Modified Shear Displacement Method for Analysis of Piles in Unsaturated Expansive Soils Considering Influence of Environmental Factors

Liu, Yunlong; Vanapalli, Sai K.

doi:10.1061/(ASCE)GM.1943-5622.0002320

Open access

Technical Papers

Apr 6, 2022

Modified Shear Displacement Method for Analysis of Piles in Unsaturated Expansive Soils Considering Influence of Environmental Factors

Authors: Yunlong Liu, Ph.D. [email protected], and Sai K. Vanapalli, M.ASCE https://orcid.org/0000-0002-3273-6149 [email protected]Author Affiliations

Publication: International Journal of Geomechanics

Volume 22, Issue 6

https://doi.org/10.1061/(ASCE)GM.1943-5622.0002320

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Abstract

Pile foundations are widely used in regions with expansive soils to transfer the loads from superstructures to a rigid soil stratum, thereby alleviating stability and deformation problems. However, current design procedures for pile foundations are typically based on saturated soil mechanics principles assuming drained conditions, despite the fact expansive soils surrounding the piles are typically in an unsaturated state. Volume changes of expansive soil induced by matric suction changes upon seasonal wetting and drying can significantly influence the pile load-transfer mechanisms. In this paper, using unsaturated soil mechanics as a tool, theoretical models are introduced for estimations of pile shaft friction and pile base resistance considering both the influences of matric suction changes and matric suction variation-induced volume changes of expansive soil around the pile. The commonly used shear displacement method for interpreting the pile mechanical behaviors is modified by including these models to extend their application to piles in unsaturated expansive soils. The proposed modified shear displacement method (MSDM) is interpreted and successfully validated using a pile infiltration test conducted in the laboratory along with another case study from published literature. The results of investigation studies using the proposed MSDM suggest that it can be a valuable tool for practicing engineers to make reasonable predictions of mechanical behaviors of piles in unsaturated expansive soils.

Introduction

Piles of significant length, strength, and stiffness that penetrate beyond a depth where there is no seasonal moisture content change with their end placed on rigid bedrock or in a stable soil stratum to perform reasonably well. Geotechnical engineers’ typical experience with pile foundations in expansive soil deposits of various regions of the world has been reasonably satisfactory in comparison with other types of foundations to alleviate stability and settlement problems and ensure safety of both the sub and superstructures (Chen 1988; Rojas et al. 2006; Dafalla et al. 2012; Nelson et al. 2015). For instance, cast in situ piles or drilled piles have been widely used to alleviate expansive soils problems in the Rocky Mountain area of the United States and in other regions of the world (e.g., Chen 1988; Al-Rawas and Goosen 2006; Nelson et al. 2015). Piles placed in expansive soils are typically designed assuming drained conditions (i.e., effective stress approach) extending saturated soil mechanics principles. However, soil surrounding the pile is typically in an unsaturated condition state. The mechanical behaviors of piles are significantly influenced by matric suction changes within the expansive soil around the pile associated with environmental factors.

Fig. 1 illustrates the influences of matric suction variations on pile load-transfer mechanisms considering volume changes associated with drying and wetting conditions in expansive soils. More specifically, as shown in Figs. 1(a)–(c), matric suction increases or decreases in expansive soil within the active zone during the drying and wetting periods, respectively from hydrostatic state. Fig. 1(e) highlights that after the pile is installed or constructed, a linear lateral earth pressure (LEP) profile develops due to the influence of soil self-weight, which typically corresponds to at-rest condition. Volume shrinkage of expansive soil in the horizontal direction causes a reduction in LEP acting on the pile due to water content changes associated with evaporation during dry seasons or drought periods. Cracks develop when the tensile stress develops due to the volume shrinkage of expansive soil. The LEP typically has a bilinear distribution when cracks develop along the pile–soil interface, as shown in Fig. 1(d). Conversely, upon wetting due to storms, flood, or snow melting, lateral swelling pressure (LSP) develops and acts as additional pressure to the conventional LEP, which is dependent on the soil unit weight, as shown in Fig. 1(f). However, the LEP should not exceed passive earth pressure (PEP) to avoid the shear strength failure of soil. Expansive soil deformation in the vertical direction (i.e., ground displacement) contributes to variations in pile shaft friction due to pile–soil relative displacement. The pile moves downward relative to the surrounding soil and exhibits a uniform distribution at hydrostatic state, as shown in Fig. 1(h), if there is no compression or tension deformation in pile in comparison with the pile displacement. The expansive soil within the active zone undergoes volume shrinkage during the drying process and contributes to ground settlement in the vertical direction, as shown in Fig. 1(g). Consequently, the pile–soil relative displacement gradually reduces from the depth of the active zone to the ground surface. The pile–soil relative displacement may reduce to zero or change its direction of displacement if there is significant soil shrinkage deformation. However, expansive soil ground heave occurs due to volume expansion in the vertical direction during the wetting process. For such a scenario, there will be a cumulative pile–soil relative displacement from the depth of the active zone to the ground surface, as shown in Fig. 1(i).

Fig. 1. Load-transfer mechanism variations of piles in expansive soils associated with drying and wetting conditions (LEP = lateral earth pressure; PEP = passive earth pressure; LSP = lateral swelling pressure; PSRD = pile-soil relative displacement; and SD = soil displacement).

Fig. 1(k) shows the load transfer mechanism of a single pile placed in expansive soil for hydrostatic condition. Prior to the water infiltration or evaporation, positive friction develops along the entire length of the pile to carry the pile head load, together with the base resistance. Upon evaporation, volume shrinkage of expansive soil arises in the active zone associated with an increase in matric suction and results in a ground settlement as well as a reduction in LEP, as discussed earlier. Ground settlement may change the pile–soil relative displacement that contributes to an upward movement of pile relative to the surrounding soil. Due to this, the pile shaft friction in the active zone changes its direction and acts as an additional load on the pile. Therefore, the pile undergoes a further settlement [Fig. 1(j)]. However, matric suction reduction associated with water infiltration triggers both ground heave and mobilization of the LSP. Ground heave in the active zone can increase the pile–soil relative displacement while the mobilized LSP acts as an additional LEP component. Under the influence of matric suction reduction, LEP increment and pile–soil relative displacement increment, the positive friction in the active zone may either increase or decrease (Liu and Vanapalli 2017). More specifically, if the contribution of increasing LEP to the pile shaft friction overweighs the reduction of pile shaft friction due to matric suction reduction, the positive shaft friction tends to increase, and vice versa. Based on the summarized discussions, two common scenarios that are typically encountered in practice are considered regarding the load-transfer mechanism variations [Fig. 1(l)]. There is a possibility of the pile to be uplifted, for a scenario in which pile shaft friction increases in the active zone. The increasing positive friction is commonly referred to as “uplift friction.” Once the pile is subjected to uplift, negative friction generates in the stable zone, and the pile base resistance reduces or even vanishes because of the possible detachment between the pile base and soil. On the contrary, the shaft friction reduction that contributes to pile settlement should be also considered as another possible scenario.

