(a) Explanation Given information: The radius of the pipe is r m . The thickness of the beam is t The beam is subjected to a vertical shear of V. Calculation: Sketch the cross section as shown in Figure 1. Refer to Figure 1. Calculate the area of the cross section as shown below. A = 2 π r m t Here, r m is the radius and t is the thickness. Calculate the polar moment of inertia as shown below. J = A r m 2 Here, A is the cross sectional area. Substitute 2 π r m t for A . J = 2 π r m t × r m 2 = 2 π t r m 3 Calculate the moment of inertia as shown below. I = J 2 Substitute 2 π t r m 3 for J . I = 2 π t r m 3 2 = π t r m 3 Calculate the distance of point P from the neutral axis as shown below. r ¯ = r m sin θ θ Calculate the area at point P as shown below. A P = 2 r m θ t Calculate the first moment of area as shown below (b) To determine Show that the maximum shearing stress occurs for θ = 90 ° is equal to 2 V A .

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Author: BEER

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Chapter 6.5, Problem 54P

(a)

To determine

The shearing stress at point P of a thin-walled pipe.

(b)

To determine

Show that the maximum shearing stress occurs for θ=90° is equal to 2VA.

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Chapter 6.5, Problem 53P

Chapter 6.5, Problem 55P

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The shearing stress at point P of a thin-walled pipe.

Show that the maximum shearing stress occurs for θ=90° is equal to 2VA.

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Chapter 6 Solutions