
Concept explainers
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.
Fig. P9.69
(a)

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
Explanation of Solution
The flexural rigidity of the beam is EI.
Show the free-body diagram of the superimposed beam as in Figure 1.
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.
Consider
When
Find the deflection at point C due to point load P at point B of the beam as follows;
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.
Consider
Find the deflection at point C due to load P at point C as follows;
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.
Consider
Find the deflection at point C due to point load P at point D of the beam as follows;
Apply the superimposition concept.
Find the deflection at point C
Substitute
Therefore, the deflection at point C of the beam is
(b)

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
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.
Consider
Find the slope at point A due to point load P at point B of the beam as follows;
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.
Consider
Find the slope at point A due to load P at point C is;
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.
Consider
Find the slope at point A due to point load P at point D of the beam is;
Apply the superimposition concept.
Find the slope at point A
Substitute
Therefore, the slope at point A of the beam is
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Chapter 9 Solutions
Mechanics of Materials, 7th Edition
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