A macro‐model with nonlinear springs for seismic analysis of URM buildings

Seismic assessment of existing unreinforced masonry buildings represents a current challenge in structural engineering. Many historical masonry buildings in earthquake regions were not designed to withstand seismic loading; thus, these structures often do not meet the basic safety requirements recommended by current seismic codes and need to be strengthened considering the results from realistic structural analysis. This paper presents an efficient modelling strategy for representing the nonlinear response of unreinforced masonry components under in‐plane cyclic loading, which can be used for practical and accurate seismic assessment of masonry buildings. According to the proposed strategy, generic masonry perforated walls are modelled using an equivalent frame approach, where each masonry component is described utilising multi‐spring nonlinear elements connected by rigid links. When modelling piers and spandrels, nonlinear springs are placed at the two ends of the masonry element for describing the flexural behaviour and in the middle for representing the response in shear. Specific hysteretic rules allowing for degradation of stiffness and strength are then used for modelling the member response under cyclic loading. The accuracy and the significant potential of the proposed modelling approach are shown in several numerical examples, including comparisons against experimental results and the nonlinear dynamic analysis of a building structure. Copyright © 2016 John Wiley & Sons, Ltd.     

Using the applied element method for modelling calcium silicate brick masonry subjected to in‐plane cyclic loading

The response of calcium silicate unreinforced masonry construction to horizontal cyclic loading has recently become the focus of experimental and numerical research, given its extensive use in some areas of the world that are now exposed to induced earthquakes (eg, north of the Netherlands). To assess the seismic behaviour of such construction, a relatively wide range of modelling methodologies are available, amongst which the discrete elements approach, which takes into account the intrinsic heterogeneity of a brick‐mortar assembly, can probably be deemed as the most appropriate computational procedure. On the other hand, however, since discrete elements numerical methods are based on a discontinuum domain, often they are not able to model every stage of the structural response adequately, and because of the high computational burden required, the analysis scale should be chosen carefully. The applied element method is a relatively recent addition to the discrete elements family, with a high potential for overcoming the aforementioned limitations or difficulties. Initially conceived to model blast events and concrete structures, its use in the earthquake engineering field is, of late, increasing noticeably. In this paper, the use of the applied element method to model the in‐plane cyclic response of calcium silicate masonry walls is discussed and scrutinised, also through the comparison with experimental results of in‐plane cyclic shear‐compression tests on unreinforced masonry walls.     

Advanced frame element for seismic analysis of masonry structures: model formulation and validation

This paper presents a masonry panel model for the nonlinear static and dynamic analysis of masonry buildings suitable for the seismic assessment of new and existing structures. The model is based on an equivalent frame idealization of the structure and stems from previous research on force‐based frame elements. The element formulation considers axial, bending, and shear deformations within the framework of the Timoshenko beam theory. A phenomenological cyclic section law that accounts for the shear panel response is coupled, through equilibrium between shear and bending forces along the element, with a fiber‐section model that accounts for the axial and bending responses. The proposed panel model traces with a low computational burden and numerical stability the main aspects of the structural behavior of masonry panels and is suitable for analyses of multi‐floor buildings with a relatively regular distribution of openings and with walls and floors organized to grant a box‐like behavior under seismic loads. The model capabilities are validated though analyses of simple unreinforced masonry panels and comparisons with published experimental results. The model accuracy is strongly dependent on the fiber and shear constitutive laws used. However, the formulation is general, and laws different from those employed in this study are easily introduced without affecting the model formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

A rate‐dependent constitutive model of high damping rubber bearings: modeling and experimental verification

