σ = F/A. (1.2) σ xy σ yy σ zy , (1.3) σ xz σ yz σ zz The use of the opposite convention should cause no problem because σ ij = σ ji.
|
|
- Megan McCarthy
- 5 years ago
- Views:
Transcription
1 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 1 Stress Strain An understing of stress strain is essential for analzing metal forming operations. Often the words stress strain are used snonmousl b the nonscientific public. In engineering, however, stress is the intensit of force strain is a measure of the amount of deformation. 1.1 STRESS Stress is defined as the intensit of force, F, at a point. where A is the area on which the force acts. If the stress is the same everwhere, σ = F/ A as A 0, (1.1) σ = F/A. (1.2) There are nine components of stress as shown in Figure 1.1. A normal stress component is one in which the force is acting normal to the plane. It ma be tensile or compressive. A shear stress component is one in which the force acts parallel to the plane. Stress components are defined with two subscripts. The first denotes the normal to the plane on which the force acts the second is the direction of the force. For eample, σ is a tensile stress in the -direction. A shear stress acting on the -plane in the -direction is denoted σ. Repeated subscripts (e.g., σ, σ, σ zz ) indicate normal stresses. The are tensile if both subscripts are positive or both are negative. If one is positive the other is negative, the are compressive. Mied subscripts (e.g., σ z,σ,σ z ) denote shear stresses. A state of stress in tensor notation is epressed as σ σ σ z σ ij = σ σ σ z, (1.3) σ z σ z σ zz The use of the opposite convention should cause no problem because σ ij = σ ji. 1 Cambridge Universit Press
2 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 2 STRESS ANDSTRAIN z σ zz σ z σ z σ z σ σ σ σ z σ 1.1. Nine components of stress acting on an infinitesimal element. where i j are iterated over,, z. Ecept where tensor notation is required, it is simpler to use a single subscript for a normal stress denote a shear stress b τ. For eample, σ σ τ σ. 1.2 STRESS TRANSFORMATION Stress components epressed along one set of aes ma be epressed along an other set of aes. Consider resolving the stress component σ = F /A onto the aes as shown in Figure 1.2. The force F in the direction is F = F cos θ the area normal to is A = A / cos θ, so Similarl σ = F /A = F cos θ/(a / cos θ) = σ cos 2 θ. (1.4a) τ = F /A = F sin θ/(a / cos θ) = σ cos θ sin θ. (1.4b) Note that transformation of stresses requires two sine /or cosine terms. Pairs of shear stresses with the same subscripts in reverse order are alwas equal (e.g., τ ij = τ ji ). This is illustrated in Figure 1.3 b a simple moment balance on an F F F A 1.2. The stresses acting on a plane, A, under a normal stress, σ. θ Cambridge Universit Press
3 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 1.2. STRESS TRANSFORMATION 3 τ 1.3. Unless τ = τ, there would not be a moment balance. τ τ τ infinitesimal element. Unless τ ij = τ ji, there would be an infinite rotational acceleration. Therefore τ ij = τ ji. (1.5) The general equation for transforming the stresses from one set of aes (e.g., n, m, p) to another set of aes (e.g., i, j, k) is 3 3 σ ij = l im l jn σ mn. (1.6) n=1 m=1 Here, the term l im is the cosine of the angle between the i m aes the term l jn is the cosine of the angle between the j n aes. This is often written as σ ij = l im l jn σ mn, (1.7) with the summation implied. Consider transforming stresses from the,, z ais sstem to the,, z sstem shown in Figure 1.4. Using equation 1.6, σ σ = l l σ + l l σ + l l zσ z + l l σ + l l σ + l l zσ z + l zl σ z + l zl σ z + l zl zσ zz (1.8a) = l l σ + l l σ + l l zσ z + l l σ + l l σ + l l zσ z + l zl σ z + l zl σ z + l zl zσ zz (1.8b) z z 1.4. Two orthogonal coordinate sstems. Cambridge Universit Press
