1 Introduction to Engineering Materials ENGR2 Chapter 6: Mechanical Properties of Metals Dr. Coates
2 6.2 Concepts of Stress and Strain tension compression shear torsion
3 Tension Tests The specimen is deformed to fracture with a gradually increasing tensile load
4 Fracture & Failure Fracture occurs when a structural component separates into two or more pieces. fracture represents failure of the component Failure is the inability of a component to perform its desired function. failure may occur prior to fracture e.g. a car axle that bends when you drive over a pothole
5 Engineering Stress & Engineering Strain Engineering stress : σ = F A Engineering strain : ε = l l = l l l F-instantaneous load applied to specimen cross section A -original cross section area before any load is applied l i -instantaneous length l -original length
6 If you double the cross sectional area, would your load needed to produce a given elongation be the same? Why not just use force F and deformation l to study mechanical behavior?
7 Compression Tests Similar to the tensile tests Force is compressive Specimen contracts along direction of the stress
8 Engineering Stress & Engineering Strain Engineering stress : F σ = A Engineering strain : ε = l l = l l l <
9 Shear stress : τ = F A Shear γ = tanθ θ Shear and Torsional Tests strain (by definition) : ( for small θ ) F-instantaneous load applied // to specimen cross section
10 Shear and Torsional Tests Shear stress due to applied torque : τ = f ( T ) Shear strain due to applied torque : γ = f ( ) φ
11 Stress Transformation A force balance in the y direction yields: A force balance in the tangential direction yields:
12 Deformation All materials undergo changes in dimensions in response to mechanical forces. This phenomenon is called deformation. If the material reverts back to its original size and shape upon removal of the load, the deformation is said to be elastic deformation. If application and removal of the load results in a permanent change in size or shape, the deformation is said to be plastic deformation.
13 Elastic Deformation 6.3 Stress-Strain Behavior Hooke's law : σ = Eε Modulus of E = σ ε Shear modulus : G = τ γ elasticity: ELASTIC DEFORMATION!
14 Modulus of Elasticity Young s modulus Elastic modulus Stiffness Material s resistance to elastic deformation greater the elastic modulus, the stiffer the material or smaller the elastic strain that results from the application of a given stress.
15 Tangent or Secant Modulus Materials with non-linear elastic behavior Gray cast iron, concrete, many polymers
16 Elastic modulus of metals, ceramics & polymers Ceramics high elastic modulus diamond, graphite (covalent bonds) Metals high elastic modulus Polymers low elastic modulus weak secondary bonds between chains Increasing E
17 Elastic Modulus on an Atomic Scale Elastic strain small changes in inter-atomic spacing stretching of inter-atomic bonds Elastic modulus resistance to separation of adjacent atoms inter-atomic bonding forces
18 Review 2.5 Bonding Forces and Energies The net force : FN = FA + FR At equilibrium : F - the centers of the two atoms are r = F F N A + R = is known as the bond length. r apart.
20 Elastic Modulus on an Atomic Scale How might E be related to df dr r? Why?
21 Effect of Temperature on the Elastic Modulus
23 Example 6.1 A piece of Cu originally 35 mm long is pulled in tension with a stress of 276 Mpa. If the deformation is entirely elastic, what will be the resultant elongation?
24 Example 6.1 Given : l = 35mm σ = 276 MPa Using Table 6.1: E Cu l =? = 11Gpa Using Hooke's law : σ = Eε = E l l l l = σ =.77 mm E
25 6.4 Anelasticity - time dependent elastic behavior Time dependent elastic strain component. Elastic deformation continues after the stress application. Upon load release, finite time is required for complete recovery.
28 Poisson s Ratio Poisson's ratio (a material constant) : ε x υ = ε υ > υ υ z theoretical max =.25 υ ε y = ε = = For most metals: z ( no net volume change).35.25
29 Elastic Properties of Materials Elastic constants (material properties) : E, G, υ For isotropic materials (same properties in all directions) : E G = 2 1+υ ( )
30 Example 6.2 A tensile stress is to be applied along the long axis of a cylindrical brass rod that has a diameter of 1 mm. Determine the magnitude of the load required to produce a 2.5 x 1-3 mm change in diameter if the deformation is entirely elastic.
