Rigid Foundation Research Articles

Analytical solution of dynamic responses of offshore wind turbine supported by monopile under combined earthquake, wave and wind

Offshore wind turbine system (OWTs) in areas of earthquake activity are potentially threatened by a combination of earthquake, wave, and wind loads. Thus, an analytical solution of OWTs supported by monopile on viscoelastic half-space is developed to investigate the dynamic responses of OWTs subjected to the earthquake, wave, and wind loads. Firstly, the OWTs supported by monopile considering a series of coupled interactions is established, in which monopile end soil-soil viscoelastic half-space interaction is solved by static spring-damping system, monopile-soil and water interaction are solved by analytical horizontal impedance, monopile-wave interaction is solved by linear wave theory, and tower drum-wind interaction is solved by wind loads applied on the turbine blades. Then, the displacement response of OWTs supported by monopile is further obtained based on the continuity conditions between the monopile and tower drum. Finally, the present solution degenerated into a rigid foundation and verified the rationality with existing analytical and substructure methods. In addition, some critical parameters are further conducted by using the present solution. The results indicate that it is necessary to consider the couple influence of earthquake, wave, and wind loads in the design of OWTs supported by monopile.

Effect of eccentricity on ultimate lateral unit earth pressure of tetrapod piles in clay

This study investigates the ultimate lateral unit earth pressure of rigid tetrapod foundation imposed by eccentric loads, based on a couple of two-dimensional adaptive finite-element limit analyses with lower-bound, upper-bound and 15-node mixed Gauss element formulations. A parameter study is conducted to emphatically explore the influence of eccentricity as well as to assess the role of additional parameters such as soil–pile interface strength and pile spacing on the normalised lateral earth pressure factor Np. Results from the numerical analyses reflect that the presence of loading eccentricity may lead to significant reduction in lateral soil resistance, the maximum diminution is up to 45% for the case that eccentricity equals to 0·5 times spacing. Three representative failure mechanisms are identified associated with various eccentricities. Pile spacing and interface strength additionally affect the variation of failure mechanisms with eccentricity increases. Finally, the eccentricity reduction factor, defined as the ratio of the Np considering the effect of loading eccentricity to the Np omitting this effect, is proposed, and the variation of this factor with eccentricity is presented for different values of pile spacing and soil–pile adhesion factor.

Specified models in the problems of the deformation of multilayer plates on a rigid foundations

The analysis and estimation of the stress-strain state (SSS) of multilayered plates on a rigid foundation under the action of stationary transverse loading is an urgent task. As its includes the calculations of strength and deformability of various homogeneous and multilayer coatings. This is the calculation of pavement on relatively rigid bridge structures, or on a non-deformable underlying layer, protective multilayer coatings of flat elements of structures of greater rigidity than roofing, parts and more. Combining materials with isotropic and transversely isotropic physical characteristics into a multilayer package allows you to create multifunctional structures. The SSS of such structures, due to their structural heterogeneity and relatively low transverse stiffness of the individual layers, is significantly associated with the influence of transverse shear deformations and transverse compression deformations. Therefore, the problem of refined modeling of SSS plates, which would take into account these types of deformations, is urgent. Based on the decomposition of the SSS plate into flexural and unflexuralcomponents, it is proposed to optimize the design scheme of deformation of a rectangular multilayer plate on a rigid foundation. The essence of optimization is to consider such a calculation scheme of the plate, in which the SSS of the plate would be fully described by only one component, namely the unflexural component of SSS. To do this, instead of the actual design of the multilayer plate, which is deformed without separation from the foundation, it is proposed to consider the design scheme of the plate, which is formed by symmetrical completion relative to the contact surface of the plate with the foundation. In this case, the plate will be bilaterally symmetrically loaded relative to the middle surface of the plate, and the thickness of the plate will double. SSS plate will be unflexural, which greatly simplifies its modeling. For unflexural SSS, a two-dimensional, high-degree iterative approximation, but three-dimensional models of deformation of multilayer rectangular plate on a rigid foundation with isotropic and transverse-isotropic layers are constructed in an elastic formulation.That models takes full account deformationof transverseshearand transverse compression at transverse loading of a plate. By solving the test problems of deformation of isotropic and transversely isotropic plates on a rigid foundation with sliding and rigid contact with the foundation, and comparing the solutions with the exact three-dimensional solutions of these problems obtained by known methods, the accuracy of the proposed refined models is estimated.The limits of admissible parameters of elastic characteristics of transversely isotropic plates for application of the offered models are established.

