Force Analogy Method Research Articles

Nonlinear Dynamic Analysis Using the Force Analogy Method and the Ibarra–Medina–Krawinkler Model

ABSTRACTA methodology to integrate the Ibarra–Medina–Krawinkler (IMK) hysteretic model with the force analogy method (FAM) is proposed. FAM is a powerful tool for performing nonlinear structural analysis that reduces analysis time, addressing the main limitation of nonlinear analysis. One area of opportunity for FAM lies in its access to material nonlinearity models, which enable the reproduction of inelastic behavior in structures. The IMK model, utilized in nonlinear analysis for updating the stiffness matrix, incorporates various rules of material nonlinear behavior and considers cyclic deterioration of stiffness and strength. By combining the IMK model with FAM, the capability to model material nonlinear behavior in a computationally efficient manner is enhanced. The proposed methodology is validated through four numerical examples, demonstrating its effectiveness in replicating experimental behaviors and enhancing FAM capabilities in inelastic structural analysis.

A new perspective into torsional inelastic response of actively controlled irregular multistorey buildings

The nonlinear dynamics of irregular structures is influenced by both local and global torsional effects. In order to capture the inelastic structural response, the concept of plastic hinges is one of the most used tools due to its numerical efficiency. However, the combined effect of bending and torsion into the inelastic domain is scarcely illustrated in literature. In this paper, the recently developed 3D Force Analogy Method is augmented with an algorithm to consider axial force – biaxial bending moment –torque interaction. A structure having both plan and elevation irregularities is considered as a case study to illustrate the benefits of the proposed algorithm. The structural control concept is further implemented in the attempt to improve structural performance of irregular structures. A non-commercial Matlab/Simulink code is developed for the scopes of this paper.

Time-Domain Structural Damage and Loss Estimates for Wind Loads: Road to Resilience-Targeted and Smart Buildings Design

A time-domain wind-induced damage evaluation for the risk- and resilience-targeted design of building structures requires nonlinear dynamic analysis (NDA) solvers to handle long duration input loads. Moreover, post-construction damage identification based on real-time vibration monitoring depends on accurate damage simulation from the design stage and available artificial intelligence (AI)-based technology. In this paper, an integrated time-domain approach for the performance-based analysis and design for wind is presented. The nonlinear dynamic analysis (NDA) is performed with the Force Analogy Method (FAM) and the state–space formulation to solve the nonlinear equations of motion. Then, the Incremental Dynamic Analysis (IDA) is used to perform probabilistic fragility analyses for a typical reinforced concrete (RC) frame structure. The analysis also provides estimates of expected losses, gives the building owners business support data to compare, e.g., the final annual building income (FABI), the cost of repair and the function recovery duration for various wind events in an integrated resilience-based building design framework. This provides the basis for mapping structural damage on large urban scales and for straightforward quantification of wind-induced damage from available response acceleration time-series acquired from real-time vibration monitoring.

A fast nonlinear dynamic analysis automated approach to produce fragility curves for 3D RC frames

This study describes an automated procedure to assess seismic fragility of 3D reinforced concrete frame structures. A number of 9600 3D nonlinear dynamic analyses are performed using the recently developed 3D Force Analogy Method, implemented in an automated Matlad-based non-commercial code. The numerical efficiency of 3D Force Analogy is first illustrated for a broad range of structures and groud motion recordings and scaling factors. The ground motion dataset consisting of 20 horizontal recordings used in the study is characteristic for soft deep soil sites (e.g Bucharest) which are affected by large magnitude earthquakes originating in the Vrancea intermediate-depth seismic source. A total of 24 RC frame structures, both regular and irregular are employed in the study. The Park-Ang damage index is employed in order to evaluate the damage states of the structures. Fragility functions corresponding to life-safety and collapse damage states are obtained for both principal directions of the structures. The results show similar median fragility values on both principal directions, and a larger variability on the transversal direction. The analyses have also shown that the standard deviations evaluated for the mean annual failure rates corresponding to the two considered damage states and on both principal directions is almost equal to the mean value. The mean value of the collapse probability evaluated for the peak ground acceleration with a mean return period of 2475 years and based on the results of all 24 models is 6.1 % for the longitudinal direction and 6.5 % for the transversal direction.

