J. Cent. South Univ. (2020) 27: 1870-1879

DOI: https://doi.org/10.1007/s11771-020-4414-y

Influence of excavation beneath existing building on dynamic impedances of underpinning pile considering stress history

LIANG Fa-yun(梁发云)¹, CAO Ping(曹平)¹, QIN Hong-yu(秦红玉)²

1. Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China;

2. College of Science and Engineering, Flinders University, Adelaide 5001, Australia

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: The addition of basement beneath existing building changes the underpinning pile from fully embedded to partially embedded, and thus influences the mechanical properties of pile. In the past, scholars paid attention to the change in the bearing capacity of pile but neglected the difference of dynamic characteristics before and after construction, and potential changes in stress history of remaining soil are also ignored. In this work, a calculation model is built to investigate the influence of excavation on dynamic impedance of underpinning pile considering the effect of stress history. The soil is simulated by the dynamic Winkler foundation, which is characterized by springs and dashpots. Properties of remaining soil after excavation are updated to consider the effect of stress history through modifying the initial shear modulus and related parameters. The dynamic impedance of pile after excavation is obtained based on the transfer matrix method. The parameter study is carried out to evaluate the dynamic impedance with various excavation depths, considering or ignoring stress history effect, and various element lengths. The results show that shallow soil plays an important role to dynamic impedance, and overestimated dynamic impedance is obtained if not considering the stress history effect.

Key words: pile; transfer matrix method; dynamic response; stress history; underground space development

Cite this article as: LIANG Fa-yun, CAO Ping, QIN Hong-yu. Influence of excavation beneath existing building on dynamic impedances of underpinning pile considering stress history [J]. Journal of Central South University, 2020, 27(6): 1870-1879. DOI: https://doi.org/10.1007/s11771-020-4414-y.

1 Introduction

The development of the underground space of existing building refers to the addition of basements beneath the building whose original designs do not contain underground space. In China, there are many related projects [1-3]. The pile foundation underpinning method is a feasible and common used technology, which can be summarized as 3 steps: 1) driving piles into soil, 2) connecting piles with superstructure by pile cap and 3) soil excavation and basement construction. The pile used to carry the load of superstructure and to ensure security of the construction is called underpinning pile, which can be used as a pillar of basement after excavation. During the construction, the shallow soil around the underpinning pile is removed in order to build the basement, and thus the pile changes from fully embedded to partially embedded. The change in mechanical properties of the underpinning pile foundation is of the most concern. In the past, researchers had concerned the change in the static bearing capacity of the underpinning pile [4-7], while limited attention has been paid to the change in the dynamic characteristics.

In fact, whether it is mechanical vibration during construction or environmental vibration from adjacent roads and subways, pile foundations are subject to various dynamic loads. Furthermore, most of China is located in seismic zone. For these reasons, the change in the dynamic characteristics of underpinning pile due to excavation beneath existing building cannot be ignored.

When using the substructure method for dynamic analysis, the soil-pile foundation- superstructure interaction can be considered, and one of the key problems is the calculation of the pile dynamic impedance. In recent decades, many methods for fully embedded piles have been presented, such as thin layer method, boundary element method, and transfer matrix method [8-12]. ZHAO et al [13] based on the finite element analysis model, analyzed the mechanical effect of pile under the dynamic load. However, the vibration characteristics of partially embedded piles are different from that of fully embedded piles. Based on the Green function, PAK [14] analyzed the internal forces and displacements of one- dimensional bar partially embedded in the elastic half-space under transverse dynamic loads. LEE et al [15] used the Runge-Kutta method to calculate the natural frequencies and the mode shapes of piles partially embedded in the Winkler type foundation. CATAL [16] analyzed the natural frequencies and relative stiffness of partially embedded pile subjected to bending moment, axial and shear force. REN et al [17, 18] took the pile as Euler-Bernoulli beam, investigated the vertical and lateral impedance of partially embedded pile. LIU et al [19] studied the lateral vibration response of partially embedded pile in saturated soil. However, the aforementioned studies mainly focused on the effect of cantilever length of pile to the dynamic response of pile but did not consider the effect of soil excavation. Soil excavation is an unloading process for deep soil, and makes changes to properties of remaining soil, including its stress history, thus changes the performance of pile. LIN et al [20, 21] found that the unloading caused by scour changes the relative density, unit weight, modulus, etc. of the remaining soil, and an overestimated lateral bearing capacity of pile would obtained if not considering stress history effect. LIANG et al [22] further studied the effects of stress history to critical buckling load of scoured pile. While the depth of removed soil caused by scouring is relatively small (commonly within 2d, where d is the diameter of pile [23]) compared with that by excavation (commonly 1/5~1/2 L, L is the length of pile [1-3]). Therefore, the studies about scoured pile are hard to represent the performance of excavation-affected pile. What is more, there is limited number of studies which have been conducted to investigate the dynamic response of unloading-affected pile. Hence, it is necessary to study the dynamic performance of underpinning pile under the larger excavation depth considering stress history.

