Pile end bearing capacity in rock mass using cavity expansion theory

Haythem Gharsallaoui, Mustafa Jafari, Alain Holeyman

Department of Civil and Environmental Engineering (GCE), Institute of Mechanics, Materials and Civil engineering (iMMC), Université Catholique de Louvain(UCLouvain), Louvain-la-Neuve, Belgium

Keywords:Hoek-Brown failure criterion Spherical cavity Limit pressure End bearing capacity

ABSTRACT Much empiricism is involved in design of rock-socketed piles in rock masses.In light of this,an analytical solution based on the cavity expansion theory is proposed for calculating the ultimate bearing capacity at the tip of a pile embedded in rock masses obeying the Hoek-Brown failure criterion. The ultimate end bearing capacity is evaluated by assuming that the pressure exerted at the boundaries of a pressure bulb immediately beneath the pile tip is equal to the limit pressure required to expand a spherical cavity. In addition, a relationship is derived to predict the pile load-settlement response. To demonstrate the applicability of the presented solution, the results of this study were compared to those of 91 field tests from technical literature.Despite the limitations,it is found that the end bearing resistance computed by the present work is in good agreement with the field test results.

1. Introduction

Piles are commonly-used forms of foundations that provide support for structures, transferring their load to layers of soil or rocks that have enough bearing capacity and suitable settlement characteristics.Nevertheless,assessment of the base resistance of a pile embedded in rock mass is a common issue in civil engineering towards which several empirical approaches have been proposed(Coates, 1965; Rowe and Armitage, 1987; Le Tirant and Marshall,1992; Zhang and Einstein,1998; Vipulanandan et al., 2007). Some of these approaches are based on either full-scale in situ tests or laboratory tests. The empirical relationships take the form of a linear/power function of the unconfined compressive strength(UCS) of the intact rock σcias

where qb.ultis the ultimate end bearing capacity,and α and k are the bearing capacity coefficients (dimensionless). Table 1 summarizes the values of α and k for some empirical approaches.

To develop these empirical relations,different interpretations of the load test data are used by original authors. According to those interpretations,the ultimate end bearing capacity is defined by way of conventions as:

(1) The bearing resistance at a certain pile head displacement,say 10%of the pile diameter;

(2) The bearing resistance at a certain pile base displacement,say 5%of the pile diameter;and

(3) The maximum applied test load.

Moreover,the ultimate end bearing capacity can be deduced by using either the finite element method or the hyperbolic load transfer function approach to analyze the load-displacement response of the rock-socketed piles. Indeed, the measured loaddisplacement curve can be matched and subsequently the ultimate end bearing capacity can be determined at a certain pile base or head settlement of 5% or 10%. Because of these different interpretations, it is hard to figure out which one gives accurate estimations of ultimate limit state (ULS). Besides, the “so-called”ultimate end bearing capacity determined from Eq. (1) is not necessarily the ‘true’ ultimate one.

On the other hand, these empirical relations have limitations related to using only the UCS of the intact rock σcito predict the end bearing capacity.σciis only one of many other parameters that affect the strength of the rock mass. With that in mind, Zhang(2010) developed a new relation for determining qb.ultby considering the effect of discontinuities, represented by the rock quality designation(RQD),on σci.Still,his empirical relation has limitation related to using only RQD to represent the effect of discontinuities.Many other factors of the rock mass rating (RMR) system such as spacing, condition and orientation of discontinuities have a significant effect on the strength of the rock mass.In light of this,rock masses can be better characterized by the Hoek-Brown (H-B)failure criterion(Hoek et al.,2002),which allows for rock structure and surface conditions using the geological strength index (GSI).Kulhawy and Prakoso (1999) stated that practicing engineers should use GSI as an alternative to the RQD in order to prescribe the rock surface state. As a result, this failure criterion has often been used in the literature to provide analytical solutions.The analytical methods for computing the end bearing capacity of piles are mostly based on two general approaches: the method of characteristic lines and the cavity expansion method.

Table 1End bearing capacity coefficients for some empirical approaches.

