Optimizing Hysteresis Parameters for RC Structural Elements Using Genetic Algorithm in OpenSees

doi:10.21203/rs.3.rs-4819174/v1

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Optimizing Hysteresis Parameters for RC Structural Elements Using Genetic Algorithm in OpenSees

https://doi.org/10.21203/rs.3.rs-4819174/v1

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An accurate numerical model is crucial for predicting the cyclic hysteresis response of Reinforced Concrete (RC) structural elements, capturing strength and stiffness degradations along with the pinched hysteresis response typical of RC members. OpenSees, an open-source fiber modeling and analysis software, is versatile for modeling RC, steel, and composite structural elements subjected to seismic loading. While the OpenSees library offers various hysteresis material models for the cyclic response of members, selecting the hysteresis parameters of each model often relies on a black-box approach. This study optimizes hysteresis parameters using a Genetic Algorithm (GA) search method. The dynamic GA search compares the energy dissipation in each cycle of the experimental specimens with the numerical models in OpenSees to optimize the hysteresis parameters. The results show an improvement in the fitness between the simulated and experimental hysteresis curves across the generations, plateauing around the 3rd generation. The modified GA methodology terminates the GA when the fitness function plateaus, saving considerable computational run time. This approach offers a reliable method for estimating hysteresis parameters which is consistent with the experimental results.

Seismic response

Genetic Algorithm

Numerical modelling

Hysteresis behavior

OpenSees

Reinforced concrete (RC) framed structures are particularly vulnerable in regions of high seismic intensity. The behavior of these structural systems under seismic loading or reversed di-rection cyclic loading can be best understood by its hysteresis response. Particularly for reinforced concrete (RC) elements the response under cyclic loading is sometimes much lower than that of the monotonic loading. This is primarily due to the crack evolution and propagation which increases the strength and stiffness degradations at each cycle of loading when the loading directions are reversed beyond the peak capacity (FEMA P440A, 2009; Megalooikonomou, 2024). This peculiar phenomenon of RC members is called the pinching behavior wherein the hysteresis curves appear to be pinched near the axes. A reliable numerical model of RC members should be capable of simulating this degradation in strength and stiffness and the pinching behavior. A typical hysteresis curve of an RC column is shown in the Fig. 1b.

OpenSees (McKenna et. al., 2000) is an open-source fiber-based modelling software and is very versatile tool for modeling and simulating the non-linear response of steel and RC structural elements subjected to monotonic and seismic loading. The vast library of OpenSees has several inherent materials models for concrete and steel which can simulate their response to external action. While the monotonic behavior of RC elements is easy to model, the modelling of the cyclic response is often challenging. Unlike the finite element modelling (FEM) where the stresses and strains are distributed over the continuum, the fiber elements in OpenSees distribute the stresses/strains over a limited number of integration points. This makes the modelling of the hysteretic behavior of RC elements difficult as the crack initiation and propagation which leads to a degradation in strength and stiffness typical of the pinching behavior cannot be modelled accurately.

OpenSees however has several hysteretic material models such as ‘Hysteretic’, ‘BWBN’ and ‘pinching4’ material models among others which can all model the strength and stiffness degradations of a pinched hysteresis response. Although all these models require the monotonic stress-strain or load-deformation response of the material/member, they also require additional hysteresis damage parameters which are often selected based on a trial-and-error basis to best fit the experimental results. This procedure is frequently tedious and time-consuming, and it lacks a scientific foundation. A reliable optimization technique must be capable of predicting these hys-teresis parameters with ease and consistent with the experimental results.

Genetic algorithms (GA) are a search technique utilized to optimize multi-dimensional pa-rameter problems within a predefined number of generations (Wright, 1991; Grefenstette 1986). Initially, the GA operates on an initial set of parameter values and progressively converges towards the optimal solution through successive mutations and crossovers in subsequent generations. This method can streamline computations by selectively mutating parameters based on their fitness with respect to the target (Xiaomin et. al., 2009). Although GA has been previously employed for parameter estimation of hysteretic models (Vigh et. al. 2012, 2014; Kamari and Gunes 2009) for RC and steel members, its application for a dynamic prediction of the hysteresis pa-rameters consistent with the experimental responses remains unexplored.

