two section telescoping cantilever beam assembly, respectively. The principle of superposition is use to obtain the eflection of the structure when subjecte to a combination of both tip an uniformly istribute loaing. In.5. the principle has been applie to loa versus isplacement but it can also be applie to connect any one of the following: loastress, loa-strain, stress-isplacement an strain-isplacement. An important caveat is that 5

16 each of the afore-mentione pairs must be linearly relate as woul be foun for a structure obeying Hooke s law within its elastic range..7 Energy Methos Chai Yoo s insightful publication was a most valuable source in the thorough grasping of the energy methos etaile in this section [8]. Energy methos provie a convenient means of formulating the governing ifferential equation an necessary natural bounary conitions. The solutions that are obtaine by solving the governing equations are eact within the framework of the theory (for classical beam theory) computing unknown forces an isplacements in elastic structures. Besies this, the energy principles are funamental to the stuy of structural stability an structural ynamics. However, one of the greatest avantages of the energy methos is its usefulness in obtaining approimate solutions in situations where eact solutions are ifficult or impossible to obtain [, ]. Unoubtely familiarity with the energy principles will be an invaluable asset in the stuy of structural mechanics. Aitional references for a more etaile treatment of energy methos may be foun in the books of Hoff [], anghaar [], Fung an Tong [5], Sokolnikoff [6] an Shames an Dym [7]. 6

17 .7. Preliminaries Consier an infinitesimal rectangular parallelepipe at point in a stresse boy an let the terms T, T an T represent the traction vectors on each face perpenicular to the coorinate aes, an respectively as shown in Figure.. The components of the stress tensor enote by σ ij are the projections of tractions T i on the face whose normal is j. T σ T σ σ σ σ T σ σ σ σ e e e Figure.: Stress tensors an their components (Aapte from [8]) Hence, each traction vector is written as T e e T T e e e e e e e (.9) Or in compact form (ine notation) T i = σ ij e j (.0) Figure. shows the traction vector T acting on an arbitrary plane ientifie by n (unit normal to the plane) along with traction vectors T i acting on the projecte plane inicate by e i an the boy force per unit volume f. The force acting on the arbitrary sloping plane ABC is T A while the force on each projecte plane is T A n n i i negative ei irection. as each has a unit normal in the 7

18 Each projecte area can be compute by A i = A i cos(n, e i ) =A i n e i (.) - T h C - T T n B f A - T so that Figure.: Stresses on an infinitesimal tetraheron (Aapte from [8]) A n = (A n )/( n e i ) = (A n )/( n i e i ) (.) where n i = n e i = cos(n, e i ) (.) Since the tetraheron is in equilibrium the resultant of all forces acting on it must vanish. Hence T n T n i i h f A n = 0 (.) Resolving T n into Cartesian components T Te an taking the limit as h 0, Equation n (.) reuces to T n = T i e i = T i n i (.5) 8 i i

19 Substituting Equation (.0) into Equation (.) yiels T i e i = T i n i = T j n j = σ ji e i n j (.6) from which the stress tensor components are: T i = σ ji n j (.7) Consier a volume of material v boune by a close surface s. et the boy force per unit volume istribute throughout the boy v be f an the stress tensors istribute over the surface s be T. If the boy is in equilibrium then the sum of all forces acting on v must vanish; that is v f v + T s = 0 (.8) s or in component form v f i v + T i s = 0 (.9) s Equation (.7) may be rewritten as s T s = i s jin j s (.0) Assuming that the components σ ji an their first erivates are continuous the surface integral in Equation (.0) can be transforme into a volume integral using the ivergence theorem as jin j s = ji, j s s v (.) From Equations (.9), (.0) an (.) it follows immeiately that v f i ji, j v = 0 (.) Equation (.) can only be satisfie if the integran is equal to ero at every point in the boy. Hence, f i ji, j 0 (.) 9

20 Equation (.) presents three equations of equilibrium written in terms of stresses an boy forces..7. Principle of Virtual Work If a structure is in equilibrium an remains in equilibrium while it is subjecte to a virtual isplacement the eternal virtual work WE one by the eternal forces acting on the structure remains equal to the internal virtual work WI one by the internal stresses [8]. The eternal virtual work is W T u s+ E s i i v f v (.) i u i Using Equation (.7) an the ivergence theorem the first term in Equation (.) can be transforme into i u i s s s T s= ijn jui s= iju i, j v= ij, ju i ijui, j v v (.5) Substituting Equation (.5) into Equation (.) yiels ij, jui fi ui ijui j W E, v (.6) v Since the structure is in equilibrium f 0. Hence Equation (.6) reuces to W E ij i, j v u i ji, j v (.7) The infinitesimal strain increment tensor is e u u / symmetrical which leas to u ij e ij i, j j, i an u i, j u j, i is i, j ij ij (.8) This transforms Equation (.8) to ij e ij v v U (.9) W I 0

21 Equation (.9) escribes the internal work one by the actual stresses (ue to real forces) an virtual strains prouce uring the virtual isplacement. The internal work W I is frequently referre to as the strain energy U store in the elastic boy. From Equations (.), (.8) an (.9) one immeiately obtains W T u s + E s i i v f v = i u i v v U (.0) ij e ij W I Equation (.0) is a mathematical statement of the principle of virtual work. The reverse of this principle is also true. That is, if W W for virtual isplacement then the boy is in E equilibrium as eplaine in Tauchert []. The principle of virtual work is vali regarless of the material stress-strain relations as shown in the erivation. I.7. Principle of Complementary Virtual Work Figure.5 shows the stress-strain iagram of a nonlinearly elastic ro. The strain energy U represents the energy store in a eforme elastic boy; however, the physical interpretation of the complementary strain energy U * is not clear. The strain energy U in the ro uner uniaial stress σ is efine by U v e 0 e e v v e (.) 0 The strain energy ensity or the strain energy per unit volume is equal to the area uner the material s stress-strain curve as is shown in Figure.5. The complementary strain energy U * in a uniform section ro is efine by U * v 0 e v 0 v e (.)

22 σ U * /v σ U/v e e Figure.5: Stress-strain curve of a non-linearly elastic ro (Aapte from [8]) Therefore, the complementary strain energy ensity correspons to the area above the stressstrain curve. For a linearly elastic material the two areas are equal an * U U. In orer to maintain the generality the structure uner consieration is assume to have arbitrary material properties. Consier an imaginary system of surface tractions prouce a state of stress ij Ti an boy forces i f that insie the structure. If these quantities are in equilibrium they must satisfy the equilibrium equations such that from Equation (.): f 0 ij, j i The work one by these virtual forces uring the actual isplacements ui is referre to as the * complementary virtual work W E an is epresse as W * T u S + E S i i V f u i i V (.) Proceeing in a manner similar to that use in the erivation of Equation (.0) with the roles of the actual an virtual quantities interchange one obtains the following T i u i s v s + v = f i u i v v (.) ij e ij

23 The right han sie of Equation (.) is enote as U * * W I e v ij ij v (.5) From Equations (.) an (.5), Equation (.) is rewritten symbolically as W U * * * E W I (.6) Equation (.5) is the principle of complementary virtual work. If a structure is in equilibrium the complementary virtual work one by the eternal virtual force system uner the actual isplacement is equal to the complementary virtual work one by the internal virtual stresses uner the actual strains..7. Principle of Minimum Potential Energy It is assume that there eists a strain energy ensity u=u/v, that is a homogeneous quaratic function of strains u such that e ij ij u e ij (.7) It is recalle that the virtual isplacement fiel u i was not relate to the stress fiel ij when applying the principle of virtual work. They are now relate through a constitutive law epresse by Equation (.7). Substituting Equation (.7) into the principle of virtual work, Equation (.0), one obtains i u i s v T s + u v = v fiu i v = ije ij v v= u e v = ij e u v v ij v U (.8) It is to be note that the variation an integration operations are interchange. The loss of potential energy of the applie loas V is now efine as a function of isplacement fiel u i an the applie loas. V T s s iu i v fiu i v (.9) Taking the first variation of Equation (.9) gives

24 u i V T i u j u s s j ui f i u j v u v j Noting that u an fori j an 0 for i j the equation leas to i u j V T i u j s i u j s v ij ij ij f v (.0) From Equations (.8) an (.0) it follows immeiately U V 0 (.) The quantity U V enote by is the total potential energy of the boy an is given as v e e v iu i s 0 Giving 0 v T s fiu i v (.a) (.b) Equation (.b) is known as the principle of minimum potential energy; an it can be state that: An elastic structure is in equilibrium if no change occurs in the total potential energy (stationary value) of the system when its isplacement is change by a small arbitrary amount [8]. In the early ays of the original evelopments of the calculus of variations the evelopers incluing Bernoulli (65-705), Euler (707-78) an agrange (76-8) i not consier the stationary value of the total potential energy as inee a minimum until egenre (75-8) postulate the so-calle egenre test seeking a mathematical rigor for a minimum [9]. A proof that actually assumes a minimum value in the case of stable equilibrium is illustrate below. From Equations (.) an (.) it follows immeiately that 0 v u e eij ij s v T s i u i v U V 0. Hence f v (.) i u i

25 Using the constitutive relations of Equation (.7) an the strain-isplacement relations for small isplacement theory (Cauchy strain) the first integral of Equation (.) is epane to v u e eij ij v ij u i, j u j, i v ij u i, j ij u j, i v v v (.) Noting that an interchanging the ummy inices j an i, the right-han sie of ij ji Equation (.) is epane to v u e eij ij v ij u i, j ij u j, i v ij u i, j v iju i, v j ij, jui v ijuin j s ij. jui v v s v where u e, e u u / u u ij ij ij i, j j, i v v v v ij u i, j, i, j i j an u j i u j i Substituting the epane form above into Equation (.) yiels ijuin j s ij jui s v which can also be epresse as s ijn j Ti ui s ij j i s, v T s v i u i, f u v = 0 i v f v = 0 This must be true for all ui. Then the equilibrium conitions follow as ij, j f i 0 an T n T which when fully epane gives; T ij j i T i u i n n n n n n n v,. n n The Euler-agrange equations are the equations of equilibrium an the necessary bounary conitions are embee into the Cauchy formula as in Equation (.7). Hence it has been prove that 0 is a sufficient conition for equilibrium as eplaine in Shames an Dym [7]. If it can be shown that the total potential energy of an amissible strain fiel 5

26 e is always greater than that of the equilibrium state, then it suffices that the total ij e ji potential energy is a local minimum for the equilibrium configuration. e ij u e ij eij ue ij e e v i u i ij ij v s Epaning ue ij e ij by a Taylor series u v T s u u e e ue e e e... ij f v (.5) ij ij ij ij kl (.6) e e e ij Substituting Equation (.6) into Equation (.5) gives ij kl i u i e ij u eij e e ij v ij v e i u i ij v s f v T i u s i e ijekl e v ij ekl u v... v u eije kl e e ij kl v v u eije kl e e ij kl v (.7) It will be emonstrate that the integran of Equation (.7) is e ij u for e 0. Eamination of Equation (.6) in association with e 0 reveals that the first term is a constant throughout the boy an is taken to be ero, so that the strain energy vanishes in the unrestraine boy. By efinition ij u eij in the secon term is stress ij. The stress in the unrestraine state must be equal to ero. Consiering up to secon orer terms it gives u eije u e ij eijekl kl e 0 ij ij Hence Equation (.7) can be written as V u e ij v Since u is a positive efinite function the secon variation of the total potential energy is positive. Hence the total potential energy is a minimum for the equilibrium state e 0 when compare to all other neighbouring amissible eformation fiels. Fung an Tong [5], ove 6 ij

27 [0], Saaa [], Shames an Dym [7] an Washiu [] use logic similar to that shown above in the proof of the nature of the total potential energy being a minimum. It appears that Sokolnikoff [6] i not impose e 0 to show that actually assumes a minimum value. ij.7.5 Principle of Minimum Complementary Potential Energy Parallel to the concept of the strain energy ensity introuce in Equation (.7) it is assume that there eists the complementary energy ensity function function of stress such that * u efine for elastic boies as e ij u * ij (.8) Substituting Equation (.8) into Equation (.) gives T i u i s f i u v * i u ij v (.9) v v ij s * As per Equation (.5), the right-han sie of Equation (.9) is U, the first variation of the complementary energy for the structure. A complementary potential energy function is efine by * V i f i v u v u s it i v for which the first variation is given by v * V u v i f i s u v (.50) i T i From Equations (.9) an (.50) it can be conclue that * * U V 0 * (.5) Equation (.5) is the principle of total complementary energy an * v ij e ij v v u v i f i s * is given by u v (.5) i T i It may be shown that the total complementary energy is a minimum for the proper stress fiel following a proceure similar to that use in the principle of minimum total potential energy. 7

28 .7.6. Castigliano s Theorem, Part The principle of minimum total potential energy can be use to erive the Castigliano theorem (which he presente in 87 in his thesis for the engineer s egree at Turin Polytechnical Institute) which is etremely useful in the analysis of elastic structures. For a structure in equilibrium uner a set of iscrete generalie forces Q i i,,... n potential energy is given by the total U n i Q i i i (.5) For the conition of equilibrium the first variation of foun by varying ero. i must be equal to n n n U U U i Qi i i Qi i Qi i 0 (.5) i i i i i Since the variations U i Q i i i are arbitrary the quantities in each parenthesis must vanish; hence, (.55) Equation (.55) is the Castigliano s theorem, part. It states that if the strain energy U store in an elastic structure is epresse as a function of the generalie isplacements i then the first partial erivative of U with respect to any one of the generalie isplacements i is equal to the corresponing generalie forceq. As the stiffness influence coefficient k ij is efine as the generalie force require at i for a unit isplacement j, while suppressing all other generalie isplacements, kij can be epresse as i k ij Q i (.56) j Using Equation (.55) it can be rewritten as k ij U i j (.57) 8

29 .7.7 Castigliano s Theorem, Part For an elastic (not necessarily linearly elastic) structure that is in equilibrium uner a system of applie generalie forces * * U V 0 Q i the principle of minimum complementary energy states that * (.58) Assuming that the complementary strain energy function of * U Q i then Equation (.58) may be rewritten as n * n * * * U U U V Q i iqi i Qi 0 i i as shown in Figure.5, is epresse as a * (.59) Qi * * V where V Qi iqi 0 Qi Qi Since Qi are arbitrary an non ero, Equation (.59) requires that U * i i i (.60) Equation (.60) is known as the Engesser theorem erive by Frierich Engesser in 889 [5] an is vali for any elastic structure. If the structure is linearly elastic the strain energy U an the complementary strain energy U Q i i i * U are equal an the Castigliano theorem, part results. (.6) Equation (.6) states that if the strain energy U in a linearly elastic structure is epresse as a function of the generalie forces Q i then the partial erivative of U with respect to the generalie force Qi is equal to the corresponing isplacement coefficient of a linearly elastic structure is given by f ij i j i. The fleibility influence U (.6) Q Q 9

30 .8 Buckling An Introuction When a structure subjecte usually to compression unergoes visibly large isplacements transverse to the loa then it is sai to buckle. Buckling may be emonstrate by pressing the opposite eges of a flat sheet of carboar towars one another. For small loas the process is elastic since buckling isplacements isappear when the loa is remove. ocal buckling of plates or shells in turn is inicate by the growth of bulges, waves or ripples an is commonly encountere in the component plates of thin walle structural members. Buckling procees in a manner which may be either () stable, in which case isplacements increase in a controlle fashion as loas are increase i.e. the structure s ability to sustain loas is maintaine or () unstable, whereupon eformations increase simultaneously, the loa carrying capacity nose ives an the structure collapses catastrophically. Neutral equilibrium is also a theoretical possibility uring buckling; this is effectively characterise by an increase in eformation with no corresponing or equivalent change in loa. From a purely linear viewpoint, buckling of struts an bening of beams are similar in that they both involve bening moments. Referring to Figure.6(a) in bening, these moments are substantially inepenent of the resulting eflections, whereas in Figure.6(b) for buckling, the moments an eflections are mutually interepenent this means that the moments, eflections an stresses are not proportional to loas. If eflections resulting from buckling become too large then the structure fails this is solely a geometric consieration an completely ifferent from any material strength consieration. If a component or part thereof is prone to buckling then its esign must satisfy both strength an buckling safety constraints, which is the why the phenomenon of buckling is eamine in etail in this literature review an applie specifically to the two section telescoping cantilever section. Buckling has become more of a problem in recent years since the avent of high strength materials requires less material for support structures an components have become generally more slener an buckling-prone. This tren has continue through technological history as is emonstrate by the following sample case stuies. 0

31 P P δ P M=P (a) P (b) M=P δ Figure.6: Differences between (a) Bening an (b) Buckling (Aapte from []).9 Buckling of thin walle structures A thin walle structure is mae from a material whose thickness is much less than other structural imensions. Into this category fall plate assemblies, common hot an col forme structural sections, tubes an cyliners an many brige an aeroplane structures. Col forme sections are increasingly supplanting traitional hot rolle I-beams an channels. They are particularly prone to buckling an in general must be esigne against several ifferent types of buckling. It is not ifficult to visualise what can happen if a beam is mae from such a col rolle channel section. One flange is in substantial compression an may therefore buckle locally at a stress lower in magnitue than the yiel stress of the channel section material. This in turn reuces the loa carrying capacity of the beam as a whole. From this it can be inferre that buckling rather than strength consieration ictates the beam s performance. The slener elastic pin-ene column is the prototype for most buckling stuies. It was first eamine by Euler in the 8 th century. The moel assumes perfection in that () the column is perfectly straight prior to loaing an () the loa, when applie, is perfectly coaial with the column.

32 λ λ Figure.7: Differentiation between the loa an lateral isplacements for buckling [] The behaviour of the buckling system is reflecte in the shape of its loa-isplacement curve referre to as the equilibrium path, in Figure.7. The lateral or out-of-plane isplacement, δ is preferre to the loa isplacement, λ, in this contet since it is more escriptive of buckling. Nothing visibly occurs when the loa on a perfect column first increases from ero- the column is stable there is no buckling an no out-of-plane isplacement. The P-δ equilibrium path is thus characterise by a vertical segment-the primary path- which lasts until the EI min increasing loa reaches the critical Euler loa P cr a constant characteristic of the column as erive in Timoshenko s oft cite master work []. When the loa reaches the Euler loa, the lateral eflections δ grow instantaneously in either equally probable irection. After buckling therefore the equilibrium path bifurcates into two symmetric seconary paths illustrate in Figure.7. Clearly the critical Euler loa limits the column s safe loa capacity. To answer why a compression member buckles if the limiting strength of the material is not reache, Chajes [8] gives creit to Salvaori an Heller in their book [9] for clearly eluciating the phenomenon of buckling by quoting the following from Structure in Architecture;

33 A slener column shortens when compresse by a weight applie to its top an in oing so lowers the weight s position. The tenency of all weights to lower their position is a basic law of nature. It is another basic law of nature that whenever there is a choice between ifferent paths, a physical phenomenon will follow the easiest path. Confronte with the choice of bening out or shortening the column fins it easier to shorten for relatively small loas an to ben out for relatively large loas. In other wors when the loa reaches its buckling value the column fins it easier to lower the loa by bening than by shortening. The bifurcation-type buckling is a purely conceptual one that occurs in a perfectly straight (geometry) homogeneous (material) member subjecte to a compressive loaing of which the resultant must pass through the centroial ais of the member (concentric loaing). The importance attache to an the consierably copious amounts of research evote to bifurcation-type loaing is justifie in that the bifurcation-type buckling loa or the critical buckling loa gives the upper boun solution for practical columns that harly satisfy any one of the three aforementione conitions. The concept of the stability of various forms of equilibrium of a compresse bar is frequently eplaine by consiering the equilibrium of a ball (rigi-boy) in various positions as shown in Figure.8 below [, ]. Although the ball is in equilibrium in each position shown, a cursory glance reveals that there are vital ifferences among the three scenarios. If the ball in Figure.8(a) is isplace slightly from its original position it will return to its initial state upon removal of the isplacing or isturbing force. A boy behaving thus is sai to be in a state of stable equilibrium. Any slight or small isplacement of the ball from its initial state of equilibrium will raise its centre of gravity an a certain amount of work is performe to bring about this isplacement. On isturbing the ball in Figure.8(b) by an infinitesimal isplacement results in it moving away from its equilibrium position an it oes not return to its initial state. The equilibrium of the ball in Figure.8(b) can be sai to be unstable. Any isplacement from the position of equilibrium will lower the centre of gravity of the ball an consequently will result in a ecrease in the potential energy of the ball. It can thus be inferre that in the case of stable equilibrium the energy of the system is minimum an in the case of unstable equilibrium the energy is at a maimum. In Figure.8(c), the ball after being isplace

34 slightly neither returns to its original equilibrium position nor continues in motion upon removal of the isturbance. This implies that the ball is in a state of neutral equilibrium. By virtue of its neutral equilibrium there is no change in energy uring a isplacement in the conservative force system. (a) (b) (c) Figure.8: Stability of Equilibrium [8] The response of the column is most similar to that of the ball in Figure.8. The straight configuration of the column is stable on application of small loas but unstable on application of larger loas. The state of neutral equilibrium is consiere to occur in the transition from stable to unstable equilibrium in the column. This leas us to conclue that the loa at which the straight configuration of the column ceases to be stable is the loa at which neutral equilibrium is possible. This loa is the critical loa. To etermine the magnitue of the critical loa it is imperative to etermine the loa uner which the member can be in equilibrium both in the straight an slight bent configuration. The magnitue of this slightly bent configuration is unetermine an is the reason why the free boy of a column must be rawn to account for this slight ben in its configuration. The metho whereby this slightly bent configuration for evaluating critical loas is accounte for is calle the metho of neutral or neighbouring or ajacent equilibrium. At critical loas the primary equilibrium path reaches a bifurcation point an branches into neutral equilibrium paths as has been shown in Figure.7. This type of behaviour is in essence the bifurcation-type buckling. Bifurcationtype buckling is in turn preicte by Eigenvalue analysis. A structural member is calle a beam when it is supporte at one or both ens an carrying transverse loas. The same member is calle a column when it is supporte at one or both ens an carries a compressive aial loa. The structural behaviours at both these conitions are ifferent an the governing equations an conitions are also ifferent. The behaviour of a crane boom when it is horiontal can be best escribe by a cantilever an by a column

35 when the boom is vertical. The members are mae up of hollow sections. iterature escribing the behaviour of such telescoping members is very limite an thus presents an opportunity for investigation..0 Rayleigh-Rit Metho Now that buckling has been eplore in as brief a manner as is possible, the reaer is irecte to the cru of the buckling problem, as applie to the two section telescoping cantilever beam assembly. The nee to etermine the critical buckling loa of the telescoping assembly was establishe from the outset, an the energy methos were eplore for that purpose in.7 above. The energy methos introuce in.7 are a convenient means of computing unknown forces an isplacements in elastic structures. They can be the basis of eriving the governing ifferential equations an require bounary conitions of the problem. They are also the starting point of many moern matri/finite element methos. The solutions that are obtaine using these methos are eact within the framework of the theory (for eample, classical beam theory). Energy methos are also use to erive approimate solutions in situations where eact solutions are ifficult or nearly impossible to obtain. The most wiely known an use approimate proceure is the Rayliegh-Rit metho in which the structure s isplacement fiel is approimate by functions that inclue a finite number of inepenent coefficients (or natural coorinates; one for the Rayleigh metho an more than one for the Rayleigh-Rit metho). The assume solution functions must satisfy the kinematic bounary conitions (otherwise the convergence is not guarantee no matter how many functions are assume) but they nee not satisfy the static bounary conitions (if they satisfy the static bounary conition a fairly goo solution accuracy can be epecte). The unknown constants in the assume functions are etermine by invoking the principle of minimum potential energy. Suppose for eample the assume function has n inepenent constants a i (i=,,.n). Since the approimate state of eformation of the structure is characterie (amplitue as well as shape) by these n constants the egrees of freeom of the structure have been reuce from to n. Invoking the principle of minimum potential energy it follows that a a... an 0 (.6) a a a n Since a are arbitrary Equation (.6) implies that 5

36 a i 0 i =,,n (.6) Equation (.6) yiels a system of n simultaneous equations that can be solve for the coefficients a for static problems an in the case of Eigenvalue problems the eterminant (characteristic eterminant) for the unknown constants is set equal to ero for the n Eigenvalues [8]. A few general observations with regars to the Rayleigh-Rit Metho are now mae. Although the accuracy is generally improve by increasing the number of inepenent functions, the computation efforts increase proportionally to the square of the number of inepenent functions. The type of functions to be selecte for a particular problem is base on an intuitive iea of what the true eformation looks like. Trignometric or polynomial functions are frequently use simply because of the ease of analysis involve. By virtue of using the principle of minimum potential energy all approimate solutions make the structure stiffer than what it is. Consequently the isplacements preicte by the Rayliegh-Rit metho are always smaller than eact ones an Eigenvalues are greater than those preicte by eact solution methos. Finally, if the approimate isplacements are use to evaluate internal forces an stresses the latter results shoul be viewe with caution because the stress components epen on the erivatives of isplacements. Although isplacements themselves may be reasonably accurate their erivatives may not be the case. In fact the higher the erivates the accuracy involve is further eteriorate. In a similar fashion the accuracy of Eigenvalues associate with higher moe Eigen vectors eviates much more rapily than those associate with lower moe Eigen vectors. 6

37 . The Rayleigh Quotient Mikhlin [50] proposes that the approimate solution of the Eigen value problem usually reuces to the integration of a ifferential equation of the form w Mw 0 (.65) where w is the isplacement that satisfies not only the ifferential equation (.65), but also certain homogenous bounary conitions (which may preclue the cantilevere enconition), an M are certain ifferential operators an λ is an unknown numerical parameter. For the stability of a column, the governing ifferential equation is w w EI P (.66) where is the length co-orinate an w is the lateral isplacement. For Equation (.66) EI (.67) M (.68) P (.69) Equations (.67) an (.68) are self-ajoint (symmetric), positive efinite operators for the usual en supports of columns. If a linear ifferential operator has the following property it is calle self-ajoint or symmetric operator: u v u, v, (.70) The inner prouct of two functions g an h over the omain V is efine as g, h inner prouct of g an h V ghv (.7) An operator is sai to be positive efinite if the following inequality is vali for any function from its fiel of efinition, u q 0 :, u 0, u, u 0 u (.7) 7

38 The reason why one is concerne whether or not a bounary-value problem has the properties of being both, a self ajoint (symmetric) an positive efinite entity, is that the bounaryvalue problems having these properties are sai to be properly pose, an there eists a unique solution to a properly pose bounary-value problem. An improperly pose bounaryvalue problem ue to haphaarly or arbitrarily assigne bounary conitions is meaningless. Multiplying both sies of Equation (.66) by w an integrating over the omain yiels l 0 l w w w EI P w 0 (.7) Integrate the left han sie of Equation (.7) by parts twice as follows: l 0 w w EI l 0 w EI w w EI l 0 w w EI For simply supporte, fie or cantilevere en conitions, the last two quantities are ero. Integrating the right-han sie of Equation (.7) gives l 0 P l w w P 0 l 0 w w Pw l 0 The last epression vanishes for fie an simple supports (not for the cantilevere en). Substituting the epane integrals back into Equation (.66) gives P l w EI 0 l w 0 (C metho) (.7) It is note that Equation (.7) works for cantilevere columns espite the fact that one of the concomitants is not ero. As mentione earlier the error involve in the approimate solution 8

39 propagates much faster in the higher orer erivatives. In orer to improve the critical value w compute from the Rayleigh quotient, M in the numerator by. Then EI P cr EI l l w' 0 0 M (C metho) (.75) The Rayleigh Quotient etaile here is in effect the basis of the theory outline in., as ientifie by Timoshenko an applie to the cantilever-column, in orer to etermine its buckling loa.. ocal Buckling ocal buckling of plates or shells is inicate by the growth of bulges, waves or ripples an is commonly encountere in the component plates of thin structural members. ocal buckling of an ege supporte thin plate oes not necessarily lea to total collapse as in the case of columns since plates can generally withstan loas greater than the critical. However the P-δ curve shown in Figure.9 below illustrates plates generally reuce stiffness after buckling so plates cannot be use in the post buckling region unless the behaviour in that region is known with confience. It shoul be emphasie that the knee in the P-λ curve is unrelate to λ λ Figure.9: ocal buckling of ege supporte thin plate with loa-loa inuce isplacement curve (P-λ) an the lateral isplacement curve (P-δ) [] 9

40 any elastic-plastic yiel transition; the systems being iscusse are totally elastic. The knee is an effect of overall geometric instability rather than material instability. A funamental ifference in the buckling characteristics of frame members an plates is that for the former, buckling terminates its ability to resist any further loa whereas in the case of the latter, this us not necessarily the case. A plate element may carry aitional loaing beyon the critical loa an this reserve strength is calle the postbuckling strength. The relative magnitue of the postbuckling strength to the buckling loa epens on various parameters such as the imensional properties, bounary conitions, types of loaing an the ratio of buckling stress to yiel stress. Plate buckling is usually referre to as local buckling [8]. Structural shapes compose of plate elements may not necessarily terminate their loacarrying capacity at the instance of local buckling of iniviual plate elements. Such an aitional strength of structural members is attributable not only to the postbuckling strength of the plate elements but also to possible stress reistribution in the member after failure of iniviual plate elements [8]. The photograph in Figure.0 below shows the local buckling of a moel bo girer constructe from thin plates. Figure.0: ocal buckling of moel bo girer []. Torsion in Structures Torsion in structures is perhaps one of the least well unerstoo subjects in structural mechanics. Purely torsional loaing rarely occurs in structures ecept in the power transmitting shafts of automobiles or generators. However torsion oes evelop in structures along with bening from unintene eccentricities as can be foun in spanrel beams. Generally thin-walle sections o not behave accoring to the law of the plane sections employe by Euler-Bernoulli-Navier [8]. A thin-walle section is referre to as rolle shape in which the thickness of an element is less than one-tenth of the with. Many stocky rolle shapes o not meet this efinition; 0

41 however the general theory of thin-walle section evelope by Vlasov [5], [5] in the 90s appears to be applicable without significant consequences [8]. A thin-walle section becomes warpe when it is subjecte to en couples (torsional moment). Hence the cross section oes not remain plane after eformation. Eceptions to this rule are tubular sections an thin walle open sections in which all elements meet at a point such as the cruciform, angle an tee section. Another istinct feature of the response of structural members to torsion is that an eternally applie twisting moment is resiste internally by some combination of uniform or pure St. Venant torsion an non uniform or warping torsion epening on the bounary conitions that is whether a member is free to warp or whether warping is restraine. Thin walle open sections are very weak against torsion an are susceptible to lateral-torsional buckling or fleural-torsional buckling, which is affecte by the torsional strength of the member even though no intentional torsional loaing is applie. Torsional buckling of columns can arise when a section uner compression is very weak in torsion, an leas to the column rotating about the force ais. More commonly where the section oes not possess two aes of symmetry as in the case of an angle section, this rotation is accompanie by bening an is known as fleural torsional buckling. (a) (b) (c) Figure.: Eamples of (a) Torsional Buckling (b) Fleural-Torsional Buckling (c) ateral Buckling [] ateral buckling of beams is possible when a beam is stiff in the bening plane but weak in the transverse plane an weak in torsion as is the I beam as shown in Figure.. If warping oes not occur or if warping is not restraine, the applie twisting moment is entirely carrie by uniform torsion. When a member is free to warp no internal normal

42 stresses evelop espite the warping eformation. This is tantamount to the fact that a heate ro will not evelop any internal stresses if it is free to epan at one or both ens espite the temperature-inuce elongation of the ro. If warping is restraine the member evelops aitional shearing stresses as well as normal stresses. Frequently warping stresses are fairly high in magnitue an they are not to be ignore.. Torsional an Fleural-Torsional Buckling The possibility of torsional column failure ha never been recognise until open thin-walle sections were use in aircraft esign in the 90s. Eperience has reveale that columns having an open section with only one or no ais of symmetry show a tenency to ben an twist simultaneously uner aial compression. The ominous nature of this type of failure lies in the fact that the actual critical loa of such columns may be less than that preicte by the generalise Euler formula ue to their small torsional rigiities. Bleich [5] gives a thorough overview of the early evelopment of the theory on the torsional buckling [8]. Bleich an Bleich [5], were among the early evelopers of the theory on torsional buckling along with Wagner an Pretschner [55], Ostenfel [56], Kappus [57], unquist an Fligg [58], Gooier [59], Hoff [60] an Timoshenko [6]. All of these authors make the funamental assumption that the plane cross sections of the column warp but that their geometry oes not change uring buckling [8]. Thus the theories consier primary failure (global buckling) of columns as oppose to local failure characterise by istortion of the cross sections. The iviing line between primary an local failure is not always sharp. Separate analysis of primary an local buckling base on governing ifferential equations without abanoning the assumption that cross sections of the column will not eform may yiel only approimate solutions since there coul be coupling of primary an local buckling. Moern finite element coes with refine moelling capabilities incorporating at least flat shell elements may be able to assess this combine buckling action.

43 .5 ateral-torsional Buckling A transversely or combine transversely an aially loae member that is bent with respect to its major ais may buckle laterally if its compression flange is not sufficiently supporte laterally. The reason buckling occurs in a beam at all is that the compression flange or etreme ege of the compression flange or the etreme ege of the compression sie of a narrow rectangular beam becomes unstable. If the fleural rigiity of the beam with respect to the plane of bening is many times greater than the rigiity of the lateral bening the beam may buckle an collapse long before the bening stresses reach the yiel point. As long as the applie loas remain below the limit value, the beam remains stable. In other wors, the beam that is slightly twiste an/or bent laterally returns to its original configuration upon the removal of the isturbing force. With increasing loa intensity the restoring forces become smaller an smaller until a loaing is reache at which in aition to the plane bening equilibrium configuration an ajacent, eflecte an twiste, equilibrium becomes equally possible. The original bening configuration is no longer stable an the lowest loa at which such an alternative equilibrium configuration becomes possible is the critical loa of the beam. At the critical loa the compression flange tens to ben laterally eceeing the restoring force provie by remaining portion of the cross section causing the section to twist. ateral buckling is a misnomer, for no lateral eflection is possible without concurrent twisting of the section. Bleich [5] gives creit to Prantl [6] an Michell [65] for proucing the first theoretical stuies on the lateral buckling of beams with long narrow rectangular sections. Similar creit is also etene to Timoshenko [66] for eriving the funamental ifferential equation of torsion of symmetrical I-beams an investigating the lateral buckling of transversely loae eep I-beams with the erive equation [8]. Since then, many investigators incluing Vlasov [5], Winter [67], Hill [68], Clark an Hill [69] an Galambos [70] have contribute on both elastic an inelastic lateral-torsional buckling of various shapes. Some of the early evelopments of the resisting capacities of steel structural members leaing to the oa an Resistance Factor Design (RFD) are summarise by Vincent [7].

