(a) Explanation Show the free-body diagram of the prismatic beam as in Figure 1. Determine the vertical reaction at point B by taking moment at point A . ∑ M A = 0 − ( w × 1.2 ) × 0.6 + B y ( 1.2 ) = 0 B y = 0.6 w Determine the vertical reaction at point A by resolving the vertical component of forces. ∑ F y = 0 A y − ( w × 1.2 ) + B y = 0 A y − 1.2 w + 0.6 w = 0 A y = 0.6 w Show the free-body diagram of the section as in Figure 2. Determine the moment at the section by taking moment about the section. ∑ M = 0 M + ( w × x ) × x 2 − 0.6 w x = 0 M + w x 2 2 − 0.6 w x = 0 M = 0.6 w x − w x 2 2 Find the height ( h ) of the tapered beam using the relation. h = a + k x Here, the width of the beam at the ends is a , the height of the beam at mid-point is h , and the length of the beam from left end is x . Substitute 300 mm for h , 120 mm for a , and 0.6 m for x . 300 = 120 + k ( 0.6 ) k = 300 − 120 0.6 = 300 mm/m Find the section modulus ( S ) of the beam using the relation. S = 1 6 b h 2 Here, the width of the beam is b and the depth of the beam is h . Substitute ( a + k x ) for h . S = 1 6 b ( a + k x ) 2 Determine the bending stress ( σ ) in the beam using the relation. σ = M S Substitute ( 0.6 w x − w x 2 2 ) for M and 1 6 b ( a + k x ) 2 for S . σ = 0.6 w x − w x 2 2 1 6 b ( a + k x ) 2 = 3.6 w x − 3 w x 2 b ( a + k x ) 2 Apply the condition d σ d x = 0 to determine the location of the maximum bending stress. d σ d x = d ( 3 (b) To determine Find the maximum normal stress.

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Chapter 5.5, Problem 149P

(a)

To determine

Find the distance of the transverse section at which the maximum normal stress occurs.

(b)

To determine

Find the maximum normal stress.

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Chapter 5.5, Problem 148P

Chapter 5.5, Problem 150P

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Find the distance of the transverse section at which the maximum normal stress occurs.

Solution Summary: The author shows the free-body diagram of the prismatic beam as in Figure 1. Determine the vertical reaction at point A by resolving vertical component of forces.

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Find the distance of the transverse section at which the maximum normal stress occurs.

Find the maximum normal stress.

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