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National Advisory Committee for Aeronautics, Technical Notes - On the Distribution of Stress in a Thin Plate Elastically Supported Along Two Edges at Loads Beyond the Stability Limit

A considerable amount of theoretical and experimental
work has been done on_the stress distribution in thin
plates at loads beyond the stability limit, as is evidenced
by the references included at the end of this report. The
two important factors which enter into the_theoretical calr
culations are the amplitude and the wave form of the buck—
led plate as a function of the ratio of the edge stress to
the buckling stress. Several approximate expressions for
the amplitude have been derived. .(See references..l. 3.
and 5.) Calculations of the exact wave form after buckling
are extremely complicated even for simple cases; for exam:
ple, a circular plate with a simply supported edge and sub-
Jectcd to a uniform compression load in_ the plane of the
plate (r-eference 6). For this reason. in the case of rec—
tangular plates, it is customary to assume that the wave
form after buckling remains constant and equal to that at_
buckling. ‘

The majority of investigators have assumed a wave
form which satisfies the boundary conditions of simple
support along all four edges. This type of edge support
is very seldom realized in airplane construction since the
edges parallel to the applied load are usually_supported
by some form of stiffener; that is, the edges are elastie
cally supported.

For the practical usage of thin plates the distribu—
tion of the median fiber stress and the.bending stress
is of importance. The first is important in the effec—
tive width calculations and the latter in the calculations“
of the maximum stresses for permanent'deformations} Since
these stresses are functions of the*plate'curvature,it'is
desirable to have a reasonably close approximation cf the
correct wave form. The condition of simple support along
the edges requires zero bending moment at the edges;
however in the actual case the bending moments along the
elastically supported edges are different from zero.
Also, the maximum comp.ression stress at the median surface
occurs at these edges. so that for the effects of combined
bending and compression stresses it is important that the
elastic support along the edges be considered.