Chapter 7. Highlights:


 Crystal Hopkins
 1 years ago
 Views:
Transcription
1 Chapter 7 Highlights: 1. Understand the basic concepts of engineering stress and strain, yield strength, tensile strength, Young's(elastic) modulus, ductility, toughness, resilience, true stress and true strain, strain exponent, and know the difference between elastic and plastic deformation.. Understand what a stressstrain curve is and what information it contains about materials properties. Be able to identify and/or calculate all the properties in #1 from a stressstrain curve, both in the elastic and plastic (before necking) regions. 3. Understand how the mechanical behavior of ceramics differs from that of metals. Be able to numerically manipulate the flexural strength and the effect of porosity on mechanical strength. 4. Understand how the mechanical behavior of polymers differs from that of metals. Understand and be able to numerically manipulate the viscoelastic modulus. Notes: Show Figures 7.1 to 7.4.
2 Define engineering stress and strain (tension and compression): Engineering stress F A Engineering strain ε l i  l l l l
3 For shear stress, Shear stress τ F A Shear strain γ tanθ ( θ is strain angle) As shown in Figure 7.4, and described in Equations 7.4a and 7.4b, an applied axial force can be geometrically decomposed into tensile and shear components. Stressstrain test: slowly increase stress and measure strain until the material fractures (show Figures 7., 7.3, 7.5, 7.11). These tests are usually performed in tension. In the linear portion of the curve, Hooke's law is obeyed, Eε. E is called the modulus of elasticity, or Young's modulus, and is a property of the material. The units of E are psi or MPa, remembering that PaN/m. Typical values for E are given in Table 7.1 for metals, ceramics and polymers.
4
5
6 A similar relationship can be written for compressive, shear, and torsional loads. For shear stress, τ G γ G is the shear modulus and is a property of the material. On an atomic scale, lengthening (compressing) a specimen during tensile (compressive) loading results in lengthening (shortening) of atomic bonds. Young's modulus is a measure of the resistance to separation of adjacent atoms. Youngs modulus df dr r Show Figure 7.7 and Table 7.1. High E W and Ni, low E Mg, Al, Au, Ag. Show Table 3.7. E can be different in different directions, and this reflects the different atomic densities in different planes and directions. When there is not a significant linear portion of an engineering stressstrain diagram, the secant and tangent modulus are sometimes employed. Show Figure 7.6. The tangent modulus is the slope of the stressstrain curve at one particular point, while the secant modulus is the slope of a secant drawn from the origin to a particular point on the stressstrain curve. By common sense, tensile strain in the z direction should yield compressive strain in the x and y directions. Show Figure 7.9. ε x ε y. Define Poisson's ratio υ ε ε y ν  x  ε z ε z υ for most metals, and is again a property of the material. It can be shown that G.4E for most metals. E G(1+ ν ) The following analogies illustrate the differences between the linear and nonlinear portions of a stressstrain curve:
7 Hooke's law Bonds stretched Elastic deformation No permanent damage Nonlinear ε Bonds broken Plastic deformation Permanent damage Example Problem A cylindrical steel bar 1 mm in diameter is to be elastically deformed. Using the data in Table 7.1, determine the force needed to produce a reduction of 3x13 mm in the diameter. From Table 7.1, E 7 GPa and ν.3. We are given information about the desired radial strain: ε r 3 3x1 mm 4 3x1 1 mm This can be related to the axial strain by the definition of Poisson s ratio, eq. 7.8: ε z 4 3x1 3 ε r 1x1 υ.3 Now use Hooke s law, eq. 7.5, to determine the stress that must be applied: 3 ( 1x1 ).7GPa MPa Eε (7GPa) 7 From the definition of stress, eq. 7.3, F 6 π A 7 x1 Pa (.1m) 1.63x1 4 4 N Tensile Properties of Materials: Show a stressstrain curve, Figure 7.1. We will now digress for a while but eventually return to this figure and try to understand it and extract useful information. The linear portion of this curve undergoes elastic deformation (linear ε), which disappears after the stress is removed. In other words, it is reversible. When the stress exceeds the linear portion of the stressstrain curve, permanent (irreversible) plastic deformation occurs. Plastic deformation Plastic deformation usually occurs near ε.5 for metals. When Hooke's law fails, the material undergoes plastic deformation. Show Figure 7.1. Either 7.1a or 7.1b can occur, usually 7.1a. Since plastic deformation involves atomiclevel breaking of bonds and reforming of new ones, it is irreversible. We want to be able to define unambiguously when a material yields, when it is permanently damaged. Show Figure 7.1a, focusing on point P. This is the proportional limit, where the stressstrain curve deviates from linearity. But this is ambiguous.
