Tensionless Foundation Research Articles

Responses of Buried Pipelines to Tunnelling Underneath Considering Effect of Gap Formation: An Analytical Approach

A gap formation underneath pipelines could be observed during tunneling beneath existing buried pipelines, thus altering the pipeline-soil interaction behavior. However, the most existing analytical approaches typically do a full contact assumption for brevity. For such purpose, we aim to present an analytical approach coupling the effect of gap formation underneath pipelines by considering the buried pipeline as an Euler-Bernoulli beam on tensionless foundation. The pipeline can be divided into three segments according to pipeline-soil interaction behavior, and then the bending behaviors of each segment are formulated. The deflection and bending moment of the pipeline caused by tunnelling underneath from the proposed approach are compared and validated with centrifuge testing data and existing analytical solutions. The effects of pipeline-soil stiffness ratio, pipeline buried depth, tunnel buried depth, tunneling induced volume loss, and tunnel-pipeline intersection angle on the pipeline responses are further discussed. The results indicate that gap formation underneath pipeline can be observed indeed for shallowly buried pipeline and/or larger pipeline-soil stiffness ratio. The proposed analytical approach provides a theoretical guideline to predict the responses of existing pipelines to tunnelling underneath.

Numerical approach based on particle swarm technique for optimization of nonlinear buried pipeline on tensionless foundation

This work aims to contribute to the development of numerical approach for optimization of buried pipeline by using finite element (FE) method and particle swarm optimization (PSO), considering static analysis and material hardening. The buried pipeline is discretized by finite element approach, constituted by Euler-Bernoulli three nodes beam element, while the soil-structure interaction is modeled by Winker-type tensionless foundation. Moreover, the tensionless foundation is discretized by spring element in the numerical model. The material nonlinearity behavior is analyzed by Von Mises isotropic hardening model. In the numerical approach, this model is solved by stress increment procedure. Furthermore, the critical loading and optimum wall thickness, that leads to optimized condition, are determined using swarm techniques. The cost function is formulated using evaluated stress and target stress, more, the calculated error is minimized by using swarm techniques for single-variable and two-variable optimization. Several applications are carried out by considering several soil conditions, boundary conditions and loading types. As a novelty, the soil random stiffness is included in the numerical model, where the optimum wall thickness and critical loading are evaluated considering material nonlinearity. The performance of finite element particle swarm technique is also compared to finite element genetic algorithms technique. The results demonstrate the effectiveness of proposed numerical approach for nonlinear buried pipeline optimization.

Steady state responses of an infinite beam resting on a tensionless visco-elastic foundation under a harmonic moving load

In this paper we study the steady state responses of an infinite beam resting on a tensionless visco-elastic foundation under a harmonic point force moving with a constant speed. The equation of motion is formulated with respect to a coordinate system moving with the point load. The beam and the foundation are defined as in contact when both the deflection and the “contact force” are negative. We replace the infinite beam by a finite beam with sufficient length and expand the beam deflection into a harmonic series. Newmark method is used to integrate the discretized equation. For a tensionless foundation, the contact condition between the beam and the foundation changes with time. Therefore, the matrices involved in the discretized equation of motion must be updated at every time step. In the bifurcation diagram we change the excitation frequency and record the beam deflection at the loading point via Poincaré sampling. When the dimensionless amplitude of the harmonic force is high, say 15, one can observe periodic, quasi-periodic, and chaotic solutions. We have also found that reducing the amplitude of the harmonic component or adding a time-invariant component can simplify the bifurcation diagram to include only periodic solutions.