31 Given : d d = 1mm = mm Using Table 6.1: E υ brass brass F =? = 97Gpa =.34
32 Compute the strain : ε x = d d = Using the Poisson's ratio : ε x ε z = = υ Using Hooke's law : σ = ε ze = 71.3MPa Compute the force : 2 d F = σ π 4 = 4 56 N
33 Plastic Deformation Elastic deformation Up to strains of.5 Plastic deformation Hooke s law is no longer valid Non-recoverable or permanent deformation Phenomenon of yielding
34 σ ε σ ys yp uts yield strength yield point strain ultimate tensile strength 6.6 Tensile Properties ( stress corresponding to the elastic limit) ε u uniform strain σ f fracture strength (stress at fracture) strain at fracture ε f ( maximum engineering stress)
35 Yielding and Yield Strength Gradual elastic-plastic transition Proportional limit (P) Initial departure from linearity Point of yielding Yield strength Determined using a.2 strain offset method
36 Yielding and Yield Strength Steels & other materials Yield point phenomenon Yield strength Average stress associated with the lower yield point
38 Example 6.3 From the tensile stress-strain behavior for the brass specimen shown, determine the following The modulus of elasticity The yield strength at a strain offset of.2 The maximum load that can be sustained by a cylindrical specimen having an original diameter of 12.8 mm The change in length of a specimen originally 25 mm long that is subjected to a tensile stress of 345 Mpa.
41 Given : d F max = 12.8mm =? Using the tensile strength : σ F uts = 45 MPa πd 4 2 max = σ uts = 57,9 N
42 Given : l = 25 mm σ = 345 MPa l =? Determine the strain (point A) : ε.6 Compute the elongation : l = εl 15mm =
43 Ductility Measure of the degree of plastic deformation that has been sustained at fracture. Brittle materials Little or no plastic deformation prior to fracture Ductile materials Significant plastic deformation prior to fracture
44 Percent elongation : l f l % EL = 1 = ε f 1 l % RA = where and where l is the original length and l is the length at fracture. f Percent reduction in area : A f A A A A f 1 Ductility is the original cross - sectional area is the final cross - sectional area of the necked region.
45 Ductility Why do we need to know the ductility of materials? The degree to which a structure will deform plastically prior to fracture (design & safety measures) The degree of allowable deformation during fabrication
47 Mechanical properties of metals Depend on prior deformation, presence of impurities heat treatments
48 Engineering stress-strain behavior for Fe at different temperatures brittle ductile
49 Resilience Capacity of a material to absorb energy when it is deformed elastically and then, upon unloading, to have this energy recovered. Modulus of resilience Strain energy per unit volume required to stress a material from an unloaded state up to the point of yielding.
50 Modulus of resilience: U r U r ε y = σdε Resilience Assuming a linear elastic region : 1 = σ yε y σ y σ y = σ y = 2 E 2E
51 Resilience Materials used in spring applications High resilience High yield strength Low elastic modulus
52 Toughness Measure of the ability of a material to absorb energy up to fracture. Area under the stress-strain curve up to the point of fracture.
55 Example Which material has the highest elastic modulus? Which material has the highest ductility? Which material has the highest toughness? Which material does not exhibit any significant plastic deformation prior to fracture?
56 6.7 True Stress and Strain Decline in stress necessary to continue deformation past the maximum point M. Is the material getting weaker?
57 6.7 True Stress and Strain Short-coming in the engineering stress-strain curve Decreasing cross-sectional area not considered
58 6.7 True Stress and Strain True stress : F = Ai where A σ t True strain : ε t = l l i dl is l l the instantaneous area. l = ln
59 6.7 True Stress and Strain Assuming no volume change : Al i i = A l Prior to necking : ε t = σ = σ t ( + ε ) ( 1+ ε ) ln 1
60 6.7 True Stress and Strain Necking introduces a complex stress state
61 Strain-hardening exponent From the onset of plastic deformation to the point of necking : σ = t K ε n t where n is the strain - hardening exponent and K is the strength coefficient.
63 Example 6.4 A cylindrical specimen of steel having an original diameter of 12.8 mm is tensile tested to fracture and found to have an engineering fracture stress of 46 Mpa. If its cross-sectional diameter at fracture is 1.7 mm, determine The ductility in terms of percent reduction in area The true stress at fracture
64 Given : = = = f f mm d MPa mm d σ 3% 1 1 % Ductility : 2 2 = = = d d A A A RA f f?? % 1.7 = = = tf f RA mm d σ 3% 1 2 = = d d d f
65 Engineering stress at fracture: σ F f = 46 MPa = The applied load : tf = σ A f F A f F A = 59,2 N True stress at fracture: σ = = 66 MPa
66 Example 6.5 Compute the strain-hardening exponent n for an alloy in which a true stress of 415 Mpa produces a true strain of.1. Assume a value of 135 Mpa for K.