In this study, a constitutive model of high damping rubber bearings (HDRBs) is developed that allows the accurate representation of the force–displacement relationship including rate‐dependence for shear deformation. The proposed constitutive model consists of two hyperelastic springs and a nonlinear dashpot element and expresses the finite deformation viscoelasticity laws based on the classical Zener model. The Fletcher–Gent effect, manifested as high horizontal stiffness at small strains and caused by the carbon fillers in HDRBs, is accurately expressed through an additional stiffness correction factor α in the novel strain energy function. Several material parameters are used to simulate the responses of high damping rubber at various strain levels, and a nonlinear viscosity coefficient η is introduced to characterize the rate‐dependent property. A parameter identification scheme is applied to the results of the multi‐step relaxation tests and the cyclic shear tests, and a three‐dimensional function of the nonlinear viscosity coefficient η with respect to the strain, and strain rate is thus obtained. Finally, to investigate the accuracy and feasibility of the proposed model for application to the seismic response assessment of bridges equipped with HDRBs, an improved real‐time hybrid simulation (RTHS) test system based on the velocity loading method is developed. A single‐column bridge was used as a test bed and HDRBs was physically tested. Comparing the numerical and RTHS results, advantage of the proposed model in the accuracy of the predicted seismic response over comparable hysteretic models is demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.     

Modelling the out‐of‐plane seismic behaviour of masonry walls by rigid elements

A computational model for evaluating the dynamical response and the damage of large masonry walls subjected to out‐of‐plane seismic actions is presented. During earthquakes, these actions are often the main cause of damage for the front wall and lateral walls of old masonry‐built churches and monuments. Since the crack patterns often tend to subdivide the plane walls into a number of blocks, the model assumes such walls as a series of quadrilateral plane rigid elements connected to each other in the middle of their adjoining sides. Only the out‐of‐plane displacements are considered, and the connections are regarded as spherical elasto‐plastic joints which allow rotations whose axis is in the plane of the undeformed wall. The hysteretic characteristics of these joints are defined so as to approximate the brittle behaviour of masonry material and the degradation due to cyclic loadings. The numerical results obtained using a limited number of elements show that the global out‐of‐plane response of the masonry walls and the mechanical degradation at each connection are in accord with the observed behaviour of real churches hit by strong earthquakes. Copyright © 2000 John Wiley & Sons, Ltd.     

Displacement‐based seismic analysis for out‐of‐plane bending of unreinforced masonry walls

This paper addresses the problem of assessing the seismic resistance of brick masonry walls subject to out‐of‐plane bending. A simplified linearized displacement‐based procedure is presented along with recommendations for the selection of an appropriate substitute structure in order to provide the most representative analytical results. A trilinear relationship is used to characterize the real nonlinear force–displacement relationship for unreinforced brick masonry walls. Predictions of the magnitude of support motion required to cause flexural failure of masonry walls using the linearized displacement‐based procedure and quasi‐static analysis procedures are compared with the results of experiments and non‐linear time‐history analyses. The displacement‐based procedure is shown to give significantly better predictions than the force‐based method. Copyright © 2002 John Wiley & Sons, Ltd.     

Multi‐directional response of unreinforced masonry walls: experimental and computational investigations

This paper describes the results of an experimental and numerical study that focused on multi‐directional behavior of unreinforced masonry walls and established the requisite of the related proposed design equations. The tests were conducted following several sets of multi‐directional loading combinations imposed on the top plane of the wall along with considering monotonic and cyclic quasi‐static loading protocols. Various boundary conditions, representing possible wall–roof connections, were also considered for different walls to investigate the influence of rotation of the top plane of the wall on the failure modes. The results of the tests were recorded with a host of high precision data acquisition systems, showing three‐dimensional displacements of a grid on the surface of the wall. Finite element models of the walls are developed using the commercial software package ABAQUS/Explicit compiled with a FORTRAN subroutine (VUMAT) written by the authors. The experimental results were then used to validate the finite element models and the developed user‐defined material models. With the utility of validated models, a parametric study was performed on a set of parameters with dominant influence on the behavior of the wall system under in‐plane and out‐of‐plane loading combinations. The experimental and numerical results are finally used to investigate the adequacy of ASCE 41 empirical equations, and some insights and recommendations are made. Copyright © 2016 John Wiley & Sons, Ltd.     

