Explanation Given information: The moment at point C of the beam is M . The flexural rigidity of the beam is E I . The length of the beam is L . Calculation: Show the free body diagram of the moment function of the beam’s cut segments as in Figure 1. Refer to Figure 1. Calculate the moment function with respect of the x 1 distance M ( x 1 ) : M ( x 1 ) = − M 0 L x 1 Calculate the moment function with respect of the x 2 distance M ( x 2 ) : M ( x 2 ) = − M 0 Elastic Curve and slope is as follows: E I d 2 υ d x 2 = M ( x ) (1) For the coordinate x 1 , Substitute ( − M 0 L x 1 ) for M ( x ) in Equation (1), E I d 2 υ 1 d x 1 2 = − M 0 L x 1 Integrate both sides of the Equation. E I d υ 1 d x 1 = − M 0 x 1 2 2 L + C 1 Integrate both sides of the Equation. E I υ 1 = − M 0 x 1 3 6 L + C 1 x 1 + C 2 (2) For the coordinate x 2 , Substitute ( − M 0 ) for M ( x ) in Equation (1), E I d 2 υ 2 d x 2 2 = − M 0 Integrate both sides of the Equation. E I d υ 2 d x 2 = − M 0 x 2 + C 3 (3) Integrate both sides of the Equation (3). E I υ 2 = − M 0 x 2 2 2 + C 3 x 2 + C 4 (4) Boundary conditions are as follows: υ 1 = 0 at x 1 = 0 υ 1 = 0 at x 1 = L υ 2 = 0 at x 2 = L Apply the first boundary condition in Equation (2). E I ( 0 ) = − M 0 ( 0 ) 3 6 L + C 1 ( 0 ) + C 2 C 2 = 0 Substitute 0 for C 2 in Equation (2). E I υ 1 = − M 0 x 1 3 6 L + C 1 x 1 + 0 = − M 0 x 1 3 6 L + C 1 x 1 Apply the second boundary condition in Equation. E I ( 0 ) = − M 0 ( L ) 3 6 L + C 1 ( L ) 0 = − M 0 L 2 6 + L C 1 L C 1 = M 0 L 2 6 C 1 = M 0 L 6 Apply the third boundary condition in Equation(4). E I ( 0 ) = − M 0 ( L ) 2 2 + C 3 ( L ) + C 4 0 = − M 0 L 2 2 + C 3 L + C 4 C 3 L + C 4 = M 0 L 2 2 (5) Consider x 1 = L and x 2 = L Continuity condition is, d υ 1 d x 1 = − d υ 2 d x 2 − M 0 x 1 2 2 L + C 1 = − ( − M 0 x 2 + C 3 ) − M 0 x 1 2 2 L + C 1 = M 0 x 2 − C 3 C 3 = M 0 x 2 − C 1 + M 0 x 1 2 2 L Substitute L for x 1 , L for x 2 , and M 0 L 6 for C 1 . C 3 = M 0 L − M 0 L 6 + M 0 L 2 2 L = M 0 L − M 0 L 6 + M 0 L 2 = 4 M 0 L 3 Substitute 4 M 0 L 3 for C 3 in Equation (5)

Statics and Mechanics of Materials Plus Mastering Engineering with Pearson eText - Access Card Package (5th Edition)

5th Edition

ISBN: 9780134301006

Author: Russell C. Hibbeler

Publisher: PEARSON

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Slope and Deflection

The slope in a deflected beam is described as an angle in radians made by the tangent of the section with the original axis of the beam. The deflection at any point on the axis of the beam is defined as the lateral displacement of the point before and af…

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Chapter 16.2, Problem 23P

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Find the equations of the elastic curve (υ1 and υ2).

Find the slope at C(θC)

Find the deflection at C(υC)

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Chapter 16.2, Problem 22P

Chapter 16.2, Problem 24P

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Statics and Mechanics of Materials Plus Mastering Engineering with Pearson eText - Access Card Package (5th Edition)

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Find the equations of the elastic curve ( υ 1 and υ 2 ) . Find the slope at C ( θ C ) Find the deflection at C ( υ C )

Solution Summary: The author calculates the moment function of the beam's cut segments with respect to the x_1 distance.

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Find the equations of the elastic curve (υ1 and υ2).

Find the slope at C(θC)

Find the deflection at C(υC)

Want to see the full answer?

Chapter 16 Solutions

Find the equations of the elastic curve ( υ 1 and υ 2 ) . Find the slope at C ( θ C ) Find the deflection at C ( υ C )

Solution Summary: The author calculates the moment function of the beam's cut segments with respect to the x_1 distance.

Concept explainers

Videos

Find the equations of the elastic curve (υ1 and υ2). Find the slope at C(θC) Find the deflection at C(υC)

Want to see the full answer?

Chapter 16 Solutions

Find the equations of the elastic curve (υ1 and υ2).

Find the slope at C(θC)

Find the deflection at C(υC)