New Approach to Nonlinear Dynamic Analysis of Reinforced Concrete 3D Frames; An Accurate and Computational Efficient Mathematical Model

The structural engineering community often deals with the issue of inelastic incursions of the structural response. Although buildings situated in seismic regions are usually designed using elastic analysis, most encounter significant inelastic deformations when major events occur. In general, material nonlinearity is the most important source of nonlinearity considered in the dynamic analysis commonly performed in structural design. In this paper, the recent concept of the force analogy method for 3D structures is developed and integrated into an accurate computational efficient algorithmic routine for nonlinear dynamic analysis of reinforced concrete frames. Moreover, a unique straightforward mathematical model for the numerical implementation of degrading cyclic behavior of structural elements is proposed and further used to simulate the response of a 10 story reinforced concrete frame structure. A set of nonlinear dynamic analyses are performed using the proposed algorithm in order to assess the structural damage in case of different peak ground accelerations seismic recordings. The seismic structural damage is evaluated using both structural response parameters expressed in terms of displacement and energy concepts.

Nonlinear dynamic response analysis of buildings for wind loads. A new frontier in the structural wind engineering

The structural engineering practitioners often deal with natural hazards that generate a structural response that is dynamic in nature. The wind loads are long duration load types and their induced effects, especially for severe events, need to be predicted through analysis by efficient numerical nonlinear solvers. In this paper the nonlinear wind-induced effects on structures are predicted using a non-commercial software based on the 3D Force Analogy Method (3D-FAM). The wind loads consist of time-histories of pressure coefficients recorded at a large number of taps in the wind tunnel test for a model corresponding to an 80 m high prototype building. A considerable number of dynamic nonlinear analyses are performed for different mean hourly wind speeds and wind blowing directions. For each case, the inelastic behavior of building is monitored through and efficient complementary energy balance-based analysis and quantified in time-domain by damage indices at each section, element, story and structural level.

Modelling nonlinear behavior of [formula omitted] frames using the Force Analogy Method

The nonlinear dynamic analysis of civil engineering structures is a topic of high significance for civil engineering community. In recent years, the linear dynamic analysis, in which no damage is taken into account, was improved to consider various sources of nonlinearity. Many commercial software packages were developed to consider both the material and geometrical nonlinearity. A new technique, entitled the Force Analogy Method, was introduced recently. In this technique, the material nonlinearity is addressed by considering a constant stiffness matrix throughout the entire dynamic analysis. Based on the principle of superposition, the displacement vector is decomposed into an elastic and inelastic component. The inelastic component of the displacement will cause perturbation forces once this term is moved to the right hand side of the equation of motion. However, although this intriguing technique has proved to be both accurate and numerical efficient over the years, no paper has addressed the 3D models case yet. In this paper, the authors develop the fundamental approach to generalize FAM for modeling the nonlinear response of three dimensional frame structures. The numerical results obtained are compared and validated using a well established commercial software.

Mass irregularity effect on seismic response of moment‐resisting steel frame by nonlinear time history analysis using force analogy method

SummaryNonlinear time history analysis (NTHA) is widely considered to be the most accurate and robust method of seismic analysis and, when performed with force analogy method (FAM), can result in significant reduction in computational time, as compared to the conventional displacement‐based finite element (FE) solution technique. Taking this advantage of efficient FAM, this paper investigates the effects of location (along building height) and intensity (150–225%) of single‐floor mass irregularity on seismic response of mid‐rise moment‐resisting steel frames. Three building heights (6, 8, and 10 storeys) have been considered, and the analysis results are presented in the form of floor displacements, storey drift ratio, and plastic energy dissipation. Based on the analysis, it has been observed that critical location (for the maximum seismic response) of mass‐irregular floor corresponds to the level that exhibited the maximum seismic response (i.e., storey drift ratio) in the reference mass‐regular frame. Further, criticality of the irregular frame to seismic excitation has been found to decrease when mass‐irregular floor moves towards the top of the building. The criticality of the irregular frame has been found to increase with the increase in mass‐irregular intensity. Collapse of the irregular frame has been seen to initial first near the storey level where mass‐irregular floor is located.

