Development of design charts to predict the dynamic response of pile supported machine foundations

Abstract

This paper proposes design charts for estimating imperative input parameters for continuum approach analysis of the nonlinear dynamic response of piles. Experimental and analytical studies using continuum approach have been conducted on single and 2 × 2 grouped piles under coupled and vertical modes of vibration, for different dynamic forces and pile depth. As these design charts are derived from model piles, the charts have been validated for prototype pile foundations using scaling law. The experimental responses of model piles are scaled up and these responses exhibit good agreement with analytical results. This study also extends to estimation of the errors in computing frequency–amplitude responses with an increase in pile length. It is found that, with an increase in pile length, the errors also increase. The effectiveness of the proposed design charts is also checked with data based on different field setups given in existing literature, and these charts are found to be valid. Thus, the developed design charts can be beneficial in estimating the input parameters for continuum approach analysis for determining the nonlinear responses of pile supported machine foundations.

Graphical abstract

Keywords

pile foundation / vertical vibration / coupled vibration / continuum approach / soil–pile separation / boundary zone parameters

1 Introduction

Pile foundations are often employed in poor soil deposits for bearing heavy structural loads, where shallow-foundations are inappropriate. One of the prime targets of pile foundations when subjected to dynamic loading is restriction of the vibration amplitude to an acceptable limit. Applications in nuclear power plant construction, petrochemical industries, offshore structures, and many elsewhere, have tended to be based on pile foundations rather than conventional block foundations as piles can resist higher loads at greater depths. Nonlinearity of soil and high soil–pile interaction, cause the behavior of pile foundations under dynamic loading to become complex. Therefore, it is typically recommended to analyze the response of piles to dynamic loading, thereby achieving secure and reliable foundations. Estimation of dynamic response of piles, with the highest possible precision, can provide a secure design by ensuring that the operating frequency of machines is outside the zone of resonance, thereby limiting the amplitude of vibration. For attainment of nonlinear frequency–amplitude response of soil–pile system analytically, an effective guideline is necessary in order to estimate the input parameters responsible for this behavior. Since these guidelines are currently scarce, input parameters may diverge from the values predicted, and designers face a challenge in analysis of pile foundations that must support heavy vibrating machines or rotating machines subjected to dynamic loading.

Many theoretical studies have been performed in the past to determine the dynamic response and impedance function of piles, and some of the prominent methods are discussed here. Earlier studies utilized a lumped soil mass approach for modeling a pile foundation under dynamic loading, neglecting damping [1]. Later, displacement of a pile under dynamic loading has been determined by including mass, spring constant and damping of the system [2]. This model has also been used to simulate lateral pile behavior under earthquake motion [3] and, recently, studies have used the same approach to develop understanding of the effectiveness of soil–structure interaction in a framed machine foundation under harmonic vibration [4]. Winkler’s [5] Foundation model depicts a soil–pile system as a beam supported by a continuous spring system, with an assumption that soil behaviors of each layer are independent of those of adjacent layers. Allotey and El Nagger [6] have performed nonlinear analysis of deep and shallow foundation using the Winkler approach. Numerical analysis has also been done by Sutaih and Aggour [7], using Winkler’s model to estimate the response of a pile subjected to horizontal dynamic loadings in elastic homogenous and inhomogeneous soil. The finite element method is considered to be an appropriate method in analyzing dynamic response of pile foundations, as it can incorporate all types of nonlinearity and damping, and as it has been used to study the effect of nonlinearity [8] on pile head response and free field response. The method has been further used in exploring several issues such as the use of under-reamed piles subjected to dynamic loading in sandy soil [9], soil–pile response under vertical vibration [10] and dynamic response of tapered piles [11] by incorporating material damping. The finite element method has also been used to analyze the influence of pile toe in group piles under harmonic vibration [12] and to comprehend the lateral response of batter piles in single and group assembly under machine induced vibration [13]. The same analytical method has been used for getting a better insight into the effect of vibration frequency, aspect ratio and soil–pile interface interaction under dynamic loading involving bored piles subjected to machine vibration [14]. A 3D finite element model has been developed by Ali et al. [15] for determining dynamic stiffness and damping, generated by soil–pile interactions for a single pile under vertical vibration. An attempt has been made using FEM [16] to understand the effect of pile size group, pile slenderness ratio and modulus of elasticity, in addition to impedance parameters under both vertical and lateral vibrations. Studies have also been done using 3D finite element analysis on a single pile under vertical cyclic loading in different topographies [17] and design recommendations have been proposed for single piles and group piles subjected to cyclic loadings [18,19]. A boundary element method has been extensively used due to its fast computational approach. Studies have been performed by Baross [20] to analyze the response of circular cross-sectional piles embedded in transversely isotropic half spaces. A 3D BEM-FEM coupling model has been developed by Pardon et al. [21] to study the dynamic behavior of a pile foundation embedded in stratified soils. Alzabeebee and Keawsawasvong [22] have made an attempt to understand the behavior of machine foundations resting on sandy soil, when different simple and advanced soil constitutive models included in the analysis.