67 Example 6.5 Given : σ = 415 MPa t ε t =.1 K = 135MPa n =? Strain - hardening exponent : σ = K n t ε K t n =.4
68 6.1 Hardness Measure of a materials resistance to localized plastic deformation (small dent or scratch)
69 Rockwell Hardness Tests A hardness number is determined Difference in depth of penetration from an initial minor load followed by a major load Combinations of indenters & loads are used
70 Hardness Testing Techniques
71 Minor load=1 kg - used for thick specimens
72 Minor load = 3 kg - thin specimens
73 Correlations between hardness & tensile strength Both are measures of resistance to plastic deformation
74 Property Variability & Design/Safety Factors Reading assignment Section 6.11 on variability of material properties Example 6.6
75 6.12 Design/Safety Factors Design stress : σ = d N' σ c σ is the calculated stress (on the basis of c N' > 1 is the design factor. Material selection criteria : σ ys σ d the estimated max load)
76 6.12 Design/Safety Factors Safe stress or working stress : σ w σ ys σ ys = N is the yield strength of the material and N > 1 is the factor of safety. Factor of safety : 1.2 N 4.
77 Design Example 6.1 A tensile-testing apparatus is to be constructed that must withstand a maximum load of 22 kn. The design calls for two cylindrical support posts, each of which is to support half of the max load. Furthermore, plain carbon steel ground and polished shafting rounds are to be used; the minimum yield and tensile strengths of this alloy are 31 Mpa and 565 Mpa respectively. Specify a suitable diameter for these support posts.
78 Given : F max σ ys σ uts d = =? MPa = Assume a N = 5 ( 22kN ) = 565 MPa factor of = 11kN safety : Working stress : ys σ w = σ = 62 MPa N Minimum diameter required : σ w F = π d max ( 2 d / 4) = 47.5mm
MME131: Lecture 13 Mechanical properties 1 Elastic behaviour of materials A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Today s Topics Deformation of material under the action of a mechanical
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.
Stress-Strain Behavior 6.3 A specimen of aluminum having a rectangular cross section 10 mm 1.7 mm (0.4 in. 0.5 in.) is pulled in tension with 35,500 N (8000 lb f ) force, producing only elastic deformation.
6.4 A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa (15.5 10 6 psi) and an original diameter of 3.8 mm (0.15 in.) will experience only elastic deformation when a tensile
ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture five mechanics www.carttalk.com of materials Mechanics of Materials 1 Mechanics of Materials MECHANICS MATERIALS
ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture five mechanics www.carttalk.com of materials Mechanics of Materials 1 Mechanics of Materials MECHANICS MATERIALS
Materials How materials work Compression Tension Bending Torsion Elemental material atoms: A. Composition a) Nucleus: protons (+), neutrons (0) b) Electrons (-) B. Neutral charge, i.e., # electrons = #
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
Elastic Properties of Solid Materials Notes based on those by James Irvine at www.antonine-education.co.uk Key Words Density, Elastic, Plastic, Stress, Strain, Young modulus We study how materials behave
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
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
6.37 Determine the modulus of resilience for each of the following alloys: Yield Strength Material MPa psi Steel alloy 550 80,000 Brass alloy 350 50,750 Aluminum alloy 50 36,50 Titanium alloy 800 116,000
Lecture 4 Rate Independent Plasticity ANSYS Mechanical Basic Structural Nonlinearities 1 Chapter Overview The following will be covered in this Chapter: A. Background Elasticity/Plasticity B. Yield Criteria
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
Chapter 12 Static Equilibrium and Elasticity Static Equilibrium Equilibrium implies that the object moves with both constant velocity and constant angular velocity relative to an observer in an inertial
PhysicsndMathsTutor.com Which of the following correctly defines the terms stress, strain and Young modulus? 97/1/M/J/ stress strain Young modulus () x (area) (extension) x (original length) (stress) /
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:
