March 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "March 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE"

Transcription

1 Chapter 4 Deflection and Stiffness 1

2 2 Chapter Outline Spring Rates Tension, Compression, and Torsion Deflection Due to Bending Beam Deflection Methods Beam Deflections by Superposition Strain Energy Castigliano s Theorem Deflection of Curved Members Statically Indeterminate Problems Compression Members General Long Columns with Central Loading Intermediate-Length Columns with Central Loading Columns with Eccentric Loading Struts or Short Compression Members Suddenly Applied Loading

3 3 Force vs Deflection Elasticity:property of a material that enables it to regain its original configuration after deformation Spring: a mechanical element that exerts a force when deformed

4 4 Force vs Deflection Linear spring Nonlinear stiffening spring Nonlinear softening spring Fig. 4 1

5 5 Spring Rate Spring rate For linear springs, k constant is constant, spring

6 6 Axially-Loaded Stiffness Total extension or contraction of a uniform bar in tension or compression Spring constant, with k = F/d

7 7 Torsionally-Loaded Stiffness Angular deflection (radians) of a uniform solid or hollow round bar subjected to a twisting moment T Torsional spring constant for round bar

8 8 Deflection Due to Bending Curvature of beam subjected to bending moment M Curvature of plane curve

9 9 Deflection Due to Bending Slope of beam at any point x along the length If slope is very small, denominator of Eq. 4.9 approaches unity:

10 10 Deflection Due to Bending Recall Eqs. 3-3 & 3-4

11 Example 4-1 For the beam in Fig. 4 2, the bending moment equation, for 0 x l, is Using Eq. (4 12), determine the equations for slope & deflection of the beam, slopes at ends, and max deflection. Fig. 4 2

12 12 HW Assignment #4-1 Problem 4-1 Due Monday 17/3/2014

13 13 Beam Deflection Methods 1. Superposition 2. Moment-area method 3. Numerical integration 4. Castigliano energy method 5. Finite element software

14 14 Beam Deflection by Superposition Table A-9 Roark s Formulas for Stress & Strain

15 Example 4-2 Consider the uniformly loaded beam with a concentrated force as shown in Fig Using superposition, determine the reactions and deflection as a function of x. Fig. 4 3

16 Example 4-3 Consider the beam in Fig. 4 4a. Determine the deflection equations using superposition. Fig. 4 4

17 Example 4-4 Figure 4 5a shows a cantilever beam with an end load. Normally we model this problem by considering the left support as rigid. After testing the rigidity of the wall it was found that the translational stiffness of the wall was k t force per unit vertical deflection, and the rotational stiffness was k r moment per unit angular (radian) deflection (see Fig. 4 5b). Determine the deflection equation for the beam under the load F.

18 Example 4-4 Fig. 4 5

19 19 HW Assignment #4-2 Problems: 4.10, 4.14, 4.41 Due Wednesday 19/3/2014

20 20 Strain Energy External work done on elastic member while deforming it is transformed into strain energy, or potential energy. Strain energy = average force deflection

21 21 Some Common Strain Energy Formulas Axial loading, k = AE/l from Eq. (4-4), Torsional loading, k = GJ/l from Eq. (4-7)

22 22 Some Common Strain Energy Formulas Direct shear loading, Bending loading,

23 23 Some Common Strain Energy Formulas Transverse shear loading, C = modifier dependent on x-sectional shape

24 24 Some Common Strain Energy Formulas Table 4 1: Strain-Energy Correction Factors for Transverse Shear

25 Example 4-8 A cantilever beam with a round cross section has a concentrated load F at the end, as shown in Fig. 4 9a. Find the strain energy in the beam. Fig. 4 9

26 26 Castigliano s Theorem Forces act on elastic systems subject to small displacements Displacement corresponding to any force along its direction = partial derivative of total strain energy wrt force For rotational displacement, in radians,

27 Example 4-9 The cantilever of Ex. 4 8 is a carbon steel bar 10 in long with a 1-in diameter and is loaded by a force F = 100 lbf. a. Find max deflection using Castigliano s theorem, including that due to shear. b. What error is introduced if shear is neglected? Fig. 4 9

28 28 Utilizing a Fictitious Force Apply a fictitious force Q at the point, and in the direction, of the desired deflection. Set up the equation for total strain energy including energy due to Q. Take derivative of total strain energy wrt Q Set Q to zero

29 29 Finding Deflection Without Finding Energy Partial derivative is moved inside the integral. For example, for bending,

30 Common Deflection Equations

31 Example 4-10 Using Castigliano s method, determine the deflections of points A and B due to the force F applied at the end of the step shaft shown in Fig The second area moments for sections AB and BC are I 1 and 2I 1, respectively. Fig. 4 10

32 Example 4-11 For the wire form of diameter d shown in Fig. 4 11a, determine the deflection of point B in the direction of the applied force F (neglect the effect of transverse shear). Fig. 4 11

33 Example 4-11

34 34 HW Assignment #4-3 Problems: 4.67, 4.70 Due Saturday 22/3/2014

35 35 Deflection of Curved Members Four strain energy terms due to: 1. Bending moment M 2. Axial force F q 3. Bending moment due to F q 4. Transverse shear F r

36 36 Deflection of Curved Members Strain energy due to bending moment M r n = radius of neutral axis

