# Understanding the cyclic response of RC walls with setback discontinuities through a finite element model and a strut-and-tie model

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## Abstract

Slender RC walls are often used in Chile and commonly, due to architectural constraint, the length of walls increases (setback) between floors designated for parking space and upper floors. These types of elements are commonly called flag walls. In this research, the behavior of slender reinforced concrete walls with a constant axial load and a cyclic lateral displacement is numerically studied, in order to compare the results obtained with previous tests. Two different model alternatives are considered: a finite element model and a strut-and-tie model. The selected models allow understanding local response, as well as, distribution of internal forces, which is also relevant information for wall design and detailing. The studied finite element model, based on quadrilateral elements with 3 degrees of freedom per node (2 translational and 1 rotation) and a model of smeared reinforced concrete material based on a rotating angle approach, is able to correctly capture the global response, showing the capacity, degradation and failure mode obtained in the tests. On the other hand, a parametric analysis is performed for models of walls with higher aspect ratio (tall buildings) with small discontinuities, showing a larger impact in deformation capacity due to the high concentration of damage at the discontinuity. These results indicate that in 25-floor high walls (or taller) a reduction of displacement capacity of 40% for discontinuities located at the first floor could be observed. In addition, by incorporating the effect of the slabs into the model, the results indicate that a pure flexure model is an adequate and sufficient tool for analysis. Finally, a strut-and-tie model is also proposed for each direction of the lateral load, whose results are compared with the estimated load calculated with the strains measured by photogrammetry. The considered strut-and-tie model for the case of lateral load with tension in the continuous wall boundary is similar to the wall without discontinuity, which is consistent with the measured strains. For both lateral loading directions, the estimated forces of the horizontally distributed bars and boundary reinforcements are consistent with photogrammetry in the lower zone of the wall, where cracking is relevant. The strut-and-tie model also adequately interprets the effect of the discontinuous bar on the discontinuous boundary of the wall. All these results can help designing and detailing flag walls.

## Keywords

Slender wall Experiment Flag walls Cyclic loading Discontinuities Setback FEM model Strut and tie model## 1 Introduction

On February 27, 2010 Chile was hit by an earthquake of magnitude Mw 8.8. While infrastructure largely performed well, several modern buildings conformed by reinforced concrete walls were damaged (concrete crushing, buckling and fracture of steel reinforcement). This was due to insufficient confinement in the wall boundary elements, as well as the relatively high axial load ratio, and discontinuities present in the walls. Due to architectural requirements, the length of several walls changes at floors destined for parking, causing an extension of the wall in the upper floors and therefore creating a setback at one edge of the wall in the first floor, which is commonly referred to as flag wall.

There is limited investigation in the literature that focuses in the response of slender walls with discontinuities, and some indirect studies of representative sections with low shear span-to-depth ratio (e.g., Kang et al. 2012). However, most of the discontinuities are focused in door or window openings (e.g., Taylor et al. 1998; Ali and Wight 1990). The work by Massone et al. (2017) presents predictive estimation of yield displacement, plastic hinge length and base curvature, based on a fiber model (flexure only) that includes a setback at the wall base (flag wall) as an extension of the work by Massone and Alfaro (2016). In general, the plastic hinge located at the base increases in height as the level of wall top displacement increases. In the case of rectangular walls, the curvature gradually decreases in height (bottom to top), whereas in walls with the presence of a discontinuity (setback) at the base, the plastic hinge tends to concentrate at the discontinuity (Massone et al. 2017). In cases where the height of the discontinuity is large enough (tall walls), the wall behavior is similar to the case of a rectangular wall, allowing the plastic hinge to fully develop. Experimental results (Massone et al. 2019) confirm that there is concentration in the discontinuity, when the discontinuity is subjected to tension given that deformations (curvature and strains) tend to concentrate below the discontinuity level.

