The assessment of the response of masonry infilled RC frame structures has been a major challenge over the last decades. While several modeling approaches have been proposed, none can cover all aspects observed in the tests. The present paper introduces a simplified model built on the approach established by Crisafulli and Carr (2007) and addresses its calibration and implementation in a nonlinear analysis software for the evaluation of the inplane lateral response of infilled RC frames. The proposed model uses a set of elements/springs to account separately for the compressive and shear behavior of masonry. The efficiency of the modeling approach is validated with available experimental data, yielding satisfactory matching. The most intricate issue encountered when attempting to represent a masonry panel is the plethora of the material parameters involved and the lack of complete and available test results. Thus, the numerical investigation is accompanied by a parametric study on the sensitivity of the model to the various parameters to identify the critical modeling quantities and provide guidance on their selection.
The evaluation of the seismic performance of masonry infilled RC frame structures is a widespread problem that has not yet been resolved despite the numerous efforts reported in the literature during the last decades. As a result, and contrary to the finding from the response of masonry infilled structures under actual seismic action, infill is often treated as nonstructural elements and is omitted by the analysis models. Nevertheless, the uncertainty associated with the interaction of the infill with the bounding frame and the different failure modes exhibited, the variability of the material properties, geometrical configurations, and construction methods reveal the complexity of the problem and justify the lack of unified, reliable, and widely accepted approaches for the design and assessment of structural systems that include infill panels.
From the computational point of view, the modeling techniques used for the analysis of infilled frames can be divided into two main categories: (i) local or micromodels and (ii) simplified macromodels. The first category is based on the nonlinear finite element method and strives to provide an accurate representation of the frameinfill interaction at the local level. Many of the researchers, who adopted this methodology, for example, Lotfi [
On the other hand, macromodels utilize the “equivalent truss” concept to provide a simple and efficient tool, able to represent the global response of the infill panel and its interaction with the surrounding RC frame. The basic idea of this family of models was first suggested by Polyakov [
Summarizing the available solutions, it seems that even if macromodeling schemes are not capable of simulating in a detailed manner all the possible failure mechanisms encountered in infilled frame structures, the limited computational effort required for their implementation makes them the best alternative, especially when analyzing large structures. The challenge is, yet, to develop a broadly applicable framework for determining the effective properties and the hysteretic behavior of the appropriate simplified model for each case study.
The aim of this paper is to provide an analytical tool for assessing the inplane behavior of masonry infilled RC frames, focusing on the establishment of a set of specific guidelines for the evaluation of the various parameters required to define the monotonic and hysteretic response of the masonry infill. The modeling scheme presented herein constitutes a modification of the “masonrypanel” approach, originally developed by Crisafulli and Carr [
As previously mentioned, the modeling scheme presented in this study is based on the “masonrypanel” model proposed by Crisafulli and Carr [
Schematic representation of the Crisafulli and Carr masonry panel element [
The configuration of the model as presented here maintains the idea of combining diagonal strut and shear spring elements with the difference that no interrelation between the individual elements that comprise the panel has been externally imposed, except the one arising from the distribution of strength and stiffness among the various components during a loading event. As the original model by Crisafulli was not available in the software framework used in this study (
Schematic representation of the proposed model (modification of [
The model has been implemented in
Generic representation of the “twonode link element,”
In this particular modeling scheme, the struts are modeled with “twonode link” objects connecting the opposite corner nodes of the panel that are active solely in the direction of the diagonal. As far as the shear spring is concerned, it is modeled with a “twonode link” object that connects two of the opposite corner nodes of the panel and acts in the horizontal direction. In the proposed formulation, the shear spring represents only part of the response of the infill panel in shear and does not contribute to the axial force assumed by the columns.
For the numerical simulation of the RC frame member, a nonlinear fiberbased element formulation was selected. A “force based beam column” element with several (usually 6 to 8) integration points along its height was used to model each beam/column member. The model takes into account the interaction between axial and bending forces, but it is incapable of representing the shear behavior of the frame members.
The material models employed in the inelastic fiberbased elements for the numerical simulation of RC frame components are available in the
The axial forcedeformation relationship developed by Crisafulli [
The constitutive law for the axial cyclic behavior of the strut is expressed in terms of stressstrain relationships (Figure
Suggested values proposed by Crisafulli [
Parameter  Suggested values  Limit values 


