J. Cent. South Univ. Technol. (2007)06-0864-06

DOI: 10.1007/s11771-007-0164-3               

Non-probabilistic fuzzy reliability analysis of pile foundation stability by interval theory

CAO Wen-gui(曹文贵), ZHANG Yong-jie(张永杰), ZHAO Ming-hua(赵明华)

(Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China)

Abstract：Randomness and fuzziness are among the attributes of the influential factors for stability assessment of pile foundation. According to these two characteristics, the triangular fuzzy number analysis approach was introduced to determine the probability-distributed function of mechanical parameters. Then the functional function of reliability analysis was constructed based on the study of bearing mechanism of pile foundation, and the way to calculate interval values of the functional function was developed by using improved interval-truncation approach and operation rules of interval numbers. Afterwards, the non-probabilistic fuzzy reliability analysis method was applied to assessing the pile foundation, from which a method was presented for non- probabilistic fuzzy reliability analysis of pile foundation stability by interval theory. Finally, the probability distribution curve of non- probabilistic fuzzy reliability indexes of practical pile foundation was concluded. Its failure possibility is 0.91%, which shows that the pile foundation is stable and reliable.

Key words: pile foundation; fuzziness; interval theory; interval-truncation approach; non-probabilistic fuzzy reliability analysis

1 Introduction

At present, certainty analysis method is widely used in the evaluation of pile foundation stability, which satisfies the requirement of bearing capacity or settlement. It is an effective method for computation and designing of pile foundation. However, the stability affecting factors of pile foundation are complicated because of many uncertainties, such as fuzziness, randomness and interval feature of mechanical parameter values. These uncertainties can not be reflected by the certainty analysis method. So the uncertainty analysis method is used to evaluate the pile foundation stability.

The uncertainty analysis methods are mostly based on probability theory. TANDIIRIA et al^[1] and BARAKAI et al^[2] analyzed the reliability of pile foundation with lateral load by response surface method and probability method. LI et al^[3] and ZAI et al^[4] studied the reliability with vertical load by probability method. Although these researches have made some progress, they still have some limitations because they cannot show the fact that there is not a define line between the instability and stability^[5], that is to say, the pile foundation instability is a fuzzy changing process. Thus, the reliability analysis method was suggested considering the fuzziness by UTKIN et al^[6] and SAWYER et al^[7]. For the pile foundation, the settlement was used as fuzzy variable to set up fuzzy membership function by LI and SHU^[8] to analyze stability reliability, which is a good attempt. However, there are still some questions unconsidered. Firstly, the method proposed by LI and SHU can only be used for the reliability analysis of pile foundation, which settlement is known. Its application range is limited. Secondly, the factors considered are not complete and cannot reflect the feature of uncertainties when determining the mechanical parameter values. Thirdly, the failure probability is only a certain calculating result and cannot show the changing process of pile foundation stability with uncertainties. But the ideal analysis method needs to reflect not only the fuzziness, randomness and interval feature of factors but also the fuzzy changing process of pile foundation instability, which will be the focus of this thesis.

In this study, for the non-probability features of the reliability analysis of pile foundation stability, the triangular fuzzy numbers were firstly used to determine the mechanical parameter, which could reflect their fuzziness, randomness, interval feature and probability distribution. Then, a reasonable functional function on reliability analysis was proposed by studying the bearing mechanism of pile foundation. Finally, in order to reflect the fuzzy changing process of pile foundation instability, a method was presented to determine the probability-distributed curve of non-probabilistic fuzzy reliability indexes and its failure possibility by interval and non-probabilistic reliability analysis theory. Thereby, a non-probabilistic fuzzy reliability evaluation approach was developed for pile foundation stability based on interval theory, which would further improve the method of uncertainty analysis of pile foundation stability.

2 Interval non-probabilistic fuzzy reliability analysis method of pile foundation

In order to establish the non-probabilistic fuzzy reliability analysis method of pile foundation stability based on interval theory, the probability distribution of load and mechanical parameters of pile and soil should be firstly determined by triangular fuzzy numbers. Then, the functional function of fuzzy reliability analysis can be constructed based on the bearing mechanism of pile foundation, and the corresponding operation technique of interval parameter values will be developed. Finally, according to the calculated interval values of functional function, a measuring way for non-probabilistic fuzzy reliability of pile foundation stability will be established by the method of non-probability analysis.

