Concept explainers
Find the reactions and sketch the shear and moment curves.
Locate the point of maximum deflection of the beam and repeat the computation considering uniform I over the length of the beam.
Answer to Problem 5P
Non-Uniform I:
The vertical reaction at C is
The moment and vertical reaction at A are
Uniform I:
The vertical reaction at C is
The moment and vertical reaction at A are
Explanation of Solution
Case 1: Non uniform I.
The moment M is 60 kip ft.
Show the free body diagram of the beam as shown in Figure 1.
Refer Figure 1.
Consider the vertical reaction at C as the redundant.
Remove the redundant at C to get the released structure.
Sketch the released structure with applied loading as shown in Figure 2.
Refer Figure 2.
Find the value of
Sketch the
Refer Figure 3.
Find the deflection
Sketch the released structure with unit loading as shown in Figure 4.
Refer Figure 4.
Find the value of
Sketch the
Refer Figure 5.
Find the deflection
Show the compatibility Equation as follows:
Thus, the vertical reaction at C is
Find the moment at A as follows:
Find the vertical reaction at A as follows:
Thus, the moment and vertical reaction at A are
Find the shear force at A and C.
Find the bending moment at A and C.
Sketch the shear force and bending moment diagram as shown in Figure 6.
Refer Figure 6.
Find the distance
Find the distance
Consider the point of maximum deflection occurs at a distance x from left support.
Find the deflection of AC as follows:
Find the slope at C as follows:
Calculate the area of the shaded region of M/EI diagram as follows:
Equate Equation (1) and (2).
Solve the above Equation.
Find the location of maximum deflection from the right support as follows:
Thus, the point of maximum deflection occurs at a distance
Case 2: Uniform I.
The moment M is 60 kip ft.
Show the free body diagram of the beam as shown in Figure 7.
Refer Figure 7.
Consider the vertical reaction at C as the redundant.
Remove the redundant at C to get the released structure.
Sketch the released structure with applied loading as shown in Figure 8.
Refer Figure 8.
Show the deflection
Sketch the released structure with unit loading as shown in Figure 9.
Refer Figure 9.
Show the deflection
Show the compatibility Equation as follows:
Thus, the vertical reaction at C is
Find the moment at A as follows:
Find the vertical reaction at A as follows:
Thus, the moment and vertical reaction at A are
Find the shear force at A and C.
Find the bending moment at A and C.
Sketch the shear force and bending moment diagram as shown in Figure 10.
Refer Figure 10.
Find the distance x as follows:
Consider the point of maximum deflection occurs at a distance x from left support.
Find the deflection of AC as follows:
Find the slope at C as follows:
Calculate the area of the shaded region of M/EI diagram as follows:
Equate Equation (1) and (2).
Solve the above Equation.
Find the location of maximum deflection from the right support as follows:
Thus, the point of maximum deflection occurs at a distance
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Chapter 9 Solutions
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