   # 6.1 through 6.6 Determine the equations for slope and deflection of the beam shown by the direct integration method. EI = constant. FIG. P6.5

#### Solutions

Chapter
Section
Chapter 6, Problem 5P
Textbook Problem
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## 6.1 through 6.6 Determine the equations for slope and deflection of the beam shown by the direct integration method. EI = constant.FIG. P6.5 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),

Consider upward force is positive and downward force is negative.

Consider clockwise moment is negative and counterclockwise moment is positive.

Since the end of the beam B is free, there will be no reaction.

Determine the support reaction at A using the relation;

V=0RA(12×w×L)=0RA=wL2

Determine the bending moment at A using the relation;

MA=0MA12×w×L×2L3=0MA=wL23

Show the reaction and moment as in Figure (2).

Draw the section at a distance x from support A as in Figure (3).

Write the equation for bending moment at x distance.

Mx=MA+RA(x)12×x×(wxL)×(13×x)=wL23+wL2x12x(wxL)(x3)=wL23+wL2xwx36L

Write the equation for MEI as follows:

d2ydx2=1EI(wL23+wL2xwx36L)        (1)

Write the equation for slope as follows:

Integrate Equation (1) with respect to x.

dydx=θ=1EI(wL23+wL2xwx36L)dx=1EI(wL23x+wL2(x22)w6L(x44))+C1        (2)

Write the equation for deflection as follows:

Integrate Equation (2) with respect to x.

y=[1EI(wL23x+wL2(x22)w6L(x44))+C1]dx=1EI[wL23(x22)+wL2(x36)w6L(x520)]+C1x+C2        (3)

Find the integration constants C1andC2:

Write the boundary conditions as follows:

A

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