Buckling forms of homogeneous and sandwich plates in pure shear in tangential directions

Problems of geometric non-linearity and stability in the mechanics of thin shells and rectilinear columns

“…3. The possibility of realizing some form of loss of stability of shells, plates and columns, of which certain non-classical FLS have been described previously 2,5,6 is completely determined by the form of the unperturbed stress-strain state, and the magnitudes of the critical loads depend very much on the values of the elastic and stiffness parameters of the material and the construction. Judging from the results obtained previously in Refs 5,6, the FLS for shells, plates and columns made of isotropic materials which have been revealed are not, as a rule, of practical interest.…”

The relations of deformation theory in the quadratic approximation and the problems of constructing improved versions of the geometrically non-linear theory of laminated structures

“…Substitution of these expressions into the second equality of (6.1) leads to the exact formulae In addition to the problems considered previously in [4,5], we will show below, using the simplest example, to what physically incorrect results and conclusions one can be led by using approximations (6.4) when solving specific problems.…”

“…(7.14) leads to the equality P2 f 0 w (E~' +ooo)dP = 0 Pl which, using the second formula of (7.7) for a thin ring, gives w(E~2 -p) = 0. Hence follows the bifurcation value of the external pressure p* = E~, similar to that obtained previously in [4,5] and related to the assumption of the approximation E0 ---eo0 in the initial geometrically non-linear problem, formulated using general considerations of the form…”

“…By virtue of this, Eqs by which the corresponding form of the loss of stability is described, after determining two of the three constants of integration. Note that, for a rectangular orthotropic plate, subjected to pure shear, the solution of the problem of the shear form of loss of stability was given previously in [5], starting from the linearized equations of the plane problem of the theory of elasticity, the basis of which is the use of the complete kinematic relation (4.3). The bifurcation value x* for an isotropic plate obtained in it is equal to E x.…”

“…"~xy = U,y at 'O x at W,xW y , ... (1.8) The need to estimate the quality of the above three versions of the kinematic relations in the quadratic approximation arose in connection with the appearance of false bifurcation points when solving specific problems, which were formulated in [4,5], starting from relations of the form (1.3). One of these estimates can be obtained by considering uniaxial extension-compression and pure shear.…”