EBK MECHANICS OF MATERIALS
EBK MECHANICS OF MATERIALS
7th Edition
ISBN: 9780100257061
Author: BEER
Publisher: YUZU
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Chapter 9.4, Problem 69P

Use the method of superposition to solve the following problems and assume that the flexural rigidity El of each beam is constant.

9.69 through 9.72 For the beam and loading shown, determine (a) the deflection at point C, (b) the slope at end A.

Chapter 9.4, Problem 69P, Use the method of superposition to solve the following problems and assume that the flexural

Fig. P9.69

(a)

Expert Solution
Check Mark
To determine

Find the deflection at point C of the beam using superposition method.

Answer to Problem 69P

The deflection at point C of the beam is yC=19Pa36EI()_.

Explanation of Solution

The flexural rigidity of the beam is EI.

Show the free-body diagram of the superimposed beam as in Figure 1.

EBK MECHANICS OF MATERIALS, Chapter 9.4, Problem 69P

Loading I:

The downward load P is acting at point B of the beam.

Refer to case 5 in Appendix D “Beam Deflections and Slopes” in the textbook.

Write the deflection equation for concentrated load acting at any point in the simply supported beam.

y=Pb6EIL[x3(L2b2)x];whenx<a

Consider L=4a;a=a;b=3a;x=2a.

When x>a, consider x as (Lx) and interchange the notation of a and b.

Find the deflection at point C due to point load P at point B of the beam as follows;

(yC)B=Pa6EIL[(Lx)3(L2a2)(Lx)]=Pa6EI(4a)[(4a2a)3((4a)2a2)(4a2a)]=P24EI[8a3(15a2)(2a)]=P24EI[8a330a3]

(yC)B=22Pa324EI=11Pa312EI

Loading II:

The downward point load P is acting at point C.

Refer to case 4 in Appendix D “Beam Deflections and Slopes” in the textbook.

Write the deflection equation for concentrated load acting at mid-point in the simply supported beam.

y=PL348EI

Consider L=4a.

Find the deflection at point C due to load P at point C as follows;

(yC)C=P(4a)348EI=64Pa348EI=4Pa33EI

Loading III:

The downward load P is acting at point D of the beam.

Refer to case 5 in Appendix D “Beam Deflections and Slopes” in the textbook.

Write the deflection equation for concentrated load acting at any point in the simply supported beam.

y=Pb6EIL[x3(L2b2)x];whenx<a

Consider L=4a;a=3a;b=a;x=2a.

Find the deflection at point C due to point load P at point D of the beam as follows;

(yC)D=P(a)6EI(4a)[(2a)3((4a)2a2)(2a)]=P24EI[8a3(15a2)(2a)]=P24EI[8a330a3]=11Pa312EI

Apply the superimposition concept.

Find the deflection at point C (yC) of the beam using the relation.

yC=(yC)B+(yC)C+(yC)D

Substitute 11Pa312EI for (yC)B, 4Pa33EI for (yC)C, and 11Pa312EI for (yC)D.

yC=11Pa312EI4Pa33EI11Pa312EI=11Pa312EI16Pa312EI11Pa312EI=38Pa312EI=19Pa36EI()

Therefore, the deflection at point C of the beam is yC=19Pa36EI()_.

(b)

Expert Solution
Check Mark
To determine

Find the slope at point A of the beam using superposition method.

Answer to Problem 69P

The slope at point A of the beam is θA=5Pa22EI(Anticlockwise)_.

Explanation of Solution

The flexural rigidity of the beam is EI.

Refer to Figure (1) in Part (a);

Loading I:

The downward load P is acting at point B of the beam.

Refer to case 5 in Appendix D “Beam Deflections and Slopes” in the textbook.

Write the slope equation for concentrated load acting at any point in the simply supported beam.

θA=Pb(L2b2)6EIL

Consider L=4a;a=a;b=3a.

Find the slope at point A due to point load P at point B of the beam as follows;

(θA)B=P(3a)((4a)2(3a)2)6EI(4a)=P(7a2)8EI=7Pa28EI

Loading II:

The downward point load P is acting at point C.

Refer to case 4 in Appendix D “Beam Deflections and Slopes” in the textbook.

Write the slope equation for concentrated load acting at mid-point in the simply supported beam.

θ=PL216EI

Consider L=4a;.

Find the slope at point A due to load P at point C is;

(θA)C=P(4a)216EI=Pa2EI

Loading III:

The downward load P is acting at point D of the beam.

Refer to case 5 in Appendix D “Beam Deflections and Slopes” in the textbook.

Write the slope equation for concentrated load acting at any point in the simply supported beam.

θA=Pb(L2b2)6EIL

Consider L=4a;a=3a;b=a;x=2a.

Find the slope at point A due to point load P at point D of the beam is;

(θA)D=Pa((4a)2a2)6EI(4a)=P(15a2)24EI=15Pa224EI

Apply the superimposition concept.

Find the slope at point A (θA) of the beam using the relation.

θA=(θA)B+(θA)C+(θA)D

Substitute 7Pa28EI for (θA)B, Pa2EI for (θA)C, and 15Pa224EI for (θA)D.

θA=7Pa28EIPa2EI15Pa224EI=21Pa224EI24Pa224EI15Pa224EI=5Pa22EI=5Pa22EI(Anticlockwise)

Therefore, the slope at point A of the beam is θA=5Pa22EI(Anticlockwise)_.

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