44 .6 Shear in Thin-Walle Close Tube Sections Rees in Mechanics of Optimal Structural Design, [] etermines from first principles the optimum imensions for a non-uniform, thin-walle, rectangular tube having sie lengths b an with wall-thicknesses t b an t when a vertical force Fy is applie at the shear centre. In 5. the optimum imensions for a uniform thin walle rectangular tube is etermine, using as a basis the following theory. y δs s E G P e F y Figure.: Net Shear Flow in a close thin walle tube The net shear flow in a close, thin walle tube uner a single vertical force F y applie through the shear centre E is written as q q b q E (.76) Here qb is the fleural shear flow epression use previously with open sections in [] as FyDb Fy q b ya (.77) I I A However unlike an open section there are no free surfaces to take as an origin so for a close tube a constant qe must be ae to q b. Thus q E is a constant of integration with an important

45 physical interpretation namely that when qe is ae to q b it ensures the rate of twist is ero at E [6]. That is s s b 0 (.78) t t q qe which allows qe to be foun. If the position of the shear centre E is not obvious it may be foun by combining Equation (.77) with the counterbalance require between the torque from the net shear flow at q q b q E an the torque with F y applie through the shear centre, each torque being referre to any convenient reference point P. This torque balance is written as follows F e y B qrs q Rs q Rs (.79) E where e is the horiontal istance require between E an P. Within the first integral R is a perpenicular istance from P to the net incremental force q s. Alternatively the split in this integral recognises qe as a constant when Rs becomes twice the area enclose by the wall s mean centre-line. 5. emonstrates the use of Equations (.76), (.77), (.78) an (.79)..7 Torsion in Thin-Walle Close Tube Sections The Bret-Batho theory provies the torque T Aq in terms of the wall shear flow q an the cross-sectional area A, which refers to that area enclose by the tube wall s mean centre line for those irregular shape tubes of varying thickness. The shear flow q t is always constant in a close tube irrespective of whether or t varies as the case may be as mentione previously. Hence in a tube with uniform thickness the torque may be epresse as T A t (.80) where an t refer to a single point in the wall. An optimum esign ensures that the limiting shear stress from torsion matches that require to cause shear buckling in the wall []. 5

46 .8 Combine Shear an Torsion in Thin-Walle Close Sections When a transverse shear force oes not act through the shear centre of a given cross-section its net shear flow is a consequence of combine torsion an shear [7]. In close sections it is more convenient to treat the effects of the St Venant s torsion an fleural shear together but if they were separate the torsional effect or the Bret-Batho shear flow amounts to taking the ifference between the net shear flow from here an that of pure fleure from 5.. The shear flow in a close tube uner a single vertical force F y applie at any position is written generally as q q b q where 0 qb is the fleural shear flow [6, 7], FyDb Fy q b ya (.8) I I A Unlike an open section there is no free surface, where q 0 0, to take as the origin for s. In the case of the close tube a constant q 0 must be ae to q b. The sum q b q0 ensures static equivalence between the net shear flow an the torque that arises when F y oes not pass through the shear centre. In the case of an asymmetric tube this equivalence is written as Fy p qbrs q0 Rs (.8) where p is the horiontal (perpenicular) istance between F y an any convenient reference point P an R is the perpenicular istance from P to the incremental force q b δs within the tube wall as shown in Figure.. A δs R q B t s p P F y Figure.: Static Equivalence between torque (F y p) an shear flow q b [5] 6

47 The secon path integral in Equation (.8) is twice the area A enclose by the mi wall. Hence the secon term in Equation (.8) becomes the contribution from the St Venant torque Aq o. It is convenient to consier the shear flows from transverse shear an St Venant torsion together rather than to separate them since we neither have pure fleure nor pure torsion F y. This allows q o to be ientifie with the shear flow at the origin chosen for the mi wall path s. In.6 the special case where F y passes through the shear centre was consiere in which the net shear flow was written as q q b q where the Equations (.76), (.77), (.78) an (.79) apply. The qb shear flows foun from Equation (.77) remain unaltere in regar to the position F y, but the net shear flow allows for its position i.e. q E is ae when F y coincies with the shear centre, q o is ae when F y is applie elsewhere. Again it woul be convenient to treat the two effects together but if nee be they can be separate with the torsional effect amounting to subtracting the shear flows given in 5. from the net shear flows obtaine here. In other wors E qb q qb qe q0 qe 0 (.8) showing that the torsional shear flow arising from a transverse force isplace from the shear centre is the ifference between the initial shear flows for each case. 7

48 .9 Applications of the Telescoping Cantilever Beam Assembly The aim of the sections that follow is to introuce to the reaer the present applications of the telescoping beam assembly in the ifferent work environments..9. Mobile Elevating Work Platforms Accoring to EN 80 [] the relevant European esign stanar the efinition of a mobile elevating work platform, or MEWP, is a machine which consists as a minimum of a work platform, an etening structure an a chassis. The work platform is a fence platform or cage that can be move uner loa to the require position an from which repair or similar work can be carrie out. The etening structure is connecte to the chassis an supports the work platform. It allows movement of the platform to its require position. The chassis is the base of the MEWP an may be pulle, pushe, self-propelle etc. MEWP s are known by a variety of names incluing hyraulic work platform, access platform an aerial work platform. For the purposes of this thesis the term access platform is use. (a) Relate Terminology The access platform inustry, like any other, has its own terminology. In any iscussion of access platform esign it is necessary to make use of various inustry specific terms. For this purpose the terminology is outline below. Figure. shoul be referre to, for clarification of the various measures. (i) Platform Floor Height The maimum height that the floor of the problem can reach with the etening structure fully unfole. This measure takes on a particular significance in the USA where it is the accepte measure of an access platform s height. (ii) Working Height In Europe it is the accepte norm to quote working height of an access platform. This is the platform floor height plus an allowance for the vertical reach of the operator(s). General practise is to a two metres to account for this reach. 8

49 (iii) Working Outreach This is the horiontal istance from the slew centre of the machine to an imaginary line 600mm beyon the outsie ege of the platform with the machine set to its maimum horiontal etension. Again, the etra 600mm is to account for the reach of the operator. (iv) Safe Working oa This refers to the maimum loa that can be safely carrie in the cage. The manufacturer specifies the SW. As woul be epecte the manufacturer is oblige to test to 5% of the working loa when evaluating a new platform esign. The accepte SW for the majority of access platforms is 5kg. This allows, accoring to EN80 stanar, for two operators at 80kg each an 65kg of tools an equipment. It also satisfies the USA requirement of 500lb cage capacity. (v) Height Stowe This is the maimum height of the machine when it is in its fully fole position. The stowe height is particularly important when esigning a machine that is etene for use inoors as well as outoors, for obvious reasons. The usual target for machine esigners is to stay below the -meter limit impose by the height of a stanar oorway. (vi) ength Stowe The corresponing figure for length is of consierable importance for trailer machines ue to the requirements of roa traffic approvals an legislation. It is equally relevant when consiering how machines are to be elivere, since there are limits to the sie of shipping containers an so forth. 9

50 Figure.: Terminology associate with the Mobile Elevating Access Platform (Taken from []) 50

51 (b) Access Platform Types Access platforms in general can be ivie into a number of groups an in the first instance the type of etening structure they employ usually classifies machines. Stick booms, articulating booms an scissors account for the majority of machines. However the stick an articulating boom types are subivie accoring to the chassis, which can be trailer mounte, vehicle mounte or self-propelle. Straight or stick boom machines (see Figure.5) elevate the operators to a given working height by means of a single straight boom. A hyraulic cyliner controls the angle of the boom. The boom can be of fie length but more usually has an etening or telescoping function. Typically this telescoping function will be achieve by means of a multi stage cyliner an a series of chains. The stick boom machine is ieal for accessing areas such as the unersie of briges but lacks the ability to eten up an over an obstruction. As the name suggests, scissors lifts (Figure.6) make use of a pantograph type linkage to achieve a given working height. In contrast to other machine types they provie minimal outreach - the maimum horiontal istance which can be reache by the operator but platform areas can be large an the smaller types are ieal for use in confine spaces inoors. An articulating boom machine as shown in Figure (.7) has a number of booms that are linke by knuckles. Again hyraulic cyliners control the angles of the booms. One or, in some cases, several of the booms may have a telescoping capability. This type of machine is less suite to the uner brige type of work mentione earlier. However inepenent operation of the booms allows the user to manoeuvre the machine up an over an obstacle. As eplaine earlier the various machine types can be sub-ivie into trailer mounte, self propelle an vehicle mounte machines. In trailer mounts as shown in Figure (.8) the chassis or base of the machine forms a roa-towable trailer. The trailer is fitte with outriggers or stabilisers that are etene once the machine forms a roa-towable trailer. The trailer with outriggers etene forms a soli base for the machine. Outriggers can be manual or hyraulic. A common feature is a retractable ale that allows the machine with to be reuce to fit through a stanar sie oorway. 5

52 Figure.5: Straight or stick boom access platform [7] Figure.6: Scissor ift [75] 5

53 Figure.7: Articulating Boom Machine (Taken from []) Figure.8: Trailer Mounte Machine (Taken from []) Self-propelle machines can be riven uner their own power whilst on site. Stick booms, articulating booms an scissors can all be self-propelle. The rive facility is generally available with the booms etene or fole although rive with the booms etene is only permitte at a reuce spee. Self propelle machines greatly reuce the amount of time neee to set up since there are no stabilisers to conten with. There have been many variations on the theme of the self-propelle machine. Rough terrain outoor machines with petrol, iesel or liqui propane gas power sources are available as are eicate inoor all electric machines. In aition, machines that make use of multiple power sources are available. The ability to switch from one power source to another instantly enables the same 5

54 machine to work inoors an outoors. This ae fleibility is a major selling point especially in the hire inustry. The term vehicle mount is self-eplanatory. In general vehicle mounte machines utilise the type of articulating boom structure foun on a trailer mounte or self propelle machine. Inee in some cases a company offering a vehicle mount will simply aapt the entire etening structure from an eisting trailer or self propelle machine as shown in Figure (.9). The etening structure is built onto an eisting vehicle. Typically the vehicle use woul be a large bo van or flat-be truck. The etening structure can be irectly attache to the chassis of the vehicle or alternatively a purpose built base is attache to the chassis an the booms are built onto the base. The increase base mass provie by a large flat-be lorry obviously allows etremely large working height an outreach figures to be achieve if require. Figure.9: Vehicle Mounte Access Platform [76].9. Telescopic Retractable Roofing Systems.I.TRA. USA [77] is one of a number of establishments engage in the esign, manufacture, installation an maintenance of retractable roofing an sliing roof systems. These offer ieal solutions for covering wie spaces such as terraces or patios of restaurants, hotels, private houses, resorts or swimming pools as is amply emonstrate in Figures.0 an.. The 5

55 telescopic coverings offer protection from the elements as a result of thermal an acoustic isolation. (a) (b) Figure.0: A retractable roof enclosure (a) before eployment an (b) after eployment [77] (a) (b) Figure.: A retractable commercial garen roof canopy (a) before eployment an (b) after eployment [77] Figure. shows a retractable pool enclosure manufacture by the same firm [78]. Naturally, covering the swimming pool offers many avantages more important of which is the increase in commercial value an prolonge usage perio, free from relying upon the effects of the elements. In winter, the retractable roofing system protects an enclose pool from ampness, thereby lowering maintenance costs an when enclose with lateral panels, provies thermal isolation an conversely in the summer, it offers protection from the sun an shelter from the win. 55

56 Figure.: A retractable pool enclosure [77] Figure. shows a retractable awning use in a commercial setting. It has all of the avantages as mentione earlier, for all other retractable type roofing systems. Figure.: A retractable awning [77].I.TRA. USA [77] also manufactures retractable roofing systems for professional sports staiums, which can be opene an close at will, as the nee emans. The retractable roof system is available in two options from the company. The first installe at the Fenway Park Staium in the USA, as shown in Figure. consists of two parts (one fie an the 56

57 other retractable) which allows for the roof to open at half its full sie. The secon is a system, mae of three parts of which one is fie an the other two are movable as is shown installe at the Churchill Downs in ouisville, Kentucky, USA, in Figure.5. This can be opene at up to two-thirs of its full sie. (a) (b) Figure.: Fenway Park retractable staium roof system consisting of two parts (a) before eployment an (b) after eployment [77] (a) (b) Figure.5: Churchill Downs Enclosure with a retractable staium roof system consisting of three parts (a) before eployment an (b) half way through full eployment [77] 57

58 .9. Telescoping Marine Assemblies Nautical Structures USA are specialists in maritime structures an a wie range of manufacturing for relate use. Of the many comple structures that they manufacture, the following two structures are of relevance to the topic at han. The first is the Sliing type Overhea Beam Crane which is a compact multi-stage telescoping crane. It is esigne to work inepenently as an aft tener garage crane or to work in synchronise matche pairs to launch large teners from a tener garage. The primary material of construction is aluminium in orer to reuce the potential for corrosion an system weight. The capacities of these cranes range from 500-0,000 kg. These compact esigns permit the crane structure to nest into the vessel s overhea an live within the confines of the tener garage. Figure.6: The S-DEX Type hyraulic overhea beam crane manufacture by Nautical Structures USA [78] The single, ouble an triple telescoping type of passerelles or gangplanks are unique esigns that increase the stowe efficiency of the plank as shown in Figures.8 (a), (b) an (c) respectively. By virtue of their esign, these telescoping gangplanks allow the ratio of stowe length to eploye length to be increase, thereby also enabling the pocketing hyraulic passerelle to stow in shorter spaces within a yacht. The gangplank may be eploye with the telescoping sections etene or retracte. The pocketing hyraulic passerelle can thus be 58

59 use in a variety of boaring conitions. As per Nautical Structures [78] the telescoping gangplanks luffs +/- twenty egrees an slews a total of ninety egrees. As a result of the hyraulic self-centring, slew-lock an float functions incorporate as stanar esign features into the telescoping gangplank, the nee for electric limit or proimity switches is eliminate. This results in long term reliable performance in a wet marine environment with reuce maintenance emans. (a) (b) (c) Figure.7: Applications of the S-DEX Type hyraulic overhea beam crane [78] 59

60 (a) (b) (c) Figure.8 (a) Single telescoping gangplank (b) Double telescoping gangplank an (c) Triple telescoping gangplank [78] 60

61 .9. Telescoping Ajustable Columns Telescoping ajustable columns are also known as tele posts, sectional columns, ouble-sectione columns, jack posts, or jacks. They come in two or more hollow steel tube sections that are assemble on site. A smaller iameter tube is fitte into a larger iameter tub an the sections are hel in place with steel supporting pins which pass through the pre-rille holes of both tubes. Telescoping ajustable columns are regularly use in construction to ajust or level a structure before installing a permanent column. They also fin use as temporary supports uring the course of builing repair. These columns also have a large screw on one en that allows the height of the column to be ajuste in site. Figure.9: Telescoping ajustable column [79].9.5 Telescoping Ajustable Wheelchair Ramps The versatile telescoping ajustable wheelchair ramps are the perfect solution for use over ifferent rises an obstacles. The ramp shown in Figure.0 can be ajuste from a minimum of si feet to a maimum of ten feet. Another avantage of these ual track ramps is their ability to accommoate any wheelchair with. These ramps come in two tracks for each set of wheels on a wheelchair or power chair an have a non-ski traction surface to prevent wheels from spinning. Also each track ramp has a hanle on it so it can be easily transporte an when fully close they have a locking tab to prevent them from acciently opening up uring storage or transport. Another avantage is the etremely light weight of the ramps for their sie ue to an aluminium boy which also means that they will not rust or corroe an will hence last the test of time. The 6mm long attaching lip has a rubber pa to 6

62 prevent slipping of the track ramp an will fit on almost any flat surface incluing steps, porches, sie oors on a minivan, siewalks an more. Figure.0: Telescoping ajustable wheelchair ramps [80].9.6 Telescoping Poles an Ajustable Masts Numerous firms are engage in the manufacture an sale of telescopic poles an ajustable masts. Of the many the one of particular importance an relevance, known as the Woner Pole [80] is etaile here. Accoring to its manufacturers [80] it is a composite material using a minimum fibre glass content of 70 % which is then circular woul with aitional fibre glass strans, using a metho that as tremenous strength in ecess of 50 MPa, which in turn also prevents linear splitting commonly associate with other fibre glass poles. UV inhibitors are ae to prevent egraation of the pole when epose to the elements. Each section of the pole is wrappe with two layers of the highest quality Neus veil which prevents fibre bloom an preserves the integrity of the outer an inner surface of the pole. The Pole has nesting sections that fit insie of each other an raw out to any esire height. Each section has eclusive stop markers to prevent the pole from too great an etension. The sections are secure into place by turning the Sure ock grips that are factory moule onto the pole. The Pole is non conuctive an oes not have any of the electrical haars commonly associate with metal poles. Any part of the pole can be replace or ae at any time. In aition to these factors are the low weight, superior strength, available accessories, 6

63 styles, moels an colours. The construction of this Woner Pole is shown in Figure.. Figure.: Construction of the Woner Pole [8] Figure.: A telescoping Woner Pole in use [8] 6

64 .9.7 Steel Telescoping Towers US Tower Corporation [8] has been engineering an manufacturing tower systems for close to twenty years. Their prouct line has since epane to inclue telescoping tubular towers with a variety of options such stanar, self supporting an rotating. (a) (b) (c) () Figure.: Eamples of trailer mounte US Tower manufacture telescoping towers [8] 6

65 Figure.: A self containe US Tower manufacture Comman, Control, Communications an Tactical Shelter or CT Trailer [8] Figures.(a)-() shows some of the many eamples of US Tower manufacture telescoping towers which are trailer mounte. Accoring to the manufacturer, US Towers [8] they are esigne for rapi eployment with minimal setup time with no earth penetration require for stabiliation an can be easily mounte an retracte on harsh terrain an unforgiving weather conitions. Figure. () shows one of these towers in a storage position whereupon it can be transporte with ease on boar both military an suitably moifie civilian aircraft. Also highlighte in Figure. is a self-containe comman, control an communications an tactical shelter also known as a CT Trailer, which are another of their grounbreaking innovations. By combining the strength an versatility of their steel telescoping towers with the freeom an portability proffere by a trailer, the company has mae it possible to access mobile communications capabilities in the most etreme conitions. As per the manufacturers [8] these telescoping towers currently fin use in conflict ones aroun the worl an also in epeitions across all manner of inhospitable climes, to maintain roun the clock communications uplinks as well as surveillance an monitoring. 65

66 (a) (b) (c) () Figure.5: Eamples of vehicle mounte US Tower manufacture telescoping towers [8] Figure.5(a)-() ehibits some of the vehicle mounte US Tower manufacture telescoping tower systems. Figures.5 (a) an (b) show two vehicle mounte systems utilise for onsite emonstrations that accentuate the capabilities of ifferent technologies use in surveillance systems. With the versatility of a high en surveillance camera mounte on an etenable telescoping tower an powere by a self containe solar system, the manufacturers [8] claim to be able to manage the most stringent security nees. Figures.5 (c) an () show vehicle mounte telescopic masts in action. The manufacturers [8] claim that these particular series of masts are operational at any intermeiate height, can be mounte onto vehicles, shelters an trailers, an are automatically operate with spinle rive for full etension or retraction within -0 minutes. 66

67 .9.8 Telescoping Storage Racks A SpaceSaver Rack as manufacture by Steel Storage Systems Inc, of Denver, Colorao, USA [8] is a series of ouble sie vertically stacke receptacles in levels up to eighteen feet high. The receptacles roll out like rawers so they can be easily accesse for loaing or retrieving the contents with an overhea crane. The racks can be structure to accommoate bars, tubing or other steel tube proucts up to si metres in length thereby offering practical means to store steel proucts as long ecept flat on the floor. The basic builing block of the SpaceSaver Rack is a telescoping tube that creates a storage rawer for steel bars, tubes an other shapes. The telescoping esign allows the rawer to be cranke out for convenient loaing or retrieval of material. The primary customers as ientifie by the company are steel istributors, heavy equipment manufacturers an steel fabricators who have large inventories of steel proucts an the necessary overhea crane equipment, who esire effective space utiliation an accessibility. Figure.6: Aerial view of a large installation of SpaceSaver Racks in a steel service center [8] 67

68 Figure.7: SpaceSaver Racks installe outoors [8] Figure.8: 8 Tall SpaceSaver Rack installe with optional electric lift cage. For maimum ensity 8Tall moels nearly si metres high are available [8] 68

69 Figure.9: 5 Tall SpaceSaver Rack storing cm tubing at Marmon/Keystone. A rolling platform laer is use to access the upper levels [8] Figure.9 shows how Marmon/Keystone, an international istributor of piping an tubing, makes use of the SpaceSaver Racks for storing carbon, stainless an aluminium tubing inventory in their Denver, Colorao branch. This instalment shown in Figure.9 as to the alreay commissione an eisting SpaceSaver racks to complete the transition from a pigeon hole rack system. The material ranges from 5-00 mm in iameter in lengths from 6-7 metres. The racks significantly reuce orer filling time ue to the selectivity provie by the telescoping roll-out rawers. They also improve safety by eliminating the proceure of manipulating heavy tubes in an out of pigeon holes. The 5 Tall SpaceSaver Rack moel maimise the available height in the plant an complimente their eisting overhea cranes. Marmon/Keystone s choice of this moel having a with of 900 mm an 00 mm high rawers, as per the manufacturer s case stuy profile [8] allowe the ual purpose of storing bulky loas of tubing an to subivie the rawers with receptacle iviers to further increase storage ensity for smaller quantity items. Each of these roll-out rawers has a 000 kg capacity an a top level capable of 8000 kgs. 69

70 CHAPTER : DEFECTION ANAYSIS. Introuction Deflection Analyses Design of cantilever beams for their many applications often requires estimates of eflections at various length positions. The evelopment of analytical methos for estimating eflection an stress for beams in bening were evelope in the 8 th century by Euler an Bernouli an are escribe in many tetbooks [5-8]. The beam eflection y is foun by four common methos: (i) irect integration [5-8], (ii) Macaulay s step function [8, 9], (iii) Mohr s theorems [5-8] an (iv) strain energy [, ]. Both (i) an (ii) are base on the fleure equation: y M EI (.) Unless otherwise mentione, the term y use throughout the course of this chapter inicates the overall tip eflection. Deflection theorems have been eamine in brief in. followe by a brief yet substantial eamination of the four common methos of eflection analysis. This chapter has in its origins the Tip Reaction Moel propose in orer to perform eflection analysis of the three section telescopic cantilever beam assembly, as etaile in []. The analysis in [] aopts the use of Macaulay s step function metho to preict the eflection at major points within the beam assembly particularly at the points of iscontinuity. The points of iscontinuity are of course those regions wherein the sectional properties change with change in the sections themselves, for eample when moving along the beam assembly from the outer beam AB to beam CD, with the emphasis being on the overlap region CB as shown in Figure. (b), for the case of the two section telescoping cantilever beam assembly. The etails of the eflection analysis of the three section telescoping cantilever beam assembly were first propagate in an interim internal report [] an then publishe [9]. (See also Appeni E). It was eeme necessary to unerstan if other establishe eflection preiction techniques coul also be applie to the two section telescoping arrangement an herein lies the basis of the sections.5,.6 an.7 that follow. A number of factors were taken into consieration such as the possibility of varying iniviual beam lengths an variations in overlap lengths. To this en a number of ratios were use to epress the sectional properties of the beams constituting the two section assembly. Parameterisation was use to great effect in [], in orer to allow for ease of comparison between eflection preiction techniques. 70

71 . Telescopic Beam Theory In a telescopic cantilever beam one or more beams are stacke insie an outer beam which is fie at one en to support the entire beam assembly. The inner pieces move out when the application nees the full span. Generally, the assembly will have three types of beams: (a) one with en fie, (b) one with en free an (c) an an overlapping section between (a) an (b). It follows that a beam of two lengths, which inclues (a), (b) an (c) an correspons to the sections marke, an, respectively, as shown in Figure.(a), is sufficiently general for the present analysis. Thus, Figure.(b) shows, schematically, a telescoping cantilever with an overlapping length a between beams with lengths an. The loaing shown is a combination of the beams self-weights w an w an a concentrate, applie en-loa P. a (a) w N/mm w N/mm N P A C B D a =ϕ (b) Figure.: Two-section, telescopic cantilever 7

72 . Tip Reactions The tip reaction moel assumes that in a telescopic cantilever beam the overlapping ens have concentrate reactions that transmit the effects of the loas applie to the top surface of the cantilever assembly. Consier the two-section beam assembly shown in Figure.(a). The fie-beam AB has an overlap of length CB with beam CD. Tip reactions eist at the contact points C an B between beams AC an BD. In aition, Figures A. an A. in Appeni A, shows the eternal loaing applie to the assembly which is a combination of self-weight an a concentrate en-loa. Thus, each of the two-sections bears the loaing shown in Figures A. an A.. That the tip reactions must remain in equilibrium with the applie loaing enables these reactions to be foun as etaile in Appeni A.. In Figures A. an A. each beam section is shown separately as a free-boy iagram. Within each iagram the tip reactions are the forces applie to the each section from its neighbour Thus, the free-en section CD, eerts upon the fie-en section AB, a ownwar force at B an an upwar force at C (see Figure A.). The fie-en section AB, on the other han eerts equal forces upon the free en-section CD, at B an C but in opposition to these (see Figure A.). Using the tip reactions etaile in Appeni A., a C program was generate using the principle outline in., wherein the eflecte shape of each portion of the beam is provie by successive integration of Equation (.). The analyses were carrie out on a telescopic cantilever assembly consisting of two hollow sections as etaile in... Macaulay s Metho for Deflection Analysis The eflecte shape of each portion of the beam is provie by successive integration of Equation (.). The first integration gives the slope y/ an the secon integration provies the eflection epression y = y(). Constants of integration are introuce to ensure compatibility within the overlapping lengths as a similar integration process is applie to each separately an in sequence. The integration shows that the eflecte shape of AC (not incluing the overlap BC) may be epresse as a polynomial: y (.) t t t t t0 7

73 in which the coefficients t 0.. t are require to match the bounary conitions. Here, as both the slope an eflection are ero at the fiing, where = 0, then t 0 an t are both ero. The remaining coefficients are seen to epen upon the length, the loaing, an the fleural rigiity EI. A further polynomial escribes the eflection for the portion of this beam which etens into the overlap CB y t t t t t0 (.) Equation (.) must match the slope an eflection impose by the ajacent beam before it (AC). This requirement also applies to a further polynomial that escribes the eflection in the same overlap CB from within the secon beam y t t t t t0 (.) The tip reactions ientifie in A. facilitate the loa transfer between the two beams. The appene sections A., A., A.5 an A.6 show how such compatibility is ensure between the t-coefficients in Equations (.) (.) for these two portions of the length ACB. The complete analysis requires aitional equation sets given in Appeni A for the remaining beam sections, which leas to the respective Equation sets (A.)-(A.5) which contribute to the eventual etermination of the overall tip eflection, represente by Equation (A.6). The sample set of Equations (.)-(.) given here are sufficient to show how they are programme to amit a specific geometry an material. The program is then applie to preict the en-eflection of a moel telescopic cantilever. Referring to [9], we now make use of two stanar sections with the specifications liste in Table. below in orer to plot the Equation set (A.6) from A.6 against varying values of the parameter α. 7

74 Table.: Nominal imensions an sectional properties of rectangular hollow sections Section Etract from ISO/FDIS 6-:0 (E), [9] Sie H mm Bmm Thickness (T mm) Weight per unit length (N/mm) Secon Moments of Area (mm ) Section (Fie- En) Section (Free-En) Assuming lengths of the fie an free sections to be 00 mm an 000 mm respectively, an utilising the etails provie from Table., it can be etermine that the values of ϕ, β an γ are 0.8, 0.9 an 0.6 respectively. The values of tip eflection were plotte (as shown in Figure.) using the parameters require in Table. which were in turn entere using the C program... The C Program The C program etaile in Appeni B, for the two section telescopic cantilever beam assembly, marries each polynomial escription of eflection within the two beam sections. Table. shows the steps leaing to the overall tip eflection. The program is able to calculate tip eflection uner various applie loaings with ifferent combinations of overlaps. To o this it requires the geometric parameters of the telescopic beam assembly entere interactively to fin specific solutions efine by the five sets of equations given in Appeni A. Specifically it applies the acquire parameters to Equation set (A.) to obtain the tip reactions. The shape of AC is provie by Equation set (A.) from which it calculates bounary conitions to efine the shape of overlap CB between beams AB an CD. This recursive process continues until the eflecte shape of every portion is efine as was emonstrate for the three section cantilever in [] an [9]. Finally, the shape of BD is use to estimate the value for the tip eflection. 7

75 Table.: Flow Chart of the C program to calculate tip eflection Define parameters like self weight, overlap length an secon Moment of Area of the two sections. Declare variables R B, R C, I, I,,, α, w, w, P, etc. Import the inclue files. Calculate the reactions R B, R C, using Equation set (A.) in A.. Consier AC in AB. Establish equation of the shapes an calculate the magnitues of g an from Equation set (A.) in A.. Calculate g an using Equation set (A.) in A.. Consier CB in AB. Establish equation of the shapes using g an which results in Equation set (A.) in A.. Consier CB in CD. Establish equation of the shapes using g an which results in Equation set (A.) in A.5. Calculate g an using Equation set (A.) in A.5. Consier BD in CD. Establish equation of the shapes using g an which results in Equation set (A.5) in A.6. Calculate Tip Deflection using Equation set (A.6) in A.6. 75

76 Figure.: En Deflection Plot obtaine from Macaulay s Theorem vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively. y/y 0 represents the ratio of the tip eflection for a given value of w/p, which in turn varies from 0.0 to 0, in multiples of 0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (Key: w/p=0; w/p=; w/p=0.; w/p=0.0) 76

77 The graph in Figure. forms the basis for the eflection plots that have been plotte in Figures.7,.0 an.. Each of the methos use to plot eflection for the two section telescoping assembly make use of normalise parameters in orer to obtain well efine plots over a range of varying tip loas whilst varying the overlap ratios from a minimum of 0. to a maimum of, in increments of 0.. Non imensionaliation is in essence the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. In the eflection analyses carrie out in this chapter, non imensionaliation is use to further simplify an parameterise the comple equations that are obtaine an to allow for comparison between the eflection methos utilise. To illustrate this point better, refer to Figure., where the normalise parameters use are y w an. The quantity y 0 P w represents the ratio of the prouct of the self weight over the single fie en section P cantilever an the length over which it acts to the tip loa acting on the same single fie en cantilever section. y y 0 w represents the ratio of the tip eflection for a given value of which in turn varies P from 0.0 to 0, in multiples of 0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I. In other wors y0 equals P. EI 77

78 .5 Mohr s Moment Area Metho = =ϕ α Figure.: Two section telescopic cantilever The thickness of the thin-walle sections that constitute the two section telescopic cantilever shown in Figure. is assume to be constant along the beam. Of course, for the free-en section to slie within the fie-en section the former will have imensions smaller than that of the latter so iniviual imensions of breath, height an thickness are to be taken into account. The Mohr s moment area metho outline here as well as the eflection analyses that follow, are applie to the telescopic beam assembly that consists of two sections each of iffering length. For the purpose of this eflection analysis the sections are numbere from the fieen towars the free-en as shown in Figure.. As can be seen from Figure. an Figure. the overlap is consiere to be a separate section in orer to account for changes in secon moment of area an the change in self weight. 78

79 Section Section (Fie En) Section (Overlap) Section (Free En) Table.: Iniviual Rectangular Section Properties ength(mm) Self Weight Secon Moments of (N/mm) Area (mm ) = w = w I =I α w =(+γ)w I =(+β)i =ϕ = ϕ w =γw I = βi The lengths of the sections have been broken own iniviually in orer to facilitate ease of calculations. The lengths, secon moments of area an self weights are all epresse in terms of ratio an each of these in turn relate to the fie-en section. The ratios α,β an γ are assume to be less than which means that the fie-en section has the larger imensions of length, secon moment of area an self weight respectively. I, w (+β) I, (+γ) w β I, γ w (-α) = α =ϕ (ϕ-α) Figure.: Cross sectional view of the two section telescopic cantilever The moment area metho evelope by Mohr is a powerful tool for fining the eflections of structures primarily subjecte to bening. Its ease of fining eflections of eterminate structures makes it ieal for solving ineterminate structures using compatibility of 79

80 isplacement. For eample in a proppe cantilever the metho is aapte to provie the isplacement at the prop..5. Mohr s Moment Area Metho applie to the two section tip loae cantilever = P A I B (+β) I C β I D (-α) α (ϕ-α) M I P ( ) I P ( ) I P I =ϕ P( ) ( ) I P( ) I Figure.5: Mohr s Moment Area Metho applie to the two section tip loae cantilever beam (a) Derivation of Slope for the two section tip loae cantilever beam Figure.5 shows the two section telescoping arrangement subject to a tip loa having magnitue P. Taking into consieration the ifferent variables etaile in Table. an shown in Figure.5, the slope of the telescopic assembly is erive as follows. Eθ = Area of the M iagram from A-D I 80

81 8 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( I P I P I P I P I P E I P I P I P I P I P E ) ( ) ( ) ( ) ( ) ( ) ( EI P (.5) Substituting, 0 in Equation (.5) we get the slope of an equivalent single section cantilever having length an uniform secon moment of area I as EI P Again substituting in Equation (.5) we get the slope of an equivalent single section cantilever having length an uniform secon moment of area I ) ( as I E P ) ( (b) Derivation of Deflection for the two section tip loae cantilever beam Having erive the slope of the two section telescopic cantilever beam assembly as shown in Equation (.5), the eflection of the telescopic assembly is erive using the relation shown below. Ey = Moment of Area of the I M iagram from A-D, about D ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( I P I P I P I P I P Ey I P I P I P I P I P Ey ) ( ) ( ) ( ) ( ) (

82 P y ( ) (.6) EI Substituting 0, in Equation (.6) we get the stanar eflection epression for an equivalent single section cantilever having length an uniform secon moment of area I as y P EI Substituting in Equation (.6) we get the eflection for an equivalent single section cantilever having length an uniform secon moment of area ( ) I as P y E( ) I 8

83 .5. Mohr s Moment Area Metho applie to the two section cantilever subjecte to uniformly istribute loaing = A w, I B (+γ)w, (+β) C γ w, β D I (-α) α (ϕ-α) w I w I =ϕ M I w ( ) ( ) I w I w ( ) I w ( ) ( ) I Figure.6: Mohr s Moment Area Metho applie to the two section cantilever beam subjecte to uniformly istribute loaing (a) Derivation of Slope for the two section cantilever beam subjecte to uniformly istribute loaing Figure.6 shows the two section telescoping arrangement subject to a uniformly istribute loa or uner the action of the iniviual self weights of the constituent beam sections. Taking into consieration the ifferent variables etaile in Table., the slope of the telescopic assembly uner the action of its self weight is erive, by summing the areas of the three iniviual M iagrams as shown in Figure.6. I 8

84 8 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( I w I w I w I w I w E ) ( ) )( ( 6 ) ( 6 EI w EI w (.7) Substituting, 0 in Equation (.7) we get the slope of an equivalent single section cantilever having length an uniform secon moment of area I as EI w 6 Substituting in Equation (.7) we get the slope of an equivalent single section cantilever having length an uniform secon moment of area I ) ( as EI w 6 (b) Derivation of Deflection for the two section cantilever beam subjecte to uniformly istribute loaing Having erive the slope of the two section telescopic cantilever beam assembly as shown in Equation (.7), the eflection of the telescopic assembly is erive by summing the moments of the three iniviual areas of the I M from A to D, about the point D as shown in Figure.6 as outline below.

85 85 6 ) ( 6 ) ( ) 6( ) ( ) 6( ) ( ) ( 6 ) ( I w I w I w I w I w Ey ) ( ) )( ( 8 ) ( 8 EI w EI w y (.8) Substituting, 0 in Equation (.8) we get the eflection for an equivalent single section cantilever having length an uniform secon moment of area I as EI w y 8 Substituting in Equation (.8) we get the eflection for an equivalent single section cantilever having length an uniform secon moment of area I ) ( as EI w y 8.5. Derivation of Deflection for the two section cantilever beam subjecte to uniformly istribute an tip loaing Knowing the eflections of the two-section cantilever uner the action of a tip loa an uner the action of a istribute loa (in effect the self-weight) as can be seen from Equations (.6) an (.8), it is imperative to fin the combine total eflection of the two-section cantilever uner their combine action. This is foun by utilising the principle of superposition. Superposition is use to solve for beam an structure eflections of combine loas when the effects are linear, in other wors each loa oes not affect the results or actions of other loas an the effect of each loa oes not alter the geometry of the structural system significantly.

86 86 Applying the principle of superposition we get the total eflection for the combine loaing as the summation of equations (.6) an (.8) ) ( ) )( ( 8 ) ( 8 ) ( EI w EI w EI P y (.9) Substituting, 0 in Equation (.9) we get the eflection for an equivalent single-section cantilever having length an uniform secon moment of area I uner the action of a combine istribute loaing an tip loa as EI w EI P y 8 ) ( Substituting in Equation (.9) we get the eflection for an equivalent single section cantilever having length an uniform secon moment of area I ) ( as EI w EI P EI w EI P y 8 8 Equation (.9) can be re written as ) ( ) )( ( ) ( 8 ) ( P EI EI w EI P y ) ( ) )( ( ) ( 8 ) ( P w EI P y (.0)

87 P The term represents the en-eflection of a cantilever of length an fleural rigiity EI EI, subjecte to a tip loa P Newton. This term is represente by the variable y0 where y y 0 y 0 P. Equation (.0) now becomes EI w 8 P ( )( ) ( ) ( ) ( ) (.) The curves in Figure.7 represent the plots of the non imensional parameter y y 0 against the overlap ratio parameter α, for w y ratios 0.0, 0., an 0. The non imensional ratio is P y0 obtaine from Equation (.) by substituting the values of ϕ, β an γ as 0.8, 0.9 an 0.6 respectively. Table. outlines how the ratios ϕ, β an γ can be erive using the specifications liste in Table.. Whilst keeping the aforementione ratios a constant, the overlap ratio parameter α is varie in increments of 0., from 0 to. 87

88 Figure.7: En Deflection Plot of Equation (.) obtaine from Mohr s Moment Area Theorem vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively. y/y 0 represents the ratio of the tip eflection for a given value of w/p which in turn varies from 0.0 to 0, in multiples of 0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (Key: w/p=0; w/p=; w/p=0.; w/p=0.0) 88

89 .6 Castigliano s Theorem Castigliano s metho is use for etermining the isplacements of a linear elastic system base on the partial erivatives of strain energy. The first of Castigliano s theorem etermines the forces in a elastic structure whereas the secon theorem etermines the isplacements in a form that is applie here. As is etaile in.7.7, if the structure is linearly elastic, then the partial erivative of the strain energy of that system with respect to the applie force system is equal to the corresponing isplacements prouce. I, w (+β) I, (+γ) w β I, γ w (-α) = α =ϕ (ϕ-α) Figure.8: Cross sectional view of the two section telescopic cantilever Referring to Figures.8 an.9, Castigliano s theorem, part, can be applie to each of the three sections that make up the composite telescopic cantilever beam assembly in orer to give the total isplacement y as follows ( ) ( ) M M M M M.. U M y.... (.) P EI P EI P EI P 0 ( ) Figure.9 makes reference to the symbol ', where ', this in turn is the basis for the Equation (.). The subscripts, an represent the free en, overlap area an the fie en respectively. Prior to evaluating the integral in Equation (.) the bening moments M, M an M at each of the three sections starting with the free en to the fie en are 89

90 evaluate an their respective erivatives with respect to the tip loa P are foun as shown in Equation sets (.)-(.5) an then substitute in Equation (.) as outline below. = P A w, I B (+γ) w, (+β) I C γ w, β I D (-α) α (ϕ-α) =ϕ = F V A B w, I (+γ) w, (+β) I C γ w, β I D (-α) α (ϕ-α) =ϕ '=/ Figure.9: Castigliano s Theorem applie to the two section cantilever beam The following moment equations are as mentione above epresse in terms of where '. ' Bening moment for the section 0 ' M P w. P w P ' w ' M Derivative of M with respect to P is ' P (.) 90

91 9 Bening moment for the section ' ) ( ' ' '. w w P w w P w w P M ' ' ' w P M Derivative of M with respect to P is ' P M Bening moment for the section ' w w P w w P M ' ' ' w P M Derivative of M with respect to P is ' P M Substituting Equation sets (.), (.), (.5) an ' in the Equation (.) for y gives: ' '. ' ' ' ' '. ' ' ' ' '. ' ' ) ( 0 w P w P w P EI y (.) (.5)

92 9 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ) ( 0 w P w P w P EI y et X then integrating the terms gives X X X X X X X X w P X X w P w P EI y 0 0 ' ' ' ' ' ' ' ' ' ' Applying the limits now yiels X X X X X X X w X P EI X X X X X w X P EI X w X P EI y

93 X X X X X X X X X X X X X EI w X X X EI P y The final equation can be written as X X X X X X X X X X X X EI w X X X EI P y (.6) where X Substituting, 0 in Equation (.6) we get the eflection for an equivalent single section cantilever having length an uniform secon moment of area I uner the action of a combine istribute loaing an tip loa as EI w EI P y 8 ) ( Substituting in Equation (.6) we get the eflection for an equivalent single section cantilever having length an uniform secon moment of inertia I ) ( as EI w EI P EI w EI P y 8 8

94 9 Equation (.6) can be re written as X X X X X X X X X X X X P EI EI w X X X EI P y X X X X X X X X X X X X P w X X X EI P y (.7)

95 95 The term EI P is represente by the variable 0 y such that EI P y 0 for similar reasons as mentione in.5.. Equation (.7) is written in its entirety as (.8) The curves in Figure.0 represent the plots of the non imensional parameter y 0 y against the overlap ratio parameter α, for P w ratios 0.0, 0., an 0, calculate using Castigliano s theorem, Part. As was the case for Macaulay s theorem the values of the constant parameters ϕ, β an γ are taken as 0.8, 0.9 an 0.6, respectively an substitute into Equation (.8), while varying the overlap ratio α from 0 to in increments of P w y y

96 Figure.0: Deflection Plot of Equation (.8) obtaine from Castigliano s Theorem vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively. y/y 0 represents the ratio of the tip eflection for a given value of w/p, which in turn varies from 0.0 to 0, in multiples of 0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (Key: w/p=0; w/p=; w/p=0.; w/p=0.0) 96

97 .7 Virtual Work Principle The principle of virtual work states that for any compatible virtual isplacement fiel impose on the boy in its state of equilibrium the total internal virtual work is equal to the total eternal virtual work [6]. The isplacement fiel impose on the boy is calle virtual because they nee not be obtaine by a isplacement that actually occurs in the system. The total virtual work is in effect the work one by the virtual isplacement fiel which can be arbitrary provie they are consistent with the constraints of the system. In the case of the two section telescopic cantilever the total virtual work one is epresse in terms of the parameter F V which is a virtual force as shown in Figure. below. This parameter along with the virtual moment M V allows for the subsequent etermination of the actual overall isplacement of the two section cantilever assembly uner the combine action of a tip loa an self weight. = P A w, I B (+γ) w, (+β) I C γ w, β I D (-α) α (ϕ-α) =ϕ F V A B w, I (+γ) w, (+β) I C γ w, β I D (-α) α (ϕ-α) ϕ =/ Figure.: Principle of Virtual Work applie to the two section cantilever beam 97

98 98 As etaile in.7., when consiering the cantilever beam assembly to be in equilibrium an on subjecting it to a virtual isplacement fiel, the internal virtual work inuce or the strain energy store within the assembly must equal the eternal virtual work. Referring to Figure., the principle of virtual work can be applie to each of the three sections that make up the composite telescopic cantilever beam assembly in orer to give the total virtual work epresse in terms of the prouct of the virtual force parameter F V an the isplacement y as ) ( 0 ) ( ) ( V V V V V EI M M EI M M EI M M EI MM yf (.9) Figure. makes reference to the symbol ', where ', this in turn is the basis for the Equation (.9). The subscripts, an represent the free-en, overlap area an the fieen respectively. As was the case in Castigliano s theorem, prior to evaluating the integral in Equation (.9) the bening moments at each of the three sections starting with the free-en to the fie-en are evaluate an their respective erivatives with respect to the tip loa P are foun as shown in Equation sets (.0)-(.) an then substitute in Equation (.9) as outline below. Also the virtual moments are evaluate an foun to be equal to the prouct of the virtual force parameter F V an the istance from the free en. Bening Moment for the section ' 0 ' '. w P w P w P M (.0) Bening Moment for the section ' ) ( ' ' '. w w P w w P w w P M ' ' ' w P M (.)