Using the characteristic lines method,Serrano and Olalla(2002)developed an analytical method for computing the bearing capacity at the tip of a pile embedded in a rock mass that obeys the 1997 version of the H-B failure criterion(Hoek and Brown,1997).Their solution was later generalized for the 2002 version (Hoek et al.,2002). Their derivation is based on the analyses of a wall in plane strain conditions.By adopting an associated flow rule,they used the characteristic lines method generalized for nonlinear failure criterion to analyze the plastic zone. The major shortcoming of their method is that the rock mass is hypothesized as rigid-plastic.Hence, the bearing capacity does not depend on the elastic parameters of the rock mass.Besides,results gained from their theory,when compared to in situ test results(Serrano et al.,2014),showed a high scatter and did not yield satisfactory results which put in doubt their methodology and its applicability to such problem.

In this paper,an analytical solution of pile end bearing capacity is proposed by employing the cavity expansion method. In this approach,the pile is assumed to be sufficiently embedded such that surface effect is neglected. It is further assumed that the pressure exerted at the boundaries of the pressure bulb of rock immediately beneath the pile tip is equal to the limit pressure required to expand a spherical cavity. Then the problem of pile end bearing capacity reduces to determination of the limit pressure.To this end,Gharsallaoui et al. (2019, 2020) developed an original relationship between cavity expansion and pressure exerted to the cavity wall for both perfectly plastic and brittle plastic H-B post failure. Besides, an analytical solution of the limit pressure was proposed by considering the problem of both cylindrical and spherical cavity expansions in an infinite elasto-perfectly plastic H-B material,and by adopting the large strain theory (Gharsallaoui et al., 2020).Therefore,a solution for pile end bearing capacity problem will be proposed based on the aforementioned analytical solutions of the limit pressure.

This paper is organized as follows. A brief recall of the generalized H-B failure criterion and its scaled form is provided in Section 2. Description of the considered problem is detailed in Section 3. Basic hypothesis for computing the limit pressure required to expand a spherical cavity and the expression of the analytical solution of the limit pressure are summarized in Section 4.Section 5 describes two failure mechanisms implemented for the estimates of the ultimate end bearing capacity. Finally, a comparison with state of the art is given in Section 6.

2. Hoek-Brown material behavior

The H-B failure criterion was first introduced in 1980 with its latest update in the 2018 edition (Hoek and Brown, 2019). It is defined as (positive compressive stresses):

It should be emphasized that the above failure criterion expression can be simplified and scaled as formulated by Gharsallaoui et al. (2019, 2020), to give the following form:

where β is a constant solely depending on the exponent a≠1, and β = a／(1 - a).

On the other hand, normal and shear stresses are related to principal stresses by the following equations (Balmer,1952):

where σ′nis the normal stress; τ is the shear stress; and

With the same manipulations,normal and shear stresses can be scaled.It is found that their expressions are related to each other in a parametric form depending exclusively on the scaled minor principal stress σ*3:

Use of Eqs. (7) and (8) rather than Eqs. (5) and (6) leads to important simplification in the analysis of stresses in rock mass.This will be detailed in Section 5,when considering the problem of the ultimate end bearing capacity of a pile embedded in H-B material.

Fig.1. Sketches of failure mechanisms around the pile tip: (a) Yasufuku and Hyde (1995), and (b) Vesic (1973).

3. Problem statement

An important assumption in the use of cavity expansion theory in the context of pile end bearing capacity is that the pressure exerted at the boundaries of the plastic bulb immediately beneath the pile tip is equal to the limit pressure Plimrequired to expand a spherical cavity. Then, the ultimate end bearing capacity qb.ultcan be related to the limit pressure based on two modes.

