The structures analyzed are frame constructions, due to the use of this structural system in many Latin American and European countries (63, 66) and the ease of inserting infill walls between the columns of frames. These structures are characterized by fast execution times and few material resources. Both aspects are directly related to the reduction of construction costs.

It has been shown in research that the structural behavior of bare frames buildings (beams and columns) (without walls, braces, among others) designed with gravitational loads, is not effective (1, 63, 64) against seismic movements, due to its low resistance. On the other hand, the structural requirements that bare frame buildings designed with seismic-resistant regulations would have in places with medium and high seismicity, make compliance very difficult, without the help of other elements (walls, bracing, among others). Consequently, it is advisable to incorporate structural (shear walls) and non-structural (infill walls) walls to mitigate the possible damage caused (63, 67), complying with the normative requirements.

In recent years, most European countries have been adapting their national (structural) codes to resemble European codes (Eurocodes). For this reason, this work has considered the earthquake resistance Eurocode 8 (EC-8) (EN, 2004) (68) for the design of building models. The elements that make up the frames (beams and columns) have been designed considering the European standard for reinforced concrete (EC-2) and the Spanish standard EHE-08. In these calculations, given the high lateral flexibility of the frames, second-order effects have been considered; however, in most cases the differences with first order analyzes are small. In addition, the calculations have considered the seismic forces obtained from EC-8, for a soil acceleration a _g = 0.20g and a hard soil (type B). Consequently, the results of this work can be considered representative for a significant percentage of existing buildings in Europe and part of Latin America.

The materials used for the structural elements (beams and columns) were concrete HA-30 corresponding to a characteristic resistance of f _ck = 300 kg/cm² and steel B-500-S, with an elastic limit of the steel f _yk = 5,000 kg/cm2. Both materials are defined in the CTE DB SE AE Standard (2006) (69).

The loads considered in the structural analysis follow the combination of actions G + 0.3Q of Eurocode 8 (2004) (68), where G determines the weight of the structure and Q the live loads (load of use of the building), considering a residential use, administrative or small business, equivalent to 2 kN/m² (2006) (69) in all the floors of the frame except for the upper floor (roof), whose load was 1 kN/m². The surface loads of the slabs have been transferred to the beams of the frames, multiplying them by the length corresponding to the length supported on it.

The present work did not consider the collaboration of the window carpentry in the frame openings due to its great fragility and low resistance. Said space is determined in the empty bays.

Finally, in all the floors of the frame, a rigid diaphragm has been considered corresponding to the effect caused by the 12 cm concrete slab on the structures, an element that limits possible displacements in the vertical axis.

As shown in Figures 1a and 1b, each frame model is made up of 4 spans of 5 meters in length (this measurement is on the column axes). The height between floors is 3 meters, with a free height per floor of 2.60 meters. The horizontal structural elements of the frames correspond to beams 30 cm wide and 40 cm high for all heights. The vertical elements of each frame are made up of five columns of different sizes that vary by 10 cm on each 3 floors. The three upper floors of both models correspond to 30x30 cm columns. The dimensioning of the columns and the beams is detailed in Figures 1a and 1b. The structural continuity of the structural elements is achieved through the longitudinal reinforcements (the steel reinforcement in the concrete that connect the beams and columns), complying with all the requirements of the European codes of structural design.

The total height of the frames is 12 and 24 meters for the 4- and 8-storey buildings, respectively. The configuration of the structures is regular and symmetrical in elevation (Figures 1a and 1b). In the different configurations of the 4- and 8-storey models used, except for the bare frames (case 4_NW and 8_NW), half of the spans are made up of infill walls and the other half are empty, as shown in Figures 2 and 3. The empty openings correspond to the window and door openings that the buildings would have.

Figures 2 and 3 show the different frame configurations analyzed in this study. The identification of each case is done by means of two terms; a first number that indicates the number of heights of the frames, an underscore (_) and the letter “W” accompanied by another number that indicates the number of the case to which it refers. For example, 4_W6 would be identified with a 4-height frame, being the configuration number 6 of the image corresponding to Figure 2. The case in which the second of the terms is “NW” corresponds to the frames without walls, in the other cases, “Wn” corresponds to the different cases with walls, where “n” is the case number.