99 99 Bening Moment for the section ' w w P w w P M ' ' ' ' ' ' w w P w w P ' ' ' w P M (.) ' F F F M M M V V V V V V (.) Substituting Equation sets (.0)-(.) an ' in the Equation (.9) for V yf gives: ' ' ' ' ' ' ' ' ' ' ' ' ' ' ) ( 0 F w P F w P F w P EI yf V V V V ' ' ' ' ' ' ' ' ' ' ' ' ' ' ) ( 0 w P F w P F w P F EI yf V V V V

100 00 et X then integrating the terms gives X X X X X X V V X X w P X X w P w P EI F yf 0 0 ' ' ' ' ' ' ' ' ' ' Applying the limits now yiels X X X X X X X w X P EI X X X X X w X P EI X w X P EI y X X X X X X X X X X X X X EI w X X X EI P y

101 0 The final equation can be written as X X X X X X X X X X X X EI w X X X EI P y (.) where X Substituting, 0 in Equation (.) we get the eflection for an equivalent single section cantilever having length an uniform secon moment of area I uner the action of a combine istribute loaing an tip loa as EI w EI P y 8 ) ( Substituting in Equation (.) we get the eflection for an equivalent single section cantilever having length an uniform secon moment of area I ) ( as EI w EI P EI w EI P y 8 8

102 0 Equation (.) can be re written as X X X X X X X X X X X X P EI EI w X X X EI P y X X X X X X X X X X X X P w X X X EI P y (.5)

103 0 Again the term EI P y 0 for reasons as mentione in.5.. Equation (.5) is written in its entirety as (.6) Once again the curves in Figure. represent the plots of the non imensional parameter y 0 y against the overlap ratio parameter α, for P w ratios 0.0, 0., an 0, calculate using the Virtual Work Metho. As before, the values of the constant parameters ϕ, β an γ are taken as 0.8, 0.9 an 0.6, respectively an substitute into Equation (.8), while varying the overlap ratio α from 0 to in increments of P w y y

104 Figure.: Deflection Plot of Equation (.6) obtaine from Virtual Work Theorem vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively. y/y 0 represents the ratio of the tip eflection for a given value of w/p, which in turn varies from 0.0 to 0, in multiples of 0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (Key: w/p=0; w/p=; w/p=0.; w/p=0.0) 0

105 Figure.: Theoretical En Deflection Plots vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively.. y/y 0 represents the ratio of the tip eflection for a given value of w/p, which in turn varies from 0.0 to 0, in multiples of 0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (w/p=0: Macaulay s Theorem; Mohr s Metho; Castigliano s Theorem an Virtual Work Metho w/p=: Macaulay s Theorem; Mohr s Metho; Castigliano s Theorem an Virtual Work Metho w/p=0.: Macaulay s Theorem; Mohr s Metho; Castigliano s Theorem an Virtual Work Metho w/p=0.0: Macaulay s Theorem; Mohr s Metho; Castigliano s Theorem an Virtual Work Metho) 05

106 Figure. (a): Theoretical En Deflection Plots vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively for a w/p ratio of 0. y/y 0 represents the ratio of the tip eflection for a w/p ratio of 0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (Key: w/p=0: Macaulay s Theorem; Mohr s Metho; Castigliano s Theorem an Virtual Work Metho) 06

107 Figure. (b): Theoretical En Deflection Plots vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively for a w/p ratio of. y/y 0 represents the ratio of the tip eflection for a w/p ratio of, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (Key: w/p=: Macaulay s Theorem; Mohr s Metho; Castigliano s Theorem an Virtual Work Metho) 07

108 Figure. (c): Theoretical En Deflection Plots vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively for a w/p ratio of 0.. y/y 0 represents the ratio of the tip eflection for a w/p ratio of 0., to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (Key: w/p=0.: Macaulay s Theorem; Mohr s Metho; Castigliano s Theorem an Virtual Work Metho) 08

109 Figure. (): Theoretical En Deflection Plots vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table., an fie an free-en lengths of 00mm an 000mm respectively for a w/p ratio of 0.0. y/y 0 represents the ratio of the tip eflection for a w/p ratio of 0.0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, such that y0 equals (P /EI). (Key: w/p=0.0: Macaulay s Theorem; Mohr s Metho; Castigliano s Theorem an Virtual Work Metho) 09

110 .8 Summary The two section telescopic cantilever beam assembly has been subjecte to eflection analyses using the four common eflection preiction techniques which have each been tailore to suit the unique nature of the structure. This has been one using a combination of ratios as outline in Table.. The curves isplaye in Figures.,.7,.0 an. represent the eflection curves, prouce as a result of subjecting the two section telescopic cantilever beam assembly, to a combination of both tip loaing an the action of its own weight, using Macaulay s theorem, Mohr s Moment Area Metho, Castigliano s Theorem an the Virtual Work Principle, respectively. It can be seen that the curves generate using Castigliano s Theorem an the Virtual Work Principle are eactly the same. It must be remembere that the four eflection preiction techniques have been applie to the caniate telescopic assembly having imensions specifie in Table., for fie an free-en lengths of 00 mm an 000 mm respectively. The reason for this was simply to see how an what effect varying lengths of the iniviual sections woul have on the overall eflection curves thus generate by the analyses. The eflection curves prouce for the caniate assembly, using the four methos, are compare in Figure.. The following are some of the observations that can be mae from Figure.:. The curves generate using Castigliano s theorem an the Virtual Work metho are one an the same.. For lower w ratios, the curves generate by Mohr s, Castigliano s an the Virtual P Work methos, are inseparable as can be evience from Figures. (b) (), with the eception of the curves obtaine from the Macaulay s theorem analyses. However for a w ratio of 0, the eflection curves are sprea apart, revealing a clear P ifference as shown in Figure. (a). The Macaulay s theorem generate curve has a greater magnitue as compare to the other eflection preiction curves, followe by Mohr s theorem an finally both, Castigliano s an the Virtual Work metho.. The eflection curve generate from Macaulay s theorem, starts not at α=0, unlike the other curves, but at α=0.. This is a consequence of the numerical implementation 0

111 scheme aopte i.e. the tip reaction moel which was aapte into an equivalent C program.. Irrespective of which metho was use to plot a given curve, for any curves meet at the same point, which coincies to the conition α=. w ratio, all the P 5. The ifference between Macaulay s theorem an the three other methos coul be eplaine as being ue to the reliance of the former on the tip reactions calculate for the varying overlap lengths, of which the latter three in turn are inepenent. Of greater interest arguably, is the ifference between the eflection curves obtaine from Mohr s metho an those generate from both Castigliano s an the Virtual Work Metho. The eflection magnitues erive using Castigliano s metho an the Virtual Work metho are the same. This is because the equations use to generate the respective curves are the same as is evient from Equations (.8) an (.6). Comparing either of these equations mentione with Equation (.) erive using Mohr s metho reveals a significant ifference in the secon term which accounts for the uniformly istribute loaing or self weight of the two section telescoping assembly. This ifference in the case of the Mohr s metho analysis is in part ue to the changing moments of areas that are taken into consieration when computing the eflection as elaborate in.5. an shown iagrammatically in Figure.6. The curves generate in Figures. an separately in Figures.(a)-() are eplaine in more etail in 9... The final objective of this chapter was to erive from first principles, eflection equations that woul preict the in-line eflection inuce in the two section telescopic cantilever beam assembly effectively, for any applie loa. Not only o these equations preict the eflections, they also take into account all the possible variable factors of the structure. Also esire was a means of isplaying the eflection curves, such that a particular configuration of the two section telescoping assembly can be selecte, to serve a given function, provie the require ata is available. ast but certainly not least, the equations use to generate the efection curves, are applie to the actual test rig, whose imensions are outline in Table 8., an then compare, with the eperimental an FEA etracte ata, for the same.

112 CHAPTER : BUCKING ANAYSIS. Introuction In the two section telescoping cantilever buckling occurs in two parts (a) local buckling prouce within the iniviual rectangular hollow sections an (b) global buckling wherein the structure in its entirety unergoes buckling. To establish a sense of orer in this thesis, global buckling phenomena of the telescoping assembly is first scrutinise here, followe by local buckling in the iniviual rectangular hollow sections in the following chapter. The critical buckling loa is a function of the section imensions, namely: (a) the effective length of the column structure; whether it is the length of the iniviual sections, or, that of the assembly an (b) the bounary conitions. In both cases the member(s) are eamine as cantilever-columns, in that one en is fie an the other is free. This is ue to the fact that the applications of the telescoping assembly relate to this bounary conition. Another point of importance is the overlap where the free en or male section fits snugly within the fie en or female member. This overlap region is of great importance, as the theory evelope in this chapter, has to account for this iscontinuity. It was etermine that to account for this overlap region, it shoul be treate as a separate section to the free-en an fie-en sections. In orer to o so, it is vital to take into consieration the beefing up or bolstering of the secon area moments in this region as being a summation of the moments of the female an male members of the telescoping assembly. In orer to arrive at the critical buckling loa for the assembly it was imperative to unerstan the concepts of bening strain energy an total potential energy systems. It was eeme appropriate to start at the simplest approimation to a single rectangular hollow section by firstly consiering tapering sections. By working from first principles a better unerstaning of the compleity of the problem was attaine. This involve a etaile unerstaning an application of energy methos to the case at han. The principles use are outline in.7. In particular, the Rayleigh quotient erive in., an referre to by Timoshenko as the energy metho [5], was use as a basis an applie in the analysis of the cantilever column in. an in subsequent critical buckling loa eterminations for the sections thereafter. This approach in turn was verifie by valiating the result obtaine against the criteria for the Euler critical buckling loa for a column having one en fie an the other free as the bounary conitions. A general form for preicting the critical buckling loa eactly is thus erive an is applie to suit each case iniviually taking into account the ifferent cross sectional secon moments an lengths.

113 . Determining the section parameters of the tapere column Figure.: Geometry of the tapere circular cantilever column Referring to Figure., the iameter at the arbitrarily chosen epth of from the free en can be epresse as follows using the principles of similar right angle triangles e o e e o e o e o e o (.) Knowing the epression for iameter at the epth from Equation (.) an once again referring to Figure., the secon moment area at a length of, along the length of the tapere circular cantilever column can be epresse as follows e o e o e

114 6 6 6 e o e e o e I e o e I I (.) Figure.: Geometry of the tapere circular cantilever column 6 P P I My (.) Substituting Equation (.) in Equation (.) we get o e o e o o e o e o P P (.) Equation (.) can also be written as o e o e o P (.5) P o e

115 5 Equation (.5) has a maimum value which is etermine later within in this section. Comparing Equation (.) an Equation (.5) we see that o = o P. et =, then ifferentiating Equation (.5) with respect to gives ' ' o e o e o ' ' ' ' ' o e o e o e o e o e o o o e o e Equating the ifferentiate quantity to ero gives 0 ' ' ' o e o e o e o o o e o e 0 ' ' ' o e o o e o e o o e o e (.6) Equation (.6) gives two quantities that can be equate to ero: Either 0 ' o e o e which gives a negative an is of no consequence to the analysis at han Or 0 ' ' o e o o e o e o Solving the above epression gives 0 ' ' o e o e o e

116 e o ' (.7) e o Figure. graphically epresses the relationship between e o an ', in Equation (.7). The curve represents the relationship between e o an ', when e o, for values of e o varying from 0. to 0.6 in increments of 0.. Figure.: Plot of e o against ' from Equation (.7), for the tapere circular cantilever column The maimum bening stress in the tapere circular column can be foun by substituting Equation (.7) in Equation (.5) as follows 6

117 7 o e o e o e o e o e o e o o e o e o e o e o o e o e P (.8) Figure (.) represents the relationship between o against ', with the curve plotting the relationship for the conition o e for values of o e varying from 0. to 0.6 in increments of 0.. Figure.: Plot of o against ' from Equation (.8), for the tapere circular cantilever column (σ/σo)

118 Figure.5: Plot of (obtaine from Equation (.5)) against ', where ' varies from 0 to, in increments of 0., for the tapere circular o cantilever column. The curves in turn represent the values of e varying from 0. to 0.9, once again in increments of 0.. o 8

119 . Section Properties of Tapere Beams It is assume that the sections at the two ens of a beam are the same in the shape an ifferent in sie. The sie of the section changes linearly with respect to the beam centre of geometry line. Figure.6 shows four tapering types for a linearly tapere beam: (a) the with (horiontal imension) changes only; (b) the height (vertical imension) changes only changes only; (c) both imensions change at the same rate; () the two imensions change at ifferent rates. The ifferent tapering types can be categorise base on the characteristics of the algebra of the changing section properties. b h b e h e y h h o b o b (a) (b) b e h e b e h e h o h o b o b o (c) () Figure.6: Section Properties of Tapere Beams (a) Section changing breath; (b) Section changing epth; (c) Section changing bi-imensionally at the same rate; () Section changing bi-imensionally at ifferent rates (Aapte from [9]) It shoul be note that the tapering type oes not necessarily correspon to the tapering category, since a tapering type may result in ifferent property changes for ifferent section 9

120 shapes. The sections are assume to be aisymmetric. The shape functions are erive in two imensions. It can also easily be epane into three imensions... Section changing breath h b e b b b e h b o b o b e h Figure.7: Geometry of the tapere rectangular cantilever column whose cross section changes in breath Referring to Figure.7, the breath/with (horiontal imension) b, at an arbitrarily chosen epth of from the free en can be epresse as follows using the principle of similar right angle triangles b b e b b o e b be bo be (.9) Knowing the epression for the breath/with (horiontal imension) at an arbitrarily chosen epth of from the free en as obtaine from Equation (.9) an once again referring to 0

121 Figure.8, the secon moment area at a length of, along the length of the tapere rectangular cantilever column from the free en can be epresse as follows e o e e o e e o e b b I b b h b b b b h h b I (.0).. Section changing epth Figure.8: Geometry of the tapere rectangular cantilever column whose cross section changes in epth Referring to Figure.8, the height/epth (vertical imension) h an arbitrarily chosen epth of from the free en can be epresse as follows using the principle of similar right angle triangles h h h h e o e e o e h h h h (.) Knowing the epression for the height/epth (vertical imension) at an arbitrarily chosen epth of from the free en as obtaine from Equation (.) an once again referring to Figure h e h h e h o h e b b h b h o

122 .8, the secon moment area at a length of, along the length of the tapere rectangular cantilever column from the free en can be epresse as follows e o e e o e e o e h h I h h bh h h h b bh I (.).. Section changing bi-imensionally at the same an at ifferent rates From Equations (.9) an (.) the variation in with an epth can be shown to be e o e b b b b e o e h h h h Referring to.. an.. as well as Figures.6(c) an.6() it is not ifficult to ascertain that the secon moment area can be epresse as e o e e o e Z h h h b b b h b I e o e o e e Z h h b b h b I e o e o e Z h h b b I I (.)

123 . The Cantilever Column The following section makes use of what was referre to by Timoshenko as the energy metho, which provies a shortcut to obtaining approimate values for the critical loa [5]. It avois solving ifferential equations an becomes very useful when applie to systems with non uniform stiffness, a case where the solution to the usual Eigen-bounary-value problem is etremely ifficult an in some cases impossible. Consier the cantilever shown in Figure.9 below uner a constant irectional thrust P applie quasistatically. As the loa is increase from ero, the work one by the force P is store into the system as stretching strain energy. On allowing a small bening formation Y(), that oes not alter the aforementione stretching strain energy, the change in the total potential energy ΔU T is given by U U U (.) T B P where U B is the bening strain energy an U P force or the work one by the loa P on the cantilever column. is the change in the potential of the eternal U B 0 EI Y' ' (.5) U P P P Y ' (.6) 0 Accoring to Timoshenko s argument, the configuration is in stable equilibrium when U 0. In aition it is require to assume a form for the amissible bening T eformation, Y(). et Y( ) cos (.7)

124 P λ Y () P δ Figure.9: Geometry of the cantilever column (Aapte from [5]) where the right han sie term cos satisfies the kinematic bounary conitions at =0. Solving Equations (.5) an (.6), for the bening strain energy an the work one on the column by the loa, respectively, we get U B EI 6 (.8) U P P 6 (.9) Equating Equations (.8) an (.9) we obtain the epression for critical buckling loa as P cr EI which is the eact solution because the eformation function happens to be the eact Eigen function.

125 .5 Determination of the Buckling oa for an Aially Symmetric Truncate Cone δ P P cr λ y o Y () e (a) (b) Figure.0: (a) A fie-free column subjecte to a tip loa (b) Cross sections of an aially symmetric truncate cone (Aapte from [9]) The analyses that follow make use of the erivations outline in the previous section. The variables use have been altere. Referring to Figure.0(a), a fie-free column is subjecte to a tip loa having cross section as shown in Figure.0(b). an U P U B is the bening strain energy is the change in the potential of the eternal force or the work one by the loa P on the tapere cantilever column. Once again the change in the total potential energy ΔU T is given by U T U B U P Taking U 0 as per the argument mae in. we get T U (.0) B U P 5

126 where U B is the bening strain energy an this time U P is the work one by the loa P on the truncate cone. It is important to note that the isplacement function Y() use here is an approimation an oes not represent the eact isplacement configuration. Hence the solution obtaine as a result is approimate. U B 0 M EI Z (.) U P Y P P (.) 0 The vertical movement of the loa P uring buckling is foun from []. 0 Y Once again as in. we make use of an assume eflecte shape that accounts for first moe buckling an satisfies the bounary conitions for a fie-free column, such that Y( ) cos (.) where Y=0 at =0 an y=δ at =. Preicting the loa which will cause the column to buckle is one either by numerical or finite element techniques. An eact metho is to solve the ifferential equation Y EI M 0, (.) with appropriate bounary conitions for the ens. Now substituting M Y EI (.5) 6

127 7 in Equation (.) an using Equation (.) yiels B I E U 0 cos (.6) The critical buckling conition occurs when Equation (.0) is satisfie an this yiels the general formula for the critical buckling loa as I E P cr 0 cos (.7) Using Equation (.) an substituting in Equation (.7) gives I E P e o e cr 0 cos (.8) EI P e o e cr 0 cos (.9) To evaluate the integral in Equation (.9) above it is assume that ' an e o. Substituting these parameters gives ' ' cos ' 0 EI P e cr ' ' cos ' 0 EI P e cr ' ' cos ' 0 EI P e cr (.0)

128 EI In Equation (.0) above it can be seen that the term e represents the Euler Critical Buckling oa for a fie-free column an so is represente as P Eu P. The quantity P obtaine is a non imensional parameter that allows for ease of plotting in terms of the variable quantity o e. cr Eu thus P P cr Eu 0 ' cos ' ' Now integrating we get the solution as P P cr Eu (.) o where. e o If 0 = = e then the ratio, which correspons to a uniform non-tapering section then e EI e EI Equation (.) woul give Pcr PEu where e for a column with one en fie an one en free. is the Critical Euler Buckling loa Figure. represents the plot of P P cr Eu erive in Equation (.), for an aially symmetric truncate cone against the non imensional parameter which varies from to 5 in increments of 0.5. The objective of plotting such a curve is to present a means to ascertain the critical buckling loa of an aially symmetric, truncate cone by making use of its imensions o an e as well as the magnitue of the loa applie P at its en. The imensions o an e inicate the iameters of the base an en of the truncate cone respectively, as shown in Figure.0 (b). The curve reveals that the greater the value of o as compare to e, the greater 8 o e

129 is the critical buckling loa that the structure can withstan. Note that P Eu is the critical buckling loa for a uniform section of iameter e having one en fie an the other en free. 9

130 Figure.: Plot of Equation (.) vs. o e, for the aially symmetric, truncate cone 0

131 .6 Determination of the Buckling oa for the Pyrami.6. Buckling oa for the Rectangular Pyrami whose sections breath changes Figure.: (a) A fie-free column subjecte to a tip loa (b) Cross sections of columns for a rectangular pyrami whose sections breath changes Referring to Figure.(a), a fie-free column is subjecte to a tip loa having cross section changing as shown in Figure.(b). Knowing that the critical buckling conition occurs when Equation (.0) is satisfie, this yiels the general formula for the critical buckling loa as I E P cr 0 cos (.) Using Equation (.0) an substituting in Equation (.) gives b b h b E P e o e cr 0 cos b b h b E e o e 0 0 cos. cos (.) P P cr Y () λ (a) δ h b o h b e y (b)

132 Integrating the Equation (.) yiels P cr EI e b o b o be be (.) EI In Equation (.) above it can be seen that the term e represents the Euler Critical Buckling oa for a fie-free column an so is represente as P Eu P. The quantity P obtaine is a non imensional parameter that allows for ease of plotting in terms of the variable b quantity b o e b o. If b 0 =b =b e then the ratio,which correspons to a uniform non-tapering be EI e EI section then Equation (.) woul give Pcr PEu where e is the Critical Euler Buckling loa for a column with one en fie an one en free. Figure. graphically represents the plot of P P cr Eu cr Eu thus erive in Equation (.), for the square pyrami whose breath b changes while its epth remains constant, against the non imensional parameter b o e which varies from to 0 in increments of. The objective of plotting such a curve is to enable a esigner a way to etermine the critical buckling loa of this structure by making use of its imensions b o an b e as well as the magnitue of the loa applie P at its en. The imensions b o an b e inicate the breath imensions of the base an en of the pyrami, as shown in Figure. (b).

133 Figure.: Plot of Equation (.5) vs b b o e, for the truncate, rectangular pyrami, whose breath changes

134 .6. Buckling oa for the Rectangular Pyrami whose sections epth changes Figure.: (a) A fie-free column subjecte to a tip loa (b) Cross sections of columns for a rectangular pyrami whose section changes epth Referring to Figure.(a), a fie-free column is subjecte to a tip loa having cross section changing as shown in Figure.(b). The general formula for the critical buckling loa is epresse as I E P cr 0 cos (.5) Using Equation (.) an substituting in Equation (.5) gives h h h b E P e o e cr 0 cos (.6) h h EI P e o e cr 0 cos (.7) To evaluate the integral in Equation (.7) above it is assume once again that ' an e o h h. Substituting these parameters yiels b h e b h o (b) y P P cr Y () λ (a) δ

135 EI e ' Pcr 0 ' cos ' EI e Pcr 0 ' cos ' ' EI e Pcr 0 ' cos ' ' (.8) EI In Equation (.8) above it can be seen that the term e represents the Euler Critical Buckling oa for a fie-free column an so is represente as P Eu obtaine is a non imensional parameter that allows for ease of plotting. P P cr Eu 0 ' cos ' ' P. The quantity P (.9) Solving this integral we get cr Eu thus P P cr Eu (.0) h o where. he ho If h 0 =h =h e then the ratio which correspons to a uniform non-tapering section then he EI e EI Equation (.8) woul give Pcr PEu where e is the Critical Euler Buckling loa for a column with one en fie an one en free. Figure.5 graphically represents the plot of P P cr Eu erive in Equation (.0), for the square pyrami whose epth changes with h constant breath, against the non imensional parameter h which varies from to 5 in increments of 0.5. The objective of plotting such a curve is to etermine the critical buckling loa of this structure by making use of its imensions h o an h e as well as the magnitue of the loa applie P at its en. The imensions h o an h e inicate the epth imensions of the base an en of the pyrami, as shown in Figure. (b). 5 o e

136 Figure.5: Plot of Equation (.) vs h h o e, for the truncate rectangular pyrami, whose epth changes 6

137 7 Comparing Figures. an.5 reveals the simple fact, that a square pyrami whose section changes in epth, will have a greater critical buckling loa or be able to withstan a greater loa just before it buckles, as compare to that pyrami whose section changes in breath. Figure.5 shows that for a square pyrami changing in epth the greater the value of h o as compare to h e, the greater is the critical buckling loa, whereas Figure. shows that even a substantial increase in b o as compare to b e oes not result in as a great a magnitue of critical buckling loa..6. Buckling oa for the Rectangular Pyramis whose sections change biimensionally at the same an ifferent rates The general formula for the critical buckling loa is epresse as I E P cr 0 cos (.) Using Equation (.) an substituting in Equation (.) gives h h b b I E P e o e o e cr 0 cos (.) h h b b I E P e o e o e cr 0 cos h h b b I E P e o e o e cr 0 cos h h b b EI P e o e o e cr 0 cos (.) To evaluate the integral in Equation (.) above it is assume once again that ' e o b b an e o h h. Substituting the above parameters yiels

138 EI e Pcr 0 ' ' cos ' ' EI e Pcr 0 ' ' cos ' ' (.) EI In Equation (.5) above it can be seen that the term e represents the Euler Critical Buckling oa for a fie-free column an so is represente as P Eu be written as. Equation (.) can now P P cr Eu 0 ' ' cos ' ' Integrating by parts yiels P P cr Eu (.5) b o h o where an b. e he b o ho If both b 0 =b =b e an h 0 =h =h e then the ratios which correspons to a uniform be he EI e EI non-tapering section then Equation (.5) woul give Pcr PEu where e is the Critical Euler Buckling loa for a column with one en fie an one en free. Figure.6 shows the plot of P P cr Eu erive in Equation (.5), for the square pyrami whose section changes bi-imensionally, either at the same or ifferent rates, against the non imensional parameters an. In Figure.6, both the parameters vary at the same rate, by an increment of 0.5, between 0 an. The objective of plotting such a curve is to etermine the 8

139 critical buckling loa of such a structure by making use of the parameters an as well as the magnitue of the loa applie at its ape, P. b o h o Knowing that an b, Table. shows that for a square pyrami, e he whose ψ changes at a much greater rate than that of η, the critical buckling loa such a structure can withstan is much greater than if a given square pyrami s η was greater than its ψ. Put simply that square pyrami whose epth varies at a greater rate as compare to its breath, is capable of bearing a greater critical buckling loa, than that pyrami whose breath variation is greater than its epth variation. 9

140 Figure.6: Plot of Equation (.6) vs.,, for the truncate rectangular pyrami, whose section changes bi-imensionally at the same rate 0

141 Table.: Comparison of Critical buckling loas for a rectangular pyrami, whose section changes bi-imensionally at ifferent rates Section changing bi-imensionally such that variation of ψ > η Section changing bi-imensionally such that variation of η > ψ η ψ P cr /P Eu η ψ P cr /P Eu

142 .7 Buckling oa for the single Thin walle Rectangular Section = Figure.7: Geometry of a thin walle rectangular section A fie-free column is subjecte to a tip loa having cross section as shown in Figure.7. The thickness of the thin-walle sections is assume to be constant along the beam. The secon moment of area of the rectangular tubular section shown in Figure.7 is epresse as I Z th tbh (.6) 6 The first parameter in Equation (.6) refers to the moment area of the vertical components an the secon is of the horiontal components. The general formula for the critical buckling loa is epresse as E Pcr IZ cos 0 Substituting Equation (.7) in the Equation (.8) gives the epression (.7) E th th b e Pcr cos 0 6

143 P cr EI Z EI Z EI Z cos cos 0 0 P cr E th 6 tbh which is the Critical Euler Buckling loa for a column with one en fie an one en free..8 Buckling oa for the Single Steppe Strut = =ϕ Figure.8: Single steppe strut The single steppe composite strut shown in Figure.8 is the first step towars the ultimate goal of ientifying the critical buckling loa for the two section thin walle telescopic cantilever. For the etermination of the buckling loa of the single steppe cantilever shown, the problem in question nees to aress the varying lengths of the sections that together make the composite assembly. To this en, the erivation of the equation for etermining the buckling loa (Equation (.7)) must be aapte to account for the (a) overall length of the composite assembly, (b) iniviual lengths of the fie an free en sections an finally, (c) the iffering secon moments of area. For ease of further calculations an also to obtain nonimensional parameters, ratios are use to relate the efining parameters of the loa-applie or

144 free-en section to the fie-en section, as emonstrate in Table. an Figure.8. The fie-en an free-en sections are enote by an respectively in Figure.8. Table.: Iniviual rectangular section properties Section ength(mm) Secon Moments of Area (mm ) Section (Fie En) = I =I= b h Section (Free En) =ϕ = ϕ I =β I = β I= b h For the purpose of this buckling analysis the sections are numbere from the fie en towars the free en as shown in Figure.8, an in the section that follows. The lengths of the sections have been consiere iniviually in orer to facilitate ease of calculations. The lengths an secon moments of area are all epresse in terms of ratios an each of these in turn relate to the fie-en section. The ratios α an β are assume to be less than which means that the fie en section has the larger imensions of length, secon moment of area an self weight respectively. Consier the single steppe composite strut as being simplifie as shown in Figure.9 below uner a constant irectional thrust P applie quasi statically. As the loa is increase from ero, the work one by the force P is store into the system as stretching strain energy. On allowing a small bening formation w(), that oes not alter the afore mentione stretching strain energy, the change in the total potential energy ΔU T is given by U T U B U P Taking U 0 as per the argument mae in. we get T U (.8) B U P

145 where U B is the bening strain energy an this time U P the single steppe composite strut. is the work one by the loa P on U B 0 M EI Z (.9) U P Y P P (.50) 0 Once again as in. we make use of an assume eflecte shape that accounts for first moe buckling an satisfies the bounary conitions for a fie-free column. It must be remembere that the overall length of the composite section is. Accounting for this gives Y cos (.5) where Y=0 at =0 an Y=δ at = (+ϕ). δ P P cr λ (+ϕ) Y () Figure.9: Geometry of the cantilever column 5

146 This means that the assume eflection shape cos satisfies the kinematic bounary conitions at both =0 an = (+ϕ). Preicting the loa which will cause the column to buckle is traitionally one by numerical or finite element technique. An eact metho is to solve the ifferential equation Y EI M 0, (.5) with appropriate bounary conitions for the ens. Now substituting M Y EI (.5) in Equation (.9) an using Equation (.50) yiels U B E I cos 0 (.5) The critical buckling conition occurs when Equation (.8) is satisfie an this yiels the general formula for the critical buckling loa as E Pcr I cos 0 (.55) I β I P ϕ Figure.0: Cross sectional view of the single steppe composite strut 6

147 7 Applying the general buckling formula epresse in Equation (.55) to the two section telescopic cantilever yiels I E I E P cr ) ( 0 cos cos (.56) where the first an secon components correspon to the sections marke an in Figure.0 respectively. sin sin I E EI P cr (.57) sin sin I E EI P cr sin sin EI P cr (.58) In Equation (.58) the term EI is the critical Euler buckling loa which yiels sin sin cr P Eu P sin sin Eu cr P P sin Eu cr P P (.59)

148 Referring to [97], we now make use of two stanar sections with the specifications liste in Table. below in orer to plot the Equation (.59) against varying values of the parameter ϕ. Table.: Nominal imensions an sectional properties of soli rectangular sections [95] Section Sie H mm B mm Secon Moments of Area (mm ) Section (Fie En) Section (Free En) Assuming the lengths of sections an to be 00 mm an 000 mm, respectively an comparing Tables. an. it can be etermine that the value of β is 0.. Using this value, the graph in Figure. is plotte with varying values of the overlap ratio ϕ, for the non imensionalise parameter Pcr erive in Equation (.59). P Eu Substituting an I I I in Equation (.60) above we get the Critical Euler Buckling oa for an equivalent single section cantilever having length an uniform secon moment of areai P P cr Eu P cr P Eu EI e EI EI 6 The curve in Figure. clearly inicates that for a single steppe strut, the critical buckling loa that the strut can withstan is not affecte to a great etent, by variations in the parameter ϕ. This can be attribute to the fact that the strut, unlike the two section telescopic cantilever beam assembly, is not a thin-walle structure an as a result the I values of both sections an are almost the same. 8

149 (Pcr/PEu) Φ Figure.: Plot of Equation (.59) vs., for the single steppe strut, having imensions outline in Table. an fie an free-en lengths of 00 an 00mm respectively 9

150 .9 Buckling oa for the Thin walle Two Section Telescoping Cantilever Beam Assembly = =ϕ α Figure.: Two section telescopic cantilever The thickness of the thin-walle sections that constitute the two section telescopic cantilever shown in Figure. is yet again assume to be constant along the beam. The secon moment of area of the rectangular tubular sections that constitute the telescopic assembly as shown in Figure. an Figure. is epresse as I th tbh (.60) 6 Of course for the free-en section to slie within the fie-en section, the former will have imensions smaller than that of the latter so iniviual imensions of breath, height an thickness are to be taken into account. The first term in Equation (.60) refers to the secon moment area of the vertical components an the secon represents the horiontal components. For ease of further calculations an also to obtain non-imensional parameters ratios are use to relate the efining characters of the free section to the fie en section. This is emonstrate in Table. an in Figure.. 50

151 Table.: Iniviual rectangular section properties Section ength(mm) Self Weight (N/mm) Secon Moments of Area (mm ) Section (Fie En) Section (Overlap) = W =w I =I α W =(+γ)w I =(+β)i Section (Free En) =ϕ =ϕ W =γw I =βi For the purpose of this buckling analysis the sections are numbere from the fie en towars the free en as shown in Figure.. As can be seen from Figure. an Figure. the overlap is consiere to be a separate section in orer to account for changes in secon moment of area an the change in self weight. The lengths of the sections have been broken own iniviually in orer to facilitate ease of calculations. The lengths, secon moments of area an self weights are all epresse in terms of ratio an each of these in turn relate to the fie-en section. The fie-en, overlap an free-en sections are enote by, an, respectively in Figures. an. respectively. The ratios α,β an γ are assume to be less than which means that the fie en section has the larger imensions of length, secon moment of area an self weight respectively. Now the general formula for the critical buckling loa is epresse as E Pcr I cos 0 (.6) For the isplacement function Y cos. 5

152 5 Figure.: Cross sectional view of the two section telescopic cantilever Applying the erive buckling formula epresse in Equation (.6) to the two section telescopic cantilever whose cross section is shown in Figure. yiels I E I E I E P cr ) ( 0 cos cos cos (.6) where the first an secon an thir components correspon to the sections marke, an in Figure. respectively. sin sin sin sin I E I E EI P cr (.6) = α =ϕ β I (+β) I I (-α) (ϕ-α) P

153 5 sin sin sin sin I E I E EI P cr sin sin sin sin Eu cr P P sin sin Eu cr P P (.6) Referring to [9], we now make use of two stanar sections with the specifications liste in Table.5 below in orer to plot the Equation (.6) against varying values of the parameter α. Table.5: Nominal imensions an sectional properties of rectangular hollow sections Etract from ISO/FDIS 6-:0 (E) [9]. Section Sie H mm B mm Thickness (T mm) Weight per unit length (N/mm) Secon Moments of Area (mm ) Section (Fie En) Section (Free En)

