INTRODUCTION

Composite sandwich shells [1–7] are widely used in modern technology, especially, in rocket and aircraft manufacturing, and in shipbuilding [8–14].

Structures often have various kinds of cutouts, near which zones of increased stresses appear, and the largest of them can be several times higher than the stresses far from such zones [15–22].

The analysis of shells weakened by holes become the topic of a large body of research. However, only a few works are devoted to the study of the stress and strain state of sandwich cylindrical shells with cutouts rectangular in plan [23–28]. These are mostly the problem-setting works. A stress-strain analysis of sandwich shells weakened by rectangular holes is, as a rule, not given.

There are works on the analysis of sandwich spherical shells with circular, elliptical and curvilinear holes [29–33], as well as with rectangular cutouts [34, 35]. Some results of an experimental study of sandwich spherical shells with holes are given in [36].

Sandwich conical shells with rectangular cutouts were studied in [37–40].

A three-layer cylinder with circular holes was considered in [41].

An analysis of publications has shown that, with the accuracy necessary for practice, studying the stress and strain state of shells (especially of a sandwich structure) weakened by rectangular holes using analytical methods is associated with large, mostly insurmountable difficulties of a mathematical nature [16, 42] and it is impossible to solve this problem without the use of numerical methods [38, 43–47].

According to the review presented in [38], the influence of the angular dimension of rectangular cutouts on the stress and strain state of sandwich cylindrical shells, despite its relevance, has not been studied and the most suitable method for solving this problem is the finite element method (FEM) [48–51].

In modern technology and especially in rocket and aircraft manufacturing, and in shipbuilding, sandwich structures with thin and rigid facesheets and a relatively thick, but less rigid core are most used [52]. When analyzing shells of such a sandwich structure, models should take into account the bending state of the facesheets and the three-dimensional stress and strain state of the core [53–55].

This paper considers a model of layer-by-layer finite element stress-strain analysis of sandwich cylindrical shells, with the help of which the influence of the angular dimension of rectangular cutouts on the stress-strain state of sandwich cylindrical compartments is studied.

A MODEL FOR CALCULATION

The layer-by-layer analysis model is constructed from two-dimensional finite elements (FEs) of momental facesheet layers and three-dimensional FEs of the core layer. These finite elements are cylindrical in shape. The coordinate systems are placed on the middle surfaces of the finite elements.

Let us introduce the numbering of layers of a sandwich cylindrical open shell of revolution—i = 1, 2, 3, starting from the inner surface of the shell. Additionally, we denote the facesheet layers with the subscript “c”, and the core layer with the subscript “f”. Let us denote the radii of the middle surfaces of the layers and their thicknesses with Ri and hi, respectively.

To study the stress-strain state in load-bearing layers, we consider the rectangular two-dimensional finite elements with nodes at corner points. When constructing a model of these finite elements, we use the classical general theory of cylindrical shells [56].

The nodal generalized displacements of the finite element for studying the stress and strain state in the facesheets will be the displacements of the points of the middle surface u, v, w and the angles of rotation of the normal ϑx, ϑy to the middle surface relative to the x and y (φ) axes. Thus, the finite element will have four nodes with five degrees of freedom per node.

The rigid-body displacement functions of the finite element (taking them into account increases the rate of convergence of solutions) are determined by integrating the relations between the components of the generalized strain vector and the displacement vector [56] at zero values of deformation. When writing these obtained displacement functions of the finite element, six indefinite coefficients were used, which are the integration constants α1, ...,α6.

The number of indefinite coefficients when recording the total displacements of the finite element is equal to the number of the FE’s degrees of freedom. Taking into account the nature of the variation in stress and strain state parameters of the facesheet layers [57], we will write down the components of the generalized strain vector using the fourteen indefinite coefficients remaining after recording the rigid-body displacement functions. Let us represent the parameters of curvature change æ1, æ2 by polynomials of higher orders than that for the other parameters of the generalized strain vector ε1, ε2, γ, χ [57]. The functions for recording the generalized strain [57] should satisfy the equations of strain compatibility [58].