37 37 Deflection of Curved Members Strain energy due to axial force F q

38 38 Deflection of Curved Members Strain energy due to bending moment from F q

39 39 Deflection of Curved Members Strain energy due to transverse shear F r

40 40 Deflection of Curved Members Combining four energy terms Deflection by Castigliano s method

41 41 Deflection of Curved Members

42 42 Deflection of Curved Members

43 43 Deflection of Thin Curved Members R/h > 10, eccentricity is small Strain energies approximated with regular energy equations: Rdq dx As R increases, bending component dominates all other terms

44 Example 4-12 The cantilevered hook shown in Fig. 4 13a is formed from a round steel wire with a diameter of 2 mm. The hook dimensions are l = 40 & R = 50 mm. A force P of 1 N is applied at point C. Use Castigliano s theorem to estimate deflection at point D at the tip. Fig. 4 13

45 45 HW Assignment #4-4 Problems: 4.78 Due Monday 24/3/2014

46 46 Statically Indeterminate Problems Redundant supports: extra constraint supports A deflection equation is required for each redundant support.

47 47 Procedure 1 for Statically Indeterminate Problems 1. Choose redundant reactions 2. Write equations of static equilibrium for remaining reactions in terms of applied loads & redundant reactions. 3. Write deflection equations for points at locations of redundant reactions in terms of applied loads and redundant reactions. 4. Solve equilibrium & deflection equations

48 Example 4-14 The indeterminate beam 11 of Appendix Table A 9 is reproduced in Fig Determine the reactions using procedure 1. Fig. 4 16

49 49 Procedure 2 for Statically Indeterminate Problems 1. Write equations of static equilibrium in terms of applied loads & unknown restraint reactions. 2. Write deflection equation in terms of applied loads and unknown restraint reactions. 3. Apply boundary conditions to deflection equation consistent with restraints. 4. Solve the set of equations.

50 Example 4-15 The rods AD & CE shown in Fig. 4 17a each have a diameter of 10 mm. The second area moment of beam ABC is I = 62.5(10 3 ) mm 4. The modulus of elasticity of the material used for the rods and beam is E = 200 GPa. The threads at the ends of the rods are singlethreaded with a pitch of 1.5 mm. The nuts are first snugly fit with bar ABC horizontal. Next the nut at A is tightened one full turn. Determine the resulting tension in each rod and the deflections of points A and C.

51 Example 4-15 Fig. 4 17

52 52 HW Assignment #4-5 Problems: 4.95, 4.97 Due Wednesday 26/3/2014

53 53 Compression Members Column: A member loaded in compression either its length or eccentric loading causes it to experience more than pure compression

54 54 Compression Members Four categories of columns 1. Long columns with central loading 2. Intermediate-length columns with central loading 3. Columns with eccentric loading 4. Struts or short columns with eccentric loading

55 55 Long Columns with Central Loading When P reaches critical load, column becomes unstable & bending develops rapidly

56 56 Euler Column Formula Pin-ended column, Other end conditions: apply a constant C for each end condition

57 57 Recommended Values for End Condition Constant Table 4-2: End-Condition Constants for Euler Columns [to Be Used with Eq. (4 43)]

58 58 Long Columns with Central Loading I = Ak 2, Euler column formula becomes l/k = slenderness ratio, to classify columns according to length categories. P cr /A = critical unit load necessary to place the column in a condition of unstable equilibrium

59 Euler Curve 59 P cr /A vs l/k, with C = 1 gives curve PQR Fig. 4 19

60 60 Long Columns with Central Loading Vulnerability to failure near point Q Buckling is sudden & catastrophic, a conservative approach near Q is desired T is defined such that P cr /A = S y /2, giving

61 61 Condition for Use of Euler Equation (l/k) > (l/k) 1, use Euler equation (l/k) (l/k) 1, use a parabolic curve between S y & T

62 62 Intermediate-Length Columns with Central Loading Intermediate-length columns, (l/k) (l/k) 1, use a parabolic curve between S y and T General form of parabola

63 63 Intermediate-Length Columns with Central Loading If parabola starts at S y, then a = S y

64 64 Columns with Eccentric Loading M = -P(e+y) d 2 y/dx 2 =M/EI Fig. 4 20

65 65 Columns with Eccentric Loading Boundary conditions: y = 0 at x = 0 and at x = l

66 66 Columns with Eccentric Loading At midspan where x = l/2

67 67 Columns with Eccentric Loading Max compressive stress: Substituting M max from Eq. (4-48)

68 68 Columns with Eccentric Loading Using S yc as the maximum value of s c, and solving for P/A, we obtain the secant column formula

69 69 Secant Column Formula ec/k 2 = eccentricity ratio Fig. 4 21

70 Example 4-16 Develop specific Euler equations for the sizes of columns having a. Round cross sections b. Rectangular cross sections

71 Example 4-17 Specify the diameter of a round column 1.5 m long that is to carry a maximum load estimated to be 22 kn. Use a design factor n d = 4 and consider the ends as pinned (rounded). The column material selected has a minimum yield strength of 500 MPa and a modulus of elasticity of 207 GPa.

72 Example 4-18 Repeat Ex for J. B. Johnson columns. (a) For round columns, Eq. (4 46) yields (b) For round columns, Eq. (4 46) yields

73 Example 4-19 Choose a set of dimensions for a rectangular link that is to carry a maximum compressive load of 5000 lbf. The material selected has a minimum yield strength of 75 kpsi and a modulus of elasticity E = 30 Mpsi. Use a design factor of 4 and an end condition constant C = 1 for buckling in the weakest direction, and design for a. a length of 15 in b. a length of 8 in with a minimum thickness of 0.5 in.

