Tensionless Foundation Research Articles

Buckling Simulation of a Plate Embedded in a Unilaterally Supported Environment#

Buckling of a von Kármán plate unilaterally connected with an elastic, Winkler-type, tensionless foundation on both sides is considered in this article. The plate is simply supported and subjected to compressive and tensile loads along its edges. A variational principle is formulated for the mechanical model. The variational problem is numerically solved by means of the spectral method and computational algorithm. The algorithm is based on Newton's scheme and a numerical continuation procedure which considers the contact conditions. The bifurcation phenomenon has been analyzed for different values of compressive and tensile loading parameters. The proposed algorithm is demonstrated with the help of numerical results. Finally, an application of the unilateral plate-bending model to layered tissues in biomechanics is considered.

Response of a finite beam on a tensionless Pasternak foundation under symmetric and asymmetric loading

The static response of a finite beam resting on a tensionless Pasternak foundation and subjected to a concentrated vertical load is assessed in this study. The concentrated vertical load may be applied at the center of the beam, or it may be offset from the center. The tensionless character of the foundation results in the creation of lift-off regions between the beam and the foundation. An analytical/ numerical solution is obtained from the governing equations of the contact and lift-off regions to determine the extent of the contact region. Although there is no nonlinear term in the equations, the problem shows a nonlinear character since the contact region is not known in advance. Due to that nonlinearity, the essentials of the problem (the coordinates of the lift-off points) are calculated numerically using the Newton-Raphson technique. The numerical results are presented in figures to illustrate the behaviours of the free-free and pinned-pinned beams under symmetric or asymmetric loading. The figures illustrate the effects of the shear foundation parameter and the symmetric and asymmetric loading options on the variation of the contact lengths and the displacement of the beam.

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.

Analysis of beams on tensionless reinforced granular fill‐soil system

AbstractIn this paper, a beam subjected to end concentrated loads has been modeled and analyzed to estimate its flexural response. The beam has been assumed to rest on reinforced earth beds with reinforcing elements having some finite bending stiffness. The reinforcing elements have been idealized as beams with smooth surface characteristics. The foundation system has been assumed to react only in compression (tensionless foundation), i.e. the separation between the upper beam and the ground surface has been taken into consideration. Winkler springs of different stiffnesses have been used to idealize the upper dense and lower poor soils. As the analysis considers the separation between the upper beam and the soil, the weight of the upper beam has been taken into account. The governing differential equations have been derived and presented in a non‐dimensional form. These equations have been solved using finite difference method with the help of appropriate boundary and continuity conditions. The response of the foundation system has been compared with the case when the beam is in perfect contact with the ground surface. The parametric study shows that the response of the model is greatly affected by the length ratio of beams, ratio of stiffness of upper and lower soil layers, ratio of flexural rigidity of upper and lower beams and weight of the upper foundation beam. Copyright © 2008 John Wiley & Sons, Ltd.

Symmetrically loaded beam on a two-parameter tensionless foundation

Static response of an elastic beam on a two-parameter tensionless foundation is investigated by assuming that the beam is symmetrically subjected to a uniformly distributed load and concentrated edge loads. Governing equations of the problem are obtained and solved by pointing out that a concentrated edge foundation reaction in addition to a continuous foundation reaction along the beam axis in the case of complete contact and a discontinuity in the foundation reactions in the case of partial contact come into being as a direct result of the two-parameter foundation model. The numerical solution of the complete contact problem is straightforward. However, it is shown that the problem displays a highly non-linear character when the beam lifts off from the foundation. Numerical treatment of the governing equations is accomplished by adopting an iterative process to establish the contact length. Results are presented in figures to demonstrate the linear and non-linear behavior of the beam-foundation system for various values of the parameters of the problem comparatively.

