Method of Strength Analysis for Doubly-Curved Stiffened Orthotropic Shells by Various Strength Theories

The paper proposes a method of strength analysis for materials of thinwalled shell structures reinforced with stiffeners. The shells under consideration were made of orthotropic materials. The authors analyze the applicability of the following seven strength criteria: the maximum stress criterion, the Mises–Hill criterion, the Fisher criterion, the Goldenblatt– Kopnov criterion, the Liu–Huang–Stout criterion, the Tsai–Wu criterion, and the Hoffman criterion. During the study, doubly-curved shallow shells square in the plan were considered. A geometrically nonlinear mathematical model of shell deformation, which considers transverse shears, was used. The calculations were based on the characteristics of T10/UPE22-27 glass-fiber-reinforced plastic. The method relies on calculating the values of several strength criteria at each step of structural loading and analyzing the development of areas failing to meet the strength conditions as the load increases.


INTRODUCTION
The study of the process of deformation of shell structures is essential for various industries, including air-craft building, shipbuilding, rocket science, and others.In construction, such structures are often used, among other things, for covering large-span structures, for example, stadiums, concert halls, markets, warehouses, hangars for machinery and equipment, and factory buildings.
Thin-walled shell structures may lose their performance due to buckling when a small change in the load results in a significant rapid increase in displacements (deflection) and irreversible changes in the material (loss of strength).
Unfortunately, most available studies analyze either buckling resistance or strength.However, in some cases, it is hard to predict what will be compromised earlier.
Thus, it needs to perform structural analysis in terms of buckling and occurrence of failure to meet the strength conditions (including using various strength theories).
Structures subjected to external loading can be analyzed based on the limited state of structural materials [1,2].
Here, loss of strength means a state when the material experiences irreversible transformations.In fact, loss of strength is determined when at least one point of the structure fails to meet the strength criterion.
With a further increase in loading, the areas failing to meet the strength conditions begin to expand, and it becomes important to analyze their distribution and development.
Much attention is paid to the study of the deformation of composite materials, which are often orthotropic.So, in the work of Smerdov [3] for the basic classes of composites (high-, medium-, and low-modulus carbon-fiber, organo-, and glass-fiber plastics), recommendations for a rational choice of their structures to obtain experimental results allowing one to identify the elastic characteristics of unidirectional composites are formulated.
Analyzing strength criteria applicable to orthotropic materials indicates the need for a generally accepted criterion.Consequently, it needs to use several criteria and ensure that the results obtained will be analyzed later.The results of such comparisons can be found in papers [4][5][6][7][8].
In [25], Korsun et al. describe in detail the key relations of the following strength criteria applicable to concrete: the Geniev criterion, the Geniev-Alikova criterion, the Leites criterion, the Yashin criterion, the Klovanich-Bezushko criterion, the Willam-Warnke criterion, and the Karpenko criterion.
A valuable extensive comparative analysis of strength criteria can be found in a paper by Oreshko et al. [8].The researchers analyzed strength criteria appli-cable to isotropic, orthotropic, and anisotropic materials.They also described the approaches used during the strength analysis of fibrous and layered composite materials.In the course of their study, they considered the following criteria: von Mises criterion, the Pisarenko-Lebedev criterion, the William-Warnke criterion, the Drucker-Prager criterion, the Bazant criterion, the Norris criterion, the Cuntze criterion, the Goldenblatt-Kopnov criterion, the Tsai-Hill criterion, the Tsai-Wu criterion, the Hashin criterion, the LaRC criterion, the Hoffman criterion, the Puck criterion, the sandwich panel strength criteria, and others.Besides, they reviewed the strength models of the materials used in the ANSYS Mechanical APDL program.
The goal of this work is to present a method to analyze the strength of thin-walled shell structures made of orthotropic materials (including the use of various strength theories).