This paper aims to propose a calculation model for pile dynamic impedance under the excavation condition and enables considering the effects of stress history. The pile is modeled by Euler beam, based on the transfer matrix method, the relationship between the load and the displacement of the pile tip and the pile top is obtained, and thus the dynamic impedance. The soil reaction is modeled in the form of dynamic Winkler foundation and the parameters are modified after soil excavation. Finally, parameter study is presented to illustrate the influence of soil excavation on the vertical and horizontal dynamic response of underpinning pile.

2 Calculation model

Figure 1 shows the schematic model for pile dynamic impedance after excavation.

The pile having length L and diameter d is embedded into an n-layered soil with an excavation depth of △h, and the pile embedded part is discretized into element per soil layer. An infinite excavation width is adopted, which means that all the soil above the excavation depth will be removed. The dynamic Winkler foundation is characterized by springs and dashpots, corresponding to the parameters of c_i and k_i (1≤i≤ n), respectively. The pile tip is embedded in the rock, which is assumed to be fixed. The dynamic load is applied at the pile top (z=0). Besides, only linear elastic deformation is considered.

Figure 1 Calculation model of pile dynamic impedance after excavation

2.1 Pile dynamic impedance of partially embedded pile

The dynamic Winkler foundation simulates soil by arranging continuously-distributed springs and dashpots along with the pile, and the parameters of the i-th layer soil can be evaluated using solutions by GAZETAS et al [24] and MAKRIS et al [25]:

k_zi≈                     (1)

c_zi≈                    (2)

k_xi≈1.2E_si                                                  (3)

c_xi≈                    (4)

where k_zi and c_zi are vertical stiffness coefficient and damping coefficient, respectively; k_xi and c_xi are lateral stiffness coefficient and damping coefficient, respectively; E_si, β_si and ρ_si are soil elastic modulus, damping ratio and density, respectively; V_si=  is shear wave velocity, G_0i is initial shear modulus; ω is the circular frequency of dynamic load, ω=2πf, f is frequency; a_0i=ωd/V_si is dimensionless frequency.

For each pile element, the relationship of displacement and force between top and bottom of the element can be described as a matrix. By applying the transfer matrix method, the relationship of displacement and force between the pile top and the pile tip can be obtained [17, 18]. The details are as follows.

For vertical vibration,

 (5)

        (6)

                 (7)

where w and N are the vertical displacement and force of the pile, respectively; [f_zi] (i>0) is the i-th matrix of the embedded part corresponding to the i-th layer soil; [f_z0] is the matrix of the cantilever section; h_i is the thickness of the i-th layer soil, especially h₀=△h; k_zi and c_zi are the vertical stiffness coefficient and vertical damping coefficient of each layer, especially k_z0 and c_z0 are equal to zero; m_p, E_p and A_p are mass per unit length, elastic modulus and cross-section area of the pile, respectively.

For lateral vibration,

 (8)

 (9)

              (10)

where u, H, φ and M are the lateral displacement, force, rotation angle and bending moment, respectively; I_p is the moment of inertial; k_1i=-h_i/λ_xi, k_2i=h_i²/(E_pI_pλ_xi²), k_3i=h_i³/(E_pI_pλ_xi³). Other variables are similar to vertical vibration.

