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Reliability analysis of allowable pressure on shallow foundation using response surface method
Computers and Geotechnics 34 (2007) 187–194 www.elsevier.com/locate/compgeo
Technical note
Reliability analysis of allowable pressure on shallow foundation using response surface method G.L. Sivakumar Babu *, Amit Srivastava Department of Civil Engineering, Indian Institute of Science, Bangalore, Karnataka 560 012, India Received 30 June 2006; received in revised form 10 November 2006 Available online 3 January 2007
Abstract The concept of response surface method (RSM) is used to generate approximate polynomial functions for ultimate bearing capacity and settlement of a shallow foundation resting on a cohesive frictional soil for a range of expected variation of input soil parameters. The response surface models are developed using available conventional equations and numerical analysis. Considering the variations in the input soil parameters, reliability analysis is performed using these response surface models to obtain an acceptable value of the allowable bearing pressure. The results of the reliability analysis are compared with the results of Monte Carlo simulation and it is demonstrated that application of response surface method in the probabilistic analysis can considerably reduce the computational efforts and memory requirements. It is also concluded that conventional analysis using available equations and numerical analysis when used in conjunction with reliability analysis enable a rational choice of allowable pressure and help in decision-making process. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Conventional analysis; Reliability analysis; Response surface method; Monte Carlo simulation; Soil parameters; Correlation
1. Introduction In the conventional design of shallow foundations resting on a cohesive frictional soil, the allowable pressure is calculated based on shear failure criterion (ultimate limit state, ULS) and settlement criterion (serviceability limit state, SLS). The allowable bearing capacity is obtained by dividing the ultimate bearing capacity of the foundation soil with a factor of safety. At the same time, it is also ensured that the magnitude of settlement of footing should not exceed a specified permissible limit from serviceability requirements [1]. Design values of input soil parameters such as cohesion (c), angle of internal friction (/), unit weight (c), modulus of elasticity (E), for estimating the allowable pressure on foundation soil, are obtained either from field tests or from laboratory tests. As soil is inherently variable, variations in measured values of those soil *
Corresponding author. Tel.: +91 80 22933124; fax: +91 80 23600404. E-mail addresses: [email protected] (G.L. Sivakumar Babu), [email protected] (A. Srivastava). 0266-352X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2006.11.002
parameters is inevitable. Phoon and Kulhawy [2] indicated that inherent variability of soil deposits, testing errors and model transformation uncertainties contribute to variability in the test data. Although factor of safety approach is simple and straightforward, it does not consider different sources of uncertainty in geotechnical design in a rational manner [3–5]. In order to incorporate these variations, reliability analysis is performed. In this approach, input soil parameters are treated as random variables and the influence of these input random variables on the output random variable is studied. Reliability analysis approaches can be used in conjunction with conventional approaches to have better insight into the choice of allowable value of bearing pressure and helps in decision-making process. Sivakumar Babu et al. [6] demonstrated this aspect with reference to the estimation of allowable bearing pressure of a shallow foundation resting on a typical stiff clay. For reliability analysis, a functional relationship (either implicit or explicit) is required between input and output random variables. Explicit relationships for reliability analysis can be obtained from conventional bearing capacity
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and settlement equations. These relationships are implicit in numerical analysis and tools such as random field finite element analysis (RFEM), response surface methodology (RSM) are used to analyze the response variability. Various researchers [7,8] used random field finite element modelling (RFEM) to study the bearing capacity of shallow foundations. Wong [9] performed reliability analysis of soil slopes using response surface method (RSM). Humphreys and Armstrong [10] analyzed a slope stability problem using results of finite difference method and regression analysis. Tandjiria et al. [11] used response surface method for reliability analysis of laterally loaded piles. A question that often arises in practice is to know ‘‘how safe is safe?’’ or to what extent factors of safety that are routinely used in conventional solutions address the question of safety adequately. It is possible to examine a degree of safety associated with analytical formulations by comparing reliability indices from conventional solutions with those from analysis of response surfaces constructed from conventional solution as well as from the results of numerical analysis. As expected, a comparative study of the results of the analysis from conventional solution and numerical analysis in terms of reliability indices demonstrates the effectiveness of conventional, numerical and probabilistic approaches in the assessment of allowable pressure on foundation soil and this is the main focus of the present study. 2. Conventional approach In a conventional approach, allowable pressure on a shallow footing is taken as lesser of the two values, i.e. (i) allowable bearing capacity based on shear failure criterion and (ii) allowable bearing pressure based on settlement criterion. (i) Shear failure criteria For a shallow strip foundation of width B resting on a horizontal ground at a depth Df and loaded with a concentric vertical loading the ultimate bearing capacity (qu) can be calculated using Meyerhof equation qu ¼ cN c sc d c þ cDf N q sq d q þ 0:5cBN c sc d c
ð1Þ
The values of bearing capacity factors Nc, Nq and Nc as well as shape factors (sc, sq, sc) and depth factors (dc, dq, dc) are obtained from available equations provided in literature [1]. Allowable bearing capacity (qa) is obtained after applying a factor of safety (FS) to the ultimate bearing capacity (qu) and usually a value between 2.5 and 3 is considered appropriate in the case of a shallow foundation. (ii) Settlement criteria For an applied pressure (q0), the magnitude of settlement of a footing can be computed using Eq. (2) and this is compared with the allowable settlement specified in design guidelines/codes to satisfy serviceability requirements
S ¼ q0 B
ð1 m2 Þ If E
ð2Þ
