Concentrated Plasticity Model Research Articles

Improved implementation of concentrated plasticity models in real-time hybrid simulation

Real-time hybrid simulation (RTHS) is a cost-effective experimental testing technique for evaluating the dynamic response of structural systems. In RTHS, a portion of the structure is modeled numerically (i.e., numerical substructure), while the remaining part is tested physically (i.e., experimental substructure). These substructures are coupled in real time. A significant challenge in RTHS is accurately computing the dynamic response of the numerical substructure in real time during testing. Constrained by this challenge, numerical substructures in RTHS are often oversimplified (e.g., spring-mass-dashpot models), failing to capture more complex dynamic behaviors and limiting the applicability of RTHS. However, with the increasing use of innovative structural systems, new materials, and advanced design guidelines to ensure resilient and sustainable civil infrastructure against natural hazards, the demand for real-time executable higher-fidelity numerical substructures in RTHS is growing. Consequently, concentrated plasticity model (CPM) has gained attention as the numerical substructure in RTHS for its good balance between numerical accuracy and computational efficiency, as well as its popularity in structural engineering applications. Currently, static condensation is utilized to prevent an increase in computational degrees of freedom (DOFs) induced by the hinges in CPM, aiming to save computational cost. However, this method proves counterproductive for nonlinear analyses. While reducing the number of computational DOFs, it complicates the stiffness representation, thereby necessitating more demanding arithmetic operations for stiffness updating. In this study, multi-point constraints are applied to enhance the computational efficiency, specifically for nonlinear analyses involving stiffness updating, and the versatility of CPM. Moreover, direct stiffness updating method and sparse matrix-vector multiplication are introduced for further enhancement in computation. An open-source MATLAB/Simulink-based computational tool, namely the Hybrid2D, is developed to promote the use of the improved CPMs in RTHS as numerical substructures. Numerical simulations are first conducted to verify the accuracy and the computational efficiency of the improved CPMs in Hybrid2D. Subsequently, two RTHS examples are experimentally conducted to demonstrate the effectiveness of the Hybrid2D in enabling RTHS with a higher-fidelity numerical substructure for more complex structural systems.

Experimental and numerical investigation on the behavior of concrete sandwich slab panel for a structural system

This paper proposes a simplified methodology based on a concentrated plasticity model to develop numerical analysis of concrete sandwich slab panels for structural systems. Also, a proposal for the determination of experimental damage and its evolution within concrete sandwich panels is formulated. To validate the proposed methodology, an experimental study was carried out on full-scale slab panels to analyze the flexural behavior. Eight specimens were subjected to three- and four-point bending tests. A cycle loading protocol with unloading up to zero load was performed. These specimens, constructed with either one or two joined panels, were used to assess the group action. Mechanical properties for structural analysis—such as strength, stiffness, degree of composite action, and effective moment of inertia—were determined. The tested specimens exhibit significant stiffness degradation, indicating a notable evolution of damage even at small plastic displacements. The proposed methodology proves to be effective and reliable for predicting the behavior of concrete sandwich panels under bending moments. The numerical simulations performed show an adequate description of slab panel behavior until the ultimate load, regardless of its simplicity.

Assessing the seismic behavior of structures controlled with a novel elastoplastic-tuned mass damper inerter considering the effects of soil-structure interactions

In this paper, an elastoplastic-tuned mass damper inerter (PTMDI) is introduced, and its practical advantages under soil-structure interaction are investigated.For this purpose, a nine-story building frame equipped with PTMDI is modeled nonlinearly in the OpenSees software using the concentrated plasticity model. The Bald Eagle Search Optimization Algorithm calculates the PTMDI parameters for a structure with a fixed base or located on three soil types (i.e., soft, medium, and stiff soils).The objective function is the minimization of the Park-Ang damage index of the controlled structure. The results show that PTMDI has dramatically reduced the Park-Ang damage index in the controlled structure using the inertia-strengthening effect and the nonlinear behavior of the PTMDI system. The results show that the PTMDI has reduced the structural damage index of the controlled structures by more than 50% compared to the uncontrolled ones since the controlled system reduces the deformation of the members and drift story. Also, more energy is dissipated because the spring of the control system goes into the nonlinearity. It has also depicted that the PTMDI increases the performance level of the structure from the collapse state to the life safety state. Due to the inertia-enhancing effect of the PTMDI, the mass ratio of the proposed control system does not significantly affect the behavior of the structure. Therefore, the dynamic responses of the structure can be controlled with a small mass.