To obtain functions that approximate the displacements determined by the deformation of FEs, the integration of the relations between the components of the generalized strain vector and the displacement vector is carried out [56, 58] with the obtained approximations for the strain functions [59].

The approximating displacement functions of the finite element of facesheet layers, written in matrix form with the use of the vector of indefinite coefficients \(\alpha _i^c\), will have the form

$$\delta _i^c = {\bf{T}}_i^c\alpha _i^c,$$
(1)

where \({\bf{T}}_i^c(3 \times 20)\) is the matrix of displacement approximating functions of the sandwich open cylindrical shell of revolution facesheet FE:

Table 1 .
Full size table

Taking into account formula (1) and writing the matrix \({\left( {{\bf{T}}_i^c} \right)^{\text{T}}}\), we obtain expressions for the rotation angles of the normal relative to the axial and circumferential coordinate axes x, y (φ) [60]

$$\displaylines{ {\vartheta _x} = \frac{{\partial w}}{{R\partial \varphi }} - \frac{v}{R} = - {\alpha _6}\frac{1}{R} + {\alpha _8}\frac{{{x^2}}}{{2{R^2}}} - {\alpha _{14}}\frac{{{x^2}}}{{2R}} - {\alpha _{15}}\frac{{{x^3}}}{{6R}} - {\alpha _{16}}R\varphi - {\alpha _{17}}Rx\varphi \cr - {\alpha _{18}}R\left( {1 + \frac{{{\varphi ^2}}}{2}} \right) - {\alpha _{19}}Rx\frac{{{\varphi ^2}}}{2} - {\alpha _{20}}x; \cr} $$
$${\vartheta _y} = - \frac{{\partial w}}{{\partial x}} = {\alpha _2}\cos \varphi + {\alpha _4}\sin \varphi - {\alpha _{10}}R + {\alpha _{12}}x + {\alpha _{13}}\frac{{{x^2}}}{2} + {\alpha _{14}}x\varphi + {\alpha _{15}}\frac{{{x^2}}}{2}\varphi + {\alpha _{17}}{R^2} + {\alpha _{19}}{R^2}\varphi .$$

It is shown in [61] that the approximating displacement functions considered are highly efficient, since they lead to a high rate of convergence of numerical solutions.

Layer-by-layer analysis models are characterized by a large number of degrees of freedom [62], which makes it difficult to carry out the finite element analysis of sandwich open shells with the required accuracy. Therefore, the construction and application of effective approximating displacement functions, leading to a decrease in the number of FEs in the calculation, is an urgent scientific problem [63], which has important applied significance [64].

Let us apply the considered approximating displacement functions (ADFs) of the finite elements of the facesheets to construct the ADFs of three-dimensional shell FEs of the core. We write the displacement vector in these FEs δf = {u, v, w}T in terms of the displacement vectors on the inner δ1 and outer δ2 cylindrical surfaces of the finite elements of the core:

$${\delta ^f} = {\delta ^1}\varphi _i^1 + {\delta ^2}\varphi _i^2,$$
(2)

where \(\varphi _i^1 = \frac{1}{2}\left( {1 - 2\frac{z}{h}} \right)\), \(\varphi _i^2 = 1 - \varphi _i^1\), z is the coordinate along the normal to the FE’s middle surface; h is the thickness of the three-dimensional FE of the core.

In expression (2) and further, the superscripts “1” and “2” correspond to the inner and outer cylindrical surfaces of the finite element of the core.

We will reduce the approximating functions of the facesheet FE displacements to the corresponding cylindrical interface with the finite elements of the core using transition matrices, similarly to what was done in [65].

When modeling the core of a sandwich shell with one three-dimensional FE along the thickness, its cylindrical surfaces will be the interface with the facesheets.

