Stochastics and Finite Elements - Challenges and

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Currently, stochastic analysis software is being developed which may be applied .... This is an example application of the PI/FEA concept. A 3D-frame as shown ...
Stochastics and Finite Elements Challenges and Chances

Christian Bucher, Dagmar Hintze and Dirk Roos Institute of Structural Mechanics, Bauhaus-Universit¨at Weimar, Germany

Summary The paper discusses several requirements for software to be successfully utilized for stochastic structural analysis. Some recent developments are presented. Their application to different problem classes is shown by means of numerical examples. Keywords Stochastics, Reliability Analysis, Monte-Carlo-Simulation, Response Surface Method, Stochastic Finite Elements.

1 INTRODUCTION Within many engineering fields, structural design must meet the demands of cost-effectiveness. This leads to lightweight, highly flexible, and consequently vulnerable structures. In order to assess potential risks associated with such a design, the structural analysis must take into account available information on the randomness of loads and structural properties. It means that the analysis should include reliability estimates in an appropriate manner. This may be seen as an additional refinement of structural analysis, like e.g. the higher degree of accuracy as introduced by the Finite Element Method compared to the traditional engineering tools of analysis. However, although the need for stochastic analysis is apparent, there is a lack of software which enables the application of reliability methods by engineers not specialized in the field. Currently, stochastic analysis software is being developed which may be applied successfully by practitioners. Within the scope of structural analysis, these tools may be classified according to the following two categories (see figs.1 and 2): • Probability Integrator (PI) with external Finite Element Analysis (FEA). Here the software can be considered as two functionally independent stand-alone applications. • Stochastic Structural Analysis (SSA). This class represents integrated software packages whose functionality mainly arises from the symbiosis of probabilistic and structural analysis. Several examples for these two classes are available. Some of them are mentioned in the following (this list is incomplete). In the first category (PI/FEA), I would like to mention as examples the packages ISPUD (Bourgund and Bucher, 1986), PROBAN (Madsen et al., 1988), RYFES (1995). In the category SSA, there are e.g. the packages NESSUS (Millwater et al., 1990), CALREL (DerKiureghian et al.), STRUREL/COMREL (Gollwitzer et al., 1994), SLang (Bucher et al., 1995).

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perform statistics

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2 SOFTWARE CONCEPT 2.1

Requirements

Software for stochastic structural analysis should encompass state-of-the-art technology in both the structural analysis as well stochastic analysis parts. This does not necessarily imply that one single program package must include everything. The following items, however, should at least be made accessible: a) Nonlinear Finite Element modeling b) Dynamic analysis and stability c) Randomly spatially distributed loads and structural properties d) FORM/SORM reliability measures e) Response Surface methodology and Monte Carlo Simulation Here the first two items fall into the traditional FEA domain and the last two are parts of most PI packages. The item c) lies somewhere in between, and is partly covered by codes in the area of Stochastic Finite Elements. In many applications of stochastic structural analysis (such as e.g. structural reliability assessment), the PIpart controls the flow of execution, i.e. the structural analysis to be performed by FEA-part. This is of specific importance with adaptive strategies (i.e. if the next step of the PI depends on the answers from the FEA), requiring continuous two-way exchange of information between PI and FEA. Apart from these action controlling messages, the involved codes may have to exchange large amounts of data. This means that whenever the PI decides to request a new FE calculation, it has to pass a new set of structural (system or loading) parameters to the FEA. If this is to work smoothly, both the PI and the FEA should agree on sharing common memory areas (communication on the basis of data files certainly is a working solution, but not really an efficient one). These requirements can be fully met by a software package falling into the SSA-class only.