The continuum approach has been used since mid-20th century to determine the impedance functions of a single pile embedded in layered soil [23]. Out of all these significant methods, the continuum approach seems to be the most promising because of its great versatility and simple computational technique. This method is distinct from others as it can easily simulate material and geometric nonlinearity. It can also automatically include radiation damping, thereby offering better determination of dynamic response of piles. Many analyses have been done, based on the continuum approach, to estimate the dynamic response of a pile-supported machine foundation by assuming soil to be linearly elastic and the soil–pile bond to be perfect [24,25]. Later, the concept of nonlinearity has been introduced into the continuum approach by incorporating boundary zones with varying soil properties [26]. Because of the advancement in industrial sectors, studies have now extended to investigation of the dynamic responses of pile groups under machine induced vibration, estimating the responses of a single and a 2 × 2 pile group under coupled and vertical vibrations [27,28]. Several studies have been performed using a continuum approach analysis in both single and group piles to understand various aspects of pile foundation under dynamic loading. Model pile tests have been conducted on piles under vertical [29] and coupled vibrations [30]. These studies have extended to theoretical continuum approach estimation of stiffness and damping of the soil–pile system. Many model pile studies have been done to find the characteristics of piles under dynamic loading, but the applicability of these findings for full scale pile foundations are dubious. Work has also been executed to bridge the gap between model piles and prototype pile results using different available scaling laws [31–33].

With the advance of research, significant studies have also been conducted to understand different aspects of pile foundations, such as under-reamed batter piles under vertical vibration [34], pile groups subjected to vertical vibration [35], effects of embedment of a pile in layered transversely isotropic unsaturated soil under vertical loading [36], the effect of laterally loading a pile in multi-layered viscoelastic soil [37], and various cross section of piles [38].

For a secure and economically advantageous design of pile foundations, analytically forecasting nonlinear frequency–amplitude response of soil–pile system is crucial. But a detailed literature review showed that there are few, if any, design guidelines for prototype piles subjected to dynamic loading. An appropriate theoretical method can be used to analyze the dynamic response of piles exposed to machine vibration by incorporating the effects of nonlinearity, soil–pile interaction and slippage between soil and pile. Since designers lack adequate guidelines in incorporating all these conditions during analysis, the accuracies of results are dubious and can be only established with ample experimental results. Field experiments are time consuming and expensive, and few tests have been performed on prototype piles. However, model pile testing can be considered to be a good choice for gauging the accuracy and applicability of theoretical approaches. By using different available scaling laws, these model pile test responses can be further used for design of prototype piles, thereby filling the existing gap in the field of acquiring safe and cost-effective design for prototype piles under dynamic loading.

Thorough review has found a wide gap in estimating input parameters required for the nonlinear analysis of pile foundation subjected to dynamic loading. This study proposes design charts for estimating input parameters, like soil–pile separation length and boundary zone parameters, required for estimating the nonlinear dynamic responses of pile foundations by continuum approach analysis. The charts are prepared for single and 2 × 2 group piles under vertical and coupled vibrations, for forecasting the nonlinear response of piles under harmonic vibration. These design charts have been proposed based on the experimental and analytical results presented in Refs. [27,28]. As these design charts are derived from model pile results, the charts have been further validated for full-scale pile foundations using the scaling law of Woods [33]. A comparative study has been performed of different analytical methods to check the trustworthiness of nonlinear analysis used for the study. The reliability of these charts is also ensured using experimental data given in various existing sources in the literature. The study has also extended estimation of errors in computing nonlinear responses of pile foundations under harmonic vibration, with an increase in pile length.

