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Structural Analysis

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

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BuyFindarrow_forward

Structural Analysis

6th Edition
KASSIMALI + 1 other
Publisher: Cengage,
ISBN: 9781337630931
Chapter 6, Problem 3P
Textbook Problem
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Determine the equations for slope and deflection of the beam shown by the direct integration method. EI = constant.

Chapter 6, Problem 3P, Determine the equations for slope and deflection of the beam shown by the direct integration method.

FIG. P6.3

To determine

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

Explanation of Solution

Calculation:

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

Refer Figure (1),

Find the reaction at support A.

Apply vertical equilibrium along y-axis.

Consider upward force as positive.

ΣFy=0Ayw(La)=0Ay=w(La)

Find the moment at A.

Consider anticlockwise moment as positive.

0=MAw×(La)×[(La)2+a]MA=w(La)[(La+2a2)]MA=w(La)(L+a2)MA=w2(L2a2)

Segment AB: 0xa.

Consider a section X1X1 in the segment AB at a distance of x from A.

Sketch the free body diagram when section X1X1 consider in the segment AB as shown in Figure 2.

Refer Figure (2),

Take the moment at section X1X1 .

M=w2(L2a2)+w×(La)(x)

Write the equation for MEI.

d2ydx2=MEI=1EI[w2(L2a2)+w(La)x]        (1)

Find the equation for slope (θ).

Integrate Equation (1) with respect to x.

dydx=θ=MEIdxθ=w2(L2a2)+w×(La)(x)EIdxEIθ=w2(L2a2)x+w×(La)(x22)+C1        (2)

Find the equation for deflection (θ).

Integrate again Equation (2) with respect to x.

EIy=w2(L2a2)x22+w×(La)(x36)+C1x+C2        (3)

Find the integration constants C1andC2:

Apply boundary conditions in Equation (2):

At x=0 and y=0.

0=M×022EI+C1×0+C2C2=0

Apply boundary conditions in Equation (1):

At x=0 and θ=0.

0=w2(L2a2)×0+w×(La)(022)+C1C1=0

Find the equation for slope of segment AB.

Substitute 0 for C1 in Equation (2).

EIθ=w2(L2a2)x+w×(La)(x22)+0EIθ=wx2(L2a2)+wx22(La)θ=wx2EI[a2L2+(La)x]

Thus, the equation for slope of segment AB is wx2EI[a2L2+(La)x]_.

Find the equation for deflection of segment AB.

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

EIy=w2(L2a2)x22+w×(La)(x36)+0×x+0EIy=w2(L2a2)x22+w×(La)(x36)EIy=wx22(L2a22)+wx36(La)y=wx22EI(a2L22+(La)x3)

Thus, the equation for deflection is wx22EI(a2L22+(La)x3)_.

Segment BC: axL.

Consider a section X2X2 in the segment BC at a distance of x from A.

Sketch the free body diagram when section X2X2 consider segment BC as shown in Figure 3.

Refer Figure 3.

Write the equation for bending moment at section X2X2.

M=w2(L2a2)+w×(La)(x)w2(xa)2

Write the equation for MEI.

d2ydx2=MEI=1EI[w2(L2a2)+w×(La)(x)w2(xa)2]        (4)

Write the equation for slope.

Integrate Equation (4) with respect to x.

dydx=θ=MEIdxθ=[w2(L2a2)+w×(La)(x)w2(xa)2]EIdxEIθ=w2(L2a2)x+w×(La)(x22)w2(x33+a2xax2)+C3        (5)

Write the equation for deflection

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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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