Module #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch.

Size: px
Start display at page:

Download "Module #3. Transformation of stresses in 3-D READING LIST. DIETER: Ch. 2, pp Ch. 3 in Roesler Ch. 2 in McClintock and Argon Ch."

Transcription

1 HOMEWORK From Dieter -3, -4, 3-7 Module #3 Transformation of stresses in 3-D READING LIST DIETER: Ch., pp Ch. 3 in Roesler Ch. in McClintock and Argon Ch. 7 in Edelglass

2 The Stress Tensor z z x O zz zy zx xz xy xx y yz yy yx x y ij yx yy xx xy xz yz zx zy zz In three dimensions the state of stress is described by the stress tensor. We can transform from one coordinate system to another in the same way that we did for two dimensions.

3 Method Lets resolve an arbitrary 3D state of stress onto an oblique plane ABC (area A). To make the problem easier, let S be parallel to the plane normal (meaning that it is a principal stress acting on a principal plane (i.e., the plane w/o shear). z z C normal yy yz xy yx xz O zx xx A S = B y x x z y y x A zy zz DIRECTION COSINES l = cos θ x m = cos θ y n = cos θ z

4 z DIRECTION COSINES l = cos θ x m = cos θ y n = cos θ z C xy xx S = yy yz yx O xz zx A B y x A zy zz z z normal x y y L* x

5 The components of S parallel to the original x-, y-, and z- axes (i.e., S x, S y, S z ) are: S x = Sl = σl To balance force we need Area COB = Al. S y = Sm = σm the areas that each stress Area AOC = Am. S z = Sn = σn components acts on Area AOB = An. z x yy A z C xy yx xz yz zy O zz zx xx A S S z S x B S y y S S S S x y z x x z y normal Recall: l = cos θ x m = cos θ y n = cos θ z y

6 All forces must balance to meet the conditions for static equilibrium (i.e., F=0): Fx Sx A SAl xxal yxam zxan Fy Sy A SAm yyam xyal zyan F S A SAn An Al Am z z zz xz yz Fx ( xx S) l yxmzxn0 Fy x yl( yy S) mzyn0 F l m ( S) n 0 z xz yz zz S S xx yx zx xy yy zy xz yz zz S l 0 m 0 n 0 When written in matrix form

7 Method cont d The solution of the determinant of the matrix on the left yields a cubic equation in terms of S. S S S 3 ( xx yy zz ) ( xx yy yyzz xxzz xy yz xz ) ( xx yyzz xy yzxz xx yz yyxz zzxy ) 0 S or I S I SI In this problem, S =. Thus, the three roots of this cubic equation represent the principal stresses, 1,, and 3.

8 Method cont d The directions in which the principal stresses act are determined by substituting 1,, and 3, each for S in: ( xx S) l yxmzxn0 xyl ( yy S) mzyn 0 l m ( S) n0 xz yz zz Then the resulting equations must be solved simultaneously for l, m, and n (using the relationship l +m +n = 1). (a) Substitute 1 for S ; solve for l, m, and n; (b) Substitute for S ; solve for l, m, and n; (c) Substitute 3 for S ; solve for l, m, and n.

9 Invariants of the Stress Tensor I I 1 xx yy zz xx xy yy yz xx xz yx yy zy zz zx zz xx yy yy zz xx zz xy yz xz I xx xy xz 3 yx yy yz xx yyzz xy yzxz xx yz yyxz zzxy zx zy zz Whenever stresses are transformed from one coordinate system to another, these three quantities remain constant.

10 Example Problem #1 Determine the principal normal stresses for the following state of stress: 0, 10, 75, xx yy zz 50, 0 xy yz xz or MPa

11 Example Problem #1 solution This problem can be solved by substituting the known state of stress into the cubic equation: where S = σ. 3 S I1S ISI3 0 This is detailed on the next page.

