BEAMS: DEFORMATION BY INTEGRATION
|
|
- Wilfrid Bartholomew Ball
- 7 years ago
- Views:
Transcription
1 LECTURE BES: DEFORTION BY INTEGRTION Third Edition. J. Clark School of Engineering Department of Civil and Environmental Engineering 16 Chapter b Dr. Ibrahim. ssakkaf SPRING 00 ENES 0 echanics of aterials Department of Civil and Environmental Engineering Universit of arland, College Park LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 1 ENES 0 ssakkaf Introduction In the previous chapter our concern with beams was to determine the fleural and transverse shear stresses is straight homogenous beams of uniform cross section. We also dealt with various tpes of composite beams, and we re able to find these stresses with some method such as the transformed section.
2 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. ENES 0 ssakkaf Introduction In this chapter our concern will be beam deformation (or deflection). There are important relations between applied load and stress (fleural and shear) and the amount of deformation or deflection that a beam can ehibit. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. ENES 0 ssakkaf Deflection of Beams w 1 P 1 Figure 1 w P (a) w >> w 1 (b) P >> P 1
3 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 4 ENES 0 ssakkaf Curvature varies linearl with t the free end, t the support B, 1 0, ρ ρ 1 0, B ρ ρ B EI PL LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 5 ENES 0 ssakkaf
4 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 6 ENES 0 ssakkaf Introduction In design of beams, it is important sometimes to limit the deflection for specified load. So, in these situations, it is not enough onl to design for the strength (fleural normal and shearing stresses), but also for ecessive deflections of beams. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 7 ENES 0 ssakkaf Introduction Figure 1 shows generall two eamples of how the amount of deflections increase with the applied loads. Failure to control beam deflections within proper limits in building construction is frequentl reflected b the development of cracks in plastered walls and ceilings.
5 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 8 ENES 0 ssakkaf Introduction Beams in man machines must deflect just right amount for gears or other parts to make proper contact. In man instances the requirements for a beam involve: given load-carring capacit, and specified maimum deflection. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 9 ENES 0 ssakkaf ethods for Determining Beam Deflections The deflection of a beam depends on four general factors: 1. Stiffness of the materials that the beam is made of,. Dimensions of the beam,. pplied loads, and 4. Supports
6 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 10 ENES 0 ssakkaf ethods for Determining Beam Deflections Three methods are commonl used to find beam deflections: 1) The double integration method, ) The singularit function method, and ) The superposition method LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 11 ENES 0 ssakkaf General Load-Deflection Relationships Whenever a real beam is loaded in an manner, the beam will deform in that an initiall straight beam will assume some deformed shape such as those illustrated (with some eaggerations) in Figure 1 In a well-designed structural beam the deformation of the beam is usuall undetectable to the naked ee.
7 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 1 ENES 0 ssakkaf General Load-Deflection Relationships On the other hand, the bending of a swimming pool diving board is quite observable. The word deflection generall refers to the deformed shape of a member subjected to bending loads. The deflection is used in reference to the deformed shape and position of the longitudinal neutral ais of a beam. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 1 ENES 0 ssakkaf General Load-Deflection Relationships In the deformed condition the neutral ais, which his initiall a straight longitudinal line, assumes some particular shape that is called the deflection curve. The derivation of this curve from its initial position at an point is called the deflection at that point.
8 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 14 General Load-Deflection Relationships Figure ENES 0 ssakkaf Deflection or Elastic Curve s LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 15 ENES 0 ssakkaf Elastic Curve The Differential Equation The differential equation that governs beam deflection will now be developed. The basis for this differential equation, plus more other approimations, is that plane sections within the beam remain plane before and after loading, and the deformation of the fibers (elongation and contraction) is proportional to the distance from N..
