Displacement Shape Functions Research Articles

Deriving Ritz formulation for the Static Flexural Solutions of Sinusoidal Shear Deformable Beams

This paper presents the Ritz variational method for the static bending analysis of sinusoidal shear deformable beams. The theory accounts for transverse shear deformation and satisfies transverse shear stress-free conditions at the top and bottom surfaces of the beam. The total potential energy functional for the thick beam bending problem is formulated and minimized using a Ritz procedure. The problem considered simply supported boundary conditions and two cases of loading – uniformly distributed load over the span and point load at the center. The function is a function of two unknown displacement functions constructed in terms of unknown generalized displacement parameters and shape functions that satisfy the boundary conditions. Ritz minimization of the functional is used to find the generalized displacement parameters and then the displacements w(x) and The displacements and stresses are found for the loading distributions considered. It is found that the results obtained agree remarkably well with the exact results obtained using the theory of elasticity. The differences between the present results and the exact solutions are less than 0.3% for maximum transverse displacement for both cases of loading considered. For uniform load over the beam, the result for l/t = 10 is 0.112% greater than the exact solution. For the point load at the center, the result for l/t = 10 is 4.468% greater than the exact solution. The increased difference in the point load case is due to the singular nature of the point load problem.

Mechanical-electric-magnetic-thermal coupled enriched finite element method for magneto-electro-elastic structures

Abstract Magneto-electro-elastic (MEE) materials possess the ability to convert mechanical, electrical, and magnetic energies, playing a critical role in smart devices. To improve the accuracy and efficiency of solving the mechanical properties of MEE structures in mechanical-electrical-magnetic-thermal (MEMT) environments, an MEMT coupled multiphysics enriched finite element method (MP-EFEM) is proposed. Based on the fundamental equations and boundary conditions of MEE materials, the interpolation coverage function is introduced into the MEMT coupled finite element method (FEM) to construct higher-order approximate interpolation displacement shape functions, electric potential shape functions, and magnetic potential shape functions. Combined with the variational principle, MP-EFEM is proposed, and the governing equations of MP-EFEM are derived. Numerical examples validate the accuracy and high efficiency of MP-EFEM in solving the mechanical properties of MEE structures in MEMT environments. When compared to the MEMT coupled FEM (MEMT-FEM), the results show that this method offers higher accuracy and efficiency. Therefore, MP-EFEM can effectively analyze the mechanical properties of MEE structures under multiphysics coupling, providing a new method for the design and development of smart devices.

Generalized strain-based finite element for non-linear stability analysis of beams with thin-walled open cross-section

Based on the generalized strain theory a shear-deformable finite element is developed for nonlinear stability analysis of thin-walled open-section beams in multibody systems. In this formulation a finite number of deformations, characterized as generalized strains, is defined which are related to dual stress resultants in a co-rotational frame. The stiffness formulation is based on a second-order approximation of the local elastic displacement field. Timoshenko’s beam theory and Vlasov’s modified thin-walled beam theory are used to include the shear strain effects due to non-uniform bending and restrained warping torsion and their mutual coupling effects. Axial shortening associated with the Wagner Hypothesis is taken into account such that the nonlinear behaviour of the beam is predicted accurately, especially under large torsion. Coupling between bending and torsional deformation due to non-coincident centroid and shear centre is modelled using a second-order cross-section transformation matrix. Cubic Hermitian polynomials are used as shape functions for the lateral displacements and twist rotation to derive the elastic and geometric stiffness matrices. Geometric nonlinearities associated with axial elongation and bending curvatures are described by additional torsion, bending and warping-related quadratic deformation terms yielding a set of modified deformations. The inertia properties of the beam are described using both consistent and lumped mass formulations. The latter is used to model rotary and warping inertias of the beam cross-section. The accuracy and the computational efficiency of the new beam element is demonstrated in several static and dynamic examples.