2.2

Software Realization

Following a long tradition in software development for structural engineering applications, it seems useful to formulate tasks in small, easily controllable steps (e.g. Kr¨atzig et al., 1977). It should be possible to combine these steps into larger segments which can be executed repeatedly. Such a solution can be achieved by implementing a problem oriented module set in which the individual modules pertain to stochastic and structural analysis. A

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dedicated software package for stochastic structural analysis along this line has conceptually been presented several years ago by Bucher et al. (1989) and substantiated subsequently by Bucher and Schu¨eller, 1994. In order to meet the above mentioned requirements regarding interaction between PI and FEA, a software system comprising data management, user interactivity, finite element analysis, and reliability tools - SLang - has been developed (Bucher et al., 1995). The name is derived from the term Structural Language. SLang basically is a command interpreter acting on a set of relatively complex commands. It allows the construction of command sequences including conditional branching and looping. SLang is a meta-language whose command set is aimed at the goals of structural analysis. SLang allows structural analysis to be performed on a step-by-step basis with transparent access to the objects it operates on at any time. In a way, SLang can be seen as a toolbox containing the basic software products for both PI and FEA (amongst others) which can interact smoothly and transparently. SLang integrates Finite Elements with stochastic modeling at a level which appears to be sufficient for a wide range of engineering problems. In addition, several advanced graphics features allow the visualization to computational results in a way suitable for the decision making process. However, the recent developments in the area of component software allow further enhancements regarding inter-process communication and parallel processing which can be utilized advantageously for stochastic analysis (Roos et al., 1999).

3 APPLICATION EXAMPLES 3.1

Failure probability of a space frame subjected to random loading

This is an example application of the PI/FEA concept. A 3D-frame as shown in Fig.3 is modeled by 24 geometrically and physically nonlinear beam elements (linear elastic-ideally plastic material law, Young’s is modulus 2.1 · 1011 N/m2, yield stress 2.4 · 108 N/m2). The cross section for the girder is a box (width 0.2m, height 0.15 m, wall thickness 0.04 m) and the columns are I-sections (flange width 0.2 m, web height 0.2 m, thickness 0.04 m). The columns are fully clamped at the supports. The loads are incremented up to total or partial collapse of the structure. This structure is analyzed with respect to its failure probability (total or partial collapse) in terms of the Response Surface Method (cf. Bucher and Bourgund, 1990; Roos et al., 1999). The problem is very well separated in a structural analysis part, and a probabilistic analysis part. Consequently, the utilization of separate software packages is not only feasible, but moreover quite convenient. Here the FEA part of SLang was utilized, but virtually any other available FE code such as ANSYS could be used instead.

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4m 5m Figure 3: Structural Model (Space Frame) with Loads Fig. 4 shows the distribution of the von-Mises stresses in the structure at the failure points for two different load combinations. It is quite clear, that a fairly large part of the structure undergoes severe plastic deformation before collapse (The deformations are magnified 10 times). Within the PI/FEA concept, the failure points can be directly loaded into the PI, and interpolation together with probability integration can be performed in an entirely separated manner. For the structure as shown in Fig. 3 the statistical data of the loads are given in Table 1. The failure criterion considered in this example is the total or partial collapse of the nonlinear structure. Algorithmically this is defined by the loss of positive definiteness of the tangential stiffness matrix. Directional sampling

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Table 1: Statistical Data for Loads Mean Std.Dev Type 120 kN 30 kN Gumbel 60 kN 10 kN Gumbel 150 kN/m 12 kN/m Log-Normal

is used to locate the finite element limit state check points. Fig. 5 shows 10.000 support points. The failure prob-

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Figure 5: Finite element collapse points in space of the random variables. ability of the structures according to the different response surfaces methods is calculated by adaptive importance sampling. Since there are three random loading variables, the calculation starts with 9 failure points for the second order polynomial and for the polyhedrons. The search directions are selected by a systematic combination of the random variables started from the basic directions (Bucher and Bourgund, 1990). Tab. 2 shows the results for this first run. The next combinations of the points are selected randomly using a directional sampling strategy. Fig. 6 shows a selected approximation using the secantial hyperplane approximation with p = 100 arbitrary chosen support points. The convergence of the failure probability obtained by using the polyhedral approximations, in dependence on the number p of points on the limit state surface, is shown in Fig. 7. The slow convergence of the secantial hyperplane method is not surprising, because it represents a consistently safe approximation It is quite important to note that re-analysis with different statistical data does not require repetition of the FE analysis. This makes the PI/FEA concept especially attractive for problems with deterministic system data but with several sets of random loading data to be considered.

