Civil Engineering Reference
In-Depth Information
extreme fiber; d
A
is the infinitesimal area under consideration;
σ
c
is the stress at
the extreme fiber of the beam;
σ
max
is the maximum stress (at the top or bottom
extremefiber);
F
y
isthespecifiedsteelyieldstress;
I
x
=
y
2
d
A
istheverticalbending
moment of inertia about the beam or girder neutral axis; and
S
x
=
I
x
/c
is the vertical
bending section modulus about the beam or girder neutral axis.
Equation 7.3 enables the determination of the section modulus based on allowable
tensile stress. However, the equation does not address instability in the compression
region. If compression region stability (usually by lateral support of compression
flange) is sustained, Equation 7.3 may be used to determine both the required net and
gross section properties of the beam. Lateral support of the compression flange may
be provided by a connected steel or concrete deck, and/or either diaphragms, cross
bracing frames, or struts at appropriate intervals. However, if the compression flange
is laterally unsupported, instability must be considered as it may effectively reduce
the allowable compressive stress.
7.2.2 B
ENDING OF
L
ATERALLY
U
NSUPPORTED
B
EAMS AND
G
IRDERS
If the compression flange of a beam or girder is not supported at adequately close
intervals, it is susceptible to lateral-torsional instability prior to yielding and may not
be able to fully participate in resisting bending moment applied to the beam or girder.
In addition to the vertical translation or deflection,
y
, simply supported doubly
symmetric elastic beams subjected to uniform bending will buckle with lateral trans-
lation,
w
,andtorsionaltranslationortwist,
φ
,asshowninFigure7.2.Itisassumedthat
I
x
I
y
sothatverticaldeformationeffectsmaybeneglectedwithrespecttothelateral
deformation. It is also assumed that vertical deformation has no effect on torsional
twist and the effect of prebuckling on in-plane deflections may be ignored because
EI
x
EI
w
/L
2
. The equilibrium equation of out-of-plane bending (in
terms of flexural resistance) is
EI
y
GJ
M
d
2
d
4
w(x)
d
x
4
(x)
d
x
2
φ
=−
EI
y
(7.4)
y
L
M
x
x
w
y'
M
w
z
y
M
d
d
x
z'
x'
z'
z
ϕ
FIGURE 7.2
Bending of a beam in buckled position.