Development of Spring Hinge Models to Simulate Structural Elements of Spherical Graphite Reinforced Concrete

The goals of the study were the formation of a model of concentrated plasticity based on spherical graphite fibers. The goal was to present indicators of the cyclic behavior of structural elements due to the lack of consideration in modern scientific research of the issue of increasing the technical level and reliability of structures due to the addition of innovative materials during the development of structural elements. Taking into account the fundamental shortcomings in construction in the conditions of installing an excess amount of reinforcement, as one of the main mechanisms for achieving strength and reliability. The aim of the study was to develop a method of improving the quality of structural elements using innovative materials through building spring hinge models. The study involved the mathematical simulation, comparison, and system analysis. The result of the work is a developed spring hinge model for simulating reinforced concrete structural elements. The work emphasizes that the obtained experimental model converges to reference data and does not reflect a significant error with the increased number of cycles. It is emphasized that the model degrades with significant calibration, while the numerical strength and hysteresis behaviour matches the experimental data at higher deformation levels. The model of concentrated plasticity based on spherical graphite fibers was developed for the range of indicators of the cyclic behaviour of structural elements. The main characteristics of the reinforced concrete structural elements under consideration in the framework of the study are provided in a table in order to visually separate the groups of structures according to the main parameters. A non-linear model of a spring hinge is graphically shown, which shows the moment of movement of the spring when a monotonous load is applied. An urgent need to build a system of indicators of the structural elements’ cyclic behaviour was emphasized. A concentrated plasticity model based on spherical graphite fibers was built for this purpose. A beam-column was chosen as a reinforced concrete structure, which is based on a zero-length spring pivot hinge at the end of each individual element. This spring pivot hinge is a uniaxial material model with a moment-rotation dependency. Such a dependency is the fundamental basis of the spring hinge model used to simulate reinforced concrete structural elements. The comparison chart of reference and cyclic strength of a spherical graphite reinforced concrete beam is presented graphically. Prospects for further research involve the development of an empirical system of equations depending on the section geometry and the properties of the structure material based on prognostic variables from the list of potential variable characteristics.

A comparative analysis of existing models and a new pushover analysis model of reinforced concrete sections

Pushover analysis is mainly carried out using the concentrated plasticity model whereby when a point reaches yield, a hinge is placed at that point. The other is the yielded block spread plasticity model, whereby when a point reaches yield, an elastic sub-element of the beam is replaced by a yielded sub-element having a reduced cross-section and second moment of area. Both of these models ignore cracking. This study aims at giving an insight into the effects of considering cracking during modelling on the accuracy of estimating deformations in reinforced concrete (RC) structures during pushover analysis by proposing a spread cracking and yielding block model. The proposed model introduces a cracked sub-element to account for the gradual spread of cracking in the beam. A single-storey RC frame is used because it doesn’t pose the challenge of lateral load distribution. A comparison between the proposed model and the existing models shows an increment in the accuracy of the rotational, displacement, moment and lateral load capacities of 63.64%, 56.86%, 64.33% and 55.56% respectively.Experimental results show that all theoretical models underestimate the ultimate floor displacements and lateral load capacities. The proposed model, however, has better accuracy on both fronts than both existing theoretical models.

Effect of prestressing corrosion on failure in bridges

AbstractInfrastructures (viaducts, bridges, tunnels, railways, and roads) are fundamental to allow people and goods move around a country. In particular, in Italy, due to its orographic and morphological complexity, the ratio between the number of viaducts and the national geographical extension is the largest in Europe. This kind of structure constitutes a weakness of the road networks, especially those over 50 years old, and among these, prestressed concrete structures are the most vulnerable. Therefore, the study of the robustness of an infrastructure network is a quite useful topic for modern societies and public mobility managers. The aim of the present work is to develop a reliable and fast analysis tool capable of processing many structures in a relatively short time. In order to calibrate the most suitable instrument, starting from the study of an experimental campaign on six prototypes of artificially corroded posttensioned concrete beams, the reliability of concentrated plasticity models was investigated to reproduce force‐displacement diagrams and in particular, the effect of corrosion on the moment‐curvature relationship. In fact, this method could provide a well‐defined nonlinear analysis of prestressed elements, especially in the case of grillage decks where external girders could be more affected by corrosion damage in opposition to internal girders or diaphragms. A numerical model by FEM analysis is also presented to reproduce a vulnerability study on an actual structure.

