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Konference ANSYS 2009

Comparison of Probabilistic Methods to Solve theReliability of Structures in ANSYS

J. Krlik

Faculty of Civil Engineering, Slovak University of Technology in Bratislava

Abstract: This paper describes some experiences from the deterministic and probabilistic analysis

of the reliability and safety of the building structure. There are presented the methods and

requirements of Eurocode EN 1990, standard ISO 2394 and JCSS. On the example of the

probability analysis of the reliability of the high rise building is demonstrated the affectivity of

the probability design of structures using SFEM. The influence of the various input parameters

(material, geometry, soil, masses,) is considered. The deterministic and probability analysis of

the seismic resistance of the structure was calculated in the ANSYS program.

Keywords: Probability, sensitivity, high rise building, earthquake, Eurocode, ANSYS.

1. Introduction

Recent advances and the general accessibility of information technologies and computingtechniques give rise to assumptions concerning the wider use of the probabilistic assessment of thereliability of structures through the use of simulation methods [5], [6], [7], [9], [10], [11], [12],

[13] and [14]. Much attention should be paid to using the probabilistic approach in an analysis ofthe reliability of structures [1], [8], [15], [16] and [17].

Most problems concerning thereliability of building structures aredefined today as a comparison oftwo stochastic values, loading effects

Eand the resistanceR, depending onthe variable material and geometriccharacteristics of the structuralelement. The variability of thoseparameters is characterized by thecorresponding functions of theprobability densityf

r(x) and f

ee(x). In

the case of a deterministic approachto a design, the deterministic Fig.1. Traditional reliability condition(nominal) attributes of those parametersRdandEdare compared.

The deterministic definition of the reliability condition has the form

d dR E (1)


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TechSoft Engineering & SVS FEM

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

0RF R E = (2)

where RF is the reliability function, which can be expressed generally as a function of thestochastic parametersX1, X2 toXn, used in the calculation ofR andE.

1 2( , ,..., )nRF g X X X = (3)

The failure function g(X) represents the condition (reserve) of the reliability, which can either bean explicit or implicit function of the stochastic parameters and can be single (defined on onecross-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:

( 0) ( 0)f dp P R E P RF p= < < < (4)

wherepdis 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(x) and fE(x) and thecorresponding distribution functions R(x) and E(x), the probability of failure can be defined inthe general form:

( ) ( ) ( ) ( )f f E R E Rp dp f x x dx x f x dx

= = = (5)

This integral can be solved analytically only for simple cases; in a general case it should be solvedusing numerical integration methods after discretization.

The index of reliability is used to define reliability of the structures on the base of the linearisedfailure function g(X). In the case of thenormal (e.g. lognormal) distribution we have following

RF

RF

= , (6)

whereRFand RFare the mean values and the standard deviation of the reliability in the form

RF R E = , 2 2 2

RF R E = + (7)

2. Types and Methods of The Reliability Analysis

The reliability analyses of the structures are differentiated from the point of view of designquantities as the deterministic and stochastic analyses [13]. On the base of stochastic methodologythe following analyses can be realized:

Stochastic analysis the mean values and the standard deviation of the variablequantities are calculated analytically or numerically using Monte Carlo simulation, whichgives us the more accurate results than deterministic values,


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Sensitivity analysis the dominant impact to the output quantities is calculated using thesensitivity analysis,

Probabilistic analysis the probability of the failure is defined in comparison with thesimulated quantities.

On the base of the evaluated input quantities the methodologies can by divided following:

Method of the allowed stresses (comparison of the maximum stresses), Method of the safety factor (capacity is defined by one factor), Method of the partial factor (action and capacity are defined by more factors).

Fig.2. Overview of reliability methods by Eurocode 1990

In the present the method of the partial factor is favorable in the practice. The Eurocode 1990 [3]and [5] recommends the use of three levels of the reliability analysis:

I. level- all base input quantities Xi are taken in the calculation by one (design) value. Thisquantity is calculated from its characteristic value and partial factors,

II. level- all base input quantities Xi are described by two statistical parameters (usually meanvalue and standard deviation). The probability of failure Pf (using the method FORM or SORM)can be used for comparison. The level II methods make use of certain well defined approximationsand lead to results which for most structural applications can be considered sufficiently accurate.

