Compatibility Equations Research Articles

Geometrically exact beam theory for gradient-based optimization

Decades of research have progressed geometrically exact beam theory to the point where it is now an invaluable resource for analyzing and modeling highly flexible slender structures. Large-scale optimization using geometrically exact beam theory remains nontrivial, however, due to the inability of gradient-free optimizers to handle large numbers of design variables in a computationally efficient manner and the difficulties associated with obtaining smooth, accurate, and efficiently calculated design sensitivities for gradient-based optimization. To overcome these challenges, this paper presents a finite-element implementation of geometrically exact beam theory which has been developed specifically for gradient-based optimization. A key feature of this implementation of geometrically exact beam theory is its compatibility with forward and reverse-mode automatic differentiation. Another key feature is its support for both continuous and discrete adjoint sensitivity analysis. Other features are also presented which build upon previous implementations of geometrically exact beam theory, including a singularity-free rotation parameterization based on Wiener-Milenković parameters, an implementation of stiffness-proportional structural damping using a discretized form of the compatibility equations, and a reformulation of the equations of motion for geometrically exact beam theory as a semi-explicit system. Several examples are presented which verify the utility and validity of each of these features.

Block-flexure toppling failure of rock slopes using an equivalent deformation compatibility method

Block-flexure toppling constitutes the predominant form of toppling failure in rock slopes. Although it has been extensively studied, the current theoretical models are often oversimplified by treating rock layers as rigid bodies that diverge from actual conditions. The proposed Equivalent Deformation Compatibility Method (EDCM) offers a fresh approach to assess the stability of rock slopes prone to block-flexure toppling. EDCM posits that blocky rock layers, with their inability to withstand significant bending and role in merely transferring forces, can be modeled as intact layers with a reduced modulus. The method simplifies the complex issue of analyzing discrete and continuous rock layers to the study of layered soft and hard rock, establishing deformation compatibility equations subsequently. Validation of the EDCM was achieved through numerical models, physical model testing, and application to an actual slope. The factor of safety (FS) for slopes corresponds with the results from both models and the actual slope, demonstrating the method's applicability for evaluating susceptibility to block-flexure toppling. When applying the EDCM, it is advised to set the elastic modulus reduction coefficient for blocky layers at a value below 0.1.

Mechanistic model for the compression strength prediction of masonry columns strengthened with fibre–polymer composites

A mechanistic model is presented for the strength prediction of squared columns made of masonry with a periodic arrangement and strengthened with a fibre–polymer composite jacketing. The formulation is based on an incremental plasticity theory that relies on equilibrium, compatibility, and kinematic equations. The strength domain of brick units and mortar joints is bounded by a multi-surface yield criterion: a Mohr–Coulomb strength domain with a linear cap in compression and a Rankine cut-off in tension. An elasto-plastic response with limited ductility is assumed for both masonry components. Differently, the FRP response is assumed elastic with a brittle failure governed by a limited tensile strain. Phenomenological-based assumptions are undertaken and justified. Details are also provided for the computational implementation of the procedure. The model accuracy is validated against experimental data on masonry squared columns and compared with existing standard-based formulas. Results demonstrate it provides real-time and accurate compressive strength solutions for squared masonry columns with or without a polymer-based wrapping and yet requiring few input parameters for the masonry constituents and reinforcement.

A Closed-Form Solution to the Mechanism of Interface Crack Formation with One Contact Area in Decagonal Quasicrystal Bi-Materials

Cracks and crack-like defects in engineering structures have greatly reduced the structural strength. An interface crack with one contact area in a combined tension–shear field of decagonal quasicrystal bi-material is investigated. Based on the deformation compatibility equation and displacement potential function, the complex representation of stress and displacement is given. Using the mixed boundary conditions, the closed-form expressions for the stresses and the displacement jumps in the phonon field and phason field on the material interface are obtained. The results show that the stress intensity factor at the crack tip is zero for the phason field. The variation in the stress intensity factor and the length of the contact zone in the phonon field is given, and the result is consistent with the properties of the crystal. The design of safe engineering structures and the formulation of reasonable quality acceptance standards may benefit from the theoretical research carried out here.

