# Stress, Strain, Mohr s Circle

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected in favour of a smooth idealization. In this idealization, each material particle is subjected to stresses that result in strains, or conversely, they are subjected to strains that result in stresses. Stresses and strains are related by material law equations. Stresses are related to external forces by equilibrium equations. Strains are related to global deformations by kinematic equations. The determination of stresses (denoted σ or τ) and strains (denoted ε or γ) is the key problem in the theory of elasticity. First consider a three-dimensional coordinate system x, y, z with a material particle. The stresses and strains that act in the coordinate directions ( coordinate stresses ) are: σ xy σ xz ε xx ε xy ε xz σ = σ ij = σ yx σ ε = ε yz ij = ε yx ε yy ε yz (1) ε σ zy ε zy ε zz zz where the first index is the direction of the normal vector to the plane where the stress acts and the second index is the direction of the stress component. Axial stresses are positive in tension. In the context of coordinate stresses, shear stresses are positive when they act in the positive axis direction on a surface that has a positive axis direction as the surface normal. Components with equal indices are axial stresses/strains and components with different indices are shear stresses/strains. Sometimes the stress and strain tensors are written in Voight notation: σ = σ zz τ xy τ yz τ zx ε = where the shear stresses τ ij σ ij and the engineering shear strains γ ij =ε ij. ε xx ε yy ε zz γ xy γ yz γ zx () D Transformation of Stresses, Principal Stresses Consider the plane stress state, in which only,, and σ xy =σ yx τ xy act. Suppose that these coordinate stresses are known. The objective in this section is to determine the stress state in rotated configurations, e.g., to determine the principal stresses. Let θ denote the angle (positive counter-clockwise) between the original coordinate system and the rotated one. The rotated plane is shown in Figure 1, where the stresses on that plane are called σ and τ. Stress, Strain, Mohr s Circle Updated January 0, 014 Page 1

2 y xx xy l y l xy l x x yy Figure 1: Stresses on an inclined plane. By noting that cos(θ)=l y /l and sin(θ)=l x /l, equilibrium in the direction of σ yields σ = cos (θ) sin (θ) + τ xy cos(θ) sin(θ) (3) The trigonometric identities cos (θ)=(1+cos(θ))/, sin (θ)=(1+sin(θ))/, and sin(θ)=sin(θ)cos(θ) lead to the modified expression σ = + Similarly, equilibrium in the direction of τ yields: τ = cos(θ) sin(θ) (4) sin(θ) τ xy cos(θ) (5) Eqs. (4) and (5) establish the basis for transformation (rotation) of stresses in twodimensional stress states. Extreme values of σ and τ and the corresponding angle θ are determined by setting the derivative of Eqs. (4) and (5) with respect to θ equal to zero. However, the graphical approach known as Mohr s circle is an appealing alternative to analytical derivations. Mohr s Circle Eqs. (4) and (5) represent a circle in the σ τ plane. To derive the expression for the circle, move the first term in the right-hand side of Eq. (4) to the left-hand side. Then square Eqs. (4) and (5) and add them. Upon using the trigonometric identity sin (θ)+cos (θ)=1 and cancelling terms, one obtains Stress, Strain, Mohr s Circle Updated January 0, 014 Page