III. level all base input quantities Xi are calculated using the theoretical model of theprobabilistic density. Value Pf is determined by calculation using simulation methods on theMonte Carlo base. Full probabilistic methods give in principle correct answers to the reliabilityproblem as stated. Level III methods are seldom used in the calibration of design codes because ofthe frequent lack of statistical data.

Empirical methodsTraditional

Level II Level III

Calibration Calibration Calibration

Level ISemiprobabilistic methods

Partial factor methods


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In both the Level II and Level III methods the measure of reliability should be identified with thesurvival probability Ps = (1 - Pf), where Pf is the failure probability for the considered failure modeand within an appropriate reference period. If the calculated failure probability is larger than apreset target value Po, then the structure should be considered to be unsafe.

Limit state Target reliability indexd50 years 1 year

Ultimate 3,8 (pf10-4) 4,7 (pf10-6)Fatigue 1,5-3,8*) (pf10-110-4) -Serviceability 1,5 (pf10

-1) 3,0 (pf10-3)

Tab.1. Target probability and reliability index by Eurocode 1990 [3]The measure of reliability in Eurocode 1990 [3] is defined by the reliability index (Table 1). Thereliability index depends on the criterion of the limited state. The standard JCSS [8] required themeasure of reliability in dependency on the safety level (Table 2).

Costs Target reliability indexd

Minor consequences Moderate consequences Large consequences(A)Large 3,1 (pf10-3) 3,3 (pf5.10-4) 3,7 (pf1.10-4)(B)Normal 3,7 (pf10-4) 4,2 (pf1.10-5) 4,4 (pf5.10-6)(C)Small 4,2 (pf10-3) 4,4 (pf5.10-5) 4,7 (pf1.10-6)

Tab.2. Target probability and reliability index by JCSS [R44]

The measure of reliability depends on the lifetime of the structure and its importance (Table 2). Inthe case of the normal distribution of the probability failure the relation between the probability

and the reliability index is defined in the form

( )f fp = (8)Design working

life category

Indicative design working

life Td(years)

Object

1 10 Temporary structures

2 10 to 25Replaceable structural parts, e.g. gantry girders,

bearings

3 15 to 30 Agricultural and similar structures

4 50 Building structures and other common structures

5 100 and moreMonumental building structures, bridges, and other

civil engineering structures

Tab.3. Indicative design working life by ENV 1990 [3]

When the main uncertainty comes from actions that have statistically independent maxima in eachyear, the values of for a different reference period can be calculated using the followingexpression

( )11 1n

np p= and ( ) ( )1

n

n = (9)


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wheren is the reliability index for a reference period ofn years,1 is the reliability index for oneyear.

3. Uncertainties of Input Data

The uncertainties of the input data action effect and resistance are for the case of thedeterministic calculation of the reliability of structure, defined in Eurocode 1990 [3]. For the caseof the probabilistic access to define the reliability of structure, the requirements stated in Europeanstandard JCSS 2000 [8] and American standard ASCE 7-95 are used.

One possible access to the definition of the load uncertainties is defined in book [14], where isdescribed the philosophy of the method SBRA based on the Monte Carlo simulations andcombination of the three components of load dead (D), live (L) and sort (S). First complexinformation about the application of the standard Eurocode 1990 and ISO 1998 from the point ofview of the probabilistic methods to analyze the reliability of structures is publication [5] with theexamples in program MathCAD and MATLAB.

The requirements of the standards ASCE 7-95 and JCSS 2000 to define the statistical input datafor probabilistic analysis are presented in the Table 4. The distribution functions of the input datacan be used for the characteristic values of the defined quantities. In the case of load combinationthe calibration of this parameters must be used [5], [6] and [13].

