Strain Energy Potential Research Articles

On energy mechanism of rate-dependent failure mode evolution in plain weave composite

The intricate failure modes and the yet unclear rate dependency of carbon fiber reinforced plain weave composite materials pose a challenge to mechanics researchers. This study establishes an energy-based evolution mechanism for the compressive failure modes of plain weave composite materials as the strain rate varies. This mechanism illustrates how the rate dependency of failure modes arises from the competitive relationship between strain potential energy and deformation kinetic energy. At low loading rates, the specimen exhibits a progressive crushing failure mode characterized by low peak stress and significant geometric deformation. As the loading strain rate increases, the energy required for this geometric deformation also increases. When the energy expenditure surpasses that needed to elevate the stress level of the specimen, it transitions to an instantaneous failure mode with high peak stress. In this mode, the specimen fractures into multiple small fragments immediately upon failure, lacking the large geometric deformations observed at lower rates. Through calculating this energy mechanism, a transition strain rate of 180 s−1 was determined for both failure modes. The accuracy of this mechanism was further verified by tests conducted near the critical strain rate. The energy-based evolution mechanism for failure modes provides a simplified and concise framework for simplifying complex models of composite material failures.

Optimizing the Location of Supports under a Monolithic Floor Slab

Monolithic reinforced concrete floor slabs are one of the most common types of building structures, and their optimization is an urgent task. The article presents the methodology for finding the optimal position of point supports under a reinforced concrete floor slab of arbitrary configuration at arbitrary load. The slab is considered thin, elastic and isotropic, with constant over-the-area stiffness, that is, the reinforcement is not taken into account or is constant. The solution is performed using the finite element method in combination with the nonlinear optimization methods. Finite element analysis is implemented by authors in MATLAB (R2024a) environment in such a way that the location of the columns may not coincide with the nodes of the finite element mesh of the slab. This allows to significantly increase the efficiency of solving the optimization problem compared to previously used algorithms, including the Monte Carlo method. Boundary conditions are taken into account using the Lagrange multiplier method. As an optimization criterion, the maximum deflection value is used, as well as the value of the potential strain energy. The effectiveness of six nonlinear optimization methods is compared in the example of a square slab under the action of a uniformly distributed load. For solutions obtained using the pattern search, simulated annealing and internal point methods, the maximum deflections are at least 1.2 times higher than for solutions obtained using the particle swarm method and genetic algorithm. An example of real object optimization is also presented. By changing the position of seven columns, it was possible to reduce the maximum deflection of the floor slab by 1.6 times.

Optimisation of Supports’ Position for a Reinforced Concrete Slab of Arbitrary Geometry

Introduction. A number of research held in recent years on rational design of buildings and structures built for different purposes study the rational position of the vertical members — columns or supports. However, the majority of up-to-date research and publications focus on elucidating the rational position of the supports in the context of improving the efficiency of the beams. Whereas the behaviour of the reinforced concrete slabs affected by changing the position of the supports is not covered well enough in the modern research. The present study is targeted exactly at revealing the behaviour of the reinforced concrete slab structures of a building in case of varying position of the main supports.Materials and methods. During the present numerical experiment, a reinforced concrete slab of arbitrary geometry with the given thickness h, the concrete grade with the respective properties and the reinforcement possessing the properties according to its class, has been investigated. Additionally, it was decided to set the distance magnitudes from the centers of gravity of the upper and lower row of reinforcement to the bottom and top surface of the slab 𝑎𝑥, 𝑎′𝑥, 𝑎𝑦, 𝑎′𝑦. The number of columns n was set for the slab. It was considered that the number of columns was known, but their position could change. Moreover, among the total number of columns, there were distinguished the certain ones having the definite (constant) position, whereas the remaining supports could change their position within the slab area. Bearing in mind the above, the researchers were to solve the problem of rational position of the moveable columns.Results. This paper presents the results of the numerical experiment on identifying the most optimal (rational) position of the reinforced concrete columns supporting a slab of the arbitrary geometry. Based on the several parameters at once, such as deflection, potential strain energy and consumption of the reinforcement products, the analysis of the rational position of the supports under the slab was made and functional dependences on changing the above mentioned parameters have been presented in the work.Discussion and conclusion. Based on the results obtained in the numerical experiment, the analysis of the behaviour of the reinforced concrete slabs of a set thickness upon changing the position of the supports under the slab was made and the changes of the slab operational parameters were analysed and compiled into the functional dependences.

