Civil Engineering Reference
In-Depth Information
(equation 3.4)
N
cr
E
t
E
E
t
N
cr,t
(equation 3.17)
E
Strain
Tangent modulus
E
t
Slenderness
L
/
i
(a) Elastic stress-strain
relationship
(b) Tangent modulus
(c) Buckling stress
Figure 3.6
Tangent modulus theory of buckling.
linear. However, the buckling of an elastic member of a non-linear material, such
as that whose stress-strain relationship is shown in Figure 3.6a, can be analysed
by a simple modification of the linear elastic treatment. It is only necessary to
note that the small bending stresses and strains, which occur during buckling, are
relatedbythetangentmodulusofelasticity
E
t
correspondingtotheaveragecom-
pressivestress
N
/
A
(Figure3.6aandb),insteadoftheinitialmodulus
E
.Thusthe
flexuralrigidityisreducedfrom
EI
to
E
t
I
,andthetangentmodulusbucklingload
N
cr
,
t
is obtained from equation 3.4 by substituting
E
t
for
E
, whence
N
cr
,
t
=
π
2
E
t
A
(
L
/
i
)
2
.
(3.15)
Thedeviationofthistangentmodulusbucklingload
N
cr
,
t
fromtheelasticbuckling
load
N
cr
is shown in Figure 3.6c. It can be seen that the deviation of
N
cr
,
t
from
N
cr
increases as the slenderness ratio
L
/
i
decreases.
3.3.2 Reduced modulus theory of buckling
The tangent modulus theory of buckling is only valid for elastic materials. For
inelasticnonlinearmaterials, thechangesinthestressesandstrainsarerelatedby
theinitialmodulus
E
whenthetotalstrainisdecreasing, andthetangentmodulus
E
t
only applies when the total strain is increasing, as shown in Figure 3.7. The
flexuralrigidity
E
r
I
ofaninelasticmemberduringbucklingthereforedependson
both
E
and
E
t
.Asadirectconsequenceofthis,theeffective(orreduced)modulus
ofthesection
E
r
dependsonthegeometryofthecross-sectionaswell.Itisshown
in Section 3.9.1 that the reduced modulus of a rectangular section is given by
4
EE
t
√
E
+
√
E
t
E
r
=
2
.
(3.16)
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