Elasticity Homework Problems 2014 Section 1. The Strain Tensor. 1. A pure shear deformation is shown. The volume is unchanged. What is the strain tensor. 2. Given a steel bar compressed with a deformation
Bulk Properties of Solids Old Exam Questions Q1. The diagram shows how the stress varies with strain for metal specimens X and Y which are different. Both specimens were stretched until they broke. Which
CIE309 : PLASTICITY THEME IS FIRST OCCURANCE OF YIELDING THE LIMIT? M M - N N + + σ = σ = + f f BENDING EXTENSION Ir J.W. Welleman page nr 0 kn Normal conditions during the life time WHAT HAPPENS DUE TO
Lab Exercise #3: Pre-lab assignment: Yes No Goals: 1. To evaluate the equations of angular displacement, shear stress, and shear strain for a shaft undergoing torsional stress. Principles: testing of round
Name: Date: Solid Mechanics Homework nswers Please show all of your work, including which equations you are using, and circle your final answer. Be sure to include the units in your answers. 1. The yield
Torsion Testing of Structural Metals Standards ASTM E143: Shear Modulus at Room Temperature Purpose To determine the shear modulus of structural metals Equipment Tinius-Olsen Lo-Torq Torsion Machine (figure
FCP Short Course Ductile and Brittle Fracture Stephen D. Downing Mechanical Science and Engineering 001-015 University of Illinois Board of Trustees, All Rights Reserved Agenda Limit theorems Plane Stress
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
Physics Including Human Applications 280 Chapter 13 ELASTIC PROPERTIES OF MATERIALS GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions
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,
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
1 Design of a fastener based on negative Poisson's ratio foam adapted from Choi, J. B. and Lakes, R. S., "Design of a fastener based on negative Poisson's ratio foam", Cellular Polymers, 10, 205-212 (1991).
Chapter -1 From Tables A-0, A-1, A-, and A-4c, (a) UNS G1000 HR: S ut = 80 (55) MPa (kpsi), S yt = 10 (0) MPa (kpsi) Ans. (b) SAE 1050 CD: S ut = 690 (100) MPa (kpsi), S yt = 580 (84) MPa (kpsi) Ans. (c)
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
Fundamentals of Durability Page 1 Your single provider of solutions System simulation solutions 3D simulation solutions Test-based engineering solutions Engineering services - Deployment services Troubleshooting
Department of Materials and Metallurgical Engineering Bangladesh University of Engineering Technology, Dhaka MME 222 Materials Testing Sessional.50 Credits Laboratory 4 Bending Test of Materials. Objective
Question 9.1: A steel wire of length 4.7 m and cross-sectional area 3.0 10 5 m 2 stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 10 5 m 2 under a given load.
5.2 The Response of Real Materials The constitutive equation as introduced in the previous section. The means by hich the constitutive equation is determined is by carrying out experimental tests on the
Book Name: NCERT Solutions Question : A steel wire of length 4.7 m and cross-sectional area 5 3.0 0 m stretches by the same 5 amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 0 m
Powder Metallurgy Progress, Vol.3 (003), No 3 119 THE DETERMINATION OF FRACTURE STRENGTH FROM ULTIMATE TENSILE AND TRANSVERSE RUPTURE STRESSES A.S. Wronski, A.Cias Abstract It is well-recognized that the
Downloaded from Class XI Physics Ch. 9: Mechanical Properties of solids NCERT Solutions Page 242 Question 9.1: A steel wire of length 4.7 m and cross-sectional area 3.0 10 5 m 2 stretches by the same amount
Failure from static loading Topics Quiz /1/07 Failures from static loading Reading Chapter 5 Homework HW 3 due /1 HW 4 due /8 What is Failure? Failure any change in a machine part which makes it unable
Simple Stresses and Strains CHPTER OJECTIVES In this chapter, we will learn about: Various types of (a) stresses as tensile and compressive stresses, positive and negative shear stresses, complementary
UNIVERSITY PHYSICS I Professor Meade Brooks, Collin College Chapter 12: STATIC EQUILIBRIUM AND ELASTICITY Two stilt walkers in standing position. All forces acting on each stilt walker balance out; neither
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
Question 9.1: A steel wire of length 4.7 m and cross-sectional area 3.0 10 5 m 2 stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 10 5 m 2 under a given load.
