Torsional Impedance Research Articles

Dynamic torsional impedance of large-diameter pipe pile for offshore engineering: 3D analytical solution

Wind-induced or current-induced torsional vibration is a long-neglected issue for offshore engineering (e.g., offshore wind turbines, offshore oil platforms, etc.). As the dimension of the pipe pile implemented offshore increases dramatically, the one-dimensional rod theory can no longer precisely compute the dynamic torsional impedance. This paper proposes a novel 3D torsional soil-pile interaction model and derives the corresponding analytical solution for the 3D dynamic torsional impedance at the pile cross-section with the adoption of the Laplace transform, variable separation, and impedance transfer method. The solution is then validated by comparisons against some widely-accepted existing solutions. The main conclusions of this study can be drawn as follows: (1). Classic one-dimensional theory overestimates the impedance of the pipe pile, which could result in severe disasters if the dynamic design is radical; (2). Even for small diameter pipe piles, the impedance would vary in the radial direction at the pile cross-section. Usually, the impedance at the outer radius would be slightly larger than that at the inner radius; (3). For large-diameter pipe piles or pipe piles with high-strength soil plugs, the most significant impedance could appear at the inner radius, indicating the resistance provided by the soil plugs can not be overlooked for large-diameter pipe piles.

Variational approach for torsional dynamic response of a single pile in multi-layered soils

ABSTRACT This paper investigates the torsional dynamic response of a single pile with a circular cross-section embedded in multi-layered soils. The energy principles and variational approach are employed to establish the governing equations of the pile-soil system subjected to dynamic torque applied at the pile head in the frequency domain. The proposed solution is based on the finite element method for a bar on an elastic foundation to provide the torsional dynamic subgrade reaction, angle of twist, torque along the pile and the torsional impedance at the pile head. The accuracy of the proposed method is verified by comparing the analysis results with those of existing analytical solutions. Parametric studies are also performed to investigate the influence of the soil properties and the excitation frequencies on the stiffness and radiation damping of the pile-soil system.

Dynamic Response of an Inhomogeneous Elastic Pile in a Multilayered Saturated Soil to Transient Torsional Load

In this paper, we solve the dynamic response of an inhomogeneous elastic pile embedded in a multilayered saturated soil and subjected to a transient torsional load via a semianalytical method. To portray the inhomogeneity of the pile and the stratification of surrounding soil, the pile-soil system is subdivided into Nth layers along the depth direction in view of the variation of shear modulus or cross-sectional dimension of the pile or differences in soil properties. Then, the vibration displacement solution with undermined constants for any saturated soil layer subjected to the time-harmonic torsional load is obtained by virtue of the separation of variables scheme. To establish the connection of adjacent longitudinal soil layers, the circumferential contact traction at the interface of the adjacent layers is treated as the distributed Winkler subgrade model independent of the radial distance. Then, by utilizing the continuity conditions of the pile-soil system and the method of recursion typically used in the transfer function technique, the torsional impedance of the pile top can be derived in the frequency domain. By virtue of inverse Fourier transform and convolution theorem, the velocity response of an inhomogeneous pile subjected to a transient half-sine exciting torque and embedded in a layered saturated soil is gained in the time domain. Finally, selected numerical results are gained to investigate the influence of typical defects in pile and soil layering on the velocity response of the pile top in the time domain.

Time-Harmonic Response of an Elastic Pile in a Radially Inhomogeneous Poroelastic Medium

The time-harmonic response of an elastic pile embedded in a radially inhomogeneous poroelastic medium and subjected to a torsional loading is studied in the present article. In engineering practice, the surrounding soil may be weakened due to the disturbance effect caused by pile driving. To simulate the weakened surrounding soil, a boundary zone model with the complex shear modulus of the inner disturbed soil changing in a parabolic form along the radial direction is proposed. In view of the axis-symmetric deformation of the surrounding soil under torsional load, the equation of motion of the saturated soil is solved in the cylindrical coordinate system. The vibration displacement and shear stress solutions for the inner disturbed soil are gained by expanding the displacement as a power series, and those for the outer undisturbed soil are obtained by solving the partial differential equation. By virtue of continuity conditions at the interface between inner and outer soil regions, the torsional impedance of the radially inhomogeneous soil is solved. Then, via the boundary and continuity conditions of the pile-soil system, the twist angle and torque of the pile are obtained in the frequency domain. Finally, selected numerical results are conducted to investigate the influence of the material damping, softening degree, and softening range of the inner soil on the distribution of the twist angle and torque of the pile along the depth direction.