Evaluating ASCE 41‐17 performance‐based provisions on optimally designed steel moment frames

SummaryIn this study, the ASCE 41‐17 nonlinear static procedure for steel moment‐resisting frames is evaluated using a three‐phase constraint handling procedure. For the first time, advanced performance measures of ASCE 41‐17 are quantified during the optimization process by constructing concentrated plasticity models of the standard. Covariance matrix adaptation in evolution strategies (CMA‐ES) is used to obtain optimal designs for three‐ and nine‐story illustrative examples. Active and inactive constraints are discussed in the current performance‐based design methodology as a guide for future research. The seismic evaluation procedure outlined in FEMA P695 is applied to 147 optimal designs. Plastic hinge models explicitly simulate cyclic deterioration in nonlinear dynamic analyses. The numerical results conclusively demonstrate that the design procedure provides an acceptable margin of safety from collapse of the treated frames. Moreover, there is no significant relationship between the structural weights and the collapse margin ratios for such optimally designed structures.

Optimal Design of Earthquake-Resistant Buildings Based on Neural Network Inversion

An effective seismic design entails many issues related to the capacity-based assessment of the non-linear structural response under strong earthquakes. While very powerful structural calculation programs are available to assist the designer in the code-based seismic analysis, an optimal choice of the design parameters leading to the best performance at the lowest cost is not always assured. The present paper proposes a procedure to cost-effectively design earthquake-resistant buildings, which is based on the inversion of an artificial neural network and on an optimization algorithm for the minimum total cost under building code constraints. An exemplificative application of the method to a reinforced-concrete multi-story building, with seismic demands corresponding to a medium-seismicity Italian zone, is shown. Three design-governing parameters are assumed to build the input matrix, while eight capacity-design target requirements are assigned for the output dataset. A non-linear three-dimensional concentrated plasticity model of the structure is implemented, and time-history dynamic analyses are carried out with spectrum-consistent ground motions. The results show the promising ability of the proposed approach for the optimal design of earthquake-resistant structures.

Resistance functions for blast fragility quantification of reinforced concrete block masonry shear walls

Reinforced masonry shear wall (RMSW) systems constructed using concrete blocks are widely used within the current North American construction practices. The limited research carried out on the performance of RMSWs under blast loading has reviled their high vulnerability. This study evaluates the blast performance of such loadbearing and non-loadbearing RMSWs considering the variability associated with different blast wavefront parameters, specifically the positive specific impulse and the peak reflected pressure. In this respect, a concentrated plasticity model is created using OpenSees that simulates the out-of-plane behavior of RMSWs. The model is first validated both statically and dynamically using different experimental datasets. Afterward, the model is utilized within an iterative framework to track the strain rates of different RMSW constituent materials, thus extracting the corresponding dynamic increase factors. Finally, the developed model is used to generate fragility curves and surfaces considering the expected variability/uncertainty in blast wavefront parameters on the response of RMSWs with different axial stress levels and reinforcement ratios. The proposed fragility quantification approach paves the way to develop new blast risk assessment tools for the next generation of blast-resistant construction standards.

Improving the nonlinear seismic performance of steel moment-resisting frames with minimizing the ductility damage index

In this paper, the parameters optimization of a tuned mass damper (TMD) is presented to enhance the seismic performance of a six-story steel structure based on the ductility damage index. Herein, the six-story frame is modeled nonlinearly in the OpenSees software by a concentrated plasticity model. Finally, the most suitable algorithm is selected among several optimization algorithms based on the convergence rate and the objective function's values. In this process, the water cycle algorithm has shown the best results. Therefore, the optimal parameters of the TMD are calculated by this algorithm in such a way that the ductility damage index is minimized in the six-story structure under earthquake loads. For this purpose, the nonlinear dynamic analysis of the structure is performed under earthquakes loads using the OpenSees software. Also, the optimum parameters of the TMD are computed to minimize the ductility damage index under the earthquake loads by linking the OpenSees and Matlab software. The results show that the optimum parameters of the TMD system obtained by the water cycle algorithm could appropriately decrease the ductility damage index. It can simultaneously increase the structure's seismic performance to reduce the displacement, stories damage, and drift ratio.

