PROBABILITY AND SENSITIVITY ANALYSIS OF SOIL … limit state. The loading effects are compared here ... (Ellingwood, B., et al., 1980; Rosovsky, D. V., 1995; Sadovský, Z., 2002)

INTRODUCTION

During the structural design process, an engineer has to consider problems of the safety, reliability and durability of a single structural element as well as the entire structure from the point of view of its planned life cycle. The reliability of structural elements and systems is analyzed using two limit states – the ultimate limit state and the serviceability limit state. The loading effects are compared here with the effects of the resistance of the structure with respect to the environmental effects. Randomness in the loading and the environmental effects, the variability of the material and geometric characteristics of structures and many other “uncertainties” affecting errors in the computing model lead to a situation where the actual behavior of a structure is different from the modeled one. Recent advances and the general accessibility of information technologies and computing techniques give rise to assumptions

concerning the wider use of the probabilistic assessment of the reliability of structures through the use of simulation methods (Marek, P., Brozzetti, J., Guštar, M., 2001; Janas, P., Krejsa, M., 2002; Soares, R. C., et al., 2002; Teplý, B., Novák, D., 2004; Králik, J., Králik, J., 2006; Králik, J., Kosa, T., 2006; Králik, J., Varga, T., 2006). Much attention should be paid to using the probabilistic approach in an analysis of the reliability of structures (Ellingwood, B., et al., 1980; Rosovsky, D. V., 1995; Sadovský, Z., 2002).Most problems concerning the reliability of building structures are defined today as a comparison of two stochastic values, loading effects E and the resistance R, depending on the variable material and geometric characteristics of the structural element. The variability of those parameters is characterized by the corresponding functions of the probability density fR(r) and fE(e). In the case of a deterministic approach to a design, the deterministic (nominal) attributes of those parameters Rd and Ed are compared.

J. KRÁLIK, J. KRÁLIK, Jr.

PROBABILITY AND SENSITIVITY ANALYSIS OF SOIL-STRUCTURE INTERACTION OF HIGH-RISE BUILDINGS

KEY WORDS

• Probability, • sensitivity, • soil-structure interaction, • high-rise building, • finite element method, • Monte Carlo, • ANSYS

ABSTRACT

This paper presents experience gained from a probabilistic and sensitivity analysis of soil-structure interaction in the case of high-rise buildings with the Response Surface Method using the ANSYS program. The advantages and disadvantages of the methods Monte Carlo, Importance Sampling, Latin Hypercube Sampling and Response Surface Method are discussed. The possibilities of utilizing the RSM method to analyze the extensive and complicated tasks in Finite Element Method are presented in the case of the CBC high-rise building in Bratislava. This analysis was made based on the design of the optimal system of the structural elements considering the soil-structure interaction. The soil conditions were complicated by aggressive groundwater and problematic ground layers. In accordance with these results, the modification of the structural systems was realized. These analyses were made using the ANSYS program.

Doc. Ing. Juraj Králik, PhD., Head of Department of Structural MechanicsResearch field: Statics and Dynamics, Safety and Reliability of StructuresSlovak University of Technology in BratislavaFaculty of Civil Engineering, Radlinského 11, 813 68, Bratislava, e-mail: [email protected]

Ing. Juraj KrálikDoctoral student of Department of Structural MechanicsResearch field: Soil –Structure InteractionSlovak University of Technology in Bratislava, Faculty of Civil Engineering, Radlinského 11, 813 68, Bratislava, e-mail: [email protected]

2006 SLOVAK UNIVERSITY OF TECHNOLOGY

2006/3 – 4 PAGES 23 – 32 RECEIVED 18. 9. 2006 ACCEPTED 27. 11. 2006

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The deterministic definition of the reliability condition has the form

Rd ≥ Ed (1)

and in the case of the probabilistic approach, it has the form

RF = R – E ≥ 0 (2)

where RF is the reliability function, which can generally be expressed as a function of the stochastic parameters X1, X2 to Xn, used in the calculation of R and E.

