Finding reaction for a moment applied to a line 1

[teleswamp]

I'm trying to find a way to determine the force distribution from a moment applied to a line. For example, suppose I have a moment of 100 lb-in applied and supported by 2 points 10 inches away. The resulting force at each point would be +10 lbs and -10 lbs. Now suppose, the support is not at 2 points but along a line that is 10 inches long. I know at the middle of the line the force will be zero. The force will vary linearly to either end of the line to some force. How do I determine the force at the end of the line?

 

[rb1957]

you're looking for a distributed force that'll represent/react a momnet ?

this is something like a bending stress distribution; as linear varying distributed force from +p to -p.
think of the =ve forces ... from +p to 0 over a distance L (making the overall distance from +p to -p as 2L)
so the resultant is a force p*L/2 acting 2/3*L from the zero point (mid-way along the overall line).
and the moment reacted by this is p*L/2*(2/3*L*2) = 2/3*L^2*p
so p = M/(2/3*L^2)

 

[charliealphabravo]

how long is the line? how is the line loaded/restrained? i don't think you have enough info yet for a FBD.

 

[teleswamp]

Thanks rb1957, that seems like a good solution.

 

[BAretired]

If the "line" is a beam which is infinitely stiff relative to the foundation material, then a moment, M applied to the line will produce reaction forces P equal and opposite in direction and separated by a distance 2/3 L where L is the length of the line. So M = 2PL/3 or P = 3M/2L. The pressure p varies from a maximum of 6M/L² at the ends to 0 at the middle.

If the "line" is a flexible beam, the forces P will move toward the middle and the pressure will not vary linearly. The precise pressure distribution will depend on the relative stiffness of the beam and the foundation material. It would be a beam on an elastic foundation and could only be solved by equating the deflection of the beam to the strain in the elastic foundation.

BA