9 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 16 ENES 0 ssakkaf Development of the Differential Equation Consider the beam shown in Fig. that is subjected to the couple shown. B Figure LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 17 ENES 0 ssakkaf Development of the Differential Equation B ρ θ Figure 4 c L L + δ B
10 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 18 ENES 0 ssakkaf Development of the Differential Equation In region of constant bending moment, the elastic curve is an arc of a circle of radius ρ as shown in Fig. 4. Since the portion B of the beam is bent onl with couples, sections and B remain plane as indicated earlier. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 19 ENES 0 ssakkaf Development of the Differential Equation From Fig. 4, we have Or L L + δ θ ρ ρ + c ρ + c L + δ ρ L (1) ()
11 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 0 ENES 0 ssakkaf Development of the Differential Equation Dividing the right left-hand side of Eq. b ρ, ields c 1+ ρ L + δ 1 L or c L + δ L + δ L 1 ρ L L Therefore, c δ ρ L () LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 1 ENES 0 ssakkaf Development of the Differential Equation But δ σ c strain ε, and σ L E I Therefore, or c ρ 1 ρ c EI EI (4)
12 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. ENES 0 ssakkaf Development of the Differential Equation Eq. 4 for the elastic curvature of the elastic curve is useful onl when the bending moment is constant for the interval of the beam involved. For most beams, however, the moment is a function of position along the beam as was seen in Chapter 8. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. ENES 0 ssakkaf Development of the Differential Equation Recall from calculus, the curvature is given b d 1 d ρ (5) d 1 + d
13 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 4 ENES 0 ssakkaf d 1 d ρ d 1 + d Development of the Differential Equation Eq. 5 is difficult to appl in real situation. However, if we realize that for most beams the slope d/d is ver small, and its square is much smaller, then term d d in Eq. 5 can be neglected as compared to unit. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 5 ENES 0 ssakkaf Development of the Differential Equation With this assumption on the slope d/d being ver small quantit, Eq. 5 becomes 1 d ρ d Combining Eqs. 4 and 6, we get d EI d ( ) (6) (7)
14 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 6 ENES 0 ssakkaf The Differential Equation of the Elastic Curve for a Beam d EI d ( ) (8) E modulus of elsticit for the material I moment of inertia about the neutral ais of cross section () bending moment along the beam as a function of LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 7 ENES 0 ssakkaf Relationship between bending moment and curvature for pure bending remains valid for general transverse loadings. 1 ( ) ρ EI Cantilever beam subjected to concentrated load at the free end, 1 P ρ EI
15 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 8 ENES 0 ssakkaf From elementar calculus, simplified for beam parameters, d 1 d d ρ d d 1 + Substituting and integrating, d 1 d EI EI ( ) ρ d d EIθ EI d 0 EI d 0 0 ( ) d + C1 ( ) d + C1 + C LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 9 ENES 0 ssakkaf Sign Convention - negative - positive d negative d d negative d Figure 5. Elastic Curve
16 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 0 ENES 0 ssakkaf Sign Convention L.H.F R.H.F V V (a) Positive Shear & oment V Figure 6 V (b) Positive Shear (clockwise) (c) Positive oment (concave upward) LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 1 ENES 0 ssakkaf Relation of the Deflection with Phsical Quantities such as V and deflection slope moment ( ) shear ( V ) load ( w) d d d EI d d d EI (for EI constant) d d 4 dv d EI (for EI constant) 4 d d (9)
17 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. ENES 0 ssakkaf Construction of Slope and Deflection Diagrams In Chapter 8, a method based on the previous relations was presented for starting from the load diagram and drawing first the shear diagram and then the moment diagram. This method can readil be etended to the construction of slope diagram and deflection diagram. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. ENES 0 ssakkaf Equation for Slope Diagram Note that from Eq. 9, the moment is given b d d d dθ EI EI EI d d d d from which Slope θ θ -B θ B dθ θ B EI d (10)
18 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 4 ENES 0 ssakkaf Equation for Deflection Diagram Note that from Eq. 9, the slope is given b d θ d from which B B θd B B EI d (11) LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 5 ENES 0 ssakkaf Load Shear (V) oment () Slope (θ) P / PL 16EI Deflection () (+) L PL 4 PL 48EI P (+) (-) P / PL 16EI Figure 7 Complete Series of Diagrams for Simpl Supported beam
19 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 6 ENES 0 ssakkaf ssumptions on Elastic Curve Equation 1. The square of the slope of the beam is negligible compared to unit.. The beam deflection due to shearing stresses is negligible (plane section remains plane). The values of E and I remain constant along the beam. If the are constant, and can be epressed as functions of, then a solution using Eq. 8 is possible. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 7 ENES 0 ssakkaf Boundar Conditions Definition: boundar condition is defined as a known value for the deflection or the slope θ at a specified location along the length of the beam. One boundar condition can be used to determine one and onl one constant of integration.