Since the boundary condition of fixed end is assumed, which means that the displacement and rotation angle at the pile tip are 0, the relation of displacement and force of pile top can be obtained. The impedance is defined as the force at the pile top divided by the displacement at the pile top as:

                              (11)

                              (12)

where R_z and R_x represent vertical impedance and lateral impedance respectively. Because the impedance is a complex number, it can be expressed as a combination of real part, namely stiffness K, and imaginary part, namely damping C.

R=K+iC                               (13)

2.2 Soil parameters before and after excavation

The characteristics of soil are expressed as stiffness coefficient and damping coefficient corresponding to spring and dash pot respectively, which are related to V_s, E_s, β_s, ρ_s and v_s. For dynamic analysis, the β_s, ρ_s and v_s are often set to constant [8-12]. Meanwhile, E_s can be calculated as follows:

E_s=2(1+v_s)G₀                              (14)

where v_s is Poison ratio. Therefore, the modified parameters are limited to G₀.

HARDIN et al [26, 27] based on a series of test, presented empirical equations for both sand and clay, which are related to the mean effective stress p′ and the void ratio e. The empirical equations of sand are as follows (in kPa):

                (15)

             (16)

where σ′ is effective stress; OCR is over-consolidated ratio; φ′ is effective frictional angle. It should be mentioned that the OCR has a limitation as follows [28]:

                 (17)

Before and after excavation, the effective stress at the depth z from the surface is

                               (18)

                          (19)

where γ′ is the effective unit weight of the soil; σ′_bef and σ′_aft correspond to the effective stress before and after excavation respectively.

The soil undergoes an unloading process by excavation. LIN et al [21] based on the Cam-clay model, derived an equation for void ratio after unloading:

e_aft=e₀+△e                              (20)

         (21)

where e_aft is the void ratio after excavation; e₀ is the initial void ratio; κ is the unloading index obtained from the isotropic oedometer test, equal to 0.434C_ur; C_ur is the unloading index obtained from the one- dimensional oedometer test, which is proportional to the loading index C_c, and generally taken as C_c/5 [29]. The loading index can be related to the initial void ratio by an empirical formula C_c=0.40(e₀-0.25) [30].

Thus, the initial shear modulus G₀ after excavation can be estimated by establishing relationship with p′ and e₀ to consider the effect of stress history.

3 Parameter studies

The transfer matrix method described previously has been used to study the dynamic response of fully embedded pile, and has been validated by more sophisticated models [9]. It also has been used to analyze partially embedded pile, and has been justified by comparing with the field test [17]. These make the transfer matrix method a powerful tool to analyze the dynamic impedance of pile under the excavation condition. Because there is no literature investigating the pile dynamic impedance considering the effect of excavation, a case study is carried out.

With the analysis procedure, a single pile embedded in sand, subjected to dynamic loads at the pile top is investigated. The diameter, elastic modulus, density of pile are 1 m, 30 GPa and 2500 kg/m³, respectively. The density, damping ratio, Poisson ratio, frictional angle, initial void ratio are 1500 kg/m³, 0.05, 0.3, 30° and 0.78, respectively. The site condition is from TU’s report [31], in which the lateral dynamic response of a composite caisson-piles foundation affected by scouring is studied. The soil and the pile embedded part are evenly discretized into 50 elements.

Firstly, the modified parameters of the initial shear modulus, the void ratio and the OCR for soil around a pile with a length-diameter ratio of 20 are determined at three excavation depths (△h=L/6, L/3 and L/2), and the vertical and lateral dynamic impedances as the frequency varying from 0 to 30 Hz are presented to evaluate the effect of excavation depth. Then the impedances of three length-diameter ratios (L/d=15, 20 and 25) at extreme excavation depth (△h=L/2) are compared under the conditions of considering and neglecting the stress history effect. Finally, the effect of element length is considered at three frequencies by changing the number of element part, while the length-diameter ratio of 20 and the excavation depth of L/3 are kept unchanged.

4 Results and discussion

4.1 Modified parameters of remaining soil after excavation

The main soil parameters (i.e., the initial shear modulus, the void ratio and the OCR) before and after excavation of a pile with the length-diameter ratio of 20 are calculated at excavation depths of L/6, L/3 and L/2, as shown in Figure 2. The OCR_max calculated by Eq. (17) is 36.