Nonlinear finite element Analysis of laterally loaded piles in Layered Soils

This work is based on Winkler’s theory to analyze laterally loaded single piles using the finite element method. Soil nonlinearity is taken into account via nonlinear springs “p-y” curves. Exact element displacement shape functions and stiffness matrix are used for the element in the case of linear Winkler’s modulus of subgrade reaction. In the nonlinear stage, an averaging technique for the element secant Winkler modulus is used to calculate the shape functions and stiffness matrix. An iterative technique is used to take into account the non-linearity of the soil. Few elements are required to simulate the pile efficiently. Unlike other analytical methods, the current method can be used to analyze piles with any load-transfer curves with arbitrary variation with depth.

Electromechanical coupling enriched finite element method for dynamic characteristic of piezoelectric materials structures

To improve the application depth and breadth of intelligent unit devices made from piezoelectric structures, their kinetic properties shall be urgently studied. Herein, based on the basic equations and boundary conditions of piezoelectric materials, an interpolation coverage function of node displacement was introduced into the displacement shape function and potential shape function of coupling electromechanical finite element method (FEM), forming an electromechanical coupling enriched FEM generalized shape function. Together with the variation principle, an electromechanical coupling enriched FEM was put forward, and its motion equations were deduced. Next, the free vibration and transient response of the piezoelectric structure were analyzed, and the accuracy and validity of the new method were validated with numerical cases. Results show this method has high application prospects for analyzing the dynamic properties of piezoelectric material structures.

A Method for Analysis of Nonlinear Deformation, Buckling, and Vibrations of Thin Elastic Shells with Inhomogeneous Structures

The formulation of the problem and the method of analysis of the stress-strain state, buckling and vibrations of elastic shells with inhomogeneous structure are considered. The modal analysis of the shells is performed at each stage of loading. The method allows one to study the behavior of shells with a complex shape of the middle surface, geometric features throughout the thickness, and a multilayer material structure under thermomechanical loading. We approximate a thin shell with one finite element (FE) over the entire thickness. At the same time, we use spatial FEs of the same type to model shell portions with stepwise-varying thickness. So we apply the universal finite element. It is based on an isoparametric 3D element with polylinear shape functions for coordinates and displacements and has additional parameters. The universal finite element can be transformed (modified) to accurately describe portions of the shell with stepped-variable thickness. This element can be eccentrically displaced relative to the average surface of the shell and change its own thickness. The side edges of neighboring FEs are in continuous contact, and the FE allows simulating sharp bends of the shell. The approach is modern and easy to implement, since it is based on the use of the relations of the three-dimensional geometrically nonlinear theory of thermoelasticity and the application of the moment finite-element scheme. The effectiveness of the method is demonstrated on classical test examples. The convergence, accuracy and reliability of the obtained solutions are investigated. Comparison of the results of calculations obtained by the moment finite-element scheme with the data of other authors shows a good agreement between the solutions.

Second-Order Collocation-Based Mixed FEM for Flexoelectric Solids

Flexoelectricity is an electromechanical coupling between the electric field and the mechanical strain gradient, as well as between the mechanical strains and the electric field gradient, observed in all dielectric materials, including those with centrosymmetry. Flexoelectricity demands C1-continuity for straightforward numerical implementation as the governing equations in the gradient theory are fourth-order partial differential equations. In this work, an alternative collocation-based mixed finite element method for direct flexoelectricity is used, for which a newly developed quadratic element with a high capability of capturing gradients is introduced. In the collocation method, mechanical strains and electric field through independently assumed polynomials are collocated with the mechanical strains and electric field derived from the mechanical displacements and electric potential at collocation points inside a finite element. The mechanical strain gradient and electric field are obtained by taking the directional derivative of the independent mechanical strain and electric field gradients. However, an earlier proposed linear element is unable to capture all mechanical strain gradient components and, thus, simulate flexoelectricity correctly. This problem is solved in the present work by using quadratic shape functions for the mechanical displacements and electric potential with fewer degrees of freedom than the traditional mixed finite element method. A Fortran user-element code is developed by the authors: first, for the linear and, after that, for the quadratic element. After verifying the linear element with numerical results from the literature, both linear and quadratic elements’ behaviors are tested for different problems. It is shown that the proposed second-order collocation-based mixed FEM can capture the flexoelectric behavior better compared to the existing linear formulations.

