Explanation Given information: The point load act at both ends of the shaft is P . The flexural rigidity of the shaft is E I . The length of the shaft is L . Calculation: Show the free body diagram of the shaft as in Figure 1. Refer to Figure 1. Calculate the vertical forces by applying the Equation of equilibrium: Sum of vertical forces is equal to 0. ∑ F y = 0 R A + R B − P − P = 0 R A + R B − 2 P = 0 R A + R B = 2 P (1) Calculate the moment about point A by applying the Equation of equilibrium: Sum of moments about point A is equal to 0. ∑ M A = 0 [ R B × ( L − 2 a ) ] − [ P × ( L − a ) ] + ( P × a ) = 0 ( L − 2 a ) R B − P L + P a + P a = 0 ( L − 2 a ) R B − P ( L − 2 a ) = 0 ( L − 2 a ) R B = P ( L − 2 a ) R B = P ( L − 2 a ) ( L − 2 a ) R B = P Substitute P for R B in Equation (1). R A + P = 2 P R A = 2 P − P R A = P Show the moment area diagram of the shaft as in Figure 2. Refer to Figure 2. Calculate the bending moment at A : M A = − [ P × ( L − a ) ] + [ P × ( L − 2 a ) ] = − P L + P a + P L − 2 P a = − P a Calculate the bending moment at B : M B = − ( P × a ) = − P a Calculate the bending moment at C : M C = − ( P × 0 ) = 0 Calculate the bending moment at D : M D = − ( P × 0 ) = 0 Show the deflected shape of the shaft as in Figure 3. Refer to Figure 3. The section is symmetry. Therefore, the slope at mid span ( θ E ) is 0. Calculate the vertical distance between the tangent at a point A and E ( t A / E ) : t A / E = ( − P a E I ) × ( L − 2 a ) 2 × ( L − 2 a ) 4 = − P a 8 E I ( L − 2 a ) 2 Calculate the deflection at E ( υ E ) using the formula: υ E = | t A / E | (2) Substitute [ − P a 8 E I ( L − 2 a ) 2 ] for t A / E in Equation (2)

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ISBN: 9780134319650

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 12.4, Problem 12.73P

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To calculate: the distance (a) of the journal bearing supports at A and B.

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Chapter 12.4, Problem 12.72P

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the distance ( a ) of the journal bearing supports at A and B .

Solution Summary: The author calculates the distance of the journal bearing supports at A and B by applying the Equation of equilibrium: Sum of vertical forces is equal to 0.

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