RF = g(X1, X2, ..., Xn) (3)

The failure function g(X) represents the condition (reserve) of the reliability, which can either be an explicit or implicit function of the stochastic parameters and can be single (defined on one cross-section) or complex (defined on several cross-sections, e.g., on a complex finite element model).The most general form of the probabilistic reliability condition is given as follows:

pf = P(R – E < 0) ≡ P(RF < 0) < pd (4)

where pd is the so-called design (”allowed” or ”acceptable”) value of the probability of failure. From the analytic formulation of the probability density by the functions fR(r) and fE(e) and the corresponding distribution functions ΦR(x) and ΦE(x), the probability of failure can be defined in the general form:

(5)

This integral can be solved analytically only for simple cases; in a general case it should be solved using numerical integration methods after discretization.Together with pf , the index of reliability β is often used for analyzing the degree of reliability. It is defined using the linearization condition of the failure function g(X). In the case of the normal distribution of this function it is valid:

(6)

where µRF and σRF are the mean value and the standard deviation of the reliability function, respectively.

RELIABILITY ANALYSIS METHODS

From the point of view of one’s approach to the values considered, structural reliability analyses can be classified in two categories, i.e., deterministic analyses and stochastic analyses. In the case of the stochastic approach, various forms of analyses (statistical analysis, sensitivity analysis, probabilistic analysis) can be performed. Considering the probabilistic procedures, Eurocode 1 recommends a 3-level reliability analysis. The reliability assessment criteria according to the reliability index are defined here. Most of these methods are based on the integration of Monte Carlo (MC) simulations. Three categories of methods have been presently realized: Direct methods (Importance Sampling - IS, Adaptive Sampling

- AS, Direct Sampling - DS) Modified methods (Conditional, Latin Hypercube Sampling

- LHS) Approximation methods (Response Surface Method - RSM)

A) Direct Monte Carlo methods are based on a simulation of the input stochastic parameters according to the expected probability distribution. The accuracy of this method depends upon the number of simulations and is expressed by the variational index:

(7)

where N is the number of simulations. If the required probability of failure is pf = 10-4, then by the number of simulations N=106, the variational index is equal to 10%, which is an acceptable degree of accuracy.Advantages of the method:• the final values of the reliability reserves can be continuously

displayed in the form of a histogram or cumulative function; the simulations are independent,

• the method is easily understandable and transparent,• the method enables the estimation of the statistical discrepancy of

the estimation on a particular relevance level.Disadvantages of the method:• large number of simulations for small probability values,• slow calculations of complex problems (for Finite Element

Method models, the calculations are expensive and ineffective).

B) The Modified LHS method is based on the same number of simulations of the function g(X) as in the Monte Carlo method; however the zone of the distributive function Φ(Xj) is divided into N intervals with identical degrees of probability. This method provides good assessments of the statistical parameters of the structural

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25PROBABILITY AND SENSITIVITY ANALYSIS OF SOIL-STRUCTURE INTERACTION ...

response when compared to the Monte Carlo method. Using the LHS strategy, we get values like the reliability reserve parameter – the mean value , the standard deviation σz, the slant index αz, the sharpness index ez, or the empirical cumulative distribution function. The reduction of the number of simulations (tens to hundreds of simulations) means a valuable benefit from this method compared to the direct Monte Carlo method (thousands to millions of simulations).

C) Approximation methods - Response Surface Methods are based on the assumption that it is possible to define the dependency between the variable input and the output data through the approximation functions in the following form:

(8)

where co is the index of the constant member; ci are the indices of the linear member and cij the indices of the quadratic member, which are given for predetermined schemes for the optimal distribution of the variables (Montgomery, Myers) or for using regression analysis after calculating the response (Neter). Approximate polynomial coefficients are given from the condition of the error minimum, usually by the “Central Composite Design Sampling” (CCD) method or the “Box-Behnken Matrix Sampling” (BBM) method.

Advantages of the method:• considerably less number of simulations than with the direct

Monte Carlo method,• it is possible to define dependencies using the "design experiments"

method or regression analysis from the defined points in the case of improper approximation functions,

• Particular simulations are independent from each other – parallel calculations can be used here.

