ABSTRACT

A probabilistic approach to the nonlinear analysis of civil engineering structures is presented. Finite element software ATENA is used for realistic simulation of concrete and reinforced concrete structures. It is combined with the stochastic package FREET into software system SARA. This system enables to take into account uncertainties and randomness of structural input parameters for the nonlinear analysis. SARA system offers a user-friendly interactive graphical environment as well as an extensive database with statistic information about material, geometrical and other structural and load properties in order to support the user during his task. The basic aim of the probabilistic-based nonlinear analysis is to obtain an estimation of the structural response statistics (failure load, deflections, cracks, stresses, etc.), which is used for sensitivity studies, safety and reliability assessment as well as for identification of structural computer models

Keywords:
probabilistic analysis, reliability assessment, computer simulation, concrete structures, fracture mechanics

1 Introduction

Optimum balance between cost and safety of concrete structures, e.g. bridges, is becoming a common problem worldwide. It has been found that reliability assessment which is going beyond the boundaries of codes can bring a significant money saving and provide a new insight into bridge administration and decision-making process (Casas et al. 2002, Enevoldsen 2001, Frangopol 2000). Methodologies for use of probabilistic based assessment are available and have been proven to work in practice. But suitable tools for use in design offices are generally missing. The authors of this paper combined efficient techniques of both fracture mechanics and reliability engineering to achieve the goal: to assess realistic behavior of concrete bridges from reliability point of view. The aim of the paper is to present applications of probabilistic-based assessment approach. It is based on recently developed integrated system of nonlinear fracture mechanics software ATENA and probabilistic module FREET called SARA (Structural Analysis and Reliability Assessment). As nonlinear analysis is computationally intensive, a suitable technique of statistical Monte Carlo simulation should be utilized which requires rather small number of simulations for accurate results. Latin Hypercube Sampling technique appeared to be efficient technique in this context. The feasibility and outcomes of the approach are documented on selected numerical examples of stochastic failure simulation of concrete structures resulting in reliability assessment.

2 NONLINEAR SIMULATION OF ENGINEERING STRUCTURES

ATENA is well-established finite element software for realistic computer simulation of damage and failure of concrete and reinforced concrete structures in deterministic way (Cervenka 2000, 2002). The constitutive relation in a material point (constitutive model) plays the most crucial role in the finite element analysis and decides how the structural model represents reality. Since concrete is a complex material with strongly non-linear response even under service load conditions, special constitutive models for the finite element analysis of concrete structures are employed (Cervenka et al. 2001).

2.1 Material models
Tensile behavior of concrete is modeled by non-linear fracture mechanics combined with the crack band method and smeared crack concept, Figure 1. Main material parameters are tensile strength, fracture energy and shape of the stress-crack opening curve.

Fig 1: Smeared crack model for tensile behavior of concrete.

Fig 2: Example - crack band in a shear wall analysis.

A real discrete crack is simulated by a band of localized strains, Figure 2. The crack strain is related to the element size. Consequently, the softening law in terms of strains for the smeared model is calculated for each element individually, while the crack-opening law is preserved. This model is objective due to the energy formulation and its dependency on the finite element mesh size is neglectable, which was confirmed by numerous studies (e.g. Cervenka & Pukl 1995).

Two alternative crack models are recognized: fixed and rotated crack model. In the fixed crack model the crack direction is determined and fixed at the time of crack initiation. In the rotating crack model the crack direction is identical with a principal strain direction at every stage and rotate if the strain direction changes. The main difference in these crack models is the absence of shear stresses on the crack plane in the rotating crack model due coincidence of principal strain directions with the crack orientation, which makes the rotating crack model more simple. In the fixed crack model the shear resistance of the cracks is modeled by means of the variable shear retention factor, which reflects the aggregate interlock effect of cracked concrete. Concrete in plane stress condition can be well described by a damage model. The model used in ATENA is based on the equivalent uniaxial law, which covers the complete range of the plane stress behavior in tension and compression. The effect of biaxial stress state on the concrete strength is captured by the biaxial failure function. For the tensile response (cracking) the crack band method described above is applied. Similar method is applied for the compressive softening. Thus the complete softening behavior is based on an objective and mesh independent approach.