The failure mechanism of mode 1,which was initially postulated by Yasufuku and Hyde (1995) for crushable sand, is illustrated in Fig.1.In this mode,active earth pressure conditions are considered to exist immediately beneath the pile tip and moment equilibrium is considered about point B. The ultimate bearing capacity of this mode can be found as a function of the friction angle φ of the soil as

The failure mechanism of mode 2 is also illustrated in Fig. 1(Vesic, 1973). This method assumes a rigid cone of soil beneath the pile tip, with an angle α = π／4+φ／2. Outside the conical region,it is assumed that the normal stress acting on the cone face is equal to the limit pressure Plim.A relationship between(qb.ult)2and Plimcan be obtained considering vertical equilibrium, i.e.

where the shear stress τ is given as per Eq. (6).

It should be emphasized that unlike the model proposed by Yasufuku and Hyde (1995), the Vesic (1973)’s model takes into account both cohesion and friction angle of the soil. Given that an H-B material is characterized by an instantaneous cohesion and an instantaneous friction angle (depending on the confining stress),the Vesic (1973)’s model is considered to be more reasonable to describe the failure mechanism of pile embedded in H-B material.A comparison between results gained from these two modes is detailed in Section 5.

For both modes, all that remains is to provide how the limit pressure of a spherical cavity is calculated for a material obeying the nonlinear H-B failure criterion.This will be summarized in the following section.

4. Limit pressure for expanding spherical cavity

In this section, we evaluated the limit pressure required to expand a spherical cavity in H-B material. Details of how the expression of the limit pressure was derived could be found in Gharsallaoui et al. (2020). Only basic hypothesis and main results are presented hereinafter.

4.1. Basic hypothesis

A spherical cavity expansion in an infinite elasto-perfectly plastic H-B material is considered. The geometry of the problem and the boundary conditions are depicted in Fig. 2. Let ribe the internal radius of the cavity and P0the far field radial pressure.Let Pibe the internal pressure applied on cavity wall that increases monotonically from its initial value P0. As the internal pressure Piincreases, the rock mass will initially behave in an elastic manner,until reaching a yield pressure Pyield.When the internal pressure Piexceeds Pyield, a plastic region will start spreading from rito the“plastic”radius rp.The remainder of the domain(r ≥rp)belongs to the elastic region.

Note that the major and minor principal stresses are assumed to be equal to the radial and circumferential stresses,respectively,i.e.σ1= σrand σ3= σθ.

Fig. 2. Geometry of the problem and boundary conditions.

4.2. Limit pressure

By adopting the large strain theory, Gharsallaoui et al. (2020)demonstrated that cavity expansion and pressure exerted to the cavity wall are related to each other by the following implicit relationship:

where ri0is the original cavity radius before cavity expansion starts;Ω is the inverse of the normalized plastic radius defined by Ω = ri／rp= (Rp／Ri)β／k;ω is the dilatancy coefficient computed by ω = (1-sin ψ)／(1+sin ψ)with ψ as the dilatancy angle;and D1,D2and η are the dimensionless coefficients defined as

The parameter δ in Eq. (11) is the normalized displacement at the elasto-plastic boundary,which is expressed as a function of the rigidity index and the scaled far field and yields pressures as

The parameters Rpand Riin Eq. (11) are the lower and upper integration limits,respectively.They are defined as solutions of the following equations:The parameter W0in Eq. (11) is the 0th branch of the Lambert function (Omega function). Note that the scaled yield pressure P*yshould be computed from the following expression:

The above equation provides an explicit solution when a = 0.5 as

Fig.3. Limit pressure chart by setting ψ = 0° and ν = 0.25(after Gharsallaoui et al.,2020).Values of =10-3,5×10-3,10-2,5×10-2,10-1,and 1 correspond to the groups of curves from top to bottom.

Fig. 4. ‘True’ ultimate bearing capacity for failure mechanisms of modes 1 and 2.

When a≠0.5,there is no explicit solution and Eq.(16)should be solved numerically. On the other hand, the scaled limit pressure P*limcan be found by putting ri／ri0→+∞in Eq. (11), i.e.

It can be noted from Eqs. (11)-(18) that P*limdepends on five main parameters, which are the far field pressure P*0, the rigidity index Ir, the Poisson’s ratio ν, the dilatancy angle ψ and the exponent a (i.e. GSI).