The discussions summarized with the aid of Fig. 1 considers two different scenarios (drying and wetting), highlighting three key factors that have significant influences on the load transfer mechanism of piles in unsaturated expansive soils, namely matric suction, matric suction variation-induced LEP, and ground displacement changes. Considering these factors, in this study, a series of theoretical equations are developed for evaluating both the pile shaft friction and pile base resistance mobilization based on the principles of unsaturated soil mechanics. These equations are then introduced into the traditional shear displacement method to extend its application to analyze and interpret the behavior of piles in unsaturated expansive soils. The methodology developed is collectively referred to as the modified shear displacement method (MSDM) in this paper. A comprehensive experiment was conducted in the laboratory to study the mechanical behavior changes of a model pile in an expansive clay from Regina, Canada, upon wetting. Experimental results were employed to interpret and validate the MSDM together with a field case study from the published literature. The results of the study present in this study suggest the proposed MSDM is a promising tool for use in routine engineering practice for the design of pile foundations in expansive soils.

Background

Traditional pile tests in conventional soils mainly focus on gathering the load-versus-deformation behavior; however, mechanical behavior variations associated with matric suction variations are required from pile tests in expansive soils. Currently, pile tests in expansive soils are typically performed either under zero pile head load, service pile head load, or displacement fixed pile head conditions (Benvenga 2005; Al-Rawas and Goosen 2006; Fan 2007; Albusoda and Abbase 2017; Sharma and Sharma 2019; Fattah et al. 2020; Abbas 2020; Jiang et al. 2020). The zero-pile head load and fixed pile head test are valuable for monitoring the pile uplift displacement and uplift force during the infiltration process, respectively. Various pile infiltration tests have been conducted under the service load to simulate the real scenarios that are typically encountered in engineering practice; only a few of them provide detailed description with respect to the pile load transfer mechanism changes, including pile axial force distributions, pile shaft friction distributions, and pile base resistance (e.g., Fan 2007; Jiang et al. 2020). Researchers are aware that matric suction affects the mechanical behavior of piles in unsaturated expansive soils; however, due to the limitations associated with gathering reliable information of matric suction over a long period of time, in most tests only water content variations of the soil around the pile were measured.

Both theoretical and numerical methods have been proposed for load transfer analyses of piles in unsaturated expansive soils by analyzing experimental results. The widely used theoretical methods for pile load transfer analysis include the shear displacement methods (Cooke 1975) and the load transfer methods (Coyle and Reese 1966). Fan et al. (2007) and Xiao et al. (2011) were the first researchers to propose a modified shear displacement method for piles in unsaturated expansive soils taking account of the influence of volume changes of expansive soils. This approach considers the pile shaft friction and pile base resistance mobilized due to the pile head load and the volume changes of expansive soils separately and then use the superposition method. Later, considering the changes associated with soil shear modulus with depth, Jiang et al. (2020) extended the shear displacement method, interpreting the behavior of piles in unsaturated expansive soils. However, due to limitations in testing techniques, these approaches failed to link the pile shaft friction variations associated with matric suction changes. The key parameter (i.e., shear modulus of soil G) required for the pile shaft friction calculations was obtained from regression analysis based on experimental results, which may be not applicable to other cases. During the last decade, several researchers focused on understanding the unsaturated soil-interface shear strength (Hamid and Miller 2009; Khoury et al. 2010; Hossain and Yin 2013; Shen et al. 2017; Cheng et al. 2018, 2019a, b). Based on these models, Liu and Vanapalli (2019a) modified the traditional load transfer method to extend its application to piles in unsaturated expansive soils, considering both the contribution of expansive soil volume expansion and matric suction to pile shaft friction. However, the proposed method is only suitable for pile load transfer analysis under wetting conditions instead of both wetting and drying conditions.

There are several numerical methods proposed for analysis of piles in expansive soils (e.g., Alonso et al. 1999; Al-Rawas and Goosen 2006; Hong 2008; Sharma 2019). However, several input model parameters must be obtained from complex experimental studies to achieve precise simulations. In addition, required for computations take time, especially for unsaturated soils. Therefore, theoretical methods are preferred in conventional engineering practice applications in comparison with presently available complex numerical methods.

MSDM

Fig. 2 shows the procedural steps used in computation of both traditional shear displacement method and MSDM. The traditional shear displacement method assumes a linear development of pile base resistance with pile base settlement and pile shaft friction with pile–soil relative displacement, as illustrated in Figs. 2(a and c), respectively. A two-phase model is employed for calculation of pile base resistance considering the possible detachment of pile base and soil due to pile uplift in the wetting process in unsaturated expansive soils [Fig. 2(b)]. Several other investigators also have employed two-phase softening models to better describe the behavior of pile shaft friction development associated with pile–soil relative displacement [Fig. 2(d)] (Fan 2007; Hamid and Miller 2009; Khoury et al. 2010; Hossain and Yin 2013; Borana et al. 2015). More information related to the MSDM are presented in the sections that follow.

