Probabilistic design model for energy piles considering soil spatial variability

Computers and Geotechnics 108 (2019) 308–318

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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Research Paper

Probabilistic design model for energy piles considering soil spatial variability

T

Zhe Luo , Biao Hu ⁎

The Key Laboratory of Road and Traffic Engineering, Ministry of Education, Tongji University, Shanghai 201804, China

ARTICLE INFO

ABSTRACT

Keywords: Energy piles Load-transfer curves Spatial variability Random finite difference method

Energy piles are sustainable green foundations. Although several geotechnical design methods have been proposed, no methods have yet been developed to quantify the uncertainty propagation in thermo-mechanical pilesoil interactions. The probabilistic model proposed in this paper, which is based on load-transfer curves, considers the effects of spatial variability for energy pile design. The developed random finite difference method is used to address the effects of vertical spatial variability on the ultimate limit state design and serviceability limit state design. This study points to the necessity of modeling the soil spatial variability in the thermo-mechanical pile-soil interactions in energy piles.

1. Introduction Energy piles are developed by installing geothermal loops in conventional piles to extract shallow geothermal energy to meet the heating and cooling needs of buildings. These piles can reduce carbon emissions in urban environments and do not require additional construction costs [27]. In practice, it is challenging to model the strains and stresses induced by the thermal expansion and contraction of energy piles [4,22]. Due to the lack of understanding of the uncertainty propagation in the design of energy piles, the current geotechnical design of energy piles is based on an enlarged factor of safety. It is reported that the factor of safety used for energy piles is more than twice that for conventional piles [27], leading to massive unnecessary costs. Numerous studies have investigated the thermo-mechanical behavior of energy piles. Energy pile tests have included full-scale in-situ tests (e.g., [4,11,22,23]) as well as laboratory model-scale tests (e.g., [19,25,40,43]). Furthermore, the numerical modeling tools—predominantly based on finite element methods (e.g., [5,10,26,30]) and the finite difference method (e.g., [7,20,28,35])—were developed to explain the observed performance of energy piles and to analyze other design scenarios. These numerical tools were shown to be capable of capturing the field observations given proper model calibration. Nevertheless, these models are intrinsically deterministic and do not address the effect of parameter uncertainty in the design and assessment of energy piles. It is known that many uncertainties are involved in the geotechnical design of pile foundations, such as design model error, random loads,

⁎

and the inherent spatial variability of soil parameters. A recent study by the authors presented some preliminary findings regarding the effect of parameter uncertainty on the geotechnical analysis of energy piles [17,41,42]. However, so far, no research has been reported on the effect of soil spatial variability—a complex and critical category of uncertainty in geotechnical design [14]—on the design of energy piles. Under combined thermo-mechanical loads, uncertainty propagation in the pile-soil interaction is a problem worthy of further investigation, in order to promote energy piles as green foundations. To this end, a probabilistic design tool for energy pile design dealing with soil spatial variability effect is urgently needed in the practice. In this paper, a probabilistic version of an energy pile model based on the random finite difference approach is presented. This model has two main components: (1) finite difference solutions based on loadtransfer curves for combined thermo-mechanical loading; and (2) random field models for simulating the spatial variability of soil parameters. These two components are integrated into a Monte Carlo simulation framework to assess both the ultimate limit state failure and the serviceability limit state failure of energy piles. The deterministic component of this model is validated by comparing the model results with in-situ measurements. Using the proposed model, this study probabilistically investigates the effect of soil spatial variability on three common failure modes of energy piles: failure due to excessive pile settlement, failure due to concrete cracking, and geotechnical bearing capacity failure. The findings in this study explicitly show the adverse effects of spatial variability and temperature change on the energy pile assessment. The developed probabilistic model is

Corresponding author. E-mail address: [email protected] (Z. Luo).

https://doi.org/10.1016/j.compgeo.2019.01.003 Received 23 August 2018; Received in revised form 10 December 2018; Accepted 6 January 2019 0266-352X/ © 2019 Elsevier Ltd. All rights reserved.

Computers and Geotechnics 108 (2019) 308–318

Z. Luo, B. Hu

Fig. 1. Load-transfer-curve-based finite difference model for thermo-mechanical analysis of energy piles (after Knellwolf et al. [20]).

