Concept explainers
A framework ABC consists of two rigid bars AB and BC. Each having a length b (see the first part of the figure part a). The bars have pin connections at A, B, and C and are joined by a spring of stiffness k. The spring is attached at the midpoints of the bars. The framework has a pin support at A and a roller support al C, and the bars are at an angle a to the horizontal.
When a vertical load P is applied at joint B (see the second part of the figure part a.) the roller support C moves to the right, the spring is stretched, and the angle of the bars decreases from a to the angle ??.
(a) Determine the angle 0 and the increase S in the distance between points A and C. Also find reactions at A and C. (Use the following data: b = 200 mm. ft = 3.2 kN/m. a = 45°. and
P = 50 N.)
(b) Repeat part (a) if a translational spring kt= kll is added at C and a rotational spring kr= kb-l2 is added at A (see figure pan b).
(a)
The angle
Answer to Problem 2.2.20P
The angle
The distance between points A and C,
The reaction at support C
The reaction at support A
Explanation of Solution
Given information:
The length of the rigid bars b = 200 mm
The stiffness of the spring k = 3.2 kN/m
The angle of bars from the horizontal =
The valueof load P = 50 N
Figure: Initial and displaced positions of the framework.
Calculation:
Let us consider the structure in its displaced position. Let us use the free body diagrams of left-hand bars and right-hand bars.
From the free body diagram of left-hand bar,
Reaction at support A,
From the free body diagram of right-hand bar,
From overall FBD of the beams,
We have two expressions for Rc, equating both of the expressions and then substitute expressions for L2, kr, k1, h and
Now,
Let us substitute the numerical values of variables and calculate the values of angle
Solving the above equation gives,
Now, let us compute the reactions at point C and A.
The reaction at point C
And the reaction at point A is
Conclusion: Thus, the angle
The distance between points A and C,
The reaction at support C
The reaction at support A
(b)
The angle
Answer to Problem 2.2.20P
The angle
The distance between points A and C,
The reaction at support C
The reaction at support A
The moment reaction at point A is
Explanation of Solution
Given information:
The length of the rigid bars b = 200 mm
The stiffness of the spring k = 3.2 kN/m
The angle of bars from the horizontal =
The value of load P = 50 N
Figure: Initial and displaced positions of the framework.
Calculation:
Let us consider the structure in its displaced position. Let us use the free body diagrams of left-hand bars and right-hand bars.
From the free body diagram of left-hand bar,
Reaction at support A,
From the free body diagram of right-hand bar,
From overall FBD of the beams,
We have two expressions for Rc, equating both of the expressions and then substitute expressions for L2, kr, k1, h and
Now,
Let us substitute the numerical values of variables and calculate the values of angle
Solving the above equation gives,
Now, let us compute the reactions at point C and A.
The reaction at point C,
And the reaction at point A is
Now, moment reaction at point A is,
Conclusion: Thus, the angle
The distance between points A and C,
The reaction at support C
The reaction at support A
The moment reaction at point A is
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