Vertical Harmonic Load Research Articles

Dynamic soil–pile interactions for machine foundations

To achieve the satisfactory performance of a machine foundation, the dynamic amplitude should be limited to a few micrometres. In using piles for the foundation, the interaction of the pile with the surrounding soils under vibratory loading will modify the pile stiffness and will generate damping. The results of a three-dimensional, finite element model of a soil–pile system with viscous boundaries were used to determine the dynamic stiffness and damping generated by soil–pile interactions for a single vertical pile subjected to a vertical harmonic loading at the pile head. The pile was embedded in a linearly elastic, homogeneous soil layer with material damping of the hysteretic type. A parametric study was undertaken to investigate the influence of the major factors of the soil–pile system that affect the vertical vibration characteristics of the pile response. These were found to include the dimensionless frequency, ao, soil properties, pile properties, and length and axial load on the pile head. Equations were developed to determine the range of such influence. Two general equations that include all of these parameters and can be used to approximate the stiffness and damping of the pile were developed so that a finite element analysis does not have to be made in every case.

An analytical model for vibration prediction of a tunnel embedded in a saturated full-space to a harmonic point load

The dynamic response of a tunnel embedded in a saturated poroelastic full-space under a vertical harmonic load was investigated by a semi-analytical method. The tunnel was modeled as a thin cylindrical shell surrounded by soil of infinite radial extent. The soil was modeled as a saturated poroelastic full-space. Biot's theory was applied to characterize the soil medium, taking the solid skeleton-pore fluid coupling effects into account. Combined with the boundary conditions at tunnel–soil interface, the coupled equations of the system were solved in the frequency domain with the aid of Fourier decomposition and Fourier transform in the circumferential and longitudinal direction, respectively. Dynamic responses of the tunnel and soil generated by a unit harmonic load applied at the tunnel invert are presented. It is found that the soil permeability and the load frequency have significant influence on the displacements, the stresses and the pore pressure in the saturated soil. In the free field, an increase of the soil permeability leads to a decrease of the soil displacement and pore pressure response. An equivalent single-phase material was used to model the saturated soil and it is found only applicable for soil with lower permeability.

Development of vertical second harmonic wave loads of a large cruise ship in short and steep head waves

The reliable prediction of springing excitation requires a detailed understanding of the origin of exciting wave loads. This study clarifies the origin of the second harmonic wave loads that can excite the springing of a large cruise ship (roughly 300m long) in short and steep head waves. The findings are based on the analysis of previously validated numerical simulations (RANS-VOF) of the flow around a rigid hull. The accumulation of the loading is presented both along the length and the depth of the hull. The results show that the second harmonic vertical loading originates mainly in the foremost part of the ship, where the whole depth of the hull matters. The analysis focuses on the effect of the phase and the amplitude of second harmonic local loading. The irregular variation of the phase of the local loading with respect to that of the freely propagating wave demonstrates a serious deformation of the waves encountered in the area where the second harmonic total loading of the hull mainly originates. The analysis of the temporal behaviour of local loads indicates that the magnitude of the local second harmonic loading relates to the rise time of the respective local loads.

Experimental study on dynamic interference effect of two closely spaced machine foundations

This paper presents an experimental study on the dynamic interaction effect of closely spaced square foundations under machine vibration. Under a dynamic condition, a number of large-scale model tests were conducted in the field, which includes a wide range of study on the isolated as well as interacting footing response resting on the local soil available at Kanpur, India. The dynamic interaction of different combinations (size) of two-footing assemblies was investigated by inducing vertical harmonic load on one of the footings (active footing), while the other footing (passive footing) was loaded with static weight only. The active footing was excited with different magnitudes of dynamic loading and the response was recorded for both footings, placed at a different clear spacing (S). The results are compiled and shown as the variation of displacement amplitude with frequency. The transmission ratio that predicts the effect of dynamic excitation of the active footing on the passive one is determined for the interacting footings and plotted with respect to frequency ratio.

