8.2 Elastic Strain Energy

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Section 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 stored energy will then be used to solve some elasticity problems using the energy methods mentioned in the previous section. 8.. Strain energy in deformed Components Bar under axial load Consider a bar of elastic material fixed at one end and subjected to a steadily increasing force, Fig The force is applied slowly so that kinetic energies are negligible. The initial length of the bar is. The work dw done in extending the bar a small amount d is dw d (8..) d Figure 8..: a bar loaded by a force It was shown in 7.. that the force and extension are linearly related through / E, Eqn. 7..5, where E is the Young s modulus and is the cross sectional area. This linear relationship is plotted in Fig The work expressed by Eqn. 8.. is the white region under the force-extension curve (line). The total work done during the complete extension up to a final force and final extension is the total area beneath the curve. The work done is stored as elastic strain energy and so (8..) E If the axial force (and/or the cross-sectional area and Young s modulus) varies along the bar, then the above calculation can be done for a small element of length dx. The energy stored in this element would be dx / E and the total strain energy stored in the bar would be the small change in force d which occurs during this small extension may be neglected, since it will result in a smaller-order term of the form ddδ 4

2 Section 8. dx E (8..3) force-extension curve dw d Figure 8..: force-displacement curve for uniaxial load The strain energy is always positive, due to the square on the force, regardless of whether the bar is being compressed or elongated. Note the factor of one half in Eqn The energy stored is not simply force times displacement because the force is changing during the deformation. Circular Bar in Torsion Consider a circular bar subjected to a torque T. The torque is equivalent to a couple: two forces of magnitude F acting in opposite directions and separated by a distance r as in Fig. 8..3; T Fr. s the bar twists through a small angle, the forces each move through a distance s r W Fs T.. The work done is therefore r F Figure 8..3: torque acting on a circular bar It was shown in 7. that the torque and angle of twist are linearly related through Eqn. 7.., T / GJ, where is the length of the bar, G is the shear modulus and J is the polar moment of inertia. The angle of twist can be plotted against the torque as in Fig The total strain energy stored in the cylinder during the straining up to a final angle of twist is the work done, equal to the shaded area in Fig. 8..4, leading to 43

3 Section 8. T T (8..4) GJ W T Figure 8..4: torque angle of twist plot for torsion gain, if the various quantities are varying along the length of the bar, then the total strain energy can be expressed as Beam subjected to a ure oment T dx GJ (8..5) s with the bar under torsion, the work done by a moment as it moves through an angle d is d. The moment is related to the radius of curvature R through Eqns , EI / R, where E is the Young s modulus and I is the moment of inertia. The length of a beam and the angle subtended are related to R through R, Fig. 8..5, and so moment and angle are linearly related through / EI. / / R Figure 8..5: beam of length under pure bending The total strain energy stored in a bending beam is then (8..6) EI and if the moment and other quantities vary along the beam, 44

4 Section 8. dx EI (8..7) This expression is due to the flexural stress. beam can also store energy due to shear stress ; this latter energy is usually much less than that due to the flexural stresses provided the beam is slender this is discussed further below. Example Consider the bar with varying circular cross-section shown in Fig The Young s modulus is Ga. m m kn r 5cm r 3cm Figure 8..6: a loaded bar The strain energy stored in the bar when a force of kn is applied at the free end is dx 9.6 Nm (8..8) E 8.. The Work-Energy rinciple The work-energy principle for elastic materials, that is, the fact that the work done by external forces is stored as elastic energy, can be used directly to solve some simple problems. To be precise, it can be used to solve problems involving a single force and for solving for the displacement in the direction of that force. By force and displacement here it is meant generalised force and generalised displacement, that is, a force/displacement pair, a torque/angle of twist pair or a moment/bending angle pair. ore complex problems need to be solved using more sophisticated energy methods, such as Castigliano s method discussed further below. Example Consider the beam of length shown in Fig. 8..7, pinned at one end () and simply supported at the other (C). moment acts at B, a distance from the left-hand end. The cross-section is rectangular with depth b and height h. The work-energy principle can be used to calculate the angle B through which the moment at B rotates. 45