Theoretical modelling of the effect of thermal delamination of an asphalt concrete pavement from a rigid foundation of a road or bridge

In the practice of road construction, one of the most common phenomena accompanied by delamination, subsequent cracking and destruction of the asphalt concrete pavement on a rigid (cement concrete or metal) base of a road or bridge is the effect of concentration of shear thermal stresses between the pavement and the base in the edge zones of the structure. They are caused by the fact that, as a rule, the coefficients of linear thermal expansion of the phases of the system have different values, which contributes to the occurrence of incompatible shrinkages and expansions in them. Under conditions of frequent temperature changes in heterogeneous asphalt concrete structures with thermomechanical incompatibility of their components, these effects can contribute to their accelerated aging. At the same time, with the thermomechanical compatibility of materials, a more favorable distribution of internal stresses of thermal and mechanical origin is achieved, which excludes premature degradation of the strength of the contacting phases and the entire system as a whole. Using the methods of strengthof materials and the finite element method, it has been established that under the conditions of a change in the temperature of the system during its seasonal and daily fluctuations, shear stresses are subjected to the highest concentration. They are localized in the edge zone of the plane of contact between the layers and increase with an increase in the thickness and modulus of elasticity of the upper layer. These stresses are the main reason for the occurrence of plastic deformations in these zones and subsequent delamination of the structure in them. It is proposed to reduce the concentration and level of generated high-gradient shear stresses by reducing the thickness of the asphalt concrete layer in these areas.

Dynamic interactions, impedance functions and elastic wave propagation due to horizontal and vertical vibrations of a rigid foundation in a transversely isotropic elastic ground

The analytical and numerical treatments of horizontal and vertical forced time-harmonic vibrations of a rigid circular foundation in a transversely isotropic elastic half-space ground are presented. A complete discussion on the boundary conditions and elastic wave propagation due to horizontal and vertical vibrations of the foundation on the half-space is presented and a comparison is made with the full-space problem. With the aid of appropriate dynamic Green’s functions reported in the literature, the horizontal and vertical vibrations of the foundation are treated analytically, and the results are expressed in terms of the solutions of several Fredholm integral equations. For these two vibration modes, the relations for the contact normal and shear stresses, the resultant forces acting on the foundation, and the dimensionless impedance and compliance functions are given. To validate the analytical formulations, the results obtained in this paper are reduced to static responses for transversely isotropic half-space, and to dynamic responses for isotropic half-space; and compared with the results reported in the literature. An efficient numerical procedure is presented to evaluate the semi-infinite integrals appeared in the analytical solutions utilizing some features of the Mathematica software. The Fredholm integral equations and impedance and compliance functions are solved and evaluated numerically using the Gaussian quadrature method. Some graphical results are provided to present the impedance and compliance functions for different transversely isotropic materials. The effects of material anisotropy, as well as the frequency of excitation on the responses are discussed. The differences between vibrations of rigid foundations in the half-space and full-space mediums are emphasized in view of the Rayleigh wave propagation. Moreover, the difficulties related to numerical procedures needed for handling the strong singularities due to propagation of Rayleigh waves in the half-space media are presented.

Analytical solution of dynamic response of horizontal vibrating pile with different soil viscoelastic half-space types

An analytical solution has been developed to investigate the dynamic responses of floating pile with different soil viscoelastic half-space types. The dynamic response of a pile is governed by 1D vibration theory, in which the pile is assumed to be viscoelastic material with a circular cross-section and can be divided into two pile segments in a soil-air system. Meanwhile, the dynamic solution for soil is achieved by the application of the potential function decomposition and the variable separation method. Moreover, an analytical solution considering the viscoelastic half-space of the soil free-field under horizontal SV wave is established. Furthermore, the dynamic pile-soil interaction is established and solved efficiently through a precise theoretical derivation based on the displacement compatibility and traction equilibrium at the interface between the pile and soil.The present solution is compared with the substructure method and analytical solution with a rigid foundation to verify the rationality of the present analytical solution. Then, the similarities and differences among several foundation types are discussed based on the present analytical solution. The results have a certain guiding significance for engineering structures.

