## International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

### Model uncertainty for the capacity of strip footings under negative and general combined loading Phoon, K. K.; Tang, Chong Jul 31, 2015

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`12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1Model Uncertainty for the Capacity of Strip Footings under Negative and General Combined Loading K.K. Phoon Professor, Dept. of Civil and Environmental Engineering, National University of Singapore, Singapore Chong Tang Research fellow, Dept. of Civil and Environmental Engineering, National University of Singapore, Singapore ABSTRACT: This paper investigates the model uncertainty of Eurocode 7 approach for estimating the bearing capacity of shallow foundations under negative and general combined loading. In the first stage, a regression equation f is derived to remove the dependency of the model factor Ms on the input parameters, where Ms is defined as the ratio between the lower bound solution and the calculated capacity from Eurocode 7 approach. The probability distribution of the residual part  of Ms is then determined. Secondly, the model uncertainty of the lower bound limit analysis is characterized by using a loading database. The result is represented as the probability distribution of the model factor MLB, which is defined as the ratio between the measured capacity and the lower bound solution. Finally, the model uncertainty of the modified model factor M defined as the ratio between the measured capacity and the capacity from Eurocode 7 approach multiplied by the regression equation f is characterized by combining the results for  and MLB. The results are validated by using another independent loading database.  1. INTRODUCTION The ultimate bearing capacity of shallow strip foundations under combined loading is usually calculated by using the Terzaghi equation with the inclination factors and the effective width rule B=B-2e (Meyerhof 1953), where e is the loading eccentricity, given by  u 0.5c c q qq cN i qN i B N i     (1a) where c=soil cohesion, q=γD=surcharge load at the ground surface, D=embedment depth of foundation; and γ=soil weight.  The bearing capacity factors Nc, Nq and Nγ are given by (CEN 2004)    tan 221 cottan 4 21 tanc qqqN NN eN N        (1b) The loading inclination factors ic, iq, and iγ are given by    231tan1 tan1 tanqc qcqii i Nii (1c) where  is the loading inclination and φ is the friction angle of soil. Phoon & Tang (2015) studied the model uncertainty of Eurocode 7 approach for the bearing capacity of shallow foundations under positive load combination (see Figure 1a). However, the effect of combined loading direction on the model uncertainty is not investigated. Therefore, the objective of the present study is to characterize the model uncertainty of Eurocode 7 approach under negative and general combined load (see Figure 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  21b) by using the finite element formulation of the lower bound limit analysis (FELA). The framework of Phoon & Tang (2015) will be used in this paper. 2. PROBLEM DEFINITION In the present study, a strip footing under a general load with eccentricity e and inclination  on granular materials is considered. In this case, the bearing capacity of foundations is related to qNq and γNγ only. Note that Nγ usually decreases as the footing width increases, which is generally known as the scale effect. One possible reason for the scale effect is the dependency of the sand friction angle φ on the mean stress σm according to (Ueno et al. 1998)   tan tan a m a       (2) where σa=100 kPa is the atmospheric pressure; φa is reference friction angle; and ξ is an empirical parameter varying from 0.02 to 0.12. FELA with the stress level effect [i.e. Eq. (2)] can also be found in Phoon & Tang (2015).   Figure 1:  (a) Positive load combination; (b) negative load combination. 