5 Section 8. B C Figure 8..7: a beam subjected to a moment at B The moment along the beam can be calculated from force and moment equilibrium, x /, x /, x x (8..9) The strain energy stored in the bar (due to the flexural stresses only) is 3 6 x x dx dx dx 3 3 EI Ebh (8..) Ebh The work done by the applied moment is / and so B 3 4 B (8..) 3 Ebh 8..3 Strain Energy Density The strain energy will in general vary throughout a body and for this reason it is useful to introduce the concept of strain energy density, which is a measure of how much energy is stored in small volume elements throughout a material. Consider again a bar subjected to a uniaxial force. small volume element with edges aligned with the x, y, z axes as shown in Fig will then be subjected to a stress only. The volume of the element is dv dxdydz. From Eqn. 8.., the strain energy in the element is dydz dx (8..) Edydz 46

6 Section 8. y dy dx dz z x volume element Figure 8..8: a volume element under stress The strain energy density u is defined as the strain energy per unit volume: u (8..3) E The total strain energy in the bar may now be expressed as this quantity integrated over the whole volume, udv, (8..4) V which, for a constant cross-section and length reads law, the strain energy density of Eqn can also be expressed as udx. From Hooke s u (8..5) s can be seen from Fig. 8..9, this is the area under the uniaxial stress-strain curve. u Figure 8..9: stress-strain curve for elastic material Note that the element does deform in the y and z directions but no work is associated with those displacements since there is no force acting in those directions. The strain energy density for an element subjected to a stress only is, by the same arguments, /, and that due to a stress is /. Consider next a shear 47

7 Section 8. stress acting on the volume element to produce a shear strain as illustrated in Fig The element deforms with small angles and as illustrated. Only the stresses on the upper and right-hand surfaces are shown, since the stresses on the other two surfaces do no work. The force acting on the upper surface is dxdz and moves through a displacement dy. The force acting on the right-hand surface is dydz and moves through a displacement dx. The work done when the element moves through angles d and d is then, using the definition of shear strain, dw dxdzd dy dydzddx dxdydz d (8..6) and, with shear stress proportional to shear strain, the strain energy density is u (8..7) d dy y dy dx dx x Figure 8..: a volume element under shear stress The strain energy can be similarly calculated for the other shear stresses and, in summary, the strain energy density for a volume element subjected to arbitrary stresses is u yz yz zx zx (8..8) sing Hooke s law, Eqns. 6..9, and Eqn. 6..5, the strain energy density can also be written in the alternative and useful forms { roblem 4} u E yz zx E ( ) yz zx yz zx (8..9) 48

8 Section 8. Strain Energy in a Beam due to Shear Stress The shear stresses arising in a beam at location y from the neutral axis are given by Eqn , ( y) Q( y) V / Ib( y), where Q is the first moment of area of the section of beam from y to the outer surface, V is the shear force, I is the moment of inertia of the complete cross-section and b is the thickness of the beam at y. From Eqns. 8..9a and 8..4 then, the total strain energy in a beam of length due to shear stress is V Q dv V I b ddx (8..) Here V, and I are taken to be constant for any given cross-section but may vary along the beam; Q varies and b may vary over any given cross-section. Expression 8.. can be simplified by introducing the form factor for shear f s, defined as so that Q f s ( x) d I (8..) b f V s dx (8..) The form factor depends only on the shape of the cross-section. For example, for a rectangular cross-section, using Eqn , h / b / bh b h 6 f ( x) (8..3) s 3 / y 4 dy dz bh 5 h / b b / In a similar manner, the form factor for a circular cross-section is found to be / 9 and that of a very thin tube is Castigliano s Second Theorem The work-energy method is the simplest of energy methods. more powerful method is that based on Castigliano s second theorem, which can be used to solve problems involving linear elastic materials. s an introduction to Castigliano s second theorem, consider the case of uniaxial tension, where / E. The displacement through which the force moves can be obtained by a differentiation of this expression with respect to that force, d d E (8..4) Casigliano s first theorem will be discussed in a later section 49

