Civil Engineering Reference
In-Depth Information
of buckling and will underestimate compression member strength. Therefore, buck-
ling direction (weak or strong axis) must be considered independently to determine
compression member strength when allowing for residual stresses.
The Column Research Council (CRC) conducted tests and analytical studies of
weak and strong axis inelastic buckling with linear and parabolic residual stress
distributions across the compression member cross section. These studies revealed
that,withintheinelasticrange,thecompressionmembercurves(see
Figure6.9a)
were
parabolic. The residual stresses used in the CRC studies were about 0.3
F
y
. However,
the value of 0.5
F
y
is used in order to conservatively represent the residual stresses
and provide a smooth transition to the Euler elastic buckling curve at
KL/r
C
c
(elastic behavior and buckling below the proportional limit of 0.5
F
y
). Therefore, for
KL/r < C
c
, the Johnson parabola (Equation 6.28) may be used to represent inelastic
behavior (Tall, 1974). The Johnson parabola is
=
B(KL/r)
2
.
F
cr
=
F
y
−
(6.28)
Thevalueoftheconstant
B
with
F
p
=
0.5
F
y
and
F
r
=
F
y
−
1952)
F
r
F
y
π
1
B
=
2
E
(F
y
−
F
r
)
=
(6.29)
2
E
4
π
and the inelastic critical buckling stress is
1
2
KL
r
F
y
F
cr
=
F
y
−
(6.30)
2
E
4
π
or
F
cr
F
y
=
2
1
−
0.25
λ
c
,
(6.31)
λ
c
=
K
r
F
y
where
2
E
.
π
Johnson parabola
F
cr
Euler curve
0.7
F
y
0.5
F
y
Strong axis, parabolic stress
Strong axis, linear stress
Weak axis, parabolic stress
Weak axis, linear stress
C
c
KL
/
r
FIGURE 6.9a
Weak and strong axis compression member curves using linear and parabolic
residual stress distributions.