where B is the width of footing, q0 is an applied pressure on the footing, If is an influence factor, m is poison’s ratio and E is modulus of elasticity of soil. As per Eurocode 7 [12], total settlements up to 50 mm are often acceptable for normal structures with isolated foundations. As per Indian codes of practice, 40 mm is considered acceptable and hence in the present study, the allowable settlement is taken as 40 mm. 3. Reliability analysis Reliability is defined as probability of safety of a system in a given environment and loading conditions and is assessed in terms of reliability index (b) values. Normally, a reliability index value in the range of 3.0–4.0 is accepted for good performance of the system [13,14]. For the estimation of reliability index, methods such as first order reliability method (FORM), second order reliability method (SORM), point estimate method (PEM), Monte Carlo simulation (MCS) are available in literature. For a linear performance function defined as g(Æ) = C D, where C is the capacity (either ultimate bearing capacity or allowable settlement) and D is the demand (either applied pressure or calculated settlement), g(Æ) > 0 denotes safety while g(Æ) < 0 represents unsafe condition. For uncorrelated normally distributed C and D, reliability index (b), representing the shortest distance of the performance function from the origin of reduced coordinate system of variables, can be calculated using Eq. (3) [13]. CD b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr2C þ r2D Þ
ð3Þ
where rC and rD are standard deviations of capacity (C) and demand (D). For non-linear performance functions, iterative procedure is used to evaluate reliability index. Reliability analysis using conventional equations for ultimate bearing capacity and settlement is complicated because calculation of derivatives for variances by first or second order methods is cumbersome. Using Monte Carlo simulation technique, probability of failure can be computed for both explicit and implicit limit state functions. This method involves the generation of N random numbers of input soil parameters with given probabilistic characteristics. These N sample points for output response are used to obtain required sample statistics, which is incorporated in probabilistic calculations. The minimum value of number N depends on percentage (%) acceptable error (e) in the estimation of sample mean and variance as well as confidence level defined in terms of parameter a [13]. As the performance function is defined as g(Æ) = C D, capacity (C) is deducted from demand (D) for each realization of output response. If Nf is the
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number of simulation cycles when g(Æ) is less than zero and N is the total number of simulation cycles, probability of N failure can be obtained as pf ¼ Nf which is approximately related to reliability index as b = /1(1 pf). On the other hand, response surface method (RSM) facilitates the formulation of explicit linear or non-linear relationships between input random variables and required output response through regression analysis and this aspect is advantageously used in the present study for reliability analysis. 4. Response surface methodology (RSM) Response surface method involves generation of polynomial equation using regression analysis and an approximate linear or non-linear functional relationship between dependent output y and input variables (x1, x2, x3, . . .) is established (Eq. (4)). y ¼ f ðx1 ; x2 ; x3 ; . . .Þ þ e
y ¼ a0 þ a1 x1 þ a2 x2 þ e y ¼ a0 þ a1 x1 þ a2 x2 þ
ð5Þ þ
a22 x22
þ a12 x1 x2 þ e
ð6Þ
where a0, a1, a2, . . . are regression coefficients and ‘e’ represents error involved in neglecting other sources of uncertainties. In RSM analysis, single replicate 2n factorial design is used to fit first order linear regression model. For this, total number of sample points is required as 2n, where n is total number of input variables. For n = 2 and n = 3, number of such sample points are 4 and 8, respectively. In the analysis, natural variables (x1, x2, x3, . . . , xk) are converted into coded variables ni using a relationship i Þþminðxi Þ=2 ni ¼ xi ½maxðx . The maximum and minimum values ½maxðxi Þminðxi Þ=2 of xi are taken as xmax = x + hr and xmin = x hr, where h is an integer and ris are the standard deviations of input variables (xis). These limits xmax and xmin cover the extent of variation of input parameters in a given situation with some level of confidence interval discussed later. Finally, the response surface model is presented in terms of natural variables. To examine the adequacy of the fitted model and to ensure that it provides a good approximation of the true system, a normal probability plot should be approximately along a straight line. In addition, computed values of coefficients of multiple determinations (R2) and adjusted R2 also give information on the adequacy of the fitted model. The non-dimensional quantity R2 is calculated as (Eq. (7)). Pn ^ 2 i¼1 y i y R2 ¼ P 2 n i¼1 y i y
^
where y , yi and y are estimated mean value, actual and predicted values of output response (y) respectively. The value of R2 lies between 0 and 1 and a value close to 1 indicates that most of the variability in y is explained by regression model. It should be noted that it is always possible to increase the value of R2 by adding more regressor variables. Therefore, adjusted R2 value is calculated using following Eq. (8): k 1 1 R2 ð8Þ R2adj ¼ 1 kp where k is total number of observations and p is number of regression coefficients. For a good model, values of R2 and adjusted R2 should be close to each other and also they should be close to 1. Myers and Montgomery [15] presented more details on the use of response surface methodology. 5. Methodology
ð4Þ
The following Eq. (5) shows multiple linear regression model having two input variables without interaction terms and Eq. (6) shows second order regression model with interaction terms containing two input variables x1, x2. a11 x21
189
ð7Þ
In the present case, the output response (i.e. the ultimate bearing capacity from shear failure consideration) is obtained using conventional Meyerhof equation corresponding to eight sample points for three input variables (c, / and c). A linear regression model is fitted between output response (qu) and input variables (c, / and c) and this approximate linear relationship is used to obtain mean and variance of qu for reliability analysis. The same procedure is adopted to obtain response surface models for settlement of the footing with input random variables E and m and for different applied pressures on the footing and reliability index values are calculated. The above results are also compared with the results of reliability analysis from response surface models obtained from numerical analysis. Using commercial finite difference code, fast Lagrangian analysis continua (FLAC) [16], response surface models corresponding to the above eight sample points are developed. For the analysis, physical domain of the problem is divided into finite difference mesh of quadrilateral elements. The unbalanced force of each node is normalized by gravitational force acting on that node. A simulation is considered to have converged when the normalized unbalanced force of every node in the mesh is less than 103. In the analysis, elastic perfectly plastic model based on Mohr–Coulomb failure criteria with associated flow rule is used for the foundation soil. After obtaining response surface models for ultimate bearing capacity and settlement from conventional approach and numerical analysis, reliability analysis is performed. The probability of failure is also obtained using Monte Carlo simulations. For an acceptable error of 5.0% and a = 0.05 (95% confidence level), minimum sample size is 1536 following the procedure explained in the literature [13,17]. In the present analysis, 20,000 random numbers are generated for simulation.
G.L. Sivakumar Babu, A. Srivastava / Computers and Geotechnics 34 (2007) 187–194
C ¼ 1191:04 þ 25:53c þ 47:64/ þ 18:29c
Sr. no.