This study focuses on a GA methodology that utilizes a numerical model developed using OpenSees to optimize hysteresis parameters. It employs a fitness function which dynamically compares the variation in the energy dissipated in each cycle of experimental and numerical models. The fitness function, incorporating strength and stiffness degradations, serves as a reliable indicator.

The test specimen which was adopted for testing the applicability of the developed GA method-ology is a rectangular RC column of dimension 300 mm x 300 mm and height 1000 mm. The grade of concrete and steel used in the specimen were M30 and Fe550 respectively. The column specimen previously tested by Gautham and Sahoo (2022) was subjected to fully reversed cyclic displacements applied at the top of the column while the bottom end was fixed to the strong floor of the laboratory. The dimensions and details of the specimen and the load vs. displacement (hysteresis) response is shown in Fig. 1 (a) and 1(b). Table 1 provides details of the material properties of the specimens and the spacing of stirrups. The loading protocol used for applying the lateral displacements were adopted from ACI Committee 374.1-05 (2005).

Table 1

Material properties and stirrup spacing of RC column specimen.
Specimen	$\:{\varvec{f}\:}_{\varvec{c}}^{\mathbf{{\prime\:}}}$ (MPa)	$\:{\varvec{f}}_{\varvec{y}\:}$ rebars (MPa)	Stirrup spacing (mm)
R1	30.1	552	ϕ8@100

The software adopted in this study for modelling the RC members was the open-source fiber-based modelling software: OpenSees. Unlike finite element models where the plasticity is continuous over the span of the member, fiber models distribute the plasticity into a finite number of integration points along the span. This numerical formulation while reducing the computational time is very effective for homogenous elements such as steel and aluminum. However, modelling of composite materials such as reinforced concrete poses a challenge particularly due to the several failure and damage modes typical of RC members. The crack propagations, subsequent spalling and opening and closing of cracks in the RC members subjected to reversed cyclic loads result in a reduction in strength and stiffness which results in the pinched hysteresis response. The OpenSees library has several material models to simulate the pinching behaviour of RC elements such as ‘pinching4’ material, ‘Hysteretic’ material and ‘BWBN’ Bouc Wen material models.

Although the use of OpenSees software to model the concrete and reinforcing steel com-ponents of the RC members subjected to cyclic loading has been investigated in the past (Gautham and Sahoo 2022), a systematic approach to estimate the hysteretic parameters commonly used to simulate the pinched hysteresis response of RC members subjected to reversed cyclic loading has not been explored in detail. In this study a simple hysteretic material model ‘Hysteretic’ having sixteen parameters was adopted. The selection of this material model was based on its restricted set of unknown parameters, which facilitates the dynamic GA to achieve faster convergence.

3.1 Modelling of RC members

The built-in characteristic of the fiber models in OpenSees software is their ability to inherently account for the interaction between axial force and moment. However, the shear force – shear deformation response and bond-slip behavior cannot be modelled implicitly and needs to be de-fined explicitly. Incorporation of the shear and bond-slip response was enabled by the ‘SectionAggregator’ command in OpenSees. Figure 2. Illustrates the fiber element modelling of the RC column and the aggregation of the shear-deformation response with the axial force moment response of the developed fiber models in OpenSees.

Figure 3 represents the modelling of the beamColumn elements (RC column) in OpenSees. Plastic hinge zones are typically located at either end of the column subjected to lateral loads and hence were modelled as displacement-based beamColumn elements near the ends of the numerical model with five integration points which ensures a higher stability in the numerical analysis. The depth of the plastic zone was assumed to be half of the depth of the column cross-section at either end (Berry et. al. 2009). The middle segment of the member was composed of a force-based beamColumn element. The shear-deformation response and bond-slip behaviors were modelled via zero-length springs located at both ends beyond the displacement beamColumn elements. The fiber section of the zero-length spring elements utilized the same material model as defined previously for the concrete, while for the steel rebars, the barSlip material was employed.