2 Experimental study

Field tests were conducted on a single pile and a 2 × 2 pile group, to determine the frequency–amplitude response of piles under coupled and vertical modes of vibration. Prior to field tests, different in situ and laboratory tests were performed to characterize the subsurface soil conditions. Forced vibration tests were executed on piles of diameter 0.166 m and length 5.8 m, under static load of 15 kN at varying eccentric moments of 0.735, 1.448, 2.117, and 2.721 N·m. The piles were installed in undersized boreholes by hammering technique. This experimental study also measured soil–pile separation length, which was predicted to be a major contributor to the soil–pile system’s nonlinear behavior. Soil–pile separation length is the distance of pile from the ground surface to the point of zero contact with the soil, due to the high stress associated with immense repeated vibrations. Before installing the piles to boreholes, eight pressure sensors were positioned on the outer surface at different locations along the peripheral region of pile length at different depth of the piles and afterwards they were driven into the ground. These pressure sensors measured the contact pressure between soil and pile using data collection hub and laptop. From the acquired readings, the depths of the sensors showing zero contact pressures were identified and separation depth was considered to be the same. As the piles were driven into undersized boreholes using a pre-boring technique, the possibility of densification of soil around pile was unlikely to occur. However, the type of soil used in the current study was more prone to disturbances due to the adopted pile installation technique, rather than densification. To eliminate or reduce the influence of the disturbances caused by the pile installation process, the piles were kept undisturbed for a period of approximately 2 months after the installation. However, the soil may have required longer time to regain its original stiffness and strength. Hence, it is possible that pre-boring and pile driving by hammering played an important role in the dynamic responses of piles, especially in existing soil conditions. Therefore, further study is required to evaluate the effect of pile installation on the dynamic pile responses. The field experimental set up for single pile under coupled vibration is demonstrated in Fig.1.

A mechanical oscillator was used to impose a harmonic vibratory load on the pile foundation. This was operated using a motor and speed control unit, so that it could run at different frequencies. Steel plates were used to exert the desired static load on the foundation. Time versus acceleration responses of the pile foundation were measured during the experiment, for all the considered eccentric moments separately, using acceleration pick-up, laptop and data logger system. Further, the frequency–amplitude response curves were derived from the acceleration–time curves, for all the eccentric moments. The plot shows nonlinear behavior of the soil–pile system with decreasing resonant frequency and disproportionally increasing resonant amplitude for increasing eccentric moments. The experimental frequency–amplitude responses and soil–pile separation length for all the piles are reported in Refs. [27,28] for coupled and vertical vibrations, respectively.

3 Continuum approach

Continuum approach is one of the broadly accepted method for estimating dynamic response of a pile because of its simple computational technique and wide versatility. This method provides solution by breaking the pile into small elements and further integrating the elemental solutions to get the generic solution. A model has been proposed [23] to establish the complex stiffness of single pile embedded in various layered medium, under varying modes of vibration. Later, a method to estimate dynamic response of pile groups has been done by introducing dynamic interaction factors [39]. All these afore-stated models assume soil to be linearly elastic and assumed soil stiffness to be a complex function, with real part indicating stiffness and imaginary part representing damping. In this study, the numerical analysis was performed [27,28] using software DYNA 5 [40] to estimate impedance function of pile foundations undergoing coupled and vertical vibrations respectively. This software works based on the continuum approach of Novak and on the superposition method for a single pile and for a pile group, respectively.