1 2 3 4 5 6 7 8
Combination
+ + + +
+ + + +
+ + + +
qu
c
/
c
6 6 6 6 4 4 4 4
27.5 27.5 22.5 22.5 27.5 27.5 22.5 22.5
18.7 15.3 18.7 15.3 18.7 15.3 18.7 15.3
628.52 547.18 361.82 318.80 568.13 486.80 320.10 277.10
600 500 400 qu 300 500-600 200
400-500 300-400
100
4.5
5 5.5
c
6
27.5
4
26.25
0
ð9Þ
Fig. 1a shows plots of actual response surface for ultimate bearing capacity using Meyerhof equation and Fig. 1b shows contours of ultimate bearing capacity calculated from conventional (Meyerhof) equation and approximated linear response surface equation. It can be seen that the results of response surface model are quite comparable with those obtained from Meyerhof solution and also the linear approximation of ultimate bearing capacity of foundation soil for the range variation of input parameters is valid. Fig. 2 shows normal probability plot for the regressed model, which is approximately a straight line. The values of R2 and adjusted R2 are 0.99 and 0.98,
Natural variables
25
A shallow strip foundation of width (B) 1.5 m at a depth (Df) of 1.0 m, resting on a cohesive frictional soil is considered. The values of input soil properties for conventional and reliability analysis are presented in Table 1. Using conventional approach, for mean values of input soil properties, the mean value of ultimate bearing capacity from shear failure criteria is obtained as 419 kPa. Using a factor of safety of 3.0, the allowable bearing capacity of foundation soil is 139.67 140 kPa. From serviceability requirements, for an applied pressure of 140 kPa, the settlement of foundation is 18.78 mm, which is less than the allowable settlement of 40 mm. Becker [18] and Orr [19] suggest that selection of characteristic values of geotechnical parameters and corresponding confidence intervals should be incorporated in reliability based designs. Response surfaces are generated corresponding to 95% confidence intervals, i.e. the lower limit and upper limit values related to mean (l) and standard deviation (r) with the relationships xu = l + 1.65r and xl = l 1.65r are used in the analysis. The upper limit (xu or xmax) and lower limit (xl or xmin) values indicated in Table 1 are based on the assumption that input soil parameters follow normal distribution and upper and lower limit values have probabilities of 5% and 95% being exceeded. Table 2 shows a single replicate 2n factorial design for constructing response surfaces for ultimate bearing capacity using RSM. Using the procedure described earlier, the response surface equation for ultimate bearing capacity (qu = C, capacity) is obtained as
Table 2 Single replicate 2n design showing eight combinations of input variables and corresponding output (qu) obtained from Meyerhof equation
23.75
6. Results and discussion
22.5
190
φ
Fig. 1a. Actual response surface for the ultimate bearing capacity of the foundation using Meyerhof equation.
respectively. Hence, the response surface model adequately represents the response variable (ultimate bearing capacity) for the range of variation of input soil parameters assumed in the analysis. The mean (lC) and standard deviation (rC) values of capacity (i.e. qu) obtained are 438.54 kPa and 76.43 kPa, respectively. For different applied pressures, reliability index (b) values are calculated.
Table 1 Input soil properties used in the analysis Input soil properties (x)
Mean (l)
Standard deviation (r)
xmax
xmin
CoV (%)
Distribution
Cohesion (c), kPa Friction angle (/) Unit weight (c), kN/m3 Modulus of elasticity (E), kPa Poison’s ratio m
5 25° 17 10,000 0.325
0.61 kPa 1.52° 1.03 kN/m3 1212 kPa 0.045
6 27.5° 18.7 12,000 0.40
4 22.5° 15.3 8000 0.25
12 6 6 12 14
Normal Normal Normal Normal Normal
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demand (D, i.e. calculated settlement) and corresponding reliability index values obtained for the allowable settlement (capacity C) of 40 mm. 6.1. Results from numerical analysis
Fig. 1b. Contour plot for ultimate bearing capacity obtained from linear response surface model and Meyerhof equation.
Following the same procedure, response surface equation for settlement (S) of the footing for an applied pressure of 200 kPa is S ¼ 62:15 0:0028E 20:31m
ð10Þ
Fig. 3 shows contour plot of settlement of footing obtained from approximated linear response surface model and conventional equation. The values of R2 and adjusted R2 are 0.99 each; and are close to 1. These results indicate the adequacy of the fitted model for settlement prediction. Similarly, response surface models are developed for different applied pressures on the footing as indicated in Table 3. The table also shows mean and standard deviation of
Normal Probability Plot
700 600 500
qu
400 300 200 100 0 0
20
40
60
80
100
Sample Percentile Fig. 2. Normal probability plot for the ultimate bearing capacity (qu).
Numerical simulations are performed using FLAC and the response surface models for ultimate bearing capacity and settlement are obtained. Fig. 4 shows load displacement curves for mean values as well as for eight combinations (sample points) of input soil parameters. These curves are used to obtain response surface models for ultimate bearing capacity and settlement predictions. For the mean value of input soil parameters, using numerical analysis, the mean ultimate bearing capacity of foundation soil is obtained as 408 kPa. This value is quite close to the value 419 kPa obtained from the conventional approach. This shows the adequacy of the numerical model used in the analysis. Using regression analysis, response surface model for the ultimate bearing capacity of foundation soil is obtained as qu ¼ 975 þ 12:5c þ 37:0/ þ 19:18c
ð11Þ
The approximately straight normal probability plot (not shown) and the calculated values of R2 and R2adj 0.992 and 0.986, respectively confirm the adequacy of the fitted model. Using Eq. (11), the values of mean and standard deviation of qu are calculated as 338 kPa and 59.64 kPa and for different applied pressures reliability index (b) values are calculated using Eq. (3). The same procedure is adopted for reliability index calculation from settlement considerations. The results of the reliability analysis using response surface method, from conventional approach and numerical analysis are compared with the corresponding reliability index values obtained from Monte Carlo simulation. The results of the analysis are summarized in Fig. 5. It can be seen from Fig. 5 that the allowable pressure on the footing varies from 140 kPa to 160 kPa in order to achieve reliability index (b) value in the range of 3–4 (shown in the shaded region). The shaded region also shows that all four different approaches provide reliability index values in the range of 3.0–4.0, if applied pressure ranges between 140 kPa and 160 kPa. It was found that the calculated values of reliability indices from settlement consideration and from all four approaches (RSM and MCS using conventional solution and numerical analysis), are well above the minimum acceptable value of b (=3.0) for the range of applied pressure 140–160 kPa. The results also indicate that the conventional procedure or analytical solution for ultimate bearing capacity, with a factor of safety of 3.0 is adequate in the present case as it is associated with required reliability index value. It can also be noted that the results obtained from RSM are comparable with the values obtained from Monte Carlo simulations. The advantage of RSM is that it requires fewer number of simulations (2n and for n = 3,
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Fig. 3. Contour plot for settlement of the footing obtained from RSM model and conventional equation.
