Explanation Given information: Consider the general case of a prismatic beam subjected to bending-moment components M y and M z when the x , y , z axes pass through the centroid of the cross section. If the material is linear elastic, the normal stress in the beam is linear function of position such that σ = a + b y + c z . Use the equilibrium conditions 0 = ∫ A σ d A , M y = ∫ A z σ d A , M z = ∫ A − y σ d A . Explanation: The given equilibrium condition is σ = a + b y + c z . Condition 1: The equilibrium condition 1 is as follows: 0 = ∫ A σ d A (1) Substitute ( a + b y + c z ) for σ in Equation (1). 0 = ∫ A ( a + b y + c z ) d A = a ∫ A d A + b ∫ A y d A + c ∫ A z d A (2) Condition 2: The equilibrium condition 2 is as follows: M y = ∫ A z σ d A (3) Substitute a + b y + c z for σ in Equation (3). M y = ∫ A z ( a + b y + c z ) d A = a ∫ A z d A + b ∫ A y z d A + c ∫ A z 2 d A (4) Condition 3: The equilibrium condition 3 is as follows: M z = ∫ A − y σ d A (5) Substitute a + b y + c z for σ in Equation (4). M z = ∫ A − y ( a + b y + c z ) d A = − a ∫ A y d A − b ∫ A y 2 d A − c ∫ A y z d A (6) Refer to Appendix A ‘Geometric properties of an Area’ in the textbook. The value of integrals is as follows: ∫ A y d A = ∫ A z d A = 0 The moment of inertia values of an geometric area is defined as follows: I y z = ∫ A y z d A ; I y = ∫ A z 2 d A ; I z = ∫ A y 2 d A Substitute 0 for ∫ A y d A and 0 for ∫ A z d A in Equation (2). 0 = a [ A ] 0 A + b ( 0 ) + c ( 0 ) 0 = a [ A − 0 ] a A = 0 A ≠ 0 ; a = 0 Thus, the value of constant a is a = 0 _

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 11.5, Problem 98P

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Find the constants a, b, and c.

Show that the normal stress can be determined from the equation σ=−(MzIy+MyIyz)y+(MyIz+MzIyz)zIyIz−Iyz2.

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Chapter 11.5, Problem 97P

Chapter 11.5, Problem 99P

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Chapter 11 Solutions

Statics and Mechanics of Materials (5th Edition)

Ch. 11.2 - In each case, the beam is subjected to the...Ch. 11.2 - In each ease, express the shear and moment...Ch. 11.2 - In each ease, express the shear and moment...Ch. 11.2 - In each ease, express the shear and moment...Ch. 11.2 - In each ease, express the shear and moment...Ch. 11.2 - Prob. 5FP Ch. 11.2 - Prob. 6FP Ch. 11.2 - In each ease, draw the shear and moment diagrams...Ch. 11.2 - Prob. 8FP Ch. 11.2 - Prob. 1P

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Find the constants a , b , and c . Show that the normal stress can be determined from the equation σ = − ( M z I y + M y I y z ) y + ( M y I z + M z I y z ) z I y I z − I y z 2 .

Solution Summary: The author explains how the normal stress can be determined from the equation sigma =-(M_y)y+

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Find the constants a, b, and c.

Show that the normal stress can be determined from the equation σ=−(MzIy+MyIyz)y+(MyIz+MzIyz)zIyIz−Iyz2.

Want to see the full answer?

Chapter 11 Solutions

Find the constants a , b , and c . Show that the normal stress can be determined from the equation σ = − ( M z I y + M y I y z ) y + ( M y I z + M z I y z ) z I y I z − I y z 2 .

Solution Summary: The author explains how the normal stress can be determined from the equation sigma =-(M_y)y+

Videos

Find the constants a, b, and c. Show that the normal stress can be determined from the equation σ=−(MzIy+MyIyz)y+(MyIz+MzIyz)zIyIz−Iyz2.

Want to see the full answer?

Chapter 11 Solutions

Find the constants a, b, and c.

Show that the normal stress can be determined from the equation σ=−(MzIy+MyIyz)y+(MyIz+MzIyz)zIyIz−Iyz2.