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National Advisory Committee for Aeronautics, Technical Notes - Secondary Stresses in Thin Walled Beams with Closed Cross Sections

An accurate method of determining secondary stresses in thin—
walled, uniform beams of closed cross section is herein presented.
The cross sections are assumed to be preserved by closely spaced rigid
diaphragms. In section I the integrodifferential equation governing
axial displacements is formulated and solved for a beam without longi-
tudinal stiffeners. In section II the corresponding summation—
difference equation is developed and solved for a beam with stiffeners
(flanges and stringers). The cross section, loading distribution, and
end conditions are assumed to be arbitrary.

By introducing generalised difference equations the mathematical
analysis for the stiffened beam may be performed in a manner exactly
analogous to the process used for the unstiffened'beam. A separation
of variables in the homogeneous equation leads to the natural stress or
displacement modes for a cross section. The solution of the nonhomo—
geneous equation is then expressed as an expansion in terms of the
natural stress modes. Particular attention is given to cross sections
with single symmetry and double symmetry.

Reference will be made only to papers which give exact solutions
for beams having rigid bulkheads with monocoque (unstiffened) or semi-
monocoque (stiffened) sections. In most cases the solutions for semi-
monocoque sections are those Obtained by replacing the true section by
an idealized section in which the normal stresses are carried on a
finite number of stiffeners and uniformly distributed shear flows are
carried on the connecting webs or wall elements.

If an idealized cross section has only three stiffeners, or flanges,
the distribution of axial stress must be planar and there are no sec-
ondary stresses. The doubly symmetrical rectangular idealized section
with four flanges was introduced and analyzed by Ebner in 1933
(reference 1). This cross section has one secondary stress mode which
occurs under torsional loading. Ebner considered such.modifications of
the problem as the effects of web buckling, shear deformation of the
ribs, and finite rib spacing.