A hollow cylinder with outer radius r0 and inner radius ri is loaded by a uniformly distributed load q across its length L. The beam is supported by a roller at its left end and a pin L5 from its right end.
What is the maximum bending moment in the beam? Express your answer in terms of q and L.
Mmax
At what point along the beam does the maximum bending moment occur? Express your answer in terms of the fraction of the length L, from the left hand support, A.
x:
To find the maximum bending moment (Mmax) in the beam, you can use the formula for bending moment in a cylindrical beam:
M = q * ((r0^2 + ri^2 + r0 * ri) / 2) * (L - x)
Where:
- q is the uniformly distributed load across the length of the beam
- r0 is the outer radius of the cylinder
- ri is the inner radius of the cylinder
- L is the length of the beam
- x is the distance from the left-hand support (A) to the point along the beam where you want to find the bending moment.
To determine the maximum bending moment, you need to find the critical point where the bending moment is at its maximum. This occurs when the distance x is equal to zero or L.
At x = 0 (left-hand support), the bending moment is zero:
M(0) = q * ((r0^2 + ri^2 + r0 * ri) / 2) * (L - 0) = 0
At x = L (right-hand support), the bending moment is also zero:
M(L) = q * ((r0^2 + ri^2 + r0 * ri) / 2) * (L - L) = 0
Therefore, the maximum bending moment (Mmax) occurs somewhere between the supports (A and B) at a certain point x.
To find the point x where the maximum bending moment occurs, you can differentiate the bending moment equation with respect to x and set it equal to zero:
dM/dx = -q * ((r0^2 + ri^2 + r0 * ri) / 2) = 0
Solving this equation, you'll find that the critical point x is independent of the values of q, r0, ri, and L. Therefore, the maximum bending moment occurs at the midpoint of the beam, which is L/2.
So, the answer to the second question is:
x = L/2
The answer to the first question, the maximum bending moment (Mmax), remains the same as it depends on q and L and not on the position along the beam. Therefore, the maximum bending moment is:
Mmax = q * ((r0^2 + ri^2 + r0 * ri) / 2) * L