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National Advisory Committee for Aeronautics, Report - The Lagrangian Multiplier Method of Finding Upper and Lower Limits to Critical Stresses of Clamped Plates

The theory of Lagrangian multipliers is applied to the prob-
lem of finding both upper and lower limits to the true compressive
buckling stress of a clamped rectangular plate. The upper and
lower limits thus bracket the true stress, which cannot be exactly
found by the difi’ere-ntial-eguation approach. The procedure for
obtaining the upper limit, which is believed to be new, presents
certain advantages over the classical Rayleigh—Ritz method of
finding upper limits. The theory of the lower-limit procedure has
been given by Trefitz but, in the present application, the method
difiers from that of Treftz in a way that makes it inherently more
quickly convergent. It is expected that in other buckling prob-
lems and in some vibration problems the Lagrangian multiplier
method of finding upper and lower limits may be advantageously
applied to the calculation of buckling stresses and natural
frequencies.

Many important problems that cannot be exactly solved by
the difi'erential—equation approach and must therefore be
analyzed by approximate methods arise in the buckling and
vibrations of thin plates. The theory of Lagrangian multi-
pliers can be a powerful tool in the analysis of many of these
problems. The present paper presents the details of applica-
tion as well as the fundamental principles of the Lagrangian
multiplier method by demonstrating the use of the method
to obtain both upper and lower limits to the true compressive
buckling stress of a rectangular plate clamped along all edges.

The procedure for obtaining the lower limit is similar to a
method used by Trefitz (reference 1) and recently described
by Reissner (reference 2). The present lower—limit method
differs from that of Trefitz, however, in a way that makes it
inherently more quickly convergent. The upper-limit pro-
cedure, which does not appear to have been presented pre-
viously, is simpler than the usual Rayleigh-Ritz method and
may be expected to permit the computation of more accurate
results with less labor.