Explanation Given information: Apply the sign conventions for calculating reactions, forces and moments using the three equations of equilibrium as shown below. For summation of forces along x -direction is equal to zero ( ∑ F x = 0 ) , consider the forces acting towards right side as positive ( → + ) and the forces acting towards left side as negative ( ← − ) . For summation of forces along y -direction is equal to zero ( ∑ F y = 0 ) , consider the upward force as positive ( ↑ + ) and the downward force as negative ( ↓ − ) . For summation of moment about a point is equal to zero ( ∑ M at a point = 0 ) , consider the clockwise moment as negative and the counter clockwise moment as positive. Calculation: Show the beam as shown in the Figure 1. Refer Figure 1. Consider the horizontal and vertical reaction at A are denoted by A x and A y . Consider the horizontal and vertical reaction at B is denoted by B y . Consider the moment at A is denoted by M A . The reactions acting in the beam is 4. The number of Equilibrium reaction is 3. The degree of indeterminacy of the beam is 1. Take the reaction at B as the redundant. Modify the Figure 1 as shown in Figure 2. Refer Figure 2. The deflection at B due to the external loading is denoted by Δ B O . The flexibility coefficient representing the deflection at B due to unit value of redundant B y is f B B . Calculate the value of Δ B O as follows: Δ B O = 1 E I [ ∫ 0 12 ( − 100 ( 6 + x ) − 5 x 2 ) x d x ] = 1 E I [ ∫ 0 12 ( − 600 x − 100 x 2 − 5 x 3 ) d x ] = 1 E I ( − 600 x 2 2 − 100 × x 3 3 − 5 × x 4 4 ) 0 12 = 1 E I ( − 600 × 12 2 2 − 100 × 12 3 3 − 5 × 12 4 4 ) = − 126 , 720 kN ⋅ m 3 E I (1) Calculate the value of f B B as follows: f B B = 1 E I ∫ 0 12 x 2 d x = 1 E I ∫ 0 12 x 3 3 d x = 1 E I × 12 3 3 = 576 kN ⋅ m 3 / kN E I (2) Show the compatibility Equation as follows: Δ B O + B y f B B = 0 Modify above Equation using Equation (1) and (2)

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ISBN: 9781337630931

Author: KASSIMALI, Aslam.

Publisher: Cengage,

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Matrix Algebra for Structural analysis

Structural analysis is a process in which the effects of different loads and internal forces, acting on structures and their elements are determined. This method of analysis is effective in detecting the failure loads of the structural elements and the d…

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Chapter 13, Problem 16P

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Find the reactions and sketch the shear and bending moment diagrams for the given beam using method of consistent deformation.

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Chapter 13 Solutions

Structural Analysis

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