Mobilization of Pile Shaft Friction

Several investigators in recent years investigated the shear behavior of unsaturated soil–structure interface and provided valuable information about the influence of soil-softening behavior (Fan 2007; Hamid and Miller 2009; Khoury et al. 2010; Hossain and Yin 2013; Borana et al. 2015). Fan (2007) proposed a softening relationship between interface shear stress and displacement based on interface direct shear tests between unsaturated expansive soils and concrete. More specifically, at the beginning of the shearing, shear stress (τ_s) increases linearly with shear displacement (w), while after reaching the peak value (τ_pk), it experiences a sudden decrease and finally stabilizes at a post-peak value (τ_pp). The philosophy suggested by Fan (2007) is extended in this paper with modifications. As shown in Fig. 3, the relationship between the shear stress and shear displacement is divided into two phases using three key parameters, namely the peak interface shear strength, (τ_pk), the post-peak interface shear strength, (τ_pp), and the critical shear displacement corresponding to the peak interface shear strength, (w_cr).

Fig. 3. Two-phase softening model relating pile shaft friction to pile displacement.

Liu and Vanapalli (2019b) proposed a series of theoretical models for the estimations of both peak (τ_pk) and post-peak shaft friction (τ_pp) of piles in unsaturated expansive soils during the wetting process, as given in Eqs. (1) and (2), respectively. These models consider the influence of matric suction reduction and LSP mobilization [Eq. (3)] on the pile shaft friction mobilization:

τ_{p k (w)} = c_{a}^{'} + [σ_{L (w e t)} + \frac{υ}{1 - υ} σ_{v s}] \tan δ^{'} + ψ_{m w} \tan δ^{b}

(1)

τ_{p p (w)} = c_{a r}^{'} + σ_{L (wet)} \tan δ_{r}^{'}

(2)

\begin{aligned} σ_{L (wet)} = & \frac{(1 - υ - 2 υ^{2}) P_{S (ψ m i - 0)}}{1 - υ^{2} - \frac{P_{S (ψ m i - 0)}}{E_{(ψ m i - 0)}} (1 + υ) (1 - υ - 2 υ^{2})} \\ - \frac{(1 - υ - 2 υ^{2}) P_{S (ψ m w - 0)}}{1 - υ^{2} - \frac{P_{S (ψ m w - 0)}}{E_{(ψ m w - 0)}} (1 + υ) (1 - υ - 2 υ^{2})} + \frac{υ}{1 - υ} σ_{v s} \end{aligned}

(3)

where

c_{a}^{'}

= effective pile–soil interface cohesion; δ′ = effective pile–soil interface friction angle; δ^b = pile–soil interface friction angle with respect to matric suction;

c_{a r}^{'}

= residual effective cohesion;

d_{r}^{'}

_r = residual interface friction angle; σ_L_(wet) = LEP considering LSP in the wetting process; P_S_(ψmi−0) = vertical swelling pressure (VSP) under constant volume condition generated from matric suction corresponding to initial condition ψ_mi to 0; P_S_(ψmw−0) = VSP under constant volume condition generated from matric suction corresponding to wetting condition ψ_mw to 0 (ψ_mi > ψ_mw); E_(ψmi−0) = elastic modulus corresponding to a matric suction variation from ψ_mi to 0; E_(ψmw−0) = elastic modulus corresponding to a matric suction variation from ψ_mw to 0; υ = Poisson’s ratio; and σ_vs = vertical stress due to soil self-weight and/or surcharge.

Experimental studies of several investigators suggest that predominant swelling or shrinkage occurs during the first wetting and drying cycle (e.g., Al-Homoud et al. 1995; Basma et al. 1996). Beyond the fourth cycle, the drying process can be assumed to be following the reverse path of the wetting process (Rosenbalm and Zapata 2017). Like the wetting process, there will be a reduction in LEP if the superposition method is employed due to the shrinkage of the soil in the drying process. Such a behavior can be attributed to matric suction increment (ψ_mi to ψ_md), which can be estimated as the difference between the LSP generated from matric suction reductions from ψ_md to 0 and from ψ_mi to 0. The LEP, peak and post-peak pile shaft friction during the drying process can be represented using Eqs. (4)–(6), respectively:

\begin{aligned} σ_{L (dry)} = & \frac{(1 - υ - 2 υ^{2}) P_{S (ψ m i - 0)}}{1 - υ^{2} - \frac{P_{S (ψ m i - 0)}}{E_{(ψ m i - 0)}} (1 + υ) (1 - υ - 2 υ^{2})} \\ - \frac{(1 - υ - 2 υ^{2}) P_{S (ψ m d - 0)}}{1 - υ^{2} - \frac{P_{S (ψ m d - 0)}}{E_{(ψ m d - 0)}} (1 + υ) (1 - υ - 2 υ^{2})} + \frac{υ}{1 - υ} σ_{v s} \end{aligned}

(4)

where ψ_md = matric suction corresponding to drying condition,

τ_{p k (d)} = c_{a}^{'} + [σ_{L (d r y)}] \tan δ^{'} + ψ_{m d} \tan δ^{b}

(5)

τ_{p p (d)} = c_{a r}^{'} + [σ_{L (dry)}] \tan δ_{r}^{'}

(6)

However, as shown in Fig. 1, cracks develop once the tensile stress that develops due to the volume shrinkage of expansive soil. The development of Rankine’s active earth pressure (AEP) at different degrees of saturation against frictionless and rough surface is shown in Fig. 4. Extending the approach detailed in Liu and Vanapalli (2017), the AEP for saturated soil against rough retaining surface (σ_ha₁), saturated soils against frictionless surface (σ_ha₂), unsaturated soil against rough retaining surface (σ_ha₃), and unsaturated soils against frictionless surface (σ_ha₄) can be estimated using Eqs. (7)–(10), respectively:

σ_{h a 1} = σ_{v s} \frac{1 - \sin ϕ^{'} \cos 2 α_{a}}{1 + \sin ϕ^{'} \cos 2 α_{a}} - c^{'} \frac{2 \cos ϕ^{'} \cos 2 α_{a}}{1 + \sin ϕ^{'} \cos 2 α_{a}} + p_{0}

(7)

σ_{h a 2} = \frac{σ_{v s} (1 - \sin ϕ^{'})}{1 + \sin ϕ^{'}} - \frac{2 c^{'} \cos ϕ^{'}}{1 + \sin ϕ^{'}}