2.1 Method to determine mechanical parameter

For the evaluation of pile foundation stability, mechanical parameter values of load as well as pile and soil should be firstly determined. There are randomness and fuzziness with them, which should be reflected. Triangular fuzzy numbers can be used to reflect the probability distribution of mechanical parameter values. For the sake of our elaboration, its definition will be firstly introduced, then the method to determine triangular fuzzy numbers of parameter values for pile foundation stability analysis will be given.

If =(a^L, a^M, a^R), where a^L≤a^M≤a^R, thencan be called as a triangular fuzzy number^[9], as shown in Fig.1. If  belongs to fuzziness set U, then

Fig.1 Explanation of triangular fuzzy number

where  A_λ=[a_λ^L, a_λ^R] is a λ cut-set of; λ is cut-set level or credible level, λ∈[0, 1]; A(u) is the membership function of. A_λ is a set consisting of the elements whose membership function values are more or equal to λ in U^[9]. It shows that triangular fuzzy number can reflect the interval feature of parameter values and their probability distribution. Hereby, according to the feature of mechanics parameters of pile foundation stability analysis, the method to determine triangular fuzzy numbers can be set up as follows.

1) For soil mechanical parameters, their average values M_soil and their variation coefficient δ_soil can be gained by statistical analysis of their test data. M_soil can be regarded as a^M of triangular fuzzy number. M_soil(1-δ_soil) and M_soil(1+δ_soil) can be respectively regarded as a^L and a^R.

2) For the load of pile foundation, its average value M_load can be calculated by upside load according to the specification of pile foundation. Its variation coefficient δ_load can also be gained according to the specification. Then, M_load, M_load(1-δ_load) and M_load(1+δ_load) can respectively be regarded as a^M, a^L and a^R.

It is known that triangular fuzzy numbers of mechanical parameters of load and pile as well as soil can be determined by the three values a^M, a^L and a^R from Fig.1 and the method proposed.

When the triangular fuzzy numbers for mechanical parameters are applied to analyzing the stability of pile foundation, they should be firstly changed into interval numbers, which can be calculated by interval analysis theory. In order to reflect the variation tendency of pile foundation stability situation with different cut-set levels to all parameter values, the reliability indexes with 10 cut-set levels 0, 0.1, 0.2,…,0.9 will be respectively calculated.

2.2 Method to determine interval functional function

The functional function of pile foundation reliability analysis should be set up when triangular fuzzy numbers are used to analyze the stability. Here, it will be decided by the equilibrium relation between bearing capacity of pile foundation and load effect under given safety factor. The limit balance equation can be shown as^[3-4]:

f_aA_c+nP_u=K(S_G+S_L+G_k)           (2)

where  f_a is ultimate bearing capacity of natural subsoil; A_c is underside area of pile cap; P_u is ultimate bearing capacity of single pile; S_G is standard value of static load; S_L is standard value of live load; G_k is gravity value of pile cap and overlying soil; K is safety factor; n is number of piles.

The ultimate bearing capacity f_a of natural subsoil can be calculated by Hansen’s formula^[10]. If the influences of foundation shape and inclining of load and earth surface as well as foundation bottom are not considered, then

f_a=0.5γbN_γ+cN_cd_c+qN_qd_q          (3)

where  γ is soil gravity; b is foundation width; c is soil cohesion; q is overload above foundation bottom, q=γd; d_c, d_q are coefficients of foundation depth; N_c, N_q and N_γ are coefficients of bearing capacity, and they can be respectively determined as follows^[10]:

N_q=tan²(45?+φ/2)exp(πtan φ)           (6)

N_c=(N_q-1)cot φ                (7)

N_γ=1.5(N_q-1)tan φ               (8)

The ultimate bearing capacity P_u of single pile can be calculated by the following equation^[10]：

             (9)

where  u_p is length of pile perimeter; q_ski is ultimate side friction resistance of pile in the ith layer soil; l_i is thickness of the ith layer soil; q_pk is ultimate bearing capacity of pile tip; A_p is area of pile tip.

