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Calibration of characteristic values of soil properties using the random finite element method
archives of civil and mechanical engineering 16 (2016) 112–124
Available online at www.sciencedirect.com
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Original Research Article
Calibration of characteristic values of soil properties using the random finite element method Ł. Zaskórski *, W. Puła Faculty of Civil Engineering, Wrocław University of Technology, Wybrzeże St. Wyspiańskiego 27, 50-370 Wrocław, Poland
article info
abstract
Article history:
In this paper the reliability assessment of the shallow strip footing was conducted using a
Received 13 May 2015
reliability index b. Therefore some approaches of evaluation of characteristic values of soil
Accepted 27 September 2015
properties were compared in order to check what reliability index b can be achieved by
Available online 30 October 2015
applying each of them. For this purpose, design values of the bearing capacity based on these approaches were referred to design values of the bearing capacity estimated by the random
Keywords:
finite element method. Design values of the bearing capacity were estimated for various
Eurocode 7
widths and depths of foundation in conjunction with design approaches defined in Eurocode
Bearing capacity
7. The cohesive soil was considered – clay from the area of Wrocław. The characteristic
Random field
values of shear strength parameters were evaluated basing on the effective values of soil
Probability analysis
parameters.
Reliability index
1.
# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
Introduction
The random character of the soil properties is an essential problem in terms of estimating parameters applied in geotechnical analysis. These parameters have a major influence on the safety of a designed structure. The randomness of the soil properties is greater than other typical materials considered by solid mechanics. Random variability of soil parameters occurs even within homogenous layers of soil [1]. The reason of this phenomenon is a natural variability of the soil itself, which cannot be unambiguously defined by mathematical equations. The uncertainty in determination of the soil parameters aside from the natural randomness is associated with measurement errors, limited amount of data based on in situ tests and an uncertainty of transformation. Therefore, a universal algorithm of estimation of soil parameters, which
would cope with all abovementioned sources of uncertainty, has not yet been created. Consequently many approaches used to define the soil properties were developed. However their goal is only the estimation of approximate values of soil parameters with a certain level of safety. The explicit method of soil parameters evaluation cannot be found in the applicable standards – Eurocode 7. In general terms Eurocode leaves wide scope for interpretation. Therefore the present study is focused on comparing commonly used approaches for the evaluation of characteristic values of soil parameters (Duncan's method, Schneider's method, Schneider's method with influence of a fluctuation scale, Orr and Breysse's method, method based on 5% quantile which is included in Eurocode 7) by assessing a shallow foundation bearing capacity in accordance with the guidelines contained in Eurocode 7. Design values of the bearing capacity were evaluated by applying design approaches – DA1.C1, DA1.C2,
* Corresponding author. Tel.: +48 668 155 089. E-mail addresses: [email protected] (Ł. Zaskórski), [email protected] (W. Puła). http://dx.doi.org/10.1016/j.acme.2015.09.007 1644-9665/# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
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DA2* and DA3. Reliability index b as adopted as a criterion for comparison of the above methods. Values of reliability index b were obtained on the basis of the bearing capacity estimated by random finite element method (RFEM).
2. Characteristic values of parameters and bearing capacity according to Eurocode 7 Eurocode 7 does not contain clarified method of evaluation of characteristic values of soil parameters. Only general guidelines can be found how these values should be estimated. Namely it is mentioned in Eurocode 7 that the characteristic value of a soil parameter shall be selected as a cautious estimation of the value affecting the occurrence of the limit state [2]. More precisely the characteristic value is a cautious estimation of a mean value over a certain zone of ground governing the behavior of a geotechnical structure at a limit state [2]. In fact this zone is more extensive than the zone of ground influenced by in situ tests. A crucial issue is the estimation of soil parameters that could represent whole area. Furthermore Eurocode 7 recommends that in case of employing statistical concept, the characteristic value should be defined as a value governing the probability of the limit state occurrence less than 5%. Abovementioned guidelines are insufficient and cause designers to rely on their experience and knowledge to evaluate characteristic properties to be taken in calculations. A design value of a soil parameter Xd can be calculated from a characteristic value Xk using an appropriate partial safety factor gM by the following equation: Xd ¼
Xk : gM
(1)
Eurocode 7 requires that for all ultimate limit state load cases the following inequality should be satisfied V d Rd ;
(2)
where Vd denotes the design load normal to the foundation base and Rd is the design bearing capacity of a foundation against vertical loads. In the sequel a strip foundation with not inclined base subjected solely to normal loads is considered. In such a case the design value of the bearing capacity Rd of a footing in drained conditions can be evaluated according to Eurocode 7 from Rd ¼ ðc0 Nc þ q0 Nq þ 0:5g 0 B0 Ng Þ A0 ;
(3)
where c0 is cohesion in terms of effective stress, q0 is the effective overburden pressure at the level of the foundation base, g0 denotes the effective unit weight of the soil below the foundation level, B0 is the effective foundation width and A0 is the effective foundation area. Nc, Nq, Ng are the bearing capacity factors which are defined as p f0 þ (4) expðp tg f0 Þ 1 ; Nc ¼ ctgf0 tg2 4 2 Nq ¼ tg2
p f0 þ expðp tg f0 Þ; 4 2
p f0 þ expðp tg f0 Þ 1 tg f0 ; Ng ¼ 2 tg2 4 2
(5)
(6)
Table 1 – Partial safety factors included in Eurocode 7. Partial factors for permanent and variable actions gF Actions
Symbol
Set A1
Set A2
gG
1.35 1.0
1.0 1.0
gQ
1.5 0
1.0 0
Permanent Unfavorable Favorable Variable Unfavorable Favorable
Partial factors for soil properties gM Soil parameters
Symbol
Set M1
Set M2
Friction angle Cohesion Undrained shear strength Soil unit weight
g f0 g c0 g cu gg
1.0 1.0 1.0 1.0
1.25 1.25 1.4 1.0
Partial factors for resistance gR Resistance
Symbol
Set R1
Set R2
Set R3
Bearing capacity
g R;v
1.0
1.4
1.0
where f0 is a friction angle in terms of effective stress. In the case of undrained conditions the bearing capacity Rd according to Eurocode 7 can be evaluated from Rd ¼ ½ðp þ 2Þcu sc ic þ q A0 ;
(7)
where cu is the undrained shear strength of soil, sc is the factor of the shape of the foundation and ic is the factor of the inclination of the load, caused by a horizontal load. The Eurocode 7 introduces three design approaches DA1, DA2 and DA3 when checking ultimate limit states. Design approaches differ in combinations of partial safety factors, which in Eurocode 7 are divided into three groups: partial factors for permanent and variable actions gF, partial factors for soil properties gM and partial factors for resistance gR. Values of partial safety factors are given in Table 1. In case DA1 two sets of partial factors were distinguished: combination DA1.C1: A1 + M1 + R1, combination DA1.C2: A2 + M2 + R1. Design approach DA2 consists of sets A1, M1 and R2, however case DA3 denotes partial safety factors: A1, M2 and R3. Case DA2* is a variation of DA2, which was introduced in national annex in Poland. In case of design approach DA2* the characteristic values of actions should be applied instead of design ones when computing a resistance. Solely drained conditions are considered in the present study.