Axisymmetric vibration of an elastic circular plate bonded on a transversely isotropic multilayered half-space

Based on the analytical layer-element method, an analytical solution is proposed to determine the dynamic interaction between the elastic circular plate and transversely isotropic multilayered half-space. The dynamic response of the elastic circular plate is governed by the classical thin-plate theory with the assumption that the contact surface between the plate and soil is frictionless. The total stiffness matrix of the transversely isotropic multilayered half-space is acquired by assembling the analytical layer-element of each soil layer with the aid of the continuity conditions between adjacent layers. According to the displacement condition of coordination between the plate and soil, the dynamic interaction problem is reduced to that of multilayered transversely isotropic half-space subjected to axisymmetric harmonic vertical loading. Some numerical examples are given to study the vertical vibration of the plate, and the results indicate that the dynamic response of elastic circular plate depends strongly on the material properties of the soils, the rigidity of the plate, the frequency of excitation and the external load form.

Analysis and design of open trench barriers in screening steady-state surface vibrations

The problem of vibration isolation by rectangular open trenches in a plane strain context is numerically studied using a finite element code, PLAXIS. The soil media is assumed to be linear elastic, isotropic, and homogeneous subjected to a vertical harmonic load producing steady-state vibration. The present model is validated by comparing it with previously published works. The key geometrical features of a trench, i.e., its depth, width, and distance from the source of excitation, are normalized with respect to the Rayleigh wavelength. The attenuation of vertical and horizontal components of vibration is studied for various trench dimensions against trench locations varied from an active to a passive case. Results are depicted in non-dimensional forms and conclusions are drawn regarding the effects of geometrical parameters in attenuating vertical and horizontal vibration components. The screening efficiency is primarily governed by the normalized depth of the barrier. The effect of width has little significance except in some specific cases. Simplified regression models are developed to estimate average amplitude reduction factors. The models applicable to vertical vibration cases are found to be in excellent agreement with previously published results.

Identifying boundary between near field and far field in ground vibration caused by surface loading

The boundary between the near and far fields is generally defined as the distance from the vibration source beyond which ground vibrations are mainly dominated by Rayleigh waves. It is closely related to the type of vibration source and the soil properties. Based on the solutions of the Lamb’s problem, the boundary at the surface between the near and far fields of ground vibration was investigated for a harmonic vertical concentrated load and an infinite line load at the surface of a visco-elastic half-space. Particularly, the variation of the boundary with the material damping was investigated for both cases. The results indicate that the material damping slightly contributes to the attenuation of vibrations in the near-source region, but significantly reduces the vibrations in the region that is at some distance away from the source. When taking the material damping into consideration, the boundary between the near and far fields tends to move towards the vibration source. Compared with the vibrations caused by a concentrated load, the vibrations induced by an infinite line load can affect a larger range of the surrounding environment, and they attenuate more slowly. This means the boundary between the near field and far field should move further away from the source. Finally, the boundaries are defined in terms of R-wave length (λ R) and Poisson ratio of the ground (gv). For the case of a point load, the boundary is located at the distance of (5.0–6.0)λ R for gv≤0.30 and at the distance of (2.0–3.0)λ R for gv≥0.35. For the case of an infinite line load, the boundary is located at the distance (5.5–6.5)λ R for gv≤0.30 and at the distance (2.5–3.5)λ R for gv≥0.35.

Reasons and laws of ground vibration amplification induced by vertical dynamic load

The phenomenon of ground vibration amplification caused by railway traffic was found and proved. In order to study the reasons which cause the amplification, a drop-weight test was performed. Then, the model for both homogeneous and layered soil subjected to a harmonic vertical load was built. With the help of this model, displacement Green’s function was calculated and the propagation laws of ground vibration responses were discussed. Results show that: 1) When applying a harmonic load on the half-space surface, the amplitude of ground vibrations attenuate with fluctuation, which is caused by the superposition of bulk and Rayleigh waves. 2) Vibration amplification can be enlarged under the conditions of embedded source and the soil layers. 3) In practice, the fluctuant attenuation should be paid attention to especially for the vibration receivers who are sensitive to single low frequencies (<10 Hz). Moreover, for the case of embedded loads, it should also be paid attention to that the receivers are located at the place where the horizontal distance is similar to embedded depth, usually 10 to 30 m for metro lines.

Dynamic Stability Analysis of Defective Piles under Vertical Harmonic Load

We established the computational mode for defective piles based on practical engineering problems,and studied the stability of the defective pile in the heterogeneous soil under vertical harmonic loads. We established the dynamical function based on the principle of Energy and Hamilton, and abtained the expressions of critical frequency of defective piles.The results show that, the instability of the defective piles relate to the degree of defect and the location of defect.