4 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 4 STRESS ANDSTRAIN These can be simplified to σ = l 2 σ + l 2 σ + l 2 z σ z + 2l l zτ z + 2l zl τ z + 2l l τ (1.9a) τ = l l σ + l l σ + l zl zσ z + (l l z + l zl )τ z + (l zl + l l z)τ z + (l l + l l )τ. (1.9b) 1.3 PRINCIPAL STRESSES It is alwas possible to find a set of aes along which the shear stress terms vanish. In this case σ 1, σ 2, σ 3 are called the principal stresses. The magnitudes of the principal stresses, σ p, are the roots of σ 3 p I 1σ 2 p I 2σ p I 3 = 0, (1.10) where I 1, I 2, I 3 are called the invariants of the stress tensor. The are I 1 = σ + σ + σ zz, I 2 = σz 2 + σ z 2 + σ 2 σ σ zz σ zz σ σ σ, (1.11) I 3 = σ σ σ zz + 2σ z σ z σ σ σz 2 σ σz 2 σ zzσ 2. The first invariant I 1 = p/3 where p is the pressure. I 1, I 2, I 3 are independent of the orientation of the aes. Epressed in terms of the principal stresses, the are I 1 = σ 1 + σ 2 + σ 3, I 2 = σ 2 σ 3 σ 3 σ 1 σ 1 σ 2, (1.12) I 3 = σ 1 σ 2 σ 3. EXAMPLE 1.1: Consider a stress state with σ = 70 MPa, σ = 35 MPa, τ = 20,σ z = τ z = τ z = 0. Find the principal stresses using equations SOLUTION: Using equations 1.11, I 1 = 105 MPa, I 2 = 2050 MPa, I 3 = 0. From equation 1.10, σ 3 p 105σ 2 p σ p + 0 = 0, so σ 2 p 105σ p + 2,050 = 0. The principal stresses are the roots σ 1 = 79.1MPa,σ 2 = 25.9MPa, σ 3 = σ z = 0. EXAMPLE 1.2: Repeat Eample 1.1 with I 3 = 170,700. SOLUTION: The principal stresses are the roots of σ 3 p 105σ 2 p +2050σ p+170,700=0. Since one of the roots is σ z = σ 3 = 40,σ p + 40 = 0 can be factored out. This gives σ 2 p 105σ p = 0, so the other two principal stresses are σ 1 = 79.1MPa,σ 2 = 25.9MPa. This shows that when σ z is one of the principal stresses, the other two principal stresses are independent of σ z. Cambridge Universit Press
5 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 1.4. MOHR S CIRCLE EQUATIONS MOHR S CIRCLE EQUATIONS In the special cases where two of the three shear stress terms vanish (e.g., τ = τ z = 0), the stress σ z normal to the plane is a principal stress the other two principal stresses lie in the plane. This is illustrated in Figure 1.5. For these conditions l z = l z = 0,τ z = τ z = 0,l = l = cos φ, l = l = sin φ. Substituting these relations into equations 1.9 results in τ = cos φ sin φ( σ + σ ) + (cos 2 φ sin 2 φ)τ, σ = (cos 2 φ)σ + (sin 2 φ)σ + 2(cos φ sin φ)τ, (1.13) σ = (sin 2 φ)σ + (cos 2 φ)σ + 2(cos φ sin φ)τ. These can be simplified with the trigonometric relations sin 2φ = 2 sin φ cos φ cos 2φ = cos 2 φ sin 2 φ to obtain τ = sin 2φ(σ σ )/2 + (cos 2φ)τ, (1.14a) σ = (σ + σ )/2 + cos 2φ(σ σ )/2 + τ sin 2φ, (1.14b) σ = (σ + σ )/2 cos 2φ(σ σ )/2 + τ sin 2φ. (1.14c) If τ is set to zero in equation 1.14a, φ becomes the angle θ between the principal aes the aes. Then tan 2θ = τ /[(σ σ )/2]. (1.15) The principal stresses, σ 1 σ 2, are then the values of σ σ σ 1,2 = (σ + σ )/2 ± (1/2)[(σ σ ) cos 2θ] + τ sin 2θ or σ 1,2 = (σ + σ )/2 ± (1/2) [ (σ σ ) 2 + 4τ] 2 1/2. (1.16) σ z τ σ τ σ σ τ τ φ 1.5. Stress state for which the Mohr s circle equations appl. Cambridge Universit Press
6 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 6 STRESS ANDSTRAIN τ σ (σ 1 -σ 2 )/2 θ σ 2 2θ (σ -σ )/2 τ σ 1 σ (σ +σ )/2 σ 1.6. Mohr s circle diagram for stress. τ σ σ 3 σ 2 τ 2θ σ 1 σ σ 1.7. Three-dimensional Mohr s circles for stresses. A Mohr s circle diagram is a graphical representation of equations The form a circle of radius (σ 1 σ 2 )/2 with the center at (σ 1 + σ 2 )/2 as shown in Figure 1.6. The normal stress components are plotted on the ordinate the shear stress components are plotted on the abscissa. Using the Pthagorean theorem on the triangle in Figure 1.6, (σ 1 σ 2 )/2 = { [(σ σ )/2] 2 + τ 2 } 1/2 (1.17) tan 2θ = τ /[(σ σ )/2]. (1.18) A three-dimensional stress state can be represented b three Mohr s circles as shown in Figure 1.7. The three principal stresses σ 1,σ 2, σ 3 are plotted on the ordinate. The circles represent the stress state in the 1 2, 2 3, 3 1 planes. EXAMPLE 1.3: Construct the Mohr s circle for the stress state in Eample 1.2 determine the largest shear stress. O. Mohr, Zivilingeneur (1882), p Cambridge Universit Press