Free–Free Beam Resting on Tensionless Elastic Foundation Subjected to Patch Load

Despite the popularity of a completely free beam resting on a tensionless foundation in the construction industry, the existing bending analysis solutions are limited to certain types of loads (mostly point and uniformly distributed loads); these are also quite complex for practicing engineers to handle. To overcome the associated complexity, a simple iterative procedure is developed in this study, which uses the Ritz method for the bending analysis of a free–free beam on a tensionless foundation subjected to a patched load. The Ritz method formulation is first presented with polynomials being used to approximate the beam deflection with unknown constants to be determined through minimization of the potential energy. To account for the tensionless action, the subgrade reaction is set to zero when the deflection is negative. The non-zero subgrade reaction zone is defined by αlL/2<x<αrL/2 where the coefficients αl and αr are to be determined iteratively. A numerical example is presented to illustrate the applicability of the proposed procedure for symmetrical and asymmetrical problems. The obtained results show high negative deflection, which proves the occurrence of separation between the beam and the supporting tensionless foundation. This location of negative deflection is called the lifted zone, while the point that separates between the negative and positive deflection is called the lift-off point. A parametric study is then performed to study the effect of the amount of load, stiffness of the beam, and the subgrade reaction on the length of the lifted zone. The results of the parametric study indicate that for the same beam stiffness to subgrade reaction modulus ratio (EI/k), the lift-off point remains the same and beams with lower stiffnesses or higher loads deflect more.

Analytical model for segmental tunnel lining with nonlinear joints

An analytical model was established through modeling the segmental lining by a curved beam, its interaction with the surrounding soils by a tensionless Winkler foundation, and the joints between segments by nonlinear rotational springs. The state space method was then combined with the Newton–Steffensen iterative procedure to formulate and solve the equations in which a tensionless foundation and nonlinear joints were engaged. Numerical examples were presented to illustrate the performance of the proposed new method through a comparison with existing methods from the literature. A parametric study was presented to reveal the influences of soil resistance coefficient and cross-sectional properties on the mechanical behavior of the lining. It has been shown that the new method is convenient and accurate to consider the interaction of lining with the surrounding soils and nonlinear behavior of joints, and thus provides an alternative approach to the design and analysis of this type of lining.

Effect of Adjacent Axle Loads on Uplift of Rails on Geocell-Stone Column Improved Tensionless Foundation

Rails resting over tensionless or unilateral foundation (responds to compressive forces only) has an uplift tendency ahead and behind the applied load. The presence of train of loads on such foundations may result in greater upward deflection of the rail. The current study investigates this possibility for rails on geocell-stone column improved tensionless foundations. Lumped parameter modelling approach has been used to idealize the rail on reinforced foundation. Granular layer has been placed between two infinite beams, one representing the rail and other the geocell with infill soil, and modelled as Pasternak shear layer. Stone columns and soft foundation soil beneath the beams have been idealized by Winkler springs of different stiffnesses. Mathematical model in the form of governing differential equations has been developed with due consideration to damping in the system. Finite difference method in combination with Gauss–Seidel iterative scheme has been used to arrive at their solution based on employed loading and the boundary conditions. Effect of adjacent axle loads and inclusion of stone columns on the response of the rail has been investigated. Influence of variations in input parameters of soil-foundation system has been studied in detail and presented in the form of non-dimensional charts along with a sensitivity study.

Analytical model of buried beams on a tensionless foundation subjected to differential settlement

As the deflection of a buried beam subjected to ground settlement is not consistent with the ground displacement, an analytical model is introduced in this study for a buried beam on a tensionless foundation subjected to differential settlement. The buried beam is divided into three segments: the left semi-infinite foundation beam, the right semi-infinite foundation beam, and the middle finite beam separated from the ground. Based on the theory of semi-infinite foundation beams, equations for the response of left and right semi-infinite segment beams are given. Explicit equations are proposed for the response of the middle segment beam; these are combined with the continuous conditions at the segment junctions, and the physical implications of the equation parameters are illustrated. The analytical approach taken in this study is then compared with, and verified against, the methods used in the existing literature. The mechanical state of a buried beam subjected to ground settlement is closely related to the foundation stiffness factor, the flexural stiffness of the beam, the characteristics of the ground settlement, and the vertical earth pressure. When the deformation coefficient is relatively large or ground settlement is relatively narrow, the buried beam may be in the partial contacting state. With an increase in the width and amplitude of ground settlement curve, the foundation stiffness factor, and the different vertical earth pressure between the ground settlement and non-settlement areas, the bending moment and shearing force of buried beams increase.