3 σ + τ = (6) This equation forms a circle in the σ τ plane, shifted along the σ-axis, as shown in Figure. All points on Mohr s circle represents stress states at planes of different angle θ. In fact, drawing the circle immediately reveals the maximum and minimum axial stresses at the locations with zero shear stress. Conversely, it is observed in Figure that the stress states with maximum shear stress are usually not associated with zero axial stresses. Although there are several methods for working with Mohr s circle, two questions appear prominently in its practical use: What is the orientation, θ, of the plane for any of the stress states on the circle? What is the direction of the shear stress at any point on the circle? These questions are answered by the following procedure: 1. Draw Mohr s circle with the radius and centre-shift as shown in Figure.. On the circle, identify the location of the stress state (, τ xy ) that acts on the right-hand side of the particle, i.e., on the plane that has the x-axis as the surface normal. This point is called the origin of planes. By definition, this stress state is associated with θ=0. Be careful with the sign of the shear stress; clockwise shear stress is positive, which contradicts the positive direction of the coordinate stress on this surface. 3. Draw a horizontal line from the origin of planes. The point where the line intersects with the circle is called the pole point. When = the origin of planes coincides with the pole point. 4. Draw a straight line from the pole point to any point of Mohr s circle and study that stress state: a. The abscissa axis provides the axial stress (tension is positive). b. The ordinate axis provides the shear stress (clockwise is positive). c. The angle between the horizontal axis and that straight line equals θ (see the identification of θ in Figure ). Stress, Strain, Mohr s Circle Updated January 0, 014 Page 3

4 yy xy "#\$%&'()\$*'+%",("-( +""*)\$,./'(#/*'##'#0( yy xy xx xx xy yy xy yy "#\$%&'()\$*'+%",("-( #1'.*(#/*'##(*'.)\$,#0( :( ( yy," xy ) ";'( <"\$,/( ( xx," xy ) 5*\$\$,("-( <;.,'#( xx xy xy xx xx + yy # \$ % xx " yy & ' ( + ) xy Figure : Mohr s circle. Notice in particular that the stress state (, τ xy ) is confirmed by drawing a vertical line from the pole point. This is the stress state that acts on the plane that has the y-axis as the surface normal. By definition, this stress state is associated with θ=90 o but on the circle it will always be 180 o away from the origin of planes. Also, notice that the direction of the principal axes are identified by drawing a straight line from the pole point to the locations on the circle with zero shear stress, as shown in Figure. For practical purposes it is useful to synthesize the earlier results for the determination of extreme stress values. Inspection of Mohr s circle for the D case shows that the maximum axial stress is Stress, Strain, Mohr s Circle Updated January 0, 014 Page 4

5 σ 1 = max 0, + and that the minimum axial stress is σ 3 = min 0, In this document the symbols σ 1 and σ 3 are reserved for the maximum and minimum stress, respectively. This explains the inclusion of zero as a possibility in Eqs. (7) and (8). This notation requires particular attention in D stress situations, where the out-of-plane stress is zero and, thus, often equals σ 3. For the D case, Mohr s circle also shows that the maximum shear stress equals the radius of the circle: (7) (8) τ max = 1 ( σ 1 σ ) = (9) 3D Transformation of Stresses, Principal Stresses Consider the stress traction t={t x t y t z } T that acts on an infinitesimal surface area with surface normal n={n x n y n z } T shown in Figure 3, where t i is the force in the i-direction. Let da denote the area of the inclined surface on which the traction acts and let da x denote the area of the side that has the negative x-axis as normal vector, and so forth. Equilibrium in the x-direction yields: t x da = da x + σ yx da y + da z (10) To refine the expression, consider the relationship between the areas da and da i. Figure 3 shows that da=0.5hl and da z =0.5h z l. Consequently, da z da = h z h = cos(θ z ) = cos(z,n) = n z (11) Stress, Strain, Mohr s Circle Updated January 0, 014 Page 5

6 z n h y h z z l Dividing Eq. (10) by da yields Figure 3: Surface on which the stress traction acts. t x = n x + σ yx n y + n z (1) Repeating this exercise for all three axis-directions, and noting that σ ij =σ ji because of equilibrium considered later in this document, yields the equilibrium equations due to Cauchy that relate a surface traction to the coordinate stresses: t = σn = t x t y t z σ xy σ xz = σ yx σ yz σ zy σ zz n x n y n z = t i = σ ij n j (13) It is noted that, because n is a unit vector, the axial stress on a plane with normal vector n is the dot product between n and the stress traction: σ n = t T n (14) Subsequently, the Pythagorean theorem determines the largest shear stress on the plane: τ n = t + σ n (15) For every material particle it is possible to find three orthogonal planes with zero shear stresses. The axial stresses on these planes are called principal stresses. They represent the extreme values of the stresses, where σ 1 is the largest and σ 3 is the smallest principal stress (in absolute value). One approach to determine the principal stresses is to employ Eq. (13). For a plane with principal stresses the traction vector is parallel with the normal x Stress, Strain, Mohr s Circle Updated January 0, 014 Page 6