affect the predicted pile-soil responses. For bored piles in clays, typical load-transfer functions include but are not limited to the simple linear curves [39], the tri-linear curves [15], the hyperbolic curves [16], and the point-by-point curves [3]. In this study, the Frank and Zhao curves [15] are selected for demonstration. Depending on the local experience, another load-transfer curves [3,15,16,39] can replace the Frank and Zhao curves in our developed probabilistic approach. Our selected loadtransfer curves require pressuremeter test data, which are often not available in the field. To this end, the pressuremeter modulus is determined with the correlation to the standard penetration number, a commonly used in-situ index (e.g., [6,31,32]). Fig. 2 shows the adopted load-transfer curves, where the mobilized shaft resistance ( s ) and base resistance ( b ) can be computed based on the pile-soil vertically relative displacement (z). The following empirical models for fine-grained soils are used to determine the slopes of the curves [1]:

demonstrated to be capable of simulating the soil spatial variability in the complex pile-soil interaction problems of energy piles. 2. Load-transfer method for piles under thermo-mechanical loading In this study, an iterative procedure [9] is adopted for the loadtransfer analysis. In this method, the pile is divided into N rigid segments with inter-spring connections that represent the pile stiffness, as shown in Fig. 1(a), where the shaft springs on the rigid segments represent the shaft resistance and the base spring represents the base resistance. It is noted that the two-way shaft springs are able to simulate negative skin friction and the resulting thermal responses. Fig. 1(b) shows that the pile expands or contracts axially due to heating or cooling, and the thermally induced shaft mobilized friction can be simulated by the two-way springs. Compared to a case with a pile under mechanical loading in Fig. 1(a), a top spring is linked to the pile head for the thermal loading scenario as shown in Fig. 1(b) in order to describe the superstructure constraint. This load-transfer model, which was developed by Knellwolf et al. [20], is used to model the combined thermo-mechanical responses of the energy piles.

Ks =

2· EM D

and

Kb =

11· EM D

(1)

where Ks and Kb are the slopes of the first linear branches for the shaft and base load-transfer curves (as illustrated in Fig. 2), respectively; EM is the Menard pressuremeter modulus, and D is the pile diameter. In Fig. 2, qs is the ultimate shaft resistance and qb is the ultimate base bearing capacity. The values for qs and qb can be determined with several empirical methods, including but not limited to the total stress method (αmethod) for clays and the effective stress method (β-method) for both sands and clays. The α-method was adopted in this study:

2.1. Load-transfer curves A great deal of effort has been dedicated to developing load-transfer curves that define the relationship between the shaft resistance and pile shaft displacement as well as the relationship between the base resistance and toe displacement. The selected load-transfer curves can 309

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Z. Luo, B. Hu

Fig. 1(a). Assuming a constant shear stress along the segment, the axial force acting at the head of Segment i (FH,i) and the updated midpoint displacement of the ith segment (zM , i ) can be calculated based on local equilibrium:

FH , i = FB, i + Ts, i = FB, i + Li · C·

zM , i = z B , i +

s , i (z M , i )

FB, i + FM , i Li 3FB, i + FH , i Li · = zB , i + · EA 2 2EA 2

(5) (6)

where FB,i is the axial force acting at the bottom of segment i (FB,N = Tb), Ts,i is the shaft friction on Segment i, Li is the length of Segment i, C is the pile circumference, s, i is the mobilized shaft resistance on Segment i computed using the load-transfer curve in Fig. 2(a), FM,i = (FB,i + FH,i)/2 is the axial force acting at the middle of Segment i, and E is the elastic modulus for the pile.

• Step 3: In the next trial, another z

M,i value is assumed, and Step 2 is repeated. This iteration substep will result in an updated midpoint displacement of the ith segment (zM , i = zM , i ). Given an acceptable discrepancy between zM,i and zM , i (e.g., |(z M , i zM , i ) zM , i | < 10 3 ), the midpoint displacement (zM,i) can converge, and this inner loop ends. The head displacement of the ith segment (zH,i) can then be obtained as:

zH , i = zB, i + Fig. 2. Load-transfer curves (based on Frank and Zhao [15]).

qs =

s ·(su ) s

and

qb =

b·(su ) b

s

= 1

(2)

•

(3)

used as the base load and base displacement, respectively, of the next upper Segment i − 1 (i.e., setting zB,i−1 = zH,i and FB,i−1 = FH,i). The load and displacement at the head of Segment i − 1 can be obtained by repeating Steps 2–3. As a result, the loads and displacements for Segment N to Segment 1 can be sequentially determined from pile base to pile head. Step 5: Lastly, an outer loop of iteration is performed by re-assigning a value for the base displacement (zB,N) and repeating Steps 1–4 until the external equilibrium is satisfied: N

Tb +

Considering that energy piles are subjected to various combinations of mechanical loading and thermal loading, three distinct cases are considered: mechanical loading only, thermal loading only, and combined mechanical and thermal loading. The algorithm for computing the thermo-geotechnical responses of energy piles using the loadtransfer method is illustrated in Fig. 3. The downward pile movements are taken as positive, and the tensile stresses are taken as positive.