A closed-form solution of the limit pressure can be provided by assuming an intact (a = 0.5), incompressible (ν = 0.5) H-B material with no plastic volume change(ψ = 0°):

A chart allowing easy and accurate estimation of the limit pressure is depicted in Fig.3.The chart is created for ψ = 0°.This value leads to conservative estimations of the limit pressure.Indeed, Manandhar and Yasufuku (2013) stated that no plastic volumetric strain should be assumed when computing Plim,i.e.the dilatancy angle will be considered to be zero for large strain analyses.

5. End bearing capacity

In this section,details of how the ultimate end bearing capacity is calculated are given below firstly for failure mechanism of mode 1 and then for mode 2. For failure mechanism of mode 1, the relationship between (qb.ult)1and Plimis given as per Eq. (9).Knowing that the friction angle φ is determined from the slope of the tangent to the H-B failure envelope as

By substituting the above expression into Eq.(9)and after some derivations,it is demonstrated that the scaled ultimate end bearing capacity ()1and the scaled limit pressureare related to each other using a parametric form depending exclusively on the scaled circumferential stressas follows:

Table 2Rock mass properties and computed end bearing resistance qb for different settlement values.

Fig. 5. Ultimate end bearing capacity as a function of the unconfined compressive strength σci: Field test vs. empirical relations from Coates (1965), Rowe and Armitage (1987),Le Tirant and Marshall (1992), Zhang and Einstein (1998), and Vipulanandan et al. (2007).

For failure mechanism of mode 2, the relationship between(qb.ult)2and Plimgiven as per Eq.(10)is further expressed in a scaled form as

where

Subsuming Eqs. (8) and (20) into Eq. (22a) results in following parametric form relating ()2andby means ofas

Fig. 6. End bearing resistance: Present work solution vs. empirical approaches from Le Tirant and Marshall (1992) and Vipulanandan et al. (2007).

On the other hand, due to the high capacity of piles in rock masses, it is rarely possible to mobilize the bearing capacity corresponding to the ultimate limit state.As a result,it is important to estimate the end bearing capacity of a pile at a certain settlement level at the pile base in the context of performance-based design.In light of this, it will be necessary to compute the ultimate bearing capacity and the load-settlement behavior of piles.To achieve this,it will be further assumed that the volume of rock mass immediately beneath the pile tip which is displaced due to the pile settlement sbis equal to the increase of spherical cavity volume,i.e.

The instantaneous cavity radius rican be expressed as a function of the pile base diameter B as (Yang, 2006):

Therefore,Eq. (24) can be further expressed as

where ri0／riis given as per Eq. (11). Referring to Eq.(20),the term cos φ in Eq. (26) can be substituted by the following expression depending exclusively on the circumferential stress(Ri) as

where

To sum up,the procedure described below should be followed to evaluate the ultimate end bearing capacity(sb／B→+∞)as well as the end bearing resistance for a given settlement sb／ B of a pile embedded in H-B material:

(1) Choose input parameters: mi,σci, D, P0, G (or E),ν and ψ =0°;

(2) Evaluate H-B derived parameters: a, mband s;

(3) Evaluate intermediate parameters:and

(4) Evaluate the scaled limit pressureas per Eq.(18)or read it approximately from spherical cavity chart (cf. Fig. 3);

(5) Evaluate the scaled ultimate end bearing capacityfor both modes using Eqs. (21) and (23); and

(6) Evaluate the end bearing resistance for a given settlement sb／Busing Eq.(11)combined with Eq.(26)for.Then,is evaluated as per Eq.(21)for mode 1 and as per Eq.(23)for mode 2.

An example is given below for computing the ultimate end bearing capacity as well as the end bearing resistance of a pile embedded in H-B material with properties similar to that of an undisturbed (D = 0) claystone (Hoek, 2006). Table 2 summarizes the rock mass properties and the computed end bearing resistance qbfor different settlement values.

Fig. 7. Load-displacement curves: Present work vs. field test.