3. Methods of determination of characteristic values of soil parameters according to Eurocode 7 Eurocode 7, as it was mentioned in Section 2, recommends that in the case of applying statistical methods, the characteristic value of a parameter should be estimated with the level of significance a = 0.05. It means that the characteristic value of a parameter should be estimated as 5% quantile basing on a probability distribution of this parameter. Properties of materials, such as concrete or steel, are often described by a
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normal distribution, hence a characteristic value of a parameter Xk can be evaluated using the equation Xk ¼ mðXÞ 1:645sðXÞ;
(8)
where m(X) and s(X) are a mean value and a standard deviation of the parameter X, respectively. The factor 1.645 is directly connected with 5% quantile and comes from a standard normal distribution table. Although an assumption of a normal distribution of soil strength parameters is not adequate and can lead to unrealistic characteristic values of these parameters. It is connected with significant values of coefficient of variation of soil parameters which can result in negative characteristic values of parameters. For comparison coefficients of variation of steel or concrete do not exceed 10%. The next reason for limited applicability of Eq. (8) is a fact that the area of ground responsible for a collapse mechanism is greater than the zone in a soil test. It should be emphasized that a value of a parameter related with the occurrence of the limit state is a certain mean value corresponding with the slip surface and not a locally measured value. Further important factor is that geotechnical designing is generally based on small number of test results. In consequence a mean value and a standard deviation of a soil parameter obtained from in situ tests cannot be similar to a mean value and a standard deviation of this parameter corresponding with the zone responsible for the occurrence of the limit state. Adequate examples were given by Orr and Breysse [3]. Orr [4] and Orr and Breysse [3] proposed method of evaluation of a characteristic values which in better way cope with abovementioned issues. This method is based on a wellknown formula for a confidence interval for a mean value. This equation takes the following form in case of an unknown standard deviation: t Xk ¼ mðXÞ pffiffiffiffi sðXÞ; N
(9)
where mðXÞ ¼
N 1X x; N i¼1 i
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u 1 X ðx mðXÞÞ2 sðXÞ ¼ t N 1 i¼1 i
(10)
(11)
are estimators of a mean value and a standard deviation (unbiased estimators), respectively. A value t is carried out from the Students' t-distribution according to sample size N and a level of confidence. It should be mentioned that in case of Orr and Breysse's method the amount of in situ tests is of vital importance. Characteristic value of property Xk is closer to m(X) together with increasing of sample size. Another method used to estimate characteristic values was introduced by Schneider [5] and is a simplified version of the Orr and Breysse's method. A characteristic value Xk is estimated as a mean value m(X) reduced by a half of a standard deviation s(X): Xk ¼ mðXÞ 0:5sðXÞ:
(12)
Schneider also proposed an algorithm whereby fluctuation scales d of shear strength parameters and the zone of ground V
responsible for the occurrence of the limit state can be considered [6]. A characteristic value Xk in this case is expressed in the following form sffiffiffiffiffiffiffi! d Xk ¼ mðXÞ 1 kV in ðXÞ (13) : jVj Vin(X) denotes a coefficient of variation of a soil parameter associated with natural (inherent) uncertainty, and k is a factor defining 5% quantile from a probability distribution of this property. In this paper it is assumed that the maximum extent of the failure area was evaluated relying on Prandtl mechanism. It can be observed that the characteristic value of the shear strength parameter, given by Eq. (13), varies with the extend of a failure area which is associated with a foundation width. The last approach considered within this study, was presented by Duncan [7], and is associated with so-called three-sigma rule. Three-sigma rule states that the occurrence of the value beyond the interval [m(X) 3s(X), m(X) + 3s(X)] is hardly possible. Therefore it can be assumed that for bounded random variables, which values are within interval [Xmin, Xmax] a standard deviation can be defined as 1 sðXÞ ¼ ðXmax Xmin Þ: 6
(14)
While a characteristic value can be evaluated from Eq. (8) using a standard deviation given by Eq. (14).
4. Random fields and random finite element method Random finite element method (RFEM) was introduced by Griffiths and Fenton in paper concerning seepage beneath water retaining structures [8]. From the very beginning it became a suitable method to consider random variability of soil. RFEM is composed of three parts – the random field theory, finite element method and Monte Carlo simulations. Although the most important in geotechnical viewpoint is the random field theory. It is responsible for describing the spatial variability of soil parameters. If a soil property is characterized by a random field then this property at each point is a separate random variable. Moreover a certain correlation structure is given between abovementioned random variables. Considering ground area consisting of infinite amount of points with different parameters it is not feasible to implement. Therefore in RFEM the random fields are generated by local average subdivision (LAS) method in order to allow effective computation. LAS allows to discretize area into finite number of elements which is equal to the number of elements of mesh applied in finite element method. The main assumption of LAS is that the mean values of considered properties are held constant. More details concerning LAS can be found in paper of Fenton and Vanmarcke [9] or in the monograph of Fenton and Griffiths [10]. Within this paper the cohesion and friction angle are characterized by random fields as their spatial variability has the greatest impact on the random bearing capacity [11]. The cohesion is described by a lognormal distribution that can be
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obtained from a normal distribution by the transformation X = exp {Z}. Z is a normally distributed random field. The probability density function of X is given by the following equation ( ) 1 1 ln x mln X 2 pffiffiffiffiffiffi exp (15) f ðxÞ ¼ ; 2 s ln X xs ln X 2p where mln X and sln X denote a mean value and a standard deviation of the underlying normally distributed random variable Z. Friction angle varies within bounded range. Consequently this property is characterized by a bounded distribution which can be generated from a standard normal random field G0(x) by the following transformation 1 sG0 ðxÞ þ m (16) : f ¼ fmin þ ðfmax fmin Þ 1 þ tanh 2 2p fmin and fmax are minimum and maximum values of friction angle, s is a scale factor correlated with a standard deviation of the property and m is a location parameter. Scale parameter is determined by equation sV
1 2s ðb aÞ : 2 pðexpð2VÞ þ expð2VÞ þ 2Þ
(17)
The probability density function of the bounded distribution takes form pffiffiffi i2 x a
pðb aÞ 1 h m ; (18) exp 2 p ln f x ðxÞ ¼ pffiffiffi 2s bx 2sðx aÞðb xÞ where x 2 (a, b). More detailed information is provided in monograph of Fenton and Griffiths [10]. Important component which has to be defined apart from probability distributions of the properties is a correlation structure. A correlation structure was described by the ellipsoidal correlation function, as follows 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2jt y j 2 A 2jt x j 2 ; rðtÞ ¼ exp@ þ (19) ux uy where tx and ty denote the distances between two points in two-dimensional space. Furthermore ux and uy are fluctuation scales along directions x and y. In this study the anisotropic case was considered and fluctuation scales were assumed relying on geotechnical literature and took following values: ux = 10.0 m and uy = 1.0 m.
5.
Computations
5.1.
General assumptions
In this paper a shallow strip foundation (plane strain conditions) is considered. Analysis was conducted for various
foundation widths B = 1.0 m; 1.2 m; 1.4 m; 1.6 m; 1.8 m; 2.0 m and for two depths H = 0.5 m and H = 1.5 m in order to test their influence on the reliability index b. Within this paper a cohesive soil – kaolin clay was considered. Statistical data concerning effective shear strength parameters were taken from PhD thesis of Thao [12], where author gathered results from 67 in situ tests made in Wrocław agglomeration. A mean value and a standard deviation of soil parameters – a friction angle and cohesion – were the main information received from abovementioned paper. It was necessary to estimate the variability interval of a friction angle, as its distribution was assumed to be bounded (Eq. (18)). It was done using the threesigma rule as in the case of Duncan's method described in Section 3. The symmetry of a variability interval with respect a mean value implies that the location parameter m in Eq. (18) is equal to zero. Other properties such as soil unit weight, Young's modulus, and Poisson's ratio were assumed as deterministic values due to their variability has insignificant influence on the random bearing capacity. Such conclusions were achieved by Puła and Zaskórski [11] who made sensitivity analysis to examine which random fields affect the randomness of the bearing capacity. Although in mentioned paper the cohesionless soil was considered, it can be assumed that in the cohesion soil the character of phenomenon would be similar. Therefore soil unit weight, Young's modulus, and Poisson's ratio were assumed as nonrandom values: E = 40 MPa, y = 0.3 and g = 19.9 kN/m3. The value of soil unit weight is based on paper of Thao [12]. Parameters of friction angle and cohesion used in further analysis are provided in Table 2.
5.2. Deterministic computations (bearing capacity according to Eurocode 7) The characteristic values of a friction angle and cohesion were determined basing on effective soil parameters from in situ tests. Table 3 presents the characteristic values of soil properties for considered approaches – the method associated with 5% quantile (the method included in Eurocode), Orr and Breysse's approach, Duncan's method and methods proposed by Schneider – the simplified method and the method related with fluctuation scales and an area responsible for a failure mechanism. The characteristic values of a friction angle differ in a small degree from the mean value based on in situ tests due to a small coefficient of variation (10%), regardless of the approach. In case of cohesion the characteristic value is considerable smaller than the mean value as a consequence of relatively large coefficient of variation (25%). Values carried out by Orr and Breysse's algorithm are the most similar to mean values of the friction angle and the cohesion. Such phenomenon is associated with impact of domain size (67 in situ tests) on the characteristic value of a given property
Table 2 – Statistical characteristics of friction angle and cohesion applied in analysis. A effective soil parameter Friction angle Cohesion
Symbol
A mean value
A standard deviation
A max. value
A min. value
Number of in situ tests
f0 c0
12.418 29 kPa
1.158 7 kPa
16.808 –
8.008 –
67 67
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Table 3 – Characteristic values of soil properties for various methods of their evaluation. Approach
Duncan's method Orr and Breysse's method 5% quantile method Simplified Schneider's method Extended Schneider's method 1.00 B [m] 1.20 1.40 1.60 1.80 2.00
Friction angle
Cohesion
f0 [8]
c0 [kPa]
10.00 12.13
21.32 27.29
10.40 11.84
19.06 25.50
10.87 11.00 11.10 11.19 11.26 11.32
21.38 22.04 22.56 22.97 23.32 23.61
within Orr and Breysse's method. Hence, for example in case of 10 in situ tests and the same values of mean values and standard deviations of strength parameters the characteristic values calculated by Eq. (9) equal: f0 = 11.608, c0 = 24.06 kPa.