154 Comparing Tables. an.5 an assuming lengths of the fie an free sections to be 00mm an 000mm respectively, it can be etermine that the values of ϕ, β an γ are 0.8, 0.9 an 0.6 respectively. Using these values the graph in Figure. is plotte. Substituting 0, an I I I in the equation (.6) above we get the Critical Euler Buckling oa for an equivalent single section cantilever having length an uniform secon moment of areai P P cr Eu P cr P Eu EI e EI EI 6 The curve in Figure. plots the non imensional parameter Pcr against the overlap ratio α, where α varies from 0 to, in increments of 0.. The conition when the ratio α=0 arises, when there is no free-en section or no telescoping arrangement, while the conition when the ratio α=, occurs when the free-en section is fully inserte or sheathe within the fieen section an it is between these two maima that the curve in Figure. is plotte. Naturally the higher value of critical buckling loa will correspon to the latter rather than the former. In Equation (.6) the parameters ϕ, β an α can be varie epening upon the imensions of the telescoping arrangement. For a esigner, this equation coul be useful in ascertaining the critical buckling loa that a particular telescoping arrangement can withstan. P Eu 5

155 (Pcr/PEu) α Figure.: Plot of Equation (.6) vs.,where varies from 0 to, in increments of 0., for the two section telescopic cantilever beam assembly, having imensions outline in Table.5, an fie an free-en lengths of 00 mm an 000 mm respectively. 55

156 (Pcr/PEu) Φ Figure.5: Plot of Equation (.6) vs. ϕ, where ϕ varies from 0 to, in increments of 0., for overlap ratios α varying from 0 to 0.6, etermine for the two section telescopic cantilever beam assembly, having imensions outline in Table.5, an fie an free-en lengths of 00 mm an 000 mm respectively. 56

157 (Pcr/PEu) Φ Figure.6: Plot of Equation (.6) vs. ϕ, where ϕ varies from 0 to, in increments of 0., for overlap ratios α varying from 0.7 to, etermine for the two section telescopic cantilever beam assembly, having imensions outline in Table.5, an fie an free-en lengths of 00mm an 000mm respectively. 57

158 .0 Summary The en objective of this chapter was to erive from first principles, an equation to etermine the critical buckling loa of the two section telescopic cantilever beam assembly. The equation thus erive ha to take into consieration, all manner of variables associate with the structure an also prove aaptable to any number of telescoping sections. Another esire objective was to generate curves that woul simply allow a esigner to select a particular configuration of the two section telescoping arrangement, for a particular function with relative ease, provie suitable ata is at han. The equation erive is compare with the values etracte using FEA software as etaile in 9... The following points are mae in summary:. For an aially truncate an symmetric cone, it has been shown that larger the value of its base iameter o as compare to its ape iameter e, the greater is the critical buckling loa that it can withstan.. A rectangular pyrami whose section changes in its epth, will have a greater critical buckling loa as compare to that pyrami whose section changes in breath. Figure.5 shows that for a square pyrami changing in epth the greater the value of h o as compare to h e, the greater is the critical buckling loa, whereas Figure. shows that even a substantial increase in b o as compare to b e oes not result in as a great a magnitue of critical buckling loa. A square pyrami, whose epth varies at a greater rate as compare to its breath, is capable of bearing a greater critical buckling loa, than that pyrami whose breath variation is greater than its epth variation, as is evience in Table... Equation (.6) effectively etermines the critical buckling loa that a particular arrangement of the two section telescopic cantilever beam arrangement, can withstan. Varying the ifferent parameters in Equation (.6) allows for a great eal of fleibility in the selection of a particular telescopic arrangement for a given application. The equation also allows for any number of lengths of telescoping sections to be taken into account, an oes not allow itself to be restricte to just the two sections. So long as the 58

159 ratios can account for a number of telescoping sections, it is entirely possible to erive an epression to etermine the critical buckling loa. Once again as in Chapter, the caniate assembly to which Equation (.6) was applie, has imensions outline in Table.5, with fie an free-en lengths of 00 mm an 000 mm, respectively.. Figure.5 presents the plot of the non imensionalise parameter P cr, against the overlap ratio α, which in turn varies from 0 to, in steps of 0.. The curve shown was plotte assuming fie an free section lengths of 00 mm an 000 mm respectively, an arrangement, for which the value of is 0.8. It can be euce from the curve that for an overlap ratio α equal to, the given arrangement will have the greatest critical buckling loa magnitue. The conition wherein the overlap ratio α equals, correspons to that arrangement of the assembly whereby the free-en section is entirely sheathe within the fie-en section. This curve assumes importance in light of the fact that it can be tailore to irectly suit the application for which a given configuration of the telescoping arrangement is require. 5. In Figures.6 an.7, for values of the overlap ratio parameter α varying from 0 to, in increments of 0., the maimum critical buckling loa, or the magnitue at which a structure just begins to unergo buckling, is observe to be maimum, for a length variation ratio of equals 0. This naturally inicates that there is no free-en section to slie within the fie-en section. However an ieal configuration of the telescoping assembly can be selecte using the curves plotte in Figures.6 an.7, for a esire combination of ratios α an. Figures.6 an.7 are in essence use to emonstrate the effect iffering lengths of the fie an free sections that constitute the two section telescoping cantilever beam assembly will have upon the critical buckling loa of sai structure. Once again, it must be remembere that the three sets of curves in Figures correspon to that telescopic arrangement having imensions outline in Table.5 an fie an free-en assume lengths of 00 mm an 000 mm respectively. P Eu 59

160 CHAPTER 5: SHEAR AND TORSION ANAYSES 5. Introuction Equilibrium an stability conitions of one imensional elements such as the tapere sections of varying cross sections, the composite single steppe strut an the two section telescoping cantilever assembly have been eamine an issecte in the previous chapter. The analysis of these members is relatively simple as bening, the essential characteristic of buckling, can be assume to take place in one plane only. The buckling of a plate on eamination involves bening in two planes an this increases the nature of the compleity. From a mathematical point of view, the main ifference between one imensional elements an plates is that quantities such as eflections an bening moments which are functions of a single inepenent variable in the former become functions of two inepenent variables in plates. As a result the behaviour of plates is governe by partial ifferential equations which increase the compleity of analysis. Another significant ifference in the buckling characteristics of plates as compare to one imensional elements is that buckling terminates the latter s ability to resist any further loa, an this critical loa is the failure or ultimate loa. The same however, oes not apply in the case of plates. A plate element may carry aitional loaing beyon the critical loa. This ability to carry aitional loa above an beyon the critical loa is the post-buckling strength. The magnitue of the post-buckling relative to the buckling loa epens on a variety of parameters such as the imensional properties, bounary conitions, type of loaing an the ratio of buckling stress to yiel stress. Plate buckling is usually referre to local buckling. Structural shapes compose of plate elements may not necessarily terminate their loa-carrying capacity at the instance of local buckling of iniviual plate elements. This aitional strength of structural members is attributable not only to the post-buckling strength of the plate elements but also to possible stress reistribution in the member after failure of iniviual plate elements. This chapter has as its basis the theories put forwar by Rees in Mechanics of Optimal Structural Design, [] an etaile in.6,.7 an.8 for shear, torsion an 60

161 combine shear an torsion in thin walle close tube sections respectively. The purpose of this chapter is to analyse the concept of shear flow, the effect of torsion an a combination of both, followe by their contribution to the local buckling of close rectangular hollow sections of uniform thickness. 5. Shear in Uniform Thin-Walle Close Rectangular Sections et a vertical force F y be applie along the vertical centre-line through the shear centre E of a rectangular tube having sie lengths b an with wall thickness t as is shown in Figure 5. below. The secon moment of this area about the ais is given as bt t bt t t b I bt 6 6 (5.) C s B s t E F y y δa δs s D s b A Figure 5.: Uniform, rectangular tube In thin-walle close tubes the net shear flow q q b q E, where the basic shear flow q b refers to any convenient origin an qe is the shear flow eisting at that origin. The qb shear flow istribution within each sie follows from Equation (.76). For sie AB with origin s at A q AB F I y A Fyt ya I F t s y s s I 0 s s (5.) 6

162 Equation (5.) gives q q 0 for s=0 an respectively. The maimum in the A B parabolic istribution occurs for s q F t y (5.) 8I In Equation (5.) the negative sign implies that q-flow opposes the s-irection as is illustrate in Figure 5.(a). For the sie BC with origin s at B q BC s Fy Fyts ts qb I (5.) I 0 Equation (5.) matches q 0 for s=0. Figure 5. (a) shows a maimum in the linear istribution for q F tb s b B y C (5.5) I q c C B C B q b q b E q ma E q ma F y F y D A D A q D (a) (b) Figure 5.: Fleural shear flows qb must be ae to q net with F y be applie at the shear centre E For sie CD with origin for s at C q CD s Fy Fyt Fytb s s qc s s I (5.6) I I 0 6

163 Equation (5.6) matches q 0 for s=0 an gives qd qc for s=. The maimum C within the parabolic shear flow istribution for CD occurs at s as is shown in Figure 5.(a). q CD F t F tb ma y y 8I I (5.7) For sie DA with origin for s at D q DA s Fy F t D I I 0 y ts q s b (5.8) Equation (5.8) gives q B 0 for s=b an matches q D for s=0 with qd being the maimum in the linear istribution as shown in Figure 5.(a). Since F y acts at the shear centre (the centroi) qe can be foun from the Equation (.78) S s s s s s q AB qbc qcd qda qe 0 (5.9) t t t t t S S S Substituting the qb istributions from Equations (5.)-(5.8) into Equation (5.9) b b b Fy F y Fy F yb Fy Fyb s s s ss s s s s ss s I I I I I I X 0 0 b qe 0 t From which q E Fybt (5.0) I Equation (5.0) escribes the constant shear flow in the walls of the tube that is to be ae to the qb istribution in Figure 5.(a). This results in the net shear flow shown in Figure 5.(b). In particular aing qe from Equation (5.0) to Equation (5.7) leas to the equal maimum net shear flows at the centres of sies AB an CD F t b (5.) y qcd qab ma ma 8I 6

164 Similarly aing Equation (5.0) to Equation (5.5) gives the maimum shear flow at C in sie BC (an at D in sie DA) F bt (5.) y qbc q ma ma DA I Net shear flows of similar magnitue to Equation (5.) apply to points A an B with a sign change. Substituting for I from Equation (5.) into Equation (5.) the greatest shear stress which lies at the mi-sie CD is given by Fyt b b F y qcd ma 8 QFY CD (5.) ma t t b b t t 6 in which the geometrical coefficient Q may again reappear within the objective function. Equation (5.) gives the lesser maimum stress at C in the sie BC q Fybt Fybt F b BC ma 6 y BC (5.) ma t I b b t t If it is require to minimise weight, the weight of the section is epresse as b W bt t t (5.5) Setting CD y ma in Equation (5.) allows the prouct t to be eliminate between Equations (5.) an (5.5). This leas to the require objective function W Q opt b F y y F f y y (5.6) in which the shape factor f follows from Equations (5.) an (5.6) f b b b b b Q (5.7) b b 6

165 Each shear stress may be limite to y uner which the limb of imension a can be arrange to buckle simultaneously, as per the general relation epresse in [], when QF t ma y KaET (5.8) at a When sies CD an AB are allowe to buckle at a limiting plastic shear stress y then CD y ma when Equation (5.8) can be epresse as follows QF t ma y KET (5.9) t The optimum imensions follow on from Equation (5.9) as t opt Q K / Fy yet / (5.0) opt Q K / F E y T y / (5.) In case of thin walle sections where the thicknesses vary all sies can be arrange to buckle from equalising their critical shear stresses. This however cannot be one in the case of a rectangular tube of uniform thickness. Here only the longer sies buckle uner their optimum imensions given in Equations (5.0) an (5.). 5. Torsion in Uniform Thin-Walle Close Rectangular Sections Buckling will occur at a lower shear stress in the longer walls if this stress is to be the limiting plastic buckling stress y then the optimisation equation becomes T t y KET (5.) bt The buckling of the rectangular wall epens upon its epth, length an thickness t an the manner of its sie supports. These an plasticity effects are subsume within the buckling stress formula epresse in Equation (5.), above []. 65

166 66 Figure 5.: Rectangular tube with uniform thin-walle thickness The optimum imensions follow from Equation (5.) as T opt E T b K t (5.) 6 6 y T opt T E b K (5.) The weight is epresse as b t t bt W (5.5) Given that the shear flow q is constant in a close tube t t t q b (5.6) It follows from Equations (5.) an (5.5) that shear buckling within the section s perpenicular sies will occur simultaneously when t b t E K t t E K q b b (5.7) A rearrangement of Equation (5.7) gives the conition as b K K b (5.8) Substituting Equation (5.8) into Equation (5.5) the weight becomes b t

167 67 K K b t W b (5.9) Substituting the optimum imensions from Equations (5.) an (5.) into Equation (5.9) an iviing by gives the optimum weight function 6 6 T E b K K K W y T b opt 6 6 T E b K K K W y T b opt (5.0) in which the least weight is foun from minimising the shape function. That is 0 6 b K K b K b (5.) Equations (5.0) an (5.) lea to the stationary values b (5.) min.8 b K b b K b K K K f (5.) Given the b ratio in Equation (5.) the optimum imensions in Equations (5.) an (5.) reuce to T T opt E K T E T K t (5.) y T y T opt T E K T E K (5.5)

168 b Hence to minimise the weight the corresponing ratio from Equation (5.) an then the optimum imensions from Equations (5.) an (5.5) are use. The proceure is assiste when one leaing imension say is known along with the length. Taking = opt we can rea K from [] at the given ratio an then iterate from an assume b value until both Equations (5.) an (5.) are satisfie. 5. Combine Shear an Torsion in Uniform Thin-Walle Rectangular Tube In Mechanics of Optimal Structural Design [], Rees etermines from first principles the optimum imensions for a non-uniform, thin-walle, rectangular tube having sie lengths b an with wall-thicknesses t b an t when a vertical force Fy is applie to the left vertical sie of a rectangular tube. Here the optimum imensions for a uniform thin walle rectangular tube uner the same conitions are etermine. The analysis that follows in this section has as its basis the theory propose within the sai tet an etaile in.8. C s B C B s t E δs q ma s D s b A D A F y (a) (b) Figure 5.: Uniform rectangular tube showing net shear flow 68

169 et a vertical force F y be applie to the left vertical sie CD of the rectangular tube having sie lengths b an with wall thickness t as is shown in Figure 5. above. The q b shear flows (see Figure 5.) were establishe previously when F y acte at the shear centre. These are from Equations (5.) (5.8): q AB Fyt I s s (5.6) q q q BC CD DA F ts y (5.7) I F t y y s s (5.8) I F tb I Fyt s b (5.9) I where the origin for s lies at the respective corners A,B,C an D. It follows from Equations (5.7) an (5.8) that maima in the linear an parabolic istributions lie along the sies BC an CD (see Figure 5.). They are for s=b an s=/ respectively q BC Fytb (5.0) I q CD Fyt Fytb (5.) 8I I where the secon moment of area about the -ais is bt t bt t t b I bt 6 6 (5.) The net shear flow in each sie as shown in Figure 5.(b) is foun by aing q 0 to q b. Here q o is foun by applying Equation (.8) in which moments are taken about corner A b Fy p qbcs b qcds Aq0 (5.)

170 Substituting into Equation (5.) the two shear flows which contribute to this moment equation from Equations (5.7) an (5.8) F b y y y y ss s s s Aq0 F b I t 0 b 0 F bt I F b t I 0 0 Fy t s Fybt s s Fyb t Fyb s bq 0 0 (5.) I I I 0 Equation (5.) leas to Fyt b q0 (5.5) I which may be checke by taking its moments about another corner. Hence Figure 5. shows that the net shear flow has its greatest value (q ma ) at the centre of sie CD. This is foun by aing q 0 from Equation (5.5) to Equation (5.8): Fyt 9 b q ma qcd (5.6) ma 6I Substituting I from Equation (5.) an with ma q ma t the maimum shear stress in CD becomes Fyt b b F 9 y 9 qcd ma 6 QFY CD (5.7) ma t t b b t t 6 ma CD imiting to y uner which buckling of sie CD is to occur provies the esign criterion QF t y KET (5.8) t from which the optimise imensions follow t opt Q K / Fy yet / (5.9) 70

171 7 / / y T y opt E F K Q (5.50) The optimum (minimum) weight is epresse as b t t bt W opt (5.5) an substituting for the prouct t from Equations (5.9) an (5.50) leas to the objective function / / / / y T y T y y opt E F K Q E F K Q b t b W F f F b Q F Q b W y y y y y y opt (5.5) In Equation (5.5) the shape factor f epens upon Q given in Equation (5.7) as follows b b b b Q f 9 (5.5)

172 5.5 Summary The concept of shear flow, torsion an the combination of the two in rectangular hollow sections of uniform thickness has been eamine. In summary;. The net shear flow in a close, uniform thin-walle close rectangular section inuce by a single, vertical force F y applie through its shear centre can be epresse as a summation of the fleural shear flow that q q B q E qb an a constant q E, such. The shear centre in turn, is that point where a shear force can act without proucing any twist in the section. The constant quantity q E is a constant of integration with an important physical interpretation, in that when it is ae to qb it ensures that the rate of twist is ero. The net shear flow istribution in the rectangular section is shown in Figure 5. (b). The greatest shear stress magnitues along the vertical an horiontal walls have been erive in Equations (5.) an (5.). 5. also etails the optimum imensions of that rectangular section, for which buckling can be arrange to occur in its longer sies first.. Torsion in uniform thin walle close tube sections has been eamine an an optimum esign has been arrive at, to ensure that the limiting shear stress from torsion matches that require to cause shear buckling in the wall. The optimum imensions generate for the rectangular section originate from Equation (5.), an the imensions are so relate that shear buckling takes place within the section s perpenicular sies simultaneously. Using these optimum imensions, the weight of the chosen rectangular section can be minimise, as outline in 5... The effects of combine shear an torsion in uniform thin-walle rectangular tubes has been analyse an a moifie net shear flow has been erive for each sie of the rectangular hollow section. The erivation for the moifie net shear flow takes into account the fleural shear flow qb an a constant q o. The sum of these two quantities ensures static equivalence between the net shear flow an the torque that arises when F y oes not pass through the shear centre. In the case where the vertical force F y oes pass through the shear centre, qo serves as a means of establishing the moifie net shear flow, as shown in Figure 5. (b). 7

173 CHAPTER 6: STRESS ANAYSIS 6. Introuction Continuous structures balance the application of eternal loas with an internal resistance within their material which is commonly calle stress. For a beam in particular, resisting moments arise from its internal stress to oppose the bening moments that the transverse loaing prouces. For eample, consier the simplysupporte beam with self- weight w/unit length subjecte to four concentrate loas P... P shown in Figure 6.. a P P P P b w R A a P P b w C T Figure 6.: Moment of resistance within section at -position To unerstan how the material in the beam resists the eternal loas it is seen that the beam sags beneath the applie loas. Sagging creates a compressive stress within longituinal fibres lying in the upper half of the section an tensile stress within fibres in the bottom half. A neutral (unstresse) plane MN ivies each half as shown in Figure 6.. The equivalent compressive force acting on the upper area MEFN is given by C. Similarly the equivalent tensile force acting on the lower area MHGN is given by T. The eternal loas applie an the effective shear force S acting on the plane EFGH are assume to be concentrate on the vertical plane of symmetry, as shown. The forces that act over length AX of the beam are therefore: (a) a vertical reaction R A at A, (b) eternal concentrate loas P an P, () uniformly istribute loa w acting over the length, (c) shear force S offere by section EFGH, () a compressive resistance C an (e) a tensile resistance T. The magnitues of the 7

174 forces C an T are equal an, since they act in opposing irections, separate by a istance, they form the section s moment of resistance: M R = C = T (6.) Taking moments about O gives the bening moment ue to the eternal forces M R w RA P ( a) P ( a b) (6.) In continuous beams we may equate Equations (6.) an (6.) when applying the principle that the moment at a given section ue to eternally applie loas equals the moment of resistance at that section. However, the same principle cannot be applie to telescopic beams within the iscontinuous region between overlapping sections, especially where there is a sieable gap between them. To overcome this, the authors propose their Tip Reaction Moel [9], the principle of which is summarise in.. The stress analysis performe in [, 9], upon the three section telescoping cantilever beam assembly forms the basis of the analysis that is etaile, within this chapter. Consequently, the internal shear force an moment within each length may be calculate from the reactions instea of the moment of resistance use normally for a continuous beam. The shear force an bening moment variations along each length are converte to their respective stresses in the following section. The stress magnitues are compare with those obtaine from a finite element analysis. The analyses were carrie out on a telescopic cantilever assembly consisting of two hollow sections, the etails of which are outline in Appenices C. an D., for both inline an offset loaing scenarios, respectively. 7

175 6. Bening Stress A A A C C C B B B D D D Figure 6.: Telescopic beam assembly with two sections The longituinal bening stress in a beam is calculate from the bening moment M by a stanar epression [6]: M y (6.) I where I is secon moment of area of the beam section an y is the istance from the neutral ais at which this stress applies. Consier the beam assembly shown in Figure 6. an assume that it is fie at en A an carries a tip loa at D. Due to self-weight an the tip loaing applie there will be tensile stresses in all two beam sections above the horiontal of symmetry (neutral plane) an compressive stresses below the plane of symmetry. For the each section epth an, the beam is represente by the vertical plane of symmetry upon which the maimum bening stress occurs at their top surfaces. These are foun from Equation (6.) as: M M an (6.a-b) I I The - an I-values are referre to a chosen geometry given in Appenices C. an in D.. The bening moment M in Equations 6.a-b varies within the length in a manner provie by an M-iagram constructe from the applie loaing an the tip reactions 75

176 as shown in Figure C. an Figure D.. Two loaing scenarios are eamine in Appenices C an D respectively. Appeni C eals with an inline loa applie to the telescopic assembly, whilst Appeni D eamines an offset loa applie to the telescopic assembly. The two loa scenarios are eamine in orer to form a base for which the comparison of eperimental, analytical an output from the software is achieve. In appene sections C. an D. the bening stresses across the entire geometry length of the telescoping assembly are attaine numerically an plotte separately in Figures C. an D., for inline an offset loaing of 0.55 N, respectively. This loa magnitue correspons to a w ratio of. In case of the P inline loaing, a tip loa of 0.55 N is applie at D as shown in Figure 6., whilst when subjecting the caniate assembly to offset loaing, the same tip magnitue is applie, but at an offset istance of 600 mm from the same point D. In similar fashion, Figures 6.5 an 6.7 presente here, within the main boy of the tet, graphically represent both inline an offset loaing, respectively, for ifferent w ratios of,, 6 an 8. P 76

177 6. Shear Stress Referring to 5.. it can be seen that the shear stress istribution an shear flow in the cross section of the rectangular tube of uniform thickness is as epicte in Figure 6.. In Figure 6. (b) the fleural shear flow must be ae to the net shear when a vertical force F y is applie at the shear centre. C s B C B s t E δs q ma q b q ma F y s D s b A D A (a) (b) Figure 6. (a) Cross section of the uniform rectangular tube (b) Net shear stress istribution in the cross section of the uniform rectangular tube From the etaile analysis performe in 5.. the following equations are utilise to etermine the maimum magnitues of shear stress along the vertical an horiontal walls for both the beam sections. Fyt b F yt b b Fy qcd ma 8I 8 CD (6.5) ma t t t b b t 6 Fybt q I Fybt F b BC ma 6 y BC (6.6) ma t t b b t t 77

178 From Equations (6.5) an (6.6) it can be seen that the shear stress formulae can be tailore to suit each of the two beam sections that make up the composite telescoping assembly simply by substituting the require value of breath b, epth an thickness t as is shown numerically for inline an offset loaing in Appenices C. an D., respectively. Once again in appene sections C. an D. the shear stresses across the entire geometry length of the telescoping assembly are attaine numerically an plotte separately in Figures C.5 an D.5, for inline an offset loaing of 0.55 N, respectively. This loa is applie in the same manner for inline an offset loaing as specifie above in 6.. Figure C.5 plots both, inline loaing inuce, maimum shear stress istribution for the assembly on faces CD an BC of the cross section shown in Figure 6.(a), which are plotte along the lines marke A B C D an A B C D along the entire length of the assembly. Figure D.5 plots both, offset loaing inuce, maimum shear stress istribution for the assembly on faces CD an BC of the cross section shown in Figure 6.(a), which are plotte along the lines marke A B C D an A B C D along the entire length of the assembly. The values plotte in Figures C.5 an D.5, were calculate at regular intervals of 500 mm along the length of the assembly. It must be note that these calculate values, correspon to those points where the shear stress istribution attains its maimum for the cross section at points C an at the mipoint of wall CD, respectively. In similar fashion, Figures 6.5, 6.6, 6.8 an 6.9 are presente here in the main boy of work, but for w ratios of,, 6 an 8. Figures 6.5 an 6.6, plot the maimum shear P stress istribution at face CD along the assembly length at A B C D an the maimum shear stress istribution at face BC along the assembly length at A B C D for inline loaing corresponing to the w ratios of,, 6 an 8, P respectively. Figures 6.8 an 6.9 on the other han, plot the maimum shear stress istribution at face CD along the assembly length at A B C D an the maimum 78

179 shear stress istribution at face BC along the assembly length at A B C D for offset loaing corresponing to the w ratios of,, 6 an 8, respectively. P Equations (6.) - (6.6) may be applie to both telescopic an continuous beams when M an F y are known. In what follows M an S are converte to their respective stress istributions from within the iagrams that show the variations in M an S over the length as in Figure C. an D.. The metho of constructing F y - an M-iagrams for continuous cantilever beams, carrying combine concentrate an istribute loaing, can be foun in many tets [5-8, ]. The F- an M-iagrams for a telescopic beam may be constructe separately once the tip reactions for each of Figures C.a-c an a-c are known an then superimpose to fin their net values within the overlaps. 79

180 Figure 6.: Inline loaing inuce bening stress (MPa) vs istance from the fie en for 00 mm overlap along A C B D as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in C. (Key: w/p=; w/p=; w/p=6; w/p=8) 80

181 Figure 6.5: Inline loaing inuce shear stress (MPa) vs istance from the fie en for 00 mm overlap along A C B D as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Appeni C. (Key: w/p=; w/p=; w/p=6; w/p=8) 8

182 Figure 6.6: Inline loaing inuce Shear Stress (MPa) vs Distance from the Fie En for 00mm overlap along A C B D as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Appeni C. (Key: w/p=; w/p=; w/p=6; w/p=8) 8

183 Figure 6.7: Offset loaing inuce bening stress (MPa) vs istance from the fie en for 00 mm overlap along A C B D as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Appeni D. (Key: w/p=; w/p=; w/p=6; w/p=8) 8

184 Figure 6.8: Offset loaing inuce shear stress (MPa) vs istance from the fie en for 00 mm overlap along A C B D as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Appeni D. (Key: w/p=; w/p=; w/p=6; w/p=8) 8

185 Figure 6.9: Offset loaing inuce shear stress (MPa) vs istance from the fie en for 00 mm overlap along A C B D as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Appeni D. (Key: w/p=; w/p=; w/p=6; w/p=8) 85

186 6. Summary Appenices C an D etail the bening an shear stress analysis of the two section telescopic cantilever beam assembly, for inline an offset loaing, respectively. The tip loa applie in both cases has a magnitue of 0.55 N. This loaing correspons to a w ratio of. P Corresponing to this ratio, bening moment an shear force istributions were create for both, inline an offset loaing scenarios, as shown in Figures C. an D. respectively. For the offset loaing inuce stress analysis of the telescopic assembly, a separate torque iagram was also prouce, in aition to the bening moment an shear force iagram. This was one to highlight the fact that the torque applie has a constant magnitue an that it inuces a constant twisting moment upon the assembly. The intention of the analysis performe in Appenices C an D, was to etract both bening an shear stresses inuce by both types of loaing an to present them in graphical form, to act as a benchmark for what has been one in this chapter. Using w ratios of,, 6 an 8, P equivalent tip loas were applie to the structure an the bening an shear stresses were plotte, in this chapter as can be seen from Figures While bening stress within the assembly attains its maimum magnitue along the line marke A B C D, as shown in Figure 6., the shear stress istribution within the hollow rectangular section, attains its maima, at points C an D on the horiontal walls, an at the mi-plane of sie CD along the perpenicular walls, as is epicte in Figure 6. (b). It is at these points, at 50 mm length intervals, that the bening an shear stresses were calculate an plotte. Further conclusions are rawn from the stress analysis unertaken in this chapter, in 9... To conclue:. For w ratios of,, 6 an 8, the corresponing inline loaing inuce bening P stress istribution is plotte in Figure 6., against the istance from the fie en, for a 00 mm overlap. For the same ratios, the relate inline loaing inuce shear stresses are calculate an plotte at the locations, as mentione above, where they attain their maimum magnitue, in Figures 6.5 an

187 . For w ratios of,, 6 an 8, the corresponing offset loaing inuce bening P stress istribution is plotte in Figure 6.7, against the istance from the fie en, for a 00 mm overlap. For the same ratios, the equivalent offset loaing inuce shear stresses are calculate an plotte at the locations, as mentione above, where they attain their maimum magnitue, in Figures 6.8 an The greatest theoretically erive bening stress magnitue of 6 MPa an 8 MPa inuce by inline an offset loaing in the two section telescoping assembly, respectively is in turn evient from Figures 6. an 6.7 for w ratio of. These P values show that the structure remains elastic given a yiel stress for a meium carbon steel of say, 00 MPa. Similarly the greatest theoretically erive shear stress magnitues for both inline an offset loaing, etermine along the horiontal an vertical walls of the assembly fall well within the yiel stress magnitue. Nowhere oes the shear stress magnitue become ero espite it having a relatively low magnitue compare to the accompanying bening stress. 87

188 CHAPTER 7: FINITE EEMENT ANAYSIS 7. Introuction This chapter covers the etensive Finite Element Analysis that has been covere over the course of unertaking this thesis. The primary aim of this chapter involving the use of the Finite Element Analysis software package ABAQUS/CAE is to act as a means of verification an valiation of theoretical results prouce in aition to the eperimental work performe, which is etaile in the following chapter. Figure 7. etails the methoology that was followe to achieve the afore-mentione objective. The three outcomes esire from the FEA are: (a) Overall Deflection of the two section telescopic assembly for both inline an offset loaing (b) Bening an Shear Stress etermination at regular intervals along the length of the assembly an (c) Determination of the Critical Buckling oa for the telescopic beam assembly whose iniviual part imensions are etaile in Table 7.. Figure 7. shows how the overall structure was ivie into two beams an wear pas, each of which was sketche an etrue. Each of the iniviual part instances thus generate were in turn iniviually assigne section an material properties followe by their assembly, to give the overall telescopic beam assembly. Iniviual steps were efine in orer to etermine the various parameters. The telescopic beam assembly now has ties efine between each of the ifferent entities that constitute the overall assembly following which the require loa an bounary conitions are applie to the moel so as to simulate as closely as is possible the real working environment of the physical assembly. Meshing the entire assembly allows for the etermination of the overall eflection an the bening an shear stresses at specifie intervals for comparisons with theoretical an eperimental results. Once meshing has been complete the telescopic beam assembly is subjecte to the first of a series of analysis runs were conucte in orer to etermine three specific outcomes as mentione earlier. Tables 7., 7. an 7. highlight the ifferent FEA approaches that were aopte for tip eflection analysis, stress analysis an critical buckling loa etermination, respectively. Appeni G elaborates in ehaustive etail how ABAQUS was use to generate the outcomes as outline in Tables 7., 7. an 7., for the two section telescopic cantilever beam assembly, whose iniviual parts imensions are outline in Table 7.. Appenices E an F provie etails of the FEA that were carrie on a three section telescopic cantilever beam assembly in orer to obtain overall eflection an stress values respectively 88

189 [9, 9]. The same approach was use ecept that the assembly in this thesis eals with a two beam rather than a three beam telescopic cantilever beam assembly. In aition to these, the critical buckling loa of the telescopic beam assembly was also etermine. Table 7.: Dimensional properties of the simulate two section telescopic cantilever beam assembly Section Material Sie H mm Bmm Thickness (T mm) ength ( mm) Unit Weight (N/mm ) Young s Moulus (N/mm ) Poisson s Ratio BEAM (Fie En) Steel BEAM (Free En) Steel Wear Pas 7 Tufnell

190 Geometry Definition of the iniviual beam shell instances an wear pas of the FEA moel Etrusion of the shell instances an wear pas Material an Element Properties of the FEA moelle parts Assembly of iniviual parts to make telescopic beam assembly Definition of iniviual Steps for etracting esire results Definition of ties between iniviual parts of the entire assembly oaing Conition Definitions Bounary Conition Definitions Mesh Definition of the iniviual parts Results Generation an Interpretation Figure 7.: ABAQUS/CAE Pictorial Methoology 90

191 Table 7.: ABAQUS/CAE Proceure for Tip Deflection Analysis Create the D Moels of the two sections in ABAQUS Sketch a 060mm rectangle with.55mm thickness an etrue 00mm Sketch a 050mm rectangle with.55mm thickness an etrue 00mm Create wear pas of rectangular section 0mm an etrue 00mm Assign Material Properties Create Materials with properties Young s Moulus, Poisson s ratio an ensity efine. Assign the create material to the parts create earlier. Create shell areas from shell volumes create The two beam sections can best be escribe as shells. ABAQUS has a special way of creating shell elements. This requires eiting the geometry. Create the telescopic beam assembly Instances of the beam sections an wear pas are assemble together to form the assembly, such that the overlap measures 00mm. This ensures that the geometry create is preserve for alternate meshing an analysis. Develop the contacts (Ties an Interactions) Two types of contacts are establishe here. They are (i) Ties or wels between the wear pas an beam an (ii) Interactions or frictional contacts where the beam can slie on top of the wear pa. Apply loas an bounary conitions Two loas are efine here. They are (i) gravity loa an (ii) tip loa. The fie en of the outer beam is constraine in all si egrees of freeom as the bounary conition. Create the Mie Element Mesh The wear pas are meshe using soli elements. The beam sections are meshe using shell elements. Submit for linear static analysis A linear static analysis is aequate for etermining the tip eflection. View the results an obtain the tip eflection The results are viewe for any obvious inications suggesting otherwise unacceptable results. If the results are vali an acceptable etract the value for Tip Deflection. Repeat the above steps with ifferent Tip oas This is carrie out to obtain tip eflection for ifferent loas. 9

192 Table 7.: ABAQUS/CAE Proceure for Bening an Shear Stress Analysis Create the D Moels of the two sections in ABAQUS Sketch a 060mm rectangle with.55mm thickness an etrue 00mm Sketch a 050mm rectangle with.55mm thickness an etrue 00mm Create wear pas of rectangular section 0mm an etrue 00mm Assign Material Properties Create Materials with properties Young s Moulus, Poisson s ratio an ensity efine. Assign the create material to the parts create earlier. Create shell areas from shell volumes create The two beam sections can best be escribe as shells. ABAQUS has a special way of creating shell elements. This requires eiting the geometry. Create the telescopic beam assembly Instances of the beam sections an wear pas are assemble together to form the assembly, such that the overlap measures 00mm. This ensures that the geometry create is preserve for alternate meshing an analysis. Develop the contacts (Ties an Interactions) Two types of contacts are establishe here. They are (i) Ties or wels between the wear pas an beam an (ii) Interactions or frictional contacts where the beam can slie on top of the wear pa. Apply loas an bounary conitions Two loas are efine here. They are (i) gravity loa an (ii) tip loa. The fie en of the outer beam is constraine in all si egrees of freeom as the bounary conition. Create the Mie Element Mesh The wear pas are meshe using soli elements. The beam sections are meshe using shell elements. Submit for linear static analysis A linear static analysis is aequate for etermining the tip eflection. View the results an obtain the bening an shear stress values The results are viewe for any obvious inications suggesting otherwise unacceptable results. The bening an shear values are rea irectly at the noes at pre-etermine istances from the fie-en to be compare with theoretical preictions. Repeat the above steps to obtain stress values for ifferent Tip oas This is carrie out to obtain stress values for (i) oaing ue to concentrate en force acting along the neutral ais of the assembly (ii) loaing ue to a concentrate en force acting through a torque arm, at a istance from the neutral ais of the assembly 9

193 Table 7.: ABAQUS/CAE Proceure for etermining Critical Buckling oa Create the D Moels of the two sections in ABAQUS Sketch a 060mm rectangle with.55mm thickness an etrue 00mm Sketch a 050mm rectangle with.55mm thickness an etrue 00mm Create wear pas of rectangular section 0mm an etrue 00mm Assign Material Properties Create Materials with properties Young s Moulus, Poisson s ratio an ensity efine. Assign the create material to the parts create earlier. Create shell areas from shell volumes create The two beam sections can best be escribe as shells. ABAQUS has a special way of creating shell elements. This requires eiting the geometry. Create the telescopic beam assembly Instances of the beam sections are assemble together to form the assembly, such that the assembly measures 00mm. This ensures that the geometry create is preserve for alternate meshing an analysis. Create the Analysis Steps Two steps are evelope namely (i) the general analysis or initial analysis step which is the efault step create an propagate in ABAQUS/CAE, in this case applie for nonlinear geometry an (ii) the linear perturbation buckling analysis step where the ancos Eigen solver is use to solve for the Eigen values an the number of Eigen values that are require are entere. Develop the contacts (Ties an Interactions) Only one type of contact is establishe here namely Ties or wels between the overlapping beam sections. Apply loas an bounary conitions Two loas are efine here. They are (i) Dea loa an (ii) ive loa. The fie en of the outer beam is constraine in all si egrees of freeom as the bounary conition. Create the Mie Element Mesh The beam sections are meshe using shell elements. Submit for linear perturbation, buckling analysis A linear perturbation buckling analysis is aequate for etermining the buckling Eigen values. View the results an obtain the buckling Eigen values Using the relation: (Dea oa + ive oa Eigen Value) the buckling loa is calculate an this value is then compare with the theoretical preictions Repeat the above steps to obtain stress values for ifferent overlap lengths This is carrie out to obtain Eigen values for ifferent overlap lengths so as to compare the results with the theoretical preictions. 9