7 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 1.5. STRAIN 7 τ τ ma = Mohr s circle for stress state in Eample 1.2. σ 3 = -40 σ = 35 σ 2 = 25.9 τ = 20 σ = 70 σ σ 1 = 79.1 A 1.9. Deformation, translation, rotation of a line in a material. A l 0 l B B SOLUTION: The Mohr s circle is plotted in Figure 1.8. The largest shear stress is τ ma = (σ 1 σ 3 )/2 = [79.1 ( 40)]/2 = 59.6MPa. 1.5 STRAIN Strain describes the amount of deformation in a bod. When a bod is deformed, points in that bod are displaced. Strain must be defined in such a wa that it ecludes effects of rotation translation. Figure 1.9 shows a line in a material that has been that has been deformed. The line has been translated, rotated, deformed. The deformation is characterized b the engineering or nominal strain, e: e = (l l 0 )/l 0 = l/l 0. (1.19) An alternative definition is that of true or logarithmic strain, ε, defined b which on integrating gives dε = dl/l, (1.20) ε = ln(l/l 0 ) = ln(1 + e). (1.21) The true engineering strains are almost equal when the are small. Epressing ε as ε = ln(l/l 0 ) = ln(1 + e) eping, ε = e e 2 /2 + e 3 /3!, so as e 0,ε e. There are several reasons wh true strains are more convenient than engineering strains. The following eamples indicate wh. First defined b P. Ludwig, Elemente der Technishe Mechanik, Springer, Cambridge Universit Press
8 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 8 STRESS ANDSTRAIN EXAMPLE 1.4: (a) (b) A bar of length l 0 is uniforml etended until its length l = 2l 0. Compute the values of the engineering true strains. To what final length must a bar of length l 0 be compressed if the strains are the same (ecept sign) as in part (a)? SOLUTION: (a) e = l/l 0 = 1.0,ε = ln(l/l 0 ) = ln 2 = (b) e = 1 = (l l 0 )/l 0, so l = 0. This is clearl impossible to achieve. ε = = ln(l/l 0 ), so l = l 0 ep(0.693) = l 0 /2. EXAMPLE 1.5: A bar 10 cm long is elongated to 20 cm b rolling in three steps: 10 cm to 12 cm, 12 cm to 15 cm, 15 cm to 20 cm. (a) (b) Calculate the engineering strain for each step compare the sum of these with the overall engineering strain. Repeat for true strains. SOLUTION: (a) e 1 = 2/10 = 0.20, e 2 = 3/12 = 0.25, e 3 = 5/15 = 0.333, e tot = = e overall = 10/10 = 1. (b) ε 1 = ln(12/10) = 0.182,ε 2 = ln(15/12) = 0.223,ε 3 = ln(20/15) = 0.288,ε tot = 0.693,ε overall = ln(20/10) = With true strains, the sum of the increments equals the overall strain, but this is not so with engineering strains. EXAMPLE 1.6: A block of initial dimensions l 0, w 0, t 0 is deformed to dimensions of l, w, t. (a) (b) Calculate the volume strain, ε v = ln(v/v 0 ) in terms of the three normal strains, ε l,ε w ε t. Plastic deformation causes no volume change. With no volume change, what is the sum of the three normal strains? SOLUTION: (a) ε v = ln[(lwt)/(l 0 w 0 t 0 )] = ln(l/l 0 ) + ln(w/w 0 ) + ln(t/t 0 ) = ε l + ε w + ε t. (b) If ε v = 0,ε l + ε w + ε t = 0. Eamples 1.4, 1.5, 1.6 illustrate wh true strains are more convenient than engineering strains. 1. True strains for an equivalent amount of tensile compressive deformation are equal ecept for sign. 2. True strains are additive. 3. The volume strain is the sum of the three normal strains. Cambridge Universit Press
9 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 1.6. SMALL STRAINS 9 EXAMPLE 1.7: Calculate the ratio of ε/e for e = 0.001, 0.01, 0.02, 0.05, 0.1, 0.2. SOLUTION: For e = 0.001, ε/e = ln(1.001)/0.001 = /0.001 = For e = 0.01,ε/e = ln(1.01)/0.01 = /0.01 = For e = 0.02,ε/e = ln(1.02)/0.02 = /0.02 = For e = 0.05,ε/e = ln(1.05)/0.05 = /0.05 = For e = 0.1,ε/e = ln(1.1)/0.1 = /0.1 = For e = 0.2,ε/e = ln(1.2)/0.2 = /0.2 = As e gets larger the difference between ε e become greater. 1.6 SMALL STRAINS Figure 1.10 shows a small two-dimensional element, ABCD, deformed into A B C D where the displacements are u v. The normal strain, e, is defined as e = (A D AD)/AD = A D /AD 1. (1.22) Neglecting the rotation e = A D d u + u + ( u/ ) d /AD 1 = 1 or d e = u/. (1.23) Similarl, e = v/ e zz = w/ z for a three-dimensional case. The shear strains are associated with the angles between AD A D between AB A B. For small deformations AD A D A B v/ AB = u/. (1.24) ( u/ )d P B C v + ( v/ )d B C D d v A Q ( v/ )d A u d D u +( u/ )d Distortion of a two-dimensional element. Cambridge Universit Press