9 Section 8. Similarly, for torsion of a circular bar, T / GJ, and a differentiation gives d / dt T / GJ. Further, for bending of a beam it is also seen that d / d. These are examples of Castigliano s theorem, which states that, provided the body is in equilibrium, the derivative of the strain energy with respect to the force gives the displacement corresponding to that force, in the direction of that force. When there is more than one force applied, then one takes the partial derivative. For example, if n independent forces,,, n act on a body, the displacement corresponding to the ith force is i (8..5) i Before proving this theorem, here follow some examples. Example The beam shown in Fig. 8.. is pinned at, simply supported half-way along the beam at B and loaded at the end C by a force and a moment. B C / / Figure 8..: a beam subjected to a force and moment at C The moment along the beam can be calculated from force and moment equilibrium, x x /, ( x), x / / x (8..6) The strain energy stored in the bar (due to the flexural stresses only) is / x dx EI 3 5 4EI 4EI 3EI / ( x) dx (8..7) In order to apply Castigliano s theorem, the strain energy is considered to be a function of the two external loads,,. The displacement associated with the force is then 5

10 Section 8. The rotation associated with the moment is Example 3 5 C (8..8) EI 4EI 5 C (8..9) 4EI 3EI Consider next the beam of length shown in Fig. 8.., built in at both ends and loaded centrally by a force. This is a statically indeterminate problem. In this case, the strain energy can be written as a function of the applied load and one of the unknown reactions. B C C / / Figure 8..: a statically indeterminate beam First, the moment in the beam is found from equilibrium considerations to be x, x / (8..3) where is the unknown reaction at the left-hand end. Then the strain energy in the left-hand half of the beam is / 3 x dx EI (8..3) 9EI 6EI 4EI The strain energy in the complete beam is double this: 3 (8..3) 96EI 8EI EI Writing the strain energy as,, the rotation at is (8..33) 8EI EI 5

11 Section 8. But and so Eqn can be solved to get / 8. Then the displacement at the centre of the beam is 3 3 B (8..34) 48EI 8EI 9EI This is positive in the direction in which the associated force is acting, and so is downward. roof of Castigliano s Theorem proof of Castiligliano s theorem will be given here for a structure subjected to a single load. The load produces a displacement and the strain energy is /, Fig If an additional force d is applied giving an additional deformation d, the additional strain energy is d d dd (8..35) If the load d is applied from zero in one step, the work done is d d/. Equating this to the strain energy d given by Eqn then gives d d. Substituting into Eqn leads to d d dd (8..36) Dividing through by d and taking the limit as d results in Castigliano s second theorem, d / d. d d Figure 8..3: force-displacement curve In fact, dividing Eqn through by d and taking the limit as d results in Castigliano s first theorem, d / d. It will be shown later that this first theorem, unlike the second, in fact holds also for the case when the elastic material is non-linear. 5

13 Section 8. where is the force acting on the bar at its imum compression. For an elastic bar, E/, or, introducing the stiffness k so that k, which is a quadratic equation in E wh k, k (8..38) and can be solved to get w hk (8..39) k w If the force w had been applied gradually, then the displacement would have been w / k, the st standing for static, and Eqn can be re-written as st st h st (8..4) If h, so that the weight is just touching the bar when released, then st roblems. Show that the strain energy in a bar of length and cross sectional area hanging from a ceiling and subjected to its own weight is given by (at any section, the force acting is the weight of the material below that section) 3 g 6E. Consider the circular bar shown below subject to torques at the free end and where the cross-sectional area changes. The shear modulus is G 8Ga. Calculate the strain energy in the bar(s). m m 6kNm 4kNm r 5cm r 3cm 3. Two bars of equal length and cross-sectional area are pin-supported and loaded by a force F as shown below. Derive an expression for the vertical displacement at point using the direct work-energy method, in terms of, F, and the Young s modulus E. 54

14 Section 8. o 45 o 45 F 4. Derive the strain energy density equations For the beam shown in Fig. 8..7, use the expression 8.. to calculate the strain energy due to the shear stresses. Take the shear modulus to be G 8Ga. Compare this with the strain energy due to flexural stress given by Eqn Consider a simply supported beam of length subjected to a uniform load w N/m. Calculate the strain energy due to both flexural stress and shear stress for (a) a rectangular cross-section of depth times height b h, (b) a circular cross-section with radius r. What is the ratio of the shear-to-flexural strain energies in each case? 7. Consider the tapered bar of length and square cross-section shown below, built-in at one end and subjected to a uniaxial force F at its free end. The thickness is h at the built-in end. Evaluate the displacement in terms of the (constant) Young s modulus E at the free end using (i) the work-energy theorem, (ii) Castigliano s theorem h F 8. Consider a cantilevered beam of length and constant cross-section subjected to a uniform load w N/m. The beam is built-in at x and has a Young s modulus E. se Castigliano s theorem to calculate the deflection at x. Consider only the flexural strain energy. [Hint: place a fictitious dummy load F at x and set to zero once Castigliano s theorem has been applied] 9. Consider the statically indeterminate uniaxial problem shown below, two bars joined at x, built in at x and x, and subjected to a force F at the join. The cross-sectional area of the bar on the left is and that on the right is. se (i) the work-energy theorem and (ii) Castigliano s theorem to evaluate the displacement at x. F 55

