Concept explainers
The overhanging beam A BCD supports two concentrated loads P and Q (see figure),
- For what ratio PIQ will the deflection at point B be zero?
- For what ratio will the deflection at point D be zero?
- If Q is replaced by a uniform load with intensity q (on the overhang), repeat parts (a) and (b), but find ratio Pl(qa).
(a)
Ratio P/Q for which deflection at B is zero .
Answer to Problem 9.5.23P
Ratio P/Q for which deflection at B is zero is
Explanation of Solution
Given Information:
The following figure is given along with relevant information,
The deflection at B is zero.
Calculation:
Consider the following free body diagram,
Take equilibrium of forces in horizontal direction as,
Take equilibrium of forces in vertical direction as,
Take equilibrium of moments about A as,
The bending moment at distance x from point A is given by,
The deflection and bending moment is related by following differential equation
Integrate differential equation (1) with respect to x by putting expression for M to get angle of rotations, as,
Integrate angle of rotation with respect to x get deflections as,
The following conditions are used to evaluate integration constants,
Since deflection at B is zero, hence
Now substitute values of constants and solve the above equation to get
Conclusion:
Therefore, the ratio P/Q for which deflection at B is zero is
(b)
Ratio P/Q for which deflection at D is zero .
Answer to Problem 9.5.23P
Ratio P/Q for which deflection at D is zero is
Explanation of Solution
Given Information:
The following figure is given along with relevant information,
The deflection at D is zero.
Calculation:
Consider the following free body diagram,
Take equilibrium of forces in horizontal direction as,
Take equilibrium of forces in vertical direction as,
Take equilibrium of moments about A as,
The bending moment at distance x from point A is given by,
The deflection and bending moment is related by following differential equation
Integrate differential equation (1) with respect to x by putting expression for M to get angle of rotations, as,
Integrate angle of rotation with respect to x get deflections as,
The following conditions are used to evaluate integration constants,
Since deflection at D is zero, hence
Now substitute values of constants and solve the above equation to get
Conclusion:
Therefore, the ratio P/Q for which deflection at D is zero is
(c)
Ratio P/Q for which deflection at B and D is zero .
Answer to Problem 9.5.23P
Explanation of Solution
Given Information:
The following figure is given along with relevant information,
The deflection at B and D is zero.
Calculation:
Consider the following free body diagram,
Take equilibrium of forces in horizontal direction as,
Take equilibrium of forces in vertical direction as,
Take equilibrium of moments about A as,
The bending moment at distance x from point A is given by,
The deflection and bending moment is related by following differential equation
Integrate differential equation (1) with respect to x by putting expression for M to get angle of rotations, as,
Integrate angle of rotation with respect to x get deflections as,
The following conditions are used to evaluate integration constants,
Substitute values of constants to get the expression for deflection.
Since deflection at B is zero, hence
Solve the above equation to get
For deflection at D is zero,
Solve the above equation to get
Conclusion:
Therefore, the ratio P/Q
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Chapter 9 Solutions
Mechanics of Materials (MindTap Course List)
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