Limit state criteria for orthotropic materials
As Tsvetkov and Kulish [23] noted, the phenomenological criterion of strength in anisotropic materials relates the possibility of structural failure to the value of stress tensor ( ij σ ) in the material and generally can be represented as follows: ( ) .
The expression includes a set of constants characterizing the structural behavior of the material.The strength criterion corresponds to the strength surface for a general case in the six-dimensional stress space .The strength surface shall pass through the points determined by the technical strength characteristics of the material, i.e., ultimate strength under uniaxial tension and compression in three mutually perpendicular directions and ultimate strength in shear in three mutually perpendicular planes.
Considers the Liu-Huang-Stout criterion (in notations accepted in the paper (Liu et al., [32])), which is the generalization of the Mises-Hill criterion (taking into account orthotropy but neglecting the difference between tensile and compressive moduli of elasticity) and the Drucker-Prager criterion (taking into account the difference between tensile and compressive moduli of elasticity but neglecting orthotropy). where As Polilov and Tatus [11] noted, the strength criteria taking into account the directional fracture of fibrous composites make it possible to interpret experimental data and perform strength analysis of composite structural members more adequately.
To analyze orthotropic structures, seven criteria for the plane stress condition case, which in uniform notations can be represented as follows, are used: Criterion 1 (the maximum stress criterion) In fact, it includes six criteria, thus providing the most information on the stress-strain state of the structure.However, it does not give any information on their cumulative effect or interaction.
Criterion 2 (the Mises-Hill criterion) . 1 Criterion 3 (the Fisher criterion (Fisher, [33])) where , 1 Criterion 4 (the Goldenblatt-Kopnov criterion (Goldenblat and Kopnov, [34])) where For criteria 2 and 3, the following condition is in place: It should be noted that some sources do not put the minus sign with ultimate compressive stresses.To ensure consistency in notation with regard to the criteria and values calculated, assumed that compressive stresses (including ultimate compressive stresses) have the minus sign.
In defining the criteria, the following designations are accepted: 3 , 2 , 1 -the orthogonal system of coordinates corresponding to the orthotropy axes of the material; shall be adjusted to the 1, 2 orthotropy directions by using equations for coordinate system rotation.In such a case, the known values of the ultimate strength of the material can be used in strength criterion equations.
From now on, it will consider that the 1 and 2 orthotropy axes coincide with the y x, axes of the accepted local coordinate system, respectively ( z y x , , the orthogonal coordinate system in the middle surface of the shell structure; y x, -the curvilinear coordinates oriented along the main shell curvature lines, z -the coordinate oriented in the direction of the concavity, perpendicular to the middle surface).

Description of structures under consideration
T-10/UPE22-27 glass-fiber-reinforced plastic [2] as the structural material is selected.Its characteristics are given in Table 1.Table 2 shows the input parameters of a doublycurved shallow shell under consideration.The orthotropic doubly-curved shell is simply supported along the contour (for Computer simulation of the process of deformation of shell structures is important for many industries, including the construction and aircraft industry [36][37][38][39].
To unambiguously interpret these input parameters, Fig. 1 presents a general view of a doubly-curved shallow shell reinforced with stiffeners with the accepted local coordinate system.The shells are square in plan, with pin support along the contour, and subjected to uniformly distributed transverse load q directed along the normal to the surface.
The shells are reinforced with an orthogonal grid of stiffeners uniformly distributed throughout the structure.The width of the stiffeners is h r r i j , and the height is . The distance between the stiffeners defined as x r , with the outer stiffeners at a distance r x 5 .0 from the edge of the structure.

Mathematical model and analysis algorithm
The functional of full potential deformation energy (the Lagrange functional) as the basis for the mathematical model of shell structure deformation is used.
In the case of static problems, the function can be represented as the difference between the potential energy of the system and the work of external forces: where 0 The part of the functional associated with the skin will be as follows: where R p E depends on the method to arrange stiffeners, taking the following general form [40]: As a rule, if the external load is applied along the normal to the shell surface, then where q 0 denotes the value of the applied load, MPa; q sv denotes the transverse shell dead load component, MPa.This paper suggests using an algorithm based on the Ritz method and the solution continuation method with respect to the best parameter for studying shell structures.
According to this algorithm, the Ritz method is applied to the function to reduce the variational problem to a system of nonlinear algebraic equations.For this purpose, the required functions are presented as follows: ( ) Having applied functions (15) to functional (11), the derivatives with respect to unknown numerical parameters kl y kl U Ψ − are founded.Thus, a system of nonlinear algebraic equations is derived.To solve this system, the solution continuation method with respect to the best parameter is used.Verification of this algorithm is considered in detail in [41].
When the solution continuation method concerning the best parameter is used, the load/deflection curve is built step by step.At each step, the stress-strain state of the structure is analyzed.In the case of isotropic structures, it is sufficient to evaluate stress intensity, but in the case of orthotropic and anisotropic structures, it is required to apply special strength criteria.These criteria use constants of the ultimate values of stresses in the material.Besides, values of ultimate strength in different directions as well as values of ultimate strength under tension/compression and in shear, are different.
In the monograph [42], the authors analyzed stresses in different layers along the z coordinate for shallow shells rectangular in plan.They showed that the maximum stresses occur on the outside of the shell at .

