ON RELIABILITY BASED PERFORMANCE DESIGN OF PILE FOUNDATIONS

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1 8 th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability PMC ON RELIABILITY BASED PERFORMANCE DESIGN OF PILE FOUNDATIONS Y. Honjo Gifu University, Gifu, Japan, M. Suzuki Izumi Research Institute, Shimizu Corporation, Chiyoda-ku, Tokyo, Japan, Abstract Performance design concept which describes the target performances of a structure for various load frequency is coming to be popular worldwide. So far, however, this method is not closely related to reliability concept, especially in the area of geotechnical engineering and foundation engineering. In this paper, taking a pile foundation as an example, an attempt is made to put together the reliability design methodology and the performance based design concept. First an performance matrix for a pile foundation is proposed: there are four factors which need to be determined to obtain a reasonable performance matrix of a structure. Namely, load frequency in given structure design life, structure performance (i.e., limit states), importance level of structure and probability of attaining each limit state under given conditions. Reliability analysis is carried out for two actual pile foundation cases. Problems are identified in processing such performance based design on pile foundations. Introduction The Vision 2000 (SEAOC, 1995) which is considered to be a monumental document to introduce the performance based design (PBD) concept to seismic design of building structures introduced the performance matrix. In this matrix, the seismic performances of a building are described based on the performance of the structure and the seismic load frequency. One of the aims of introducing this matrix was to improve the communications between the building owners and the designers so that they can start from the same understanding on the seismic performance of buildings. The concept arose from the problems felt after the consequences of the major two earthquakes in California, Loma Prieta in 1989 and Northridge in It was found that there had been considerable gap in understanding on the seismic design for the owners and the designers (Hamburger, 1997). However, it is also coming to be known that the information provided by the performance matrix is not sufficient (Ellingwood, 1998). One of the main drawbacks is that performance defined in the matrix is not 100% attainable in reliability sense. It is necessary to put additional information to the performance matrix, and that is proposed in this study. The final aim of this study is to propose Level I limit state design format for pile foundation design based on the performance design concept, which implies the determination of partial factors for the forces and the resistances. The procedure taken in this study is as follows: 1

2 1. Define the required performance levels of a pile foundation. 2. Specify the design calculation model and procedure. 3. Reliability analysis is performed for the existing structure so as to obtain target reliability level. 4. Code calibration procedure is taken to obtain appropriate partial factors for Level I format. 5. Reflections on further study are made. Reliability based performance matrix and reliability analysis The reliability based performance matrix need to be based on four factors, namely structure performance level which can be defined by limit states, load levels and their frequency distribution, importance of structure under consideration, and probability of attaining the performance levels under given conditions. The matrix is conceptually different form the one proposed in Vision 2000 (SEAOC 1995) where only performance levels, various load magnitudes and importance of structures were taken into account. An example of such performance matrix is illustrated in Table. 1. Table 1. Reliability based performance matrix for a structure of given importance level External Load Limit States Serviceability Limit State Repairable Limit State Ultimate Limit State Frequency distribution for 50 years maximum seismic load β T ( serv.) β T ( rep.) β T ( ult.) Limit State Performance level of structures are defined by the limit states in this study. The wordings which directly describe performances of a structure, e.g., life safe limit state, are not adopted, but definitions describe states of the structure during/after loadings are employed. Thus, the traditional wordings presented in Table 1 are used in this study. Uncertainty in Seismic Load The seismic force is the major target load in this study. Since a pushover analysis is performed, the horizontal load and the moment induced by it applied to pile foundations are the major concern here. The load frequency distribution is taken from a draft of A Guidelines of Limit State Design of Buildings prepared by Architectural Institute of Japan (AIJ, 1999), which gives the frequency distribution of the 50 years maximum seismic force on building foundations. It is claimed in the report that the distribution follows the lognormal distribution. It is assumed that the seismic force employed in the design based on current Highway Bridge Standard Specification (JRA, 1996) in Japan be equivalent to the mean of the 50 year maximum seismic force, and the coefficient of variation is set to