194 7. Deflection Analysis using ABAQUS Tip eflections for varying inline tip loas are etracte from the FEA software an graphs of the non imensionalise parameter y against the overlap ratios α varying from 0. to 0.8, y 0 in increments of 0. are plotte for iffering w ratios of 0,, 0. an 0.0, an shown in P Figure 7.. The methoology an reasoning behin this graphical representation are eplaine in etail in... Following the proceure for tip eflection etermination as eplaine in etail in Table 7., the en eflections for increasing loa magnitues are compute for the varying w ratios, for iffering overlap ratios α. This involves repeate P simulations using ABAQUS, an etracting the tip eflection of the assembly, for all values of the overlap ratio parameter α, for the ifferent tip loaings corresponing to their equivalent w ratios. The proceure etaile in Table 7., is repeate till this objective is P achieve. The results are etracte using the techniques outline in G.9.. In similar fashion, eflection curves were generate to match the eperimentally an theoretically erive plots, as is shown in comprehensive etail in Figure 9.. 9

195 Figure 7.: FEA generate eflection curves vs. overlap ratio α, where varies from 0. to 0.8, in increments of 0., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7.. (Key: w/p=0; w/p=; w/p=0.; w/p=0.0) 95

196 7. Stress Analysis using ABAQUS The bening stress values along the top line of symmetry of the assembly an the corresponing shear stress values along the top face an sie walls are obtaine at regular intervals of 50 mm from the fie en. The results are isplaye for both loaing scenarios namely the inline an offset loaing cases. The two loa scenarios are eamine in orer to form a base for which the comparison of eperimental, analytical an output from the software is achieve. The bening stresses across the entire geometry length of the telescoping assembly are attaine at intervals of 50 mm an represente graphically in Figures 7. an 7.6 for both inline an offset loaing, respectively, for ifferent w ratios of,, 6 an 8. The etracte shear stress values are in turn plotte in P Figures 7., 7.5, 7.7 an 7.8, for w ratios of,, 6 an 8. Figure 7. plots the maimum P value of shear stress inuce by inline loaing at the sie walls of the telescoping assembly along the line marke as A B C D. Figure 7.5 plots the maimum value of shear stress inuce by inline loaing along the top face of the telescoping assembly along the line marke A B C D. Figures 7.7 an 7.8 were similarly plotte but for offset loaing. The stress analysis results are obtaine using the strategy outline in Table 7. whilst the etraction of stress values is accomplishe using the techniques etaile in Appeni G

197 Figure 7.: Inline loaing inuce bening stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key: w/p=; w/p=; w/p=6; w/p=8) 97

198 Figure 7.: Inline loaing inuce shear stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key: w/p=; w/p=; w/p=6; w/p=8) 98

199 Figure 7.5: Inline loaing inuce shear stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key: w/p=; w/p=; w/p=6; w/p=8) 99

200 Figure 7.6: Offset loaing inuce bening stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key: w/p=; w/p=; w/p=6; w/p=8) 00

201 Figure 7.7: Offset loaing inuce shear stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key: w/p=; w/p=; w/p=6; w/p=8) 0

202 Figure 7.8: Offset loaing inuce shear stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key: w/p=; w/p=; w/p=6; w/p=8) 0

203 7. Buckling Analysis using ABAQUS Critical Buckling loas for the two section telescoping cantilever beam assembly are etermine using the strategy eplaine in etail in Table 7.. The values were etermine for overlap ratios from 0. to 0.8 in increasing intervals of 0. an etracte as eplaine in G.9. an plotte as shown in Figure 7.9. The graph in Figure 7.9 plots the non imensionalise parameter Pcr against the increasing overlap ratio α. cr P Eu P is the critical buckling loa whilst assembly as eplaine in.9. PEu represents the Euler buckling loa of the two section telescoping 0

204 Pcr Figure 7.9: FEA etracte values of vs. Overlap ratio α, where varies from 0. to 0.8, in increments of 0., for the two section PEu telescopic cantilever beam assembly, having imensions outline in Table 7.. 0

205 7.5 Summary The two section telescopic cantilever beam assembly, having imensions specifie in Table 7. has been subjecte to a rigorous an comprehensive series of analyses, using the FEA software package ABAQUS. The three outcomes esire from the FEA were ientifie in 7. an successfully attaine. The results of the FEA performe are in turn compare an contraste with the numerical an eperimentally erive results in Chapter 9. The results of the analyses performe are summarise briefly as follows:. The eflections inuce for inline loas equivalent to their respective w ratios, P enote by y an irectly etracte from ABAQUS simulations, were ivie by the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I, enote by y 0 such that y0 equals P EI, to give the normalise parameter y. It is this parameter in turn that is plotte against the y 0 overlap ratio α, varying from 0. to 0.8, in increments of 0., for w ratios of 0,, P 0. an 0.0, to give the resultant eflection preictions shown in Figure 7.. These curves serve as a benchmark for the final objective of comparing the eperimentally an theoretically etracte eflection plots as is amply etaile in Inline an Offset loaing inuce bening an shear stress istributions are plotte against the istance measure along the assembly, at 50 mm intervals, from the fie en. These curves were in turn plotte for w ratios of,, 6 an 8. These curves are P finally compare an contraste against the theoretical preictions for the same as outline in 9.. an graphically presente in Figures (9.) (9.8).. The critical buckling loas of the two section telescopic cantilever beam assembly having imensions outline in Table 7., enote by P cr an etrapolate from ABAQUS as etaile in G.9., are ivie by the Euler Critical Buckling loa corresponing to a column having one en fie an the other free, enote 05

206 EI by P Eu such that PEu equals, to give the normalise parameter P P cr Eu. This parameter in turn is once again plotte against the overlap ratio α, varying from 0. to 0.8, in increments of 0.. The curve thus generate is compare against the theoretically generate curve as shown in Figure 9. an etaile in

207 CHAPTER 8: EXPERIMENTA ANAYSIS 8. Introuction This chapter contains the results an analyses carrie out on a two section telescopic assembly consisting of two thin walle rectangular steel sections having Tufnell wear pas at locations specifie subject to a combination of bening, shear an torsion. The test rig has imensions as specifie in Table 8.. In earlier chapters the two section telescoping cantilever beam assembly was analyse theoretically as well as moelle using Finite Element Analysis an the results etracte. This chapter aims to take the analysis that has been performe a step further an not only valiate the theory an the software but in its own right highlight, if any, iscrepancies that may arise in practice. The importance of eperimentation in this thesis cannot be highlighte enough; it coul be argue that it forms the berock of this boy of work. The aim of the eperimental tests carrie out can be outline as follows: (i) etermination of tip eflections inuce by loaing the assembly; an finally (ii) stress an strain analysis in the beam cross sections at two ifferent locations along the telescopic beam assembly, which in turn leas onto; (iii) calculation of principal stresses at the four locations. 8. The Test Specimen BEAM BEAM Figure 8.: The Eperimental Test Rig The telescopic beam assembly moel (test specimen) was assemble using two mil steel sections as shown in Figure 8. an the properties of which are outline in the Table 8. below. 07

208 Table 8.: Sectional Properties of the iniviual beams of the test specimen Section Sie H mm B mm Thickness (T mm) ength ( mm) Calculate Weight per unit length (N/mm) Calculate Secon Moments of Area (mm ) BEAM (Fie En) BEAM (Free En) Wear Pas The loa was applie onto the test specimen by means of a loaing arm shown in Figure 8., through which loas coul be applie to simulate pure bening (inline loaing) or a twisting moment (offset loaing). oa Arm Point on loaing arm where offset loaing is applie Point on loaing arm where inline loaing is applie Figure 8.: oaing arm through which loas are applie Figure 8. (a) an (b) shows the loa configurations for inline loaing an offset loaing. 08

209 (a) (b) Figure 8.: oaing arm configuration for (a) Inline oaing (b) Offset oaing Figure 8.: Tufnell wear pas attache to beam. These four wear pas are locate on the four walls of beam, at the en opposite to that where loas are applie 09

210 Position A Position B Position C Figure 8.5: Unattache wear pas, inserte into the gap at the three positions A, B an C, at the start of the overlap between beam an beam The wear pas are place in seven locations in the eperimental test rig. Figure 8. shows the Tufnell wear pas that are attache to beam or the free-en section of the assembly, whilst Figure 8.5 shows the locations where the unattache wear pas are inserte into the gap between the beam sections an, when the latter is sheathe or inserte into the fomer. There are four attache wear pas an three unattache wear pas within the assembly. The four attache wear pas are situate at all four walls of beam, at the en opposite to where the loaing is applie, such that they are always enclose within the overlap region. The three unattche wear pas on the other han are inserte into the gaps, create between both the vertical walls of Beam s an, at positions A an B respectively an into the gap create between the horiontal walls of Beam s an, at position C. This placement is possible only when beam is sheathe within beam. 0

211 Position W Position Y Position X Position Z Figure 8.6: Position where the strain gauges were bone onto the telescopic assembly. Positions W an X are 00 mm from the fie en of beam, whilst positions Y an Z are 00 mm from the inner en of beam Table 8.: Position of gauges along the telescopic beam assembly Position of strain gauge along the telescopic beam assembly Section to which the strain gauge is bone Distance of the gauge (mm) Position W Beam Position X Beam Position Y Beam Position Z Beam 00 mm from fie en along the top surface of the assembly 00 mm from fie en along the sie surface of the assembly 00 mm from en encase insie the overlap along the top surface of the assembly 00 mm from en encase insie the overlap along the sie surface of the assembly The strain gauge rosettes attache at positions W an X, as shown in Figure 8.7 (a) an (b) are bone to beam, whilst the rosettes attache at positions Y an Z as shown in Figure 8.7(c) an (), are bone to beam.

212 The rosettes bone at positions W an X are at the top an the sie of beam along the neutral aes, respectively. The rosettes bone at positions Y an Z, are at the top an the sie of beam along the neutral aes, respectively. The gauges bone at positions Y an Z are place so that they are always sheathe within beam in the overlap region where beam slies within beam. Measurements of strains of the beam sections in the teste segments at positions W, X, Y an Z were carrie out using the strain gauging methos that are highlighte in etail in Appeni H. Measurements of strains were carrie out using the Scorpio ata acquisition system which in turn was integrate with a esktop PC. (a) (b) (c) () Figure 8.7: Strain gauge rosettes bone at (a) position W (b) position X (c) position Y an () position Z, as shown in Figure 8.6. The overlap region is of particular significance in that the combination of the two sections (one sheathe within the other) bolsters the magnitue of its secon moments of area. Also of

213 importance are the imensions of the wear pas. Depening on the imensions of the wear pa, a significant reuction in bening an shear stresses is registere as compare to away from the overlap as seen in Figure 6.. Also of note is the fact that ue to the increase moments of area within the overlap, buckling always takes place in front of the overlap in that section of smaller imensions. Naturally this can be attribute to the susceptibility of that section to buckle first ue to it having the lesser value of I or secon moment of area. Position where eflection reaings are taken Figure 8.8: Front view of the telescopic assembly. The arrow inicates the position where ial gauge reaings of eflection for ifferent loa magnitues were taken.

214 The objectives of the stress an stain analysis is to recor the strain state at the four positions on the telescoping assembly subjecte to combine bening an torsion. Using the strain ata collecte the principal strains an their irections can be calculate. With the principal strains in turn the principal stresses can be obtaine using the stress-strain transformation equations as outline in Appeni H. It is also possible to preict the stress state of the shaft using theoretical techniques consiering such factors as bening, shear stress ue to torsion an the effects of shear force. This eperiment will be a measure of how well the theory moels reality. Apart from measurements of strains, the measurements of tip eflections was carrie out at the point shown in Figure 8.8 for increasing loas using ial gauges of accuracy 0.0mm. 8. The Eperimental Mounting Stan Figure 8.9: Frontal view of the eperimental mounting jig clampe to support column. The eperimental jig use to mount the telescopic beam assembly in orer to simulate an encastre fiing or a rigily fie en is shown in Figure 8.9. The jig is attache to columns in the laboratory environment an is mounte onto the same using a spigot which is screwe

215 onto a threae ro which in turn is screwe into the fie beam. The spigot as well as the threae ro an the location where it screws into beam are shown in Figure 8.0. Spigot ocation in Beam where the threae ro screws into, to mount onto column Threae Ro Insert Figure 8.0: Details of the mounting mechanism (a) (b) Figure 8.: (a) Front view of the mounting jig (b) Rear view of the mounting jig Figures 8. (a) an (b) show the ifferent views of the mounting rig an Figure 8. shows the left han view of the mounting jig. Figure 8. also shows the metho by which the jig is clampe to the column. Once the threae ro-insert slies into the hole in the support column, the spigot centres the insert, which in turn is what the front view of the rig slies onto, through the through-hole shown in both Figures 8.(a) an (b). 5

216 Figure 8.: eft han view of the mounting jig showing the metho by which the same is clampe to the support column. The assembly rawing entitle Mounting Fiture on the page that follows shows the etaile measurements of the mounting jig an the iniviual components that comprise the same. 6

217 7

218 8. Eperimental Tip Deflection Analysis Figure 8. represents the eperimentally erive eflection curves obtaine by plotting the en loa applie to the corresponing tip eflection, whilst varying the overlap lengths. These curves were plotte for the eperimental test rig, the imensions of which are elaborate in Table 8.. These curves were plotte first by initially loaing the assembly in the inline loaing configuration as shown in Figure 8. (a) with a nominal loa of 0 N an setting the ial gauge to ero before commencing the eflection testing. This was one in orer to reuce any errors that may creep in an remove any slack within the assembly. Tip eflections were measure for increasing magnitues of en loa, in the inline loaing configuration as shown in Figure 8. (a). Figure 8., in turn represents the moifie eflection curves plotte, in similar fashion to those in Figures.,.7,.0 an. respectively. Knowing the tip eflection inuce in the assembly by a particular en loa eperimentally (enote here as y ), an the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I (here enote by the term y 0 such that y0 equals P EI ) we obtain the normalise orinate parameter y. This y 0 orinate parameter is in turn plotte for values of w which can be calculate given the P imensions of the eperimental rig as outline in Table 8., against the overlap ratio parameter α. The intention of plotting this curve was to act as a means of comparison against the theoretical preictions an the FEA generate curves. 8

219 Figure 8.: oa applie in Newtons vs tip eflection in mm for the eperimental test rig having imensions outline in Table 8. (Key: Overlap length 00 mm; Overlap length 500 mm; Overlap length 600 mm; Overlap length 700 mm) 9

220 Figure 8.: Etrapolate eflection curves vs. overlap ratio α for the eperimental test rig having imensions outline in Table 8. (Key: Applie oa 0 N; Applie oa 0 N; Applie oa 50 N; Applie oa 80 N) 0

221 8.5 Eperimental Stress Analysis The results of the eperimental analysis performe on the eperimental test rig having imensions as etaile in Table 8. are presente below in Figures Using the Scorpio Data Acquisition system, the principal strains are measure using the strain gauges bone as outline in the strain gauging techniques mentione in H.5, H.6 an H.7. The gauges are shown in Figures 8.7 (a) () an are bone at each of the locations shown in Figure 8.6 an Table 8.. The strains recore are converte to their corresponing principal strain entities using the equations (H.a) an (H.b). Using the principal strain values calculate from equations (H.a) an (H.b), the equivalent principal stresses are obtaine using the stress transformation equations, (H.6a) an (H.6b). Equation (H.c) gives the principal strain irections. The values of principal stresses are plotte against the loa applie, in either the inline or offset loaing configurations, as shown in Figures 8. (a) (b), for varying overlap lengths. Figures show the inline loaing inuce principal stresses at each of the four locations shown in Figure 8.6 for an overlap length of 00 mm. Figures shows the offset loaing inuce principal stresses at each of the four locations, once again for an overlap length of 00 mm. Figures show inline loaing inuces principal stresses for an overlap length of 500 mm whilst Figures show the principal stresses inuce through offset loaing for the same overlap length. Inline an offset loaing inuce principal stresses are shown for an overlap length of 600 mm in Figures an Figures , respectively. Finally the principal stresses plotte for inline an offset loaing configurations, as inuce within the assembly having an overlap length of 700 mm are presente in Figures an Figures , in that orer. These plots are compare with both theoretical an FEA obtaine preictions of principal stresses, an shown in 9...

222 Figure 8.5: Principal stresses (σ, σ (MPa)) at position W vs Inline loa applie (kg) with 00 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.6: Principal stresses (σ, σ (MPa)) at position X vs Inline loa applie (kg) with 00 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa))

223 Figure 8.7: Principal stresses (σ, σ (MPa)) at position Y vs Inline loa applie (kg)with 00 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.8: Principal stresses (σ, σ (MPa)) at position Z vs Inline loa applie (kg)with 00 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa))

224 Figure 8.9: Principal stresses (σ, σ (MPa)) at position W vs Offset loa applie (kg) with 00 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.0: Principal stresses (σ, σ (MPa)) at position X vs Offset loa applie (kg) with 00 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa))

225 Figure 8. Principal stresses (σ, σ (MPa)) at position Y vs Offset loa applie (kg) with 00 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8. Principal stresses (σ, σ (MPa)) at position Z vs Offset loa applie (kg) with 00 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 5

226 Figure 8. Principal stresses (σ, σ (MPa)) at position W vs Inline loa applie (kg) with 500 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.: Principal stresses (σ, σ (MPa)) at position X vs Inline loa applie (kg) with 500 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 6

227 Figure 8.5: Principal stresses (σ, σ (MPa)) at position Y vs Inline loa applie (kg) with 500 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.6: Principal stresses (σ, σ (MPa)) at position Z vs Inline loa applie (kg)with 500 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 7

228 Figure 8.7: Principal stresses (σ, σ (MPa)) at position W vs Offset loa applie (kg) with 500 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.8: Principal stresses (σ, σ (MPa)) at position X vs Offset loa applie (kg) with 500 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 8

229 Figure 8.9: Principal stresses (σ, σ (MPa)) at position Y vs Offset loa applie (kg) with 500 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.0: Principal stresses (σ, σ (MPa)) at position Z vs Offset loa applie (kg) with 500 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 9

230 Figure 8.: Principal stresses (σ, σ (MPa)) at position W vs Inline loa applie (kg) with 600 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.: Principal stresses (σ, σ (MPa)) at position X vs Inline loa applie (kg) with 600 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 0

231 Figure 8.: Principal stresses (σ, σ (MPa)) at position Y vs Inline loa applie (kg) with 600 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.: Principal stresses (σ, σ (MPa)) at position Z vs Inline loa applie (kg) with 600 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa))

232 Figure 8.5: Principal stresses (σ, σ (MPa)) at position W vs Offset loa applie (kg) with 600 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.6: Principal stresses (σ, σ (MPa)) at position X vs Offset loa applie (kg) with 600 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa))

233 Figure 8.7: Principal stresses (σ, σ (MPa)) at position Y vs Offset loa applie (kg) with 600 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.8: Principal stresses (σ, σ (MPa)) at position Z vs Offset loa applie (kg) with 600 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa))

234 Figure 8.9: Principal stresses (σ, σ (MPa)) at position W vs Inline loa applie (kg) with 700 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.0: Principal stresses (σ, σ (MPa)) at position X vs Inline loa applie (kg) with 700 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa))

235 Figure 8.: Principal Stresses (σ, σ (MPa)) at Position Y vs Inline oa Applie (Kg) with 700 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.: Principal stresses (σ, σ (MPa)) at position Z vs Inline loa applie (kg) with 700 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 5

236 Figure 8.: Principal stresses (σ, σ (MPa)) at position W vs Offset loa applie (kg) with 700 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.: Principal stresses (σ, σ (MPa)) at position X vs Offset loa applie (kg) with 700 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 6

237 Figure 8.5: Principal stresses (σ, σ (MPa)) at position Y vs Offset loa applie (kg) with 700 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) Figure 8.6: Principal stresses (σ, σ (MPa)) at position Z vs Offset loa applie (kg) with 700 mm overlap (Key: Eperimental σ (MPa); Eperimental σ (MPa)) 7

238 8.6 Summary The eperimental analyses an results conucte on the eperimental test rig having imensions etaile in Table 8. are ocumente in this chapter. The esire outcomes of this eperimental section were attaine an are briefly summarise.. Tip eflections for varying loa magnitues were physically measure an plotte as shown in Figure 8., for iffering overlap lengths. From eperimentally etermine tip eflections, the imensionless parameter y is etrapolate, an plotte against y 0 the equivalent overlap ratio parameter α, in turn obtaine by iviing the overlap length by the length of the fie-en section of the telescoping assembly. These curves are plotte 8... Strain gauges were bone at four locations along the telescopic assembly an the principal strains inuce were recore for increasing magnitues of inline an offset loaing, for varying overlap lengths of 00 mm, 500 mm, 600 mm an 700 mm. These strains were converte using relevant stress-strain transformation equations outline in Appeni H, to their equivalent principal stresses an plotte against the loa applie. 8

239 CHAPTER 9: DISCUSSION AND CONCUSIONS 9. Results This chapter aims to ocument the results of the theoretical, eperimental an Finite Element Analysis conucte thus far, upon the two section telescopic cantilever beam assembly an comparisons between them. The unerlying principle of bening cantilever beams has been revisite in which the moment of resistance is ientifie as the mechanism for transferring the effects of eternal loas within continuous beams. Since this cannot be use as the mechanism for iscontinuous telescopic beams an alternative Tip Reaction Moel is propose in which the eternal loaing is reacte at the tips of the overlaps. The two section telescoping cantilever beam assembly has been investigate an analyse in etail over the course of this boy of work. Using the four common methos of tip eflection preiction as etaile in Chapter, unique epressions for en-loa versus eflection for the two section telescoping cantilever beam assembly are erive. Each of these epressions, although unique, compares remarkably well, not only with each other, but with both eperimental an Finite Element Analysis results, in the form of eflection curves plotte against increasing magnitues of tip loa, for ifferent presente in 9... w ratios. All of these results are P.9 etails the epression that epresses the critical buckling loa, that the two section telescoping cantilever beam assembly can withstan. This epression was erive from energy methos an has been tailore to suit a number of variables within the structure such as iffering sectional lengths, overlap lengths an sectional properties. In aition to the theoretical erivation of this formula, the etermination of critical buckling loa for the same structure using Finite Element Analysis has been achieve an then compare an use to valiate the theory mentione earlier. 6. an 6. etail the bening an shear stress formulation for the two section telescopic cantilever beam assembly for both inline an offset loaing scenarios. Theoretical bening an shear stress values were etermine along the relevant cross sections of the assembly an plotte for ifferent values w ratios. Finite Element Analysis software was also use to P generate both bening an shear stress values along the cross section of the assembly, for 9

240 comparison with the afore mentione theoretical values. Chapter 8 etails the stress analysis that was performe on the two section telescoping cantilever beam assembly an graphs of principal stresses at preetermine locations along the structure are shown in Figures The graphs plot the principal stresses at given positions against increasing magnitues of either inline or offset applie loaings for ifferent overlap lengths. 9.. Deflection Results Figure. shows the eflection plot obtaine from Macaulay s theorem, plotte against increasing overlap ratios for ifferent w ratios. Tip eflections for iffering overlap lengths P an increasing tip loas are generate using a C program as evelope an shown in its entirety in Appeni B. Entering the variables as outline in Table. the require tip eflections are obtaine. The orinate magnitue is obtaine by iviing the ratio of tip eflection obtaine from the C program, for a given value of w which in turn varies from P 0.0 to 0, in multiples of 0, to the tip eflection of a single fie en section cantilever having length an uniform secon moment of area I. In other wors y equals P 0 EI. The quantity w represents the ratio of the prouct of the self weight over the single fie en P section cantilever an the length over which it acts to the tip loa acting on the same single fie en cantilever section. The abscissa magnitue, as mentione earlier, represents the increasing values of the overlap ratios, increasing in steps of 0., from 0. to. Similar eflection curves are plotte for the epressions erive using Mohr s Moment Area Theorem, Castigliano s Theorem an the Virtual Work Metho as shown in Figures.7,.0 an., respectively. The only eception to the curves generate by the latter three methos an those generate in the case of Macaulay s theorem are the abscissa magnitues varying from 0. to in steps of 0.. The reason for this is the fact that the C program is unable to compute values for the tip reactions as etaile in Appeni A. for an overlap ratio of 0. On comparing the eflection curves generate for lower w ratios of 0.0, 0. an in Figures P.,.7,.0 an. it is evient that not much scope is available for comparison purely because the curves overlay upon each other. However of interest here is the fact that for a 0

241 w ratio of 0 the curves are more apart an separate from each other as is clearly evient in P Figure.. The ifference between Macaulay s theorem an the three other methos coul be eplaine as being ue to the reliance on the former on the tip reactions calculate for the varying overlap lengths, which the latter three in turn are inepenent of. Of greater interest arguably, is the ifference between the eflection curves obtaine from Mohr s metho an those generate from both Castigliano s an the Virtual Work Metho. The eflection magnitues erive using Castigliano s metho an the Virtual Work metho are the same. This is because the equations use to generate the respective curves are the same as is evient from Equations (.8) an (.6). Comparing either of these equations mentione with Equation (.) erive using Mohr s metho reveals a significant ifference in the secon term which accounts for the uniformly istribute loaing or self weight of the two section telescoping assembly. This ifference in the case of the Mohr s metho analysis is in part ue to the changing moments of areas that are taken into consieration when computing the eflection as elaborate in.5. an shown iagrammatically in Figure.6. An important point to be mae here is that the eflection curves plotte in Chapter were plotte for a caniate assembly having imensions as specifie in Table., with the fie an free section lengths assume to be 00 mm an 000 mm respectively. Figure 7. represents the eflection curves generate using Finite Element Analysis. The orinate an abscissa magnitues are plotte as mentione above. It must be highlighte that the use of non imensionalise ratios is conucive to plotting presentable values an also allows for effective comparisons. The eflection curves generate from the software use as the caniate assembly the same imensions as that of the eperimental test rig, the imensions of which are etaile in Table 8., for w ratios of 0,, 0.0 an 0.. The P curves thus generate an isplaye in Figure 7., serve merely as a benchmark, for further comparison against the eperimentally obtaine eflection curves shown in Figure 8.. Figure 8. represents the eperimentally erive eflection curves obtaine by plotting the en loa applie to the corresponing tip eflection prouce for increasing overlap lengths as inicate by the key. Again these curves were obtaine for the eperimental test rig the imensions of which are elaborate in Table 8.. These curves were plotte first by initially loaing the assembly in the inline loaing configuration as shown in Figure 8. (a) with a

242 pre-loa of 0 N an setting the ial gauge to ero before commencing the eflection testing. This was one in orer to reuce any errors that may creep in an remove any slack within the assembly. Figure 8. in turn represents the eflection curves of y plotte against the y 0 overlap ratio α for applie loas of 0, 0, 50 an 80 N. The ratio y was etrapolate using y 0 the ata recore in Figure 8.. When similar loaing an eperimental test rig set up conitions an are applie to generate eflection curves, using the four eflection preiction equations, they compare most favourably with the eflection curves generate from eperimental results, such that there is no means of ifferentiation between the curves themselves for the ifferent loa magnitues. Again this is ue to the fact that the curves overlap upon each other as can be evience in Figure 9.. Another point to be consiere, with regars to the eperimental results is eplaine using Figure 9.. The caption for Figure 8. etails the key for each curve that has been obtaine eperimentally. Each of the curves meets the abscissa at certain points, each of which is a specific an unique value of tip eflection. These values correspon to the eflection of the assembly for a particular overlap length, uner the action of its own weight. Also to be note in Figure 9. is that all the curves irrespective of overlap lengths meet at a fie orinate having a magnitue of 0 N. This orinate magnitue can be taken to be the self weight of the assembly, irrespective an inepenent of the overlap length or the overall length of the assembly.

243 Figure 9.: Comparison of Deflection Curves vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 8. (Applie oa 0 N: Macaulay; Mohr; Castigliano an Virtual Work; FEA; Eperimental Applie oa 0 N: Macaulay; Mohr; Castigliano an Virtual Work; FEA; Eperimental Applie oa 50 N: Macaulay; Mohr; Castigliano an Virtual Work; FEA; Eperimental Applie oa 80 N: Macaulay; Mohr; Castigliano an Virtual Work; FEA; Eperimental)

244 Figure 9. (a): Comparison of Deflection Curves vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 8., for an applie loa of 80 N. (Applie oa 80 N: Eperimental; Macaulay; Mohr; Castigliano an Virtual Work; FEA)

245 Figure 9. (b): Comparison of Deflection Curves vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 8., for an applie loa of 50 N. (Applie oa 50 N: Eperimental; Macaulay; Mohr; Castigliano an Virtual Work; FEA) 5

246 Figure 9. (c): Comparison of Deflection Curves vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 8., for an applie loa of 0 N. (Applie oa 0 N: Eperimental; Macaulay; Mohr; Castigliano an Virtual Work; FEA) 6

247 Figure 9. (): Comparison of Deflection Curves vs. Parameter α, for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 8., for an applie loa of 0 N. (Applie oa 0 N: Eperimental; Macaulay; Mohr; Castigliano an Virtual Work; FEA) 7

248 Figure 9.: Eperimental Deflection linear Plots showing oa Applie in Newton vs Tip Deflection in mm, etene such that they meet the orinate at 0 N (Key: Overlap length 00 mm; Overlap length 500 mm; Overlap length 600 mm; Overlap length 700 mm) [Refer to Figure 8.] 8

249 9.. Stress Analysis Results The results of stress analysis from theoretical preictions are presente in Figures Figures 6. represents the inline loaing inuce bening stress along the top surface of the beam assembly. Figure 6.5 etails the inline loaing inuce shear stress along the sie wall of the beam assembly whilst Figure 6.6 plots the inline loaing inuce shear stress along the top, left han corner line of the beam assembly. Similarly Figures represent the offline loaing inuce bening an shear stress values. The graphs are plotte for the assembly having the same imensions as that of the eperimental test rig an outline in Table 8., as well as in Appenices D an E, for an overlap of 00 mm. The results of stress analysis from FEA preictions are presente in Figures The orer of these graphs is similar to that of those plots erive from theoretical preictions, as has been etaile above. Once again the curves have been plotte for the two section telescoping assembly having the same imensions as that of the eperimental test rig an outline in Table 8., for an overlap of 00mm. Comparisons of both the theoretically an FEA generate bening an shear stress curves, in general, was once again ifficult as they overlai upon one other to the point, where there was no means to ifferentiate between the generate sets of curves at all, in some instances. Figure 9. plots the theoretically an FEA etracte inline loaing inuce bening stress in MPa, along the top surface of the beam assembly against the istance from the fie en in mm. Figure 9. isplays the theoretically an FEA erive inline loaing inuce shear stress along the mile of the sie wall of the beam assembly whilst Figure 9.5 epicts the inline loaing inuce shear stress along the top, left han corner line of the beam assembly. In similar fashion, Figures represent the offset loaing inuce bening an shear stress values. These comparison graphs were plotte, once again for the test rig having imensions as specifie in Table 8.. In case of the bening stress curves erive for both inline an offset loaing scenarios, stress values were obtaine at 50 mm intervals along the top of the beam sections as is etaile in Appenices C an D, respectively. FEA values were similarly obtaine in a manner as outline in Appeni G. The unreinforce beam lengths AB an CD are shown for which the stress ais refers to the maimum bening stress at their mi, outer surfaces. In [9] stress ips from FEA at lengths of 600 mm an 500 mm were eplaine by the presence of wear 9

250 pas. Similarly in the caniate assembly imensions of which are outline in Appenices D an E, wear pas were use in the simulation in FEA, as is outline in Appeni I. In the caniate assembly, from Figures 6. an 6.7, it can be seen that at a length of 900 mm there is a stress ip ue in part to the set of wear pas at sai length. They ecrease the stress concentration in the overlap area between sections. Before an beyon each overlap the bening stress in each is seen to iminish from its greatest value at the fie-en to ero at the free-en. Figures 6. an 6.7 show that there are two further free-ens within this telescopic assembly where the bening stresses are also ero. The greatest theoretically erive bening stress magnitue of 6MPa an 8MPa inuce by inline an offset loaing in the two section telescoping assembly, respectively is in turn evient from Figures 6. an 6.7 for w ratio of. Similarly, the greatest FEA etracte values of bening stresses inuce by P inline an offset loaing, is roughly 7 MPa an 7.5 MPa, respectively. A significant ifference between the curves generate by FE an theory for inline loaing bening stresses can be seen in Figure 9. for Beam ACB. This in part, is ue to the bounary conitions use while moelling the FE moel in ABAQUS/CAE. The fie en beam is constraine in si egrees of freeom, an this in turn contributes to a reuction in stress values note up to 50 mm along its length after which there is a sharp spike in the stress values registere. This may be ue to the bounary conition making its effect felt along the assembly, albeit over a small length. In case of the shear stress curves erive for both inline an offset loaing scenarios, stress values were obtaine at 50 mm intervals along the top an sies of the beam sections as is etaile in Appenices C an D, respectively. FEA values were similarly obtaine in a manner as outline in Appeni G. Shear stress values were erive an obtaine from both the mi wall position of the top, horiontal an nearest corner to it, on the sie, vertical walls respectively as is mae clear in Figures 6.5 an 6.6 for inline loaing an Figures 6.8 an 6.9 for offset loaing. In comparison FEA generate curves were also prouce in similar fashion an are comprehensively etaile in Figures 7. an 7.5 for inline loaing an Figures 7.7 an 7.8 for offset loaing respectively. The reversal in the tip reaction between beams AB an CD is responsible for the alternation in signs of the shear stress as is shown in Figures 6.5, 6.6, 6.8, 6.9, 7., 7.5, 7.7 an 7.8. Nowhere oes the shear stress magnitue become ero espite it having a relatively low magnitue compare to the accompanying bening stress. 50

251 Figure 9.: Inline loaing inuce bening stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key to FEA Preictions: w/p=; w/p=; w/p=6; w/p=8) (Key to Theoretical Preictions: w/p=; w/p=; w/p=6; w/p=8) 5

252 Figure 9.: Inline loaing inuce shear stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key to FEA Preictions: w/p=; w/p=; w/p=6; w/p=8) (Key to Theoretical Preictions: w/p=; w/p=; w/p=6; w/p=8) 5

253 Figure 9.5: Inline loaing inuce shear stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key to FEA Preictions: w/p=; w/p=; w/p=6; w/p=8) (Key to Theoretical Preictions: w/p=; w/p=; w/p=6; w/p=8) 5

254 Figure 9.6: Offset loaing inuce bening stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key to FEA Preictions: w/p=; w/p=; w/p=6; w/p=8) (Key to Theoretical Preictions: w/p=; w/p=; w/p=6; w/p=8) 5

255 Figure 9.7: Offset loaing inuce shear stress (MPa) vs istance from the fie en along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00 mm (Key to FEA Preictions: w/p=; w/p=; w/p=6; w/p=8) (Key to Theoretical Preictions: w/p=; w/p=; w/p=6; w/p=8) 55

256 Figure 9.8: Offset loaing inuce Shear Stress (MPa) vs Distance from the Fie En along A C B D, as marke in Figure 6., for the two section telescopic cantilever beam assembly having iniviual part imensions outline in Table 7., an an overlap of 00mm (Key to FEA Preictions: w/p=; w/p=; w/p=6; w/p=8) (Key to Theoretical Preictions: w/p=; w/p=; w/p=6; w/p=8) 56

257 Eperimentally etermine values of stress are obtaine using strain gauging techniques an ata acquisition techniques outline at Appeni G, at points as shown in Figure 8.6. Using ata acquisition techniques the principal strain values are etermine at sai points an converte to their equivalent principal stresses using equations erive in Appeni G, the results of which are plotte in Figures Theoretically etermine principal stresses are then use as a comparison tool against the eperimentally etermine values of principal stress an these are compare against each other in Figures The principal stress values obtaine both theoretically an eperimentally compare favourably with each other for both inline an offset loaing scenarios. The broa question of why inulge in eperimentation in the age of highly avance an ever increasing computer an numerical moelling is aresse an five primary motives are put forwar: (i) to gain a better unerstaning of stress an stress inuce behaviour within the telescopic structure; (ii) to iscover new phenomena for the overlap region; (iii) to obtain improve input ata for further computer moelling; (iv) to etermine the correlation factors between analysis an test embracing material effects; an finally (v) to fine tune an buil further confience in general computer software. 57

258 Figure 9.9: Comparison of principal stresses (σ, σ (MPa)) at position W vs Inline loa applie (kg) with 00 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.0: Comparison of principal stresses (σ, σ (MPa)) at position X vs Inline loa applie (kg) with 00mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 58

259 Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position Y vs Inline loa applie (kg) with 00 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position Z vs Inline loa applie (kg) with 00 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 59

260 Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position W vs Offset loa applie (kg) with 00 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9. Comparison of principal stresses (σ, σ (MPa)) at position X vs Offset loa applie (kg) with 00 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 60

261 Figure 9.5 Comparison of principal stresses (σ, σ (MPa)) at position Y vs Offset loa applie (kg) with 00 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.6: Comparison of principal stresses (σ, σ (MPa)) at position Z vs Offset loa applie (kg) with 00 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 6

262 Figure 9.7: Comparison of principal stresses (σ, σ (MPa)) at position W vs Inline loa applie (kg) with 500 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.8: Comparison of principal stresses (σ, σ (MPa)) at position X vs Inline loa applie (kg) with 500 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 6

263 Figure 9.9: Comparison of principal stresses (σ, σ (MPa)) at position Y vs Inline loa applie (kg) with 500 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.0: Comparison of principal stresses (σ, σ (MPa)) at position Z vs Inline loa applie (kg) with 500 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 6

264 Figure 9.: Comparison of Principal Stresses (σ, σ (MPa)) at Position W vs Offset oa Applie (Kg) with 500 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position X vs Offset loa applie (kg) with 500 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 6

265 Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position Y vs Offset loa applie (kg) with 500 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position Z vs Offset loa applie (kg) with 500 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 65

266 Figure 9.5: Comparison of principal stresses (σ, σ (MPa)) at position W vs Inline loa applie (kg) with 600 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.6: Comparison of principal stresses (σ, σ (MPa)) at position X vs Inline loa applie (kg) with 600 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 66