The results indicate that excavation makes change to soil parameters, and the closer to the excavation surface, the greater the change. The void ratio and OCR are constant before excavation, and decrease with the increase of soil depth after excavation. The OCR is more affected, which is larger than 10 after excavation near the surface. The initial shear modulus increases with the soil depth, and decreases significantly after excavation. Compared with that before excavation, the initial shear modulus within 0.5 m from the excavation surface is reduced by more than 50%, while that of pile tip is reduced by less than 10%. The curves of initial shear modulus changing with the soil depth at different excavation depths tend to be parallel in the deep layer, because the void ratio and OCR in the deep layer change little compared with that before excavation, making that the initial shear modulus mainly depends on the vertical effective stress.

4.2 Effect of excavation depth

The effects of excavation depth on the dynamic impedance of a pile with length-ratio of 20 at excavation depths of L/6, L/3 and L/2, and compared with the impedance before excavation, are presented in Figures 3 and 4 for vertical and lateral vibrations respectively.

As plotted in Figure 3, the vertical stiffness before excavation changes little with frequency,whereas the vertical damping before excavation increases significantly with frequency (0-30 Hz). Furthermore, excavation reduces the vertical impedance, and the reduction increases with frequency. At the extreme excavation depth of L/2, the stiffness and damping are decreased by 25%-50% and 83%-88% respectively.

Figure 2 Calculated parameters before and after excavation:

Figure 4 reflects that the lateral impedance before excavation has a similar trend with vertical impedance before excavation. When the excavation depth is small (L/6), the effect of excavation is to reduce the impedance, and the stiffness becomes negative when the frequency exceeds 12 Hz, whereas it keeps positive before excavation. However, deeper excavation complicates the curve shape. When the excavation depth exceeds L/6, it can be observed that the removal of shallow soil makes a significant reduction of lateral impedance at low frequencies (0-10 Hz), while the peaks and valleys occur as the frequency varies from 10 to 30 Hz for both stiffness and damping. It is worth noting that the impedance after excavation is less than that before excavation, even at the peak. For the excavation depth of L/3 and L/2, the frequencies of stiffness at first valley are 22 and 12 Hz, respectively, and those of damping at first peak are 26 and 14 Hz respectively. Thus it can be inferred that the frequency at which the impedance first reaches peak or valley values decreases with the increase of excavation depth.

Figure 3 Vertical impedance before and after excavation (L/d=20):

Figure 4 Lateral impedance before and after excavation (L/d=20):

Accordingly, the effect of excavation depth is to reduce the impedance at both modes of vertical and lateral vibration, and it further changes the curve shape of impedance varying with frequency for lateral vibration.

4.3 Effect of stress history

Based on the calculation model discussed earlier, the stress history caused by excavation can be considered by modifying the parameters of remaining soil. In order to evaluate the effect of stress history, the dynamic impedance of a pile with length-diameter ratio of 15, 20 and 25 at extreme excavation depth of L/2, considering and ignoring stress history is determined for vertical and lateral vibrations respectively.

Figure 5 illustrates the vertical impedance considering and ignoring stress history. It can be seen that neglecting stress history results in 2%-16% higher vertical stiffness and 3%-60% higher damping compared with the cases that the stress history effect is considered. The results show that the stiffness increases with the increase of the length-diameter ratio. This is because the boundary condition of fixed end is adopted, the longer the pile length is, the larger the deformation of the pile body is, the smaller the stiffness of the pile top is. Whereas the damping decreases with the increase of the length to diameter ratio.

Figure 6 displays the effect of stress history on lateral impedance of 3 different length to diameter ratios. The result shows that ignoring stress history amplifies the impedance value of peaks and valleys. Compared with the impedance of considering stress history, the amplified value is even more than 100% (the damping for the length to diameter ratio of 25 at the first peak). On the other hand, considering stress history results in a left movement of the locations of peaks and valleys. That is to say that the effect of stress history reduces the frequency at which the impedance reaches its peak or valley by approximately 1-2 Hz. What is more, a larger length to diameter ratio makes a smaller frequency of first peak or valley.

In general, the results presented in Figures 5 and 6 indicate that the effect of stress history is decreasing the impedance. Thus a design ignoring stress history caused by excavation may be unsafe, especially for the lateral vibration. As a consequent, it is important to consider the effects of stress history due to excavation.