74 74 Struts or Short Compression Members Strut: short member loaded in compression If eccentricity exists, max stress is at B with axial compression and bending.

75 75 Struts or Short Compression Members Differs from secant equation in that it assumes small effect of bending deflection If bending deflection is limited to 1% of e, then from Eq. 4-44, the limiting slenderness ratio for strut is

76 Example 4-20 Figure 4 23a shows a workpiece clamped to a milling machine table by a bolt tightened to a tension of 2000 lbf. The clamp contact is offset from the centroidal axis of the strut by a distance e = 0.10 in, as shown in part b of the figure. The strut, or block, is steel, 1 in square and 4 in long, as shown. Determine the maximum compressive stress in the block.

77 Example 4-20 Fig. 4 23

78 78 HW Assignment #4-6 Problems: 4.107, Due Saturday 29/3/2014

79 79 Suddenly Applied Loading Weight falls from distance h and suddenly applies a load to a cantilever beam Find deflection and force applied to beam due to impact

80 80 Suddenly Applied Loading Abstract model considering beam as simple spring Table A-9: beam 1, k= F/y =3EI/l 3 Assume beam to be massless, so no momentum transfer, just energy transfer Loss of potential energy from change of elevation is W(h + d)

81 81 Suddenly Applied Loading Increase in potential energy from compressing spring is kd 2 /2 Conservation of energy W(h + d) = kd 2 /2

82 82 Suddenly Applied Loading Maximum deflection Maximum force

Mechanical Design in Optical Engineering

Mechanical Design in Optical Engineering OPTI Buckling Buckling and Stability: As we learned in the previous lectures, structures may fail in a variety of ways, depending on the materials, load and support conditions. We had two primary concerns:

More information

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction

More information

Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE

Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE 1 Chapter 3 Load and Stress Analysis 2 Chapter Outline Equilibrium & Free-Body Diagrams Shear Force and Bending Moments in Beams Singularity Functions Stress Cartesian Stress Components Mohr s Circle for

More information

Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method

Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Objectives In this course you will learn the following Deflection by strain energy method. Evaluation of strain energy in member under

More information

3. BEAMS: STRAIN, STRESS, DEFLECTIONS

3. BEAMS: STRAIN, STRESS, DEFLECTIONS 3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets

More information

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods

More information

Chapter 3. Load and Stress Analysis

Chapter 3. Load and Stress Analysis Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3

More information

7.5 Elastic Buckling Columns and Buckling

7.5 Elastic Buckling Columns and Buckling 7.5 Elastic Buckling The initial theory of the buckling of columns was worked out by Euler in 1757, a nice example of a theory preceding the application, the application mainly being for the later invented

More information

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad

INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad -00 04 CIVIL ENGINEERING QUESTION BANK Course Name : STRENGTH OF MATERIALS II Course Code : A404 Class : II B. Tech II Semester Section

More information

UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation.

UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation. UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The magnitude

More information

CHAPTER -6- BENDING Part -1-

CHAPTER -6- BENDING Part -1- Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and

More information

UNIT IV FLEXIBILTY AND STIFFNESS METHOD

UNIT IV FLEXIBILTY AND STIFFNESS METHOD SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : SA-II (13A01505) Year & Sem: III-B.Tech & I-Sem Course & Branch: B.Tech

More information

Unit III Theory of columns. Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE, Sriperumbudir

Unit III Theory of columns. Dr.P.Venkateswara Rao, Associate Professor, Dept. of Civil Engg., SVCE, Sriperumbudir Unit III Theory of columns 1 Unit III Theory of Columns References: Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength of Materials", Tata

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS Third E CHAPTER 2 Stress MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University and Strain Axial Loading Contents Stress & Strain:

More information

Mechanics of Solids. Mechanics Of Solids. Suraj kr. Ray Department of Civil Engineering

Mechanics of Solids. Mechanics Of Solids. Suraj kr. Ray Department of Civil Engineering Mechanics Of Solids Suraj kr. Ray (surajjj2445@gmail.com) Department of Civil Engineering 1 Mechanics of Solids is a branch of applied mechanics that deals with the behaviour of solid bodies subjected

More information

If the number of unknown reaction components are equal to the number of equations, the structure is known as statically determinate.

If the number of unknown reaction components are equal to the number of equations, the structure is known as statically determinate. 1 of 6 EQUILIBRIUM OF A RIGID BODY AND ANALYSIS OF ETRUCTURAS II 9.1 reactions in supports and joints of a two-dimensional structure and statically indeterminate reactions: Statically indeterminate structures

More information

CHAPTER 5 Statically Determinate Plane Trusses

CHAPTER 5 Statically Determinate Plane Trusses CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS TYPES OF ROOF TRUSS ROOF TRUSS SETUP ROOF TRUSS SETUP OBJECTIVES To determine the STABILITY and DETERMINACY of plane trusses To analyse

More information

Only for Reference Page 1 of 18

Only for Reference  Page 1 of 18 Only for Reference www.civilpddc2013.weebly.com Page 1 of 18 Seat No.: Enrolment No. GUJARAT TECHNOLOGICAL UNIVERSITY PDDC - SEMESTER II EXAMINATION WINTER 2013 Subject Code: X20603 Date: 26-12-2013 Subject