3 Uncertainty in Pile Resistance The resistance of pile foundation is calculated based on a rigid frame-subgrade reaction model which is conceptually presented in Fig. 1. One of the model cases employed in this study is presented in Fig. 2 for comparison. Fig. 1 Rigid frame-subgrade reaction model Fig. 2. An example of pile foundation The vertical force as well as horizontal and moment forces induced by a seismic motion are applied at the top center of the foundation. Both vertical and horizontal subgrade reactions are mobilized to account for the applied forces. The springs are estimated based on the subgrade reaction coefficients assigned in each part of the pile. The springs are of the bilinear type, which generate reactions to a defined value, then yields and would not sustain additional force. Uncertainties concerning calculated resistance of pile foundations are evaluated based on the research done by Okahara et al.. This research assesses the uncertainties of various calculated resistance values of piles by comparing them to the pile loading tests results. The values adopted in this study are tabulated in Table 2. Reliability Analysis The reliability method employed in this study is advance FORM, and Hasofer-Lind reliability index, β, is calculated to evaluate the performances (Thoft-Christensen and Baker, 1982). The sensitivity factors, α s, also play important role in the analysis and the discussions. Results of the analysis and the code calibration Description of the case Two actual design of highway bridge pire pile foundations are considered. The sections and plan of the first case (Case 1) is presented in Fig. 2. In this case, the foundation consists of 12 piles each of which has diameter 1m and length 23m. These are cast in place concrete piles with EI= tf/m 3 and EA= tf. The pile is installed in a gravelly sand layer with SPT N value of 36 which are covered by soft stratified layers of known properties. 3

4 Table 2. Input values for reliability analysis on resistance side factors Notation Definition Mean c.o.v. Comments δ kv Vertical spring constant at pile top Based on Okahara et al. δ kh.horizontal subgrade reaction coefficient estimated by N Based on Okahara et al. δ f Circumference friction resistance: f=0.4n for sand and f=n for clay Based on Okahara et al. Pile top bearing resistance qd=10n δ qd (<300) Based on Okahara et al. δ PHU Passive earth pressure Based on the estimation by the authors δ N Variation of N averaged over pile length Seismic coefficient Based on Vanmarcke (1977) (Note) Upper row: values for cast in place piles, Lower row: values for steel driven piles. N: SPT N-value Results of the pushover analysis The result of the pushover analysis of the first case is presented in Fig. 3. To the horizontal displacement of 3.5cm, the force-displacement curve keeps almost linear. At 3.5cm, i.e., the horizontal seismic force of about 0.5, all the vertical springs on the left most pile row yield, and the pile row is pulled out. Thereafter, at the horizontal displacement of 6.5cm, all the vertical spring, including one at the pile top, yields, and this pile row is actually in the failure condition. After this point, the foundation cannot sustain any additional horizontal load. This type of behavior is known to be common for most of the pushover analysis on pile foundations. It may be worse while to know that the design seismic force in the highway bridge standard specification (JRA, 1996) for this case is This was assumed to be the mean of the 50 year maximum seismic force in this study as mentioned previously. Results and discussions on serviceability limit state In this study, a pile foundation is considered to reach the serviceability limit state when the horizontal displacement at top center of the foundation reaches 1% of diameter of pile in case of concrete piles, or 5% in Horizontal Displacement (m) Fig. 3. A pushover curve: horizontal seismic coefficient vs. horizontal displacement steel piles. These are the quantities proposed in Okahara et al., which would not course any damage to the piles. 4

