   # Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P13.1–P13.4 using the method of consistent deformations. Select the reaction at the roller support to be the redundant. FIG. P13.3, P13.7

#### Solutions

Chapter
Section
Chapter 13, Problem 3P
Textbook Problem
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## Determine the reactions and draw the shear and bending moment diagrams for the beams shown in Figs. P13.1–P13.4 using the method of consistent deformations. Select the reaction at the roller support to be the redundant. FIG. P13.3, P13.7

To determine

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

### Explanation of Solution

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 (Fx=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 (Fy=0), consider the upward force as positive (+) and the downward force as negative ().
• For summation of moment about a point is equal to zero (Matapoint=0), consider the clockwise moment as negative and the counter clockwise moment as positive.

Calculation:

Show the free body diagram of the given beam as shown in Figure 1.

Refer Figure 1.

Consider the horizontal and vertical reaction at C are denoted by Cx and Cy.

Consider the moment at C is denoted by MC.

Consider the vertical reaction at C is denoted by Cy.

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 vertical reaction at A as the redundant.

Modify the Figure 1 as shown in Figure 2.

Refer Figure 2.

The deflection at A due to the external moment is denoted by ΔA.

The deflection coefficient representing the deflection at A due to unit value of redundant Ay is fAA Then,

The deflection at A due to unknown redundant Ay is fAAAy.

Show the deflection at the free end of the cantilever beam with a point load P acting at a distance from the fixed end as follows:

Δ=Pa26EI(3la) (1)

Here, l is the length of the beam, E is modulus of Elasticity of the beam, I is the moment of inertia.

Find the deflection at the free end of the beam due to the load of 120kN as follows:

Substitute 6m for a, 12m for l and 120kN for P in Equation (1).

Δ1=120×626EI(3×126)=21600EI

Show deflection at the free end of the beam due to the clockwise moment M at the free end as follows:

Δ2=ML22EI (2)

Substitute 12m for L and 75kNm for M in Equation (2).

Δ2=75×1222EI=5400EI

Calculate the deflection at A due to the loading on the beam as follows:

ΔA=Δ1+Δ2=21600EI+5400EI=16200kNm3EI (3)

Calculate the value of fAA as follows:

Substitute 12m for a, 12m for l, and 1kN for P in Equation (1).

fAA=1×1226EI(3×1212)fAA=576kNm3/kNEI (4)

Show the compatibility Equation as follows:

ΔA+fAA×Ay=0

Modify the above Equation using Equation (3) and (4)

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