Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method. Version 2 CE IIT, Kharagpur


 Tobias Harmon
 11 months ago
 Views:
Transcription
1 odule Analysis of Statically Indeterminate Structures by te Displacement etod Version E IIT, Karagpur
2 Lesson 17 Te SlopeDeflection etod: rames wit Sidesway Version E IIT, Karagpur
3 Instructional Objectives After reading tis capter te student will be able to 1. Derive slopedeflection equations for te frames undergoing sidesway.. Analyse plane frames undergoing sidesway., Draw sear force and bending moment diagrams. 4. Sketc deflected sape of te plane frame not restrained against sidesway Introduction In tis lesson, slopedeflection equations are applied to analyse statically indeterminate frames undergoing sidesway. As stated earlier, te axial deformation of beams and columns are small and are neglected in te analysis. In te previous lesson, it was observed tat sidesway in a frame will not occur if 1. Tey are restrained against sidesway.. If te frame geometry and te loading are symmetrical. In general loading will never be symmetrical. Hence one could not avoid sidesway in frames. or example, consider te frame of ig In tis case te frame is symmetrical but not te loading. Due to unsymmetrical loading te beam end moments and are not equal. If b is greater tan a, ten >. In Version E IIT, Karagpur
4 suc a case joint and are displaced toward rigt as sown in te figure by an unknown amount Δ. Hence we ave tree unknown displacements in tis frame: rotations θ, and te linear displacement Δ. Te unknown joint rotations θ θ and θ are related to joint moments by te moment equilibrium equations. Similarly, wen unknown linear displacement occurs, one needs to consider forceequilibrium equations. Wile applying slopedeflection equation to columns Δ in te above frame, one must consider te column rotation ψ as unknowns. It is observed tat in te column, te end undergoes a linear displacement Δ wit respect to end A. Hence te slopedeflection equation for column is similar to te one for beam undergoing support settlement. However, in tis case Δ is unknown. or eac of te members we can write te following slopedeflection equations. EI + [ θ A + θ ψ ] were ψ Δ ψ is assumed to be negative as te cord to te elastic curve rotates in te clockwise directions. A D D EI A + [ θ + θ A ψ ] EI + [ θ + θ ] EI + [ θ + θ ] EI Δ D + [ θ + θ D ψ D ] ψ D EI D + [ θ D + θ ψ D ] (17.1) As tere are tree unknowns ( θ, θ and Δ ), tree equations are required to evaluate tem. Two equations are obtained by considering te moment equilibrium of joint and respectively (17.a) A (17.b) Now consider free body diagram of te frame as sown in ig Te orizontal sear force acting at A and of te column is given by D Version E IIT, Karagpur
5 H1 (17.a) A + Similarly for memberd, te sear force is given by H H + D D (17.b) Now, te required tird equation is obtained by considering te equilibrium of member, X H1+ H 0 0 A + + D + D 0 (17.4) Substituting te values of beam end moments from equation (17.1) in equations (17.a), (17.b) and (17.4), we get tree simultaneous equations in tree unknowns θ, θ and Δ, solving wic joint rotations and translations are evaluated. Version E IIT, Karagpur
