Explanation Consider a section at a distance x from left end A of the beam. Show the free-body diagram of the section AB as in Figure 1. Determine the moment at the section by taking moment about the section. ∑ M = 0 M + M A − A y ( x ) + ( w x × x 2 ) = 0 M = A y x − w x 2 2 − M A Write the second order differential equation as follows; d 2 y d x 2 = M ( x ) E I Here, the moment at the corresponding section is M ( x ) , the modulus of elasticity of the material is E , and the moment of inertia of the section is I . Substitute ( A y x − w x 2 2 − M A ) for M ( x ) in Equation (1). d 2 y d x 2 = A y x − w x 2 2 − M A E I E I d 2 y d x 2 = A y x − w x 2 2 − M A Integrate the equation with respect to x ; E I d y d x = A y x 2 2 − w x 3 6 − M A x + C 1 (1) Integrate the Equation (1) with respect to x . E I y = A y x 3 6 − w x 4 24 − M A x 2 2 + C 1 x + C 2 (2) Boundary condition 1: At the point A ; x = 0 ; y = 0 . Substitute 0 for x and 0 for y in Equation (2). E I ( 0 ) = A y ( 0 ) 3 6 − w ( 0 ) 4 24 − M A ( 0 ) 2 2 + C 1 ( 0 ) + C 2 0 = 0 − 0 − 0 + 0 + C 2 C 2 = 0 Boundary condition 2: At the point A ; x = 0 , d y d x = 0 . Substitute 0 for x and 0 for d y d x in Equation (1). E I ( 0 ) = A y ( 0 ) 2 2 − w ( 0 ) 3 6 − M A ( 0 ) + C 1 0 = 0 − 0 − 0 + C 1 C 1 = 0 Boundary condition 3: At the point B ; x = L ; y = 0 . Substitute 0 for y , L and x , 0 for C 1 , and 0 for C 2 in Equation (2). E I ( 0 ) = A y L 3 6 − w L 4 24 − M A L 2 2 + ( 0 ) x + ( 0 ) 0 = A y L 3 6 − w L 4 24 − M A L 2 2 0 = A y L 3 − w L 2 12 − M A M A = A y L 3 − w L 2 12 (3) Boundary condition 4: At the point B ; x = L ; d y d x = 0

7th Edition

ISBN: 9780073398235

Author: Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf, David F. Mazurek

Publisher: McGraw-Hill Education

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Bending Moment

In general, we can define the bending moment as the reaction induced when a certain amount of load or force is applied to an element which makes it bend.

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Chapter 9.2, Problem 33P

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Find the reaction at the roller support of the beam.

Draw the bending moment diagram of the beam.

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Chapter 9.2, Problem 32P

Chapter 9.2, Problem 34P

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Find the reaction at the roller support of the beam. Draw the bending moment diagram of the beam.

Solution Summary: The author analyzes the reaction at the roller support of the beam.