Concrete Damaged Plasticity Model Research Articles

Concrete Damage Plastic Model for High Strength-Concrete: Applications in Reinforced Concrete Slab and CFT Columns

Concrete Damage Plastic Model for High Strength-Concrete: Applications in Reinforced Concrete Slab and CFT Columns

Mechanical properties and damage plastic model for cement-soil mortar: Laboratory tests and numerical analysis

Cement-soil mortar is a composite material that provides an efficient and cost-effective solution for a wide range of construction applications. This study analyzes the mechanical properties of cement-soil mortar through experimental investigation and explores the application and parameter calibration of the Concrete Damage Plasticity (CDP) model in the finite element simulation of cement-soil mortar. Additionally, an innovative Initial Defect Generation (IDG) method is proposed to enhance the accuracy in simulation of failure mode. The research findings provide a simulation framework that balances simplicity and accuracy for cement-soil mortar. Uniaxial compressive tests are first conducted on cement-soil mortar specimens with water contents ranging from 40 % to 70 %, and cement-to-soil proportions from 30 % to 300 %. Based on the experimental results, the regression relationships correlating unconfined compressive strength (UCS) with elastic modulus, peak strain and stress-strain curves are established. Then, the simplified equations for calculating CDP model parameters from the UCS of cement-soil mortar are further proposed following damage mechanics. A parameter table summarizing the calculation methods for all relevant CDP parameters is provided to streamline the model calibration process. Simulations incorporating the simplified calibration method and IDG method successfully replicated the stress-strain responses and failure modes observed in uniaxial compressive strength tests of cement-soil specimens with varied strength. The results demonstrate the reliability and broad applicability of this simulation framework in predicting the mechanical performance of cement-soil mortar.

Evaluating the impact of temperature variations and subgrade reactions under traffic-load on airport concrete pavement performance

This study delves into the intricate dynamics of airport concrete pavement slabs under the influence of temperature fluctuations and traffic loads. It presents a comprehensive analysis of the stress distribution and subsequent damage in concrete slabs, factoring in the critical role of dowels in load transfer. The research utilizes advanced finite element simulation ABAQUS to model concrete slabs with varying subgrade reactions, subjected to both temperature gradients and aircraft gear wheel load. Findings reveal that lower subgrade reactions mitigate stress levels under temperature variations, while higher subgrade reactions amplify the stress, particularly around dowel areas and leading to potential damage. The application of the Concrete Damage Plasticity (CDP) model demonstrates that temperature-induced stress combined with traffic load can lead to complete damage around the dowel, compromising the load transfer capability. This paper underscores the necessity of integrating temperature effects and traffic loads into pavement design and maintenance strategies to enhance durability and reduce repair costs.

Numerical Study on the Retrofitting of Exterior RC Beam-column Joints with CFRP Composites Using the Grooving Method

Retrofitting seismically deficient beam-column joints (BCJs) in reinforced concrete (RC) structures is crucial to preventing collapse during high-intensity earthquakes. Fiber reinforced polymer (FRP) composites have proven effective for retrofitting these components, but debonding of the FRP from concrete surfaces remains a challenge. Recently, the grooving method (GM) has emerged as a promising solution to mitigate this issue, enhancing the bond between FRP and concrete. This paper presents a numerical investigation of BCJs retrofitted with FRP composites using the GM, focusing on various design parameters. The study utilized finite element models developed in ABAQUS, incorporating the concrete damage plasticity (CDP) model to simulate the nonlinear behavior of concrete. The interface between the concrete and FRP was modeled as a perfect bond, simulating the strong adhesion provided by the GM. The numerical results were validated against experimental data from two BCJ specimens, showing good agreement in terms of load-displacement behavior, peak loads, and failure modes. Key parameters studied include the beam longitudinal reinforcement ratio, column axial load ratio, and joint aspect ratio. The findings reveal that these parameters significantly affect the shear capacity of retrofitted joints, impacting the efficiency of the retrofitting.