CH 4: Deflection and Stiffness

CH 4: Deflection and Stiffness Stress analyses are done to ensure that machine elements will not fail due to stress levels exceeding the allowable values. However, since we are dealing with deformable

Deflections. 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

Mechanics of Materials Stress-Strain Curve for Mild Steel

Stress-Strain Curve for Mild Steel 13-1 Definitions 13-2a Hooke s Law Shear Modulus: Stress: Strain: Poisson s Ratio: Normal stress or strain = " to the surface Shear stress = to the surface Definitions

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK. UNIT I STRESS STRAIN DEFORMATION OF SOLIDS PART- A (2 Marks)

KINGS COLLEGE OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING QUESTION BANK SUB CODE/NAME: CE1259 STRENGTH OF MATERIALS YEAR/SEM: II / IV 1. What is Hooke s Law? 2. What are the Elastic Constants?

Stress 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 load-carrying

Laboratory Weeks 9 10 Theory of Pure Elastic Bending

Laboratory Weeks 9 10 Theory of Pure Elastic Bending Objective To show the use of the Sagital method for finding the Radius of Curvature of a beam, to prove the theory of bending, and find the elastic

MCE380: 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.

Nonlinear analysis and form-finding in GSA Training Course

Nonlinear analysis and form-finding in GSA Training Course Non-linear analysis and form-finding in GSA 1 of 47 Oasys Ltd Non-linear analysis and form-finding in GSA 2 of 47 Using the GSA GsRelax Solver

MECHANICS OF MATERIALS

2009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 4 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Pure Bending Lecture

Approximate 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

OUTCOME 3 - TUTORIAL 1 STRAIN ENERGY

UNIT 6: Unit code: QCF level: 5 Credit value: 5 STRENGTHS OF MATERIALS K/6/49 OUTCOME - TUTORIAL STRAIN ENERGY Be able to determine the behavioural characteristics of loaded structural members by the consideration

STRESS 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

Mechanics of Materials Summary

Mechanics of Materials Summary 1. Stresses and Strains 1.1 Normal Stress Let s consider a fixed rod. This rod has length L. Its cross-sectional shape is constant and has area. Figure 1.1: rod with a normal

Stresses 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

BENDING STRESS. Shear force and bending moment

Shear force and bending moment BENDING STESS In general, a beam can be loaded in a way that varies arbitrarily along its length. Consider the case in which we have a varying distributed load w. This will

Chapter 9 Deflections of Beams

Chapter 9 Deflections of Beams 9.1 Introduction in this chapter, we describe methods for determining the equation of the deflection curve of beams and finding deflection and slope at specific points along

Figure 12 1 Short columns fail due to material failure

12 Buckling Analysis 12.1 Introduction There are two major categories leading to the sudden failure of a mechanical component: material failure and structural instability, which is often called buckling.

Bending 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

When 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

EML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss

Problem Statement: Project 1 Finite Element Analysis and Design of a Plane Truss The plane truss in Figure 1 is analyzed using finite element analysis (FEA) for three load cases: A) Axial load: 10,000

An introduction to the relationship between bending moments and shear forces using the Push Me Pull Me models on Expedition Workshed

Worksheet 6 Bending moments & shear force An introduction to the relationship between bending moments and shear forces using the Push Me Pull Me models on Expedition Workshed National HE STEM Programme

M x (a) (b) (c) Figure 2: Lateral Buckling The positions of the beam shown in Figures 2a and 2b should be considered as two possible equilibrium posit

Lateral Stability of a Slender Cantilever Beam With End Load Erik Thompson Consider the slender cantilever beam with an end load shown in Figure 1. The bending moment at any cross-section is in the x-direction.

