To determine The deflection at the center of the beam ( υ C ) . The slope at B of the beam ( θ B ) . Explanation Given information: The moment at left end of the beam is M 0 . The flexural rigidity of the beam is EI . The length of the beam is L . Calculation: Show the free body diagram of the beam as in Figure 1. Refer to Figure 1. Calculate the support reactions A and B ( A y and B y ) : Calculate the vertical forces by applying the Equation of equilibrium: Sum of vertical forces is equal to zero. ∑ F y = 0 A y + B y = 0 (1) Calculate the moment about point A by applying the Equation of equilibrium: Sum of moments about point A is equal to zero. ∑ M A = 0 ( B y × L ) − M 0 = 0 ( B y × L ) = M 0 B y = M 0 L Substitute M 0 L for B y in Equation (1). A y + M 0 L = P A y = − M 0 L A y = M 0 L ( ↓ ) Show the free body diagram of the moment function of the beam’s cut segment as in Figure 2. Refer Figure 2. Calculate the moment function with respect of the x distance M ( x ) : M ( x ) = M 0 ( 1 − x L ) Write the bending equation to determine the bending moment as follows: E I d 2 υ d x 2 = M ( x ) (2) For the coordinate x , Substitute M 0 ( 1 − x L ) for M ( x ) in equation (2). E I d 2 υ d x 2 = M 0 ( 1 − x L ) (3) Integrate both the sides of the Equation (3). E I d υ d x = M 0 ( x − x 2 2 L ) + C 1 (4) Integrate both the sides of the Equation (4). E I υ = M 0 ( x 2 2 − x 3 6 L ) + C 1 x + C 2 (5) Boundary conditions are, Condition 1: υ = 0 at x = 0 Condition 2: υ = 0 at x = L Apply the first boundary condition in Equation (5). Substitute 0 for υ and 0 for x in Equation (5). E I ( 0 ) = M 0 ( 0 2 2 − 0 3 6 L ) + C 1 ( 0 ) + C 2 C 2 = 0 Apply the second boundary condition in Equation (5). Substitute 0 for υ and L for x in Equation (5). E I ( 0 ) = M 0 ( L 2 2 − L 3 6 L ) + C 1 ( L ) + C 2 0 = M 0 ( L 2 2 − L 2 6 ) + C 1 L + C 2 (6) Substitute 0 for C 2 in Equation (6). 0 = M 0 ( L 2 2 − L 2 6 ) + C 1 L + 0 C 1 L = − M 0 ( L 2 2 − L 2 6 ) C 1 = − M 0 L 2 3 L = − M 0 L 3 Substitute ( − M 0 L 3 ) for C 1 in Equation (4)

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ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 12.2, Problem 12.11P

To determine

The deflection at the center of the beam (υC) .

The slope at B of the beam (θB) .

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Chapter 12.2, Problem 12.10P

Chapter 12.2, Problem 12.12P

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Mechanics of Materials

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