Tensionless Elastic Foundation Research Articles

Response of an infinite beam resting on a tensionless elastic foundation subjected to arbitrarily complex transverse loads

This paper presents an investigation into the static response of an infinite beam supported on a unilateral (tensionless) elastic foundation and subjected to arbitrary complex loading, including self-weight. A new numerical method is developed to determine the initially unknown lengths that remain in contact. Based on the continuity conditions at the junctions of contact and non-contact segments, the response of the whole beam may be expressed through the displacement constants of the initial segment, reducing the contact problem to two nonlinear algebraic equations with two unknowns. The technique has been named the transfer displacement function method (TDFM). Comparison with the exact results of a particular limiting case shows the expected complete agreement. Finally, an example of a beam with several contact segments is presented and verified by the application of equilibrium conditions.

Initial Compressive Buckling of Clamped Plates Resting on Tensionless Elastic or Rigid Foundations

The unilateral contact buckling problem of thin plates resting on tensionless foundations is investigated. Three different plate models are considered. For a plate of limited length on a tensionless elastic foundation, the plate is first simplified to a one-dimensional mechanical model by assuming a buckling mode in terms of transverse coordinates, after which a new method is employed to determine the initially unknown boundaries of the areas in contact. Based on the continuity condition on the borderline between contact and noncontact regions, the buckling mode displacements of the whole plate may be expressed through the critical load coefficient and the first half-wavelength, reducing the buckling problem to two nonlinear algebraic equations with two unknowns. This procedure has been named the transfer function method. For a very long plate with a symmetric buckling mode, an infinite plate model with two half-waves is presented. For a plate on a rigid foundation, a single half-wave buckling model is shown to be appropriate. Comparison of limiting cases with exact solutions and with ABAQUS results showed good agreement. Finally, the influences of aspect ratio and foundation stiffness are presented.

Thermomechanical postbuckling of unilaterally constrained shear deformable laminated plates with temperature-dependent properties

This paper presents a study on the postbuckling responses of shear deformable laminated plates resting on a tensionless foundation of the Pasternak-type and subjected to combined axial and thermal loads. Two different postbuckling cases are considered, namely (1) the compressive postbuckling of initially heated plates and (2) the thermal postbuckling of initially compressed plates. The postbuckling analysis of laminated plates is based on the higher order shear deformation plate theory with a von Kármán-type of kinematic non-linearity. It is assumed that the foundation reacts in compression only. The thermal effects are also included and the material properties are assumed to be temperature dependent. The initial geometric imperfection of the plates is taken into account. The analysis uses a two-step perturbation technique to determine the postbuckling response of the plates. An iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region. Numerical solutions are presented in tabular and graphical forms to study the postbuckling behavior of antisymmetric angle-ply and symmetric cross-ply laminated plates resting on tensionless elastic foundations of the Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results reveal that the unilateral constraint has a significant effect on the postbuckling response of the plates subjected to combined axial and thermal loads when the foundation stiffness is sufficiently large. The results also confirm that the postbuckling responses are significantly influenced by temperature dependency and initial membrane stress as well as initial thermal stress.

Dynamic responses of Reissner–Mindlin plates with free edges resting on tensionless elastic foundations

Dynamic response analysis is presented for a Reissner–Mindlin plate with four free edges resting on a tensionless elastic foundation of the Winkler-type and Pasternak-type. The mechanical loads consist of transverse partially distributed impulsive loads and in-plane static edge loads while the temperature field is assumed to exhibit a linear variation through the thickness of the plate. The material properties are assumed to be independent of temperature. The two cases of initially compressed plates and of initially heated plates are considered. The formulations are based on Reissner–Mindlin first-order shear deformation plate theory and include the plate–foundation interaction and thermal effects. A set of admissible functions is developed for the dynamic response analysis of moderately thick plates with four free edges. The Galerkin method, the Gauss–Legendre quadrature procedure and the Runge–Kutta technique are employed in conjunction with this set of admissible functions to determine the deflection-time and bending moment–time curves, as well as shape mode curves. An iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region. The numerical illustrations concern moderately thick plates with four free edges resting on tensionless elastic foundations of the Winkler-type and Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results confirm that the plate will have stronger dynamic behavior than its counterpart when it is supported by a tensionless elastic foundation.