The bricks that make up the infill walls of the frames are located under the upper beam of each story without any type of anchoring to the structural elements. These non-structural walls are made up of 23 cm x 12 cm x 10 cm bricks, separated by 1 cm thick mortar with ladder-type steel transverse reinforcements every four rows of bricks, as shown in Figure 4.

The elements that make up each of the frames were modelled using finite bar elements (70, 71) made up of 2 nodes. For each structural element (columns and beams), its mechanical properties were individually specified following the prescriptions proposed by Mander et al. (1988), for concrete (72) and Ferrara’s bilinear model (73) for reinforced steel bars.

In particular, and due to the simplicity and speed of the calculations, the beams and columns that make up the frames are represented by non-linear finite bar elements (74), where the non-linearities are concentrated in the plastic hinges located at the ends, corresponding to 15% of the total length of the element (75, 76). Furthermore, according to Scott et al. (75), it was considered that the joints/connections between the columns and the concrete beams were rigid assuming that the reinforcements used are satisfactorily anchored (connectivity) between the structural elements (beams and columns), while the hysteretic behavior that represents the stress distribution was calculated with fiber models based on the properties of the material and the cross-section of the structural elements (discretized with 300 fibers). In the model, the gravitational loads are applied on the beams, and the horizontal load increment to carry out the non-linear incremental static calculations (push-over), applied laterally on the nodes corresponding to each of the frame heights, follows a triangular loading pattern.

The structural analyzes carried out have been considered for a damping of 5%, specified by most of the earthquake resistant Standards (EC-8, NCSE-02, NSR-10, among others) and existing studies.

The simulation of the mechanical behavior of each of the materials that make up the elements of the frame (concrete, steel, and ceramic bricks) requires the entry of various data corresponding to the properties of the material and the requirements established by FEMA and ASCE in relation to the hysterical behavior of the elements that make up the frame were considered sufficient. For this reason, the experimental plasticization and rotation values obtained from the capacity curves of the materials and the structural elements (beams and columns) that make up the frames were considered sufficient. The characteristics of the infill walls are determined in the next section.

The existence of infill panels modifies the behavior of RC structures. The modelling of the infill wall has been established considering the non-linear inelastic behavior, the determination of the mechanical properties of the materials and the interaction with the frame. In research there are many techniques for analyzing these elements. This work has considered the studies of Crisafulli et al. (76) for computational analysis. To do this, after a detailed review of the different existing analysis, this work adopts the double-strut approach proposed by Crisafulli (77) and implemented by Calvi et al. (78) using Seismosoft® software. The selection of this model has been based on the good results offered by the panel‒frame interaction in the modelling and the reasonable computational calculation times. In addition, this type of analysis has been successfully applied for the seismic response of reinforced concrete frames with multi-story infill walls, with verified results (79).

The Crisafulli approach proposes a macro model for the evaluation of the global response of this system. The model is implemented as a four-node panel element, which is connected to the frame at the beam‒column joints (Figure 5). Internally, the infill panel element considers the compressive and shear behavior of the masonry panel using two parallel struts and a shear spring in each direction, as indicated in Figures 5a and 5b. This model allows adequate consideration of the lateral stiffness of the panel and the strength of the masonry panel, particularly when shear failure along mortar joints or diagonal stress failure is expected. In the bibliographic reference (79), the numerical analysis on the transformation of the forces in the internal and fictitious nodes into the external forces in the four nodes that make up the panels can be observed in detail.

For the modelling of the brick elements (infill walls) that make up the masonry panels, a four-node panel element has been considered, developed by the studies carried out by Crisafulli et al. (76). For the modelling of the non-linear response of these panels, the SeismoStruct software has been used, through the studies carried out by Smyrou et al. (80). Each panel is represented by six strut members. Each diagonal direction has two parallel struts to transport axial loads through the two opposite diagonal corners and a third to carry the shear from the top to the bottom of the panel. This last prop only acts through the diagonal that is in compression, so its ‘activation’ depends on the deformation of the panel. Axial load struts use the masonry strut hysteresis model, while the shear strut uses a bilinear hysteresis structural behavior rule.

