Problem 1 Compare the maximum bending and shear stresses in a cantilever of dimensionsL × b × d subjected to uniform load q o per unit length, and draw appropriate conclusions
Question
Problem 1 Compare the maximum bending and shear stresses in a cantilever of dimensionsL × b × d subjected to uniform load q o per unit length, and draw appropriate conclusions
Solution
The maximum bending stress in a cantilever beam subjected to a uniform load can be calculated using the formula:
σ_max = M_max / S
where M_max is the maximum moment and S is the section modulus. For a rectangular cross section, S = bd^2 / 6, and for a cantilever beam with a uniform load, M_max = qo * L^2 / 2. Substituting these values in, we get:
σ_max = qo * L^2 / (2 * bd^2 / 6) = 3qo * L^2 / (2 * bd^2)
The maximum shear stress in a cantilever beam subjected to a uniform load can be calculated using the formula:
τ_max = V_max / A
where V_max is the maximum shear force and A is the cross-sectional area. For a rectangular cross section, A = bd, and for a cantilever beam with a uniform load, V_max = qo * L. Substituting these values in, we get:
τ_max = qo * L / bd
Comparing these two equations, we can see that the maximum bending stress is proportional to the square of the length of the beam and inversely proportional to the square of the depth, while the maximum shear stress is directly proportional to the length and inversely proportional to the depth. This means that for long, shallow beams, the bending stress will be much greater than the shear stress, while for short, deep beams, the shear stress will be more significant.
In terms of design, this means that if you are designing a beam to resist bending, you should make it as deep as possible, while if you are designing a beam to resist shear, you should make it as short as possible. However, in most practical applications, both bending and shear stresses need to be considered, and a balance must be struck between the two.
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