1.5–2.5  ≥0 

0.2–0.4  ≥0 

0.3–0.6  0.1–0.7 

1.5–2.0  ≥0 

0.60.7  0.5–0.9 

0.5–0.7  0–1.0 

1.1–1.5  ≥1 

1.5–2.0  ≥0 

1.0–1.5  ≥0 
Cyclic response of axially loaded masonry proposed by Crisafulli [
The stressstrain relationship can be easily expressed in terms of the forcedeformation quantities developed in the equivalent strut according to the following formulas:
An interesting characteristic of the model by Crisafulli is that the area of the equivalent strut is assumed to decrease with the increase in the lateral displacement (and consequently in the axial deformation of the strut), due to the reduction of the contact length between the panel and the frame and the development of extensive cracking in the panel. To account for this effect, the area (or equally the width) of the diagonal strut is assumed to vary as a function of the axial displacement,
Variation of the area of the strut as a function of the axial displacement.
The cyclic response of the shear spring used in the present model differs from the one adopted by Crisafulli et al. [
The envelope of the modified IK model (Figure
The Modified IbarraKrawinkler (IK) deterioration model: (a) monotonic curve and (b) basic modes of cyclic deterioration.
The hysteretic response of the model follows a bilinear curve accounting for three modes of deterioration with respect to the backbone curve: (a) basic strength deterioration, (b) postcapping strength deterioration, and (c) unloading/reloading stiffness deterioration.
The rate of cyclic deterioration relates to the hysteretic energy dissipated during a cyclic event according to the proposal by Fardis [
Cyclic strength deterioration (either basic or postcapping) is modeled by translating the two strength bounds towards the origin at a rate equal to
The same concepts apply to modeling unloading stiffness deterioration.
In the present application of the model basic strength and stiffness deterioration are considered for capturing the shear behavior of the panel, by evaluating the appropriate values of
For the evaluation of the initial stiffness of the masonry panel, the following formula suggested by Bertoldi et al. [
The total initial stiffness (defined in the horizontal direction) of the panel is shared between the shear spring and the diagonal struts according to the following expressions:
Nevertheless, the initial stiffness of the diagonal strut element, expressed in the horizontal direction, is equal to
The initial modulus of elasticity assigned to the strut elements is derived through equating (
In the early stages of the response of a laterally loaded infilled frame, the infill panel acts as a shear wall, exhibiting substantially high lateral stiffness. However, as the infill panel deforms and the contact with the surrounding frame is limited to the vicinity of the beamcolumn joints, the lateral stiffness starts to degrade. In the present model, this high initial stiffness is attributed to the elastic branch of the shear spring response. Yielding of the shear spring is reached at a force level equal to a
Parameter
The compressive strength of the diagonal strut is estimated taking into account four possible failure mechanisms, as it was suggested by Pauley and Priestley [
Parameters for the calculation of the compressive strength of the strut [






1.3  0.707  0.47 

−0.178  0.010  0.04 
The final compressive strength of the strut components is determined as the minimum of the four resistances calculated by (
A suitable value for the effective strut width has been sought by many researchers. Holmes [
The area reduction factor used in the definition of the strut model is, therefore, determined as the ratio
Interstorey drift level (IDR) is taken as the unifying parameter to connect the global deformation level to those of the diagonal struts and the shear spring, according to the following assumptions.
There exists a characteristic interstorey drift level, IDR_{1}, for which extensive cracks have been formed in the panel indicating the start of a higher rate of deterioration. Thus, it is assumed that IDR_{1} corresponds to the starting point of both the falling branch of the shear spring backbone curve and the reduction of the area of the compressive strut.
Interstorey drift level, IDR_{2}, is related to the point where the area of the diagonal reaches its residual value,
Ultimate state conditions are considered at an interstorey drift level named IDR_{3}, for which both the shear spring and the compressive diagonal strut have exhausted their resistance. It should be emphasized that the value of IDR_{3} is usually higher than the one observed during test events. This issue can be attributed to the fact that either some other kind of failure mechanism, related to the frame response (e.g., shear failures in the beamcolumn connections), has preceded the development of excessive damage on the infill or the experiment has been interrupted before the actual collapse of the panel.
Dependence of shear spring and diagonal strut deformation quantities on the specified IDRlevels.
IDRlevel  Shear spring  Diagonal strut 

IDR_{1}  — 

IDR_{2} 


IDR_{3} 


For completing the formulation of the model, two additional deformation quantities remain to be defined: (a) the axial strain at maximum stress and (b) the closing strain on the strut elements.
The axial strain,
As it was previously mentioned, the closing strain,
Figures
Compressive envelope in the diagonal strut, characteristic points of the forcedeformation response, and variation of the effective area.
Shear spring envelope curve and characteristic points of the forcedeformation response.
The predictions of the proposed approach are compared to a number of test results from masonry infilled RC frames subjected to quasistatic or cyclic displacement histories. Five of the investigated experiments are presented here to demonstrate the efficiency of the model. The first three are 2/3scale models of onestorey single bay masonry infilled frames tested by Pires and Carvalho [
Geometry of the investigated specimens.
Specimen 



