INTERNATIONAL JOURNAL OF SCIENTIFIC & ENGINEERING RESEARCH, VOLUME 4, ISSUE ş, ȱ2013

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Geometrical Nonlinear Analysis Considering

Probabilistic Material Nonlinearity

Iffat Siddique

1Dipartimento di Ingegneria Strutturale, Edile e Geotecnica(DISEG), Politecnico Di Torino, Torino, Italy.

(*) Email: iffat.siddique@polito.it

AbstractThe statistics reflecting the contribution of the material parameters to the total variability of the response parameter are presented by carrying out sensitivity analysis (SA). To accomplish SA a numerical model for the geometrical plus material nonlinear analysis of 2D structural elements is developed. The model employs corotational formulation combined with numerical integration and hence is suitable for many commonly used cross sectional shapes. The accuracy of the proposed algorithm is validated through examples from previous literature. Then material uncertainties are addressed in probabilistic fashion through Monte Carlo simulations.

Keywords — co-rotational formulation, geometrical and material nonlinear structural analysis, Monte Carlo Simulations, numerical integration, sensitivity analysis.

1INTRODUCTION

one of the fundamental assumptions of linear analysis is that neither the shape nor the material properties change during all the deformation process i.e. change in the stiffness is really small. Such linear analysis provides merely an approximation of the real behavior of the structures. Most of the challenging problems call for nonlinear analysis to have a real picture of the structural behavior. Although changing stiffness is common in all types of nonlinear analysis, nonlinearities are classified depending upon the principle origin. A lot of research work is done on the nonlinear analysis (section2.1&2.2) but considering both types of nonlinearities together is reported in very few works [1]. Uncertainties are an inevitable part of analysis and it’s really important to account for them in analysis. In contrast to old safety factor method new and better approach is to address these uncertainties in the probabilistic way (section 2.3). In order to check how reliable our analysis is, it is always fruitful to conduct sensitivity analysis for a number of important decisive results (section 3).

In this paper a sensitivity analysis for the material properties of 2D RC structures is presented. For it a MATALAB code addressing nonlinear geometrical and a nonlinear elasto-plastic material behavior of structures is developed. The code is then validated through examples from previous texts at various stages. Further the uncertainties in the material properties are simulated using crude Monte Carlo method. Then a detailed sensitivity analysis is carried out for compressive strength of concrete and yield strength of steel as these are the two most influencing material properties and effect of other
properties like young’s modulus on the system can be easily inferred from the results of these two properties.

2 PROBABILISTIC ANALYSIS AND NONLINEARITIES

2.1 GEOMETRICAL NONLINEARITIES

Geometric nonlinearities results when the forces producing structural deformations are a nonlinear function of the displacements and leads to change in shape of the structure. Geometric nonlinearities are extremely important in collapse simulation because they capture the effects of buckling, large changes in structural shape and the changes in internal forces necessary to keep the structure in static equilibrium. Except for very simple problems, it is extremely difficult to obtain a closed form solution. Hence incremental iterative techniques are to be used based on computer simulations. Various formulations address these nonlinearities depending upon the kinematic description and the choice of the reference frame. In the context of geometrically nonlinear FEM analysis, three kinematic descriptions have been extensively used. Total Lagrangian formulation, Updated Lagrangian formulation [2], and corotational formulation (CR) [3]. In the analysis done here CR formulation has been used because of the several advantages as described in the next sections. A summary of majority of the important research works about geometrical nonlinear analysis conducted in the past is also summarized in some papers. [4].

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2.1.1 COROTATIONAL CONCEPT

When a frame element is loaded it will deform from its original configuration. During this process each element undergoes the following three things: translation, rotation and deformation. The rotation and translation being the rigid body motions can be removed from the beam element leaving behind the strain producing deformations. The strain producing local deformations are the ones related to the forces induced in the beam elements.
A corotational formulation separates these two components at the local element level by attaching a local element reference coordinate system, which rotates and translates with the beam element. The rigid body rotations and translations are zero with respect to this local co-rotating coordinate system. The derivation of corotational formulation to get relationship between global and local variables, the angle of rotation and a variationally consistent tangent stiffness matrix can be found in literature like for example [5]. Many of the structural materials experience large rotations but small strains. CR formulation can very effectively treat such problems. It can decouple small-strain material nonlinearities from geometric nonlinearities. CR is very well suited to the treatment of structural elements having rotational degree of freedom for arbitrary large rotations e.g; beams, plates and shells etc. It is extremely difficult to treat with such problems with TL description which is the main competitor of CR formulation [6]. Keeping in view all stated above it is preferred in code developed here.

