Eight identical 500 × 750-mm rectangular plates, each of mass = 40 kg, are held in a vertical plane as shown. All connections consist of frictionless pins, rollers, or short links. In each case, determine whether (a) the plate is completely, partially, or improperly constrained, (b) the reactions are statically determine or indeterminate, (c) the equilibrium of the plate is maintained the position shown. Also, wherever possible, compute the reactions.
(a)
Find whether the plate is completely, partially, or improperly constrained.
Answer to Problem 4.59P
The plate in figure 1 is
The plate figure 2 is
The plate figure 3 is
The plate figure 4 is
The plate figure 5 is
The plate figure 6 is
The plate figure 7 is
The plate figure 8 is
Explanation of Solution
Given information:
The size of the identical plates is
Number of plates is 8.
The mass of each plate is
Calculation:
Find the weight (W) of the plate using the relation.
Here, the acceleration due to gravity is g.
Consider the acceleration due to gravity as
Substitute 40 kg for m and
Figure 1:
Show the free-body diagram of the Figure 1.
The three reactions in the plate behave like non-concurrent and non-parallel force system.
The plate in figure 1 is
Figure 2:
Show the free-body diagram of the Figure 2.
The three reactions in the plate behave like non-concurrent and non-parallel force system.
The plate figure 2 is
Figure 3:
Show the free-body diagram of the Figure 3.
The four reactions in the plate behave like non-concurrent and non-parallel force system.
The plate figure 3 is
Figure 4:
Show the free-body diagram of the Figure 4.
The three reactions in the plate behave like concurrent force system.
The plate figure 4 is
Figure 5:
Show the free-body diagram of the Figure 5.
The two reactions in the plate behave like concurrent force system.
The plate figure 5 is
Figure 6:
Show the free-body diagram of the Figure 6.
The three reactions in the plate behave like non-concurrent and non-parallel force system.
The plate figure 6 is
Figure 7:
Show the free-body diagram of the Figure 7.
The two reactions in the plate behave like concurrent force system.
The plate figure 7 is
Figure 8:
Show the free-body diagram of the Figure 8.
The four reactions in the plate behave like non-concurrent and non-parallel force system.
The plate figure 8 is
(b)
Find whether the reactions are statically determinate or indeterminate.
Answer to Problem 4.59P
The reactions in figure 1 is
The reactions in figure 2 is
The reactions in figure 3 is
The reactions in figure 4 is
The reactions in figure 5 is
The reactions in figure 6 is
The reactions in figure 7 is
The reactions in figure 8 is
Explanation of Solution
Refer Figure 1:
The equilibrium equations are;
The equilibrium equations are enough to determine the unknown reactions.
The reactions in figure 1 is
Refer Figure 2:
The equilibrium equations are;
The equilibrium equations are enough to determine the unknown reactions.
The reactions in figure 2 is
Refer Figure 3:
The equilibrium equations are;
The equilibrium equations are not enough to determine the unknown reactions.
The reactions in figure 3 is
Refer Figure 4:
The equilibrium equations are;
The equilibrium equations are enough to determine the unknown reactions.
But the plate is improperly constrained and the plate is not in equilibrium.
The reactions in figure 4 is
Refer Figure 5:
The equilibrium equations are;
The equilibrium equations are enough to determine the unknown reactions.
The reactions in figure 5 is
Refer Figure 6:
The equilibrium equations are;
The equilibrium equations are enough to determine the unknown reactions.
The reactions in figure 6 is
Refer Figure 7:
The equilibrium equations are;
The equilibrium equations are enough to determine the unknown reactions.
But the plate is improperly constrained and the plate is not in equilibrium.
The reactions in figure 7 is
Refer Figure 8:
The equilibrium equations are;
The equilibrium equations are not enough to determine the unknown reactions.
The reactions in figure 8 is
(c)
Find whether the equilibrium of the plate is maintained.
Answer to Problem 4.59P
The reactions in the plate 1 are
The plate 1 is in
The reactions in the plate 2 are
The plate 2 is in
The reactions in the plate 3 are
The plate 3 is in
The plate 4 is in
The reactions in the plate 5 are
The plate 5 is in
The reactions in the plate 6 are
The plate 6 is in
The plate 7 is in
The reactions in the plate 8 are
The plate 8 is in
Explanation of Solution
Refer Figure 1:
The equilibrium equations are;
Take moment about point A.
Resolve the horizontal component of forces.
Resolve the vertical component of forces.
Therefore, the reactions in the plate 1 are
The plate 1 is in
Refer Figure 2:
The equilibrium equations are;
Take moment about point B.
Resolve the horizontal component of forces.
Resolve the vertical component of forces.
Therefore, the reactions in the plate 2 are
The plate 2 is in
Refer Figure 3:
The equilibrium equations are;
Take moment about point A.
Resolve the horizontal component of forces.
Resolve the vertical component of forces.
Therefore, the reactions in the plate 3 are
The plate 3 is in
Refer Figure 4:
The equilibrium equations are;
The moment about point D is not equal to zero.
The plate 4 is in
Refer Figure 5:
The equilibrium equations are;
Take moment about point A.
Resolve the vertical component of forces.
Therefore, the reactions in the plate 5 are
The plate 5 is in
Refer Figure 6:
The equilibrium equations are;
Take moment about point A.
Resolve the vertical component of forces.
Resolve the horizontal component of forces.
Find the resultant force at D;
Find the angle
Therefore, the reactions in the plate 6 are
The plate 6 is in
Refer Figure 7:
The equilibrium equations are;
The plate 7 is in
Refer Figure 8:
The equilibrium equations are;
Take moment about point D.
Resolve the vertical component of forces.
Resolve the horizontal component of forces.
Therefore, the reactions in the plate 8 are
The plate 8 is in
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