Disadvantages of the method:• The number of simulations depends on the number of variable

input parameters; in the case of a large number of input parameters, the method is ineffective,

• the method is unsuitable in the case of discontinuous changes in the dependencies between the input and output values (e.g., the method is not suitable for resolving the stability of ideal elasto-plastic materials beyond the failure limit...).

The ANSYS Program belongs among the complex programs for solving potential problems. It contains a postprocessor, which enables the execution of the probabilistic analysis of structures. In Fig. 2, the procedural diagram sequence is presented from the structure of the model through the calculations, up to an evaluation of the probability of structural failure. The ANSYS postprocessor enables the modeling of a structure as a solid body having a general shape (solid modeling), using Boolean operations, general spline planes (non-uniform rational B-splines), automated meshing and adaptive meshing.

Fig. 1 Distribution schemes of the stochastic numbers of the RSM method for three input variables

a) Central Composite Design b) Box-Behnken Matrix

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The postprocessor enables the displaying of numerical results and, using APDL language (ANSYS Parametric Design Language), the compiling of the numerical results obtained. The postprocessor for the probabilistic design of structures enables the definition of random variables using standard distribution functions (normal, lognormal, exponential, beta, gamma, weibull, etc.), or externally (user-defined sampling) using other statistical programs like AntHILL or FREET. The probabilistic calculation procedures are based on Monte Carlo simulations (DS, LHS, user-defined sampling) and „Response Surface Analysis Methods (RSM)“ (CCD, BBM, user-defined sampling). The statistical postprocessor compiles the results numerically and graphically in the form of histograms and cumulative distributional functions. The sensibility postprocessor processes the data numerically and graphically and provides information about the sensitivity of the variables and about the correlation matrices.

CBC HIGH-RISE BUILDING IN BRATISLAVA

During the process of the structural design of the CBC high-rise building in Bratislava, several problems occurred concerning the complicated hydrogeological conditions and uncertainties in the geometric and physical characteristics of the subgrade layers under the given building as well as their unfavorable effect upon stress of the main structural elements, especially the base slab. The CBC office buildings Nos. 1 and 2 are situated on Karadžičova Street in Bratislava between the building of the VUB Bank headquarters and an unused civil defense shelter. The No. 1 CBC

Tab. 1 Summary of the stochastic numbers depending on the number of stochastic input variablesCentral Composite Design Box-Behnken Design

Number of stochastic variables

Number of quadratic function coefficients

Number of factorials f

Number of stochastic numbers

Number of quadratic function coefficients

Number of stochastic numbers

1 3 N/A N/A - N/A2 6 0 9 6 N/A3 10 0 15 10 124 15 0 25 15 255 21 1 27 21 416 28 1 45 28 497 36 1 79 36 578 45 2 81 45 659 55 2 147 55 12110 66 3 149 66 161

ANSYSdatabase

„File.DB“

PROBdatabase

„File.PDS“

PROBProb.command„File.PDAN“

PROBCycle command

„File.LOOP“

Modeldatabase

Probabilitydatabase

SAVE RESUME

ANSYS

/EXIT

/CLEAR

PDSAVE PDRESU

PDEXE PDEXE

Fig. 2 Procedural diagram of probabilistic calculations using the ANSYS software system

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high-rise building has groundplan proportions of 40 x 22 m and height 98.3 m. The building has three underground floors with a base gap on a level of -11.1 m.The structure rests on a monolithic base slab with a thickness of 1.9 m. The foundation pit was stabilized with a reinforced concrete bath realized by Keller pile systems, and the ground properties were improved. The structural system of CBC building No 1 is designed as a combined system consisting of two reinforced-concrete monolithic cores and a system of columns and concrete plates. In the bottom part of the building, on its edge, reinforced-concrete walls are designed along three sides of the building.

