Semi-analytical Finite Element Method Research Articles

CALCULATED RELATIONS OF THE SEMI-ANALYTICAL FINITE ELEMENT METHOD OF PRISMATIC BODIES FOR A FINITE ELEMENT BASED ON THE REPRESENTATION OF DIS-PLACEMENTS BY POLYNOMIALS

In article [8, 10], a variant of the semi-analytical finite element method for the calculation of prismatic bodies was developed using the Fourier series function as a coordinate system. The use of trigonometric series ensures maximum efficiency of the semi-analytical finite element method, however, at the ends of the body it is possible to satisfy only the boundary conditions corresponding to the support of the object on an absolutely rigid in its plane and flexible diaphragm. As a result of the performed researches the basis of representation of movements by polynomials is received that allows to expand considerably a range of boundary conditions on end faces of a body. In this case, it is not possible to reduce the solution of the original spatial boundary value problem to a sequence of two-dimensional problems, so a reasonable choice of appropriate polynomials becomes especially important. Their correct choice depends on the conditionality of the matrix of the system of separate equations and, consequently, the convergence of integration algorithms for its solution, and the universality of the approach to the possibility of satisfying different variants of boundary conditions at the ends of the body.

Semi-Analytical Finite Element Method for the Analysis of Guided Wave Dispersion in the Pre-stressed Composite Plates

Ultrasonic guided wave techniques offer an accurate and efficient procedure for damage monitoring in the structures. To develop reliable damage monitoring systems, it is essential to have a thorough understanding of the quantitative nature of ultrasonic guided waves that can be transmitted in composite laminates. In the present paper, the Semi-Analytical Finite Element (SAFE) method is employed due to its efficiency in the treatment of wave propagation problems involving complex materials and geometry. SAFE method is considered for the analysis of dispersion behaviour of guided waves in composite laminates by accounting the effect of in-plane load. The present study considers an infinite width plate such that the cross-section of the waveguide is modelled using 3 noded isoparametric 1-D elements representing the thickness of the plate. Equation of motion is formulated by using Hamilton’s equation. Finally, various parametric studies are carried out. The study includes analysing the effect of wave velocity in the plate subjected to applied in-plane load, and understanding the effect of dispersion characteristics on the direction of propagation (slowness curve). The study shows that at the lower frequency thickness product, the group velocity of the wave increases with the increase in the applied tensile load and vice-versa.

SECONDARY BUCKLING OF A HEATED CIRCULAR PLATE

Upon heating, a circular plate with immovable edge exhibits buckling, which results in ax-isymmetric postcritical bending. Progressive axisymmetric bending due to increasing thermal load in the postbuckling range leads to substantial redistribution of the stresses in the plate and occurrence of high compressive circumferential stresses in a narrow zone adjacent to the plate edge. In this case, secondary buckling can occur resulting in unsymmetric stress-strain state. The aim of the present work is to study stability behavior of the postbuckling axisymmetric equilibrium configurations of a circular plate subjected to uniform temperature rise. The plate edge is assumed to be immovable in the radial and transverse directions and elastically re-strained against bending rotation, which allows one to model the boundary conditions between two extremes of clamped and simply supported. The stability analysis is performed by the semi-analytical finite element method, where unknown displacements are approximated by truncated Fourier series in the circumferential coordinate. The geometrical nonlinearity is taken into ac-count in a quadratic approximation using the Fӧppl – von Karman nonlinear plate theory. To find equilibrium states of the plate, an iterative algorithm is proposed which requires determination of the coefficients of the first and second variations of the total potential energy. Stability of the equilibrium states is examined using the criterion of positive defineteness of the Hess matrix of the plate finite-element model. The critical temperature rise at which secondary buckling occurs is determined. The post-buckling nonlinear deformation characterized by local winkling near the plate edge is studied. The effect of boundary conditions and temperature-dependent material properties on the critical thermal load and secondary buckling modes is discussed.