Introduction, Method of Sections

Lecture #1 Introduction, Method of Sections Reading: 1:1-2 Mechanics of Materials is the study of the relationship between external, applied forces and internal effects (stress & deformation). An understanding

Section 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

MECHANICS 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

Statics and Mechanics of Materials

Statics and Mechanics of Materials Chapter 4 Stress, Strain and Deformation: Axial Loading Objectives: Learn and understand the concepts of internal forces, stresses, and strains Learn and understand the

MECHANICS 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.

Introduction 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

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 STRESS AND STRAIN

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 STRESS AND STRAIN 1.1 Stress & Strain Stress is the internal resistance offered by the body per unit area. Stress is represented as force per unit area. Typical

Problem 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

ENGINEERING 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

Design Analysis and Review of Stresses at a Point

Design Analysis and Review of Stresses at a Point Need for Design Analysis: To verify the design for safety of the structure and the users. To understand the results obtained in FEA, it is necessary to

Finite 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

Structural 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

STRENGTH OF MATERIALS - II

STRENGTH OF MATERIALS - II DEPARTMENT OF CIVIL ENGINEERING INSTITUTE OF AERONAUTICAL ENGINEERING UNIT 1 TORSION TORSION OF HOLLOW SHAFTS: From the torsion of solid shafts of circular x section,

Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

COMPLEX STRESS TUTORIAL 5 STRAIN ENERGY

COMPLEX STRESS TUTORIAL 5 STRAIN ENERGY This tutorial covers parts of the Engineering Council Exam D Structural Analysis and further material useful students of structural engineering. You should judge

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

odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 21 The oment- Distribution ethod: rames with Sidesway Instructional Objectives After reading this chapter the student

MATERIALS SELECTION FOR SPECIFIC USE

MATERIALS SELECTION FOR SPECIFIC USE-1 Sub-topics 1 Density What determines density and stiffness? Material properties chart Design problems LOADING 2 STRENGTH AND STIFFNESS Stress is applied to a material

Civil Engineering. Strength of Materials. Comprehensive Theory with Solved Examples and Practice Questions. Publications

Civil Engineering Strength of Materials Comprehensive Theory with Solved Examples and Practice Questions Publications Publications MADE EASY Publications Corporate Office: 44-A/4, Kalu Sarai (Near Hauz

Course 1 Laboratory. Second Semester. Experiment: Young s Modulus

Course 1 Laboratory Second Semester Experiment: Young s Modulus 1 Elasticity Measurements: Young Modulus Of Brass 1 Aims of the Experiment The aim of this experiment is to measure the elastic modulus with

10 Space Truss and Space Frame Analysis

10 Space Truss and Space Frame Analysis 10.1 Introduction One dimensional models can be very accurate and very cost effective in the proper applications. For example, a hollow tube may require many thousands

Chapter 3 THE STATIC ASPECT OF SOLICITATION

Chapter 3 THE STATIC ASPECT OF SOLICITATION 3.1. ACTIONS Construction elements interact between them and with the environment. The consequence of this interaction defines the system of actions that subject

Shear 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

MECHANICS 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

Analysis 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

Chapter 9 - Plastic Analysis

Chapter 9 lastic Analysis Chapter 9 - lastic Analysis 9.1 Introduction... 3 9.1.1 Background... 3 9.2 Basis of lastic Design... 4 9.2.1 aterial Behaviour... 4 9.2.2 Cross Section Behaviour... 6 9.2.3 lastic

Analysis of Plane Frames

Plane frames are two-dimensional structures constructed with straight elements connected together by rigid and/or hinged connections. rames are subjected to loads and reactions that lie in the plane of

Worked Examples of mathematics used in Civil Engineering

Worked Examples of mathematics used in Civil Engineering Worked Example 1: Stage 1 Engineering Surveying (CIV_1010) Tutorial - Transition curves and vertical curves. Worked Example 1 draws from CCEA Advanced

Strength of Materials

Strength of Materials 1. Strain is defined as the ratio of (a) change in volume to original volume (b) change in length to original length (c) change in cross-sectional area to original cross-sectional

CHAPTER MECHANICS OF MATERIALS

CHPTER 4 Pure MECHNCS OF MTERLS Bending Pure Bending Pure Bending Other Loading Tpes Smmetric Member in Pure Bending Bending Deformations Strain Due to Bending Stress Due to Bending Beam Section Properties

Impact Load Factors for Static Analysis

Impact Load Factors for Static Analysis Often a designer has a mass, with a known velocity, hitting an object and thereby causing a suddenly applied impact load. Rather than conduct a dynamic analysis

Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II

Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 23 Bending of Beams- II Welcome to the second lesson of the fifth module which is on Bending of Beams part

The 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

Mechanics 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.