8 Usually the stressstrain curve deviates from linearity quite gradually, so we need to define an arbitrary but universal definition of the elastic region. Starting from point, ε., draw a parallel line to the linear portion of the stressstrain curve. The stress at which this line intersects with the stressstrain curve is called the yield strength, y. This is one measure of a material's strength, its ability to resist plastic (permanent) deformation. In the case of Figure 7.1b the yield strength is the lower yield point. Show Figure The tensile strength is defined as the maximum stress prior to fracture. After M, "necking" occurs, the crosssectional area shrinks abruptly. The ductility is defined as the % plastic deformation at fracture. Ductility A low ductility material is brittle. Show Figure l f  l l x1 Resilience is the capability of a material to absorb energy when it is deformed elastically. The modulus of resilience U r is ε y U r dε where ε y is the strain at fracture. Show Figure For linear stressstrain curve, U r 1 y ε y y E Toughness is the ability of a material to absorb energy up to fracture. It is defined by the same integral as U r above, but the upper limit of integration is now fracture instead of yielding. Show Table 7.3. High ductility corresponds somewhat with low strength. Example Problem For a given set of data, which I will not reproduce here, you should be able to: a) Create a plot of engineering stress versus engineering strain, b) determine the elastic modulus, c) determine the yield strength, d) determine the tensile strength, e) determine the ductility, and f) determine the modulus of resilience. a) The plots below were obtained by taking: F( N ) F( N ) ( Pa ) π π 3 D ( 1.8x1 m) F( N )
9 ε l i i l l l e8 Stress (Pa).e8 1.e8.e Strain b) The elastic modulus is taken as the slope of the linear portion of the curve. Looking at the nd plot, which focuses on the linear portion of the curve, take the average slope only from the 1 st four points. This yields: x1 Pa 1.173x1 Pa 1.795x1 Pa E ε x1 Pa.398 Taking the average of these 4 values yields E 5.88x1 1 Pa. c) On the nd plot, draw a line from ε. with a slope of E 5.88x1 1 Pa. The intersection of this line with the ε curve gives y.8x1 8 Pa. d) From the 1 st plot, the maximum in is the tensile strength 3.7x1 8 Pa. e) From the 1 st plot, the strain at failure is about.145, subtract out the elastic strain (.5) to get the plastic strain of.14. Thus the ductility is 14%.
10 f) Determine the modulus of resilience from equation 7.14: U r y E Elastic recovery after plastic deformation: Show figure 7.17 and discuss. 8 (.8x1 Pa) (5.88x1 1 Pa) 6.67x1 5 Pa True Stress and Strain: True stress( T ) and true strain(ε T ) are defined similarly to engineering stress and strain. T F A i Force Instantaneous area ε l ln l T i l i is the instantaneous length If volume is conserved, then A i l i A l and T (1 + ε ) ε T ln (1 + ε ) The above two equations are only true until necking. After necking, true stress and true strain can only be
11 determined by actual measurements. Show Figure 7.16, where the true stress is corrected to account for nontensile components. For some metals, from the onset of plastic deformation to the point at which necking begins, the true stress is approximately: n T K ε T K and n, the strain hardening exponent, vary across different metals and alloys. Example Problem You are given that just prior to necking, (, ε) (35 MPa,.194) and (5 MPa,.96). What value of will produce ε.5. In order to use equation (7.19), we need to convert from engineering stress and strain to true stress and strain. From equations (7.18) 1 T (1+ ε ) MPa T (1+ ε ) 34 MPa 1 ε T ln(1+ ε ).1773 ε T ln(1+ ε ) ε T ln(1+ ε ).31 You need to use equation (7.19) to setup two equations with two unknowns. Taking the ln of both sides: ln(34 ln ln K + nlnε T MPa ) ln K + n ln(.593) T