6 Knowing joint rotations and translations, beam end moments are calculated from slopedeflection equations. Te complete procedure is explained wit a few numerical examples. Example 17.1 Analyse te rigid frame as sown in ig. 17.a. Assume EI to be constant for all members. Draw bending moment diagram and sketc qualitative elastic curve. Solution In te given problem, joints and rotate and also translate by an amount Δ. Hence, in tis problem we ave tree unknown displacements (two rotations and one translation) to be evaluated. onsidering te kinematically determinate structure, fixed end moments are evaluated. Tus, 0 ; 0 ; + 10 kn. m ; 10kN. m ; 0 ; 0. (1) A Te ends A and D are fixed. Hence, θ A θ D 0. Joints and translate by te same amount Δ. Hence, cord to te elastic curve ' and D' rotates by an amount (see ig. 17.b) Δ ψ ψ D () ords of te elastic curve ' and D' rotate in te clockwise direction; enceψ and ψ D are taken as negative. D D Version E IIT, Karagpur
7 Now, writing te slopedeflection equations for te six beam end moments, EI + [ θ + θ ψ ] A 0 ; θ A 0 ; ψ Δ. A D EIθ + EIΔ 4 EIθ + EIΔ 10 + EIθ EIθ EIθ + EIθ 4 EIθ + EIΔ Version E IIT, Karagpur
8 D EIθ + EIΔ () Now, consider te joint equilibrium of and (vide ig. 17.c) (4) A D (5) Te required tird equation is written considering te orizontal equilibrium of te entire frame i.e. 0 (vide ig. 17.d). X H H 0 H + H 10. (6) 1 Version E IIT, Karagpur
9 onsidering te equilibrium of te column andd, yields H 1 A + and H D + D (7) Te equation (6) may be written as, (8) A + D D Substituting te beam end moments from equation () in equations (4), (5) and (6).EIθ + 0.5EIθ EIΔ 10 (9).EIθ + 0.5EIθ EIΔ 10 (10) Version E IIT, Karagpur
10 8 EIθ + EIθ + EIΔ 0 (11) Equations (9), (10) and (11) indicate symmetry and tis fact may be noted. Tis may be used as te ceck in deriving tese equations. Solving equations (9), (10) and (11), EIθ 9.57 ; EIθ 1.55 and EI Δ Substituting te values of EIθ, EIθ and EIΔ in te slopedeflection equation (), one could calculate beam end moments. Tus, 5. kn.m (counterclockwise) 1.14 kn.m(clockwise) A 1.10 kn.m kn.m kn.m D kn.m. D Te bending moment diagram for te frame is sown in ig. 17. e. And te elastic curve is sown in ig 17. f. te bending moment diagram is drawn on te compression side. Also note tat te vertical atcing is used to represent bending moment diagram for te orizontal members (beams). Version E IIT, Karagpur
11 Version E IIT, Karagpur
12 Example 17. Analyse te rigid frame as sown in ig. 17.4a and draw te bending moment diagram. Te moment of inertia for all te members is sown in te figure. Neglect axial deformations. Version E IIT, Karagpur
13 Solution: In tis problem rotations and translations at joints and need to be evaluated. Hence, in tis problem we ave tree unknown displacements: two rotations and one translation. ixed end moments are kn. m ; A 9 kn. m 6 0 ; 0 ; 0 ; 0. D D ; (1) Te joints and translate by te same amount Δ. Hence, te cord to te elastic curve rotates in te clockwise direction as sown in ig. 17.b. and ψ D Δ 6 Δ ψ () Now, writing te slopedeflection equations for six beam end moments, (EI ) θ Δ EIθ + 0. EIΔ A EIθ + 0. EIΔ Version E IIT, Karagpur
14 EIθ EIθ 0. 5EIθ + EIθ D 1.EIθ EIΔ D 0.667EIθ EIΔ () Now, consider te joint equilibrium of and (4) A (5) Te required tird equation is written considering te orizontal equilibrium of te entire frame. onsidering te free body diagram of te member (vide ig. 17.4c), D H + H 0. 1 (6) Version E IIT, Karagpur