Fig 3: Concrete failure surface in 3D-stress state.

Concrete in the three-dimensional stress state is covered by the theory of plasticity with a non-associated flow rule, Figure 3. An important phenomenon in the three-dimensional behavior of concrete, which is reflected in the implemented material model according to Menétrey & Willam (1995) is a strong influence of the lateral stresses to the compressive strength, so called confinement. The strength is increasing with the hydrostatic compressive stress. The plasticity theory describes also the plastic flow and volume change due to a distorsion and can model the volume increase of concrete under plastic deformations. Cervenka & Cervenka (1999) implemented a predictor-corrector scheme with return mapping algorithm covering this behavior. The general three-dimensional behavior of concrete can be also well described by the microplane models based on a micro-structural damage idea: constitutive laws are formulated on several planes with an arbitrary orientation - microplanes. Stress-strain equations are defined independently for projection of the macroscopic strain tensor to the microplanes. The macroscopic material response (stress tensor in a material point) is consequently integrated over all microplanes by principle of virtual work. Microplane model by Bažant et al. (2000) is implemented in ATENA and can be used for solution of practical cases. Many other material models are implemented in ATENA in order to support a successful simulation (reinforcement, steel, bond slip of reinforcing bars, interface, soil, etc.).

2.2 Features and structure of ATENA software
An efficient solution of engineering problems based on the described material models is supported by user-friendly graphical environment (GUE). Native ATENA GUE is available for 2D and rotationally symmetrical problems. It supports the user during pre- and postprocessing and enables real-time graphical tracing and control during the analysis.

The preprocessing includes an automatic meshing procedure, which generates Q10, isoparametric quadrilateral and triangular elements. Reinforcement can be treated in form of reinforcing bars, prestress-ing cables or as smeared reinforcement given by re-inforcement ratio and direction. The discrete rein-forcement is fully independent on the finite element mesh. The structure can be loaded with various actions: body forces, nodal or linear forces, supports, pre-scribed deformations, temperature, shrinkage, pre-stressing. These loading cases are combined into load steps, which are solved utilizing advanced solution methods: Newton-Raphson, modified Newton-Raphson or arc-length. Secant, tangential or elastic material stiffness can be employed in particular models. Line-search method with optional parameters accelerates the convergence of solution, which is controlled by energy-based and residua-based criteria. The graphical postprocessing can show cracks in concrete with their thickness, shear and residual normal stresses. User-defined crack filter is available for obtaining of realistic crack patterns. Other im-portant values (strains, stresses, deflections, forces, reactions etc.) can be represented graphically as ren-dered areas, isoareas, and isolines, in form of vector or tensor arrow fields. All values can be also ob-tained in well-arranged numerical form. In the necessary modifications of ATENA for implementation into SARA system it was a crucial point to keep all of the features described in the concise survey above available also for the repeated stochastic analysis. It was enabled due to versatile programming architecture and build up of ATENA system, which supersedes usual finite element packages.

3 PROBABILISTIC-BASED ASSESSMENT

A multi-purpose probabilistic software for statistical, sensitivity and reliability analysis of engineering problems FREET (Feasible Reliability Engineering Efficient Tool) is based on efficient reliability techniques. In general, it is designed in the form suitable for relatively easy assessment of any user-defined computational problem written in C++ or FORTRAN programming languages, but it is focused especially on the computationally intensive problems, which do not allow performing thousands of samples. Software FREET and the methods utilized in this program are subject of another papers submitted to this conference by Novák et al. (2003) and Vorechovský & Novák (2003). Therefore, only some basic ideas and principles are comprehensively mentioned here.

3.1 Latin hypercube sampling
A special type of numerical probabilistic simulation called Latin Hypercube Sampling (LHS) makes it possible to use only a small number of Monte Carlo simulations. LHS uses the stratification of the theoretical cumulative probability distribution function (CPDF) of input random variables. CPDF for all random variables are divided into N equivalent intervals, where N is the number of simulations. Centroids of intervals are then used in simulation process. The representative parameters of variables are selected randomly based on random permutations of integers 1, 2, ..., j, ..., N. Every interval of each variable must be used only once during the simulation (Novák et al. 2003).