(8)

\begin{aligned} σ_{h a 3} = & σ_{v s} \frac{1 - \sin ϕ^{'} \cos 2 α_{a}}{1 + \sin ϕ^{'} \cos 2 α_{a}} - [c^{'} + (u_{a f} - u_{w f}) \tan ϕ^{b}] \\ \times \frac{2 \cos ϕ^{'} \cos 2 α_{a}}{1 + \sin ϕ^{'} \cos 2 α_{a}} \end{aligned}

(9)

σ_{h a 4} = \frac{σ_{v s} (1 - \sin ϕ^{'})}{1 + \sin ϕ^{'}} - \frac{2 [c^{'} + (u_{a f} - u_{w f}) \tan ϕ^{b}] \cos ϕ^{'}}{1 + \sin ϕ^{'}}

(10)

α_{a} = \frac{1}{2} \arcsin \frac{C}{\sqrt{A^{2} + B^{2}}} + \frac{1}{2} \arctan \frac{B}{A}

{\begin{matrix} A = σ_{s} \sin ϕ^{'} + [c^{'} + (u_{a f} - u_{w f}) \tan ϕ^{b}] \cos ϕ^{'} \\ \begin{matrix} B = [{c^{'}}_{a} + (u_{a f} - u_{w f}) \tan δ^{b}] \sin ϕ^{'} - σ_{s} \sin ϕ^{'} \tan δ^{'} \\ - 2 [c^{'} + (u_{a f} - u_{w f}) \tan ϕ^{b}] \cos ϕ^{'} \tan δ^{'} \end{matrix} \\ C = σ_{s} \tan δ^{'} + [{c^{'}}_{a} + (u_{a f} - u_{w f}) \tan δ^{b}] \end{matrix}

where c′ = effective cohesion of soil; ϕ′ = effective internal friction angle of soil; ϕ^b = internal friction angle of soil with respect to matric suction; p₀ = pore water pressure and (u_af – u_wf) = matric suction on the failure plane at failure.

Fig. 4. Development of Rankine’s active earth pressure in unsaturated soils against frictionless and rough surfaces.

Shrinkage deformation that arises after reaching a fully mobilized AEP condition can lead to detachment between the backfill expansive soil and retaining structure. The freestanding height of the soil is usually referred to as the “depth of tension crack” (Das 2015). The expansive soil-retaining structure interface tensile strength may vary from zero to the tensile strength of the expansive soil. In this paper, models for the estimation of tensile strength of unsaturated soil [Eq. (11)] proposed by Morris et al. (1992) were used since it is derived based on principles of unsaturated soil mechanics considering the matric suction as an independent stress-state variable. The depth of the tension cracks can be estimated by combining Eq. (11) to various AEP estimation models [i.e., Eqs. (7)–(10)]. For example, for unsaturated expansive soil against a frictionless surface, the minimum crack depth [Eq. (12)] can be estimated by equaling Eqs. (11) to (7), while the maximum crack depth [Eq. (13)] can be estimated assuming the interface tensile strength is negligible. In the calculation of LEP, self-weight of the soil within the depth of crack should be treated as surcharge:

σ_{t s} = α_{T} [c^{'} + (u_{a} - u_{w}) \tan ϕ^{b}] \cot ϕ^{'}

(11)

where c′ = effective cohesion; and α_T = modified coefficient for tensile stress in unsaturated soils, within the range of 0.5–0.7:

\begin{aligned} z_{c (min)} = \frac{[α_{T} \cot ϕ^{'} (1 + \sin φ^{'}) + 2 \cos ϕ^{'}] [c^{'} + (u_{a} - u_{w}) \tan ϕ^{b}]}{(1 - \sin φ^{'}) γ} \end{aligned}

(12)

z_{c (max)} = \frac{2 [c^{'} + (u_{a} - u_{w}) \tan ϕ^{b}] \cos ϕ^{'}}{(1 - \sin ϕ^{'}) γ}

(13)

where γ = unit weight of soil.

The critical shear displacement (w_cr) corresponding to the peak interface shear strength is influenced by various factors, including soil type, structure, stratigraphy, and loading procedure. Several investigators suggested that w_cr is to be determined experimentally or from back-analysis of field tests because of the complexities associated in understanding the independent contribution of each of these factors (Krasiński 2012; Zhang and Zhang 2012). A brief summary of various values of w_cr suitable for different scenarios is available in Liu and Vanapalli (2019b).

Mobilization of Pile Base Resistance

As discussed earlier, the pile base can be detached from the soil once the pile is uplifted in the wetting process, as shown in Fig. 1. Therefore, Eq. (14) proposed by Randolph and Wroth (1978) for pile base resistance estimation is revised to Eq. (15), considering the possible appearance of negative pile base settlement (w_b). However, it should be noted that Eq. (15) is only suitable for idealized scenarios that assume the supporting soil is in elastic state; however, in engineering practice the mobilization of pile base resistance is much more complex (Kuwajima et al. 2009; Zhang et al. 2013; Malik et al. 2017). Influencing factors, such as the failure mechanism and deformation properties of supporting soils as well as the piling technology (e.g., the mud surrounding the pile base is not cleaned up after installation of in situ cast piles), combined determine the pile base resistance mobilization. Therefore, necessary modifications should be introduced to Eq. (15) considering the influence of these factors to provide reasonable pile base resistance estimations:

P_{z b} = \frac{4 G_{s b} r_{0}}{1 - υ_{b}} w_{b}

(14)

P_{z b} = {\begin{matrix} 0 & w_{b} < 0 \\ \frac{4 G_{s b} r_{0}}{1 - υ_{b}} w_{b} & w_{b} \geq 0 \end{matrix}

(15)

Pile Load-Transfer Mechanism Analysis

The basic assumptions used in the traditional shear displacement method are extended in the MSDM, which are: (i) An expansive soil layer that has approximately the same water content is assumed homogeneous, isotropic, and linear elastic; (ii) The cross section is constant along the pile length, without considering the nonlinear compressive behavior, and the pile material can withstand both the compressive and tensile stresses.