Defect Detection inside a Rail Head by Ultrasonic Guided Waves

Early detection of defects inside a rail is of great significance to ensure the safety of rail transit. This work investigated the ability of ultrasonic guided waves (UGWs) to detect internal defects in a rail head. First, the model of UGW propagation in rail, which has an irregular cross-section, was constructed based on the semi-analytical finite element (SAFE) method. Fundamental characteristics, such as wavenumber, phase or group velocity, and wave structure inside the rail, were then calculated. Following modal and vibration energy distribution analysis, a guided wave mode that is sensitive to transverse fissure (TF) defects was selected, and its excitation method was proposed. The effectiveness of the excitation method was confirmed by simulations performed in the ABAQUS software. According to the simulation data, the dispersion curve calculated by using the two-dimensional Fourier fast transform (2D-FFT) coincided well with that of the SAFE method. After that, the sensitivity of the selected mode to internal rail defects was validated and its ability to locate defects was also demonstrated. Finally, the effects of excitation frequency, defect size, and vertical and horizontal defect depth on the reflection waveforms were investigated.

The effect of axial boundary conditions on breakout noise from finite cylindrical ducts

Ductborne noise within HVAC and exhaust systems may transfer into the surrounding environment through the walls of the duct. This breakout noise normally needs to be lowered in order to meet health and safety guidelines. This means that the ability of the duct walls to lower breakout noise needs to be predicted during the design stage of the duct systems. Significant work has been conducted into the prediction of breakout noise for infinite length ducts, however there is little to be found on finite length ducts. In particular there are very few studies on the difference between finite and infinite length duct breakout, especially where the noise source lies within the internal fluid. The aim of this work is therefore to investigate the difference in breakout noise between finite length and equivalent infinite length ducts when excited by an internal noise source. This is performed through numerical experiments using the semi analytical finite element method which enables the equations of elasticity for the duct wall, as well as a surrounding fluid, to be accommodated. Axial continuity equations at each end of a finite length elastic duct are then enforced through the point collocation method. It is observed that high sound power levels are emitted at axial resonances of the finite length duct. In the low frequency region these resonances have a narrow bandwidth and are expected to be of little practical significance. However, above the critical frequency the resonance bandwidth increases and this is observed to significantly lower the transverse transmission loss of the duct wall when compared to an infinite length duct. This phenomenon is observed for both clamped and simply supported ducts, as well as for two different internal sound sources. It is concluded that breakout noise from axial resonances in finite length ducts should be examined in design calculations in order to avoid excessive breakout noise.

Algorithms analysis for solving geometrically nonlinear mechanics problems in the scheme of the semi-analytical finite element method

The effectiveness of using the semi-analytical finite element method (SAFEM) to research geometrically nonlinear construction mechanics problems for 3D bodies of revolution under the arbitrary loads based on a basic ring finite element is considered. An estimate of the rational application parameters of algorithms for taking into account the geometric nonlinearity of a defined above structures class, which has been obtained.
 In the process of numerically solving spatial problems of the theory of elasticity and plasticity with finite displacements, the choice of rational algorithms for solving systems of nonlinear equations is of fundamental importance. It is conditioned by the need of determining the coordinates of the discrete model in the actual configuration and changing the metric characteristics of the finite elements, which, in its turn, leads to the necessity for multiple solutions of systems of nonlinear equations of high order. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation: the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFEM), it is possible, on the one hand, to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element, on the other hand- to maintain a high efficiency of spatial discretization. An important stage in the implementation of computer systems for solving spatial problems is the selection of optimal, from the point of the solution convergence speed and algorithms for solving equilibrium equations. The specificity of the algebraic equations of the SAFEM is conditioned by violation of the trigonometric function orthogonality in the space of the elasticity operator for bodies with variable stiffness and mass parameters along the guide. The clearly defined block structure of the stiffness matrix became the basis for using algorithms combining direct and iterative methods of solving. The reliability of the obtained results and the effectiveness of the approach are confirmed by the solution of a wide range of control examples under various boundary conditions and external loads.

Research of the stressed and deformed state of metal stripin the broaching process

In papers [2,4,5] the solution relations and the algorithm of the method of block iterations of solving linear and nonlinear equations by the semivanalytic finite element method for curvilinear inhomogeneous prismatic bodies are realized. In paper [1], a numerical study of the convergence of solutions was performed, and a wide range of test problems for bodies with smoothly and abruptly changing physical and geometric characteristics in elastic and resilient-plastic formulation was considered. In paper [6], to confirm the reliability of the results obtained on the basis of the semi-analytical finite element method, the effectiveness of this approach for the calculation of curvilinear inhomogeneous prismatic objects is shown. Solving control problems of the theory of elasticity, thermoelasticity and thermoplasticity, as well as problems of shape change makes it possible to draw conclusions about the reliability of the results of the study of a selected class of objects based on the developed methodology and implements its application package.
 In this work, using the methodology outlined in the above works, the solution of the spatial problem of plastic deformation of a prismatic body was solved on the basis of the semi-analytical method of finite elements, and the process of broaching a strip in the process of metal embossing was analyzed. The purpose of this study is to determine the spatial picture of the stress-strain state of the strip during drawing and to compare the spatial resolution with the flat one. A comparison of the parameters of the stress-strain state of the workpiece during the broaching process, obtained in flat and spatial settings, allows us to conclude that the research must be carried out on the basis of spatial calculation.

