The effective stiffness of modern unreinforced masonry walls

Code design of unreinforced masonry (URM) buildings is based on elastic analysis, which requires as input parameter the effective stiffness of URM walls. Eurocode estimates the effective stiffness as 50% of the gross sectional elastic stiffness, but comparisons with experimental results have shown that this may not yield accurate predictions. In this paper, 79 shear‐compression tests of modern URM walls of different masonry typologies from the literature are investigated. It shows that both the initial and the effective stiffness increase with increasing axial load ratio and that the effective‐to‐initial stiffness ratios are approximately 75% rather than the stipulated 50%. An empirical relationship that estimates the E‐modulus as a function of the axial load and the masonry compressive strength is proposed, yielding better estimates of the elastic modulus than the provision in Eurocode 6, which calculates the E‐modulus as a multiple of the compressive strength. For computing the ratio of the effective to initial stiffness, a mechanics‐based formulation is built on a recently developed analytical model for the force‐displacement response of URM walls. The model attributes the loss in stiffness to diagonal cracking and brick crushing, both of which are taken into account using mechanical considerations. The obtained results of the effective‐to‐initial stiffness ratio agree well with the test data. A sensitivity analysis using the validated model shows that the ratio of effective‐to‐initial stiffness is for most axial load ratios and wall geometries around 75%. Therefore, a modification of the fixed ratio of effective‐to‐initial stiffness from 50% to 75% is suggested.     

Out‐of‐plane seismic response of vertically spanning URM walls connected to flexible diaphragms

A simplified numerical model was used to investigate the out‐of‐plane seismic response of vertically spanning unreinforced masonry (URM) wall strips. The URM wall strips were assumed to span between two flexible diaphragms and to develop a horizontal crack above the wall mid‐height. Three degrees of freedom were used to accommodate the wall displacement at the crack height and at the diaphragm connections, and the wall dynamic stability was studied. The equations of dynamic motion were obtained using principles of rocking mechanics of rigid bodies, and the formulae were modified to include semi‐rigid wall behaviour. Parametric studies were conducted that included calculation of the wall response for different values of diaphragm stiffness, wall properties, applied overburden, wall geometry and earthquake ground motions. The results of the study suggest that stiffening the horizontal diaphragms of typical low‐rise URM buildings will amplify the out‐of‐plane acceleration demand imposed on the wall and especially on the wall–diaphragm connections. It was found that upper‐storey walls connected to two flexible diaphragms had reduced stability for applied earthquake accelerograms having dominant frequency content that was comparable with the frequency of the diaphragms. It was also found that the applied overburden reduced wall stability by reducing the allowable wall rotations. The results of this study suggest that the existing American Society of Civil Engineers recommendations for assessment of vertically spanning walls overestimate the stability of top‐storey walls in multi‐storey buildings in high‐seismic regions or for walls connected to larger period (less stiff) diaphragms. Copyright © 2015 John Wiley & Sons, Ltd.     

Analytical and empirical models for predicting the drift capacity of modern unreinforced masonry walls

Displacement‐based seismic assessment of buildings containing unreinforced masonry (URM) walls requires as input, among others, estimates of the in‐plane drift capacity at the considered limit states. Current codes assess the drift capacity of URM walls by means of empirical models with most codes relating the drift capacity to the failure mode and wall slenderness. Comparisons with experimental results show that such relationships result in large scatter and usually do not provide satisfactory predictions. The objective of this paper is to determine trends in drift capacities of modern URM walls from 61 experimental tests and to investigate whether analytical models could lead to more reliable estimates of the displacement capacity than the currently used empirical models. A recently developed analytical model for the prediction of the ultimate drift capacity for both shear and flexure controlled URM walls is introduced and simplified into an equation that is suitable for code implementation. The approach follows the idea of plastic hinge models for reinforced concrete or steel structures. It explicitly considers the influence of crushing due to flexural or shear failure in URM walls and takes into account the effect of kinematic and static boundary conditions on the drift capacity. Finally, the performance of the analytical model is benchmarked against the test data and other empirical formulations. It shows that it yields significantly better estimates than empirical models in current codes. The paper concludes with an investigation of the sensitivity of the ultimate drift capacity to the wall geometry, static, and kinematic boundary conditions.     

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.     

Analytical model for the out‐of‐plane response of vertically spanning unreinforced masonry walls

An analytical model describing the flexural response of vertically spanning out‐of‐plane loaded unreinforced masonry walls is presented in this paper. The model is based on the second‐order Euler‐Bernoulli beam theory and captures important characteristics of the out‐of‐plane response of masonry walls that have been observed in experimental tests and from numerical studies but for which an analytical solution was still lacking: the onset and the evolution of cracking, the peak strength of the out‐of‐plane loaded walls, and the softening of the response due to P ?Δ effects. The model is validated against experimental results, and the comparison shows that the model captures both the prepeak and postpeak response of the walls. From the analytical model of the force‐displacement curve, a formula for the maximum out‐of‐plane strength of the walls is derived, which can be directly applied in engineering practice.     

