Assessment of a force–displacement based multiple-vertical-line element to simulate the non-linear axial–shear–flexure interaction behaviour of reinforced concrete walls
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A comprehensive assessment of a new version of the macroscopic force–displacement based multiple-vertical-line element (SFI-MVLEM-FD), which can be used to simulate non-linear axial–shear–flexure interaction in RC walls, is presented. The element models the shear response taking into account all the basic physical mechanisms that transfer shear forces over cracks: (a) the dowel effect of vertical bars, (b) the axial resistance of horizontal/shear bars, and (c) the interlocking of aggregate particles in cracks. In order to provide a wide range of its use, and to enable the analysis of various types of buildings, the SFI-MVLEM-FD element was included in the local version of the OpenSees program system. The element was assessed with respect to already performed quasi-static cyclic experiments of various RC shear walls. In this paper, the results of numerical analyses of two representative rectangular walls, where the influence of shear on the overall response was of particularly significance, are presented and compared with those obtained in the experiments. In order to estimate the efficiency of the new element in more general cases, it was also assessed by means of a large-scale shake-table test of a typical non-planar lightly reinforced RC coupled wall. The test examples showed that the SFI-MVLEM-FD model can capture all the important mechanisms of the response, as well as being able to efficiently describe the axial–shear–flexure interaction in various types of RC walls: (a) rectangular and non-planar, (b) cantilever and coupled, and (c) subjected to different types of excitation, uni-axial or bi-axial. It was found that the model is capable of clearly identifying the three fundamental mechanisms, which contribute to shear resistance. This is one of the few models, which are able to describe the significant deterioration of the (shear) strength of RC walls that are near to collapse for different reasons: e.g. the buckling of their longitudinal bars, rupture of the horizontal reinforcement, and other significant degradation of different types of shear mechanism. This makes it suitable for the analysis of different types of RC walls, which are subjected to different levels of seismic excitations. It is even able to simulate the near collapse response influenced by very different collapse mechanisms.
KeywordsShear–flexure interaction RC structural walls MVLEM Nonlinear response Seismic response
After several failures of RC walls during the strong earthquakes in Chile (Boroschek et al. 2014; Massone 2013) and New Zealand (Elwood et al. 2014), the interest of the engineering community in the seismic response of such walls has considerably increased. It was observed that the near collapse response of RC walls is a very complex phenomenon and difficult to predict by means of standard numerical models, particularly if the shear properties have an important effect on the overall response. Thus, the need for improvements to existing numerical models has been recently expressed. Appropriate modelling of the axial–shear–flexure interaction in the nonlinear range has also been identified as one of the research priorities within the NSF Virtual Wall Institute (Wallace 2016), which join together different universities all over the world (i.e. the USA, Europe, Japan, New Zealand, and South America) and leading world researchers at the field, in order to make progress in the seismic analysis and design of RC structural walls.
Currently, there are various modelling techniques available that can be used to capture the axial–shear–flexure interaction of RC walls subjected to reversed cyclic loads in the nonlinear range. They can be divided into two major groups: microscopic and macroscopic models.
Refined FEM element models monitoring the inelastic stress–strain relationship (“microscopic” models) offer a relatively good solution for the modelling of shear–flexure interaction. However, their implementation may still involve difficulties due to the lack of a completely reliable model and data, the complexities involved in the numerical solution, and the need for expert users.
The alternative macroscopic group of elements attempts to describe the overall behaviour introducing different types of idealizations and simplifications, which are mainly based on the experimentally observed response and experiences obtained during past earthquakes. This group of elements includes different types of beam–column elements, which can be classified as: (a) multiple vertical line elements (MVLEM), where the basic axial–flexure interaction is extended to a coupled axial–shear–flexure response in the nonlinear range (e.g. Kolozvari et al. 2015a, b), (b) a nonlinear truss or beam–truss elements, as proposed by (Panagiotou et al. 2012; Lu et al. 2016), and (c) distributed-plasticity fibre-based beam–column numerical models (e.g. Neuenhofer and Filippou 1998). The SFI-MVLEM-FD element, which is assessed in this paper, can be classified as a MVLEM type of element.