Exact Solution for the Torsional Vibration of an Elastic Pile in a Radially Inhomogeneous Saturated Soil

An exact solution is proposed to study the time-harmonic torsional vibration of an elastic pile embedded in a radially inhomogeneous saturated soil. The radially inhomogeneous saturated soil is composed of inner disturbed and outer semi-infinite undisturbed concentric annular regions, with the shear modulus of the inner region changing in an exponential form along the radial direction. The governing equation of each region of the saturated soil is solved through rigorous mathematical derivation and the soil torsional impedance is derived with an exact and explicit expression. Making use of the boundary and continuity conditions of the pile-soil system, the torsional complex stiffness at the pile top is obtained in an exact closed form in the frequency domain. Selected numerical results are presented to investigate the influence of the radial inhomogeneity of the surrounding soil on the vibration characteristics of the pile-soil system.

Torsional dynamic response of tapered pile considering compaction effect and stress diffusion effect

Considering both the compaction effect of pile surrounding soil and the stress diffusion effect of pile end soil, this paper theoretically investigates the torsional vibration characteristics of tapered pile. Utilizing the complex stiffness transfer model to simulate compaction effect and tapered fictitious soil pile model to simulate stress diffusion, the analytical solution for the torsional impedance at tapered pile top is obtained by virtue of Laplace transform technique and impedance transfer method. Based on the present solution, a parametric study is conducted to investigate the rationality of the present solution and the influence of soil and pile properties on the torsional vibration characteristics of tapered pile embedded in layered soil. The results show that, both the compaction effect and stress diffusion effect have significant influence on the torsional vibration characteristics of tapered pile, and these two factors should be considered during the dynamic design of pile foundation.

Torsional dynamic impedance of a stepped pile based on the wedged soil model

Stepped piles have been increasingly used as foundation elements for bridges and soft ground improvement in China. This paper derives an analytical solution of the torsional impedance of a stepped pile based on a wedged soil model. The accuracy of the developed solution is verified by degenerating the stepped pile into a straight pile. A parametric study is then conducted to investigate the influence of the wedged soil boundary and geometrical parameters on the torsional impedance of the stepped pile. A comparison between the present solution and the solution of a Voigt model is also presented. The findings demonstrate that the wedged soil boundary has a substantially influence on the torsional impedance for stepped piles with a short and stout upper pile segment. The difference in torsional impedance between the wedged soil model and the Voigt model is mainly reflected at high frequencies, while in a low frequency range these two solutions are almost identical, which is greatly divergent from the results of a vertical vibration case.

Torsional vibration solution of tapered pile considering stress diffusion effect of pile end soil

Based on the torsional plane strain model and tapered fictitious soil pile model, the torsional vibration characteristics of tapered pile considering stress diffusion effect of pile end soil is investigated theoretically. Considering the coupling conditions of pile-soil system, the torsional impedance at the tapered pile head is obtained by means of Laplace transform and impedance transfer method. A parametric study is performed, the rationality of the tapered fictitious soil model is verified and the influence of the cone angle of tapered fictitious soil pile on the torsional vibration characteristics of tapered pile embedded in layered soil is studied. The results show that: the cone angle of tapered fictitious soil pile has obvious effect on the torsional vibration characteristics of tapered pile with the shorter length.

Impedance functions of three-dimensional rectangular foundations embedded in multi-layered half-space

Although most of the building foundations are approximate to embedded three-dimensional rectangular foundations of different length-to-width ratios, the two-dimensional models or axis-symmetric models which ignore the influence of length-to-width ratios, are usually used for dynamic analysis due to less computational effort on impedance functions. The impedance functions of foundations characterized by different length-to-width ratios and depth-to-width ratios embedded in multi-layered half-space, are obtained by a boundary element method, combined with non-singular Green's functions of high efficiency. Results of high precision based on dense element meshing are presented on complete eight impedance functions (two horizontal, two coupled, two rocking, vertical and torsional impedance functions) for the practical values in the dynamic analysis of the buildings.

Dynamic torsional response of an elastic pile in a radially inhomogeneous soil

Surrounding soil may be weakened or strengthened due to construction disturbance of the pile, resulting in soil properties varying gradually in the radial direction. In this paper, an analytical solution is developed to investigate the torsional vibration of an elastic pile embedded in a radially inhomogeneous soil and subjected to a time-harmonic torsional loading. The radially inhomogeneous soil is subdivided into many thin concentric annular sub-zones, with each having constant complex shear modulus in radial direction. The dynamic equilibrium equation of each sub-zone of the soil is then solved and the circumferential displacement and shear stress along the pile-soil interface are obtained by a newly developed method that enforces the continuity conditions between the two adjacent sub-zones. By virtue of the boundary and continuity conditions of the pile-soil system, the torsional impedance at the pile head is derived in an exact closed form in the frequency domain. Some numerical results are presented to study the influence of the soil layer rigidities on the vibration characteristic of the pile-soil system.