3.2 Material Models

The cover concrete and core concrete regions of the column were modelled using ‘Con-crete02’ uniaxial material model in OpenSees. Confined concrete strength and peak strain were estimated using Mander’s model (Mander et. al. 1988). Tensile strength of the concrete was assumed to be 10% of the peak compressive stress. The material model adopted for reinforcement steel was the hysteretic material ‘Hysteretic’. Although the pinched hysteresis response is a property of the concrete (opening and closing of cracks and cover concrete spalling), the hysteretic properties are usually assigned to the reinforcement steel material (Oinam and Sahoo 2018). The idealized backbone stress-strain curve for the reinforcement steel for the Hysteretic material model was adopted from the experimental results of uniaxial tensile test on steel bars (Gautham and Sahoo 2022). Figure 4 shows the stress-strain response of the adopted material models in OpenSees.

The material model which was adopted for modelling the shear force vs. shear deformation relationship was the ‘RotationShearCurve’ limit state material. By using the limit state material, damage is triggered only if the shear capacity of the member is exceeded and the degradation follows the shear limit curve as proposed by Elwood (2008). The shear capacity of the RC column was estimated using ACI 318 − 19 (2019) and the degrading slope was estimated using Elwood (2008) and the residual shear strength was assumed to be 20% of the shear capacity. The shear deformation response was appended to the beamColumn element by using the ‘sectionAggregator’ command (Fig. 3).

In real-world experiments, longitudinal reinforcement bars often begin to slip out of the concrete once cracking initiates near the junctions of the beam-column and column-footing. This slipping results in additional joint rotations. To replicate this behavior, the 'Barslip' material model was implemented in the zeroLengthElement added at both ends. This material model is specifically applied to the rebars in the fiber section, while the material model for concrete remains unchanged.

For isotropic materials such as steel, fatigue fracture occurs when loaded multiple times at the same level of stress. This behavior was simulated using the ‘Fatigue’ material model which triggers a fracture in the material when the maximum value of strain is reached for the material. Fatigue material model was applied to the reinforcement bars in addition to the ‘Hystereic’ material model defined previously.

Genetic Algorithm (GA) is a search optimization tool used to estimate the best fit parameters of multi-dimensional problems working on the principle of natural selection. The optimization tool works on a predefined pattern of mutations and cross-overs to converge to the optimal solution within a few generations. GA methodology have been successfully implemented to estimate the hysteresis parameters of Bouc Wen hysteresis models in the past (Xiaomin et. al. 2009).

In the current study, a genetic algorithm is developed using MATLAB (2022) software for the estimation of the parameters of the ‘Hysteresis’ material model for the steel rebars in OpenSees software while the rest of the material models were as defined previously. The parameters of the ‘Hysteretic’ model for which GA model was used were ‘pinchx’, ‘pinchy’, ‘damage1’ and ‘damage2’ while the remaining parameters (e1p, s1p, e2p, s2p, e3p, s3p) were adopted from an idealized bilinear uniaxial tensile stress-strain curve for reinforcement steel. The values of stress and strain in the negative loading direction (e1n, s1n, e2n, s2n, e3n, s3n) were assumed to equal but opposite sign as that of the corresponding values in the positive loading direction.

Figure 5 illustrates the flowchart of the proposed GA for estimating the hysteresis parameters. The energy dissipated at the end of each cycle of the lateral drift was calculated through numerical integration for both the experiment and the simulation. The difference between the dissipated energy of experiment and simulation served as the fitness function for assessing the hysteretic parameters. The chromosome's representation in a specific generation improves as the fitness function value decreases. This iterative process continued until the optimal values of the parameters were ultimately identified.