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Table 2: Results of computing failure probabilities using 9 points selected by systematic search. pf Finite element limit state 1.5455e-05 Polynomial response surface 1.9551e-07 Normal hyperplane response surface 3.4379e-07 Secantial hyperplane response surface 0.015386

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Figure 6: Secantial hyperplane approximation using 9 limit state check points.

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Figure 7: Convergence of probability of failure vs. number p of points on limit state surface.

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3.2

Parabolic Shell under Vertical and Horizontal Load

This example is intended to show the application of ANSYS as Finite Element Analyzer within the PI/FEA concept. Herein the mechanical system is a parabolic shell, as sketched in this Fig.8, subjected to horizontal (12 x H) and vertical loads (40 x V ). The constitutive relation of the shell material is von Mises plasticity without hardening. The

Figure 8: Structural Model (Parabolic Shell) with Loads structure is modeled by 120 geometrically linear shell elements (SHELL93). By means of incremental analysis 5 limit state points were determined. The failure in equilibrium iteration was used to determine collapse of the structure. The distribution of the von-Mises stress in the elastic range (loads at their mean values) is shown in Fig.9.

Figure 9: von Mises strain in elastic range The distribution of the equivalent plastic strain (von-Mises strain) at collapse under predominantly vertical load is shown in Fig.10. Under predominantly horizontal load the final stress pattern develops according to Fig.11. In obtaining these result, the loads were incremented proportionally up to collapse of the structure. Hence no path dependence was considered in the analysis. A set of 6 failure points was calculated as shown in Table 3. These points can be used to construct a polyhedral response surface approximation of the actual failure surface as shown in Fig.12. For statistical data as given in Table 4 the failure probability is calculated as p f = 1 · 10−5 . This result is obtained from adaptive sampling with 1000 samples as shown in Fig.13.

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Figure 10: Equivalent plastic strain at plastic collapse under vertical load

Figure 11: Equivalent plastic strain at plastic collapse under horizontal load Table 3: Failure Points Number H (MN) V (MN) 1 4.64 3.53 2 5.41 2.40 3 5.61 1.48 4 2.92 3.85 5 0.68 3.84 6 5.54 0.98 V 4.0E0 3.5 3.0 2.5 2.0 1.5 1.0

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Table 4: Statistical Data for Loads (set 1) Mean Std.Dev Type H 3 MN 2.5 MN normal V 0.3 MN 0.25 MN lognormal 5.0E0

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Figure 13: Adaptive sampling (random variable set 1) A repetition of the failure analysis with different load data (see Table 5 is shown in Fig.14. The failure probability in this case is p f = 3 · 10−6. Table 5: Statistical Data for Loads (set 2) Mean Std.Dev Type H 4.2 MN 1.5 MN normal V 0.3 MN 0.25 MN lognormal

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Figure 14: Adaptive sampling (random variable set 2)

3.3

Natural Frequencies of a Structure with Randomly Distributed Elastic Properties

In the following, a spherical shell structure (cf. Fig. 15) in free vibration is considered. This is an example application of the SSA concept. The shell is modeled by 288 triangular elements of the type SHELL3N (Schorling, 1997). The shell is assumed to be fully clamped along the edge. The material of the shell is assumed to be elastic (in plain stress) and the elastic modulus E(x) is modeled as a log-normally distributed, homogeneous, and isotropic random field. Its autocovariance function CEE (x, y) is assumed to be of the form CEE (x, y) = σ2EE exp(−

||x − y) ) Lc

(1)

In the above equations, the 3D vectors x and y are the coordinates of points within the plate structure. The basis diameter of the shell is 20 m , and the correlation length Lc is 10 m. The coefficient of variation of the random field is assumed to be 0.2. The question to be answered is what magnitude of randomness may be expected in the calculated natural frequencies. Quite clearly, this is very important for structural elements designed to carry rotating machinery which