Damage shape recognition algorithm of composite woven fabric plate based on guided waves

Modern lightweight woven fabric composite structures play an important role in industrial field. The occurrence and accumulation of structure damages can threaten the integrity of in-service structures. With the high damage sensitivity, guided wave-based structural health monitoring has been regarded as the effective tool for damage detection. In this study, a damage shape recognition algorithm has been developed to continuously identify the damage extent in composite woven fabric structures, which is necessary for further analysis of remaining life estimation. The proposed algorithm is established relying on the accurate guided wave propagation information and precise damage scattered wave sources identification, and consists of the following stages: guided wave velocity calculation, guided wave signals collection, damage location identification, extraction of time of flight (TOF), damage boundary estimation, noise points filtering and damage shape and size recognition. The semi-analytical finite element method is presented to model guided wave propagation in multi-ply woven fabric panels. The numerical predictions are experimentally validated using the novel omnidirectional wave velocity measurement of guided wave collected by mobile PZT transducer. The precise TOF extraction and noise point elimination method are first proposed for improving effectiveness of damage identification. The approach was validated by several cases of damage at arbitrary positions. The estimated numerical and experimental results show the potential of the proposed algorithm in identifying various damage features in anisotropic composite plates, such as its location, shape and size, and achieve good agreement with the actual damage.

Scattering of the Fundamental Lamb Modes in Bent Metallic Plates

Abstract Guided elastic waves, propagating through curved waveguides, have attracted significant attention in the recent past, both from the perspectives of assessment of structural integrity and generating novel designs of acoustic waveguides. This article presents investigation of the interaction of the fundamental Lamb modes with a cylindrical bend in thin metallic plates. A hybrid numerical method is exposited, which combines the computational efficacy of the semi-analytical finite element method in modeling the long straight portions of the plate and the versatility of the conventional finite element method in modeling the bent portion. The predictive capabilities of the proposed method are validated using transient finite element simulations. Appropriate modifications to the hybrid method, needed for simulating the multimodal incidence resulting from the point-force actuation, are discussed. Using the hybrid method, the scattering and mode-conversion behavior, imparted by the cylindrical bend, is studied when the two fundamental Lamb modes are used as the incident interrogation signals. The extents of equi-modal and cross-modal contributions in both the reflected as well as transmitted waveforms are quantified in terms of the respective modal energy ratios. Explicable contour charts are presented for comprehending the scattering behavior over a wide range of frequencies and bend angles that span from 0 deg to 180 deg. For two representative cases, the modal displacement patterns inside the bend region are presented and discussed. The present investigation can find its potential use in the analysis of geometrically irregular structures leading to the design of novel acoustic waveguides.

Application of the finite element method to atmospheric sound propagation over impedance discontinuities

Outdoor noise propagation typically involves the propagation of sound over mixed ground conditions, including variations in the surface impedance. Diffraction occurs at the interface between different ground conditions, and this can significantly affect local sound attenuation. Popular approaches to modelling impedance discontinuities include the boundary integral method, parabolic equations, and semi-empirical methods. This paper presents an alternative approach that takes advantage of the generality of the finite element method. It was shown recently that the semi analytic finite element method can be used to solve the exact governing wave equation for atmospheric sound propagation in a complex vertically stratified medium. This allows for the inclusion of arbitrary temperature and wind profiles provided the problem is range independent. It is shown here that it is relatively straightforward to extend this approach to include multiple surface impedance discontinuities, provided each discrete surface is uniform in the axial direction. This is achieved using point collocation to enforce the axial boundary conditions over each discontinuity. Predictions are generated for single and multiple discontinuities, and comparisons are made against predictions obtained using other computational methods.

