Concept explainers
Repeat Prob. 3–8 using singularity functions exclusively (including reactions).
The reaction at the supports.
The shear force diagram.
The bending moment diagram.
Answer to Problem 12P
The reaction force
The reaction force
The reaction force
The shear force diagram is shown is below.
The bending moment diagram is shown below.
Explanation of Solution
The following figure shows the load applied to a beam.
Figure-(1)
Refer to Table 3-1 “Singularity (Macaulay) functions”.
Write the expression for load intensity acting on the beam.
Here, the reaction force at point O is
Write the expression for the shear force.
Here, the net shear force is
Substitute the value from Equation (I) to Equation (II).
Write the expression for the moment.
Here, the net bending moment is
Substitute the value from Equation (III) to Equation (IV).
Conclusion:
Substitute
Thus, the reaction force
Substitute
Substitute
Thus, the reaction force
Substitute
Thus, the reaction force
Substitute the value of
The magnitude of shear force remains constant.
Substitute the value of
At
Substitute
At
Substitute
Similarly for the other values of
The Table-1 shows the variation of moment.
Table-1
Substitute the value of
The magnitude of shear force remains constant.
Substitute the value of
At
Substitute
At
Substitute
Similarly for the other values of
Table-2 shows the variation of moment.
Table-2
Substitute the value of
At
Substitute
At
Substitute
Similarly for the other values of
The Table-3 shows the variation of shear force.
Table-3
Substitute the value of
At
Substitute
At
Substitute
Similarly for the other values of
Differentiate Equation (X).
Equate the above expression to zero to obtain the location of maxima.
The Table-4 shows the variation of moment.
Table-4
The shear force diagram is shown in Figure-(2).
Figure-(2)
The bending moment diagram is shown in Figure-(3).
Figure-(3)
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Chapter 3 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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