7 vector of that plane; i.e., there are no shear stresses on that plane and the traction is the scaled normal vector: t = σn = λ n (16) This is an eigenvalue problem in the unknown scalar λ, i.e., ( σ λ I)n = 0. Solutions are obtained by setting the determinant of the coefficient matrix equal to zero: where the stress variants are defined as λ 3 I 1 λ + I λ I 3 = 0 (17) I 1 = + σ zz (18) I = σ yz σ zy σ zz + σ xz σ zz + σ xy σ yx (19) I 3 = σ (0) where vertical bars indicate the determinant operation. Upon solving for the eigenvalues, λ, i.e., σ 1, σ, and σ 3, the eigenvectors yield the principal directions. The quantities I 1, I, and I 3 are called stress invariants because they retain the same value regardless of the orientation of the coordinate system. These stress invariants are somewhat different from the stress invariants J 1, J, and J 3 that are mentioned in the section below on stress-based failure criteria and extensively used in the theory of plasticity, which is described in another document on material nonlinearity. Note that the maximum shear stress in 3D is τ max = 1 ( σ σ 1 3) (1) Stress, Strain, Mohr s Circle Updated January 0, 014 Page 7

### 1 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

### Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

50 Module 4: Lecture 1 on Stress-strain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin p-q space; Mohr-Coulomb failure

### Stress transformation and Mohr s circle for stresses

Stress transformation and Mohr s circle for stresses 1.1 General State of stress Consider a certain body, subjected to external force. The force F is acting on the surface over an area da of the surface.

### Strain Transformation equations

Strain Transformation equations R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents 1. Stress transformation

### By drawing Mohr s circle, the stress transformation in 2-D can be done graphically. + σ x σ y. cos 2θ + τ xy sin 2θ, (1) sin 2θ + τ xy cos 2θ.

Mohr s Circle By drawing Mohr s circle, the stress transformation in -D can be done graphically. σ = σ x + σ y τ = σ x σ y + σ x σ y cos θ + τ xy sin θ, 1 sin θ + τ xy cos θ. Note that the angle of rotation,

### Properties of the stress tensor

Appendix C Properties of the stress tensor Some of the basic properties of the stress tensor and traction vector are reviewed in the following. C.1 The traction vector Let us assume that the state of stress

### Bone Tissue Mechanics

Bone Tissue Mechanics João Folgado Paulo R. Fernandes Instituto Superior Técnico, 2016 PART 1 and 2 Introduction The objective of this course is to study basic concepts on hard tissue mechanics. Hard tissue

### Combined Stresses and Mohr s Circle. General Case of Combined Stresses. General Case of Combined Stresses con t. Two-dimensional stress condition

Combined Stresses and Mohr s Circle Material in this lecture was taken from chapter 4 of General Case of Combined Stresses Two-dimensional stress condition General Case of Combined Stresses con t The normal

### ARC 341 Structural Analysis II. Lecture 10: MM1.3 MM1.13

ARC241 Structural Analysis I Lecture 10: MM1.3 MM1.13 MM1.4) Analysis and Design MM1.5) Axial Loading; Normal Stress MM1.6) Shearing Stress MM1.7) Bearing Stress in Connections MM1.9) Method of Problem

### 9. Stress Transformation

9.7 ABSOLUTE MAXIMUM SHEAR STRESS A pt in a body subjected to a general 3-D state of stress will have a normal stress and shear-stress components acting on each of its faces. We can develop stress-transformation