In the next step the characteristic values of soil parameters (Table 3) were applied to evaluate the deterministic values of the bearing capacity. The results for all footing widths and embedments are presented in Tables 4–7. In this paper, the simplification is made to allow comparison of results obtained by means of each design approach with results from numerical analysis. Accordingly, solely permanent load is applied to a shallow foundation, for which partial safety factors are equal to 1.35 or 1.00 depending on design approach. In case of design approaches where a permanent load factor is equal to 1.35, the bearing capacity was divided by 1.35. Such procedure causes that deterministic values of the bearing capacity become comparable regardless of the design approach. The correlation between the smallest design values of the bearing capacity and methods which gave the safest estimation of characteristic soil parameters can be easily observed regardless of the design approach. Comparing the design approaches the smallest design values of the bearing capacity were achieved for DA2* and DA3. It is caused by the fact that the bearing capacity is reduced with two groups of partial safety factors – in case of DA2* partial factors are used for
Table 4 – Design values of the bearing capacity for various methods of evaluation of characteristic values of soil properties – design approach DA1.C1. Footing depth hz [m]
Footing width
5% quantile method
Duncan's method
Orr & Breysse's method
B [m]
Simplified Schneider's method
Extended Schneider's method
Rd [kN/m]
0.50
1.00 1.20 1.40 1.60 1.80 2.00
143.39 173.08 203.11 233.48 264.19 295.24
153.75 185.42 217.39 249.67 282.26 315.15
217.44 262.46 307.98 354.02 400.56 447.61
201.25 242.93 285.09 327.72 370.83 414.41
162.74 203.34 244.97 287.53 330.95 375.16
1.50
1.00 1.20 1.40 1.60 1.80 2.00
181.18 218.43 256.01 293.94 332.21 370.81
190.17 229.12 268.37 307.93 347.80 387.97
261.79 315.67 370.07 424.97 480.38 536.30
244.40 294.71 345.50 396.76 448.50 500.72
202.20 251.27 301.44 352.57 404.59 457.44
Table 5 – Design values of the bearing capacity for various methods of evaluation of characteristic values of soil properties – design approach DA1.C2. Footing depth hz [m]
Footing width
5% quantile method
Duncan's method
Orr and Breysse's method
B [m]
Simplified Schneider's method
Extended Schneider's method
Rd [kN/m]
0.50
1.00 1.20 1.40 1.60 1.80 2.00
141.76 170.92 200.34 230.03 259.99 290.21
152.52 183.76 215.23 246.95 278.91 311.11
209.90 253.07 296.64 340.60 384.96 429.72
195.05 235.17 275.67 316.54 357.79 399.40
159.80 199.05 239.18 280.10 321.73 364.03
1.50
1.00 1.20 1.40 1.60 1.80 2.00
184.27 221.93 259.86 298.05 336.51 375.24
193.77 233.25 272.97 312.94 353.14 393.59
258.39 311.26 364.53 418.20 472.26 526.72
242.46 292.07 342.05 392.41 443.14 494.23
203.84 252.44 301.98 352.33 403.42 455.20
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Table 6 – Design values of the bearing capacity for various methods of evaluation of characteristic values of soil properties – design approach DA2*. Footing depth hz [m]
Footing width
5% quantile method
Duncan's method
Orr and Breysse's method
B [m]
Simplified Schneider's method
Extended Schneider's method
Rd [kN/m]
0.50
1.00 1.20 1.40 1.60 1.80 2.00
102.42 123.63 145.08 166.77 188.71 210.88
109.82 132.44 155.28 178.34 201.61 225.11
155.32 187.47 219.99 252.87 286.11 319.72
143.75 173.52 203.64 234.09 264.88 296.01
116.25 145.24 174.98 205.38 236.39 267.97
1.50
1.00 1.20 1.40 1.60 1.80 2.00
129.41 156.02 182.87 209.96 237.29 264.86
135.83 163.65 191.69 219.95 248.43 277.12
186.99 225.48 264.33 303.55 343.13 383.07
174.57 210.51 246.79 283.40 320.36 357.65
144.43 179.48 215.31 251.84 289.00 326.74
Table 7 – Design values of the bearing capacity for various methods of evaluation of characteristic values of soil properties – design approach DA3. Footing depth hz [m]
Footing width
5% quantile method
Duncan's method
Orr and Breysse's method
B [m]
Simplified Schneider's method
Extended Schneider's method
Rd [kN/m]
0.50
1.00 1.20 1.40 1.60 1.80 2.00
105.01 126.60 148.40 170.39 192.58 214.97
112.98 136.12 159.43 182.92 206.60 230.45
155.48 187.46 219.73 252.30 285.16 318.31
144.48 174.20 204.20 234.48 265.03 295.85
118.37 147.44 177.17 207.48 238.32 269.65
1.50
1.00 1.20 1.40 1.60 1.80 2.00
136.50 164.39 192.49 220.78 249.27 277.95
143.53 172.78 202.20 231.80 261.59 291.55
155.48 187.46 219.73 252.30 285.16 318.31
179.60 216.35 253.37 290.67 328.25 366.10
150.99 186.99 223.69 260.98 298.83 337.18
actions and resistance, and in case of DA3 partial factors are used for actions and soil parameters. It should be emphasized that the choice of an approach to evaluate the characteristic values is of great importance as it can lead to drastically different design values of the bearing capacity which are crucial in geotechnical engineering. The difference between the design values of the bearing capacity evaluated by various methods can reach 35% within one design approach. It should be regarded as significant discrepancy since the effective values of soil parameters were characterized by relatively small coefficients of variation. Therefore differences between the approaches used for the estimation of soil parameters can be more substantial considering more variable shear strength properties.
5.3. Stochastic computations (bearing capacity according to RFEM) The stochastic analysis was carried out in RBEAR2D software (free access on the website http://www.engmath.dal.ca/rfem/) which utilizes RFEM. In this study the plane strain condition was assumed (2D case). Mesh size was selected using calibration analysis. Consequently it was specified so that
the impact of boundary conditions on the random bearing capacity could be neglected. The mesh consists of 4800 rectangular 8-nodal elements of size 0.1 m 0.1 m. The boundary conditions used in finite element method are as follows: right and left sides of the mesh are constrained against horizontal displacement and bottom boundary is fixed (see Fig. 1). Application of RBEAR2D code required the calibration of other program parameters such as the displacement increment, maximal displacement step and maximal number of iterations. These features are related with computation of the bearing capacity of the soil by displacing the footing into the soil. Parameters used in RFEM simulations are provided in Table 8. An example of the deformed mesh with particular realization of the cohesion random field is given in Fig. 2. Numerical analysis in conjunction with RFEM was carried out to assess reliability of a shallow foundation and specify which method of evaluation of characteristic soil parameters gives most appropriate results in terms of the Eurocodes. The reliability index b is used as it is one of measures of structure reliability described in Eurocode 0 [13] and is related to the failure probability pf by equation p f ¼ F0 ðbÞ;
(20)
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Fig. 1 – Soil model.
Table 8 – Parameters applied in RFEM analysis. Property Friction angle A mean value A standard deviation A scale factor A minimum value A maximal value Cohesion A mean value A standard deviation A soil unit weight Young's modulus Poisson's ratio Displacement increment Maximal displacement step Maximal iterations
Symbol
Unit
mf0 s f0 s f0min f0max
8 8 8 8 8
m c0 s c0 g E y
kPa kPa kN/m 3 MPa – m – –
Value 12.41 1.15 1.72 8.00 16.80 29.00 7.00 19.90 40 0.3 0.007 50 240
where F0 is the standard normal cumulative distribution function. The probability of failure can be defined as a probability that the random bearing capacity qf takes a lower value than the deterministic design value of the bearing capacity Qd based on RFEM analysis (Qd corresponds to exact value of reliability index). Therefore pf is given by following form p f ¼ Pjq f < Q d j:
(21)
An estimation of the reliability index b and the related deterministic design value of the bearing capacity Qd is based on probability distribution of the random bearing capacity qf. The probability distribution of the bearing capacity was estimated by Monte Carlo simulation with 10,000 runs. It is necessary mentioning that computation time for one case (for fixed width and depth of foundation) is rather large. In the present study 10,000 runs required about 6 days of work of one CPU (Intel Core i7 4.00 GHz). Empirical distributions for considered widths and depths of foundation were obtained from conducted simulations. Subsequently several commonly used theoretical probability distributions were tested to find the most accurate one which fits the empirical distributions of the bearing capacity. Lognormal distribution turned out to fit empirical data obtained from stochastic analysis regardless of the width and depth of a foundation. Similar results were presented in other papers [14] which focused on cohesive soil. The results of RFEM analysis and the parameters of lognormal distribution are gathered in Table 9. It is worth mentioning that assumption of lognormal distribution in case of cohesionless soil is inadvisable. The Weibull distribution is the best-fitted empirical distribution of the random bearing capacity [11] in case of soil described solely by the friction angle random field. This phenomenon can be
Fig. 2 – Deformed mesh and realization of cohesion random field.
