Civil Engineering Reference
In-Depth Information
However, since the allowable bending stress may not be the same in each plane of
bending, Equation 8.6 may be expressed as
f
b
x
F
b
x
+
f
b
y
F
b
y
=
(M
x
/I
x
)y
F
b
x
(M
y
/I
y
)x
F
b
y
+
≤
1,
(8.7)
where
F
b
x
is the allowable bending stress in the direction of the
x
axis and
F
b
y
is the
allowable bending stress in the direction of the
y
axis.
The interaction Equation 8.7 may be used for design considering both tensile and
compressive flexural stresses by using the appropriate allowable bending stress for
flexural tension (
F
b
x
,
F
b
y
=
F
tall
=
0.55
F
y
)
or compression (
F
b
x
,
F
b
y
=
F
call
)
.
8.3 UNSYMMETRICAL BENDING (COMBINED BENDING
AND TORSION)
The best design strategy is to avoid torsion. However, in some cases it is unavoidable.
Torsion is combined with bending when transverse loads are not applied through the
shear center of the member.
Whenatorsionalmomentisapplied,puretorsionalwaysexists.Puretorsioncreates
shearing stresses in the flanges and webs of structural shapes such as channels and
I-shaped beams. However, warping torsion also exists when cross sections do not
remain plane due to some form of restraint. Warping torsion creates shearing and
normal stresses in the flanges of I shapes and normal stresses in the flanges and web
of channels. These torsional shear and normal stresses must be superimposed on the
shear and normal stresses in flanges and webs due to flexure.
The torsional moment resistance,
T
, of a cross section to a constant torsional
moment is
T
=
T
t
+
T
w
,
(8.8)
where
T
t
is the pure torsional (or St. Venant) moment resistance and is given by
GJ
d
d
z
,
T
t
=
(8.9)
T
w
is the warping torsional moment resistance and is given by
d
3
θ
d
z
3
,
T
w
=−
EC
w
(8.10)
z
is the longitudinal axis of the beam or girder,
G
is the shear modulus of elasticity
of steel (
29,000 ksi),
J
is the torsional constant of the cross section, and
C
w
is the warping constant of
the cross section (see Table 7.1).
The shear stresses from pure torsion effects of the applied torsional moments are
∼
11,200 ksi),
E
is the tensile modulus of elasticity of steel (
∼
T
t
t
J
τ
t
=
(8.11)