More information

Chapter 2: Deflections of Structures

Chapter 2: Deflections of Structures Chapter 2: Deflections of Structures Fig. 4.1. (Fig. 2.1.) ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 1 (2.1) (4.1) (2.2) Fig.4.2 Fig.2.2 ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 2

More information

CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS

CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS CHAPTER 5 Statically Determinate Plane Trusses TYPES OF ROOF TRUSS 1 TYPES OF ROOF TRUSS ROOF TRUSS SETUP 2 ROOF TRUSS SETUP OBJECTIVES To determine the STABILITY and DETERMINACY of plane trusses To analyse

More information

1-1 Locate the centroid of the plane area shown. 1-2 Determine the location of centroid of the composite area shown.

1-1 Locate the centroid of the plane area shown. 1-2 Determine the location of centroid of the composite area shown. Chapter 1 Review of Mechanics of Materials 1-1 Locate the centroid of the plane area shown 650 mm 1000 mm 650 x 1- Determine the location of centroid of the composite area shown. 00 150 mm radius 00 mm

More information

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain - Axial Loading Statics

More information

CHAPTER 6: Shearing Stresses in Beams

CHAPTER 6: Shearing Stresses in Beams (130) CHAPTER 6: Shearing Stresses in Beams When a beam is in pure bending, the only stress resultants are the bending moments and the only stresses are the normal stresses acting on the cross sections.

More information

ENCE 455 Design of Steel Structures. III. Compression Members

ENCE 455 Design of Steel Structures. III. Compression Members ENCE 455 Design of Steel Structures III. Compression Members C. C. Fu, Ph.D., P.E. Civil and Environmental Engineering Department University of Maryland Compression Members Following subjects are covered:

More information

BE Semester- I ( ) Question Bank (MECHANICS OF SOLIDS)

BE Semester- I ( ) Question Bank (MECHANICS OF SOLIDS) BE Semester- I ( ) Question Bank (MECHANICS OF SOLIDS) All questions carry equal marks(10 marks) Q.1 (a) Write the SI units of following quantities and also mention whether it is scalar or vector: (i)

More information

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3

Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6.2, 6.3 M9 Shafts: Torsion of Circular Shafts Reading: Crandall, Dahl and Lardner 6., 6.3 A shaft is a structural member which is long and slender and subject to a torque (moment) acting about its long axis. We

More information

2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at A and supported at B by rod (1). What is the axial force in rod (1)?

2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at A and supported at B by rod (1). What is the axial force in rod (1)? IDE 110 S08 Test 1 Name: 1. Determine the internal axial forces in segments (1), (2) and (3). (a) N 1 = kn (b) N 2 = kn (c) N 3 = kn 2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at

More information

Torsion Stresses in Tubes and Rods

Torsion Stresses in Tubes and Rods Torsion Stresses in Tubes and Rods This initial analysis is valid only for a restricted range of problem for which the assumptions are: Rod is initially straight. Rod twists without bending. Material is

More information

14. *14.8 CASTIGLIANO S THEOREM

14. *14.8 CASTIGLIANO S THEOREM *14.8 CASTIGLIANO S THEOREM Consider a body of arbitrary shape subjected to a series of n forces P 1, P 2, P n. Since external work done by forces is equal to internal strain energy stored in body, by

More information

6. Bending CHAPTER OBJECTIVES

6. Bending CHAPTER OBJECTIVES CHAPTER OBJECTIVES Determine stress in members caused by bending Discuss how to establish shear and moment diagrams for a beam or shaft Determine largest shear and moment in a member, and specify where

More information

MECHANICAL PROPERTIES OF SOLIDS

MECHANICAL PROPERTIES OF SOLIDS Chapter Nine MECHANICAL PROPERTIES OF SOLIDS MCQ I 9.1 Modulus of rigidity of ideal liquids is (a) infinity. (b) zero. (c) unity. (d) some finite small non-zero constant value. 9. The maximum load a wire

More information

CHAPTER OBJECTIVES CHAPTER OUTLINE. 4. Axial Load

CHAPTER OBJECTIVES CHAPTER OUTLINE. 4. Axial Load CHAPTER OBJECTIVES Determine deformation of axially loaded members Develop a method to find support reactions when it cannot be determined from euilibrium euations Analyze the effects of thermal stress

More information

MECE 3321: MECHANICS OF SOLIDS CHAPTER 5

MECE 3321: MECHANICS OF SOLIDS CHAPTER 5 MECE 3321: MECHANICS OF SOLIDS CHAPTER 5 SAMANTHA RAMIREZ TORSION Torque A moment that tends to twist a member about its longitudinal axis 1 TORSIONAL DEFORMATION OF A CIRCULAR SHAFT Assumption If the

More information

7.4 The Elementary Beam Theory

7.4 The Elementary Beam Theory 7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. s with pressure vessels, the geometry of the beam, and the specific type of loading which will be

More information

FIXED BEAMS IN BENDING

FIXED BEAMS IN BENDING FIXED BEAMS IN BENDING INTRODUCTION Fixed or built-in beams are commonly used in building construction because they possess high rigidity in comparison to simply supported beams. When a simply supported