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.     

In‐plane and out‐of‐plane behavior of confined masonry walls for various toothing and openings details and prediction of their strength and stiffness

Eight half‐scale brick masonry walls were tested to study two important aspects of confined masonry (CM) walls related to its seismic behavior under in‐plane and out‐of‐plane loads. Four solid wall specimens tested to investigate the role of type of interface between the masonry and tie‐columns, such as toothing varying from none to every course. The other four specimens with openings were tested to study the effectiveness of various strengthening options around opening to mitigate their negative influence. In the set of four walls, one wall was infilled frame while the other three were CM walls of different configurations. The experimental results were further used to determine the accuracy of various existing models in predicting the in‐plane response quantities of CM walls. Confined masonry walls maintained structural integrity even when severely damaged and performed much better than infill frames. No significant effect of toothing details was noticed although toothing at every brick course was preferred for better post‐peak response. For perforated walls, provision of vertical elements along with continuous horizontal bands around openings was more effective in improving the overall response. Several empirical and semi‐empirical equations are available to estimate the lateral strength and stiffness of CM walls, but those including the contribution of longitudinal reinforcement in tie‐columns provided better predictions. The available equations along with reduction factors proposed for infills could not provide good estimates of strength and stiffness for perforated CM walls. However, recently proposed relations correlating strength/stiffness with the degree of confinement provided reasonable predictions for all wall specimens. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical study on factors that influence the in‐plane drift capacity of unreinforced masonry walls

Displacement‐based assessment procedures require as input reliable estimates of the deformation capacity of all structural elements. For unreinforced masonry (URM) walls, current design codes specify the in‐plane deformation capacity as empirical equations of interstory drift. National codes differ with regard to the parameters that are considered in these empirical drift capacity equations, but the inhomogeneity of datasets on URM wall tests renders it difficult to validate the hypotheses with the currently available experimental data. This paper contributes to the future development of such empirical relationships by investigating the sensitivity of the drift capacity to the shear span, the aspect ratio, the axial load ratio, and the size of the wall. For this purpose, finite element models of URM walls are developed in Abaqus/Explicit and validated against a set of experimental results. The results show that the axial load ratio, the shear span, and the wall size are among the factors that influence the drift capacity the most. Empirical equations are mainly derived from test results on small walls, and the numerical results suggest that this can lead to a significant overestimation of the drift capacity for larger walls.     

Out‐of‐plane strength reduction of unreinforced masonry walls because of in‐plane damages

There are numerous studies on the behavior of Unreinforced Masonry (URM) walls in both in‐plane (IP) and out‐of‐plane (OP) directions; however, few aimed at understanding the simultaneous contribution of these intrinsic responses during earthquakes. Undoubtedly, even a strong URM wall shows weakened capacity in the OP direction because of minor cracks and other damages in the IP direction, and this capacity reduction has not yet been accounted for in seismic codes. In this study, performance of three URM walls is evaluated by several numerical analyses in terms of the OP capacity reduction because of IP displacements and failure modes. Several parameters influencing the OP capacity have been studied including aspect ratio, roof boundary condition, IP displacement and IP loading patterns. The results indicate that reduction in the OP capacity of URM walls varies from negligible to very high depending on boundary conditions, IP failure mode and IP damage severity. Moreover, IP loading pattern is more important in walls with higher aspect ratios because of their IP failure modes. Copyright © 2015 John Wiley & Sons, Ltd.     

Axially equilibrated displacement‐based beam element for simulating the cyclic inelastic behaviour of RC members

Distributed plasticity beam elements are commonly used to evaluate limit state demands for performance‐based analysis of reinforced concrete (RC) structures. Strain limits are often preferred to drift limits because they directly relate to damage and are therefore less dependent on member geometry and boundary conditions. However, predicting accurately strain demands still represents a major simulation challenge. Tension shift effects, which induce a linear curvature profile in the plastic hinge region of RC columns and walls, are one of the main causes for the mismatch between experimental and numerical estimates of local level quantities obtained through force‐based formulations. Classical displacement‐based approaches are instead suitable to simulate such linear curvature profile. Unfortunately, they verify equilibrium only on an average sense due to the wrong assumption on the axial displacement field, leading to poor deformation and force predictions. This paper presents a displacement‐based element in which axial equilibrium is strictly verified along the element length. The assumed transversal displacement field ensures a linear curvature profile, connecting accurately global displacement and local strain demands. The proposed finite element is validated against two sets of quasi‐static cyclic tests on RC bridge piers and walls. The results show that curvature and strain profiles for increasing ductility demands are significantly improved when axially equilibrated rather than classical displacement‐based or force‐based elements are used to model the structural members. Copyright © 2017 John Wiley & Sons, Ltd.     

