Civil Engineering Reference
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Distortion deflections
Warping normal stresses
Warping shear stresses
Figure 10.36
Warping stresses in non-uniform distortion.
loadingcorrespondstotheappliedbeamloadsintheanalogy,andtheout-of-plane
resistance of the plate elements corresponds to the stiffness
k
of the elastic foun-
dation, whilethein-planeresistanceoftheseelementscorrespondstotheflexural
rigidity
EI
oftheanalogousbeam.TheBEFanalogycanaccountforconcentrated
and distributed distortional loads, for rigid diaphragms which either permit or
prevent warping, and for flexible diaphragms and stiffening rings, as indicated in
Figure 10.35. Many solutions for problems of beams on elastic foundations have
beenobtained[31],andthesecanbeusedtofindtheout-of-planebendingstresses
intheplateelements(whicharerelatedtothefoundationreactions),andthewarp-
ing and normal shear stresses due to non-uniform distortion (which are related
to the bending moment and shear force in the beam on the elastic foundation).
Thedistributionsofthewarpingstressesinarectangularboxsectionareshownin
Figure 10.36.
10.7 Appendix - uniform torsion
10.7.1 Thin-walled closed sections
Theshearflow
τ
t
t
aroundathin-walledclosed-sectionmemberinuniformtorsion
can be determined by using the condition that the total change in the warping
displacement
u
around the complete closed section must be zero for continuity.
The change in the warping displacement due to the twist rotations
φ
is equal to
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