20 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 8 Eample Boundar Conditions ENES 0 ssakkaf Figure 8 (a) Slope 0 at 0 Deflection 0 at 0 (b) Slope at L/ 0 Deflection 0 at 0, and L (c) Slope at rollers? Deflection at rollers 0 (d) Slope 0 at 0 Deflection 0 at 0 and L LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 9 ENES 0 ssakkaf General Procedure for Computing 1. Select the interval or intervals of the beam to be used; net, place a set of coordinate aes on the beam with the origin at one end of an interval and then indicate the range of values of in each interval.. List the variable boundar and matching
21 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 40 ENES 0 ssakkaf General Procedure for Computing (cont d) Conditions for each interval selected.. Epress the bending moment as a function of for each interval selected and equate it to EI(d /d ). 4. Solve the differential equation from step and evaluate all constants of integration. Calculate at specific points when required LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 41 ENES 0 ssakkaf Eample 1 beam is loaded and supported as shown in the figure. a) Derive the equation of the elastic curve in terms of P, L,, E, and I. b) Determine the slope at the left end of the beam. c) Determine the deflection at L/.
22 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 4 Eample 1 (cont d) ENES 0 ssakkaf P B L L FBD P P R P R LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 4 Eample 1 (cont d) ENES 0 ssakkaf P R s V P P + S 0; + 0 P for 0 L / (1a) P R L/ s V P L P for L / L (1b)
23 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 44 ENES 0 ssakkaf Eample 1 (cont d) Boundar conditions: θ 0 at L/ (from smmetr) 0 at 0 and L Using Eqs 8 and 1a: d P EI ( ) d d P EI EIθ ( ) d d P P EIθ + C1 + C1 4 (1c) LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 45 ENES 0 ssakkaf Eample 1 (cont d) Epression for the deflection can be found b integrating Eq. 1c: P EI EIθd 4 P EI + C1 + C 4 P EI + C1 + C 1 + C1 (1d)
24 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 46 ENES 0 ssakkaf Eample 1 (cont d) The objective now is to evaluate the constants of integrations C 1 and C : EIθ ( L / ) 0 P 4 P L + C1 0 C PL C (1e) LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 47 ENES 0 ssakkaf Eample 1 (cont d) ( 0) 0 P + C1 + C 1 C 0 P 0 1 (1f) ( 0) + C ( 0) 0 (a)the equation of elastic curve from Eq. 1d P EI + C1 + C 1 1 P PL EI C 0 (1g)
25 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 48 ENES 0 ssakkaf Eample 1 (cont d) (b) Slope at the left end of the beam: From Eq. 1c, the slope is given b P P PL EIθ + C P PL θ EI 4 16 Therefore, θ θ 1 P EI 4 ( 0) ( 0) PL PL LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 49 ENES 0 ssakkaf Eample 1 (cont d) (c) Deflection at L/: From Eq. 1g of the elastic curve: 1 P EI 1 ( L / ) PL 16 1 P L EI 1 PL 48EI PL 16 L
26 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 50 ENES 0 ssakkaf Eample beam is loaded and supported as shown in the figure. a) Derive the equation for the elastic curve in terms of w, L,, E, and I. b) Determine the slope at the right end of the beam. c) Find the deflection at L. LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 51 Eample (cont d) ENES 0 ssakkaf w L B FBD wl L wl 6 wl R
27 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 5 Eample (cont d) w Find an epression for a segment of the distributed load: w Equation of Straight Line w w w L ENES 0 ssakkaf w L w L w L ( ) w L L w w L (1a) LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 5 Eample (cont d) w wl 6 wl V R wl wl ( ) ( w w ) + s 0; + w 0 6 or w ( w w ) ENES 0 ssakkaf wl wl w ( ) + (1b) 6
28 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 54 ENES 0 ssakkaf Eample (cont d) The solution for parts (a), (b), and (c) can be completed b substituting for w into Eq. 1b, equating the epression for () to the term EI(d /d ), and integrating twice to get the elastic curve and epression for the slope. Note that the boundar conditions are that both the slope and deflection are zero at 0. i.e.; d EI EI d ( ) LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 55 ENES 0 ssakkaf Eample For the overhanging steel beam BC that subjected to concentrated load of 50 kips as shown, (a) derive an epression for the elastic curve, (b) determine the maimum deflection, and (c) find the slope at point. The modulus of elasticit E is psi and the moment of inertia of the cross section of the beam was found as 7 in 4.