1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may
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
Fatigue Problems Solution Problem 1. (a) Given the values of σ m (7 MPa) and σ a (1 MPa) we are asked t o compute σ max and σ min. From Equation 1 Or, σ m σ max + σ min 7 MPa σ max + σ min 14 MPa Furthermore,
Question. A steel wire of length 4.7 m and cross-sectional area 3.0 x 10-5 m 2 stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 x 10-5 m 2 under a given load.
University of Sulaimani School of Pharmacy Dept. of Pharmaceutics Pharmaceutical Compounding Pharmaceutical compounding I Colloidal and Surface-Chemical Aspects of Dosage Forms Dr. rer. nat. Rebaz H. Ali
3.032 Problem Set 2 Solutions Fall 2007 Due: Start of Lecture, 09.21.07 1. In the beam considered in PS1, steel beams carried the distributed weight of the rooms above. To reduce stress on the beam, it
Course: Higher Diploma in Civil Engineering Unit: Structural Analysis I Experiment: Expected Duration: 1.25 Hours Objective: 1. To determine the shear modulus of the metal specimens. 2. To determine the
Introduction to Composite Materials and Structures Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 16 Behavior of Unidirectional Composites Lecture Overview Mt Material ilaxes in unidirectional
9:00-10: 00 (last four digits) 32-141 Unified Quiz M4 May 7, 2008 M - PORTION Put the last four digits of your MIT ID # on each page of the exam. Read all questions carefully. Do all work on that question
Lecture 6 Brittle Deformation Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm WW Norton, unless noted otherwise Brittle deformation EarthStructure (2 nd
MECH 401 Mechanical Design Applications Dr. M. O Malley Master Notes Spring 008 Dr. D. M. McStravick Rice University Updates HW 1 due Thursday (1-17-08) Last time Introduction Units Reliability engineering
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
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
STRUCTURAL MECHANICS: CE203 Chapter 5 Torsion Notes are based on Mechanics of Materials: by R. C. Hibbeler, 7th Edition, Pearson Dr B. Achour & Dr Eng. K. El-kashif Civil Engineering Department, University
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:
Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical
April 7, 2017 UNIVERSITY OF RHODE ISAND Department of Electrical, Computer and Biomedical Engineering BE 207 Introduction to Biomechanics Spring 2017 Homework 7 Problem 14.3 in the textbook. In addition
Reference material Reference books: Y.C. Fung, "Foundations of Solid Mechanics", Prentice Hall R. Hill, "The mathematical theory of plasticity", Oxford University Press, Oxford. J. Lubliner, "Plasticity
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
Rutgers University Department of Physics & Astronomy 01:750:271 Honors Physics I Fall 2015 Lecture 19 Page 1 of 36 12. Equilibrium and Elasticity How do objects behave under applied external forces? Under
CIVL222 STRENGTH OF MATERIALS Chapter 6 Torsion Definition Torque is a moment that tends to twist a member about its longitudinal axis. Slender members subjected to a twisting load are said to be in torsion.
Mechanics of Materials MENG 70 Fall 00 Eam Time allowed: 90min Name. Computer No. Q.(a) Q. (b) Q. Q. Q.4 Total Problem No. (a) [5Points] An air vessel is 500 mm average diameter and 0 mm thickness, the
Shafts. Power transmission shafting Continuous mechanical power is usually transmitted along and etween rotating shafts. The transfer etween shafts is accomplished y gears, elts, chains or other similar
Pressure Vessels Stresses Under Combined Loads Yield Criteria for Ductile Materials and Fracture Criteria for Brittle Materials Pressure Vessels: In the previous lectures we have discussed elements subjected
CHAPTER 1 ENGINEERING MECHANICS I 1.1 Verification of Lame s Theorem: If three concurrent forces are in equilibrium, Lame s theorem states that their magnitudes are proportional to the sine of the angle
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
Chapter 4 Deflection and Stiffness 1 2 Chapter Outline Spring Rates Tension, Compression, and Torsion Deflection Due to Bending Beam Deflection Methods Beam Deflections by Superposition Strain Energy Castigliano
ME 383S Bryant February 17, 2006 CONTACT 1 Mechanical interaction of bodies via surfaces Surfaces must touch Forces press bodies together Size (area) of contact dependent on forces, materials, geometry,
HW#5 Due 2/13 (Friday) Lab #1 Due 2/18 (Next Wednesday) For Friday Read: pg 130 168 (rest of Chpt. 4) 1 Poisson s Ratio, μ (pg. 115) Ratio of the strain in the direction perpendicular to the applied force