In the traditional shear deformation method Cooke (1975) proposed, the soil around the pile was considered as a series of concentric cylinders (Fig. 5). The vertical settlement of soil around the pile at a certain depth z can be given as

w (z, r) = \frac{τ_{0} r_{0}}{G_{s}} \int_{r_{0}}^{\infty} \frac{\partial r}{r} = \frac{τ_{0} r_{0}}{G_{s}} \ln (\frac{r_{m}}{r})

(16)

where w = vertical settlement of soil around the pile; G_s = shear modulus of soil around the pile; r₀ = radius of the pile; r = horizontal distance between the calculated point and the pile axis; and r_m = maximum influencing radius of the pile on the soil. Generally, r_m can be estimated using (Xiao et al. 2011)

r_{m} = 2.5 L (1 - υ_{s})

(17)

where L = length of the pile; and υ_s = Poisson’s ratio of the soil around the pile.

Fig. 5. Analytical model for pile and soil around the pile.

The relationship between axial force and the shaft friction around the pile is given as

\frac{\partial P_{z}}{\partial z} = 2 π r_{0} τ_{0}

(18)

The relationship between the axial force of the pile and the pile displacement is given as

\frac{\partial w_{p z}}{\partial z} = \frac{P_{z}}{E_{p} A_{p}}

(19)

Eq. (20) can be acquired by combining Eqs. (16), (18), and(19):

\frac{\partial^{2} w_{p z}}{\partial z^{2}} - \frac{k}{A_{p} E_{p}} w_{p z} = 0

(20)

The pile displacement can be estimated from Eq. (21), which is obtained by solving Eq. (20):

\begin{aligned} w_{p z} = A e^{r_{1} z} + B e^{r_{2} z} \\ r_{1} = \sqrt{\frac{k}{A_{p} E_{p}}}; r_{2} = - \sqrt{\frac{k}{A_{p} E_{p}}}; k = 2 π G_{s} / \ln (\frac{r_{m}}{r_{0}}) \end{aligned}

(21)

However, Eq. (21) is only suitable for certain scenarios for saturated soils. For unsaturated expansive soils, possible ground displacement in the active zone should also be considered. The heave prediction equation presented by Adem and Vanapalli (2016) [Eq. (22)] for the elastic range is employed in this paper for estimating the ground heave. Since the soil is assumed elastic, this equation can be also used to estimate the ground settlement during the drying process. The matric suction variations can be different for soil layers at different levels of depth. Thus, the vertical displacement calculated using Eq. (22) for different soil layers would be different. In this paper, within a certain soil layer, the development of the vertical displacement is assumed to have a linear distribution [Eq. (23)], as shown in Fig. 6. The soil displacement in vertical direction is cumulative from bottom to the top soil layer in the active zone:

Δ h = h [\frac{(1 + υ) (1 - 2 υ)}{E_{a} (1 - υ)}] Δ (u_{a} - u_{w})

(22)

where Δh = heave of soil; h = thickness of the calculated soil layer; E_a = average elastic modulus over the matric suction variation range; and Δ(u_a–u_w) = variation in matric suction:

Δ h = H z + a

(23)

Fig. 6. Simplified ground displacement calculation model.

The pile displacement considering the influences of ground displacement induced by expansive soil volume changes can be expressed as Eq. (24):

w_{p z} + (H z + a) = A e^{r z} + B e^{- r z}

(24)

Then the pile axial force can be calculated using Eq. (25) by combining Eq. (24) with Eq. (19).

P_{z} = - A_{p} E_{p} \frac{\partial w_{p z}}{\partial z} = A_{p} E_{p} [- A r e^{r z} + B r e^{- r z} + H]

(25)

Eq. (26) can be obtained by expressing Eqs. (24) and (25) in a matrix form, which can be further simplified as Eq. (27).

{\begin{matrix} w_{p z} \\ P_{z} \end{matrix}} = {\begin{matrix} e^{r z} & e^{- r z} \\ - A_{p} E_{p} r e^{r z} & A_{p} E_{p} r e^{- r z} \end{matrix}} {\begin{matrix} A \\ B \end{matrix}} + {\begin{matrix} - H z - a \\ A_{p} E_{p} H \end{matrix}}

(26)

{\begin{matrix} w_{p z} \\ P_{z} \end{matrix}} = [T (z)] {\begin{matrix} A \\ B \end{matrix}} + {\begin{matrix} - H z - a \\ A_{p} E_{p} H \end{matrix}}

(27)

The soil properties can experience significant variations at different depth levels. In this paper, both the pile and soil are divided into several segments, as shown in Fig. 5. For a typical pile segment that is numbered as segment n in Fig. 5, the pile axial force and displacement equation at the top and bottom of the segment are given as Eqs. (28) and (29), respectively:

{\begin{matrix} w_{p z n} \\ P_{z n} \end{matrix}} = [T (z)_{(n) bot}] {\begin{matrix} A \\ B \end{matrix}}_{(n) bot} + {\begin{matrix} - H_{n} z_{n} - a_{n} \\ A_{p} E_{p} H_{n} \end{matrix}}

(28)

\begin{aligned} {\begin{matrix} w_{p z (n - 1)} \\ P_{z (n - 1)} \end{matrix}} = [T (z)_{(n - 1) top}] {\begin{matrix} A \\ B \end{matrix}}_{(n - 1) top} + {\begin{matrix} - H_{n} z_{(n - 1)} - a_{n} \\ A_{p} E_{p} H_{n} \end{matrix}} \end{aligned}

(29)

For the same pile segment n, Eq. (30) is valid; thus, Eq. (31) can be deduced:

{\begin{matrix} A \\ B \end{matrix}}_{(n) bot} = {\begin{matrix} A \\ B \end{matrix}}_{(n - 1) top}

(30)

\begin{aligned} {\begin{matrix} w_{p z n} \\ P_{z n} \end{matrix}} = & [T (z)_{(n) bot}] {[T (z)_{(n - 1) top}]}^{- 1} {\begin{matrix} w_{p z (n - 1)} \\ P_{z (n - 1)} \end{matrix}} \\ + [T (z)_{(n) bot}] {[T (z)_{(n - 1) top}]}^{- 1} {\begin{matrix} H_{n} z_{(n - 1)} + a_{n} \\ - A_{p} E_{p} H_{n} \end{matrix}} \\ - {\begin{matrix} H_{n} z_{n} + a_{n} \\ - A_{p} E_{p} H_{n} \end{matrix}} \end{aligned}