Acoustic emission based damage source localization for structural digital twin of wind turbine blades

An improved acoustic emission (AE) based structural damage source localization method for wind turbine blade is proposed in this paper. Firstly, the dispersion relations of wind turbine blade structures are achieved by a semi analytical finite element (SAFE) method. Propagation characteristics of AE waves in key structures of wind turbine blade is then investigated in terms of the dispersion relations. Based on the propagation characteristics of AE waves, including wave attenuation, wave reflection and transmission, and direction-dependent velocities, the improved structural damage source localization method is developed. Furthermore, five methods are used to determine the most reasonable time difference of arrival (TDOA) for improving the numerical accuracy of damage source localization. Finally, an example application of damage source localization for a “T” shaped structure is performed and the robustness of the developed method under Gaussian white noise conditions is investigated. This work elucidates the propagation characteristics of AE waves in typical wind turbine blade structures and the contribution to structural damage source localization. The results show that the developed structural damage source localization method is efficient and reliable and it may be used as a technical basis for establishing the structural digital twin (DT) of wind turbine blades.

Crack Detection of Threaded Steel Rods Based on Ultrasonic Guided Waves.

Fatigue cracks are typical damage of threaded steel rods under dynamic loads. This paper presents a study on ultrasonic guided waves-based, fatigue-crack detection of threaded rods. A threaded rod with given sizes is theoretically simplified as a cylindrical rod. The propagation characteristics of ultrasonic guided waves in the cylindrical rod are investigated by semi-analytical finite element method and the longitudinal L(0, 1) modal ultrasonic guided waves in low frequency band is proposed for damage detection of the rod. Numerical simulation on the propagation of the proposed ultrasonic guided waves in the threaded rod without damage shows that the thread causes echoes of the ultrasonic guided waves. A numerical study on the propagation of the proposed ultrasonic guided waves in the threaded rod with a crack on the intersection of the smooth segment and the threaded segment shows that both linear indexes ( and ARS) and nonlinear indexes ( and ) are able to detect the crack. A constant-amplitude tensile fatigue experiment was conducted on a specimen of the threaded rod to generate fatigue cracks in the specimen. After every 20,000 loading cycles, the specimen was tested by the proposed ultrasonic guided waves and evaluated by the linear indexes and nonlinear indexes. Experimental results show that both the linear and nonlinear indexes of the ultrasonic guided waves are able to identify the crack before it enters the rapid growth stage and the nonlinear indexes detect the crack easier than the linear indexes.

Dispersion of Guided Waves in Complex Waveguides: A Hybrid Modeling Technique Combining Gauss–Lobatto–Legendre Node Collation and Semi-Analytical Finite Element Method

Guided waves (GW) are massively used for structural health monitoring and defect evaluation in plate, pipe, and rail structures. To accurately and efficiently calculate the dispersive natures of GW in complex waveguides, this study proposes a novel Gauss–Lobatto–Legendre-based high-order semi-analytical finite element method (GLL-SAFE). Combining the GLL node collation and Lobatto quadrature into SAFE, the mass matrix in the developed GLL-SAFE is diagonal, which enables a faster solution speed and a reduced error of matrix inversion. Firstly, the GWs in the single-layer isotropic material, composite lamina, and composite laminates are calculated with both GLL-SAFE and the conventional Gauss-SAFE featuring an equidistant node collation and Gaussian quadrature. Before reaching the convergence limit, the calculated average relative errors for GLL-SAFE are smaller than those for Gauss-SAFE, and can reach an order of 10[Formula: see text] and 10[Formula: see text] for the phase and group velocity, respectively. Then a novel mesh automatic reconstruction with arbitrary element polynomial order is developed to calculate GW propagation in waveguides of complex cross section. As a hollow cylinder for validation, the calculated average relative errors reach below [Formula: see text] and [Formula: see text] for the phase and group velocity, respectively. Finally, with a complex rail track as the waveguide, the calculated dispersion characteristics with GLL-SAFE show an excellent match with those from the time-domain finite element analysis, and GLL-SAFE shows its higher calculation efficiency over Gauss–SAFE.