Damage correlation in flexural walls with a displacement approach method for boundary detailing

The 2010 Chile earthquake of magnitude Mw 8.8 showed the importance of using correct detailing at the wall boundary. Most design methodologies for wall boundary special detailing have been compared to testing, but at the building level have not been validated with evidence from actual earthquakes. For this purpose, two buildings that exhibited flexural wall damage were studied. An additional building that was not damaged, standing less than 100 m from one of the damaged buildings, was also included. The buildings were analyzed using linear–elastic models, as is common practice among designers, and the walls were analyzed in terms of compressive strain demands resulting from roof displacement. The strain demands were determined using a concentrated plasticity model, as well as a model that included the elastic and plastic components of deformation. In this last case, the elastic component was determined by pushing the building model with a pattern consistent with the first mode of lateral vibration until the first wall reached its moment capacity, including a reduced stiffness in the walls and slabs. Results indicate that the damaged walls were expected to reach moment capacity earlier than the other walls, obtaining compressive strain values greater than 0.003. Thus, the walls damaged during the 2010 earthquake correlated well with the estimations of compressive strains, since these were also the first walls to reach maximum capacity. On the other hand, the undamaged building presented generally lower levels of compressive strain values in the walls than those of the damaged buildings.

Damage detection of steel moment frames under earthquake excitation

This paper investigates the effects of uncertainties and modeling errors on the performance of nonlinear updating process through an experimental study on two 1/8-scale four-story steel moment-resisting frames. To examine the accuracyo of the models calibrated using different data features and modeling assumptions, the actual structural responses obtained from shaking table tests and the response of the models have been compared. The modeling uncertainties are investigated by applying frames with different modeling assumptions, including concentrated and distributed plasticity, and different hysteretic material behaviors. To calibrate the models, four data features are considered and combined: (1) acceleration time history, (2) displacement time history, (3) instantaneous characteristics of the decomposed monocomponent of the response, and (4) dissipated energy. In this paper, a sensitivity analysis has been performed to investigate the effect of various parameters of hysteretic materials on the response of reference steel moment-resisting frames. To this end, the optimum values of the unknown parameters are determined through minimizing the objective function defined based on the residuals of data features for the actual structure and finite element (FE) model. By comparing the structural responses of the model with concentrated plasticity calibrated by instantaneous characteristics with those of the other models, it can be concluded that the concentrated plasticity model predicts a more accurate response compared to the other models. Consequently, the aleatory uncertainty was taken into account by evaluating the performance of the system with concentrated plasticity under 40 seismic ground motions, and the structural response has been obtained through the calibrated model.

Performance of battery rack auxiliary power systems under FEMA 461 quasi-static seismic loading protocol

The performance of nonstructural components in nuclear power plants (NPPs), which is primarily based on experience and historical data, has been attracting increased interest from researchers following the Fukushima Daiichi nuclear disaster in 2011. This disaster demonstrated the importance of using batteries in NPPs as an auxiliary power system, where such systems can provide the necessary power to mitigate the risk of serious accidents. However, little research has been conducted on such nonstructural components (e.g., auxiliary battery power systems) to evaluate their performance following the post-Fukushima safety requirements, recommended by several nuclear regulators worldwide [e.g., Nuclear Regulatory Commission (NRC), and Nuclear Safety Commission (NSC)]. To address this research gap, the current study investigates the lateral performance of an auxiliary battery power system similar to those currently existing/operational in NPPs in Canada. The rack system was experimentally tested under displacement-controlled quasi-static cyclic fully-reversed loading that simulates lateral seismic demands, following the FEMA 461 guidelines “Interim testing protocol for determining the seismic performance characteristics of structural and nonstructural components”. Following a brief summary of the experimental program, the test results are presented in terms of the rack hysteretic response, damage sequence, stiffness degradation, ductility capacity, member strains, and local deformations. Subsequently, a simplified mechanistic model and a concentrated plasticity model in OpenSees have been developed and calibrated using the experimental results. The results show that without detailed modeling of the rack system connections (i.e., L-shaped connection and sliding nuts), incorrect performance prediction of such systems may result. The findings of the current study can be utilized, within the next generation of performance-based seismic design approaches, to enhance the robustness and improve the reliability of damage state predictions of auxiliary battery power systems in critical facilities.