3. LOADING DATABASE Laboratory model tests were conducted recently by Patra et al. (2012a, b) to determine the bearing capacity of eccentrically and obliquely loaded strip foundations. The results were illustrated in Patra et al. (2012a, b), which are not reproduced here. The loading database for positive load combination in Patra et al. (2012a), has been used by Phoon & Tang (2015) to evaluate the model uncertainty of Eurocode 7 approach. The loading database for negative load combination was presented in Patra et al. (2012b) consisting of 72 cases, which will be used to investigate the combined loading direction on the model uncertainty of Eurocode 7 approach. This database will be divided into two parts evenly. Part I and II will be used to characterize the model uncertainty of FELA and Eurocode 7 approach, respectively.   Figure 2: The variation of the averaged value of lnMs with input parameters.  Table 1: Coefficients in the regression equation f of the model factor Ms for general load combination. Coefficients Positive Negative General b0 0.28 0.1 0.1 b1 5.05 4.5 4.5 b2 11.4 10.4 10.25 b3 -0.26 -0.25 -0.15 b4 -0.09 -0.12 0.05 b5 0.21 -1.03 -0.93 b6 -1.12 -0.45 -0.05 b7 -0.98 -1.81 -2.53 R2 0.9 0.9 0.9 4. RESULTS AND DISCUSSION This section will focus on the case of negative loading combination. The procedure is similar to the positive loading combination as presented in Phoon & Tang (2015). The correspond results for the general combined loading, which includes (a)B(b)BQCLCL Q0 0.25 0.5 0.75 10.10.20.30.40.5D/Bln Ms  -0.3 -0.2 -0.1 0-0.500.5e/Bln Ms  0 0.1 0.2 0.3 0.4 0.50.40.60.8/aln Ms  0.5 0.6 0.7 0.8 0.9 1-101tanaln Ms  0.02 0.04 0.06 0.08 0.1 0.120123ln Ms  0 0.1 0.2 0.3 0.4 0.5-2-101B/aln Ms  averaged Linear fit (R2=0.92)averaged Linear fit (R2=0.95)averaged Linear fit (R2=0.92) averaged Linear fit (R2=0.95)averaged Linear fit (R2=0.98)avearged Linear fit (R2=0.9)(b)(a)(c) (d)(e) (f)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3the positive and negative load combination, are also summarized to explain the effect of combined loading direction on the model uncertainty. 4.1. Comparison between FELA and Eurocode 7 approach In this case, the model factor Ms is defined as the ratio between the FELA calculated capacityLBu_calcq  and the predicted capacity u_calcq from Eurocode 7 approach LBu_calc u_calcsM q q                      (3) Phoon & Tang (2015) have shown that the model factor Ms is a function of the following dimensionless parameters: (1) D/B; (2) tanφa; (3) ξ; (4) γB/σa; (5) e/B; and (6) /φa. The same ranges for each parameter as the positive load combination (Phoon & Tang 2015) will be used; however, the load eccentricity e is taken as negative. The minus of e denotes the direction of moment. Consequently, a total of 128 orthogonal parameter sets are designed.  Table 2: Spearman rank correlation analysis.  Negative General  MLB M  MLB M D/B 0.32 0.36 0.4 0.23 0.31 0.37 /φa 0.21 0.25 0.2 0.19 0.2 0.18 tanφa 0.35 0.4 0.38 0.3 0.37 0.34 e/B 0.18 0.2 0.15 0.12 0.16 0.13 ξ 0.12 0.16 0.17 0.1 0.14 0.15 γB/σa 0.6 0.54 0.58 0.55 0.5 0.45  Table 3: Statistics of the residual part , MLB, and M for negative load combination  Positive Negative General Mean S.D Mean S.D Mean S.D  1.01 0.06 1.01 0.06 1.02 0.09 MLB 1.03 0.09 1.06 0.09 1.04 0.09 M 1.04 0.11 1.06 0.11 1.06 0.13  The variation of the averaged value of Ms with each parameter is plotted Figure 2. It shows that lnMs varies linearly with D/B, e/B, lnγB/σa, /φa, ξ, and tanφa with a coefficient of determination (R2) that is larger than 0.9. Consequently, these variation trends of lnMs can be fitted approximately by a linear function of these input parameters, given by          10 2 36 74 5ln tan       a aa ab Bb b bb b e Bb D B b e Bf e e e ee e e e                 (4) The model coefficients   0, ,7i ib  L are determined by using the MATLAB function ‘regress’ to carry out multiple linear regression analysis. The results for positive, negative and general load combination are given in Table 1. It can be seen that the combined loading direction significantly affect the regression equation f. In fact, in the case of the positive load combination, the rotation induced by moment exacerbates the displacement induced by the horizontal load, leading to smaller failure loads. In contrast, for negative load combination, the induced rotations counteract the horizontal displacements leading to higher failure loads. The effect of loading direction on the bearing capacity of shallow foundations under eccentric and inclined loading has been discussed by Tang et al. (2014).   