Closed-form expressions for the effective properties of piezoelastic composites reinforced with cylindrical fibers by Maxwell scheme

In this work, the Maxwell scheme is implemented to characterize the effective properties of piezoelastic fiber composites reinforced by unidirectional cylindrical fibers distributed along the x3-axis. The piezoelastic composites are assumed to have transversely isotropic constituent materials. Analytical expressions for the effective properties (elastic, piezoelectric, and dielectric) are reported. The Mori–Tanaka approach is also developed for piezoelastic composites and it is shown that their analytical results coincide with the Maxwell scheme and the effective field method. Numerical analysis and comparisons with the two-scale asymptotic homogenization method and the semi-analytical finite element method are shown and discussed. As a summary, the novelty value of this work is that the Maxwell and Mori–Tanaka schemes have been extended to heterogeneous piezoelectric structures. Closed formulas are derived for the effective elastic, piezoelectric, and dielectric moduli. A numerical implementation of the effective moduli is carried out and the overall properties are validated with other methods.

On the Use of High-Order Shape Functions in the SAFE Method and Their Performance in Wave Propagation Problems

In this research, high-order shape functions commonly used in different finite element implementations are investigated with a special focus on their applicability in the semi-analytical finite element (SAFE) method being applied to wave propagation problems. Hierarchical shape functions (p-version of the finite element method), Lagrange polynomials defined over non-equidistant nodes (spectral element method), and non-uniform rational B-splines (isogeometric analysis) are implemented in an in-house SAFE code, along with different refinement strategies such as h-, p-, and k-refinement. Since the numerical analysis of wave propagation is computationally quite challenging, high-order shape functions and local mesh refinement techniques are required to increase the accuracy of the solution, while at the same time decreasing the computational costs. The obtained results reveal that employing a suitable high-order basis in combination with one of the mentioned mesh refinement techniques has a notable effect on the performance of the SAFE method. This point becomes especially beneficial when dealing with applications in the areas of structural health monitoring or material property identification, where a model problem has to be solved repeatedly.

Disbond contour estimation in aluminum/CFRP adhesive joint based on the phase velocity variation of Lamb waves

Adhesive lap joints between composite and metal plates have been widely used in industrial fields including the automotive industry, marine manufacturing and aerospace engineering. Low quality of operation, harsh environment, adhesive aging and other disadvantages may lead to disbonding. To assess the disbond contour at an adhesive interface, this study proposes a detection method based on the phase velocity variation of Lamb waves. First, the dispersion curves of Lamb waves in both single-layer and bonded multi-layer areas are acquired using the semi-analytical finite element (FE) method. Subsequently, numerical models of Lamb wave propagation in intact and disbonded joints are established. Due to the difference in phase velocity between relevant modes, the phase difference of Lamb wave between disbonded and intact joints is quantitatively linear with the disbond length under specific excitation, which is verified by the simulated signals based on FEs. Then, a probabilistic reconstruction algorithm based on phase delay is employed to localize the disbond center. On this basis, the edge points of the disbond are acquired, and the convex envelope of these points is sketched for disbond contour estimation. As a result, both the location and shape of the disbond can be obtained, thereby providing information for subsequent assessment. The experiment is carried out on an adhesive lap joint specimen composed of an aluminum plate and a quasi-isotropic carbon fiber reinforced plastic laminate, and the results demonstrate the effectiveness of the proposed method.

An online stress monitoring strategy based on Wigner–Ville time–frequency energy extraction of single-frequency dual mode Lamb waves

In this paper, a new online uniaxial stress monitoring methodology has been developed. Based on the acoustoelastic effect of Lamb waves, multimodal information at a single frequency, in which the approximate linear fitting between group velocity ratio of two modes and stress is obtained, is fused to characterize stress. The transceiver distance can be eliminated, which is essential knowledge for existing methods, in stress coefficient calibration and practical application by using the ratio of acoustic time. The semi-analytical finite element method(SAFE) is adopted to calibrate the stress coefficient. Next, the time–energy assignment algorithm derived from the smoothed pseudo-Wigner–Ville distribution(SPWVD) is evaluated on the ideal dataset obtained by time-domain finite element(TDFE) simulation. The arrival times of different modes are extracted accurately, and the fitting results are consistent with the theoretical analysis. Finally, the laboratory experiments employing a pair of piezoelectric wafer active sensors(PWAS) are conducted to calibrate unknown coefficients. The same calibration process is carried out at three different distances, and almost the same calibration results are obtained, indicating the distance-dependent nature of the stress coefficients. According to the completed calibration, the actual measurements from the same batch show the maximum absolute error of stress measurement is 4.2MPa within the applied stress range from 20MPa to 100MPa and the standard deviation is less than 3MPa, demonstrating good measurement accuracy and consistency. It shows the great potential of the proposed strategy to measure stress at unknown propagation distances.