Numerical simulation on structural behavior of FRP profile‐concrete hybrid beams with bolt shear connector

AbstractFRP profile‐concrete hybrid beam, composed of a FRP profile beam, a concrete slab made of normal concrete (NC) or ultra‐high‐performance‐concrete (UHPC), and shear connectors, shows significant potential for employment in large‐span bridges due to its excellent corrosion resistance, lightweight, and easy construction. The structural behavior of such hybrid beam was simulated using a three‐dimensional non‐linear finite element model (FEM) in Abaqus. The Hashin failure model and concrete damage plastic (CDP) model were employed to characterize the progressive failure of FRP, NC, and UHPC, respectively. The FEM considered partial shear connection by utilizing shear load‐slip model and the orthotropic behavior of the FRP profile. The proposed FEM was validated through comparisons with experimental data in terms of load‐midspan deflection curve, load‐end slip curve, and failure modes. A parametric study was subsequently conducted to identify the effects of various factors, including concrete strength, bolt spacing, concrete slab width, concrete slab height, FRP flange and web thickness, and shear span. Based on the results yielded from FEM analysis, the accuracy of the calculation method in Chinese standard (GB‐50608) for predicting the loading capacity of hybrid beams was evaluated and found too conservative.

Damage mechanism of model arch dam subjected to far-field underwater explosion

The dynamic response and damage modes of arch dams subjected to explosions are key topics having been extensively researched in recent years. However, the majority of research employs numerical simulation, and there is a lack of experimental investigations to further support those numerical simulation results. Given this, small-scale model experiments were conducted on the arch dam to investigate its dynamic response to underwater explosions. The findings of these experiments indicated a potential overall damage mode along the crown cantilever profile. A numerical investigation of the experimental arch dam was executed employing the CEL method in combination with the Concrete Damage Plasticity model. The damage development process and cracking mechanism of the arch dam under a far-field underwater explosion were examined. According to the numerical findings, the damage process to the arch dam resulting from a far-field underwater explosion might be divided into two phases: the shock wave action phase and the overall response phase. During the shock wave action phase, slight damage first appeared on the downstream dam face. Later, during the overall response phase, a crack spread from the dam base to dam crest and from the downstream side to the upstream side, eventually penetrating through the crown cantilever profile. The damage mechanism in the shock wave action phase is the reflected rarefaction wave off the downstream dam face causing localized tensile damage to surrounding concrete. While the cracking mechanism in the overall response phase is structural bending-induced tensile damage driven by the overall response.

Nature-inspired approach for enhancing the fracture performance of 3D printed strain-hardening cementitious composites (3DP-SHCC)

3D printing technology enhances design flexibility, enables the creation of complex geometrical structures, facilitates rapid prototyping, and allows the fabrication of micro- and macroscopic structures. This study investigates the use of 3D concrete printing to create Bouligand structures inspired by mantis shrimp. Bouligand-printed strain-hardening cementitious composites (SHCC) with different pitch angles (15°, 30°, 45°, 60°, 75°, and 90°) are fabricated and compared with traditional parallel-printed and mold-cast SHCC. The fracture behavior of these specimens is investigated through three-point bending tests on notched beams. The results reveal that Bouligand-printed specimens exhibit enhanced flexural strength, fracture energy, and fracture toughness, ranging from 0.97 to 1.29 times, 0.99 to 1.63 times and 0.87 to 1.47 times that of the mold-cast specimens, respectively. Bouligand-printed specimens with a pitch angle of 30° exhibits the highest flexural strength, fracture energy, and fracture toughness. The Bouligand-printed SHCC enhance toughness through the synergistic effects of crack twisting and bridging. A finite element model integrating cohesive elements with the concrete plastic damage model is developed to numerically replicate the experimental process. This study highlights the potential of 3D concrete printing for the fabrication of biomimetic concrete structures with superior fracture performance.

Shear behavior of RC beams with openings under impact loads: unveiling the effects of HSC and RECC