Stability analysis of three-dimensional thick rectangular plate using direct variational energy method

This study investigated the elastic static stability analysis of homogeneous and isotropic thick rectangular plates with twelve boundary conditions and carrying uniformly distributed uniaxial compressive load using the direct variational method. In the analysis, a thick plate energy expression was developed from the three-dimensional (3-D) constitutive relations and kinematic deformation; thereafter the compatibility equations used to resolve the rotations and deflection relationship were obtained. Likewise, the governing equations were derived by minimizing the equation for the potential energy with respect to deflection. The governing equation is solved to obtain an exact deflection function which is produced by the trigonometric and polynomial displacement shape function. The degree of rotation was obtained from the equation of compatibility which when equated to the deflection function and put into the potential energy equation formulas for the analysis were obtained after differentiating the outcome with respect to the deflection coefficients. The result obtained shows that the non-dimensional values of critical buckling load decrease as the length-width ratio increases (square plate being the highest value), this continues until failure occurs. This implies that an increase in plate width increases the probability of failure in a plate. Hence, it can be deduced that as the in-plane load on the plate increase and approaches the critical buckling, the failure in a plate structure is abound to occur. Meanwhile, the values of critical buckling load increase as the span-thickness ratio increases for all aspect ratios. This means that, as the span-thickness ratio increases an increase in the thickness increases the safety in the plate. It also indicates that the capacity of the plate to resist buckling decreases as the span-depth ratio increases. To establish the credibility of the present study, classical plate theory (CPT), refined plate theory (RPT) and exact solution models from different studies were employed to validate the results. The present works critical buckling load varied with those of CPT and RPT with 7.70% signifying the coarseness of the classical and refined plate theories. This amount of difference cannot be overlooked. The average total percentage differences between the exact 3-D study (Moslemi et al., 2016), and the present model using polynomial and trigonometric displacement functions is less than 1.0%. These differences being so small and negligible indicates that the present model using trigonometric and polynomial produces an exact solution. Thus, confirming the efficacy and reliability of the model for the 3-D stability analysis of rectangular plates.

Three-dimensional nonlinear mixed 6-DOF beam element for thin-walled members

This paper presents a three-dimensional mixed beam element formulation for fully nonlinear distributed plasticity analysis of members composed of sections with no significant torsional warping such as steel angles and tees. This formulation is presented using a corotational total Lagrangian approach and implemented in the OpenSees corotational framework. In this context, a basic coordinate system is lined up with the element chord and translates and rotates as the element deforms. The element tangent stiffness matrix and resisting forces in the basic system are derived through linearization of the two-field Hellinger-Reissner variational principle. The displacement shape functions are cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation. The generalized stress resultant shape functions are linear for moments and constant for axial force and torque with the P - δ effect considered, which are developed from equilibrium equations. The fiber section method with uniaxial constitutive laws is adopted to account for material nonlinearity. Since the degrees-of-freedom in the basic system are defined with respect to different reference points, all element responses are transformed to acting about the shear center before conducting the corotational transformation. The mixed element is validated through a number of experimental and numerical examples.

LINEAR AND NONLINEAR FREE VIBRATION ANALYSIS OF RECTANGULAR PLATE

The major assumption of the analysis of plates with large deflection is that the middle surface displacements are not zeros. The determination of the middle surface displacements, u0 and v0 along x- and y- axes respectively is the major challenge encountered in large deflection analysis of plate. Getting a closed-form solution to the long standing von Karman large deflection equations derived in 1910 have proven difficult over the years. The present work is aimed at deriving a new general linear and nonlinear free vibration equation for the analysis of thin rectangular plates. An elastic analysis approach is used. The new nonlinear strain displacement equations were substituted into the total potential energy functional equation of free vibration. This equation is minimized to obtain a new general equation for analyzing linear and nonlinear resonating frequencies of rectangular plates. This approach eliminates the use of Airy’s stress functions and the difficulties of solving von Karman's large deflection equations. A case study of a plate simply supported all-round (SSSS) is used to demonstrate the applicability of this equation. Both trigonometric and polynomial displacement shape functions were used to obtained specific equations for the SSSS plate. The numerical results for the coefficient of linear and nonlinear resonating frequencies obtained for these boundary conditions were 19.739 and 19.748 for trigonometric and polynomial displacement functions respectively. These values indicated a maximum percentage difference of 0.051% with those in the literature. It is observed that the resonating frequency increases as the ratio of out–of–plane displacement to the thickness of plate (w/t) increases. The conclusion is that this new approach is simple and the derived equation is adequate for predicting the linear and nonlinear resonating frequencies of a thin rectangular plate for various boundary conditions.

