Explanation Given information: The point load act at each end of the beam ( P ) is 20 kN. The length of region AB is 3 m. 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 reactions by applying the Equation of equilibrium: Sum of vertical forces is equal to zero. ∑ F y = 0 A y + B y − 20 − 20 = 0 A y + B y − 40 = 0 A y + B y = 40 kN (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 × 3 ) − ( 20 × 4.5 ) + ( 20 × 1.5 ) = 0 3 B y − 60 = 0 B y = 20 kN Substitute 20 kN for R B in Equation (1). A y + 20 = 40 A y = 40 − 20 A y = 20 kN Calculate the moment function with respect of the x distance M ( x ) : M ( x ) = − 20 ( x − 0 ) − ( − 20 ) ( x − 1.5 ) − ( − 20 ) ( x − 4.5 ) = − 20 x + 20 〈 x − 1.5 〉 + 20 〈 x − 4.5 〉 Write the bending equation to find the bending moment as follows: E I d 2 υ d x 2 = M ( x ) (2) For the coordinate x . Substitute − 20 x + 20 〈 x − 1.5 〉 + 20 〈 x − 4.5 〉 for M ( x ) in equation (2). E I d 2 υ d x 2 = − 20 x + 20 〈 x − 1.5 〉 + 20 〈 x − 4.5 〉 (3) Integrate both sides of the Equation (3). E I d υ d x = − 20 x 2 2 + 20 〈 x − 1.5 〉 2 2 + 20 〈 x − 4.5 〉 2 2 + C 1 = − 10 x 2 + 10 〈 x − 1.5 〉 2 + 10 〈 x − 4.5 〉 2 + C 1 (4) Integrate both sides of the Equation (4)

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Author: HIBBELER

Publisher: PEARSON

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Chapter 12.3, Problem 12.44P

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The equation of the elastic curve of the beam (υ) .

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Chapter 12.3, Problem 12.43P

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The equation of the elastic curve of the beam ( υ ) .

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