Since both these methods assumed soil to be linearly elastic and the experimental response showed nonlinear behavior, an equivalent linear model [26] was considered for theoretically estimating nonlinear dynamic response of pile foundation (Fig.2). In this soil model, a cylindrical annulus soil boundary zone was considered around the pile, with reduced shear modulus and increased damping compared to the outer soil medium. These individual layers were considered to be linearly elastic, and by amending the boundary zone soil properties, the stiffness of the inner soil zone was reduced. As a result, the overall stiffness of the soil–pile system was reduced, which could effectively simulate the effect of soil nonlinearly, soil–pile slippage, pile installation technique in the dynamic pile response. A boundary zone was defined by boundary zone parameters such as shear modulus reduction ratio (G_{\mathrm{m}} /G), thickness ratio (t_{\mathrm{m}} /R) and inner zone damping \left(D_{\mathrm{m}}\right), where G_{\mathrm{m}}, D_{\mathrm{m}} are shear modulus and damping of boundary zone. G, D are shear modulus and damping of outer zone, t_{\mathrm{m}} is the thickness of boundary zone and R is radius of pile. The soil model is dimensionless as input parameters are used in the form of ratios.

A continuum approach analysis was performed and variations of boundary zone parameters with eccentric moment and depth are detailed in Refs. [27,28]. These boundary zone parameters were adapted in order to incorporate effects of pile installation techniques, soil nonlinearity, soil–pile slippage and change of soil parameters during vibration on the dynamic stiffness of the soil–pile system. A step variation trend was considered to define the parameters of boundary zone relative to depth, and the trend approximately followed a parabolic pattern. It was observed that, with an increase in depth, shear modulus increased. However, thickness ratio along with damping decreased, because at greater depth profound disturbance due to vibration is minimal. Furthermore, it was found that, with an increase in eccentric moment, shear modulus decreased and thickness ratio and damping increased, because of the enlarged weakened zone associated with high vibration.

Plots of boundary zone parameters with depth exhibited that the inner zone development for the single pile was superior to that of the 2 × 2 pile group for any specific eccentric moment, as the load distributed on a pile in a group was less than that on a single pile. It was also noted that the extent of boundary zone was higher in case of coupled vibration than in case of vertical vibration, for any eccentric moment. This trend indicated that, the effect of coupled vibration on sounding soil was more than the effect of vertical vibration under same dynamic force.

Besides boundary zone parameters, one of the main aspects concerned with nonlinearity is considered to be soil–pile separation length. Soil–pile separation length measured during field tests were included in theoretical analysis by assuming shear modulus \left(G_{\mathrm{m}}\right) for inner weak zone to have been zero till separation depth. By considering appropriate influencing parameters, experimental and analytical responses were observed to be in harmony, and analytical responses also showed nonlinear behavior, with a decrease of resonant frequency and disproportional increase of resonant amplitude.

4 Design charts

For a secure and economical design of pile foundation subjected to dynamic loads, an effective estimation, involving input parameters, of the nonlinear response of a pile is vital. Since the guidelines for these parameters are scarce, the predictions of dynamic response of piles are generally considered to be unreliable.

Therefore, this section provides design charts for input parameters such as soil–pile separation length and boundary zone parameters, for a single and a 2 × 2 pile group under coupled and vertical vibrations, prepared following the results presented in Refs. [27,28]. The charts are presented for variation of soil–pile separation length and different boundary zone parameters—shear modulus reduction ratio (G_{\mathrm{m}}/G), thickness ratio (t_{\mathrm{m}} /R) and inner zone damping \left(D_{\mathrm{m}}\right)—with eccentric moments and depth.

4.1 Soil pile separation length

Soil–pile separation can be considered as one of the governing factors of soil nonlinearity, as the upper part of a pile may become separated from soil due to vibration, thereby reducing the stiffness of soil–pile system. Therefore, precise estimation of separation length is vital because its value is necessary for estimating dynamic response of a pile subjected to harmonic vibration. Based on the soil–pile separation length presented in Refs. [27,28], design charts are proposed to estimate the same.

Fig.3 illustrates the variation of separation length with eccentric moments for both single and group piles under vertical and coupled vibrations; it was found that with the increase in eccentric moment, both single and group piles exhibited a linear increase in separation length. In the case of a single pile, separation length reached a highest value of 0.9d, where d denotes the diameter of pile, under vertical vibration, whereas in coupled vibration soil–pile separation length approximately varied from 1.8d to 2.7d as eccentric moments increased. However, for 2 × 2 group pile, highest eccentric moment exhibited a value of 0.5d under vertical vibration and 1.2d to 2.1d under coupled vibration for the range of eccentric moments considered. Therefore, this plot indicates that soil–pile separation was more predominant in coupled vibration than in vertical vibration. Both figures showed that, with an increase of eccentric moment, separation length increased for both single and group piles. The plot also showcased that, the separation length in single pile was greater than in a pile group at a particular eccentric moment, for both vertical and coupled vibrations. This can be interpreted as distribution of dynamic load on each pile in 2 × 2 group pile being much lower than that on a single pile. Therefore, this plot can be used for predicting the value of separation length for various eccentric moments. Aside from dynamic loading, soil–pile separation can vary for a few reasons, such as pile inclination during installation, reduction of soil resistance due to experimental activities around the pile and many more. Therefore, this parameter should always be considered in the analysis, as soil–pile separation can be likely to occur in actual field condition.