### 7. STRESS ANALYSIS AND STRESS PATHS

7-1 7. STRESS ANALYSIS AND STRESS PATHS 7.1 THE MOHR CIRCLE The discussions in Chapters and 5 were largely concerned with vertical stresses. A more detailed examination of soil behaviour requires a knowledge

### Tensor Transformations and the Maximum Shear Stress. (Draft 1, 1/28/07)

Tensor Transformations and the Maximum Shear Stress (Draft 1, 1/28/07) Introduction The order of a tensor is the number of subscripts it has. For each subscript it is multiplied by a direction cosine array

### ANALYSIS OF STRAINS CONCEPT OF STRAIN

ANALYSIS OF STRAINS CONCEPT OF STRAIN Concept of strain : if a bar is subjected to a direct load, and hence a stress the bar will change in length. If the bar has an original length L and changes by an

### Tensor Visualization. CSC 7443: Scientific Information Visualization

Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its

### Classical Mechanics. Luis Anchordoqui

1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body

### A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#\$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y \$ Z % Y Y x x } / % «] «] # z» & Y X»

### 2. Mechanics of Materials: Strain. 3. Hookes's Law

Mechanics of Materials Course: WB3413, Dredging Processes 1 Fundamental Theory Required for Sand, Clay and Rock Cutting 1. Mechanics of Materials: Stress 1. Introduction 2. Plane Stress and Coordinate

### Chapter 3: Stress and Equilibrium of Deformable Bodies

Ch3-Stress-Equilibrium Page 1 Chapter 3: Stress and Equilibrium of Deformable Bodies When structures / deformable bodies are acted upon by loads, they build up internal forces (stresses) within them to

### Lecture 3 The Concept of Stress, Generalized Stresses and Equilibrium

Lecture 3 The Concept of Stress, Generalized Stresses and Equilibrium Problem 3-1: Cauchy s Stress Theorem Cauchy s stress theorem states that in a stress tensor field there is a traction vector t that

### Strain analysis.

Strain analysis ecalais@purdue.edu Plates vs. continuum Gordon and Stein, 1991 Most plates are rigid at the until know we have studied a purely discontinuous approach where plates are

### * Many components have multiaxial loads, and some of those have multiaxial loading in critical locations

Why do Multiaxial Fatigue Calculations? * Fatigue analysis is an increasingly important part of the design and development process * Many components have multiaxial loads, and some of those have multiaxial

### CONSTITUTIVE 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

### Stress Transformation Equations: u = +135 (Fig. a) s x = 80 MPa s y = 0 t xy = 45 MPa. we obtain, cos u + t xy sin 2u. s x = s x + s y.

014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently 9 7. Determine the normal stress and shear stress acting

### Problem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1

Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate

### 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

### Lecture Notes 5

1.5 Lecture Notes 5 Quantities in Different Coordinate Systems How to express quantities in different coordinate systems? x 3 x 3 P Direction Cosines Axis φ 11 φ 3 φ 1 x x x x 3 11 1 13 x 1 3 x 3 31 3

### 3D and Planar Constitutive Relations

3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace

### Mohr's Circle for 2-D Stress Analysis

Mohr's Circle for 2-D Stress Analysis If you want to know the principal stresses and maximum shear stresses, you can simply make it through 2-D or 3-D Mohr's cirlcles! You can know about the theory of

### Lecture 3: The Concept of Stress, Generalized Stresses and Equilibrium

Lecture 3: The Concept of Stress, Generalized Stresses and Equilibrium 3.1 Stress Tensor We start with the presentation of simple concepts in one and two dimensions before introducing a general concept

### Basic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008

Basic Concepts of Strain and Tilt Evelyn Roeloffs, USGS June 2008 1 Coordinates Right-handed coordinate system, with positions along the three axes specified by x,y,z. x,y will usually be horizontal, and

### EAS MIDTERM EXAM

Ave = 98/150, s.d. = 21 EAS 326-03 MIDTERM EXAM This exam is closed book and closed notes. It is worth 150 points; the value of each question is shown at the end of each question. At the end of the exam,