Also, as can be observed in Figure 5, four internal nodes are employed to account for the actual points of contact between the frame and the infill panel (i.e. to account for the width and height of the columns and beams, respectively), whilst four dummy nodes are introduced with the objective of accounting for the contact length between the frame and the infill panel. All the internal forces are transformed to the exterior four nodes (which, as noted here, need to be defined in anti-clockwise sequence) where the element is connected to the frame.

The masonry element type is combination of a 3D, force-based, plastic hinge element type employed in modelling mainly the bending behavior of the masonry member (herein referred to as the ‘internal sub-element’) with two links at the two edges that are employed to simulate the shear behavior of the member (herein referred to as the ‘external links’ or the ‘link sub-elements’). The internal sub-element and the external links are connected in series, ensuring equilibrium in bending moment and shear force. The only ‘active’ degrees-of-freedom of the link sub-elements are the two translational ones in the shear directions (in-plane and out-of-plane), whilst the other four DOFs (axial and 3 rotational) remain perfectly rigid links. Both masonry walls and spandrels can be accurately modelled with such configuration. The shear DOFs of the link sub-elements feature a hysteretic curve that is based on SeismoStruct’s built-in MIMK pinched nonlinear curve (Modified Ibarra‒Medina‒Krawinkler deterioration curve with bilinear hysteretic rules and pinching), according to a phenomenological law that describes the shear behavior of the entire member. Simultaneously, in the internal sub-element the fiber-section modelling allows for a relatively accurate description of the coupled axial-flexural behavior. The sectional stress‒strain state is obtained through the integration of the nonlinear uniaxial material response of the individual fibers, in which the section has been subdivided, fully accounting for the spread of inelasticity along the member length and across the section depth. The determination of the shear strength of the member is crucial for the model’s accuracy, and is automatically carried out by the model, based on the masonry’s material properties, the dimensions of the member, and the selected Structural Code. The following expressions are employed for the calculation of the member’s shear capacity (it is noted that different equations are employed in the different Standards).

The parameters required for the full definition of the element properties are the following tables (Table 1 and 2):

The nonlinear static analysis of incremental thrust is used to estimate the maximum horizontal capacity (base shear) of a structure, considering the deformation and the frequency content of the dynamic response movement. For the evaluation and calculations carried out, the references used were Antoniou and Pinho (81, 82) and Ferracuti et al. (83).

In carrying out the calculations, the lateral load distribution is not kept constant, but is continually updated during the analysis, in accordance with the modal forms and the participation factors derived from the eigenvalue analysis at each step of the calculation process. This method is multimodal, explaining the softening of the structure, the lengthening of its period and the modification of the inertial forces due to spectral amplification.

The constant updating of the triangular lateral load patterns is carried out according to the modal properties that constantly change the system, which provide better response estimates than conventional methods, especially in cases where there are strength or stiffness irregularities in the structure and/or the highest mode effects, according to Pinho, R.et al. (84).

The adaptive algorithm that SeismoStruct implements is very flexible and can accept several different parameters that adapt to the specific requirements of each project. Examples are the SRSS and CQC modal combination methods (Clough and Penzien (85); Chopra (86)).

Non-linear analysis (push-over) allow calculation of the maximum horizontal strength capacity of structures whose dynamic response is not significantly affected by the levels of deformation experienced. That is, the distribution of horizontal forces that simulate the dynamic response can be assumed constant. NLSA is one of four analysis procedures embodied in FEMA 356 (87) and ASCE 41 (88, 89) and is commonly used in performance-based design approaches. For interested readers, a complete description of the method can be found in (87-89).

