probabilistic sensitivity of limit states of structures. the monte carlo simulation

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PAMM · Proc. Appl. Math. Mech. 9, 549 – 550 (2009) / DOI 10.1002/pamm.200910247 © 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Probabilistic sensitivity of limit states of structures. The Monte Carlo simulation M. Skowronek *,1 1 Gdansk University of Technology, Department of Structural Mechanics and Bridge Structures, Narutowicza 11/12 80-233 Gdansk The main issue of the paper is the probabilistic sensitivity of the limit states of structures with respect to selected input design variables. Attempt to the problem is done by the dedicated Monte Carlo simulation procedure. Basic design variables are random variables of given probability distributions, presented in the form of random numbers. Uni-parametrical increment of the dominant basic variable (basic variables) is done, finally achieving the structural limit state. The simulation procedure restuls in a set of limit multipliers. Statistical analysis leads to the estimate of the probability density function of the limit state. Thus the numerical image is presented of the probabilistic sensitivity of the structural limit state. Reliability or the probability of failure are to be estimated, as statistical parameters of the histogram. Numerical examples of engineering structures illustrate the method introduced in the paper, conclusions are formulated eventually. © 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Theory of sensitivity is a recognized branch of mechanics, including the methodology of assessing the impact of given input parameters to structural response. Development of sensitivity analysis was generally performed in a deterministic way ([1]). Due to research of the past two decades sensitivity problems were subject to nondeterministic approach ([2]). The paper is focused on the problems of civil engineering only, namely: to provide a tool to estimate the sensitivity of structural limit states to a given input design parameter (group of parameters). The general numerical procedure to deal with random variables represented by populations of generated numbers is a Monte Carlo simulation method. In the problem of the paper the Monte Carlo simulation method was implemented to form a dedicated algorithm, described in the following sections. 2 The general idea Limit state analysis is a branch of mechanics to be performed with a couple of initial assumptions, concerning the failure modes involved. The zero-length plastic hinges in critical cross-sections is a widely recognized model of failure of framed structures, where bending is the dominant action. However, it may bye simply generalized, to a limit combination of cross- sectional forces, in planar or three-dimensional problems. Standards often make use of semi-empirical simplified limit state formulae including components of the cross-sectional forces vector. To make an engineer’s simplificatiom, the formulae used in codes are often linearised. Serviceability limit states make up another group to be considered. Definition of structural limit state may be otherwise: brittle crack in one or a group of cross-sections, loss of global or local stability, etc. The general formulation of the limit state problem of a structure does not allow us to provide general analytical solutions. It is required to specify the problem and the relevant failure mode(s). The problem of limit states of structures may be described by a general stochastic operator L to provide a relation between structural input p and output u, as follows ( ) ( ) ( ) , , L s s ω ω ω ω = u p (1) where s is a one-dimensional coordinate on the element axis, ω is an elementary event. Discretization of the problem turns the random vector functions p and u into random vectors defined in discretization points. Such a problem cannot be solved analytically, thus the effort is concentrated on the development of numerical procedures. The Monte Carlo simulation algorithm to assess the structural sensitivity of a limit state is based on the designer’s decision to choose the dominant input variable or a group of variables. They can be action parameters – loads, temperature increment, restraint forced deflections, or the resistance parameters – dimensions of members or material characteristics, like yield stress of steel or compressive strength of concrete. Simulation algorithm to assess the structural sensitivity of a structure, with respect to chosen dominant variable(s), consists of several steps. First of all, random structural and action parameters are gererated due to the assumed probability distributions, thus every basic variable is represented by a set of random numbers. The general idea lies in the operations of a single simulation step, as follows: ____________________ * Corresponding author: e-mail [email protected], Phone: +48 50 3471891, Fax: +48 58 3471670

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Page 1: Probabilistic sensitivity of limit states of structures. The Monte Carlo simulation

PAMM · Proc. Appl. Math. Mech. 9, 549 – 550 (2009) / DOI 10.1002/pamm.200910247

© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Probabilistic sensitivity of limit states of structures. The Monte Carlo simulation

M. Skowronek*,1 1 Gdansk University of Technology, Department of Structural Mechanics and Bridge Structures,

Narutowicza 11/12 80-233 Gdansk

The main issue of the paper is the probabilistic sensitivity of the limit states of structures with respect to selected input design variables. Attempt to the problem is done by the dedicated Monte Carlo simulation procedure. Basic design variables are random variables of given probability distributions, presented in the form of random numbers. Uni-parametrical increment of the dominant basic variable (basic variables) is done, finally achieving the structural limit state. The simulation procedure restuls in a set of limit multipliers. Statistical analysis leads to the estimate of the probability density function of the limit state. Thus the numerical image is presented of the probabilistic sensitivity of the structural limit state. Reliability or the probability of failure are to be estimated, as statistical parameters of the histogram. Numerical examples of engineering structures illustrate the method introduced in the paper, conclusions are formulated eventually.

© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Theory of sensitivity is a recognized branch of mechanics, including the methodology of assessing the impact of given input parameters to structural response. Development of sensitivity analysis was generally performed in a deterministic way ([1]). Due to research of the past two decades sensitivity problems were subject to nondeterministic approach ([2]). The paper is focused on the problems of civil engineering only, namely: to provide a tool to estimate the sensitivity of structural limit states to a given input design parameter (group of parameters). The general numerical procedure to deal with random variables represented by populations of generated numbers is a Monte Carlo simulation method. In the problem of the paper the Monte Carlo simulation method was implemented to form a dedicated algorithm, described in the following sections.

2 The general idea

Limit state analysis is a branch of mechanics to be performed with a couple of initial assumptions, concerning the failure modes involved. The zero-length plastic hinges in critical cross-sections is a widely recognized model of failure of framed structures, where bending is the dominant action. However, it may bye simply generalized, to a limit combination of cross-sectional forces, in planar or three-dimensional problems. Standards often make use of semi-empirical simplified limit state formulae including components of the cross-sectional forces vector. To make an engineer’s simplificatiom, the formulae used in codes are often linearised. Serviceability limit states make up another group to be considered. Definition of structural limit state may be otherwise: brittle crack in one or a group of cross-sections, loss of global or local stability, etc. The general formulation of the limit state problem of a structure does not allow us to provide general analytical solutions. It is required to specify the problem and the relevant failure mode(s). The problem of limit states of structures may be described by a general stochastic operator L to provide a relation between structural input p and output u, as follows

( ) ( ) ( ), ,L s sω ω ω ω=⎡ ⎤⎣ ⎦u p (1)

where s is a one-dimensional coordinate on the element axis, ω is an elementary event. Discretization of the problem turns the random vector functions p and u into random vectors defined in discretization points. Such a problem cannot be solved analytically, thus the effort is concentrated on the development of numerical procedures. The Monte Carlo simulation algorithm to assess the structural sensitivity of a limit state is based on the designer’s decision to choose the dominant input variable or a group of variables. They can be action parameters – loads, temperature increment, restraint forced deflections, or the resistance parameters – dimensions of members or material characteristics, like yield stress of steel or compressive strength of concrete. Simulation algorithm to assess the structural sensitivity of a structure, with respect to chosen dominant variable(s), consists of several steps. First of all, random structural and action parameters are gererated due to the assumed probability distributions, thus every basic variable is represented by a set of random numbers. The general idea lies in the operations of a single simulation step, as follows: ____________________ * Corresponding author: e-mail [email protected], Phone: +48 50 3471891, Fax: +48 58 3471670

Page 2: Probabilistic sensitivity of limit states of structures. The Monte Carlo simulation

550 Short Communications 14: Applied Stochastics

© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com

- generation of a structure subjected to external actions, - uni-parametrical increment of the dominant variable(s), while the other basic variable(s) are constant, until the limit

state of a structure is reached, due to the assumed failure mode - the outcome of a single simulation step is the limit multiplier of a dominant variable (group of variables).

The result of the whole simulation process is the set of limit multipliers. The histogram is the numerical estimate of the probability distribution of the limit state with respect to dominant variable(s). Estimators of the distribution parameters can be obtained: mean, variance and the parameters of engineering importance: reliability or the probability of failure.

3 Numerical examples

3.1 Monumental structure - Basilica

The monumental structure – Lichen Basilica is situated close to the geographical centre of Poland. The major load-carrying part of the building is the cantilever structure which contains of a circular foundation, columns, colonnade and a dome (Fig. 1). The colonnade is the structural part to be analysed.

Fig. 1 The basilica, featuring the colonnade. Fig. 2 Results of the three variants of calculations

Dominant variables are the wind load intensities on both storeys of the colonnade. Wind velocity on each storey was assumed to be Gumbel-distributed with parameters: u = 0.15, α = 12.4. Three variants of calculations were provided, they differ in the correlation coefficient ρ between the random wind load intensities on both storeys, W1 and W2. The simulation procedure was provided, as described in Section 2, for the following three variants of calculations: a) ρ = 0, b) ρ = 0.62, c) ρ = 1. The results – histograms of relative frequencies of the wind load multipliers λ, are shown in Fig. 2. In every case the minimum limit load value is above 1, min 1λ∆ = − . The conclusion comes that the probability of failure is estimated to be less than 10-4, because 10000 trials were made in the simulation process.

3 Conclusions

Probabilistic sensitivity of the structural limit states is described in general terms, illustrated by an engineering example. The algorithm is based on the problem-oriented Monte Carlo simulation procedure, being part of the methodology, not only the computational tool. Thus the method can be described as an example of computational science application ([3]). The probabilistic sensitivity analysis of limit states of structures, shown on specific examples, may be implemented to a variety of problems, different basic variables can be taken as dominant. Thus the histogram of the limit state of the structure has to interpreted in a relevalt way, with specific interpretation of structural reliability or failure probability.

References [1] M. Kleiber, T.T. Hien, H. Antunes, P. Kowalczyk, Parameter sensitivity in nonlinear mechanics (Wiley, Chichester, 1997). [2] P. Bjerager, S. Krenk, Parametric Sensitivity in first order reliability theory, J.of Eng. Mech.115,1577-1582 (1989). [3] P. Thagard, Computational philosophy of science (MIT Press, Cambridge, 1993).