10 Cambridge Universit Press Metal Forming: Mechanics Metallurg, Third Edition Ecerpt 10 STRESS ANDSTRAIN The total shear strain is the sum of these two angles, Similarl, γ = γ = u + v. γ z = γ z = v z + w γ z = γ z = w + u z. (1.25a) (1.25b) (1.25c) This definition of shear strain, γ, is equivalent to the simple shear measured in a torsion of shear test. 1.7 THE STRAIN TENSOR If tensor shear strains ε ij are defined as ε ij = (1/2)γ ij, (1.26) small shear strains form a tensor, ε ε ε z ε ij = ε ε ε z ε z ε z ε zz. (1.27) Because small strains form a tensor, the can be transformed from one set of aes to another in a wa identical to the transformation of stresses. Mohr s circle relations can be used. It must be remembered, however, that ε ij = γ ij /2 that the transformations hold onl for small strains. If γ z = γ z = 0, ε = ε l 2 + ε l 2 + γ l l (1.28) γ = 2ε l l + 2ε l l + γ (l l + l l ). (1.29) The principal strains can be found from the Mohr s circle equations for strains, ε 1,2 = ε + ε ± (1/2) [ (ε ε ) 2 + γ 2 1/2. 2 ] (1.30) Strains on other planes are given b ε, = (1/2)(ε 1 + ε 2 ) ± (1/2)(ε 1 ε 2 ) cos 2θ (1.31) γ = (ε 1 ε 2 ) sin 2θ. (1.32) 1.8 ISOTROPIC ELASTICITY Although the thrust of this book is on plastic deformation, a short treatment of elasticit is necessar to underst springback residual stresses in forming processes. Cambridge Universit Press
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 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 informationModule #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 informationKINEMATIC RELATIONS IN DEFORMATION OF SOLIDS
Chapter 8 KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS Figure 8.1: 195 196 CHAPTER 8. KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS 8.1 Motivation In Chapter 3, the conservation of linear momentum for a
More informationStrain 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
More informationx y plane is the plane in which the stresses act, yy xy xy Figure 3.5.1: non-zero stress components acting in the x y plane
3.5 Plane Stress This section is concerned with a special two-dimensional state of stress called plane stress. It is important for two reasons: () it arises in real components (particularl in thin components
More informationStress, 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 information4 Strain true strain engineering strain plane strain strain transformation formulae
4 Strain The concept of strain is introduced in this Chapter. The approimation to the true strain of the engineering strain is discussed. The practical case of two dimensional plane strain is discussed,
More informationMECHANICS 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
More informationAnd 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;
More informationSymmetry and Properties of Crystals (MSE638) Stress and Strain Tensor
Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Somnath Bhowmick Materials Science and Engineering, IIT Kanpur April 6, 2018 Tensile test and Hooke s Law Upto certain strain (0.75),
More informationStress and Strain ( , 3.14) MAE 316 Strength of Mechanical Components NC State University Department of Mechanical & Aerospace Engineering
(3.8-3.1, 3.14) MAE 316 Strength of Mechanical Components NC State Universit Department of Mechanical & Aerospace Engineering 1 Introduction MAE 316 is a continuation of MAE 314 (solid mechanics) Review
More informationStress 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 informationGG303 Lab 10 10/26/09 1. Stresses
GG303 Lab 10 10/26/09 1 Stresses Eercise 1 (38 pts total) 1a Write down the epressions for all the stress components in the ' coordinate sstem in terms of the stress components in the reference frame.