The version of MVLEM which has been developed at the University of Ljubljana for many years (Fischinger et al. 1990, 2004) is force–displacement based (MVLEM-FD). This makes it different from other available MVLEM elements, which are mostly stress–strain based.
The MVLEM-FD element has been consistently proved to be efficient in the cases of a predominantly flexural response. For example, it was successfully used in the best benchmark predictions for “San Diego” (Panagiotou et al. 2011) and “E-Defense” (Nagae and Wallace 2011; Tuna et al. 2014) RC cantilever walls. It was extended (Rejec 2011) so as to be able to capture complex axial–shear–flexure interaction (SFI-MVLEM-FD) in the nonlinear range. To provide a wide range of its use and to enable the analysis of various types of buildings, the element was included in the local University of Ljubljana (UL) version of the OpenSees program system.
In this paper, an assessment of the capabilities of the SFI-MVLEM-FD element to simulate the non-linear axial–shear–flexure interaction behaviour of reinforced concrete walls is presented. It was recently evaluated with respect to several quasi-static cyclic experiments of various RC rectangular shear walls (Kolozvari et al. 2018). In this paper, only the numerical analysis of two representative walls with heavily reinforced boundary regions, where the influence of shear on the overall response was particularly significant, is presented and compared with the results of experiments. In order to estimate the efficiency of the new element in more general cases, an assessment of a completely different type of wall was also performed. The SFI-MVLEM-FD element was also used in the simulation of a large-scale shake-table test of a typical non-planar lightly reinforced RC coupled wall, which failed in shear. The main properties of this element are presented in Sect. 2. Its assessment is described in Sects. 3 and 4.
2 Description of the SFI-MVLEM-FD element
2.1 MVLEM-FD: the basic version of the force–displacement based MVLEM, developed at UL
MVLEM-FD consists of several vertical springs. Each spring correspond to a certain part of the wall’s cross section-segment. The springs are rigidly connected at both of the element’s nodes. In the local coordinate system this element has three degrees of freedom. They correspond to axial deformations of the element (Δu) and the flexural response (Δφ). The first versions of MVLEM-FD were intended for 2D analyses. Later, the element was also extended to a 3D version, enabling biaxial bending analysis (see Fig. 1b).
The vertical springs simulate the axial–flexure behaviour, taking into account Bernoulli’s rule (“plane sections remain plane”). The response is described by means of force–displacement hysteretic rules. For the springs, which correspond to the weakly reinforced parts of the wall cross-section, the response of the vertical springs can be modelled by an experimentally observed force–displacement relationship (uniaxial material VertSpringType 2 included in the local UL version of OpenSees), presented in Fig. 2a. The envelope of these hysteretic loops can be defined taking into account the combined properties of the concrete and the flexural reinforcement. The hysteretic response presented in Fig. 2a depends on four parameters: α, β, γ, δ. The recommended values of these parameters, defined based on empirical/experimental observations, are: α = 1.0, β = 1.5 α, γ = 1.05, δ = 0.5.
When the wall boundary elements are heavily reinforced, it is more appropriate to model the concrete and the reinforcement separately, by means of two different material models, which are connected together in parallel. Any appropriate material model available in OpenSees (McKenna 2019) can be used for this purpose. For example, in one of the studies presented in Sect. 3, VerticalSpringType 2 and Steel02 (Filippou et al. 1983) material were used for the concrete and steel of the boundary regions, respectively. The typical response of the vertical spring defined by the combination of these two materials is presented later on in Sect. 3, in Fig. 7.
