Winkler Springs Research Articles

Elastic-brittle-plastic analysis of the radial subgrade modulus for a circular cavity based on the generalized nonlinear unified strength criterion

When using the bedded-beam model based on Winkler springs for cavity lining design, it is always difficult for engineers to select an appropriate value for the radial subgrade modulus (k) due to the lack of related theoretical research. The solutions of k for a circular tunnel proposed by Wood (1975) and Zhang et al. (2014) have been found to be applicable only when a tunnel is in elastic ground. However, a rock mass will exhibit elastic, brittle and plastic states under different stress conditions. Considering that the plastic state can effectively improve the bearing capacity and reduce the engineering cost, to investigate the radial subgrade modulus in an elastic-brittle-plastic rock mass, modified analytical solutions are derived that are compatible with a generalized nonlinear unified strength criterion. Using the maximum shear stress modulus, these solutions consider the comprehensive influences of the intermediate principal stress, the dilation characteristic and brittle plastic. Then, a comparative verification of the new solutions using a nonlinear Hoek-Brown yield criterion is conducted. The effects of the parameters are discussed in detail. The calculation results show that k is a comprehensive evaluation index for multiple factors, is significantly influenced by the geo-stress and support pressure, and increases with the geological strength index, elastic strain and Young’s modulus in the plastic zone. During cavity contraction, the intermediate principal stress improves the magnitude of k more significantly than dilation, whereas dilation has a greater effect during cavity expansion. Using the perfectly plastic model without considering brittle-plastic behaviour will lead to an overestimation of k, and using different strength criteria directly affects the timing of the plastic deformation of a rock mass. The influence of various factors should be comprehensively considered to select a reasonable radial subgrade modulus to take full advantage of the latent potentialities of a rock mass. The deep-buried diversion tunnel at the Jinping Ⅱ hydropower station is studied as an example to demonstrate the practical application of the proposed model.

Spatial distribution of Winkler spring stiffness for rectangular mat foundation analysis

Structural design of mat foundations of buildings is often done by performing static analysis of a slab resting on vertical uncoupled Winkler springs. It is already well established that the simplifying assumption of a uniform modulus of subgrade reaction throughout the mat foundation leads to inaccurate results that significantly underestimate the bending moments in the mat. This paper examines the spatial variation of the Winkler spring stiffness constants that is necessary for the mat-on-springs analysis to produce the same slab deflections and bending moment diagrams as finite element analysis that treats the soil as continuum. For this purpose, three-dimensional parametric analyses of slabs resting on elastic soil are performed using the finite element method for various values of soil elastic properties, slab geometrical characteristics and column load configurations. The finite element analysis results were used for back-calculating analytically the equivalent Winkler spring constants at each node of the mat. Based on the numerical results, equations describing the spatial distribution of spring stiffness are proposed. The performance of the proposed equations is compared against existing spring stiffness spatial distribution approaches used in practice.

ANALYSIS OF COMBINED FOOTINGS ON EXTENSIBLE GEOSYNTHETIC-STONE COLUMN IMPROVED GROUND

Analysis of combined footings resting on an extensible geosynthetic reinforced granular bed on stone column improved ground has been carried out in the present work. Various components of soil-foundation system have been idealized using lumped parameter modeling approach as: combined footing as finite length beam, granular layer as nonlinear Pasternak shear layer, geosynthetic reinforcement as elastic extensible membrane, stone columns as nonlinear Winkler springs and foundation soil as nonlinear Kelvin body. Hyperbolic constitutive relationships have been adopted to represent the nonlinear behavior of various elements of a soil-foundation system. Finite difference method has been employed to solve developed governing differential equations with the help of appropriate boundary and continuity conditions. A detailed parametric study has been conducted to study the effect of model parameters like applied load, flexural rigidity of footing, configuration of stone columns, ultimate bearing resistance of foundation soil and stone columns, tensile stiffness of geosynthetics and degree of consolidation on response of soil-foundation system by means of deflection and bending moment in the footing and mobilized tension in geosynthetic layer. These parameters have been found to have significant influence on the response of footing and the geosynthetic reinforcement layer. To quantify this, results have been nondimensionalized to produce design charts for ready use for the analysis of combined footings resting on such a soilfoundation system.