Ts, i + P = 0 i=1

(8)

2.3. Case II: Pile under thermal loading only Due to the heating or cooling effect, an energy pile expands or contracts about a null point with a zero displacement [4]. It should be noted that the null point, as the location of force equilibrium, is consistent with the definition of neutral plane in the negative friction analysis of traditional piles (e.g., [13]). The right portion of Fig. 3 illustrates the algorithm for the load-transfer analysis of an energy pile under thermal loading only. The iteration procedure developed by Knellwolf et al. [20] is adopted to model the load-transfer mechanism of the energy piles in this study, as described in the subsequent steps.

2.2. Case I: Pile under mechanical loading only The left portion of the flow chart in Fig. 3 summarizes the iterative procedure for the load-transfer analysis for a pile subjected only to mechanical loading (P) [9]:

• Step 1: As a first step, the pile is divided into N segments [Fig. 1(a)].

• Step 6: The null point is assumed to be located on Segment N

The pile segments are numbered as i = 1, 2, 3, … N. Assuming a small base displacement (zB,N), the base load (Tb) can be computed using Eq. (4):

Tb = A· b (zB, N )

(7)

• Step 4: Next, the load and displacement at the head of Segment i are

where αs is the adhesion coefficient, su is the undrained shear stress; and αb is the bearing capacity coefficient, which is equal to 9.0 [33]. As suggested by American Petroleum Institute [2], values for αs are taken as follows:

1.0 for su 25 kPa (su 25) 90 for 25 kPa < su <70 kPa 0.5 for su 70 kPa

FB, i + FH , i ·Li 2· EA

0 (i.e., z t , N0 = 0 ), and this loop begins with N0 = 1. When a pile is heated or cooled, the thermally induced displacement of Segment i (zt,i) can be calculated by:

(4)

where A is the pile cross-sectional area; b is the mobilized reaction at the base of the pile, as computed with the load-transfer curve shown in Fig. 2(b).

zt ,i =

zt ,i+1 · Li · t , i for 1 i < N0 z t , i 1 + ·Li · t , i for N0 < i N

(9)

where ς is the related parameter (ς = 1 during a heating cycle or ς = −1 during a cooling cycle) and εt,i is the thermally induced axial strain of Segment i. In the first trial, εt,i is taken as free axial strain (i.e., εt,i = εt,free = α · ΔT), where α is the thermal expansion coefficient of the pile and ΔT is the temperature change.

• Step 2: Starting with i = N, the midpoint displacement of the ith

segment (zM,i) is determined by the internal loops of iteration in Fig. 3. In the first trial, zM,i is assumed to equal the base displacement (zB,i). The local free body diagram of Segment i is shown in 310

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Z. Luo, B. Hu

Fig. 3. Flow chart for computational thermo-geotechnical analysis of energy piles.

• Step 7: Next, the mobilized stress due to the thermal effect for each

N

Tt , b +

segment can be readily obtained using the load-transfer curves (Fig. 2); thus, the corresponding thermally induced axial load of Segment i (Pt,i) can be computed. The blocked thermal strain (εt,b,i) and the observed strain (εt,o,i) of Segment i are:

t , b, i

=

Pt , i = EA

t , o, i

=

t , free

Tt , b +

i j =N

EA t , b, i

Tt , s, j

b (z t , N )· A

=

+

i j=N

EA

b (z t , N )·A

i=1

+

s, i (z t , i )· C· Li

+ Kh·z t ,1·A = 0

i=1

(12) where Th is the thermally induced force by the upper structure at the pile head, and Kh is the spring constant of the pile-superstructure interaction.

s, j (z t , j )·C ·Lj t , free

2.4. Case III: Pile under combined mechanical and thermal loading

(10) (11)

In this scenario, the load-transfer analysis is initialized by the mechanical loading. Following the procedure for mechanical loading [Eqs. (4)–(8)], the resulting initialized displacement and strain are treated as the starting displacement and starting strain in the thermal loading analysis. Thus, Eqs. (10) and (12), which were used in the previous thermal loading scenario, are updated to incorporate the effect of mechanical loading:

where Tt,b is the thermally induced pile base load and where Tt,s,i is the thermally induced shaft friction on Segment i.

• Step 8: Steps 6–7 are repeated with another initial strain value by •

N

Tt , s, i + Th =

setting εt,i = εt,o,i. This procedure is repeated until the discrepancy between εt,i and εt,o,i falls within an acceptable tolerance (e.g., 3 |( t , o, i t , i ) t , i | < 10 ). Step 9: By changing the location of the null point (N0) from Segment 1 to Segment N, Steps 6 to 8 are repeated until the required precision of the external equilibrium [Eq. (12)] is reached:

t , b, i [ b (z B , N + z t , N )

= 311

=

Pt , i EA

b (zB, N )]·A +

Tt , b +

=

i j =N

EA

i j =N

Tt , s, j

EA

[ s , j (z M , j + z t , j )

s, j (zM , j )]·C·Lj

t , free

(13)

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Z. Luo, B. Hu N

Tt, b +

Table 2 Soil parameters used to model EPFL test pile.