Quantity JCSS 2000 ASCE 7-95

Category Type Symbol Distribution

MeanX

Stand.dev.X

Symbol

Distribution

MeanX

Stand.dev.X

Load Dead G N 1 0,03-0,1 D N 1,05 0,10Live (50years) Q GU 0,6 0,35 L Typ I 1 0,25

Wind (50years) W GU 0,7 0,35 W Typ I 0,78 0,37Snow (50years) S GU 1,1 0,33 S LN 0,2 0,87Earthquake AE N 1 0,20 AE Typ II 0,5-1,0 0,5-1,4

Strength Yield steel strength fay LN 1+2 0,07-0,1 fay LN 1,05 0,11Ultimate steelstrength

fau LN .fay 0,05 fau LN 1,03-1,11 0,11-0,14

Concrete strength fc LN 1+2 0,1-0,18 fc LN 1,05 0,11Yieldreinfor.strength

fsy LN 1+2 0,08-0,1 fsy LN 1,11 0,13

Geometry IPE profile A,J N 0,99Xn 0,01-0,04 A,J LN 1 0,05steel profile L profile, bar A,J N 1,02Xn 0,01-0,02 A,J LN 1 0,05Geometry Dimension of

sectionb,h N 1 0,005-0,01 b,h - - -

concrete Conrete covers a BET 1 0,005-0,015 a - - -profile Accident excentricity e N 0 0,003-0,01 e - - -Model

Action effect E N 1 0,05-0,10 E LN 1 0,05-0,10uncertainties Resistance R N 1-1,25 0,05-0,20 R LN 1,05 0,11

Tab.4. Comparison of the base parameters for the probabilistic design


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4. Reliability Criteria for Seismic Resistance of Structure

Reliability of the bearing structures is designed in accordance of standard requirements STN731201 and ENV 1998-1-1 to 3 [4, 13 and 15] for ultimate and serviceability limit state.Horizontal reinforced plane structures are designed on the bending and shear loads for ultimatelimit state function (10) in the next form

( ) 1 0E R

g M M M = , ( ) 1 0E R

g V V V = (10)

where ME, VE are design bending moment and design shear force of the action and MR, VR are

resistance bending moment and resistance shear force of the structure element.Vertical plane reinforced concrete structures are designed to the tension or pressure and shearresistance for function of failure [13] in the form

( ) 1 0E R

g N N N = , ( ) 1 0E R

g V V V = (11)

where NE, VE are normal and shear design forces of action and NR, VR are resistance normal andshear forces to unit length.

In the case of the combination of the action of the normal forces and bending moments the yieldfunction F(.) must be used as follows

( ) ( )( , ) 1 / 0, ,E E R Rg N M F F N M N M =

(12)

Damage limitation of the reinforced concrete structures depend on the criterion of the maximum

interstorey drifts. The standard ENV 1998 defines the function of failure in the form( ) 1 0E Rg d d d = (13)

where dE is interstorey horizontal displacement, dR is limit value of interstorey horizontaldisplacement defined (for non-structural elements of brittle materials attached to the structure) inthe form

0,005. / R

d h = (14)

where h is storey height ( h = 3m ) and is reduction factor to take into account the lower returnperiod of the seismic action associated with the damage limitation requirement (= 0,4).


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5. Simulation Methods

From the point of view of ones approach to the values considered, structural reliability analysescan be classified in two categories, i.e., deterministic analyses and stochastic analyses. In the caseof the stochastic approach, various forms of analyses (statistical analysis, sensitivity analysis,probabilistic analysis) can be performed. Considering the probabilistic procedures, Eurocode 1recommends a 3-level reliability analysis. The reliability assessment criteria according to thereliability index are defined here. Most of these methods are based on the integration of MonteCarlo (MC) simulations. Three categories of methods have been presently realized [13]:

Straight methods (Importance Sampling - IS, Adaptive Sampling - AS, Direct Sampling -DS)

Modified methods (Conditional, Latin Hypercube Sampling - LHS)Approximation methods (Response Surface Method - RSM)

A) Straight Monte Carlo methods are based on a simulation of the input stochastic parametersaccording to the expected probability distribution. The accuracy of this method depends upon thenumber of simulations and is expressed by the variation index:

1fp

fNp

= (15)

whereNis the number of simulations. If the required probability of failure is pf = 10-4, then by the

number of simulations N=106, the variation index is equal to 10%, which is an acceptable degreeof accuracy.

