Book cover for Mechanics of Materials

Mechanics of Materials

Ferdinand P. Beer, E. Russell Johnston, Jr., John T. DeWolf

ISBN #9781260113273

8th Edition

1,507 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter has provided an in?depth examination of the transformation of stress and strain in two-dimensional (plane stress and plane strain) settings. Key techniques include deriving rotated stress components using transformation equations and employing Mohr's circle to graphically determine principal stresses, maximum shearing stresses, and strain orientations. In addition, applications to thin-walled pressure vessels show how these principles guide the design of cylinders and spheres under internal pressure. Understanding yield criteria and the use of strain rosettes further bridges theory with experimental practice, ensuring that engineers can confidently analyze complex loading scenarios in practical structures.

Learning Objectives

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Key Concepts

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Example Problems

Example 1

For the given state of stress, determine the normal and shearing stresses exerted on the oblique face of the shaded triangular element shown. Use a method of analysis based on the equilibrium of that element, as was done in the derivations of Sec. $7.1 \mathrm{A}$

Example 2

For the given state of stress, determine the normal and shearing stresses exerted on the oblique face of the shaded triangular element shown. Use a method of analysis based on the equilibrium of that element, as was done in the derivations of Sec. $7.1 \mathrm{A}$

Example 3

For the given state of stress, determine the normal and shearing stresses exerted on the oblique face of the shaded triangular element shown. Use a method of analysis based on the equilibrium of that element, as was done in the derivations of Sec. $7.1 \mathrm{A}$

Example 4

For the given state of stress, determine the normal and shearing stresses exerted on the oblique face of the shaded triangular element shown. Use a method of analysis based on the equilibrium of that element, as was done in the derivations of Sec. $7.1 \mathrm{A}$

Example 5

For the given state of stress, determine $(a)$ the principal planes, ( $b$ ) the principal stresses.
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Step-by-Step Explanations

QUESTION

Given a state of plane stress with \u03c3x, \u03c3y, and \u03c4xy, how do you find the stress components \u03c3x\u2032, \u03c3y\u2032, and \u03c4x\u2032y\u2032 for an element rotated by an angle \u03b8?\nStep-by-step Answer:\nStep 1: Begin with the transformation equations: \u03c3x\u2032 = (\u03c3x+\u03c3y)/2 + (\u03c3x\u2212\u03c3y)/2 cos 2\u03b8 + \u03c4xy sin 2\u03b8.\nStep 2: Similarly, compute \u03c3y\u2032 = (\u03c3x+\u03c3y)/2 \u2212 (\u03c3x\u2212\u03c3y)/2 cos 2\u03b8 \u2212 \u03c4xy sin 2\u03b8.\nStep 3: The transformed shear stress is given by: \u03c4x\u2032y\u2032 = \u2212(\u03c3x\u2212\u03c3y)/2 sin 2\u03b8 + \u03c4xy cos 2\u03b8.\nStep 4: Substitute the given values and the rotation angle \u03b8 into these equations to obtain the new stress components.\nFinal Answer: The rotated stresses are found directly from the above formulas.\n\n- Topic: Using Mohr\u2019s Circle to Find Principal Stresses \nQuestion: How can you determine the principal stresses from a given state of plane stress using Mohr\u2019s circle?\nStep-by-step Answer:\nStep 1: Plot the point X corresponding to (\u03c3x, \u2212\u03c4xy) and point Y corresponding to (\u03c3y, +\u03c4xy) on a stress coordinate system.\nStep 2: Draw a line connecting X and Y; its midpoint is the average stress \u03c3ave = (\u03c3x+\u03c3y)/2.\nStep 3: The radius R of Mohr\u2019s circle is computed as R = \u221a[((\u03c3x\u2212\u03c3y)/2)\u00b2 + \u03c4xy\u00b2].\nStep 4: The principal stresses are then given by \u03c3max = \u03c3ave + R and \u03c3min = \u03c3ave \u2212 R.\nStep 5: The angle that rotates the element to the principal plane is given by tan 2\u03b8p = 2\u03c4xy/(\u03c3x\u2212\u03c3y).\nFinal Answer: Principal stresses and orientation are obtained using the above relationships.\n\n- Topic: Determining Maximum Shearing Stress \nQuestion: How is the maximum in-plane shearing stress determined from a given state of plane stress?\nStep-by-step Answer:\nStep 1: Recognize that the maximum shearing stress is the radius of Mohr\u2019s circle, i.e., R.\nStep 2: Compute R using R = \u221a[((\u03c3x \u2212 \u03c3y)/2)\u00b2 + \u03c4xy\u00b2].\nStep 3: The corresponding normal stress on the plane where maximum shear occurs is \u03c3ave = (\u03c3x+\u03c3y)/2.\nFinal Answer: \u03c4max = R represents the maximum in-plane shearing stress.\n\n- Topic: Transformation of Strain \nQuestion: How do you obtain the normal strain \u03b5(\u03b8) in a direction at an angle \u03b8 from the x axis in terms of \u03b5x, \u03b5y, and \u03b3xy?\nStep-by-step Answer:\nStep 1: Start with a deformed square element and apply the law of cosines to its sides to relate the deformed length in an arbitrary direction.\nStep 2: For small strains, derive the linearized equation: \u03b5(\u03b8) = \u03b5x cos\u00b2\u03b8 + \u03b5y sin\u00b2\u03b8 + \u03b3xy sin\u03b8 cos\u03b8.\nStep 3: Validate by checking for \u03b8=0\u00b0 (\u03b5(0)=\u03b5x) and \u03b8=90\u00b0 (\u03b5(90)=\u03b5y).\nFinal Answer: The normal strain in any direction is given by \u03b5(\u03b8)= \u03b5x cos\u00b2\u03b8 + \u03b5y sin\u00b2\u03b8 + \u03b3xy sin\u03b8 cos\u03b8.\n\n"