Table 3 Response surface models obtained from conventional settlement equation for different applied pressure on the footing and corresponding reliability indices (b) from settlement criterion Response surface equation
Mean (D)
Std. dev. (lD)
140 160 180 200
43.50–0.0019E14.22m 49.72–0.0022E16.25m 55.93–0.0025E18.28m 62.15–0.0025E20.31m
19.88 22.44 24.99 27.55
2.37 2.76 3.14 3.51
4
b 3 RSM - Meyerhof Equation
β
Applied pressure (kPa)
5
Allow. settlement (40 mm)
8.50 6.35 4.78 2.87
MCS - Meyerhof Equation
Applied pressure (kPa) 100
200
300
400
120
140
160
180
200
Applied pressure (kPa)
500
600
0 100
settlement (mm)
MCS - Numerical analysis
1
0 100
0
RSM - Numerical analysis
2
Fig. 5. Results of the reliability analysis using RSM and MCS from shear failure criterion.
the number of simulations is 8) as compared to 20,000 Monte Carlo simulations. Hence, a distinct advantage of RSM is in terms of computational time and memory requirements, notwithstanding the availability of high-end computers and simulation procedures.
200
300
6.2. Effect of correlation among input variables 400
In order to study the effect of correlation among input soil parameters on the reliability index values, negative
500
600
For 8 combination runs mean soil properties Fig. 4. Load displacement curve for mean values and eight combinations of input soil parameters obtained through numerical analysis of the soil foundation system.
Table 4 Correlation coefficient between input soil parameters Correlation coefficient (q)
c–/
c–c
/–c
Case 1 Case 2 Case 3
0.25 0.50 0.75
+0.25 +0.50 +0.75
+0.25 +0.50 +0.75
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Table 5 Reliability index values from shear failure consideration (using response surface model developed from Meyerhof equation)
Table 6 Reliability index values from shear failure consideration using RSM approach
Applied pressure (kPa)
Applied pressure (kPa)
(1)
(2)
(3)
120 140 160 180 200
3.64 3.30 2.97 2.64 2.31
6.94 6.16 5.38 4.60 3.81
6.38 5.64 4.90 4.19 3.38
120 140 160 180 200
Reliability index (b) Uncorrelated normally distributed input soil parameters
Correlated normally distributed input soil parameters Case 1
Case 2
Case 3
3.94 3.70 3.45 3.20 2.95
4.12 3.86 3.57 3.31 3.05
4.08 3.82 3.57 3.31 3.05
4.03 3.78 3.53 3.27 3.02
(1) From MV FOSM, variables follow normal distribution. (2) From MV FOSM, variables follow log-normal distribution. (3) Iterative procedure, variables follow log- normal distribution.
correlation between cohesion (c) and friction angle (/) varying from (0.25 to 0.75) and positive correlation between bulk density (c) and shear strength parameters, viz, cohesion (c) and friction angle (/) varying from 0.25 to 0.75 are considered in accordance with the reported values in literature [20]. The values of correlation coefficients used in this study are presented in Table 4. From shear failure consideration, the response surface models developed from Meyerhof equation (Eq. (10)) and numerical analysis (Eq. (12)) are used to calculate reliability index values for different applied pressures. The results of the analysis are summarized in Table 5. It can be noted that the consideration of correlation between input soil parameters marginally affects the reliability index values.
Reliability index values for different applied pressures are calculated using Eq. (3) and also using FORM. Table 6 shows the reliability index values calculated using Eq. (3). It can be noted that there is large difference between the two reliability index values computed, based on the assumption of normal distribution and log-normal distribution of input random variables for lower applied pressures but this difference reduces for higher loads. It can also be noted that the allowable pressure on the footing would be higher if the input soil parameters are log-normally distributed provided it also satisfies the serviceability requirements. For comparison, reliability index is also calculated using iterative procedure described in literature [17]. As the performance function is linear, only two iterations were required for convergence. 7. Concluding remarks
6.3. Consideration of non-normal distributions in response surface methodology In a further study, it is assumed that input soil parameters are uncorrelated log-normally distributed. For the development of response surface model (RSM) for uncorrelated log-normally distributed input soil parameters and calculation of reliability index values, the following steps are used: (i) Log-normally distributed input soil parameters are transformed into equivalent normal variable using Rackwitz and Fiessler procedure [21] described in [17]. (ii) Using the parameters of equivalent normal transformation (leq, req) of input soil parameters, sample points are selected (leq ± kreq). (In the present case k = 1.65.) (iii) Response surface model is generated using regression analysis and least square error approach and the adequacy of the fitted model is checked as described earlier. (iv) The response surface model for ultimate bearing capacity of the foundation soil is developed for eight combination of sample points and corresponding output (i.e. ultimate bearing capacity) as C ¼ 1164:81 þ 25:01c þ 46:63/ þ 17:43c
ð12Þ
The work presented in this paper demonstrates the applicability of response surface methodology in the assessment of allowable bearing pressure on shallow foundation resting on cohesive frictional soil. The results show that reliability analysis results from conventional approaches and numerical analysis help in deciding the range of allowable pressures on the footing. Although the reliability based approach is not a substitute for the conventional approach, it can be noted from the present study that the choice of allowable pressure on the footing can be made rationally. The use of response surface method in establishing an approximate functional relationship between input soil properties and output response, viz. ultimate bearing capacity or settlement of the foundation, reduces computational efforts to a great extent for the probabilistic analysis of the foundation. Acknowledgments The authors thank the reviewers for their critical comments which have been very useful in improving the work presented in this paper. References [1] Bowles JE. Foundation analysis and design. 5th ed. McGraw-Hill Book Company; 1996.
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[2] Phoon KK, Kulhawy FH. Characterization of geotechnical variability. Canad Geotech J 1999;36:612–24. [3] Whitman RV. Organizing and evaluating uncertainty in geotechnical engineering. Geotech Geol Eng 2000;126(7):583–93. [4] Duncan JM. Factors of safety and reliability in geotechnical engineering. J Geotech Geoenviron Eng 2000;126(4):307–16. [5] Schweiger HF, Thurner R, Pottler R. Reliability analysis in geotechnics with deterministic finite elements. Int J Geomech ASCE 2001;1(4):389–413. [6] Sivakumar Babu, Amit Srivatsava, Murthy DSN. Reliability analysis of bearing capacity of a shallow foundation resting on cohesive soil. Canad Geotech J 2006;43:1–7. [7] Griffiths DV, Fenton GA. Seepage beneath water retaining structures founded on spatially random soil. Geotechnique 1993;43(4): 577–87. [8] Paice GM, Griffiths DV, Fenton GA. Influence of spatially random soil stiffness on foundation settlements. In: Yeung AT, Felio GY, editors. ASCE specialty conference ‘‘Settlement’94’’. Texas A&M University; 1994. p. 628–39. [9] Wong FS. Slope stability and response surface method. J Geotech Geoenviron Eng ASCE 1985;111(1):32–53. [10] Humphreys MP, Armstrong LW. Assessing the sensitivity of numerical models using response surface methodology. In: Proc of the conf of probabilistic methods in geotechnical engineering, Canberra, Australia, 1993. p. 135–44.
