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Structural Analysis

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

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Section
BuyFindarrow_forward

Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 6, Problem 1P
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.

Chapter 6, Problem 1P, 6.1 through 6.6 Determine the equations for slope and deflection of the beam shown by the direct

FIG. P6.1

To determine

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

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).

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).

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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Chapter 6 Solutions

Structural Analysis
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Ch. 6 - Determine the slope and deflection at point B of...Ch. 6 - Determine the slope and deflection at point B of...Ch. 6 - Determine the slope and deflection at point A of...Ch. 6 - Use the moment-area method to determine the slopes...Ch. 6 - 6.14 through 6.17 Use the moment-area method to...Ch. 6 - 6.14 through 6.17 Use the moment-area method to...Ch. 6 - 6.14 through 6.17 Use the moment-area method to...Ch. 6 - Determine the smallest moment of inertia I...Ch. 6 - Determine the smallest moment of inertia I...Ch. 6 - Determine the smallest moment of inertia I...Ch. 6 - 6.18 through 6.22 Determine the smallest moment of...Ch. 6 - 6.18 through 6.22 Determine the smallest moment of...Ch. 6 - 6.23 through 6.30 Determine the maximum deflection...Ch. 6 - 6.23 through 6.30 Determine the maximum deflection...Ch. 6 - 6.23 through 6.30 Determine the maximum deflection...Ch. 6 - 6.23 through 6.30 Determine the maximum deflection...Ch. 6 - Determine the maximum deflection for the beam...Ch. 6 - 6.23 through 6.30 Determine the maximum deflection...Ch. 6 - 6.23 through 6.30 Determine the maximum deflection...Ch. 6 - 6.23 through 6.30 Determine the maximum deflection...Ch. 6 - 6.31 and 6.32 Use the moment-area method to...Ch. 6 - Use the moment-area method to determine the slope...Ch. 6 - Use the moment-area method to determine the slopes...Ch. 6 - 6.33 and 6.34 Use the moment-area method to...Ch. 6 - Use the conjugate-beam method to determine the...Ch. 6 - Use the conjugate-beam method to determine the...Ch. 6 - Use the conjugate-beam method to determine the...Ch. 6 - Use the conjugate-beam method to determine the...Ch. 6 - 6.39 Determine the slope and deflection at point A...Ch. 6 - 6.40 through 6.43 Use the Conjugate-beam method to...Ch. 6 - Use the conjugate-beam method to determine the...Ch. 6 - 6.40 through 6.43 Use the Conjugate-beam method to...Ch. 6 - 6.40 through 6.43 Use the Conjugate-beam method to...Ch. 6 - 6.44 through 6.48 Using the conjugate-beam method,...Ch. 6 - 6.44 through 6.48 Using the conjugate-beam method,...Ch. 6 - 6.44 through 6.48 Using the conjugate-beam method,...Ch. 6 - Using the conjugate-beam method, determine the...Ch. 6 - Using the conjugate-beam method, determine the...Ch. 6 - 6.49 through 6.56 Determine the maximum deflection...Ch. 6 - 6.49 through 6.56 Determine the maximum deflection...Ch. 6 - 6.49 through 6.56 Determine the maximum deflection...Ch. 6 - Determine the maximum deflection for the beams...Ch. 6 - Determine the maximum deflection for the beams...Ch. 6 - 6.49 through 6.56 Determine the maximum deflection...Ch. 6 - 6.49 through 6.56 Determine the maximum deflection...Ch. 6 - 6.49 through 6.56 Determine the maximum deflection...Ch. 6 - 6.57 and 6.58 Use the conjugate-beam method to...Ch. 6 - Use the conjugate-beam method to determine the...Ch. 6 - 6.59 and 6.60 Use the conjugate-beam method to...Ch. 6 - 6.59 and 6.60 Use the conjugate-beam method to...

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