SYSTEMS OF COORDINATE FUNCTIONS DURING THE DECOMPOSITION OF DISPLACEMENTS BY POLYNOMIALS

The formulas obtained in [10, 12] for the calculation of nodal reactions and stiffness matrix coefficients allow to use different systems of coordinate functions constructed on the basis of polynomials to represent displacements. A distinctive feature of these relations in comparison with similar ones derived when using Fourier series to representthe displacements is that the coefficients of not only diagonal, but also peripheral submatrices do not equal zero and the solution of systems of equations obtained on their basis by direct methods becomes impractical. The factors that determine the efficiency of the semi-analytical finite element method include, first of all, the task of fixing the conditions at the ends of the body and the amount of calculations due to the rate of convergence of the integration process of solving systems of equations.Of particular importance in the semi-analyticalvariant of the finite element method is the choice of the appropriate system of coordinate functions to represent the displacements along the length of the prismatic element. In deriving the relations of a universal finite element with the exception of Fourier series, the direct use of functions used in the above works to calculate prismatic bodies based on the semianalytic finite element method seems irrational, because each of them satisfies only individual cases of boundary conditions. from the standpoint of the theory of shells. In addition, among thefunctions considered, only Fourier series provide a strict separation of variables and reduce the original spatial problem to a series of two-dimensional for each contained harmonic.Based on the above research, it is concluded that mixed coordinate function systems based on Michlin polynomials allow the simplest formulation of different conditions of fixation at the ends, while ensuring a high rate of convergence of the iterative process.

Inspection of nuclear assets with limited access using Feature Guided Waves

Resistance Seam Welding (RSW) is frequently used in many industries where low-cost mass manufacturing of gas/fluid-tight seals are required. One particular use case is in the fabrication of canisters to store nuclear material. Given the expected long lifetime and high replacement cost associated with nuclear infrastructure, it is common for an increase in the design life to be sought during operation. This leads to a need to integrate Non-Destructive Testing (NDT) techniques in a complex environment which frequently has accessibility issues. For the inspection of RSW nuclear canisters, one promising solution is to use Feature Guided Waves (FGWs) to rapidly screen the whole circumferential RSW joint from only partial circumferential access to ensure safe storage and targeted repackaging. FGWs can travel long distances while locally confining their energy to a topological feature such as a weld or stiffener. This paper explores such use of FGWs through both theoretical and practical experimental methods. To understand which FGW modes exist in the RSW joint, the Semi-Analytical Finite Element (SAFE) method was used, revealing four suitable wave modes. An anti-symmetric flexural RSW guided mode was down-selected due to its ease of excitation, intense energy concentration around the RSW, low dispersion and attenuation. Three-dimensional Finite Element (FE) simulations were conducted to explore the sensitivity of the flexural FGW to transverse cracks of differing dimensions. Experimental results on simplified and realistic geometries to transverse crack defects of ≥1 mm deep show the efficacy of using this method to efficiently screen nuclear canister RSW joints for such defects in-situ.

Investigation of the influence of flange thickness on the nature of the development of zones of plasticity in casing detail

In papers [11, 15 18] the solution relations and the algorithm of the method of block iterations of solving linear and nonlinear equations by the semivanalytic finite element method for curvilinear inhomogeneous prismatic bodies are realized. In paper [1], a numerical study of the convergence of solutions was performed, and a wide range of test problems for bodies with smoothly and abruptly changing physical and geometric characteristics in elastic and resilient-plastic formulation was considered. In paper [21], to confirm the reliability of the results obtained on the basis of the semi-analytical finite element method, the effectiveness of this approach for the calculation of curvilinear inhomogeneous prismatic objects is shown. Solving control problems of the theory of elasticity, thermoelasticity and thermoplasticity, as well as problems of shape change makes it possible to draw conclusions about the reliability of the results of the study of a selected class of objects based on the developed methodology and implements its application package.
 In this work, using the method described in the above works, a numerical analysis of the stress-strain state of a spatial object was performed, namely the study of the influence of flange thickness on the nature of the development of plasticity zones in the casing detail. It should be noted that the use of a thickened flange allowed to localize the zone of plasticity and its length in this case does not exceed half the length of the bell. In this case, the additional cost of material for the manufacture of thickened flange is fully justified. This reduces the level of plastic deformations and stresses in the hazardous area and prolongs the life of the casing detail.