Incorporating transverse openings in reinforced concrete (RC) beams reduces their load-bearing capacity and stiffness, making them prone to premature failure. This highlights the need for a more nuanced understanding of their behavior to ensure structural integrity, particularly under impact loads. This research delves into a relatively unexplored area, investigating the impact performance of RC beams containing openings through numerical analysis. By comparing different concrete types, the study seeks to identify optimal materials for such applications. The concrete damage plasticity model, accounting for strain rate effects, will be employed to simulate the material behavior of normal-strength concrete (NSC), high-strength concrete (HSC), and a novel eco-friendly alternative: rubberized engineered cementitious composite (RECC). RECC incorporates recycled tire rubber as a partial substitute for traditional concrete aggregates, offering a sustainable solution while mitigating environmental hazards associated with waste tire incineration. The finite element models are validated with experimental results, accurately predicting ultimate capacities, failure modes, and post-cracking response in RC beams (with/without openings) under static/impact loads. A comprehensive parametric analysis investigates the effects of concrete strength, impact energy, impactor mass, drop height, and opening location, providing valuable insights into how these factors influence the impact behavior under drop-weight testing. The results reveal that openings in RC beams under impact loads significantly reduce strength and stiffness, with detrimental effects observed for dual shear-zone openings. Surprisingly, a small mid-span opening can enhance impact response. HSC beams exhibit lower initial displacements but higher residual values, while RECC improves overall behavior in beams with openings, reducing maximum displacement and promoting energy dissipation for improved post-impact recovery.

Study on the bond-slip numerical simulation in the analysis of reinforced concrete wall-beam-slab joint under cyclic loading

This paper presents the finite element analysis (FEA) of a full-scale RC wall-beam-slab joint under reversed cyclic loading. Five different approaches in ABAQUS are introduced to simulate the bond-slip effect between reinforcement and concrete. The concrete damaged plasticity (CDP) model and the combined hardening constitutive model are illustrated and adopted for concrete and reinforcement, respectively. By comparing the overall mechanical behavior of the wall-beam-slab joint predicted by FEA models to that of the test results, the predictive capability of FEA models with different bond-slip simulation methods are studied. In general, the analysis results indicate that the numerical results without the bond-slip effect present a gross overestimation of the overall mechanical behavior, while the numerical results with the bond-slip effect are in good agreement with the test results. Wherein, the FEA models with the bond-slip effect simulated by spring elements or implemented by user-defined subroutine predict more consistent results with the test results. Moreover, the disadvantages of each method utilized to simulate the bond-slip effect have also been described. A comprehensive study on material parameters of concrete is accomplished to obtain the influence of parameters such as the dilation angle(ψ), the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian(Kc), and the viscosity parameters(υ).

Research on the Gap Effect of Circular Concrete-Filled Steel Tubes Using the Improved Cohesive Zone Model

Understanding the influence of gap distribution characteristics on the mechanical properties of circular concrete-filled steel tubes (CCFSTs) under bending load is important for stability and support design in engineering projects. In this study, the improved cohesive zone model considering friction was used to describe the mechanical behavior of mortar interfaces. Meanwhile, the concrete damage plastic model and isotropic elastoplastic model were applied for core concrete and steel tubes. The improved cohesive zone model has a unified potential function that governs the Mode I and Mode II failure processes of mortar interfaces to realize the mechanical interaction between concrete and steel. A smooth frictional function was utilized in the elastic stage to calculate the accurate frictional effect. Furthermore, the capability of the model in addressing unloading and reloading was verified, and the fracture energy varied accordingly during the cyclic loading. Then, the mechanical response of CCFSTs was investigated under bending loads by setting different gap sizes and angles between the gap and loading direction. The results show that under three-point bending, the equivalent plastic strains at the middle part of CCFSTs are much larger and the peak bearing forces are much lower than the other degrees when the angles between the coronal gap axis and loading direction equal 0° and 180°. In addition, the order of the peak bearing forces, from highest to lowest, is when the height of the coronal-cap gap increases from 0.0 mm to 2.5 mm, 5.0 mm, and 7.5 mm. The significant effect makes it inappropriate to ignore the weakening of the structural performance caused by coronal gaps in structural design.

Experimental and Numerical Study on the Performance of Steel–Coarse Aggregate Reactive Powder Concrete Composite Beams with Uplift-Restricted and Slip-Permitted Connectors under Negative Bending Moment