Modeling of Rail Tracks on Stone Column Reinforced Tensionless Foundations

This research investigates the response of rails on geocell-stone column composite reinforced foundation beds under a moving load. Improved earth bed has been considered to respond only to compressive forces. The granular mat below the rail has been idealized as a Pasternak shear layer and geocell reinforcement as an infinite beam with finite bending stiffness. Soft soil and stone columns have been symbolized by Winkler springs of different stiffnesses. Analysis has been carried out with due consideration to viscous damping in the system. The governing differential equations have been established and simplified for general use with the help of dimensionless parameters. These equations have been solved in presence of appropriate boundary conditions by utilizing Finite difference method in combination with iterative Gauss-Seidel procedure. Inclusion of stone columns has been observed to significantly affect the onset of separation between rail and the soil layer underneath. Various parameters namely, applied load and its velocity, stiffnesses of top, bottom soil layers and stone columns, damping ratio, relative flexural rigidity, depth of placement of geocell, configuration of stone columns have been found to affect the response of soil-foundation system significantly. Improvement in the properties of soil by means of higher value of relative compressibility resulted in typical reduction of 50% in maximum deflection. It has been observed that the region of detachment reduces on increasing the depth of placement of the bottom beam. Sensitivity analysis highlighted the greater sensitivity of upward deflection as compared to the downward deflection of rail with respect to all the parameters except for relative compressibility of the soil and relative stiffness of the stone columns.

Finite element approach of the buried pipeline on tensionless foundation under random ground excitation

This work presents an investigation in the numerical performance of finite element approach in the dynamic elastoplastic analysis of buried pipeline, which is subjected to random ground motion. The surrounding soil is modeled by Winkler and Pasternak type foundation, while the random ground motion is generated synthetically by generalized nonstationary Kanai–Tajimi model. The governing equation is formulated by Euler–Bernoulli beam theory and discretized by beam-pipe element. Moreover, the von Mises isotropic hardening model is employed for material behavior modeling. The Hilber–Hughes–Taylor (HHT) method and the Newton–Raphson method are adopted as the time incremental iterative algorithm to solve the global equilibrium equation. Several applications are carried out to investigate the numerical performance of the present numerical model in dealing with the dynamic elastoplastic analysis of buried pipe. The norm L2 of stress and displacement error are determined for different time intervals and the factors that contribute to the error reduction are investigated in present work.

Double Beam Model for Reinforced Tensionless Foundations under Moving Loads

An attempt is made herein to analyse the rails resting on earth beds with geocell inclusion under a moving load. Reinforced soil system is assumed to react to compressive forces only, i.e., the tensionless foundation. It comprises of a granular fill layer (modelled as Pasternak shear layer) followed by a geocell inclusion (represented by an infinite beam with finite bending stiffness) overlying the foundation soil (modelled as a series of spring and dashpot connected in parallel). The governing differential equations are developed and converted for general use in non-dimensional form. These equations are solved using suitable boundary conditions and employing Finite Difference Scheme along with iterative Gauss-Seidel technique. Critical velocity of tensionless system is determined and compared with perfect contact case. The parametric study conducted shows the influence of applied load, relative compressibility of soils, relative flexural rigidity, depth of placement of geocell, damping in the system and velocity of load on the response of rail and geocell reinforcement. Also, a sensitivity study is carried out which suggests that uplift of the rail is quite sensitive to more number of parameters than to its downward deflection.

Buckling analysis of steel jacking pipes embedded in elastic tensionless foundation based on spline finite strip method

Buckling of steel jacking pipes subjected to axial compression and surrounded by soil is investigated. The jacking pipes are simplified as simply-supported circular cylinders embedded in tensionless Winkler-Pasternak foundation that reacts to soil compression only. The governing equation for tensionless buckling analysis is established based on the spline finite strip method (SFSM), and the critical buckling load is obtained by solving the governing equation through an iterative procedure. Numerical examples are analyzed, and the comparison of buckling loads with available solutions and finite element results shows good accuracy of the SFSM. Based on a parametric study, the variation of tensionless buckling loads with slenderness ratio of jacking pipes is first studied. Moreover, the effects of diameter-thickness ratio and foundation stiffness are also investigated. The present SFSM-based tensionless buckling analysis is applicable to cylinders of any length and on tensionless foundation, and it is capable of efficiently and effectively predicting buckling of steel jacking pipes subjected to axial compression and surrounded by soil.