15 Te forces H1 and H are calculated from te free body diagram of column andd. Tus, and H 1 H 6 A D + D (7) Substituting te values of and H into equation (6) yields, H (8) A + D D Substituting te beam end moments from equation () in equations (4), (5) and (8), yields.eiθ + 0.5EIθ + 0.EIΔ 9.EIθ + 0.5EIθ EIΔ 0 EIθ + 4EIθ +.EIΔ 6 (9) Solving equations (9), (10) and (11), EIθ.76 ; EIθ 4.88 and EI Δ Substituting te values of EIθ, EIθ and EIΔ in te slopedeflection equation (), one could calculate beam end moments. Tus, kn.m (counterclockwise) 0.5 kn.m(clockwise) A 0. kn.m.50 kn.m.50 kn.m D 6.75 kn.m. D Te bending moment diagram for te frame is sown in ig d. Version E IIT, Karagpur
16 Version E IIT, Karagpur
17 Example 17. Analyse te rigid frame sown in ig a. oment of inertia of all te members are sown in te figure. Draw bending moment diagram. Under te action of external forces, te frame gets deformed as sown in ig. 17.5b. In tis figure, cord to te elastic curve are sown by dotted line. ' is perpendicular to and " is perpendicular to D. Te cords to te elastic Version E IIT, Karagpur
18 curve " rotates by an angle ψ D as sown in figure. Due to symmetry, D figure, ψ, "" rotates by ψ and D rotates by ψ ψ. rom te geometry of te ψ " Δ 1 L L ut Tus, Δ ψ 1 Δ cosα L Δ Δ cosα 5 ψ D Δ 5 Δ Δ tanα ψ Δ tan Δ α (1) 5 We ave tree independent unknowns for tis problem A and D are fixed. Hence, θ θ 0. ixed end moments are, A D θ, θ and Δ. Te ends 0 ; 0 ; +.50 kn. m ;.50kN. m ; 0 ; 0. A Now, writing te slopedeflection equations for te six beam end moments, D D E(I) 5.1 [ θ ψ ] A 0.784EIθ EIΔ A 1.568EIθ EIΔ.5 + EIθ + EIθ 0. EIΔ EIθ + EIθ 0. 6 EIΔ D 1.568EIθ EIΔ D 0.784EIθ EIΔ () Now, considering te joint equilibrium of and, yields Version E IIT, Karagpur
19 0 + 0 A.568EIθ + EIθ 0.19EIΔ.5 () D.568EIθ + EIθ 0.19EIΔ.5 (4) Sear equation for olumn olumn D 5H1 1 A + (1) V 0 (5) 5H D D + (1) V 0 (6) eam 0 V (7) Version E IIT, Karagpur
20 X 0 H 1 + H 5 (8) Y V 1 V 10 0 (9) 0 rom equation (7), V rom equation (8), H1 5 H rom equation (9), 1 10 V V 10 Substituting te values of V, H 1 1and V in equations (5) and (6), 60 10H (10) A H (11) Eliminating H in equation (10) and (11), D D (1) + A D D Substituting te values of equation. Tus,,,, A D D in (1) we get te required tird Simplifying, 0.784EIθ EIΔ EIθ EIΔ EIθ EIΔ EIθ EIΔ (.5 + EIθ + EIθ 0. EIΔ ) (.5 + EIθ + EIθ 0. EIΔ ) EIθ 0.648EIθ +.084EIΔ 5 (1) Solving simultaneously equations () (4) and (1), yields EIθ ; EIθ 1.05 and EI Δ Substituting te values of EIθ, EIθ and EIΔ in te slopedeflection equation (), one could calculate beam end moments. Tus,.8 kn.m Version E IIT, Karagpur
21 A.70 kn.m.70 kn.m 5.75 kn.m 5.75 kn.m D 4.81 kn.m. (14) D Te bending moment diagram for te frame is sown in ig d. Summary In tis lesson, slopedeflection equations are derived for te plane frame undergoing sidesway. Using tese equations, plane frames wit sidesway are analysed. Te reactions are calculated from static equilibrium equations. A couple of problems are solved to make tings clear. In eac numerical example, te bending moment diagram is drawn and deflected sape is sketced for te plane frame. Version E IIT, Karagpur
Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Analysis of Statically Indeterminate Structures by the Matrix Force Method esson 11 The Force Method of Analysis: Frames Instructional Objectives After reading this chapter the student will be able
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationDeflections. Question: What are Structural Deflections?
Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the
More informationSurface Areas of Prisms and Cylinders
12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of
More informationChapter 5: Indeterminate Structures Force Method
Chapter 5: Indeterminate Structures Force Method 1. Introduction Statically indeterminate structures are the ones where the independent reaction components, and/or internal forces cannot be obtained by
More informationReinforced Concrete Beam
Mecanics of Materials Reinforced Concrete Beam Concrete Beam Concrete Beam We will examine a concrete eam in ending P P A concrete eam is wat we call a composite eam It is made of two materials: concrete
More informationShear Forces and Bending Moments
Chapter 4 Shear Forces and Bending Moments 4.1 Introduction Consider a beam subjected to transverse loads as shown in figure, the deflections occur in the plane same as the loading plane, is called the
More informationModule 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur
Module Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version CE IIT, haragpur esson 7 The Direct Stiffness Method: Beams Version CE IIT, haragpur Instructional Objectives
More informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More informationApproximate Analysis of Statically Indeterminate Structures
Approximate Analysis of Statically Indeterminate Structures Every successful structure must be capable of reaching stable equilibrium under its applied loads, regardless of structural behavior. Exact analysis
More informationThe elements used in commercial codes can be classified in two basic categories:
CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for
More informationChapter 5: Indeterminate Structures SlopeDeflection Method
Chapter 5: Indeterminate Structures SlopeDeflection Method 1. Introduction Slopedeflection method is the second of the two classical methods presented in this course. This method considers the deflection
More informationPressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:
Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force
More informationIndeterminate Analysis Force Method 1
Indeterminate Analysis Force Method 1 The force (flexibility) method expresses the relationships between displacements and forces that exist in a structure. Primary objective of the force method is to
More informationDetermine the perimeter of a triangle using algebra Find the area of a triangle using the formula
Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using
More information8.2 Elastic Strain Energy
Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for
More informationNew Vocabulary volume
. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding
More informationBending of Beams with Unsymmetrical Sections
Bending of Beams with Unsmmetrical Sections Assume that CZ is a neutral ais. C = centroid of section Hence, if > 0, da has negative stress. From the diagram below, we have: δ = α and s = αρ and δ ε = =
More information4.2 Free Body Diagrams
CE297FA09Ch4 Page 1 Friday, September 18, 2009 12:11 AM Chapter 4: Equilibrium of Rigid Bodies A (rigid) body is said to in equilibrium if the vector sum of ALL forces and all their moments taken about
More informationSimilar interpretations can be made for total revenue and total profit functions.
EXERCISE 37 Tings to remember: 1. MARGINAL COST, REVENUE, AND PROFIT If is te number of units of a product produced in some time interval, ten: Total Cost C() Marginal Cost C'() Total Revenue R() Marginal
More informationStructural Analysis  II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture  02
Structural Analysis  II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay Lecture  02 Good morning. Today is the second lecture in the series of lectures on structural
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationStatics of Structural Supports
Statics of Structural Supports TYPES OF FORCES External Forces actions of other bodies on the structure under consideration. Internal Forces forces and couples exerted on a member or portion of the structure
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More informationDESIGN OF BEAMCOLUMNS  I
13 DESIGN OF BEACOLUNS  I INTRODUCTION Columns in practice rarely experience concentric axial compression alone. Since columns are usually parts of a frame, they experience both bending moment and axial
More information4.1 Rightangled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53
ontents 4 Trigonometry 4.1 Rigtangled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65
More information7.6 Complex Fractions
Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are
More informationSTRUCTURAL ANALYSIS II (A60131)
LECTURE NOTES ON STRUCTURAL ANALYSIS II (A60131) III B. Tech  II Semester (JNTUHR13) Dr. Akshay S. K. Naidu Professor, Civil Engineering Department CIVIL ENGINEERING INSTITUTE OF AERONAUTICAL ENGINEERING
More informationArea of a Parallelogram
Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make
More informationArea of Trapezoids. Find the area of the trapezoid. 7 m. 11 m. 2 Use the Area of a Trapezoid. Find the value of b 2