The functional function Z of reliability analysis of pile foundation stability can be set up by substituting Eqns.(3)-(9) into Eqn.(2), shown as follows:

   (10)

The fuzziness of parameters φ, c, q_pk, q_ski, S_G and S_L are only considered, when the non-probabilistic fuzzy reliability analysis of pile foundation stability is carried out according to Eqn.(10). The parameters in Eqn.(10) should be written as triangular numbers while analyzing the reliability of pile foundation stability, and they need to be changed into interval numbers by level-cut technique. Then Eqn.(10) can be changed into the following formula:

Z(λ)={0.5γb[N_γλ^L, N_γλ^R]+[c_λ^L, c_λ^R][N_cλ^L, N_cλ^R]d_c+q[N_qλ^L, N_qλ^R][d_q^L, d_q^R]}A+n{u_p∑[]l_i+

 [q_pkλ^L,q_pkλ^R]A_p}-K{[S_Gλ^L, S_Gλ^R]+ [S_Lλ^L, S_Lλ^R]+G_k} (11)

where 

 [N_γλ^L, N_γλ^R], [N_cλ^L, N_cλ^R], [N_qλ^L, N_qλ^R] and [d_q^L,d_q^R] can be calculated by Eqn.(4)-(8) with [φ_λ^L, φ_λ^R].

So the functional function of reliability analysis of pile foundation stability can be put forward based on the interval theory.

2.3 Interval functional function computation method

In general, interval numbers can be usually calculated by the operation rules of interval numbers^[11]. But if one interval number exists more than once in an equation, the width expanding problem of interval number will appear. This problem also exists during the non-probabilistic reliability analysis of pile foundation. For example, φ exists more than twice when calculating N_c, N_q and N_γ. This problem must be solved. Therefore, the improved interval-truncation approach^[12-13] was introduced to determine the interval value of functional function for non-probabilistic reliability analysis of pile foundation stability.

Supposing A_iλ=[](i=1, 2, …, n) as the interval value of triangular fuzzy numberwith cut-level λ and B_λ=[b_λ^L, b_λ^R] as the interval value of the functional function, which can be gained by the operation rules of interval numbers with some cut-level. The middle value B_λ0 of interval number B_λ can be calculated by the middle value A_iλ0=()/2 of interval number A_iλ. When B_λ0 is near zero, the interval number expending cannot be reduced by truncation approach. When B_λ0 is far from zero, the relative deviations ₁ and ₂ between b_λ^L, b_λ^R and middle value B_λ0 can be respectively calculated as follows:

₁=∣(b_λ^L-B_λ0)/B_λ0∣              (12)

₂=∣(b_λ^R-B_λ0)/B_λ0∣              (13)

For conservative analysis, the total relative deviation =₁+₂ is always expected to be more than the true value. Now the maximal relative deviation of the interval value of functional function is supposed as 2t when analyzing non-probabilistic reliability of pile foundation stability, where t is the maximal deviation of all the inputting interval variation values to their middle values. If the finally interval value B_λ can be expressed by the truncation interval [c_λ^L, c_λ^R], then the method to determine it can be described as follows:

1) when ₁≤t and ₂≤t，then

c_λ^L＝b_λ^L，c_λ^R＝b_λ^R                 (14)

2) when ?₁>t and ?₂>t，then

           (15)

3) when ₁≤t and ₂＞t，then

c_λ^L＝b_λ^L，c_λ^R＝B_λ0+t(b_λ^R-B_λ0)/₁       (16)

4) when ₁＞t and ₂≤t，then

c_λ^L＝B_λ0+t(b_λ^L-B_λ0)/₁，c_λ^R＝b_λ^R       (17)

The interval value of the functional function gained through the above process can match the practical situation to a larger extent, thus it will improve the veracity of reliability evaluation of pile foundation stability.