In this paper, slender reinforced concrete walls with varying degrees of edge discontinuity at the base are studied. A comparison between a theoretical study by a finite element model and a strut-and-tie model with experimental results of reinforced concrete specimens is performed. In the case of the finite elements model, which allows incorporating flexure and shear, the ability of the model to capture strength, degradation and hysteretic response is evaluated. Based on this model, the development of plastic hinge is reviewed, as well as the impact of varying parameters such as the wall slenderness in order to understand the behavior of full-sized walls, which is relevant for wall detailing. On the other hand, a strut-and-tie model is developed for both lateral loading directions, and forces are compared with estimates determined based on strain values captured with photogrammetry and constitutive material laws. The selected model is able to describe the distribution of forced inside the wall, which is a useful tool for force design.

## 2 Test program

_{c}A

_{g}(0.081f’

_{c}A

_{g}for specimen W1, and 0.071f’

_{c}A

_{g}for specimens W2, W3 and W4) and cyclic lateral increasing drift levels. The lateral displacement was applied at a height of 2.8 m (measured from the base of the wall). Actual concrete compressive strength was 33.0 MPa for W1 and 38.3 MPa for W2, W3, and W4. The average measured yield stress for the φ8 bars was 493 MPa and for the φ16 bars was 496 MPa. More details can be found elsewhere (Massone et al. 2019).

The capacity of all 4 specimens (W1, W2, W3, and W4) was similar, but they differed in the location and concentration of damage, which was noticeable at large drift levels (Massone et al. 2019). The presence of the discontinuity in the specimens W2 and W3 causes degradation in different cycles of 4% drift, before it happens with specimen W1 (continuous wall). In the case of specimen W4 degradation is observed towards the end of the 3% drift cycle. In general, all specimens began to lose concrete cover during the first cycle of 3% drift (with the discontinuity zone in tension for flag walls). Diagonal cracks also extended over half the height of the wall, although concentrated at the bottom of the wall. Strength degradation for specimens W1, W2 and W3 occurred during the 4% drift, at which point the bars were exposed and presented significant buckling, while concrete at the base of the wall boundary was crushed. In the case of specimen W4, in the first cycle of 4% drift there was spalling of concrete where the boundary bar was discontinued, due to insufficient anchoring of the boundary reinforcement, causing the reinforcement to slide and concentrate damage at this point.

## 3 Finite element model

For each model, the as-measured properties of the materials are used. It must be noted that for the discontinuous longitudinal boundary reinforcement, a length value of 700 mm (from the pedestal), 300 mm less than the actual length, was used, considering a development length of 600 mm (assuming 50% effectiveness). The finite element layout is selected in such a way to cover the boundaries with two elements, while the central area will be covered by six elements. For the vertical distribution, the height-to-length aspect ratio of the elements is set close to 1. The pedestal is modeled as an elastic element for simplicity.

### 3.1 Load versus roof displacement

### 3.2 Distribution of vertical strains along the height

Figure 5a shows a symmetric behavior with a strain concentration at the base for W1, which is consistent with the model results, although the model shows larger strain values (Fig. 5b). Specimen W3 (Fig. 5c, d) shows similar strain magnitude and distribution to specimen W1 in the positive side of the plot for the test and the model, indicating that the continuous side in tension behaves the same as for the continuous wall. In W3, the strain concentration in the negative side (discontinuity in tension) is larger than in the positive side, and with larger tensile strains implying larger damage under cyclic loading. A large strain value is also observed at the location of the end of the discontinuous longitudinal reinforcement (1 m) for the experimental results for W3, which is replicated in the model, but at a lower location since the model considered a shorter reinforcement to capture the strain reduction towards the reinforcement tip (adherence). The strain at that location results also in cracks that develop towards the hanging part of the flag wall, which indicates that the strain distribution is different for both loading directions and would be considered in a following section (strut-and-tie model). In the case of W4, such strain concentration provided the initiation of degradation that is observed in Fig. 4d, which is also captured by the model.