Seismic capacity of irregular unreinforced masonry walls with openings

Masonry buildings are often characterized by geometric irregularities. In many cases, such buildings meet global regularity requirements provided by seismic codes, but they are composed by irregular walls with openings. The latter are masonry walls characterized by (i) openings of different sizes, (ii) openings misaligned in the horizontal and/or vertical direction, or (iii) a variable number of openings per story. An irregular layout of openings can induce not only a nonuniform distribution of gravity loads among masonry piers but also unfavorable damage localizations resulting in a premature collapse of the wall and hence a higher seismic vulnerability. This paper is aimed at providing a simplified methodology to assess the effects of irregularities on the in‐plane seismic capacity of unreinforced masonry (URM) walls with openings. To this end, a macroelement method was developed and validated through experimental results available in the literature. The proposed methodology was based on the quantification of wall irregularities by means of geometric indices and their effects on seismic capacity of URM walls with openings through both sensitivity and regression analyses. Sensitivity analysis was based on a high number of static pushover analyses and allowed to assess variations in key seismic capacity parameters. Regression analysis let to describe each capacity parameter under varying irregularity index, providing empirical models for seismic assessment of irregular URM walls with openings. The in‐plane seismic capacity was found to be significantly affected by wall irregularities, especially in the case of openings with different heights. Copyright © 2012 John Wiley & Sons, Ltd.     

Cyclic testing of unreinforced masonry walls in two‐way bending

An experimental programme was conducted in which eight full‐scale unreinforced masonry walls were subjected to cyclic face loading using a system of airbags. Of the eight walls, six contained a window opening and four were subjected to vertical pre‐compression. Combined supports at the vertical and horizontal edges ensured that under face loading the walls underwent two‐way bending. The test walls were found to possess good post‐peak strength and displacement capacity as well as reasonable energy dissipation characteristics. Significant strength and stiffness degradation and non‐symmetry of strength in the positive and negative displacement directions were also evident. Discussion of the causes of the aforementioned trends and their implications towards the seismic response of masonry walls is provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Seismic behaviour of confined masonry walls

The results of tests of plain and confined masonry walls with h/l ratio equal to 1·5, made at 1:5 scale, have been used to develop a rational method for modelling the seismic behaviour of confined masonry walls. A trilinear model of lateral resistance–displacement envelope curve has been proposed, where the resistance is calculated as a combination of the shear resistance of the plain masonry wall panel and dowel effect of the tie-columns’ reinforcement. Lateral stiffness, however, is modelled as a function of the initial effective stiffness and damage, occurring to the panel at characteristic limit states. Good correlation between the predicted and experimental envelopes has been obtained in the particular case studied. The method has been also verified for the case of prototype confined masonry walls with h/l ratio equal to 1·0. Good correlation between the predicted and experimental values of lateral resistance indicates the general validity of the proposed method. © 1997 John Wiley & Sons, Ltd.     

Towards error‐free hybrid simulation using mixed variables

Two procedures are developed and implemented in a hybrid simulation system (HSS) with the aim of enhancing the accuracy and reliability of the online, i.e. pseudo‐dynamic, test results. The first procedure aims at correcting the experimental systematic error in executing the displacement command signal. The error is calculated as the difference between command and feedback signals and correlated to the actuator velocity using the least‐squares method. A feed‐forward error compensation scheme is devised leading to a more accurate execution of the test. The second procedure employs mixed variables with mode switching between displacement and force controls. The newly derived force control algorithm is evaluated using a parametric study to assess its stability and accuracy. The implementation of the mixed variables procedure is designed to adopt force control for high stiffness states of the structural response and displacement control otherwise, where the resolution of the involved instruments may favour this type of mixed control. A simple pseudo‐dynamic experiment of steel cantilever members is used to validate the HSS. Moreover, two experiments as application examples for the two developed procedures are presented. The two experiments focus on the seismic response of (a) timber shear walls and (b) reinforced concrete frames with and without unreinforced masonry infill wall. Copyright © 2007 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.     

