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A beam of length L is designed to support a uniform load of intensity q (see figure). If the supports of the beam are placed at the ends, creating a simple beam, the maximum bending moment in the beam is qL 2 /8. However, if the supports of the beam are moved symmetrically toward the middle of the beam (as shown), the maximum bending moment is reduced. Determine the distance a between the supports so that the maximum bending moment in the beam has the smallest possible numerical value. Draw the shear-force and bending-moment diagrams for this condition. Repeat part (a) if the uniform load is replaced with a triangularly distributed load with peak intensity q 0 = q at mid-span (see Fig. b).
Mechanics of Materials (MindTap Co...
Solutions
Mechanics of Materials (MindTap Co...
A beam of length L is designed to support a uniform load of intensity q (see figure). If the supports of the beam are placed at the ends, creating a simple beam, the maximum bending moment in the beam is qL2/8. However, if the supports of the beam are moved symmetrically toward the middle of the beam (as shown), the maximum bending moment is reduced.
- Determine the distance a between the supports so that the maximum bending moment in the beam has the smallest possible numerical value. Draw the shear-force and bending-moment diagrams for this condition.
- Repeat part (a) if the uniform load is replaced with a triangularly distributed load with peak intensity q0= q at mid-span (see Fig. b).
a.
The distance a between the support by having the small numerical value of maximum bending moment and to draw the diagrams of shear force and bending moment.
Explanation of Solution
Given:
The given figure
The length of the beam L supports the load which is uniform having the intensity as q. The bending moment that is maximum is given as
Concept Used:
Resultant force,
Calculation:
As the forces are symmetry,
At a distance from x to C, the section between CA is considered,
When x=0,
When
At a distance from x to C, the section between AB is considered,
When
When,
b.
The distance ‘a’ between the support by having the small numerical value of maximum bending moment and to draw the diagrams of shear force and bending moment.
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