To determine The state of stress ( σ A ) at point A of rod. The state of stress ( τ A ) at point A of rod. Explanation Given information: The diameter of rod is 1 in. The radius of the rod is 0.5 in. Calculation: Show the diameter of rod as shown in Figure 1. M z is moment at the section about z axis, N x is normal force at the section about x axis, V y , V z is shear force at the section about y , z axis. Refer to Figure 1: The equilibrium forces along z direction. ∑ F z = 0 V z + 100 = 0 V z = − 100 lb The equilibrium forces along x direction. ∑ F x = 0 N x − 75 = 0 N x = 75 lb The equilibrium forces along y direction. ∑ F y = 0 V y = 80 lb Take a moment about z axis. ∑ M z = 0 M z + 80 ( 8 ) = 0 M z + 640 = 0 M z = − 640 lb ⋅ in Take a moment about x axis. ∑ M x = 0 T x + ( 80 × 3 ) = 0 T x = − 240 lb ⋅ in Take a moment about y axis. ∑ M y = 0 M y + 100 ( 8 ) − 75 ( 3 ) = 0 M y + 800 − 225 = 0 M y = − 575 lb ⋅ in Calculate the area ( A ) of the rod as shown in below: A = π 4 d 2 (1) Here d is the diameter of the shaft. A = π 4 ( 1 ) 2 = 1 4 π = 0.78539 in 2 Calculate the polar moment of inertia ( J ) of the rod as follows: J = 1 2 π ( c 4 ) (2) Here, c is the radius of the shaft. Substitute 0.5 in. for c in Equation (2). J = 1 2 π ( 0.5 4 ) = 1 2 π ( 0.5 ) 4 = 0.09817 in 4 Show the cross section of the rod as shown in Figure 2. Refer Figure 2: Find the first moment area at point A along y axis. A is located at the top of the surface. Since A is zero. ( Q y ) A = 0 Find the first moment area at point A as follows: ( Q A ) A = y ¯ ′ A (3) Modify Equation (3). ( Q A ) A = 4 r 3 π 1 2 ( π × r 2 ) (4) Substitute 0.5 in. for r in Equation (4). ( Q A ) A = 4 ( 0.5 ) 3 π 1 2 ( π × 0.5 2 ) = 0

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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 8.53P

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The state of stress (σA) at point A of rod.

The state of stress (τA) at point A of rod.

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Chapter 8.2, Problem 8.52P

Chapter 8.2, Problem 8.54P

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The state of stress ( σ A ) at point A of rod. The state of stress ( τ A ) at point A of rod.

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The state of stress (σA) at point A of rod.

The state of stress (τA) at point A of rod.