Seismic behavior and modeling of steel reinforced concrete (SRC) walls

The steel reinforced concrete (SRC) wall consists of structural steel embedded at the boundary elements of a reinforced concrete (RC) wall. The use of SRC walls has gained popularity in the construction of high‐rise buildings because of their superior performance over conventional RC walls. This paper presents a series of quasi‐static tests used to examine the behavior of SRC walls subjected to high axial force and lateral cyclic loading. The SRC wall specimens showed increased flexural strength and deformation capacity relative to their RC wall counterpart. The flexural strength of SRC walls was found to increase with increasing area ratio of embedded structural steel, while the section type of embedded steel did not affect the wall's strength. The SRC walls under high axial force ratio had an ultimate lateral drift ratio of approximately 1.4%. In addition, a multi‐layer shell element model was developed for the SRC walls and was implemented in the OpenSees program. The numerical model was validated through comparison with the test data. The model was able to predict the lateral stiffness, strength and deformation capacities of SRC walls with a reasonable level of accuracy. Finally, a number of issues for the design of SRC walls are discussed, along with a collection and analysis of the test data, including (1) evaluation of flexural strength, (2) calculation of effective flexural stiffness, and (3) inelastic deformation capacity of SRC walls. Copyright © 2014 John Wiley & Sons, Ltd.     

Procedures for calibration of linear models for damage limitation in design of masonry‐infilled RC frames

Recent earthquakes have confirmed the role played by infills in the seismic response of reinforced concrete buildings. The control and limitation of damage to such nonstructural elements is a key issue in performance‐based earthquake engineering. The present work is focused on modeling and analysis of damage to infill panels, and, in particular, it is aimed towards linear analysis procedures for assessing the damage limitation limit state of infilled reinforced concrete frames. First, code provisions on infill modeling and acceptance criteria at the damage limitation limit state are reviewed. Literature contributions on damage to unreinforced masonry infill panels and corresponding displacement capacity are reported and discussed. Two procedures are then proposed aiming at a twofold goal: (i) the determination of ‘equivalent’ interstory drift ratio limits for a bare frame model and (ii) the estimation of the stiffness of equivalent struts representing infill walls in a linear model. These two quantities are determined such that a linear model ensures a reliable estimation of seismic capacity at the damage limitation limit state, providing the same intensity level as that obtained from nonlinear analyses carried out on structural models with infills. Finally, the proposed procedures are applied to four‐story and eight‐story case study‐infilled frames, designed for seismic loads according to current technical codes. The results of these application examples are presented and discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

Shake‐table experiment on reinforced concrete structure containing masonry infill wall

A hypothetical 5‐storey prototype structure with reinforced concrete (RC) frame and unreinforced masonry (URM) wall is considered. The paper focuses on a shake‐table experiment conducted on a substructure of this prototype consisting of the middle bays of its first storey. A test structure is constructed to represent the selected substructure and the relationship between demand parameters of the test structure and those of the prototype structure is established using computational modelling. The dynamic properties of the test structure are determined using a number of preliminary tests before performing the shake‐table experiments. Based on these tests and results obtained from computational modelling of the test structure, the test ground motions and the sequence of shakings are determined. The results of the shake‐table tests in terms of the global and local responses and the effects of the URM infill wall on the structural behaviour and the dynamic properties of the RC test structure are presented. Finally, the test results are compared to analytical ones obtained from further computational modelling of the test structure subjected to the measured shake‐table accelerations. Copyright © 2006 John Wiley & Sons, Ltd.     

Estimation of load reduction factors for clay masonry walls

The in‐plane cyclic behaviour of three types of unreinforced clay masonry was characterized by means of laboratory tests on full‐scale specimens. The masonry walls were assembled with various bonding arrangements (head joints made with mortar pockets, dry head joints with mechanical interlocking, thin‐layer mortar bed joints), which are not yet inserted in seismic codes. Experimental behaviour was modelled with an analytical hysteretic model able to predict lateral load–displacement curves in case of shear failure of the unreinforced walls. According to the experimental results and those of the selected analytical model, parametric study to evaluate the reduction in lateral strength demand produced by non‐linear behaviour in masonry walls, i.e. the load reduction factor was carried out by non‐linear dynamic analyses. The calculated values of the load reduction factor were modest. The differences in values found for the three masonry types, although consistent with them, were not great. This may indicate that, in the ultimate limit state, the type of masonry cannot significantly affect the behaviour of an entire building. Copyright © 2009 John Wiley & Sons, Ltd.     

A nonlinear macroelement model for the seismic analysis of masonry buildings

The macroelement technique for modelling the nonlinear response of masonry panels is particularly efficient and suitable for the analysis of the seismic behaviour of complex walls and buildings. The paper presents a macroelement model specifically developed for simulating the cyclic in‐plane response of masonry walls, with possible applications in nonlinear static and dynamic analysis of masonry structures. The model, starting from a previously developed macroelement model, has been refined in the representation of flexural–rocking and shear damage modes, and it is capable of fairly simulating the experimental response of cyclic tests performed on masonry piers. By means of two internal degrees of freedom, the two‐node macroelement permits to represent the coupling of axial and flexural response as well as the interaction of shear and flexural damage. Copyright © 2013 John Wiley & Sons, Ltd.