STEP-BY-STEP ANSWER:

Step 1: Begin with the transformation equations: \u03c3x\u2032 = (\u03c3x+\u03c3y)/2 + (\u03c3x\u2212\u03c3y)/2 cos 2\u03b8 + \u03c4xy sin 2\u03b8.\nStep 2: Similarly, compute \u03c3y\u2032 = (\u03c3x+\u03c3y)/2 \u2212 (\u03c3x\u2212\u03c3y)/2 cos 2\u03b8 \u2212 \u03c4xy sin 2\u03b8.\nStep 3: The transformed shear stress is given by: \u03c4x\u2032y\u2032 = \u2212(\u03c3x\u2212\u03c3y)/2 sin 2\u03b8 + \u03c4xy cos 2\u03b8.\nStep 4: Substitute the given values and the rotation angle \u03b8 into these equations to obtain the new stress components.\nFinal Answer: The rotated stresses are found directly from the above formulas.\n\n- Topic: Using Mohr\u2019s Circle to Find Principal Stresses \nQuestion: How can you determine the principal stresses from a given state of plane stress using Mohr\u2019s circle?\nStep-by-step Answer:\nStep 1: Plot the point X corresponding to (\u03c3x, \u2212\u03c4xy) and point Y corresponding to (\u03c3y, +\u03c4xy) on a stress coordinate system.\nStep 2: Draw a line connecting X and Y; its midpoint is the average stress \u03c3ave = (\u03c3x+\u03c3y)/2.\nStep 3: The radius R of Mohr\u2019s circle is computed as R = \u221a[((\u03c3x\u2212\u03c3y)/2)\u00b2 + \u03c4xy\u00b2].\nStep 4: The principal stresses are then given by \u03c3max = \u03c3ave + R and \u03c3min = \u03c3ave \u2212 R.\nStep 5: The angle that rotates the element to the principal plane is given by tan 2\u03b8p = 2\u03c4xy/(\u03c3x\u2212\u03c3y).\nFinal Answer: Principal stresses and orientation are obtained using the above relationships.\n\n- Topic: Determining Maximum Shearing Stress \nQuestion: How is the maximum in-plane shearing stress determined from a given state of plane stress?\nStep-by-step Answer:\nStep 1: Recognize that the maximum shearing stress is the radius of Mohr\u2019s circle, i.e., R.\nStep 2: Compute R using R = \u221a[((\u03c3x \u2212 \u03c3y)/2)\u00b2 + \u03c4xy\u00b2].\nStep 3: The corresponding normal stress on the plane where maximum shear occurs is \u03c3ave = (\u03c3x+\u03c3y)/2.\nFinal Answer: \u03c4max = R represents the maximum in-plane shearing stress.\n\n- Topic: Transformation of Strain \nQuestion: How do you obtain the normal strain \u03b5(\u03b8) in a direction at an angle \u03b8 from the x axis in terms of \u03b5x, \u03b5y, and \u03b3xy?\nStep-by-step Answer:\nStep 1: Start with a deformed square element and apply the law of cosines to its sides to relate the deformed length in an arbitrary direction.\nStep 2: For small strains, derive the linearized equation: \u03b5(\u03b8) = \u03b5x cos\u00b2\u03b8 + \u03b5y sin\u00b2\u03b8 + \u03b3xy sin\u03b8 cos\u03b8.\nStep 3: Validate by checking for \u03b8=0\u00b0 (\u03b5(0)=\u03b5x) and \u03b8=90\u00b0 (\u03b5(90)=\u03b5y).\nFinal Answer: The normal strain in any direction is given by \u03b5(\u03b8)= \u03b5x cos\u00b2\u03b8 + \u03b5y sin\u00b2\u03b8 + \u03b3xy sin\u03b8 cos\u03b8.\n\n"