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 half-analytical elastic solution for 2D analysis of cracked pavements

This paper presents a half-analytical elastic solution convenient for parametric studies of 2D cracked pavements. The pavement structure is reduced to three elastic and homogeneous equivalent layers resting on a soil. In a similar way than the Pasternak's modelling for concrete pavements, the soil is modelled by one layer, named shear layer, connected to Winkler's springs in order to ensure the transfer of shear stresses between the pavement structure and the springs. The whole four-layer system is modelled using a specific model developed for the analysis of delamination in composite materials. It reduces the problem by one dimension and gives access to regular interface stresses between layers at the edge of vertical cracks allowing the initial debonding analysis. In 2D plane strain conditions, a system of twelve-second order differential equations is written analytically. This system is solved numerically by the finite difference method (Newmark) computed in the free Scilab software.The calculus tool allows analysis of the impact of material characteristics changing, loads and locations of cracks in pavements on the distribution of mechanical fields. The approach with fracture mechanic concepts is well suited for practical use and for some subsequent numerical developments in 3D.

Stability and free vibration analyses of double-bonded micro composite sandwich cylindrical shells conveying fluid flow

• Development of stability and vibration of sandwich small-scale axisymmetric cylindrical shell model. • Presenting formulation based on the modified couple stress double-bonded Reddy shells theory. • Solving formulation using generalized differential quadrature method (GDQM). • Study of the effects of h f / h c , u f , L / h and H z on the damping and natural frequency. In this study, based on Reddy cylindrical double-shell theory, the free vibration and stability analyses of double-bonded micro composite sandwich cylindrical shells reinforced by carbon nanotubes conveying fluid flow under magneto-thermo-mechanical loadings using modified couple stress theory are investigated. It is assumed that the cylindrical shells with foam core rested in an orthotropic elastic medium and the face sheets are made of composites with temperature-dependent material properties. Also, the Lorentz functions are applied to simulation of magnetic field in the thickness direction of each face sheets. Then, the governing equations of motions are obtained using Hamilton's principle. Moreover, the generalized differential quadrature method is used to discretize the equations of motions and solve them. There are a good agreement between the obtained results from this method and the previous studies. Numerical results are presented to predict the effects of size-dependent length scale parameter, third order shear deformation theory, magnetic intensity, length-to-radius and thickness ratios, Knudsen number, orthotropic foundation, temperature changes and carbon nanotubes volume fraction on the natural frequencies and critical flow velocity of cylindrical shells. Also, it is demonstrated that the magnetic intensity, temperature changes and carbon nanotubes volume fraction have important effects on the behavior of micro composite sandwich cylindrical shells. So that, increasing the magnetic intensity, volume fraction and Winkler spring constant lead to increase the dimensionless natural frequency and stability of micro shells, while this parameter reduce by increasing the temperature changes. It is noted that sandwich structures conveying fluid flow are used as sensors and actuators in smart devices and aerospace industries. Moreover, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In fact, the present study can be provided a valuable background for more research and further experimental investigation.

Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory

In this paper, free vibration characteristics of functionally graded (FG) nanobeams embedded on elastic medium are investigated based on third order shear deformation (Reddy) beam theory by presenting a Navier type solution for the first time. The material properties of FG nanobeam are assumed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on third order shear deformation beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The obtained results are presented for the vibration analysis of the FG nanobeams such as the influences of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

Effect of different parameters on the behavior of strip footing resting on weak soil improved by granular piles

This paper presents a method to analyze strip footing resting on granular layer over weak soil improved by end bearing or floating granular piles. The granular layer beneath the strip footing idealized as incompressible shear layer. The weak soil idealized as soft Winkler springs and the granular piles idealized as stiff Winkler springs. These springs connected at their ends by a thin membrane under uniform tension to overcome the drawbacks of Winkler model related to the shear effect or the continuity of the granular piles-weak soil composite. Hyperbolic relationships used to represent the nonlinear behavior of weak soil and granular piles. Granular piles of different diameters, lengths, stiffness and arrangements can be modeled. The numerical solution obtained by the finite element method. The validity of the present analysis verified through comparisons with the field measurements, other existing analysis method and PLAXIS finite element program. A parametric study for a strip footing subjected to seven concentrated column loads carried out to study the effect of various parameters on the behavior of the soil-strip footing system. These parameters include number of granular piles, modular ratio, stiffness of granular layer, diameter of granular piles, length of granular piles and arrangements of granular piles. It is found that stiffness of granular layer has little effect compared to a significant effect of other parameters on the vertical and differential displacements and the induced bending moment of the strip footing.