When comparing the various geological profiles, one can see that northward from the VUB Bank building, the amount of fine-grained deposits decreases in favour of sand. Hence, two geotechnical models were designed, to respect the geological evolution as well as the dominant service load. Model No. 1 considers the relief from the VS-1 – VS-2 – VS-3 bores and model No.2, the relief from the VS-9 – VS-10 bores (Fig.6).In order to calculate the dimensioning forces on the structural elements from the static loading, we start from the mean values of the mechanical soil properties defined in VS1 to VS3 and VS9 to VS10 bores (Tab.3). The improvement of the subgrade by the Keller grouting system was realized under the building (Tab.4), whereby the preferable values of the physical and mechanical properties of the sub-soil were achieved.

Tab. 2 Mechanical properties of the soil layers after the reduction of the number of layers

Point Z hi γ Edef ν Number of

material [m] [m] [kNm-3] [kPa]

1 -1.97 1.97 26.25 155,625 0.26 112 -3.36 1.39 25.30 11,314 0.49 123 -11.10 7.74 25.98 9452 0.51 1323 -13.80 2.70 23.05 15,331 0.43 1424 -17.30 3.50 22.59 17,810 0.42 1525 -18.30 1.00 23.97 12,729 0.46 167 -30.60 12.30 19.00 10,466 0.38 178 -35.10 4.50 19.00 18,692 0.35 189 -37.80 2.70 20.00 102,383 0.2 22

Fig. 3 CBC high-rise building and its calculation model

Fig. 4 Calculation models of the CBC building with subsoil and embedded concrete walls

Fig. 5 Calculation models of the CBC building with subsoil and embedded concrete walls

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LOADING AND LOAD COMBINATION

The loading and load combination in the case of the deterministic assessment of the ultimate limit state of the structure is considered according to STN 73 0035 (STN P ENV 1991-1), part 9.4.2, as follows:A) Permanent and temporary design situations

(9)

B) Seismic design situation

(10)

where Gkj is the characteristic value of the permanent loads, Pk - the characteristic value of the prestress loading, Qk1 - the characteristic value of the dominant variable loading, Qki - the characteristic value of the i-variable loading, AEd - the design value of the seismic loading, γGj - the partial coefficient for the permanent load, γP – the value coefficient for the prestress loading, γQi – the partial coefficient for the varying load i, γ1 – the value coefficient (of the building structure), ψ - the combination coefficients (according to part.9.4.4, STN 730035) according to Tab.4.If the sensitivity of the structure affects the uncertainties in the subgrade’s stiffness, by using the deterministic approach, one can consider the alternative values of stiffness ko,min, ko,mean, ko,max (where ko,mean/ko,min ≅ ko,max/ko,mean ≅ 1.5) and solve the effect of the loads on several models from the defined load combinations. However, in these cases it is often difficult to analyze the effects of the individual factors upon the structural strain for more sophisticated structures. Therefore, in practice, we usually start from the vabsolute extremes of such solutions, and no special attention is paid to an analysis of the input values of the uncertainty effects.

VARIABILITY OF THE INPUT PARAMETERS

The uncertainty of loading, the stiffness of the sub-soil and the resistance of the structure can be effectively analyzed by a sensitivity analysis in the case of a probabilistic approach. Previously, the large number of necessary simulations as well as the high demands for the time and capacity of computers represented serious handicaps for using the direct Monte Carlo method. As long as the dependencies between the monitored values can be expressed by a continuous function, it is appropriate to use the RSM method for large-scale problems.In the probabilistic and sensitivity analyses of the CBC high-rise building, the uncertainties of the loading and resistance are expressed depending on the characteristic values of the input values and the variable parameters based on the recommendations of JCSS [3] and ASCE 7/95 [1] (Tab.5).The variable values of the ground’s stiffness come from geological survey results at the VS1 to VS3 and VS9 to VS10 bores and the given relief under the base slab as well as from the design of the ground improvement using the KELLER system (kz,k =1.5MPa/m). The variability of the subgrade’s stiffness on a horizontal plane is expressed through the variable values of the global rotations kxx_var, kyy_var of the building as a solid body. The uncertainties of the calculation model were taken into account by the various parameters of the model’s uncertainties θR and loading effects θE (µ = 1, σ = 0.1) according to a Gaussian normal distribution.