12 MPa ( xx yy zz ) ( xx yy yyzz xxzz xy yz xz ) ( xx yy zz xy yz xz xx yz yy xz zz xy ) ( 75 ) [(0 10) (10 75) (0 75) ( 50) (0) (0) ] [0 ( 1075) ( 50 00) (00 ) (10 0 ) ( 7550 )] 0 3 ( 65) [ 350] [187500] 0 I 1 I I

13 You can determine the principal stresses by plotting this equation OR you can solve it using more traditional means (10 6 MPa) = -75 = = (MPa)

14 Example Problem #1 solution This problem is easier than most because there are no shear stresses along the z-axis. It should have been obvious toyou that one of the principal stresses is = -75 MPa (since zx = xz = 0 and zy = yz = 0). Can you determine the directions in which the principal stresses act? (I RECOMMEND THAT YOU TRY IT)

15 Resources on the Web There are many useful eigenvalue calculators on the world wide web. Here are a few: /MohrCircles-3D/Applets/applet.htm H*

16 Example Problem # p. 1/11 Determine (a) the principal stresses, (b) maximum shear stress, and (c) the orientations of the principal planes for the state of stress provided below: MPa ( xx yy zz ) ( xx yy yyzz xxzz xy yz xz ) ( xx yy zz xy yz xz xx yz yy xz zz xy ) ) 3 ( 0 0) [(80 40) ( 40 60) (80 60) (0) (30) (50 [( ) ( ) (80 30 ) ( ) (60 0 )] 0 ] ( 100) [ 4600] [ 48000]

17 p. / (10 6 MPa) = 11 MPa; = 36 MPa; 3 =-57 MPa = 36 1 = 11 3 = (MPa)

18 MPa ( ) l m n 0 xx yx zx l ( ) m n 0 xy yy zy l m( ) n 0 xz yz zz (80 ) l 0m50n0 0 l ( 40 ) m30n0 50l 30 m(60 ) n0 substitute,, and in place of and solve simultaneous equations p. 3/ (80 ) l 0m50n0 41l 0m50n l ( 40 ) m30n 0 0l 161m30n 0 50l 30 m(60 ) n0 50l 30m61n 0 3 (3 [1]) + (5 []) yields: 13l 60m150n0 100l 805m150n0 3l 745m0n0; m0.0309l

19 Sample Problem # cont d p. 4/11 substitute,, and in place of and solve simultaneous equations (80 ) l 0m50n0 41l 0m50n l ( 40 ) m30n 0 0l 161m30n 0 50l 30 m(60 ) n0 50l 30m61n (3 [1]) + (- [3]) yields: 13l 60m150n0 100l 60m1n0 3l 0m8n0; n0.81l

20 Sample Problem # cont d p. 5/11 l m n 1 substitute expresions for m & n l 0.031l 0.81l 1.673l l m0.031l 0.04 n0.81l Orientations of principal planes associated with 1

21 Sample Problem # cont d p. 6/11 substitute,, and in place of and solve simultaneous equations (80 ) l 0m50n 0 44l 0m50n0 [4] 0 l ( 40 ) m30n 0 0l 76m30n0 [5] 50l 30 m(60 ) n 0 50l 30m4n0 [6] 5 (3 [4]) + (5 [5]) yields: 13l 60m150n0 100l 380m150n0 3l 30m0n0; m0.75l

22 Sample Problem # cont d p. 7/11 substitute,, and in place of and solve simultaneous equations (80 ) l 0m50n 0 44l 0m50n0 [4] 0 l ( 40 ) m30n 0 0l 76m30n0 [5] 50l 30 m(60 ) n 0 50l 30m4n0 [6] 6 (3 [4]) + (- [6]) yields: 13l 60m150n0 100l 60m48n0 3l 0m198n0; n1.17l

23 Sample Problem # cont d p. 8/11 l m n 1 substitute expresions for m & n l 0.75l 1.17l.899l 7 1 l m0.75l 0.46 n1.17l Orientations of principal planes associated with

24 Sample Problem # cont d p. 9/11 substitute,, and in place of and solve simultaneous equations (80 ) l 0m50n0 137l 0m50n l ( 40 ) m30n 0 0l 17m30n l 30 m(60 ) n 0 50l 30m117n (3 [7]) + (5 [8]) yields: 411l 60m150n0 100l 85m150n0 511l 145m0n0; m3.54l

25 Sample Problem # cont d p. 10/11 substitute,, and in place of and solve simultaneous equations (80 ) l 0m50n0 137l 0m50n l ( 40 ) m30n 0 0l 17m30n l 30 m(60 ) n 0 50l 30m117n (3 [7]) + (- [6]) yields: 411l 60m150n0 100l 60m34n0 511l 0m384n0; n1.331l