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Table 9 – A mean value, a standard deviation of the random bearing capacity obtained in RFEM analysis and the parameters of lognormal distribution. A footing depth
A footing width
The bearing capacity mean value
The bearing capacity standard deviation
B [m]
mq [kN/m]
sq [kN/m]
0.5
1.0 1.2 1.4 1.6 1.8 2.0
298.57 358.70 420.20 483.22 547.97 614.04
46.47 53.48 60.17 66.64 72.92 79.10
1.5
1.0 1.2 1.4 1.6 1.8 2.0
359.72 431.94 505.41 580.45 657.16 735.16
48.31 55.60 62.52 69.22 75.80 82.20
hz [m]
Qdet (by FEM) [kN/m]
Parameters of lognormal distribution (Eq. (15)) mln X [–]
sln X [–]
319.82 386.21 453.24 520.75 589.05 658.06
5.687 5.872 6.031 6.171 6.297 6.412
0.155 0.148 0.143 0.137 0.133 0.128
381.33 460.06 539.45 619.33 700.00 781.30
5.876 6.060 6.218 6.357 6.481 6.594
0.134 0.128 0.123 0.119 0.115 0.112
Table 10 – The deterministic design values of the bearing capacity Qd corresponding to selected values of reliability index b. B [m]
b
2.6
3.0
hz [m]
3.4
3.8
Qd [kN/m]
1.0 1.2 1.4 1.6 1.8 2.0
0.5
207.16 255.25 302.58 352.08 403.96 457.94
194.07 240.19 285.31 332.64 382.46 434.45
181.73 225.94 268.93 314.22 361.99 412.05
170.16 212.51 253.47 296.76 342.85 390.76
1.0 1.2 1.4 1.6 1.8 2.0
1.5
265.17 322.17 381.40 442.15 505.25 570.46
251.05 305.65 362.58 421.01 481.89 544.97
237.62 289.89 344.59 400.78 459.48 520.49
224.88 274.91 327.44 381.48 438.08 497.06
explained by fact that in the cohesive soil two random fields of both strength parameters (f and c) are included in the computational model. This is in contrast to the cohesionless soil where solely the random field of internal friction angle was included. In the present study the coefficient of variation in the cohesion field (24%) was significantly greater than the coefficient of variation of angle of internal friction. Moreover the effect of the cohesion to the bearing capacity is ‘‘almost linear’’. Therefore one can expect that the probability distribution of the bearing capacity can be similar to the lognormal distribution that is the distribution of cohesion. The Kolmogorov–Smirnov goodness-of-fit test was introduced in order to examine which theoretical probability distribution fits the best an empirical probability distribution of the bearing capacity. The criterion of Kolmogorov–Smirnov test can be expressed as DN ¼
sup
1 < x < þ1
jFðxÞ SN ðxÞj;
(22)
where F(x) is the theoretical cumulative distribution function, and SN(x) is the empirical cumulative distribution function. The details of Kolmogorov–Smirnov test can be found in a classical monograph of mathematical statistics [15].
Selected theoretical probability distribution can be used to estimate reliability indices b and corresponding deterministic design values of the bearing capacity Qd which are included in Table 10 for various foundation widths and depths.
5.4. Comparison of deterministic and stochastic computations RFEM method was applied to estimate reliability indices b which were used to compare the design values of the bearing capacity obtained for various evaluation methods of characteristic values of soil parameters (friction angle and cohesion). Design values were carried out in accordance with Eurocode for the described design approaches – see the results included in Section 5.2. Thus the most efficient method in terms of abovementioned standards was specified. Guidelines concerning minimum values of reliability index (ultimate limit states) for various types of structures and reference periods are indicated in Eurocode 0. In this study reliability class RC2 was considered as a criterion. Reliability class RC2 corresponds to a recommended minimum value of b equal to 3.8 for a 50 year reference period. Comparison of bearing capacity design values in reference to reliability indices b = 2.6; 3.0; 3.4; 3.8 is illustrated in Figs. 3–10.
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Fig. 3 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 0.50 m; the design approach DA1.C1.
Fig. 4 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 0.50 m; the design approach DA1.C2.
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Fig. 5 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 0.50 m; the design approach DA2*.
Fig. 6 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 0.50 m; the design approach DA3.
6.
Closing remarks
In the present study the new approach to the estimation of the design values of the bearing capacity in drained condition by random finite element method was proposed. The new methodology allows comparison of reliability indices obtained by different methods of selecting characteristic values of soil properties.
The lowest values of the bearing capacity were achieved for methods, which gave the safest (the most conservative) estimation of characteristic values of shear strength parameters regardless the design approach – method of 5% quantile and Duncan's method. Application of design approaches DA2* and DA3 leads to the lowest values of the bearing capacity. The reason is directly connected with types and amount of unfavorable partial safety factors used in these design approaches. Unfavorable safety
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Fig. 7 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 1.50 m; the design approach DA1.C1.
Fig. 8 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 1.50 m; the design approach DA1.C2.
factors concern actions and a resistance in case of DA2*, actions and soil parameters in case of DA3. Design approaches DA1.C1 and DA1.C2 give similar results basing on stochastic analysis. In both approaches extended method of Schneider (method concerning an influence of fluctuation scales) gives the most reliable evaluation of the bearing capacity with respect to the index b = 3.8.
The conducted deterministic analysis has shown that the choice of a estimation method for characteristic values of soil parameters is of great importance, as the evaluated characteristic values of soil properties can vary considerably. It can lead to significantly different values of the bearing capacity. Difference between bearing capacities obtained in one design approach can reach up to 35%. It should be considered as a
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Fig. 9 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 1.50 m; the design approach DA2*.
Fig. 10 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 1.50 m; the design approach DA3.
high discrepancy, bearing in a mind a small variability of effective values of internal friction angle. These results suggest that the selection of evaluation method of characteristic value should be associated with adopted design approach. In the presented study solely the drained conditions were discussed. The undrained bearing capacity by random finite element method was evaluated by Griffiths and Fenton [16].
In the end it should be emphasized that the use of RFEM approach has to be supported by credible statistical data. In geotechnical practice quite often amount of this data coming from testing is not satisfactory sufficient. In order to overcome this difficulty in several countries databases of soil properties are created. However, appropriate elaboration of data gained from CPT testing can give a satisfactory result [17].
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To conclude, the application of RFEM allowed for a random spatial variability of soil properties to be considered and to confront the design values of the bearing capacity obtained by deterministic methods with reliability indices. It can be used as a supporting tool in the process of foundation design that can help to select the most adequate characteristic values of soil strength parameters.
references
[1] D. Łydżba, A. Róźański, Microstructure measures and the minimum size of a representative volume element: 2D numerical study, Acta Geophysica 62 (5) (2014) 1060–1086. [2] EN 1997-1:2004, Eurocode 7: Geotechnical Design, Part 1: General Rules, CEN, European Committee for Standardization, Brussels. [3] T.L.L. Orr, D. Breysse, Eurocode 7 and reliability-based design, in: Reliability Based Design in Geotechnical Engineering, Taylor and Francis, London/New York, 2008. [4] T.L.L. Orr, Selection of characteristic values and partial factors in geotechnical designs to Eurocode 7, Computers and Geotechnics 26 (3) (2000) 263–279. [5] H.R. Schneider, Definition and determination of characteristic soil properties, in: Proc. XII Intern. Conf. on Soil Mech. and Geotech. Engineering, Hamburg, Germany, (1997) 2271–2274. [6] H.R. Schneider, Dealing with uncertainties in EC7 with emphasis on characteristic values, in: Proc. of Workshop on Safety Concepts and Calibr. of Partial Factors in Eur. and N. Amer. Codes of Prac., Delft, Holland, 2011. [7] J.M. Duncan, Factors of safety and reliability in geotechnical engineering, Journal of Geotechnical and Geoenvironmental Engineering 126 (4) (2000) 307–316.
[8] D.V. Griffiths, G.A. Fenton, Seepage beneath water retaining structures founded on spatially random soil, Géotechnique 43 (6) (1993) 577–587. [9] G.A. Fenton, E.H. Vanmarcke, Simulation of random fields via local average subdivision, Journal of Engineering Mechanics 116 (8) (1990) 1733–1749. [10] G.A. Fenton, D.V. Griffiths, Risk Assessment in Geotechnical Engineering, John Wiley & Sons, New York, 2008. [11] W. Puła, Ł. Zaskórski, Estimation of the probability distribution of the bearing capacity of cohesionless soil using the random finite element method, Structure and Infrastructure Engineering 11 (5) (2014) 707–720. [12] N.T.P. Thao, Geotechnical analysis of a chosen region of the Wrocław city by statistical method, (unpublished doctoral dissertation), Wrocław University of Technology, Wrocław, Poland, 1984. [13] EN 1990:2002, Eurocode: Basis of Structural Design, CEN, European Committee for Standardization, Brussels. [14] J.M. Pieczyńska-Kozłowska, W. Puła, D.V. Griffiths, F.A. Fenton, Influence of embedment, self-weight and anisotropy on bearing capacity reliability using the random finite element method, Computers and Geotechnics 67 (2014) 229–238. [15] M. Fisz, Probability Theory and Mathematical Statistics, John Wiley & Sons, New York, 1960. [16] D.V. Griffiths, G.A. Fenton, Bearing capacity of spatially random field: the undrained clay Prandtl problem revisited, Géotechnique 51 (8) (2001) 731. [17] F. Cafaro, C. Cherubini, F. Cotecchia, Use of the scale of fluctuation to describe the geotechnical variability of an Italian clay, in: Melchers, Stewart (Eds.), Proceeding of the 8th International Conference on Applications of Statistics and Probability in Civil Engineering, Rotterdam, (2000) 481–486.