267 Figure 9.7: Comparison of principal stresses (σ, σ (MPa)) at position Y vs Inline loa applie (kg) with 600 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.8: Comparison of principal stresses (σ, σ (MPa)) at position Z vs Inline loa applie (kg) with 600 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 67

268 Figure 9.9: Comparison of principal stresses (σ, σ (MPa)) at position W vs Offset loa applie (kg) with 600 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.0: Comparison of principal stresses (σ, σ (MPa)) at position X vs Offset loa applie (kg) with 600 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 68

269 Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position Y vs Offset loa applie (kg) with 600 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position Z vs Offset loa applie (kg) with 600 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 69

270 Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position W vs Inline loa applie (kg) with 700 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.: Comparison of principal stresses (σ, σ (MPa)) at position X vs Inline loa applie (kg) with 700 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 70

271 Figure 9.5: Comparison of principal stresses (σ, σ (MPa)) at position Y vs Inline loa applie (kg) with 700 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.6: Comparison of principal stresses (σ, σ (MPa)) at position Z vs Inline loa applie (kg) with 700 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 7

272 Figure 9.7: Comparison of principal stresses (σ, σ (MPa)) at position W vs Offset loa applie (kg) with 700 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.8: Comparison of principal stresses (σ, σ (MPa)) at position X vs Offset loa applie (kg) with 700 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 7

273 Figure 9.9: Comparison of principal stresses (σ, σ (MPa)) at position Y vs Offset loa applie (kg) with 700 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) Figure 9.0 Comparison of principal stresses (σ, σ (MPa)) at position Z vs Offset loa applie (kg)with 700 mm overlap (Key: Theoretical σ (MPa); Eperimental σ (MPa); Eperimental σ (MPa); Theoretical σ (MPa)) 7

274 9.. Buckling Results Of all the curves relate to buckling of sections ealt with in comprehensive etail in Chapter, of prime importance to this boy of work are those curves generate for the two section telescopic cantilever beam assembly. The curves in question are represente in Figures.,.5 an.6. On eamination, Equation (.6) reveals that, the higher the value of the overlap ratio parameter α, the closer the structure tens to be a single section strut, in that the free en is subsume within the fie en, an naturally ue to the subsequent increase in the secon moment area magnitue, the greater will be the critical buckling loa for sai structure. Figure. presents the plot of the non imensionalise parameter P P cr Eu, against the overlap ratio α, which in turn varies from 0 to, in steps of 0.. The curve shown was plotte assuming fie an free section lengths of 00 mm an 000 mm respectively, an arrangement, for which the value of is 0.8. It can be euce from the curve that for an overlap ratio α equal to, the given arrangement will have the greatest critical buckling loa magnitue. The conition wherein the overlap ratio α equals, correspons to that arrangement of the assembly, whereby the free-en section is entirely sheathe within the fie-en section. This curve assumes importance in light of the fact that it can be tailore to irectly suit the application for which a given configuration of the telescoping arrangement is require. In Figures.5 an.6, for values of the overlap ratio parameter α varying from 0 to, in increments of 0., the maimum critical buckling loa, or the magnitue at which a structure just begins to unergo buckling, is observe to be maimum, for a length variation ratio of equals 0. This inicates that there is no free-en section to slie within the fie-en section. However an ieal configuration of the telescoping assembly can be selecte using the curves plotte in Figures.5 an.6, for a esire combination of ratios α an. Figures.5 an.6 are in essence use to emonstrate the effect iffering lengths of the fie an free sections that constitute the two section telescoping cantilever beam assembly will have upon the critical buckling loa of sai structure. Once again, it must be remembere that the three sets of curves in Figures. -.6 correspon to that telescopic arrangement having 7

275 imensions outline in Table.5 an fie an free-en assume lengths of 00 mm an 000 mm respectively. The eperimental test rig was simulate using FEA an subjecte to theoretical analysis using Equation (.6) to generate the curves shown in Figure 9.. Both the curves compare most favourably an thus the theory generate has been valiate, but these plots await eperimental verification. 75

276 Figure 9.: Comparison between buckling curves generate from theoretical preictions an FEA, for the telescopic assembly whose iniviual part imensions are outline in Table 7. (Key: FEA Preictions; Theoretical Preictions) 76

277 9. Contributions to Knowlege Despite the increasing use of telescoping sections all aroun us from simple househol appliances to comple retractable staium roofing, as etaile in.9, there is an alarming lack of literature in the public omain, as compare to the ata available in the private omain. This is a eterrent to the further use of this simple yet highly effective structure. This thesis has attempte to brige this gap, in its consistent application to esigning with telescoping beams an struts. In Chapter, the four most common eflection preiction theorems are applie to the telescoping cantilever beam assembly an in each instance a unique eflection equation was erive. These equations not only preict the eflection that is inuce within the assembly for an inline applie loaing effectively, but also take into account the variables within the arrangement, such as the length of the overlap, ifference in sectional lengths, changing secon moments of area an finally varying self weights of the iniviual constituent sections. A user looking to attain a particular configuration of a telescoping assembly (proviing imensional ata is available) for a task at han woul be able to plot the eflections anticipate, using a C program, base on the calculations etaile in Appeni A, which performs the eflection calculations. This program was aapte to preict eflections in a three section telescopic cantilever beam assembly [9] an is now in commercial application. Chapter also allues to the Tip Reaction Moel, first propose in [9] an applie in this instance to the two section cantilever beam assembly. Chapter erives from the energy principles, the critical buckling loa for the two section telescoping cantilever beam assembly. Not only oes this equation preict the critical buckling loa, that a particular configuration of the telescopic cantilever beam assembly can withstan, but it also accounts for variables within the assembly, as state above. The variables use in the erivation of both the tip eflections an in the etermination of critical buckling loa, are the same, hence lening to the continuity in the analyses of the two section telescoping arrangement. Non imensionalisation is use throughout the analysis conucte, to simplify the comple equations that were generate as well as to allow for relative ease in etermining solutions, for a particular buckling problem. The equations thus erive using these techniques also len themselves to esign curves an further comparisons with results obtaine from other avenues. The equation use to etermine the critical buckling loa is not limite to just the two section telescoping arrangement. Provie ratios can be erive to 77

278 account for more telescoping sections within the assembly, it is entirely possible to etermine the critical buckling loa for an assembly having more than two telescoping sections. What was referre to by Timoshenko as the energy metho is use as a base an applie to the cantilever column as etaile in [5]. This approach in turn is verifie by valiating the result obtaine against the criteria for the Euler Critical Buckling loa for a column having one en fie an the other free as the bounary conitions. A general form for preicting the critical buckling loa eactly is thus erive an is applie to suit each case iniviually taking into account the ifferent cross sectional secon moments of area an lengths. Chapter 5 eamines the phenomena of local buckling as applie to the iniviual rectangular hollow section members that comprise the overall telescoping assembly. It has been shown that the best saving in material can be attaine when the force is applie irectly at the shear centre of the chosen rectangular hollow section. The analysis of shear loaing in the constituent, hollow rectangular sections of the telescopic assembly, is complicate by the istribution in shear stress that a section suffers uner a transverse force. In that case where both torsion an fleural shear stress are consiere, the combination contributes to a raising of the beam s weight, thereby offsetting the greatest weight saving that can be attaine, when the force is applie irectly at the shear centre. However in the case of the latter, it is not necessary to separate the two effects when etermining the position an magnitue of the maimum shear stress upon which the optimum esign is base. When consiering either the former or the latter, it is possible to optimise the rectangular sections imensions, to provie minimum weight, as is shown in this chapter. This has been achieve by limiting the greatest shear stress to a pre etermine esign stress value, at which buckling will also take place, so as to ensure that the most highly stresse material, is use to its full loa-bearing capability. This chapter etails optimum esign criteria wherein the longer limbs of the rectangular cross section beam, can be ajuste to raise the stress to a similar esign stress value, as mentione, thereby saving weight, allowing for greater efficiency in loa bearing an simultaneous buckling at the loa limit. Chapter 6 escribes the bening an shear stress analysis of the two section telescoping assembly when it is subjecte to both inline an offset loaing. The stress analysis performe 78

279 on the two section telescoping assembly, in this chapter, has in its basis the work [9], unertaken with respect to the three section telescopic cantilever beam assembly. Chapter 7 provies a numerical analyses of the two section telescopic assembly using FEA software package, ABAQUS. The analysis performe here valiate the eflection, stress an buckling analyses performe on the same. Chapter 8 etails the eperimental work that was conucte on a two section telescoping cantilever beam an the results obtaine for eflection an stress analysis. The assumptions mae in the eflection theory, were valiate in eperimental work. Despite ehaustive ocumentation of the eperimental stress analysis of the telescoping arrangement, a more effective unerstaning of the stress istributions within the overlap region, is highly esire. This is because the overlap region has been ientifie as being of great importance to the overall structural strength. The test rig in itself was subjecte to both inline an offset loaing in orer to mimic the possible loaing configurations of the telescopic assembly in its working environment. The mounting fiture use to simulate the fie en, is perhaps the closest approimation available, besies physically encasing the fie-en beam within an enclosure. The nature of the beam sections an the fleibility provie allowe the free-en section to be sli in an out of the fie-en section at will, thereby allowing one to vary the overlap length, as esire. A wie variety of testing was thus accomplishe as a result, along with the fleibility affore by repeatability. Chapter 9 compares an iscusses the results obtaine from all the analyses that have been performe on the two section telescopic cantilever beam arrangement. The eflection resulting from en loaing prouce eperimentally, theoretically an through FEA have compare well. The bening an shear stress analysis unertaken both theoretically an through FE has been compare an the theory propagate has been valiate. Comparison of the theoretical an eperimentally measure principal stresses has been achieve. In terms of buckling, the results obtaine from FE matches the theory propagate in Chapter, an hence the theory has been partially valiate. Two papers have been publishe [9, 9], the first of which proposes the Tip Reaction Moel to provie the eflection of a telescopic cantilever beam. The moel uses the reactions at the tips of the overlapping portions as the mechanism of transfer of the eternal loas between 79

280 sections. A generalise, three-section, telescopic beam is analyse in which the irect integration metho is applie repeately to provie eflection. The theory is evelope then aapte to a convenient C program liste here. The program is applie to provie eflections uner en-loaing in a moel beam consisting of three hollow, thin-walle sections. The accuracy in the moel s en-eflection is checke from a further finite element analysis of the beam. The fact that the two eflections compare valiates the tip reaction moel of en-eflection arising from self-weight an eternal loaing in a telescopic cantilever. A linear structural response between loa an eflection appears consistently from both preictions. This paper is appene to the thesis, as Appeni E. In the secon paper, appene to this thesis, Appeni F, the bening an shear stresses for the three-section cantilever, are obtaine both analytically an numerically. A check upon stress levels is provie from a parallel stuy upon an equivalent, two-steppe, continuous beam. Graphical presentations of the beam stresses, foun from applying the two methos to each structure, are self-valiating. That is, the continuous beam theory provies a check upon numerical stress levels from FEA an, in turn, FEA provies a check upon the analytical stresses calculate from tip reactions within a telescopic beam. The fact that comparable stress levels were foun confirms that the analytical technique propose is perfectly aequate for a telescoping beam, just as the classical theory is aequate for continuous beams. Taken together, the two papers provie an analytical theory for bening of a iscontinuous beam that i not eist heretofore, thereby obviating the nee for a numerical solution. 80

281 9. imitations. Although the buckling analysis has been verifie from the FEA moels, the buckling theory has to be fully valiate by conucting buckling on telescopic antenna sections. To fulfil this comparison satisfactorily it will be necessary to aapt the eisting test rig to moel the bounary conitions, such that one en of the telescopic antenna is fie whilst the loa applie en is to be unconstraine, or free to move. The test rig available has en conitions wherein the loa applie en is equivalent to a pinne en. A isplacement function woul nee to be conceive to account for this conition.. Eperimentally conucte stress analyses of the test rig have been ocumente in etail in this thesis. Theoretically an eperimentally etrapolate values of principal stresses have been plotte an compare against each other. However, these stress values have been collecte at four strategic locations. Ieally, gauges woul be place all along the length of the assembly to further valiate the moels erive. The access particularly to the overlap region is physically constraine. The overlap region is probably a critical area of the assembly an the effect on the overall strength of the assembly has yet to be assesse. With respect to this overlap region a variable of the assembly that has not been investigate is the effect of wear pas. The stress analysis conucte use Tufnell sections, but eperimentation with other composites nee to be eamine. 8

282 9. Recommenations for Future Work. A test rig that accommoates a number of gauges at regular intervals along the length of the assembly, especially within the overlap woul greatly enhance the unerstaning of the actual mechanics of the telescoping arrangement.. A long slener telescoping strut, susceptible to buckling is essential to valiate the theory that has been evelope. A test rig that can accommoate this specimen an also moel the bounary conitions wherein one en is fie an the loa applie en is free is preferential to valiate the analytical moel.. Aapting an applying the methoology unertaken in thesis to a three section telescoping cantilever beam assembly an valiating with equivalent FEA moels an eperimental set ups woul emonstrate an epan the use of the Tip Reaction Moel.. Ascertaining the critical effect of wear pas present within the assembly an the effects the associate variables can have on the structural behaviour of the assembly. 5. The effect of using beams consisting of non homogeneous anisotropic materials (i.e. composites), upon the characteristics of the assembly nees to be eamine. This woul be of great relevance as the engineering sector implements the use of composite material wiely in structural application. 8

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288 8. Steel Storage Systems, Denver, Colorao, USA Prouct Photo Gallery: Racks [online] Available at: < [Accesse December 0] 8. Steel Storage Systems, Denver, Colorao, USA Marmon/Keystone Case Stuy [online] Available at: < [Accesse December 0] 85. Wikipeia, the free encyclopaeia Article: Pont u Gar [online] Available at: < [Accesse July 0] 86. Wikipeia, the free encyclopaeia Article: Royal Borer Brige [online] Available at: < [Accesse July 0] 87. Ol UK Photos, Article: Ol Photos of Crumlin in Monmouthshire / Sir Fynwy in South Wales, Unite Kingom of Great Britain, Photo of the Crumlin Viauct [online] Available at: < in%0viauct%0-%080pi.jpg> [Accesse July 0] 88. Wikipeia, the free encyclopaeia Article: Humber Brige [online] Available at: < [Accesse July 0] 89. Wikipeia, the free encyclopaeia article about present ay twin suspension briges: Tacoma Narrows Brige [online] Available at: < [Accesse July 0] 90. Wikipeia, the free encyclopaeia article about the first Tacoma Narrows Brige which collapse in 90: Tacoma Narrows Brige (90) [online] Available at: < [Accesse July 0] 9. Abraham J, Sivaloganathan S, Rees D.W.A. The Telescopic Cantilever Beam: Part Deflection Analysis, Engineering Integrity, February Abraham J, Sivaloganathan S, Rees D.W.A. The Telescopic Cantilever Beam: Part Stress Analysis, Engineering Integrity, September 0 9. International Stanars ISO/FDIS 6- [Hot-finishe structural hollow sections of non alloy an fine grain steels Part : Dimensions an sectional properties]. 9. Tang, Xiaong, Shape Functions of Tapere Beam-Column Elements, Computers an Structures, 6(5), 99, pp

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290 . Fagan M.J., Finite Element Analysis: Theory an Practice, ongman Group UK imite 99.. Kulatunga M. Stress Analysis of a Telescopic Boom Assembly of an MEWP using Direct Measurements, MSc Engineering Design Dissertation submitte to the Avance Manufacturing an Enterprise Engineering Group of the School of Engineering an Design, Brunel University (March 008). Robert Reis Strain Gauge- Pieoelectric Gauges Department of Agricultural an Biological Engineering February 0, 006 pp -5. Dobie, W.B., an Isaac, P.C.G., Electric Resistance Strain Gauges, English Universities Press, Perry, C.C., an issner, H.R., The Strain Gauge Primer, McGraw Hill, Dove, R.C., an Aams, P.H., Eperimental Stress Analysis an Motion Measurement, Merrill Frocht, M.M., Photoelasticity Vols. &, John Wiley an Sons, Coker, E.G., an Filon,.N.G., A treatise on photoelasticity, CUP, 9, Revise Henry, A.W., Elements of Eperimental Stress Analysis, Mawell Dally, J.W. an Riley, W.F., Eperimental Stress Analysis, McGraw Hill Durelli, A.J. an Riley,W.F., Introuction of Photo mechanics, Prentice-Hall Holister, G.S., Eperimental Stress Analysis, CUP, Bush, A. J., Strain Gauge Installation for Avance Environment April Doebelin, E. O., Measurement Systems-Application an Design Aes. C. S an. H. N. ee Strain gauge measurements in regions of high stress graient vol pp Strain gauge technology-prouct catalogue [online] Available at < [Accesse February 00] 7. ABAQUS/CAE User's Manual, The Part Moule, ABAQUS Documentation, Abaqus Inc, ABAQUS/CAE User's Manual, The Property Moule, ABAQUS Documentation, Abaqus Inc, ABAQUS/CAE User's Manual, The Assembly Moule, ABAQUS Documentation, Abaqus Inc,

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292 APPENDIX A 9

293 APPENDIX A THE TWO SECTION TEESCOPIC CANTIEVER BEAM ASSEMBY A. Deflection in the two section telescoping assembly Deflection of the assembly is consiere as the combination of eflection in the three beams AB, CD an EF in a three-section, telescopic cantilever beam. The eflecte shapes of the ifferent beams however are assume to be the same in the overlappe regions. Beam AB has two eflecte portions AC an C B. Similarly Beam CD has two eflecte portions C B an B D. The equations of the eflecte shapes of the beams can be erive by integrating the fleure equation twice. There are four ifferent lengths having ifferent bening moments in this assembly. They are ientifie within Fig. A as follows i. AC in beam AB ii. CB in beam AB iii. CB in beam CD iv. BD in beam CD A R R M M l l w N N/mm m C C B B l l w N/mm N m E R E a D P R D a (a ) D R C R C a B l w N/mm w N /m m P R D C R B R B E R E D C B D (b ) Figure A.: Deflecte shapes of the two-section telescoping cantilever beam assembly 9

294 Equations escribing the bent shape equations of the four segments are erive by integrating y EI M twice, where M is the sagging bening moment. The integration starts with AC with integration constants foun from the known bounary conition at A. Using the equation so erive the slope an eflection at C are calculate. These then become the bounary conitions for the overlap CB in beam AB. This process of matching the iniviual equations to the bounary conitions calculate from the ajoining section is continue to establish the full beam s eflection curve AC B D in Figure A.. A. Tip Reactions The tip reactions ientifie here facilitate the loa transfer between the two beams. To show this, consier the beam assembly shown in Figure A.. Since a part of beam CD lies insie beam AB it will prouce an upwar reaction at C in beam AB an a ownwar reaction at B in beam AB. The applie forces an moments acting upon the fie-en beam AB are as shown in Figure A.. R C R w B M A A C l B a Figure A.: Fie-en beam loaing These inclue: the tip forces R B an R C, the self-weight loaing w an the fiing moment M. Similarly, when beam CD is consiere, at C there will be a ownwar reaction an at B there will be an upwar reaction, ue to its contacts with beam AB. Thus the forces upon CD will be those shown in Figure A.. 9

295 95 Figure A.: Free-en beam loaing Consier the beam section CD as shown in Figure A.. Taking moments about C gives l w l l P a R B But l a From the above equations w l P R B In a similar fashion taking moments about B gives ) ( ) ( l l w l l P l R c Hence ) ( ) ( l w P R C ) ( ) ( w l P R w l P R C B Equation Set (A.) Thus, in the propose Tip Reaction Moel the internal reactions are use to transmit the forces. The effects of the eternal loas applie to the telescopic cantilever beam can then be calculate using tip reactions instea of the bening moment or, moment of resistance, use in the continuous beam. This technique allows the equilibrium an compatibility requirements for each beam to be consiere separately as the free-boy iagrams given in Figure A. an A.. In this way the normal tip reactions at the beginning an en of the a l B C w R C R B D P

296 overlap are establishe. Once the reactions are known, the eflection of each beam can be calculate as shown in Appenices A., A. an A.5. A. Derivation of the eflection curve for the section AC in beam AB R A w R B A C B M R C a l Figure A.: A section in AC Consier section AC shown in Figure A.. The bening moment at the section at a istance from A is ( l ) M RB ( l ) RC ( l a ) w ( l ) for 0 ( l a ) assuming the sign convention Sagging is positive. But y EI M for the beam portion taking I to be the secon moment of area of beam ACB. Integrating this twice will give M y. C C EI To fin C an C substitute the bounary conitions at A i.e. when 0 y 0 an slope y 0 y EI y RB ( l ) R C EI R B ( l ) R ( l ) ( l a ) w ( l ) c [( l w a) ] ( l l C 0 because when y 0 the slope 0. Integrating again ) C 96

297 97 ) ( ] 6 ) ( [ ) 6 ( C C l l w a l R l R EI y c B 0 C because when 0 0 y Thus, if the eflection equation for the section AC in the beam AB is given by 0 t t t t t y then it follows that the coefficients t are ( w EI t w l R R EI t w l l l l R l R EI t C t C t C B C B Equation Set (A.) et g y AC C where AC C y means the slope of section AC at C an ) ( l a Also let y AC C where AC C y means the eflection of section AC at C. k t k t k t g k t k t k t

298 98 A. Derivation of the eflection curve for the section CB in beam AB Figure A.5: A section in CB Consier the section CB shown in Figure A.5. The methoology is similar to the one aopte for the portion AC but the bounary conitions applie correspon to point C ( g an ) calculate earlier in A.. Consier the section at a istance from A as shown in Figure A.5. Bening moment ) ( ) ( ) ( l l w l R M B for l a l But M y EI Integrating this twice will give. C C EI M y ) ( ) ( C l l w l R EI y B When k a l slope g y ] [ ] [ k k l k l w k k l R EI g C B Integrating again ) ( ) 6 ( C C l l w l R EI y B a B C A w R C R B R A M l

299 When ( l a ) k y C EI l k RB ( w l k ( k ) 6 l k C k k ) Thus if the equation of the section CB in the beam AB is given by Then the coefficients t become y t t t t t0 t0 C t C R B l wl t EI RB wl t EI 6 6 w t EI Equation Set (A.) A.5 Derivation of the eflection curve for the section CB in beam CD A R R M M l l ww N N/mm m C C B B l l w N N/mm m E R E a D P R D P (a ) a D R C R C a B l l w N/mm w N /m m P R D P C R B R B E R E D C B D (b ) Figure A.6: Deflection of beams AB an CD 99

300 00 Again, the methoology is similar to the one aopte earlier an the bounary conitions applie correspon to point C ( g an ) calculate earlier in A.. When, instea of beam AB, the beam CD is consiere, the bening moment is from Figure A.6(b): ) ( ) ( ) ( a l a l w a l R M C for l a l Substituting ) ( k a l this limit becomes l k But M y EI where I is the secon moment of area of beam CBD. Integrating once gives the slope 5 [ ] [ C k k w k R EI y C Integrating again gives the equation 6 5 [ ] 6 [ C C k k w k R EI y C When ) ( k a l, g y 5 C k w k R EI g C 5 k w k R EI g C C When ) ( k a l y an when l y See Figure A.6(a). 6 5 ] [ ] 6 [ C k C k k k w k k R EI C 5 6 C k k w k R EI C C Thus if the equation of the section CB in the beam CD 6 5 [ ] 6 [ C C k k w k R EI y C is given by 0 t t t t t y Where the coefficients t are

301 t t 0 C t C5 RC k wk EI RC wk t EI 6 6 w t EI The eflection of section CB at B is. 6 Equation Set (A.) tl tl tl tl t0 Also at B, when l, the slope is y g g tl tl tl t A.6 Derivation of the Deflection Curve for the Section BD in Beam CD A R R M M l l l w N N/mm m w N m C B N/mm C B l E R E a P R D D R C (a ) a a R C B l l w w N N/mm /m m D P R D C R B R B E R E D C B D (b ) Figure A.7: Deflection of Beam CD Again the methoology is similar to the one aopte earlier an the bounary conitions applie correspon to point B ( g an ) calculate earlier in A.5. 0

302 0 From Figure A.7(b) bening moment ) ( ) ( ) ( ) ( l R a l a l w a l R M B C for ) ( l a l l,which is ) ( l k l. But M y EI where I is the secon moment of area of beam CBD. l R k w k R y EI B C Integrating once gives the slope 7 C l R k k w k R EI y B C Integrating again gives the equation C C l R k k w k R EI y B C When l, g y an when l y 7 C l R k k w k R EI y B C 7 C l R l l k l k w l k l R EI g B C 7 l R l l k l k w l k l R EI g C B C C C l R k k w k R EI y B C C C l l R l l k l k w l k l R EI B C C l l R l l k l k w l k l R EI C B C

303 0 Thus if the equation of the section BE in the beam CD C C l R k k w k R EI y B C is given by 0 t t t t t y Here the coefficients t are Equation Set (A.5) The full length of the beam assembly is given by: l k l a l The total tip eflection or the eflection at D is foun by substituting l k l a l 0 ) ( ) ( ) ( ) ( t l k t l k t l k t l k t (A.6) ) ( ) ( ) ( t l k t l k t l k t g (A.7) w EI t R w k R EI t l R w k k R EI t C t C t B C B C

304 APPENDIX B 0

305 APPENDIX B THE C PROGRAM #inclue <stio.h> #inclue <math.h> #inclue <string.h> #inclue <stlib.h> float w, w, P, alpha; float l,l, Rb, Rc; float I, I,r,r,r; floattn,t,tn,t,tn,t,tn,t,t0n,t0,t0n,t0,tn,t,tn,t,t n,t,tn,t; float t0,t,t,t,t,t0,t,t,t,t,g,,k,m,g,; float k,,,,g,g,g; float C,C,C,C,C5,C6,C7,C8; floatt,t,t,t,t0,t,t,t,t,t0,t50,t5,t5,t5,t5,t60,t6,t6,t6,t6,t70,t7,t7,t7,t7; float tmp,tmp,n,n,n,n; float n; int main() { printf(" CACUATION OF REACTIONS\n"); printf(" \n"); printf("\n"); printf("\n"); printf("enter the Self Weight in N/mm for Section : \n"); scanf( "%f", &w ); printf("\n"); printf("enter the Self Weight in N/mm for Section : \n"); scanf( "%f", &w ); printf("\n"); printf("enter the Tip oa: \n"); scanf( "%f", &P ); printf("\n"); printf("enter the overlap between beams an as a ecimal fraction of the length of beam : \n"); scanf( "%f", &alpha ); printf("\n"); printf("enter the length of beam : \n"); scanf( "%f", &l ); printf("\n"); printf("enter the length of beam : \n"); scanf( "%f", &l ); printf("\n"); Rb=(P+((w*l)/))/alpha; Rc=((P*(-alpha))+((w*l*(-*alpha))/))/alpha; printf("reaction at B = %f\n", Rb); printf("\n"); printf("reaction at C = %f\n", Rc); printf("\n"); printf( "CACUATION OF REACTIONS IS COMPETE. Press any character to continue\n"); printf("\n"); printf("\n"); getchar(); 05

306 printf(" OBTAINING SECOND MOMENTS OF AREA\n"); printf(" \n"); printf("\n"); printf("\n"); printf("enter the Secon Moment of Area of Section : \n"); scanf( "%f", &I); printf("\n"); printf("enter the Secon Moment of Area of Section : \n"); scanf( "%f", &I); printf("\n"); printf( "CACUATION OF SECOND MOMENTs OF AREA IS COMPETE. Press any character to continue\n"); printf("\n"); printf("\n"); getchar(); printf(".cacuation OF THE COEFFICIENTS OF THE TH ORDER EQUATION FOR AC IN BEAM AB\n"); printf(" \n"); printf("\n"); printf("\n"); t0=0; t=0; k=l-alpha*l; r= 00000*I; tn=((rb*l/)-(rc*((l-(alpha*l))/))+((w*pow(l,))/)); t=(tn)/r; tn=(-rb/6+rc/6-(w*l)/6); t=(tn)/r; t=(w/)/r; g= *t*pow(k,)+ *t*pow(k,) + *t*k; = t*pow(k,)+t*pow(k,)+t*pow(k,); printf("the equation of AC in the beam AB is y= t^+t^+t^+t+t0 : \n"); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t0 = %f\n", t0); printf("slope at C is %f\n",g); printf("deflection at C in Beam AB through AC is printf("\n"); printf("\n"); getchar(); %f\n",); 06

307 printf(". CACUATION OF THE COEFFICIENTS OF THE TH ORDER EQUATION FOR CB IN BEAM AB\n"); printf(" \n"); printf("\n"); printf("\n"); C=g-(Rb*(l*k - (pow(k,)/)) + ((w/)*(((pow(l,)*k) - (l*pow(k,)) + (pow(k,)/)))))/(r); C=-(C*k)-(Rb*(((l*pow(k,))/) - (pow(k,)/6)) + (w/)*(((pow(l,)*(pow(k,)))/) - ((l*pow(k,))/) + (pow(k,)/)))/(r); printf("c is %f\n",c); printf("c is %f\n",c); t0=c; t=c; t=(((rb*l)/)+((w*pow(l,))/))/r; t=(-rb/6-((w*l)/6))/r; t=(w/)/r; printf("the equation of CB in the beam AB is y= t^+t^+t^+t+t0 : \n"); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t0 = %f\n", t0); n=t*pow(k,)+t*pow(k,)+t*pow(k,)+t*(k)+t0; printf("deflection at C in Beam AB through CB is %f\n",n); printf("\n"); printf("\n"); =t*pow(l,)+t*pow(l,)+t*pow(l,)+t*(l)+t0; printf("deflection at B in Beam AB is %f\n",); printf("\n"); printf("\n"); getchar(); printf(". CACUATION OF THE COEFFICIENTS OF THE TH ORDER EQUATION FOR CB IN BEAM CD\n"); printf(" \n"); printf("\n"); printf("\n"); r=00000*i; C5=g-((Rc*(-pow(k,))/) + (w/6)*(pow(k,)))/(r); C6=-(C5*k)-((Rc*(-(pow(k,)/)))+((w/8)*pow(k,)))/(r); t0=c6; t=c5; t=(((-rc*k)/)+((w*pow(k,))/))/(r); t=(rc/6-((w*k)/6))/(r); t=(w/)/(r); printf("the equation of CB in the beam CD is y= t^+t^+t^+t+t0 : \n"); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t = %f\n", t); 07

308 printf( "t0 = %f\n", t0); printf("\n"); printf("\n"); n=t*pow(k,)+t*pow(k,)+t*pow(k,)+t*k+t0; printf("deflection at C in the Beam CD through CB is %f\n",n); printf("\n"); printf("\n"); =t*pow(l,)+t*pow(l,)+t*pow(l,)+t*l+t0; g=*t*pow(l,)+*t*pow(l,)+*t*l+t; printf("slope at B is %f\n",g); printf("deflection at B in the Beam CD through CB is %f\n",); getchar(); printf(". CACUATION OF THE COEFFICIENTS OF THE TH ORDER EQUATION FOR BE IN BEAM CD\n"); printf(" \n"); printf("\n"); printf("\n"); C7=g-(Rc*((pow(l,)/)-(k*l))+((w/)*((pow(k,)*l)- (k*pow(l,))+pow(l,)/))+((rb*pow(l,))/))/(r); C8=-(C7*l)-(Rc*(((pow(l,))/6)- ((k*(pow(l,)))/))+(w/)*(((pow(k,)*pow(l,))/)- ((k*pow(l,))/)+((pow(l,)/)))+ Rb*((pow(l,)/)))/(r); t0=c8; t=c7; t=((-rc*k)/ + (w*pow(k,))/ + (Rb*l)/)/(r); t=(rc/6 - (w*k)/6 - (Rb/6))/(r); t=(w/)/r; printf("the equation of BE in the beam CD is y= t^+t^+t^+t+t0 : \n"); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t = %f\n", t); printf( "t0 = %f\n", t0); printf("\n"); printf("\n"); } =t*pow((k+l),)+t*pow((k+l),)+t*pow((k+l),)+t*(k+l )+t0; g=*t*pow((k+l),)+*t*pow((k+l),)+ *t*(k+l)+ t; printf("slope at D is %f\n",g); printf("the overall tip eflection at D is %f\n",); getchar(); return 0; 08

309 APPENDIX C 09

310 APPENDIX C PART ININE OADING ANAYSIS OF INDUCED STRESS IN THE TWO SECTION TEESCOPIC ASSEMBY C. Calculation of tip reactions P A w w B C D l a l (a) w R C R B A C B l a (b) R C w R B P C B a l D Figure C.: Tip Reaction Moel Beam assembly an reactions on iniviual beams (c) The eperimental telescopic cantilever beam assembly consists of two hollow rectangular steel sections, each.55 mm thick, with outer imensions: 60mm 0mm 00mm an 50mm 0mm 00mm. In the first scenario a loa of 0.55 N is applie at the en of the beam assembly. Beams CD an AB have an overlap of 00 mm. The secon moment of area about the neutral ais for the cross-section of beams AB an CD are 6700 mm an mm respectively. Their linear ensities (istribute self-weights) are 0.05 N/mm 0

311 an N/mm respectively. Referring to Figure C. (a), the tip loa is P = 0.55 N an the istribute self-weights are once again w N/mm an w N/mm. Consier Figure C. an the Equation Set (A.) erive in [] for the following calculations an taking moments about C gives R B wl P N R C RC P( ) wl 8.97 N ( ( 0.076) ) 0.55( 0.076) Thus when l = 00mm, l =00mm an α = the reactions are RB RC 8.8 N 8.97 N Reaction R at A RA N Moment at A M N mm C. Shear force an bening moment iagram for beam ACB R B =8.8N w =0.05N/mm R C =8.97N w =0.087N/mm ength AC Referring to Figure D. (b): Shear force is where is the istance from A. Bening moment 78 ( )

312 A 85. N Therefore Shear force at C 6.6 N A 78 N mm Bening moment at C N mm ength CB Referring to Figure D. (b): Shear force is where is the istance from A. Bening moment 78 ( ( 900) C 8. N Therefore Shear force at B 8.8 N C N mm Bening moment at B 0 C. Shear force an bening moment iagram for beam CBD ength CB Referring to Figure C. (c): Shear force is ( 900) where is the istance from A. ( 900) Bening moment 8.97( 900) 0.087( 900) ) C Therefore Shear force at B C Bening moment at B ength BD 8.97 N 9.5 N N mm Referring to Figure C. (c): Shear force is ( 900) 8. 8 where is the istance from A. Bening moment ( 900) 8.97 ( 900) ( 900) 8.8 ( 00) B 7.8 N Therefore Shear force at D 0.55 N

313 Bening moment at B D NN mm mm N/mm R C = 8.97 N R B =8.8 N A C 00 mm B 00 mm R C = 8.97N N/mm R B =8.8 N P=0.55 N C B D 00 mm 85. N 8. N 6.6 N 8.8 N BEAM ACB Nmm -78 Nmm 7.8 N 0.55 N BEAM CBD N -9.5N Nmm Figure C.: Shear force an bening moment iagrams for the iniviual sections

314 C. Calculation of bening an shear stresses for beam ACB The bening stress will be maimum along the vertical plane of symmetry at the top of the beam assembly marke as shown in Figure C.. The net shear stress istribution will vary across the cross section as etaile in 5.. an epicte in Figure 5. (b). A A C C B B D D Figure C.: A Telescopic beam assembly with two sections an the vertical an horiontal planes of symmetry shown ength AC Consier, firstly, the uniform section within the length portion AC as shown in Figure C.. The bening moment at the section at a istance from A M 78 ( ) Nmm N mm Taking sagging moments as positive, the maimum bening stress from Equation (6.) as Ma M y I M ma N / MPa mm The shear force in AC is given by S N N

315 The maimum shear stress magnitue in sies CD an AB follows from Equation (6.5) as follows b F y b t N MPa mm CD ma / an the maimum shear stress magnitue in sies BC an DA follows from Equation (6.6) as follows N MPa mm t F y b BC / ma b where b=0 mm, =60 mm an t=.55 mm for the section length AC within beam ACB with F y =0.55 N. ength CB Consiering the uniform section within the length portion CB as shown in Figure C. the bening moment at the section at a istance from A M 78 ( ( 900) Nmm N mm Taking sagging moments as positive, the maimum bening stress follows from Equation (6.) Ma M y I M ma N / MPa mm The shear force in AC is given by S N N Once again the maimum shear stress magnitue in sies CD an AB follows from Equation (6.5) to give 5

316 b F y b t N mm CD ma / MPa an the maimum shear stress magnitue in sies BC an DA follows from Equation (6.6) as N mm t F y b BC / ma b MPa where b=0 mm, =60 mm an t=.55 mm for the section length CB within beam ACB with F y =0.55 N. C.5 Calculation of Bening an Shear Stresses for Beam CBD ength CB Taking into consieration the uniform section within the length portion CB as shown in Figure C. the bening moment at the section at a istance from A ( 900) M 8.97 ( 900) ( 900) Nmm N mm Taking sagging moments as positive, the maimum bening stress from Equation (6.) as Ma M y I M ma N / MPa mm The shear force in AC is given by S ( 900) NN The maimum shear stress magnitue in sies CD an AB follows from Equation (6.5) as follows b F y b t N mm CD ma / MPa 6

317 an the maimum shear stress magnitue in sies BC an DA follows from Equation (6.6) as follows N MPa mm t F y b BC / ma b where b=0 mm, =50 mm an t=.55 mm for the section length CB within beam ACB with F y =0.55 N. ength BD Consiering the uniform section within the length portion BD as shown in Figure C. the bening moment at the section at a istance from A ( 900) M 8.97 ( 900) ( 900) 8.8 ( 00) Nmm N mm Taking sagging moments as positive, the maimum bening stress follows from Equation (6.) Ma M y I M ma N / MPa mm The shear force in AC is given by S ( 900) 8. 8N N Once again the maimum shear stress magnitue in sies CD an AB follows from Equation (6.5) to give b F y b t N mm CD ma / MPa an the maimum shear stress magnitue in sies BC an DA follows from Equation (6.6) as N mm t F y b BC / ma b MPa 7