Advantages of the method:

the final values of the reliability reserves can be continuously displayed in the form of ahistogram 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.

Drawbacks of the method:

large number of simulations for small probability values,

slow calculations of complex problems (for Finite Element Method models, the calculationsare expensive and ineffective).

B) The Modified LHS methodis based on the same number of simulations of the function g(X) asin the Monte Carlo method; however the zone of the distributive function (Xj) is divided intoNintervals with identical degrees of probability. This method provides good assessments of thestatistical parameters of the structural response when compared to the Monte Carlo method. Using

the LHS strategy, we get values like the reliability reserve parameter the mean valueZ, thestandard deviation z, the slant index z, the sharpness index ez, or the empirical cumulativedistribution function.


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The reduction of the number of simulations (tens to hundreds of simulations) means a valuablebenefit from this method compared to the straight Monte Carlo method (thousands to millions ofsimulations).

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

1 1 1

.NRV NRV NRV

o i i ij i j

i i j

Y c c X c X X = = =

= + + (16)

where co is the index of the constant member; ci are the indices of the linear member and cij theindices of the quadratic member, which are given for predetermined schemes for the optimaldistribution of the variables (Montgomery, Myers) or for using regression analysis after calculatingthe response (Neter).

a) Central Composite Design b) Box-Behnken Matrix

Fig.3 Distribution schemes of the stochastic numbers of the RSM method for three input

variables

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 MatrixSampling" (BBM) method.

Advantages of the method:

considerably less number of simulations than with the straight Monte Carlo method,

it is possible to define dependencies using the "design experiments" method or regressionanalysis from the defined points in the case of improper approximation functions,


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Particular simulations are independent from each other parallel calculations can be usedhere.

Drawbacks of the method:

The number of simulations depends on the number of variable input parameters; in the caseof a large number of input parameters, the method is ineffective,

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

Central Composite Design Box-Behnken Design

Number ofstochastic variables

Number ofquadratic

functioncoefficients

Number offactorials f

Number ofstochastic

numbers

Number ofquadratic

functioncoefficients

Number ofstochastic

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 55 121

10 66 3 149 66 161

Tab.5. Summary of the stochastic numbers depending on the number of stochastic input

variables

6. Probabilistic Postprocessor in ANSYS Program

For reliability analysis of the complicated structures the displacement-based FEM is favorable touse with the one from defined simulation methods. In this work the licensed program ANSYS [13]with probabilistic postprocessor was utilized for probability analysis of the reliability of NPPstructures for various action effects.

The ANSYS Program belongs among the complex programs for solving potential problems. Itcontains a postprocessor, which enables the execution of the probabilistic analysis of structures.

In Figure 4, 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 postprocessor for the probabilistic design of structures enables the definition of randomvariables using standard distribution functions (normal, lognormal, exponential, beta, gamma,weibull, etc.), or externally (user-defined sampling) using other statistical programs like AntHILLor 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).


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Fig.4. Procedural diagram of probabilistic calculations using the ANSYS

7. Probabilistic Analysis of NBS in ANSYS

In the case of the design of high rise buildings in the seismic active locality and in the complicatedgeological conditions the engineer must to consider the various action and resistance uncertainties.The variable material and geometric properties of the structural elements determine the resistance

of the structures. The soil stiffness can be variable through to depth below foundation base andduring the seismic even too. Bratislava locality is known for complicated hydrogeologicalconditions due to flood along the Danube river. The seismic hazard in Bratislava locality isdetermined by the source of the earthquake excitation in Austria ( design acceleration ag = f.ar=1,1.0,41 = 0,451ms-2). One from the interesting high rise building in Bratislava is the building ofthe Slovak National Bank (NBS). On the example of this building the importance of theprobabilistic analysis of the seismic resistance of building will be presented.