A Nonlinear Model to Analyze the Structures Pounding Forces During Earthquakes

The estimation of the pounding effects occurring during earthquakes is still a hot topic for structural engineers. The multi-degree of freedom models replaced of stick or single-degree of freedom models in recent years. Consequently, a fast and robust solver is required to accurately calculate the nonlinear structural behavior when the pounding forces are accounted. A sufficiently accurate impact model is required to estimate both the approach period and the restitution period during impact. In this paper, the Force Analogy Method is employed to model the physical nonlinear behavior of the structure. Also, a nonlinear viscoelastic impact element is used to model the pounding forces occurring during earthquakes. The equation of motion is written in a state-space form and integrated with variable timestep in a Simulink routine.

Experimental evaluation of force analogy method (FAM) by element type

The force analogy method (FAM) is considered as one of the most time-saving and cost-efficient methods for analyzing frames. Through a set of assumptions and restoring forces, FAM analyzes nonlinear frames responses through Hooke’s law. This study evaluates the effect of element type on FAM through numerical and experimental tests. The conventional Euler Bernoulli (EB) element is replaced by the Timoshenko (TS) beam-column element and the results are interpreted and compared with experimental findings. Three experimental tests were conducted for benchmarking and comparison purposes via 2D aluminum frames. The results indicated that during the analysis, when the frame response is in the linear region, there is no difference between the element types in the response of the frames. When the frame entered the plastic region, the frames which were analyzed by the TS element revealed closer responses to the experimental outcomes. The gap between the results of the frame which was analyzed by EB and TS enlarged especially when the frame experienced a sharp or huge rotation of more than 0.2 rad. The final recorded deformations based on the TS element revealed an accuracy between 98.05% to 98.65%, while the EB element showed 14.66% to 45.14% for rotations of more than 0.2 rad at plastic hinge locations (PHLs).

A study on the accuracy of force analogy method in nonlinear static analysis

SummaryIn this paper, the accuracy of the force analogy method (FAM) for nonlinear static (pushover) analysis is investigated. Hence, after explanation of the concept of FAM by a new approach, 12 2D steel special moment resisting frames with different configurations were derived from two different 3D archetypes and modeled using four various methods of capturing nonlinearity including SAP2000 super elements, OpenSees force‐based and displacement‐based fiber method, and FAM. In addition, a MATLAB code was developed for modeling and analyzing the frames by FAM. The accuracy of FAM in predictions of pushover curves, base shear forces, and plastic rotation of the critical hinges was investigated. The results indicated that in general, the predictions obtained using the FAM had a good agreement and compatibility with those from other methods of analysis currently used in practice for seismic performance assessment of structures.

Retracted] Multidimensional Fragility Analysis for a NEES Frame Structure by Integrating a New Energy Damage Index: Cumulative Plastic Strain

Cumulative plastic strain (CPS) damage index is proposed in this study for seismic fragility analysis by integrating the force analogy method into the energy balance equation, and CPS can be defined as the ratio of the demand of plastic dissipation energy to its capacity. The cumulative plastic strain can indicate the structural damage cumulative effect under earthquakes, which makes it especially suitable to be selected as the damage index for the structural component. Threshold values of cumulative plastic strain for different performance limit state (PLS) levels are obtained through the degree of consistency of interstory drift‐based fragility curves and CPS‐based fragility curves. Regarding the multidimensional fragility evaluation, CPS and the floor acceleration will be selected as the quantification indices for performance limit state of the structural component and nonstructural component, respectively. The probabilistic seismic demand model (PSDM) following multivariate logarithmic normal distribution will be developed, and taking PLS uncertainty and correlation into consideration, multidimensional PLS function is constructed to identify the structural failure domain. A full‐scale 2‐bay 2‐story frame structure for the Network for Earthquake Engineering Simulation (NEES) project is employed as the case study structure to demonstrate the proposed theory. Nonlinear dynamic time‐history analysis is carried out for the structure to obtain its maximum responses under earthquakes. Consequently, the multidimensional fragility curves can be derived on the basis of CPS. Besides, the influence of PLS threshold value, engineering demand parameter (EDP) correlation, and PLS correlation on the multidimensional fragility is investigated. Results show that (1) CPS damage index can fully consider the cumulative effect of damage under earthquakes, which makes up for the deficiency of the interstory drift damage index in this aspect, (2) the multidimensional fragility framework can deal with the PLS correlation and EDP correlation simultaneously, which will generate a more precise seismic damage assessment result, and (3) multidimensional fragility is sensitive to PLS threshold values and PLS correlation parameters.