Tt ,s, i + Th=

i=1

[ b (zB, N + z t ,N )

b (zB, N )]· A

+

N

[

s, i (zM , i

+ z t, i)

s, i (zM , i )]· C · Li

Soil layer

+ Kh· z t,1· A = 0

i=1

Depth (m) Elastic slope for pile shaft friction Ultimate shaft resistance Elastic slope for pile tip resistance Ultimate base bearing capacity

(14) The complete flow for the load-transfer analysis of energy piles that considers the combined thermo-mechanical loading is summarized in Fig. 3. MATLAB subroutines are developed for use in the subsequent probabilistic analysis in this study. 2.5. Comparison between load-transfer analysis results with field observations

1

2

3

4

5

Ks (MPa/m)

0–5.5 16.7

5.5–12 10.8

12–22 18.2

22–25 121.4

25–25.8 –

qs (kPa) Kb (MPa/m)

102 –

70 –

74 –

160 –

300 667.7

qb (MPa)

–

–

–

–

11

Note: data from Knellwolf et al. [20] and Mimouni [24].

The simulation results using the developed routines for geotechnical design of energy piles in this study are compared with field test data. The results from comprehensive energy pile tests conducted at École Polytechnique Fédérale de Lausanne (EPFL) in Switzerland [22], which included seven different tests corresponding to seven successive stages of construction, are used in this study. Test 1 was conducted prior to the construction of the building (i.e., Case II with thermal loading only), and Test 7 was conducted at the end of the building construction (i.e., Case III with combined mechanical and thermal loading). Using the parameters for the EPFL pile in Tables 1 and 2, the axial strains simulated using the developed subroutines are compared with the field test results, as shown in Fig. 4. Fig. 4(a) shows that for the thermal loading conditions, the load-transfer method can yield results that are comparable to the experimental data for various levels of temperature changes (ΔT from 7.5 °C to 21.8 °C). Similarly, Fig. 4(b) shows that for combined loading, the simulations and field observations agree well with each other for various temperature changes (ΔT from 9.1 °C to 14.3 °C). It is demonstrated that the load-transfer-curve-based geotechnical design model can rationally predict the responses of energy piles under complex loadings. 3. Limit state design of energy piles Two limit states in energy pile design are addressed in this study: ultimate limit state (ULS) and the serviceability limit state (SLS). In ULS, we focus on the bearing capacity of a single energy pile; in SLS, the excessive pile settlement and concrete tensile cracking of the pile are considered. It should be noted that only the immediate pile settlement is considered in the developed probabilistic model. To explore the three common failure modes (excessive pile settlement, concrete cracking, and bearing capacity failure), a hypothetical floating bored energy pile in monolayer clays is adopted in this study. This energy pile has a length of 30 m and a diameter of 0.5 m, and the building load on the top of the pile is 2000 kN. A complete list of the geotechnical and structural parameters for the hypothetical energy pile are given in Table 3. 3.1. SLS: Settlement and cracking of a single energy pile The limit state equations for SLS design against excessive pile settlement ([g (x )]SLS _settlement ) and cracking ([g (x )]SLS _cracking ) are those Table 1 Pile parameters used to model EPFL test pile. Parameter

Notation

Value

Diameter Length Young’s modulus Thermal expansion/contraction coefficient Mechanical load Pile-superstructure interaction stiffness

D L E α P Kh

0.88 m 25.8 m 29.2 GPa 10−5 °C 1000 kN 2 GPa/m

Fig. 4. Comparisons between simulated axial strains and field observed axial strains in EPFL energy pile.

Note: data from Laloui et al. [22]. 312

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Z. Luo, B. Hu

parameters in the ground. Thus, the probabilistic analysis can be conservative if N60 is used. It is somewhat desirable to perform a conservative design. In addition to N60, other laboratory or in-situ index, such as the undrained shear strength or cone penetration resistance, can also be implemented under our probabilistic framework.

Table 3 Parameters used in the hypothetical energy pile. Parameter

Notation

Value

COV

Pile diameter Pile length Young’s modulus of pile Thermal expansion/contraction coefficient Normalized SPT-N value Mechanical load Temperature change Pile-head-structure contact stiffness Ultimate base bearing capacity

D L E α N60 P ΔT Kh qb

0.5 m 30 m 30 GPa 10−5 °C−1 25 2000 kN −20 to 20 °C 10 GPa/m 0

– – – – 0.05–0.4 – – –

4.1. Random field modeling of N60 In this paper, the Cholesky decomposition method is adopted to generate the lognormal random field of N60 [14]. The elements in the random field follow the discretization of the pile as shown in Fig. 1 (i.e., the number of elements in the random field is N). For each soil layer element, the spatial average effect is addressed by multiplying a variance reduction factor (Γ2) to the variance of the equivalent normal distribution for N60. In this study, the variance reduction function Γ2 is computed following the exponential correlation [38]:

proposed by the China Academy of Building Research [8]:

[g (x )]SLS _settlement = sult

(15)

z top

[g (x )]SLS _cracking = fult

(

m

(16)

pc )

2

where sult is the limiting immediate pile settlement, ztop is the immediate displacement at the pile top, fult is the concrete axial tensile strength, σm is the maximum axial tensile stress in the pile, and σpc is the pre-compressive stress of concrete. Considering that the pile is a bored pile, the pre-compressive stress in the pile is zero (i.e., σpc = 0). The values for ztop and σm can be obtained using the aforementioned loadtransfer analysis, and example fult values for several strength grades of concrete are provided in Table 4.

Qside =

(17)

i=1

s· su (z i )· C · Li

C40

C45

C50

Axial tensile strength (fult/MPa)

1.78

2.01

2.20

2.39

2.51

2.64

(23)

i = 1, 2, ...,N

(24)

4.2. Empirical correlations to address spatial variability of load-transfer curves In the geotechnical assessment of energy piles, multiple soil parameters are involved in the determination of the load-transfer curves as shown in Eqs. (1) and (2). However, generating multiple random fields with cross-correlations among parameters is complicated and can require a considerably large number of iterations of Monte Carlo simulation (MCS). To this end, the empirical equations that correlate the key soil parameters (su and EM) with N60 are used in this study. Several studies have focused on developing relationships between su and N60 (e.g., [34,37]). In this study, the empirical equation for estimating su of clays from Kulhawy and Mayne [21] is selected:

Table 4 Concrete tensile strengths for various concrete strength grades. C35

(22)

where Zj is the sequence of independent standard normalized random variables. Using this procedure, the example random fields of N60 with scales of fluctuation θ = 0.5, 4 and 16 m are shown in Fig. 5. In this simulation, the mean value and coefficient of variation (COV) for N60 are 25 and 0.3, respectively. Fig. 5 shows that a smaller θ corresponds to more dramatic fluctuation around the mean N60, indicating a more significant spatial variability.

To explicitly understand the effect of soil spatial variability on energy pile design, random field modeling [38] is implemented into the load-transfer model (Fig. 1). In this study, a commonly used in-situ index, the standard penetration test (SPT) penetration number N60 (the SPT blowcount value corrected for field procedures and apparatus), is selected as the random variable with spatial variability effect. In some scenario, the variability of N60 can be larger than that of some soil

C30

Mij Zj j=1

4. Random field modeling of energy piles

C25

(21)

i

G (x i ) =

where Qmob is the cumulative mobilized side resistance along the pilesoil interface (i.e., for a floating pile), which is computed using loadtransfer analysis. For the three limit states—as presented in Eqs. (15), (16) and (19)—failure is said to occur when g(x) is less than zero.

Concrete strength grade

(20)

With the matrix M, the correlated standard normal random field is obtained as follows:

(19)

Qmob

2l

= M × MT

where As,i is the lateral surface area of Segment i. The limit state equation for the ULS design ([g (x )]ULS ) is defined as:

[g (x )]ULS = Qult

1 + exp

where = |x i xj | represents the absolute distance between any two points in the random field. The correlation matrix M built by the correlation function is further reduced by the Cholesky decomposition (e.g., [14]).

(18)

i=1

2l

2

( ) = exp

N

qs (z i)·As, i =

·

where xi is the position at which N60 is modeled; G(xi) is a standard normal random field; and σn and μn are the standard deviation and mean, respectively, of the equivalent normal distribution of N60. The exponential correlation function can be expressed as follows [18]:

where Qside is the side resistance, Qbase is the tip resistance, and W is the pile weight. Considering that the hypothetical pile is a floating pile embedded in homogeneous undrained clays, Qbase can be ignored, and the α-method can be used to compute Qside: N

2

N60 (x i ) = exp[µn + · n· G (x i )]

The ultimate bearing capacity (Qult) of a pile under axial loading can be determined with Eq. (17):

W

1 · 2 l

where θ is the vertical scale of fluctuation and l is the characteristic length. Given a variance reduction factor Γ2, the lognormal random field of N60 can be generated using the following equation [14]:

3.2. ULS: Bearing capacity of a single energy pile

Qult = Qside + Qbase

=

su pa = 0.06cs N60

Note: data from China Academy of Building Research [8]. 313

(25)

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Z. Luo, B. Hu

Fig. 6. Relative frequency of the computed immediate pile settlement using random field modeling (COV = 0.3, ΔT = −20 °C and θ = 4 m).