Available online at www.sciencedirect.com
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Original Research Article
Calibration of characteristic values of soil properties using the random finite element method Ł. Zaskórski *, W. Puła Faculty of Civil Engineering, Wrocław University of Technology, Wybrzeże St. Wyspiańskiego 27, 50-370 Wrocław, Poland
article info
abstract
Article history:
In this paper the reliability assessment of the shallow strip footing was conducted using a
Received 13 May 2015
reliability index b. Therefore some approaches of evaluation of characteristic values of soil
Accepted 27 September 2015
properties were compared in order to check what reliability index b can be achieved by
Available online 30 October 2015
applying each of them. For this purpose, design values of the bearing capacity based on these approaches were referred to design values of the bearing capacity estimated by the random
Keywords:
finite element method. Design values of the bearing capacity were estimated for various
Eurocode 7
widths and depths of foundation in conjunction with design approaches defined in Eurocode
Bearing capacity
7. The cohesive soil was considered – clay from the area of Wrocław. The characteristic
Random field
values of shear strength parameters were evaluated basing on the effective values of soil
Probability analysis
parameters.
Reliability index
1.
# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
Introduction
The random character of the soil properties is an essential problem in terms of estimating parameters applied in geotechnical analysis. These parameters have a major influence on the safety of a designed structure. The randomness of the soil properties is greater than other typical materials considered by solid mechanics. Random variability of soil parameters occurs even within homogenous layers of soil [1]. The reason of this phenomenon is a natural variability of the soil itself, which cannot be unambiguously defined by mathematical equations. The uncertainty in determination of the soil parameters aside from the natural randomness is associated with measurement errors, limited amount of data based on in situ tests and an uncertainty of transformation. Therefore, a universal algorithm of estimation of soil parameters, which
would cope with all abovementioned sources of uncertainty, has not yet been created. Consequently many approaches used to define the soil properties were developed. However their goal is only the estimation of approximate values of soil parameters with a certain level of safety. The explicit method of soil parameters evaluation cannot be found in the applicable standards – Eurocode 7. In general terms Eurocode leaves wide scope for interpretation. Therefore the present study is focused on comparing commonly used approaches for the evaluation of characteristic values of soil parameters (Duncan's method, Schneider's method, Schneider's method with influence of a fluctuation scale, Orr and Breysse's method, method based on 5% quantile which is included in Eurocode 7) by assessing a shallow foundation bearing capacity in accordance with the guidelines contained in Eurocode 7. Design values of the bearing capacity were evaluated by applying design approaches – DA1.C1, DA1.C2,
* Corresponding author. Tel.: +48 668 155 089. E-mail addresses: [email protected] (Ł. Zaskórski), [email protected] (W. Puła). http://dx.doi.org/10.1016/j.acme.2015.09.007 1644-9665/# 2015 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
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DA2* and DA3. Reliability index b as adopted as a criterion for comparison of the above methods. Values of reliability index b were obtained on the basis of the bearing capacity estimated by random finite element method (RFEM).
2. Characteristic values of parameters and bearing capacity according to Eurocode 7 Eurocode 7 does not contain clarified method of evaluation of characteristic values of soil parameters. Only general guidelines can be found how these values should be estimated. Namely it is mentioned in Eurocode 7 that the characteristic value of a soil parameter shall be selected as a cautious estimation of the value affecting the occurrence of the limit state [2]. More precisely the characteristic value is a cautious estimation of a mean value over a certain zone of ground governing the behavior of a geotechnical structure at a limit state [2]. In fact this zone is more extensive than the zone of ground influenced by in situ tests. A crucial issue is the estimation of soil parameters that could represent whole area. Furthermore Eurocode 7 recommends that in case of employing statistical concept, the characteristic value should be defined as a value governing the probability of the limit state occurrence less than 5%. Abovementioned guidelines are insufficient and cause designers to rely on their experience and knowledge to evaluate characteristic properties to be taken in calculations. A design value of a soil parameter Xd can be calculated from a characteristic value Xk using an appropriate partial safety factor gM by the following equation: Xd ¼
Xk : gM
(1)
Eurocode 7 requires that for all ultimate limit state load cases the following inequality should be satisfied V d Rd ;
(2)
where Vd denotes the design load normal to the foundation base and Rd is the design bearing capacity of a foundation against vertical loads. In the sequel a strip foundation with not inclined base subjected solely to normal loads is considered. In such a case the design value of the bearing capacity Rd of a footing in drained conditions can be evaluated according to Eurocode 7 from Rd ¼ ðc0 Nc þ q0 Nq þ 0:5g 0 B0 Ng Þ A0 ;
(3)
where c0 is cohesion in terms of effective stress, q0 is the effective overburden pressure at the level of the foundation base, g0 denotes the effective unit weight of the soil below the foundation level, B0 is the effective foundation width and A0 is the effective foundation area. Nc, Nq, Ng are the bearing capacity factors which are defined as p f0 þ (4) expðp tg f0 Þ 1 ; Nc ¼ ctgf0 tg2 4 2 Nq ¼ tg2
p f0 þ expðp tg f0 Þ; 4 2
p f0 þ expðp tg f0 Þ 1 tg f0 ; Ng ¼ 2 tg2 4 2
(5)
(6)
Table 1 – Partial safety factors included in Eurocode 7. Partial factors for permanent and variable actions gF Actions
Symbol
Set A1
Set A2
gG
1.35 1.0
1.0 1.0
gQ
1.5 0
1.0 0
Permanent Unfavorable Favorable Variable Unfavorable Favorable
Partial factors for soil properties gM Soil parameters
Symbol
Set M1
Set M2
Friction angle Cohesion Undrained shear strength Soil unit weight
g f0 g c0 g cu gg
1.0 1.0 1.0 1.0
1.25 1.25 1.4 1.0
Partial factors for resistance gR Resistance
Symbol
Set R1
Set R2
Set R3
Bearing capacity
g R;v
1.0
1.4
1.0
where f0 is a friction angle in terms of effective stress. In the case of undrained conditions the bearing capacity Rd according to Eurocode 7 can be evaluated from Rd ¼ ½ðp þ 2Þcu sc ic þ q A0 ;
(7)
where cu is the undrained shear strength of soil, sc is the factor of the shape of the foundation and ic is the factor of the inclination of the load, caused by a horizontal load. The Eurocode 7 introduces three design approaches DA1, DA2 and DA3 when checking ultimate limit states. Design approaches differ in combinations of partial safety factors, which in Eurocode 7 are divided into three groups: partial factors for permanent and variable actions gF, partial factors for soil properties gM and partial factors for resistance gR. Values of partial safety factors are given in Table 1. In case DA1 two sets of partial factors were distinguished: combination DA1.C1: A1 + M1 + R1, combination DA1.C2: A2 + M2 + R1. Design approach DA2 consists of sets A1, M1 and R2, however case DA3 denotes partial safety factors: A1, M2 and R3. Case DA2* is a variation of DA2, which was introduced in national annex in Poland. In case of design approach DA2* the characteristic values of actions should be applied instead of design ones when computing a resistance. Solely drained conditions are considered in the present study.
3. Methods of determination of characteristic values of soil parameters according to Eurocode 7 Eurocode 7, as it was mentioned in Section 2, recommends that in the case of applying statistical methods, the characteristic value of a parameter should be estimated with the level of significance a = 0.05. It means that the characteristic value of a parameter should be estimated as 5% quantile basing on a probability distribution of this parameter. Properties of materials, such as concrete or steel, are often described by a
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normal distribution, hence a characteristic value of a parameter Xk can be evaluated using the equation Xk ¼ mðXÞ 1:645sðXÞ;
(8)
where m(X) and s(X) are a mean value and a standard deviation of the parameter X, respectively. The factor 1.645 is directly connected with 5% quantile and comes from a standard normal distribution table. Although an assumption of a normal distribution of soil strength parameters is not adequate and can lead to unrealistic characteristic values of these parameters. It is connected with significant values of coefficient of variation of soil parameters which can result in negative characteristic values of parameters. For comparison coefficients of variation of steel or concrete do not exceed 10%. The next reason for limited applicability of Eq. (8) is a fact that the area of ground responsible for a collapse mechanism is greater than the zone in a soil test. It should be emphasized that a value of a parameter related with the occurrence of the limit state is a certain mean value corresponding with the slip surface and not a locally measured value. Further important factor is that geotechnical designing is generally based on small number of test results. In consequence a mean value and a standard deviation of a soil parameter obtained from in situ tests cannot be similar to a mean value and a standard deviation of this parameter corresponding with the zone responsible for the occurrence of the limit state. Adequate examples were given by Orr and Breysse [3]. Orr [4] and Orr and Breysse [3] proposed method of evaluation of a characteristic values which in better way cope with abovementioned issues. This method is based on a wellknown formula for a confidence interval for a mean value. This equation takes the following form in case of an unknown standard deviation: t Xk ¼ mðXÞ pffiffiffiffi sðXÞ; N
(9)
where mðXÞ ¼
N 1X x; N i¼1 i
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u 1 X ðx mðXÞÞ2 sðXÞ ¼ t N 1 i¼1 i
(10)
(11)
are estimators of a mean value and a standard deviation (unbiased estimators), respectively. A value t is carried out from the Students' t-distribution according to sample size N and a level of confidence. It should be mentioned that in case of Orr and Breysse's method the amount of in situ tests is of vital importance. Characteristic value of property Xk is closer to m(X) together with increasing of sample size. Another method used to estimate characteristic values was introduced by Schneider [5] and is a simplified version of the Orr and Breysse's method. A characteristic value Xk is estimated as a mean value m(X) reduced by a half of a standard deviation s(X): Xk ¼ mðXÞ 0:5sðXÞ:
(12)
Schneider also proposed an algorithm whereby fluctuation scales d of shear strength parameters and the zone of ground V
responsible for the occurrence of the limit state can be considered [6]. A characteristic value Xk in this case is expressed in the following form sffiffiffiffiffiffiffi! d Xk ¼ mðXÞ 1 kV in ðXÞ (13) : jVj Vin(X) denotes a coefficient of variation of a soil parameter associated with natural (inherent) uncertainty, and k is a factor defining 5% quantile from a probability distribution of this property. In this paper it is assumed that the maximum extent of the failure area was evaluated relying on Prandtl mechanism. It can be observed that the characteristic value of the shear strength parameter, given by Eq. (13), varies with the extend of a failure area which is associated with a foundation width. The last approach considered within this study, was presented by Duncan [7], and is associated with so-called three-sigma rule. Three-sigma rule states that the occurrence of the value beyond the interval [m(X) 3s(X), m(X) + 3s(X)] is hardly possible. Therefore it can be assumed that for bounded random variables, which values are within interval [Xmin, Xmax] a standard deviation can be defined as 1 sðXÞ ¼ ðXmax Xmin Þ: 6
(14)
While a characteristic value can be evaluated from Eq. (8) using a standard deviation given by Eq. (14).