More information

MECH 344/X Machine Element Design

MECH 344/X Machine Element Design 1 MECH 344/X Machine Element Design Time: M 14:45-17:30 Lecture 2 Contents of today's lecture Introduction to Static Stresses Axial, Shear and Torsional Loading Bending in Straight and Curved Beams Transverse

More information

Equilibrium of a Particle

Equilibrium of a Particle ME 108 - Statics Equilibrium of a Particle Chapter 3 Applications For a spool of given weight, what are the forces in cables AB and AC? Applications For a given weight of the lights, what are the forces

More information

MECE 3321 MECHANICS OF SOLIDS CHAPTER 3

MECE 3321 MECHANICS OF SOLIDS CHAPTER 3 MECE 3321 MECHANICS OF SOLIDS CHAPTER 3 Samantha Ramirez TENSION AND COMPRESSION TESTS Tension and compression tests are used primarily to determine the relationship between σ avg and ε avg in any material.

More information

Comparison of Euler Theory with Experiment results

Comparison of Euler Theory with Experiment results Comparison of Euler Theory with Experiment results Limitations of Euler's Theory : In practice the ideal conditions are never [ i.e. the strut is initially straight and the end load being applied axially

More information

Experiment Two (2) Torsional testing of Circular Shafts

Experiment Two (2) Torsional testing of Circular Shafts Experiment Two (2) Torsional testing of Circular Shafts Introduction: Torsion occurs when any shaft is subjected to a torque. This is true whether the shaft is rotating (such as drive shafts on engines,

More information

7 TRANSVERSE SHEAR transverse shear stress longitudinal shear stresses

7 TRANSVERSE SHEAR transverse shear stress longitudinal shear stresses 7 TRANSVERSE SHEAR Before we develop a relationship that describes the shear-stress distribution over the cross section of a beam, we will make some preliminary remarks regarding the way shear acts within

More information

Initial Stress Calculations

Initial Stress Calculations Initial Stress Calculations The following are the initial hand stress calculations conducted during the early stages of the design process. Therefore, some of the material properties as well as dimensions

More information

Chapter 5 Compression Member

Chapter 5 Compression Member Chapter 5 Compression Member This chapter starts with the behaviour of columns, general discussion of buckling, and determination of the axial load needed to buckle. Followed b the assumption of Euler

More information

CHAPTER 3 THE EFFECTS OF FORCES ON MATERIALS

CHAPTER 3 THE EFFECTS OF FORCES ON MATERIALS CHAPTER THE EFFECTS OF FORCES ON MATERIALS EXERCISE 1, Page 50 1. A rectangular bar having a cross-sectional area of 80 mm has a tensile force of 0 kn applied to it. Determine the stress in the bar. Stress

More information

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.

More information

Chapter 7: Bending and Shear in Simple Beams

Chapter 7: Bending and Shear in Simple Beams Chapter 7: Bending and Shear in Simple Beams Introduction A beam is a long, slender structural member that resists loads that are generally applied transverse (perpendicular) to its longitudinal axis.

More information

MARKS DISTRIBUTION AS PER CHAPTER (QUESTION ASKED IN GTU EXAM) Name Of Chapter. Applications of. Friction. Centroid & Moment.

MARKS DISTRIBUTION AS PER CHAPTER (QUESTION ASKED IN GTU EXAM) Name Of Chapter. Applications of. Friction. Centroid & Moment. Introduction Fundamentals of statics Applications of fundamentals of statics Friction Centroid & Moment of inertia Simple Stresses & Strain Stresses in Beam Torsion Principle Stresses DEPARTMENT OF CIVIL

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method Module 2 Analysis of Statically Indeterminate Structures by the Matrix Force Method Lesson 8 The Force Method of Analysis: Beams Instructional Objectives After reading this chapter the student will be

More information

Mechanics of Materials

Mechanics of Materials Mechanics of Materials 2. Introduction Dr. Rami Zakaria References: 1. Engineering Mechanics: Statics, R.C. Hibbeler, 12 th ed, Pearson 2. Mechanics of Materials: R.C. Hibbeler, 9 th ed, Pearson 3. Mechanics

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS CHTER MECHNICS OF MTERILS 10 Ferdinand. Beer E. Russell Johnston, Jr. Columns John T. DeWolf cture Notes: J. Walt Oler Texas Tech University 006 The McGraw-Hill Companies, Inc. ll rights reserved. Columns

More information

MECHANICS OF MATERIALS. Prepared by Engr. John Paul Timola

MECHANICS OF MATERIALS. Prepared by Engr. John Paul Timola MECHANICS OF MATERIALS Prepared by Engr. John Paul Timola Mechanics of materials branch of mechanics that studies the internal effects of stress and strain in a solid body. stress is associated with the

More information

STATICALLY INDETERMINATE STRUCTURES

STATICALLY INDETERMINATE STRUCTURES STATICALLY INDETERMINATE STRUCTURES INTRODUCTION Generally the trusses are supported on (i) a hinged support and (ii) a roller support. The reaction components of a hinged support are two (in horizontal

More information

SECOND ENGINEER REG. III/2 APPLIED MECHANICS

SECOND ENGINEER REG. III/2 APPLIED MECHANICS SECOND ENGINEER REG. III/2 APPLIED MECHANICS LIST OF TOPICS Static s Friction Kinematics Dynamics Machines Strength of Materials Hydrostatics Hydrodynamics A STATICS 1 Solves problems involving forces

