Theory of Thin-Walled Rods Page: 3 of 54
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NACA UM 1322
There is first of all investigated a particular integral and it
is shown that the transverse load in the case, and only in the case,
where it varies linearly over the length of the shell-rod may be repre-
sented in the form of two components R = R1 + R2 in such manner that
for each of them separately an approximate value of the particular inte-
gral may be found by an elementary method; R1, the part of the total
load statically equivalent at each cross-section to the entire load R,
gives the particular integral for which on the longitudinal sections
only shearing stresses arise, and R2, the remaining statically self-
balanced part of the load in each section, gives the particular inte-
gral for which the normal and transverse stresses in the longitudinal
sections play the fundamental part.
There are further sought such particular integrals of the
homogeneous system in which
(a) The transverse forces and moments on the longitudinal edges
may with sufficient accuracy be assumed equal to zero.
(b) The intensity of the stresses in the cross-sections drops at
a considerably slower rate than in the case of other integrals.
The class of such particularly slowly damped stress states is
found to be wider than that which results from the principle of Saint-
Venant because there is added the statically self-balanced solution in
which the.normal stresses in the cross-section are distributed according
to the law of sectorial areas. This stress state does not reduce to
zero in a shell type rod that is not too long and must therefore be
considered as a principal one. It is in the very presence of this
stress state that open section rods differ from solid closed section
rods. When the length of the rod-shell exceeds a certain limit, however,
a thin-walled rod in the fundamental character of its stress state
ceases to differ from a solid rod although of course a difference is
maintained in the character and rate of reduction of the local stress
These considerations lead to the imposing of restrictions on the
upper limit of the length of a thin-walled rod if we wish to consider
it as a structure which behaves in a fundamentally different manner
from a solid rod. The small rate of damping of the local stress states
imposes restrictions also on the lower limit of length of the rod-shell,
namely the length must be sufficient for the local stresses to reduce
to a sufficient degree at the center sections.
Assuming that both these limitations are satisfied and that the
transverse load R varies linearly over the longitudinal direction,
a theory of computation of thin-walled rods may be constructed on the
assumption that their principal stress state is with a certain accuracy
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Goldenveizer, A. L. Theory of Thin-Walled Rods, report, October 1951; (digital.library.unt.edu/ark:/67531/metadc62968/m1/3/: accessed January 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.