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Bulletin of Earthquake Engineering

, Volume 17, Issue 12, pp 6547–6563 | Cite as

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

  • Leonardo M. MassoneEmail author
  • Ignacio Manríquez
  • Sebastián Díaz
  • Fabián Rojas
  • Ricardo Herrera
S.I.: Nonlinear Modelling of Reinforced Concrete Structural Walls

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

Figure 1 shows the reinforcement details of four reinforced concrete wall specimens, all 265 cm high and 15 cm thick. In order to study the effect of the discontinuity at the base of these flag walls, a rectangular benchmark wall was designed with a length of 90 cm (W1), which was progressively extended in length or height for the 3 remaining specimens. Two specimens were extended over an additional length of 25 cm and 50 cm, above a height of 30 cm measured from the wall base (W2 and W3), whereas the last specimen had a length extension of 25 cm, but above a height of 60 cm (W4). Boundary reinforcement was composed by 4φ16 (φ, deformed bar diameter in mm) on both sides, confined by stirrups φ6 spaced at 70 mm, along a height of 1 m measured from the base. Web distributed vertical and horizontal reinforcement was composed of φ8 bars spaced at 200 mm. All specimens were subjected to a nominal axial constant compression load equal to 0.1f’cAg (0.081f’cAg for specimen W1, and 0.071f’cAg 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).
Fig. 1

Schematic of specimens (units in mm) (after 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

In this article, a quadrilateral layered membrane element with drilling degrees of freedom (DOF) developed by Rojas et al. (2016) is used. It is a micro model which incorporates the coupling of axial, flexure and shear forces observed in RC walls. The drilling degrees of freedom refer to the incorporation of the in-plane rotation as a degree of freedom at each node of the element. The membrane element consists of a quadrilateral element with a total of 12 DOF, 3 per node (2 displacements and 1 in-plane rotation, see Fig. 2a), and uses a blended field interpolation for the displacements over the element. The modeling of the section of the element consists of a layered system of fully bonded, smeared steel reinforcement and smeared orthotropic concrete material with a rotating angle formulation (Fig. 2b).
Fig. 2

Model scheme—a element global degree of freedom and b layered membrane section (after Rojas et al. 2016)

In this formulation, it is assumed that the principal axes of concrete stresses coincide with the principal axes of strain, and the average stress–strain relationship in each principal direction can be represented by a uniaxial constitutive concrete model (Rojas et al. 2016; Rojas 2012). In addition, the orthotropic formulation incorporates the following features: Poisson’s ratio under biaxial loads and compression strength enhancement due to biaxial compression (Vecchio 1992), compression softening effect (Belarbi and Hsu 1995), damage due to cyclic or reversal loading (Palermo and Vecchio 2003), and enhancement of the compression peak strength of the concrete due to confinement (Saatcioglu and Razvi 1992). Also, in each principal direction, the uniaxial constitutive concrete model proposed by Massone (2006) is used (Fig. 3a). Each steel layer is modeled considering the steel bars as homogeneous material within the wall, fully bonded with the concrete, and working in the longitudinal direction of the bar only. The variation of stresses due to cracking over an area is modeled using the average stresses and strains in the steel, and it is represented by a uniaxial constitutive model, with apparent yield stress determined as suggested by Belarbi and Hsu (1994). Two uniaxial models are used for the constitutive model of steel, these two models are selected depending if the steel is prone to buckling or not. For the steel layer that is not expected to buckle, the model proposed by Menegotto and Pinto (1973) and later modified by Filippou et al. (1983) is used (Fig. 3b), whereas for the steel layers that are susceptible to buckling, the model proposed by Massone and Moroder (2009) is used.
Fig. 3

Material constitutive laws—a concrete and b steel

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

Figure 4 shows the lateral load versus lateral displacement response of the specimens tested under lateral cyclic loading. All test cycles were started in the negative direction, i.e., with the discontinuous side in tension. The results show a good correlation between the model and the experimental results, especially regarding capacity and ductility. The presence of discontinuity in specimens W2 and W3 causes degradation in previous cycles at 4% drift, when compared with specimen W1 (no discontinuity). The onset of degradation is due to the slightly higher concentration of strains at the base of flag walls (with concrete crushing and boundary reinforcement buckling, similar to the rectangular wall—Massone et al. 2019). Concrete crushing and longitudinal boundary bar buckling at wall base is observed in the models, which is consistent with the experimental observation. In the case of specimen W4, the numerical results show good agreement with the test, despite the fact that this wall had a different failure mode (earlier crushing at the end of the discontinuous longitudinal reinforcement—also captured by the model), showing degradation towards the end of the 3% drift cycle.
Fig. 4

Load displacement response of specimens—a W1, b W2, c W3, and d W4

3.2 Distribution of vertical strains along the height

In order to provide a local comparison between the model and the test specimens, the results from photogrammetry (Massone et al. 2019) are also used to estimate the distribution of vertical strains. Figure 5 shows vertical strains on the edge of specimens W1 and W3 at three drift levels. Strains in the left side (negative) correspond to the tensile strains on the discontinuous side for W3, whereas positive strains correspond to the tensile strains on the continuous edge for W3 (for W1, due to the symmetry, the definition is not relevant).
Fig. 5

Vertical strain distribution—a test W1, b model W1, c test W3, and d model W3

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.