An innovative form of steel–concrete composite beam, the steel–coarse aggregate reactive powder concrete (CA-RPC) composite beam with uplift-restricted and slip-permitted (URSP) connectors, is introduced in this paper. The aim is to enhance the cracking resistance under negative bending moments, which is a difficult problem for traditional composite beams, and to make the cost lower than using ordinary reactive powder concrete (RPC). An experimental investigation of the behavior of six specimens of simply supported steel–CA-RPC composite beams with URSP connectors under negative bending moments is presented in this paper. The test results validated that the cracking load of steel–CA-RPC composite beams could be approximately three times that of the ordinary steel–concrete composite beams while the bearing capacity and stiffness are almost the same. A numerical model, using the concrete damaged plasticity (CDP) model to simulate the behavior of the CA-RPC material, was proposed and successfully calculated the overall load–displacement relationship of the composite beams with sufficient accuracy compared with the experimental results, and the distribution of cracks and the failure mode of the beams could also be captured by this model. Furthermore, a parametric analysis was carried out to find out how the application of prestress, CA-RPC, and URSP connectors could affect the cracking resistance of the composite beams, and the results indicated that using CA-RPC and prestress made the main contributions and that the usage of URSP could boost the effect of the other two factors. The plastic resistance moment of the beams was also compared with the calculation results using the methods introduced in Eurocode 4, and it was proved that the calculation results were lower than the experimental results by approximately 10%, which meant that the method was reliable for this kind of composite beam.

Influence of the Diameter Size on the Deformation and Failure Mechanism of Shield Precast Segmental Tunnel Lining under the Same Burial Depth

With the development of large-diameter shield tunnels, how to realize effective security and stability control of shield tunnel lining has become a significant research topic. This paper investigates the deformation and failure mechanism of lining large diameter shield tunnels in depth and discusses the deformation characteristics and influencing factors of the lining of the shield tunnel with various diameters through the software of finite element analysis ABACUS. A set of models with varying diameters is built under identical stress conditions in order to maintain control over the variable. The utilization of the elastic–plastic model is observed in the application of bolts and rebar. The utilization of the Concrete Damage Plasticity model has been taken into account for the concrete lining. For the sake of comparison, the crown displacement of the shield tunnel, strain in tension and compressive zones, bolt stress and strain, deformation and intemal force distribution around the shield tunnel, and cracks in the tension zone, are carefully studied. An in-depth analysis is conducted to elucidate the variations in damage evolution mechanisms across linings of different sizes, within the framework of plastic hinge theory. The results indicate that the convergence deformation of large-diameter tunnel lining increases significantly during loading compared with that of small-diameter tunnel. Moreover, the probability of brittle failure is higher in big-diameter shield tunnels compared to small-diameter tunnels, indicating that these larger tunnel structures are more prone to suffering geometric instability.

Study on damage of CRTS II slab tracks in coated-uncoated transition zones subjected to temperature and train loads

The CRTS II ballastless track is sensitive to temperature due to its longitudinal continuity. The application of reflective insulation coating can efficiently lower the temperature of the track slab, thereby decelerating track deterioration. Given the considerable length of high-speed railway lines, the application of reflective insulation coatings leads to numerous transitions between coated and uncoated sections. This study investigates the influence of reflective insulation coatings on the structural damage of coated-uncoated transition zones in railways. To this end, a CRTS II slab track model involving the bilinear cohesive zone model and the concrete plastic damage model is established. The model examines the combined impact of high temperature and train loading, aiming to investigate the damage patterns within the coated-uncoated transition zone of the track. The results indicate: (1) During high temperature loading, the center of the track slab layer is predisposed to warping compared to the sides. (2) If the cohesion parameter falls below 0.04 MPa, augmenting the cohesion model parameter effectively alleviates track slab arching. (3) In instances where the track slab lacks complete coverage of reflective insulation coatings, train loading may shift the location of the maximum vertical displacement and interface failure mode.

Effectiveness of Different Configurations of Ferrocement Retrofitting for Seismic Protection of Confined Masonry: A Numerical Study

A ferrocement layer, which consists of a wire mesh and cement mortar, is a popular retrofitting method for existing structural elements, particularly wall or slab panels. This paper presents a study on the effectiveness of different configurations of ferrocement for seismic retrofitting of confined masonry through finite element analysis. The masonry panel was modeled using expanded brick-unit elements, where the element was expanded in size by as much as half of the mortar thickness, and an interacting zero-thickness interface was applied to mimic the elastic-plastic and damage behavior during tension, shear, and compression. The concrete damage plasticity (CDP) model was used to model the confining reinforced concrete frame and overlay mortar in the ferrocement layer, and the reinforcing bars and wire mesh were modeled using elastic-plastic behavior. In the present numerical study, nine models were subjected to cyclic and pushover shear test simulations, considering the effects of the number of ferrocement layers and the wire mesh orientation. The volumetric ratio of the wire mesh to the masonry (ρwm) ranged from 0.48% to 1.92%, whereas the ratio of the mortar overlay to the masonry (ρmo) varies from 10.42% to 41.66%. Based on the increase in the lateral strength, the model with the largest volume of the ferrocement layer exhibited the largest increase in strength. However, the most cost-effective retrofitting configuration was presented by model DS-1-45, in which a single layer of ferrocement was applied on both sides of the wall using 45° of wire mesh orientation. The DS-1-45 model provided a lateral strength increase of more than 6 times compared to the original unreinforced model. Doi: 10.28991/CEJ-2024-010-09-02 Full Text: PDF