Tab. 3 Mechanical properties of the soil layers after improving the ground

Point Z hi γ Edef ν Number of

material [m] [m] [kNm-3] [kPa]

1 -1.97 1.97 26.3 155,625 0.3 112 -3.36 1.39 25.3 11,314 0.5 123 -11.10 7.74 26.0 9452 0.5 134 -13.80 2.70 20.0 75,735 0.2 195 -17.30 3.50 19.0 33,748 0.4 206 -18.30 1.00 25.0 900,000 0.2 217 -30.60 12.30 19.0 10,466 0.4 178 -35.10 4.50 19.0 18,692 0.4 189 -37.80 2.70 20.00 102,383 0.2 22

Fig. 6 Geotechical investigation of the locality

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Tab. 4 Probabilistic model of the principal values

Characteristic

Subsoil Load

Stiffness characteristics Permanent Variables

direction Z around X around Y Dead load Live load Wind

Characteristic value kz,k kxx,k kyy,k Gk Qk1 Wk

Variable kz_var kxx_var kyy_var G_var Q_var W_var

Type of histogram Normal Normal Normal Normal Beta (Typ I) Beta (Typ I)

Mean value 1 1 1 1 0.57 0.67

Direction of deviation 0.200 0.033 0.033 0.033 0.28 0.14

Min. value 0.148 0.851 0.853 0.569 0.858 0.40

Max. value 1.867 1.163 1.135 1.385 1.127 1.20

Tab. 5 Comparison of the foundation slab’s strain depending on various calculation models

Model Extreme bending moment on base slab [kNm]Number of

cyclesCPU

Min Max Mean St.dev [sec]

Deterministic

CBC2-CBC9 -12,316 -4974 - - - -

Probabilistic

CBC9p -16,404 -10,782 -13,563 651 15 5175

CBC9p2 -15,313 -4419 -9756 1132 81 27,855

CBC9L -11,664 -8044 -9761 769 100 34,385

Fig. 7 Distribution of points of the RSM shape for the moment M_max and displacement D_max

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The probabilistic calculation was realized on the three CBC9P models (five variable parameters kz, kxx, kyy, θR, θE), CBC9P2 and CBC9L (eight variable parameters kz, kxx, kyy, Gk, Qk, Wk, θE, θR). The output parameters were resolved by 106 simulations using the Monte Carlo method. The CBC9P and CBC9P2 models were resolved using the RSM method, and the CBC9L model was resolved using the LHS method for 100 simulations.In Figs. 7 and 8, there the distribution of the points of uncertainty for the parameters M_max (bending moment) and D_max (max. displacement) are displayed depending on the input parameters, which were obtained using the RSM method. From these discrete points, the development of the uncertain parameters is made, based on the relation (8) using the Monte Carlo simulations. The sensitivity analysis is based on a matrix of the correlation coefficients of the input and output stochastic parameters. The probability of failure

or the reliability of the structure is obtained from the probability density of the failure functions, or reliability (Fig. 9). The results of the calculations, which are displayed in Tab.5 and Figs. 8 and 9 show that the RSM method is more conservative than the LHS method. On the other hand, the RSM method is more effective with a smaller number of variable parameters than the LHS method.

CONCLUSION

The advantages and disadvantages of the methods using the probabilistic approach for analyses of robust structures using FEM were presented in the paper. The effectiveness of the RSM method in combination with the Monte Carlo simulations is shown on the

Fig. 8 Sensitivity analysis of the foundation plate’s bending resistance on the CBC9P2 and CBC9L models

Fig. 9 Sensitivity analysis of the maximal horizontal displacement on the CBC9P2 and CBC9L models

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example of the sensitivity analysis of the CBC high-rise building in Bratislava, and on the uncertainties in the subgrade’s stiffness and loading values, using the ANSYS software system. It should be mentioned that the RSM method is preferable to the LHS method only if the number of variables is less than 8; otherwise, the benefits of the RSM method are lost. Moreover users should be warned of the limitations in using the method (mentioned in the paper), i.e., that it is only suitable for solving problems with

continuous bonds among the variable parameters (not suitable for stability problems…).

ACKNOWLEDGEMENT

The project was realized with the financial support of the Grant Agency of the Slovak Republic (VEGA). The project registration number is VEGA 1/2136/05.