( 
( 




(mm)  (mm)  (mm)  (mm) × (mm)  (mm) × (mm)  
Pires M2  150  2250  1825  150 × 150  150 × 200 



 
Pires M3  150  2250  1825  150 × 150  150 × 200 



 
Pires M6  150  2250  1825  150 × 150  150 × 200 



 
Colorado T1  190.5  3600  1930  280 × 280  240 × 280 



 
Koutas U1  110  2500  2000  170 × 230  170 × 330 




Material properties in the experimental studies considered.
Specimen 








(MPa)  (MPa)  (MPa)  (MPa)  (MPa)  (MPa)  (MPa)  
Pires M2  28.3  434  523  590  0.40  0.35  2.10 
Pires M3  33.2  434  523  590  0.40  0.35  2.10 
Pires M6  35.2  434  523  1000  0.51  0.40  2.30 
Colorado T1  30.1  458  458  2900  1.0*  —  17.93 
Koutas U1  28.0  500  220  1620**  0.90**  —  5.07 
Note: *the value of the parameter
Note: **
The results obtained from the analysis of the above specimens are compared with the experimental results in terms of total base shear and interstorey drift, except for the test by Koutas et al. [
Definition of IDRlevels for the selected specimens.
Specimen  IDR_{1} (%)  IDR_{2} (%)  IDR_{3} (%) 

Pires M2  0.050  1.20  8.0 
Pires M3  0.021  0.60  8.0 
Pires M6  0.035  1.10  8.0 
Colorado T1  0.50  0.55  2.8 
Koutas U1  1.0  2.50  8.0 
Material properties assigned to the strut/spring elements for the selected specimens.
Specimen 


( 
( 





(MPa)  (MPa)  
Pires M2  −0.97  0.05  0.25  0.175  −0.00013  0.002  0.6  0.3 
Pires M3  −0.97  0.05  0.25  0.163  −0.00010  0.002  0.6  0.3 
Pires M6  −1.25  0.05  0.25  0.163  −0.00013  0.002  0.6  0.3 
Colorado T1  −2.55  0.05  0.27  0.189  −0.00013  0.002  0.7  0.3 
Koutas U1  −2.14*  0.05  0.28  0.196  −0.00050  0.002  0.7  0.3 
Empirical parameters assigned to the strut/spring elements for the selected specimens.
Specimen 