26 Sample Problem # cont d p. 11/11 l m n 1 substitute expresions for m & n l 3.54l 1.331l l 10 1 l m3.54l n1.331l 0.34 Orientations of principal planes associated with max 11 ( 57) 89 MPa

27 5 minute break

28 General Method for Triaxial States of Stress [p in Dieter] In previous example/method, we assumed that the stress on the inclined plane was a principal stress. What if the stress on the new plane is not a principal stress? The math is nearly the same. S S S S n s x yy A yz x y z C yx zy O zz z xy xz zx xx S s n B x y x z z y normal y

29 Triaxial Stress States cont d From summation of forces parallel to the x, y, z axes, we find the components S x, S y, S z : S l m n x xx yx zx S l m n y xy yy zy S l m n z xz yz zz The normal stress on the oblique plane equals the sum of the components S x, S y, S z parallel to the plane normal. SlSmSn n x y z l m n lm mn nl xx yy zz xy yz zx

30 Triaxial Stress States cont d From the expression = σ + τ S n s, the shear stress can be obtained. When written in terms of principal axes, it becomes: s 1 lm 1 3 ln 3 mn The maximum shear stress occurs when: max 1 3 max min

31 Mohr s s Circle in 3-D3 We can use a 3-D Mohr s circle to visualize the state of stress and to determine principal stresses. Essentially three -D Mohr s circles corresponding to the x-y, x-z, and y-z faces of the elemental cubic element. z zz zx zy yz xz yy xy yx xx y x

32 1 max Uniaxial Tension = = 0 3 = max 1 max = = 3 3 max = 1 Biaxial Tension = 3 max = = = = 0 σ 3 1 Uniaxial Compression = = 3 Triaxial Tension (unequal) Adapted from G.E. Dieter, Mechanical Metallurgy, 3 rd ed., McGraw-Hill (1986) p. 37 σ = - = - 3 = 3 1 Uniaxial Tension plus Biaxial Compression

Unit IV State of stress in Three Dimensions

Unit IV State of stress in Three Dimensions Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength

More information

Stress transformation and Mohr s circle for stresses

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.

More information

Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle 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

More information

1 Stress and Strain. Introduction

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

More information

Strain Transformation equations

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

More information

CHAPTER 4 Stress Transformation

CHAPTER 4 Stress Transformation CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z

More information

Equilibrium of Deformable Body

Equilibrium of Deformable Body Equilibrium of Deformable Body Review Static Equilibrium If a body is in static equilibrium under the action applied external forces, the Newton s Second Law provides us six scalar equations of equilibrium

More information

19. Principal Stresses

19. Principal Stresses 19. Principal Stresses I Main Topics A Cauchy s formula B Principal stresses (eigenvectors and eigenvalues) C Example 10/24/18 GG303 1 19. Principal Stresses hkp://hvo.wr.usgs.gov/kilauea/update/images.html

More information

Continuum mechanism: Stress and strain

Continuum mechanism: Stress and strain Continuum mechanics deals with the relation between forces (stress, σ) and deformation (strain, ε), or deformation rate (strain rate, ε). Solid materials, rigid, usually deform elastically, that is the

More information

Properties of the stress tensor

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

More information

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

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»

More information

ME 243. Lecture 10: Combined stresses

ME 243. Lecture 10: Combined stresses ME 243 Mechanics of Solids Lecture 10: Combined stresses Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil6791@gmail.com Website: teacher.buet.ac.bd/sshakil

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Surface and body forces Tensors, Mohr circles. Theoretical strength of materials Defects Stress concentrations Griffith failure

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

More information

Module #4. Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST. DIETER: Ch. 2, Pages 38-46

Module #4. Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST. DIETER: Ch. 2, Pages 38-46 HOMEWORK From Dieter 2-7 Module #4 Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST DIETER: Ch. 2, Pages 38-46 Pages 11-12 in Hosford Ch. 6 in Ne Strain When a solid is

More information

Principal Stresses, Yielding Criteria, wall structures

Principal Stresses, Yielding Criteria, wall structures Principal Stresses, Yielding Criteria, St i thi Stresses in thin wall structures Introduction The most general state of stress at a point may be represented by 6 components, x, y, z τ xy, τ yz, τ zx normal

More information

Bone Tissue Mechanics

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

More information

GG612 Lecture 3. Outline

GG612 Lecture 3. Outline GG61 Lecture 3 Strain and Stress Should complete infinitesimal strain by adding rota>on. Outline Matrix Opera+ons Strain 1 General concepts Homogeneous strain 3 Matrix representa>ons 4 Squares of line

More information

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ APPLIED MECHANICS Study Support Leo Václavek Ostrava 2015 Title:Applied Mechanics Code: Author: doc. Ing.