In this study, an effort has been made to predict soil–pile separation length using the plot of maximum vibration amplitude with separation length for both vertical and coupled modes of vibration. The available actual data points are used to build best fit curves, which are then mathematically described using Eqs. (1)–(4). These equations are able to estimate the separation length of single and pile group using known value of maximum resonant amplitude under vertical and coupled modes of vibration.

For single pile under vertical vibration,

(1)

for pile group under vertical vibration,

(2)

for single pile under coupled vibration,

(3)

for pile group under coupled vibration,

(4)

where l_{\mathrm{s}} is the soil–pile separation length, d is the diameter of pile, {\mathrm{\Delta }}_{\mathrm{v}} and {\mathrm{\Delta }}_{\mathrm{h}} represent maximum vertical and horizontal resonant amplitude of vibrations, respectively.

The above equations are based on the soil–pile conditions of the present study and they can be used for obtaining separation length for similar types of soil conditions.

4.2 Boundary zone parameters

This section briefly describes various proposed design charts for predicting the links between all boundary zone parameters or input parameters and the eccentric moments at different depths of the pile, as presented in the analytical studies from Refs. [27,28]. Design charts are proposed for a single and a 2 × 2 pile group with both vertical and coupled vibrations, up to a l/d ratio of 35. Fig.4–Fig.9 demonstrate the variation of shear modulus reduction ratio ( G_{\mathrm{m}}/G), thickness ratio (t_{\mathrm{m}}/R) and inner zone soil damping (D_{\mathrm{m}}) with eccentric moments and depth for a single pile and a 2 × 2 pile group under vertical and coupled vibrations. From these charts, the values of input parameters that are needed to perform nonlinear continuum approach analysis can be determined for different slenderness ratios of the pile and magnitudes of the dynamic force.

4.2.1 Shear modulus reduction ratio

Fig.4 shows that the shear modulus reduction ratio varies from 0.21 to 0.75 for a single pile and 0.22 to 0.80 for a 2 × 2 pile group, over the range of eccentric moments from 0.735 to 2.721 N·m under vertical vibration up to the z/d ratio of 35, where z denotes depth of soil. Thus it has been noticed that the pile group exhibits higher value of shear modulus reduction ratio, i.e. approximately 10% higher than that of a single pile, which demonstrated that reduction of boundary zone soil shear modulus is greater in a single pile than in a pile group. The plot also shows that, shear modulus reduction ratio \left(G_{\mathrm{m}}/G\right), decreases with an increase in eccentric moment and an increasing of depth.

Fig.5 illustrates the variation of shear modulus reduction ratio \left(G_{\mathrm{m}}/G\right) with eccentric moments for a single and a pile group under coupled vibration. The values varies from 0.07 to 0.65 for a single pile and 0.11 to 0.70 for a pile group, over a range of eccentric moments varying from 0.735 to 2.721 N·m. This indicates that coupled vibration leads to approximately 30% lower value than vertical vibration, showing that decrease in boundary zone soil shear modulus is more significant in the former case. However, the trends of the curves for coupled vibration is similar to that for vertical vibration.