Response of a finite beam in contact with a tensionless foundation under symmetric and asymmetric loading

In this work a general analytical model is developed for the static response of a beam resting on a tensionless elastic foundation subjected to a lateral point load. This load may either be located at the center of the beam or may be offset. An analytical/numerical solution is obtained to the governing equations; this solution makes no assumption about either the contact area or the kinematics associated with the transverse deflection of the beam. This is in contrast to previous work in which, for an infinite beam (where the load is symmetric by definition), implicit assumptions about the contact area and the response kinematics were made. Because these assumptions are dropped, the contact behavior differs in several fundamental ways from its infinite counterpart. Specifically, it is shown that (i) the contact area is a sensitive function of the beam length and that this function may change nonmonotonically, (ii) the contact area may depend on the magnitude of the load, (iii) asymmetric loads, which cannot exist in the infinite problem, have a dramatic influence the contact area for the finite system. These features are demonstrated with specific examples and explained in terms of the fundamental physics of the system. The implications for these behaviors are also discussed.

Postbuckling of shear deformable laminated plates resting on a tensionless elastic foundation subjected to mechanical or thermal loading

Postbuckling responses of shear deformable laminated plates supported by a tensionless elastic foundation and subjected to in-plane compressive edge loads or a uniform temperature rise are investigated. The formulations are based on the higher order shear deformation plate theory with a von Kármán-type of kinematic non-linearity and include the plate-foundation interaction, for which the foundation reacts in compression only. The thermal effects are also included and the material properties are assumed to be independent of temperature. The initial geometric imperfection of the plate is taken into account. The analysis uses a two step perturbation technique to determine the postbuckling response of the plate. An iterative scheme is developed to obtain numerical results without using any prior assumption for the shape of the contact region. The numerical illustrations concern the postbuckling behavior of antisymmetric angle-ply and symmetric cross-ply laminated plates resting on tensionless elastic foundations of the Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results reveal that the unilateral constraint has a significant effect on the postbuckling response of the plate subjected to either mechanical or thermal loading when the foundation stiffness is sufficiently large.

Nonlinear bending behavior of Reissner–Mindlin plates with free edges resting on tensionless elastic foundations

A nonlinear bending analysis is presented for a Reissner–Mindlin plate with four free edges subjected to thermomechanical loads and resting on a tensionless elastic foundation of the Pasternak-type. The mechanical loads consist of transverse partially distributed loads and in-plane edge loads while the temperature field is assumed to exhibit a linear variation through the thickness of the plate. The material properties are assumed to be independent of temperature. The two cases of initially compressed plates and of initially heated plates are considered. The formulations are based on Reissner–Mindlin first order shear deformation plate theory and include the plate-foundation interaction and thermal effects. A set of admissible functions, which satisfy both geometrical and natural boundary conditions, are developed for the nonlinear bending analysis of moderately thick plates with four free edges. A two step perturbation technique is employed in conjunction with this set of admissible functions to determine the load-deflection and load-bending moment curves. An iterative scheme is developed to obtain numerical results without using any prior assumption for the shape of the contact region. The numerical illustrations concern moderately thick plates with four free edges resting on tensionless elastic foundations of the Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results show that the nonlinear bending responses for the conventional and tensionless elastic foundation are quite different.

Postbuckling Analysis of Plates Resting on a Tensionless Elastic Foundation

The buckling and large deflection postbuckling behavior of plates laterally constrained by a tensionless foundation and subjected to in-plane compressive forces are investigated. A nonlinear finite-element formulation based on Marguerre’s nonlinear shallow shell theory, modified by Mindlin’s hypothesis, is employed to model the plate response. To overcome difficulties in solving the plate–foundation equilibrium equations together with the inequality constraints due to the unilateral contact condition, two different approaches are used: (1) the unilateral constraint is accounted for indirectly by a bilinear constitutive law and (2) the problem is formulated as a mathematical programming problem with inequality constraints from which a linear complementarity problem is derived and solved by the Lemke algorithm. To obtain the nonlinear equilibrium paths, the Newton–Raphson algorithm is used together with path-following strategies. Plate–foundation interaction leads to interesting deformation sequences, chara...