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The NBS building has 32 floors and the height isequal to 98,3m. The structure in plan is in the formof the quarter-circle with radius of 41,5m. Thebuilding has 3 underground floors with thefoundation base on the level -11,5 m. The structuralwall system is in the form L. The system issymmetrical about to diagonal axis. The planedimension is variable through the building height.The structural system is composed from thecoupling walls, flooring plates and the compositesteel concrete columns. Building is based on theconcrete foundation plate with the thickness of2,3m.

The calculation model is composed from the beamelements (BEAM44), shell elements (SHELL43)and solid elements (SOLID45) number of theelement is equal to 5.580 and nodes 3.464 (Fig.5). Fig.5. Calculation model of NBS

7.1 Loading and Load Combination

The loading and load combination in the case of the deterministic as well as the probabilisticcalculation is different due to requirements of Eurocode 1990 [3] and JCSS 2000 [8], too. Theseismic load was taken in accordance with ENV 1998 [4] as a design acceleration responsespectrum for B type of soil and design acceleration ag = 1,0ms

-2.

In the case of deterministic calculation and the ultimate limit state of the structure the loadcombination is considered according to ENV 1990 as follows:

Seismic design situation deterministic method

2.d k Q k Ed E G Q A= + (7)

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

4

SahSavSahSav

Frequency [Hz]

Spectruma

cceleration[m/s^2]

Fig.6. Response spectrum acceleration


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where Gk is the characteristic value of the permanent loads, Qk - the characteristic value of thevariable loading,AEd (=1AEd.k) - the design value of the seismic loading,AEd.k - the characteristicdesign value of the seismic loading, I importance factor (of the building structure), 2.Q - thecombination factor according to ENV 1990 (2.Q = 0,3).

In the case of probabilistic calculation and the ultimate limit state of the structure the loadcombination we take following:

Seismic design situation probability method

var var var .E k k E k E G Q A g G q Q a A= + + = + + (8)

where gvar, qvar, avar are the variable parameters defined in the form of the histogram calibrated tothe load combination in compliance with Eurocode [3].

7.2 Spectrum analysis

The seismic analysis of the high rise building structure was realized using the linearized responsespectrum method. This method allows an approximate determination of the maximum response ofan MDOF system without performing a time history analysis.

The response spectrum of the displacements and forces from the excitation in direction a=1, 2, 3 iscalculated from the modal response by method square root of sum of squares mode (SRSS). Thetotal response spectrum is calculated from three base acceleration spectra (in space) alternativelyfrom the combination SRSS or standard combination rule

Rtot=R1+0,3R2+0,3R3 or Rtot=0,3R1+0,3R2+R3 or Rtot=0,3R1+R2+0,3R3 (8)

whereRi (i=1,2,3) are response values from the acceleration excitation in the direction 1, 2, 3.

The modal analysis of these two models of buildings was realized on the software ANSYS usingLanczos method.

7.3 Uncertainties of input variables

The stiffness of the structure is determined with the characteristic value of Youngs modulus Ekand variable factor evar. A load is taken with characteristic values Gk, Qk, AE,k and variable factorsgvar, qvarand avar.

The effect of soil-structure interaction can be investigated in the case of probabilistic assessment

by sensitivity analysis of the influence of variable properties of soil. A soil stiffness variability inthe vertical direction is defined by the characteristic stiffness value kz from the geologicalmeasurement and the variable factor kz.var. The variability of the soil stiffness in the horizontalplane is determined by the variability of the global rotation kxx.var, kyy.var . The random distributionof the soil stiffness under foundation plate is approximated with bilinear function on the slab planein dependency on three parameters kz.var, kxx.var, kyy.var


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( ) ( ) ( ).var .var .var ,, 2 2o o

z yy xx z k

x y

x x y yk x y k k k k L L

= + +

(14)

where kz.k is characteristic value of soil stiffness, xo, yo are coordinates of foundation structuregravity centre,Lx andLy are the plane dimensions of the slabs in directionsx andy.