Thermoelastic response of multilayer cylinders: The direct integration and single-solid approach

An approach for solving the plane nonaxisymmetric thermoelasticity problems in multilayer cylinders is suggested from the single-solid standpoint, which implies considering a multilayer structure as a single unity with a step-wise variation of material properties. The direct integration method was used to the advantage of our approach by applying toward the compatibility equation in terms of strains instead of the one in terms of stresses. As a result, the governing equation is derived without involving generalized derivatives by the radial coordinate and then reduced to an integral equation of the second kind. Using the resolvent-kernel for solving the latter equation, the thermal stresses are computed as the Fourier series whose constituents are explicitly determined by the corresponding constituents of the thermal loading.

Interaction diagrams of thin-walled, open cross section PC beams

Thin-walled, open cross section PC (prestressed concrete) beams are widely used in precast structures. Nevertheless, the analytical computation of their load carrying capacity under bending and torsion is still an unsolved problem. This is because of the markedly non-linear behaviour of both steel and concrete and of the asymmetry of the behaviour of concrete in tension and compression. This involves a search for the correct solution from a physical point of view (among those mathematically admissible) based on trial-and-error methods of numerical analysis.This paper suggests a method to evaluate the interaction diagrams under bending and torsion. According to this method the solving system is made by two equilibrium plus two compatibility equations. Some examples show the outcomes of this approach.

Experimental Investigation of a Rapid Calculation and Damage Diagnosis of the Quasistatic Influence Line of a Self-Anchored Suspension Bridge Based on Deflection Theory

AbstractBased on deflection theory, the basic differential equations and deformation compatibility equations of self-anchored suspension bridges are established. The calculation formula for the stiffening girder deflection of a three-span self-anchored suspension bridge is derived, and the validity of the formula is confirmed through an example. In addition, a fast calculation method for the theoretical deflection influence line is proposed by using the deflection calculation formula, and a calculation method for the actual deflection influence line is founded on the concept of time and space coordinate transformation. By loading the bridge with a load car and using real-time monitoring of the displacement response of the bridge test section, combined with the proposed intelligent damage evaluation program, the abnormal situation of the damage of the three-span self-anchored suspension bridge can be quickly and accurately identified. The research results of this paper can provide flexible, economical, and effective guidance for detecting the capacity of real bridges.

The Application of the Modified Lindstedt–Poincaré Method to Solve the Nonlinear Vibration Problem of Exponentially Graded Laminated Plates on Elastic Foundations

The solution of the nonlinear (NL) vibration problem of the interaction of laminated plates made of exponentially graded orthotropic layers (EGOLs) with elastic foundations within the Kirchhoff–Love theory (KLT) is developed using the modified Lindstedt–Poincaré method for the first time. Young’s modulus and the material density of the orthotropic layers of laminated plates are assumed to vary exponentially in the direction of thickness, and Poisson’s ratio is assumed to be constant. The governing equations are derived as equations of motion and compatibility using the stress–strain relationship within the framework of KLT and von Karman-type nonlinear theory. NL partial differential equations are reduced to NL ordinary differential equations by the Galerkin method and solved by using the modified Lindstedt–Poincaré method to obtain unique amplitude-dependent expressions for the NL frequency. The proposed solution is validated by comparing the results for laminated plates consisting of exponentially graded orthotropic layers with the results for laminated homogeneous orthotropic plates. Finally, a series of examples are presented to illustrate numerical results on the nonlinear frequency of rectangular plates composed of homogeneous and exponentially graded layers. The effects of the exponential change in the material gradient in the layers, the arrangement and number of the layers, the elastic foundations, the plate aspect ratio and the nonlinearity of the frequency are investigated.