Out‐of‐plane seismic behaviour of rocking masonry walls

The evaluation of the out‐of‐plane behaviour of unreinforced walls is one of the most debated topics in the seismic assessment of existing masonry buildings. The discontinuous nature of masonry and its interaction with the remainder of the building make the dynamic modelling of out‐of‐plane response troublesome. In this paper, the results of a shaking table laboratory campaign on a tuff masonry, natural scale, U‐shaped assemblage (façade adjacent to transverse walls) are presented. The tests, excited by scaled natural accelerograms, replicate the behaviour of external walls in existing masonry buildings, from the beginning of rocking motion to overturning. Two approaches have been developed for modelling the out‐of‐plane seismic behaviour: the discrete element method and an SDOF analytic model. Both approaches are shown to be capable of reproducing the experimental behaviour in terms of maximum rotation and time history dynamic response. Finally, test results and numerical time history simulations have been compared with the Italian seismic code assessment procedures. Copyright © 2011 John Wiley & Sons, Ltd.     

Bidirectional behavior of unreinforced masonry walls

Most of the studies related to the modeling of masonry structures have by far investigated either the in‐plane (IP) or the out‐of‐plane (OP) behavior of walls. However, seismic loads mostly impose simultaneous IP and OP demands on load‐bearing or shear masonry walls. Thus, there is a need to reconsider design equations of unreinforced masonry walls by taking into account bidirectional effects. The intent of this study is to investigate the bidirectional behavior of an unreinforced masonry wall with a typical aspect ratio under different displacement‐controlled loading directions making use of finite element analysis. For this purpose, the numerical procedure is first validated against the results of the tests on walls with different failure modes conducted by the authors. Afterward, the response of the wall systems is evaluated with increasing top displacement having different orientations. A set of 19 monotonic and three cyclic loading analyses are performed, and the results are discussed in terms of the variation of failure modes and load–displacement diagrams. Moreover, the results of wall capacity in each loading condition are compared with those of the ASCE41‐06 formulations. The results indicate that the direction of the resultant force, vectorial summation of IP and OP forces, of the wall is initially proportional to the ratio of stiffness in the IP and the OP directions. However, with the increase of damage, the resultant force direction inclines towards the wall's longitudinal direction regardless of the direction of the imposed displacement. Finally, recommendations are made for applicability of ASCE41‐06 formulations under different bidirectional loading conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of shear‐dominated RC columns using the nonlinear truss analogy

This paper uses nonlinear truss models for the analysis of shear‐dominated reinforced concrete (RC) columns subjected to cyclic loading. A previously established method, aimed to the analysis of RC walls, is enhanced to allow simulations of column members. The concrete constitutive equations are modified to account for the contribution of the aggregate interlock to the shear resistance. Additionally, an equation is proposed to determine the inclination angle of the diagonal members in the truss models. The modeling approach is validated using the results of quasi‐static and dynamic tests on shear‐dominated RC columns. The combination of predictive capabilities and conceptual simplicity establishes truss‐based models as an attractive approach for the systematic analysis of shear‐dominated RC frame construction. Copyright © 2014 John Wiley & Sons, Ltd.     

Practice‐oriented probabilistic seismic performance assessment of infilled frames with consideration of shear failure of columns

The objective of the study presented in this paper is to investigate the effects of masonry infills on the shear demand and failure of columns for the case when reinforced concrete frames with such infills are modeled by means of simplified nonlinear models that are not capable of the direct simulation of these effects. It is shown that an approximate simulation of the shear failure of columns can be achieved through an iterative procedure that involves pushover analysis, post‐processing of the analysis results using limit‐state checks of the components, and model adaptation if shear failure of columns is detected. The fragility parameters and the mean annual frequency of limit‐state exceedance are computed on the basis of nonlinear dynamic analysis by using an equivalent SDOF model. The proposed methodology is demonstrated by means of two examples. It was shown that the strength of the four‐story and seven‐story buildings and their deformation capacity are significantly overestimated if column shear failure due to the effects of masonry infills is neglected, whereas the mean annual frequency of limit‐state exceedance for the analyzed limit states is significantly larger than that estimated for the case if the shear failure of columns is neglected. Copyright © 2012 John Wiley & Sons, Ltd.     

