Concept explainers
5.128 and 5.129 The beam AB, consisting of a cast-iron plate of uniform thickness b and length L, is to support the distributed load w(x) shown. (a) Knowing that the beam is to be of constant strength, express h in terms of x, L, and h0. (b) Determine the smallest value of h0 if L = 750 mm, b = 30 mm, w0 = 300 kN/m, and σall = 200 MPa.
Fig. P5.129
(a)
Express h in terms of x, L, and
Answer to Problem 129P
The expression for h in terms of x, L, and
Explanation of Solution
Given information:
The uniform thickness of the beam is b.
The length of the beam is L.
The rise of the beam is
Calculation:
By definition;
Substitute
Integrate the equation to find V.
Apply the boundary condition;
Substitute 0 for V and 0 for x in Equation (1).
Substitute
By definition;
Substitute
Apply the boundary condition;
Substitute 0 for M and 0 for x in Equation (2).
Substitute 0 for
Determine the section modulus (S) of the beam using the relation.
Here, the allowable stress in the beam is
Substitute
Determine the section modulus (S) of the rectangular cross section using the relation.
Here, the width of the beam is b and the depth of the beam is h.
Substitute
When
Substitute
Substitute
Therefore, the expression for h in terms of x, L, and
(b)
The smallest value of
Answer to Problem 129P
The smallest value of
Explanation of Solution
Given information:
The maximum allowable stress is
The length of the beam is
The distributed load is
The width of the beam is
Calculation:
Substitute
Therefore, the smallest value of
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Chapter 5 Solutions
EBK MECHANICS OF MATERIALS
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