In the case of the rectangular wall (W1), there is a certain similarity when comparing the complete model (flexure and shear) and the flexure model, which is reflected in Fig. 6. The capacity that is controlled by bending does not show differences. The differences are observed in the initiation of degradation, where the complete model degrades earlier. The implementation of rigid beams in the rectangular wall increases its deformation capacity, where for the taller wall studied (aspect ratio of 9) it reaches a drift of 10% (initiation of degradation is fixed at 10% strength reduction), while when using the complete model it reaches 6.7% (Fig. 6a, Table 1), and similarly for the narrower wall studied (aspect ratio of 6), the flexure model (Fig. 6b) has a more ductile response, reaching in this case a drift of 6.7%, while the complete model reaches only 4.9%. This is explained by the increase in compressions at the base of the wall in the most compressed zone for the wall with the complete model (flexure and shear).
Fig. 6

Lateral load versus lateral displacement W1 (continuous)—a taller, and b narrower

Table 1

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

Similar to the rectangular case, the failure in the flag-wall case is controlled by bending and therefore both models are able to predict the lateral strength with similar precision. In this case, the beginning of degradation is very similar in both models, and for both conditions, the taller and narrower wall, which is reflected in Fig. 7. In the case of the former, both models present initiation of degradation at 4% drift, like the base flag wall, which shows that this parameter is less relevant for this case (Fig. 7a). This is because the strains are concentrated in the zone of discontinuity, while the upper part of the wall moves as a rigid body. In the case of the narrower wall, there is also no significant difference between the complete model and the one that considers only flexure, which has a degradation beginning at a 4.8% drift (Fig. 7b). In the case of the flag wall, the deformations (curvatures) are concentrated in the zone of discontinuity, without a significant increase in the compression strains in the wall boundary.
Fig. 7

Lateral load versus lateral displacement W3 (discontinuous)—a taller, and b narrower

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.

Figure 8 shows the strain profiles for the rectangular wall and the wall with discontinuity for the case with aspect ratio 9, highlighting in the complete model larger tensile strains present in the wall with discontinuity in the continuous area, as well as the larger compressive strains in the continuous zone. On the other hand, the largest compressive strains are observed for the flexure model of the flag wall in the discontinuous zone. The large compressive and tensile strains in the discontinuous wall explain the lower displacement capacity of the flag wall. For the rectangular wall, the strain profile is similar for small drift levels (under 2%) for the complete model (flex. + shear) and flexure model (flex.). However, for roof drifts above 2%, significant differences are observed: for the compressive strains, the complete model (flex. + shear) presents a slight increase (~ 10%) with respect to the flexure model (flex.); for the tensile strains, the flexure model (flex.) gives much larger values than the complete model (flex. + shear) due to the presence of rigid beams that prevent cracking from growing in height, concentrating larger strains at the base. On the other hand, for the discontinuous wall, the compression strains increase for the pure flexure model (flex.) compared to the compression strains of the complete model (flex. + shear) in the zone of discontinuity.
Fig. 8

Strain profile for taller walls (aspect ratio 9)—a continuous wall positive direction, b discontinuous wall positive direction, c continuous wall negative direction, and d discontinuous wall negative direction

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

For the generation of the strut-and-tie model for the flag wall specimens when the zone of discontinuity is in compression, few considerations must be taken into account, since the response is similar to the rectangular wall (Massone et al. 2019). That is, strain concentration or curvature distribution at the wall base is similar for all specimens (Fig. 5). For the model of the rectangular wall, ties and longitudinal struts are defined at the boundaries of the wall, located on the centroid of the edge reinforcement and the center of the compressed boundary, respectively (see Figs. 9, 10b, 11b). A diagonal strut is also placed near the base, where the cracks are distributed in a fan shape. The direction of these cracks points towards the lower compressed corner, where the wall joins the foundation. This distribution forms on average a 45° angle with the horizontal (e.g., Fig. 9d). This strut is joined to a horizontal tie in the lower part that represents the horizontal reinforcement within a tributary area. Similarly, the upper section of the wall presents another horizontal reinforcement that covers representative reinforcement (tributary area) and whose location depends also of a middle strut. Finally, a top diagonal strut joins the model with the applied lateral load. The axial load is transferred to a node consistent with the location of the actual force (e.g., nodes A and B in Fig. 9a).
Fig. 9

Strut-and-tie model for W1—a model and steel reinforcement, b model with tie T3 at 2075 mm height, c model with tie T3 at 1750 mm height, and d principal strain for specimen W1 by photogrammetry at 3% drift

Fig. 10

Strut-and-tie model for W3—a lateral load with discontinuous side under tension, and b lateral load with continuous side under tension

Fig. 11

Strut-and-tie model for W4—a lateral load with discontinuous side under tension, and b lateral load with continuous side under tension

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%.

Figure 12 summarizes the comparison of results obtained from photogrammetry and from the ties of the strut-and-tie model for all specimens. Figure 12a shows the ties for a lateral load that generates tension in the discontinuous side. The results show that elements T3, T5, T7 and T8 remain close to the diagonal line, that is, they have consistency between the values obtained from photogrammetry and the strut-and-tie model. On the other hand, the upper element T2 shows the worst correlation. Similarly, Fig. 12b shows the ties for a lateral load that generates compression in the discontinuous boundary. The lower ties T6 and T8 show close correlation between the model and photogrammetry, with only one case outside the general trend. The upper tie T5 has a lower correlation, which indicates, as previously discussed, that a refined discretization for this element could better represent how the tension decreases in height. All these results confirm that the proposed model, which in the case of flag walls varies with loading direction, is a good and simple alternative for dimensioning the elements, in particular, the vertical and horizontal reinforcement of flag walls.
Fig. 12

Force comparison for ties in the model under lateral load and photogrammetry—a discontinuous side under tension, and b continuous side under tension

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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringUniversity of ChileSantiagoChile

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