318 where b=0 mm, =50 mm an t=.55 mm for the section length BD within beam ACB with F y =0.55 N. The bening an shear stresses inuce by inline loaing for a tip loa of 0.55 N or w/p ratio of are shown for both beams ACB an CBD in Figures C. an C.5 that follow. 8

319 Figure C.: Telescopic beam bening stresses inuce by inline loaing, from tip reaction analysis. 9

320 Figure C.5: Telescopic beam shear stresses inuce by inline loaing, from tip reaction analysis ( Key: Sie BC/DA; Sie CD/AB; refer to Figure 6. (b)) 0

321 APPENDIX D

322 APPENDIX D - PART OFFSET OADING ANAYSIS OF INDUCED STRESSIN THE TWO SECTION TEESCOPIC ASSEMBY D. Calculation of Tip Reactions P A w w B C D l l a l (a) w R C R B A C B l a (b) P R C w R B C B a l D l (c) Figure D.: Tip Reaction Moel Beam Assembly an Reactions on Iniviual Beams The eperimental telescopic cantilever beam assembly consists of two hollow rectangular steel sections, each.55 mm thick, with outer imensions: 60 mm 0 mm 00 mm an 50 mm 0 mm 00 mm. As in Appeni D Part, a loa of 0.55 N is applie at the

323 en of the beam assembly, but offset from the neutral ais of the assembly by 600 mm. In other wors the loa of 0.55 N is applie as a torque through a loa arm of 600 mm length. Beams CD an AB have an overlap of 00 mm. The secon moment of area about the neutral ais for the cross-section of beams AB an CD are 6700 mm an mm respectively. Their linear ensities (istribute self-weights) are 0.05 N/mm an N/mm respectively. Referring to Figure D. (a), the tip loa is P = 0.55 N which in turn acts through a istance of 600mm an the istribute self-weights are once again w N/mm an w N/mm. Consier Figure D. an the Equation Set (A.) erive in A. for the following calculations an taking moments about C gives wl P 0.55 R B 8. 8NN R C P( ) wl R C 8. 97NN ( ( 0.076) ) 0.55( 0.076) Thus when l 00, l 00, an the reactions are R R B C 8.8NN 8.97NN Reaction R at A R A NN Bening Moment at A M NNmm Twisting Moment inuce at A M NNmm

324 D. Shear Force, Bening Moment an Torque Diagram for Beam ACB R R B C 8.8 N 8.97NN w w 0.05N N/mm / 0.087NN/mm / ength AC Referring to Figure D. (b): Shear force is where is the istance from A. Bening moment 78 ( ) A 85. N Therefore Shear force at C 6.6 N A 78 NNmm Bening moment at C NNmm A 80 NNmm Twisting moment at C 80 NNmm ength CB Referring to Figure D. (b): Shear force is where is the istance from A. Bening moment 78 ( ( 900) C 8. N Therefore Shear force at B 8.8 N C NNmm Bening moment at B 0 C 80 NNmm Twisting moment at B 80 NNmm

325 D. Shear Force, Bening Moment an Torque Diagram for Beam CBD ength CB Referring to Figure D. (c): Shear force is ( 900) where is the istance from A. ( 900) Bening moment 8.97( 900) 0.087( 900) ) C Therefore Shear force at B C Bening moment at B C Twisting moment at B 8.97 N 9.5 N NNmm 80 NNmm 80 NNmm ength BD Referring to Figure D. (c): Shear force is ( 900) 8. 8 where is the istance from A. Bening moment ( 900) 8.97 ( 900) ( 900) 8.8 ( 00) B 7.8 N Therefore Shear force at D 0.55 N B NNmm Bening moment at D 0 B 80 NNmm Twisting moment at D 80 NNmm 5

326 0.05 N/mm R C = 8.97 N R B =8.8 N A C 00 mm B 00 mm R C = 8.97 N N/mm R B =8.8 N P=0.55 N C B D 00 mm 600 mm 85. N 8. N 6.6 N 8.8 N BEAM ACB Nmm -78 Nmm 7.8 N 0.55 N BEAM CBD N -9.5 N Nmm 80 Nmm Figure D.: Shear force, bening moment an torque iagrams for the iniviual sections 6

327 D. Calculation of Bening an Shear Stresses for Beam ACB The bening stress will be maimum along the vertical plane of symmetry at the top of the beam assembly marke as shown in Figure D.. The net shear stress istribution will vary across the cross section as etaile in 5.. an epicte in Figure 5. (b). A A C C B B D D Figure D.: A telescopic beam assembly with two sections an the vertical an horiontal planes of symmetry shown ength AC Consier, firstly, the uniform section within the length portion AC as shown in Figure D.. The bening moment at the section at a istance from A M 78 ( ) Nmm N/mm Taking sagging moments as positive, the maimum bening stress from Equation (6.) is Ma M y I M ma N / MPa mm The shear force in AC is given by S NN 7

328 The maimum shear stress magnitue in sies CD an AB follows from Equation (6.5) as follows b F y b t NMPa mm CD ma / an the maimum shear stress magnitue in sies BC an DA follows from Equation (6.6) as follows NMPa mm t F y b BC / ma b where b=0 mm, =60 mm an t=.55 mm for the section length AC within beam ACB with F y =0.55 N. ength CB Consiering the uniform section within the length portion CB as shown in Figure D. the bening moment at the section at a istance from A M 78 ( ( 900) Nmm Taking sagging moments as positive, the maimum bening stress from Equation (6.) Ma M y I M ma N / MPa mm The shear force in AC is given by S N Once again the maimum shear stress magnitue in sies CD an AB follows from Equation (6.5) to give b F y b t NMPa mm CD ma / 8

329 an the maimum shear stress magnitue in sies BC an DA follows from Equation (6.6) as NMPa mm t F y b BC / ma b where b=0 mm, =60 mm an t=.55 mm for the section length CB within beam ACB with F y =0.55 N. D.5 Calculation of Bening an Shear Stresses for Beam CBD ength CB Taking into consieration the uniform section within the length portion CB as shown in Figure D. the bening moment at the section at a istance from A ( 900) M 8.97 ( 900) ( 900) Nmm Taking sagging moments as positive, the maimum bening stress follows from Equation (6.) Ma M y I M ma N / MPa mm The shear force in AC is given by S ( 900) NN The maimum shear stress magnitue in sies CD an AB follows from Equation (6.5) as follows b F y b t NMPa mm CD ma / an the maimum shear stress magnitue in sies BC an DA follows from Equation (6.6) as follows NMPa mm t F y b BC / ma b 9

330 where b=0 mm, =50 mm an t=.55 mm for the section length CB within beam CBD with F y =0.55 N. ength BD Consiering the uniform section within the length portion BD as shown in Figure D. the bening moment at the section at a istance from A ( 900) M 8.97 ( 900) ( 900) 8.8 ( 00) Nmm Taking sagging moments as positive, the maimum bening stress follows from Equation (6.) Ma M y I M ma N / The shear force in AC is given by MPa mm S ( 900) 8. 8N Once again the maimum shear stress magnitue in sies CD an AB follows from Equation (6.5) to give b F y b t NMPa mm CD ma / an the maimum shear stress magnitue in sies BC an DA follows from Equation (6.6) as NMPa mm t F y b BC / ma b where b=0 mm, =50 mm an t=.55 mm for the section length BD within beam CBD with F y =0.55 N. The bening an shear stresses inuce by offset loaing for a tip loa of 0.55 N or w/p ratio of are shown for both beams ACB an CBD in Figures D. an D.5 that follow. 0

331 Figure D.: Telescopic beam bening stresses inuce by offset loaing, from tip reaction analysis

332 Figure D.5: Telescopic beam shear stresses, inuce by offset loaing from tip reaction analysis ( Key: Sie BC/DA; Sie CD/AB; refer to Figure 6. (b))

333 APPENDIX E

334 . APPENDIX F

335 APPENDIX G 5

336 APPENDIX G FINITE EEMENT ANAYSIS PROCEDURE G. Part Moule The Part moule is use to create, eit, an manage the parts in the current moel. ABAQUS/CAE stores each part in the form of an orere list of features. The parameters that efine each feature such as etrue epth, iameter, sweep path, etc; combine to efine the geometry of the part. The functions of the Part moule are liste as follows [7]:. Create eformable, iscrete rigi or analytical rigi parts. The part tools also allows the eiting an manipulation of the eisting parts efine in the current moel.. Create those features such as solis, shells, wires, cuts, an rouns that efine the geometry of the part.. Use the Feature Manipulation toolset to eit, elete, suppress, resume, an regenerate a part's features.. Assign the reference point to a rigi part. 5. Use the Sketcher to create, eit, an manage the two-imensional sketches that form the profile of a part's features. These profiles can be etrue, revolve, or swept to create part geometry; or they can be use irectly to form a planar or aisymmetric part. 6. Use the Set toolset, the Partition toolset, an the Datum toolset. These toolsets operate on the part in the current viewport an allow the creation of sets, partitions, an atum geometry, respectively. 6

337 Sketcher tools Figure G.: Sketcher Winow in ABAQUS/CAE Figure G.: Etrusion of the Part Instance sketche in Figure I., the arrow inicates the epth to which the part is etrue 7

338 Figures G. an G. etail the creation of the first or fie beam instance of the telescopic beam assembly followe by its etrusion, respectively. The two beam instances are create as shell sections having thicknesses of.55mm each. This proceure is repeate for the secon beam instance as etaile in Figures G. (a)-(b). Figures G. (c) an () illustrate the creation an etrusion of the wear pa instance. The imensions of the sketche an etrue parts are etaile in Table 7.. (a) (b) (c) Figure G. (a) Dimensione Sketch of the secon or free en beam instance, (b) Etrusion of the secon beam instance (as sketche in G. (a)), (c) Dimensione Sketch of the wear pa instance, () Etrusion of the wear pa instance (as sketche in G. (c)) () 8

339 G. Material an Element Properties Definition In this step the material an element properties are efine an assigne to the parts iniviually. The two materials that are efine are steel an Tufnell; which in turn are use to represent metal structures an wear pas respectively. These materials are specifie by their Density, Elastic Mouli an respective values of Poisson s ratio, which have been specifie in Table 7.. When a part with shell regions or an aisymmetric part with wire regions is encountere, ABAQUS assigns a irection for the normals of the regions. It is possible to reverse the irections of the normals for these regions. In aition, it is also possible to reverse the irections for the normals of importe parts or selecte elements of an orphan mesh. In the figure shown on the net page, the normal is assigne to the given shell component in the Property Moule by selecting the following options: Assign>Normal. As shown in the Figure G. the thickness is ae onto the inner surface of the shell, brown being the positive irection along which the thickness is ae while purple represents the negative or irection in which the thickness is not ae. Figure G.: Assigning normals to the shell elements (Purple is the negative irection while Brown is the positive irection) 9

340 Also, more importantly in this moule, the materials an element section use are efine an assigne to the parts iniviually. Changing the Moule to Property, as shown in the Figures G.5 an G.6, the relevant manager tabs will be pop up on the left han tool bar. Material Manager>Create, allows the input of the require material properties. Property moule tools Figure G.5: Property moule tools Materials use in this stuy are Steel an Tufnell to represent the structure an wear pas respectively. Material properties consist of Moulus of elasticity an Poisson s ratio, which are inputte by selecting the following options within the Property moule: Material Manager > Create > Mechanical > Elasticity > Elastic an applying relevant figures in the boes as it is shown in Figures G.6, G.7 an G.8 [8]. Figure G.6: Steps to create an efine material properties 0

341 Figure G.7: Tabs to be fille in orer to create material section having properties of Steel Figure G.8: Tabs to be fille in orer to create material section having properties of Tufnel With respect to element assignments, soli-homogeneous element is chosen for all wear pas; while shell-homogeneous element is allocate to the shell sections in the assembly. The etails of the material an element sections are elaborate in the Table G..

342 Table G.: Materials an Elements efine in the analysis S.No Name Material Element Specification. Tufnell Tufnell Soli-Homogeneous NA. Shell.55mm Steel Shell-Homogeneous.55mm thick Both the shell an soli sections are efine by selecting the Section Manager tab an setting the require sections as is shown in Figures G.8 an G.9. Figure G.9: Creating a homogeneous, shell section of thickness.55 mm, having properties of steel Finally elements are assigne to each part iniviually through Section Assignment Manager>Create, an selecting the target part an assigning the relevant section to the part in the blank bo as shown in Figures G.0 an G. for the soli Tufnell an shell steel sections respectively.

343 Figure G.0: Creating a homogeneous, soli section having properties of Tufnell Figure G.: Assigning the homogeneous, soli Tufnell section to the part highlighte

344 Figure G.: Assigning the homogeneous, shell steel section to the part highlighte

345 G. Assembling the Two Section Telescopic Cantilever Beam Assembly When a part is create, it eists in its own coorinate system, inepenent of other parts in the moel. In contrast, the Assembly moule is use to create instances of the beam instances an the wear pa an to position the instances relative to each other in a global coorinate system, thus creating the assembly. The wear pa instance that has been create is repeately instance seven times in orer to position the iniviual instances in seven ifferent preetermine locations within the assembly. The overall assembly, in turn replicates the eperimental test rig. The part instances are positione by sequentially applying position constraints that align selecte faces, eges or vertices or by applying simple translations an rotations. Assembly moule tools Figure G.: Assembly moule tools An instance maintains its association with the original part. If the geometry of a part changes ABAQUS/CAE automatically upates all instances of the part to reflect these changes. It is not possible to eit the geometry of a part instance irectly. A moel can contain many parts an a part can be instance many times in the assembly (as is the case with the wear pa as mentione earlier) however a moel contains only one assembly. oas, bounary conitions, preefine fiels an meshes are all applie to the assembly. Even if the moel 5

346 consists of only a single part, it is imperative to create an assembly that consists of just a single instance of that part. A part instance can be thought of as a representation of the original part. It is possible to create either inepenent or epenent part instances. An inepenent instance is effectively a copy of the part. A epenent instance is only a pointer to a part, partition or virtual topology; an as a result, it is not possible to mesh a epenent instance. However, the original part from which the instance was erive can be meshe in which case ABAQUS/CAE applies the same mesh to each epenent instance of the part [9]. Figure G.: Creation of part instances an their assembly to constitute the overall two section telescopic cantilever assembly 6

347 G. The Step Moule An ABAQUS/CAE moel uses the following two types of steps: The Initial Step ABAQUS/CAE creates a special initial step at the beginning of the moel's step sequence. ABAQUS/CAE creates only one initial step for the moel, an it cannot be rename, eite, replace, copie, or elete. The initial step allows the efinition of bounary conitions, preefine fiels, an interactions that are applicable at the very beginning of the analysis. For eample, if a bounary conition or interaction is applie throughout the analysis, it is usually convenient to apply such conitions in the initial step. ikewise, when the first analysis step is a linear perturbation step, conitions applie in the initial step form part of the base state for the perturbation. Analysis Steps The initial step is followe by one or more analysis steps. Each analysis step is associate with a specific proceure that efines the type of analysis to be performe uring the step, such as a static stress analysis or a transient heat transfer analysis. It is possible to change the analysis proceure from step to step in any meaningful way, so as to have greater fleibility in performing analyses. Since the state of the moel (stresses, strains, temperatures, etc.) is upate throughout all general analysis steps, the effects of previous history are always inclue in the response for each new analysis step. There is no limit to the number of analysis steps that can be efine, but there are restrictions on the step sequence. The Step moule is use to perform the following tasks: () Create analysis steps, () Specify output requests () Specify aaptive meshing () Specify analysis controls [0]. 7

348 G.. Creation of analysis steps Within a moel it is possible to efine a sequence of one or more analysis steps. The step sequence provies a convenient way to capture changes in the loaing an bounary conitions of the moel, changes in the way parts of the moel interact with each other, the removal or aition of parts, an any other changes that may occur in the moel uring the course of the analysis. In aition, steps allow for any change to the analysis proceure, the ata output, an various controls. Steps can also be use to efine linear perturbation analyses about nonlinear base states [0]. inear Perturbation Proceures in ABAQUS/CAE An analysis step uring which the response can be either linear or nonlinear is calle a general analysis step. An analysis step uring which the response can be linear only is calle a linear perturbation analysis step. General analysis steps can be inclue in an ABAQUS/Stanar or ABAQUS/Eplicit analysis; linear perturbation analysis steps are available only in ABAQUS/Stanar. A clear istinction is mae in ABAQUS/Stanar between general analysis an linear perturbation analysis proceures. oaing conitions are efine ifferently for the two cases, time measures are ifferent, an the results shoul be interprete ifferently. These istinctions are efine in this section. ABAQUS/Stanar treats a linear perturbation analysis as a linear perturbation about a preloae, preeforme state. ABAQUS/Founation, a subset of ABAQUS/Stanar, is limite entirely to linear perturbation analysis but oes not allow preloaing or preeforme states [0]. inear Perturbation Analysis Steps inear perturbation analysis steps are available only in ABAQUS/Stanar (ABAQUS/Founation is essentially the linear perturbation functionality in ABAQUS/Stanar). The response in a linear analysis step is the linear perturbation response 8

349 about the base state. The base state is the current state of the moel at the en of the last general analysis step prior to the linear perturbation step. If the first step of an analysis is a perturbation step, the base state is etermine from the initial conitions. In ABAQUS/Founation the base state is always etermine from the initial state of the moel [0]. inear perturbation analyses can be performe from time to time uring a fully nonlinear analysis by incluing the linear perturbation steps between the general response steps. The linear perturbation response has no effect as the general analysis is continue. The step time of linear perturbation steps, which is taken arbitrarily to be a very small number, is never accumulate into the total time. A simple eample of this metho is the etermination of the natural frequencies of a violin string uner increasing tension. The tension of the string is increase in several geometrically nonlinear analysis steps. After each of these steps, the frequencies can be etracte in a linear perturbation analysis step [0]. If geometric nonlinearity is inclue in the general analysis upon which a linear perturbation stuy is base, stress stiffening or softening effects an loa stiffness effects (from pressure an other follower forces) are inclue in the linear perturbation analysis. oa stiffness contributions are also generate for centrifugal an Coriolis loaing. In irect steay-state ynamic analysis Coriolis loaing generates an imaginary anti symmetric matri. This contribution is accounte for currently in soli an truss elements only an is activate by using the unsymmetric matri storage an solution scheme in the step [0]. Eigenvalue buckling analysis:. is generally use to estimate the critical (bifurcation) loa of stiff structures;. is a linear perturbation proceure;. can be the first step in an analysis of an unloae structure, or it can be performe after the structure has been preloae if the structure has been preloae, the buckling loa from the preloae state is calculate;. can be use in the investigation of the imperfection sensitivity of a structure; 5. works only with symmetric matrices (hence, unsymmetric stiffness contributions such as the loa stiffness associate with follower loas are symmetrie); an 6. cannot be use in a moel containing substructures. 9

350 Step moule tools Figure G.5: Step moule tools Figure G.6: Creation of buckling step as outline in [0] an G.. 50

351 G.5 Interaction Definitions Relation between the piece parts shoul be establishe through interactions. Interaction moule provies relevant options to ientify the piece parts interfaces. Tie constraint is efine between connecte/wele surfaces; whereas the type of interaction efine throughout the analysis is surface-to-surface. This approach optimies the stress accuracy for a given surface pairing. The improve stress accuracy with respect to the surface-to-surface approach is realise if an only if neither surface of tie pairing is noe base. The surface-to-surface option generally involves more master noes per constraint than the noe-to-surface approach, which in turn results in an aitional increase in the solver banwith in the ABAQUS/Stanar software, thereby increasing the solver cost. The following factors, especially in combination, can lea to the surface-to-surface approach being costly []:. A large fraction of tie noes (egrees of freeom) in the moel. The master surface being more refine than the slave surface. Multiple layers of tie shells, such that the master surface of one tie constraint acts as the slave surface of another constraint. Interaction moule tools Figure G.7: Interaction moule tools 5

352 In orer to set up a Tie constraint between two surfaces, the type of interaction shoul be efine as being either surface-to-surface or noe-to-surface by choosing the master surface as being either Surface or Noe region. On choosing the surface option the selecte surface becomes the master region an in a similar fashion the slave surface is also efine. In a tie interaction between shell elements ABAQUS allows the user to choose either the inner or outer surfaces with the help of a colour coe, with brown representing the outer surface an purple representing the inner surface. One of the problems that may arise at the time of efining interactions is that of the over constraint. An over constraint can be efine as the application of multiple consistent or inconsistent kinematics constraints. Many moels have moels have noal egrees of freeom that are over constraine. Such over constraints may lea to inaccurate or nonconvergent solution generation. Common eamples of situations that may lea to over constraints inclue []:. Contact slave noes that are involve in bounary conitions or multi-point constraints. Eges of surfaces involve in a surface-base tie constraint that are inclue in contact slave surfaces or have symmetry bounary conitions. Bounary conitions applie to noes alreay involve in coupling or rigi boy constraints. One of the most important steps within the interaction moule, is the selection of the Master an Slave surfaces. Attention is require in selecting slave surfaces as one slave surface cannot have two master surfaces. If such a constraint is present in the moel the simulation will eit with over constraint errors. However the converse of this is true i.e. a master surface can have more than one slave surfaces. This conflict usually happens in the selection of slave surfaces which are sharing at least one ege. That ege provies common noes between two slave surfaces which causes the error at the time of running the simulation. In aition to the Tie constraints, a coupling constraint is also create. A Coupling constraint is use to constrain the motion of a surface to the motion of a reference noe. This constraint 5

353 is create at the free en or the loaing en in orer to apply loaing as is shown in the net section. Figure G.8: Definition of constraints (tie contacts) between surfaces Figure G.9: Definition of coupling constraint The establishe contacts can be viewe an eite from Interaction Manager tab by selecting the relevant interaction an pressing eit as it is shown in Figure G.0. Also the step 5

354 in which each interaction was create whether it be in the initial step or otherwise is also isplaye in the Eit Interaction Manager. Figure G.0: Eisting tie efinitions that can be controlle an eite from the constraint manager tab At the en of this process all the interactions between consiere parts will be efine an the assembly is reay to be meshe an loae. Those parts that are not vital to the analysis results are ignore i.e. are not efine with interactions. These parts will no oubt be eclue from the simulation altogether at the time of solving. 5

355 G.6 oaing Conitions oas an constraints are applie as per the actual assembly scenario as in the eperimental test rig. The effect of self-weight is first applie by entering the gravity specifications. Eternal loas are applie by efining them as concentrate loas, which in turn are applie at the coupling constraint as specifie in Figure G.9. Torque is also applie at this constraint point, as a twisting moment. In orer to etermine the critical buckling loa, a separate analysis is conucte an the steps for applying the require loa conitions are specifie in Figures G.5 an G.6. oaing an bounary conition tool sets Figure G.: oaing an bounary conition tool sets First of all the effect of self weight is applie by inputting the gravity. By selecting the Create oa tab from left han menu an selecting the gravity from the provie list the net winow opens to allow for the input of the relevant amount. The magnitue of gravity is entere into the ialog bo an the selection is accepte as shown in Figure G.. As has been mentione earlier in this chapter the three outcomes esire from the FEA are: (a) Overall Deflection of the telescopic assembly for increase loaing (b) Bening an Shear Stresses at regular intervals of 50mm along the top face an sie wall respectively an (c) Determination of the Critical Buckling oa for the telescopic beam assembly. 55

356 Figure G.: Application of self weight or gravity on the assembly To this effect in orer to achieve the first outcome loaing is applie in the form of concentrate force acting at the constraint point shown in Figure G.9. Bening an shear stress values are etracte at regular intervals along the length of the assembly for two types of loaing scenarios namely (i) a concentrate en force acting the constraint point (shown in Figure G.9) an (ii) a concentrate en force acting on a torque arm attache to the free en of the telescopic assembly which constitutes both a concentrate en loa an a twisting moment. The former an the latter are shown in Figure G. an G. respectively. Figure G.: Application of the concentrate en force at the free en of the assembly 56

357 Figure G.: Application of the twisting moment at the free en of the assembly To etermine the critical buckling loa of the assembly for ifferent overlap lengths requires a separate analysis to be run for which the values of both ea an live loas are to be entere [07]. The buckling analysis in turn generates the buckling Eigen values which in turn are substitute in the relation shown in Table 7., to obtain the critical buckling loa. Figures G.5 an G.6 show how the magnitues of the ea an live loas, inputte as 5000 N an 500 N respectively. Figure G.5: Application of Dea loa on the assembly 57

358 Figure G.6: Application of ive loa on the assembly For linear buckling analysis - in the elastic state of the structure static/general an buckling are the two steps efine in orer to etermine the critical buckling loa of the assembly [07]. The initial Static/General step is use to primarily show the first eformations on the moel for buckling, whilst the Buckling step will calculate the Eigen-values from which the Buckling oa Factor can be calculate. inear-buckling analysis is often associate with the Eigen-value buckling or the Euler buckling analysis because it preicts the theoretical buckling strength. The Eigen-values are values of loa at which the buckling takes place, while the Eigenvectors are the corresponing buckling shapes associate with the Eigen-values. The Eigen-value buckling analysis shows that the buckling takes place when the resultant structure stiffness rops to ero [07]. At linear-buckling an moal analysis, FEA provies a large number of buckling moes, but from this analysis, the only buckling moe of practical importance is the first one with a positive Eigen-value, from which the buckling loa is etermine. The buckling moe shows what shape the structure assumes when it is subject to buckling, in that particular moe. 58

359 G.7 Bounary Conitions Bounary conitions are conitions that must be place on a moel in other to represent everything about the system that will not be moelle. The bounary conitions are use to constrain the motion of the moel at the time of running the simulation. Bounary conitions are conitions aroun the bounary of the geometry that are known. The locations where the constraints are applie, as well as the egrees of freeom that are restricte, are shown in the Figure G.7 below. Select Create Bounary Conition>Symmetry/Antisymmetry/Encastre >Continue., as shown in Figure G.7, an select the region where the constraint shoul be applie (surface, ege, noe etc) an confirm accoringly. The resultant winow provies the options to set the type of constraints. Figure G.7: Position where the telescopic assembly is constraine as inicate by the arrow, in all egrees of freeom to simulate an encastre type fiing. 59

360 G.8 Meshing Definitions The Mesh moule allows for the generation of meshes on parts an assemblies create within ABAQUS/CAE. Various levels of automation an control are available so that it is possible to create a mesh that meets the nees of the analysis. As with creating parts an assemblies, the process of assigning mesh attributes to the moel such as sees, mesh techniques, an element types is feature base. As a result it is possible to moify the parameters that efine a part or an assembly, an the mesh attributes that are specifie within the Mesh moule can be regenerate automatically. Parts are selecte iniviually an meshe accoring to their sie an amount of fine etails they contain. Therefore the mesh sie is specifie eclusively for each part. To ease the process the parts can be appear in the screen iniviually by switching of the other parts. The mesh sie shoul be efine fine enough to cover the parts feature, however very fine mesh sie can result in complicate an time consuming analysis process. Therefore fining the optimum mesh sie coul reuce the process time an maintain the accuracy. In ABAQUS, the following element shapes are efine in the Mesh moule. Depening on whether the region to be meshe was either D or D, the element shapes were assigne accoringly by efault in the Mesh moule. As mentione earlier the element shapes that were use in this analysis were the following linear elements namely: quarilateral, triangular, tetraheral an wege shape elements. For Shell parts, a miture of linear quarilateral elements of type SR an linear triangular elements of type S were use to create the resultant mesh. The SR element shape is efine as a -noe oubly curve thin or thick shell, reuce integration, hourglass control, finite membrane strains []. For Soli parts, a miture of linear wege elements of type CD6 an linear tetraheral elements of type CD were use on iniviual parts. For those soli parts that have hole features or are of a circular or cylinrical isposition, CD6 type elements were use. The CD6 element shape is efine as a 6-noe linear triangular prism []. Those soli parts that have no such features an have no circular cross section are assigne the CD element shape. The CD element shape is efine as a -noe linear tetraheron []. 60

361 As far as was possible elements of a linear geometric orer were chosen at the time of analysis, as oppose to ieally using elements of a quaratic geometric orer. This was because the use of quaratic elements severely affecte the analysis run time resulting in the ABAQUS program to hang thereby not proviing the require results. The proceure for the creation of an acceptable mesh is as follows []:. Assign Mesh Attributes an Set Mesh Controls The Mesh moule provies a variety of tools that allow you to specify ifferent mesh characteristics, such as mesh ensity, element shape, an element type.. Generate the Mesh The Mesh moule uses a variety of techniques to generate meshes. The ifferent mesh techniques provie you with ifferent levels of control over the mesh.. Refine the Mesh The Mesh moule provies a variety of tools that allows the refining of the mesh: a. The seeing tools allow you to ajust the mesh ensity in selecte regions. b. The Partition toolset allows you to partition comple moels into simpler sub regions. c. The Virtual Topology toolset allows you to simplify your moel by combining small faces an eges with ajacent faces an eges.. The Eit Mesh toolset allows you to make minor ajustments to your mesh.. Optimie the Mesh It is possible to assign re-meshing rules to regions of the moel. Re-meshing rules enable successive refinement of the mesh where each refinement is base on the results of an analysis. 6

362 5. Verify the Mesh The verification tool provies information concerning the quality of the elements use in a mesh. G.8. Structure Meshing The structure meshing technique generates structure meshes using simple preefine mesh topologies. ABAQUS/CAE transforms the mesh of a regularly shape region, such as a square or a cube, onto the geometry of the region you want to mesh. For eample, Figure G.8 illustrates how simple mesh patterns for triangles, squares, an pentagons are applie to more comple shapes. Figure G.8: Two-imensional structure mesh patterns [] The structure meshing technique can be applie to simple two-imensional regions (planar or curve) or to simple three-imensional regions that have been assigne the He or Heominate element shape option. 6

363 G.8. Swept Meshing ABAQUS/CAE uses swept meshing to mesh comple soli an surface regions. As per the The Mesh Moule within the ABAQUS Documentation, the swept meshing technique involves two phases []:. ABAQUS/CAE creates a mesh on one sie of the region, known as the source sie.. ABAQUS/CAE copies the noes of that mesh, one element layer at a time, until the final sie, known as the target sie, is reache. ABAQUS/CAE copies the noes along an ege, an this ege is calle the sweep path. The sweep path can be any type of ege a straight ege, a circular ege, or a spline. If the sweep path is a straight ege or a spline, the resulting mesh is calle an etrue swept mesh. If the sweep path is a circular ege, the resulting mesh is calle a revolve swept mesh. For eample Figure G.9 shows an etrue swept mesh. To mesh this moel, ABAQUS/CAE first creates a two-imensional mesh on the source sie of the moel. Net, each of the noes in the two-imensional mesh is copie along a straight ege to every layer until the target sie is reache. Figure G.9: The swept meshing technique for an etrue soli [] To etermine if a region is swept meshable, ABAQUS/CAE tests if the region can be replicate by sweeping a source sie along a sweep path to a target sie. In general, ABAQUS/CAE selects the most comple sie (for eample, the sie that has an isolate ege 6

364 or verte) to be the source sie. In some cases it is possible to use the mesh controls to select the sweep path. If some regions of a moel are too comple to be swept meshe, ABAQUS/CAE asks if these regions shoul be remove from the selection before it generates a swept mesh on the remaining regions. The free meshing technique can be use to mesh the comple regions, or the regions can be partitione into simplifie geometry that can be structure or swept meshe. When you assign mesh controls to a region, ABAQUS/CAE inicates the irection of the sweep path an allows you to control the irection. If the region can be swept in more than one irection, ABAQUS/CAE may generate a very ifferent two-imensional mesh on the faces that it can select as the source sie. As a result, the irection of the sweep path can influence the uniformity of the resulting three-imensional swept mesh, as shown in Figure G.0. Figure G.0: The sweep irection can influence the uniformity of the swept mesh [] G.8. Free Meshing Unlike structure meshing, free meshing uses no pre establishe mesh patterns. When meshing a region using the structure meshing technique, it is possible to preict the pattern of the mesh base on the region topology. In contrast, it is impossible to preict a free mesh pattern before creating the mesh. Because it is unstructure, free meshing allows more fleibility than structure meshing. The topology of regions allows for meshing with the free mesh technique can be very comple. 6

365 This technique can be use to mesh a region with the Tri, Qua, or Qua-ominate element shape options for two-imensional regions or the Tet element shape option for threeimensional regions. The final meshe assembly is meshe using an amalgamation of the meshing techniques outline in the [] an is shown in Figure G.. To ease the process of meshing of the iniviual parts, those parts that are not require within the current view can be iniviually switche on or off. Select View>Assembly Display option from main menu. By selecting the instance tab from the pop up winow the require parts can be selecte properly. The steps are shown in Figure G.. Select See Part Instance from the left han tool bar an select the target part an confirm it. The Global Sees winow will be opene which provies the option to select the mesh sie, isplaye in Figure G.. Figure G.: Controlling the screen view by switching of the irrelevant parts which provies more control in selecting parts in meshing process 65

366 Figure G.: Ajustment of mesh sie The other settings in this winow remaine untouche in this eample. The mesh sie will be confirme an will be assigne to the part by pressing Assign Mesh Control. The meshing process will be finishe by selecting Mesh Part Instance an selecting the part to be meshe. All the engage parts have to be meshe in the same manner. The unmeshe parts will not involve in the analysis hence the parts which nee to be temporarily isolate can be remaine unmeshe. 66

367 Figure G.: Meshe telescopic beam assembly 67

368 G.9 Generation an Interpretation of results Once all of the tasks involve in creating a moel (such as specifying the geometry of the moel, assigning section properties, an efining contact), the Job moule is entere into to analyse the moel. The Job moule allows for the creation of a job, to submit it to ABAQUS for analysis, an to monitor its progress. It is also possible to create multiple moels an jobs an run an monitor these jobs simultaneously. In aition, the option of creating only the analysis input file for a moel is available which allows the viewing an eiting of the input file before submitting it for analysis. Figure G.: Submitting a job for analysis 68

369 G.9. Tip Deflection Results Etraction Once the job has been processe an is complete as inicate in the job manager winow as shown in Figure G., clicking on the Results button runs the Visualiation Moule an the assembly will be shown in its meshe format. Selecting Results>Fiel Output, the esire quantity to be etracte, whether it be stress or isplacement is selecte from the main tab winow, followe by the component of either, in the appropriate coorinate aes as is require. In Figure G.5, the isplacement tab is selecte along with the component of isplacement require. Figure G.5: Selecting the isplacement tab an its component in the negative y- irection The overall tip eflection inuce by any given loa upon the assembly can be euce from the contour plot legen as shown in the top left han corner in Figure G.6. In orer to etract the eflections at iniviual noes in the assembly, noe numbers or noe symbols are generate on the cantilever. This is one by selecting the Options>Common tab an within the Common Plot Options ialog bo selecting the noe labels tab. Once this tab has been selecte an accepte by clicking OK an the noe numbers will appear as shown in the Figure G.8. 69

370 Figure G.6: The isplacement at each of the noes as is plotte along the assembly 70

371 (a) (b) Figure G.7: Noe abel Display Options Figure G.8: Noe abels isplaye on Part In orer to generate reports to etermine the eflections at particular sections or points in the cantilever assembly, select Report>Fiel Output from the Main Menu bar an in the Report Fiel Output ialog bo select the options that are neee. In orer to obtain results at particular noes select the Unique Noal option within Position, in the Variable section of the Report Fiel Output ialog bo. It is also possible to give a unique name to the report file be generate as well as to arrange output accoring to the noe label, element label an so on within the Setup section of the Report Fiel Output ialog bo. The report 7

372 generation proceure outline above is illustrate in Figure G.9, an the generate fiel report in turn is shown in Figure G.0. Figure G.9: Report Generation Proceure for eflection magnitue etraction at iniviual noes Figure G.0: Report arrange accoring to Noe abels 7

373 Knowing the noe numbers an their istance from the fie en, it is possible to etermine the vertical eflection values at sai noes either by means of Report Generation Proceure outline in Figure G.0 as shown above or by using the Probe function available in ABAQUS as shown in the Figure G. below. Figure G.: Obtaining eflection values irectly using the Probe function available in ABAQUS 7

374 G.9. Stress Analysis Results Etraction Once again in orer to etract the esire stress values from the assembly the Results>Fiel Output tab is selecte an the esire stress component is selecte as shown in Figure G.. Figure G. isplays the stress istribution inuce by loaing the cantilever. As has been outline in the eflection etraction section in G.9., the stress results can be etracte by either using report generation techniques specific to each noe as outline in Figures G. an G.5 or by using the Probe function available in ABAQUS as emonstrate in Figure G.6. It must be note that to etract stress magnitues using the former metho, isplaying noe labels along the cantilever assembly as shown in Figures G.7 an G.8 is vital. Figure G.: Selecting the esire stress component tab 7

375 Figure G.: The stress istribution is shown after the analysis is complete 75

376 Figure G.: Report Generation Proceure for stress etermination at iniviual noes Figure G.5: Report arrange accoring to Noe abels 76

377 Figure G.6: Obtaining Stress values irectly using the Probe function available in ABAQUS 77

378 G.9. Buckling Analysis Results Etraction Following completion of the FEA proceure to etermine the critical buckling loa of the telescoping assembly for varying overlap lengths as etaile in Table 7., the results are plotte as shown in Figure 7.0. To avoi uplication, only the etermination of the critical buckling loa for the telescoping assembly for an overlap ratio of 0. is etracte here using the Eigen values generate as is shown in Figure G.7. From Figure G.7, the first Eigen value of 5.68 is etracte an substitute in the formulae for generating the Critical buckling loa [07] as follows: Critical Buckling oa = Dea oa + ive oa Eigen Value As per Figures G.5 an G.6, the ea an live loas are entere as 5000 N an 500 N respectively, an the buckling loa is compute as,8 N. Figure G.7: Determination of critical buckling loa for the telescopic arrangement, for an overlap ratio of 0.. The single Eigen value generate is highlighte an substitute in the equation above to etermine the critical buckling loa. 78