A State Space Formulation for the Evaluation of the Pounding Forces During Earthquake

Abstract In recent years, the pounding effect during earthquake is a subject of high significance for structural engineers. In this paper, a state space formulation of the equation of motion is used in a MATLAB code. The pounding forces are calculated using nonlinear viscoelastic impact element. The numerical study is performed on SDOF structures subjected by 1940 EL-Centro and 1977 Vrancea N-S recording. While most of the studies available in the literature are related to Newmark implicit time integration method, in this study the equations of motion in state space form are direct integrated. The time domain is chosen instead of the complex one in order to catch the nonlinear behavior of the structures. The physical nonlinear behavior of the structures is modeled according to the Force Analogy Method. The coupling of the Force Analogy Method with the state space approach conducts to an explicit time integration method. Consequently, the collision is easily checked and the pounding forces are taken into account into the equation of motion in an easier manner than in an implicit integration method. A comparison with available data in the literature is presented.

A force analogy method (FAM) assessment on different static condensation procedures for frames with full Rayleigh damping

SummaryIn this paper, influence of different static condensation methods on nonlinear time history response of structures in presence of full Rayleigh damping matrix has been studied. Equation of motion is derived in details for 2 newly introduced cases; Full Static Condensation Model and Reduced Static Condensation Model (RSCM or RSCM‐1). Nonlinear analyses are performed with state‐space form of equation of motion along with force analogy method. A novel aspect of this study in comparison with previous researches is to consider full Rayleigh damping in formulations for static condensation in nonlinear structural analysis. Numerical examples are presented for the proposed methods. In addition, the same structure was modeled in finite element method and analyzed by the OpenSEES program. Comparisons demonstrate that results, which are obtained from introduced methods in this article, are in good agreement with the finite element method results.In addition to explaining the equation derivations in main article, proofs for some equations have been demonstrated in appendices. The comparisons show that both RSCM and Full Static Condensation Model lead to almost the same results but in case of using RSCM‐1, the responses have some deviation from other condensation methods.

Seismic Damage Detection of Moment Resisting Frame Structures Using Time‐Frequency Features

To detect seismic damage of moment resisting frame (MRF) structures, a data‐driven method using the fractal dimension (FD) of time‐frequency feature (TFF) of structural seismic dynamic responses at measured stories is extended and refined. The TFF is defined as the real part of Gabor wavelet transform of translational interstory displacement, and FD is used to give a quantitative value to describe the calculated TFF. Static condensation method is first used to reduce the degrees‐of‐freedom (DOFs) of MRF and to express the rotational displacements using translational displacements. For linear MRF, the FDs of TFFs at all stories are the same using the definition of TFF and modal superposition principle. For damaged MRF with plastic hinges at the ends of beams and columns, the force analogy method is implemented to establish transformation matrix from plastic hinge rotations to translational interstory inelastic displacements. Due to the sparseness of the transformation matrix, plastic hinges only generate interstory inelastic displacements, which are low‐frequency contents, in the vicinity of plastic hinges. Correspondingly, the FDs of TFFs of interstory displacements with inelastic component are different from the FDs of TFFs of the interstory displacements that do not contain inelastic component. A numerical simulation on a 16‐story MRF was conducted. The simulation included 10 cases such as no damage or linear structure, plastic hinges in single‐story beams, plastic hinges in single‐story columns, plastic hinges in single‐story beams and columns, and plastic hinges in multiple story beams and columns. The robustness to measurement noise was also investigated. The seismic damage detection results demonstrated that the proposed method was capable of locating the stories where the plastic hinges occurred.

A novel approach to enhance the accuracy of vibration control of Frames

All structures built within known seismically active regions are typically designed to endure earthquake forces. Despite advances in earthquake resistant structures, it can be inferred from hindsight that no structure is entirely immune to damage from earthquakes. Active vibration control systems, unlike the traditional methods which enlarge beams and columns, are highly effective countermeasures to reduce the effects of earthquake loading on a structure. It requires fast computation of nonlinear structural analysis in near time and has historically demanded advanced programming hosted on powerful computers. This research aims to develop a new approach for active vibration control of frames, which is applicable over both elastic and plastic material behavior. In this study, the Force Analogy Method (FAM), which is based on Hook’s Law is further extended using the Timoshenko element which considers shear deformations to increase the reliability and accuracy of the controller. The proposed algorithm is applied to a 2D portal frame equipped with linear actuator, which is designed based on full state Linear Quadratic Regulator (LQR). For comparison purposes, the portal frame is analysed by both the Euler Bernoulli and Timoshenko element respectively. The results clearly demonstrate the superiority of the Timoshenko element over Euler Bernoulli for application in nonlinear analysis.