2.2 MATERIAL NONLINEARITIES

Large deformations lead to post-elastic response in the structures. In order to simulate these large deformations it is generally necessary to account for material nonlinearities. In literature there are a lot of ways of considering these nonlinearities in analysis. Among them noticeable ones include: through the development of concentrated plastic hinges [7], studying gradual development of inelasticity across the beam depth referred to as distributed plasticity approach [8]. Sivaselvan and Reinhorn [9] presented a flexibility based approach to the collapse of plane frames in contrast to the previous mentioned displacement based approach. One of the key benefits of flexibility based approach is the ability to use single frame member compared to multiple element discretization used in the displacement-based approach. We can also find in literature
fiber-beam element using flexibility [10].The fiber beam
element during its incremental global analysis, adopts integration of fiber across the beam depth and hence the designer can keep track of the state of the distributed plasticity.
By combining material and geometrical nonlinearities it becomes possible to model plastic and geometrical instabilities, which can be found in many of the previously cited works [1]. In our analysis material nonlinearities are in-cooperated through numerical integration across the volume of the element along with the geometrical nonlinearities.

2.3 PROBABILISTIC ANALYSIS AND ITS NEED

Presence of uncertainties in the analysis and design due to measurement, physical, mechanical or statistical constraints has been recognized by the engineers since years. In the past the tradition was to simplify the problem by considering the uncertain parameters as deterministic ones and then accounting for the uncertainties by using empirical safety factors. As these factors were determined based on past experience and hence do not guarantee fully the safety of structures. Also, they do not give any idea about how different parameters influence the structural safety as they do not take into account the underlying distribution of the random variables involved in the system. Also, the deterministic safety factor approach does not provide adequate information to achieve optimal use of the available resources while maximizing safety at the same time. A new increasing fashion of addressing these uncertainties is the probabilistic analysis which takes into account the parameter variability along with its underlying distribution. Hence it provides more information about the system behavior, the influence of various parameters on system performance along with their interaction among themselves.
Material uncertainties due to skill and experience of workmanship, various manufacturing procedures and plants, environmental impact, existing structures etc contribute a significant part to the overall uncertainties of the system. In our analysis only material uncertainties have been addressed. The main source used for the probabilistic input data was JCSS model code. Table-1 summarizes all the data recommended in the code.

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3 SENSITIVITY ANALYSIS

Sensitivity analysis is the study of how uncertainty in the input affects the uncertainty in the output of a mathematical model or system. Before stepping into the detailed reliability analysis it is always fruitful to conduct sensitivity analysis for a number of important decisive results. Among them includes: which parameter require additional research for concrete knowledge about the system and hence reducing output uncertainty, parameters that have a minor impact on the system output and therefore can be eliminated resulting in model simplification, correlations between the input parameters and output and many other.

3.1 METHODS OF SENSITIVITY ANALYSIS

There are several ways of undertaking sensitivity analysis. These are generally grouped as: one way sensitivity analysis which allows reviewer to assess the impact of changes in one specific parameter on the model output. Then we may have multiway sensitivity analysis in which it may be necessary to examine the relationship of two or more different parameters changing simultaneously. However, the presentation and interpretation of multiway sensitivity analysis becomes increasingly difficult and complex as the number of parameters involved increases [11]. Also there are probabilistic sensitivity analyses that provide a useful technique to quantify the level of confidence that a designer has in its decisions. In probabilistic sensitivity analysis, rather than assigning a single value to each parameter, a probabilistic distribution is assigned to all the parameters of the mode. Hence a range of data is assigned through mean value, standard deviation and ‘shape’ of the data spread.
In this work several methods have been selected covering two important broad categories of sensitivity analysis mentioned above. Among the one way sensitivity analysis sensitivity index and parameter uncertainty factor were calculated [11]. Also Pearson correlation coefficient has been determined. Thereafter probabilistic SA has been performed by random sampling following probabilistic distribution of the parameters.