Numerical and laboratory experimental analysis of the movement of silicone cleaning pigs through 90-degree bends pipe

Analysis of the stress-strain state of cylindrical cleaning pigs manufactured of hyperelastic material during their movement through the pipe bends of the pipelines, identification of maximum contact forces' points, and places of their separation from the inner wall. The effect of the dynamic friction coefficient between the pig's lateral surface and the pipeline's inner wall on the value of the maximum equivalent von Mises stresses in the pig is investigated – determination of the required pressure for the passage of the pig through pipeline bends.The finite element method performs numerical modelling of the pigs' movement through pipeline bends is performed. The non-linear properties of a hyperelastic pig material (a silicone compound with a hardness of 30 units on the Shore scale) are described by the potential Yeoh strain energy model. The contact interaction of the pig with the inner wall of the pipe bend is modelled by a surface-to-surface contact with dynamic friction coefficients of 0.1, 0.2, and 0.3 between them. For research, pigs with a length of 60 mm, 75 mm and 90 mm with a convex front and a concave rear end were manufactured and modelled. The experimental unit was designed and mounted from metal and glass pipes, between which pipe bends were placed with a bending angle of 90 and a bending radius of 1.5 DN. The metal pipeline has an internal diameter of 49 mm and a total length of 5.3 m, and the glass pipeline is 54 mm and 5 m, respectively. The experimental installation made of glass pipes was designed for visual observation of the dynamics of the pig movement through the glass pipeline bend and from metal – for measuring the pressure during the pig movement along the straight sections and the pipeline bend. To verify the correctness of the numerical modelling, the fully calculated deformations of the cleaning pigs in the bends of the pipeline were visualized and compared with photographs of the deformations of the pigs during their movement through the glass bends of the pipelines.The bending of the pig in the pipe bend and contact forces increase equivalent von Mises stresses in the pig. Moreover, during the movement of the pig in a pipe bend, the distribution of equivalent von Mises stresses in it, as well as its deformations, changes continuously. It depends on the stage at which it is located. Numerical modelling and experiments have established that when the pig is at the stage of movement in the middle part of the pipe bend, due to bending, its lateral surface is partially separated from the inner wall of the pipe bend. With this, on the convex side of the pipe bend, the pig is separated from the wall in the front and rear parts and on the concave side – in the middle part. This separation of the lateral surface of the pig from the inner wall of the pipe bend results in the formation of a gap and cross-flows through the pig, reducing the pressure drop on it, which can lead to its sticking. For pigs made of silicone compound with a hardness of 30 units on the Shore scale to pass pipeline bends with a bending angle of 90 and a bending radius of 1.5 DN, the pressure in the pig space shall be at least twice as high as the pressure required for the pig to move along a straight section of the pipeline.Modelling and experimental studies were performed for pigs made of silicone compound. Therefore, subsequent studies will establish the influence of other hyperelastic materials' physical and mechanical properties on the pig movement along the pipeline bends.The results obtained in this study allow for determining the possibility of pigs passing through the pipeline bends of gas gathering systems at gas fields and gas networks, determining the necessary pressure for this and, if necessary, optimising the geometric shape and dimensions of the pigs passing through the pipe bends.The influence of the location of a solid pig made of hyperelastic material in the pipeline bend and the coefficient of dynamic friction on the stress-strain state of the pig is studied. The article contains original experimental units designed and installed to study the movement of pigs through pipeline bends.