4.2.2 Thickness ratio

Fig.6 illustrates the variation of thickness ratio with eccentric moment under vertical vibration. It is found that, thickness ratio ( t_{\mathrm{m}}/R) varies from 0.48 to 0.86 for a single pile and from 0.45 to 0.86 for a pile group, for eccentric moment varying from 0.735 to 2.721 N·m. But in the case of coupled vibration, notable differences are observed between the single and the group pile, as is demonstrated in Fig.7. From the figures, variation of thickness ratio ( t_{\mathrm{m}}/R) is found to vary from 0.67 to 0.98 for a single pile, whereas for a group pile it varies from 0.61 to 0.93 over the same range of eccentric moments. This indicates that thickness ratio of a single pile is higher than for a group pile, which implies that development of boundary zone is more advanced in that case of a single pile than for a group pile. On comparing vertical and coupled vibrations, coupled vibration exhibits approximately 25% higher thickness ratio than vertical case, which indicates that the effect of coupled vibration is more prominent in surrounding soil. Additionally, it has been noted that, with an increase in eccentric moment, thickness ratio (t_{\mathrm{m}} /R) increases, and it decreases with an increasing depth for both types of piles and vibration modes.

4.2.3 Inner zone damping

Fig.8 illustrates the variation of inner zone damping {(D}_{\mathrm{m}})with eccentric moments for single and group pile under vertical vibration. It is observed that inner zone damping (D_{\mathrm{m}}) values vary from 0.14 to 0.28 for a single pile and from 0.12 to 0.26 for pile group, which indicates that the single and the pile group exhibit almost the same value in vertical mode of vibration. The figure also depicts that, an increase in eccentric moments increases inner zone damping and it decreases as depth of pile increases.

From Fig.9, it can be noted that inner zone damping varies from 0.20 to 0.38 for a single pile and 0.18 to 0.34 for a pile group under coupled vibration. The values of coupled vibration are almost 70% higher than in the case of vertical vibration, which implies that the effect of coupled vibration in sounding soil is more predominant than vertical vibration under the same dynamic force.

5 Validation of design charts

The aforementioned design charts, prepared based on model tests, need to be validated for prototype foundations, which will ensure that these charts can be more pragmatic. This has been achieved by using scaling law [33], where all model pile parameters were scaled up for different lengths of full-scale prototype piles using different scaling factors. The study was conducted for three different lengths of pile, with the intention of estimating errors in evaluating the dynamic behavior of pile with increasing length. Tab.1 represents the scaling law and scaling factors used in the study, by which parameters of prototype piles have been fixed. Three different scaling factors (n = 3, 4, 5) are used to simulate three different lengths of the pile. The following expression is utilized in computing flexural rigidity of pile:

(5)\frac{E_{\mathrm{p}}{I}_{\mathrm{P}}}{E_{\mathrm{m}}{I}_{\mathrm{m}}}=n^{4+\propto },

where E_{\mathrm{p}}, E_{\mathrm{m}} are the Young’s modulus and I_{\mathrm{P}},I_{\mathrm{m}} are the moment of inertia of prototype and model pile, respectively. The \propto value quantifies the dependency of stress level on the system stiffness and in this study, as the soil is characterized as silt, the value is considered to be 0.5. Thickness of the hollow steel pile wall is computed for all the lengths of prototype piles using Eq. (5).

The study has been extended by including a comparative study for a single and a 2 × 2 group pile under both vertical and coupled modes of vibration using different analytical methods to check the reliability of analytical method used in this study. Further, an attempt has been made to check the effectiveness of the proposed design charts using available experimental data from different literature.

5.1 Comparison of model and prototype pile responses

Validation of design charts was implemented by comparing analytical prototype pile responses with scaled up experimental responses. Using the parameters of the proposed design charts, numerical analysis was performed on prototype piles with DYNA 5 [40] software based on the continuum approach, thereby acquiring its dynamic response under different eccentric moments. All the basic soil properties and pile properties required for the analysis were scaled up using scaling factors as mentioned in Tab.1. Tab.2 illustrates all the parameters used for analysis of all three prototype piles (length = 17.4, 23.2, and 29.0 m)

This study proceeded by scaling up the experimental frequency–amplitude responses of model piles using the scaling law shown in Tab.1, for obtaining prototype pile responses. Further, analytical responses of prototype piles were compared with this scaled up experimental responses of the prototype piles. The studies were extended for both single and 2 × 2 pile group under vertical and coupled modes of vibration for three different lengths.

Fig.10 demonstrates the comparison curve of analytical and experimental responses of both single and 2 × 2 pile group prototypes under vertical vibration, for 23.2 m length pile. It can be observed from the figure that the responses are well matched.