Shear behaviour and the axial–flexural response are uncoupled in the basic version of MVLEM-FD. Shear response is modelled with one horizontal spring located at the centroid of the cross-section (in the horizontal plane) and at the centre of the rotation of the corresponding element (in the vertical plane). The response of the shear spring is defined by special “ShearSlip” material, the behaviour of which is presented by the hysteretic loop shown in Fig. 2b. This behaviour does not depend on the variation of the axial force in the spring.
2.2 SFI-MVLEM-FD: an extension of the element that can simulate complex axial–shear–flexure interaction in the nonlinear range
The MVLEM-FD element, which is described in Sect. 2.1 above, was extended so as to be able to simulate the complex axial–shear–flexure interaction of the nonlinear response of RC walls all the way up to the near-collapse phase of the response (Rejec 2011).
Cracks are straight and equally spaced. The (constant) spacing between the cracks should be evaluated according to empirical procedures (e.g. Collins and Mitchell 1991).
The shear displacements of the element caused by the compression deformation of the diagonal struts are neglected. It is assumed that the tensile and shear deformations in the cracked strips are localized in the cracks.
The inclination of cracks and the displacement occurring within different cracks is assumed to be the same along the total height of the wall element.
The inclination of the cracks is time-varying, and is updated at each load step (the rotating-crack model) according to the corresponding average strain state in the element.
the dowel effect of the vertical bars (HSD), described by the relationships proposed by Vintzeleou and Tassios (1986);
the axial resistance of the horizontal/shear bars (HSS), defined by using the model proposed by Elwood and Moehle (2003);
The properties of each spring component are variable and depend on deformations/displacements at the effective cracks of the element that are linked to the displacements of the element nodes, which enables coupling between the axial/flexural and the shear behaviour at the element level of the model. It is assumed that the contribution of the shear to the displacements along the crack wz,i (see Fig. 4c) is constant along the crack, whereas the vertical components wx,i follow Bernoulli’s rule (“plane sections remain plane”).
The constitutive behaviour of the shear resisting mechanisms (HSA, HSD, and HSS) can be represented by any appropriate uniaxial material, included in OpenSees. In the presented study (see Sect. 3) “Hysteretic material” was used in the majority of cases, since it can adequately describe the pinching, typical for different response mechanisms influencing the shear response, as well as to take into account the strength degradation. Typical hysteretic response of shear resisting mechanisms is presented in Fig. 4.
It should be emphasized that the SFI-MVLEM-FD element differs significantly from the SFI-MVLEM-SS element (Kolozvari et al. 2015a, b), which is included in the official version of OpenSees. SFI-MVLEM-SS is stress–strain based, so that axial–shear–flexure coupling is achieved at the panel level considering stress–strain rather than the characteristic force–displacement constitutive models. The FSAM (Fixed-Strut-Angle-Model) model is used to define the constitutive RC panel model (for more details see Kolozvari et al. 2015a, b).
3 Evaluation of the SFI-MVLEM-FD element based on the results of quasi-static cyclic tests of rectangular RC walls with heavily reinforced boundary regions
3.1 A brief overview of the specimens’ properties
Both walls were subjected to a vertical and a horizontal load. Quasi-static cyclic tests were performed, gradually increasing the horizontal displacements demand at the top of specimens up to their failure. More details about the loading protocols can be found in the references provided in Table 1.
3.2 Description of the numerical models
3.2.1 Mesh information
126.96.36.199 Discretization of the wall in the vertical direction
Both walls were modelled by means of several SFI-MVLEM-FD elements of different lengths. In general, shorter elements were used to model the bottom segments of the walls in order to be able to take into consideration considerable gradients of curvature in these regions (see Fig. 6a). The length of the upper elements was gradually increased as is illustrated in Figs. 5a and 6a. In wall RW-A15-P10-S78 the length of the elements was defined based on the position of the instrumentation used in the experiment.
