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.

Question

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:

Assume that the length of the beam is 5m and the uniformly distributed load is 1kN/m. Draw the shear force and bending moment diagrams, then indicate the value of the shear force and bending moment at x=0m, x=1m, x=2m, x=3m, x=4m and x=5m.
Shear forces:

V(x=0m) in (kN):

V(x=1m) in (kN):

V(x=2m) in (kN):

V(x=3m) in (kN):

V just before the support B, at (x=4m) in (kN):

V just after the support B, at (x=4m) in (kN):

V(x=5m) in (kN):

Bending moments:

M(x=0m) in (kNm):

M(x=1m) in (kNm):

M(x=2m) in (kNm):

M(x=3m) in (kNm):

M(x=4m) in (kNm):

M(x=5m) in (kNm):

Assume that the inner radius of the cylinder ri is 10cm, and the thickness of the cylinder wall is 2mm, and, as before, the length L is 5m and the distributed load q is 1kN/m. What is the magnitude of the maximum stress in the cylinder? (Please remember that common moments of inertia can be easily looked up.)

Magnitude of σmax (in MPa):

Answer 1

M_max=(9/128)*q*L^2
x=3/8 * L

V(x=0m) in (kN):1.88

V(x=1m) in (kN):0.875

V(x=2m) in (kN):-0.125

V(x=3m) in (kN):-1.13

V just before the support B, at (x=4m) in (kN):-2.13

V just after the support B, at (x=4m) in (kN):1

V(x=5m) in (kN):0

Bending moments:

M(x=0m) in (kNm):0

M(x=1m) in (kNm):1.38

M(x=2m) in (kNm):1.76

M(x=3m) in (kNm):1.13

M(x=4m) in (kNm):-0.50

M(x=5m) in (kNm):0