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National Advisory Committee for Aeronautics, Report - Deflection and Stress Analysis of Thin Solid Wings of Arbitrary Plan Form with Particular Reference to Delta Wings

One of the present trends in the development of high—speed
airplanes and missiles is toward the use of thin low—aspect—
ratio wings. The structural analysis of these wings often
cannot be based on beam theory since the structural defor—
mations may vary considerably from those of a beam and,
indeed, may more closely approach those of a plate. In
cases where the wing construction is solid or nearly solid the
use of plate theory in the analysis is particularly valid, and
it is this type of wing which is considered in the present
report.

Exact solutions to the partial—differential equation of plate
theory are not readily obtained, especially for plates of
arbitrary shape and loading; however, a number of approxi—
mate solutions to specific problems on cantilever plates have
appeared in the literature (see, for example, refs. 1 to 7).
Of the approaches used in these references, only the one in
references 6 and 7 is readily applicable to plates of arbitrary
plan form, thickness distribution, and load distribution; thus
it is the most useful one for the analysis of actual wings.

In reference 6 the cantilever—plate problem is simplified by
the assumption that the deformations of the plate in the
chordwise direction (parallel to the root) are linear. By
minimizing the potential energy of the plate, the partial-
diflerential equation of plate theory is replaced by two
simultaneous ordinary differential equations for the spanwise
variations of the bending deflection and twist. In reference 7
the same ordinary differential equations are obtained in a
different manner. Refinement of the analysis by inclusion
of the effect of parabolic, cubic, or higher—order chordwise
camber terms is indicated in reference 6, and as the order of
refinement is increased a corresponding increase in the num-
ber of ordinary differential equations is obtained.

In the present report, which is an extension of reference 6,
a general set of ordinary differential equations is presented
which may be used to obtain any desired degree of approxi-
mation to the deflection of the plate. These equations are
solved exactly for several cases of delta plates under uniform
load first by considering linear chordwise deformation only
and second by including the effect of parabolic chordwise
camber. Comparisons are drawn between the stresses and
deflections computed from the equations of each approxi4
mation and also with some experimental results.