4.4 Effect of element length

As depicted in Figure 2, the soil parameters vary with soil depth, therefore, the pile embedded part needed to be segmented to obtain an accurate dynamic impedance, and so is the surrounding soil. Normalized impedance is defined as the impedance at n elements divided by the impedance at 50 elements. The normalized impedances of a pile before excavation with a length-diameter ratio of 20 at three frequencies of 5, 15 and 25 Hz are plotted in Figures 7 and 8. The effect of element length is assessed by changing the number of element from 1 to 50.

Figure 5 Vertical impedance considering and ignoring stress history (△h=L/2):

Figure 6 Lateral impedance considering and ignoring stress history (△h=L/2):

Figure 7 Vertical normalized impedance varying with segment part (L/d=20):

Figure 8 Lateral normalized impedance vary with segment part (L/d=20):

Figure 7 portrays the effect of element number to vertical normalized impedance at three frequencies. The frequency has negligible influence on the vertical normalized impedance. The normalized impedance is greater than 1 until the element number exceeds 10, which means that the element length of L/10 is sufficient for the accurate calculation of vertical impedance. In contrast, the effect of element length to lateral normalized impedance shown is different as shown in Figure 8. The trends are different at different frequencies, and the lateral normalized impedance is not stable until the number of element excesses 30.

Thus from Figures 7 and 8, the effects of element length on dynamic impedance are revealed. With the decease of element length, the normalized impedance tends to 1, verifying the stability of the calculation model. It is found that lateral vibration is more sensitive to element length compared with vertical vibration. An element length shorter than L/30 is recommended.

5 Conclusions

The dynamic response of underpinning pile before and after excavation due to the construction of basement beneath existing building is studied. The dynamic impedance of partially embedded pile is calculated by transfer matrix method in which the pile is divided into two parts of embedded and cantilever sections. The parameters of soil are modified after excavation to consider the effect of stress history. The modified parameters are firstly calculated, and the effects of excavation depth, the stress history, and the element length on vertical and lateral dynamic impedance are discussed in parameter study. The following conclusions can be obtained:

1) Excavation reduces the initial shear modulus, whereas the void ratio and the OCR increase with the excavation depth. Hence, an overestimate and unsafe result for design is obtained when ignoring the effect of stress history. The lateral vibration is more sensitive to stress history than vertical vibration.

2) Shallow soil contributes much to the dynamic impedance, especially for lateral vibration at low frequencies. The effect of excavation depth is to reduce the impedance. The curve shape of impedance with frequency changes little for vertical vibration, while for lateral vibration, peaks and valleys appear after excavation.

3) The pile of embedded part needs to be segmented evenly to obtain accurate impedance. Although the smaller the element length, the more accurate the impedance, L/30 is enough for accurate analysis of vertical and lateral vibration.

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(Edited by FANG Jing-hua)

中文导读

考虑应力历史的增层开挖对托换桩动力阻抗的影响

摘要：在既有建筑物下增设地下室会使得托换桩从完全埋入状态变为部分埋入状态，从而改变桩基的力学性能。过去学者们主要研究桩的承载力变化，而对施工造成的动力特性变化关注较少，也没有考虑土体开挖造成深层土的应力历史变化。本文建立了能够考虑应力历史效应的开挖后桩基动力阻抗计算模型，并研究了土体开挖的影响。在动力Winkler地基上用弹簧和阻尼器模拟土体。通过计算初始剪切模量和其他相关参数，对开挖后残余土体的力学参数进行修正，以考虑应力历史的影响。基于传递矩阵法获得了开挖后桩的动力阻抗。随后对开挖深度、应力历史以及分段长度等参数进行了分析。结果表明，浅层土体的约束作用对动阻抗起到了重要作用，如不考虑应力历史可能会高估动力阻抗。

关键词：桩；传递矩阵法；动力响应；应力历史；地下空间开发

Foundation item: Projects(51878487, 41672266) supported by the National Natural Science Foundation of China

Received date: 2020-01-30; Accepted date: 2020-03-13

Corresponding author: LIANG Fa-yun, PhD, Professor; Tel: +86-21-65986072; E-mail: fyliang@tongji.edu.cn; ORCID: 0000-0002- 3740-1110