spatial variability on the pile responses due to thermo-mechanical loading, the immediate settlements (i.e., pile head displacements) for various combinations of COV and θ for N60 are simulated using the aforementioned random field procedure. The Monte Carlo simulation results in distributions of immediate pile settlement for a given mean, COV and θ of N60. For a pile subjected to cooling (ΔT = −20 °C), the mean, COV and θ of N60 are set to be 25, 0.3 and 4, respectively, in this case study. Fig. 6 shows the resulting histogram of immediate pile settlement when using load-transfer analysis; it indicates that the immediate pile settlement approximately follows a lognormal distribution. For other combinations of COV and θ of N60, the fitted lognormal distributions of immediate pile settlements during cooling (ΔT = −20 °C) are shown in Fig. 7. The mean values of immediate pile settlement are comparable (i.e., 7.6 mm) regardless of COV and θ, where the mean value of the cooling-induced immediate settlement is approximately 4.2 mm and the immediate settlement due to mechanical load is close to 3.4 mm. Thus, the thermal effect plays a significant role in the pile settlement analysis. Fig. 7(a) shows that for a constant COV, a smaller θ results in a smaller variation in settlement, which is due to the spatial averaging effect: a smaller scale of fluctuation causes a larger reduction in variance in the soil parameters. Fig. 7(b) shows that for a constant θ, the resulting variation in immediate pile settlement increases with COV. Hence, it is important to statistically characterize spatial variability (COV and θ) during the site investigation.

Fig. 5. Example of simulated spatial variability of N60 by means of random field modeling.

where pa is the atmospheric pressure (i.e., 1 atm = 100 kPa), and cs is the model bias factor. Bozbey and Togrol [6] provided a power relationship between the Menard pressuremeter modulus [EM (MPa)] and N60, which is given as:

EM = 1.61cE (N60)0.71

(26)

where cE is the model bias factor. It is noted that in this study cs and cE are first assumed to be unity (i.e., unbiased models) and the effect of transformation model biases on the energy pile assessment is explored in a subsequent section. 5. Effect of spatial variability on the energy pile assessment In the random field modeling of energy piles, the spatially varied random soil parameter N60 is first generated using Eq. (21). Next, the related soil parameters (su and EM) are readily determined with empirical correlations [i.e., Eqs. (25) and (26)] and the model bias factors (cs and cE) are assumed to be 1.0. The input random soil parameters are then mapped onto the discretized pile segments, as shown in Fig. 1. These soil parameters are used to determine the load-transfer curves for each pile segment, using Eq. (1) and following Fig. 2. Load-transfer analysis following the flowchart shown in Fig. 3 is then performed to estimate the pile responses induced by the combined thermo-mechanical loadings. This procedure is iterated in a Monte Carlo framework, and the resulting pile responses can be assessed in a probabilistic manner. In this case study, the number of pile segments and the number of elements in the random field N is 1500 (i.e., each pile segment has a length of 0.02 m). A series of scales of fluctuation (θ = 0.5, 1, 2, 4, 8, 16, 1000 m) and coefficients of variation (COV = 0.05, 0.2, 0.3, 0.4) are selected in the probabilistic analysis of a single energy pile subjected to different temperature changes (ΔT = −20, −15, −10, −5, 0, 5, 10, 15, and 20 °C). The number of iterations of MCS for each combination of θ, COV and ΔT is 106.

5.2. Effect of spatial variability on the SLS failure of an energy pile It is known that an energy pile can exhibit excessive settlement during a cooling cycle. Similarly, structural cracking failure can occur in an energy pile during the cooling cycle as thermally induced tensile strains develop. Based on the limit state function in Eqs. (15) and (16), the geotechnical SLS (settlement and cracking) failure probability is further assessed herein. To illustrate the effect of soil spatial variability on the SLS failure of energy piles, the limiting criteria in these two limit state functions (i.e., sult and fult) are taken as 8 mm and 1.78 MPa, respectively. Given a temperature drop of −20 °C, Fig. 8 shows the probability of SLS failure at various combinations of COV and θ. It is obvious from Fig. 8 that Pf increases with both COV and θ of N60. When θ is less than 10 m, Pf increases significantly with θ. The effect of spatial variability is obvious: take failure due to excessive pile settlement as an example, when COV = 0.30, Pf for θ = 1.0 m is 9.7 × 10−2, while Pf for θ = 1000 m is 4.1 × 10−1. These results indicate that when the effect of spatial variability is ignored (e.g., using θ = 1000 m instead of 1.0 m), the probability of geotechnical SLS failure can be overestimated.