Galerkin-Kantorovich variational method for solving saint venant torsion problems of rectangular bars

The unrestrained torsional analysis of bars is an important theme in elasticity theory, first solved by Saint-Venant using semi-inverse methods. It has been considered and solved by several others using analytical methods and numerical procedures due to the importance in the design of machine parts under torsional moments. In this paper, the Saint Venant torsion problem is solved for rectangular prismatic bars using Galerkin-Kantorovich variational method (GKVM). The work presents a detailed theoretical framework of the problem, deriving using first principles considerations the stress compatibility equation in terms of the Prandtl stress function ϕ(x,y). The derived domain equation which is required to be satisfied over the rectangular cross-sectional domain is a partial differential equation of the Poisson type. GKVM is adopted as the solution method for finding the solution to the domain equation. The unknown Prandtl stress function ϕ(x,y) is assumed, following Kantorovich method to be a product of an unknown function for fx sought to minimize the Galerkin-Kantorovich variational functional (integral) (GKVF) and a known function (y2-b2) which satisfies the boundary conditions at all boundary points in the y-direction, that is, at y=±b. The resulting GKVF is a simplified functional whose integral is a second order inhomogeneous ordinary differential equation (ODE) in fx. The integrand is solved to find fx leading in a full determination of the Prandtl stress function. The expression for stresses, torsional moments and torsional parameters are then found and they satisfy the boundary conditions and the domain equation. The results for the torsional moments and torsional parameters are identical to previous results obtained using double finite sine transform method (DFSTM), and analytical methods. The merit of GKVM is that it has led to the exact solution of the unrestrained torsion problems.

Solving Saint Venant torsion problems for rectangular beams using single finite Fourier sine transform method

This research presents the single Fourier sine transform method (SFSTM) for solving the Saint Venant torsion problem of rectangular prismatic bars. The problem is a common theme in the theory of elasticity of unrestrained torsion which was previously expressed by Prandtl using Prandtl stress functions ϕ(x,y) as a Poisson type nonhomogeneous partial differential equation (PDE) called the stress compatibility equation. In this work the SFSTM was applied to the stress compatibility equation, converting the PDE to an easier to solve ordinary differential equation (ODE) in the transformed domain. The boundary conditions were used to find the integration constant and inversion was used to find the solution in the physical domain. The non vanishing stresses and torsional moments were thus found as a single series of infinite terms with rapid convergence. The maximum stresses and moments were found in standard form in terms of torsional parameters which were tabulated for various ratios of the cross-sectional dimensions. A comparison of the torsional parameters with previous results show that the present results are identical with previous results illustrating the accuracy of the SFSTM used. The sine kernel of the SFSTM satisfies the boundary conditions of the problem and contributed to the exact solution obtained. The SFSTM simplified the PDE to an ODE which is simpler to solve.

Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems

Lagrangian multiforms provide a variational framework to describe integrable hierarchies. The case of Lagrangian 1-forms covers finite-dimensional integrable systems. We use the theory of Lie dialgebras introduced by Semenov-Tian-Shansky to construct a Lagrangian 1-form. Given a Lie dialgebra associated with a Lie algebra g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}$$\\end{document} and a collection Hk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_k$$\\end{document}, k=1,⋯,N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=1,\\dots ,N$$\\end{document}, of invariant functions on g∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}^*$$\\end{document}, we give a formula for a Lagrangian multiform describing the commuting flows for Hk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_k$$\\end{document} on a coadjoint orbit in g∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}^*$$\\end{document}. We show that the Euler–Lagrange equations for our multiform produce the set of compatible equations in Lax form associated with the underlying r-matrix of the Lie dialgebra. We establish a structural result which relates the closure relation for our multiform to the Poisson involutivity of the Hamiltonians Hk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_k$$\\end{document} and the so-called double zero on the Euler–Lagrange equations. The construction is extended to a general coadjoint orbit by using reduction from the free motion of the cotangent bundle of a Lie group. We illustrate the dialgebra construction of a Lagrangian multiform with the open Toda chain and the rational Gaudin model. The open Toda chain is built using two different Lie dialgebra structures on sl(N+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}(N+1)$$\\end{document}. The first one possesses a non-skew-symmetric r-matrix and falls within the Adler–Kostant–Symes scheme. The second one possesses a skew-symmetric r-matrix. In both cases, the connection with the well-known descriptions of the chain in Flaschka and canonical coordinates is provided.

