huangyhg

bending and torsion on curved monorail
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.
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
vx, vy: distributed lateral loads
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...
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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