The methodology followed in this work concentrates on the failures of the frames in the plastic hinges that appear in the areas near the nodes of each structural element (beams and columns). The analysis was made assuming triangular load distributions. This load pattern is increased proportionally with a factor (λ·p) until structural instability is reached. Additionally, a response control corresponding to an increase in the top floor nodes displacement is used.

The criteria considered in the deformations and failures in this analysis, corresponding to concrete and steel, use the standard values implemented in Seismostruct (90-94), being; concrete cracking (0.0001), concrete detachment (−0.002), crushing the core concrete (0.006), creep (0.0025) and steel fracture (0.06). In addition, the criteria referring to curvature and rotations were verified through the rotational capacity of Mergos (95), and the shear base capacity established in the Eurocode 8 (EC-8). The tolerances used for the displacements and rotations were of the order of 10-5 in all cases, with a maximum number of 300 iterations.

The maximum displacement capacity is predicted considering geometric non-linearities such as inelasticity of materials. The greater or lesser displacement that the structure has is determined based on the number of plastic hinges that are formed in the structure, until it becomes totally unstable.

The legend corresponding to the capacity curves of Figures 6 and 7 that appear on the right side of the same, corresponds to the cases considered in Figures 2 and 3 of section 3.1

Figures 6 and 7 show the ‘non-linear’ static behavior of the analyzed 4- and 8-storey models, respectively. The graphs represented allow us to see the elastoplastic behavior of the frames prior to collapse. The beginning of each curve makes it possible to locate the elastoplastic behavior of the structures before the major fault. In the elastic area of the curve (linear part), small differences were observed between the different cases analyzed, except for the bare frames. However, as the applied horizontal force increases, the displacement differences are greater between models for the same load level. During the load increase from 0 kN to the first major failure, it is the infill walls that resist the shear forces applied to the structures. These elements increase the initial stiffness of the structure, causing the main structural elements (beams and columns) to fail later. This is observed with a significant reduction in the structural rigidity of the frames once their maximum strength is reached. On the other hand, it can also be observed how the structural behavior of buildings, once the maximum shear is reached, tends to equalize. This is due to the collapse of the infill walls in the initial part of the graph’s capacity curves, the main elements (beams and columns) being those that support the frame structure once the maximum resistance produced by the walls has been reached.

The plasticity in the capacity curves is mainly generated by the cracks produced in the concrete and bricks that make up the infill walls and the plasticity of the existing steel reinforcement in the primary structural elements. These effects are determined by the formation of plastic hinges in each element. The greater or lesser number of hinges that are formed in each case analyzed prior to the collapse will determine the length of these curves and the ductility of the frames.

The structural performance of brick-walled frames generates a significant increase in shear force when compared to the structure without any type of wall. The maximum value reached for both frame structures is between 800 kN and 1000 kN in the case of frames with infill walls, this being approximately 20% higher in most of the cases analyzed than frames without walls. After these failures, a slight decrease in the structural performance of the cases can be observed. The displacement and resistance in the frames with infill walls is associated with the location of the walls in the frames, which generates more ductile structures.

Finally, due to the increased ductility, it is possible to see that the first major failure in most of the 4-storey frames occurs with an approximate displacement of 100 mm in structures with walls, being slightly higher in frames without walls (134 mm). On the other hand, in the 8-storey frames, the first major failure in most of the structures with walls occurs with a displacement of 200 mm, being slightly lower in the frames without walls (269 mm). This proportional relationship of displacements is produced because they are structures with similar stiffnesses.

Table 3 show the values of the ductility, fundamental period, deformation and plastic and ultimate resistance in the frames of 4 and 8 stories. Obviously, the results show greater ductility in frames without walls. The percentages of values included in parentheses in table 2 correspond to the increase in these values with respect to the frames without any walls.

Analysis of the best behaviors ranks cases 4_W8 and 8_W10 as the most resistant and cases 4_W9 and 8_W2 as the most ductile. Similarly, cases 4_NW and 8_NW are the least strong and cases 4_W12 and 8_W8 are the least ductile.

By selecting the 6 most resistant and ductile frames, they confirm that the walls are concentrated in the external openings and in the upper part of them.