Dynamic analysis of beam-soil interaction systems with material and geometrical nonlinearities

In this paper a Hybrid Domain Boundary Element Method is developed for the geometrically nonlinear dynamic analysis of inelastic Euler-Bernoulli beams of arbitrary doubly symmetric simply or multiply connected constant cross-section, resting on viscous inelastic Winkler foundation. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse dynamic loading and bending moments in both directions as well as to axial loading, while its edges are subjected to the most general boundary conditions. A displacement based formulation is developed and inelastic redistribution is modelled through a distributed plasticity (fibre) approach. A uniaxial hysteretic law is considered for the evolution of the plastic part of the normal stress following the phenomenological hysteresis model, while hysteretic force-displacement model is also employed in order to describe the inelastic behaviour of the Winkler springs. Numerical integration over the beam cross sections is performed in order to resolve the hysteric parts of the stress resultants. Application of the boundary element technique yields a system of nonlinear Differential-Algebraic Equations, which are written in state-space form and solved by an incremental–iterative solution strategy. Numerical examples are worked out confirming the accuracy and the computational efficiency of the proposed beam formulation, as well as the significant influence of material and geometrical nonlinearities in the response of beam-soil interaction systems.

Parametric study of turbo generator foundation considering different types of beam column joint

In the current work, turbo generator foundation system is simulated in SAP: 2000 V17.1 software and dynamic response of foundation are analyzed. Parametric study of Strong Column Weak Beam (with and without haunches) and Weak Column Strong Beam has been carried out considering Winkler spring soil model, solid finite element modelling and dynamic analysis of Turbo machinery foundations. The comparison of frequency and time periods at various mode shapes has been performed. The effects of damping (0%, 3% and 5%) on the response spectra curve of the foundation top deck are investigated considering the dynamic machine load.It has been found that Strong Column Weak Beam remains more elastic during seismic action compared to Weak Column Strong Beam. Hence Strong Column Weak Beam is recommended while designing a turbo machine foundation as it can provide strength and stability to the entire structure. The presence of haunches did not show any influence on vertical stiffness, however, it showed the increase in lateral Stiffness of the frame thus enhancing the stability of the foundation.

Vibration of a Cylindrical Tunnel under a Centric Point-Source Explosion

Underground tunnels are vulnerable to terrorists’ bombing attacks, which calls for studies on tunnel’s response to internal explosive loading. In this paper, the dynamic response of a cylindrical tunnel to an ideal centric point explosion was treated as an axisymmetric 2-dimensional problem, in which the tunnel was modeled with a continuous anisotropic shell, while the ground medium’s effect was accounted for with linear elastic Winkler springs and the explosive loading described by a temporal and spatial function. The governing equation of the motion is a fourth-order partial differential equation, for which a numerical method combining finite difference with the implicit Newmark-β method was adopted. This method avoided complicated integral transform and numerical inverse transformation, thus allowing efficient parameter study. The maximum radial displacement was found on the cricle of the center of explosive, where hoop stress is the maximum principal stress. The anisotropy showed little influence on maximum hoop stress. Within the range of ground medium’s modulus, minor influence on maximum hoop stress was incurred. This research may be helpful to hazard assessment and protective design for some critical subway tunnels.

Nonlinear Deformation Analysis of Geocell Reinforcement in Pavements

Geocell reinforcement has found a large applicability in the area of pavement engineering, and its deformation characteristics play an important role in the analysis and design of pavements. In the present work, a simple analytical approach that idealizes geocell reinforcement as a beam subjected to vertical symmetrical loads and foundation soil as nonlinear Winkler springs are proposed. A detailed parametric study indicated the influence of the weight of infill material, applied loads, the modulus of subgrade reaction, flexural rigidity of reinforcement, interface resistance, and the ultimate bearing resistance of foundation soil on deformation characteristics of reinforcement. Charts that are useful in the analysis and design of pavements reinforced with geocell are presented.

Horizontal Stiffness and Damping of Piles in Inhomogeneous Soil

AbstractA practically oriented analytical procedure for determining the dynamic stiffness and damping (impedance coefficients) of a laterally loaded pile in soil exhibiting different types of inhomogeneity with depth, is presented. To this end, an energy method based on the Winkler model of soil reaction in conjunction with pertinent shape functions for the deflected shape of the pile are employed. A new elastodynamic model for the wave field around a pile is also introduced. The method is self-standing and free of empirical formulas or constants. Dimensionless closed-form solutions are derived for (1) the distributed (Winkler) springs and dashpots along the pile; (2) dynamic stiffness and damping coefficients at the pile head; (3) active length, beyond which the pile can be treated as infinitely long; and (4) relative contributions to the overall head stiffness and damping of the soil and the pile media. Swaying, rocking, and cross swaying-rocking impedances are considered for parabolic, exponential, and...