Review of Nonlinear Analysis and Modeling of Steel and Composite Structures

Structural steel frames exhibit significantly geometric and material nonlinearities which can be captured using the second-order inelastic analysis, also known as advanced analysis. Current specifications of most modern steel design codes, e.g. American code AISC360, European code EC3, Chinese code GB50017 and Australian code AS4100 permit the use of advanced analysis methods for the direct design of steel structures to avoid tedious member capacity checks. In the past three decades, a huge number of advanced analysis and modeling methods have been developed to predict the behavior of steel and composite frames. This paper presents a comprehensive review of their developments, which focus on beam-column elements with close attention to the way to capture geometric and material nonlinearity effects. A brief outline of analysis methods and analysis tools for frames was presented in the initial part of the paper. This was followed by a discussion on the development of displacement-based, force-based and mixed beam elements with distributed plasticity and concentrated plasticity models. The modeling of frames subjected to fire and explosion was also discussed. Finally, a review of the beam-column models for composite structures including concrete-filled steel tubular (CFST) columns, composite beams and composite frames was presented.

The Effect of Axial Force Variations on Nonlinear Modeling and Seismic Response of Reinforced Concrete Structures

In order to increase the accuracy of evaluating seismic response of structures, it is critical to conduct dynamic analyses based upon precise nonlinear models being as consistent as possible with the real conditions of corresponding structures. The concentrated plasticity model including one elastic element and two nonlinear spring elements at both ends has been considered within the research community for simulating beams and columns, counting the effect of strength and stiffness degradation. In this type of simulation, the axial force ratio generated in each structural component, which is a major factor in introducing nonlinear springs, has always been considered constant in the literature. The main objective of the present research is, therefore, to modify the fundamental weakness in this type of modeling approach; indeed, any variation of element’s axial effort, owing to redistribution of axial forces during an earthquake, is applied in the calculation of parameters of the concentrated plasticity model as a decisive step toward the development of nonlinear dynamic analysis. Moreover, an algorithm is presented for implementing this approach in the OpenSees software. Verification is established and the efficiency of the proposed method is illustrated through a reinforced concrete moment frame subjected to a specific record, as a case study building. Regarding the results, it is confirmed that the proposed algorithm is an appropriate tool for achieving quite a realistic nonlinear model and estimating reasonably accurate responses of structural systems with cyclic degrading behavior under earthquake loading.

Efficiency of employing fiber-based finite-length plastic hinges in simulating the cyclic and seismic behavior of steel hollow columns compared with other common modelling approaches

The accuracy and efficiency of the modelling techniques utilized to model the nonlinear behavior of structural components is a significant issue in earthquake engineering. In this study, the sufficiency of three different modelling techniques that can be employed to simulate the structural behavior of columns is investigated. A fiber-based finite length plastic hinge (FB-FLPH) model is calibrated in this study. In order to calibrate the FB-FLPH model, a novel database of the cyclic behavior of hollow steel columns under simultaneous axial and lateral loading cycles with varying amplitudes is used. By employing the FB-FLPH model calibrated in this study, the interaction of the axial force and the bending moment in columns is directly taken into account, and the deterioration in the cyclic behavior of these members is implicitly considered. The superiority of the calibrated FB-FLPH modelling approach is examined compared with the cases in which conventional fiber-based distributed plasticity and concentrated plasticity models are utilized. The efficiency of the enumerated modelling techniques is probed when they are implemented to model the columns of a typical special moment frame in order to prove the advantage of the FB-FLPH modelling approach.

A new damage index for steel MRFs based on incremental dynamic analysis

The present study introduced a damage index based on a relationship between two values: the maximum interstory drift ratio and spectral acceleration at the first-mode period of the structure. The proposed damage index was defined so as it equals 0 in the yield point of the structural system, while it equals 1 in the collapse point. The incremental dynamic analysis was used to compute the constituent parameters of the damage index in both yield and collapse points. It was also used to observe the change trend of the index through varying intensities of the earthquake ground motion records. A number of steel models, including single degree of freedom models, stick models and moment resisting frames served as different examples to exhibit the performance of the damage index as well as to compare it with a number of recognized damage indices. Non-linear models were developed in OpenSees. Stiffness and strength deterioration in these models were included using the concentrated plasticity model. The results indicated that unlike the comparative damage indices, the proposed damage index was not influenced by successive hardening and instantaneous softening. In addition, once the intensity of ground motion records increased, the index maintained its ascending trend. Besides, increasing the period as well as the number of floors along with the presence of some levels of inconsistency in structures caused a decline in the correlation between the proposed damage index and the comparative indices.