单调及低周往复荷载作用下砌体非线性分析

带构造柱和圈梁的约束砌体结构在四川灾区乡镇房屋重建中被广泛采用,其抗震性能是人们所关心的.基于绵竹市土门镇当地重建房屋常用建筑材料的实验数据以及通用有限元软件ANSYS中Solid 65单元的性质和特点,用有限元模型模拟了粘土砖砌体在不同压应力状态(σ-/fm)下沿通缝截面抗剪强度试验,给出了相关单元在模拟砖砌体开裂中闭合及开口剪力传递系数的建议值;利用这些结果,分别建立了带约束(构造柱、圈梁等)和不带约束砌体墙的有限元模型,进而分析了他们在单调荷载以及低周往复荷载作用下的抗震性能.结果表明,与不带约束的墙体相比,带约束墙体在单调水平荷载作用下的初裂性能、极值荷载和延性都有很大的提高,在低周往复荷载作用下其耗能能力得到了改善.所得结果可供相应结构抗震设计的参考.     

A modified parallel IWAN model for cyclic hardening behavior of sand

A modified parallel IWAN model, which includes a cyclic hardening function, is proposed and verified. The proposed model consists of elasto-perfect plastic and isotropic hardening elements. The model is able to predict cyclic hardening behavior through the adjustment of the internal slip stresses of its elements beyond the cyclic threshold, and satisfies Bauschinger's effect and the Masing rule with its own behavior characteristics. The cyclic hardening function is developed based on the irrecoverable plastic strain (accumulated shear strain) of dry sand during shearing, which is assumed to be a summation of shear strain beyond the cyclic threshold. Symmetric-limit cyclic loading and irregular loading tests were performed to determine model parameters and to verify the behavior of the proposed model. Finally, a one-dimensional site response analysis program (KODSAP) is developed by using the proposed model. The effects of cyclic hardening behavior on site response are evaluated using KODSAP.     

Three‐dimensional beam‐truss model for reinforced concrete walls and slabs – part 1: modeling approach,validation, and parametric study for individual reinforced concrete walls

A three‐dimensional beam‐truss model for reinforced concrete (RC) walls developed by the first two authors in a previous study is modified to better represent the flexure–shear interaction and more accurately capture diagonal shear failures under static cyclic or dynamic loading. The modifications pertain to the element formulations and the determination of the inclination angle of the diagonal elements. The modified beam‐truss model is validated using the experimental test data of eight RC walls subjected to static cyclic loading, including two non‐planar RC walls under multiaxial cyclic loading. Five of the walls considered experienced diagonal shear failure after reaching their flexural strength, while the other three walls had a flexure‐dominated response. The numerically computed lateral force–lateral displacement and strain contours are compared with the experimentally recorded response and damage patterns for the walls. The effects of different model parameters on the computed results are examined by means of parametric analyses. Extension of the model to simulate RC slabs and coupled RC walls is presented in a companion paper. Copyright © 2016 John Wiley & Sons, Ltd.     

Cyclic shearing and liquefaction of soil under irregular loading: an incremental model for the dynamic earthquake-induced deformation

The paper presents a mathematical model for the deformation of soil under irregular cyclic loading in the simple-shear conditions. The model includes the possible change in the effective pressure in saturated soil due to the cyclic shearing, the reciprocal influence of the effective pressure on the response of the soil to the shear loading, and the pore pressure dissipation due to the seepage of the pore fluid. The hysteresis curves for the strain–stress relationship are constructed in such a way that they produce both the required backbone curve and the required damping ratio as functions of the strain amplitude. At the same time, the approach enables the constitutive functions involved in the model to be specified in various ways depending on the soil under study. The constitutive functions can be calibrated independently of each other from the conventional cyclic shear tests. The constitutive model is incorporated in the boundary value problem for the dynamic site response analysis of level ground. A numerical solution is presented for the dynamic deformation and liquefaction of soil at the Port Island site during the 1995 Hyogoken-Nambu earthquake.     