5.1. Effect of spatial variability on the pile responses Pile displacement, as a direct outcome of the load-transfer analysis, can then be used to calculate other pile responses (e.g., axial strain, shaft friction, axial force). As a result, to investigate the effect of soil 314

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Fig. 8. Effect of soil spatial variability on geotechnical serviceability limit state (SLS) failure for an energy pile (ΔT = −20 °C).

excessive pile settlement failure can be overestimated or underestimated if the soil spatial variability is ignored. It is worth noting that the limiting immediate pile settlement that distinguishes the overestimation and the underestimation is approximately the nominal mean value of the pile displacement (e.g., 7.6 mm, as shown in Fig. 7). Considering that a larger limiting settlement with a range between 10 mm and 20 mm is generally used in practice, this study shows that the probability of excessive pile settlement failure is unduly conservative if the soil spatial variability is ignored. Similarly, based on fult values as listed in Table 4, the effect of soil spatial variability on probability of failure due to pile cracking at various concrete strength grades is shown in Fig. 9(b). From this figure, it is apparent that a higher tensile strength results in a lower probability of failure against cracking at the same θ. Thus, it is critical to determine the optimum strength grade of concrete to ensure the structural SLS of an energy pile. In addition, as can be seen in Fig. 9(b), a smaller θ value leads to a smaller Pf for all levels of concrete tensile strength grade. In short, the soil’s inherent variability (COV and θ) has a profound impact on the SLS failure of energy piles. Based on the estimated probability of SLS (settlement and cracking) failure, it is shown that if the effect of soil spatial variability is ignored, Pf can be unduly overestimated in this case study.

Fig. 7. Fitted lognormal distribution curves of the computed energy pile settlement distributions (ΔT = −20 °C).

The selected limiting criteria (i.e., sult and fult) has considerable impact on the predicted Pf. At an extreme condition where ΔT = −20 °C, the probability of SLS failure for each level of θ at a COV of 0.3 under various limiting criteria is presented in Fig. 9. Considering the limiting immediate pile settlement (sult) is case-specific, the effect of soil spatial variability on probability of exceeding various limiting immediate settlements for an energy pile is shown in Fig. 9(a). As can be noticed in this figure, Pf decreases with the limiting immediate settlement for all levels of θ. The Pf values at different levels of θ are approximately 0.50 when the limiting immediate settlement is equal to 7.6 mm, which is consistent with the nominal mean value of the pile displacement shown in Fig. 7. From Fig. 9(a), it can also be noticed that depending on the limiting immediate pile settlement, the probability of 315

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(a) Excessive pile settlement failure (limiting immediate settlement = 8 mm)

Fig. 9. Effect of soil spatial variability on geotechnical serviceability limit state (SLS) failure under various limiting criteria (ΔT = −20 °C and COV of N60 = 0.3).

(b) Pile concrete cracking failure (tensile strength = 1.78 MPa) Fig. 11. Probability of structural and geotechnical serviceability limit state (SLS) failure for various combinations of temperature changes and scales of fluctuation (COV of N60 = 0.3).

in the previous thermo-mechanical load-transfer analysis [20]. By determining the ultimate bearing capacity (Qult) with Eqs. (17) and (18), the probability of bearing capacity failure at various combinations of COV and θ can be estimated. As shown in Fig. 10, Pf increases with the increase of both COV and θ. The effect of θ on bearing capacity is very obvious: for example, when COV is 0.3, it is observed that the Pf value is about 2.2 × 10−3 at θ = 1.0 m and is about 0.20 at θ = 1000 m during the heating. Thus, the bearing capacity failure probability of 0.20 obtained from the reliability analysis is almost a spatial constant (θ = 1000 m) is likely to be overestimated by a few orders of magnitude. It should be noted that this observation may only be applicable for this case study, as the simulation results may differ if another mechanical load is applied on the energy pile. It is noted that in practice, energy piles are designed without considering spatial variability, based on an enlarged factor of safety. The factor of safety itself cannot address the effect of soil spatial variability. Thus, the developed probabilistic model in this study provides a robust tool to quantify various categories of uncertainty in the energy pile design.

Fig. 10. Effect of soil spatial variability on geotechnical bearing capacity failure for an energy pile (ΔT = +20 °C).

5.3. Effect of spatial variability on the ULS failure of an energy pile Next, the effect of spatial variability on geotechnical ULS assessment is presented. Eq. (19) is employed as the limit state function to predict the probability of ULS failure. Since the temperature effect on soil bearing capacity (QULT) is complicated and worthy of future investigation, QULT is reasonably assumed to be temperature-independent

5.4. Effect of temperature on the three failure modes Finally, the effect of temperature on the ULS and SLS assessment is investigated. Fig. 11 shows the geotechnical SLS failure at various levels 316

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Fig. 12. Probability of bearing capacity failure for various combinations of temperature changes and scales of fluctuation (COV of N60 = 0.3).