29 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 56 Eample (cont d) ENES 0 ssakkaf E psi 50 kips I 7in 4 B C 15 ft 4 ft LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 57 Eample (cont d) We first find the reactions as follows: 50 kips B C ENES 0 ssakkaf 15ft 4ft R -1. kips R B 6. kips + + R B R 0; R B 50 R ( 15) + 50( 4) 1. kips F 0; R + R B ( 1.) 6. kips
30 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 58 Eample (cont d) 1. kips s V + B s 15 V 1. kips 6. kips + S S 0; ; ENES 0 ssakkaf (14a) ( 15) (14b) LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 59 ENES 0 ssakkaf Eample (cont d) Boundar conditions: (0) 0, and (15)0 EI ( ) 1. for 0 15 EI EIθ 1. + C EI C1 + C (0) EI( 0) C1(0) + C C 0 + C 1 (14e) (14c) (14d)
31 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 60 ENES 0 ssakkaf Eample (cont d) EI C ( 15) ( 15) (a) The elastic curve is + C 1 ( 15) + 0 (14f) 1 ( ) EI ft (14g) LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 61 ENES 0 ssakkaf Conversion factor Eample (cont d) (b) aimum deflection occurs when the slope is zero. So setting d/d θ 0 in Eq. 14c, gives 1 θ 0 ( ) EI 8.66 ft Using Eq. 14g with ma 8.66 ft, gives ( 1) 10 ( 7) (8.66) ma (8.66) ft 0.8 in 9 +
32 LECTURE 16. BES: DEFORTION BY INTEGRTION (9.1 9.) Slide No. 6 ENES 0 ssakkaf Eample (cont d) (c) The slope at point ( 0) can be computed from Eq. 14c b substituting zero for as follows: 1 ( 0) θ ( 0) ( ) θ EI (1) 9 10 [ 6.67( 0) ] rad
BEAMS: 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 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 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 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 informationBending Stress in Beams
936-73-600 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 informationBeam Deflections: Second-Order Method
10 eam Deflections: Second-Order Method 10 1 Lecture 10: EM DEFLECTIONS: SECOND-ORDER METHOD TLE OF CONTENTS Page 10.1 Introduction..................... 10 3 10.2 What is a eam?................... 10 3
More informationIntroduction to Plates
Chapter Introduction to Plates Plate is a flat surface having considerabl large dimensions as compared to its thickness. Common eamples of plates in civil engineering are. Slab in a building.. Base slab
More informationIntroduction to Mechanical Behavior of Biological Materials
Introduction to Mechanical Behavior of Biological Materials Ozkaya and Nordin Chapter 7, pages 127-151 Chapter 8, pages 173-194 Outline Modes of loading Internal forces and moments Stiffness of a structure
More informationCHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES INTRODUCTION
CHAP FINITE EEMENT ANAYSIS OF BEAMS AND FRAMES INTRODUCTION We learned Direct Stiffness Method in Chapter imited to simple elements such as D bars we will learn Energ Method to build beam finite element
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 informationINTRODUCTION TO BEAMS
CHAPTER Structural Steel Design LRFD Method INTRODUCTION TO BEAMS Third Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part II Structural Steel Design and Analysis
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 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 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 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 informationMATERIALS AND MECHANICS OF BENDING
HAPTER Reinforced oncrete Design Fifth Edition MATERIALS AND MEHANIS OF BENDING A. J. lark School of Engineering Department of ivil and Environmental Engineering Part I oncrete Design and Analysis b FALL
More informationBEAMS: SHEAR FLOW, THIN WALLED MEMBERS
LECTURE BEAMS: SHEAR FLOW, THN WALLED MEMBERS Third Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering 15 Chapter 6.6 6.7 by Dr. brahim A. Assakkaf SPRNG 200 ENES
More informationMCE380: Measurements and Instrumentation Lab. Chapter 9: Force, Torque and Strain Measurements
MCE380: Measurements and Instrumentation Lab Chapter 9: Force, Torque and Strain Measurements Topics: Elastic Elements for Force Measurement Dynamometers and Brakes Resistance Strain Gages Holman, Ch.
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 three-dimensional solid deformable
More informationStress Strain Relationships
Stress Strain Relationships Tensile Testing One basic ingredient in the study of the mechanics of deformable bodies is the resistive properties of materials. These properties relate the stresses to the
More informationMECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS
MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS This is the second tutorial on bending of beams. You should judge your progress by completing the self assessment exercises.