3.2 PRESENTING SENSITIVITY ANALYSIS:

Sensitivity Analysis methods can be classified in a variety of ways and accordingly results can be presented. Mathematical methods used for the validation and
verification purposes as they do not address the variation in the output due to the variation in the input. Then there are statistical methods which involve running simulations in which inputs are assigned probability distributions and assessing the effect of variance in inputs on the output distributions [11]. They allow identifying the effect of interactions among the individual inputs. Also we have some graphical methods which give representation of sensitivity in the form of graphs, charts or surfaces[12](Christopher Frey and Patil, 2002). They can be used as a screening method before the further analysis of a model to represent strong dependencies among input and output variables [12].
The mathematical results of the sensitivity analysis is presented herein Table-3. Also graphical representation of the results in the form of histograms and scatter plots are included in the Section-6 (Results and Discussion).

4 COMPUTATIONAL METHODOLOGY

The work was started by taking a model problem of a beam and developing a MATLAB code based on corotational formulation for the geometrical nonlinear analysis. The code was based on some mathematical expressions for calculating the internal force vector taken from lecture notes of Yaw and the results in the form of load deflection curve were compared with the curve given by Yaw [5]. The code so far could perform geometrical non linear analysis for the linear elastic 2D elements only. After verifying that the results were an exact match the code was then modified. These expressions were afterwards replaced by generalized expressions in the form of integrals over the volume of the element [13] and the code was modified to compute internal force vector by numerical integration. This modified code results for the linear elastic material model was then verified against the previous results. The Load deflection curves for both the codes are shown in fig-
1. Afterwards the constitutive law was changed from linear
elastic to elasto-plastic to in cooperate material non linearity along with geometrical non linearity for the analysis of the line elements. Uncertainties are always an inevitable part of the data also of material characteristics so to use nominal strength values in analysis becomes questionable. Hence probabilistic approach was adopted. A small code was written to simulate the probabilistic characteristics of the material properties (compressive strength of concrete and yield strength of steel) using
Monte Carlo simulation method [14]. And finally

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sensitivity analysis was carried out for the two random parameters considered i.e. compressive strength of concrete and yield strength of steel.

4.1 SENSITIVITY ANALYSIS:

4.1.1 SENSITIVITY INDEX (SI):

It is a simple method to calculate %age difference when varying one input parameter from its minimum to its maximum value. For this we vary the parameter, whose sensitivity is to be evaluated by increasing it by a percentage of standard deviation while keeping the other parameter constant at its mean value. This helps us to calculate output %age difference (Table-3) as:
To understand the overall system behavior we consider the randomness of both the variables involved in our system. According to the probabilistic distribution of both the parameters random samples were obtained and used in the calculation of the ultimate load. This is the most important of all the analysis as it is most close to the reality problems and takes into account the interaction of both parameters as well. The results of this analysis are shown in Fig-4.

5 EXAMPLE PROBLEM:

5.1 CONFIGURATION:

As an example problem, in order to validate the authenticity of the developed code, a cantilever column with a fixed vertical compressive load of 1280 KN and a
variable horizontal load (CEB/FIP manual of Buckling and

SensitivityIndex(SI )= Dmax Dmin

Dmax

(1)

instability; pg. 29) was selected. The sectional and material properties are given in Table-2. All the probabilities data used in the analysis is provided in Table-1.
Dmax/min= maximum/minimum value of ultimate load.