Analytical solution for nonlocal buckling characteristics of higher-order inhomogeneous nanosize beams embedded in elastic medium

In this paper, buckling characteristics of nonhomogeneous functionally graded (FG) nanobeams embedded on elastic foundations are investigated based on third order shear deformation (Reddy) without using shear correction factors. Third-order shear deformation beam theory accounts for shear deformation effects by a parabolic variation of all displacements through the thickness, and verifies the stress-free boundary conditions on the top and bottom surfaces of the FG nanobeam. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam in deformation, which acts in tension as well as in compression. The material properties of FG nanobeam are supposed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. Nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. Comparison between results of the present work and those available in literature shows the accuracy of this method. The obtained results are presented for the buckling analysis of the FG nanobeams such as the effects of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

Analytical Solution to Analyze LTP on Column-Improved Soft Soil Considering Soil Nonlinearity

AbstractIn this paper, a mechanical model to idealize the load-settlement response of the load transfer platform (LTP) on column-improved soft soil is proposed. This model simultaneously considers the nonlinear and time-dependent stress-strain behavior of soft soil and the negligible tensile strength of the granular material in LTP. The reinforced Timoshenko beam is adopted to model LTP to consider the shear and flexural deformations. Soft soil is idealized by a spring-dashpot system that includes nonlinear and time-dependent behaviors. The columns and geosynthetics are modeled with linear Winkler springs in the applied range of stresses and rough elastic membrane, respectively. The response function of LTP has been derived for distributed pressure loading in the plane strain condition. The principle of superposition is used to solve the fourth-order differential equations. Parametric studies indicate that the spacing of columns, thickness of LTP, degree of consolidation of the soft soil, and tensile stif...

Surface stress effects on the free vibration and bending analysis of the nonlocal single-layer graphene sheet embedded in an elastic medium using energy method

In this article, using energy method, the effect of surface stress on the free vibration and bending analysis of single-layer graphene sheet embedded in an elastic medium based on nonlocal elasticity theory is studied. Surface stress plays a more important role due to the high surface-to-volume ratio in nanoscale materials. For this purpose, Gurtin–Murdoch continuum mechanics approach is used. The effects of surface properties including surface elasticity, surface residual stresses and surface mass density are considered. In this research, the effects of Winkler spring constant, Pasternak shear constant, aspect ratio, Young’s modulus of surface layer, surface residual stress, nonlocal parameter on the natural frequency ratio and deflection of single-layer graphene sheet are investigated. The results indicate that the natural frequency ratio decreases with an increase in the surface residual stress and vice versa for deflection of nonlocal single-layer graphene sheet. Also, the natural frequency ratio increases with an increase in Young’s modulus of surface layer and vice versa for deflection of nonlocal single-layer graphene sheet. Moreover, the elastic medium causes to stiffen the single-layer graphene sheet. The influence of the small-scale parameter on the natural frequency ratio and deflection of single-layer graphene sheet is more significant for lower values of foundation parameters and vice versa for higher values of foundation parameters.

Equation to predict maximum pipe stress incorporating internal and external loadings on buried pipes

Pipelines used for water and other services are very important lifelines in modern society. Commonly, these buried pipes are subjected to significant stresses due to external (traffic and earth) and internal (water pressure) loads. As many of these pipelines were laid sometime in the last century or earlier, in most cases their condition has deteriorated primarily by electrochemical and (or) microbiological corrosion. Corrosion activity (internal and external) can manifest in various forms, but in many cases will lead to reduced pipe thickness, which in turn leads to an increase in pipe stresses induced by the external and internal loads. Currently available analytical procedures to estimate pipe stresses are based on oversimplifications such as the two-dimensional (2-D) analysis based on Winkler springs, limiting their application to general pipe burial conditions. This paper describes the application of a three-dimensional (3-D) finite element method to analyse a buried pipe subjected to external and internal loads. Firstly, the finite element model is validated against the data from field tests conducted on the basis of a cast iron pipe that was laid in 1930 at Strathfield, Sydney, Australia. The results of these 3-D finite element analyses are then used to develop a closed-form expression to predict maximum stresses in pipes of different sizes buried in different soil types. Having obtained a good agreement between the proposed model outcomes and the 3-D finite element analysis results, the proposed model has been validated against the field test data under different internal and external loadings. The verified outcomes of the model reveal that it can be used to predict maximum stresses without conducting a full-scale finite element analysis, which often requires specific computational resources and computational skills. Furthermore, the proposed model can be used in probabilistic analyses, where a large number of calculations need to be carried out to account for the uncertainty of the input variables. The applications of the model are also discussed in relation to the assessment of pipe performance and remaining safe life.

