The continuous frame ABC has a pin support at A, roller supports at B and C and a rigid corner connection at B (see figure). Members AB and BC each have flexural rigidity EI. A moment Müacts counterclockwise at A. Note: Disregard axial deformations in member AB and consider only the effects of bending.
- Find all reactions of the frame.
Find the required new length of member AB in terms of L, so that
(a)
All the reactions of the frame.
Answer to Problem 10.4.9P
The reactions are :
Explanation of Solution
Given Information:
The following figure is given with all relevant information,
Calculation:
Consider the following free body diagram,
Take equilibrium of horizontal forces as,
Take equilibrium of vertical forces as,
Take equilibrium of moments about B as,
The bending moment at distance x from A along AB in part AB is given by,
Use second order deflection differential equation,
Integrate above equation to get rotation as,
Integrate above equation to get rotation as,
The bending moment at distance x from B along BC in part BC is given by,
Use second order deflection differential equation,
Integrate above equation to get rotation as,
Integrate above equation to get rotation as,
The constraint equations are,
Solve equations (1-8) to get integration constants and reactions.
So the reactions are
Conclusion:
Therefore, the reactions are:
(b)
Rotations at joints A, B, and C.
Answer to Problem 10.4.9P
Rotations at joints A, B, and C are
Explanation of Solution
Given Information:
The following figure is given with all relevant information,
Calculation:
Consider the following free body diagram,
Take equilibrium of horizontal forces as,
Take equilibrium of vertical forces as,
Take equilibrium of moments about B as,
The bending moment at distance x from A along AB in part AB is given by,
Use second order deflection differential equation,
Integrate above equation to get rotation as,
Integrate above equation to get rotation as,
The bending moment at distance x from B along BC in part BC is given by,
Use second order deflection differential equation,
Integrate above equation to get rotation as,
Integrate above equation to get rotation as,
The constraint equations are,
Solve equations (1-8) to get integration constants and reactions.
So the reactions are
Substitute the integration constants and reactions in expressions of rotations to get,
Conclusion:
Therefore, rotations at joints A, B, and C are
(c)
Length of AB.
Answer to Problem 10.4.9P
Length of AB is
Explanation of Solution
Given Information:
The following figure is given with all relevant information,
Also
Calculation:
Consider the following free body diagram,
Take equilibrium of horizontal forces as,
Take equilibrium of vertical forces as,
Take equilibrium of moments about B as,
The bending moment at distance x from A along AB in part AB is given by,
Use second order deflection differential equation,
Integrate above equation to get rotation as,
Integrate above equation to get rotation as,
The bending moment at distance x from B along BC in part BC is given by,
Use second order deflection differential equation,
Integrate above equation to get rotation as,
Integrate above equation to get rotation as,
The constraint equations are,
Solve equations (1-8) to get integration constants and reactions.
So the reactions are
Substitute the integration constants and reactions in expressions of rotations to get,
Solve above equation to get
Conclusion:
Therefore, length of AB is
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Chapter 10 Solutions
Mechanics of Materials (MindTap Course List)
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