4.1.2 PARAMETER UNCERTAINTY FACTOR (PUF):

Another statistical method of evaluating the importance of parameters is the parameter uncertainty index (PUF) and is given as:

5.2 COMPUTER MODELING:

The modeling has been done using MATLAB by dividing the column into eight equal finite elements along its length with nine nodes. A constant compressive force of 1280KN acts on it and a variable horizontal load is applied in
increasing incremental manner to get the maximum

PUF = 2Std change output

Change input

4.1.3 PEARSON CORRELATION COEFFICIENT (RX,Y):

(2)

horizontal load that can be applied.
For carrying out numerical integration; the section was divided into varying no. of strips to achieve integration along the cross section. Further to get integration along the
In order to have a better idea of the parameter randomness
on the system output, random samples of one of the parameters according to its probabilistic distribution were simulated using Monte Carlo simulation technique while the other sample taken as a constant value(either equal to its Mean, Mean ±Std, Mean±2Std). The Pearson correlation coefficient is then computed to give an estimate of the correlation between the input and output and is given by:

𝑁𝑁

� �Xi −X �(Yi −Y )

entire volume of the element, Gauss quadrature was adopted for the integration along the length of the element. Here we have taken four gauss points.

TABLE 1

STATISTICAL PARAMETERS FOR COMPRESSIVE AND YIELD

STRENGTHS

Rx, y = 𝑖𝑖 =1

(3)

𝑛𝑛

2 𝑛𝑛 2

�(�𝑖𝑖 =1(Xi −X ) − �𝑖𝑖 =1(Yi −Y ) )

X’ and Y’ represent the mean values of the input and output parameters. The results are shown in Table-3 along with the associated graphical representation in the section-6 (results and discussion).

4 .1.4 PROBABILISTIC SENSITIVITY ANALYSIS:

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TABLE 2

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SECTIONAL AND MATERIAL PROPERTIES OF EXAMPLE PROBLEM

Es

6 RESULTS & DISCUSSION

The response of the structure (ultimate load) obtained from both analytical method and numerical integration techniques are in close agreement showing the accuracy of the developed base code (Fig.1).

Fig.1 Comparison of Analytical & Numerical Integration

Fig.2 Load Deflection Curve

The Fig.2 shows the load deflection curve of the example problem and is found to in exact synchronization with the one given in the CEB/FIP manual. The results of the sensitivity analysis are presented in the table below:

TABLE 3

STATISTICS OF SENSITIVITY ANALYSIS

Quantity

SI

PUF

Rx,y

fc

0.23

6

-0.984

fy

0.12

3

-0.993

Though all these are one way sensitivity analysis procedures. The first two methods compares output variability at some specified points of the input parameter. Both of them give quite similar results. That compressive strength of concrete has a much larger contribution to the variability of the response variable (i.e; ultimate load. The Pearson correlation coefficient of both the variables is close to -1. This means that the there is a strong linear relationship between either of the input variables and the ultimate load capacity of the structural element being investigated. And hence ANOVA methods have not been investigated in this study so far. The scatter of the data with respect to two variables considering one of them as variable is shown in Fig.3. The scatter plots show that the due to randomness of the compressive strength the data is scattered evenly within 80 to 105 KN (for this specified problem) while in case of yield strength the data is clustered at specific values though this clustering is in the same range of 80 to 105 KN. Hence it may be concluded that though a slightly different distribution may fit the results of both the variables but the statistics will be quite similar (e.g. mean and Std). The results of the probabilistic sensitivity analysis considering both the variables as random along with the effect of their interaction are plotted as a histogram. Various distributions were tried to fit in the ultimate load data and it was found that lognormal distribution is the one that comes out to be the best fit as can be seen in the next figure (Fig. 4).

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Fig.3 Scatterplots showing effect of parameter variability on the distribution of structural response

Fig.4 Probabilistic SA and best fit PDF for ultimate Load

7 CONCLUSIONS

A sensitivity analysis for the material properties has been conducted. For this firstly, a MATLAB code for the geometrical nonlinear analysis based on corotational
formulation was developed. In the model elasto-plastic nonlinear material behavior has been considered.
Uncertainty in the material properties are addressed in a probabilistic fashion simulated using Monte Carlo simulations. And finally sensitivity analysis has been performed though which it is concluded that: both the yield strength of steel and compressive strength bear a strong linear relationship with the failure load. And ultimate load is found to have a log normal probabilistic distribution.

ACKNOWLEDGMENT

The constant support of my professor Prof. Mancini and his assistant Allaix Diego for all my Phd research is greatly acknowledged.

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