[11] Tandjiria VI, Teh CI, Low BK. Reliability analysis of laterally loaded piles using response surface methods. Struct Safety 2000;22:335–55. [12] Eurocode 7 Geotechnical Design – Part 1: General rules, EN 19971:2004(E). [13] Baecher GB, Christian JT. Reliability and statistics in geotechnical engineering. Chichester, NJ: John Wiley Publications; 2003. [14] USACE. Risk-based analysis in Geotechnical Engineering for Support of Planning Studies, Engineering and Design. US Army Corps of Engineers, Department of Army, Washington, DC, 1997. p. 20314-100. [15] Myers RH, Montgomery D. Response surface methodology. John Wiley & Sons; 2002. [16] FLAC 4.0. Reference Manual. Itasca Consulting Group Inc., Minneapolis, 2000. [17] Haldar A, Mahadevan S. Probability, reliability and statistical methods in engineering design. NY: John Wiley & Sons; 2000. [18] Becker DE. Limit state design for foundations: Part I. An overview of the foundation design process. Eighteenth Canadian geotechnical colloquium. Canad Geotech J 1996;33:956–83. [19] Orr TLL. Selection of characteristic values and partial factors in geotechnical designs to Eurocode 7. Comput Geotech 2000;26:263–79. [20] Chowdhury RN, Xu DW. Reliability index for slope stability assessment–two methods compared. Reliab Eng Syst Safety 1992;37:99–108. [21] Rackwitz R, Fiessler B. Note on discrete safety checking when using non-normal stochastic models for basic variables. Load project working session. Cambridge, MA: MIT; 1976. June.
Technical note
Reliability analysis of allowable pressure on shallow foundation using response surface method G.L. Sivakumar Babu *, Amit Srivastava Department of Civil Engineering, Indian Institute of Science, Bangalore, Karnataka 560 012, India Received 30 June 2006; received in revised form 10 November 2006 Available online 3 January 2007
Abstract The concept of response surface method (RSM) is used to generate approximate polynomial functions for ultimate bearing capacity and settlement of a shallow foundation resting on a cohesive frictional soil for a range of expected variation of input soil parameters. The response surface models are developed using available conventional equations and numerical analysis. Considering the variations in the input soil parameters, reliability analysis is performed using these response surface models to obtain an acceptable value of the allowable bearing pressure. The results of the reliability analysis are compared with the results of Monte Carlo simulation and it is demonstrated that application of response surface method in the probabilistic analysis can considerably reduce the computational efforts and memory requirements. It is also concluded that conventional analysis using available equations and numerical analysis when used in conjunction with reliability analysis enable a rational choice of allowable pressure and help in decision-making process. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Conventional analysis; Reliability analysis; Response surface method; Monte Carlo simulation; Soil parameters; Correlation
1. Introduction In the conventional design of shallow foundations resting on a cohesive frictional soil, the allowable pressure is calculated based on shear failure criterion (ultimate limit state, ULS) and settlement criterion (serviceability limit state, SLS). The allowable bearing capacity is obtained by dividing the ultimate bearing capacity of the foundation soil with a factor of safety. At the same time, it is also ensured that the magnitude of settlement of footing should not exceed a specified permissible limit from serviceability requirements [1]. Design values of input soil parameters such as cohesion (c), angle of internal friction (/), unit weight (c), modulus of elasticity (E), for estimating the allowable pressure on foundation soil, are obtained either from field tests or from laboratory tests. As soil is inherently variable, variations in measured values of those soil *
Corresponding author. Tel.: +91 80 22933124; fax: +91 80 23600404. E-mail addresses: [email protected] (G.L. Sivakumar Babu), [email protected] (A. Srivastava). 0266-352X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2006.11.002
parameters is inevitable. Phoon and Kulhawy [2] indicated that inherent variability of soil deposits, testing errors and model transformation uncertainties contribute to variability in the test data. Although factor of safety approach is simple and straightforward, it does not consider different sources of uncertainty in geotechnical design in a rational manner [3–5]. In order to incorporate these variations, reliability analysis is performed. In this approach, input soil parameters are treated as random variables and the influence of these input random variables on the output random variable is studied. Reliability analysis approaches can be used in conjunction with conventional approaches to have better insight into the choice of allowable value of bearing pressure and helps in decision-making process. Sivakumar Babu et al. [6] demonstrated this aspect with reference to the estimation of allowable bearing pressure of a shallow foundation resting on a typical stiff clay. For reliability analysis, a functional relationship (either implicit or explicit) is required between input and output random variables. Explicit relationships for reliability analysis can be obtained from conventional bearing capacity
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and settlement equations. These relationships are implicit in numerical analysis and tools such as random field finite element analysis (RFEM), response surface methodology (RSM) are used to analyze the response variability. Various researchers [7,8] used random field finite element modelling (RFEM) to study the bearing capacity of shallow foundations. Wong [9] performed reliability analysis of soil slopes using response surface method (RSM). Humphreys and Armstrong [10] analyzed a slope stability problem using results of finite difference method and regression analysis. Tandjiria et al. [11] used response surface method for reliability analysis of laterally loaded piles. A question that often arises in practice is to know ‘‘how safe is safe?’’ or to what extent factors of safety that are routinely used in conventional solutions address the question of safety adequately. It is possible to examine a degree of safety associated with analytical formulations by comparing reliability indices from conventional solutions with those from analysis of response surfaces constructed from conventional solution as well as from the results of numerical analysis. As expected, a comparative study of the results of the analysis from conventional solution and numerical analysis in terms of reliability indices demonstrates the effectiveness of conventional, numerical and probabilistic approaches in the assessment of allowable pressure on foundation soil and this is the main focus of the present study. 2. Conventional approach In a conventional approach, allowable pressure on a shallow footing is taken as lesser of the two values, i.e. (i) allowable bearing capacity based on shear failure criterion and (ii) allowable bearing pressure based on settlement criterion. (i) Shear failure criteria For a shallow strip foundation of width B resting on a horizontal ground at a depth Df and loaded with a concentric vertical loading the ultimate bearing capacity (qu) can be calculated using Meyerhof equation qu ¼ cN c sc d c þ cDf N q sq d q þ 0:5cBN c sc d c
ð1Þ
The values of bearing capacity factors Nc, Nq and Nc as well as shape factors (sc, sq, sc) and depth factors (dc, dq, dc) are obtained from available equations provided in literature [1]. Allowable bearing capacity (qa) is obtained after applying a factor of safety (FS) to the ultimate bearing capacity (qu) and usually a value between 2.5 and 3 is considered appropriate in the case of a shallow foundation. (ii) Settlement criteria For an applied pressure (q0), the magnitude of settlement of a footing can be computed using Eq. (2) and this is compared with the allowable settlement specified in design guidelines/codes to satisfy serviceability requirements