Numerical Simulation for Strength and Stability of RC Tapered Columns

This study investigates the strength and stability of Reinforced Concrete (RC) linearly tapered square columns. Evaluating the slenderness ratio of an RC column requires the radius of gyration of its cross-section, which is well-defined for a prismatic column but not for a non-prismatic one. This study primarily investigates the application of the ACI Code formulae to evaluate the slenderness ratio of RC columns with the studied geometry. Validated numerical models, using Abaqus, were employed to perform nonlinear first- and second-order analyses on the investigated columns subjected to eccentric axial loads. The concrete damaged plasticity model was employed to simulate the nonlinear behavior of concrete. The static Riks solver, available in Abaqus, was utilized for nonlinear analyses: first, with an inactivated geometric nonlinearity for a first-order analysis, and second, with an activated geometric nonlinearity to consider the effects of secondary moments (p-δ effects). The findings indicate the reliability of defining the slenderness ratio of an RC linearly tapered column based on the ACI Code formulae, using the average cross-section of the tapered column.

Numerical Analysis on Fatigue Behavior of Plain Concrete and Alkali-activated Concrete

Abstract Alkali-activated concrete (AAC) has recently gained a lot of potential to become one of the most recommended sustainable replacements for Ordinary Portland cement concrete (OPCC). In the present investigation, an attempt has been made to study the static and fatigue flexural behavior of AAC compared to that of OPCC by using numerical modeling FEA software, ABAQUS. The nonlinear behavior of the stress-strain curve of the concrete has been studied using the concrete damage plasticity (CDP) model. A 2D notched beam was modelled using plane stress condition and three-point bending tests were performed under monotonic loading to obtain the static behavior of concrete. The result obtained has been utilized to fix the loading range for cyclic loads in fatigue analysis and at different loading frequencies. The load-CMOD curves and load-deflection curves were obtained for both static and fatigue loading, and the number of cycles to failure during fatigue. From the results, it has been observed that the Ordinary Portland cement concrete specimens sustain more load than that of Alkali-activated concrete under monotonic loading. However, AAC has shown more resistance to fatigue than that of the OPCC and the frequency of loading significantly influences the fatigue performance.

Numerical Study on Static and Fatigue Behavior of Alkali-Activated Fly Ash Concrete Modelled using Concrete Damage Plasticity (CDP) Model

Abstract Alkali-activated concretes (AAC) are considered as an alternative to Portland cement concretes because of their much lower carbon emissions. To study its characteristics is of utmost importance. In the present research, an attempt is made to create a numerical model to analyze the static and fatigue flexural behavior of alkali-activated fly ash concrete (FC) and compare them with ordinary Portland cement concrete (PCC). Three-point bending tests are simulated for the monotonic load to obtain the static behavior of a 2D notched rectangular beam. A 2D beam specimen of size 262.5 mm x 75 mm, with a notch of size 15 mm x 3 mm was provided at the bottom middle. Fly ash-based alkali-activated concrete (FC) and corresponding PCC of similar strength are modeled in ABAQUS CAE using the concrete damage plasticity (CDP) model. The ultimate loads, maximum deflection, and CMOD for each of them were obtained from the history output manager. The results of monotonic tests are utilized for creating a sinusoidal load cycle for fatigue tests with various loading frequencies and stress ratios. Different structural responses are examined under the fatigue test. The results showed that FC performs better than PCC under static and fatigue tests. Fatigue performance of both FC and PCC is influenced by the frequency of loading and fatigue life increases at higher frequencies. It is also found that stress reversal negatively impacts the fatigue life of both FC and PCC.