188.8.131.52 Discretization of the wall in the horizontal direction
The cross-section was divided into segments considering the location, distribution and amount of the longitudinal reinforcement. Each segment was represented by the corresponding vertical spring. The influence of the segments in the boundary region on the strength of the wall is larger in comparison with the inner segments due to the typically stronger reinforcement and larger distance from the centre of rotation. Consequently the outer segments were shorter than the inner segments (see Figs. 5a, 6a).
These segments typically included two longitudinal reinforcement bars. For example, in wall RW-A15-P10-S78 the boundary regions were divided into four segments. In each segment, a pair of longitudinal bars was located in the centre of the segment (see Fig. 5a).
The inner segments were larger (see Figs. 5a, 6a). Their length was defined considering the location of the longitudinal reinforcement. Each segment typically included two to four bars. For example, the inner part of wall RW-A15-P10-S78 was divided into six segments of equal length, which included two reinforcing bars each. The number and location of the vertical springs were the same in all the elements.
3.2.2 Constitutive models of the vertical springs
When modelling the concrete part, the properties of confined concrete were taken into account according to Mander’s model (Mander et al. 1988). All the materials were applied using appropriate force–displacement relationships (instead of stress–strain relationships).
3.2.3 Constitutive models of the horizontal springs
The semi-empirical models, which have been proposed in the literature (see Sect. 2.2), were used to define the constitutive relationships for the horizontal springs. The factors which define pinching, damage, and degradation of the unloading stiffness in the Hysteretic model were specified based on the typical shapes of the hysteretic loops corresponding to the HSA, HSD and HSS mechanisms, as reported in the literature (Rejec 2011). The hysteretic loops which were considered in the case study are presented in Fig. 4. More details about the response of the horizontal springs are provided in Sect. 3.3.
3.3 Comparison of the results of the analyses and the experimental results
3.3.1 Shear force–displacement relationships: flexural and shear response
The analysis accurately simulated the strength and the stiffness of both walls at almost all intensity levels. The only exception was the initial stiffness, which was somewhat overestimated. This was, however, expected, since the authors had found before (e.g. Fischinger et al. 2017) that in the majority of the past experiments the initial stiffness was smaller than the theoretical one (corresponding to the gross cross-section). Note, however, that in the SFI-MVLEM-FD element the initial stiffness can be simply reduced if the initial branches of the force–displacement envelopes of the vertical springs are appropriately modified.
3.3.2 Components of the shear resistance mechanism
The aggregate interlock is the predominant shear mechanism at lower loading intensities. Therefore the initial stiffness and the cracking force in the wall were predominantly influenced by the HSA spring behaviour.
After cracking, the HSA mechanism is typically considerably degraded. Consequently, the importance of the other two components contributing to the shear response (dowel mechanism and the shear reinforcement) is increased. Their role and the contribution to the shear strength depend on the configuration of the reinforcement.
In lightly reinforced walls, the role of the HSD mechanism is, in general, less important (for more details see Sect. 4) and the contribution of the shear reinforcement HSS to the overall shear strength is more significant.
4 Evaluation of the SFI-MVLEM-FD element based on the shake-table tests of non-planar walls
4.1 A brief overview of the experiment
The reliability and accuracy of the SFI-MVLEM-FD element was also evaluated simulating the response of a more complex RC lightly reinforced non-planar wall, which was tested on the shake table at LNEC in Lisbon, Portugal within the scope of the ECOLEADER project, which was coordinated by the University of Ljubljana team (Fischinger et al. 2017).
4.2 Analysis and comparison with the experiment
Masses of the specimen were lumped at each floor level. They included mass of the slab, tributary mass of the wall (half of the mass of the wall above and below the slab), and additional masses. The area of each slab forming part of the model was 2.50 m2. In actual structures the walls carry part of the slabs from adjacent spans. To address this, a realistic pertinent slab area was added to the self-weight of the test specimen (1.05 t per story, including the tributary mass of the walls). The total mass at each floor was large but fully realistic. It amounted to 5.93 t per story. The total mass of the whole specimen was 31.9 t (considering 2.25 t of the mass of the foundation beam).