More informationModule 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 informationType of Force 1 Axial (tension / compression) Shear. 3 Bending 4 Torsion 5 Images 6 Symbol (+ -)
Cause: external force P Force vs. Stress Effect: internal stress f 05 Force vs. Stress Copyright G G Schierle, 2001-05 press Esc to end, for next, for previous slide 1 Type of Force 1 Axial (tension /
More informationMECHANICS OF SOLIDS - BEAMS TUTORIAL 1 STRESSES IN BEAMS DUE TO BENDING. On completion of this tutorial you should be able to do the following.
MECHANICS OF SOLIDS - BEAMS TUTOIAL 1 STESSES IN BEAMS DUE TO BENDING This is the first tutorial on bending of beams designed for anyone wishing to study it at a fairly advanced level. You should judge
More informationMECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS
MECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS This the fourth and final tutorial on bending of beams. You should judge our progress b completing the self assessment exercises.
More informationFinite Element Simulation of Simple Bending Problem and Code Development in C++
EUROPEAN ACADEMIC RESEARCH, VOL. I, ISSUE 6/ SEPEMBER 013 ISSN 86-48, www.euacademic.org IMPACT FACTOR: 0.485 (GIF) Finite Element Simulation of Simple Bending Problem and Code Development in C++ ABDUL
More informationDesign of reinforced concrete columns. Type of columns. Failure of reinforced concrete columns. Short column. Long column
Design of reinforced concrete columns Type of columns Failure of reinforced concrete columns Short column Column fails in concrete crushed and bursting. Outward pressure break horizontal ties and bend
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
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 informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationModeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method
Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method Yun-gang Zhan School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang,
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 informationSection 16: Neutral Axis and Parallel Axis Theorem 16-1
Section 16: Neutral Axis and Parallel Axis Theorem 16-1 Geometry of deformation We will consider the deformation of an ideal, isotropic prismatic beam the cross section is symmetric about y-axis All parts
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 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 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 information3D Stress Components. From equilibrium principles: τ xy = τ yx, τ xz = τ zx, τ zy = τ yz. Normal Stresses. Shear Stresses
3D Stress Components From equilibrium principles:, z z, z z The most general state of stress at a point ma be represented b 6 components Normal Stresses Shear Stresses Normal stress () : the subscript
More informationES240 Solid Mechanics Fall 2007. Stress field and momentum balance. Imagine the three-dimensional body again. At time t, the material particle ( x, y,
S40 Solid Mechanics Fall 007 Stress field and momentum balance. Imagine the three-dimensional bod again. At time t, the material particle,, ) is under a state of stress ij,,, force per unit volume b b,,,.
More informationSheet metal operations - Bending and related processes
Sheet metal operations - Bending and related processes R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Table of Contents 1.Quiz-Key... Error! Bookmark not defined. 1.Bending
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More information2.5 Library of Functions; Piecewise-defined Functions
SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More information9 Stresses: Beams in Bending
9 Stresses: Beams in Bending The organization of this chapter mimics that of the last chapter on torsion of circular shafts but the stor about stresses in beams is longer, covers more territor, and is
More informationENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P
ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 3 BENDING MOMENTS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P This material is duplicated in the Mechanical Principles module H2 and those
More informationPRESTRESSED CONCRETE. Introduction REINFORCED CONCRETE CHAPTER SPRING 2004. Reinforced Concrete Design. Fifth Edition. By Dr. Ibrahim.
CHAPTER REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach - Fifth Edition Fifth Edition PRESTRESSED CONCRETE A. J. Clark School of Engineering Department of Civil and Environmental
More informationPlane Stress Transformations
6 Plane Stress Transformations ASEN 311 - Structures ASEN 311 Lecture 6 Slide 1 Plane Stress State ASEN 311 - Structures Recall that in a bod in plane stress, the general 3D stress state with 9 components
More informationA Resource for Free-standing Mathematics Qualifications
To find a maximum or minimum: Find an expression for the quantity you are trying to maximise/minimise (y say) in terms of one other variable (x). dy Find an expression for and put it equal to 0. Solve
More informationLecture 8 Bending & Shear Stresses on Beams
Lecture 8 Bending & hear tresses on Beams Beams are almost always designed on the asis of ending stress and, to a lesser degree, shear stress. Each of these stresses will e discussed in detail as follows.