Soil Actions

Characteristics stiffness permanent variable accidental

translationin axis Z

rotationabout X

Rotationabout Y

deadload

liveload

windload

seismicload

Character. values kz,k kxx,k kyy,k Gk Qk1 Wk AEd,.kVariable kz_var kxx_var kyy_var g_var q_var w_var a_var

Histogram type Normal Normal Normal Normal Beta (T. I) Weibul (T.III) Beta (T. I)

Mean value 1 1 1 1 0,57 0,37 0,67

Stand. deviation 0,200 0,033 0,033 0,033 0,28 0,18 0,14Min. value 0,148 0,851 0,853 0,569 0,858 0,021 0,40

Max. value 1,867 1,163 1,135 1,385 1,127 1,335 1,20

Tab.6. Probabilistic model of input parameters

The uncertainties of the calculation model are considered by variable model factorR

(Mu.var) for

Beta distribution ( 1,05; 0,1 = = ) and variable load factorE

for Gausss normal distribution

( 1; 0,05 = = ).

In the case of the probabilistic analysis the variability of the building stiffness and masses has nonneglected influences to the building frequency characteristics. The principal frequencies in thedirection X (e.g. Y) are in the interval from 0,50HZ to 0,78Hz. This interval of the

eigenfrequencies impacts to the seismic input accelerations from the design response spectrumacceleration (Fig.6).

7.4 Sensitivity Analysis

Sensitivity analysis of the influence of the variable input parameters to the reliability of thestructures depends on the statistical independency between input and output parameters. Matrix ofthe correlation coefficients of the input and output parameters is defined by Spearman. The figures8 and 9 show you the results of the sensitivity analysis of the interstorey drift and the shear forcesin models RSM2 and LHS2 in the case of the seismic action. The variability of the seismic actionsis dominant. The influency of the other input parameters is different in the method RSM and LHS.

In the case of RSM method the other influencies are the variability of the stiffness (Evar) and themasses (Gvar) for the maximum displacement. In the case of the LHS method is the variability ofthe soil stiffness (kz.var). The same situation is in the case of the shear forces. The variability of thesoil stiffness is in the fourth (e.g. second) place in the case of the RSM (e.g. LHS) analysis. Thesedifferencies results from the differencies of the algorithms LHS and RSM described in the chap.5.

The sensitivity analysis gives the valuable information about the influence of uncertainties of inputvariables (load, material, model,) to engineer for optimal design of the structures.


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Fig.7. The results from the sensitivity analysis of the Dmaxand Vmax in the model RSM2

Fig.8. The results from the sensitivity analysis of the Dmax and Vmax in model LHS2

7.5 Comparison of Deterministic and Probabilistic Analyses

Deterministic calculation of the seismic resistance of the high rise building NBS was realized ontwo models different in dependency on the soil stiffness. From the point of view of probabilisticanalysis, type and number of the variable input data the probabilistic analysis were realized by

methods RSM and LHS for 100 simulations. On examples of the free models - RSM1, LHS1 (5var.data - kz, Gk, Qk, Ak, R ), RSM2, LHS2 (8 var.data - kz, kxx, kyy, Gk, Qk, Ak, R, E) and

RSM3, LHS3 (10 and 11 var.data - kz, kxx, kyy, Gk, Ak, Ek, qz.k, qxx.k, qyy.k,R, E) were realized thedifferences in the reliability conditions and the time of calculations were analyzed. The results ofthe deterministic and probabilistic analysis are described in the table 7. The results from thedeterministic analysis are more conservative as probabilistic. This fact is related to the variouscombination rule for action impact in the case of the deterministic and probabilistic solution. The


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results from the RSM analysis for 81 and 149 simulation give us the simile results than in the caseof the LHS method for 100 simulations. From the point of view of the time and space calculationrequirements the RSM method is more effective than LHS (see Tab.7 ) for the small number of theinput parameters .

