Explanation Given information: The uniformly varying load of the beam is w 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. Convert the load: P = 1 2 × w 0 × L = w 0 L 2 Calculate the support reactions A ( A y ) : Calculate the vertical forces by applying the Equation of equilibrium: Sum of vertical forces is equal to zero. ∑ F y = 0 A y − w 0 L 2 = 0 A y = w 0 L 2 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 M A − ( w 0 L 2 × 2 L 3 ) = 0 M A − w 0 L 2 3 = 0 M A = w 0 L 2 3 Show the free body diagram of the moment function of the beam’s cut segment as in Figure 1. Refer Figure 1. Calculate the moment function with respect of the x distance M ( x ) : Sum of moments about point O is equal to zero. ∑ M O = 0 M ( x ) + [ 1 2 ( w 0 L x ) x ] ( x 3 ) + w 0 L 2 3 − 1 2 w 0 L ( x ) = 0 M ( x ) + w 0 x 3 6 L + w 0 L 2 3 − w 0 L x 2 ( x ) = 0 M ( x ) + w 0 6 L ( x 3 + 2 L 3 − 3 L 2 x ) = 0 M ( x ) = − w 0 6 L ( x 3 + 2 L 3 − 3 L 2 x ) = w 0 6 L ( − x 3 − 2 L 3 + 3 L 2 x ) 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 w 0 6 L ( − x 3 − 2 L 3 + 3 L 2 x ) for M ( x ) in equation (2). E I d 2 υ d x 2 = w 0 6 L ( − x 3 − 2 L 3 + 3 L 2 x ) (3) Integrate both the sides of the Equation (3). E I d υ d x = w 0 6 L ( − x 4 4 − 2 L 3 x + 3 L 2 x 2 2 + C 1 ) (4) Integrate both sides of the Equation (4). E I υ = w 0 6 L ( − x 5 20 − 2 L 3 x 2 2 + 3 L 2 x 3 6 + C 1 x + C 2 ) = w 0 6 L ( − x 5 20 − L 3 x 2 + L 2 x 3 2 + C 1 x + C 2 ) (5) Boundary conditions are, Condition 1: d υ d x = 0 at x = 0 Condition 2: υ = 0 at x = 0 Apply the first boundary condition in Equation (4)

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ISBN: 9789810694364

Author: Russell C Hibbeler

Publisher: Pearson Education

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Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial…

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Chapter 12.2, Problem 12.27P

To determine

The elastic curve of the cantilever beam (υ) .

The maximum deflection of the cantilever beam (υmax) .

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Chapter 12.2, Problem 12.26P

Chapter 12.2, Problem 12.28P

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The elastic curve of the cantilever beam ( υ ) . The maximum deflection of the cantilever beam ( υ max ) .

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The elastic curve of the cantilever beam (υ) . The maximum deflection of the cantilever beam (υmax) .

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Chapter 12 Solutions

The elastic curve of the cantilever beam (υ) .

The maximum deflection of the cantilever beam (υmax) .