4. Random fields and random finite element method Random finite element method (RFEM) was introduced by Griffiths and Fenton in paper concerning seepage beneath water retaining structures [8]. From the very beginning it became a suitable method to consider random variability of soil. RFEM is composed of three parts – the random field theory, finite element method and Monte Carlo simulations. Although the most important in geotechnical viewpoint is the random field theory. It is responsible for describing the spatial variability of soil parameters. If a soil property is characterized by a random field then this property at each point is a separate random variable. Moreover a certain correlation structure is given between abovementioned random variables. Considering ground area consisting of infinite amount of points with different parameters it is not feasible to implement. Therefore in RFEM the random fields are generated by local average subdivision (LAS) method in order to allow effective computation. LAS allows to discretize area into finite number of elements which is equal to the number of elements of mesh applied in finite element method. The main assumption of LAS is that the mean values of considered properties are held constant. More details concerning LAS can be found in paper of Fenton and Vanmarcke [9] or in the monograph of Fenton and Griffiths [10]. Within this paper the cohesion and friction angle are characterized by random fields as their spatial variability has the greatest impact on the random bearing capacity [11]. The cohesion is described by a lognormal distribution that can be
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obtained from a normal distribution by the transformation X = exp {Z}. Z is a normally distributed random field. The probability density function of X is given by the following equation ( ) 1 1 ln x mln X 2 pffiffiffiffiffiffi exp (15) f ðxÞ ¼ ; 2 s ln X xs ln X 2p where mln X and sln X denote a mean value and a standard deviation of the underlying normally distributed random variable Z. Friction angle varies within bounded range. Consequently this property is characterized by a bounded distribution which can be generated from a standard normal random field G0(x) by the following transformation 1 sG0 ðxÞ þ m (16) : f ¼ fmin þ ðfmax fmin Þ 1 þ tanh 2 2p fmin and fmax are minimum and maximum values of friction angle, s is a scale factor correlated with a standard deviation of the property and m is a location parameter. Scale parameter is determined by equation sV
1 2s ðb aÞ : 2 pðexpð2VÞ þ expð2VÞ þ 2Þ
(17)
The probability density function of the bounded distribution takes form pffiffiffi i2 x a
pðb aÞ 1 h m ; (18) exp 2 p ln f x ðxÞ ¼ pffiffiffi 2s bx 2sðx aÞðb xÞ where x 2 (a, b). More detailed information is provided in monograph of Fenton and Griffiths [10]. Important component which has to be defined apart from probability distributions of the properties is a correlation structure. A correlation structure was described by the ellipsoidal correlation function, as follows 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2jt y j 2 A 2jt x j 2 ; rðtÞ ¼ exp@ þ (19) ux uy where tx and ty denote the distances between two points in two-dimensional space. Furthermore ux and uy are fluctuation scales along directions x and y. In this study the anisotropic case was considered and fluctuation scales were assumed relying on geotechnical literature and took following values: ux = 10.0 m and uy = 1.0 m.
5.
Computations
5.1.
General assumptions
In this paper a shallow strip foundation (plane strain conditions) is considered. Analysis was conducted for various
foundation widths B = 1.0 m; 1.2 m; 1.4 m; 1.6 m; 1.8 m; 2.0 m and for two depths H = 0.5 m and H = 1.5 m in order to test their influence on the reliability index b. Within this paper a cohesive soil – kaolin clay was considered. Statistical data concerning effective shear strength parameters were taken from PhD thesis of Thao [12], where author gathered results from 67 in situ tests made in Wrocław agglomeration. A mean value and a standard deviation of soil parameters – a friction angle and cohesion – were the main information received from abovementioned paper. It was necessary to estimate the variability interval of a friction angle, as its distribution was assumed to be bounded (Eq. (18)). It was done using the threesigma rule as in the case of Duncan's method described in Section 3. The symmetry of a variability interval with respect a mean value implies that the location parameter m in Eq. (18) is equal to zero. Other properties such as soil unit weight, Young's modulus, and Poisson's ratio were assumed as deterministic values due to their variability has insignificant influence on the random bearing capacity. Such conclusions were achieved by Puła and Zaskórski [11] who made sensitivity analysis to examine which random fields affect the randomness of the bearing capacity. Although in mentioned paper the cohesionless soil was considered, it can be assumed that in the cohesion soil the character of phenomenon would be similar. Therefore soil unit weight, Young's modulus, and Poisson's ratio were assumed as nonrandom values: E = 40 MPa, y = 0.3 and g = 19.9 kN/m3. The value of soil unit weight is based on paper of Thao [12]. Parameters of friction angle and cohesion used in further analysis are provided in Table 2.
5.2. Deterministic computations (bearing capacity according to Eurocode 7) The characteristic values of a friction angle and cohesion were determined basing on effective soil parameters from in situ tests. Table 3 presents the characteristic values of soil properties for considered approaches – the method associated with 5% quantile (the method included in Eurocode), Orr and Breysse's approach, Duncan's method and methods proposed by Schneider – the simplified method and the method related with fluctuation scales and an area responsible for a failure mechanism. The characteristic values of a friction angle differ in a small degree from the mean value based on in situ tests due to a small coefficient of variation (10%), regardless of the approach. In case of cohesion the characteristic value is considerable smaller than the mean value as a consequence of relatively large coefficient of variation (25%). Values carried out by Orr and Breysse's algorithm are the most similar to mean values of the friction angle and the cohesion. Such phenomenon is associated with impact of domain size (67 in situ tests) on the characteristic value of a given property
Table 2 – Statistical characteristics of friction angle and cohesion applied in analysis. A effective soil parameter Friction angle Cohesion
Symbol
A mean value
A standard deviation
A max. value
A min. value
Number of in situ tests
f0 c0
12.418 29 kPa
1.158 7 kPa
16.808 –
8.008 –
67 67
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Table 3 – Characteristic values of soil properties for various methods of their evaluation. Approach
Duncan's method Orr and Breysse's method 5% quantile method Simplified Schneider's method Extended Schneider's method 1.00 B [m] 1.20 1.40 1.60 1.80 2.00
Friction angle
Cohesion
f0 [8]
c0 [kPa]
10.00 12.13
21.32 27.29
10.40 11.84
19.06 25.50
10.87 11.00 11.10 11.19 11.26 11.32
21.38 22.04 22.56 22.97 23.32 23.61
within Orr and Breysse's method. Hence, for example in case of 10 in situ tests and the same values of mean values and standard deviations of strength parameters the characteristic values calculated by Eq. (9) equal: f0 = 11.608, c0 = 24.06 kPa.