3.2 Statistical correlation by simulated annealing
Statistical correlation among input random variables can be considered. Stochastic optimization technique called Simulated Annealing (Vorechovský & Novák 2003) is utilized to adjust random samples in such a way that the resulting correlation matrix is as close as possible to the target (user-defined) correlation matrix. Note that the approach allows working also with non-positive definite matrix on input, which can be the result of lack of knowledge of the user. This technique generates samples as close as possible to a positive definite matrix (mathematically and physically correct).

3.3 Sensitivity analysis
An important task in the structural reliability analysis is to determine the significance of random variables, i.e. how they influence a response function of a specific problem. The dominating and non-dominating random variables can be distinguished. Sensitivity analysis approach based on nonparametric rank-order statistical correlation with Spearman correlation coefficient or Kendall's tau is employed (Novák et al. 2003). This technique is distribution free and quite robust. Parallel coordinates representation in FREET graphical user environment gives an insight into statistical structure of relationship between random input variables and response output variables.

3.4 Reliability assessment
Cornell´s reliability index could be calculated in FREET from the limit state function under assumption of normal probability distribution for both structural resistance and acting load. Reliability index is estimated from mean value and standard deviation of the limit state function. Histogram of safety margin as specified in limit state function definition can be visualized. The results can be compared with the target reliability index, e.g. 4.7 for 1 year as specified by Eurocode (2001).

4 NONLINEAR STOCHASTIC SIMULATION

The programs FREET and ATENA are integrated in software package SARA (Structural Analysis and Reliability Assessment) in order to allow for a probabilistic nonlinear analysis of concrete structures (Bergmeister et al. 2002, Pukl et al. 2003). It enables also degradation analysis of reinforced concrete structures as shown by Teplý et al. (2003).

4.1 SARA Studio
An interactive graphical shell SARA Studio was developed in order to assure well-arranged data exchange and management as well as control of both mentioned programs and additional supporting tools. The whole process of the nonlinear stochastic simulation is controlled by the user due to commands and interfaces available in SARA Studio.

4.2 Randomization of input variables
The material properties and other input parameters used in ATENA deterministic analysis are firstly defined. These values are exported to FREET, where they will be used as mean values for random distributions of the corresponding variables. Further stochastic parameters (variance, type of the probability density function) for selected variables are defined directly in FREET. The randomness of input variables reflects uncertainties and randomness of the input values regarding material properties, geometry of the structure, prestressing etc. Integrated database of stochastic parameters for various structural and material properties (concrete, reinforcing steel, prestressing, geometrical imperfections) is available in order to support the user in preparing of the stochastic input data. Correlation between random input variables can be introduced in form of the correlation matrix. Number of samples is selected according to complexity of the problem to be solved and required quality of expected results. Already 8 samples could give a reasonable estimation of stochastic parameters of the structural response and reliability index prediction.

4.3 Repeated nonlinear solution
In the next step, sets of input parameters for the required number of samples are generated by FREET. SARA Studio prepares input data for the multiple analysis using ATENA. The single samples are consequently solved in ATENA under SARA Studio control. Selected results from the structural response from ATENA solution (ultimate load, deflection, maximum crack width etc.) are collected. Finally, obtained results are transferred to FREET and evaluated in form of histograms of structural response and sensitivity plots. Reliability index can be assessed.

4.4 Solution procedure
The whole solution procedure can be itemized as follows:

1. Deterministic model of the structure is prepared and checked within ATENA.
2. Uncertainties and randomness of the input parameters are modeled as random variables described by their probability density functions (PDF). The result of this step is the set of input parameters for ATENA computational model - random variables described by mean value, variance and other statistical parameters (generally by PDF).
3. Random input parameters are generated according to their PDF using LHS sampling. Statistical correlation among the parameters is imposed using simulated annealing.
4. Generated samples of random parameters are used as inputs for ATENA computational model. The complex nonlinear solution is performed and selected results (structural response) are saved.
5. Previous two steps are repeated for all samples.
6. The resulting set of structural responses from the whole simulation process is statistically evaluated. The results are: histogram, mean value, variance, coefficient of skewness, empirical cumulative probability density function of structural response, sensitivity evaluation, reliability index assessment.