Fragility functions for masonry infill walls with in‐plane loading

Recent seismic events have provided evidence that damage to masonry infills can lead not only to large economic losses but also to significant injuries and even fatalities. The estimation of damage of such elements and the corresponding consequences within the performance‐based earthquake engineering framework requires the construction of reliable fragility functions. In this paper, drift‐based fragility functions are developed for in‐plane loaded masonry infills, derived from a comprehensive experimental data set gathered from current literature, comprising 152 masonry infills with different geometries and built with different types of masonry blocks, when tested under lateral cyclic loading. Three damage states associated with the structural performance and reparability of masonry infill walls are defined. The effect of mortar compression strength, masonry prism compression strength, and presence of openings is evaluated and incorporated for damage states where their influence is found to be statistically significant. Uncertainty due to specimen‐to‐specimen variability and sample size is quantified and included in the proposed fragility functions. It is concluded that prism strength and mortar strength are better indicators of the fragility of masonry infills than the type of bricks/blocks used, whose influence, in general, is not statistically significant for all damage states. Finally, the presence of openings is also shown to have statistically relevant impact on the level of interstory drift ratio triggering the lower damage states.     

Seismic fragility of lightly reinforced concrete frames with masonry infills

Seismic fragility of lightly reinforced concrete frames with masonry infills is assessed through numerical simulations considering uncertainty in ground motion and building materials. To achieve this aim, a numerical model of the components is developed, a rational approach to proportion and locate individual struts in the equivalent three‐strut model is proposed, and an explicit nonlinear column shear response model accounting for the infill–column interaction and soft‐story mechanism is employed. The proposed numerical model is used to (1) generate probabilistic seismic demand models accounting for a wide range of ground motion intensities with different frequency content and (2) determine limit state models obtained from nonlinear pushover analysis and incremental dynamic analysis. Using the demand and limit state model, fragility curves for the masonry‐infilled frames are developed to investigate the impact of various infill properties on the frame vulnerability. It is observed that the beneficial effect of the masonry infill diminishes at more severe limit states because of the interaction with the boundary frame. In some cases, this effect almost vanishes or switches to an adverse effect beyond a threshold of ground motion intensities. Copyright © 2015 John Wiley & Sons, Ltd.     

Force–displacement response of in‐plane‐loaded URM walls with a dominating flexural mode

This article presents a new mechanical model for the non‐linear force–displacement response of unreinforced masonry (URM) walls developing a flexural rocking mode including their displacement capacity. The model is based on the plane‐section hypothesis and a constitutive law for the masonry with zero tensile strength and linear elastic behaviour in compression. It is assumed that only the compressed part of the wall contributes to the stiffness of the wall and therefore the model accounts for a softening of the response due the reduction of the effective area. Stress conditions for limit states are proposed that characterise the flexural failure. The new model allows therefore linking local performance levels to global displacement capacities. The limit states criteria describe the behaviour of modern URM walls with cement mortar of normal thickness and clay bricks. The model is validated through comparison of local and global engineering demand parameters with experimental results. It provides good prediction of the effective stiffness, the force capacity and the displacement capacity of URM walls at different limit states. Copyright © 2015 John Wiley & Sons, Ltd.     

Steel slit shear walls with double‐tapered links capable of condition assessment

The concept of using a hysteretic damper as a condition assessment device that functions immediately after a damaging earthquake is realized by making use of the residual out‐of‐plane deformation of links that are arranged in slit shear walls. According to the proposed inspection procedure, the maximum drift ratio experienced by the slit wall is estimated based on the number of torsionally deformed links whose dimensions are determined so that the links would exhibit notable torsional deformation at the target deformations. The adoption of a double‐tapered shape for the links enables us to significantly increase the amount of out‐of‐plane deformation. The relationship between the dimensions and the torsional deformation of the links is established using numerical simulations. The effectiveness of the proposed condition assessment scenario is verified by using a series of cyclic loading tests for individual links and groups of links. As a hysteretic damper, the strength and stiffness of the links predicted by design equations matched well with test results. Copyright © 2014 John Wiley & Sons, Ltd.