In the next step the characteristic values of soil parameters (Table 3) were applied to evaluate the deterministic values of the bearing capacity. The results for all footing widths and embedments are presented in Tables 4–7. In this paper, the simplification is made to allow comparison of results obtained by means of each design approach with results from numerical analysis. Accordingly, solely permanent load is applied to a shallow foundation, for which partial safety factors are equal to 1.35 or 1.00 depending on design approach. In case of design approaches where a permanent load factor is equal to 1.35, the bearing capacity was divided by 1.35. Such procedure causes that deterministic values of the bearing capacity become comparable regardless of the design approach. The correlation between the smallest design values of the bearing capacity and methods which gave the safest estimation of characteristic soil parameters can be easily observed regardless of the design approach. Comparing the design approaches the smallest design values of the bearing capacity were achieved for DA2* and DA3. It is caused by the fact that the bearing capacity is reduced with two groups of partial safety factors – in case of DA2* partial factors are used for
Table 4 – Design values of the bearing capacity for various methods of evaluation of characteristic values of soil properties – design approach DA1.C1. Footing depth hz [m]
Footing width
5% quantile method
Duncan's method
Orr & Breysse's method
B [m]
Simplified Schneider's method
Extended Schneider's method
Rd [kN/m]
0.50
1.00 1.20 1.40 1.60 1.80 2.00
143.39 173.08 203.11 233.48 264.19 295.24
153.75 185.42 217.39 249.67 282.26 315.15
217.44 262.46 307.98 354.02 400.56 447.61
201.25 242.93 285.09 327.72 370.83 414.41
162.74 203.34 244.97 287.53 330.95 375.16
1.50
1.00 1.20 1.40 1.60 1.80 2.00
181.18 218.43 256.01 293.94 332.21 370.81
190.17 229.12 268.37 307.93 347.80 387.97
261.79 315.67 370.07 424.97 480.38 536.30
244.40 294.71 345.50 396.76 448.50 500.72
202.20 251.27 301.44 352.57 404.59 457.44
Table 5 – Design values of the bearing capacity for various methods of evaluation of characteristic values of soil properties – design approach DA1.C2. Footing depth hz [m]
Footing width
5% quantile method
Duncan's method
Orr and Breysse's method
B [m]
Simplified Schneider's method
Extended Schneider's method
Rd [kN/m]
0.50
1.00 1.20 1.40 1.60 1.80 2.00
141.76 170.92 200.34 230.03 259.99 290.21
152.52 183.76 215.23 246.95 278.91 311.11
209.90 253.07 296.64 340.60 384.96 429.72
195.05 235.17 275.67 316.54 357.79 399.40
159.80 199.05 239.18 280.10 321.73 364.03
1.50
1.00 1.20 1.40 1.60 1.80 2.00
184.27 221.93 259.86 298.05 336.51 375.24
193.77 233.25 272.97 312.94 353.14 393.59
258.39 311.26 364.53 418.20 472.26 526.72
242.46 292.07 342.05 392.41 443.14 494.23
203.84 252.44 301.98 352.33 403.42 455.20
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Table 6 – Design values of the bearing capacity for various methods of evaluation of characteristic values of soil properties – design approach DA2*. Footing depth hz [m]
Footing width
5% quantile method
Duncan's method
Orr and Breysse's method
B [m]
Simplified Schneider's method
Extended Schneider's method
Rd [kN/m]
0.50
1.00 1.20 1.40 1.60 1.80 2.00
102.42 123.63 145.08 166.77 188.71 210.88
109.82 132.44 155.28 178.34 201.61 225.11
155.32 187.47 219.99 252.87 286.11 319.72
143.75 173.52 203.64 234.09 264.88 296.01
116.25 145.24 174.98 205.38 236.39 267.97
1.50
1.00 1.20 1.40 1.60 1.80 2.00
129.41 156.02 182.87 209.96 237.29 264.86
135.83 163.65 191.69 219.95 248.43 277.12
186.99 225.48 264.33 303.55 343.13 383.07
174.57 210.51 246.79 283.40 320.36 357.65
144.43 179.48 215.31 251.84 289.00 326.74
Table 7 – Design values of the bearing capacity for various methods of evaluation of characteristic values of soil properties – design approach DA3. Footing depth hz [m]
Footing width
5% quantile method
Duncan's method
Orr and Breysse's method
B [m]
Simplified Schneider's method
Extended Schneider's method
Rd [kN/m]
0.50
1.00 1.20 1.40 1.60 1.80 2.00
105.01 126.60 148.40 170.39 192.58 214.97
112.98 136.12 159.43 182.92 206.60 230.45
155.48 187.46 219.73 252.30 285.16 318.31
144.48 174.20 204.20 234.48 265.03 295.85
118.37 147.44 177.17 207.48 238.32 269.65
1.50
1.00 1.20 1.40 1.60 1.80 2.00
136.50 164.39 192.49 220.78 249.27 277.95
143.53 172.78 202.20 231.80 261.59 291.55
155.48 187.46 219.73 252.30 285.16 318.31
179.60 216.35 253.37 290.67 328.25 366.10
150.99 186.99 223.69 260.98 298.83 337.18
actions and resistance, and in case of DA3 partial factors are used for actions and soil parameters. It should be emphasized that the choice of an approach to evaluate the characteristic values is of great importance as it can lead to drastically different design values of the bearing capacity which are crucial in geotechnical engineering. The difference between the design values of the bearing capacity evaluated by various methods can reach 35% within one design approach. It should be regarded as significant discrepancy since the effective values of soil parameters were characterized by relatively small coefficients of variation. Therefore differences between the approaches used for the estimation of soil parameters can be more substantial considering more variable shear strength properties.
5.3. Stochastic computations (bearing capacity according to RFEM) The stochastic analysis was carried out in RBEAR2D software (free access on the website http://www.engmath.dal.ca/rfem/) which utilizes RFEM. In this study the plane strain condition was assumed (2D case). Mesh size was selected using calibration analysis. Consequently it was specified so that
the impact of boundary conditions on the random bearing capacity could be neglected. The mesh consists of 4800 rectangular 8-nodal elements of size 0.1 m 0.1 m. The boundary conditions used in finite element method are as follows: right and left sides of the mesh are constrained against horizontal displacement and bottom boundary is fixed (see Fig. 1). Application of RBEAR2D code required the calibration of other program parameters such as the displacement increment, maximal displacement step and maximal number of iterations. These features are related with computation of the bearing capacity of the soil by displacing the footing into the soil. Parameters used in RFEM simulations are provided in Table 8. An example of the deformed mesh with particular realization of the cohesion random field is given in Fig. 2. Numerical analysis in conjunction with RFEM was carried out to assess reliability of a shallow foundation and specify which method of evaluation of characteristic soil parameters gives most appropriate results in terms of the Eurocodes. The reliability index b is used as it is one of measures of structure reliability described in Eurocode 0 [13] and is related to the failure probability pf by equation p f ¼ F0 ðbÞ;
(20)
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Fig. 1 – Soil model.
Table 8 – Parameters applied in RFEM analysis. Property Friction angle A mean value A standard deviation A scale factor A minimum value A maximal value Cohesion A mean value A standard deviation A soil unit weight Young's modulus Poisson's ratio Displacement increment Maximal displacement step Maximal iterations
Symbol
Unit
mf0 s f0 s f0min f0max
8 8 8 8 8
m c0 s c0 g E y
kPa kPa kN/m 3 MPa – m – –
Value 12.41 1.15 1.72 8.00 16.80 29.00 7.00 19.90 40 0.3 0.007 50 240
where F0 is the standard normal cumulative distribution function. The probability of failure can be defined as a probability that the random bearing capacity qf takes a lower value than the deterministic design value of the bearing capacity Qd based on RFEM analysis (Qd corresponds to exact value of reliability index). Therefore pf is given by following form p f ¼ Pjq f < Q d j:
(21)
An estimation of the reliability index b and the related deterministic design value of the bearing capacity Qd is based on probability distribution of the random bearing capacity qf. The probability distribution of the bearing capacity was estimated by Monte Carlo simulation with 10,000 runs. It is necessary mentioning that computation time for one case (for fixed width and depth of foundation) is rather large. In the present study 10,000 runs required about 6 days of work of one CPU (Intel Core i7 4.00 GHz). Empirical distributions for considered widths and depths of foundation were obtained from conducted simulations. Subsequently several commonly used theoretical probability distributions were tested to find the most accurate one which fits the empirical distributions of the bearing capacity. Lognormal distribution turned out to fit empirical data obtained from stochastic analysis regardless of the width and depth of a foundation. Similar results were presented in other papers [14] which focused on cohesive soil. The results of RFEM analysis and the parameters of lognormal distribution are gathered in Table 9. It is worth mentioning that assumption of lognormal distribution in case of cohesionless soil is inadvisable. The Weibull distribution is the best-fitted empirical distribution of the random bearing capacity [11] in case of soil described solely by the friction angle random field. This phenomenon can be
Fig. 2 – Deformed mesh and realization of cohesion random field.
