Pile load test results as a basis for reliability calculation with an open polygon response surface

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1 Proceedings of the XVI ECSMGE Geotechnical Engineering for Infrastructure and Development ISBN The authors and ICE Publishing: All rights reserved, 2015 doi: /ecsmge Pile load test results as a basis for reliability calculation with an open polygon response surface Résultats de test de charge pile en tant que la base pour le calcule de la fiabilité avec la methode d open réponse polygone surface M.Wyjadlowski *1, J.Bauer 1,and W.Puła 1 1 Wroclaw University of Technology, Wrocław, Poland * Corresponding Author ABSTRACT The aim of this report is to assess a method of analysing intermediate foundation load test results with an open polygon, used for the purposes of pile design employing probabilistic calculations. This method of approximating measurement results ensures that the whole range of loading force will be free of areas with unphysical load-displacement relation, while employing elementary continuous functions like a parabola in ranges of low force values usually results in a relation of a decreasing and negative displacement for a growing load value. RÉSUMÉ Le but de ce article est d'évaluer une méthode d'analyse des résultats d'essais de fondation de charge intermédiaires avec un polygone ouvert, utilisé aux fins de la conception de pile utilisant des calculs probabilistes. Cette méthode d'approximation des résultats de mesure engarantit que l'ensemble de la force de chargement sera exempt de zones de relation déplacement de la charge non physique, tout en employant des fonctions élémentaires continues comme une parabole dans des gammes de valeurs de force faible se traduit généralement par une relation d'un déplacement en baisse et négative pour une valeur de charge de plus en plus. 1 INTRODUCTION The methodology of load test result analysis will be presented based on real measurements of lateral displacements of a pile head under a statically applied lateral load. To achieve greater clarity of the discourse and reduce some calculations to simple arithmetic operations, the presented analysis uses only load test results for two pairs of piles. However, one could easily employ the proposed calculation method for an arbitrarily large group of pile pairs subjected to a load test (ASTM 1997; Polish Standard 1983; Eurocode 7, 1997). The report will propose a method of defining the values of allowable lateral loads that can be applied to pile heads. As specified before, calculations will be based on load test results for two piles, analysed with probabilistic methods using open polygon response surfaces. This approach calculates the value of allowable lateral load in such a way that the displacement of a pile head does not, with a determined probability (safety level), exceed a predefined value. The response surface obtained from a set of load test has one random variable in the form of a standard approximation error, which grows with an increase of load force value. This approximation error random variable is assigned to every displacement value. What follows is a transfer of random variability from the range of high load values towards that of low and medium ones. The value of the standard error random variable, dependent on the range of load force and consequently on the number of segments in the open polygon forming the response surface, results in the fact that the process of building the response surface can be non-objective. The report proposes a criterion, whose satisfaction will ensure that the process of determining the number of segments in the open 1223

2 Geotechnical Engineering for Infrastructure and Development polygon making the response surface is an objective one. 2 LOAD TESTS OF LATERALLY LOADED PILES The measurement results specified in chapter 2, obtained during the load tests of two pairs of piles, contain a stochastic uncertainty resulting from many different causes, but what had a crucial impact on obtaining different load curves for different piles was the spatial variation in soil conditions. Lateral load tests were performed on concrete piles with 500 mm diameter. The length of the piles was 8.0 m. The maximum lateral load value, 187 kn, was determined during the load test of the first pile. In accordance with a Polish standard recommendation, this load was divided into 11 increments of 17 kn each. Pile foundation was adopted by cause of unfavourable soil conditions. The displacements measured during the load tests have been compiled in Table 1 and presented in Figure 1. Only the results relating to the first load cycle are included, as only these ones will be used in the next part of the paper. The displacements listed in Table 1 form the basis for probabilistic calculations performed further in the report. Table 1. Test results of mean lateral displacements of two pairs of pile heads. Displacement of Pile No 1 Displacement of Pile No 2 Force Sensor 1 Sensor 2 Mean Mean Sensor 1 Sensor 2 reading reading [kn] [mm] [mm] [mm] [mm] [mm] [mm] The graph (Figure 1) demonstrates that piles installed in the same geotechnical conditions display almost comparable values of lateral displacement of their heads for low and medium load values. However, with high load values, differences in head displacement are very well-defined. 3 OPEN POLYGON RESPONSE SURFACE Figure 1. Mean lateral displacements of two pairs of pile heads as a function of the applied force. Less significant factors are differences between piles which arise during their fabrication, as well as measurement errors. An open polygon will be used as a regression model: i 11 U P ( ) { ai bi[ P ( i 1) dp ]} Zi err (1) i 1 where U stands for the pile head lateral displacement, P is the applied lateral force, a i and b i are parameters describing segments of the open polygon, dp is a 17 kn increment of the force loading the piles, and err represents the random matching error with the expected value equal to zero and standard deviation s e. In the above formula, dummy variables Z i were used to denote the value range of force P credited to a particular segment of the open polygon. Using the symbol P i to denote the force values for which pile head displacements were read, the dummy variables define the following inequalities: P 0 =0, Z i =1 for P i-1 P P i, (2) Z i =0 for P P i-1 and for P i P. 1224

