To determine The state of stress acting at point A on the cross section of the post at a section a - a . To indicate: the results on a differential element at the point. Explanation Given information: The width ( b ) of the cross section is 100 mm. The depth ( d ) of the cross section is 100 mm. Calculation: Show the free body diagram of the post upper segment as shown in Figure 1. Here, V y , V z is the shear force in a member is developed by a shear stress distribution in y and z direction respectively and M y , M z is the internal bending moment is developed by a normal stress distribution in y and z direction respectively. Refer Figure 1. Find the shear force values along y direction using equilibrium equation. ∑ F y = 0 V y + 3 = 0 V y = − 3 kN Take a moment about x axis. ∑ M x = 0 T = 0 Take a moment about y axis. ∑ M y = 0 M y + ( 4 × 0.4 ) = 0 M y + 1.6 = 0 M y = − 1.6 kN ⋅ m Take a moment about z axis. ∑ M z = 0 M z − ( 3 × 0.4 ) = 0 M z − 1.2 = 0 M z = 1.2 kN ⋅ m Calculate the moment of inertia ( I ) about y and z axis as follows: I y = I z = b d 3 12 (1) Substitute 100 mm for b, and 100 mm for d Equation (1). I y = I z = 100 × 100 3 12 = 8 , 333 , 333.33 mm 4 ( 1 m 4 10 12 mm 4 ) = 8.333 × 10 − 6 m 4 Show the cross section as shown in Figure 2. Find ( Q y ) A . Refer Figure 2. The point A is located at top. Therefore, the value of ( Q y ) A is zero. Find the ( Q z ) A as follows: ( Q z ) A = y ′ ¯ A (2) Refer Figure 2. Substitute 0.025 m for y ′ ¯ and ( 0.05 × 0.1 ) m for A in Equation (2). ( Q z ) A = 0.025 × 0.05 × 0.1 = 0.125 × 10 − 3 m 3 Calculate the normal stress ( σ A ) at point A as follows: ( σ A ) = − M z y I z + M y z I y (3) Here, I z , I y is the moment of inertia about z , y axis, M z , M y is the internal bending moment is developed by a normal stress distribution in z and y direction respectively, z is the distance at z axis, and y is the centroid distance. Substitute 1.2 kN ⋅ m for M z , 0 for z, − 0.05 m for y , − 1.6 kN ⋅ m for M y , and 8.333 × 10 − 6 m 4 for I z , I y in Equation (3)

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial…

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Chapter 8.2, Problem 8.55P

To determine

The state of stress acting at point A on the cross section of the post at a section a-a.

To indicate: the results on a differential element at the point.

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Chapter 8.2, Problem 8.54P

Chapter 8.2, Problem 8.56P

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The state of stress acting at point A on the cross section of the post at a section a - a . To indicate: the results on a differential element at the point.

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The state of stress acting at point A on the cross section of the post at a section a-a. To indicate: the results on a differential element at the point.

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The state of stress acting at point A on the cross section of the post at a section a-a.

To indicate: the results on a differential element at the point.