More information

Solution of Matrix Eigenvalue Problem

Solution of Matrix Eigenvalue Problem Outlines October 12, 2004 Outlines Part I: Review of Previous Lecture Part II: Review of Previous Lecture Outlines Part I: Review of Previous Lecture Part II: Standard Matrix Eigenvalue Problem Other Forms

More information

3D Elasticity Theory

3D Elasticity Theory 3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

More information

16.20 Techniques of Structural Analysis and Design Spring Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T

16.20 Techniques of Structural Analysis and Design Spring Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T 16.20 Techniques of Structural Analysis and Design Spring 2013 Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T February 15, 2013 2 Contents 1 Stress and equilibrium 5 1.1 Internal forces and

More information

Numerical Modelling in Geosciences. Lecture 6 Deformation

Numerical Modelling in Geosciences. Lecture 6 Deformation Numerical Modelling in Geosciences Lecture 6 Deformation Tensor Second-rank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system): - First invariant trace:!!

More information

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

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

More information

ANALYSIS OF STERSSES. General State of stress at a point :

ANALYSIS OF STERSSES. General State of stress at a point : ANALYSIS OF STERSSES General State of stress at a point : Stress at a point in a material body has been defined as a force per unit area. But this definition is some what ambiguous since it depends upon

More information

M E 320 Professor John M. Cimbala Lecture 10

M E 320 Professor John M. Cimbala Lecture 10 M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body

More information

CHAPER THREE ANALYSIS OF PLANE STRESS AND STRAIN

CHAPER THREE ANALYSIS OF PLANE STRESS AND STRAIN CHAPER THREE ANALYSIS OF PLANE STRESS AND STRAIN Introduction This chapter is concerned with finding normal and shear stresses acting on inclined sections cut through a member, because these stresses may

More information

Solid State Theory Physics 545

Solid State Theory Physics 545 olid tate Theory hysics 545 Mechanical properties of materials. Basics. tress and strain. Basic definitions. Normal and hear stresses. Elastic constants. tress tensor. Young modulus. rystal symmetry and

More information

9. Stress Transformation

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

More information

12. Stresses and Strains

12. Stresses and Strains 12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

More information

Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems

Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems Bishakh Bhattacharya & Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 19 Analysis of an Orthotropic Ply References

More information

Eigen decomposition for 3D stress tensor

Eigen decomposition for 3D stress tensor Eigen decomposition for 3D stress tensor by Vince Cronin Begun September 13, 2010; last revised September 23, 2015 Unfinished Dra ; Subject to Revision Introduction If we have a cubic free body that is

More information

Applications of Eigenvalues & Eigenvectors

Applications of Eigenvalues & Eigenvectors Applications of Eigenvalues & Eigenvectors Louie L. Yaw Walla Walla University Engineering Department For Linear Algebra Class November 17, 214 Outline 1 The eigenvalue/eigenvector problem 2 Principal

More information

Continuum Mechanics. Continuum Mechanics and Constitutive Equations

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

More information

Failure from static loading

Failure 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 information

Mohr's Circle for 2-D Stress Analysis

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

More information

Algebraic Expressions

Algebraic Expressions Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly

More information

Tensor Visualization. CSC 7443: Scientific Information Visualization

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

More information

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why

More information

Chapter 9: Differential Analysis of Fluid Flow

Chapter 9: Differential Analysis of Fluid Flow of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known

More information

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST

More information

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. 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

More information

Determination of Locally Varying Directions through Mass Moment of Inertia Tensor

Determination of Locally Varying Directions through Mass Moment of Inertia Tensor Determination of Locally Varying Directions through Mass Moment of Inertia Tensor R. M. Hassanpour and C.V. Deutsch Centre for Computational Geostatistics Department of Civil and Environmental Engineering