Out-of-plane behaviour of imperfect beam-column with sinusoidally corrugated web

This paper presents an analytical formula for the second-order normal stresses in flanges of the imperfect I beam-columns with sinusoidal web corrugation. To deduce the analytical expression, the beam is separated into three parts. The flanges are modelled as thin-walled beam-columns and the corrugated web is modelled as a thin equivalent orthotropic plate with uniform thickness. The indispensable elastic critical loads are derived from the total potential strain energy of the imperfect beam-column using an energy method. To assess the accuracy of the proposed analytical formula, full shell finite element models were created in Abaqus software. The result of the benchmark example showed that the derived expression is accurate and applicable for geometrically nonlinear analysis of beam-columns with sinusoidal web corrugation. Based on the proposed analytical expression, the Ayrton-Perry type solutions for corrugated web columns and beams are also presented in this paper.

Parameter influence analysis of stochastic resonance and stochastic P-bifurcation for the shape-memory alloy laminate

This paper investigates the transverse vibration of an axially moving shape-memory alloy (SMA) fiber hybrid laminations under the combined action of transverse harmonic excitation and stochastic disturbance. Considering the shape memory alloy (SMA) fiber volume fraction random field, the kinetic energy and strain potential energy of SMA laminates are solved, and the axial motion equation of SMA laminates is derived according to Hamilton principle, a non-dimensional differential equation of the transverse vibration of the axially moving SMA laminated plates is obtained by adopting the Galerkin integral method. Considering the volume-fraction stochastic field, the stochastic natural vibration of a SMA laminate is analyzed. Considering the stochastic resonance of the SMA laminate under the action of Gaussian white noise and harmonic excitation, the effects of geometric parameters, physical parameters of materials, and temperature on the stochastic resonance behavior of the SMA laminate is analyzed. Finally, the effect of the stochastic P-bifurcation phenomenon of the parameters on the steady-state response of the SMA laminate is analyzed.

The mechanics of thin elastic sheet reinforced with fiber mesh subjected to combined flexure and extension

We present a three-dimensional continuum model aimed at investigating the mechanical performance of elastomeric materials reinforced with bidirectional fibers undergoing the combined out-of-plane bending moment and bilateral extension. To achieve this, we consider the Neo-Hookean strain energy model for the matrix material of the fiber composite and incorporate the strain energy of bidirectional fibers into the strain energy potential of the fiber composite. The bidirectional fibers’ strain energy is rigorously modeled by configuring the fiber kinematics on the fiber composite surface using differential geometry and accounting for stretching, bending, and twisting responses of the fibers via the computation of the first-order and second-order gradient of deformation. To establish equilibrium equations, we apply the variational principle to derive the constitutive relations, resulting in the normal and tangential shape equations of the fiber composite, and boundary conditions of bending and stretching. The model implementation emphasizes analyzing the combined out-of-plane bending and bilateral stretching effects on the fiber composite material, and the numerical results include the out-of-plane and in-plane deformation, strain-loading responses, bending, twisting, and stretching of fibers within the matrix material, as well as the deformation of the fiber network. Specifically, it is found that a larger bending moment results in intensified out-of-plane deformation while larger bilateral stretch decreases the out-of-plane deformation of fiber composite. In particular, bilateral stretches reduce the Lagrange strain (ɛ1) near the bilateral boundaries, unlike the intensity distribution of Lagrange strain (ɛ2) maintains almost unchanged due to no external tension applied on the upper and lower boundaries of the fiber while bending moments persist. These findings highlight the role of bilateral stretching acting as the prestress in reducing deflection, enhancing tensile strength and efficient use, and controlling crack generation of fiber-reinforced polymer (FRP) composites. More importantly, despite the scarcity of experimental data on fiber composite deformation at the microscopic, the theoretical demonstration of the extension and flexure of fiber units provide reasonable explanations for the resulting deformation of the overall fiber meshwork, validating the mechanism that the fiber microstructure deformation determines the overall deformation of fiber composites when subjected to combined bending moment and tension. Special attention should be given to the potential dislocation between the embedded meshwork and matrix material located in the center of the domain. Furthermore, the proposed model provides qualitatively reasonable explanations on the shaping of Bamboo Polylactic acid (PLA) composites, out-of-plane deformation of woven fabric, Sikken-type stiffener deformation, and fiber-reinforced thermoplastics via a series of numerical results.

