A hollow cylinder with outer radius r_0 and inner radius r_i 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 L/5 from its right end.
What is the maximum bending moment in the beam? Express your answer in terms of q and L.
what is the force Fp up at the pin?
Fp(4/5)L = q L(L/2)
Fp = q L (5/8)
so the force up at he roller on the left is
Fr = q L - (5/8) q L = (3/8) q L
in the left sector
shear= (3/8) q L - integral from 0 to x of q dx
= (3/8) q L - q x
moment = integral of shear * dx
= integral [ (3/8) q L dx - q x dx ]
= (3/8) q L x - q x^2/2 + constant
when x = 0 bending moment must be 0 so constant is 0
so look for max of that
where (3/8) q L - q x = 0
x = (3/8) L
and there the BM is
(3/8) q L x - q x^2/2
= (3/8) q L(3/8) L - q (9/64)L^2/2
= (9 /128) q L^2
check my arithmetic !
Hey thanks for your guidance this is correct.
Can you help me with the step where you did:
moment = integral of shear * dx
= integral [ (3/8) q L dx - q x dx ]
why does the derivative of x just become x?
My apologies for separate messages.. but I'm also trying to figure out how the second term has a 2 in the denominator now:
M = (3/8) q L x - q x^2/2 + constant
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To find the maximum bending moment in the beam, we need to first determine the reaction forces at the supports. In this case, the beam is supported by a roller at its left end and a pin L/5 from its right end.
1. Calculate the reaction forces:
- Since the roller support at the left end can only provide a vertical reaction, the reaction force at the left end will be vertical and is equal to the total load, q multiplied by the length of the beam, L.
- At the right end, the beam is supported by a pin. The reaction force at this support will have both vertical and horizontal components. However, since the beam is loaded uniformly across its length and the pin is located at L/5 from the right end, the horizontal reaction force at the pin will be zero. Therefore, the reaction force at the pin will only have a vertical component, which can be calculated by taking the total load, q multiplied by L/5.
2. Once the reaction forces are determined, we can calculate the maximum bending moment using the following formula:
- Maximum Bending Moment = Reaction Force at the Pin * Distance from the Pin to the Maximum Bending Point
In this case, the maximum bending moment occurs at the point where the load is maximum, which is at the middle of the beam.
3. Calculate the distance from the pin to the maximum bending point:
- The position of the maximum bending point can be found by taking half the length of the beam, L/2.
Now we have all the necessary values to calculate the maximum bending moment:
Maximum Bending Moment = (Reaction Force at the Pin) * (Distance from the Pin to the Maximum Bending Point)
= (q * L/5) * (L/2)
Therefore, the maximum bending moment in the beam is given by the expression:
Maximum Bending Moment = (q * L^2) / 10