(31)

The pile axial force and displacement at the top of segment n can be calculated based on the pile base axial force and displacement using Eq. (31). In addition, with Eqs. (22)–(24), the influence of ground displacement induced by matric suction variations has been considered. Furthermore, as discussed earlier, the unsaturated soil shear modulus, G_s, is sensitive to matric suction variations. Note that in Eq. (21), a constant shear modulus has been used. To consider the influence of matric suction on the pile–soil interface shear behavior, a two-phase relationship between the shear stress and shear displacement is used instead. As shown in Eq. (32), the magnitude of the shear modulus in phase I is determined by the slope of τ_su over w_cr. Then T_(z1) and T_(z2b) in Eqs. (28) and (29) can be calculated using Eqs. (33) and (34), respectively. Substituting Eqs. (33) (34) into Eq. (31), the displacement at the top of segment n (w_pz₂) can be calculated. Comparisons can be made between the calculated displacement (w_pz₂) and the critical displacement (w_cr). A calculated displacement (w_pz₂) higher than critical displacement suggests that shear softening has occurred. In this scenario, the interface shear strength dropped from the peak value to the post-peak value.

k = 2 π G_{s} / \ln (\frac{r_{m}}{r_{0}}) = 2 π r_{0} (\frac{τ_{s u}}{w_{c r}})

(32)

T (z)_{(n) bot} = {\begin{matrix} e^{r_{n} z_{n}} & e^{- r_{n} z_{n}} \\ - A_{p} E_{p} r_{n} e^{r_{n} z_{n}} & A_{p} E_{p} r_{n} e^{- r_{n} z_{n}} \end{matrix}}

(33)

T (z)_{(n - 1) top} = {\begin{matrix} e^{r_{n} z_{(n - 1)}} & e^{- r_{n} z_{(n - 1)}} \\ - A_{p} E_{p} r_{n} e^{r_{n} z_{(n - 1)}} & A_{p} E_{p} r_{n} e^{- r_{n} z_{(n - 1)}} \end{matrix}}

(34)

The pile axial force and displacement can be calculated for this scenario using Eq. (35) instead of Eq. (31):

\begin{aligned} r_{n} = & \sqrt{\frac{k_{n}}{A_{p} E_{p}}}; k_{n} = \frac{2 π τ_{s u (n)} r_{0}}{w_{c r}} \\ {\begin{matrix} w_{p z n} \\ P_{z n} \end{matrix}} = {\begin{matrix} w_{p z (n - 1)} \\ P_{z (n - 1)} \end{matrix}} + {\begin{matrix} (\frac{2 P_{z (n - 1)} + τ_{s p (n)} L π d}{2}) (\frac{L}{A_{p} E_{p}}) \\ τ_{s p (n)} L π d \end{matrix}} \end{aligned}

(35)

Eqs. (28)–(35) can be used to calculate the pile segment head axial force and displacement from the pile segment base axial force and displacement. This procedure can be followed to calculate the load transfer from the bottom segment to the top segment of the entire pile. In other words, the pile head load displacement that arises for a certain pile base displacement can be computed using the MSDM. The proposed MSDM only requires information of the matric suction profile, soil water characteristic curve (SWCC), and limited number of soil properties (saturated elastic modulus, saturated interface shear strength properties, Poisson’s ratio, plasticity index, and maximum dry density of expansive soil). For this reason, the proposed approach is a valuable tool for practicing engineers to make quick and reasonable estimations of pile mechanical behaviors in expansive soils considering the influence of environmental factors.

Validation of the Modified Shear Displacement Method (MSDM)

Model Pile Infiltration Test Conducted in the Geotechnical Laboratory of University of Ottawa

A model pile infiltration test (PIT) was performed in unsaturated expansive soil in the geotechnical laboratory of the University of Ottawa for validation of the proposed MSDM. Schematics of the experimental test setup used in the study are provided in Figs. 7(a and b). The PIT was conducted in an aluminum tank with an internal diameter of 300 mm and a height of 700 mm. The model pile made of an aluminum pipe with strain gauges pasted inside has a diameter of 25.4 mm and a height of 600 mm. A rough surface is achieved on the pile shaft by pasting a thin layer of fine sand using epoxy, whose roughness information can be derived from interface shear strength parameters. The pile was installed in a similar way as cast in situ pile. The soil that was already mixed with a water content of 27% was compacted around the model pile using a specially designed compaction apparatus to achieve a wet density of 1,736 kg/m³ (i.e., dry density of 1,367 kg/m³). The ultimate bearing capacity of the model pile was estimated to be 1,000 N according to the Chinese Technical Code for Building Pile Foundations (JGJ 94-2008, MOC 2008) through pile shaft friction and pile base resistance estimations. Using a factor of safety of two, in the loading process an allowable load of 500 N was applied on the pile head prior to wetting. Once the pile head settlement was observed to be constant for 24 h, water was added manually, as and when required, to ensure the ground water table was always slightly higher than the ground surface. The experiment was terminated when the pile head displacement stabilized and all buried GS-3 water content sensors and MPS-6 suction sensors indicated that the soil has been fully saturated. Pile mechanical behavior information, which includes the axial force distribution, pile base resistance, pile head displacement, and the soil behaviors that include the soil displacement, soil volumetric water content variations, and suction variations were monitored during the infiltration process.

Fig. 7. Experimental settings of pile infiltration test (PIT).

Soil index properties and various parameters required in the computation of MSDM, including soil and pile-soil interface shear strength properties, elastic modulus of pile and soil, were measured from laboratory tests and are summarized in Tables 1 and 2, respectively. It should be noted that sand was poured into the testing tank and no attempt was made to compact it. This sand may have been disturbed to certain extent while setting the earth pressure cell at the bottom of the pile. Therefore, the sand at the bottom of the pile can be considered to be in a relative loose state. The elastic modulus was back-calculated to have an approximate value of 800 kPa according to the pile base resistance–settlement response prior to wetting. The entire SWCC of Regina clay (i.e., in the suction range of 0 to 10⁶ kPa), as shown in Fig. 8, were measured using four different methods, including the chilled-mirror hygrometer (WP4) method, pressure plate method, filter paper method, and desiccator method. Data points of SWCC obtained through different methods were also fitted using the Fredlund and Xing (1994) model, and corresponding fitting parameters are shown in Fig. 8.

