Based on the established equations, we study the phenomenon of the jump zone of the stress field in the structure. The effects of boundary conditions, relative thickness and relative shell length are investigated. Many researchers have presented the results of studies on laminated composite plates and shells using first and higher order shear deformation theories [11-17].

The final plate formulation was obtained using the first order shear deformation theory (FSDT). Based on the obtained results, the effects of the properties of the matrix polymer on the concentration of the surface tension of the fibers were carried out. The main direction of fiber reinforcement of each layer coincides with the direction of the local coordinate system O123.

The angle between the fiber reinforcement direction and the vertical axis O of the general coordinate system is. The displacement field of the shell in orthogonal curvilinear Ozis coordinate systems is analyzed as follows. The linear relationship between the stress and strain fields of the laminated composite cylindrical shell in the curvilinear orthogonal coordinate systems Oz is defined as follows [6].

Due to the local coordinate system, O123 does not coincide with the general coordinate systemz (see Figure 1), so we need to perform the coordinate transfer for equation (7).

## Characteristic equation and analysis of root forms

Substituting the root displacement expression in equations (3), (4) and (5) we obtain the deformation of the shell structure. To find the stresses , and , we use Hooke's principle equation (9) by then integrating the equilibrium equation of the 3D elasticity theory. To satisfy the cyclic boundary condition respected by the coordinate, we need to expand the displacement field and the load according to the single trigonometric series as follows.

Substituting equation (16) into equation (15) and then performing some simple mathematical transformations, we obtain the differential equations to determine the functions Ui0 , Wj0 as follows. In equation (18), we ignore the superscript indices 1 and 2 of the quantities Uim, Vim and Wim. In shell theories, the strain-stress field is divided into the fundamental strain-stress field and the boundary strain-stress field.

In addition, the tensile stress field of the shell structure depends on root types characteristic equation of systems (17) and (18). Therefore, to analyze the stress-stress field, we need to study the solutions of the characteristic equation of systems (17) and (18) in some specific cases K2,K3. The characteristic equation corresponding to each. Let the right-hand factor determinant of the corresponding system of equations received be equal to 0.

To illustrate the results obtained above, we consider the roots of the characteristic equation for a cylindrical laminated composite shell under symmetric axial loading. From Table 1-3, we understand that the roots of the characteristic equation are divided into two groups: small roots and large roots. The small roots represent the fundamental strain-stress field of the shell structure, and the large roots describe the boundary phenomena of the shell structure.

## Numerical analysis and discussion 1. Verification examples

Non-dimensional deflection w at the center of the laminated composite shell for different relative thicknesses;L R4;S R h;m1;n4. The non-dimensional deflection and stress results at the midpoint of single layer [90o] of the laminated composite shall be compared with the results of exact solution by Varadan-Bhaskar [38] are shown in Table 5. The relative horizontal normal stresseszat the midpoint of the shell is much smaller than the maximum deflection.

Moreover, the suitability of calculation results between different theories in the middle of the shell can be explained from the roots of the characteristic equation (16) and (19). Far away from the boundary region, the stress field is mainly determined by the small roots of the characteristic equation (16) and (19), the other large roots only have a large influence in the surrounding areas of the jump stress zone. Table 6-8 shows non-dimensional deflection w and stress z, z with different relative thicknesses and boundary conditions for the laminated composite [0/90] shell.

For shell structures whose length is greater than the average value, boundary conditions in the area far from the boundary zone have a slight effect on the deflection and stress. The area near boundary zone, the maximum stress of the shell structure depends on the type of boundary conditions. The shorter the length of the shell structure, the more boundary conditions affect the deflection and stress in the center of the structure.

Now the following figures show the non-dimensional stress distribution at the thickness of the composite isotropic shell (R/h=10, L/R=4, h=0.1) at the ambient C-C boundary condition. The structure is affected by the distributed load on the inner surface. a) voltage at the boundary position b) voltage is 5 hours away from the boundary. From the calculated results of the stress distribution at a distance h from the boundary, we can see that the proposed theory and calculation program meet a very good agreement.

This can be explained as follows: The voltage at the far points from the boundary is determined by the small roots p1 iq1 in characteristic equations (16) and (19). In the boundary zone, the stress field is strongly influenced by the large roots; the size of the affected area of this stress jump zone is approximately equal to the thickness of the shell. The above analysis shows that for thin composite cylindrical shells, using higher order shear deformation shell theory gives higher results at the ambient boundary condition compared to the classical theory.

Next, consider a C-C laminated composite shell [0/90] of graphite-epoxy (AS with thickness h = 0.1 and relative length L/R=4; the shell is under distributed loading on the inner side. Furthermore, from Tables 6 and 8 we understand that at L R1 the boundary conditions strongly influence the deflection and stress;

## Conclusion

The mathematical model and the calculated results are compared with those according to 3D elastic theory, thereby confirming the reliability of the results obtained in this work. The models presented in this paper take into account the effects of the horizontal normal stress (which is neglected in terms of first-order shear deformation theory and high-order deformation theory), and thus we can extend the scope of these models. The roots of the characteristic equation are divided into a small root group and a large root group.

The small characterizes the stress field far from the area of the metamorphic stress field (the boundary area, structural jumping, power jumping and so on), while the other has a large influence on the stress field of the metamorphic stress field. The parameter studies show that both the thickness of the shell and the boundary conditions and the order of the lamination have a strong influence on the deflection and stress of the shell structure. In the boundary zone, the stress field is affected by the marginal effects, corresponding to the major roots of the characteristic equation, so in determining the stress field in the boundary region, it is necessary to use the higher-order shear-deformation scale theory.

Analytical solutions for the static analysis of laminated composite and sandwich panels based on sophisticated higher-order theory. Bending and thermal buckling of non-symmetric functionally graded sandwich beams in a high-temperature environment based on a new third-order shear deformation theory. A finite element model for the dynamic analysis of three-layer composite panels with layers connected by shear connectors subjected to dynamic loading.

A refined simple first-order shear deformation theory for static bending and free vibration analysis of advanced composite plates. An efficient beam element based on quasi-3d theory for static bending analysis of functionally graded beams. Free vibration analysis of functionally graded shells using an edge-based smoothed finite element method.

Taking the stress concentration of fiber surfaces into account when predicting the tensile strength of unidirectional carbon fiber reinforced plastic composites. Free vibration analysis of saturated porous FG round plates integrated with piezoelectric actuators via differential quadrature method. Investigation of the statics and free vibrations of cylindrical shells based on a non-classical theory.