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Теория устойчивости пластин, основанная на асимптотическом анализе…
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Theory of plates stability, based on asymptotic analysis
of stability theory equations
for three-dimensional elastic bodies
© Yu.I. Dimitrienko
Bauman Moscow State Technical University, Moscow, 105005, Russia
The objective of this research is to develop a theory of elastic stability of thin multilayer
plates. The theory is based on general equations of three-dimensional theory of elastic
stability by means of introducing the asymptotic expansion over a small parameter,
which represents a thickness to length of plate ratio, without any hypothesis about dis-
placements and stress distributions. Within the research, we stated local problems of sta-
bility, as well as the averaged equations of plate equilibrium for the ground states and
the varied states of the plate. Consequently, we obtained the analytical solution of the lo-
cal problems, which helped deduce relations for all six components of the stress tensor,
including throw-thickness normal stresses and shear stresses for the ground and varied
states. Moreover, we found that the averaged equations of plates’ stability differ from the
classic equations of Kirchoff—Love and Timoshenko's plate theory of stability. It is de-
termined that for orthotropic plates the constitutive relations simplify and become similar
to classical relations of thin plates. However, the membrane and flexural stiffness of
plates depends on stresses of the ground state. The study is illustrated with an example of
calculating a thin orthotropic plate under uniaxial compression. As a result, we obtained
an expression for the critical buckling force, which differs from the classical Euler for-
mula in expression for flexural stiffness, which depends on the parameters of the ground
state of the plate. The findings of the research show that the difference of the critical
force values is the most significant for the plates with strong anisotropic layers.
Keywords
: theory of plates’ stability, three-dimensional stability theory, thin multi-layer
plates, orthotropic plates, asymptotic expansion.
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