### 3.3 Parametric study of the slenderness

The previous results show good correlation between FEM (finite element model) and test results. However, few wall characteristics are captured with the test results. The aspect ratio has been shown to impact the response of flag walls (Massone et al. 2017), and it is studied here in more detail. In the research work by Massone et al. (2017), a fiber model is used for the predictions. The fiber model, as a column-type formulation, applies the Euler–Bernoulli hypothesis capturing flexure and axial behavior. In the current work, aside from presenting results for the FEM, an alternative formulation is implemented that mimics the fiber model behavior. In this case, rigid beams are included in the FEM model between each layer of finite elements in order to impose the Euler–Bernoulli hypothesis. Thus, the modified model does not capture shear and would be called “flexure model”. The full FEM formulation (without the rigid beams) can capture flexure and shear and would be called “flexure and shear model”. This consideration produces a restriction in the deformations and in the extension of the diagonal cracks in height, impacting both the maximum compressive and tensile strains. This effect might be important for flag-type walls, since the largest strains are concentrated in the area of the opening (Fig. 5) and the use of rigid beams preclude the extension of cracks, causing strain concentration, which in turns results in an accelerated degradation of the capacity under lateral load of walls with discontinuities. In this way, it can be expected that the response of the studied walls will present a greater concentration of damage in the opening if the slenderness is larger for the same discontinuity size. For a better understanding of this phenomenon, it was decided to triple the height of the W1 and W3 models, going from 2.8 to 8.4 m in height to the point of application of the lateral load, as well as dividing the length of the wall by half (together with the flag-hanging portion), going from 900 to 450 mm at the base and 500 (W3) to 250 mm in the extension of the flag, but maintaining the distribution of the boundary and web reinforcement. The axial load ratio was also maintained and the lateral load uses the same loading protocol as in the tested specimens.

Drift at initiation of degradation

Models/degradation drift | W1 (%) | W3 (%) |
---|---|---|

Base model (F + S) | 4.0 | 4.0 |

Taller model (F + S) | 6.7 | 4.0 |

Taller model (F) | 10.0 | 4.0 |

Narrower model (F + S) | 4.9 | 4.8 |

Narrower model (F) | 6.7 | 4.8 |

As it is observed, in the taller walls the concentration of deformations in the discontinuity is accentuated, producing differences in drift capacity close to 40% as compared to the rectangular wall when the model that includes flexure and shear is considered (Table 1, 6.7% and 4.0% for continuous and discontinuous wall, respectively), and even larger differences when considering the flexure model.

Finally, with all the aforementioned information, it can be concluded that for buildings with low slenderness (with aspect ratio less than 3) or buildings lower than 9 floors (opening of 1 floor height is considered), the discontinuity generates little impact, while with a high slenderness (aspect ratio over 9) or buildings of 25 floors or more, walls with discontinuity degrade at a considerably smaller drift than the rectangular wall. The model with rigid beams or flexure model increases the concentration of strains in the discontinuous zone, causing a slightly more pronounced damage in discontinuous walls, but with similar drift capacity compared to the complete model. Therefore, a flexure model is adequate to represent the flag wall response. On the other hand, the largest difference between the complete model and flexure model is observed for the continuous walls.

The previous results provide valuable information regarding deformation capacity of flag walls, which is fundamental information for wall boundary detailing. However, force distribution can be studied with simpler formulations (than a FEM model), based on stress or strain flow that allows designing reinforcement quantities (vertical and horizontal). The following section provides strut-and-tie models capable of capturing the force distribution within flag walls.

## 4 Strut-and-tie model

One of the advantages of strut-and-tie models is that it allows designing an element with discontinuities through an isostatic lattice. Schlaich et al. (1987) presented a lattice design methodology, where one of the characteristics was that the arrangement of each bar (strut or tie) that is part of it must be adapted to follow the flow of stresses or strains. For those bars that were subjected to tension, the strength of the steel reinforcement was attributed to it, and for those with compressive forces, they were represented by the strength of concrete. Struts and ties are connected in nodes. Tests in the literature have shown that the strut-and-tie models can be adapted to reproduce the element response (Brown et al. 2006) where the use of strain gauges indicate conservative predictions by the model.

In this research, a strut-and-tie model was created in each direction of loading for all tests. In the experiments, each connecting node on the upper face of the wall was loaded vertically downward with 178 kN, which represents the axial load. Additionally, the lateral load was applied at 2.8 m above the wall base. In the model, the diagonal struts were oriented, in most cases, according to the crack directions observed in the strain field at 3% drift. The ties on the other hand, were located in the position of representative reinforcement. The horizontal ties correspond to the horizontally distributed reinforcement within a tributary area. Similarly, the vertical reinforcement is assigned to representative vertical elements, which could be either ties or struts.

