Explanation Given information: The length of the beam is 2 L . EI is constant. Calculation: Sketch the Free Body Diagram of the beam as shown in Figure 1. Refer to Figure 1. Find the support reactions as shown below. Apply the Equations of Equilibrium as shown below. Summation of forces along horizontal direction is Equal to zero. ∑ F x = 0 B x = 0 Summation of forces along vertical direction is Equal to zero. ∑ F y = 0 − A y + B y + C y − P = 0 (1) Take moment about A is Equal to zero. ∑ M A = 0 B y L + C y × 2 L − P × 3 2 L = 0 B y L + 2 C y L − 3 P L 2 = 0 (2) Sketch the elastic curve of the beam as shown in Figure 2. Refer to Figure 2. t A / C = ( t A / C ) 1 + ( t A / C ) 2 (3) t B / C = ( t B / C ) 1 + ( t B / C ) 2 (4) t A / C = 2 t B / C (5) Sketch the M E I diagram as shown in Figure 3. Refer to Figure 3. Apply the moment area theorem for deflection as shown below. Deflection at A with respect to C due to the applied load. ( t A / C ) 1 = 1 2 ( 3 P L 8 E I ) ( 3 L 2 ) ( 2 3 × 3 L 2 ) + 1 2 ( 3 P L 8 E I ) ( L 2 ) ( 3 L 2 + L 6 ) = 9 P L 3 32 E I + 3 P L 3 32 E I × 9 + 1 6 = 9 P L 3 + 5 P L 3 32 E I = 7 P L 3 16 E I Deflection at A with respect to C due to the reaction at B . ( t A / C ) 2 = 1 2 ( − B y L 2 E I ) 2 L × L = − B y L 3 2 E I Deflection at B with respect to C due to the applied load. ( t B / C ) 1 = ( 1 2 ( P L 8 E I ) ( L 2 ) ( 2 3 × L 2 ) + ( P L 4 E I ) ( L 2 ) ( L 4 ) + 1 2 ( 3 P L 8 E I ) ( L 2 ) ( L 2 + L 6 ) ) = P L 3 96 E I + P L 3 32 E I + 3 P L 3 32 E I × 4 6 = P L 3 + 3 P L 3 + 6 P L 3 96 E I = 10 P L 3 96 E I ( t B / C ) 1 = 5 P L 3 48 E I Deflection at B with respect to C due to the reaction at B . ( t B / C ) 2 = 1 2 ( − B y L 2 E I ) L × L 3 = − B y L 3 12 E I Substitute 7 P L 3 16 E I for ( t A / C ) 1 and − B y L 3 2 E I for ( t A / C ) 2 in Equation (3). t A / C = 7 P L 3 16 E I + ( − B y L 3 2 E I ) = 7 P L 3 16 E I − B y L 3 2 E I Substitute 5 P L 3 48 E I for ( t B / C ) 1 and − B y L 3 12 E I for ( t B / C ) 2 in Equation (4)

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ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 12.8, Problem 12.115P

To determine

The vertical reactions at the bearing supports A, B, and C.

To draw: The shear and moment diagrams.

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Chapter 12.8, Problem 12.114P

Chapter 12.8, Problem 12.116P

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Mechanics of Materials

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