More informationConsider a slender rod, fixed at one end and stretched, as illustrated in Fig ; the original position of the rod is shown dotted.
4.1 Strain If an object is placed on a table and then the table is moved, each material particle moves in space. The particles undergo a displacement. The particles have moved in space as a rigid bod.
More informationUNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 20, 2011 Professor A. Dolovich
UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 20, 2011 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS LAST NAME (printed): FIRST NAME (printed): STUDENT
More informationStrain 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 informationMAE4700/5700 Finite Element Analysis for Mechanical and Aerospace Design
MAE4700/5700 Finite Element Analsis for Mechanical and Aerospace Design Cornell Universit, Fall 2009 Nicholas Zabaras Materials Process Design and Control Laborator Sible School of Mechanical and Aerospace
More informationAPPENDIX D Rotation and the General Second-Degree Equation
APPENDIX D Rotation and the General Second-Degree Equation Rotation of Aes Invariants Under Rotation After rotation of the - and -aes counterclockwise through an angle, the rotated aes are denoted as the
More informationPeriod #5: Strain. A. Context. Structural mechanics deals with the forces placed upon mechanical systems and the resulting deformations of the system.
Period #5: Strain A. Contet Strctral mechanics deals with the forces placed pon mechanical sstems and the reslting deformations of the sstem. Solid mechanics on the smaller scale relates stresses in the
More informationCombined 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 informationTensor 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 information9.2. Cartesian Components of Vectors. Introduction. Prerequisites. Learning Outcomes
Cartesian Components of Vectors 9.2 Introduction It is useful to be able to describe vectors with reference to specific coordinate sstems, such as the Cartesian coordinate sstem. So, in this Section, we
More informationAPPLIED MECHANICS I Resultant of Concurrent Forces Consider a body acted upon by co-planar forces as shown in Fig 1.1(a).
PPLIED MECHNICS I 1. Introduction to Mechanics Mechanics is a science that describes and predicts the conditions of rest or motion of bodies under the action of forces. It is divided into three parts 1.
More information8 Properties of Lamina
8 Properties of Lamina 8- ORTHOTROPIC LAMINA An orthotropic lamina is a sheet with unique and predictable properties and consists of an assemblage of fibers ling in the plane of the sheet and held in place
More informationTrigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric
Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric
More informationboth an analytical approach and the pole method, determine: (a) the direction of the
Quantitative Problems Problem 4-3 Figure 4-45 shows the state of stress at a point within a soil deposit. Using both an analytical approach and the pole method, determine: (a) the direction of the principal
More informationSolution ME 323 EXAM #2 FALL SEMESTER :00 PM 9:30 PM Nov. 2, 2010
Solution ME 33 EXAM # FALL SEMESTER 1 8: PM 9:3 PM Nov., 1 Instructions 1. Begin each problem in the space provided on the eamination sheets. If additional space is required, use the paper provided. Work
More informationEMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 4 Pure Bending
EA 3702 echanics & aterials Science (echanics of aterials) Chapter 4 Pure Bending Pure Bending Ch 2 Aial Loading & Parallel Loading: uniform normal stress and shearing stress distribution Ch 3 Torsion:
More informationy R T However, the calculations are easier, when carried out using the polar set of co-ordinates ϕ,r. The relations between the co-ordinates are:
Curved beams. Introduction Curved beams also called arches were invented about ears ago. he purpose was to form such a structure that would transfer loads, mainl the dead weight, to the ground b the elements
More informationENR202 Mechanics of Materials Lecture 12B Slides and Notes
ENR0 Mechanics of Materials Lecture 1B Slides and Notes Slide 1 Copright Notice Do not remove this notice. COMMMONWEALTH OF AUSTRALIA Copright Regulations 1969 WARNING This material has been produced and
More information3D Stress Tensors. 3D Stress Tensors, Eigenvalues and Rotations
3D Stress Tensors 3D Stress Tensors, Eigenvalues and Rotations Recall that we can think of an n x n matrix Mij as a transformation matrix that transforms a vector xi to give a new vector yj (first index
More informationSurvey of Wave Types and Characteristics
Seminar: Vibrations and Structure-Borne Sound in Civil Engineering Theor and Applications Surve of Wave Tpes and Characteristics Xiuu Gao April 1 st, 2006 Abstract Mechanical waves are waves which propagate
More informationExercise solutions: concepts from chapter 5
1) Stud the oöids depicted in Figure 1a and 1b. a) Assume that the thin sections of Figure 1 lie in a principal plane of the deformation. Measure and record the lengths and orientations of the principal
More informationTENSOR TRANSFORMATION OF STRESSES
GG303 Lecture 18 9/4/01 1 TENSOR TRANSFORMATION OF STRESSES Transformation of stresses between planes of arbitrar orientation In the 2-D eample of lecture 16, the normal and shear stresses (tractions)
More informationLecture Triaxial Stress and Yield Criteria. When does yielding occurs in multi-axial stress states?