### 4.1 Strut-and-tie model selection

The inversion of direction of the lateral load results in an impact in the discontinuity of the wall and the longitudinal reinforcement that exists in the discontinuous wall edge (Figs. 10a, 11a). Previous findings indicate that there is strain or curvature concentration at the discontinuity, especially for W2 and W3, with also diagonal cracks in the hanging part of the flag (Massone et al. 2019). Similar observation was pointed out from the analysis in Fig. 5. For this configuration there are three horizontal ties at different heights and a fourth one that covers the discontinuous boundary in specimens W2 and W3. The lower tie is used as a connection between the longitudinal reinforcement in the opening and the longitudinal reinforcement of the opposite boundary. At the height of the discontinuity of the longitudinal reinforcement (1.0 m) another horizontal tie is placed. Finally, a third tie is located in the upper zone of the wall, covering the rest of the horizontally distributed reinforcement that has not been considered. According to the cracking pattern, observed during testing, from 1.0 m in height to the height of the opening, struts are located to better resemble the observed 45° cracking angle. In the case of specimens W2 and W3, this area was discretized with two struts joined by a tie at half height of this section (Fig. 10a). In test W4, only one strut was used since the length of that section was smaller (Fig. 11a). In the upper zone, similar to the other lateral loading direction, there is a strut that connects the upper horizontal tie with the point of application of the axial load. Similarly, a diagonal strut joins the main upper horizontal tie with the horizontal tie that meets the discontinuous longitudinal reinforcement (1.0 m in height), in the direction of the cracks.

This choice of model ensures consistent behavior with the results obtained in the tests, that is, regardless of the size of the opening, elements under tension are represented by ties and compressive elements by struts. Thus, a consistent model can be defined for each loading direction in the case of cyclic loading that can help dimensioning all elements (e.g., steel reinforcement quantity).

### 4.2 Comparison of experimental results

Forces in strut-and-tie models for all specimens are compared for a lateral load at 3% drift (to ensure that there is concrete cracking distributed in the wall and that high stresses in the reinforcement have been reached) in both directions with the forces in ties obtained by means of photogrammetry. Photogrammetry allows tracking the position of an object through a sequence of images. In this case, the walls were whitewashed and then painted with a random pattern of varying-size black dots (objects). Using images, the displacement of regions of the objects in two orthogonal axes perpendicular to the normal to the image can also be calculated, and from there strains can be estimated. Photogrammetry, by means of two cameras, was used to monitor global and local (discontinuity region) displacements and strains of walls, and results from the global monitoring were used in this work. More details of photogrammetry measurements for the test program can be found in Massone et al. (2019). After obtaining strains with photogrammetry and considering the material constitutive law of the bars, it is possible to estimate the force carried by each tie. This is done exclusively with the ties that present strains larger than the measurement error. The following sections describe representative results.

#### 4.2.1 Specimen W1

In the case of the rectangular wall, only one model is created given the symmetry. In Fig. 9a the strut-and-tie model is shown superimposed with the reinforcement layout. Figure 9b shows the comparison between the force (in tonf) results obtained by the strut-and-tie model and photogrammetry. In this figure, the bars drawn with red lines are the ties (tension elements), while those with blue lines represent the struts (compression elements). The data in parentheses are the forces obtained from solving the lattice, whereas those without the parentheses are the forces calculated based on the strain values obtained by photogrammetry. The first observation that can be made is that the difference between the strut-and-tie model and the photogrammetry results is smaller in the lower ties (less than 20%) than in the upper ties, where tie 2 (T2) shows the largest difference (the value from the strut-and-tie model is only 2% of the value from photogrammetry). This element, given its size and location, is more influenced by the axial load in this section than the lateral load that produces bending. This is illustrated in Fig. 9c, where moving tie 3 (T3) down causes the value obtained by the resolution of the lattice to grow from 3.2 to 82.3 kN (reaching 56% of the photogrammetry value). Thus, for a better result, the longitudinal boundary reinforcement could be discretized in more elements. Another relevant force difference is shown in tie 3 (T3). For this element, the resolution of the model yields a value of 203.8 kN compared to 31.4 kN calculated from photogrammetry. This is explained by the fact that this area of the wall is not severely cracked (vertically or diagonally), implying that the concrete is still supporting tensile loads under low strains.