S ¼ q0 B
ð1 m2 Þ If E
ð2Þ
where B is the width of footing, q0 is an applied pressure on the footing, If is an influence factor, m is poison’s ratio and E is modulus of elasticity of soil. As per Eurocode 7 [12], total settlements up to 50 mm are often acceptable for normal structures with isolated foundations. As per Indian codes of practice, 40 mm is considered acceptable and hence in the present study, the allowable settlement is taken as 40 mm. 3. Reliability analysis Reliability is defined as probability of safety of a system in a given environment and loading conditions and is assessed in terms of reliability index (b) values. Normally, a reliability index value in the range of 3.0–4.0 is accepted for good performance of the system [13,14]. For the estimation of reliability index, methods such as first order reliability method (FORM), second order reliability method (SORM), point estimate method (PEM), Monte Carlo simulation (MCS) are available in literature. For a linear performance function defined as g(Æ) = C D, where C is the capacity (either ultimate bearing capacity or allowable settlement) and D is the demand (either applied pressure or calculated settlement), g(Æ) > 0 denotes safety while g(Æ) < 0 represents unsafe condition. For uncorrelated normally distributed C and D, reliability index (b), representing the shortest distance of the performance function from the origin of reduced coordinate system of variables, can be calculated using Eq. (3) [13]. CD b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr2C þ r2D Þ
ð3Þ
where rC and rD are standard deviations of capacity (C) and demand (D). For non-linear performance functions, iterative procedure is used to evaluate reliability index. Reliability analysis using conventional equations for ultimate bearing capacity and settlement is complicated because calculation of derivatives for variances by first or second order methods is cumbersome. Using Monte Carlo simulation technique, probability of failure can be computed for both explicit and implicit limit state functions. This method involves the generation of N random numbers of input soil parameters with given probabilistic characteristics. These N sample points for output response are used to obtain required sample statistics, which is incorporated in probabilistic calculations. The minimum value of number N depends on percentage (%) acceptable error (e) in the estimation of sample mean and variance as well as confidence level defined in terms of parameter a [13]. As the performance function is defined as g(Æ) = C D, capacity (C) is deducted from demand (D) for each realization of output response. If Nf is the
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number of simulation cycles when g(Æ) is less than zero and N is the total number of simulation cycles, probability of N failure can be obtained as pf ¼ Nf which is approximately related to reliability index as b = /1(1 pf). On the other hand, response surface method (RSM) facilitates the formulation of explicit linear or non-linear relationships between input random variables and required output response through regression analysis and this aspect is advantageously used in the present study for reliability analysis. 4. Response surface methodology (RSM) Response surface method involves generation of polynomial equation using regression analysis and an approximate linear or non-linear functional relationship between dependent output y and input variables (x1, x2, x3, . . .) is established (Eq. (4)). y ¼ f ðx1 ; x2 ; x3 ; . . .Þ þ e
y ¼ a0 þ a1 x1 þ a2 x2 þ e y ¼ a0 þ a1 x1 þ a2 x2 þ
ð5Þ þ
a22 x22
þ a12 x1 x2 þ e
ð6Þ
where a0, a1, a2, . . . are regression coefficients and ‘e’ represents error involved in neglecting other sources of uncertainties. In RSM analysis, single replicate 2n factorial design is used to fit first order linear regression model. For this, total number of sample points is required as 2n, where n is total number of input variables. For n = 2 and n = 3, number of such sample points are 4 and 8, respectively. In the analysis, natural variables (x1, x2, x3, . . . , xk) are converted into coded variables ni using a relationship i Þþminðxi Þ=2 ni ¼ xi ½maxðx . The maximum and minimum values ½maxðxi Þminðxi Þ=2 of xi are taken as xmax = x + hr and xmin = x hr, where h is an integer and ris are the standard deviations of input variables (xis). These limits xmax and xmin cover the extent of variation of input parameters in a given situation with some level of confidence interval discussed later. Finally, the response surface model is presented in terms of natural variables. To examine the adequacy of the fitted model and to ensure that it provides a good approximation of the true system, a normal probability plot should be approximately along a straight line. In addition, computed values of coefficients of multiple determinations (R2) and adjusted R2 also give information on the adequacy of the fitted model. The non-dimensional quantity R2 is calculated as (Eq. (7)). Pn ^ 2 i¼1 y i y R2 ¼ P 2 n i¼1 y i y
^
where y , yi and y are estimated mean value, actual and predicted values of output response (y) respectively. The value of R2 lies between 0 and 1 and a value close to 1 indicates that most of the variability in y is explained by regression model. It should be noted that it is always possible to increase the value of R2 by adding more regressor variables. Therefore, adjusted R2 value is calculated using following Eq. (8): k 1 1 R2 ð8Þ R2adj ¼ 1 kp where k is total number of observations and p is number of regression coefficients. For a good model, values of R2 and adjusted R2 should be close to each other and also they should be close to 1. Myers and Montgomery [15] presented more details on the use of response surface methodology. 5. Methodology
ð4Þ
The following Eq. (5) shows multiple linear regression model having two input variables without interaction terms and Eq. (6) shows second order regression model with interaction terms containing two input variables x1, x2. a11 x21
189
ð7Þ
In the present case, the output response (i.e. the ultimate bearing capacity from shear failure consideration) is obtained using conventional Meyerhof equation corresponding to eight sample points for three input variables (c, / and c). A linear regression model is fitted between output response (qu) and input variables (c, / and c) and this approximate linear relationship is used to obtain mean and variance of qu for reliability analysis. The same procedure is adopted to obtain response surface models for settlement of the footing with input random variables E and m and for different applied pressures on the footing and reliability index values are calculated. The above results are also compared with the results of reliability analysis from response surface models obtained from numerical analysis. Using commercial finite difference code, fast Lagrangian analysis continua (FLAC) [16], response surface models corresponding to the above eight sample points are developed. For the analysis, physical domain of the problem is divided into finite difference mesh of quadrilateral elements. The unbalanced force of each node is normalized by gravitational force acting on that node. A simulation is considered to have converged when the normalized unbalanced force of every node in the mesh is less than 103. In the analysis, elastic perfectly plastic model based on Mohr–Coulomb failure criteria with associated flow rule is used for the foundation soil. After obtaining response surface models for ultimate bearing capacity and settlement from conventional approach and numerical analysis, reliability analysis is performed. The probability of failure is also obtained using Monte Carlo simulations. For an acceptable error of 5.0% and a = 0.05 (95% confidence level), minimum sample size is 1536 following the procedure explained in the literature [13,17]. In the present analysis, 20,000 random numbers are generated for simulation.