11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

Lecture 4: Basic Review of Stress and Strain, Mechanics of Beams

MECH 466 Microelectromechanical Sstems Universit of Victoria Dept. of Mechanical Engineering Lecture 4: Basic Review of Stress and Strain, Mechanics of Beams 1 Overview Compliant Mechanisms Basics of Mechanics

Strength of Materials

FE Review Strength of Materials Problem Statements Copyright 2008 C. F. Zorowski NC State E490 Mechanics of Solids 110 KN 90 KN 13.5 KN A 3 = 4.5x10-3 m 2 A 2 = 2x10-3 m 2 A 1 = 5x10-4 m 2 1. A circular

MECHANICS OF SOLIDS - BEAMS TUTORIAL 3 THE DEFLECTION OF BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL THE DEECTION OF BEAMS This is the third tutorial on the bending of beams. You should judge your progress by completing the self assessment exercises. On completion

2. 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,

Solid 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

METHODS FOR ACHIEVEMENT UNIFORM STRESSES DISTRIBUTION UNDER THE FOUNDATION

International Journal of Civil Engineering and Technology (IJCIET) Volume 7, Issue 2, March-April 2016, pp. 45-66, Article ID: IJCIET_07_02_004 Available online at http://www.iaeme.com/ijciet/issues.asp?jtype=ijciet&vtype=7&itype=2

Torsion Tests. Subjects of interest

Chapter 10 Torsion Tests Subjects of interest Introduction/Objectives Mechanical properties in torsion Torsional stresses for large plastic strains Type of torsion failures Torsion test vs.tension test

Module 6 : Design of Retaining Structures. Lecture 28 : Anchored sheet pile walls [ Section 28.1 : Introduction ]

Lecture 28 : Anchored sheet pile walls [ Section 28.1 : Introduction ] Objectives In this section you will learn the following Introduction Lecture 28 : Anchored sheet pile walls [ Section 28.1 : Introduction

Statics 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

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 29 Stresses in Beams- IV

Strength of Materials Prof: S.K.Bhattacharya Dept of Civil Engineering, IIT, Kharagpur Lecture no 29 Stresses in Beams- IV Welcome to the fourth lesson of the sixth module on Stresses in Beams part 4.

Shear Forces and Bending Moments in Beams

Shear Forces and ending Moments in eams ending Stress: Moment of Inertia: σ = My I I x = y 2 d Parallel xis Theorem: I y = x 2 d I x = I xc + d 2 eam Classifications: I y = I yc + d 2 eams are also classified

MATERIALS AND SCIENCE IN SPORTS. Edited by: EH. (Sam) Froes and S.J. Haake. Dynamics

MATERIALS AND SCIENCE IN SPORTS Edited by: EH. (Sam) Froes and S.J. Haake Dynamics Analysis of the Characteristics of Fishing Rods Based on the Large-Deformation Theory Atsumi Ohtsuki, Prof, Ph.D. Pgs.

11 Vibration Analysis

11 Vibration Analysis 11.1 Introduction A spring and a mass interact with one another to form a system that resonates at their characteristic natural frequency. If energy is applied to a spring mass system,

Mechanics of Materials

Mechanics of Materials Notation: a = acceleration A = area (net = with holes, bearing = in contact, etc...) ASD = allowable stress design d = diameter of a hole = calculus symbol for differentiation e

There are three principle ways a load can be applied:

MATERIALS SCIENCE Concepts of Stress and Strains Stress-strain test is used to determine the mechanical behavior by applying a static load uniformly over a cross section or a surface of a member. The test

DESIGN OF BEAM-COLUMNS - I

13 DESIGN OF BEA-COLUNS - 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

P4 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

6 1. Draw the shear and moment diagrams for the shaft. The bearings at A and B exert only vertical reactions on the shaft.