#### 4.2.2 Specimen W3

For specimen W3, the model maintains the same main characteristics of specimen W2, and therefore only the results of this case are shown, modifying only some distances or angles in order to adapt the opening dimension. Figure 10 shows the scheme with lattice adaptations in both loading directions. Figure 10a shows the strut-and-tie model with the reinforcement of the wall and the forces in each element for a load that pulls the discontinuity zone. Similarly to the rectangular wall, a good correlation is observed in most of the lower ties, with an average error under 20%. The differences in the upper ties are due to the coarse discretization of the vertical elements and the limited cracking at the location of the horizontal ties. When studying the behavior with the opposite lateral load (Fig. 10b), again the model manages to detect the flow of the forces until reaching the foundation with close correlation in tie 6 (lower horizontal tie). It is also capable of capturing that element 2 (T2) maintains a compressive force.

#### 4.2.3 Specimen W4

Configurations similar to specimen W3 are established for specimen W4. The model arrangement in the specimen reinforcement and the comparison of the forces between the resolution of the lattice and by photogrammetry is shown in Fig. 11. The greatest discrepancy in lower ties is produced due to the arrangements of the horizontal reinforcement for ties 3 (T3) and 7 (T7) of Fig. 11a (discontinuous boundary in tension). Being very close to each other, a tributary area was considered to pass through the centroid of these two elements and then the reinforcement was divided into two equal parts. The errors in the other ties are less than 5%. In the case of the load in the other direction (Fig. 11b), similarly to the previous cases, good correlation is seen in ties closer to the base (mainly tie 6 and 8) with an error close to 1%.

## 5 Conclusions

Walls with setback discontinuities (flag walls) are common in Chile, but little information on their behavior is available. In the case of rectangular walls, the curvature gradually increases in height, whereas in walls with the presence of a setback discontinuity at the base, the plastic hinging tends to concentrate at the base as the aspect ratio of the wall increases.

The finite element model showed that in all cases the models were capable of capturing the initial stiffness, maximum capacity, and initiation of degradation. For the taller walls modeled either by tripling the height or by reducing its length, the results obtained showed that the taller walls studied concentrated more damage at the base, causing a degradation at an earlier drift for the walls with discontinuities, with reductions of up to 40%. This reduction in deformation capacity was observed in very slender walls (aspect ratio 9) that represent buildings of approximately 25 floors where a large part of the deformations were concentrated in the discontinuity zone. In the case of a wall consisting of a 9-story building, the differences are much smaller. On the other hand, the degradation of the capacity is similar in the complete models (flexure and shear) and flexure models for the walls with discontinuities. Hence, a simple flexure model is sufficient to reproduce the overall response of slender flag walls. Larger differences are observed for rectangular walls.

With the experimental information of the distribution of strain, a generic isostatic strut-and-tie model was proposed for each direction of the lateral load. The force in each tie was determined through the resolution of the lattice and compared with the force calculated from the photogrammetry data. The results showed that the ties in the upper part of the wall present larger error due to either distribution of elements in the longitudinal reinforcement or limited cracking in the location of horizontal reinforcement. In the lower part of the wall a better correlation was found given that the concrete was cracked and therefore better approaches the conditions assumed by the strut-and-tie model. When the lateral load pulled the discontinuous boundary, it was possible to appreciate the importance of the discontinuous bar. Two ties and three struts extend over this node, distributing the loads in the same way that it was presented and observed in the test. In the case with the opposite lateral load, it was possible to capture the fact that all walls behave in a similar way to the rectangular wall. Finally, it is concluded that the strut-and-tie model allows dimensioning the ties of a flag wall under axial and lateral loads.

## Notes

### Acknowledgements

This work was financially supported by FONDECYT regular 2013 No. 1130219 “Analytical and experimental study of RC walls with discontinuities”. Also, the help with the specimens testing by Mr. Ernesto Inzunza, Mr. Victor González and Mr. Pedro Soto are also thanked.

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