Pires M2  0.3  1.5  1.0  0.9  0.7  1.0  1.1  3.0  1.0  0.3  1000  0.5  1.0 
Pires M3  0.3  1.5  1.0  0.9  0.5  1.0  1.1  3.0  1.0  0.3  1000  0.5  1.0 
Pires M6  0.3  1.5  1.0  0.9  0.7  1.0  1.1  3.0  1.0  0.3  0.20  0.9  1.0 
Colorado T1  0.3  0.8  1.0  0.9  0.5  1.0  1.1  3.0  1.0  0.4  0.20  0.6  0.25 
Koutas U1  0.3  0.8  1.0  0.9  0.7  1.0  1.1  3.0  1.0  0.3  0.25  0.5  1.0 
The comparison of model predictions and experimental results is quantified via a dimensionless index,
the median of the positive
the median of the negative
the maximum and the minimum
Another way to assess the efficiency of the proposed model is to compare the strain energy dissipated during each test with the dissipated energy calculated analytically. The cumulative dissipated strain energy,
In the following paragraphs, the comparative results obtained after interpreting the experimental and analytical data referring to each case study are presented and discussed. Generally, a smaller stepsize was considered for the evaluation of the strain energy dissipation to allow a more accurate estimation of the area enclosed in the “Base shearDisplacement” loops, which explains the difference in the “pseudotime” measures in the relevant figures compared to the ones depicting the variation of index
Comparison of the analytically derived response for Pires M2 specimen with the corresponding experimentally obtained results (Figure
Pires M2 specimen.
Variation of index
The range of
As far as the cumulative strain energy dissipated by the infilled frame is concerned, it is generally underestimated in the numerical model (Figure
Cumulative strain energy comparison for Pires M2 specimen.
Similar conclusions are drawn after evaluating the analytical results of specimen M3 by Pires. As depicted in Figure
Pires M3 specimen.
Variation of index
The cumulative strain energy, presented in Figure
Cumulative strain energy comparison for Pires M3 specimen.
Test M6 by Pires was also used, which, compared to the previous two specimens of the same research, features lighter reinforcement in the columns and stronger infill. The comparative results (Figures
Pires M6 specimen.
Variation of index
The cumulative strain energy dissipated in the numerical model is overestimated until a drift level of around 3% is reached and underestimated in the following cycles (Figure
Cumulative strain energy comparison for Pires M6 specimen.
The response of an even stronger masonry infill is examined by the case of Colorado T1 specimen. A satisfactory fit is achieved in the numerical model as depicted in Figures
Colorado T1 specimen.
Variation of index
It is important to note that a sheardominated failure mechanism appeared in the frame members during the test at quite early stages of the loading history, limiting the ability of the proposed model to accurately capture the response of the system. More specifically, the rapid decrease of the strength of the structure noticed at +0.5% drift is a result of the formation of a wide shear crack in the top east column of the specimen, followed by another shear crack at the lower east and top west columns during the next half cycle. Such failure mechanism cannot be reproduced with the fiber elements used in the present study. Nonetheless, the general behavior of the infilled frame system is—up to that point—represented adequately.
The comparison of cumulative strain energies derived by the experimental and analytical data is illustrated in Figure
Cumulative strain energy comparison for Colorado T1 specimen.
Finally, the response of the 3storey infilled frame tested by Koutas et al. is compared to the analytically obtained results. The final half cycle of the response is excluded from the comparison, as a shear failure mechanism that occurred to the west column is responsible for the divergence between the experimental and the numerical results from that point onwards. As presented in Figures
Koutas et al. U1 specimen.
Variation of index
The strain energy dissipated in the numerical simulation seems to overestimate the actual energy dissipated during the experiment (Figure
Cumulative strain energy comparison for Koutas et al. U1 specimen.
An interesting outcome of the present comparison is that not only is the global behavior of the infilled frame system adequately reproduced, but also the individual response of each floor is captured quite well. The relations between the shear force at the 1st, 2nd, and 3rd floors and the corresponding (absolute) floor displacement are illustrated in Figure
Koutas et al. U1 specimen. Floor shear force versus floor displacement (absolute) comparative results: (a) 1st floor, (b) 2nd floor, and (c) 3rd floor.
A parametric study has been conducted to investigate the sensitivity of the numerical results to the various material parameters involved in model. The study considered all the empirical parameters as well as some of the mechanical properties and was implemented using the numerical simulation of Pires M2 specimen. The parameters were individually varied, keeping all the other quantities fixed to the value used in the calibrated model presented in the previous section. Some of the modeling parameters, although examined during the parametric process, are not included in the following discussion as their influence was found to be negligible.
In order to quantify the influence of a certain parameter on the structural response, the index
Parametric investigation of M2 specimen by Pires.
The parameters that were found to have an important impact on the response are (i) the strut area reduction factor,
As expected, the reduction of the effective width of the diagonal strut seems to be one of the most influential parameters. Nevertheless, the selection of the area reduction factor,
The estimated response is also very sensitive to the selection of parameter
Parameter
The ultimate drift, IDR_{3}, is also of importance for the determination of the response. Applying higher values of ultimate drift capacity, index
Among the various empirical parameters used to define the hysteretic response of the diagonal struts, the global response of the infilled frame system is more sensitive in
Parameters
Finally, based on the parametric investigation of M2 specimen by Pires and the numerical results obtained from the analysis of all the considered specimens, generalized conclusions are drawn regarding the selection of the material parameters incorporated in the proposed model.
A number of parameters can be characterized as noncritical and thus assigned a fixed value:
The values of the parameters
Appropriate values for parameter
For parameter
Parameter
Higher values than the ones suggested by Crisafulli (Table
The suggested values for parameter
Parameter
Parameter
The variance of IDR_{1} (0.021%–1.0%) and IDR_{2} (0.55%–2.5%) levels used in the present study is indicative of their high dependence on the characteristics of the masonry infill and the surrounding frame and impedes the suggestion of a feasible range of values for practical applications.
IDR_{3}parameter, controlling the ultimate state of the infill response, can be assigned a value between 5% and 8%.
The paper presents an approach for assessing the nonlinear response of masonry infilled RC frames under inplane lateral loads. The model, based on the masonry panel by Crisafulli, combines three elements to provide a relatively accurate representation of the global behavior of the infill panel. The constitutive models adopted for each element involve a number of parameters, either empirical or linked to the mechanical properties of the infill. A procedure for the evaluation of the various parameters is established, followed by the validation of the proposed methods with available experimental data.
The comparison between experimental data and numerical results is conducted on the basis of two quantities: (i) the dimensionless index
It is concluded that the model can adequately reproduce the loaddisplacement response exhibited by infilled RC frames. The medians of the positive and negative
In addition, a parametric study, carried out to evaluate the sensitivity of the numerical results to the selection of the modeling parameters (especially those with empirical nature), shows that only few of the parameters show a significant impact on the overall response, providing evidence that a possible reduction of the unknown and noneasily identified parameters could be considered.
Based on the results of both the parametric investigation and the analysis of the individual specimens considered, proposals regarding appropriate selection of the parameters employed in the model are provided.
The authors declare that there is no conflict of interests regarding the publication of this paper.