### MECHANICS OF MATERIALS

00 The McGraw-Hill Companies, Inc. All rights reserved. T Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Teas Tech Universit

### Continuum 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

### ELASTICITY (MDM 10203)

LASTICITY (MDM 10203) Lecture Module 5: 3D Constitutive Relations Dr. Waluyo Adi Siswanto University Tun Hussein Onn Malaysia Generalised Hooke's Law In one dimensional system: = (basic Hooke's law) Considering

### The hitch in all of this is figuring out the two principal angles and which principal stress goes with which principal angle.

Mohr s Circle The stress basic transformation equations that we developed allowed us to determine the stresses acting on an element regardless of its orientation as long as we know the basic stresses σx,

### Dynamics of Glaciers

Dynamics of Glaciers McCarthy Summer School 01 Andy Aschwanden Arctic Region Supercomputing Center University of Alaska Fairbanks, USA June 01 Note: This script is largely based on the Physics of Glaciers

### Composites Design and Analysis. Stress Strain Relationship

Composites Design and Analysis Stress Strain Relationship Composite design and analysis Laminate Theory Manufacturing Methods Materials Composite Materials Design / Analysis Engineer Design Guidelines

### 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

### (MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e

EN10: Continuum Mechanics Homework : Kinetics Due 1:00 noon Friday February 4th School of Engineering Brown University 1. For the Cauchy stress tensor with components 100 5 50 0 00 (MPa) compute (a) The

### 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

### 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

### STRESS. Bar. ! Stress. ! Average Normal Stress in an Axially Loaded. ! Average Shear Stress. ! Allowable Stress. ! Design of Simple Connections

STRESS! Stress Evisdom! verage Normal Stress in an xially Loaded ar! verage Shear Stress! llowable Stress! Design of Simple onnections 1 Equilibrium of a Deformable ody ody Force w F R x w(s). D s y Support

### Elements of Rock Mechanics

Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

### MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS

1 MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS Version 2011-01-14 Stress tensor Definition of traction vector (1) Cauchy theorem (2) Equilibrium (3) Invariants (4) (5) (6) or, written in terms of principal

### Lagrange Multipliers

Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each

### 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

### Theory of Plasticity. Lecture Notes

Theory of Plasticity Lecture Notes Spring 2012 Contents I Theory of Plasticity 1 1 Mechanical Theory of Plasticity 2 1.1 Field Equations for A Mechanical Theory.................... 2 1.1.1 Strain-displacement

### CIVL4332 L1 Introduction to Finite Element Method

CIVL L Introduction to Finite Element Method CIVL L Introduction to Finite Element Method by Joe Gattas, Faris Albermani Introduction The FEM is a numerical technique for solving physical problems such

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

### ME 202 STRENGTH OF MATERIALS SPRING 2014 HOMEWORK 4 SOLUTIONS

ÇANKAYA UNIVERSITY MECHANICAL ENGINEERING DEPARTMENT ME 202 STRENGTH OF MATERIALS SPRING 2014 Due Date: 1 ST Lecture Hour of Week 12 (02 May 2014) Quiz Date: 3 rd Lecture Hour of Week 12 (08 May 2014)

### The Mohr Stress Diagram. Edvard Munch as a young geologist!

The Mohr Stress Diagram Edvard Munch as a young geologist! Material in the chapter is covered in Chapter 7 in Fossen s text The Mohr Stress Diagram A means by which two stresses acting on a plane of known

### A FAILURE CRITERION FOR POLYMERS AND SOFT BIOLOGICAL MATERIALS

Material Technology A FALURE CRTERON FOR POLYMERS AND SOFT BOLOGCAL MATERALS Authors: William W. Feng John O. Hallquist Livermore Software Technology Corp. 7374 Las Positas Road Livermore, CA 94550 USA

### Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA UNESCO EOLSS

MECHANICS OF MATERIALS Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA Keywords: Solid mechanics, stress, strain, yield strength Contents 1. Introduction 2. Stress