Strength analysis of a glass-fiber-reinforced plastic shell
Next, the fulfillment of the strength conditions for a T-10/UPE22-27 glass-fiber-reinforced plastic shell with the 4 4 × orthogonal grid of stiffeners by using several criteria.The author suggests building criterion/load curves and criteria fields under various loads as well as areas failing to meet the strength conditions in the postbuckling state.Fig. 2 shows a strength criterion vs. load diagram.The load corresponding to the point where a criterion curve exceeds 1 on the horizontal axis indicates the onset of failure to meet the strength conditions.In the diagram below, the maximum stress criterion is represented by six curves corresponding to its components.The sharp curve bends because the criterion's maximum value is calculated over the entire structure (i.e., those may be different points in different moments of loading).When one point with the maximum values changes to another, the curve changes direction since the criterion values in different points of the structure increase at different rates.
Table 3 shows the maximum stresses obtained based on various strength criteria.Among other things, such a spread in values is also due to the fact that, in terms of some criteria, maximum stresses are achieved before buckling and, in terms of other criteria -after buckling.In Fig. 2, this is represented by the loops.This load value was chosen because it exceeds the values of all ultimate loads q pr obtained according to various criteria, and allows us to analyze the areas of the structure where the strength conditions were violated.In addition, this value is close to the values of ultimate loads, which makes it possible to estimate exactly where the beginning of irreversible changes in the shell material occurred before these areas had time to expand.
According to the components of the criterion of maximum stresses, the places of stress concentration are visible, which together can be seen in the images of other strength criteria.The highest values correspond to the sections at the edges of the structure (in the middle of the span).
All criteria show the absence of stress concentration in the center of the shell; also, the minimum stress values are observed at the corner points.
The Mises -Hill and Fisher criteria fields look almost the same, corresponding to the minimum difference in their formulas (the difference lies in only one coefficient).
The nature of the Liu -Huang -Stout criterion field is similar to the Mises -Hill, and Fisher criteria but gives higher values and shows a smaller difference in values between the concentrations at the x and y edges.
The Tsai -Wu, and Hoffman criteria show stress concentrations at the edges of the structure to a lesser extent.It can be seen from them that the strength conditions are violated at the boundary of the structure along the x axis, while the stresses at the boundary along the y axis are practically insignificant.The extension of areas failing to meet the strength conditions, based on some of the described criteria, at a load of 0.5869 MPa and a load of 1.056 MPa, are shown in Fig. 4.These load values were chosen due to the fact that it allows analyzing the process of increasing the areas of irreversible changes in the material: the first load value is taken slightly more than the founded maximum allowable load q pr , and the second -before the strength conditions cease to be satisfied at all points of the structure.
The development of areas failing to meet the strength conditions indicate, first of all, the occurrence of damage to the material on the edge of the structure along the x axis.This can be explained as follows: despite the structure under consideration being symmetrical in terms of geometry, reinforcement, and the applied load, orthotropic material is used.In the case of T-10/UPE22-27 glass-fiber-reinforced plastic, the ultimate strength along the 1 orthotropy axis (which coincides with the x axis) is twice as high as that along direction 2. Thus, when tensile stresses occur along direction 2, areas failing to meet the strength conditions occur closer to the edges of the structure.As the load increases, areas are expanding and supplemented by similar areas near the other two edges.These areas develop similarly but with a "delay".
The results obtained based on different strength criteria are very similar.However, it should be noted that the Liu-Huang-Stout criterion comes into action earlier and is indicative of more significant changes in the material under the same loads.In terms of the Hoffman criterion, the areas expand more slowly and differ in shape.Besides, they even fail to develop along the 1 orthotropy axis under the loads under consideration.
The maximum stress criterion shall be considered as a set of all its components: in this case, only + y F and + x F are significant, and the cumulative area failing to meet the strength conditions can be obtained by their combination.

CONCLUSION
The paper described a method of strength analysis with regard to the materials of thin-walled shell structures reinforced with stiffeners, using doubly-curved shallow shells as an example.The applicability of the following seven strength criteria was analyzed: the maximum stress criterion, the Mises-Hill criterion, the Fisher criterion, the Goldenblatt-Kopnov criterion, the Liu-Huang-Stout criterion, the Tsai-Wu criterion, and the Hoffman criterion.Three-dimensional graphs for strength criteria under a given load are also presented.Besides, the development of areas failing to meet the strength conditions by different criteria is analyzed.The suggested presentation format with regard to data on areas failing to meet the strength conditions and a strength criterion diagram is new and makes it possible to evaluate the state of the structure visually.
The sequence of actions indicated in work and, especially, the format for presenting data on the state of the structure (fields of strength criteria, graphs of dependence of strength criteria on load, graphs of the development of areas of non-fulfillment of strength conditions) are new and represent a scientific novelty.The practical significance of the work lies in the fact that the proposed method and the developed computational computer program for studying the stress-strain state, strength, and buckling of shells stiffened with ribs made of orthotropic materials under static loading can be used in design organizations and further scientific research.deformations of elongation along the x , y coordinates of the middle surface

Figure 1 .
Figure 1.Accepted local coordinate system

sE
-the component of the statics functional, associated with the skin; R p E -the potential energy of the system, associated with the stiffeners; 0 p E -the potential energy of the system, associated with the skin; Α -the work of external forces.
components), and their transverse component can be determined as follows:

FFigure 2 .
Figure 2. Strength criterion curves for a doubly-curved shallow shell made of T-10/UPE22-27 glass-fiber-reinforced plastic with the 4 x 4 grid of stiffeners a load of 0.3883 MPa are shown in Fig.3.

Figure 3 .
Figure 3. Strength criterion under a load of 0.3883 MPa for a doubly-curved shallow shell

FFigure 4 .
Figure 4. Development of areas failing to meet the strength conditions at a load of 0.5869 MPa and a load of 1.056 MPa for a doubly-curved shallow shel