379 APPENDIX H 79

380 APPENDIX H STRAIN GAUGING PRINCIPES AND PROCEDURES Strain is a funamental engineering phenomenon. It eists in all matter at all times, ue either to eternal loa or to the weight of the matter itself. Strains vary in magnitue from automatic imensions to istances easily iscernible by the nake eye, epening upon the materials an loas involve. Scientists an engineers have worke for centuries in the attempt to measure strain accurately, but only the last few ecaes have seen outstaning avancement in the art of strain measurement. The term strain an linear eformation are synonymous an, as use in engineering, refer to change in any linear imension of a boy, usually ue to application of eternal forces. The strain in a piece of rubber when loae is orinarily apparent to the eye but strain in a steel structure or a rigi boy may not be. Average unit strain is the total eformation of the boy in a given irection ivie by the original length in that irection an as such because of its imensionless character has much greater significance than total strain. Strain gauges are use to etermine unit strain. Keeping in min the relationship between stress an strain, it becomes apparent that we can etermine the average intensity of stress in a boy uner some given eternal loa by measuring strains an multiplying by the moulus of elasticity. This is one metho in which stress can be etermine, since stress is not a funamental physical quantity like strain but only a erive quantity. It is no woner then that a great eal of effort has been epene towars prefacing a universal strain gauge. In attempting to evelop such a strain gauge an ieal might be set up as a goal. This ieal strain gauge woul be. Etremely small in sie. Of significance mass. Easy to attaché to the member being analye. Highly sensitive to the strain 5. Unaffecte by the temperature, vibration, humility, or other ambient conitions such as to be encountere in testing machine parts uner service loa 6. Capable of inicating both static an ynamic strains 7. Capable of remote inication an recoring 8. Characterie by the gauge length 80

381 Eternal force applie to an elastic material generates stress, which subsequently generates eformation of the material. the length of the material etens to +Δ if applie force is a tensile force. The ratio of Δ to, that is Δ/, is calle strain. (Precisely, this is calle normal strain or longituinal strain.) On the other han, if compressive force is applie, the length is reuce to - Δ. Strain this time is (- Δ)/. Strain is usually represente as ε. Supposing the cross sectional area of the material to be A an the applie force to be P, stress σ will be P/A, since a stress is a force working on a efinite cross sectional area. In a simple uniaial stress fiel as illustrate below, strain ε is proportional to stress σ, thus an equation σ = E ε is satisfie, provie that the stress σ oes not ecee the elastic limit of the material. E in the equation is the elastic moulus (Young's moulus) of the material. ε = Δ/ ε : Strain : Original length : Change ue to force P Figure H.: Simple illustration for the strain measurement. Because longituinal strain is a ratio between lengths of two wires, it is a quantity having no imension. Usually it is represente in a unit of 0-6, since the ratio of eformation is often very small. Strain may be compressive or tensile an is typically measure by strain gauges. It was or Kelvin who first reporte in 856 that metallic conuctors subjecte to mechanical strain ehibit a change in their electrical resistance. This phenomenon was first put to practical use in the 90s. Funamentally, all strain gauges are esigne to convert mechanical motion into an electronic signal. A change in capacitance, inuctance, or resistance is proportional to the strain eperience by the sensor. If a wire is hel uner tension, it gets slightly longer an its cross-sectional area is reuce. This changes its resistance (R) in proportion to the strain sensitivity (S) of the wire's resistance. In a strain gauge S is the gauge factor efine as R R 8

382 R R S S S R R H. The Strain Gauge The electrical resistance of a length of wire varies in irect proportion to the change in any strain applie to it. That s the principle upon which the strain gauge works. The most accurate way to measure this change in resistance is by using the Wheatstone brige. This is a balance electrical circuit which isplays any resistance change on an inicator or fees it into a process. The main component of a strain gauge is a strain sensitive alloy. The most common is constantan at a thickness of inch, which is use in the foil gri. Constantan also has the best combination of properties necessary for many strain gauge applications. The gri consists of a photo-etche pattern mounte on a very thin backing mae from a plastic such as polyimie, epoy or glass-fiber reinforce epoy-phenolic approimately 0.00 inch thick. This backing allows the strain gauge to be hanle uring installation. It also provies a reay-to-bon surface for cementing the gauge to the specimen, an electrical insulation between the metal foil an the test piece. Depening on the particular application, there s a wie range of foils an backings to choose from. There are also many factors to consier when selecting a gauge. These inclue temperature range, test frequency, elongation, environment an nominal resistance, an so on. Depening on the purpose an accuracy strain gauges can be classifie as follows. General Purpose Gauges This type of gauges are use in general applications such as measuring strain in a simple cantilever, pressure vessel etc. epening on the orientation of gri pattern it can be classifie as follows. 8

383 (a) Uniaial gauge (Figure H.) Gauge with a single gri for measuring strain in the gri irection. A typical uniaial strain gage pattern esigne to measure strains in the irection of the grilines. Gauge lengths for Micro-Measurements strain gauges range from in to.000 in (0.0 mm to 0.6 mm). Figure H.: Uniaial strain gauge [] (b) Biaial Rosettes (Figure H.) Gauge with two perpenicular gris use to etermine principal strains when their irections are known. Figure H.: Biaial rosette [] The biaial ("Tee") rosette pattern has two measuring gris perpenicular to one another. Planar rosettes, like the one shown here, are constructe with all gris on the same plane. Stacke rosettes are also available with separate gris "stacke" on top of one another. At the gauge position two inepenent measurements are mae in perpenicular irections these normally being aligne with strain irections. (c) Three-Element Rosettes (Figure H.) Gauge with three inepenent gris in three ifferent irections are use for ascertaining the principal strains an their irections. The typical "rectangular" rosette pattern shown here has 8

384 its three inepenent gris oriente at 0, 5, an 90 egrees. "Delta" patterns, with gris at 0, 60, an 0 egrees, are also available. Planar rosettes, like the one shown here, are constructe with all gris on the same plane. Stacke rosettes are also available with separate gris "stacke" on top of one another. With three inepenent strain measurements at the gauge installation, the principal strains an their irections can be calculate. Figure H.: Three element rosette [] () Shear Patterns (Figure H.5) Gauges having two chevron gris are use in half-brige circuits for irect inication of shear strain arising uner torsion an shear loaings. Figure H.5: Shear patterns [] Shear gauges have two gris in a chevron pattern that sense normal strains in perpenicular irections 0 5 to the torque ais. The gris often have a common connection for use in halfbrige circuits which yiel the shear strain (ifference in normal strains) irectly when shear pattern gauges are use.. Transucer-Class Gauges Transucer-Class strain gauges are a select group of gauge patterns esigne specifically for transucer applications. The main objective is optimum gauge performance at lower cost in high-volume prouction quantities. 8

385 Eclusive features of Transucer-Class gauges inclue:. Optimum backing thickness tolerance - This is particularly important to minimie creep variations between gage installations.. Uniform backing trim imensions - Matri imensions liste have a tolerance of ±0.005 in (±0. mm) on any ege (measure from gri centrelines).. Multiple creep compensation choices for most gauge patterns - A close inspection of the gage pattern will reveal a small letter on the gauge matri net to the gri.. Special pattern refinement for improve gauge-to-gauge reproucibility the creep variation ue to operating temperature changes is reuce.. Special-Purpose Sensors This inclues variety of proucts esigne to meet special nees an perform special functions in eperimental stress analysis. These inclue:. Bonable Temperature Sensors - With nickel-foil gris, these sensors are use for general-purpose temperature measurements over the range of -0 to +500 F (-95 to +60 C).. Crack Detection Gauges - Convenient, economical metho of etecting cracks or crack growth.. Crack Propagation Gauges - Multiple conucting gris on a single backing accurately inicate the rate of crack propagation.. Welable Gauges - Strain gauges an temperature sensors bone to a metal carrier for spot weling to a test structure when ahesive cannot be use or minimum installation time is require. 5. Shear Moulus Gauges - Special gauges for accurately etermining the shear moulus of composite materials within stanar obsolesce an compact test specimens. 6. Embement Strain Gauges - Special strain sensors for embeing in concrete. 85

386 H. Strain Transformation an Rosette Gauge Theory It is often esire to measure the full state of strain on the surface of a part, that is to measure not only the two etensional strains, an y, but also the shear strain, γ y, with respect to co-orinate,y. It is clear that a single gauge is capable only of measuring the etensional strain in the irection that the gage is oriente. Assuming that the an y aes are specifie, it woul be possible to mount two gauges in the an y irections, respectively to measure the associate etensional strains in these irections. However, there is no irect way to measure the shear strain, γ y. Nor is it possible to measure the principal strains since the principal irections are not generally known. The solution to this problem lies in recogniing that the D state of strain at a point (on a surface) is efine by three inepenent quantities which can be taken as either: (a), y, an γ y, or (b),, an θ, case (a) refers to strain components with respect to an arbitrary - y ais system, an case (b) refers to the two principal strains an their perpenicular irections. Either case efines the state of D strain on the surface an can be use to compute strains with respect to any other co-orinate system. This situation implies that it shoul be possible to etermine these inepenent quantities if it is possible to make three inepenent measurements of strain at a point on the surface. The most obvious approach is to place three strain gauges together in a rosette with each gage oriente in a ifferent irection an with all of them locate as close together as possible to approimate to measurements at a point. As will be shown below, if the three strains an the gauge irections are known, it is possible to solve for the principal strains an their irections or equivalently, the state of strain with respect to an arbitrary -y coorinate system. The relations neee are the strain transformation equations for which Mohr s Circle construction provies a goo visualiation of this process. D Strain Transformation an Mohr s Circle The two imensional strain transformation equations are very similar to stress transformation equations an are represente below., cos ' sin y sin cos y, sin cos sin cos (H.) ' y y y 86

387 , y '' sin cos cos sin y y These transformation equations involve squares an proucts of sine an cosine functions an these can be replace with ouble-angle results to yiel the ouble-angle form of the transformation equations:, ' y y y cos sin, y ' y y cos sin (H.) y sin cos, '' y y y Given the reay availability of powerful calculators an spreasheets, evaluation of these transformation equations is a relatively simple matter toay an involves little more than a few secons to enter the formula in a calculator or a spreasheet. This was not always so simple a task an the Mohr s Circle graphical representation as in Figure H.6, was evelope long ago to ai in this process. Toay, Mohr s Circle is not neee for graphical calculation, but it oes provie a goo visualiation of the transformation equations an the geometry can be use to infer the actual form for the equations neee for eecution in a calculator or computer. ε y ε ε ε, ε y ϕ γ y / γ y / (ε +ε y )/ ε (ε -ε y )/ Figure H.6: Basic Mohr s circle geometry 87

388 The ouble-angle form of the transformation relations involve simple sine an cosine terms along with a constant that shoul suggest equations of a circle with centre away from the coorinate origin. The trick here is to ientify the appropriate new an y values to plot to construct a circle. This is perhaps easier to o by eplaining the Mohr s Circle than it is to actually euce the form irectly. Figure H.6 above shows Mohr s Circle for a state of strain efine by y ais. Assume for the moment that the coorinates of the opposite ens (X,Y) of the inicate iameter of the circle efine the strain state,, y, an γ y, in the y ais system. Note that both the an y etensional strains are plotte on the abscissa ( ais) while one-half the shear strain, γ y /, is plotte along the orinate. The positive irection for γ y, is taken to be ownwar. One can construct the circle by first plotting the (ε, γ y /) pair as point X on the iameter. Net, the pair (ε, -γ y /) are plotte as the opposite en Y of the iameter, an the circle can be constructe with X-Y as the iameter. This is the basic Mohr s circle an it always has its centre on the abscissa at a point given by the value / compute as: D. y y y. The circle iameter is easily So far, there is no clear connection to the ouble-angle transformation equations above, but this will become evient in a moment. To calculate a new state of strain, ε, ε y an γ y in an,y co-orinates rotate θ counterclockwise from the,y ais system, one must construct a new iameter for Mohr s Circle rotate θ counterclockwise from the initial X-Y iameter as shown below in Figure H.7. The coorinates of the new iameter enpoints, X -Y, represent the new strain state as compute from the ouble-angle strain transformation equations. This shoul be clear by inspection of the Mohr s Circle an by evaluation of the circle geometry as shown. Note that rotation in Mohr s Circle is always twice the geometric rotation so that the state of strain efine by opposite ens of a iameter of the circle (e.g., 80 apart) correspons to strains that are at 80 /=90 in geometric aes. 88

389 ε y Y ε ε X γ y / γ y / ε, ε y (ε +ε y )/ (ε -ε y )/ ε ε γ y / Figure H.7: Strain transformation of It shoul be pointe out also that this particular form for Mohr s Circle uses an inverte orinate (y ais) an positive θ is counterclockwise. Another popular form for Mohr s Circle uses an upwar positive orinate ais an positive clockwise θ. It is strikingly apparent from Figure H.7 that there is a state of strain that oes not involve any shear strain an this correspons to the circle iameter lying along the abscissa an efine by the two points, an which are the etremities of the iameter as shown. These are calle the principal strains an they are associate with an y ais system rotate, φ, counterclockwise from the y ais (φ in Mohr s Circle). This efines not only the principal strain magnitues but also their irections (a,b) torsion an shear, plane strain an () equi-biaial tension. Figure H.8 shows other useful Mohr s Circle configurations. Strain Gauge Rosettes Strain gauge rosettes consist of two or more co-locate strain gauges oriente at a fie angle with respect to each other. Strict co-location of the gauges requires mounting each iniviual gauge on top of the others in what is calle a stacke rosette, but this leas to a complicate an often inaccurate type of gauge. The more common approach is to place the gauges in a tightly packe pattern as close as possible to the rosette center. Rosettes typically involve, or strain gauges with relative orientations of 0, 5, 60 or 90. Figure H.9 shows several eamples. At least inepenent strain reaings are neee to efine the D state of strain if no other information is available so the -gauge rosettes are the most popular (the 90 - gauge rosette can be use to measure principal strains when the principal irection is known 89

390 an the gage can be oriente accoringly). The rectangular rosette an the elta rosette are the most commonly use -gauge rosettes because of their simple geometry. Figure H.8: Some useful Mohr s circle configurations. (a) (b) (c) () Figure H.9: Typical strain gauge rosettes (a) Rectangular rosette (b) Delta rosette (c) Delta rosette () Stacke elta rosette [] 90

391 Normal Strain n an Shear Strain s at a plane incline at y B C B C s N o rm a l to A C y E E A D A D Figure H.0: Normal an Shear Strains Consier Figure H.0. The objective is to etermine the irect strain n an shear strain s for irections normal an tangential to a plane incline at ue to,, The figure y y shows an unstraine rectangular element ABCD an its straine conition A B C D. Normal strain n is obtaine by consiering the length change along AC. Shear Strain s is obtaine by consiering the rotation of BE. It can be assume that A C D is forme by AD having an etension D C. to become A D an DC having an etension y to become The shear strain s is the rotation angle between the normal to A C at E an B E. The shear strain y. is the rotation angle between the normal to A D at D an D C. Now the following equations can be written A' D' AD AD( ) AD( ) AD y C' D' CD y CD( ) CD( y ) CD In a similar fashion it can sai that A' C' AC( ) n Now using cosine rule in A C D gives 9

392 ( A' C') ( A' D') ( C' D') A' D' C' D' cos( y ) AC ( ) n AD ( ) CD ( ) y AD( ) CD ( ) sin y y Neglecting higher powers of, an an using n y AC sin AD CD an y y this can be reuce to AC ( ) AD ( ) CD ( ) AD CD n y y ( AC) n ( AD) ( CD) AD CD y y Diviing both sies by AC an introucing cos sin sin cos n y y sin an cos gives Rectangular Rosette Gauge Equations Given the measurement of inepenent strains from the gauges in a rectangular rosette it is possible to calculate the principal strains an their orientation with respect to the rosette gauge. It is also reaily possible to calculate the state of strain at the gauge location with respect to any particular y ais system using either the rosette reaings or the principal strains an their ais orientation. To illustrate this, consier a situation in which the rosette is oriente with gauges labele A, B an C at 5 apart as shown in Figure H.. Figure H.: Rectangular rosette strain orientation Assume also that the principal strains at the rosette are oriente at an angle, φ, to the rosette s gauge A. For this case, it is easy to use the strain transformation equations to calculate the strain in each rosette gage in terms of the principal strains an the angle, φ, (simply assume 9

393 an γ y =0 an compute ε =ε A, ε B an ε C for angles of rotation, +5º, an, y +90º to yiel three equations: A cos (H.a) 0 B cos 5 (H.b) 0 C cos 90 (H.c) In a plane strain situation the full state of strain on the surface of a part can either be efine with three measurements, to incluing two normal strains, an y, an one the shear strain, γ y, with respect to co-orinates -y; or two the principal strains an. Gauges however are capable only of measuring the etensional strain in the irection that the gauge is oriente. Therefore a rosette with three gauges is neee for the measurements. In Equations (H. (a-c)), these are simultaneous equations relating, to,,. It is a relatively simple matter to invert these an solve for yieling: A, B C,, in terms of A, B, C A C [( A B) ( B C ) (H.a) A C [( A B ) ( B C ) (H.b) B A C tan (H.c) A C The equations obtaine above can be use to compute the principal strains an the principal ais orientation irectly from the rectangular rosette gauge reaings. Note that there are many ifferent possible gauge numbering arrangements besies the particular A, B, C layout here, an they can lea to forms for the final results shown above but with A, B an C interchange. It shoul be note that the above results can also be evelope irectly from the Mohr s Circle representation with about the same amount of effort an perhaps a bit more visualiation of the results alternatively. It is somewhat simpler to arrange the rosette such 9

394 that gauge A is along the ais an gauge C is along the y ais. Here, y are not principal irections when from Equation H. it follows that: A B C 0 0 y y cos 0 y sin 0 0 y y cos 5 y sin 5 y y y y cos 90 y sin 90 y These equations can reaily be solve for the strain in the -y co-orinates: A (H.5a) y C (H.5b) y B A C (H.5c) This efines the strain state at the rosette with respect to co-orinates -y. It is a simple matter to now construct a Mohr s Circle an from this to compute the principal strains an their orientation with respect to the -y ais (an therefore the rosette). Figure H. summaries the results an reveals clearly that the maimum shear strain is given by twice the raius of the Mohr s Circle, an in this case it can be compute in terms of principal strains as: / ma. 9

395 ε C ε ε ε, ε y ϕ ε A - ε B -ε C γ y / (ε A +ε C )/ ε A (ε A -ε C )/ Figure H.: Mohr s circle for rectangular rosette Alternatively the principal strain equations may be applie to Equations (H.5a-c) to give A C [( A B) ( B C ) A C [( A B ) ( B C ) tan B A C Principal Stresses A C It shoul be pointe out that the above results involve strain only an o not escribe the state of stress at the rosette. In orer to etermine the stress state, it is necessary to use the stress strain relations to epress the stress components in terms of the strain components. For linearly elastic (Hookean) behavior, it follows that the principal stresses can be compute from the principal strains (shear strain is ero for principal irections). Principal stresses are obtaine using the following equations: (H.6a) E E (H.6b) E E 95

396 From Equations (H.6a, b): E [ ] ( ) E [ ] ( ) H. Instrumentation an Data Acquisition System The basic principles for accurate measurement were establishe many years ago. Moern technology offers better techniques, better resolution, the potential for higher accuracies, but primary much higher spees for ata acquisition an analysis. Almost all strain measurement systems can be broken own into components as shown in below figure. Brige Wheatstone brige Supply Amplifier Display &/or Analogue Storage Processor/ Computer Display &/or Digital Storage Figure H.: Schematic strain measurement system The correct choice an the proper installation are very important an it is assume here that the gauges perform correctly. A gauge can therefore consier as a passive resistor which requires a power source. Changes in resistance cause by mechanical strain are measure in a brige circuit which prouces an out of balance voltage. This voltage nees to be amplifie an after processing isplaye or store or both, after processing it to represent the require units. This manipulation may be by means of controls in harware e.g. gauge factor control. 96

397 In a computer base system, all the manipulation may occur in the software (igital) either in the test programming, or in the ata reuction an analysis, before or after storage. For straight forwar measurements of static strains from a small number of strain gauges, the commercially available self containe strain inicators are robust, reliable, accurate an almost foolproof. Similarly, for ynamic measurements from a few channels the choice of suitable strain gauge amplifiers an recorers is quite straightforwar. For large test programmes, either static or elaborate ynamic systems can be built up from commercially available instrumentation an peripherals or purchase as a buil system. Wheatstone brige circuit an strain gauges The strain gauge is connecte into a Wheatstone brige circuit with a combination of four active gauges (full brige), two gauges (half brige), or, less commonly, a single gauge (quarter brige). In the half an quarter circuits, the brige is complete with precision resistors. The complete Wheatstone brige is ecite with a stabilie DC supply an with aitional conitioning electronics, can be eroe at the null point of measurement. As strain is applie to the bone strain gauge, a resistive change takes place an unbalances the Wheatstone brige. This results in a signal output, relate to the strain value. As the signal value is small, (typically a few millivolts) the signal conitioning electronics provies amplification to increase the signal level to 5 to 0 volts, a suitable level for application to eternal ata collection systems such as recorers or PC Data Acquisition an Analysis Systems. Typical strain gauge resistances range from 0 Ohms to k Ohms (unstresse). This resistance may change only a fraction of a percent for the full force range of the gauge, given the limitations impose by the elastic limits of the gauge material an of the test specimen. Forces great enough to inuce greater resistance changes woul permanently eform the test specimen an/or the gauge conuctors themselves, thus ruining the gauge as a measurement evice. Hence, in orer to use the strain gauge as a practical instrument, we must measure etremely small changes in resistance with high accuracy. Such emaning precision calls for a brige measurement circuit. Using a Wheatstone brige with a nullbalance etector an a human operator to maintain a state of balance, a strain gauge brige circuit measures strain by the egree of imbalance. A precision voltmeter in the centre of the brige provies an accurate measurement of that imbalance. 97

398 Quarter brige strain gauge circuit Typically, the rheostat arm of the brige (R in the Figure H.) is set at a value equal to the strain gauge resistance with no force applie. The two ratio arms of the brige (R an R) are set equal to each other. Thus, with no force applie to the strain gauge, the brige will be symmetrically balance an the voltmeter will inicate ero volts, representing ero force on the strain gauge. Figure H.: Quarter brige strain gauge circuit [6] As the strain gauge is either compresse or tense, its resistance will ecrease or increase, respectively, thus unbalancing the brige an proucing an output at the voltmeter. This arrangement, with a single element of the brige changing resistance in response to the measure variable (mechanical force), is known as a quarter-brige circuit. As the istance between the strain gauge an the three other resistances in the brige circuit may be substantial, wire resistance has a significant impact on the operation of the circuit. To illustrate the effects of wire resistance, the same schematic iagram is shown in Figure H.5 but with the aition of two resistor symbols in series with the strain gauge to represent the wires. Figure H.5: Quarter brige strain gauge circuit with aition of two resistors [6] 98

399 The strain gauge's resistance (R gauge ) is not the only resistance being measure: the wire resistances R wire an R wire, being in series with R gauge, also contribute to the resistance of the lower half of the rheostat arm of the brige, an consequently contribute to the voltmeter's inication. This, of course, will be falsely interprete by the meter as physical strain on the gauge. While this effect cannot be completely eliminate in this configuration, it can be minimie with the aition of a thir wire, connecting the right sie of the voltmeter irectly to the upper wire of the strain gauge: Figure H.6: Three-wire, quarter-brige strain gauge circuit [6] Because the thir wire carries practically no current (ue to the voltmeter's etremely high internal resistance), its resistance will not rop any substantial amount of voltage corresponingly. Notice how the resistance of the top wire (R wire ) has been by-passe now that the voltmeter connects irectly to the top terminal of the strain gauge, leaving only the lower wire's resistance (R wire ) to contribute any stray resistance in series with the gauge. There is a way, however, to reuce wire resistance error far beyon the metho just escribe, an also help mitigate another kin of measurement error ue to temperature. An unfortunate characteristic of strain gauges is that of resistance change with changes in temperature. This is a property common to all conuctors, some more than others. Thus, the quarter-brige circuit shown (either with two or with three wires connecting the gauge to the brige) works as a thermometer just as well as it oes a strain inicator. 99

400 Availability of Equipment Post processing Security Environment Repetitive Testing Non-Repeatable Tests Accuracy/Resolution Attene or unattene test Real Time processing Number of Gauges Test Duration Signal frequency Factors which affect the choice of an instrumentation system Selecting a precise acquisition system epen on several factors. Each factor capable of much finer subivision an infinite number of possible system types can be imagine. Below test conitions stretch the selecting appropriate gauge for the right application. (a) Test conitions Choice of instrumentation system Figure H.7: Fishbone iagram showing factors which affect the selection of an instrumentation system [] 00

401 (b) Typical systems A static test usually consists of loaing a component or structure in increment with the strains being recore for each increment. The choice of instrumentation epen on the number of channels, the available time or manpower, the time over which each loa increment can be hel constant for the all the channels to be scanne, an the amount of real time processing require. Testing may be carrie out uner laboratory, workshop or site conitions. () The simplest equipment woul be a manually operate portable strain inicator as shown in Figure H.8. The equipment is portable an robust an ieal for on-site work. There are channel selectors which can be connecte to the inicator so that, typically, one operator can log up to 0 reaings per minute on han written sheets. Figure H.8: Strain inicator [] () More elaborate instrumentation systems have much higher scanning spees an sophisticate computer control of test, real-time processing an storage. The system utilies a laptop which is only eicate to the system whilst actually scanning an recoring test ata. The scanning spee of this system is 5 channels per secon with a maimum of 000 channels. Post processing an test programming can be one with the laptop off site uner office or laboratory conitions. () A Data ogger is a evice use mainly in measurement application. Data logging is the measurement of any physical or electrical parameters over a perio of time. The ata can be in the form of voltage, strain, temperature, current, etc. It acts as an interface between the strain gauge an the ABVIEW software. abview is a ata acquisition software package mainly use with harware acquisition boars abview has many features like processing of measure ata or simulate signals an use for ata acquisition. ABVIEW (short for aboratory Virtual 0

402 Instrumentation Engineering Workbench) is a graphical programming environment use to evelop sophisticate measurement, test an control systems using block iagrams that resemble a flowchart. Using ab VIEW the strain parameters can be measure. The wires of strain gauge are connecte to the ata logger an measurements will be collecte via the ABVIEW given that the ABVIEW is connecte to the ata logger. A typical ata logger uses a combination of analog an igital filtering to make an accurate representation of in-ban signals while giving out-of-ban signals. The filters iffer between signals base on the frequency range, or banwith, of the signal. arger centralie computers can be use for post processing either using ata recore on site, or with irect inputs from a remote ata acquisition system using a ata link, such as a MODEM operating on GPO telephone lines. Figure H.9: National Instruments Data acquisition system [] 0

403 H. Strain Gauge Selection Figure H.0: Characteristic of a strain gauge []. Consier the Gauge Pattern a) Uni-aial strain gauge shoul be consiere if: A single strain is to be measure an the irection is known or low cost is a priority. Figure H.: Uni-aial strain gauge [] b) Bi-aial (0, 90 T-Rosette) shoul be consiere if: Principal strains (, ) are to be measure an irection is known (also applicable to torques). Figure H.: Bi-aial strain gauge [] c) Tri-aial/ Three-Element ( rectangular rosette, elta rosette) shoul be consiere if: Principal strains (, ) are to be measure an irection is unknown. 0

404 Figure H.: Rectangular rosette & Delta Rosette [] ) Stacke rosette gauge configuration shoul be consiere if: There isn t much space available for mounting or if localie strain is to be measure where large strain graient eists. Figure H.: Stacke Strain Gauge Configuration [] e) Planar gauge configuration shoul be consiere if: Heat effects are likely to be an issue, where accuracy an stability is critical. Figure H.5: Planar gauge configuration []. Gauge ength Shorter gauges are use (l mm) when. There isn t much space available for mounting.. ocalie strain is to be measure (e. near a fillet, hole, notch etc.).. A large strain graient eists. 0

405 . Accuracy of the measurement is less critical. onger gauges are use (l 6mm) when. Easier installation is a priority.. Heat effects are likely to be an issue.. Accuracy an stability are critical.. The surface is non-homogeneous. 5. ow cost is a priority (5- mm lengths are usually the cheapest).. Wire Material There are a number of ifferent metal alloys that are use in strain gauges. Each one has its own unique properties that make it more suitable for particular applications. a) Nickel/Copper Alloy (Gauge Factor.). Most common material in gauges, an therefore low cost.. Better suite to static strains rather than ynamic.. Gauge factor remains nearly constant even through large eformations.. Ehibits self-temperature compensation. 5. Temp range -0 C to 9 C (though can eperience a lot of rift above 65 C). b) Nickel/Chromium/Iron/Aluminum (Gauge Factor.0). Best suite to low temperature environments (as low as -65 C).. More stable over etene perios of strain.. Very ifficult to soler. c) Iso-Elastic: Iron/Nickel/Chromium/Manganese Alloy (Gauge Factor.6). High sensitivity.. High resistance.. Well suite for ynamic strain reaings (has a goo fatigue life).. Does not ehibit temperature compensation. 5. Non-linear response beyon -5000με. ) Platinum Base Alloys, alloye with Tungsten or Iriium (Gauge Factor -.0 to 5.). High sensitivity.. Well suite to high temperature environments (in ecess of 0 C ). 05

406 e) Semi-Conuctor Gauges (Gauge Factor 70 to 5).. Very high sensitivity (-50 times that of wire).. High resistance.. Typically more epensive than wire.. Can be mae smaller than wire/foil gauges for lower cost. 5. More likely to rift with temperature changes. 6. Resistance oesn t change linearly with strain (making ata analysis more ifficult). 7. Typically have lower strain limits than a comparable wire gauge.. Backing Material a) Polyimie. Most common backing material an therefore low cost.. Better suite to static strains rather than ynamic.. Capable of large elongations an is very fleible.. Not suitable in etreme temperature conitions. b) Epoy. Minimies errors cause by the backing.. Brittle an require special skill to install.. Maimum elongation is limite. c) Glass Fibre Enforce Epoy. Performs well over wiest temperature range (up to 00 C).. Well suite to ynamic strains an fatigue loaing.. Maimum elongation is limite. ) Strippable Backing. Backing is remove uring installation an the ahesive serves as an insulator between the gage an the mounting surface.. Best for use in etremely high temperature applications.. Installation requires special skill. 06

407 5. Consier Ahesives a) Cyanoacrylate Cement. Very common / Inustry stanar.. Fast boning -0min.. Gentle clamping require for - minutes.. Does not last for etene perios of time (months). b) Epoy. Ehibits high boning strength.. Shoul be use when high strains (e.g. to failure) are to be measure.. Require a clamping pressure (-5 to 0psi) uring cure.. Has a long cure time, can be ecrease by applying heat (-0 C). 07

408 H.5 Surface Preparation Steps For a strain gauge to rea properly an reliably it must be installe correctly. This means first preparing the surface to which you will be boning the gauge later. The proceures for preparing the surface are simple an easy to follow, yet will result in consistent, strong, an stable bons. The proceures outline below are generalie for all metals. The sequence of steps that follows is illustrate in Figures H.6-8 [].. Surface clean-degreasing Use a solvent (such as acetone or alcohol) to remove any grease or oils from the surface to which the stain gauge will be bone. This is to prevent any contaminants from being riven into the surface while performing subsequent steps. Clean an area significantly larger than the gauge ( to 6 inches on all sies) to prevent any contaminants from the surrouning area from being introuce into the gauge area. Figure H.6: Use a liberal amount of egreaser. Figure H.7: Wipe the specimen surface thoroughly with a gaue sponge. 08

409 Figure H.8: To avoi recontamination, iscar soile sponges an continue until the sponge comes up clean.. Abrae surface Remove any oiation, paint or coating from the surface finishing the abraing with a 00 grit silicone-carbie paper to ensure a proper teture for ahesion. A cross-hatche abrasion pattern is preferable. Be careful not to over-abrae the surface resulting in change of either imensions or mechanical properties. Figure H.9: Floo the gagging area with conitioner. Wet lap with the 0-grit silicon carbie paper. Do not allow conitioner to ry on the surface. Figure H.0: A oen strokes are usually aequate. 09

410 Figure H.: Wipe ry with a gaue sponge. Use only once through the gauging area. With a refole or fresh sponge, wipe away from the gauging area. Figure H.: Remove any ecess chemicals from the work surface.. Mark layout lines-burnishing Use a clean rule an a har pencil or pen to mark the esire position of the gage. Perpenicular lines crossing at the center of the gage area is stanar, so that they can be line up with reference marks on the gage. Figure H.: With a clean straight ege, an a H pencil firmly burnish a layout line. Hol the pencil perpenicular to the surface.. Cleaning Scrub the area with a solvent or markete aci conitioner with a cotton-tippe applicator until the tip no longer comes up iscolore. Do not allow the conitioner to ry on the surface, use a gaue sponge to wipe it off in a single slow stroke, then again with a clean sponge in the opposite irection. This prevents ragging any of the contaminates back into the gage area. 0

411 Figure H.: Use a liberal amount of aci conitioner to remove all graphite from the burnishe layout line by scrubbing along the line with a cotton-tippe applicator. Figure H.5: Keep scrubbing, but check the applicator tip for soile appearance. Continue until the tip comes up clean. Figure H.6: Now, floo an re-clean the entire gagging area. Figure H.7: Replace the applicators when they become soile. As before, continue scrubbing until the tip comes up clean.

412 Figure H.8: Refol, an ry the remaining area. H.6 Boning Proceure. Using tweeers to remove the gage from the transparent envelope, place the gauge (boning sie own) on a chemically clean glass plate or gauge bo surface. If a soler terminal will be use, position it on the plate ajacent to the gauge as shown. Place a gage installation tape over the gauge an terminal. Take care to center the gauge on the tape. Carefully lift the tape at a shallow angle (about 5 egrees to specimen surface), bringing the gage up with the tape as illustrate above. Figure H.9: Removing the gauge from transparent envelope.. Position the gauge/tape assembly so that the triangle alignment marks on the gauge are over the layout lines on the specimen. If the assembly appears to be misaligne, lift one en of the tape at a shallow angle until the assembly is free of the specimen. Realign properly, an firmly anchor at least one en of the tape to the specimen.

413 Figure H.0: Positioning the gauge on the layout line.. ift the gauge en of the tape assembly at a shallow angle to the specimen surface (about 5 egrees) until the gage an terminal are free of the specimen surface. Continue lifting the tape until it is free from the specimen approimately / in [0 mm] beyon the terminal. Tuck the loose en of the tape uner an press to the specimen surface so that the gauge an terminal lie flat, with the boning surface epose. Figure H.: ift the tape to allow applying catalyst.. A cyano acrylate can now be applie to the boning surface of the gage an terminal. Very little of this catalyst is neee, an it shoul be applie in a thin, uniform coat. Wipe the brush approimately 0 strokes against the insie of the neck of the bottle to wring out most of the catalyst. Move the brush to the ajacent tape area prior to lifting from the surface. Allow the catalyst to ry at least one minute uner normal ambient conitions of +75 F [+ C] an 0% to 65% relative humiity before proceeing.

414 Figure H.: Applying cyano acrylate. 5. ift the tucke-uner tape en of the assembly, an, holing in the same position, apply one or two rops of ahesive at the fol forme by the junction of the tape an specimen surface. This ahesive application shoul be approimately / in [ mm] outsie the actual gauge installation area. This will insure that local polymeriation that takes place when the ahesive comes in contact with the specimen surface will not cause unevenness in the gauge glue line. Figure H.: Applying ahesive. 6. Immeiately rotate the tape to approimately a 0-egree angle so that the gage is brige over the installation area. While holing the tape slightly taut, slowly an firmly make a single wiping stroke over the gage/tape assembly with a piece of gaue bringing the gauge back own over the alignment marks on the specimen. Use a firm pressure with your fingers when wiping over the gage. A very thin, uniform layer of ahesive is esire for optimum bon performance.

415 Figure H.: Applying gauge on the test specimen. 7. Immeiately upon completion of wipe-out of the ahesive, firm thumb pressure must be applie to the gage an terminal area. This pressure shoul be hel for at least one minute. In low-humiity conitions (below 0%), or if the ambient temperature is below +70 F [+0 C], this pressure application time may have to be etene to several minutes. Figure H.5: Applying uniform pressure. 8. The gage an terminal strip are now solily bone in place. It is not necessary to remove the tape immeiately after gage installation. The tape will offer mechanical protection for the gri surface an may be left in place until it is remove for gage wiring. To remove the tape, pull it back irectly over itself, peeling it slowly an steaily off the surface. This technique will prevent possible lifting of the foil on open-face gauges or other amage to the installation. 5

416 Figure H.6: Remove the tape. H.7 ea Wire Attachment When attaching lea wire, the most important factor is to prevent overheating of the gauge. This can melt the gauge an rener it useless.. Mask the strain gauge with rafting tape leaving only the soler tabs epose.. Clean the tip of the solering iron on the wet sponge pa.. Tin the solering tip with some rosin core soler.. ay the en of the soler across the soler pa an apply the iron tip onto the soler. Apply firm pressure for no more than one secon an remove both the soler an the iron simultaneously. 5. You shoul now have a bright even moun of soler on the pa. Repeat the above proceure until you have a nice even moun of soler on each soler pa an each strain relief terminal. 6. Strip an tin the wires. Ben the ens of the wire such that there will be a small ben between the strain relief an the gauge when the wire is solere to the gauge. 7. Tape the wire assembly in place using the provie rafting tape. Make sure the wires are in the eact place they will be once they are installe. If you are installing a rosette or multiple gauges, labeling the wires at this point will allow you to ientify them once they are solere into place. 8. Press the solering iron onto the wire over a pa while feeing a small amount of soler between the iron an the wire. The wire shoul slie into the melting soler. Allow the connection to cool thoroughly before hanling. 9. Repeat for each soler tab an the strain relief terminals. 6

417 0.Remove any leftover flu from the soler with a gaue sponge soake in rosin solvent. Use a abbing action to prevent amage to the gauge..check the connections an their resistances. If there are any unepecte resistances or the wires are not solily attache, re-soler the connections. 7