Determination of Plane Stress-Strain States of the Plates on the Basis of the Three-Dimensional Theory of Elasticity
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DETERMINATION OF PLANE STRESS-STRAIN STATES OF THE PLATES ON THE BASIS OF THE THREE-DIMENSIONAL THEORY OF ELASTICITY V. P. Revenko1,2 and A. V. Revenko3
UDC 539.3
We construct a theory of plates loaded only on their sides by loads parallel and symmetric to the median surface. We use the general representation of three-dimensional stress-strain state and exactly satisfy the trivial boundary conditions on the plane surfaces of the plate. The numerical analysis of the threedimensional stressed state of the plate is reduced to the determination of its two-dimensional state under the assumption that the normal stresses perpendicular to the median surface are negligible. The displacements and stresses are expressed via two two-dimensional harmonic functions. We obtain the homogeneous solutions and develop a numerical-analytic algorithm for the solution of the boundary-value problems posed for rectangular plates. Keywords: plate, plane stressed state, stress tensor, homogeneous solutions.
Plates loaded only on their sides by loads parallel and symmetric to the median surface are widely used in building and engineering structures [1–7]. The stressed state of thin plates is, as a rule, determined by using the equations of the plane problem of the theory of elasticity [2–4]. For thick plates, it is customary to use the homogeneous solutions and the symbolic method [5, 8], harmonic and biharmonic functions [2–4], hypothesis on the behavior of normal to the median surface [1, 8], or expansions of the Papkovich–Neuber representation in the variable normal to the median surface [5, 9]. Statement of the Problem and Its Solution Consider a three-dimensional static problem of the theory of elasticity for a plate with constant thickness h whose median surface occupies a domain S with contour L and coincides with the plane Oxy of a Cartesian coordinate system: x1 = x , x 2 = y , x 3 = z . The general stressed state of the plate can be split [10] into the states of bending and symmetric pressure:
ui (x, y, − z) = ui (x, y, z) ,
i = 1, 2 ,
u 3 (x, y, − z) = − u 3 (x, y, z) ,
(1)
where ui are displacements in the directions of the corresponding axes of the Cartesian coordinate system. We now consider a special case of the second problem where the plane surfaces of the plate ( z = h j ,
j = 1, 2 , h1 = h/2 , h2 = − h/2 ) are free of normal and tangential loads and its sides (lateral surfaces) are sub-
1 2 3
Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine. Corresponding author; e-mail: [email protected].
Karpenko Physicomechanical Institute, Ukrainian National Academy of Sciences, Lviv, Ukraine.
Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 52, No. 6, pp. 63–68, November–December, 2016. Original article submitted January 11, 2016. 1068-820X/17/5206–0811
© 2017
Springer Science+Business Media New York
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jected to the action of loads symmetric and parallel to the median surface S :
σ n (x, y
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