Verification of bounding surface plasticity model with reversal surfaces for the analysis of liquefaction problems

This paper presents the verification of a new bounding surface plasticity model with reversal surfaces when used in boundary value problems related to liquefaction. The model is implemented in a commercial finite-difference code using an explicit stress integration algorithm. After validating the model against element tests on Nevada sand, the paper employs the same set of values of model parameters for ascertaining the model's simulative potential at the system level, via comparisons with data from 8 dynamic centrifuge tests performed on the same sand. These pertain to: a) the free-field response of a horizontal liquefiable layer, b) the lateral spreading response of a gently sloping liquefiable layer and c) the seismic response of rigid foundations on a single-layered and two-layered horizontal liquefiable ground profiles. Parametric analyses emphasize the effects of non-constant sand permeability coefficient and excitation intensity on the liquefied system response. Similar analyses highlight the importance of post-liquefaction strain accumulation and the avoidance of stress-strain overshooting, both constitutive attributes of the new model.

On a frictional unilateral contact problem in nonlinear elasticity-existence and smoothing procedure.

This note is devoted to a novel frictional unilateral contact problem in finite-strain elasticity. Here, we adopt the Tresca friction model from linear elasticity. Our analysis relies on the polyconvexity approach to nonlinear elasticity due to J. Ball. We include the delicate case where the elastic body neither is fixed nor has a deformation prescribed along some part of its boundary, but rests on a rigid foundation with a free boundary and is only submitted to forces and loads acting in the interior (like gravity) and at the boundary, respectively. This leads to a loss of coercivity and necessitates an extra condition that prevents the body from escaping by the geometry of the obstacle. This new condition extends a similar condition of Ciarlet and Nečas from the frictionless case to the case of Tresca friction. In addition, as a first step towards a numerical treatment of such nonlinear problems, we present a smoothing procedure that tackles the non-smooth term from Tresca friction and provide a convergence result for the novel smoothing method. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Large-scale shake table testing of pile group-bridge model in inclined liquefiable soils with overlying crusts

Large scale shaking table tests were performed to investigate the failure mechanism of pile group-bridge system subjected to liquefaction-induced lateral spreading. The tests comprised two cases: Case L which refers to a bridge model supported by a 2 × 2 pile group embedded into the inclined liquefiable soils with crusts; and case R which refers to the same bridge model fixed to a rigid foundation. The experimental details are provided and typical test results are elucidated. The failure mechanism of the pile group-bridge system under strong earthquakes is revealed. The results indicate that the lateral peak soil displacement profile has a linear distribution, while the lateral permanent displacement profile has a cosine distribution pattern. It was also found that the pile is prone to damage and bending failure both at its head and at the bottom of saturated sand where it is fixed into bearing stratum. For the pile group-bridge system, the liquefaction-induced lateral spreading causes the failure position to move from the pier bottom to the pile head. In addition, the pier bottom is more likely to experience damage under strong earthquakes in case R than in case L, while the opposite is true during weak earthquake excitation.

Experimental Study on Influence of Soil–Structure Interaction on Seismic Performance of UHV Transformer

ABSTRACT Ultra-high voltage (UHV) transformer has a high seismic vulnerability. The transformers are usually directly connected to the shaking table to conduct an experiment to study its seismic performance, without considering the effect of the foundation soil on the upper equipment. In the present study, the transformer was scaled down based on the dimensional theory. Then, a soil box that can effectively eliminate the boundary effect is designed and a method that can simulate the site category is proposed. Finally, shaking table tests were conducted on rigid foundations and type I site for the scaled-down model, the seismic response of equipment under different site types is obtained. The results imply that under the rigid foundation, the transformer body response is larger with the seismic wave input in the X+Z direction compared to the seismic wave input in the X-direction; as the seismic acceleration increase, the acceleration and strain of bushings gradually ascend, and the response of bushing with Y direction earthquake wave input is significant than that with X direction earthquake wave input. Under type I site condition, with the earthquake excitation increase, the acceleration amplification effect for transformer body caused by the soil–structure interaction decreases gradually; the response of high voltage bushing under earthquake waves is more significant compared with medium and low voltage bushing; with the earthquake excitation increases, the acceleration of high voltage bushing is 1.13 to 1.40 times that of rigid foundation connection, and the strain response is 1.01 to 1.26 times that of rigid foundation connection. The research findings can be applied in the seismic field of UHV transformers to improve the level of seismic design.