The Effect of Element Types on Force Analogy Method Analysis

In this study, the seismic performance of a 2D portal frame subjected to the recorded seismic ground motions of the Northridge 1994 earthquake was evaluated by the force analogy method (FAM) with different element types. To increase the accuracy of FAM, Timoshenko (TS) elements were employed instead of the classical Euler Bernoulli (EB) elements, to revert the shear deformations that are neglected in EB elements. To perform evaluation, the same material and section properties were considered and the same portal frame was analyzed with different element lengths, from 0.5 to 7.0 m in 0.5 m steps.

Robust market-based control method for nonlinear structure

For a nonlinear control system, there are many uncertainties, such as the structural model, controlled parameters and external loads. Although the significant progress has been achieved on the robust control of nonlinear systems through some researches on this issue, there are still some limitations, for instance, the complicated solving process, weak conservatism of system, involuted structures and high order of controllers. In this study, the computational structural mechanics and optimal control theory are adopted to address above problems. The induced norm is the eigenvalue problem in structural mechanics, i.e., the elastic stable Euler critical force or eigenfrequency of structural system. The segment mixed energy is introduced with a precise integration and an extended Wittrick-Williams (W-W) induced norm calculation method. This is then incorporated in the market-based control (MBC) theory and combined with the force analogy method (FAM) to solve the MBC robust strategy (R-MBC) of nonlinear systems. Finally, a single-degree-of-freedom (SDOF) system and a 9-stories steel frame structure are analyzed. The results are compared with those calculated by the <TEX>$H{\infty}$</TEX>-robust (R-<TEX>$H{\infty}$</TEX>) algorithm, and show the induced norm leads to the infinite control output as soon as it reaches the critical value. The R-MBC strategy has a better control effect than the R-<TEX>$H{\infty}$</TEX> algorithm and has the advantage of strong strain capacity and short online computation time. Thus, it can be applied to large complex structures.

Seismic Performance Assessment of Offshore Reinforced Concrete Bridges Using the Force Analogy Method

In this paper, the seismic performance of offshore reinforced concrete (RC) bridges during their life cycle periods is assessed by the pushover analysis based on the concept of the force analogy method (FAM). The governing equations and implementation process of the proposed pushover method are first derived. The material nonlinearity is modeled by the local plastic mechanism, which is capable of simulating the monotonic strength hardening and softening behaviors of RC piers. The chloride ions corrosion effect for the RC bridges located in coastal areas is considered by using the deterioration model for the mechanical property of reinforcement steel. Besides, structural stability against overturning is considered by incorporating the geometric nonlinearity with the FAM. Since the initial stiffness matrices remain constant through the computation process, the advantages of the FAM, such as high efficiency and stability, are retained. A numerical example is carried out to illustrate the process of seismic performance assessment for offshore RC bridges with the FAM.

Nonlinear Seismic Analysis of Reinforced Concrete Bridges Using the Force Analogy Method

Abstract In this paper, a three-dimensional nonlinear dynamic analysis framework for RC bridges is established based on the force analogy method (FAM). Two biaxial local plastic mechanisms, rotation hinge (RH) and slide hinge (SH), are proposed based on the fundamental concept of FAM to simulate the nonlinear flexure-shear interactive behavior of RC piers, a critical consideration for seismic performance assessment of RC bridges. The RHs located at the ends of RC piers express the relationships of moment versus plastic rotation, whereas the SHs are assigned to simulate the shear versus plastic shear deformation behaviors of RC piers. Coupling the presented biaxial local plastic mechanisms with FAM, the nonlinear response of RC bridges under bidirectional earthquakes can be evaluated. Because only constant initial stiffness matrices are used throughout the whole analyzing process, the state space integration formulation can be used for solving the equations of motion, which makes the computational process ...