Let us write down the displacement vectors of the facesheet FE on the cylindrical interface surfaces with the core FEs

$$\bar \delta _1^c = {\left\{ {\bar u_1^c,\bar v_1^c,\bar w_1^c} \right\}^T};\quad \bar \delta _3^c = {\left\{ {\bar u_3^c,\bar v_3^c,\bar w_3^c} \right\}^T},$$
(3)

where \(\bar u_1^c = {{u_1^c + \vartheta _{y1}^ch_1^c} \mathord{\left/ {\vphantom {{u_1^c + \vartheta _{y1}^ch_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}\), \(\bar v_1^c = {{v_1^c - \vartheta _{x1}^ch_1^c} \mathord{\left/ {\vphantom {{v_1^c - \vartheta _{x1}^ch_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}\), \(\bar w_1^c = w_1^c\), \(\bar u_3^c = {{u_3^c - \vartheta _{y3}^ch_3^c} \mathord{\left/ {\vphantom {{u_3^c - \vartheta _{y3}^ch_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}\), \(\bar v_3^c = {{v_3^c + \vartheta _{x3}^ch_3^c} \mathord{\left/ {\vphantom {{v_3^c + \vartheta _{x3}^ch_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}\), \(\bar w_3^c = w_3^c\); \(h_1^c\), \(h_3^c\) are the thicknesses of the inner and outer facesheets of the sandwich open cylindrical shell of revolution.

We use the approximating functions of the facesheet FE displacement fields as the ADFs of the core FEs on the inner and outer cylindrical surfaces. Then we will get

$${\delta ^1} = \bar \delta _1^c;\quad {\delta ^2} = \bar \delta _3^c.$$
(4)

Taking into account expressions (1) – (4) and knowing the ADFs of the facesheet finite elements of a sandwich cylindrical open shell of revolution, we obtain the approximating displacement functions of a three-dimensional core FE, written using forty indefinite coefficients α1, ..., α40:

$$\displaylines{ u = {u^1}\varphi _i^1 + {u^2}\varphi _i^2 = \left( {{\alpha _2}\left( {{R_1} + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\cos \varphi + {\alpha _4}\left( {{R_1} + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\sin \varphi + {\alpha _5} + {\alpha _7}x} \right. \cr + {\alpha _8}x\varphi - {\alpha _{10}}{{{R_1}h_1^c} \mathord{\left/ {\vphantom {{{R_1}h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2} + {\alpha _{11}}{R_1}\varphi + {{{\alpha _{12}}h_1^cx} \mathord{\left/ {\vphantom {{{\alpha _{12}}h_1^cx} 2}} \right. \kern-\nulldelimiterspace} 2} + {\alpha _{13}}{{h_1^c{x^2}} \mathord{\left/ {\vphantom {{h_1^c{x^2}} 4}} \right. \kern-\nulldelimiterspace} 4} + {\alpha _{14}}h_1^c{{x\varphi } \mathord{\left/ {\vphantom {{x\varphi } 2}} \right. \kern-\nulldelimiterspace} 2} \cr + {\alpha _{15}}h_1^c{x^2}{\varphi \mathord{\left/ {\vphantom {\varphi 4}} \right. \kern-\nulldelimiterspace} 4} + {\alpha _{17}}{R_1}^2{{\left( {h_1^c - {R_1}{\varphi ^2}} \right)} \mathord{\left/ {\vphantom {{\left( {h_1^c - {R_1}{\varphi ^2}} \right)} 2}} \right. \kern-\nulldelimiterspace} 2} + {\alpha _{19}}R_1^2\varphi \left. {\left( {{{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2} + {R_1}\left( {1 - {{{\varphi ^2}} \mathord{\left/ {\vphantom {{{\varphi ^2}} 6}} \right. \kern-\nulldelimiterspace} 6}} \right)} \right) - {\alpha _{20}}R_1^2\varphi } \right)\varphi _i^1 \cr + \left( {{\alpha _{22}}\left( {{R_3} - {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\cos \varphi + {\alpha _{24}}\left( {{R_3} - {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\sin \varphi + {\alpha _{25}} + {\alpha _{27}}x} \right. \cr + {\alpha _{28}}x\varphi + {\alpha _{30}}{{{R_3}h_3^c} \mathord{\left/ {\vphantom {{{R_3}h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2} + {\alpha _{31}}{R_3}\varphi - {\alpha _{32}}{{h_3^cx} \mathord{\left/ {\vphantom {{h_3^cx} 2}} \right. \kern-\nulldelimiterspace} 2} - {\alpha _{33}}h_3^c{{{x^2}} \mathord{\left/ {\vphantom {{{x^2}} 4}} \right. \kern-\nulldelimiterspace} 4} - {\alpha _{34}}h_3^c{{x\varphi } \mathord{\left/ {\vphantom {{x\varphi } 2}} \right. \kern-\nulldelimiterspace} 2} \cr \left. { - {\alpha _{35}}h_3^c{{{x^2}\varphi } \mathord{\left/ {\vphantom {{{x^2}\varphi } 4}} \right. \kern-\nulldelimiterspace} 4} - {\alpha _{37}}{R_3}^2{{\left( {h_3^c + {R_3}{\varphi ^2}} \right)} \mathord{\left/ {\vphantom {{\left( {h_3^c + {R_3}{\varphi ^2}} \right)} 2}} \right. \kern-\nulldelimiterspace} 2} - {\alpha _{39}}R_3^2\varphi \left( {{{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2} - {R_3}\left( {1 - {{{\varphi ^2}} \mathord{\left/ {\vphantom {{{\varphi ^2}} 6}} \right. \kern-\nulldelimiterspace} 6}} \right) - {\alpha _{40}}R_3^2\varphi } \right)} \right)\varphi _i^2; \cr} $$
$$\displaylines{ v = {v^1}\varphi _i^1 + {v^2}\varphi _i^2 = \left( {\left( {{\alpha _1} + {\alpha _2}x} \right)\sin \varphi - \left( {{\alpha _3} + {\alpha _4}x} \right)\cos \varphi + {\alpha _6}\left( {1 + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} {2{R_1}}}} \right. \kern-\nulldelimiterspace} {2{R_1}}}} \right)} \right. \cr - {\alpha _8}\left( {1 + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} {2{R_1}}}} \right. \kern-\nulldelimiterspace} {2{R_1}}}} \right){{{x^2}} \mathord{\left/ {\vphantom {{{x^2}} {2{R_1}}}} \right. \kern-\nulldelimiterspace} {2{R_1}}} + {{{\alpha _{14}}h_1^c{x^2}} \mathord{\left/ {\vphantom {{{\alpha _{14}}h_1^c{x^2}} {4{R_1}}}} \right. \kern-\nulldelimiterspace} {4{R_1}}} + {{{\alpha _{15}}h_1^c{x^3}} \mathord{\left/ {\vphantom {{{\alpha _{15}}h_1^c{x^3}} {12{R_1}}}} \right. \kern-\nulldelimiterspace} {12{R_1}}} \cr + {\alpha _{16}}{R_1}\left( {{R_1} + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\varphi + {\alpha _{17}}{R_1}\left( {{R_1} + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)x\varphi + {\alpha _{18}}{R_1}{{\left( {\left( {{R_1} + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right){\varphi ^2} + h_1^c} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {{R_1} + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right){\varphi ^2} + h_1^c} \right)} 2}} \right. \kern-\nulldelimiterspace} 2} \cr \left. { + {\alpha _{19}}{R_1}\left( {\left( {{R_1} + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right){{{\varphi ^2}} \mathord{\left/ {\vphantom {{{\varphi ^2}} {2 - {R_1}}}} \right. \kern-\nulldelimiterspace} {2 - {R_1}}}} \right)x + {\alpha _{20}}\left( {{R_1} + {{h_1^c} \mathord{\left/ {\vphantom {{h_1^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)x} \right)\varphi _i^1 \cr} $$
$$\displaylines{ + \left( {\left( {{\alpha _{21}} + {\alpha _{22}}x} \right)\sin \varphi - \left( {{\alpha _{23}} + {\alpha _{24}}x} \right)\cos \varphi + {\alpha _{26}}\left( {1 + {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} {2{R_3}}}} \right. \kern-\nulldelimiterspace} {2{R_3}}}} \right)} \right. \cr + {\alpha _{28}}\left( {{{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} {2{R_3} - 1}}} \right. \kern-\nulldelimiterspace} {2{R_3} - 1}}} \right){{{x^2}} \mathord{\left/ {\vphantom {{{x^2}} {2{R_3}}}} \right. \kern-\nulldelimiterspace} {2{R_3}}} - {\alpha _{34}}h_3^c{{{x^2}} \mathord{\left/ {\vphantom {{{x^2}} {4{R_3}}}} \right. \kern-\nulldelimiterspace} {4{R_3}}} - {\alpha _{35}}h_3^c{{{x^3}} \mathord{\left/ {\vphantom {{{x^3}} {12{R_3}}}} \right. \kern-\nulldelimiterspace} {12{R_3}}} \cr + {\alpha _{36}}{R_3}\left( {{R_3} - {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\varphi + {\alpha _{37}}{R_3}\left( {{R_3} - {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)x\varphi + {\alpha _{38}}{R_3}{{\left( {\left( {{R_3} - {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right){\varphi ^2} - h_3^c} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {{R_3} - {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right){\varphi ^2} - h_3^c} \right)} 2}} \right. \kern-\nulldelimiterspace} 2} \cr \left. { + {\alpha _{39}}{R_3}\left( {\left( {{R_3} - {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right){{{\varphi ^2}} \mathord{\left/ {\vphantom {{{\varphi ^2}} 2}} \right. \kern-\nulldelimiterspace} 2} - {R_3}} \right)x + {\alpha _{40}}{R_3}\left( {1 - {{h_3^c} \mathord{\left/ {\vphantom {{h_3^c} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)x} \right)\varphi _i^2; \cr} $$
$$\displaylines{ w = {w^1}\varphi _i^1 + {w^2}\varphi _i^2 = \left( { - \left( {{\alpha _1} + {\alpha _2}x} \right)\cos \varphi - \left( {{\alpha _3} + {\alpha _4}x} \right)\sin \varphi + {\alpha _9}{R_1} + {\alpha _{10}}{R_1}x} \right. \cr - {\alpha _{12}}{{{x^2}} \mathord{\left/ {\vphantom {{{x^2}} 2}} \right. \kern-\nulldelimiterspace} 2} - {\alpha _{13}}{{{x^3}} \mathord{\left/ {\vphantom {{{x^3}} 6}} \right. \kern-\nulldelimiterspace} 6} - {\alpha _{14}}{{{x^2}\varphi } \mathord{\left/ {\vphantom {{{x^2}\varphi } 2}} \right. \kern-\nulldelimiterspace} 2} - {\alpha _{15}}{{{x^3}\varphi } \mathord{\left/ {\vphantom {{{x^3}\varphi } 6}} \right. \kern-\nulldelimiterspace} 6} - {\alpha _{16}}R_1^2 - {\alpha _{17}}R_1^2x - {\alpha _{18}}R_1^2\varphi \left. { - {\alpha _{19}}R_1^2x\varphi } \right)\varphi _i^1 \cr + \left( { - \left( {{\alpha _{21}} + {\alpha _{22}}x} \right)\cos \varphi - \left( {{\alpha _{23}} + {\alpha _{24}}x} \right)\sin \varphi + {\alpha _{29}}{R_3} + {\alpha _{30}}{R_3}x - } \right.{\alpha _{32}}{{{x^2}} \mathord{\left/ {\vphantom {{{x^2}} 2}} \right. \kern-\nulldelimiterspace} 2} \cr - \left. {{\alpha _{33}}{{{x^3}} \mathord{\left/ {\vphantom {{{x^3}} 6}} \right. \kern-\nulldelimiterspace} 6} - {\alpha _{34}}{{{x^2}\varphi } \mathord{\left/ {\vphantom {{{x^2}\varphi } 2}} \right. \kern-\nulldelimiterspace} 2} - {\alpha _{35}}{{{x^3}\varphi } \mathord{\left/ {\vphantom {{{x^3}\varphi } 6}} \right. \kern-\nulldelimiterspace} 6} - {\alpha _{36}}R_3^2 - {\alpha _{37}}R_3^2x - {\alpha _{38}}R_3^2\varphi - {\alpha _{39}}R_3^2x\varphi } \right)\varphi _i^2. \cr} $$

