EBK STRUCTURAL ANALYSIS
EBK STRUCTURAL ANALYSIS
6th Edition
ISBN: 9780357030974
Author: KASSIMALI
Publisher: VST
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Chapter 6, Problem 1P
To determine

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

Expert Solution & Answer
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Answer to Problem 1P

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).

EBK STRUCTURAL ANALYSIS, Chapter 6, Problem 1P , additional homework tip  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).

EBK STRUCTURAL ANALYSIS, Chapter 6, Problem 1P , additional homework tip  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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