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National Advisory Committee for Aeronautics, Technical Notes - A Deflection Formula for Single Span Beams of Constant Section Subjected to Combined Axial and Transverse Loads

In this paper there is presented a deflection formula
for single-span beams of constant section subjected to
combined axial and transverse loads of the types commonly
encountered in airplane design. The form of the equation
is obtainable by dimensional analysis. Tables and curves
of the nondimensional coefficients are appended to facili~
tate the use of the formula.
The equation is applied to the determinatien of the
spring constant of a beam. Tables and curves are present—
ed to show the variation of the spring constant with
changes in the axial load and position along the beam.
In reference 1 deflection formulas are presented for
single~span beams of constant section subjected to axial
compression:and transverse loading. These formulas are
considerably different for the various loading conditions
treated and must be altered when the axial load is tension.
The purpose of this report is to present a simple for-
mula that includes all the above—mentioned cases of ref—
erence 1 and is valid when the axial load is either ten-
sion or compression.
The deflection formula derived in this report makes
it possible to make the suggested study and to show very
clearly the effect of changes in the axial load. The gen—
eral deflection formula presented herein is derived by the
strain-energy method of analysis which is believed to lead
to a more simple form of solution, A comprehensive treat-
ment of this method has been presented by Timoshenko.
The following derivation applies to the case of a
single—span beam of constant section subjected to the com—
bination of loadings shown in figure 1. The-loads. mo—
ments. and deflections are shown in the positive direc—
tions. The conventions adopted here are such that positive
lateral loads and positive and moments produce positive de—
flections; also positive axial load increases deflections.