Dynamic Torsional Response of a Pile Partially Embedded in Saturated Soil

The dynamic response of an elastic supporting pile partially embedded in a saturated soil and subjected to a time-harmonic torsional loading is investigated. At first, the pile is divided into two parts along the vertical direction, pile part above the soil and pile part embedded in the soil. Then, based on boundary and continuity conditions of the pile-soil system, the torsional impedance at the top end of the pile part embedded in the soil is obtained. By utilizing the transfer technique of impedance function, the admittance function of the pile top is defined in the frequency domain. By virtue of inverse Fourier transform and convolution theorem, a semi-analytical solution for the velocity response of a pile subjected to a semi-sine wave exciting torque is obtained in the time domain. Finally, selected numerical results are obtained to analyze the influence of main parameters on the torsional vibration characteristics of the pile.

Dynamic torsional response of pre-strained end bearing pile embedded in pre-strained isotropic saturated soil medium

The influence of initial strain state on the dynamic response of an end bearing pile embedded in isotropic saturated soil is investigated through the linearized theory of small elastic perturbation superposed on largely stressed bodies. The governing equations for soil, based on Biot’s poroelasticity theory, are derived in the cylindrical coordinates, and the pile is modeled by using the one-dimensional elastic theory. The analytical solutions of pile impedance, frequency response of both twist angle and time history of velocity response are obtained by using of separation of variables technique. Finally, a parametric study of the influence of initial strains on the torsional impedance, twist angle, and velocity response at the top of the pile is carried out.

Torsional Dynamic Impedance of Pipe Pile Embedded in Viscoelastic Soil Considering Soil-Pile Interaction

Abstract. Based on the three-dimensional wave theory,the coupled torsion vibration of the end bearing pile embedded in viscoelastic soil is researched by using the separation of variables technique. Calculation formulas of the torsional complex stiffness and admittance of pipe pile embedded in viscoelastic soil are obtained and the influence of shear modulus ratio, viscous damping coefficient ratio and the ratio of inner and outer radius on the torsional complex stiffness and admittance of pipe pile are discussed.Finally, the following conclusions are drawn.(1) Shear modulus ratio of soil inside and outside the pile, viscous damping coefficient ratio and the ratio of inner and outer radius have no effect on dynamic damping of pipe pile. The impact of shear modulus ratio of soil inside and outside the pile and the ratio of inner and outer radius on pipe pile in low frequency range is little. With the increase of frequency, the influence is increasing. Viscous damping coefficient ratio does not cause real stiffness of pipe pile. (2) Shear modulus ratio and the ratio of inner and outer radius have a significant influence on the admittance of pipe pile top.The greater shear modulus ratio and the ratio of inner and outer radius, the amplitude of the speed admittance curve of pipe pile top is larger .

The Parameters Research on Considering the Influence of Dynamic Torsion Impedance of the Radial Inhomogeneous Viscoelastic Soil

Based on the recursion method of shearing ragidity of soil layer, dynamic torsional impedance of the radial inhomogeneous viscoelastic soil is studied.Under the certain fundamental assumption,combined with displacement basic solution of the plane axial symmetry strain question,the displacement the continuous and boundary of the soil layers, the formula of dynamic torsional impedance of the pile-soil contact surface is deduced.Degenerated into homogeneous viscoelastic model,the model is further confirmed accurate.Finally, the conclusion is obtained:(1) The bigger is G2/G1,the larger the soil body actual stiffness is. The dynamic damping presents the trend of increase;(2)In the area of the radial disturbance soil,as ‘d’increases, the soil body actual stiffness is increasing. When G2/G1 is smaller than 1,the dynamic damping is reducing along with increases of ‘d’,when G2/G1 is bigger than 1,the dynamic damping is increasing along with increases of ‘d’.(3)Actual stiffness and dynamic damping assumes the linear variation along with increasing of material damping.

Dynamic torsional response of an end bearing pile in saturated poroelastic medium

A comprehensive analytical solution is developed in this paper to investigate the torsional vibration of an end bearing pile embedded in a homogeneous poroelastic medium and subjected to a time-harmonic torsional loading. The poroelastic medium is modeled using Biot’s two-phased linear theory and the pile using one-dimensional elastic theory. By using the separation of variables technique, the torsional response of the soil layer is solved first. Then based on perfect contact between the pile and soil, the dynamic response of the pile is obtained in a closed form. Numerical results for torsional impedance of the soil layer are presented first to portray the influence of wave modes, slenderness ratio, pile–soil modulus ratio and poroelastic properties. A comparison with the plane strain theory is performed. The selected numerical results are obtained to analyze the influence of the major parameters on the torsional impedance at the level of the pile head. Finally, the dynamic torsional impedance of this study is compared with that for floating pile in elastic soil.