4.1 Chromosome

The parameters to be estimated makeup the form of the chromosome. In the current study sixteen parameters are required to accurately model the hysteretic material in OpenSees out of which the first twelve parameters were adopted from the idealized stress-strain backbone curve of the reinforcement steel. The stress-strain response for compression was assumed to be the same as that of the uniaxial tension. The remaining four parameters i.e. ‘pinchx’, ‘pinchy’, ‘damage1’ and ‘damage2’ were related to the pinching and degradation of the hysteresis response. Identification of these parameters is crucial as they affect the cyclic response of the member. The minimum and maximum limits for these parameters were specified as inputs and the chromosomes generated a random value between these limits for each parameter. Table 2 gives the adopted lower and upper bounds for each parameter. The chromosome is thus defined as:

$$\:{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{\Theta\:}}_{i}=\left\{{pinch}_{x}^{i},{pinch}_{y}^{i},{damage}_{1}^{i},{damage}_{2}^{i}\right\}\:\:\:,\:i=1,\:.\:.\:.\:.,\:N$$

Table 2

Upper and lower bounds for the hysteresis parameters
Parameter	Lower bound	Upper bound
pinchx	0.3	1.0
pinchy	0.3	1.0
damage1	0.0	0.1
damage2	0.0	0.1

In which N is the maximum number of chromosomes which in the present case was taken as 4 and $\:i\:$is the individual chromosome. The objective of the present study is to optimize the parameters with a limited number of chromosomes thereby reducing the run time of the algorithm. Traditional GA requires the algorithm to run for a specific number of generations. However, to optimize the solution as well as the time, an additional check was created which assessed the similarity of the fitness function across the generations to either proceed with the next generation or to terminate the algorithm.

4.2 Fitness function and Selection

The reliability of a chromosome can be tested using the fitness function which compares the variance of the predicted and experimental results. The chromosomes must pass through the fitness function sequentially to evaluate their variance. The energy dissipated at the end of each cycle of the lateral loading was estimated from the hysteresis curves of the experiment and simulation and the mean square error (MSE) between the two was computed as the fitness function. The adopted fitness function is defined as follows:

$$\:{fitness}_{i}=\:\frac{1}{n}{\left({E}_{sim}^{i}-{E}_{exp}^{i}\right)}^{2}$$

Where $\:\varvec{n}\:$represents the total number of displacement cycles in the loading protocol, $\:{\varvec{E}}_{\varvec{s}\varvec{i}\varvec{m}}^{\varvec{i}}$ and $\:{\varvec{E}}_{\varvec{e}\varvec{x}\varvec{p}}^{\varvec{i}}$ are the cumulative energy dissipated at the end of $\:{\varvec{i}}^{\varvec{t}\varvec{h}}$ cycle in the simulation and experiment respectively. As the fitness function represents the error or the variance between the simulated and experimental results, a lower value of fitness would indicate a more reliable model and the corresponding chromosomes are selected and passed on to the next generation.

The Genetic Algorithm replicates the best chromosomes (lower fitness) identified from the fitness function in a particular generation. The available chromosomes in a particular generation are phased into a roulette wheel based on the fitness function and is used to locate and identify the better performing chromosomes. This methodology was employed for its simplicity.

4.3 Crossover and Mutation

To further assess the reliability of the best fit chromosomes and to optimize their values, a random pair of chromosomes (variables) from the roulette wheel are swapped in their locations creating two new individuals. Following the cross-over process, another operation randomly mutates the values of one of the variables and produces a new chromosome. This process prevents any impetuous convergence of the solution to local optimum values rather than converging to the true optimal solution. In this article the variable to be mutated is selected by a random number generation within the total number of variables in a chromosome.