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Figure 15: Spherical Shell Structure produces almost harmonic excitation, and possibly resonance. Hence the probability of obtaining high deviations from the mean natural frequency needs to be calculated. This example shows quite typically the close connection required between the PI and the FEA. Within the Finite Element model, the random field E(x) is represented by its values in the integration points of the elements. The shell elements as utilized here have two layers of each 13 integration points, so there is a total of 26x288 = 7488 integration points. In order to reduce this rather high number of random variables the following strategy is applied (see e.g. Bucher and Brenner, 1992 or Brenner, 1994 for details). First, the elastic modulus is represented by one value per element (given the correlation length of 10 m this is not a severe simplification). Second, the remaining random field is represented in terms of independent random variables and corresponding space dependent shape functions. These independent variables are obtained by applying the Nataf joint density model (Liu and DerKiureghian, 1986) along with an eigenvalue decomposition of the covariance matrix of the random field. This can be seen as an extension of the spectral representations expanded on by Ghanem and Spanos, 1991. Due to the spatial correlation structure, the covariances depend on the relative location of the integration points within the structure. In order to match the random field discretization and the Finite Element discretization the availability of the topological information on the Finite Element model is essential. Some of the SLang -commands required to perform the analysis are given in the following: • Create a random field with lognormal distribution, isotropic exponential correlation function with correlation length 0.5 and coefficient of variation 0.2: ranfield create, replace lognormal exponential_corr_iso, 1 .2 .5 all_elements, e_mod_field/ • Decompose the covariance matrix of the 288 random variables utilizing a representation of 15 independent Gaussian variables, prepare the Nataf transformations: ranfield build, replace, e_mod_field 15,

rf_eigenvalues rf_var/

• Perform Monte Carlo simulation in a loop: #label simulation object modify, add, num_sim 1,/ output print, , num_sim,/ control gosub, , new_sample, / control if, integer less, num_sim max_sim simulation,/ • Generate samples of the random field (including transformation fron Gaussian to non-Gaussian space) and to update the structure accordingly: #label new_sample ranfield sample, , e_mod_field, / element modify, material_data ranfield, e_mod_field 1, / element build, stiffness total,,/ global matrix, stiffness,, stiffnesses/

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compact factorize, , stiffnesses, sky/ compact iteigengesy, skyline, stiffnesses glob_m sky num_values, values vectors/ control return,,,/ The commands as shown above require a highly integrated software environment. For example, the command ranfield sample... generates a realization of the values of the random field (1118 values) and the command element modify... assigns these values as random deviations from the mean elastic modulus to each element. This is used for the computation of the element stiffness matrices and the eigenfrequencies of the structure, whose samples can finally be handled by data analysis commands (statistics) as available in SLang . The simulation results in terms of histograms are given in Figs.16 and 17. The results indicate a relatively high scatter of the two fundamental frequencies (the deterministic systems has two equal lowest natural frequencies at 42 Hz as indicated in fig.16). The coefficient of variation is approximately 15%. 8.0E-2

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Figure 16: Histogram of two fundamental Natural Frequencies Lowest Frequencies of Deterministic System

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Figure 17: Histogram of lowest 10 Natural Frequencies

4 CONCLUDING REMARKS The examples, as presented above, indicate that the need for an integrated software package containing both structural analysis and probabilistic (reliability) analysis very much depends on the specifics of the problem under consideration. Traditional reliability issues, involving randomness of a small number of loading parameters only, can be handled satisfactorily by separate PI/FEA packages. In contrast, problems involving spatial structural randomness, especially if described in terms of random fields or processes, require a highly integrated SSA software environment allowing close communication between FE analysis and probability integration.

ACKNOWLEDGEMENT The results as presented here were partially obtained within the research project “Probabilistic numerical analysis based on component software” which is supported by the Deutsche Forschungsgemeinschaft under Contract No. Be-2160/3-1. This support is gratefully acknowledged.