More information

Computer Applications in Engineering and Construction Programming Assignment #9 Principle Stresses and Flow Nets in Geotechnical Design

Computer Applications in Engineering and Construction Programming Assignment #9 Principle Stresses and Flow Nets in Geotechnical Design CVEN 302-501 Computer Applications in Engineering and Construction Programming Assignment #9 Principle Stresses and Flow Nets in Geotechnical Design Date distributed : 12/2/2015 Date due : 12/9/2015 at

More information

CH.4. STRESS. Continuum Mechanics Course (MMC)

CH.4. STRESS. Continuum Mechanics Course (MMC) CH.4. STRESS Continuum Mechanics Course (MMC) Overview Forces Acting on a Continuum Body Cauchy s Postulates Stress Tensor Stress Tensor Components Scientific Notation Engineering Notation Sign Criterion

More information

Homework 1/Solutions. Graded Exercises

Homework 1/Solutions. Graded Exercises MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both

More information

Basic Equations of Elasticity

Basic Equations of Elasticity A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ

More information

Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth

Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth References: Turcotte and Schubert, Geodynamics, Sections 2.1,-2.4, 2.7, 3.1-3.8, 6.1, 6.2, 6.8, 7.1-7.4. Jaeger and Cook, Fundamentals of

More information

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

* 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

More information

Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

More information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

More information

Topics. GG612 Structural Geology Sec3on Steve Martel POST 805 Lecture 4 Mechanics: Stress and Elas3city Theory

Topics. GG612 Structural Geology Sec3on Steve Martel POST 805 Lecture 4 Mechanics: Stress and Elas3city Theory GG612 Structural Geology Sec3on Steve Martel POST 805 smartel@hawaii.edu Lecture 4 Mechanics: Stress and Elas3city Theory 11/6/15 GG611 1 Topics 1. Stress vectors (trac3ons) 2. Stress at a point 3. Cauchy

More information

Stress/Strain. Outline. Lecture 1. Stress. Strain. Plane Stress and Plane Strain. Materials. ME EN 372 Andrew Ning

Stress/Strain. Outline. Lecture 1. Stress. Strain. Plane Stress and Plane Strain. Materials. ME EN 372 Andrew Ning Stress/Strain Lecture 1 ME EN 372 Andrew Ning aning@byu.edu Outline Stress Strain Plane Stress and Plane Strain Materials otes and News [I had leftover time and so was also able to go through Section 3.1

More information

MAE 323: Lecture 1. Review

MAE 323: Lecture 1. Review This review is divided into two parts. The first part is a mini-review of statics and solid mechanics. The second part is a review of matrix/vector fundamentals. The first part is given as an refresher

More information

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

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay 51 Module 4: Lecture 2 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

More information

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT SEISMOLOGY Lecture 2: Continuum mechanics PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

More information

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Stress and Strain

More information

CE 240 Soil Mechanics & Foundations Lecture 7.1. in situ Stresses I (Das, Ch. 8)

CE 240 Soil Mechanics & Foundations Lecture 7.1. in situ Stresses I (Das, Ch. 8) CE 240 Soil Mechanics & Foundations Lecture 7.1 in situ Stresses I (Das, Ch. 8) Class Outline Stress tensor, stress units Effective stress, Stresses in saturated soil without seepage Stresses in saturated

More information

Classical Mechanics. Luis Anchordoqui

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

More information

MECH 401 Mechanical Design Applications

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

More information

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

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

More information

L8. Basic concepts of stress and equilibrium

L8. Basic concepts of stress and equilibrium L8. Basic concepts of stress and equilibrium Duggafrågor 1) Show that the stress (considered as a second order tensor) can be represented in terms of the eigenbases m i n i n i. Make the geometrical representation

More information

FINAL EXAMINATION. (CE130-2 Mechanics of Materials)

FINAL EXAMINATION. (CE130-2 Mechanics of Materials) UNIVERSITY OF CLIFORNI, ERKELEY FLL SEMESTER 001 FINL EXMINTION (CE130- Mechanics of Materials) Problem 1: (15 points) pinned -bar structure is shown in Figure 1. There is an external force, W = 5000N,

More information

LESSON 7.1 FACTORING POLYNOMIALS I

LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of

More information

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, ; 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