More information

Flexure: Behavior and Nominal Strength of Beam Sections

Flexure: Behavior and Nominal Strength of Beam Sections 4 5000 4000 (increased d ) (increased f (increased A s or f y ) c or b) Flexure: Behavior and Nominal Strength of Beam Sections Moment (kip-in.) 3000 2000 1000 0 0 (basic) (A s 0.5A s ) 0.0005 0.001 0.0015

More information

Influence of residual stresses in the structural behavior of. tubular columns and arches. Nuno Rocha Cima Gomes

Influence of residual stresses in the structural behavior of. tubular columns and arches. Nuno Rocha Cima Gomes October 2014 Influence of residual stresses in the structural behavior of Abstract tubular columns and arches Nuno Rocha Cima Gomes Instituto Superior Técnico, Universidade de Lisboa, Portugal Contact:

More information

MECE 3321 MECHANICS OF SOLIDS CHAPTER 1

MECE 3321 MECHANICS OF SOLIDS CHAPTER 1 MECE 3321 MECHANICS O SOLIDS CHAPTER 1 Samantha Ramirez, MSE WHAT IS MECHANICS O MATERIALS? Rigid Bodies Statics Dynamics Mechanics Deformable Bodies Solids/Mech. Of Materials luids 1 WHAT IS MECHANICS

More information

Lecture 8: Flexibility Method. Example

Lecture 8: Flexibility Method. Example ecture 8: lexibility Method Example The plane frame shown at the left has fixed supports at A and C. The frame is acted upon by the vertical load P as shown. In the analysis account for both flexural and

More information

PLAT DAN CANGKANG (TKS 4219)

PLAT DAN CANGKANG (TKS 4219) PLAT DAN CANGKANG (TKS 4219) SESI I: PLATES Dr.Eng. Achfas Zacoeb Dept. of Civil Engineering Brawijaya University INTRODUCTION Plates are straight, plane, two-dimensional structural components of which

More information

Beam Bending Stresses and Shear Stress

Beam Bending Stresses and Shear Stress Beam Bending Stresses and Shear Stress Notation: A = name or area Aweb = area o the web o a wide lange section b = width o a rectangle = total width o material at a horizontal section c = largest distance

More information

UNIT II SLOPE DEFLECION AND MOMENT DISTRIBUTION METHOD

UNIT II SLOPE DEFLECION AND MOMENT DISTRIBUTION METHOD SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : SA-II (13A01505) Year & Sem: III-B.Tech & I-Sem Course & Branch: B.Tech

More information

Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal Unbraced Lengths

Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal Unbraced Lengths Proceedings of the Annual Stability Conference Structural Stability Research Council San Antonio, Texas, March 21-24, 2017 Flexural-Torsional Buckling of General Cold-Formed Steel Columns with Unequal

More information

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Elmer E. Marx, Alaska Department of Transportation and Public Facilities Michael Keever, California Department

More information

Beams. Beams are structural members that offer resistance to bending due to applied load

Beams. Beams are structural members that offer resistance to bending due to applied load Beams Beams are structural members that offer resistance to bending due to applied load 1 Beams Long prismatic members Non-prismatic sections also possible Each cross-section dimension Length of member

More information

A concrete cylinder having a a diameter of of in. mm and elasticity. Stress and Strain: Stress and Strain: 0.

A concrete cylinder having a a diameter of of in. mm and elasticity. Stress and Strain: Stress and Strain: 0. 2011 earson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copyright laws as they currently 8 1. 3 1. concrete cylinder having a a diameter of of 6.00

More information

Materials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon.

Materials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon. Modes of Loading (1) tension (a) (2) compression (b) (3) bending (c) (4) torsion (d) and combinations of them (e) Figure 4.2 1 Standard Solution to Elastic Problems Three common modes of loading: (a) tie

More information

two structural analysis (statics & mechanics) APPLIED ACHITECTURAL STRUCTURES: DR. ANNE NICHOLS SPRING 2017 lecture STRUCTURAL ANALYSIS AND SYSTEMS

two structural analysis (statics & mechanics) APPLIED ACHITECTURAL STRUCTURES: DR. ANNE NICHOLS SPRING 2017 lecture STRUCTURAL ANALYSIS AND SYSTEMS APPLIED ACHITECTURAL STRUCTURES: STRUCTURAL ANALYSIS AND SYSTEMS DR. ANNE NICHOLS SPRING 2017 lecture two structural analysis (statics & mechanics) Analysis 1 Structural Requirements strength serviceability

More information

Strength Of Materials/Mechanics of Solids

Strength Of Materials/Mechanics of Solids Table of Contents Stress, Strain, and Energy 1. Stress and Strain 2. Change in length 3. Determinate Structure - Both ends free 4. Indeterminate Structure - Both ends fixed 5. Composite Material of equal

More information

SERVICEABILITY OF BEAMS AND ONE-WAY SLABS

SERVICEABILITY OF BEAMS AND ONE-WAY SLABS CHAPTER REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach - Fifth Edition Fifth Edition SERVICEABILITY OF BEAMS AND ONE-WAY SLABS A. J. Clark School of Engineering Department of Civil

More information

Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING )

Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) 5.1 DEFINITION A construction member is subjected to centric (axial) tension or compression if in any cross section the single distinct stress