Application of intelligent controller to speed up vibration attenuation of a sandwich smart structure subjected to external excitation

Intelligent controllers represent a class of advanced control systems designed to adapt, learn, and make decisions based on changing environments and dynamic conditions. These controllers leverage artificial intelligence and computational techniques to enhance traditional control methods, providing a level of adaptability and sophistication that is particularly valuable in complex and uncertain systems. In this work for the first time, the Proportional–integral–derivative (PID) controller the intelligent controller is used to speed up vibration attenuation of a sandwich smart structure subjected to external excitation. The sandwich smart structure is made of a composite core and sensor and actuator patches. Using shear deformation theory, Hamilton’s principle, and compatibility equations, the governing equations of the sandwich smart structure under external excitations are obtained. After that using numerical solver and time-dependent solution procedure, the equations are solved. After obtaining the datasets via mathematical modeling, this kind of dataset is used as the input and output data to train, validate, and test the machine learning method to use it with low computational cost. After that, using this kind of machine learning method, future researchers can use it to estimate a control vibration of the sandwich structure after mechanical excitation. The results suggest some applicable recommendations to speed up the vibration attenuation of a sandwich smart structure subjected to external excitation.

Elastostatic stiffness modeling and performance evaluation of a 2RPU-UPU parallel mechanism with two operation modes

This article presents a parallel mechanism (PM) that is consists of a UPU and two RPU limbs (U, P, R denote universal joint, prismatic joint, and revolute joint, respectively). It can switch into two different configurations with three degrees of freedom (DOFs), in which one configuration outputs two rotations and one translation, and the other outputs two translations and one rotation. Four joints should be actuated to fully control the PM. Inverse position analysis is carried out. Taking the compliances of links into consideration, wrench analysis of the RU chain is conducted by using screw theory. Stiffness matrix and compliance matrix of the limb are derived through the strain energy method. The PM’s overall stiffness matrix is then obtained according to the deformation compatibility equations. The theoretical results are compared with simulation results obtained in ANSYS, which shows a maximum deviation less than 0.6%. Static stiffness performance evaluation is carried out using the virtual work stiffness index. Performance distributions of the PM in two operation modes under external loads are obtained. The proposed stiffness modeling and performance evaluation could be helpful for optimal design and application investigation of the PM.

Accelerate the numerical convergence of incompressible flows: Novel preconditioned characteristic boundary conditions

For the first time, the locally power-law preconditioning method (LPLPM) is used to formulate the preconditioned characteristic boundary conditions (CBCs). Then, it is implemented to solve the numerical modeling of unsteady and steady flows from viscous to turbulent regimes. The compatibility equations and Riemann invariants are mathematically derived and then utilized to the incompressible flow solvers as suitable boundary conditions. This method discretizes time derivative and governing equations' space terms by applying the four-stage, fourth-order Runge–Kutta method, and a finite volume, respectively. The preconditioning matrix in the LPLPM is automatically derived by local velocity sensors through a power-law formulation. The baseline k−ω is applied as an appropriate turbulence model. Several test cases are conducted around airfoils of Office National d'Etudes et de Recherches Aerospatiales, NACA0012 (National Advisory Committee for Aeronautics), and S809 at varied angles of attack of 0–20 and Reynolds numbers of 500 to 5.25 × 106 to examine the effectiveness and accuracy of the LPLPM employing preconditioned CBCs. A sensitivity analysis is also performed to examine how numerical parameters affect the simulation. The results show that using preconditioned CBCs in conjunction with LPLPM at the artificial boundary is precise, reliable, and computationally efficient in simulating viscous/turbulent flows. Furthermore, it is also concluded that the present approach considerably improves the convergence speed contrasted to the simplified boundary conditions.