5 APPLICATION EXAMPLE

The feasibility and outcomes of the stochastic fracture analysis are documented on a practical example of statistical failure simulation and reliability assessment of existing bridge structure: cantilever beam bridge on the Brenner Motorway in Italy with a length of 167.5 m. Another example of segmental prestressed bridge was presented by Pukl et al. (2002).

Fig 4: Bridge scheme.

5.1 Highway bridge - cantilever beam
The fully post-tensioned box-girder bridge built in 1969 is one of the objects of the Brenner Highway in Italy. The Brenner Highway consists of two separated lanes, each of them includes several objects of this type. The bridges are cast-in-place balanced cantilever beams with varying girder depth, Figure 4. The mid-span has a length of 91 m, the cantilever beams have a lengths of 59 m and 17.5 m. Total length of the bridge is 167.5 m. The lane slab has a width of 10.60 m and a thickness is of about 0.20 m. The lower girder slab has a width of 6.00 m and a thickness of about 0.20 m. The height of the box girder varies from 10.80 m over the middle support to 2.85 m in the mid-span. The bridge is cast from concrete B500 and is reinforced with mild steel BST 500. The post-tension tendons system consists of 211 strands of St 1350/1500. The bridge was pre-stressed and then loaded with a prescribed line load along the whole lane slab. The ultimate failure load and the descending branch were obtained in the analyses.

5.2 Stochastic simulations
Stochastic simulations with 8 and 30 samples were performed. Mean values of the concrete properties for nonlinear analysis were generated from the cubic compressive strength using ATENA defaults, i.e. built-in recommendations by CEB, fib, RILEM etc. Statistical properties of the random variables (Table 1) originated mostly from the database available in the SARA system. The coefficient of variation for the yield strength of the prestressed cable was estimated rather large due to high uncertainties.

Correlation between material parameters was introduced. The prescribed correlation matrix is shown in the upper triangle of Table 2. The lower triangle of Table 2 shows the correlation matrix generated by simulating annealing for 30 samples.

In the nonlinear simulations (samples) the relationship between the applied line load and the vertical displacement at selected points has been monitored. The ultimate load and postpeak behavior (descending branch) have been obtained. The bridge girder failed typically next to the middle support. First the prestressed tendons yielded, tensile cracks developed in the upper flange of the box girder and finally shear failure occurs. Two different failure modes occurred in the analyses - with negative and positive vertical displacement at the end of the cantilever beams. Histograms of monitored values were obtained and evaluated in FREET, statistical characteristic were estimated. Estimated statistical characteristics of the ultimate load (resistance of the structure) for 8 and 30 samples are compared in Table 3. They show that already a sample size of about 8 samples gives good results for this problem.

5.3 Reliability assessment
Calculated reliability index for the ultimate load under consideration of different levels of the line load (mean values) is plotted in Figure 7. Various values of coefficient of variation are compared. The horizontal line represents the target reliability index as specified by Eurocode (2001) (4.7 for 1 year).

Fig 7: Reliability assessment of the ultimate line load / Reliability assessment of displacement at the cantilever north.

Furthermore, reliability of the bridge related to the vertical displacements was investigated and compared with JCSS (2001) regulations, see Figures 7. It resulted that the end of the north cantilever is the critical point regarding the vertical displacements.

CONCLUSIONS

A Methodology for probabilistic-based assessment of concrete structures is introduced and documented on practical example. The programs FREET and ATENA are combined in the software package SARA in order to support stochastic nonlinear analysis. This approach enables stochastic assessment of the analyzed structure, which is going beyond the boundaries of design codes. It can lead to considerable cost saving as the reliability requirements can be targeted more precisely. Additional to this performed stochastic calculation a monitoring system is established at this investigated object. So it is looked ahead to compare the information extracted of the simulation with the monitoring data. This scheme allows to adjust in more detail the stochastic models - some of the parameters are assumed - and furthermore to evaluate the current probability of failure and the remaining lifespan.

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