Earthquake response analysis of nuclear facilities subjected to incoherent seismic waves based on the random-vibration-theory methodology

In this study, based on the random-vibration-theory (RVT) methodology, an earthquake response analysis of a nuclear facility with a rigid foundation on rock/soil sites is considered when incoherent seismic waves are incident on the system. Using the coherency function of seismic waves, the power spectral density (PSD) matrix for six-degree-of-freedom motions of the rigid foundation is calculated. The stochastic earthquake responses of the soil–structure interaction system are obtained, and the PSD function for the pseudo-acceleration of equipment on one floor of the structure is derived. Peak factors of random vibrations are employed to estimate the spectral acceleration or in-structure response spectrum (ISRS) of the equipment. The accuracy of the proposed RVT methodology is verified for nuclear power plant structures on hard rock, soft rock, and layered soil acted upon by coherent and incoherent seismic waves. Because the ISRS outcomes are found to decline at high frequencies, reduction factors due to incoherency are obtained from the proposed RVT methodology. These factors can be applied to earthquake responses to coherent ground motions to consider the effects of incoherent seismic waves.

Calculation and Analysis of Nonlinear Algorithm for Stability of Nanosilica Powder Soft Soil Pile Foundation.

In order to reasonably evaluate the stability of the embankment supported by the rigid pile composite foundation, a nonlinear algorithm calculation and analysis method for the stability of the nanosilica powder soft soil pile foundation is proposed. First, the soft soil foundation reinforced by the composite structure of “piles (prestressed pipe piles, (cement fly-ash gravel, CFG) piles)—pile caps—geotextile pads” is analyzed through the soft soil embankment test section of the Wenfu high-speed railway. Second, the compressive modulus under one-dimensional confining compression is calculated using the layered summation method. Finally, the deep lateral deformation of the pile foundation soil is measured by an inclinometer to evaluate the actual displacement of the pile foundation soil. In the standard method, the empirical coefficient 1.1–1.7 is multiplied on the basis of the calculation result of this method to correct the calculation error. Combined with the test results, it is shown that nanosilica powder can give full play to its excellent characteristics: promoting hydration speed and hydration degree through the reaction of pozzolan refining and consuming Ca(OH)2 crystals produced by cement hydration. Soft soils and modified soft soils: a certain amount of nanosilica powder can significantly improve the strength of soft soil at different ages.

Solution of the static contact problem with Coulomb friction between an elastic body and a rigid foundation

We consider a static contact problem between an elastic body and an absolutely rigid foundation with Signorini contact conditions and Coulomb friction law. The problem can be formulated as a quasi-variational inequality in which the normal stress is proportional with the friction coefficient to the friction force in the contact zone. In this case, the normal stress itself depends on the desired solution, and the existence of a solution to the problem reduces to the existence of a fixed point of a certain mapping. Consideration of Coulomb friction naturally leads to the non-differentiability of the objective functional in the auxiliary problem of the method of successive approximations and, thereby, to a significant complication of the algorithms for solving the constrained minimization problem. We propose a method of solving an auxiliary problem with given friction, which is based on the Uzawa algorithm and modified Lagrange functionals. The main advantage of the proposed method is that it allows to effectively solve auxiliary problems and to prove the theoretical convergence of the method of successive approximations to a fixed point. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.

