   # Determine the equations for slope and deflection of the beam shown by the direct integration method. EI = constant. FIG. P6.4

#### Solutions

Chapter
Section
Chapter 6, Problem 4P
Textbook Problem
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## Determine the equations for slope and deflection of the beam shown by the direct integration method. EI = constant. FIG. P6.4

To determine

Find the equations for slope and deflection of the beam using direct integration method.

### Explanation of Solution

Calculation:

Consider flexural rigidity EI of the beam is constant.

Draw the free body diagram of the beam as in Figure (1).

Refer Figure (1),

Find the reaction at support B.

MA=0M+By×L+M=0By=2ML

Find the reaction at support A.

MB=0Ay×L+M+M=0Ay=2ML

Draw the section at a distance x from support A as in Figure (2).

Refer Figure (2),

Write the equation for bending moment at x distance.

Mx=M+Ay(x)=M+2ML(x)=2MLxM=M(2xL1)

Write the equation for MEI as follows:

d2ydx2=MEI(2xL1)        (1)

Write the equation for slope as follows:

Integrate Equation (1) with respect to x.

dydx=θ=MEI(2xL1)dx=MEI(2x22Lx)+C1=MxEI(xL1)+C1        (2)

Write the equation for deflection as follows:

Integrate Equation (2) with respect to x.

y=[MxEI(xL1)+C1]dx=[(Mx2EILMxEI)+C1]dx=Mx33EILMx22EI+C1x+C2        (3)

Find the integration constants C1andC2:

Write the boundary conditions as follows:

Atx=0;y=0:

Apply the above boundary conditions in Equation (3):

0=M(0)2EI(03L12)+C1(0)+C2C2=0

Write the boundary condit

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