4.4 Termination

In each generation the chromosome having the least variance (fitness) is stored into a global variable and the algorithm automatically exits after executing the methodology for a fixed number of generations in the traditional GA method. However, since each generation requires at the least eight independent simulations in OpenSees, running the algorithm for a fixed number of generations is computationally expensive and ineffective especially if the methodology converges to the optimal solution within a few generations. Hence an appropriate terminal rule was established which assessed the difference in the fitness values across subsequent generations. A threshold was established for this variance, and once this threshold is reached, the GA terminates its operation without needing to iterate through the entire population. Figure 6 shows that the fitness function plateaus after 3 generations rendering subsequent runs of simulations unnecessary.

To ascertain the optimal hysteresis parameters through the Genetic Algorithm process, experimental data from an RC column subjected to reverse cyclic loading was utilized (Gautham and Sahoo 2022). The comparison between the experimental hysteresis curve and the numerical model's hysteresis curve is illustrated in Fig. 7, revealing a close match between the two. The parameters of the hysteretic function optimized using the GA methodology accurately modeled the hysteresis response of the RC beam. Table 3 gives the optimized values of the hysteresis parameters.

Table 3

Hysteresis parameters optimized using Genetic Algorithm
Parameter	Optimized value
pinchx	0.775
pinchy	0.830
damage1	0.037
damage2	0.0196

Figure 8 displays the comparison of cumulative energy dissipated at the end of each cycle of the lateral displacement protocol, where a satisfactory match was observed for all cycles barring the last cycle of the loading protocol. However, further reduction in fitness was hindered by certain experimental phenomena. The experimental test specimen exhibited differential crack patterns along a preferred direction post-peak strength, unattainable in numerical models. Consequently, a slight asymmetry in the experimental hysteresis curves occurred, unlike the symmetrical hysteresis produced by the numerical models in both push and pull loading directions (Fig. 1b). Despite this discrepancy, the hysteretic numerical model predicted by the Genetic Algorithm (GA) provided a reasonably accurate representation of the experimental hysteresis curve. This underscores the effectiveness of the GA methodology in optimizing hysteresis parameters to closely match experimental data, even in scenarios where exact replication of experimental phenomena may not be feasible.

The utilization of the Genetic Algorithm proved successful in identifying optimal hysteresis parameters by utilizing experimental data from an RC column under reverse cyclic loading. A close alignment between the experimental and numerical hysteresis curves was observed, indicating that the parameters optimized through the GA accurately represented the RC beam's hysteresis response. This underscores the effectiveness of the GA methodology in fine-tuning the hysteresis parameters to closely correspond with experimental data. Additionally, the implementation of the termination rule in the modified GA contributed to a reduction in computation time.

CONFLICT OF INTEREST

The authors declare that there is no conflict of interest that could be perceived as prejudicing the impartiality of the research reported.

FINANCIAL INTERESTS

The authors declare they have no financial interests.

Author Contribution

Gautham A. wrote the main manuscript text and all the figures. All authors reviewed the manuscript.

Acknowledgement

The author would like to thank Prof. D. R. Sahoo, Department of Civil Engineering, IIT Delhi, for his invaluable guidance and the technical staff of the Heavy Structures lab for their contribution in the experiments.

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No competing interests reported.

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Optimizing Hysteresis Parameters for RC Structural Elements Using Genetic Algorithm in OpenSees

Status:

Version 1

Abstract

Figures

1. Introduction

2. Experimental DATA

3. MODELLING TECHNIQUE

3.1 Modelling of RC members

3.2 Material Models

4. GENETIC ALGORITHM FOR ESTIMATING THE HYSTERETIC PARAMETERS

4.1 Chromosome

4.2 Fitness function and Selection

4.3 Crossover and Mutation

4.4 Termination

5. RESULTS AND DISCUSSION

5. CONCLUSIONS

Declarations

CONFLICT OF INTEREST

FINANCIAL INTERESTS

Author Contribution

Acknowledgement

References

Additional Declarations

Status:

Version 1