Page 194 - Aircraft Stuctures for Engineering Student

178 Structural instability

carry load even though a portion of the plate has buckled. In fact, th~ ultimate load is
not reached until the stress in the majority of the plate exceeds the elastic limit. The
theoretical calculation of the ultimate stress is diffcult since non-linearity results from
both large deflections and the inelastic stress-strain relationship.
Gerard' proposes a semi-empirical solution for flat plates supported on all four
edges. After elastic buckling occurs theory and experiment indicate that the average
compressive stress, Fa, in the plate and the unloaded edge stress, ne, are related by the
following expression
(6.59)
DCR = 12(1 k2E - d) u2
where

b
and al is some unknown constant. Theoretical work by Stowell' and Mayers and
Budianskyg shows that failure occurs when the stress along the unloaded edge is
approximately equal to the compressive yield strength, u,.+ of the material. Hence
substituting uCy for oe in Eq. (6.59) and rearranging gives
1 -n
*f (6.60)

where the average compressive stress in the plate has become the average stress at
failure af. Substituting for uCR in Eq. (6.60) and putting
a12('
-4
[12(1 - d)]'-" =a
yields

(6.61)

or, in a simplified form

(6.62)

where 0 = aKnI2. The constants ,6' and m are determined by the best fit of Eq. (6.62) to
test data.
Experiments on simply supported flat plates and square tubes of various alumi-
nium and magnesium alloys and steel show that p = 1.42 and m = 0.85 fit the results
within f10 per cent up to the yield strength. Corresponding values for long clamped
flat plates are p = 1.80, m = 0.85.
extended the above method to the prediction of local failure stresses
for the plate elements of thin-walled columns. Equation (6.62) becomes

(6.63)