Locking-Proof Tetrahedra

The simulation of incompressible materials suffers from locking when using the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio ν gets close to 0.5 and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating incompressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure, and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young’s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean, and we focus on adapting it to our method. For ν = 0.5, we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking, which, in turn, affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis.

Mixed formulation for geometric and material nonlinearity of shear-critical reinforced concrete columns

This study presents a new beam element formulation following a Hellinger-Reissner variational principle for shear critical reinforced concrete members, and considering large displacement and rotation effects. The corotational formulation is used to describe the large displacement at the element nodal level and degenerated Green-Lagrange strain measures are used at the basic element level. A new displacement shape function has been developed considering section kinematics and compatibility conditions to satisfy stability criteria. New explicit equations for the internal geometric stiffness matrix to consider large deformation effect has been developed. A robust state determination algorithm for the large deformation mixed-based formulation is proposed. The Timoshenko beam theory has been adopted to consider shear deformations in deriving the section kinematics equations. The multi-axial stress state in reinforced concrete fibre has been simulated through the fixed crack smeared softened membrane model. The developed shear fibre beam element is capable of accurately reproducing the experimentally-observed large displacement effect on the post-peak shear strength degradation in the softening region. The accuracy and efficiency of the mixed-based formulation was evaluated by examining the responses at local and global levels for both monotonic and reverse cyclic loading conditions along with the process of reproducing experimentally-observed failure modes.

Forced-based Shear-flexure-interaction Frame Element for Nonlinear Analysis of Non-ductile Reinforced Concrete Columns

An efficient frame model with inclusion of shear-flexure interaction is proposed here for nonlinear analyses of columns commonly present in reinforced concrete (RC) frame buildings constructed prior to the introduction of modern seismic codes in the Seventies. These columns are usually characterized as flexure-shear critical RC columns with light and non-seismically detailed transverse reinforcement. The proposed frame model is developed within the framework of force-based finite element formulation and follows the Timoshenko beam kinematics hypothesis. In this type of finite element formulation, the internal force fields are related to the element force degrees of freedom through equilibrated force shape functions and there is no need for displacement shape functions, thus eliminating the problem of displacement-field inconsistency and resulting in the locking-free Timoshenko frame element. The fiber-section model is employed to describe axial and flexural responses of the RC section. The modified Mergos-Kappos interaction procedure and the UCSD shear-strength model form the core of the shear-flexure interaction procedure adopted in the present work. Capability, accuracy, and efficiency of the proposed frame element are validated and assessed through correlation studies between experimental and numerical responses of two flexure-shear critical columns under cyclic loadings. Distinct response characteristics inherent to the flexure-shear critical column can be captured well by the proposed frame model. The computational efficiency of the force-based formulation is demonstrated by comparing local and global responses simulated by the proposed force-based frame model with those simulated by the displacement-based frame model.

Buckling and vibrations of the shell with the hole under the action of thermomechanical loads

The paper outlines the fundamentals of the method of solving static problems of geometrically nonlinear deformation, buckling, and vibrations of thin thermoelastic inhomogeneous shells with complex-shaped midsurface, geometrical features throughout the thickness, under complex thermomechanical loading. The technique is based on the geometrically nonlinear equations of three-dimensional thermoelasticity, the finite element formulation of the problem in increments, and the use of the moment finite-element scheme. A thin shell is considered by this method as a three-dimensional body. We approximate a shell by one spatial universal finite element (FE) throughout the thickness. The universal FE is based on an isoparametric spatial FE with polylinear shape functions for coordinates and displacements. The universal element has additional variable parameters introduced to expand its capabilities. The method of modal analysis of the shell is based on an approach that at each current stage of thermomechanical loading takes into account the stresses accumulated at the previous stages. The developed algorithm allows one to study geometric nonlinear deformation and buckling of elastic shells of an inhomogeneous structure with a thin and medium thickness, as well as to study small vibrations of the shells relative to the reference deformed state caused by static loading, taking into account large displacements and a prestressed state. An analysis of the stability and vibration of the spherical panel with the hole is carried out. The effect on the frequencies and mode shapes of the shell of the sequential action of thermal and mechanical loads is investigated.