Numerical methods for analysis of plates on tensionless elastic foundations

A numerical methodology for analysis of plates resting on tensionless elastic foundations, described either by the Winkler model or as an elastic half-space, is presented in this paper. The contact surface is assumed unbonded and frictionless. The finite element method is used to discretize the plate and foundation. To overcome the difficulties in solving the plate–foundation equilibrium equations together with the inequality constraints due to the frictionless unilateral contact condition, a variational formulation equivalent to these equations is presented from which three alternative linear complementary problems (LCP) are derived and solved by Lemke’s complementary pivoting algorithm. In the first formulation, the LCP variables are the plate displacements and the elastic foundation reaction, in the second, the LCP is derived in terms of the elastic foundation reaction and, in the third formulation, the variables are the elastic foundation displacements and the gap between the bodies. Once the LCP is solved the no-contact regions where the plate lifts up away from the foundation and the sub-grade reaction, as well as the plate displacements and stresses, can be easily obtained. The methodology is illustrated by three examples and the results are compared with existing analytical and numerical results found in the literature.

Axisymmetric shells and plates on tensionless elastic foundations

This paper first describes a finite element method for the large deflection analysis of axisymmetric shells and plates on a nonlinear tensionless elastic foundation. Through the use of discrete data points, any form of nonlinear elastic foundation behaviour can be easily modelled. The analysis is then validated by comparison with existing results for circular plates and beams as the only existing results for shells on tensionless foundations are found to be in error. Following this verification, the analysis is applied to investigate the behaviour of shallow spherical shells subject to a central concentrated load on tensionless linear elastic foundations. A number of insightful conclusions regarding the behaviour of such structure-foundation systems are drawn. The numerical results for shells are believed to be the first correct results, which may be useful in benchmarking results from other sources in the future.

Three-dimensional behavior of geosynthetic tubes

Geosynthetic tubes filled with pressurized slurry are used for various purposes (e.g., as dikes or breakwaters). The three-dimensional behavior of such structures is investigated. The tubes are assumed to be comprised of two rectangular sheets connected at their edges, and to rest on a tensionless elastic foundation. As the slurry is pumped in, the top surface rises and the outer portion of the bottom surface lifts off the foundation. For tubes with aspect ratios of 2:1 and 5:1, the deflected shapes, mid-surface stresses, and contact regions with the foundation are determined with the use of the finite element method and thin shell elements. Wrinkling occurs close to the long edges of the tubes.

Shear and attachment effects on the behaviour of rectangular plates resting on tensionless elastic foundation

The behaviour of flexible plates of rectangular shapes supported on tensionless elastic foundation is analysed using FEM techniques. Transverse shear deformations have been included which significantly improve the calculations for plate central deformations and stress-resultants over the calculations made using thin plate theory. A nine-noded Mindlin element has been adopted for modelling the plate to account for transverse shear effects. The tensionless character of the foundation is incorporated by using a binary multiplier ( 0 1 ) for uncorrelated foundation spring stiffness depending upon the displacement of the spring. A dimensionless quantity K, which relates the stiffness characteristics of the foundation and plate, is introduced. Effects of realistic design parameters (e.g. edge boundary conditions, plate thickness and attachment) have been studied considering thick plate behaviour, and observations have been made regarding governing design parameters while comparing with the existing thin plate analysis method.

Stability Of Arches And Beams UnderUnilateral Constraints

A numerical methodology is presented in this paper for the buckling and post-buckling analysis of slender structural elements such as arches and beams subjected to unilateral contact constraints. A geometrically non-linear finite element model is used together with an updated Lagrangian formulation to obtained the non-linear equilibrium paths for perfect and imperfect structural models. The unilateral constraints are imposed by tensionless elastic or rigid foundations. At each load step, in order to obtain the equilibrium configuration and contact regions, the equilibrium equations are linearized and the contact problem is treated as a minimization problem with inequality constraints. The resulting linear programming problem is solved by the use of Lemke's algorithm and the contact regions are identified. Next, the Newton-Raphson method is used with path following methods to obtain the new contact forces and equilibrium configurations. Some structural systems are analyzed and the computational results show that the proposed methodology is feasible and efficient and that the results compare well with experimental and other numerical results found in literature.

Rectangular Plates Resting on Tensionless Elastic Foundation: Some New Results

The behavior of flexible rectangular plates resting on tensionless elastic foundations is analyzed using finite-element method (FEM) techniques. A nine-noded Mindlin element has been adopted for modeling the plate to account for transverse shear effects. Effects of realistic design parameters have been studied.