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C ¼ 1191:04 þ 25:53c þ 47:64/ þ 18:29c
Sr. no.
1 2 3 4 5 6 7 8
Combination
+ + + +
+ + + +
+ + + +
qu
c
/
c
6 6 6 6 4 4 4 4
27.5 27.5 22.5 22.5 27.5 27.5 22.5 22.5
18.7 15.3 18.7 15.3 18.7 15.3 18.7 15.3
628.52 547.18 361.82 318.80 568.13 486.80 320.10 277.10
600 500 400 qu 300 500-600 200
400-500 300-400
100
4.5
5 5.5
c
6
27.5
4
26.25
0
ð9Þ
Fig. 1a shows plots of actual response surface for ultimate bearing capacity using Meyerhof equation and Fig. 1b shows contours of ultimate bearing capacity calculated from conventional (Meyerhof) equation and approximated linear response surface equation. It can be seen that the results of response surface model are quite comparable with those obtained from Meyerhof solution and also the linear approximation of ultimate bearing capacity of foundation soil for the range variation of input parameters is valid. Fig. 2 shows normal probability plot for the regressed model, which is approximately a straight line. The values of R2 and adjusted R2 are 0.99 and 0.98,
Natural variables
25
A shallow strip foundation of width (B) 1.5 m at a depth (Df) of 1.0 m, resting on a cohesive frictional soil is considered. The values of input soil properties for conventional and reliability analysis are presented in Table 1. Using conventional approach, for mean values of input soil properties, the mean value of ultimate bearing capacity from shear failure criteria is obtained as 419 kPa. Using a factor of safety of 3.0, the allowable bearing capacity of foundation soil is 139.67 140 kPa. From serviceability requirements, for an applied pressure of 140 kPa, the settlement of foundation is 18.78 mm, which is less than the allowable settlement of 40 mm. Becker [18] and Orr [19] suggest that selection of characteristic values of geotechnical parameters and corresponding confidence intervals should be incorporated in reliability based designs. Response surfaces are generated corresponding to 95% confidence intervals, i.e. the lower limit and upper limit values related to mean (l) and standard deviation (r) with the relationships xu = l + 1.65r and xl = l 1.65r are used in the analysis. The upper limit (xu or xmax) and lower limit (xl or xmin) values indicated in Table 1 are based on the assumption that input soil parameters follow normal distribution and upper and lower limit values have probabilities of 5% and 95% being exceeded. Table 2 shows a single replicate 2n factorial design for constructing response surfaces for ultimate bearing capacity using RSM. Using the procedure described earlier, the response surface equation for ultimate bearing capacity (qu = C, capacity) is obtained as
Table 2 Single replicate 2n design showing eight combinations of input variables and corresponding output (qu) obtained from Meyerhof equation
23.75
6. Results and discussion
22.5
190
φ
Fig. 1a. Actual response surface for the ultimate bearing capacity of the foundation using Meyerhof equation.
respectively. Hence, the response surface model adequately represents the response variable (ultimate bearing capacity) for the range of variation of input soil parameters assumed in the analysis. The mean (lC) and standard deviation (rC) values of capacity (i.e. qu) obtained are 438.54 kPa and 76.43 kPa, respectively. For different applied pressures, reliability index (b) values are calculated.
Table 1 Input soil properties used in the analysis Input soil properties (x)
Mean (l)
Standard deviation (r)
xmax
xmin
CoV (%)
Distribution
Cohesion (c), kPa Friction angle (/) Unit weight (c), kN/m3 Modulus of elasticity (E), kPa Poison’s ratio m
5 25° 17 10,000 0.325
0.61 kPa 1.52° 1.03 kN/m3 1212 kPa 0.045
6 27.5° 18.7 12,000 0.40
4 22.5° 15.3 8000 0.25
12 6 6 12 14
Normal Normal Normal Normal Normal
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demand (D, i.e. calculated settlement) and corresponding reliability index values obtained for the allowable settlement (capacity C) of 40 mm. 6.1. Results from numerical analysis
Fig. 1b. Contour plot for ultimate bearing capacity obtained from linear response surface model and Meyerhof equation.
Following the same procedure, response surface equation for settlement (S) of the footing for an applied pressure of 200 kPa is S ¼ 62:15 0:0028E 20:31m
ð10Þ
Fig. 3 shows contour plot of settlement of footing obtained from approximated linear response surface model and conventional equation. The values of R2 and adjusted R2 are 0.99 each; and are close to 1. These results indicate the adequacy of the fitted model for settlement prediction. Similarly, response surface models are developed for different applied pressures on the footing as indicated in Table 3. The table also shows mean and standard deviation of
Normal Probability Plot
700 600 500
qu
400 300 200 100 0 0
20
40
60
80
100
Sample Percentile Fig. 2. Normal probability plot for the ultimate bearing capacity (qu).
Numerical simulations are performed using FLAC and the response surface models for ultimate bearing capacity and settlement are obtained. Fig. 4 shows load displacement curves for mean values as well as for eight combinations (sample points) of input soil parameters. These curves are used to obtain response surface models for ultimate bearing capacity and settlement predictions. For the mean value of input soil parameters, using numerical analysis, the mean ultimate bearing capacity of foundation soil is obtained as 408 kPa. This value is quite close to the value 419 kPa obtained from the conventional approach. This shows the adequacy of the numerical model used in the analysis. Using regression analysis, response surface model for the ultimate bearing capacity of foundation soil is obtained as qu ¼ 975 þ 12:5c þ 37:0/ þ 19:18c
ð11Þ
The approximately straight normal probability plot (not shown) and the calculated values of R2 and R2adj 0.992 and 0.986, respectively confirm the adequacy of the fitted model. Using Eq. (11), the values of mean and standard deviation of qu are calculated as 338 kPa and 59.64 kPa and for different applied pressures reliability index (b) values are calculated using Eq. (3). The same procedure is adopted for reliability index calculation from settlement considerations. The results of the reliability analysis using response surface method, from conventional approach and numerical analysis are compared with the corresponding reliability index values obtained from Monte Carlo simulation. The results of the analysis are summarized in Fig. 5. It can be seen from Fig. 5 that the allowable pressure on the footing varies from 140 kPa to 160 kPa in order to achieve reliability index (b) value in the range of 3–4 (shown in the shaded region). The shaded region also shows that all four different approaches provide reliability index values in the range of 3.0–4.0, if applied pressure ranges between 140 kPa and 160 kPa. It was found that the calculated values of reliability indices from settlement consideration and from all four approaches (RSM and MCS using conventional solution and numerical analysis), are well above the minimum acceptable value of b (=3.0) for the range of applied pressure 140–160 kPa. The results also indicate that the conventional procedure or analytical solution for ultimate bearing capacity, with a factor of safety of 3.0 is adequate in the present case as it is associated with required reliability index value. It can also be noted that the results obtained from RSM are comparable with the values obtained from Monte Carlo simulations. The advantage of RSM is that it requires fewer number of simulations (2n and for n = 3,
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Fig. 3. Contour plot for settlement of the footing obtained from RSM model and conventional equation.