An improved inverse finite element method for shape sensing using isogeometric analysis

The inverse finite element method (iFEM) is advisable as accurate and efficient method for structure health monitoring (SHM). The previous standard iFEM uses 0th-order and 1st-order shape functions to reconstruct displacement fields under concentrated load and distributed load, respectively. However, in engineering environmental loads, it is difficult to determine which external load is mentioned above. To solve this problem, this paper proposes an improved iFEM based on isogeometric analysis (IGA), which can achieve a unified description of the displacement fields under the effect of concentrated load, distributed load, or concentrated-distributed-hybrid load. Initially, B-spline functions of order two are employed to reconstruct the strain fields. Then, the shape function order of translations and rotations are deduced based on the section strain theory. Next, these deduced displacement shape functions are further used to reconstruct the deformation field using strain measurement. Finally, a wing structure is subjected to concentrated and distributed loads in simulation analysis and experimental tests. The result shows that improved iFEM is more accurate and effective than standard iFEM in displacement reconstruction.

A Fibre Smart Displacement Based (FSDB) beam element for the nonlinear analysis of reinforced concrete members

Beam finite elements for non linear plastic analysis of beam-like structures are formulated according to Displacement Based (DB) or Force Based (FB) approaches. DB formulations rely on modelling the displacement field by means of displacement shape functions. Despite the greater simplicity of DB over FB approaches, the latter provide more accurate responses for those requiring a coarser mesh. In order to fill the existing gap between the two approaches, improvement of the DB formulation without the introduction of mesh refinement, is necessary. To this aim, the authors recently provided a contribution to the improvement of the DB approach by proposing new enriched, adaptive displacement shape functions, leading to the Smart Displacement Based (SDB) beam element.In this paper the SDB element is extended to include the axial force-bending moment interaction, crucial for the analysis of reinforced concrete (r/c) cross sections. The proposed extension requires the formulation of discontinuous axial displacement shape functions which are dependent on the diffusion of plastic deformations. The stiffness matrix of the extended smart element is provided explicitly and is dependent on the displacement shape functions updating. The axial force-bending moment interaction is approached by means of a fibre discretisation of the r/c cross section. The extended element, addressed as Fibre Smart Displacement Based (FSDB) beam element, is shown to be accurate furthermore, it is accompanied by an optional procedure proposal in order to verify the strong equilibrium of the axial force along the beam element, which is not usually accomplished by DB beam elements. Given a fixed mesh discretisation, the performance of the FSDB beam element is compared with the DB approach to show the higher degree of accuracy in the proposed element.

Modal analysis of thin parabolic shells

The modal analysis of parabolic shells of revolution is based on using the finite-element model of inhomogeneous shell. The shells can have complex-shaped midsurface, geometrical features throughout the thickness, or multilayer structure. To develop the finite-element shell model we approximate a thin shell by one spatial finite element throughout the thickness. The structural elements of an inhomogeneous shell require the finite element to be universal: it should be eccentrically arranged relative to the mid-surfaces of the casing, it should be possible to vary the thickness of the lateral edges of the finite element and ets. The universal finite element is based on an isoparametric spatial finite element with polylinear shape functions for coordinates and displacements. Additional variable parameters are introduced to enhance the capabilities of the modified finite element. Two hypotheses are used to describe the features of the stress–strain state of a thin inhomogeneous shell. The static hypothesis assumes that the compressive stresses in the fibers throughout the thickness are constant. The nonclassical kinematic hypothesis of deformed straight line is used: a straight segment along the thickness remains straight though stretched or shortened during deformation. This segment is not necessarily normal to the mid-surface of the shell.The stress–strain state of a shell and its structural elements is determined using the geometrically nonlinear equations of the three-dimensional theory of thermoelasticity. A linear elastic continuous medium with large displacements and small strains is used as a model whose properties correspond to the generalized Duhamel–Neumann law. To derive the governing finite-element equations for displacements the moment finite-element scheme is used. The moment finite-element scheme approximations of displacements and strains guarantee a correct description of the rigid-body displacements of finite elements, which enhances the convergence and accuracy of solutions on coarse meshes.The natural vibrations of parabolic shells with various heights have been investigated. The convergence of solutions has been studied and compared with the results obtained by other authors.During operation, realistic shell structures often undergo various changes in the temperature field. This can significantly affect their dynamic characteristics. Extension of this work to modal analysis of parabolic shells considering heating is currently being pursued.

