Explanation Given information: The beam is provided with unsymmetrical cross-section. The couple M 0 is applied on the beam acting along the horizontal plane x z . Calculation: The variation of stress ( σ A ) at A with the coordinates y and z are linear. The y and z axis are the centroidal axes as the axial force acting on the cross-section is zero. Then, Show the expression for the stress at A as follows: σ A = C 1 y + C 2 z (`1) Here, C 1 and C 2 are constants. Calculate the moment of the couple, M z using the relation: M z = − ∫ y σ A d A (2) Substitute C 1 y + C 2 z for σ A in Equation (2). M z = − ∫ y ( C 1 y + C 2 z ) d A = − C 1 ∫ y 2 d A − C 2 ∫ y z d A Equate the moment M z to zero as follows: M z = 0 − C 1 ( ∫ y 2 d A ) − C 2 ( ∫ y z d A ) = 0 − C 1 ( I z ) − C 2 ( I y z ) = 0 C 1 I z = − C 2 I y z C 1 = − C 2 I y z I z (3) Calculate the moment of the couple, M y using the relation: M y = ∫ z σ A d A (4) Substitute C 1 y + C 2 z for σ A in Equation (4). M y = ∫ z ( C 1 y + C 2 z ) d A = C 1 ( ∫ y z d A ) + C 2 ( ∫ z 2 d A ) = C 1 ( I y z ) + C 2 ( I y ) (5) Substitute ( − C 2 I y z I z ) for C 1 in Equation (5)

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ISBN: 8220102804487

Author: BEER

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Chapter 4.9, Problem 158P

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Show the stress at point A, of coordinate y and z is

σA=zIz−yIyzIyIz−Iyz2

Here, Iy,Iz and Iyz are the moments and product of inertia of the cross-section with respect to coordinate axes, and My is the moment of the couple.

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Chapter 4.9, Problem 157P

Chapter 4.9, Problem 159P

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Show the stress at point A , of coordinate y and z is σ A = z I z − y I y z I y I z − I y z 2 Here, I y , I z and I y z are the moments and product of inertia of the cross-section with respect to coordinate axes, and M y is the moment of the couple.

Solution Summary: The author shows the stress at point A with the coordinates y and z are linear, and the moment of the couple.

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Show the stress at point A, of coordinate y and z is

σA=zIz−yIyzIyIz−Iyz2

Here, Iy,Iz and Iyz are the moments and product of inertia of the cross-section with respect to coordinate axes, and My is the moment of the couple.

Want to see the full answer?

Chapter 4 Solutions

Show the stress at point A , of coordinate y and z is σ A = z I z − y I y z I y I z − I y z 2 Here, I y , I z and I y z are the moments and product of inertia of the cross-section with respect to coordinate axes, and M y is the moment of the couple.

Solution Summary: The author shows the stress at point A with the coordinates y and z are linear, and the moment of the couple.

Concept explainers

Videos

Show the stress at point A, of coordinate y and z is σA=zIz−yIyzIyIz−Iyz2 Here, Iy,Iz and Iyz are the moments and product of inertia of the cross-section with respect to coordinate axes, and My is the moment of the couple.

Want to see the full answer?

Chapter 4 Solutions

Show the stress at point A, of coordinate y and z is

σA=zIz−yIyzIyIz−Iyz2

Here, Iy,Iz and Iyz are the moments and product of inertia of the cross-section with respect to coordinate axes, and My is the moment of the couple.