Fig. 10 Probability density of the reliability function of the foundation plate’s failure on the CBC9P2 and CBC9L models

REFERENCES

• Ellingwood, B., Galambos, T.B., MacGregor, J.C., and Cornell, C.A. (1980) Development of a Probability Based Load Criterion for American National Standard A58, NBS Special Publication SP577, National Bureau of Standards, Washington, D.C.

• Janas, P., Krejsa, M. (2002) Numerical probabilistic calculation with cut off histograms, In: Third international conference on Reliability of structures, 2002, pp. 33-38, ISBN 80-02—01489-8.

• JCSS-OSTL/DIA/VROU-10-11-2000, Probabilistic Model Code, Part 1, Basis of Design, Working material, http://www.jcss.ethz.ch/, 2001.

• Kohnke, P. (2005) ANSYS, Inc., Theory, Theory Manual, 001369, Twelfth Edition. SAS IP, Inc. 2005.

• Králik, J., Varga, T. (2006) Deterministic and Probability

Analysis of the Fire Resistance of Steel Portal Frames with Tapered Members. In: Proceedings of the European Safety and Reliability Conference 2006, ESREL 2006, Safety and Reliability for Managing Risk. Estoril, Portugal, 18-22 Sept. 2006. Balkema, Taylor & Francis Group, London, pp. 2081-2086. ISBN 978-0-415-42315-1.

• Králik, J., Kosa,T. (2006) Pravdepodobnostná a citlivostná analýza seizmickej odolnosti výškovej budovy (Probability and Sensitivity Analysis of the Seismic Resistance of High-Rise Buildings), In: 5th International Conference on New Trends in the Statics and Dynamics of Buildings. SSM SAS Bratislava, STU Bratislava, Slovakia, October 19-20, 2006. pp. 267-271, ISBN 80-227-2479-3. (In Slovak)

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REFERENCES

• Králik, J., Králik, J. (2006) Pravdepodobnostná a citlivostná analýza seizmickej odolnosti konštrukcií výškových budov metódou RSM. (Probability and Sensitivity Analysis of the Seismic Resistance of High-Rise Buildings using the RSM Method) In: 2nd Conference PPK2006. FS VUT Brno October 3 – 4, 2006. pp. 37-46. ISBN 80-214-3251-9. (In Slovak)

• Marek,P.-Brozzetti,J.-Guštar,M. (2001) Probabilistic Assessment of Structures using Monte Carlo Simulations, Background, Exercises and Software, ITAM Academy of Sciences of the Czech Republic, GLOS s.r.o., Semily Czech Republic, Prague 2001.

• Mrázik, A. (1986) Teória spoľahlivosti oceľových konštrukcií, (Reliability of steel structures), VEDA SAV Bratislava, 1986. (In Slovak)

• Rosovsky, D.V. (1995) Structural Reliability. Part of publication W.Chen, The Civil Engineering Handbook, CRC Press, W.F.Chen Ed. 1995, Clemson University, SC.

• Sadovský Z. (2002) Probability based design codes in Slovakia: historical development and new challenges. JCSS workshop on Reliability Based Code Calibration.

• Saito, T. et al. (1997) Seismic damage analysis of reinforced concrete buildings based on statistics of structural lateral resistance. Structural Safety, Vol.19, No.1, pp.141-151.

• Soares, R.C. et al. (2002) Reliability analysis of non-linear reinforced concrete frames using the response surface method. Journal of Reliability Engineering &System Safety, vol. 75, pp. 1-16.

• Teplý, B., Novák, D. (2004) Spolehlivost stavebných konstrukcí. Teórie, numerické metódy, navrhování, software (Reliability of Building Structures. Theories, numerical methods, design, software). University textbook. CERM, s.r.o. Brno, 2004. (In Czech).

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Documents

PROBABILITY AND SENSITIVITY ANALYSIS OF SOIL … limit state. The loading effects are compared here ... (Ellingwood, B., et al., 1980; Rosovsky, D. V., 1995; Sadovský, Z., 2002)