Civil Engineering Reference
In-Depth Information
have already been yielded before the initiation of instability. Therefore, the effective
modulus of elasticity is less than the initial value. This nonlinear behavior occurs
primarily as a result of residual stresses
∗
but may also be a result of initial curvature
and force eccentricity.
These material and/or geometric imperfections (or nonlinearities) are considered
by replacing the elastic modulus,
E
, with an effective modulus,
E
eff
.Therefore, inelas-
elastic modulus,
E
, replaced with the effective modulus,
E
eff
, so that
2
E
eff
I
(KL)
2
.
P
cr
=
π
(6.24)
Engesser (see Chapter 1) proposed both the tangent modulus,
E
t
(Equation 6.25),
and the reduced modulus,
E
r
(Equation 6.26), for the effective modulus. The tangent
modulus is
E
F
y
− σ
F
y
−
.
d
d
E
t
=
=
(6.25)
ε
c
σ
The reduced modulus, for symmetric I-sections (and neglecting web area), is
(Timoshenko and Gere, 1961)
2
EE
t
E
E
r
=
E
t
,
(6.26)
−
where d
σ
is the change in stress, d
ε
is the change in strain,
σ
is the applied stress
=
P/A,
c
0.99 for structural steel.The reduced modulus is less than the tangent
modulus,
E
t
, as shown in Figure 6.7.
An inelastic compression member theory was also proposed by Shanley (Tall,
1974). The theory indicates that actual inelastic compression member behavior lies
between that of the tangent and the reduced modulus load curves. However, because
test results are closer to the tangent modulus curve values (Chen and Lui, 1987), the
=
0.96
−
Euler curve
f
cr
Curve with
E
t
Curve with
E
r
KL
/
r
FIGURE 6.7
Typical compression member curves.
∗
Since residual stresses are most affected by size, the use of high-strength steel can make their effect
relatively smaller. Annealing to reduce residual stresses (heat treatment) may increase the strength of a
compression member.