Fig. 8. SWCC of Regina clay measured over the entire suction range using multiple methods.

Table 1. Index properties of Regina clay

Index properties of Regina clay	Value
Liquid limit, LL (%)	89
Plastic limit, PL (%)	32
Plastic index, PI	57
Specific gravity, G	2.85
Maximum dry unit weight, γ_d,max (kN/m³)	13.82
Optimum water content, w (%)	29
Vertical swelling pressure under constant volume condition (from gravimetric water content of 27% to fully saturation), (kPa)	136
Free swell index = [(V_d – V_k)/V_k] × 100% where V_d = volume of soil specimen read from the graduated cylinder containing distilled water; V_k = volume of soil specimen read from the graduated cylinder containing kerosene.	100.6

Table 2. Summary of various parameters used in the computation of MSDM for PIT

Mechanical properties of expansive soil and pile	Value
Effective internal friction angle with respect to net normal stress, ϕ′ (°)	15.6
Effective internal friction angle with respect to matric suction, ϕ_b (°)	12.3
Effective interface friction angle with respect to net normal stress, δ′ (°)	12.4
Effective interface friction angle with respect to matric suction, δ_b (°)	8.5
Effective cohesion, c′ (kPa)	14
Effective interface cohesion, $c_{a}^{'}$ (kPa)	8
Critical pile–soil relative displacement corresponding to peak interface shear strength, w_cr (mm) (Liu and Vanapalli 2019b)	10
The ratio of residual interface shear strength to peak interface shear strength, β_s	0.83
Elastic modulus of saturated Regina clay, E (kPa)	12,500
Poisson’s ratio of Regina clay, υ (Adem and Vanapalli 2016)	0.4
Elastic modulus of the aluminum model pile, E_p (GPa)	69
Elastic modulus of sand below the model pile, E_sand (kPa)	800

Fig. 9 summarizes the suction changes during the testing period. Water was manually added to keep the water table slightly higher than the ground surface. A period of 40 h was required to saturate the expansive soil layer in the test tank with a thickness of 400 mm. The expansive soil volume increases significantly during the infiltration process; for this reason, both the void ratio and the coefficient of permeability increase. Fig. 10 shows the variation of the soil and pile head displacement during the infiltration process. The pile experienced a settlement of around 3 mm under the applied load of 500 N, while no noticeable soil settlement was detected during the pile loading. During the saturation process, ground heave quickly increased and eventually reached 25 mm. The pile head settlement increased from around 3 mm to around 7 mm.

Fig. 9. Variation of water potential (suction) distribution with respect to time.

Fig. 10. Variation of soil and pile head displacement with respect to time.

The pile axial force distribution and pile shaft friction distribution for four different time periods of testing (i.e., 160, 170, 180, and 200 h) were used to validate the MSDM. These four distinct times were selected because they provide representative suction values along the depth (Fig. 9), which indicate the wetting front just passed the depth of certain suction sensors. The real-time suction profile cannot be recorded since the readings of suction sensors show no changes when the wetting front passes the distance between two adjacent sensors. Fig. 10 presents the pile head displacement at four time points, with reasonably good comparisons between estimations using the MSDM and experimental data. Fig. 11 presents the comparisons of the pile axial force distribution between the experimental data and the estimations using MSDM. The estimated pile axial forces at different depth were higher than the experimental data for all four time points. This is because in the MSDM the interface shear strength drops rapidly from the peak value to the post-peak value after the critical pile–soil relative displacement.

Fig. 11. Comparisons of the pile axial force distribution using MSDM (a) 160 h; (b) 170 h; (c) 180 h; and (d) 200 h.

Field Investigation Case Study Presented by Benvenga (2005)

The Colorado State University (CSU) expansive soils test site at which several full-scale piles were tested has been widely documented in the literature (Abshire 2002; Benvenga 2005; Nelson et al. 2012). Fig. 12 shows the schematic of the piles along with instrumentation details. Field measurement on pile axial strains, volumetric water content of soil around the pile, and pile head and soil displacement were conducted for almost 10 years from September 1995 to April 2004 (Benvenga 2005). Four pile tests were conducted on reinforced concrete piles that were installed at this site. The properties of soil around the pile were determined from laboratory tests by Abshire (2002) and are summarized in Table 3. Since all four piles demonstrated similar behavior, only one pile, labeled as D1.130 (Nelson et al. 2012), was selected for validation purposes using the MSDM. According to Benvenga (2005), the soil experienced ground heave up to 64 mm from February 1997 to August 1997; however, in September 2003, the soil suffered a settlement and the ground heave reduced to around 6 mm.

Fig. 12. Diagram of the drilled reinforced concrete pier at CSU expansive soil test site.
(Data from Benvenga 2005.)

Table 3. Index properties of expansive soil at CSU test site

Depth (m)	Material	Water content (%)	Dry density (kN/m³)	S_%(%)	σ′_cs(kPa)	σ′_i(kPa)
0–1.5	Clay	14	17.3	1	39.9	23.9
1.5–1.8	Clay	13.9	17.7	2.9	205.9	47.9
1.8–2.4	Weathered claystone	15.6	17.4	2.5	215.5	47.9
2.4–3.2	Weathered claystone	20.6	18.1	0.9	124.5	47.9
3.2–4.3	Weathered claystone	10.4	17.6	3.8	430.9	29.7
4.3–4.7	Weathered claystone	11.5	17.9	1.9	210.7	29.7
4.7–7	Weathered claystone	11.5	17.9	3.8	287.3	29.7

Source: Data from Nelson et al. (2011).

Note: S_% is the percentage of expansive soil swell in oedometer test;

s_{c s}^{'}

is vertical swelling pressure acquired from consolidation-swell oedometer test;

s_{i}^{'}

is the confining pressure in oedometer test.

