For torsion of the I-beam shown below, determine the effective polar
area moment of the cross section, the maximum shearing stress
( neglecting stress concentrations) and the warping displacement of
the center lines. All distances are center line distances
Assume:
tw < t f
tf
tw
h
tf
b
b
tw
tf
1
1
J eff = bt 3f × 2 + htw3
3
3
2bt 3f + htw3
=
3
h
tf
τ max
Ttmax Tt f
=
=
J eff
J eff
sectorial area function
ω = 2Ω
ω = bh / 4
ω = −bh / 4
shear center (on axes of symmetry)
take initial point of integration here
with ω0 = 0
ω = −bh / 4
Note:
∫ ω dA = 0
ω=0
for the web
( r⊥ = 0 )
ω = bh / 4
is satisfied automatically by our choice
ω0 = 0
(If we had let the constant to be non-zero, we could show it is zero by this
condition)
Since the warping displacement
u x = −φ ′ω p
the warping of the center lines of the flanges looks like:
T