Fig.11 and Fig.12 illustrate the comparison of analytical and experimental prototype pile responses of single and 2 × 2 pile group under coupled vibration for both horizontal and rocking modes respectively. In coupled vibration, a second peak can be observed in both modes for a single pile and it has shown a minimal deviation between both responses. However, for group piles, a second peak is not observed within the frequency limit considered.

These comparative graphs explained above clearly depict that the proposed design charts from model tests can foresee the dynamic response of prototype pile foundation, and so these charts can be used for ascertaining the frequency–amplitude responses of the pile foundation under rotating-machine induced vibration.

Tab.3 illustrates the analytical and experimental responses for single piles with different lengths, under vertical mode of vibration. Similarly, Tab.4 condenses responses of group piles under coupled vibration and all responses showcased decreasing resonant frequency and disproportionally increasing resonant amplitude with increasing eccentric moments, thereby indicating the nonlinear response of the soil–pile system.

5.2 Estimation of average error

The comparison of analytical responses of prototype pile and scaled up experimental test results expressed marginal variations and these variations were observed to increasing with an increase in pile length. Therefore, estimation of error with increasing pile length is decisive for a detailed and precise estimation of dynamic response of pile. Analyses were performed with different pile lengths 17.4, 23.2, and 29 m for estimating average error in computing resonant frequencies and amplitudes. At first, errors in computing resonant frequency and amplitude values for varying eccentric moments were computed from both analytical and scaled experimental results for a particular pile length. Further, error values obtained from different eccentric moments were averaged out for all the pile lengths. Fig.13 and Fig.14 illustrate the variation of average error in estimating resonant frequency and amplitude with different pile lengths for coupled and vertical vibrations respectively.

From Fig.13, it can be observed that, for a single pile under coupled vibration, the average error for resonant frequencies and resonant amplitudes were found to deviate by 25% and 12% for the maximum pile length (29 m) considered. However, as per Fig.14, for 2 × 2 pile group under vertical vibration, the average errors for resonant frequencies and amplitudes in both responses were found to vary by 8% and 6% respectively. Similarly, studies have also been extended to all other sets considered, and average errors were estimated for different pile length. As this deviation in values was within the reasonable limit, this chart can be considered for assessing the error associated with pile length increment. This discrepancy in responses can be optimized by choosing different available scaling laws and therefore, for future use, their suitability can be chosen as per the acceptable limit of error.

5.3 Comparison of different analytical methods

A comparative study was performed for single and 2 × 2 group pile under both vertical and coupled mode of vibration using three different analytical methods: 1) Novak [41], 2) Novak and El Sharnouby [42], 3) DYNA 5 [40]. The purpose was to check the reliability of analytical methods used in this study. The analytical method described as in Refs. [41,42] performed linear analysis of the soil–pile system using frequency independent stiffness and damping. The frequency–amplitude responses were obtained using the above-mentioned analytical methods considering averaged dynamic soil properties. To compute the responses of pile groups by former two methods, interaction factors [43] were used for coupled vibrations respectively. The frequency–amplitude responses of both single and group piles obtained from three different analytical methods were compared with experimental results for vertical and coupled vibrations which is illustrated in Fig.15 and Fig.16, respectively.

The comparison graph (Fig.15) for a single pile under vertical vibration with different analytical method showcases that, DYNA 5 [40] responses was more similar to the experimental responses than were the other two methods considered in this study. The graph also shows that other two methods overestimate resonant frequency and amplitude values relative to experimental responses. Similarly, comparison plot of a 2 × 2 pile group under coupled vibration (Fig.16) with different analytical methods exhibits a reasonably well-matched response of DYNA 5 [40] with experimental response, as compared to other methods. From this comparative study, it is clearly understandable that DYNA 5 showcased better consistency with experimental data than other analytical methods.

5.4 Validation of design chart using literature

With a focus on checking the effectiveness of the proposed design charts, these charts were also validated using available experimental data from different literature. The equivalent linear analysis was performed using the proposed design charts to acquire frequency–amplitude responses of different soil–pile setups, as described in Refs. [29,30], for vertical and coupled vibrations. Further, these analytical frequency–amplitude responses were contrasted with experimental responses presented in the literature to get a better insight of the efficiency and plausibility of the proposed design charts. The studies were extended for both a single and a group pile under vertical and coupled vibrations. Fig.17 compares the experimental frequency–amplitude responses of a pile group from vertical vibration tests [29] with equivalent linear analysis utilizing the proposed design charts. Fig.18 shows the comparative curve of frequency–amplitude response of single pile acquired from the coupled vibration tests [30] and equivalent linear analysis done for the same setup using the proposed design charts.