Lecture 5.11 Triaial Stress and Yield Criteria When does ielding occurs in multi-aial stress states? Representing stress as a tensor operational stress sstem Compressive stress sstems Triaial stresses:
More information3D 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 informationEquilibrium 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 informationMathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
More informationMcKinney High School AP Calculus Summer Packet
McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work
More informationTrigonometry. Pythagorean Theorem. Force Components. Components of Force. hypotenuse. hypotenuse
Pthagorean Theorem Trigonometr B C α A c α b a + b c a opposite side sin sinα hpotenuse adjacent side cos cosα hpotenuse opposite side tan tanα adjacent side AB CB CA CB AB AC orce Components Components
More informationA short review of continuum mechanics
A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material
More informationContinuum 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 informationBone 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 informationAircraft Structures Structural & Loading Discontinuities
Universit of Liège Aerospace & Mechanical Engineering Aircraft Structures Structural & Loading Discontinuities Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationREVIEW FOR EXAM II. Dr. Ibrahim A. Assakkaf SPRING 2002
REVIEW FOR EXM II. J. Clark School of Engineering Department of Civil and Environmental Engineering b Dr. Ibrahim. ssakkaf SPRING 00 ENES 0 Mechanics of Materials Department of Civil and Environmental
More information1 HOMOGENEOUS TRANSFORMATIONS
HOMOGENEOUS TRANSFORMATIONS Purpose: The purpose of this chapter is to introduce ou to the Homogeneous Transformation. This simple 4 4 transformation is used in the geometr engines of CAD sstems and in
More informationMAE 323: Chapter 4. Plane Stress and Plane Strain. The Stress Equilibrium Equation
The Stress Equilibrium Equation As we mentioned in Chapter 2, using the Galerkin formulation and a choice of shape functions, we can derive a discretized form of most differential equations. In Structural
More information6. Bending CHAPTER OBJECTIVES
CHAPTER OBJECTIVES Determine stress in members caused by bending Discuss how to establish shear and moment diagrams for a beam or shaft Determine largest shear and moment in a member, and specify where
More informationBEAMS: SHEAR AND MOMENT DIAGRAMS (FORMULA)
LETURE Third Edition BEMS: SHER ND MOMENT DGRMS (FORMUL). J. lark School of Engineering Department of ivil and Environmental Engineering 1 hapter 5.1 5. b Dr. brahim. ssakkaf SPRNG 00 ENES 0 Mechanics
More informationME 7502 Lecture 2 Effective Properties of Particulate and Unidirectional Composites
ME 75 Lecture Effective Properties of Particulate and Unidirectional Composites Concepts from Elasticit Theor Statistical Homogeneit, Representative Volume Element, Composite Material Effective Stress-
More informationFigure 1: General Plane Motion (Translation and Rotation)
STRIN ND TH TRNSFORMTION OF STRIN INTRODUCTION - DFORMBL BODY MOTION ) Rigid Bod Motion T T Translation Rotation Figure : General Plane Motion (Translation and Rotation) Figure shows the general plane
More informationModule 2 Stresses in machine elements. Version 2 ME, IIT Kharagpur
Module Stresses in machine elements Version M, IIT Kharagpur Lesson 3 Strain analsis Version M, IIT Kharagpur Instructional Objectives At the end of this lesson, the student should learn Normal and shear
More informationCHAPER 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 informationLecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity
Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity by Borja Erice and Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling
More informationMechanics: Scalars and Vectors
Mechanics: Scalars and Vectors Scalar Onl magnitude is associated with it Vector e.g., time, volume, densit, speed, energ, mass etc. Possess direction as well as magnitude Parallelogram law of addition
More informationELASTICITY (MDM 10203)
ELASTICITY () Lecture Module 3: Fundamental Stress and Strain University Tun Hussein Onn Malaysia Normal Stress inconstant stress distribution σ= dp da P = da A dimensional Area of σ and A σ A 3 dimensional
More informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part I
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Comple Analsis II Lecture Notes Part I Chapter 1 Preliminar results/review of Comple Analsis I These are more detailed notes for the results
More informationModule 2 Stresses in machine elements
Module 2 Stresses in machine elements Lesson 3 Strain analysis Instructional Objectives At the end of this lesson, the student should learn Normal and shear strains. 3-D strain matri. Constitutive equation;
More informationTrigonometric Functions
TrigonometricReview.nb Trigonometric Functions The trigonometric (or trig) functions are ver important in our stud of calculus because the are periodic (meaning these functions repeat their values in a
More informationBy 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,
More informationCrystal Micro-Mechanics
Crystal Micro-Mechanics Lectre Classical definition of stress and strain Heng Nam Han Associate Professor School of Materials Science & Engineering College of Engineering Seol National University Seol
More informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown.