Blind competition on the numerical simulation of slabs reinforced with conventional flexural reinforcement and fibers subjected to punching loading configuration

AbstractThis paper describes the 3rd Blind Simulation Competition (BSC) organized by the fib WP 2.4.1 which aims to assess the predictive performance of models based on the finite element method (FEM) for analysis and design of fiber reinforced concrete (FRC) structures submitted to loading and support conditions that promote punching failure mode. Fiber reinforcement is used in an attempt to eliminate conventional punching reinforcement and provide technical and economic advantages. The two tested real‐size prototypes represent a column‐slab interior region of an elevated steel‐fiber reinforced concrete (E‐SFRC) slab where anti‐progressive collapse reinforcement is disposed in the alignment of columns/piles. Despite a punching failure surface being formed in both experimentally tested prototypes at the rupture stage, fiber reinforcement was able to mobilize the yield capacity of the conventional flexural reinforcement, providing high deformation capacity, and ductility to the prototypes. The average post‐peak load‐carrying capacity of the tested prototypes at a deflection seven times higher than the deflection at yield initiation of the conventional reinforcement was still 90% of the average peak load. Regarding the BSC, a total of 26 proposals were received and involved 94 participants from 29 institutions and 17 countries, with 53.9% using smeared crack models (SCMs), 30.8% a concrete damage plasticity (CDP) model, 3.8% discrete crack models (DCMs) and 11.5% considered as “other models.” From these simulations it was verified, in average terms, that SCM assured the best predictive performance apart from the average strain in the SFRC and the maximum crack width which were better predicted by DCM. More accurate predictions were obtained by using in‐house software than by adopting commercial software.

Mesoscale modeling of new-to-old concrete interface under combined shear and compressive loads

A mesoscale model of new-to-old concrete interface under combined shear and compressive loads is established, and a parameter study is conducted to evaluate the influence of interface roughness, bond strength, and fracture energy. The mesoscale model is numerically generated using a random aggregate technique, in which the number of aggregates is calculated using the Fuller grading curve and Walraven formula and the aggregates are randomly placed using the Monte Carlo simulation. The concrete damaged plasticity model is employed to simulate the mechanical behaviors of new and old mortars, as well as the interfacial transition zone between the mortar and aggregate. The linear elastic model is utilized to simulate the behavior of aggregates, while the cohesive-friction model is employed to represent the behavior of new-to-old concrete interface. The effect of interface roughness, bond strength, and fracture energy on the interface shear strength and post-peak shear strength using the double-shear test specimens is investigated. The results indicate that increasing the roughness of old concrete surface leads to significant increases in both the interface shear strength and post-peak shear strength, but the larger interface roughness is not necessarily better. As the interface bond strength increases, the interface shear strength remains the same, whereas the post-peak shear strength shows a large increase with the lower interface roughness. Moreover, the interface fracture energy imparts the minimal influence on the interface shear strength and post-peak shear strength, but the higher interface fracture energy can improve the overall constitutive behavior of new-to-old concrete interface if the roughness of old concrete surface is low. The numerical mesoscale model presented can provide the theoretical guidance and technical support for effective design and construction of new-to-old concrete interface and structures.

Numerical Modeling of Steel Fiber Reinforced Recycled Concrete Filled Steel Tube Column Under Cyclic Loading

A finite element model (FEM) was created with the aim of analyzing the behavior of steel fiber reinforced recycled concrete (SFRRC)-filled steel tube columns under combined cyclic loading and monotonic axial load. The FEM considered the effect of steel tube confinement on the inner concrete behavior under cyclic loading. The numerical model was described in detail, with a focus on modeling the materials involved (normal concrete, SFRRC, and steel) under cyclic loading. A constitutive concrete model - with and without considering confinement - was based on utilizing a concrete damaged plasticity (CDP) model. The steel tube - concrete core interface was modeled by a surface-to-surface contact. A stress-strain constitutive concrete model, confined by circular steel tubes, was implemented, and validated using experimental results from the literature. The developed FEM considered various parameters: steel tube thickness, volume ratios of steel fibers, besides strengths of both concrete core and the steel tube. The FEM results showed great similarity to the under- cyclic- loading tested columns. The results indicated that the concrete confining pressure must be considered in CDP model. A good correlation between numerical and experimental findings was obvious, including failure modes, and hysteretic curves of load-displacement.