Two rigid bars are connected to each other by two linearly elastic springs. Before loads are applied, the lengths or the springs are such, that the bars are parallel and the springs are without stress.
(a) Derive a formula for the displacement E4at point 4 when the load P is applied at joint 3 and moment PL is applied at joint 1. as shown in the figure part a. (Assume that the bars rotate through very small angles under the action of load P.)
(b) Repeat part (a) if a rotational spring, kr= kL2, is now added at joint 6. What is the ratio of the deflection d4 in the figure part a to that in the figure part b ?
(a)
The formula for the displacement at point
Answer to Problem 2.2.13P
The formula for the displacement at point
Explanation of Solution
Given information:
The load applied at point
The moment art point
The following figure shows the free body diagram of the bar:
Figure-(1)
Write the expression for the moment equilibrium about support
Here, the reaction at forces in spring
Write the expression for the moment equilibrium about support
Figure-(2)
Write the expression for deflection
Here, the deflection at point
Figure-(3)
Write the expression for deflection
Here, the deflection at point
Write the expression for the spring stiffness.
Here, the spring constant is
Write the expression for the elongation of spring
Write the expression for the elongation of spring
Calculation:
Substitute
Substitute
Substitute
Substitute
Conclusion:
The formula for the displacement at point
(b)
The ratio of deflection at point
Answer to Problem 2.2.13P
The ratio of deflection at point 4 is
Explanation of Solution
Given Information:
The spring constant for rotational spring is
The following figure shows the free body diagram of the bar having rotational spring:
Figure-(4)
Write the expression for the moment equilibrium about support
Here, angle of deflection of beam is
Write the expression for the net elongation of spring
Write the expression for the displacement at point
Write the expression for the displacement at point
Write the expression for the net elongation of spring
Write the expression for displacement at point 3.
Write the expression for displacement at point 5.
The figure below shows the free body diagram of the bar.
Figure-(5)
Write the expression for the vertical force equilibrium.
Write the expression for the moment equilibrium about point 2.
Write the expression for the angle of deflection of the beam
The following figure shows the forces acting on point
Figure 6
Write the expression for the ratio of deflection of part
Calculation:
Substitute
Substitute
Substitute
Substitute
Conclusion:
The ratio of deflection at point
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Chapter 2 Solutions
Mechanics of Materials (MindTap Course List)
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- Mechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage Learning