D : SOLID MECHANICS Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown. Q.2 Consider the forces of magnitude F acting on the sides of the regular hexagon having
More information2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
More informationUniversity of Pretoria Department of Mechanical & Aeronautical Engineering MOW 227, 2 nd Semester 2014
Universit of Pretoria Department of Mechanical & Aeronautical Engineering MOW 7, nd Semester 04 Semester Test Date: August, 04 Total: 00 Internal eaminer: Duration: hours Mr. Riaan Meeser Instructions:
More informationLesson 3: Free fall, Vectors, Motion in a plane (sections )
Lesson 3: Free fall, Vectors, Motion in a plane (sections.6-3.5) Last time we looked at position s. time and acceleration s. time graphs. Since the instantaneous elocit is lim t 0 t the (instantaneous)
More informationMEG 741 Energy and Variational Methods in Mechanics I
MEG 74 Energ and Variational Methods in Mechanics I Brendan J. O Toole, Ph.D. Associate Professor of Mechanical Engineering Howard R. Hghes College of Engineering Universit of Nevada Las Vegas TBE B- (7)
More informationCH.7. PLANE LINEAR ELASTICITY. Multimedia Course on Continuum Mechanics
CH.7. PLANE LINEAR ELASTICITY Multimedia Course on Continuum Mechanics Overview Plane Linear Elasticit Theor Plane Stress Simplifing Hpothesis Strain Field Constitutive Equation Displacement Field The
More informationAn angle in the Cartesian plane is in standard position if its vertex lies at the origin and its initial arm lies on the positive x-axis.
Learning Goals 1. To understand what standard position represents. 2. To understand what a principal and related acute angle are. 3. To understand that positive angles are measured by a counter-clockwise
More informationTwo Dimensional State of Stress and Strain: examples
Lecture 1-5: Two Dimensional State of Stress and Strain: examples Principal stress. Stresses on oblique plane: Till now we have dealt with either pure normal direct stress or pure shear stress. In many
More informationVYSOKÁ 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 informationBasic 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 informationMECHANICS 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,
More informationResearch Article Equivalent Elastic Modulus of Asymmetrical Honeycomb
International Scholarl Research Network ISRN Mechanical Engineering Volume, Article ID 57, pages doi:.5//57 Research Article Equivalent Elastic Modulus of Asmmetrical Honecomb Dai-Heng Chen and Kenichi
More information18 Equilibrium equation of a laminated plate (a laminate)
z ν = 1 ν = ν = 3 ν = N 1 ν = N z 0 z 1 z N 1 z N z α 1 < 0 α > 0 ν = 1 ν = ν = 3 Such laminates are often described b an orientation code [α 1 /α /α 3 /α 4 ] For eample [0/-45/90/45/0/0/45/90/-45/0] Short
More informationStress Functions. First Semester, Academic Year 2012 Department of Mechanical Engineering Chulalongkorn University
Lecture Note 3 Stress Functions First Semester, Academic Year 01 Department of Mechanical Engineering Chulalongkorn Universit 1 Objectives #1 Describe the equation of equilibrium using D/3D stress elements
More informationTHE GENERAL ELASTICITY PROBLEM IN SOLIDS
Chapter 10 TH GNRAL LASTICITY PROBLM IN SOLIDS In Chapters 3-5 and 8-9, we have developed equilibrium, kinematic and constitutive equations for a general three-dimensional elastic deformable solid bod.