Figure 3: Calculated capacity from Eurocode 7 approach and its value multiplied by the regression equation f (modified Eurocode 7 approach) versus the FELA calculated capacity.  Figure 3 plots the modified capacityu_calc u_calcq q f   against the FELA results. The discrepancy between u_calcq and LBu_calcq is relatively small, compared to the greater discrepancy between u_calcq and LBu_calcq . It suggests that the 0 50 100 150 200 250 300050100150200250300qu (kPa) (FELA)f*qu (kPa) (Eurocode 7)12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4performance of the modified Eurocode 7 approach multiplied by the regression equation f is better than that of the original Eurocode 7 approach.    Figure 4: (a) the residual part of the model factor Ms versus the FELA calculated capacity; (b) empirical distribution of ln.  The residual part LBu_calc u_calcq q  is plotted against the FELA calculated capacity in Figure 4a. In contrast to Ms,  is independent of the input parameters. This can be verified by p-values of the Spearman rank correlation as shown in Table 2, which are largely higher than 0.05. Therefore,  can be treated as a random variable. The mean and standard deviation of  is 1.01 and 0.06, as summarized in Table 3. The probability model for  is identified by using Kolmogorov-Smirnov goodness-of-fit hypothesis test (i.e. KS test). This is performed by using the MATLAB function ‘h=kstest(x)’. It returns a test decision for the null hypothesis that the data in vector x comes from a standard normal distribution, against the alternative that it does not come from such a distribution. The KS test for ln with h=0 indicates ln is a normally distributed random variable. Therefore, the lognormal distribution model with the above mean and standard deviation is a reasonable model to describe . An empirical distribution of  is presented in Figure 4b. 4.2. Comparison between FELA and model tests In this case, the model factor MLB is expressed as the ratio between the measured capacity u_expq and the FELA capacity LBu_calcq , namely LBLB u_exp u_calcM q q                      (5)  Figure 5: FELA calculated capacity versus the measured capacity  For negative combined load, the 36 bearing capacities calculated from FELA are plotted against the 36 measured capacities from Part I database given by Patra et al. (2012b) in Figure 5. As the mean trend line of the FELA results is quite close to the 45 trend line, it is visually verified that the FELA methodology is unbiased. Figure 6a plots the model factor MLB for each case against the corresponding measured capacity, which appears to be randomly distributed. It is quantitatively validated by Spearman rank correlation p-values, which are higher than 0.05, as shown in Table 2. This indicates that MLB is not a function of the input parameters. Therefore, MLB can be treated as a random variable directly. The same KS test procedure as used for the remaining residual part  is performed to 0 50 100 150 200 250 3000.70.80.911.11.21.3qu (FELA) (kPa)(a)-0.2 -0.1 0 0.1 0.2051015202530lnFrequency(b) KS test for lnn=128, h=0Mean of =1.01S.D of =0.060 50 100 150 200 250 300050100150200250300qu,exp (kPa)q u (FELA) (kPa)  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5identify the probability distribution of MLB. The h-value for the normality of lnMLB is 0. It suggests that MLB can be modeled as a lognormal random variable. The mean value and standard deviation of MLB is 1.06 and 0.09, respectively. This can be revised with more exact description of the variation of friction angle with stress level. The empirical distribution of MLB is plotted in Figure 6b.    