Explanation Given information: The point load act at center of the beam ( P ) is 40 kN . The moment act at center of the beam ( M ) is 18 kN ⋅ m . The modulus of elasticity of the beam ( E ) is 200 GPa . The moment of inertia of the beam ( I ) is 65 × 10 6 mm 4 . The length of the beam ( L ) is 6 m . The length of the segments AB and BC ( L A B and L B C ) is 3 m . Calculation: Show the free body diagram of the beam as in Figure 1. Refer to Figure 1. 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 − 40 = 0 R A + R B = 40 (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 × 6 ) − ( 40 × 3 ) + 18 = 0 6 R B − 102 = 0 6 R B = 102 R B = 102 6 R B = 17 kN Substitute 17 kN for R B in Equation (1). R A + 17 = 40 R A = 40 − 17 R A = 23 kN Show the moment area diagram of the beam as in Figure 2. Refer to Figure 2. Calculate the bending moment at A : M A = ( 17 × 6 ) − ( 40 × 3 ) + 18 = 0 Calculate the bending moment at C (only Point load): ( M C ) P = ( 17 × 3 ) − ( 40 × 0 ) = 51 kN ⋅ m Calculate the bending moment at C : M C = ( 17 × 3 ) − ( 40 × 0 ) + 18 = 69 kN ⋅ m Calculate the bending moment at B : M B = ( 17 × 0 ) = 0 Show the deflected shape of the beam as in Figure 3. Refer to Figure 3. Calculate the vertical distance between the tangent at a point C and B ( t C / B ) : t C / B = ( 1 3 × 3 ) × [ 1 2 × 51 E I × 3 ] = 76.5 E I kN ⋅ m 3 Calculate the vertical distance between the tangent at a point A and B ( t A / B ) : t A / B = { [ 1 3 ( 3 ) + 3 ] × ( 1 2 × 51 E I × 3 ) } + [ ( 2 3 × 3 ) × ( 1 2 × 69 E I × 3 ) ] = 306 E I + 207 E I = 513 E I kN ⋅ m 3 Calculate the slope at B of the beam ( θ B ) using the formula: θ B = | t A / B | L (2) Substitute 513 E I kN ⋅ m 3 for t A / B and 6 m for L in Equation (2)

9th Edition

ISBN: 9780133254426

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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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 12.75P

To determine

To calculate:

Theslope at B of the beam (θB).

The deflection at C of the beam (υC).

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Chapter 12.4, Problem 12.74P

Chapter 12.4, Problem 12.76P

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Theslope at B of the beam ( θ B ) . The deflection at C of the beam ( υ C ) .

Solution Summary: The author calculates the vertical forces by applying the equation of equilibrium.

Concept explainers

Videos

To calculate:

Theslope at B of the beam (θB).

The deflection at C of the beam (υC).

Want to see the full answer?

Chapter 12 Solutions

Theslope at B of the beam ( θ B ) . The deflection at C of the beam ( υ C ) .

Solution Summary: The author calculates the vertical forces by applying the equation of equilibrium.

Concept explainers

Videos

To calculate: Theslope at B of the beam (θB). The deflection at C of the beam (υC).

Want to see the full answer?

Chapter 12 Solutions

To calculate:

Theslope at B of the beam (θB).

The deflection at C of the beam (υC).