# 6.1 through 6.6 Determine the equations for slope and deflection of the beam shown by the direct integration method. EI = constant. FIG. P6.1

### Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931

### Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931

#### Solutions

Chapter 6, Problem 1P
Textbook Problem

## 6.1 through 6.6 Determine the equations for slope and deflection of the beam shown by the direct integration method. EI = constant.FIG. P6.1

Expert Solution
To determine

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

The equation for slope is M6EIL(3x26Lx+2L2)_.

The equation for deflection is M6EIL[x33Lx2+2L2x]_.

### Explanation of Solution

Calculation:

Consider the 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=0By×LM=0By=ML()

Find the reaction at support A.

MB=0Ay×L+M=0Ay=MLAy=ML()

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

Refer Figure (2),

Write the equation for bending moment at x distance.

Mx=MAy(x)=MML(x)=ML(Lx)

Write the equation for MEI.

d2ydx2=MEI=MEIL(Lx)        (1)

Integrate Equation (1) to find the equation of the slope.

dydx=θ=MEIL(Lx)dx=MEIL(Lxx22)+C1        (2)

Integrate Equation (2) to find the equation of the deflection.

y=[MEIL(Lxx22)+C1]dx=[MEIL(Lx22x36)+C1x+C2]        (3)

Find the integration constants C1andC2:

Apply boundary conditions in Equation (3):

At x=0 and y=0.

0=[MEIL(L×0206)+C1×0+C2]C2=0

At x=L and y=0.

0=[MEIL(L×L22L36)+C1×L+0]0=MEIL(L32L36)+(C1×L)0=ML3EIL(6212)+(C1×L)C1L=ML23EIC1=ML3EI

Find the equation of the slope.

Substitute ML3EI for C1 in Equation (2).

θ=MEIL(Lxx22)ML3EI=M6EIL(6Lx6x22)2L=M6EIL(6Lx3x22L2)=M6EIL(3x26Lx+2L2)

Thus, the equation of the slope is M6EIL(3x26Lx+2L2)_.

Find the equation of the deflection.

Substitute ML3EI for C1 and 0 for C2 in Equation (3).

y=MEIL(Lx22x36)ML3EIx+0=M6EIL[(6Lx226x36)2Lx]=M6EIL[x33Lx2+2L2x]

Thus, the equation of the deflection is M6EIL[x33Lx2+2L2x]_.

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