### 6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant

DE2-EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Three-Dimensional

### MECH 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 (1-17-08) Last time Introduction Units Reliability engineering

### EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants

EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For

### Code_Aster. Calculation algorithm of the densities of reinforcement

Titre : Algorithme de calcul des densités de ferraillage Date : 05/08/2013 Page : 1/11 Calculation algorithm of the densities of reinforcement Summary: One presents the calculation algorithm of the densities

### Content. Department of Mathematics University of Oslo

Chapter: 1 MEK4560 The Finite Element Method in Solid Mechanics II (January 25, 2008) (E-post:torgeiru@math.uio.no) Page 1 of 14 Content 1 Introduction to MEK4560 3 1.1 Minimum Potential energy..............................

### Example 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph:

Section 10.3: Polar Coordinates The polar coordinate system is another way to coordinatize the Cartesian plane. It is particularly useful when examining regions which are circular. 1. Cartesian Coordinates

### PEAT SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II

PEAT8002 - SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II Nick Rawlinson Research School of Earth Sciences Australian National University Waveform modelling P-wave first-motions

### Constitutive models: Incremental plasticity Drücker s postulate

Constitutive models: Incremental plasticity Drücker s postulate if consistency condition associated plastic law, associated plasticity - plastic flow law associated with the limit (loading) surface Prager

### 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

### Spherical 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

### Problem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:

Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein

### And similarly in the other directions, so the overall result is expressed compactly as,

SQEP Tutorial Session 5: T7S0 Relates to Knowledge & Skills.5,.8 Last Update: //3 Force on an element of area; Definition of principal stresses and strains; Definition of Tresca and Mises equivalent stresses;

### MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Isotropic Elastic Models: Invariant vs Principal Formulations

MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #2: Nonlinear Elastic Models Isotropic Elastic Models: Invariant vs Principal Formulations Elastic

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

### Phys 7221 Homework # 8

Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with

### 2 Introduction to mechanics

21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy

### Strain Transformation and Rosette Gage Theory

Strain Transformation and Rosette Gage Theor It is often desired to measure the full state of strain on the surface of a part, that is to measure not onl the two etensional strains, and, but also the shear

### MECHANICS OF MATERIALS

CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Teas Tech Universit Transformations of Stress and Strain 006 The McGraw-Hill Companies,

### 10.2,3,4. Vectors in 3D, Dot products and Cross Products

Name: Section: 10.2,3,4. Vectors in 3D, Dot products and Cross Products 1. Sketch the plane parallel to the xy-plane through (2, 4, 2) 2. For the given vectors u and v, evaluate the following expressions.

### C:\Users\whit\Desktop\Active\304_2012_ver_2\_Notes\4_Torsion\1_torsion.docx 6

C:\Users\whit\Desktop\Active\304_2012_ver_2\_Notes\4_Torsion\1_torsion.doc 6 p. 1 of Torsion of circular bar The cross-sections rotate without deformation. The deformation that does occur results from

### Stress Integration for the Drucker-Prager Material Model Without Hardening Using the Incremental Plasticity Theory

Journal of the Serbian Society for Computational Mechanics / Vol. / No., 008 / pp. 80-89 UDC: 59.74:004.0 Stress Integration for the Drucker-rager Material Model Without Hardening Using the Incremental

### COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction

COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS Hsiang-Chuan Tsai, National Taiwan University of Science and Technology, Taipei, Taiwan James M. Kelly, University of California,

### 2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3).

Circle 1. (i) Find the equation of the circle with centre ( 7, 3) and of radius 10. (ii) Find the centre of the circle 2x 2 + 2y 2 + 6x + 8y 1 = 0 (iii) What is the radius of the circle 3x 2 + 3y 2 + 5x

### Closed-Form Solution Of Absolute Orientation Using Unit Quaternions

Closed-Form Solution Of Absolute Orientation Using Unit Berthold K. P. Horn Department of Computer and Information Sciences November 11, 2004 Outline 1 Introduction 2 3 The Problem Given: two sets of corresponding