More information

Chem8028(1314) - Spin Dynamics: Spin Interactions

Chem8028(1314) - Spin Dynamics: Spin Interactions Chem8028(1314) - Spin Dynamics: Spin Interactions Malcolm Levitt see also IK m106 1 Nuclear spin interactions (diamagnetic materials) 2 Chemical Shift 3 Direct dipole-dipole coupling 4 J-coupling 5 Nuclear

More information

Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India AIR WALK PUBLICATIONS

Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India AIR WALK PUBLICATIONS Advanced Mechanics of Solids (APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY) And As per Revised Syllabus of Leading Universities in India Dr. S. Ramachandran Prof. R. Devaraj Professors School of Mechanical

More information

3.22 Mechanical Properties of Materials Spring 2008

3.22 Mechanical Properties of Materials Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 3.22 Mechanical Properties of Materials Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Quiz #1 Example

More information

Mechanics of Materials Lab

Mechanics of Materials Lab Mechanics of Materials Lab Lecture 5 Stress Mechanical Behavior of Materials Sec. 6.1-6.5 Jiangyu Li Jiangyu Li, orce Vectors A force,, is a vector (also called a "1 st -order tensor") The description

More information

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles.

Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles. » ~ $ ) 7 x X ) / ( 8 2 X 39 ««x» ««! «! / x? \» «({? «» q «(? (?? x! «? 8? ( z x x q? ) «q q q ) x z x 69 7( X X ( 3»«! ( ~«x ««x ) (» «8 4 X «4 «4 «8 X «x «(» X) ()»» «X «97 X X X 4 ( 86) x) ( ) z z

More information

Name (Print) ME Mechanics of Materials Exam # 2 Date: March 29, 2016 Time: 8:00 10:00 PM - Location: PHYS 114

Name (Print) ME Mechanics of Materials Exam # 2 Date: March 29, 2016 Time: 8:00 10:00 PM - Location: PHYS 114 Name (Print) (Last) (First) Instructions: ME 323 - Mechanics of Materials Exam # 2 Date: March 29, 2016 Time: 8:00 10:00 PM - Location: PHYS 114 Circle your lecturer s name and your class meeting time.

More information

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

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

More information

A critical analysis of the Mises stress criterion used in frequency domain fatigue life prediction

A critical analysis of the Mises stress criterion used in frequency domain fatigue life prediction Focussed on Multiaxial Fatigue and Fracture A critical analysis of the Mises stress criterion used in frequency domain fatigue life prediction Adam Niesłony Opole University of Technology, Poland a.nieslony@po.opole.pl,

More information

THE CAUSES: MECHANICAL ASPECTS OF DEFORMATION

THE CAUSES: MECHANICAL ASPECTS OF DEFORMATION 17 THE CAUSES: MECHANICAL ASPECTS OF DEFORMATION Mechanics deals with the effects of forces on bodies. A solid body subjected to external forces tends to change its position or its displacement or its

More information

EE C247B ME C218 Introduction to MEMS Design Spring 2017

EE C247B ME C218 Introduction to MEMS Design Spring 2017 247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 Outline C247B M C28 Introduction to MMS Design Spring 207 Prof. Clark T.- Reading: Senturia, Chpt. 8 Lecture Topics:

More information

Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis. Supplementary Document

Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis. Supplementary Document Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis Supplementary Document Shan Yang, Vladimir Jojic, Jun Lian, Ronald Chen, Hongtu Zhu, Ming C.

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Thermodynamics Derivation Hooke s Law: Anisotropic Elasticity

More information

Introduction to Condensed Matter Physics

Introduction to Condensed Matter Physics Introduction to Condensed Matter Physics Elasticity M.P. Vaughan Overview Overview of elasticity Classical description of elasticity Speed of sound Strain Stress Young s modulus Shear modulus Poisson ratio

More information

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort - 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

More information

Chapter 2 Governing Equations

Chapter 2 Governing Equations Chapter Governing Equations Abstract In this chapter fundamental governing equations for propagation of a harmonic disturbance on the surface of an elastic half-space is presented. The elastic media is

More information

Two Posts to Fill On School Board

Two Posts to Fill On School Board Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

More information

σ = F/A. (1.2) σ xy σ yy σ zx σ xz σ yz σ, (1.3) The use of the opposite convention should cause no problem because σ ij = σ ji.