The stress state of a plate with several holes under the action of axial tensile load

The purpose of the research is to develop a numerical method for determining the stress state of plates with several holes. The research approach is based on the development of the superposition method for singular solutions. The study objective is to solve a practical problem of determining the stress concentration coefficients in a plate with two holes of the same diameter. The study uses the numerical method of elasticity theory in the form of a superposition of four special solutions, each of which lies entirely within the corresponding hole contour. The resulting contour integrals in the displacement formula are replaced by their approximate values using the M-point trapezoidal formula or the Gaussian quadrature formula. Equating to zero the expression in variations of unknown constants (the principle of minimum potential strain energy), we have obtained a system of linear algebraic equations 4MN of unknown distribution densities. After determining the constants using known formulas, the stress-strain state of the plate with holes can be calculated. The error of the developed method is no more than 0.6% compared to the known solution.

Deformation as a process to transform shape and volume of protective structures of the development mine workings during coal-rock mass off-loading

Purpose is to assess deformation characteristics of protective structures while coal-rock mass off-loading to ensure wall rock stability as well as operating conditions of the development mine workings in coal mines. Methods. In a laboratory environment, uniaxial compression of protective structures has been applied on the models to identify the influence by deformation processes on the changes in their rigidness resulting from the shape and volume transformation. Findings. Under the deformation of rigid structures in the context of a safe strain resource, potential energy of their changes in shape is 4.1-6.5 times higher than the one of changes in volume. Beyond the safe deformation resource when critical level of the specific potential strain energy has been exceeded, strength of protective structures is not sufficient to restrict wall rock movement limiting their use. If relative volume variation in the rigid protective structures is δV > 0.06-0.082 then they lose their stability. Under such conditions, structural rigidity decreases by 14-22%. If pliable wooden protective structures are used then relative 0.62 ≤ δV ≤ 0.72 volume change doubles their rigidity. In the circumstances, the potential shape change energy is 2.1 times higher than the volume change energy; the abovementioned favours temporary compaction of wooden components of the compressive structure while improving its resistivity. Originality. Regularities of changes in the specific potential deformation energy of protective structures depending upon their shape and volume variation in terms of uniaxial compression have been identified. Practical implications. To ensure stability of wall rocks and maintain operating conditions of the development mine wor-kings, it is reasonable to apply pliable wooden protective structures which will help restrict roof and floor movements after their compaction. Insufficient residual strength of rigid protective structures, resulting if they lose their stability, provokes rock failure within the working areas of coal mines.

DEFORMATION FEATURES OF EMBEDDED MATERIALS UNDER COMPRESSION CONDITIONS

Purpose. The purpose of the article is t o study the deformation characteristics of the backfill materials made of crushed rock of different granulometric composition under compression conditions to assess the stability of the side rocks in a coal massif with preparatory workings when selecting the parameters of protective structures. Methods. Compression tests were carried out in the laboratory to model the behaviour of crushed rock fill materials under load. Cylindrical specimens were subjected to axial static compression without the possibility of lateral expansion. The results. It has been experimentally established that the maximum value of the compaction coefficient of the crushed rock kс=1.57 is achieved under conditions of compression, when the relative change in the volume of the source material is 35-36% and the relative void is 5-6%. Under such conditions, the ultimate compressibility of the deformable body is achieved at its relative deformation λ = 0.36 and the presence of particles of different sizes (diameters) in the total volume of the rock layer (before compression). At the minimum value of the compaction coefficient kс=1.14, which corresponds to the small fraction of the crushed rock, the relative change of the void is 1.5%. For a large fraction of crushed rock at kс=1.43, the relative change in voids is 6.2%. This situation is associated with the destruction of rock particles and the reduction of voids between them. Compliance with certain requirements for backfill materials made of crushed rock, taking into account its particle size distribution, increases the efficiency of using the backfill of the excavated space. It is advisable to evaluate the bearing capacity of backfill materials by the value of the compaction coefficient of crushed rock. Scientific novelty. A functional relationship between the change in the specific potential strain energy and the relative deformation of the rock layer has been established, which allows us to estimate the degree of compaction of crushed rock under compression. Practical significance. To ensure maximum compaction of the crushed rock when using partial or complete backfilling of the worked-out space used for the protection of mine workings, it is advisable to use a heterogeneous (in terms of particle size) source material. Keywords: crushed rock, backfill material, stiffness, particle size distribution, compaction, deformation, compression.