More informationChapter 8 BIAXIAL BENDING
Chapter 8 BAXAL BENDN 8.1 DEFNTON A cross section is subjected to biaial (oblique) bending if the normal (direct) stresses from section are reduced to two bending moments and. enerall oblique bending is
More informationI. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.
0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:.
More informationAlgebra/Pre-calc Review
Algebra/Pre-calc Review The following pages contain various algebra and pre-calculus topics that are used in the stud of calculus. These pages were designed so that students can refresh their knowledge
More informationMechanics 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 informationGeometric Stiffness Effects in 2D and 3D Frames
Geometric Stiffness Effects in D and 3D Frames CEE 41. Matrix Structural Analsis Department of Civil and Environmental Engineering Duke Universit Henri Gavin Fall, 1 In situations in which deformations
More informationForce and Stress. Processes in Structural Geology & Tectonics. Ben van der Pluijm. WW Norton+Authors, unless noted otherwise 1/9/ :35 PM
Force and Stress Processes in Structural Geology & Tectonics Ben van der Pluijm WW Norton+Authors, unless noted otherwise 1/9/2017 12:35 PM We Discuss Force and Stress Force and Units (Trigonometry) Newtonian
More informationPLATE AND PANEL STRUCTURES OF ISOTROPIC, COMPOSITE AND PIEZOELECTRIC MATERIALS, INCLUDING SANDWICH CONSTRUCTION
PLATE AND PANEL STRUCTURES OF ISOTROPIC, COMPOSITE AND PIEZOELECTRIC MATERIALS, INCLUDING SANDWICH CONSTRUCTION SOLID MECHANICS AND ITS APPLICATIONS Volume 10 Series Editor: G.M.L. GLADWELL Department
More informationLECTURE 13 Strength of a Bar in Pure Bending
V. DEMENKO MECHNCS OF MTERLS 015 1 LECTURE 13 Strength of a Bar in Pure Bending Bending is a tpe of loading under which bending moments and also shear forces occur at cross sections of a rod. f the bending
More informationHigher. Functions and Graphs. Functions and Graphs 15
Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 5 Set Theor 5 Functions 6 Inverse Functions 9 4 Eponential Functions 0 5 Introduction to Logarithms 0 6 Radians 7 Eact Values
More informationMathematics Trigonometry: Unit Circle
a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Trigonometr: Unit Circle Science and Mathematics Education Research Group Supported b UBC Teaching and
More information1 Differential Equations for Solid Mechanics
1 Differential Eqations for Solid Mechanics Simple problems involving homogeneos stress states have been considered so far, wherein the stress is the same throghot the component nder std. An eception to
More informationTrigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
More informationMechanics 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 information9. 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 information16.20 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 informationSolutions to Assignment #01. (a) Use tail-to-head addition and the Parallelogram Law to nd the resultant, (a + b) = (b + a) :
Solutions to Assignment # Puhalskii/Kawai Section 8. (I) Demonstrate the vector sums/di erences on separate aes on the graph paper side of our engineering pad. Let a = h; i = i j and b = h; i = i + j:
More informationConsider an elastic spring as shown in the Fig.2.4. When the spring is slowly
.3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original
More information5.3 Properties of Trigonometric Functions Objectives
Objectives. Determine the Domain and Range of the Trigonometric Functions. 2. Determine the Period of the Trigonometric Functions. 3. Determine the Signs of the Trigonometric Functions in a Given Quadrant.
More informationSolutions to Problem Sheet for Week 11
THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Solutions to Problem Sheet for Week MATH9: Differential Calculus (Advanced) Semester, 7 Web Page: sydney.edu.au/science/maths/u/ug/jm/math9/
More informationMECHANICS OF MATERIALS REVIEW
MCHANICS OF MATRIALS RVIW Notation: - normal stress (psi or Pa) - shear stress (psi or Pa) - normal strain (in/in or m/m) - shearing strain (in/in or m/m) I - area moment of inertia (in 4 or m 4 ) J -
More informationThe aircraft shown is being tested to determine how the forces due to lift would be distributed over the wing. This chapter deals with stresses and
The aircraft shown is being tested to determine how the forces due to lift would be distributed over the wing. This chapter deals with stresses and strains in structures and machine components. 436 H P
More information1.1 The Equations of Motion
1.1 The Equations of Motion In Book I, balance of forces and moments acting on an component was enforced in order to ensure that the component was in equilibrium. Here, allowance is made for stresses which
More information