12 ln(8.59 MPa ) ln K + n ln(.1773) Subtracting the nd from the 1 st ln n ln Substituting back into above: n.378 ln (8.59 MPa ) ln K + (.378) ln(.1773) ln K 6.91 or K 54 MPa Now find where ε T is.31: n T K ε T.378 T (54 MPa)(.31) MPa Convert back to engineering stress: T 1 + ε 36.3 MPa MPa Mechanical behavior: Ceramics Ceramic materials are less mechanically useful than metals because they are generally quite brittle. Their mechanical strength is not normally assessed by tensile stressstrain measurements. It is extremely difficult to prepare the proper sample geometry, the samples generally crack when gripped, and their strain at failure is too small to accurately measure. Show Figure For these reasons, the strength of ceramic materials is normally assessed by a traverse bending test. Show Figure The failure point is described by the flexural strength ( fs ), also known as the modulus of rupture( mr ). For rectangular cross sections, and for circular crosssections: fs 3 F f L b d F f L π R fs 3
13
14 Example Problem A 3point bending test is performed on Al O 3 with a circular crosssection of radius 3.5 mm. The specimen fractures at a load of 95 N when the support points are 5 mm apart. Consider a square sample of the same material with a square crosssection of 1mm on each edge. If the support points are 4 mm apart, at what load will the sample fracture? First, determine the flexural strength of this material from equation (7.b): fs F L 3 f (95 N )(5x1 m) 3.53x πr π (3.5x1 m) 8 Pa Now determine the failure force from equation (7.a): F bd 3 3 fs (1x1 m) 3 3L 3(4x1 (3.53x1 m) 8 Pa) 1,17 N Ceramic solids are often fabricated by compaction or foaming of ceramic particles. In this case, the ceramic solid may have considerable porosity, which degrades its strength. The modulus of elasticity (E) and the flexural strength ( fs ) depend on the volume fraction porosity (P) according to: ( 1 1.9P. P ) E E + 9 fs exp( np) where and n are experimental constants. Example Problem Using the data in Table 7., a) Determine the flexural strength of nonporous MgO assuming n 3.75 b) Calculate the volume fraction porosity when fs 6 MPa. a) Taking the ln of both sides of equation 7.: ln fs ln np From Table 7., fs 15MPa with 5% porosity, so ln(15 MPa ) ln (3.75)(.5) ln MPa
15 b) Rearranging the above equation: P ln ln fs n ln(6 MPa) P or 19vol.% Note that 5% porosity reduces fs by 17%, and 19% porosity reduces fs by 51%.
16 Mechanical behavior: Polymers Typical stressstrain diagrams for polymers are shown in Figure 7.. The behavior ranges from strong and brittle (curve A) to rubbery (curve C). The mechanical behavior of polymers is far more temperaturedependent than that of metals and polymers. Show Figure 7.4. Polymethyl methylacrylate behaves like a brittle metal at low temperature, but like an elastomer (rubber band) at low temperature. At intermediate temperatures, this polymer behaves like a viscoelastic material, which combines the behavior of a viscous and an elastic material. The most famous viscoelastic material is silly putty, which is intermediate between a liquid and a solid. Show Figure 7.6, which shows the strain behavior with time following instantaneous application of a load. Alternatively, one can study the mechanical behavior of a viscoelastic material at constant strain, in which case the stress that is initially applied will gradually decrease. Show Figure 7.7, which shows the stress as a function of time and temperature. How can we compare different materials?? One way is to choose an arbitrary time (1 sec is common) and define the relaxation modulus, E r (t), at that time: E r ( t) ( t) ε Remember that the value of E r (1) depends on the temperature. Show Figure 7.7 again.
17 Safety factor Engineering with a safety factor accounts for material variability, nonideal conditions, human error, etc. and is common engineering practice. The working stress ( w ) is reduced from the yield stress ( y ) by the safety factor (N) according to: The safety factor N is commonly taken as. y w N
Introduction to Engineering Materials ENGR2000. Dr. Coates
Introduction to Engineering Materials ENGR2 Chapter 6: Mechanical Properties of Metals Dr. Coates 6.2 Concepts of Stress and Strain tension compression shear torsion Tension Tests The specimen is deformed
More informationOutline. TensileTest Specimen and Machine. StressStrain Curve. Review of Mechanical Properties. Mechanical Behaviour
TensileTest Specimen and Machine Review of Mechanical Properties Outline Tensile test True stress  true strain (flow curve) mechanical properties:  Resilience  Ductility  Toughness  Hardness A standard
More informationStressStrain Behavior
StressStrain 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.
