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# Bending and torsion on curved monorail

## Bending and torsion on curved monorail

(OP)
I am designing a curved monorail and would like anyone's input on the best method to use to calculate bending and shear stress in the beam when the load is applied on a curve between supports.

Thanks.

### RE: Bending and torsion on curved monorail

I would suggest you find the principal coordinates of the section of your monorail (including shear center). In this case you get for uncoupled equations:

EA W'' = - n
EIxx V'''' =  mx + vy
EIyy U'''' = - my + vx
EIww F'''' - GJs F'' = - mw' + t'

where

E: Young Modulus of elasticity
G: shear modulus
A: Cross sectional area
Ixx, Iyy: second moments of area in shear center principal coordinates (x,y)
Iww: warping constant in principal coordinates (in^6)
Js: St. Venant torsional rigidity
W'' = d^2 W/dz^2 where W is the axial deformation
V'''' = d^4 V/dz^4 where V is dislacement in y dir
U'''' = d^4 U/dz^4 where U is dislacement in x dir
F: twist of the section
n: distributed axial force
mx, my: distributed bending moments
mw: distributed warping moment
t': distributed torquwe

Solve these equations with appropriate boundary conditions.

Most of the work is in finding the principal axis and the various cross sectional properties...

### RE: Bending and torsion on curved monorail

If that is too hard, simply calculate the torsion load in the beam with the load at mid-span. Then resist that torsion by warping alone. If your beam is not strong enough to do that, it probably isn't stiff enough either.

If that doesn't work, make a composite section of a Rectangular Hollow Section (RHS) above a Wide Flange Beam, with the RHS welded to the Top Flange with a pair of fillet welds. Use the RHS to resist the torsion, and the Beam + RHS to resist the vertical bending.

Russell Keays

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