Integrity Assessment of Offshore Subsea Wells: Evaluation of Wellhead Finite Element Model Against Monitoring Data Using Different Soil Models

Understanding well conductor–soil interaction mechanism plays an important role in well integrity assessment. Evaluation of field data obtained from monitored offshore wells can provide valuable insights into the matter. This paper presents a comparison of the numerical results obtained from a full three-dimensional (3D) finite element (FE) analyses model of the blowout preventer (BOP), wellhead (WH), conductor, and surface casing versus the field measured data obtained during drilling operations. Sensitivity studies were also performed for several parameters that were considered important in the local well response analysis under observed sea state conditions. The conductor–soil analyses were simulated using the Winkler spring p–y curves obtained by two different approaches: American Petroleum Institute (API) RP2GEO and a recently developed model by Zakeri et al. The results of bending moments in conductor and surface casing have indicated good agreement between FE model results and the field measurements.

Winkler load-transfer analysis for laterally loaded piles

The Winkler modulus for a vertical beam buried in elastic soil is reassessed for the problem of a horizontal loaded pile in the framework of linear elasticity. By matching the integral solution in elastic continuum and the expression with the elastic Winkler modulus, the subgrade modulus of an infinitely long pile embedded into elastic space is obtained first. Then the influence of embedment depth and pile rigidity on the subgrade modulus is evaluated by virtue of Mindlin’s and Kelvin’s solutions, which gives the variation of the Winkler spring stiffness along the pile length. Comparison of the results by the present method for single piles in homogeneous and nonhomogeneous soils has shown good agreement with those obtained from the more rigorous elastic continuum solutions and boundary element method, which also revealed the disadvantage of the conventional Winkler expression in evaluating the displacement of the pile. Finally, the present method is used to analyze the pile–soil–pile interaction of a pile group, which also shows good agreement with the finite element method and the elastic continuum method, and proves the feasibility of extending the active subgrade modulus into the realm of the horizontal loaded pile group.

Nonlinear magneto-electro-mechanical vibration analysis of double-bonded sandwich Timoshenko microbeams based on MSGT using GDQM

In the present study, the nonlinear magneto-electro-mechanical free vibration behavior of rectangular double-bonded sandwich microbeams based on the modified strain gradient theory (MSGT) is investigated. It is noted that the top and bottom sandwich microbeams are considered with boron nitride nanotube reinforced composite face sheets (BNNTRC-SB) with electrical properties and carbon nanotube reinforced composite face sheets (CNTRC-SB) with magnetic fields, respectively, and also the homogenous core is used for both sandwich beams. The connections of every sandwich beam with its surrounding medium and also between them have been carried out by considering Pasternak foundations. To take size effect into account, the MSGT is introduced into the classical Timoshenko beam theory (CT) to develop a size-dependent beam model containing three additional material length scale parameters. For the CNTRC and BNNTRC face sheets of sandwich microbeams, uniform distribution (UD) and functionally graded (FG) distribution patterns of CNTs or BNNTs in four cases FG-X, FG-O, FG-A, and FG-V are employed. It is assumed that the material properties of face sheets for both sandwich beams are varied in the thickness direction and estimated through the extended rule of mixture. On the basis of the Hamilton's principle, the size-dependent nonlinear governing differential equations of motion and associated boundary conditions are derived and then discretized by using generalized differential quadrature method (GDQM). A detailed parametric study is presented to indicate the influences of electric and magnetic fields, slenderness ratio, thickness ratio of both sandwich microbeams, thickness ratio of every sandwich microbeam, dimensionless three material length scale parameters, Winkler spring modulus and various distribution types of face sheets on the first two natural frequencies of double-bonded sandwich microbeams. Furthermore, a comparison between the various beam models on the basis of the CT, modified couple stress theory (MCST), and MSGT is performed. It is illustrated that the thickness ratio of sandwich microbeams plays an important role in the vibrational behavior of the double-bonded sandwich microstructures. Meanwhile, it is concluded that by increasing H/lm, the values of first two natural frequencies tend to decrease for all amounts of the Winkler spring modulus.