Optimization and performance of metafoundations for seismic isolation of small modular reactors

AbstractThis paper aims to study the seismic mitigation of a typical nuclear small modular reactor (SMR) where extreme loading conditions are considered by the safe shutdown earthquake. For this purpose, to reproduce the main dynamic properties of the reactor's reinforced concrete system, a detailed structural model was synthetized, also taking into account the presence of the reactor pools. Thus, to protect the reactor from strong earthquakes, finite locally resonant multiple degrees of freedom metafoundations were developed; and resonator parameters were optimized by means of an improved frequency domain multivariate and multiobjective optimization procedure. Also, the stochastic nature of the seismic input was taken into account. It is proposed: (i) a linear metafoundation endowed with multiple cells, linear springs, and linear viscous dampers; and (ii) a foundation equipped with additional nonlinear vertical quasi‐zero stiffness (QZS) cells. QZS cells were obtained by horizontally precompressed springs in an unstable state with vertical springs in parallel. With this arrangement, additional flexibility and dissipation against nonsymmetrical modes of the SMR and vertical seismic loadings are proposed. It was shown in both cases, how each metafoundation was successfully optimized via a sensitivity‐based parameter grouping strategy and a hybrid grid searching algorithm. Thus, the performance of the optimized metafoundations was assessed by means of frequency and time history analyses; and finally, results were compared with an SMR endowed with both rigid foundation and conventional base‐isolation solutions.

An analytical solution of a horizontally vibrating pile considering water-pile-soil interaction on viscoelastic half-space

The water-pile-soil model on viscoelastic half-space is developed to investigate the dynamic responses of an end-bearing pile subjected to the horizontal dynamic loads. The pile is assumed to be viscoelastic material with a circular cross-section and can be divided into several pile segments in a water-pile-soil system, in which the soil and water are modeled as linear viscoelastic media and linear acoustic media, respectively. The pile-soil with viscoelastic half-space and pile-water with rigid ground vibration are solved by the potential function decomposition and the variable separation method, respectively. The displacement response of the pile is then obtained based on the continuity conditions between the pile and soil and water. The predictions of the present solution are compared with numerical method. Moreover, the present solution can be degenerated into a rigid foundation and verified the rationality with existing analytical solutions and numerical methods. Parametric analyses are further conducted by using the present solution to evaluate the relationships between water and pile-soil parameters, especially focus on the influence of the soil viscoelastic half-space shear modulus. The results indicate that it is necessary to consider the influence of the viscoelastic half-space at the pile end soil when the pile elastic modulus is relatively small in the design of offshore cast-in-place piles.

An analytical model for 2D seismic tunnel-aboveground structure interaction with considering the influence of tunnel longitudinal shear deformation

The influence of tunnel longitudinal shear deformation on the response of its adjacent ground building is investigated based on a 2D analytical model considering a flexible circular tunnel and a shear wall standing on a semi-circular rigid foundation under SH-waves excitation. Soil-tunnel relative stiffness is introduced, and a closed-form analytical solution is presented for the system response. Tunnel stiffness significantly affects the foundation impedance function of the building. Rayleigh scattering frequency can accurately predict the position of the fluctuation peaks of the foundation impedance function. At locations where the tunnel–building interaction is strong, the tunnel cannot be considered as rigid even if the tunnel relative stiffness to the soil reaches 100. At locations where constructive and destructive wave interference occurs, the building relative response increases or decreases accordingly. Considering the influence of tunnel relative stiffness, the amplitude of the building relative response can be amplified by 27.0% or reduced by approximately 28.0%, respectively. This analytical model is also used to qualitatively predict the amplitude of the relative response transfer function of the Hollywood Storage Building under the influence of the tunnel longitudinal shear deformation, and its accuracy is verified.

Equivalent dynamic stiffnesses and 3D wave propagations of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation

An advanced theoretical formulation along with a robust numerical evaluation method are employed to study the 3 D wave propagations and equivalent dynamic stiffnesses of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation. Using appropriate dynamic Green’s functions presented in the literature, the rocking and torsional harmonic oscillations of the foundation are formulated analytically in terms of several Fredholm integral equations of the second kind. Then the dynamic stiffnesses and flexibilities are presented analytically for rocking and torsional oscillations. To validate the analytical results, they are reduced to two simpler cases reported in the literature: the static problem and the case of isotropic materials. The semi-infinite integrals appeared in the solutions are evaluated numerically by utilizing the contour integration and residue theorem along with several commands of the Mathematica software. A complete discussion on the elastic wave propagations due to oscillations of the foundation are presented and a detailed comparison is performed between different waves propagate within the half-space and full-space elastic grounds including the P, SV, SH and the Rayleigh waves. Our results showed that due to rocking oscillation of the foundation, the Rayleigh wave does not propagate in the full-space medium but it propagates in the half-space medium. However due to torsional oscillation, the Rayleigh wave does not propagate neither in the full-space nor in the half-space elastic medium. In addition, due to rocking oscillation, the body waves including P, SV and SH propagate within both half-space and full-space elastic mediums; but due to torsional oscillation, only SH-wave propagates within both full-space and half-space elastic mediums. In some tables, numerical values and mathematical formulas for the dimensionless wave numbers and wave velocities are provided for the P, SV, SH and the Rayleigh waves. The Gaussian quadrature method is implemented for definite integrals appeared in the dynamic stiffnesses. Some graphical figures are provided to represent the dimensionless rocking and torsional dynamic stiffnesses and flexibilities versus the dimensionless frequency for different transversely isotropic materials. The effects of anisotropy of materials and the frequency of excitations on the responses are discussed. The results presented in this paper including the dynamic stiffnesses and wave propagations have direct applications in dynamic soil-structure interaction analyses in civil engineering.

2D dynamic tunnel-soil-aboveground building interaction Ⅰ: Analytical solution for incident plane SH-waves based on rigid tunnel and foundation model

The structure-soil-structure interaction (SSSI) problem between the tunnel and its adjacent aboveground building is investigated by using a 2D model with a rigid circular tunnel and a shear wall supported by a semi-circular rigid foundation embedded in a homogenous half-space. The analytical solution is presented for the model system response under the SH-wave excitation based on the wave function expansion method. The analysis focuses on the influence of the SSSI features between the tunnel and its adjacent aboveground building on the system response. The resonance of the ground structure will have a significant effect on the tunnel response, which is manifested by the sudden change of the tunnel's response transfer function due to structural resonance. The SSSI effect will amplify or reduce the structural relative response at some specific frequencies and tunnel-building distances. The structural relative response amplitude can be amplified by about 40% compared to that of the single structure. The group effect that exists in the interaction of multiple ground structures does not necessarily apply to the tunnel-building interaction. The results in this paper may help to assess the impact of SSSI effects on the seismic response of the tunnel and its aboveground structure, as well as to interpret the observed seismic responses.

3D Finite Element Analysis of Seismic Response of Rigid Pile Composite Foundation with Microprobe Group.

In order to study the seismic performance of the rigid pile composite foundation, a three-dimensional finite element analysis of the seismic response of the rigid pile composite foundation with microprobe group was proposed. The three-dimensional finite element method is used to analyze the dynamic response law of pile group rigid pile composite foundation under the action of the earthquake, especially the internal force response of the pile body. It is worth noting that when the internal force response of pile decreases significantly, the maximum internal force of the whole pile is also significantly smaller than the maximum internal force of pile foundation. However, when the pile body is located on a layered foundation and the modulus of adjacent soil layers differs greatly, the earthquake action will cause the pile body to generate a large internal force. The seismic performance of the rigid pile composite foundation is feasible, and it can be extended and applied to the similar seismic performance test of other composite foundations.

Response of rigid circular plate in a functionally graded transversely isotropic, half-space under horizontal steady-state excitation

In this paper, a theoretical formulation for lateral translation and impedance function of a rigid foundation in a transversely isotropic functionally graded (TIFG) half-space is presented. Accordingly, the medium is considered linear elastic and the elastic constants of materials vary exponentially along the depth for considering the effect of elastic non-homogeneities. The relax treatment of mixed boundary value problem is formulated by integrating the transformed kernels of Green’s functions over the foundation area with unknown surface traction function, which can be reduced into a Fredholm integral equation. The results represent the impact and treatment of frequency and material anisotropy through the medium. In addition, static and dynamic distribution of tangential shear stress over the foundation for surface and embedded disk extracted and the cutoff frequency is found out explicitly. The proposed solution gives exact concurrence with accessible literature.