Linear composite curvature MITC3+ flat shell elements

This paper introduces a novel linear composite curvature MITC3+ flat shell element, named κMITC3+, that combines the advantages of the MITC3+ theory for flexural (bending and shear) behavior and the Allman-like triangular element (which ensures a true rotation of drilling degrees of freedom and eliminates a spurious energy mode) for membrane behavior. To enhance the bending behavior of the MITC3+ approach, we propose an assumed composite curvature field that is constructed using a polynomial projection technique derived from the orthogonality condition of the compatibility equation in the Hu-Washizu principle. Since two additional internal cross-section rotations are interpolated by cubic bubble functions, a linear polynomial function of the admissible space is chosen that provides a significant enhancement of bending behavior. The κMITC3+ element successfully passes all critical tests and exhibits superior performance compared to the original MITC3+ element and the recent CS-MITC18+ element.

Modeling and Simulation of Elevation Dynamics of Main Battle Tank Weapon System with Linear Graph Method

The use of main battle tanks in armies goes back many years. The most crucial component is undoubtedly the weapon system. In this study, it is aimed to model the elevation dynamics of the weapon system that provides the movement of the weapon system in the vertical axis. The elevation dynamics consists of three parts: an electric motor, a driver, and a barrel. Both parts are expected to have similar vibration characteristics. The linear graph method was preferred as the modeling method. First of all, the system components and types were determined, and the linear graph, normal tree, and tree links of the system were drawn. Then the state variables, primary and secondary variables, as well as the number of branches, nodes, and sources, were determined, respectively. Elemental, continuity, and compatibility equations were written using the obtained graphs. By organizing these equations, state equations were obtained in standard form and matrix form. With the simulation studies carried out in MATLAB/Simulink, the displacement, velocity, and acceleration responses of the breech and muzzle sections were obtained. It is seen that obtained angular displacement, velocity, and acceleration results for both breech and muzzle sections are almost the same. As a result, the breech and muzzle sections have similar vibration motions and have proven to be able to work in harmony. This result shows that the barrel can be designed from two parts without additional stress and tension, which makes it difficult to focus on the target. By forming the barrel in two parts, the risk of damage can be minimized by better distributing the stress and tension occurring in the barrel during firing.

Time domain modeling of elastic waves using a stress-based unsplit-field perfectly matched layer with enhanced numerical stability

A stress-based unsplit-field perfectly matched layer (PML) method with enhanced numerical stability is proposed to simulate elastic waves in heterogeneous unbounded domains. The proposed PML method attenuates the waves by forcing the satisfaction of mixed-field equations constructed using a complex coordinate stretching in the frequency domain. Unlike other unsplit-field PMLs, which introduce coordinate stretching before combining the constitutive and compatibility equations, the proposed method first combines the two equations and then stretches the resulting equation. The stretched equation is converted back to the time domain using the inverse Fourier transform to yield transient elastic wave equations in the PML-truncated domain. Semi-discrete equations of motion for displacement and stresses were obtained using standard Lagrange-family finite elements. In a series of numerical examples involving semi-infinite elastic media, the accuracy of the wave solutions calculated using the proposed PML method was evaluated using a displacement L2-norm, a normalized L2-error, and an energy norm. The examples further validated the wave absorption performance of the PML and evaluated the long-term stability of solutions utilizing the Ricker pulse excitation with various central frequencies. The numerical experiments indicate that the solutions obtained using the proposed PML are significantly more stable than those obtained using the existing strain-based unsplit-field PMLs. The proposed method is particularly useful for applications requiring explicit time integration of semi-discrete equations of motion because the mass matrix can be diagonalized. The PML method and its associated solution strategies can be applied to various engineering problems, such as soil-structure interaction analysis, monitoring of structures, and geophysical probing.