06 Solutions 46060_Part1 5/27/10 3:51 PM Page 329 6 1. Draw the shear and moment diagrams for the shaft. The bearings at and exert only vertical reactions on the shaft. 250 mm 800 mm 24 kn 6 2. Draw the

The Mathematics of Simple Beam Deflection

The Mathematics of Simple Beam Laing O Rourke Civil Engineering INTRODUCTION Laing O Rourke plc is the largest privately owned construction firm in the UK. It has offices in the UK, Germany, India, Australia

Bending of Thin-Walled Beams. Introduction

Introduction Beams are essential in aircraft construction Thin walled beams are often used to lower the weight of aircraft There is a need to determine that loads applied do not lead to beams having ecessive

8.2 Continuous Beams (Part I)

8.2 Continuous Beams (Part I) This section covers the following topics. Analysis Incorporation of Moment due to Reactions Pressure Line due to Prestressing Force Introduction Beams are made continuous

New 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 beam-column FEM Ferenc Papp* and József Szalai ** * Associate Professor, Department of Structural

cos 2u - t xy sin 2u (Q.E.D.)

09 Solutions 46060 6/8/10 3:13 PM Page 619 010 Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copyright laws as they currently 9 1. Prove that

Geometric Stiffness Effects in 2D and 3D Frames

Geometric Stiffness Effects in D and 3D Frames CEE 41. Matrix Structural Analsis Department of Civil and Environmental Engineering Duke Universit Henri Gavin Fall, 1 In situations in which deformations

superimposing the stresses and strains cause by each load acting separately

COMBINED LOADS In many structures the members are required to resist more than one kind of loading (combined loading). These can often be analyzed by superimposing the stresses and strains cause by each

UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS

UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS Rigid and Deformable bodies Strength, stiffness and stability Stresses: Tensile, compressive and shear Deformation of simple and compound bars under axial load

Laterally Loaded Piles 1 Soil Response Modelled by p-y Curves In order to properly analyze a laterally loaded pile foundation in soil/rock, a nonlinear relationship needs to be applied that provides soil

w Technical Product Information Precision Miniature Load Cell with Overload Protection 1. Introduction The load cells in the model 8431 and 8432 series are primarily designed for the measurement of force

MECHANICS 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.

Statics 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

Experimental Question 1: Levitation of Conductors in an Oscillating Magnetic Field SOLUTION ( )

a. Using Faraday s law: Experimental Question 1: Levitation of Conductors in an Oscillating Magnetic Field SOLUTION The overall sign will not be graded. For the current, we use the extensive hints in the

Statics of Bending: Shear and Bending Moment Diagrams

Statics of Bending: Shear and Bending Moment Diagrams David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 November 15, 2000 Introduction

Chapter 28 Fluid Dynamics

Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example

Reinforced Concrete Design SHEAR IN BEAMS

CHAPTER Reinforced Concrete Design Fifth Edition SHEAR IN BEAMS A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part I Concrete Design and Analysis 4a FALL 2002 By Dr.

Structures and Stiffness

Structures and Stiffness ENGR 10 Introduction to Engineering Ken Youssefi/Thalia Anagnos Engineering 10, SJSU 1 Wind Turbine Structure The Goal The support structure should be optimized for weight and

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

Shear Center in Thin-Walled Beams Lab

Shear Center in Thin-Walled Beams Lab Shear flow is developed in beams with thin-walled cross sections shear flow (q sx ): shear force per unit length along cross section q sx =τ sx t behaves much like

B.TECH. (AEROSPACE ENGINEERING) PROGRAMME (BTAE) Term-End Examination December, 2011 BAS-010 : MACHINE DESIGN

No. of Printed Pages : 7 BAS-01.0 B.TECH. (AEROSPACE ENGINEERING) PROGRAMME (BTAE) CV CA CV C:) O Term-End Examination December, 2011 BAS-010 : MACHINE DESIGN Time : 3 hours Maximum Marks : 70 Note : (1)

Beam Structures and Internal Forces

RCH 331 Note Set 7 Su2011abn Beam Structures and Internal Forces Notation: a = algebraic quantity, as is b, c, d = name for area b = intercept of a straight line d = calculus symbol for differentiation

Mechanics of Materials. Chapter 6 Deflection of Beams

Mechanics of Materials Chapter 6 Deflection of Beams 6.1 Introduction Because the design of beams is frequently governed by rigidity rather than strength. For example, building codes specify limits on

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