Response of an infinite beam on a bilinear elastic foundation: Bridging the gap between the Winkler and tensionless foundation models

The response of an infinite beam on an elastic foundation depends on the property/modeling of the foundation. There is a qualitative difference between the responses of a beam when it is on the Winkler foundation and on the tensionless foundation. A bilinear elastic foundation model, which describes the different behaviors of the elastic foundation in the tensile and compressive zones with two different foundation moduli, is proposed and a straightforward computational method is also formulated. The Winkler and tensionless foundations are shown to be the two special cases of the bilinear elastic foundation model. With this bilinear elastic foundation model, a more general method of modeling an elastic foundation is provided, which can be of some help to the modeling of the support of the ballastless high-speed railway.

Unilateral contact buckling behaviour of orthotropic plates subjected to combined in-plane shear and bending

This paper addresses the buckling behaviour of a long thin orthotropic plate on a tensionless rigid foundation under combined in-plane shear and bending, where the sheet can only buckle away from the foundation due to the tensionless foundation support. Two boundary conditions at the longitudinal edges (y = 0 and b) are investigated, i.e. clamped and simply supported. The energy method is used to derive the explicit expression for the lateral buckling mode function. The buckling response is then obtained by solving the governing differential equation after substituting the lateral buckling mode function. Based on the analytical solutions developed in the paper, fitted formulae are given for buckling coefficients and the interaction curve. Finally, a numerical example of a corrugated sheet is given to verify the presented model through comparing the results with finite element (FE) analysis.

Contact buckling behaviour of corrugated plates subjected to linearly varying in-plane loads

An analytical method is developed for analysing the contact buckling response of infinitely long, thin corrugated plates and flat plates restrained by a Winkler tensionless foundation and subjected to linearly varying in-plane loadings, where the corrugated plates are modelled as orthotropic plates and the flat plates are modelled as isotropic plates. The critical step in the presented method is the explicit expression for the lateral buckling mode function, which is derived through using the energy method. Simply supported and clamped edges conditions on the unloaded edges are considered in this study. The acquired lateral deflection function is applied to the governing buckling equations to eliminate the lateral variable. Considering the boundary conditions and continuity conditions at the border line between the contact and non-contact zones, the buckling coefficients and the corresponding buckling modes are found. The analytical solution to the buckling coefficients is also expressed through a fitted approximate formula in terms of foundation stiffness, which is verified through previous studies and finite element (FE) method.

Efficient analysis of plates on nonlinear foundations

This paper presents efficient analysis of plates on nonlinear foundations. The Reissner plate theory is used to model plates. Foundations are presented as the Winkler springs or the elastic half space. The developed analysis is mainly presented for tensionless foundation; however as demonstrated, it is straightforward extended to analysis of elastic-plastic foundations. The plate is analyzed using the boundary element method (BEM). Unlike the traditional BEM which uses equations in form ([H] {u} = [G] {t}), the presented formulation uses finite element like equations, in the form of ([K] {u} = {P}). An innovative formulation is presented to derive the relevant plate stiffness matrix [K] and load vector {P} from the BEM integral equation. Iterative procedures together with condensation process are used to eliminate degree of freedom at failed zones. Results of the present analysis are more accurate than those obtained from previously published results. The main advantages of the presented technique are its simplicity and accuracy and it gains both advantages of the boundary element and the finite element methods.