More informationChapter 6: Mechanical Properties of Metals. Dr. Feras Fraige
Chapter 6: Mechanical Properties of Metals Dr. Feras Fraige Stress and Strain Tension Compression Shear Torsion Elastic deformation Plastic Deformation Yield Strength Tensile Strength Ductility Toughness
More information6.4 A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa ( psi) and
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
More informationMECE 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 informationMechanical properties 1 Elastic behaviour of materials
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
More informationEMA 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 information4.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 stressstrain diagram
More informationCHAPTER 3 THE EFFECTS OF FORCES ON MATERIALS
CHAPTER THE EFFECTS OF FORCES ON MATERIALS EXERCISE 1, Page 50 1. A rectangular bar having a crosssectional area of 80 mm has a tensile force of 0 kn applied to it. Determine the stress in the bar. Stress
More informationUNIT I SIMPLE STRESSES AND STRAINS
Subject with Code : SM1(15A01303) Year & Sem: IIB.Tech & ISem SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) UNIT I SIMPLE STRESSES
More informationMECHANICS 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 informationHow materials work. Compression Tension Bending Torsion
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 = #
More informationLaboratory 4 Bending Test of Materials
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
More informationMATERIALS. Why do things break? Why are some materials stronger than others? Why is steel tough? Why is glass brittle?
MATERIALS Why do things break? Why are some materials stronger than others? Why is steel tough? Why is glass brittle? What is toughness? strength? brittleness? Elemental material atoms: A. Composition
More information2. 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 informationElastic Properties of Solid Materials. Notes based on those by James Irvine at
Elastic Properties of Solid Materials Notes based on those by James Irvine at www.antonineeducation.co.uk Key Words Density, Elastic, Plastic, Stress, Strain, Young modulus We study how materials behave
More informationMechanics 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 informationGeology 229 Engineering Geology. Lecture 5. Engineering Properties of Rocks (West, Ch. 6)
Geology 229 Engineering Geology Lecture 5 Engineering Properties of Rocks (West, Ch. 6) Common mechanic properties: Density; Elastic properties:  elastic modulii Outline of this Lecture 1. Uniaxial rock
More informationLab Exercise #3: Torsion
Lab Exercise #3: Prelab 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
More informationLecture 7, Foams, 3.054
Lecture 7, Foams, 3.054 Opencell foams StressStrain curve: deformation and failure mechanisms Compression  3 regimes  linear elastic  bending  stress plateau  cell collapse by buckling yielding
More informationMaterials and Structures. Indian Institute of Technology Kanpur
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
More informationfive Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture
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
More informationfive Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture
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
More informationStructural Metals Lab 1.2. Torsion Testing of Structural Metals. Standards ASTM E143: Shear Modulus at Room Temperature
Torsion Testing of Structural Metals Standards ASTM E143: Shear Modulus at Room Temperature Purpose To determine the shear modulus of structural metals Equipment TiniusOlsen LoTorq Torsion Machine (figure
More informationChapter 26 Elastic Properties of Materials
Chapter 26 Elastic Properties of Materials 26.1 Introduction... 1 26.2 Stress and Strain in Tension and Compression... 2 26.3 Shear Stress and Strain... 4 Example 26.1: Stretched wire... 5 26.4 Elastic
More informationANSYS Mechanical Basic Structural Nonlinearities
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
More informationA 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 informationDesign of a fastener based on negative Poisson's ratio foam adapted from
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, 205212 (1991).
More information6.37 Determine the modulus of resilience for each of the following alloys:
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
More informationFundamentals of Durability. Unrestricted Siemens AG 2013 All rights reserved. Siemens PLM Software
Fundamentals of Durability Page 1 Your single provider of solutions System simulation solutions 3D simulation solutions Testbased engineering solutions Engineering services  Deployment services Troubleshooting
More informationExperiment 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 informationX has a higher value of the Young modulus. Y has a lower maximum tensile stress than X
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
More informationCHAPTER 6: ULTIMATE LIMIT STATE
CHAPTER 6: ULTIMATE LIMIT STATE 6.1 GENERAL It shall be in accordance with JSCE Standard Specification (Design), 6.1. The collapse mechanism in statically indeterminate structures shall not be considered.
More informationTuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE
1 Chapter 3 Load and Stress Analysis 2 Chapter Outline Equilibrium & FreeBody Diagrams Shear Force and Bending Moments in Beams Singularity Functions Stress Cartesian Stress Components Mohr s Circle for
More informationTHE DETERMINATION OF FRACTURE STRENGTH FROM ULTIMATE TENSILE AND TRANSVERSE RUPTURE STRESSES
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 wellrecognized that the
More information1. Demonstrate that the minimum cationtoanion radius ratio for a coordination number of 8 is
1. Demonstrate that the minimum cationtoanion radius ratio for a coordination number of 8 is 0.732. This problem asks us to show that the minimum cationtoanion radius ratio for a coordination number
More informationUNIT 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 information3. 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 informationChapter 12. Static Equilibrium and Elasticity
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
More informationSolid Mechanics Homework Answers
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
More informationChapter 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 informationBME 207 Introduction to Biomechanics Spring 2017
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
More informationTINIUS OLSEN Testing Machine Co., Inc.
Interpretation of StressStrain Curves and Mechanical Properties of Materials Tinius Olsen has prepared this general introduction to the interpretation of stressstrain curves for the benefit of those
More informationPharmaceutical compounding I Colloidal and SurfaceChemical Aspects of Dosage Forms Dr. rer. nat. Rebaz H. Ali
University of Sulaimani School of Pharmacy Dept. of Pharmaceutics Pharmaceutical Compounding Pharmaceutical compounding I Colloidal and SurfaceChemical Aspects of Dosage Forms Dr. rer. nat. Rebaz H. Ali
More informationFCP Short Course. Ductile and Brittle Fracture. Stephen D. Downing. Mechanical Science and Engineering
FCP Short Course Ductile and Brittle Fracture Stephen D. Downing Mechanical Science and Engineering 001015 University of Illinois Board of Trustees, All Rights Reserved Agenda Limit theorems Plane Stress
More informationMSE 383, Unit 33. Joshua U. Otaigbe Iowa State University Materials Science & Engineering Dept.
Dynamic Mechanical Behavior MSE 383, Unit 33 Joshua U. Otaigbe Iowa State University Materials Science & Engineering Dept. Scope Why DMA & TTS? DMA Dynamic Mechanical Behavior (DMA) Superposition Principles
More informationFlexure: 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 (kipin.) 3000 2000 1000 0 0 (basic) (A s 0.5A s ) 0.0005 0.001 0.0015
More informationExternal Work. When a force F undergoes a displacement dx in the same direction i as the force, the work done is
Structure Analysis I Chapter 9 Deflection Energy Method External Work Energy Method When a force F undergoes a displacement dx in the same direction i as the force, the work done is du e = F dx If the
More information1 Stress and Strain. Introduction
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
More informationElasticPlastic Fracture Mechanics. Professor S. Suresh
ElasticPlastic Fracture Mechanics Professor S. Suresh Elastic Plastic Fracture Previously, we have analyzed problems in which the plastic zone was small compared to the specimen dimensions (small scale
More informationMechanics of Materials MENG 270 Fall 2003 Exam 3 Time allowed: 90min. Q.1(a) Q.1 (b) Q.2 Q.3 Q.4 Total
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
More informationChapter 8. Shear and Diagonal Tension
Chapter 8. and Diagonal Tension 8.1. READING ASSIGNMENT Text Chapter 4; Sections 4.14.5 Code Chapter 11; Sections 11.1.1, 11.3, 11.5.1, 11.5.3, 11.5.4, 11.5.5.1, and 11.5.6 8.2. INTRODUCTION OF SHEAR
More informationThe Frictional Regime
The Frictional Regime Processes in Structural Geology & Tectonics Ben van der Pluijm WW Norton+Authors, unless noted otherwise 1/25/2016 10:08 AM We Discuss The Frictional Regime Processes of Brittle Deformation
More information22 Which of the following correctly defines the terms stress, strain and Young modulus? stress strain Young modulus
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) /
More informationSymmetric Bending of Beams
Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications
More informationIntroduction to Fracture
Introduction to Fracture Introduction Design of a component Yielding Strength Deflection Stiffness Buckling critical load Fatigue Stress and Strain based Vibration Resonance Impact High strain rates Fracture
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139
MASSACHUSES INSIUE OF ECHNOLOGY DEPARMEN OF MAERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSES 02139 3.22 MECHANICAL PROPERIES OF MAERIALS PROBLEM SE 5 SOLUIONS 1. (Hertzber 6.2) If it takes 300 seconds
More informationStrainBased Design Model for FRPConfined Concrete Columns
SP230 57 StrainBased Design Model for FRPConfined Concrete Columns by N. Saenz and C.P. Pantelides Synopsis: A constitutive strainbased confinement model is developed herein for circular concrete columns
More informationCONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS
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
More information5.2 The Response of Real Materials
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
More informationMECH 401 Mechanical Design Applications
MECH 401 Mechanical Design Applications Dr. M. O Malley Master Notes Spring 008 Dr. D. M. McStravick Rice University Updates HW 1 due Thursday (11708) Last time Introduction Units Reliability engineering
More informationSeismic 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 informationLecture 2: Deformation in the crust and the mantle. Read KK&V chapter 2.10
Lecture 2: Deformation in the crust and the mantle Read KK&V chapter 2.10 Tectonic plates What are the structure and composi1on of tectonic plates? Crust, mantle, and lithosphere Crust relatively light
More informationBending and Shear in Beams
Bending and Shear in Beams Lecture 3 5 th October 017 Contents Lecture 3 What reinforcement is needed to resist M Ed? Bending/ Flexure Section analysis, singly and doubly reinforced Tension reinforcement,
More informationOutline. Organization. Stresses in Beams
Stresses in Beams B the end of this lesson, ou should be able to: Calculate the maimum stress in a beam undergoing a bending moment 1 Outline Curvature Normal Strain Normal Stress Neutral is Moment of
More informationBTECH MECHANICAL PRINCIPLES AND APPLICATIONS. Level 3 Unit 5
BTECH MECHANICAL PRINCIPLES AND APPLICATIONS Level 3 Unit 5 FORCES AS VECTORS Vectors have a magnitude (amount) and a direction. Forces are vectors FORCES AS VECTORS (2 FORCES) Forces F1 and F2 are in
More informationBrittle Deformation. Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm
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
More informationFracture Behavior. Section
Section 6 Fracture Behavior In January 1943 the oneday old Liberty Ship, SS Schenectady, had just completed successful sea trials and returned to harbor in calm cool weather when... "Without warning and
More informationSpherical Pressure Vessels
Spherical Pressure Vessels Pressure vessels are closed structures containing liquids or gases under essure. Examples include tanks, pipes, essurized cabins, etc. Shell structures : When essure vessels
More informationA Critical Planeenergy Model for Multiaxial Fatigue Life Prediction. of Homogeneous and Heterogeneous Materials. Haoyang Wei
A Critical Planeenergy Model for Multiaxial Fatigue Life Prediction of Homogeneous and Heterogeneous Materials by Haoyang Wei A Thesis Presented in Partial Fulfillment of the Requirements for the Degree
More informationCHEMC2410: Materials Science from Microstructures to Properties Composites: basic principles
CHEMC2410: Materials Science from Microstructures to Properties Composites: basic principles Mark Hughes 14 th March 2017 Today s learning outcomes To understand the role of reinforcement, matrix and
More informationChapter 13 ELASTIC PROPERTIES OF MATERIALS
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
More information1. A pure shear deformation is shown. The volume is unchanged. What is the strain tensor.
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
More informationRheology and the Lithosphere
Rheology and the Lithosphere Processes in Structural Geology & Tectonics Ben van der Pluijm WW Norton+Authors, unless noted otherwise 3/8/2017 16:51 We Discuss Rheology and the Lithosphere What is rheology?
More informationProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
56 Module 4: Lecture 7 on Stressstrain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin pq space; MohrCoulomb failure
More information3.032 Problem Set 2 Solutions Fall 2007 Due: Start of Lecture,
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
More informationReference material Reference books: Y.C. Fung, "Foundations of Solid Mechanics", Prentice Hall R. Hill, "The mathematical theory of plasticity",
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
More informationFatigue Problems Solution
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,
More informationTherefore, for all members designed according to ACI 318 Code, f s =f y at failure, and the nominal strength is given by:
5.11. Underreinforced Beams (Read Sect. 3.4b oour text) We want the reinforced concrete beams to fail in tension because is not a sudden failure. Therefore, following Figure 5.3, you have to make sure
More informationDEFORMATION CAPACITY OF OLDER RC SHEAR WALLS: EXPERIMENTAL ASSESSMENT AND COMPARISON WITH EUROCODE 8  PART 3 PROVISIONS
DEFORMATION CAPACITY OF OLDER RC SHEAR WALLS: EXPERIMENTAL ASSESSMENT AND COMPARISON WITH EUROCODE 8  PART 3 PROVISIONS Konstantinos CHRISTIDIS 1, Emmanouil VOUGIOUKAS 2 and Konstantinos TREZOS 3 ABSTRACT
More informationTesting and Analysis
Testing and Analysis Testing Elastomers for Hyperelastic Material Models in Finite Element Analysis 2.6 2.4 2.2 2.0 1.8 1.6 1.4 Biaxial Extension Simple Tension Figure 1, A Typical Final Data Set for Input
More informationRheology. What is rheology? From the root work rheo Current: flow. Greek: rhein, to flow (river) Like rheostat flow of current
Rheology What is rheology? From the root work rheo Current: flow Greek: rhein, to flow (river) Like rheostat flow of current Rheology What physical properties control deformation?  Rock type  Temperature
More informationMECHANICS 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 informationFailure from static loading
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
More informationQuestion Figure shows the strainstress curve for a given material. What are (a) Young s modulus and (b) approximate yield strength for this material?
Question. A steel wire of length 4.7 m and crosssectional area 3.0 x 105 m 2 stretches by the same amount as a copper wire of length 3.5 m and crosssectional area of 4.0 x 105 m 2 under a given load.
More informationEFFECT OF SHEAR REINFORCEMENT ON FAILURE MODE OF RC BRIDGE PIERS SUBJECTED TO STRONG EARTHQUAKE MOTIONS
EFFECT OF SHEAR REINFORCEMENT ON FAILURE MODE OF RC BRIDGE PIERS SUBJECTED TO STRONG EARTHQUAKE MOTIONS Atsuhiko MACHIDA And Khairy H ABDELKAREEM SUMMARY Nonlinear D FEM was utilized to carry out inelastic
More informationMaterials: 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 informationNE 125 L. Title Page
NE 125 L Title Page Name: Rajesh Swaminathan ID Number: 20194189 Partners Names: Clayton Szata 20193839 Sarvesh Varma 20203153 Experiment Number: 1 Experiment: Date Experiment was Started: Date Experiment
More informationContinuum Mechanics. Continuum Mechanics and Constitutive Equations
Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform
More informationQuestion 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 informationTorsion of Solid Sections. Introduction
Introduction Torque is a common load in aircraft structures In torsion of circular sections, shear strain is a linear function of radial distance Plane sections are assumed to rotate as rigid bodies These
More informationTask 1  Material Testing of Bionax Pipe and Joints
Task 1  Material Testing of Bionax Pipe and Joints Submitted to: Jeff Phillips Western Regional Engineer IPEX Management, Inc. 20460 Duncan Way Langley, BC, Canada V3A 7A3 Ph: 6045348631 Fax: 6045347616
More informationDetermine 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 informationCIVL222 STRENGTH OF MATERIALS. Chapter 6. Torsion
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.
More informationExperiment: Torsion Test Expected Duration: 1.25 Hours
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
More informationStrength 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 informationSERVICEABILITY OF BEAMS AND ONEWAY SLABS
CHAPTER REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach  Fifth Edition Fifth Edition SERVICEABILITY OF BEAMS AND ONEWAY SLABS A. J. Clark School of Engineering Department of Civil
More informationMarch 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE
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
More informationUnified Quiz M4 May 7, 2008 M  PORTION
9:0010: 00 (last four digits) 32141 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
More information