3 Wyjadlowski, Bauer and Puła It is noteworthy that for open polygon segments starting in the origin of a coordinate system, the following relations occur: a 1 =0, a i =a i-1 +b i-1 dp, dp =17 kn, for i = 2, (3) For an open polygon composed of r segments, k=2r-1 parameters are determined, with n=2r test measurements for the two pairs of tested piles available. There are r-1parameters A i, and the number of parameters B i is equal to r. The values A i and B i of parameters a i and b i, as well as the approximating error standard deviation s e will be determined by regression analysis in the process of matching the open polygon to the results of pile head displacement measurements presented in Figure 1. Parameters A j and B j are determined in the process of minimising the sum of the squares of differences between the given quantities U i and the predicted quantities U i (a j,b j ): n 2 U i Uˆ i (4) i for the set of n data: (U i ), i = 1, 2,..., n. The estimation error random variable err takes the expected zero value and its standard deviation is equal to: s e n k min (5) Using a computer programme facilitates obtaining the results, but the values of parameters A i, B i and s e can be also obtained with a calculator. The calculation of the standard approximation error is simplified by using relation n-k =1. However, this relation is valid only for a set of measurement results consisting of two pairs of piles. A statistical analysis of the results was performed by using the non-linear regression NLIN2 programme, which is based on a numerical algorithm employing the so-called Marquardt s compromise (Marquardt D. W. 1963; Marquardt D. W. 1966). The goal of each of the presented result analyses was to obtain stochastic measures of the variability present in a particular set of measurement results. In this case, this measure is the random matching error err from equation (1), with the expected zero value and standard deviation s e, obtained in the process of determining the values of A i and B i. The values of parameters A i and B i received from regression model (1) are shown in Table 2. Table 2. Values A i and B i of coefficients a i and b i for open polygon segments describing the averaged (estimated) displacement depending on load force values. No i Values A i Values B i P i U(P i) estimation [kn] [cm] E E E E E E E E E E E The A i, B i parameters listed in Table 2 enable building 11 response surfaces, each composed of 11 segments associated with unique estimation errors. The loads in segment 11 range from 170 kn to infinity. These surfaces differ only in the value of the standard estimation error. For instance, response surface L10 has a standard error obtained after removing the read values for a force of 187 kn from the set of measurement data, while for surface L1, the error is determined based on displacements induced by a force of 17 kn, only. Thus, surface L1 is composed of only one segment - a straight line with a force range 0-17 kn and a standard error obtained while determining parameter B 1. The remaining 10 segments are assigned to this response surface as its extrapolation and they lie outside its validity range. The results of probabilistic analyses depend on selecting one of the eleven potential response surfaces. The absence of a criterion of choosing the appropriate response surface leads to random results of probabilistic calculations. To clarify the problem of getting random allowable force values, calculation of the forces, on the base of six response surfaces L6-L11 are carried out. As considered before, the response surface model L11 was developed taking into account all data included in Table 1 and presented on Figure 1. Parameters A i, 1225

4 Geotechnical Engineering for Infrastructure and Development B i of all eleven response surface are included in Table 2. Assuming that only the standard estimation error err is a random variable, while the applied force P and the obtained parameters A i and B i are deterministic parameters, all the random variability contained in the data set affects the expected value of this error. Table 3 demonstrates that the standard estimation error decreases almost tenfold when test readings with the greatest variation are removed from the data set. To which extend pile allowable forces depend on the adopted response surface is shown in Table 3 and on Figure 2. The allowable forces were calculated assuming their reliability index beta is equal 2.3. Table 3. Values of allowable load for reliability index beta 2.3, depending on the adopted response surface. Response L11 L10 L9 L8 L7 L6 surface Estimation error s e Force range R [kn] Allowable forces [kn] no data , Taking up randomly number of segments in a response surface (range of test load force), used to work out pile test data, produces random value of allowable force in the range kn. Comparing the force ranges R shown in the table with the allowable forces, one can observe that the values of allowable forces lie beyond the force ranges used to define response surfaces L6 and L7. Additionally, there is no allowable force value for L11. The obtained response surfaces L1-L11 could only become a basis for engineering calculations after the discussed non-objectivity (arbitrariness) is eliminated in the process of adopting the response surface for further calculations. The criterion making it possible to choose the appropriate response surface out of the possible eleven could be proposed after analysing the SORM and FORM methods (Hochenblicher 1987), which are used by the computer program Comrel (STRUREL, 2003) supporting probabilistic calculations performed in the example analyses discussed in this report. These analyses define the values of allowable loads with the assumed value of reliability index beta, satisfying the inequality of serviceability limit state: U(P, err) u all = 1 cm, (6) where U a response surface equal to one of the surfaces L1- L11, P random variable of pile head load, err - approximation error random variable. In following example calculation, it was assumed for the calculations that a pile was loaded with random force P with lognormal distribution and variation coefficient of 15%, while error err had a rectangular distribution. The computer calculation results comprise the expected force value P all and force value P * at the design point. The analysis of the SORM and FORM methods demonstrated that 'the main probability mass ' of force P all lies in the range 0 - P *. This observation makes it possible to formulate a criterion whose satisfaction will unmistakably point to the appropriate response surface chosen from the possible eleven (L1- L11). The appropriate response surface is the one that has the most segments and additionally satisfies the inequality: P * R, (7) where R is the force range used to define the standard deviation of the approximation error. Fulfilment of this criterion maximally limits the transfer of random variability from the range of high forces to that of medium and low ones. An example application of the response surfaces obtained by means of inequality (7) can be determining the bearing capacities of piles for the preset levels of probability of failure recommended by the standard (ISO 2394:1998). These recommendations are listed in Table. 4. Table 4. Recommended values of reliability index β in limit states of serviceability, C can be applied in ultimate limit states. Relative safety provision costs Damage effects Minor Noticeable Moderate Heavy Low (C) Moderate (C) High (C) For highly responsible structures designed with the use of ultimate limit states, the value of reliability index beta can be much higher (e.g. 7) 1226