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
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 informationObjectives. Experimentally determine the yield strength, tensile strength, and modules of elasticity and ductility of given materials.
Lab 3 Tension Test Objectives Concepts Background Experimental Procedure Report Requirements Discussion Objectives Experimentally determine the yield strength, tensile strength, and modules of elasticity
More informationINTRODUCTION TO BEAMS
CHAPTER Structural Steel Design LRFD etho INTRODUCTION TO BEAS Thir Eition A. J. Clark School of Engineering Department of Civil an Environmental Engineering Part II Structural Steel Design an Analsis
More informationFinite Element Formulation for Beams - Handout 2 -
Finite Element Formulation for Beams - Handout 2 - Dr Fehmi Cirak (fc286@) Completed Version Review of Euler-Bernoulli Beam Physical beam model midline Beam domain in three-dimensions Midline, also called
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second
More informationMECHANICS OF MATERIALS
T dition CHTR MCHNICS OF MTRIS Ferdinand. Beer. Russell Johnston, Jr. John T. DeWolf ecture Notes: J. Walt Oler Texas Tech University Stress and Strain xial oading - Contents Stress & Strain: xial oading
More informationAffine Transformations
A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates
More informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
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 informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationQualitative Influence Lines. Qualitative Influence Lines. Qualitative Influence Lines. Qualitative Influence Lines. Qualitative Influence Lines
IL 32 Influence Lines - Muller-reslau Principle /5 In 886, Heinrich Müller-reslau develop a method for rapidly constructing the shape of an influence line. Heinrich Franz ernhard Müller was born in Wroclaw
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationAluminium systems profile selection
Aluminium systems profile selection The purpose of this document is to summarise the way that aluminium profile selection should be made, based on the strength requirements for each application. Curtain
More informationShear Force and Moment Diagrams
C h a p t e r 9 Shear Force and Moment Diagrams In this chapter, you will learn the following to World Class standards: Making a Shear Force Diagram Simple Shear Force Diagram Practice Problems More Complex
More informationChapter 5: Indeterminate Structures Slope-Deflection Method
Chapter 5: Indeterminate Structures Slope-Deflection Method 1. Introduction Slope-deflection method is the second of the two classical methods presented in this course. This method considers the deflection
More informationME 343: Mechanical Design-3
ME 343: Mechanical Design-3 Design of Shaft (continue) Dr. Aly Mousaad Aly Department of Mechanical Engineering Faculty of Engineering, Alexandria University Objectives At the end of this lesson, we should
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationReinforced Concrete Design
FALL 2013 C C Reinforced Concrete Design CIVL 4135 ii 1 Chapter 1. Introduction 1.1. Reading Assignment Chapter 1 Sections 1.1 through 1.8 of text. 1.2. Introduction In the design and analysis of reinforced
More informationABSTRACT 1. INTRODUCTION 2. DESCRIPTION OF THE SEGMENTAL BEAM
Ninth LACCEI Latin American and Caribbean Conference (LACCEI 11), Engineering for a Smart Planet, Innovation, Information Technology and Computational Tools for Sustainable Development, August 3-, 11,
More informationSECTION 5 ANALYSIS OF CONTINUOUS SPANS DEVELOPED BY THE PTI EDC-130 EDUCATION COMMITTEE LEAD AUTHOR: BRYAN ALLRED
SECTION 5 ANALYSIS OF CONTINUOUS SPANS DEVELOPED BY THE PTI EDC-130 EDUCATION COMMITTEE LEAD AUTHOR: BRYAN ALLRED NOTE: MOMENT DIAGRAM CONVENTION In PT design, it is preferable to draw moment diagrams
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationRecitation #5. Understanding Shear Force and Bending Moment Diagrams
Recitation #5 Understanding Shear Force and Bending Moment Diagrams Shear force and bending moment are examples of interanl forces that are induced in a structure when loads are applied to that structure.
More informationIII. Compression Members. Design of Steel Structures. Introduction. Compression Members (cont.)