Page 1 of. Area of Trapezoids Goal Find te area of trapezoids. Recall tat te parallel sides of a trapezoid are called te bases of te trapezoid, wit lengts denoted by and. base, eigt Key Words trapezoid
More informationA NOVEL PASSIVE ENERGY DISSIPATION SYSTEM FOR FRAMECORE TUBE STRUCTURE
Te Sevent AsiaPacific Conference on Wind Engineering, November 8, 009, Taipei, Taiwan A NOVEL PASSIVE ENERGY DISSIPATION SYSTEM FOR FRAMECORE TUBE STRUCTURE Zengqing Cen and Ziao Wang Director, Wind
More informationStress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t
Stress and Deformation Analysis Material in this lecture was taken from chapter 3 of Representing Stresses on a Stress Element One main goals of stress analysis is to determine the point within a loadcarrying
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationBEAMS: SHEAR AND MOMENT DIAGRAMS (GRAPHICAL)
LECTURE Third Edition BES: SHER ND OENT DIGRS (GRPHICL). J. Clark School of Engineering Department of Civil and Environmental Engineering 3 Chapter 5.3 by Dr. Ibrahim. ssakkaf SPRING 003 ENES 0 echanics
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationNonlinear analysis and formfinding in GSA Training Course
Nonlinear analysis and formfinding in GSA Training Course Nonlinear analysis and formfinding in GSA 1 of 47 Oasys Ltd Nonlinear analysis and formfinding in GSA 2 of 47 Using the GSA GsRelax Solver
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationAnalysis of Stresses and Strains
Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we
More information6.3 Trusses: Method of Sections
6.3 Trusses: Method of Sections 6.3 Trusses: Method of Sections xample 1, page 1 of 2 1. etermine the force in members,, and, and state whether the force is tension or compression. 5 m 4 kn 4 kn 5 m 1
More informationFinite Volume Discretization of the Heat Equation
Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te onedimensional variable coefficient eat equation, wit Neumann boundary conditions u t x
More informationThe Mathematics of Beam Deflection
The athematics of eam Deflection Scenario s a structural engineer you are part of a team working on the design of a prestigious new hotel comple in a developing city in the iddle East. It has been decided
More informationMETHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION
International Journal of Civil Engineering and Technology (IJCIET) Volume 7, Issue 2, MarchApril 2016, pp. 4566, Article ID: IJCIET_07_02_004 Available online at http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=7&itype=2
More informationThe differential amplifier
DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c
More informationBending Stress in Beams
93673600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending
More information4 Shear Forces and Bending Moments
4 Shear Forces and ending oments Shear Forces and ending oments 8 lb 16 lb roblem 4.31 alculate the shear force and bending moment at a cross section just to the left of the 16lb load acting on the simple
More information2. Axial Force, Shear Force, Torque and Bending Moment Diagrams
2. Axial Force, Shear Force, Torque and Bending Moment Diagrams In this section, we learn how to summarize the internal actions (shear force and bending moment) that occur throughout an axial member, shaft,
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationThe Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)
More informationSTRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION
Chapter 11 STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION Figure 11.1: In Chapter10, the equilibrium, kinematic and constitutive equations for a general threedimensional solid deformable
More informationShear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams. Shear and Moment Diagrams
CI 3 Shear Force and Bending oment Diagrams /8 If the variation of and are written as functions of position,, and plotted, the resulting graphs are called the shear diagram and the moment diagram. Developing
More informationOptimum proportions for the design of suspension bridge
Journal of Civil Engineering (IEB), 34 (1) (26) 114 Optimum proportions for the design of suspension bridge Tanvir Manzur and Alamgir Habib Department of Civil Engineering Bangladesh University of Engineering
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationVerifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationUnit 21 Influence Coefficients
Unit 21 Influence Coefficients Readings: Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Have considered the vibrational behavior of
More informationThe Basics of FEA Procedure
CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring
More informationP4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 3 Statically Indeterminate Structures
4 Stress and Strain Dr... Zavatsky MT07 ecture 3 Statically Indeterminate Structures Statically determinate structures. Statically indeterminate structures (equations of equilibrium, compatibility, and
More informationAdvanced Structural Analysis. Prof. Devdas Menon. Department of Civil Engineering. Indian Institute of Technology, Madras. Module  5.3.
Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras Module  5.3 Lecture  29 Matrix Analysis of Beams and Grids Good morning. This is
More informationAdvanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras. Module
Advanced Structural Analysis Prof. Devdas Menon Department of Civil Engineering Indian Institute of Technology, Madras Module  2.2 Lecture  08 Review of Basic Structural Analysis2 Good morning to you.
More informationPLANE TRUSSES. Definitions
Definitions PLANE TRUSSES A truss is one of the major types of engineering structures which provides a practical and economical solution for many engineering constructions, especially in the design of
More informationNotes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.
Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But
More informationDifferential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)
OpenStaxCNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStaxCNX an license
More informationStresses in Beam (Basic Topics)
Chapter 5 Stresses in Beam (Basic Topics) 5.1 Introduction Beam : loads acting transversely to the longitudinal axis the loads create shear forces and bending moments, stresses and strains due to V and
More informationMath Test Sections. The College Board: Expanding College Opportunity
Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt
More information13 PERIMETER AND AREA OF 2D SHAPES
13 PERIMETER AND AREA OF D SHAPES 13.1 You can find te perimeter of sapes Key Points Te perimeter of a twodimensional (D) sape is te total distance around te edge of te sape. l To work out te perimeter
More information2.1: The Derivative and the Tangent Line Problem
.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position
More informationDesign of pile foundations following Eurocode 7Section 7
Brussels, 1820 February 2008 Dissemination of information workshop 1 Workshop Eurocodes: background and applications Brussels, 1820 Februray 2008 Design of pile foundations following Eurocode 7Section
More informationCHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS
1 CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS Written by: Sophia Hassiotis, January, 2003 Last revision: February, 2015 Modern methods of structural analysis overcome some of the
More informationProf. Dr. Hamed Hadhoud. Design of Water Tanks: Part (1)
Design of Water Tanks: Part (1) 1 Types of Tanks Elevated Tanks Resting on Soil & Underground Tanks Tank Walls Walls Shallow Medium Deep L/ L/ < & L/ >0.5 /L L L L 3 Shallow Walls L 1 m L/ (for same continuity
More informationAnalysis of Statically Determinate Trusses
Analysis of Statically Determinate Trusses THEORY OF STRUCTURES Asst. Prof. Dr. Cenk Üstündağ Common Types of Trusses A truss is one of the major types of engineering structures which provides a practical
More informationStructural Analysis: Space Truss
Structural Analysis: Space Truss Space Truss  6 bars joined at their ends to form the edges of a tetrahedron as the basic noncollapsible unit  3 additional concurrent bars whose ends are attached to
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More informationTheoretical calculation of the heat capacity
eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: DulongPetit, Einstein, Debye models Heat capacity of metals
More informationIntroduction to Beam. Area Moments of Inertia, Deflection, and Volumes of Beams
Introduction to Beam Theory Area Moments of Inertia, Deflection, and Volumes of Beams Horizontal structural member used to support horizontal loads such as floors, roofs, and decks. Types of beam loads
More informationChapter 6 Tail Design
apter 6 Tail Design Moammad Sadraey Daniel Webster ollege Table of ontents apter 6... 74 Tail Design... 74 6.1. Introduction... 74 6.. Aircraft Trim Requirements... 78 6..1. Longitudinal Trim... 79 6...