Torsional impedance effect in Fe-rich amorphous wires

An amorphous ferromagnetic wire with a highly positive saturation magnetostriction constant, made of Fe 73.5Si 13.5B 9Nb 3Cu 1, was simultaneously submitted to both, an AC current flowing through it and a torsional stress, in order to induce a helical magnetic anisotropy that modifies the magnetic domain structure and therefore the magnetic response of the sample. We study the torsion-impedance effect in an Fe-rich amorphous wire when it is submitted to different applied torsional stresses in both senses, up to 0.8 π rad/cm, as a function of the drive current frequency, in the range of 10 Hz–2 MHz. The experimental results are explained on the basis of the core-shell magnetic structure model combined with the skin effect.

Vertical and torsional soil reactions for radially inhomogeneous soil layer

The response of an embedded body to dynamic loads is greatly influenced by the reactions of the soil to the motion of the body. The properties of the soil surrounding embedded bodies (e.g., piles) may be different than those of the far-field for a variety of reasons. It may be weakened or strengthened according to the method of installation of piles, or altered due to applying one of the soil strengthening technique (e.g., electrokinetic treatment of soil, El Naggar ct al. 1998). In all these cases, the shear strength of the soils and its shear modulus vary gradually in the radial direction, resulting in a radially inhomogeneous soil layer. This paper describes an analysis to compute vertical and torsional dynamic soil reactions of a radially inhomogeneous soil layer with a circular hole. These soil reactions could then be used to model the soil resistance in the analysis of the pile vibration under dynamic loads. The soil layer is considered to have a piecewise, radial variation for the complex shear modulus. The model is developed for soil layers improved using the electrokinetic technique but can be used for other situations where the soil properties vary gradually in the radial direction (strengthened or weakened). The soil reactions (impedance functions) are evaluated over a wide range of parameters and compared with those obtained from other solutions. A parametric study was performed to examine the effect of different soil improvement parameters on vertical and torsional impedance functions of the soil. The effect of the increase in the shear modulus and the width of the improved zone is investigated.

The Torsional Impedance test: short-time, large-strain stress response in tough polymers

Tough polymers (especially crystalline polyolefins) tend to show brittle behaviour, with a pronounced drop in fracture toughness, at high crack speeds. Since this effect is of special industrial importance, the High Speed Double Torsion test has been developed to measure it. Extracting fracture resistance data from the test demands accurate dynamical stress analysis of the specimen, but bulk strains are usually large enough to induce pronounced stress–strain non-linearity. The Torsional Impedance test was devised to acquire shear stress–strain data for corresponding geometries and loading times. The original analysis is here shown to have been flawed and it is corrected according to a more complete model, implemented as a numerical finite-difference scheme. Data are presented for modified high density polyethylene, and for pure and rubber-toughened grades of both polyoxymethylene and polypropylene.

A simplified procedure for torsional impedance functions of embedded foundations in a soil layer

A simplified calculation procedure for torsional impedance functions of circular embedded foundations in a soil layer with rigid base is presented. The solution is constructed by means of linear combinations of natural modes of propagation and participation factors. The unknown coefficients of such expansions are obtained from the continuous treatment of appropriate boundary conditions in the least-squares sense. The influence of soil material damping is investigated by using hysteretic and viscous models. Dynamic stiffnesses of soil-foundation systems are computed for different foundation embedments and soil layer depths. Finally, the practical importance of introducing viscous instead of hysteretic damping in linear interaction analyses in the time domain is discussed.

Torsional vibrations of elastic foundations embedded in an elastic half‐space

AbstractThis paper examines the axisymmetric torsional vibrations of an elastic pile and a hemispherical foundation embedded in a homogeneous elastic half‐space. The embedded foundation–half‐space system is decomposed into an extended half‐space and a fictitious foundation. The deformations of the fictitious system are specified by an admissible function containing a set of generalized coordinates. The Lagrangian equations of motion are used to determine these coordinates associated with the assumed displacement function. Numerical results are presented for torsional impedance of an elastic pile and a hemisphere to illustrate the effects of relative flexibility and geometry. By employing certain simplifications on the pile–half‐space system an approximate closed form solution is presented for the torsional impedance of an elastic pile.