Fig. 13. Effect of transformation model biases on the mean value and one standard deviation bounds of the immediate pile settlement (COV of N60 = 0.3).

of θ for each temperature change (COV of N60 = 0.3). For both excessive pile settlement failure and pile concrete cracking failure, a larger θ indicates a larger Pf at the same level of ΔT. At the same θ during cooling cycles (shown in Fig. 11), Pf against cracking and excessive settlement increases noticeably with a decreasing temperature (−ΔT), which indicates that a large decrease in temperature is the worst-case scenario in the SLS design of a single energy pile. Hence, the SLS design of an energy pile should be determined based on the maximum temperature drop during the service period. For ULS failure, it can be noticed from Fig. 12 that the probability of bearing capacity failure Pf increases noticeably with an increase in temperature (+ΔT). For instance, at the condition of θ = 8 m, the Pf values for ΔT = 5 and 20 °C are 4.7 × 10−4 and 4.5 × 10−2, respectively. Under heating, a floating pile expands about a null point with zero movement. Due to the superstructure constraint, the upward side friction along the lower pile segment is larger than the downward side friction along the upper pile segment, indicating that there is some bearing capacity loss due to the soil mobilization. Given constant QULT, the mechanical load (P) that a pile under heating can carry is smaller, and thus a higher probability of bearing capacity failure is expected. Similarly, these results indicate that the maximum temperature increase dominates the ULS design of a single energy pile. As a summary, an extreme temperature change leads to a worse-case scenario in the geotechnical design of a single energy pile. The random field modeling of energy piles in this study explicitly demonstrates the effect of soil spatial variability on ULS and SLS design. Since SLS design and ULS design of energy piles are dominated by different failure modes, both the heating and cooling scenarios need to be addressed.

ΔT = ± 20 °C) in this case study is selected and the COV value of N60 is set as 0.3. For comparison, the mean values and one standard deviation bounds of the immediate pile settlement with and without considering model biases are shown in Fig. 13. It is obvious that both the mean values and upper bounds considering model biases are slightly larger than those ignoring the model biases, which can influence the estimated failure probability. Hence, whatever laboratory or field index is used to determine the load-transfer curves, it is important to properly calibrate the transformation model in the assessment of energy piles. 7. Concluding remarks Focusing on the effect of soil spatial variability on energy piles, a probabilistic model based on the load-transfer method was developed in this study. The developed MCS-based random finite difference approach is used to investigate the effect of vertical spatial variability of the soil parameter N60 on the ULS (bearing capacity) and SLS (settlement and cracking) design of a single energy pile. The following conclusions are drawn:

• The soil spatial variability has a significant impact on the pile re-

•

6. Effect of transformation model uncertainty on probabilistic analysis of energy piles

•

The transformation models [i.e., Eqs. (25) and (26)] to determine the load-transfer curves, are key elements in the developed probabilistic approach for energy piles. It is noteworthy that the model bias factors (cs and cE) for Eqs. (25) and (26) are simply set as unities in the previous analysis. Here, the effect of cs and cE on the energy pile assessment is further explored. A model bias is defined as a ratio of the observed quantity over the predicted quantity. Using the database from Sivrikaya and Toğrol [31,32] as an example, the mean value and COV for cs are estimated to be 1.0 and 14% under a normal assumption. Similarly, the mean value and COV for cE are determined to be 1.0 and 26%, based on a set of data from Bozbey and Togrol [6]. Using the calibrated model bias factors, the immediate pile settlements for various combinations of θ for N60 are simulated again. For demonstration, the worst-case scenario of energy piles (i.e.,

sponses under thermo-mechanical loading. The mean value of immediate pile settlement remains constant regardless of COV and θ for N60, whereas the COV of immediate pile settlement increases with COV and θ of N60. Hence, it is important to statistically characterize spatial variability (COV and θ) during the site investigation. In this case study, the probability of both SLS failure and ULS failure without considering soil spatial variability can be overestimated by a few orders of magnitude, compared to that considering the spatial effect. These results underscore the importance of quantifying the uncertainty in the SLS and ULS assessment of an energy pile. The effect of temperature change is also obvious: for this case study, a larger temperature drop results in a higher probability of SLS failure, while a larger temperature increase induces a higher probability of ULS failure. Thus, an extreme temperature change typically leads to a worse-case scenario in the geotechnical design of a single energy pile. Since SLS design and ULS design of energy piles are dominated by different failure modes, both the heating and cooling scenarios need to be addressed.

The developed probabilistic model in this study provides a robust tool to quantify various categories of uncertainty in the energy pile design. The predictions of this probabilistic model are limited to the short-term settlement due to the selected load-transfer curves. It should be noted that the down drag loads induced by soil consolidation could 317

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further impair the geotechnical performance of piles (e.g., [12,13,29]). It is advisable to further address the soil consolidation settlement and to implement cyclic load-transfer approach (e.g., [36]). The long-term thermo-mechanically induced negative side friction and the consolidation settlement of energy piles still need further investigation.

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