Page 175 - Aircraft Stuctures for Engineering Student

6.2 Inelastic buckling 159

or
I1
E,=E--+EtT I2 (6.16)
I
where E, is known as the reduced modulus, gives
d2v
E,I-+ PV = 0
d22
Comparing this with Eq. (6.2) we see that if P is the critical load PCR then
~E,I
PCR = - (6.17)
1:
and

(6.18)

The above method for predicting critical loads and stresses outside the elastic range is
known as the reduced modulus theory. From Eq. (6.13) we have
J:
E y1 dA - Et y2dA = 0 (6.19)

which, together with the relationship d = dl + d2, enables the position of nn to be
found.
It is possible that the axial load P is increased at the time of the lateral disturbance
of the column such that there is no strain reversal on its convex side. The compressive
stress therefore increases over the complete section so that the tangent modulus
applies over the whole cross-section. The analysis is then the same as that for
column buckling within the elastic limit except that Et is substituted for E. Hence
the tangent modulus theory gives

(6.20)

and

(6.21)

By a similar argument, a reduction in P could result in a decrease in stress over the
whole cross-section. The elastic modulus applies in this case and the critical load and
stress are given by the standard Euler theory; namely, Eqs (6.7) and (6.8).
In Eq. (6.16), I1 and 12 are together greater than I while E is greater than Et. It
follows that the reduced modulus E, is greater than the tangent modulus Et.
Consequently, buckling loads predicted by the reduced modulus theory are greater
than buckling loads derived from the tangent modulus theory, so that although we
have specified theoretical loading situations where the different theories would
apply there still remains the difficulty of deciding which should be used for design
purposes.