5 Wyjadlowski, Bauer and Puła As mentioned before, it was assumed for the calculations that the test pile is loaded with a random force with a lognormal distribution and a variation coefficient of 15 percent. The employed criterion of choosing the appropriate response surface (8) pointed to surfaces L7 and L8, which should be used for determining the allowable forces. The results are shown in Table 5. Values of forces P all in bold are those that fulfil inequality (7). Table 5. Values of allowable forces and forces associated with them at a calculation point. β L8 L7 P all P* P all P* R=136 kn R=119 kn It should be noted that inequality (8) chooses the allowable forces in bold type from beta range 0-2.3, obtained by means of surface L8, and from range , obtained by using surface L7. The values bold marked of allowable forces make it possible to compile Table 6 for reliability indices recommended by the ISO standard. Table 6. Values of allowable forces P all, expressed in kn, determined for the reliability indices recommended in Table 4. Relative costs to provide safety Damage effects Minor Noticeable Moderate Heavy Low Moderate High The values of allowable lateral pile loads obtained in this way can be used for designing intermediate foundations. 4 CONSLUSIONS The non-linear regression method employed for load test result analysis proposes estimating approximation errors by means of one random variable with a normal distribution. The expected value of the matching error equals zero. The standard deviation value is determined in the course of the approximation process and it depends on the number of segments in the open polygon. The standard error deviation is constant in whole the load range. Unfortunately, this constancy of the standard estimation error transfers the large displacement variation recorded in the range of high loads onto low and medium load values. Decreasing this transfer of random variability from the range of high loads to low loads by removing the results for high loads from result data sets may lead to the non-objectivity of the obtained probabilistic results. Taking the above into account, the following conclusions referring to using an open polygon as a response surface could be drawn: - The problem of stochastic uncertainty transfer was minimised through introducing the criterion of selecting the appropriate number of segments in the response surface (7). - In the case of an open polygon, it is easy to introduce the selection criterion (7), compared to models based on continuous functions, - Open polygon regression models do not exhibit unphysical displacement-load relations, unlike models based on continuous functions, - All the response surface models have the same displacement-load relation. The only difference between them is the number of active (interpolation) segments for which the standard approximation error is determined. Models based on elementary continuous functions have different functions within the active (interpolation) and extrapolation range. - Despite a higher workload involved in analysing test results compared to other, e.g. parabolic models, the discussed advantages of open polygon models outweigh the drawbacks. The discussed method of allowable pile load estimation, based on load test results and using polygon line response surfaces, employed in structural reliability theory calculations, is marked by the easiness of obtaining a response surface, but due to a large number of coefficients it requires a bit more work when performing probabilistic calculations. 1227

6 Geotechnical Engineering for Infrastructure and Development REFERENCES ASTM International Standard test method for piles under lateral load. ASTM standard D (reapproved 1995). ASTM International, West Conshohocken, PA. EN Eurocode 7. Geotechnical Design. Part 1. General Rules. CEN, Brussels. Hochenbichler M., Gollwitzer S., Kruse W., Rackwitz R New light on first and second-order reliability methods, Structural Safety, Vol. 4, ISO (International Standards Organisation) General principles on reliability for structures. International Standard ISO 2394:1998. Marquardt D.W NLIN2 for least-squares estimation of nonlinear parameters computer code. Distribution no , IBM Share Library, 77 p. Marquardt D. W An algorithm for least-squares estimation of non-linear parameters, J. Soc. Indust. Appl. Math, 11, No. 2, June Polish Standard PN-B 02482:1983 Bearing capacity of piles and pile foundations. STRUREL A structural reliability analysis program system COMRREL&SYSTREL: Users Manual, RCP Consult, Germany. 1228