Table 3 Response surface models obtained from conventional settlement equation for different applied pressure on the footing and corresponding reliability indices (b) from settlement criterion Response surface equation
Mean (D)
Std. dev. (lD)
140 160 180 200
43.50–0.0019E14.22m 49.72–0.0022E16.25m 55.93–0.0025E18.28m 62.15–0.0025E20.31m
19.88 22.44 24.99 27.55
2.37 2.76 3.14 3.51
4
b 3 RSM - Meyerhof Equation
β
Applied pressure (kPa)
5
Allow. settlement (40 mm)
8.50 6.35 4.78 2.87
MCS - Meyerhof Equation
Applied pressure (kPa) 100
200
300
400
120
140
160
180
200
Applied pressure (kPa)
500
600
0 100
settlement (mm)
MCS - Numerical analysis
1
0 100
0
RSM - Numerical analysis
2
Fig. 5. Results of the reliability analysis using RSM and MCS from shear failure criterion.
the number of simulations is 8) as compared to 20,000 Monte Carlo simulations. Hence, a distinct advantage of RSM is in terms of computational time and memory requirements, notwithstanding the availability of high-end computers and simulation procedures.
200
300
6.2. Effect of correlation among input variables 400
In order to study the effect of correlation among input soil parameters on the reliability index values, negative
500
600
For 8 combination runs mean soil properties Fig. 4. Load displacement curve for mean values and eight combinations of input soil parameters obtained through numerical analysis of the soil foundation system.
Table 4 Correlation coefficient between input soil parameters Correlation coefficient (q)
c–/
c–c
/–c
Case 1 Case 2 Case 3
0.25 0.50 0.75
+0.25 +0.50 +0.75
+0.25 +0.50 +0.75
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Table 5 Reliability index values from shear failure consideration (using response surface model developed from Meyerhof equation)
Table 6 Reliability index values from shear failure consideration using RSM approach
Applied pressure (kPa)
Applied pressure (kPa)
(1)
(2)
(3)
120 140 160 180 200
3.64 3.30 2.97 2.64 2.31
6.94 6.16 5.38 4.60 3.81
6.38 5.64 4.90 4.19 3.38
120 140 160 180 200
Reliability index (b) Uncorrelated normally distributed input soil parameters
Correlated normally distributed input soil parameters Case 1
Case 2
Case 3
3.94 3.70 3.45 3.20 2.95
4.12 3.86 3.57 3.31 3.05
4.08 3.82 3.57 3.31 3.05
4.03 3.78 3.53 3.27 3.02
(1) From MV FOSM, variables follow normal distribution. (2) From MV FOSM, variables follow log-normal distribution. (3) Iterative procedure, variables follow log- normal distribution.
correlation between cohesion (c) and friction angle (/) varying from (0.25 to 0.75) and positive correlation between bulk density (c) and shear strength parameters, viz, cohesion (c) and friction angle (/) varying from 0.25 to 0.75 are considered in accordance with the reported values in literature [20]. The values of correlation coefficients used in this study are presented in Table 4. From shear failure consideration, the response surface models developed from Meyerhof equation (Eq. (10)) and numerical analysis (Eq. (12)) are used to calculate reliability index values for different applied pressures. The results of the analysis are summarized in Table 5. It can be noted that the consideration of correlation between input soil parameters marginally affects the reliability index values.
Reliability index values for different applied pressures are calculated using Eq. (3) and also using FORM. Table 6 shows the reliability index values calculated using Eq. (3). It can be noted that there is large difference between the two reliability index values computed, based on the assumption of normal distribution and log-normal distribution of input random variables for lower applied pressures but this difference reduces for higher loads. It can also be noted that the allowable pressure on the footing would be higher if the input soil parameters are log-normally distributed provided it also satisfies the serviceability requirements. For comparison, reliability index is also calculated using iterative procedure described in literature [17]. As the performance function is linear, only two iterations were required for convergence. 7. Concluding remarks
6.3. Consideration of non-normal distributions in response surface methodology In a further study, it is assumed that input soil parameters are uncorrelated log-normally distributed. For the development of response surface model (RSM) for uncorrelated log-normally distributed input soil parameters and calculation of reliability index values, the following steps are used: (i) Log-normally distributed input soil parameters are transformed into equivalent normal variable using Rackwitz and Fiessler procedure [21] described in [17]. (ii) Using the parameters of equivalent normal transformation (leq, req) of input soil parameters, sample points are selected (leq ± kreq). (In the present case k = 1.65.) (iii) Response surface model is generated using regression analysis and least square error approach and the adequacy of the fitted model is checked as described earlier. (iv) The response surface model for ultimate bearing capacity of the foundation soil is developed for eight combination of sample points and corresponding output (i.e. ultimate bearing capacity) as C ¼ 1164:81 þ 25:01c þ 46:63/ þ 17:43c
ð12Þ
The work presented in this paper demonstrates the applicability of response surface methodology in the assessment of allowable bearing pressure on shallow foundation resting on cohesive frictional soil. The results show that reliability analysis results from conventional approaches and numerical analysis help in deciding the range of allowable pressures on the footing. Although the reliability based approach is not a substitute for the conventional approach, it can be noted from the present study that the choice of allowable pressure on the footing can be made rationally. The use of response surface method in establishing an approximate functional relationship between input soil properties and output response, viz. ultimate bearing capacity or settlement of the foundation, reduces computational efforts to a great extent for the probabilistic analysis of the foundation. Acknowledgments The authors thank the reviewers for their critical comments which have been very useful in improving the work presented in this paper. References [1] Bowles JE. Foundation analysis and design. 5th ed. McGraw-Hill Book Company; 1996.
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