MCNN-DIC: a mechanical constraints-based digital image correlation by a neural network approach

Digital image correlation (DIC) is a widely used photomechanical method for measuring surface deformation of materials. Practical engineering applications of DIC often encounter challenges such as discontinuous deformation fields, noise interference, and difficulties in measuring boundary deformations. To address these challenges, a new, to the best of our knowledge, DIC method called MCNN-DIC is proposed in this study by incorporating mechanical constraints using neural network technology. The proposed method applied compatibility equation constraints to the measured deformation field through a semi-supervised learning approach, thus making it more physical. The effectiveness of the proposed MCNN-DIC method was demonstrated through simulated experiments and real deformation fields of nuclear graphite material. The results show that the MCNN-DIC method achieves higher accuracy in measuring non-uniform deformation fields than a traditional mechanical constraints-based DIC and can rapidly measure deformation fields without requiring extensive pre-training of the neural network.

Symmetry Properties of Models for Reversible and Irreversible Thermodynamic Processes

The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible processes, an algorithm based on the principle of stationarity of the functional is incorrect. In this paper, to formulate a variational model of irreversible deformation processes with an expanded range of coupled effects, an approach is developed based on the idea of the introduction of the non-integrable variational forms that clearly separate dissipative processes from reversible deformation processes. The fundamental nature of the properties of symmetry and anti-symmetry of tensors of physical properties in relation to multi-indices characterizing independent arguments of bilinear forms in the variational formulation of models of thermomechanical processes has been established. For reversible processes, physical property tensors must necessarily be symmetric with respect to multi-indices. On the contrary, for irreversible thermomechanical processes, the tensors of physical properties that determine non-integrable variational forms must be antisymmetric with respect to the permutation of multi-indices. As a result, an algorithm for obtaining variational models of dissipative irreversible processes is proposed. This algorithm is based on determining the required number of dissipative channels and adding them to the known model of a reversible process. Dissipation channels are introduced as non-integrable variational forms that are linear in the variations of the arguments. The hydrodynamic models of Darcy, Navier–Stokes, and Brinkman are considered, each of which is determined by a different set of dissipation channels. As another example, a variational model of heat transfer processes is presented. The equations of heat conduction laws are obtained as compatibility equations by excluding the introduced thermal potential from the constitutive equations for temperature and heat flux. The Fourier and Maxwell–Cattaneo equations and the generalized heat conduction laws of Gaer–Krumhansl and Jeffrey are formulated.

Application of stress function and Fourier’s series for calculation of multilayer reinforced road structure

The article is devoted to the development of a method for static load calculation of a multilayer reinforced road structure. Unlike the traditional calculation practice of such structures, in which both reinforced and unreinforced layers are considered as isotropic layers, in this paper reinforced layers are presented as orthotropic layers, and unreinforced layers as isotropic. The computational model of the structure with isotropic and orthotropic layers is considered in the framework of the plane problem of elastic theory. The width of all layers in the crosswise direction is the same. The resolving equations are derived from the equations of strain compatibility involving the stress function and Fourier’s series. The unknown constants are obtained from the boundary conditions and layer coupling conditions. Formulas for determining the maximum deflections are derived. As an example, the two-layer structure calculation is given, the upper layer of which is reinforced with a geocell and the lower layer is unreinforced. A uniform loading is applied on a limited part of the upper layer surface. The paper presents a maximum deflections dependence diagram of the two-layer system on the number of terms of the series. The calculated deflections of the two-layer structure are compared with the maximum deflections of an equally thick unreinforced single-layer structure, as well as with the results of field experiments. The effect of reinforcing the upper layer of the structure is defined as the ratio of the difference between the unreinforced and reinforced maximum deflections of structures and the unreinforced structure maximum deflection. The reinforcement effect calculated according to the proposed theory was 16 %, while the effect obtained as a result of field experiment was 10 %. The difference is explained by the difference in the size and shape of the load deck, as well as the difference between the plane and volume deformation models of the theoretical model and the real structure.