Knowing the ADFs of the finite elements of the open cylindrical shell of revolution facesheets and core and following the algorithms from [40], we calculate the FE stiffness matrices. Further solution of the problem is carried out using the procedures of the FEM displacement method [48–50].

THE INFLUENCE OF THE ANGULAR DIMENSION OF RECTANGULAR CUTOUTS ON THE STRESS AND STRAIN STATE OF SANDWICH CYLINDRICAL COMPARTMENTS

The stress-strain state of a three-layer circular cylindrical compartment with two diametrically opposite rectangular cutouts located symmetrically relative to the ends of the shell was studied. The shell was loaded with an axial compressive load p (resulting axial compressive force P) uniformly distributed at the ends (Fig. 1).

Fig. 1
figure 1

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The boundary conditions at the ends corresponded to the hinged support of the facesheets.

The shell has the following geometric parameters: the compartment length is L = 80 cm; the radius of the middle surface of the core is R2 = 17.65 cm; the thicknesses of the facesheets and core are h1 = h3 = 0.2 cm and h = 0.7 cm; the length of cutouts is lc = 20 cm.

We considered a sandwich compartment with facesheets made of composite materials: (E1 = 2.1×105 kg/cm2, E2 = 1.9×105 kg/cm2 are the axial and circumferential moduli of elasticity; G12 = 0.36×105 kg/cm2 is the shear modulus; μ1 = 0.1 is Poisson’s ratio) and a lightweight core (E3 = 240 kg/cm2 is the transverse modulus of elasticity; G13 = G23 = 100 kg/cm2 are the transverse shear moduli).

The range of angular dimensions of the cutouts was 30 ≤ βc ≤ 50 deg. A 1/8 symmetrical part of the compartment was considered with a variation in φ from 0 to 90 deg and in x from 0 to L/2. The mesh size was refined in the area of the cutout.

As a result of the calculations carried out, the stresses in the facesheets and the core of a sandwich circular cylindrical compartment were determined.

The highest stresses in the facesheets are axial compressive membrane stresses. The change in these stresses in dimensionless form \(\left( {\bar \sigma = {{\sigma 2\pi \left( {{R_1}{h_1} + {R_3}{h_3}} \right)} \mathord{\left/ {\vphantom {{\sigma 2\pi \left( {{R_1}{h_1} + {R_3}{h_3}} \right)} P}} \right. \kern-\nulldelimiterspace} P}} \right)\) in the internal facesheet of a sandwich compartment along a section passing near the curvilinear edge of the cutout for the rectangular cutout angular dimensions of 30, 40, and 50 deg is shown in Fig. 2.

Fig. 2
figure 2

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Near the corner point of the cutout, the axial bending stresses increase significantly in magnitude, reaching their highest values, while being in the negative region, then decrease and stabilize towards the middle of the curved edge of the cutout (Fig. 3).

Fig. 3
figure 3

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Bending stresses \(\bar \sigma _1^M\) are significantly lower than the membrane stresses \({\bar \sigma _1}^T\) that is observed for other stresses in the facesheets of the sandwich compartment.

The difference between the stress values in the outer and the inner facesheets is insignificant.

The change in the maximum stresses in the core of a sandwich compartment, which are the transverse shear stresses τ23, in dimensionless form \(\left( {{{\bar \tau }_{23}} = {{{\tau _{23}}2\pi {R_2}{h_2}} \mathord{\left/ {\vphantom {{{\tau _{23}}2\pi {R_2}{h_2}} P}} \right. \kern-\nulldelimiterspace} P}} \right)\) in a section passing near the straight edge of the cutout is shown in Fig. 4.

Fig. 4
figure 4

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As evident from Fig. 4, the transverse shear stresses \({\bar \tau _{23}}\) in the core of the sandwich compartment are almost completely in the negative region, reaching maximum values in the area of the corner point of the cutout.

The maximum stress values τ13 in the core of a sandwich compartment are almost 2.5 times lower than the maximum stress values τ23, and the maximum values of radial stresses τ33 are more than three times less than the maximum stress values τ13.

Thus, for the first time, the important scientific and applied research was carried out on the effect of the angular dimension of rectangular cutouts on the stresses in the layers of a sandwich circular cylindrical compartment.