BEAMS ON TENSIONLESS ELASTIC FOUNDATION: APPROXIMATE QUANTIFIER ELIMINATION WITH CHEBYSHEV SERIES

The well-known Sturm's theorem (based on Sturm's sequences) for the determination of the number of distinct real zeros of polynomials in a finite or infinite real interval has been already used in elementary quantifier elimination problems including applied mechanics and elasticity problems. Here it is further suggested that this theorem can also be used for quantifier elimination, but in more complicated problems where the functions involved are not simply polynomials, but they may contain arbitrary transcendental functions. In this case, it is suggested that the related transcendental equations/inequalities can be numerically approximated by polynomial equations/inequalities with the help of Chebyshev series expansions in numerical analysis. The classical problem of a straight isotropic elastic beam on a tensionless elastic foundation, where the deflection function (incorporating both the exponential function and trigonometric functions) should be continuously positive (this giving rise to a quantifier elimination problem along the length of the beam) is used as an appropriate vehicle for the illustration of the present mixed (symbolic–numerical) approach. Two such elementary beam problems are considered in some detail (with the help of the Maple V computer algebra system) and the related simple quantifier-free formulae are established and seen to coincide with those already available in the literature for the same beam problems. More complicated problems, probably necessitating the use of more advanced computer algebra techniques (together with Sturm's theorem), such as the Collins well-known and powerful cylindrical algebraic decomposition method for quantifier elimination, can also easily be employed in the present approximate (because of the use of Chebyshev series expansions) symbolic–numerical computational environment.

Effect of anisotropy of plates on the contact region resting on tensionless elastic foundation

Effect of anisotropy of plates on the contact region resting on tensionless elastic foundation

Static and dynamic responses of a circular plate on a tensionless elastic foundation

Abstract This study deals with the static and dynamic responses of a thin circular plate on an elastic foundation of Winkler type that reacts in compression only. The plate is assumed to be subjected to time dependent external moment and vertical load at its center. The non-dimensionless governing equations of motion are derived by application of Galerkin’s method. At this step the governing equations are reduced to the time domain by the use of the modal functions of the completely free plate. This gives rise to a set of coupled differential equations of second order having time dependent coefficients. Numerical results are presented in figures to illustrate static and dynamic behaviours of the circular plate. The figures indicate the extent of the contact region and the deflections of the plate depending on the loading combination and on time.

Nonaxisymmetric Unbonded Contact of Plates on Tensionless Winkler Foundations*

ABSTRACT The unbonded contact problem for an annular plate resting on a tensionless Winkler elastic foundation is investigated. The problem is solved by minimizing the total potential energy of the plate-foundation system. Two alternative functions that include either Bessel or Kelvin functions are used to define the plate displacement. By satisfying a set of four boundary conditions and using the Ritz method, the complete displacement function for the plate is obtained iteratively. To improve the accuracy and convergence of the iterative approach, the solution requires a complete revision of the assumed displacement functions after every iteration. It is proven that the selected displacement functions and the suggested procedure make the approach highly accurate. Comparisons with results obtained by other approaches, in addition to other examples, are presented.

Dynamic response of a column with foundation uplift

The effects of transient foundation uplift on the dynamic response of a column with attached masses supported on a Winkler foundation are investigated. The analysis developed here is based on the assumption that the column, having continuously distributed mass and rigidity, is supported on a tensionless elastic foundation through a rigid circular base plate. The method employed includes use of series expansion of the lateral displacement of the column, application of the Galerkin technique and step-wise solution of the governing equations of the problem. Besides the free vibrations of the column, the forced vibrations of the system are investigated for an assumed harmonic ground motion excitation. Additionally, the dynamic response of the system to the El Centro 1940 ground motion is presented.

Axisymmetric Vibrations of Circular Plates on Tensionless Elastic Foundations

The present study deals with the axisymmetric vibrations of a circular plate subjected to a concentrated dynamic load at its center. The plate is assumed to be supported on an elastic foundation that reacts in compression only. At first the static solution is obtained and contact radius is determined. Later, this is used as an initial configuration of the forced vibrations. The forced vibrations are assumed to be due to the time dependency of the load. The solution is carried out by Galerkin’s method and by using modal functions of the completely free plate. Numerical results are illustrated in figures for stepwise change of the loading.