Final Answer: The rotated stresses are found directly from the above formulas.\n\n- Topic: Using Mohr\u2019s Circle to Find Principal Stresses \nQuestion: How can you determine the principal stresses from a given state of plane stress using Mohr\u2019s circle?\nStep-by-step Answer:\nStep 1: Plot the point X corresponding to (\u03c3x, \u2212\u03c4xy) and point Y corresponding to (\u03c3y, +\u03c4xy) on a stress coordinate system.\nStep 2: Draw a line connecting X and Y; its midpoint is the average stress \u03c3ave = (\u03c3x+\u03c3y)/2.\nStep 3: The radius R of Mohr\u2019s circle is computed as R = \u221a[((\u03c3x\u2212\u03c3y)/2)\u00b2 + \u03c4xy\u00b2].\nStep 4: The principal stresses are then given by \u03c3max = \u03c3ave + R and \u03c3min = \u03c3ave \u2212 R.\nStep 5: The angle that rotates the element to the principal plane is given by tan 2\u03b8p = 2\u03c4xy/(\u03c3x\u2212\u03c3y).\nFinal Answer: Principal stresses and orientation are obtained using the above relationships.\n\n- Topic: Determining Maximum Shearing Stress \nQuestion: How is the maximum in-plane shearing stress determined from a given state of plane stress?\nStep-by-step Answer:\nStep 1: Recognize that the maximum shearing stress is the radius of Mohr\u2019s circle, i.e., R.\nStep 2: Compute R using R = \u221a[((\u03c3x \u2212 \u03c3y)/2)\u00b2 + \u03c4xy\u00b2].\nStep 3: The corresponding normal stress on the plane where maximum shear occurs is \u03c3ave = (\u03c3x+\u03c3y)/2.\nFinal Answer: \u03c4max = R represents the maximum in-plane shearing stress.\n\n- Topic: Transformation of Strain \nQuestion: How do you obtain the normal strain \u03b5(\u03b8) in a direction at an angle \u03b8 from the x axis in terms of \u03b5x, \u03b5y, and \u03b3xy?\nStep-by-step Answer:\nStep 1: Start with a deformed square element and apply the law of cosines to its sides to relate the deformed length in an arbitrary direction.\nStep 2: For small strains, derive the linearized equation: \u03b5(\u03b8) = \u03b5x cos\u00b2\u03b8 + \u03b5y sin\u00b2\u03b8 + \u03b3xy sin\u03b8 cos\u03b8.\nStep 3: Validate by checking for \u03b8=0\u00b0 (\u03b5(0)=\u03b5x) and \u03b8=90\u00b0 (\u03b5(90)=\u03b5y).\nFinal Answer: The normal strain in any direction is given by \u03b5(\u03b8)= \u03b5x cos\u00b2\u03b8 + \u03b5y sin\u00b2\u03b8 + \u03b3xy sin\u03b8 cos\u03b8.\n\n"