Benvenga (2005) estimated the average interface shear strength (shaft friction) between the pile and soil using Eq. (36) and suggested that the range of α varies 0.6 to 1.0 in October 1997. Benvenga (2005) also mentioned that the calculated values of α can be greater than 1 for some cases; however, such cases were not discussed. The interface shear strength properties corresponding to the peak interface shear strength can be even higher. By comparing the estimated pile shaft friction using Eq. (36) with the experimental data, Liu and Vanapalli (2019b) suggested α equaling to 1.2 could be assumed and introduced to calculate the peak interface shear strength from average shaft friction estimated using

f_{s} = α σ_{c v}^{'}

(36)

where f_s = average ultimate shaft friction; and α = empirical adhesion coefficient.

Other parameters required for the computation of MSDM were derived from properties summarized in Table 4; some of these properties were based on semiempirical equations using the soil properties summarized in Table 3 as well as volumetric water content and vertical swelling pressure distributions given in Fig. 13 (Benvenga 2005; Liu and Vanapalli 2019b). The SWCC and matric suction variations of the expansive soil can be back-calculated; these details are summarized in Figs. 14 and 15, respectively.

Fig. 13. Volumetric water content in February 1997 and October 1997.
(Data from Benvenga 2005.)

Fig. 14. Estimated SWCC using model proposed by Fredlund and Xing (1994).

Fig. 15. Matric suction variations in February 1997; October 1997; and September 2003.

Table 4. Various parameters used in the computation of MSDM for CSU pile test

Mechanical properties of expansive soil and pile	Value
Critical pile–soil relative displacement corresponding to peak interface shear strength, w_cr (mm)	10
The ratio of residual interface shear strength to peak interface shear strength, β_s	0.85
Average elastic modulus of soil, E_a (kPa)	5,000
Elastic modulus of the pile, E_p (MPa)	1,820
Poisson’s ratio, υ	0.4

The MSDM was used to simulate the mechanical behavior of a pile that was recorded in October 1997 and September 2003. Figs. 16(a and b) provide comparisons of the pile axial force distribution in October 1997 and September 2003. Several strain gauges that were pasted on the pile shaft were damaged during the testing process, as reported by Benvenga (2005). Therefore, only a limited number of data points (e.g., three data points for October 1997 and two data points for September 2003) were available from the Benvenga (2005) investigations. The reason for the appearance of the polynomial curve can be attributed to negative friction that generates in the stable zone as the pile gets uplifted with ground heave development. Nelson et al. (2015) proposed a finite-element program and analyzed the mechanical behavior of the pile in October 1997 as a case study. Fig. 16 shows that the proposed MSDM accurately estimates the mechanical behavior of the piles, taking account of the influence of environmental factors. The estimated pile head displacement in October 1997 and September 2003 using MSDM are 20.8 and 3.7 mm, respectively, which are closer to the field measurements of 9.2 and 3.1 mm compared with the simulations of around 27 and 26 mm by Nelson et al. (2015). Some discrepancies between estimations and experimental data may be attributed to the uncertainties involved in the SWCC that is estimated based on a limited number of data points from the transition zone.

Fig. 16. Comparison of pile axial force distribution for pile test by Benvenga (2005): (a) October 1997; and (b) September 2003.

Summary and Conclusions

Load transfer mechanism of piles placed in unsaturated expansive soils is sensitive to changes associated with environmental factors, such as the wetting and drying conditions. In the active zone, not only does the matric suction change, but these matric suction changes-induced volume changes that occur in expansive soils around the pile contribute to the pile shaft friction changes. Volume changes of expansive soil in the horizontal direction change the normal stress acting on the pile shaft while volume changes in the vertical direction changes the pile–soil relative displacement. In turn, variations in pile shaft friction in the active zone change the load transfer mechanism of piles.

In this study, using unsaturated soil mechanics as a tool, a two-phase softening model was proposed for calculations of pile shaft friction considering the influence of matric suction changes and matric suction variations-induced expansive soil volume changes. Furthermore, theoretical models for pile base-resistance estimations were also proposed considering the possible pile base detachment from the subsoil due to uplift friction generated in the active zone in the wetting process. By introducing these models into the traditional shear displacement method, its application can be extended for interpreting the pile behavior in unsaturated expansive soils. A model pile infiltration test was conducted in the geotechnical laboratory of the University of Ottawa, Canada, and presented for the interpretation and validation of the proposed MSDM, together with a field investigation presented by Benvenga (2005). Reasonably good comparisons were achieved between the estimations using the proposed MSDM and measured data for both cases. In engineering practice applications, the matric suction profile can be obtained by direct measurement using various suction sensors or estimated from the water-content profile and the SWCC, which can be used in estimating approximate suction values in uniform soils. If there is no site investigation data available, engineers may use commercial numerical software, such as the VADOSE-W (Geo-slope Ltd), to predict a matric suction profile based on the information of SWCC, environmental data that include infiltration and evaporation and hydraulic boundary conditions. The proposed MSDM can be used together with the suction profile data and soil and/or pile–soil interface shear strength properties that facilitate practicing engineers to provide rational design for pile foundations in expansive soils by estimating the pile load-transfer mechanism.

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request. These items include detailed output for all analyzed scenarios.

Acknowledgments

The first author, Yunlong Liu, thanks the department of Science and Technology of Henan Province for funding the project entitled “Multiscale study on the shear behavior of expansive soil-structure interface under drying-wetting and freezing-thawing cycles” – Fund code: 212300410280 for the period 2021-01 to 2022-12. The second author thanks the NSERC for the Discovery Grant support in 2020.

References

Abbas, H. O. 2020. “Laboratory evaluation of effective parameters on uplift force under reamed pile in expansive soil.” Geotech. Geol. Eng. 38 (4): 4243–4252. https://doi.org/10.1007/s10706-020-01292-8.

Abstract

Introduction

Background

MSDM

Mobilization of Pile Shaft Friction

Mobilization of Pile Base Resistance

Pile Load-Transfer Mechanism Analysis

Validation of the Modified Shear Displacement Method (MSDM)

Model Pile Infiltration Test Conducted in the Geotechnical Laboratory of University of Ottawa

Field Investigation Case Study Presented by Benvenga (2005)

Summary and Conclusions

Data Availability Statement

Acknowledgments

References

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