On the use of the Lagrange Multiplier Technique for the unilateral local buckling of point-restrained plates, with application to side-plated concrete beams in structural retrofit

Reinforced concrete beams can be strengthened in a structural retrofit process by attaching steel plates to their sides by bolting. Whilst bolting produces a confident degree of shear connection under conditions of either static or seismic overload, the plates are susceptible to local buckling. The aim of this paper is to investigate the local buckling of unilaterally-restrained plates with point supports in a generic fashion, but with particular emphasis on the provision of the restraints by bolts, and on the geometric configuration of these bolts on the buckling loads. A numerical procedure, which is based on the Rayleigh-Ritz method in conjunction with the technique of Lagrange multipliers, is developed to study the unilateral local buckling of rectangular plates bolted to the concrete with various arrangements of the pattern of bolting. A sufficient number of separable polynomials are used to define the flexural buckling displacements, while the restraint condition is modelled as a tensionless foundation using a penalty function approach to this form of mathematical contact problem. The additional constraint provided by the bolts is also modelled using Lagrange multipliers, providing an efficacious method of numerical analysis. Local buckling coefficients are determined for a range of bolting configurations, and these are compared with those developed elsewhere with simplifying assumptions. The interaction of the actions in bolted plates during buckling is also considered.

Axisymmetric forced vibrations of an elastic free circular plate on a tensionless two parameter foundation

The static and dynamic responses of a circular elastic plate on a two-parameter tensionless foundation are investigated by assuming that the external load is rotationally symmetric and the plate experiences axially symmetric displacements. In the solution procedure, the vertical displacement of the foundation is determined by the corresponding governing equation, whereas the vertical displacement of the plate is expressed in series in terms of the mode shapes of the completely free circular plate. For the case of complete contact, the corresponding governing equation of the plate incorporated with the edge reaction from the foundation is satisfied through the Galerkin's approximation technique. The contact radius is obtained by requiring that the surface of the foundation satisfies the corresponding continuity equations. It is shown that the problem displays a highly nonlinear character due to the lift-off of the plate from the foundation and the numerical treatment of the governing equation is accomplished by adopting iterative processes in terms of the contact radius. The governing equation of the problem is solved numerically for the static and dynamic cases and the results are presented in figures to demonstrate the nonlinear behavior of the plate for various values of the parameters of the problem comparatively.

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.

Response of a rectangular plate-column system on a tensionless Winkler foundation subjected to static and dynamic loads

The response of a plate-column system having five-degree-of-freedom supported by an elastic foundation and subjected to static lateral load, harmonic ground motion and earthquake motion is studied. Two Winkler foundation models are assumed: a conventional model which supports compression and tension and a tensionless model which supports compression only. The governing equations of the problem are obtained, solved numerically and the results are presented in figures to demonstrate the behavior of the system for various values of the system parameters comparatively for the conventional and the tensionless Winkler foundation models.

Steady State Response of Beams on a Tensionless Geosynthetic-Reinforced Granular Fill-Soft Soil System Subjected to Moving Loads

ABSTRACT In this paper, geosynthetic reinforced granular fill-soft soil system subjected to a moving load with constant velocity over an infinitely long beam is analyzed. The foundation is assumed as tensionless foundation (to react only in compression). The upper reinforced granular bed is modeled by a rough elastic membrane embedded in Pasternak shear layer overlying a series of compressible Winkler springs representing the underlying soft soil. The parametric studies reveal that the flexural response of the infinite beam as well as of the reinforcement, are greatly affected by the velocity, intensity of load and compressibility of granular fill. The interfacial frictional coefficients, shear modulus of granular layers and coefficient of lateral earth pressure do not significantly affect the response of the beam and the reinforcement for the range of parameters studied.

Response of beams on a tensionless extensible geosynthetic-reinforced earth bed subjected to moving loads

In this paper analysis of an infinite beam resting on reinforced granular bed overlying a soft soil strata subjected to a moving load with constant velocity is presented. The separation between the beam and ground surface is taken into the account while analyzing the system. The upper reinforced granular bed is modeled by a rough membrane embedded in Pasternak shear layer overlying a series of compressible Winkler springs representing the underlying soft soil. The extensible geosynthetic is assumed to deform such that at interface the geosynthetic and the soil have same deformation. This condition eliminates the two interfacial shear stress parameters and at the same time also incorporates the stiffness of geosynthetic in the model. The lift up of the beam and mobilized tension in the geosynthetic layer are observed to be more for the tensionless foundation as compared to that reacting in tension and compression. It has been further observed that the flexural response of the infinite beam as well as of the reinforcement, are greatly affected by the velocity, intensity of load, relative compressibility of granular fill and stiffness of the geosynthetic. For the range of parameters studied the shear moduli of granular layers do not significantly affect the response of the beam.