Proposed Relationship to Design the Waffle Floor under Harmonic Vertical Loading

A B S T R A C T The design codes of building are mainly related to the strength of the building and there are no specific codes for the design of building with waffle floors for vibration sensitive equipment. The finite element model, ANSYS, is capable to consider the effects of floor thickness, the size of the bays and stiff ness of the columns for analyzing the vibration of the waffle floors and vibration transmission along the waffle floor. Because the finite element analysis is time consuming and it needs enough expertise for modeling, in this study, an approximate relationship is proposed to design the waffle floor based on the comprehensive investigations of different effective parameters in the response of waffle floor. The results obtained fr om finite element analysis. This proposed relationship comforts the designers of industrial buildings and vibration sensitive equipment to attain a preliminary and appropriate outline to design the waffle floor considering the effective parameters on the floor vibration.

Dynamic Behavior of Pile Foundation in Frozen and Thawing Soil

Permafrost is widespread in China, especially in Northeast China and the Qinghai-Tibet Plateau. Regions like Qinghai-Tibet Plateau have the most strenuous crustal movement. Therefore, earthquake-resistance of structures in permafrost region is an important issue. Furthermore, the permafrost will degenerate gradually as global warming mounts up. In some regions permafrost thickness tends to attenuate. Most bridge designs adopt pile foundation in order to reduce the effects of instable frost. The deterioration of frost leads to degradation of anti-seismic performance of bridges’ pile foundations. Pile-soil dynamic interaction numerical analysis models are established based on data of indoor low-temperature dynamic triaxial tests. Studies are performed on the dynamic stiffness and damping characters and the influencing factors of pile foundation under vertical harmonic load in frozen and thawing soil. The result shows that the dynamic response of the pile decreases along the depth, and the frictional resistance around the pile mainly distributes along the upper half of the pile, and the dynamic stiffness and damping of the pile are affected by temperature. Dynamic stiffness increases as temperature goes down, whereas the decrease of the temperature of frozen soil can notably lower the dynamic damping of the head of the pile. As the frequency of the dynamic load augments, the dynamic stiffness of the head increases marginally, whereas frequency has little influence on damping. The relative thickness of the frozen and thawing soil layer has considerable influence on dynamic stiffness, but negligible on damping.

Analysis on Dynamic Stability of Cuneiform Pile under Vertical Harmonic Loads

We discussed the problems of dynamic stability of cuneiform pile under vertical dynamic load, and studied the influence of parameters resonance and soil resisting force on dynamic stability of foundation pile.

A Semi-Analytical Model for the Simulation of the Isolation of the Vibration Due to a Harmonic Load Using Pile Rows Embedded in a Saturated Half-Space

The isolation of the vibration due to a harmonic vertical load using pile rows embedded in a saturated poroelastic half-space is investigated in this study. Using the fundamental solution for a circular patch load and Muki’s method, the second kind of Fredholm integral equations describing the dynamical interaction between the pile rows and the saturated poroelastic half-space are obtained. Numerical solution of the integral equations yields the dynamic response of the pile-half-space system. The vibration isolation effect of the pile rows is investigated via the proposed semi-analytical model. Numerical results indicate that stiffer piles have better isolation vibration effect than flexible piles. Moreover, the pile length and the spacing between neighboring piles in one pile row have significant influence on the isolation vibration effect of pile rows, while the influence of the spacing between neighboring pile rows is relatively smaller.

Ground vibration due to a high-speed moving harmonic rectangular load on a poroviscoelastic half-space

The transmission of vibrations in the ground, due to a high-speed moving vertical harmonic rectangular load, is investigated theoretically. The problem is three-dimensional and the interior of the ground is modelled as a totally or partially saturated porous viscoelastic half-space, using the complete Biot theory. The solutions in the transformed domain are obtained using a double Fourier transform on the surface spatial variables. A modified hysteretic damping model defined in the wavenumber domain is used, first presented by Lefeuve-Mesgouez et al. [Lefeuve-Mesgouez, G., Le Houédec, D., Peplow, A.T., 2000. Vibration in the vicinity of a high-speed moving harmonic strip load. Journal of Sound and Vibration 231(5) 1289–1309]. Numerical results for the displacements of the solid and fluid phases, over the surface of the ground and in depth, are presented for loads moving with speeds up to and beyond the Rayleigh wave speed of the medium.