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Table 9 – A mean value, a standard deviation of the random bearing capacity obtained in RFEM analysis and the parameters of lognormal distribution. A footing depth
A footing width
The bearing capacity mean value
The bearing capacity standard deviation
B [m]
mq [kN/m]
sq [kN/m]
0.5
1.0 1.2 1.4 1.6 1.8 2.0
298.57 358.70 420.20 483.22 547.97 614.04
46.47 53.48 60.17 66.64 72.92 79.10
1.5
1.0 1.2 1.4 1.6 1.8 2.0
359.72 431.94 505.41 580.45 657.16 735.16
48.31 55.60 62.52 69.22 75.80 82.20
hz [m]
Qdet (by FEM) [kN/m]
Parameters of lognormal distribution (Eq. (15)) mln X [–]
sln X [–]
319.82 386.21 453.24 520.75 589.05 658.06
5.687 5.872 6.031 6.171 6.297 6.412
0.155 0.148 0.143 0.137 0.133 0.128
381.33 460.06 539.45 619.33 700.00 781.30
5.876 6.060 6.218 6.357 6.481 6.594
0.134 0.128 0.123 0.119 0.115 0.112
Table 10 – The deterministic design values of the bearing capacity Qd corresponding to selected values of reliability index b. B [m]
b
2.6
3.0
hz [m]
3.4
3.8
Qd [kN/m]
1.0 1.2 1.4 1.6 1.8 2.0
0.5
207.16 255.25 302.58 352.08 403.96 457.94
194.07 240.19 285.31 332.64 382.46 434.45
181.73 225.94 268.93 314.22 361.99 412.05
170.16 212.51 253.47 296.76 342.85 390.76
1.0 1.2 1.4 1.6 1.8 2.0
1.5
265.17 322.17 381.40 442.15 505.25 570.46
251.05 305.65 362.58 421.01 481.89 544.97
237.62 289.89 344.59 400.78 459.48 520.49
224.88 274.91 327.44 381.48 438.08 497.06
explained by fact that in the cohesive soil two random fields of both strength parameters (f and c) are included in the computational model. This is in contrast to the cohesionless soil where solely the random field of internal friction angle was included. In the present study the coefficient of variation in the cohesion field (24%) was significantly greater than the coefficient of variation of angle of internal friction. Moreover the effect of the cohesion to the bearing capacity is ‘‘almost linear’’. Therefore one can expect that the probability distribution of the bearing capacity can be similar to the lognormal distribution that is the distribution of cohesion. The Kolmogorov–Smirnov goodness-of-fit test was introduced in order to examine which theoretical probability distribution fits the best an empirical probability distribution of the bearing capacity. The criterion of Kolmogorov–Smirnov test can be expressed as DN ¼
sup
1 < x < þ1
jFðxÞ SN ðxÞj;
(22)
where F(x) is the theoretical cumulative distribution function, and SN(x) is the empirical cumulative distribution function. The details of Kolmogorov–Smirnov test can be found in a classical monograph of mathematical statistics [15].
Selected theoretical probability distribution can be used to estimate reliability indices b and corresponding deterministic design values of the bearing capacity Qd which are included in Table 10 for various foundation widths and depths.
5.4. Comparison of deterministic and stochastic computations RFEM method was applied to estimate reliability indices b which were used to compare the design values of the bearing capacity obtained for various evaluation methods of characteristic values of soil parameters (friction angle and cohesion). Design values were carried out in accordance with Eurocode for the described design approaches – see the results included in Section 5.2. Thus the most efficient method in terms of abovementioned standards was specified. Guidelines concerning minimum values of reliability index (ultimate limit states) for various types of structures and reference periods are indicated in Eurocode 0. In this study reliability class RC2 was considered as a criterion. Reliability class RC2 corresponds to a recommended minimum value of b equal to 3.8 for a 50 year reference period. Comparison of bearing capacity design values in reference to reliability indices b = 2.6; 3.0; 3.4; 3.8 is illustrated in Figs. 3–10.
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Fig. 3 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 0.50 m; the design approach DA1.C1.
Fig. 4 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 0.50 m; the design approach DA1.C2.
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Fig. 5 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 0.50 m; the design approach DA2*.
Fig. 6 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 0.50 m; the design approach DA3.
6.
Closing remarks
In the present study the new approach to the estimation of the design values of the bearing capacity in drained condition by random finite element method was proposed. The new methodology allows comparison of reliability indices obtained by different methods of selecting characteristic values of soil properties.
The lowest values of the bearing capacity were achieved for methods, which gave the safest (the most conservative) estimation of characteristic values of shear strength parameters regardless the design approach – method of 5% quantile and Duncan's method. Application of design approaches DA2* and DA3 leads to the lowest values of the bearing capacity. The reason is directly connected with types and amount of unfavorable partial safety factors used in these design approaches. Unfavorable safety
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Fig. 7 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 1.50 m; the design approach DA1.C1.
Fig. 8 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 1.50 m; the design approach DA1.C2.
factors concern actions and a resistance in case of DA2*, actions and soil parameters in case of DA3. Design approaches DA1.C1 and DA1.C2 give similar results basing on stochastic analysis. In both approaches extended method of Schneider (method concerning an influence of fluctuation scales) gives the most reliable evaluation of the bearing capacity with respect to the index b = 3.8.
The conducted deterministic analysis has shown that the choice of a estimation method for characteristic values of soil parameters is of great importance, as the evaluated characteristic values of soil properties can vary considerably. It can lead to significantly different values of the bearing capacity. Difference between bearing capacities obtained in one design approach can reach up to 35%. It should be considered as a
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Fig. 9 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 1.50 m; the design approach DA2*.
Fig. 10 – Comparison of design values of the bearing capacity Qd – the depth of foundation hz = 1.50 m; the design approach DA3.
high discrepancy, bearing in a mind a small variability of effective values of internal friction angle. These results suggest that the selection of evaluation method of characteristic value should be associated with adopted design approach. In the presented study solely the drained conditions were discussed. The undrained bearing capacity by random finite element method was evaluated by Griffiths and Fenton [16].
In the end it should be emphasized that the use of RFEM approach has to be supported by credible statistical data. In geotechnical practice quite often amount of this data coming from testing is not satisfactory sufficient. In order to overcome this difficulty in several countries databases of soil properties are created. However, appropriate elaboration of data gained from CPT testing can give a satisfactory result [17].
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To conclude, the application of RFEM allowed for a random spatial variability of soil properties to be considered and to confront the design values of the bearing capacity obtained by deterministic methods with reliability indices. It can be used as a supporting tool in the process of foundation design that can help to select the most adequate characteristic values of soil strength parameters.
references
[1] D. Łydżba, A. Róźański, Microstructure measures and the minimum size of a representative volume element: 2D numerical study, Acta Geophysica 62 (5) (2014) 1060–1086. [2] EN 1997-1:2004, Eurocode 7: Geotechnical Design, Part 1: General Rules, CEN, European Committee for Standardization, Brussels. [3] T.L.L. Orr, D. Breysse, Eurocode 7 and reliability-based design, in: Reliability Based Design in Geotechnical Engineering, Taylor and Francis, London/New York, 2008. [4] T.L.L. Orr, Selection of characteristic values and partial factors in geotechnical designs to Eurocode 7, Computers and Geotechnics 26 (3) (2000) 263–279. [5] H.R. Schneider, Definition and determination of characteristic soil properties, in: Proc. XII Intern. Conf. on Soil Mech. and Geotech. Engineering, Hamburg, Germany, (1997) 2271–2274. [6] H.R. Schneider, Dealing with uncertainties in EC7 with emphasis on characteristic values, in: Proc. of Workshop on Safety Concepts and Calibr. of Partial Factors in Eur. and N. Amer. Codes of Prac., Delft, Holland, 2011. [7] J.M. Duncan, Factors of safety and reliability in geotechnical engineering, Journal of Geotechnical and Geoenvironmental Engineering 126 (4) (2000) 307–316.
[8] D.V. Griffiths, G.A. Fenton, Seepage beneath water retaining structures founded on spatially random soil, Géotechnique 43 (6) (1993) 577–587. [9] G.A. Fenton, E.H. Vanmarcke, Simulation of random fields via local average subdivision, Journal of Engineering Mechanics 116 (8) (1990) 1733–1749. [10] G.A. Fenton, D.V. Griffiths, Risk Assessment in Geotechnical Engineering, John Wiley & Sons, New York, 2008. [11] W. Puła, Ł. Zaskórski, Estimation of the probability distribution of the bearing capacity of cohesionless soil using the random finite element method, Structure and Infrastructure Engineering 11 (5) (2014) 707–720. [12] N.T.P. Thao, Geotechnical analysis of a chosen region of the Wrocław city by statistical method, (unpublished doctoral dissertation), Wrocław University of Technology, Wrocław, Poland, 1984. [13] EN 1990:2002, Eurocode: Basis of Structural Design, CEN, European Committee for Standardization, Brussels. [14] J.M. Pieczyńska-Kozłowska, W. Puła, D.V. Griffiths, F.A. Fenton, Influence of embedment, self-weight and anisotropy on bearing capacity reliability using the random finite element method, Computers and Geotechnics 67 (2014) 229–238. [15] M. Fisz, Probability Theory and Mathematical Statistics, John Wiley & Sons, New York, 1960. [16] D.V. Griffiths, G.A. Fenton, Bearing capacity of spatially random field: the undrained clay Prandtl problem revisited, Géotechnique 51 (8) (2001) 731. [17] F. Cafaro, C. Cherubini, F. Cotecchia, Use of the scale of fluctuation to describe the geotechnical variability of an Italian clay, in: Melchers, Stewart (Eds.), Proceeding of the 8th International Conference on Applications of Statistics and Probability in Civil Engineering, Rotterdam, (2000) 481–486.