Buckling of Stiffened Heterogeneous Shells Taking into Account Material Creep

The authors propose a mathematical model of shell structures taking into account the linear theory of hereditary material creep. This model is based on the total potential strain energy functional. Shells are reinforced with stiffeners, and the contact between the stiffener and the shell skin along a strip is taken into account. The Ritz method is applied to the functional, and a system of algebraic or integro-algebraic equations is found. The resulting system is first solved as a linear system (without account for the terms reflecting material creep), and the state of the structure under the specified load is determined. Then the iterative method is used to solve the creep problem for a given value of time. The paper presents the results of studying stiffened shallow shells of double curvature made of reinforced concrete under uniformly distributed load. The buckling of shells occurs over time as a result of the development of creep strains. Values of the buckling load as a result of material creep are found. It is shown that stresses are redistributed over the shell field over time and the maximum stress is observed near the contour of the structure.

Mechanical behaviour of composite materials including waste rubber chips: experimental and numerical investigations

ABSTRACT Converting scrap tires to a useful form is becoming an acceptable technique for a broad range of applications. This study focuses on the mechanical characterisation of composites containing a high amount of waste tire rubber which are reinforced with different content of aluminium powder. Therefore, a set of experimental test is performed to highlight the effect of metallic fillers reinforcement on the mechanical response of elaborated composites under static and quasi-static tests. Results of this study reveal the high improvement of material properties for the elastomeric matrix reinforced with metallic fillers. In fact, the ultimate strength of composite material has increased to 1.8 MPa for a high reinforcement content. In the second part, a modified visco-hyperelastic model is developed on the basis of experimental results which demonstrate the prominence of the strain-rate-dependent behaviour consideration. Coupled with the Prony’s series approximation, the Mooney Rivlin model is actually applied to refine the strain energy potential in order to examine the mechanical behaviour of the composite material under different loadings conditions. Finally, the good concurrence between the numerical and experimental results highlights the impressive efficacy of the proposed model in accurately predicting the mechanical characteristics of the materials under investigation.

Novel formula for the elastic critical moment of beams with corrugated web

AbstractThis paper presents a novel analytical expression to determine the elastic lateraltorsional buckling resistance of I‐beams with corrugated web. The corrugated web beams have higher torsional stiffness than the flat web beams; therefore, the self‐weight of the beam can be reduced. The appropriate hand calculation methods to determine the elastic critical moment of corrugated web beams are based on the modification of warping constant of the cross‐section. In this paper, instead of modifying the warping constant the approximate expression is derived from the total potential strain energy of the beam using an energy method. During the solution, the beam is separated into three parts. The flanges are modelled as thin‐walled beams and the corrugated web is modelled as a thin equivalent orthotropic plate with uniform thickness. To assess the accuracy of the proposed analytical formula full shell finite element models were created in Abaqus software. The accuracy of the proposed expression is examined for sinusoidal and trapezoidal corrugation also. The results of the parametric study showed that the derived formula is accurate and applicable for elastic lateral‐torsional buckling analysis of I‐beams with web corrugation.

Benchmarking physics-informed frameworks for data-driven hyperelasticity.