Figure 6: (a) model factor MLB versus the measured capacity; (b) empirical probability distribution model for MLB. 4.3. Comparison between model tests and Eurocode 7 approach In this case, the model factor is defined as the ratio between the measured capacity qu_exp and the calculated capacity qu_calc from Eurocode 7 approach, given by u_exp u_calcM q q                      (6) Introducing the modified capacityu_calc u_calcq q f   with Eq. (3) and (5) into Eq. (6), the modified model factor M is given by LBM M                          (7) According to Eq. (7), the modified model factor M can be fully characterized by using the results for MLB and the residual part. The residual part  and MLB follow the lognormal distribution. It is known that the product M of these two statistically independent lognormal random variables MLB and  is also a lognormal variable. According to the equations presented in Phoon & Tang (2015), the mean and standard deviation of M is 1.04 and 0.09.   Figure 7: Calculated capacity from Eurocode 7 approach and its value multiplied by the regression equation f (modified Eurocode 7 approach) versus the measured capacity.  On the other hand, the Part II database will be used to validate the probability model of M obtained before. The modified model factor M is plotted against the measured capacity in Figure 7a. It appears to be randomly distributed with the measured capacity. The Spearman rank correlation analysis with p-values being greater than 0.05 also confirms that M is independent of the input parameters. Consequently, M can also be modeled as a random variable directly. The computed results for the mean value and standard deviation of M are 1.06 and 0.11. The empirical distribution of M is illustrated in Figure 8b, which is also lognormal. It indicates that the above framework to characterize the model uncertainty by using the FELA methodology is reasonable. 0 50 100 150 200 2500.60.811.21.4qu,exp (kPa)M LB  (a)-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.40246810lnMLBFrequency(b) KS test for lnMLBn=36, h=0Mean of MLB=1.06S.D of MLB=0.090 40 80 120 160 200 240 280 32004080120160200240280320qu,exp (kPa)f*qu (kPa) (Eurocode 7)  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  65. CONCLUSION The model uncertainty of Eurocode 7 approach for estimating the bearing capacity of shallow foundations on granular materials under negative and general load combination is characterized.     Figure 8: (a) modified model factor M versus the measured capacity; (b) empirical probability distribution of the modified model factor M.  It can be seen that the combined loading direction has a significant effect on the regression equation f expressing the dependency of the model factor Ms on the input parameters, while the effect on the probability models of the residual part, MLB and M can be negligible, as shown in Table 3. However, the model uncertainty arising from the empirical shape and base inclination factors is not investigated, which should be done in future. 6. REFERENCES CEN (2004). “EN 1997-1: Eurocode 7-Part 1: Geotechnical design-part 1: General rules.” Brussels, Belgium: CEN. Meyerhof, G. G. (1953). “The bearing capacity of foundations under eccentric and inclined loads.” In Proceedings of the 3rd International Conference on Soil Mechanics and Foundation Engineering (ICSMFE), Zurich, vol. I., pp. 440-445. Patra, C. R., Behara, R. N., Sivakugan, N., and Das, B. M. (2012a). “Ultimate bearing capacity of shallow strip foundation under eccentrically inclined load, Part I.” International Journal of Geotechnical Engineering, 6, 343-352. Patra, C. R., Behara, R. N., Sivakugan, N., and Das, B. M. (2012b). “Ultimate bearing capacity of shallow strip foundation under eccentrically inclined load, Part II.” International Journal of Geotechnical Engineering, 6, 507-514. Phoon, K. K., and Tang, C. (2015). “Model uncertainty for the capacity of strip footings under combined loading.” Submitted to Geotechnical Special Publication in honor of Wilson. H. Tang (ASCE). Tang, C., Phoon, K. K., and Toh, K. C. (2014). Effect of footing width on Nγ and failure envelope of eccentrically and obliquely loaded strip footings on sand. Can. Geotech. J., 10.1139/cgj-2013-0378. 0 40 80 120 160 200 240 280 3200.40.60.811.21.41.6qu,exp (kPa)M'  (a)-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4051015lnM'FrequencyKS test for lnM'n=36, h=0Mean of M'=1.04S.D of M'=0.11(b)`

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