Considering the large additional masses at each floor, the distribution of masses of wall piers (lumped at each floor level) did not have a considerable influence to the overall response. Note, that if needed, the mass of the wall piers can be also lumped in MVLEM nodes in between the slabs.
Moderate inelastic response of the specimen was observed in the 5th run (corresponding to maximum accelerations of 0.42 g and 0.73 g applied to the shake table in the direction parallel to the wall web, and the direction perpendicular to the wall web, respectively), which was the last but one run. Considerable lifting of the piers due to strong coupling was observed. The vertical bars in the flanges yielded, the cracks in the flanges expanded, and shear cracks formed in the webs of the both piers in the first storey.
Failure of the wall was observed after 3rd second of the last, i.e. 6th run, corresponding to maximum accelerations of 0.52 g and 1.02 g applied in the direction parallel to the wall web and in the direction perpendicular to the wall web, respectively. The failure of the wall was brittle and sudden. It occurred due to the simultaneous shear failure of both piers. The damage was mostly confined to the first storey (see Fig. 18a).
A large crack opened up in the web at the moment of its failure. Its upper part was pushed against the flange. Consequently, the dowel capacity of the weak longitudinal reinforcement in the central part of the flange was exhausted. As a result of this, local damage to the flange was observed, due to the punching of the web through it.
The residual resistance of the wall observed in the test after failure (and not numerically verified) can be attributed to the frame action of the flanges and slabs, which was not considered in the model. This is the reason for the discrepancy between the analysis and the experiment after the 3rd second.
Simultaneous failure of the webs in both piers was attributed to the bi-axial loading. The results of the analysis confirmed that at the time of the failure the webs in both piers were in net tension, which explains the same inclination of the crack in both piers (see Fig. 18a).
Recent earthquakes have clearly demonstrated that many RC walls, designed before the modern principles of earthquake engineering were known (and even some new walls), did not provide adequate shear resistance. It has been observed that many key mechanisms of the seismic response of RC structural walls are still not adequately understood. For this reason the development of new models for RC walls, particularly those that can describe the complex axial–shear–flexure interaction in the nonlinear range, has been intensified.
One of such models, i.e. SFI-MVLEM-FD was developed at UL. Its basic features are: (a) it is a multiple-vertical-line type element, (b) it is a force–displacement based element, (c) it can be used for 2D and 3D analyses, (d) it models the shear response taking into account all the basic physical mechanisms that transfer shear forces over cracks, and (e) the shear response is fully coupled with the axial–flexure response.
The element was evaluated by a range of different experiments, including rectangular and non-planar walls, as well as quasi-static cyclic tests and shake table tests. The main goals of the assessment were: (a) to test the element capabilities to model the complex axial–shear–flexure interaction, (b) to adequately simulate the contributions of flexure and shear to the overall response, and (c) to properly identify the type of failure, particularly in walls where the shear response was important.
It has been found that the proposed element can well describe axial–shear–flexure interaction and can simulate all the important mechanisms of the global response for very different types of RC walls: (a) rectangular and non-planar, (b) cantilever and coupled, and (c) subjected to different types of excitation, uni-axial or bi-axial.
The model is able to clearly identify the three fundamental mechanisms contributing to shear resistance. Consequently, this is one of the few models that is able to describe the significant deterioration of the (shear) strength of RC walls near collapse due to different reasons: e.g. buckling of the longitudinal bars, rupture of the horizontal reinforcement, and other significant degradation of different types of shear mechanism. This makes it suitable for the analysis of different types of RC walls subjected to different levels of seismic excitation. It is even able to simulate the near collapse response influenced by a number of very different collapse mechanisms.
The assessed element was developed by Klemen Rejec, extending the UL FGG version of MVLEM. The research was funded by Slovenian National Research Agency.
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