Concept explainers
A block of weight W is dropped from a height h onto the horizontal beam AB and hits point D. (a) Denoting by ym the exact value of the maximum deflection at D and by y’m the value obtained by neglecting the effect of this deflection on the change in potential energy of the block, show that the absolute value of the relative error is (y’m — ym)/ym, never exceeding y’m /2h. (b) Check the result obtained in part a by solving part a of Prob. 11.52 without taking ym into account when determining the change in potential energy of the load, and comparing the answer obtained in this way with the exact answer to that problem.
11.52 The 2-kg block D is dropped from the position shown onto the end of a 16-mm-diameter rod. Knowing that E = 200 GPa, determine (a) the maximum deflection of end A, (b) the maximum bending moment in the rod, (c) the maximum normal stress in the rod.
Fig. P11.52
(a)
The absolute value of the relative error
Answer to Problem 57P
The absolute value of the relative error
Explanation of Solution
Calculation:
Sketch the loading diagram as shown in Figure 1.
Refer to Figure 1.
Apply the spring constant k for the load applied at point D.
The load
Calculate the maximum strain energy
Substitute
Calculate the work of the block exactly as shown below.
Calculate the work of the block approximately as shown below.
Equating work and strain energy as shown below.
Equating work and strain energy exactly as shown below
Substitute
Equating work and strain energy exactly as shown below
Substitute
Here,
Subtracting Equation (3) from Equation (2) as shown below.
Here,
Substitute
Hence, the absolute value of the relative error
(b)
The relative error of the block.
Answer to Problem 57P
The relative error of the block is
Explanation of Solution
Given information:
The mass of the block is
The modulus of elasticity is
The diameter of the rod is
The length of the beam is
The dropping height is
Calculation:
Refer to part (a).
The relative error is
Consider the acceleration due to gravity as
Calculate the weight of the block as shown below.
Substitute
Calculate the moment of inertia
Substitute
Calculate the moment of inertia
Substitute
Calculate the centroid (c) of the rod as shown below.
Substitute
Sketch the deformation diagram as shown in Figure 2.
Refer to Figure 2.
Refer to Appendix D “Beam Deflections and Slope” in the text book,
Calculate the maximum deflection
Substitute
Calculate the maximum strain energy
Substitute
Calculate the work of the block as shown below.
Substitute
Calculate the maximum deflection
Substitute
Calculate the spring constant
Substitute
Calculate the approximate value of
Substitute
Calculate the relative error as shown below.
Substitute
To check:
Therefore, the relative error is
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Chapter 11 Solutions
Mechanics of Materials, 7th Edition
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