Mixed beam formulation with cross-section warping for dynamic analysis of thin-walled structures

This paper presents the formulation of a three-dimensional beam finite element (FE) that accounts for cross-section warping and dynamic inertia effects. The model is the extension of an existing mixed formulation, originally developed for the static analysis of thin-walled beams, to the case of dynamic loading conditions.Four independent fields are considered to derive the element governing equations, i.e. material rigid displacements, strains and stresses and an additional displacement field, describing the out-of-plane warping displacement of the beam cross-sections. The latter is independently interpolated in the element volume by including additional degrees of freedom (DOF) to the nodal translations and rotations classically considered in beam formulations.To obtain a consistent form of the element mass matrix, the cross-section displacement shape functions are computed, relating the generalized cross-section displacement fields to the element nodal variables. In mixed FE formulations, these are not assigned a priori, as in displacement-based approaches, but are derived on the basis of material stiffness and element geometry, together with compatibility conditions. Thus, the Unit Load method is applied to deduce the expressions of the shape functions consistent with the force-based approach, assuming the simply-supported beam as reference element configuration.As opposed to the original FE model, the additional warping DOFs are not condensed-out with the definition of the element quantities but are treated as additional global unknowns. This permits a correct description of the inertia effects and ensures continuity of the warping displacement fields between adjacent FEs.Correlation studies are presented to validate the proposed model and investigate the effects of cross-section warping on the dynamic behavior of thin-walled structures. For selected specimens, the studies compare solutions obtained adopting the proposed beam element with those resulting from shell or brick FE models. Modal decompositions and time-history analyses are conducted, assuming both linear elastic and nonlinear constitutive behavior for the latter.

Dynamic Analysis of Thick Plates Resting on Winkler Foundation Using a New Finite Element

The aim of this paper is to study parametric earthquake analysis of thick plates resting on Winkler foundation. The governing equation of the plate is prepared with Mindlin’s thick plate theory. In the dynamic analysis, Newmark method is used for the time integration of the governing dynamic equation. This study investigates the effects of the thickness/span ratio, the aspect ratio, and the boundary conditions on the linear responses of thick plates subjected to earthquake excitations. In the analysis, finite element method is used for spatial integration. For the formulation of the equations of the thick plate theory, a finite element is derived by using higher-order displacement shape functions which is named MT17. This is a fourth-order finite element that shows excellent performance for thick plate analysis. A computer program using finite element method is coded in C++ to analyze the plates clamped or simply supported along all four edges. Graphs are presented that should help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that MT17 element can be effectively used in the earthquake analysis of thick plates. It is also concluded that in general, the changes in the thickness/span ratio are more effective on the maximum responses considered in this study than the changes in the aspect ratio.

Dynamic behaviour of thick plates resting on Winkler foundation with fourth order element

This paper focuses on the study of dynamic analysis of thick plates resting on Winkler foundation. The governing equation is derived from Mindlin\'s theory. This study is a parametric analysis of the reflections of the thickness / span ratio, the aspect ratio and the boundary conditions on the earthquake excitations are studied. In the analysis, finite element method is used for spatial integration and the Newmark-B method is used for the time integration. While using finite element method, a new element is used. This element is 17-noded and it\'s formulation is derived from using higher order displacement shape functions. C++ program is used for the analyses. Graphs are presented to help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that the 17-noded finite element is used in the earthquake analysis of thick plates. It is shown that the changes in the aspect ratio are more effective than the changes in the aspect ratio. The center displacements of the reinforced concrete thick clamped plates for b/a=1, and t/a=0.2, and for b/a=2, and t/a=0.2, reached their absolute maximum values of 0.00244 mm at 3.48 s, and of 0.00444 mm at 3.48 s, respectively.