6. Comparison with field test results

To demonstrate the applicability of the proposed analytical solution,91 field tests from the technical literature are summarized in Table A1 in the Appendix. For each case, the rock layer characteristics, measured end bearing resistance and settlements were identified.

It is further assumed that all rocks listed in Table A1 obey the HB failure criterion.Indeed,these type of rocks were characterized as H-B material and their parameters are published in Hoek (2006).

In order to compute the bearing capacity, the elastic and mechanical properties of the rock mass (deformation modulus Erm,UCS,etc.)were retrieved from the original papers.However,it was still necessary to estimate the parameters of the H-B criterion:GSI,miand the elastic modulus for some particular cases. The parameter miwas estimated based on published guidelines and tables for different rock types(Hoek, 2006).Similarly,the elastic modulus of intact rock was also estimated based on rock type when necessary(Hoek and Diederichs, 2006).

In order to profit from the large amount of test data accumulated using the traditional RMR system, the GSI values can be calculated after Hoek et al. (2013) as

where JCond89 denotes the joint condition rating that varies from 0 to 30.When RQD was not available,the GSI values were estimated based on the rock surface description in the original papers.

Finally,the analytical and measured end bearing resistances are compared in Table A1. The results indicate the crucial role of incorporating the allowable pile settlement in the bearing capacity computations.

A comparison between measured end bearing capacity and those predicted from the empirical relations(Eq.(1))is depicted in Fig. 5. As can be seen, significant scatter exists. This may be attributed to the fact that these empirical relations rely only on the UCS of the intact rock mass σciand do not take into account the rock structure and surface conditions represented by GSI.The measured end bearing resistance is plotted against the computed bearing resistance in Fig. 6. For analytical solution, the mean values of modes 1 and 2 were considered. As can be seen, the present work solution provides better estimations in comparison with the empirical method (Eq. (1)).

To better demonstrate the benefits of the proposed method,the measured load-displacement curve is visualized against the computed curve according to the present work.Fig.7 displays these curves for cases 2, 3, 4, 6, 7, 8, 74, 86, 87 and 89. As observed, the general trends of the curves agree well.

7. Conclusions

An analytical method for computing the ultimate end bearing capacity qb.ultof a pile embedded in a rock mass that obeys the H-B failure criterion was proposed. This method was developed based on the assumption that the pressure exerted at the boundaries of the bulb of rock immediately beneath the pile tip is equal to the limit pressure Plimrequired to expand a spherical cavity. To compute the limit pressure,an analytical solution which required a numerical integration over the plastic region was presented.Thereafter,qb.ultwas evaluated as a function of Plimby adopting two failure mechanisms around the pile tip. The limit pressure was computed by assuming a no dilatant H-B material (ψ = 0°). This assumption was on the conservative side since Plimincreased for an increasing ψ value.

Results derived from the present work were compared to field load test results of 91 specific cases. It is found that developed solution agreed well with experimental results.

However, the existing empirical relations for estimating the ultimate end bearing capacity do not give reliable estimates.These empirical relations have limitations related to using only the UCS of the intact rock σcito predict the end bearing capacity and this parameter is only one of many other parameters that affect the strength of the rock.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2020.03.004.

Nomenclature

riCavity radius

r Radial coordinate

rpPlastic radius

PiInternal cavity pressure

PyYield pressure

P0Far field pressure

PlimLimit pressure

qbEnd bearing resistance

qb.ultUltimate end bearing capacity

GSI Geological strength index of the rock

RMR Rock mass rating

miStrength parameter of the intact rock

D Disturbance factor of the rock

s， a， mbHoek-Brown derived parameters

EiDeformation modulus of the intact rock

ErmDeformation modulus of the rock

G Shear modulus

IrRigidity index

σ′1Major principal stress

σ′3Minor principal stress

σrRadial stress assumed equal to σ′1

σθCircumferential stress assumed equal to σ′3

σciUniaxial compressive stress of the intact rock

σhActive earth pressure

τ Shear stress

σ′nNormal stress

ν Poisson’s ratio

φ Friction angle of the rock mass

ψ Dilatancy angle of the rock mass

ω Dilatancy coefficient