"- Topic: Determining Rotated Stress Components \nQuestion: Given a state of plane stress with \u03c3x, \u03c3y, and \u03c4xy, how do you find the stress components \u03c3x\u2032, \u03c3y\u2032, and \u03c4x\u2032y\u2032 for an element rotated by an angle \u03b8?\nStep-by-step Answer:\nStep 1: Begin with the transformation equations: \u03c3x\u2032 = (\u03c3x+\u03c3y)/2 + (\u03c3x\u2212\u03c3y)/2 cos 2\u03b8 + \u03c4xy sin 2\u03b8.\nStep 2: Similarly, compute \u03c3y\u2032 = (\u03c3x+\u03c3y)/2 \u2212 (\u03c3x\u2212\u03c3y)/2 cos 2\u03b8 \u2212 \u03c4xy sin 2\u03b8.\nStep 3: The transformed shear stress is given by: \u03c4x\u2032y\u2032 = \u2212(\u03c3x\u2212\u03c3y)/2 sin 2\u03b8 + \u03c4xy cos 2\u03b8.\nStep 4: Substitute the given values and the rotation angle \u03b8 into these equations to obtain the new stress components.\nFinal Answer: The rotated stresses are found directly from the above formulas.\n\n- Topic: Using Mohr\u2019s Circle to Find Principal Stresses \nQuestion: How can you determine the principal stresses from a given state of plane stress using Mohr\u2019s circle?\nStep-by-step Answer:\nStep 1: Plot the point X corresponding to (\u03c3x, \u2212\u03c4xy) and point Y corresponding to (\u03c3y, +\u03c4xy) on a stress coordinate system.\nStep 2: Draw a line connecting X and Y; its midpoint is the average stress \u03c3ave = (\u03c3x+\u03c3y)/2.\nStep 3: The radius R of Mohr\u2019s circle is computed as R = \u221a[((\u03c3x\u2212\u03c3y)/2)\u00b2 + \u03c4xy\u00b2].\nStep 4: The principal stresses are then given by \u03c3max = \u03c3ave + R and \u03c3min = \u03c3ave \u2212 R.\nStep 5: The angle that rotates the element to the principal plane is given by tan 2\u03b8p = 2\u03c4xy/(\u03c3x\u2212\u03c3y).\nFinal Answer: Principal stresses and orientation are obtained using the above relationships.\n\n- Topic: Determining Maximum Shearing Stress \nQuestion: How is the maximum in-plane shearing stress determined from a given state of plane stress?\nStep-by-step Answer:\nStep 1: Recognize that the maximum shearing stress is the radius of Mohr\u2019s circle, i.e., R.\nStep 2: Compute R using R = \u221a[((\u03c3x \u2212 \u03c3y)/2)\u00b2 + \u03c4xy\u00b2].\nStep 3: The corresponding normal stress on the plane where maximum shear occurs is \u03c3ave = (\u03c3x+\u03c3y)/2.\nFinal Answer: \u03c4max = R represents the maximum in-plane shearing stress.\n\n- Topic: Transformation of Strain \nQuestion: How do you obtain the normal strain \u03b5(\u03b8) in a direction at an angle \u03b8 from the x axis in terms of \u03b5x, \u03b5y, and \u03b3xy?\nStep-by-step Answer:\nStep 1: Start with a deformed square element and apply the law of cosines to its sides to relate the deformed length in an arbitrary direction.\nStep 2: For small strains, derive the linearized equation: \u03b5(\u03b8) = \u03b5x cos\u00b2\u03b8 + \u03b5y sin\u00b2\u03b8 + \u03b3xy sin\u03b8 cos\u03b8.\nStep 3: Validate by checking for \u03b8=0\u00b0 (\u03b5(0)=\u03b5x) and \u03b8=90\u00b0 (\u03b5(90)=\u03b5y).\nFinal Answer: The normal strain in any direction is given by \u03b5(\u03b8)= \u03b5x cos\u00b2\u03b8 + \u03b5y sin\u00b2\u03b8 + \u03b3xy sin\u03b8 cos\u03b8.\n\n"

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Common Mistakes

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