More information

CHIEF ENGINEER REG III/2 APPLIED MECHANICS

CHIEF ENGINEER REG III/2 APPLIED MECHANICS CHIEF ENGINEER REG III/2 APPLIED MECHANICS LIST OF TOPICS A B C D E F G H I J Vector Representation Statics Friction Kinematics Dynamics Machines Strength of Materials Hydrostatics Hydrodynamics Control

More information

APPENDIX 1 MODEL CALCULATION OF VARIOUS CODES

APPENDIX 1 MODEL CALCULATION OF VARIOUS CODES 163 APPENDIX 1 MODEL CALCULATION OF VARIOUS CODES A1.1 DESIGN AS PER NORTH AMERICAN SPECIFICATION OF COLD FORMED STEEL (AISI S100: 2007) 1. Based on Initiation of Yielding: Effective yield moment, M n

More information

The bending moment diagrams for each span due to applied uniformly distributed and concentrated load are shown in Fig.12.4b.

The bending moment diagrams for each span due to applied uniformly distributed and concentrated load are shown in Fig.12.4b. From inspection, it is assumed that the support moments at is zero and support moment at, 15 kn.m (negative because it causes compression at bottom at ) needs to be evaluated. pplying three- Hence, only

More information

EE C245 ME C218 Introduction to MEMS Design

EE C245 ME C218 Introduction to MEMS Design EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 16: Energy

More information

REVIEW FOR EXAM II. Dr. Ibrahim A. Assakkaf SPRING 2002

REVIEW FOR EXAM II. Dr. Ibrahim A. Assakkaf SPRING 2002 REVIEW FOR EXM II. J. Clark School of Engineering Department of Civil and Environmental Engineering b Dr. Ibrahim. ssakkaf SPRING 00 ENES 0 Mechanics of Materials Department of Civil and Environmental

More information

DES140: Designing for Lateral-Torsional Stability in Wood Members

DES140: Designing for Lateral-Torsional Stability in Wood Members DES140: Designing for Lateral-Torsional Stability in Wood embers Welcome to the Lateral Torsional Stability ecourse. 1 Outline Lateral-Torsional Buckling Basic Concept Design ethod Examples In this ecourse,

More information

SIMPLE STRESSES AND STRAINS

SIMPLE STRESSES AND STRAINS SIMPLE STRESSES ND STRINS Mechanics of Materials Mechanics of materials deals with the elastic behavior of materials and the stability of members. Mechanics of materials concepts are used to determine

More information

Basic Energy Principles in Stiffness Analysis

Basic Energy Principles in Stiffness Analysis Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting

More information

Strength of Material. Shear Strain. Dr. Attaullah Shah

Strength of Material. Shear Strain. Dr. Attaullah Shah Strength of Material Shear Strain Dr. Attaullah Shah Shear Strain TRIAXIAL DEFORMATION Poisson's Ratio Relationship Between E, G, and ν BIAXIAL DEFORMATION Bulk Modulus of Elasticity or Modulus of Volume

More information

Presented By: EAS 6939 Aerospace Structural Composites

Presented By: EAS 6939 Aerospace Structural Composites A Beam Theory for Laminated Composites and Application to Torsion Problems Dr. BhavaniV. Sankar Presented By: Sameer Luthra EAS 6939 Aerospace Structural Composites 1 Introduction Composite beams have

More information

THE INFLUENCE OF THERMAL ACTIONS AND COMPLEX SUPPORT CONDITIONS ON THE MECHANICAL STATE OF SANDWICH STRUCTURE

THE INFLUENCE OF THERMAL ACTIONS AND COMPLEX SUPPORT CONDITIONS ON THE MECHANICAL STATE OF SANDWICH STRUCTURE Journal of Applied Mathematics and Computational Mechanics 013, 1(4), 13-1 THE INFLUENCE OF THERMAL ACTIONS AND COMPLEX SUPPORT CONDITIONS ON THE MECHANICAL STATE OF SANDWICH STRUCTURE Jolanta Błaszczuk

More information

Theory and Analysis of Structures

Theory and Analysis of Structures 7 Theory and nalysis of Structures J.Y. Richard iew National University of Singapore N.E. Shanmugam National University of Singapore 7. Fundamental Principles oundary Conditions oads and Reactions Principle

More information

Determine the resultant internal loadings acting on the cross section at C of the beam shown in Fig. 1 4a.

Determine the resultant internal loadings acting on the cross section at C of the beam shown in Fig. 1 4a. E X M P L E 1.1 Determine the resultant internal loadings acting on the cross section at of the beam shown in Fig. 1 a. 70 N/m m 6 m Fig. 1 Support Reactions. This problem can be solved in the most direct

More information

Failure interaction curves for combined loading involving torsion, bending, and axial loading

Failure interaction curves for combined loading involving torsion, bending, and axial loading Failure interaction curves for combined loading involving torsion, bending, and axial loading W M Onsongo Many modern concrete structures such as elevated guideways are subjected to combined bending, torsion,

More information

Engineering Mechanics Department of Mechanical Engineering Dr. G. Saravana Kumar Indian Institute of Technology, Guwahati

Engineering Mechanics Department of Mechanical Engineering Dr. G. Saravana Kumar Indian Institute of Technology, Guwahati Engineering Mechanics Department of Mechanical Engineering Dr. G. Saravana Kumar Indian Institute of Technology, Guwahati Module 3 Lecture 6 Internal Forces Today, we will see analysis of structures part