Concentrated-plasticity modelling of circular concrete-filled steel tubular members under flexure

The research reported herein aims at the proposal of an accurate and efficient simplified numerical modelling approach for circular concrete-filled steel tubular (CFST) members under flexural loading. Experimental tests were carried out to characterize the bending behaviour of CFST members under monotonic and cyclic loading. The observed behaviour was characterized by strength and stiffness deterioration effects, as a result of the development of local buckling of the steel tube and cracking of the concrete core. Numerical simulations of these tests were conducted by resorting to existing modelling approaches, namely through Distributed Plasticity (DP) and Concentrated Plasticity (CP) models. It was found that existing modelling approaches failed to accurately capture the levels of strength deterioration and pinching effects observed in the tests. Thus, a novel CP-based simplified model, designated by matCFSTdet, was implemented in OpenSees. The hysteretic response of the CP model is based on a novel rotational spring model. An advanced calibration framework was introduced with targets to calibrate the accuracy of the model. The validation analyses indicate that the model is able to capture well the deterioration in both strength and stiffness of CFST members under cyclic flexural loading. Furthermore, the elastic stiffness, ultimate strength and the pinching effects of the hysteretic loops were also well simulated. The proposed CP model, coupled with the advanced calibration framework, thus results in a more realistic simulation of the cyclic flexural response of circular CFST members.

Collapse capacity of reinforced concrete skewed bridges retrofitted with buckling-restrained braces

This study assesses the collapse capacity and failure modes of skewed bridges retrofitted with buckling-restrained braces (BRBs) at the column bent. The asymmetric configuration of these bridges requires a three-dimensional numerical model that considers uncertainties in the seismic ground motions. The factors controlling the seismic performance of these bridges are obtained from a case study of a three-span reinforced concrete box girder skewed bridge with varying skew angles of 0°, 18°, 36°, and 54°. Numerical models with distributed plasticity and concentrated plasticity are created considering material deterioration and plastic deformation properties. A numerical distributed plasticity model with strength and stiffness deterioration is calibrated with a concentrated plasticity model by performing two-dimensional incremental dynamic analyses (IDAs) on the straight bridges (i.e., 0° skew angle), using 21 far-field ground motions. Then, the collapse capacity of original and retrofitted skewed bridges is obtained from three-dimensional IDAs using the distributed plasticity model. The collapse capacity and failure modes of BRB components are summarized based on investigations from experimental data. Nonlinear time history analyses indicate that the BRB retrofit greatly improves the seismic performance of skewed bridges, but it has negligible effects on the bridge’s collapse capacity after BRB failure. However, the use of BRB greatly reduces the bridge’s probability of failure, and the mean annual frequency of global collapse, since the BRB components dissipate seismic energy as structural fuses. The proposed method of parameter calibration, including BRB failure, is found to be sufficiently reliable to perform satisfactory results.

The Effects of Viscous Damping Modeling Methods on Seismic Performance of RC Moment Frames Using Different Nonlinear Formulations

Nonlinear response history analysis, contributing to seismic performance assessment, is a constructive tool for evaluating buildings' behavior and damages due to the earthquake. Applying an accurate viscous damping model constitutes a crucial part in this regard. The seismic response and performance of two reinforced concrete moment frames considering various damping modeling techniques and nonlinear elements are investigated. Two basic approaches to consider nonlinearity are selected: distributed and concentrated plasticity, using force and displacement-based fiber elements and elements that contain two end springs and elastic parts to which zero and modified initial stiffness proportional damping are allocated respectively. Rayleigh damping with mass and four different stiffness matrices are applied in fiber elements. The results of Incremental Dynamic Analysis and loss estimation, considering performance-based earthquake engineering methodology, indicate that Rayleigh damping with mass and initial stiffness overestimates collapse capacity and underestimates performance parameters, while, the model with mass and tangent stiffness with updated proportionality terms shows the opposite trend. In concentrated plasticity models, the more flexible the springs, the more conservative the evaluation of building performance.