More informationChapter 10: Refrigeration Cycles
Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators
More informationGeoActivity. 1 Use a straightedge to draw a line through one of the vertices of an index card. height is perpendicular to the bases.
Page of 7 8. Area of Parallelograms Goal Find te area of parallelograms. Key Words ase of a parallelogram eigt of a parallelogram parallelogram p. 0 romus p. GeoActivity Use a straigtedge to draw a line
More informationIntroduction, Method of Sections
Lecture #1 Introduction, Method of Sections Reading: 1:12 Mechanics of Materials is the study of the relationship between external, applied forces and internal effects (stress & deformation). An understanding
More informationRigid and Braced Frames
Rigid Frames Rigid and raced Frames Rigid frames are identified b the lack of pinned joints within the frame. The joints are rigid and resist rotation. The ma be supported b pins or fied supports. The
More informationMechanics of Materials. Chapter 4 Shear and Moment In Beams
Mechanics of Materials Chapter 4 Shear and Moment In Beams 4.1 Introduction The term beam refers to a slender bar that carries transverse loading; that is, the applied force are perpendicular to the bar.
More informationStability Of Structures: Basic Concepts
23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More informationPerimeter, Area and Volume of Regular Shapes
Perimeter, Area and Volume of Regular Sapes Perimeter of Regular Polygons Perimeter means te total lengt of all sides, or distance around te edge of a polygon. For a polygon wit straigt sides tis is te
More informationResearch on the Antiperspective Correction Algorithm of QR Barcode
Researc on te Antiperspective Correction Algoritm of QR Barcode Jianua Li, YiWen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic
More informationIMPORTANT NOTE ABOUT WEBASSIGN:
Week 8 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationChapter 18 Static Equilibrium
Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationNote nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1
Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te
More informationChapter 7 Numerical Differentiation and Integration
45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea
More informationDistribution of Forces in Lateral Load Resisting Systems
Distribution of Forces in Lateral Load Resisting Systems Part 2. Horizontal Distribution and Torsion IITGN Short Course Gregory MacRae Many slides from 2009 Myanmar Slides of Profs Jain and Rai 1 Reinforced
More informationSAT Math Facts & Formulas
Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:
More informationCLASSICAL STRUCTURAL ANALYSIS
Table of Contents CASSCA STRUCTURA ANAYSS... Conjugate beam method... External work and internal work... 3 Method of virtual force (unit load method)... 5 Castigliano s second theorem... Method of consistent
More informationNew approaches in Eurocode 3 efficient global structural design
New approaches in Eurocode 3 efficient global structural design Part 1: 3D model based analysis using general beamcolumn FEM Ferenc Papp* and József Szalai ** * Associate Professor, Department of Structural
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More informationBeam Deflections: 4th Order Method and Additional Topics
11 eam Deflections: 4th Order Method and dditional Topics 11 1 ecture 11: EM DEFECTIONS: 4TH ORDER METHOD ND DDITION TOICS TE OF CONTENTS age 11.1. Fourth Order Method Description 11 3 11.1.1. Example
More informationProblem 1: Computation of Reactions. Problem 2: Computation of Reactions. Problem 3: Computation of Reactions
Problem 1: Computation of Reactions Problem 2: Computation of Reactions Problem 3: Computation of Reactions Problem 4: Computation of forces and moments Problem 5: Bending Moment and Shear force Problem
More informationAn inquiry into the multiplier process in ISLM model
An inquiry into te multiplier process in ISLM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 0062763074 Internet Address: jefferson@water.pu.edu.cn
More informationTruss Structures. See also pages in the supplemental notes. Truss: Mimic Beam Behavior. Truss Definitions and Details
Truss Structures Truss: Mimic Beam Behavior Truss Definitions and Details 1 2 Framing of a Roof Supported Truss Bridge Truss Details 3 4 See also pages 1215 in the supplemental notes. 1 Common Roof Trusses
More information