A contact element for dynamic analysis of beams to a moving oscillator on tensionless elastic foundation

This paper presents the contact elements for dynamic analysis of Euler–Bernoulli beams to a moving oscillator on tensionless elastic foundation considering discontinuous contact. The elastic foundation is modeled by linear elastic springs accounting for discontinuous contact between the beam and foundation. Three types of contact element including a full-bonded element, full-unbonded element and half-bonded element in finite element method expressing the relation displacement between the beam and foundation are suggested in this paper. The contact force of the beam and foundation during vibration as external force of the beam is established by the elastic foundation’s reaction distributed per length of the element. The governing equations of motion are derived by means of finite element method and solved by Newmark’s time integration procedure in the time domain. The effects of the discontinuous contact phenomenon due to tensionless foundation on dynamic responses of the beam are discussed. The comparisons between present solution and ordinary solution with continuous contact show that the dynamic responses of the beam are quite different and increasing than those of the ordinary solution.

Mechanical Characterization of Beams on Tensionless Foundation Materials

The deformation and internal forces of beams on tensionless foundation materials were studied. The reaction force between the beam the foundation was fitted as a cubic polynomial about the deflection based on the experimental data, and the corresponding control equations of beams were derived by the finite difference method. Results show there are significant differences between tensionless and tensional foundation materials for the deformation and internal forces of beams. The difference is varying with the length of beams. Both the relative errors of the maximum of deflection and slope can be over 20%, and the relative errors of the maximum of shearing force and bending moment are smaller comparatively, so the tensionless effect of foundation materials can not be neglected for the stiffness verification and the strength verification of beams.

Nonlinear FEA of higher order beam resting on a tensionless foundation with friction

A novel higher order shear-deformable beam model, which provides linear variation of transversal normal strain and quadratic variation of shearing strain, is proposed to describe the beam resting on foundation. Then, the traditional two-parameter Pasternak foundation model is modified to capture the effects of the axial deformation of beam. The Masing.s friction law is incorporated to deal with nonlinear interaction between the foundation and the beam bottom, and the nonlinear properties of the beam material are also considered. To solve the mathematical problem, a displacement-based finite element is formulated, and the reliability of the proposed model is verified. Finally, numerical examples are presented to study the effects of the interfacial friction between the beam and foundation, and the mechanical behavior due to the tensionless characteristics of the foundation is also examined. Numerical results indicate that the effects of tensionless characteristics of foundation and the interfacial friction have significant influences on the mechanical behavior of the beam-foundation system.

A Review of Skin Buckling Theory in Composite Members

This paper provides a comprehensive review of various methods used for skin buckling analysis in composite components. The skin buckling phenomenon is one of the governing criteria in composite design. It is a kind of contact buckling in which partial sections of skin buckle away from the filler material. In general, the problem can be modelled as a thin plate (skin) in unilateral contact with elastic medium (filler material). The theoretical analysis of contact buckling is complicated due to the nonlinearity arising from changing contact regions. To simplify the calculations, the filler material was usually modelled as a tensionless elastic foundation. The skin buckling coefficient varies in terms of the relative foundation stiffness factors. Because the Eigen-value method is not applicable to nonlinear systems, the finite element (FE) method was usually employed for post-buckling analysis, while initial buckling performance was investigated through analytical or semi-analytical methods such as rigid foundation model, infinite plate model and finite plate model. The compressive buckling and shear buckling problems for thin plates resting on tensionless foundations have been solved successfully. However, there are still urgent needs for future research on the topic. For example, the load carrying capacity of the buckling plates needs to be formulated for practical application. Complicated problems with complex loadings and/or corrugated skins need further investigation as well.

Response of Cylindrical Storage Tank Foundation Resting on Tensionless Stone Column-Improved Soil

AbstractIn the present paper, the response of cylindrical storage tank foundation resting on stone column-reinforced ground has been studied considering the separation between the bottom raft and ground under liftoff condition. The bottom raft of the tank is idealized by a circular plate. The stone column-reinforced ground is modeled as a tensionless foundation. Soft foundation soil and granular fill are idealized as a spring-dashpot element and Pasternak shear layer, respectively. Stone columns are modeled as stiff nonlinear springs. The stone columns are converted into equivalent concentric rings with one stone column at the center to simulate the axisymmetric condition for the analysis. The governing differential equations are derived from the classical theory of thin plate and solved using the finite-difference technique. A parametric study is performed to investigate the differences between the response of tensionless and conventional elastic foundation systems. It is observed that tensionless founda...