The plots illustrated above in both cases exhibit a nonlinear pattern, with a decreased resonant frequency and disproportionately increased resonant amplitude. For a single pile under coupled vibration, a second peak is observed in both horizontal and rocking modes of vibration for the frequency range considered. From the above comparison curves, it can be seen that analytically attained response curves are well matched with experimental results for both a single pile and a group pile, which establishes that the design chart can predict the dynamic responses reasonably well for similar types of soil–pile condition.

5.5 Practical application and scope of the study

This study is an attempt to acquire a simplified approach toward predicting the nonlinear frequency–amplitude response of pile foundations subjected to machine induced harmonic motions using continuum approach analysis. This method requires values of boundary zone parameters as its prerequisites, but guidelines for acquiring an exact values are not given in the past studies. Therefore, this study developed a new concept of design charts to obtain values of boundary zone parameters with variation of dynamic load intensity and depth of the pile. This study also provides a novel technique to utilize scaling laws in finding prototype pile responses from small scale model responses.

These charts will be beneficial to design the pile foundations that support machines, as used in many industries including petrochemical, petroleum, power, cement, steel, nuclear power, oil refining etc. These industries use machines, such as turbo generators, rotary compressors, steam turbines, and pumps, which produce high magnitude harmonic loads, so there is demand for pile foundations rather than other type of foundations. Hence, proposed design charts can be utilized in acquiring safe and secure design of pile foundations supporting such machines.

From the charts, soil–pile separation and boundary zone parameters can be estimated for other pile foundations for different eccentric moments and depths. These charts were developed based on site-specific as well as limited number of dynamic test results. Hence these charts are applicable for akin types of soil–pile condition and should be wisely used in future studies. The study exhibits that errors in estimating responses were found to increase with increase in pile length; therefore, the studies can be widened to different aspect ratios, group arrangements, and spacing of piles and design charts can be modified accordingly. It is also recommended that, design charts should be used with proper engineering judgement for longer piles. Furthermore, these charts can be made more universal by extending the studies to different soil conditions. The charts can also be fine-tuned by broadening the studies to different pile group arrangements and spacing. In addition, the effect of pile installation technique on the dynamic response can also be studied separately.

6 Conclusions

This paper presents various design charts to evaluate soil–pile separation length and boundary zone parameters, which are the essential prerequisites for determination of the nonlinear dynamic behavior of pile foundations using continuum approach analysis. The usability of the proposed charts is verified for different length of prototype pile foundations using scaling laws. The effectiveness of the analytical method used in this study is further gauged using different analytical methods, to make it more trustworthy. The reliability of the charts is also verified by utilizing experimental data of different existing literature. The major conclusions are stated below.

1) It is observed that analytical response curves of prototype pile foundations are reasonably well matched with the scaled up experimental response curves for both modes of vibration. Hence it is concluded that the design charts proposed in the study for estimating influencing parameters can be used for full-scale pile foundations.

2) It is found that analytical response of all the prototype pile foundations exhibit nonlinear response by a decrease in resonant frequency and disproportionate increase in resonant amplitude with an increasing eccentric moment.

3) The error in estimating resonant amplitude and resonant frequency is observed to increase for both experimental and analytical response with an increase in pile length.

4) The comparison of analytical response curve of various soil–pile setups in the literature, obtained from proposed design charts with experimental response, exhibits a well-matched response for both modes of vibration, which further makes the chart more reliable.

5) A comparative study has been done for a single and a 2 × 2 pile group under both modes of vibration using three different popular analytical methods to check the reliability of the chosen method. It is observed that DYNA 5 [40] shows better response than do the other two methods [41,42].

6) A chart has been developed for estimating separation length and it has been noted that for a single and a 2 × 2 pile group under both modes of vibration, separation length exhibits an increase with increased eccentric moment.

References

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