2.4 Measuring method of non-probabilistic fuzzy reliability

For pile foundation stability, the functional function interval value of non-probabilistic reliability analysis can be determined through the above method. However, the measuring method of fuzzy reliability must be developed when evaluating the reliability of pile foundation. The interval functional function can be changed into standard form as follows:

            (18)

Then, the measuring method of pile foundation reliability can be set up by the definition of non-probabilistic reliability index η_λ as follows^[14]:

η_λ=min(‖δ_λ‖_∞)               (19)

Meanwhile it should satisfy the condition:

Z(λ)=G(δ_1λ, δ_2λ,…, δ_6λ)=0          (20)

where  δ_λ={δ_1λ, δ_2λ,…, δ_6λ} is a vector which consists of standardized interval numbers of triangular fuzzy number set with cut-level λ. If one interval number is c_λ=[c_λ^L, c_λ^R], its standardized interval number can be written as

c_λ= c_λ0±δ_cλc_λ^r                (21)

where  c_λ0=(c_λ^L+c_λ^R)/2 is middle value of interval number; c_λ^r=( c_λ^R-c_λ^L)/2 is deviation of interval number; δ_cλ∈[-1,1] is standardized interval variable of c with cut-level λ.

The functional functionof reliability analysis for pile foundation stability is complicated and with many variables during practical analysis. According to the optimization calculating method of interval function^[14], Eqn.(19) can be equivalent to the following equation:

η_λ=Z_0λ/Z_λ^′                 (22)

where Z_0λ and Z_λ^′ are respectively the mean and deviation of functional function interval value Z(λ) with cut-level λ.

It is known from Eqn.(19) that the shortest distance between the coordinate origin and failure surface (calculated by infinite normal number‖?‖_∞) can be regarded as non-probabilistic reliability index η_λ of pile foundation stability analysis. The boundary of failure region is fuzzy, so non-probabilistic reliability index η_λ is a fuzzy variable. Then it can be called as non-probabilistic fuzzy reliability index and written as η_λF. Moreover, η_λF increases monotonously with cut-level λ. If η_λF>1, stability region of pile foundation does not intersect with its failure region, and the pile foundation is stable, or it is instable. The larger η_λF is, the more stable pile foundation is. This can reflect the changing process of pile foundation stability situation with different parameter values.

So far, the probability-distributed curve can be got through solving the non-probabilistic fuzzy reliability index of pile foundation stability with different cut-levels, which can reflect the changing process of pile foundation stability situation. That is to say, in whatever situation the pile foundation is in stable state, limit state or instable state. Furthermore, it can reflect the whole failure probability of pile foundation with different parameter values. In order to be convenient to get the instability probability, the definition of possibility estimation is first shown as follows.

It is known by the possibility theory^[15] that the probability distribution function π_X of fuzzy variable is equal to the value of membership function u_A in numerical value. If A is a fuzzy set to the field U and Π_X is probability distribution of variable X in U, then the possibility estimation π(A) to fuzzy set A can be defined as^[15]

Poss{X∈A}=π(A)={u_A(u)∧π_X(u)}    (23)

where  π_X(u) is probability-distributed function of Π_X; “∧”stands for the operation of taking smaller value; u is variable value. Then the possibility to x≤u is

Poss{x≤u}=sup{π_X(u)∣x≤u, x∈U}      (24)

According to the above definition, the failure possibility π_f and stable inevitability N_r of pile foundation stability can be got, where π_f is the cut-level value when the non-probabilistic fuzzy reliability index is 1 on the probability-distributed curve, shown as

π_f=Poss(η_λF≤1)=λ∣η_λF=1  λ∈[0, 1], η∈R    (25)

N_r=Ness(η_λF＞1)=1-π_f            (26)

From the study, it is known that the probability distribution curve of pile foundation stability can be got through its non-probability fuzzy reliability curve index with different cut-levels, which reflects the distributed situation of fuzzy reliability index. The failure possibility is a value, which reflects the maximal failure possibility of pile foundation. Compared with the traditional reliability index, the results gained by non-probabilistic fuzzy reliability analysis method can give much more information to assess the pile foundation stability.

3 Practical engineering analysis

3.1 General situation

The data concerning the following engineering project are from Ref.[3]. Composite pile foundation was used to one multi-story building on soft ground. The calculating parameter values of load are S_G＝34.5 kN, S_L＝8.2 MN, δ_G＝0.07, δ_L＝0.29. The designed raft dimensions are 10 m×30 m and its depth is 1.3 m. The groundwater level is 0.5 m. The designed section dimensions of prefabricated concrete square pile are  200 mm×200 mm, its compressive strength is C30, length is 16 m, total number n is 154, safety factor K is 2.0. The parameters of soil mechanics and their variation coefficients of ground foundation are shown in Table 1. Pile layout and soil distribution are separately shown in Figs.2 and 3.