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REFERENCES Brenner, C.E. 1994. On methods for nonlinear problems including system uncertainties, Proc. ICOSSAR’93, August 9-13, 1993, Innsbruck, Austria, A.A. Balkema Publ., Rotterdam, pp 311 - 317. Bourgund, U. and Bucher, C.G. 1986. Importance sampling procedure using design points - ISPUD - a user’s manual, Report 8-86, Institute of Engineering Mechanics, University of Innsbruck, Austria. Bucher, C.G. 1988. Adaptive Sampling - An Iterative Fast Monte Carlo Procedure, Struct. Safety, Vol.5, No.2, pp 119 - 126. Bucher C.G. and Bourgund U. 1990. A fast and efficient response surface approach for structural reliability problems, Structural Safety, 7 , pp 57-66. Bucher C.G. and Brenner C.E. 1992. Stochastic Response of Uncertain Systems, Archive of Applied Mechanics, 62, 507–516. Bucher C.G., Pradlwarter H.J., Schu¨eller G.I. 1989. COSSAN - Ein Beitrag zur Software-Entwicklung f¨ur die Zuverl¨assigkeitsbewertung von Strukturen, VDI-Bericht Nr. 771, D¨usseldorf, Germany, 1989, pp 271 - 281. Bucher C.G. and Schu¨eller, G.I. 1994. Software for Reliability Based Analysis, Structural Safety 16, pp 13 - 22. Bucher C., Schorling, Y., Wall, W. 1995. SLang - the Structural Language, a tool for computational stochastic structural analysis, Proc., 10th ASCE Eng.Mech.Conf, Boulder, CO, May 21-24, 1995, pp 1132 - 1126. Bucher C. 1997. ISPUD4.1 for the Apple Power Macintosh, User’s manual, Institut f¨ur Strukturmechanik, Bauhaus-Universit¨at Weimar, Germany. Ghanem R.,Spanos P. 1991. Stochastic Finite Elements - A Spectral Approach, Springer-Verlag, New York. Gollwitzer, S., Zverev A., Cuntze, R., Grimmelt M. 1994. Structural reliability applications in aerospace engineering, Proc. ICOSSAR’93, August 9-13, 1993, Innsbruck, Austria, A.A. Balkema Publ., Rotterdam, pp 1265 - 1272. Kr¨atzig, W.B, Metz, H., Schmid, G., Weber, B. 1977. MISS-SMIS, Ein Matrizeninterpretationssystem der Strukturmechanik f¨ur Praxis, Forschung und Lehre, Mitteilung Nr. 77-5, Institut f¨ur Konstruktiven Ingenieurbau, Ruhr-Universit¨at Bochum. Liu P.-L. and DerKiureghian A. 1989. Finite-Element Reliability Methods for Geometrically Nonlinear Stochastic Structures, Report No. UCB/SEMM-89/05, Dept. of Civil Engineering, University of California, Berkeley, USA. Madsen, H.O., 1988. PROBAN: Theoretical Manual for External Release, Technical Report No 88-2005, AS Veritas, Oslo, Norway. Millwater H.R., Wu Y.-T., Dias J.B., McClung R.C., Raveendra S.T.,Thacker B.H. 1990. The NESSUS Software System for Probabilistic Structural Analysis, A.H-S. Ang et al. (eds.): Structural Safety and Reliability, ASCE, New York, N.Y., Vol III, pp 2283 - 2290. Roos, D., Bucher, C., Bayer, V. 1999. Polyhedral response surfaces for structural reliability assessment, paper accepted for presentation, ICASP8, 13 - 16 December 1999. RYFES - Reliability Software Combined with your Finite Element Software, Software Information Sheet, Institut Francais de mechanique avancee, 1995. Schorling Y. 1997. Beitrag zur Stabilit¨atsuntersuchung von Strukturen mit r¨aumlich korrelierten geometrischen Imperfektionen (Contribution to stability analysis of structures with spatially correlated geometrical imperfections, in German), doctoral dissertation, Bauhaus Universit¨at Weimar, Germany.

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