Model Extreme interstorey drifts [mm] Number of

the variable

input data

Number

cykl.CPU

Min Mean Max St.dev [sec]

DeterministicMKP 5,72 5,73 5,74 - - 3 147

Probabilistic RSM

RSM1 2,14 2,19 3,92 0,83 5 27 891RSM2 1,79 3,99 7,28 0,86 8 81 2670RSM3 1,79 4,08 7,46 0,88 10 149 4805

Probabilistic LHS

LHS1 2,90 4,40 7,01 0,89 5 100 3145LHS2 2,80 4,32 6,89 0,88 8 100 4852LHS3 2,50 4,08 6,38 0,86 11 100 4965

Probabilistic MC

MC2 2,72 4,21 6,21 0,90 8 100 2777

Tab.7. Comparison of the analysis for maximum interstorey drift

The efficiency and demandingness of the probabilistic methods is compared in the Tab.8. The timedemands up to 8 input parameters evidently increase in the method RSM. The method LHS ismore demanding to calculation time than the MC for the same number of the input variables. Inthe case of the same number of simulations the differences of the mean values of the interstoreydrift is maximum equal to 2,5% between the LHS and MC. The differences in the output data forthe 4 time more simulations are 15% for LHS method and 11% for the MC method (see Tab. 8).From the comparison of the probabilistic methods (Tab. 8) results, than the RSM method is themost effective fro the number of the input parameters less than 8.

Model Extreme of the interstorey drift [mm] Number of

the variable

input data

Number

cykl.

CPU

Min Mean Max St.dev Min Mean

DeterministickyMKP 5,72 5,73 5,74 - - 3 147

Pravdepodobnostne RSMRSM2 1,79 3,99 7,28 0,86 8 81 2670

Pravdepodobnostne LHSLHS2a 2,80 4,32 6,89 0,88 8 100 4852LHS2b 2,22 3,68 6,76 0,82 8 200 15243LHS2c 2,00 3,68 5,43 0,80 8 400 30403

Pravdepodobnostne MC

MC2a 2,72 4,21 6,21 0,90 8 100 2777MC2b 2,16 3,59 6,21 0,85 8 200 8741MC2c 2,08 3,75 7,55 0,83 8 400 17567

Tab.8. Comparison of the probabilistic method efficiency

If we have the results from experimental measurements of the material and geometric properties ofthe system, the method MC is more effective and accurate solution of the structural reliability. The


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sensitivity analysis of the influencies of the variability of the input parameters give to engineersthe tools to design the optimal structures.

8. Conclusion

This paper presented the methodology of the seismic analysis of the high rise building structure onthe base of deterministic and probabilistic assessment. This analysis was realized on the exampleof the one symmetrical structure of the NBS in Bratislava. There were presented the advantagesand disatvanges of the probabilistic methods to solve soil- structure interaction. Three probabilistic

methods (MC, LHS and RSM) were investigated from the point of the accuracy and affectivity incomparison with the deterministic methods. On the example of this building the sensitivityanalysis of the influence of the variable input parameters were considered. From these analysisresults than the RSM method is the most effective method in the case of limited input variableparameters. In the case of the experimental input data the MC analysis is the most credible methodfor solution of the structural reliability.

Acknowledgement

The project was realized with the financial support of the Grant Agency of the Slovak Republic

(VEGA). The project registration number is VEGA 1/0849/08.

9. References

1. Bolotin,V.V.: Pouit metd terie pravd podobnosti a teorie spolehlivosti pi navrhovnkonstrukc, SNTL, Praha, 1978.

2. Chopra, A.N., 2001,Dynamics of Structures, 2nd ed., Prentice-Hall, New York.

3. EN 1990, Eurocode Basis of structural design. CEN Bruxelles. 2002.

4. Eurocode 8, Design of structures for earthquake resistance, Part 1: General rules, seismicactions and rules for buildings, CEN may 2000.

5. Hanbook 2, Implementation of Eurocodes Reliability Backgrounds. Guide of the basis ofstructural reliability and risk engineering related to Eurocodes. Development of Skills

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