Table 1 Parameters of soil mechanics for ground foundation

Fig.2 Arrangement of pile foundation

Fig.3 Soil distribution of pile foundation

3.2 Implementing course and analytical results

1) When the ultimate bearing capacity of ground foundation is calculated, the randomness and fuzziness of the mechanics parameters of only silt clay are considered. While calculating the limited bearing capacity of pile, only soil layers 2－5 are considered. Triangular fuzzy numbers of load and soil mechanics parameters can separately be gained by the above method proposed in this paper. They are shown as follows: =(32 085, 34 500, 36 915) kN, =(5 822,

8 200, 10 578) kN, =(13.3, 16.8, 20.2) kPa, =(8.3?, 10.4?, 12.5?),=(600, 800, 1 000) kPa, =(10.7, 12,13.3) kPa, =(10, 12, 14) kPa, =(10.1, 12, 13.9) kPa, =(27.6, 33.6, 39.6) kPa. The certainty parameter values are γ=18.0 kN/m³, b=10 m, q=23.5 kN/m², A=294 m², n=156, u_p=0.04 m, A_p=0.04 m², K=2.0, G_K=7.02 MN.

    2) For the triangular fuzzy numbers decided, the corresponding interval numbers with cut-level of 0.1 are determined according to the definition of cut-level. They are shown as follows: S_G0.1=[32 327, 36 674] kN, S_L0.1=

[6 060, 10 340] kN, c_0.1=[13.8, 19.9] kPa, φ_0.1=[8.5?, 12.3?], q_pk0.1=[620, 980] kPa, q_sk2-0.1=[10.9, 13.1] kPa, q_sk3-0.1=[10.2, 13.8] kPa, q_sk4-0.1=[10.3, 13.7] kPa, q_sk5-0.1= [28.2,39.0] kPa.

3) Calculating interval functional function of pile foundation stability by Eqn.(11) with operation rules of interval numbers and improved interval-truncation approach, the interval value [10, 120] kN with cut-level of 0.1 can be got.

4) The middle value Z_0-0.1=110 kN and deviation Z′_0-0.1=100 kN can be got by interval theory. Then it is known that the non-probability fuzzy reliability index η_0.1F of pile foundation stability is 1.1 with cut-level of 0.1 by Eqn.(22).

5) The interval values of functional function with cut-level of 0 and 0.2-0.9 are separately calculated, so the non-probabilistic fuzzy reliability indexes and their probability-distributed curve can be got, as shown in Fig.4.

6) Finally, the failure possibility π_f and the stable inevitability N_r of pile foundation non-probabilistic fuzzy reliability analysis can be respectively gained by Eqns.(25) and (26). The result is that N_r is equal to  99.09％ and π_f is equal to 0.91％.

It is known from the above results that the design of the concerned engineering project is reliable. In fact, the pile foundation is stable and safe, which has no failure phenomenon since its completion^[3].

Fig.4 Probability distribution of non-probabilistic fuzzy reliability index

4 Conclusions

1) Triangular fuzzy numbers are used to stand for the values of affecting factors of pile foundation stability, which can reflect their randomness, fuzziness, interval feature and probability distribution, accords well with engineering practice.

2) The interval functional function of pile foundation reliability analysis is analyzed by improved interval-truncation approach, which can avoid the problem that the reliability analysis result dose not accord with practical situation due to interval expanding.

3) The probability distribution curve of non- probabilistic fuzzy reliability index can reflect the fuzzy changing process of pile foundation stability, and the failure probability can reflect the fail possibility of the whole pile foundation. Compared with random reliability analysis method, the method proposed in this paper can give much more information for pile foundation stability evaluation.

References

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(Edited by YANG Hua)

Foundation item:Project(50378036) supported by the National Natural Science Foundation of China；Project(03JJY5024) supported by the Natural Science Foundation of Hunan Province, China

Received date: 2007-03-18; Accepted date: 2007-06-20

Corresponding author: CAO Wen-gui, Professor, PhD; Tel: +86-731-8821659; E-mail: cwglyp@public.cs.hn.cn