More information

Structures. Shainal Sutaria

Structures. Shainal Sutaria Structures ST Shainal Sutaria Student Number: 1059965 Wednesday, 14 th Jan, 011 Abstract An experiment to find the characteristics of flow under a sluice gate with a hydraulic jump, also known as a standing

More information

Finite Element Modelling with Plastic Hinges

Finite Element Modelling with Plastic Hinges 01/02/2016 Marco Donà Finite Element Modelling with Plastic Hinges 1 Plastic hinge approach A plastic hinge represents a concentrated post-yield behaviour in one or more degrees of freedom. Hinges only

More information

4. SHAFTS. A shaft is an element used to transmit power and torque, and it can support

4. SHAFTS. A shaft is an element used to transmit power and torque, and it can support 4. SHAFTS A shaft is an element used to transmit power and torque, and it can support reverse bending (fatigue). Most shafts have circular cross sections, either solid or tubular. The difference between

More information

BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE 2 ND YEAR STUDENTS OF THE UACEG

BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE 2 ND YEAR STUDENTS OF THE UACEG BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE ND YEAR STUDENTS OF THE UACEG Assoc.Prof. Dr. Svetlana Lilkova-Markova, Chief. Assist. Prof. Dimitar Lolov Sofia, 011 STRENGTH OF MATERIALS GENERAL

More information

= 50 ksi. The maximum beam deflection Δ max is not = R B. = 30 kips. Notes for Strength of Materials, ET 200

= 50 ksi. The maximum beam deflection Δ max is not = R B. = 30 kips. Notes for Strength of Materials, ET 200 Notes for Strength of Materials, ET 00 Steel Six Easy Steps Steel beam design is about selecting the lightest steel beam that will support the load without exceeding the bending strength or shear strength

More information

LECTURE 14 Strength of a Bar in Transverse Bending. 1 Introduction. As we have seen, only normal stresses occur at cross sections of a rod in pure

LECTURE 14 Strength of a Bar in Transverse Bending. 1 Introduction. As we have seen, only normal stresses occur at cross sections of a rod in pure V. DEMENKO MECHNCS OF MTERLS 015 1 LECTURE 14 Strength of a Bar in Transverse Bending 1 ntroduction s we have seen, onl normal stresses occur at cross sections of a rod in pure bending. The corresponding

More information

Now we are going to use our free body analysis to look at Beam Bending (W3L1) Problems 17, F2002Q1, F2003Q1c

Now we are going to use our free body analysis to look at Beam Bending (W3L1) Problems 17, F2002Q1, F2003Q1c Now we are going to use our free body analysis to look at Beam Bending (WL1) Problems 17, F00Q1, F00Q1c One of the most useful applications of the free body analysis method is to be able to derive equations

More information

Structural Steelwork Eurocodes Development of A Trans-national Approach

Structural Steelwork Eurocodes Development of A Trans-national Approach Structural Steelwork Eurocodes Development of A Trans-national Approach Course: Eurocode Module 7 : Worked Examples Lecture 0 : Simple braced frame Contents: 1. Simple Braced Frame 1.1 Characteristic Loads

More information

Rigid and Braced Frames

Rigid and Braced Frames RH 331 Note Set 12.1 F2014abn Rigid and raced Frames Notation: E = modulus of elasticit or Young s modulus F = force component in the direction F = force component in the direction FD = free bod diagram

More information

Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7

Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7 Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7 Dr. Colin Caprani, Chartered Engineer 1 Contents 1. Introduction... 3 1.1 Background... 3 1.2 Failure Modes... 5 1.3 Design Aspects...

More information

4.MECHANICAL PROPERTIES OF MATERIALS

4.MECHANICAL PROPERTIES OF MATERIALS 4.MECHANICAL PROPERTIES OF MATERIALS The diagram representing the relation between stress and strain in a given material is an important characteristic of the material. To obtain the stress-strain diagram

More information

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 4 COLUMNS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P

ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL 4 COLUMNS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL 4 H1 FORMERLY UNIT 21718P ENGINEERING SCIENCE H1 OUTCOME 1 - TUTORIAL COLUMNS EDEXCEL HNC/D ENGINEERING SCIENCE LEVEL H1 FORMERLY UNIT 21718P This material is duplicated in the Mechanical Principles module H2 and those studying

More information

Due Monday, September 14 th, 12:00 midnight

Due Monday, September 14 th, 12:00 midnight Due Monday, September 14 th, 1: midnight This homework is considering the analysis of plane and space (3D) trusses as discussed in class. A list of MatLab programs that were discussed in class is provided

More information

Question 1. Ignore bottom surface. Solution: Design variables: X = (R, H) Objective function: maximize volume, πr 2 H OR Minimize, f(x) = πr 2 H

Question 1. Ignore bottom surface. Solution: Design variables: X = (R, H) Objective function: maximize volume, πr 2 H OR Minimize, f(x) = πr 2 H Question 1 (Problem 2.3 of rora s Introduction to Optimum Design): Design a beer mug, shown in fig, to hold as much beer as possible. The height and radius of the mug should be not more than 20 cm. The

More information

Indeterminate Analysis Force Method 1

Indeterminate 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 information

Static Equilibrium; Elasticity & Fracture

Static Equilibrium; Elasticity & Fracture Static Equilibrium; Elasticity & Fracture The Conditions for Equilibrium Statics is concerned with the calculation of the forces acting on and within structures that are in equilibrium. An object with

More information