Three-dimensional analysis of the screening effectiveness of hollow pile barriers for foundation-induced vertical vibration

The study applies the three-dimensional boundary element method in frequency domain to investigate the screening effectiveness of circular piles in a row for a massless square foundation subject to harmonic vertical loading. Four types of piles were studied: steel pipe piles, concrete hollow piles, concrete solid piles and timber piles. A parametric study was undertaken to examine the effects of pile dimensions, operational frequency, and source distance on the screening effectiveness. The results showed that screening effectiveness of steel pipe piles is generally better than that of solid piles, and that a concrete hollow pile barrier can be ineffective due to its stiffness. The influence of pile length on screening effectiveness is more significant than that of pile spacing and the distance between the vibrating foundation and the pile barrier.

地表面の鉛直方向調和振動荷重による地盤の層厚・弾性定数の推定方法とその数値実験による検証

This paper presents a method of estimating the elastic properties of the soils and their variation with depth using the characteristics of P-SV wave fields caused in the surface of an elastic multi-layered half space by the vertical harmonic point load applied also on the soil surface. To demonstrate the capability of the proposed method, the two numerical examples are presented, for the estimation of S wave velocity, P wave velocity and thickness of each soil layer, using a three-layered half space and a five-layered half space. In the numerical examples, the stiffness matrix method is used for an accurate simulation of all P-SV wave fields.

Dynamic Response of Footing and Machine with Spring Mounting Base

The effect of the spring mounting cushion inserted in between a machine base and its concrete footing has been examined experimentally by conducting a number of block vibrations tests. The machine was subjected to steady state vertical harmonic loading. Experiments were performed with two different stiffness values of the spring mounting cushion. The employment of the spring mounting cushion, with the stiffness much smaller than that of soil strata, offers a drastic reduction in the resonant displacement amplitudes of the footing. It also results in a significant decrease in the resonant frequency of the foundation. The resonant displacement amplitudes of both the footing and the machine were found to become lower with the smaller stiffness value of the springs. The resonant frequency for the machine base, in all the experiments, was found to be invariably the same as that of the footing.

調和振動荷重によるＰ‐ＳＶ波の地盤振動・波動伝播特性

Elastic waves generated by a controlled dynamic load applied on the surface of soil deposits yield useful tools for non destructive testing of the systems to determine the elastic properties of the soils and their variation with depth. A number of methods for the non destructive testing have been developed and are being used in practice. The objective of this paper is to examine some of the basic concepts related to P-SV wave propagation in layered media and their dynamic response due to the vertical harmonic point load applied on the soil surface. A proper understandings of these concepts is necessary for a correct interpretation of the field data and for improving the accuracy of the estimation of the soil properties at low strain values by using the spectral analysis of surface waves or the Rayleigh wave method.

Vertical vibration analysis of soil-pile interaction systems considering the soil-pile interface behavior

A numerical method is presented herein to study soil-pile interaction systems in a stratified media. The method is based on a coupling of the finite element method and the boundary element method. The near field is modeled by the finite elements whereas the far field is modeled by the boundary element formulation using the multi-layer dynamic fundamental solution that satisfies the radiation condition of the waves. The two fields are coupled by imposing the displacement compatibility conditions at the interface between the near and far fields. Special attention is paid to the soil-pile interface using elastic springs to model the discontinuity at the interface. In order to validate the analytical approach, a forced vibration test is simulated and the response under vertical harmonic loads at the pile cap in the layered half plane is calculated. The results are compared with the published theoretical and experimental results. Finally, numerical analyses of various soil-pile systems are performed to examine the dynamic behavior of the system depending on parameters such as the elastic modulus of the pile and the stiffness of the interface springs.

3D seismic response of a limited valley via BEM using 2.5D analytical Green's functions for an infinite free-rigid layer

This paper presents analytical solutions for computing the 3D displacements in a flat solid elastic stratum bounded by a rigid base, when it is subjected to spatially sinusoidal harmonic line loads. These functions are also used as Greens functions in a boundary element method code that simulates the seismic wave propagation in a confined or semi-confined 2D valley, avoiding the discretization of the free and rigid horizontal boundaries. The models developed are then used to simulate wave propagation within a rigid stratum and valleys with different dimensions and geometries, when struck by a spatially sinusoidal harmonic vertical line load. Simulations are performed in the frequency domain, for varying spatial wave numbers in the axial direction of the valley. Time results are obtained by means of inverse Fourier transforms, to help understand how the geometry of the valley may affect the variation of the displacement field.