Explanation Given information: The flexural rigidity of the beam is E I . The length of the beam ( L ) is 2 a . Calculation: Show the free body diagram of the beam as in Figure 1. Refer Figure 1. Calculate the support reactions A and B ( R A and R B ) : Calculate the vertical forces by applying the equation of equilibrium: Sum of vertical forces is equal to 0. ∑ F y = 0 R A + R B − ( w × a ) = 0 R A + R B − w a = 0 R A + R B = w a (1) Calculate the moment about point A by applying the equation of equilibrium: Sum of moments about point A is equal to 0. ∑ M A = 0 ( R B × 2 a ) − ( w × a × a 2 ) = 0 2 a R B − w a 2 2 = 0 2 a R B = w a 2 2 R B = w a 2 2 2 a R B = w a 4 Substitute w a 4 for R B in Equation (1). R A + w a 4 = w a R A = w a − w a 4 R A = 3 w a 4 Show the free body diagram of the moment function of the beam’s cut segment as in Figure 2. Refer to Figure 2. Calculate the moment function with respect of the x distance M ( x ) : M ( x ) = 3 4 w a ( x − 0 ) − w 2 ( x − 0 ) 2 − ( − w 2 ) ( x − a ) 2 = 3 w a x 4 − w x 2 2 + w 2 ( x − a ) 2 Elastic Curve and slope is as follows: E I d 2 υ d x 2 = M ( x ) (2) Substitute [ 3 w a x 4 − w x 2 2 + w 2 ( x − a ) 2 ] for M ( x ) in Equation (2). E I d 2 υ d x 2 = 3 w a x 4 − w x 2 2 + w 2 ( x − a ) 2 (3) Integrate both sides of the Equation (3). E I d υ d x = 3 w a x 2 8 − w x 3 6 + w 6 ( x − a ) 3 + C 1 (4) Integrate both sides of the Equation (4). E I υ = 3 w a x 3 24 − w x 4 24 + w 24 ( x − a ) 4 + C 1 x + C 2 = w a x 3 8 − w x 4 24 + w 24 ( x − a ) 4 + C 1 x + C 2 (5) Boundary conditions are as follows: υ = 0 at x = 0 υ = 0 at x = 2 a Apply the first boundary condition in Equation (5). E I ( 0 ) = [ 0 − 0 + 0 + 0 + C 2 ] C 2 = 0 Substitute 0 for C 2 in Equation (5). E I υ = w a x 3 8 − w x 4 24 + w 24 ( x − a ) 4 + C 1 x + 0 = w a x 3 8 − w x 4 24 + w 24 ( x − a ) 4 + C 1 x (6) Apply the second boundary condition in Equation (6)

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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 12.3, Problem 47P

To determine

The equation of the elastic curve (υ).

The slope at A(θA).

The slope at B(θB).

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Chapter 12.3, Problem 46P

Chapter 12.3, Problem 48P

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Solids: Lesson 53 - Slope and Deflection of Beams Intro; Author: Jeff Hanson;https://www.youtube.com/watch?v=I7lTq68JRmY;License: Standard YouTube License, CC-BY

The equation of the elastic curve ( υ ) . The slope at A ( θ A ) . The slope at B ( θ B ) .

Solution Summary: The author explains the equation of the elastic curve (upsilon) and the slope at B.

Concept explainers

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The equation of the elastic curve (υ).

The slope at A(θA).

The slope at B(θB).

Want to see the full answer?

Chapter 12 Solutions

The equation of the elastic curve ( υ ) . The slope at A ( θ A ) . The slope at B ( θ B ) .

Solution Summary: The author explains the equation of the elastic curve (upsilon) and the slope at B.

Concept explainers

Videos

The equation of the elastic curve (υ). The slope at A(θA). The slope at B(θB).

Want to see the full answer?

Chapter 12 Solutions

The equation of the elastic curve (υ).

The slope at A(θA).

The slope at B(θB).