Explanation Given information: The point load act atpoint D of the beam is P . The flexural rigidity of the beam is E I . The length of the beam is 2 L . The length of segment AB is L . Calculation: Show the free bodydiagram of the beam 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 + R C − P = 0 R A + R B + R C = 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 ) + ( R C × 2 L ) − ( P × 3 L 2 ) = 0 L R B + 2 L R C − 3 P L 2 = 0 L R B = 3 P L 2 − 2 L R C R B = 3 P 2 − 2 R C (2) Substitute ( 3 P 2 − 2 R C ) for R B in Equation (1). R A + 3 P 2 − 2 R C + R C = P R A + 3 P 2 − R C − P = 0 R A + P 2 − R C = 0 R A = R C − P 2 (3) Substitute ( R C − P 2 ) for R A and ( 3 P 2 − 2 R C ) for R B in Equation (1). R C − P 2 + 3 P 2 − 2 R C + R C = P R C + P − R C − P = 0 0 R C = 0 R C = 0 Substitute 0 for R C in Equation (2). R B = 3 P 2 − 2 ( 0 ) = 3 P 2 Substitute 0 for R C in Equation (3). R A = 0 − P 2 = − P 2 = P 2 ↓ Show the moment area of the beam as in Figure 2. Refer to Figure 2. Show the bending monument calculation follows: Calculate the bending moment at A : M A = ( 0 × 2 L ) − ( P × 3 L 2 ) + ( 3 P 2 × L ) = 0 − 3 P L 2 + 3 P L 2 = 0 Calculate the bending moment at B : M B = ( 0 × L ) − ( P × L 2 ) = 0 − P L 2 = − P L 2 Calculate the bending moment at C : M C = ( 0 × 0 ) = 0 Calculate the bending moment at D : M D = ( 0 × L 2 ) = 0 Show the deflected shape of the shaft as in Figure 3

9th Edition

ISBN: 9789810694364

Author: Russell C Hibbeler

Publisher: Pearson Education

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Slope and Deflection

The slope in a deflected beam is described as an angle in radians made by the tangent of the section with the original axis of the beam. The deflection at any point on the axis of the beam is defined as the lateral displacement of the point before and af…

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Chapter 12.4, Problem 12.81P

To determine

Theslope at A(θA).

The deflection at D(υD).

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Chapter 12.4, Problem 12.80P

Chapter 12.4, Problem 12.82P

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