5 In Table 3, the obtained reliability indices and the sensitivity factors are presented for Case 1. It is understood from this table that α s for horizontal and vertical spring constants are lager, whereas those are smaller for the limiting values of side frictions and top bearing capacity. This fact implies that the servceability limit state is more influenced by factors controlling the stiffness characteristics of pile foundation rather than the strength characteristics. The obtained safety index distributed between -0.9 to 0.0 for the all the cases analyzed. Therefore, it is expected that the serviceability limit state is quite certainly reached more than once during 50 years of reference period of the pile foundation. Table 3. Results of the reliability analysis: safety index and sensitivity factors Serviceability Limit State β= 0.89 Repairable Limit State β= 1.64 Variables α Variables α SPT N value(1st layer) E-02 SPT N value(1st layer) E-02 SPT N value(2nd layer) E+00 SPT N value(2nd layer) E-11 SPT N value(3rd layer) E-01 SPT N value(3rd layer) E-02 SPT N value(4th layer) E-02 SPT N value(4th layer) E-02 SPT N value(5th layer) E-04 SPT N value(5th layer) E-02 SPT N value(6th layer) E-05 SPT N value(6th layer) E-01 SPT N value(7th layer) E+00 SPT N value(7th layer) E-11 Cohesion (3rd layer) E-08 Cohesion (3rd layer) E-02 Cohesion (4th layer) E+00 Cohesion (4th layer) E+00 Cohesion (6th layer) E+00 Cohesion (6th layer) E+00 δ kv E+00 δ kv E-01 δ kh E+00 δ kh E-01 δ f E+00 δ f E+00 δ qd E-11 δ qd E-01 δ PHU δ E-03 δ PHU E-03 hori. seismic force E+00 hori. seismic force E+00 Results and discussions on repairable limit state The repairable limit state is assumed to be reached when the front pile row is pulled into the ground, i.e. loss of the vertical bearing resistance of the front pile row. It is observed from Table 3 that α of the side friction resistance is the most influential for δ 5

6 this limit state. This is because the side friction controls both pull out and push in of the back and the front pile rows, whereas the pile tip bearing resistance only affects the push in of the front piles. It is also understood that both the horizontal stiffness and the yielding limit do not affect the repairable limit state very much. The safety index obtained for this limit sate is about 1.6 in the all cases. Code calibrations Calibrations of partial factors are attempted in the limited cases. The results will be presented at the time of the conference. Conclusions A proposal is made which combines the performance design concept with the limit state design approach. A pile foundation design is taken as an example, and reliability analysis is carried out to illustrate the idea. Although not discussed in details due to the limitation of the space, some issues are identified by doing this excise. Further research on the topic is underway, which is expected to give more thorough results on the topic. Acknowledgments The authors acknowledge the effort of Mr. Naruhito Funazu of Emori Corporation, who has carried out most of the calculation presented here. References SEAOC (1995), Vision A Framework for Performance-based Design of buildings (4 volumes), Structural Engineering Association of California. Sacramento, CA, USA. Ellingwood, B. (1998), Reliability-based performance concept for building construction, Proc. Structural Engineers World Congress, San Francisco, CA, USA. Hamburger, R.O. (1997), The development of performance-based structural design in the United States of America, Proc. Int. Workshop on Harmonization in Performance Based Building Structure Design in Countries surround Pacific Ocean, Tsukuba, Japan JRA (Japan Road Association) (1996), Design Specification of Highway Bridges, Part IV Foundation Design. RTRI (Railway Technical Research Institute) (1997), Railway Structure Design Standard (in Japanese). AIJ (Architectural Institute of Japan) (1999), A draft of a guide for limit state design of buildings. Thoft-Christensen, P., and M.J. Baker (1982), Structural Reliability Theory and Its Applications, Springer- Verlag. Okahara, M., A. Takagi,M. Nakatani, and Y. Kimura, A study on bearing capacity of a single pile and design method of column shaped structures, Research Report of Public Works Research Institute No.2919 (in Japanese). Honjo, Y., H. Ishihara, M. Suzuki, K. Matsui, and Y. Kimura (1999), A basic study on reliability based performance design of pile foundation, Proc. ICASP 8,