ENCE 455 Design of Steel Structures III. Compression Members C. C. Fu, Ph.D., P.E. Civil and Environmental Engineering Department University it of Maryland Compression Members Following subjects are covered:
More informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More informationFinite Element Formulation for Plates - Handout 3 -
Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the
More informationHow To Calculate Tunnel Longitudinal Structure
Calculation and Analysis of Tunnel Longitudinal Structure under Effect of Uneven Settlement of Weak Layer 1,2 Li Zhong, 2Chen Si-yang, 3Yan Pei-wu, 1Zhu Yan-peng School of Civil Engineering, Lanzhou University
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 informationTechnical Notes 3B - Brick Masonry Section Properties May 1993
Technical Notes 3B - Brick Masonry Section Properties May 1993 Abstract: This Technical Notes is a design aid for the Building Code Requirements for Masonry Structures (ACI 530/ASCE 5/TMS 402-92) and Specifications
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 3 LOADED COMPONENTS
EDEXCEL NATIONAL CERTIICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQ LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 3 LOADED COMPONENTS 1. Be able to determine the effects of loading in static engineering
More informationREINFORCED CONCRETE. Reinforced Concrete Design. A Fundamental Approach - Fifth Edition. Walls are generally used to provide lateral support for:
HANDOUT REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach - Fifth Edition RETAINING WALLS Fifth Edition A. J. Clark School of Engineering Department of Civil and Environmental Engineering
More informationIdeal Cable. Linear Spring - 1. Cables, Springs and Pulleys
Cables, Springs and Pulleys ME 202 Ideal Cable Neglect weight (massless) Neglect bending stiffness Force parallel to cable Force only tensile (cable taut) Neglect stretching (inextensible) 1 2 Sketch a
More informationENGINEERING COUNCIL CERTIFICATE LEVEL
ENGINEERING COUNCIL CERTIICATE LEVEL ENGINEERING SCIENCE C103 TUTORIAL - BASIC STUDIES O STRESS AND STRAIN You should judge your progress by completing the self assessment exercises. These may be sent
More informationZero and Negative Exponents and Scientific Notation. a a n a m n. Now, suppose that we allow m to equal n. We then have. a am m a 0 (1) a m
0. E a m p l e 666SECTION 0. OBJECTIVES. Define the zero eponent. Simplif epressions with negative eponents. Write a number in scientific notation. Solve an application of scientific notation We must have
More informationSHAFTS: TORSION LOADING AND DEFORMATION
ECURE hird Edition SHAFS: ORSION OADING AND DEFORMAION A. J. Clark Shool of Engineering Department of Civil and Environmental Engineering 6 Chapter 3.1-3.5 by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220
More informationCompression Members: Structural elements that are subjected to axial compressive forces
CHAPTER 3. COMPRESSION MEMBER DESIGN 3.1 INTRODUCTORY CONCEPTS Compression Members: Structural elements that are subjected to axial compressive forces onl are called columns. Columns are subjected to axial
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. Dr Tay Seng Chuan
Ground Rules PC11 Fundamentals of Physics I Lectures 3 and 4 Motion in One Dimension Dr Tay Seng Chuan 1 Switch off your handphone and pager Switch off your laptop computer and keep it No talking while
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
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 information135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the
More informationEQUILIBRIUM STRESS SYSTEMS
EQUILIBRIUM STRESS SYSTEMS Definition of stress The general definition of stress is: Stress = Force Area where the area is the cross-sectional area on which the force is acting. Consider the rectangular
More informationWhen the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.
Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationPROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers
More informationSTEEL-CONCRETE COMPOSITE COLUMNS-I
5 STEEL-CONCRETE COMOSITE COLUMNS-I 1.0 INTRODUCTION A steel-concrete composite column is a compression member, comprising either a concrete encased hot-rolled steel section or a concrete filled tubular
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationMECHANICAL PRINCIPLES HNC/D PRELIMINARY LEVEL TUTORIAL 1 BASIC STUDIES OF STRESS AND STRAIN
MECHANICAL PRINCIPLES HNC/D PRELIMINARY LEVEL TUTORIAL 1 BASIC STUDIES O STRESS AND STRAIN This tutorial is essential for anyone studying the group of tutorials on beams. Essential pre-requisite knowledge
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 information16. Beam-and-Slab Design
ENDP311 Structural Concrete Design 16. Beam-and-Slab Design Beam-and-Slab System How does the slab work? L- beams and T- beams Holding beam and slab together University of Western Australia School of Civil
More information2.7 Applications of Derivatives to Business
80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.
More informationSolid Mechanics. Stress. What you ll learn: Motivation
Solid Mechanics Stress What you ll learn: What is stress? Why stress is important? What are normal and shear stresses? What is strain? Hooke s law (relationship between stress and strain) Stress strain
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More information