Data-driven methods have changed the way we understand and model materials. However, while providing unmatched flexibility, these methods have limitations such as reduced capacity to extrapolate, overfitting, and violation of physics constraints. Recently, frameworks that automatically satisfy these requirements have been proposed. Here we review, extend, and compare three promising data-driven methods: Constitutive Artificial Neural Networks (CANN), Input Convex Neural Networks (ICNN), and Neural Ordinary Differential Equations (NODE). Our formulation expands the strain energy potentials in terms of sums of convex non-decreasing functions of invariants and linear combinations of these. The expansion of the energy is shared across all three methods and guarantees the automatic satisfaction of objectivity, material symmetries, and polyconvexity, essential within the context of hyperelasticity. To benchmark the methods, we train them against rubber and skin stress-strain data. All three approaches capture the data almost perfectly, without overfitting, and have some capacity to extrapolate. This is in contrast to unconstrained neural networks which fail to make physically meaningful predictions outside the training range. Interestingly, the methods find different energy functions even though the prediction on the stress data is nearly identical. The most notable differences are observed in the second derivatives, which could impact performance of numerical solvers. On the rich data used in these benchmarks, the models show the anticipated trade-off between number of parameters and accuracy. Overall, CANN, ICNN and NODE retain the flexibility and accuracy of other data-driven methods without compromising on the physics. These methods are ideal options to model arbitrary hyperelastic material behavior.

Interface Effects of Strain-Energy Potentials on Phase Transition Characteristics of VO2 Thin-Films.

Metal-insulator-transition (MIT) of VO2 has attracted strong attention as a potential phenomenon to be utilized in nanostructured devices. Dynamics of MIT phase transition determines the feasibility of VO2 material properties in various applications, for example, photonic components, sensors, MEMS actuators, and neuromorphic computing. However, conventional interface strain model predicts the MIT effect accurately for bulk, but fairly for the thin films, and thus, a new model is needed. It was found that the VO2 thin film-substrate interface plays a crucial role in determining transition dynamics properties. In VO2 thin films on different substrates, coexistence of insulator-state polymorph phases, dislocations, and a few unit cell reconstruction layer form an interface structure minimizing strain energy by the increase of structural complexity. As a consequence, MIT temperature and hysteresis of structure increased as the transition enthalpy of the interface increased. Thus, the process does not obey the conventional Clausius-Clapeyron law anymore. A new model is proposed for residual strain energy potentials by implementing a modified Cauchy strain. Experimental results confirm that the MIT effect in constrained VO2 thin films is induced through the Peierls mechanism. The developed model provides tools for strain engineering in the atomic scale for crystal potential distortion effects in nanotechnology, such as topological quantum devices.

Series solution for dynamical characteristic of spatial beam system structure in offshore platform

In engineering, the beam system is often used as a support element for large shell structures to improve the support stiffness of the shell and its vibration characteristics are worth being investigated. In this paper, the dynamics of the beam system is modeled and its vibration characteristics are investigated. This paper develops a 6-degree-of-freedom beam theory to describe the displacements of the basic cells of a beam system. This facilitates the solution of the problem of coupling failure of the beam system. The strain potential energy, the boundary spring potential energy, Energy generated by external forces and the structural kinetic energy of the beam system are derived from the energy principle. The above energy expression is then used to construct a Lagrangian energy functional. The displacement function of the beam cells is constructed by improving the Fourier series. It is substituted into the energy functional. The functional is also processed using the Rayleigh Ritz method to finally obtain a dynamic analysis model of the beam system. Then convergence analysis and correctness verification are carried out to ensure the correctness of the model. Finally, the parametric analysis is carried out to investigate the effect of the geometry on its frequency characteristics.

