PSI - Issue 72

Alexander Kamenskikh et al. / Procedia Structural Integrity 72 (2025) 252–259

256

T   f B c ε e

th d , V

e

V



 T

th 1,1,1,0,0,0 T  ε is the vector of temperature strains.

where

The problems (8) and (9) are solved using the ANSYS Mechanical APDL software. In order to reduce the probability of obtaining unphysical results, the temperature load is prescribed in the form of separate increments. At each step, the solution of the nonlinear static problem is found using the Newton – Raphson method. The natural frequencies of vibration are found based on the obtained displacement and stress fields, which are taken into account via the matrices K  and K  . The shell is discretized using three-dimensional twenty-node finite elements in the form of a hexahedron with quadratic approximation of displacements (solid186) (Fig. 1 b ). 3. Numerical results As an example, we consider a shell of L =300mm, C = 200mm, h =1 mm made of aluminum alloy 2024 (Young's modulus E =69GPa, Poisson's ratio ν =0.3, coefficient of linear thermal expansion  =22.9  10 – 6 °C – 1 ). For representation of the obtained results we use the following notation for the boundary conditions set at the edges of the structure: F — free edge, C — rigid clamping   0 . i u  The notation is introduced in the clockwise direction, starting from the edge at x = 0 (Fig. 1 a ). Figure2 shows the dependence of the maximum normal displacement w max of the shell rigidly clamped at all edges (CCCC) on the temperature load at different ratios H / h and the lowest natural frequency of vibration  1 . The effect of the temperature load is most pronounced in the plate ( H / h =0). It experiences a sudden buckling when the load reaches  T cr .

Fig. 2. Normal displacements ( a ) and lowest natural frequencies ( b ) of hollow shells for different ratios H/h (CCCC).

A fragment of the curve H/h = 0 in the vicinity of  T c r is shown on the enlarged scale in Fig. 3. Each point on the graph corresponds to the solution of the static and modal problems. Fig.3 shows that the displacements begin to increase sharply and nonlinearly after the temperature load has reached a certain value. Note however that its exact value cannot be calculated from the solution of the static problem. On the other hand, the lowest natural frequency of the plate reaches a minimum at Δ T =2.835°C (Fig.3 b ). This value corresponds to the critical temperature obtained from the solution of the linear buckling problem within the accuracy of the selected increment Δ T. In the literature, the method of determining the critical load responsible for the loss of structure stability which is based on the behavior o f the lowest natural frequency of vibration, is called “Vibration Correlation Technic” ( Franzoni et al., 2019). Figure4 shows the first three mode shapes of vibration for a plate rigidly clamped at all edges (H/h =0) at different values of the temperature load, which correspond to certain points on the graph in Fig. 2 b . These points are in different ranges of Δ T, in which the behavior of the natural frequency is monotonic: Δ T =0 °C, Δ T =10°C and Δ T =40 °C.