### Math 263 Final. (b) The cross product is. i j k c. =< c 1, 1, 1 >

Math 63 Final Problem 1: [ points, 5 points to each part] Given the points P : (1, 1, 1), Q : (1,, ), R : (,, c 1), where c is a parameter, find (a) the vector equation of the line through P and Q. (b)

### Chapter 2 CONTINUUM MECHANICS PROBLEMS

Chapter 2 CONTINUUM MECHANICS PROBLEMS The concept of treating solids and fluids as though they are continuous media, rather thancomposedofdiscretemolecules, is one that is widely used in most branches

### Lecture 7. Properties of Materials

MIT 3.00 Fall 2002 c W.C Carter 55 Lecture 7 Properties of Materials Last Time Types of Systems and Types of Processes Division of Total Energy into Kinetic, Potential, and Internal Types of Work: Polarization

### Rotational & Rigid-Body Mechanics. Lectures 3+4

Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions

### 3/31/ Product of Inertia. Sample Problem Sample Problem 10.6 (continue)

/1/01 10.6 Product of Inertia Product of Inertia: I xy = xy da When the x axis, the y axis, or both are an axis of symmetry, the product of inertia is zero. Parallel axis theorem for products of inertia:

### Math review. Math review

Math review 1 Math review 3 1 series approximations 3 Taylor s Theorem 3 Binomial approximation 3 sin(x), for x in radians and x close to zero 4 cos(x), for x in radians and x close to zero 5 2 some geometry

### S12.1 SOLUTIONS TO PROBLEMS 12 (ODD NUMBERS)

OLUTION TO PROBLEM 2 (ODD NUMBER) 2. The electric field is E = φ = 2xi + 2y j and at (2, ) E = 4i + 2j. Thus E = 2 5 and its direction is 2i + j. At ( 3, 2), φ = 6i + 4 j. Thus the direction of most rapid

### chapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?

chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "

### TRESS - STRAIN RELATIONS

TRESS - STRAIN RELATIONS Stress Strain Relations: Hook's law, states that within the elastic limits the stress is proportional to t is impossible to describe the entire stress strain curve with simple

### MAT 419 Lecture Notes Transcribed by Eowyn Cenek 6/1/2012

(Homework 1: Chapter 1: Exercises 1-7, 9, 11, 19, due Monday June 11th See also the course website for lectures, assignments, etc) Note: today s lecture is primarily about definitions Lots of definitions

### 4: birefringence and phase matching

/3/7 4: birefringence and phase matching Polarization states in EM Linear anisotropic response χ () tensor and its symmetry properties Working with the index ellipsoid: angle tuning Phase matching in crystals

### Pressure Vessels Stresses Under Combined Loads Yield Criteria for Ductile Materials and Fracture Criteria for Brittle Materials

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

### Symmetric 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

### Math 20C Homework 2 Partial Solutions

Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we

### Brittle 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

### Lecture Notes 3

12.005 Lecture Notes 3 Tensors Most physical quantities that are important in continuum mechanics like temperature, force, and stress can be represented by a tensor. Temperature can be specified by stating

### Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive

### The Sphere OPTIONAL - I Vectors and three dimensional Geometry THE SPHERE

36 THE SPHERE You must have played or seen students playing football, basketball or table tennis. Football, basketball, table tennis ball are all examples of geometrical figures which we call "spheres"

### Concept Question Comment on the general features of the stress-strain response under this loading condition for both types of materials

Module 5 Material failure Learning Objectives review the basic characteristics of the uni-axial stress-strain curves of ductile and brittle materials understand the need to develop failure criteria for

### SLOPE-DEFLECTION METHOD

SLOPE-DEFLECTION ETHOD The slope-deflection method uses displacements as unknowns and is referred to as a displacement method. In the slope-deflection method, the moments at the ends of the members are