Lateral-torsional buckling assessment of I-beams with sinusoidally corrugated web

This paper assesses the elastic and inelastic lateral-torsional buckling resistance of I-beams with sinusoidal corrugated web through an extended 22 degrees of freedom thin-walled beam-column finite element method. The appropriate hand calculation methods to determine the elastic critical moment of corrugated web beams are based on the modification of the warping constant of the cross-section. Instead of modifying the warping constant, the corrugated web is modelled as a thin, equivalent orthotropic plate with uniform thickness. An approximate expression is also derived from the total potential strain energy of the beam using the Ritz method. The result of the parametric study showed that the developed finite element is accurate and applicable for linear buckling analysis of beams with sinusoidal web corrugation. With the results of the linear buckling analysis the plastic resistance of the corrugated web beam can also be determined. Based on advanced finite element simulations' results the parametric study showed that the standard-based imperfection factor leads to conservative results to predict the inelastic lateral-torsional buckling resistance of sinusoidally corrugated beams, so a more precise but still safe imperfection factor is proposed for sinusoidally corrugated web beams.

Shock Wave Formation and Cloaking in Hyperelastic Rods

The analysis of propagating an initially harmonic acoustic pulse in a semi-infinite hyperelastic rod obeying the Yeoh strain energy potential reveals attenuation with distance of the wave amplitudes caused by the elastic energy dissipation due to forming and propagation of the shock wave fronts and heat production. The observed attenuation of harmonic waves results in a broadband cloaking of fairly remote regions. The analysis is based on solving a nonlinear equation of motion by an explicit Lax–Wendroff time-difference scheme combined with the finite element discretization in the spatial domain. The revealing phenomena are applicable to studies of acoustic wave propagation in various elastomeric rubberlike materials modeled by the Yeoh hyperelastic potentials.

Application of the Variational Method to the Large Deformation Problem of Thin Cylindrical Shells with Different Moduli in Tension and Compression

In this study, the variational method concerning displacement components is applied to solve the large deformation problem of a thin cylindrical shell with its four sides fully fixed and under uniformly distributed loads, in which the material that constitutes the shell has a bimodular effect, in comparison to traditional materials, that is, the material will present different moduli of elasticity when it is in tension and compression. For the purpose of the use of the displacement variational method, the physical equations on the bimodular material model and the geometrical equation under large deformation are derived first. Thereafter, the total strain potential energy is expressed in terms of the displacement component, thus bringing the possibilities for the classical Ritz method. Finally, the relationship between load and central deflection is obtained, which is validated with the numerical simulation, and the jumping phenomenon of thin cylindrical shell with a bimodular effect is analyzed. The results indicate that the bimodular effect will change the stiffness of the shell, thus resulting in the corresponding change in the deformation magnitude. When the shell is relatively thin, the bimodular effect will influence the occurrence of the jumping phenomenon of the cylindrical shell.

REFINED DISCRETE METHOD FOR CALCULATING STIFFENED ORTHOTROPIC SHELLS

The author proposes a refined discrete method for taking into account stiffeners in the simulationof thin-walled shell structures. According to the method, it is necessary to add different reduction factors along different coordinate axes. For ribs directed perpendicular to the considered direction, a reduction factor is introduced equal to the ratio of the width of the ribs in this direction to the linear size of the shell in the considered direction. This method supplements the previously developed geometrically nonlinear mathematical model, which takes into account transverse shears and material orthotropy. The model is written as a functional of the total potential strain energy and can be used for different types of shells by specifying the Lame parameters and the radii of principal curvatures. The computational algorithm is based on the Ritz method and the method of continuation of the solution with respect to the best parameter. The software imple-mentation was carried out in the Maple software package. The applicability of the refined discrete method is shown by the example of orthotropic shallow shells of double curvature, simply supported along the contour and under the action of an external uniformly distributed transverse load. Material parameters were selected for T-10/UPE22-27 and 0/90 Woven Roving E-Glass/Vinyl Ester fiberglass. A comparison was made of the values of critical buckling loads for different stiffening options (a grid of ribs from 0 to 12 ribs in each direction) and a comparison of the values with the conventional discrete method, which showed that with the conventional discrete method, the values of buckling loads are significantly overestimated, especially with an increase in the number stiffening ribs. Comparison of the results of the test problem with the results of experiments obtained by other authors showed good agreement between the refined discrete method.