Explanation Given information: The flexural rigidity of the beam is E I . The moment act at a distance from the left end of the beam is M 0 . The length of the beam ( L ) is ( a + b ) . The length of the segment AC is a . The length of the segment CB is b . Calculation: Show the free body diagram of the beam as in Figure 1. Refer Figure 1. Calculate the vertical forces by applying the equation of equilibrium: Sum of vertical forces is equal to 0. ∑ F y = 0 A y + B y = 0 (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 B y ( a + b ) − M 0 = 0 B y ( a + b ) = M 0 B y = M 0 a + b Substitute M 0 a + b for B y in Equation (1). A y + M 0 a + b = 0 A y = − M 0 a + b A y = M 0 a + b ↓ Show the moment area diagram of the beam as in Figure 2. Refer to Figure 2. Calculate the bending moment at A : M A = [ M 0 ( a + b ) × ( a + b ) ] − M 0 = M 0 − M 0 = 0 Calculate the bending moment at B : M B = M 0 a + b × 0 = 0 Calculate the bending moment at C from right side of the beam: M C = M 0 a + b × b = M 0 b a + b Calculate the bending moment at C from left side of the beam: M C = − M 0 a + b × a = − M 0 a a + b Calculate the vertical distance between the tangent at a point A and B ( t A / B ) : t A / B = { 1 2 × [ − M 0 a E I ( a + b ) × a ] × ( 2 3 × a ) } + { 1 2 × [ M 0 b E I ( a + b ) × b ] × ( a + b 3 ) } = − 2 M 0 a 3 6 E I ( a + b ) + M 0 b 2 2 ( 3 a + b 3 ) = M 0 ( − 2 a 3 + 3 a b 2 + b 3 ) 6 E I ( a + b ) = M 0 ( b 3 + 3 a b 2 − 2 a 3 ) 6 E I ( a + b ) Calculate the vertical distance between the tangent at a point C and B ( t C / B ) : t C / B = { 1 2 × [ M 0 b E I ( a + b ) × b ] × ( 1 3 × b ) } = M 0 b 2 2 E I ( a + b ) × b 3 = M 0 b 3 6 E I ( a + b ) Show the deflected shape of the beam as in Figure 3. Refer to Figure 3. Calculate the slope at B ( θ B ) : θ B = t A / B L (2) Substitute M 0 ( b 3 + 3 a b 2 − 2 a 3 ) 6 E I ( a + b ) for t A / B and ( a + b ) for L in Equation (2)

9th Edition

ISBN: 9789810694364

Author: Russell C Hibbeler

Publisher: Pearson Education

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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 12.4, Problem 12.64P

To determine

The slope at B (θB) .

The deflection at C (ΔC) .

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Chapter 12.4, Problem 12.63P

Chapter 12.4, Problem 12.65P

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The slope at B ( θ B ) . The deflection at C ( Δ C ) .

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The slope at B (θB) . The deflection at C (ΔC) .

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Chapter 12 Solutions

The slope at B (θB) .

The deflection at C (ΔC) .