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 analysis of unilaterally constrained laminated thin plates

Postbuckling responses of laminated thin plates supported by a tensionless foundation and subjected to in-plane compressive edge loads are investigated. The postbuckling analysis presented here is based on the classical laminated plate theory, and includes the plate-foundation interaction, for which the foundation reacts in compression only. The plate is assumed to be simply supported at four edges, while the two longitudinal unloaded edges may be either movable or immovable. 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 any prior assumption for the shape of the contact region. The numerical examples concern the postbuckling behavior of antisymmetric angle-ply and symmetric cross-ply laminated plates with unilateral constraints, from which results for the same unconstrained plates are obtained as comparators. The numerical results show that the unilateral constraint has a significant effect on the postbuckling response of laminated thin plates.

Static and dynamic responses of a rigid circular plate on a tensionless Winkler foundation

Static and dynamic responses of a rigid circular plate on a tensionless Winkler foundation

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...

Buckling tests on steel plates restrained at discrete points in the retrofit of reinforced concrete beams

Steel plates may be bolted to the sides of reinforced concrete beams in order to both strengthen and stiffen them. When these steel plates are subjected to their design actions, bending, shear and axial stresses may be produced in the plate and this can precipitate local buckling. The local buckles are unilateral, since the concrete beam constrains the steel to buckle locally away from the concrete core. This type of behaviour is referred to as a contact problem, and research into its structurally based applications has only accelerated over the last decade or so. This paper presents the description and results of a so-called local buckling push test undertaken on bolted plates of various configurations to calibrate the theoretical solutions developed elsewhere by the authors. This analytical method, based on the Rayleigh–Ritz method with polynomial displacements and a tensionless foundation, is found to be in good agreement with the test results, and furthermore these test results provide benchmark data for future studies.

Buckling tests on steel plates restrained at discrete points in the retrofit of reinforced concrete beams

Steel plates may be bolted to the sides of reinforced concrete beams in order to both strengthen and stiffen them. When these steel plates are subjected to their design actions, bending, shear and axial stresses may be produced in the plate and this can precipitate local buckling. The local buckles are unilateral, since the concrete beam constrains the steel to buckle locally away from the concrete core. This type of behaviour is referred to as a contact problem, and research into its structurally based applications has only accelerated over the last decade or so. This paper presents the description and results of a so-called local buckling push test undertaken on bolted plates of various configurations to calibrate the theoretical solutions developed elsewhere by the authors. This analytical method, based on the Rayleigh–Ritz method with polynomial displacements and a tensionless foundation, is found to be in good agreement with the test results, and furthermore these test results provide benchmark data for future studies.

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.

Local Buckling of Side-Plated Reinforced-Concrete Beams. I: Theoretical Study

Steel plates can be bolted to the sides of reinforced-concrete beams to both strengthen and stiffen them. When these side-plated beams are subject to their intended design actions, compression, bending, and shear stresses are produced in the steel plates and this can lead to instability by local buckling of the plates. This type of buckling problem is known as a contact problem, since the plate is constrained to buckle in only one lateral direction. The paper presents a Rayleigh-Ritz method of the local buckling analysis of rectangular unilaterally restrained plates in compression, bending, and shear. Polynomials are used to define the displacement function, while the restraining medium is modeled as a tensionless foundation. The method is shown to be very efficient computationally, and elastic local buckling coefficients are presented for a variety of restraint cases for various plate aspect ratios, in terms of interaction diagrams. The buckling model is shown to be in good agreement with experimental results from the full-scale testing of two side-plated reinforced-concrete beams, as presented in a companion paper.

Elastic buckling of unilaterally constrained rectangular plates in pure shear

For a certain class of systems, the buckling response may be unilateral rather than bilateral. Unilateral buckling is a contact problem whereby buckling is confined to take place in only one direction. For plate structures, this can occur when a thin steel plate is juxtaposed with a rigid concrete medium, and the steel may only buckle locally away from the concrete core. Examples of this include composite profiled beams, walls and concrete-filled steel tubes, as well as reinforced concrete beams that are strengthened and stiffened by gluing and/or bolting steel plates to their sides. This paper presents a Rayleigh–Ritz method of the local buckling analysis of rectangular unilaterally restrained plates in pure shear. The displacement functions are modelled as polynomials, and the restraining medium as a tensionless foundation. The method presented is shown to be very efficient computationally, and elastic local buckling coefficients are presented for a variety of restraint cases for various plate aspect ratios. The use of these coefficients in determining limiting width to thickness ratios is demonstrated.