σ = F/A. (1.2) σ xy σ yy σ zx σ xz σ yz σ, (1.3) The use of the opposite convention should cause no problem because σ ij = σ ji. Cambridge Universit Press 978-1-107-00452-8 - Metal Forming: Mechanics Metallurg, Fourth Edition Ecerpt 1 Stress Strain An understing of stress strain is essential for the analsis of metal forming operations.

More information

Useful Formulae ( )

Useful Formulae ( ) Appendix A Useful Formulae (985-989-993-) 34 Jeremić et al. A.. CHAPTER SUMMARY AND HIGHLIGHTS page: 35 of 536 A. Chapter Summary and Highlights A. Stress and Strain This section reviews small deformation

More information

Mechanics of materials Lecture 4 Strain and deformation

Mechanics of materials Lecture 4 Strain and deformation Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum

More information

6. SCALARS, VECTORS, AND TENSORS (FOR ORTHOGONAL COORDINATE SYSTEMS)

6. SCALARS, VECTORS, AND TENSORS (FOR ORTHOGONAL COORDINATE SYSTEMS) (FOR ORTHOGONAL COORDINATE SYSTEMS) I Main Topics A What are scalars, vectors, and tensors? B Order of scalars, vectors, and tensors C Linear transformaoon of scalars and vectors (and tensors) D Matrix

More information

GG611 Structural Geology Sec1on Steve Martel POST 805

GG611 Structural Geology Sec1on Steve Martel POST 805 GG611 Structural Geology Sec1on Steve Martel POST 805 smartel@hawaii.edu Lecture 5 Mechanics Stress, Strain, and Rheology 11/6/16 GG611 1 Stresses Control How Rock Fractures hkp://hvo.wr.usgs.gov/kilauea/update/images.html

More information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In this section, mathematical description of the motion of fluid elements moving in a flow field is Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

More information

L bor y nnd Union One nnd Inseparable. LOW I'LL, MICHIGAN. WLDNHSDA Y. JULY ), I8T. liuwkll NATIdiNAI, liank

L bor y nnd Union One nnd Inseparable. LOW I'LL, MICHIGAN. WLDNHSDA Y. JULY ), I8T. liuwkll NATIdiNAI, liank G k y $5 y / >/ k «««# ) /% < # «/» Y»««««?# «< >«>» y k»» «k F 5 8 Y Y F G k F >«y y

More information

Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps

More information

Chapter 5. The Differential Forms of the Fundamental Laws

Chapter 5. The Differential Forms of the Fundamental Laws Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations

More information

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 7 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Transformations of

More information

Constitutive Equations

Constitutive Equations Constitutive quations David Roylance Department of Materials Science and ngineering Massachusetts Institute of Technology Cambridge, MA 0239 October 4, 2000 Introduction The modules on kinematics (Module

More information

Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector

Review for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector Calculus 3 Lia Vas Review for Exam 1 1. Surfaces. Describe the following surfaces. (a) x + y = 9 (b) x + y + z = 4 (c) z = 1 (d) x + 3y + z = 6 (e) z = x + y (f) z = x + y. Review of Vectors. (a) Let a

More information

1. Background. is usually significantly lower than it is in uniaxial tension

1. Background. is usually significantly lower than it is in uniaxial tension NOTES ON QUANTIFYING MODES OF A SECOND- ORDER TENSOR. The mechanical behavior of rocks and rock-like materials (concrete, ceramics, etc.) strongly depends on the loading mode, defined by the values and

More information

Multivariable Calculus and Matrix Algebra-Summer 2017

Multivariable Calculus and Matrix Algebra-Summer 2017 Multivariable Calculus and Matrix Algebra-Summer 017 Homework 4 Solutions Note that the solutions below are for the latest version of the problems posted. For those of you who worked on an earlier version

More information

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS

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

More information

LB 220 Homework 4 Solutions

LB 220 Homework 4 Solutions LB 220 Homework 4 Solutions Section 11.4, # 40: This problem was solved in class on Feb. 03. Section 11.4, # 42: This problem was also solved in class on Feb. 03. Section 11.4, # 43: Also solved in class

More information

Lecture 7. Properties of Materials

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

More information