Bulletin of Earthquake Engineering

, Volume 17, Issue 12, pp 6359–6368 | Cite as

Guest editorial: Nonlinear modelling of reinforced concrete structural walls

  • M. FischingerEmail author
  • T. Isaković
  • K. Kolozvari
  • J. Wallace
S.I. : Nonlinear Modelling of Reinforced Concrete Structural Walls

1 Introduction

Structural systems including reinforced concrete (RC) walls are common and efficient in the design of buildings in seismic areas all over the world. RC structural walls have large (over)strength and stiffness, as well as adequate ductility if appropriate design procedures and detailing are used. The resulting seismic energy dissipating capacity has typically proved to be adequate even in the case of very strong earthquakes, for example during the Chile 1985 earthquake (Wood 1991; Wallace and Moehle 1993) and Montenegro 1979 earthquake (Fajfar et al. 1981). After the 2010 Chile earthquake significant damage was observed only in 2% of buildings with RC structural walls (Massone et al. 2012; Ugalde et al. 2019).

However, if the wall over-strength is exhausted and the adequate design measures for (flexural) ductile mechanisms are not provided, highly inelastic and dangerous complex mechanisms may develop including bar buckling and fracture, lateral instability of wall edges, nonlinear shear–flexural interaction, compression shear failure, and unexpected 3D interaction between the components of the system. Some cases that can initiate such behaviour are listed below:
  • Architects and investors seek for more attractive but irregular structural concepts;

  • The traditional architectural and structural concepts are sometimes pushed over their limits. This happened in Chile, for example, where similar structural layouts, thickness of walls, wall-to-floor-ratios, and detailing originally developed for buildings of moderate height were continued to be used for much higher structures;

  • Discontinuities in wall elevation are frequently introduced, particularly in the first/basement level where the functionality of the floor plan changes (typically for the need of parking places). This configuration leads to high concentrations of the shear demand and compression demand in boundary areas (Massone et al. 2019; Fischinger et al. 2011);

  • Increased shear demand during the inelastic response has been underestimated in the design and numerical models as well as inelastic shear–flexure–axial interaction was typically not considered;

  • In many countries walls are predominantly used as RC cores to provide lateral stiffness and strength to frame systems, which are used to allow more architecturally flexible floor plans. In high-rise buildings such cores can be heavily loaded and reinforced;

  • While majority of the research has been done on cantilever walls with rectangular cross section loaded in the in-plane direction only, multi-directional loading, and more complex geometry are more common in real structures but less commonly investigated;

  • The complex 3D inelastic interactions between all components of the structural system is far from being understood. For example, the transfer of the loads through coupling elements (e.g., slab, beams) can lead to unexpected overloading of individual walls in the structural system.

Such complex nonlinear mechanisms are typically not considered in the design, which may lead to potential heavy damage and even collapses, which were observed in several buildings after the Chile 2010 (Boroschek et al. 2011) and Christchurch 2011 earthquakes (Elwood et al. 2011). Although not widespread, such significant damage could not be tolerated in the engineered environment. Due to this as well as due to the fact that nonlinear structural analysis and performance-based evaluation have become common practice in regions with moderate-to-high seismicity, it has become evident that elastic and simplified nonlinear models, which have predominantly worked fine in the design and research of RC shear walls with moderate flexural inelastic demand, are not sufficient. The ability to accurately predict complex inelastic seismic behaviour of RC walls is essential. The key objectives of this Special Issue of BEE on Nonlinear Modelling of Reinforced Concrete Structural Walls, described in the following section, are in line with this need.

2 Objectives and the concept of the Special Issue

A great number of macroscopic (phenomenological) and microscopic (FE or continuum-based) nonlinear models has been developed over the past 20 years with various formulations and capabilities. However, most of the models have not been validated over a wide range of wall characteristics and responses, and their capabilities and range of applicability have not been compared. In addition, the ability of the models to simulate wall strength degradation due to various failure mechanisms observed in recent earthquakes (e.g., concrete crushing and rebar buckling, instability of wall boundaries, shear failure) was not systematically assessed. Finally, interaction between components of a structural system (e.g., effect of wall slab/beam coupling on component demands) is not well understood, motivating researchers to assess current ability to reliably simulate system-level responses and validate model results against available test data. In line with these observations the primary objective of this Special Issue in Bulletin of Earthquake Engineering is to provide a systematic overview of state-of-the-art macroscopic and microscopic modelling approaches for RC walls available in commercial and research-oriented software, and to report results of detailed model validation studies for a range of analytical formulations and wall characteristics.

Most contributions in this Special Issue are based on the results of a collaborative effort toward this objective entitled “Virtual International Institute for Performance Assessment of Structural Wall Systems”—NSF SAVI Wall Institute (Wallace 2016). The Institute, which includes more than thirty leading researchers in the field from all over the world, has been active in various fields of research, where Modelling and Analysis working group members have been collaborating on addressing issues mentioned above. A comprehensive summary of the results obtained through this international effort are reported in overview papers by Kolozvari et al. (2018, 2019a). However, given the large number of models and specimens considered in this study, some of the detailed responses, representative of each of the models considered, are not presented in these papers. Therefore, this Special Issue provides the needed, more detailed validation studies. To provide even more complete overview of the-state-of-the-art some other leading researchers outside the Wall Institute group contributed to this Special Issue, which provides a balanced discussion of different architectural, structural and detailing solutions used all over the world.

Common benchmark examples, defined within Wall Institute, were analysed in the majority of the papers included in the Special Issue. This provides systematic overview and clearer picture of the capabilities and shortcomings of the very different numerical models nowadays available to the international engineering community. Model improvements and directions of future research are identified.

3 Overview of the contributions

A representative number of 12 papers is published in this Special Issue:

(A) The first four papers are analysing macro-element models for RC walls defined as an assemblage of springs, macro-fibres or trusses controlled by phenomenological rules, defined based on the observed response during the past earthquakes and experiments. The strength of these elements is not only in their relative simplicity and applicability in design offices but first of all in their robustness, which is essential due to many uncertain parameters involved and data needed to describe the complex nonlinear response of structural RC walls. However, macro elements are typically limited to cases for which the assumptions implemented in the model formulations are valid.

Papers of Isakovic and Fischinger (2019) and Kolozvari et al. (2019b) are analysing the advanced versions of the multiple-vertical-line-element-model (MVLEM) with incorporated refinements, such as modelling of axial–shear–flexural interaction and 3D analysis of non-planar walls under multi-directional excitation. The models were validated using experimental results for a number of RC wall specimens, which were selected to cover different important parameters, such as wall aspect ratio (slender, intermediate, squat), level of compressive axial load, level of shear stresses up to very high values, wall cross-section (planar and non-planar), loading regime (in-plane, multidirectional, cyclic, shake-table), individual walls, and coupled walls. A number of common benchmark specimens defined within the Wall Institute is used in both papers to provide more meaningful comparison.

While the models in both papers use the same MVLEM concept, they are in fact quite different. Vertical elements in the model developed by Isakovic and Fischinger (2019) are non-linear force–displacement controlled springs. Additional horizontal springs are incorporated into the vertical springs to model three fundamental mechanisms of shear resistance (aggregate interlock, dowel action and horizontal reinforcement) and their interaction with flexural response.

Each vertical macro fibre in the model developed by Kolozvari et al. (2019b) consists of a RC panel constitutive model. The behaviour of the panel is described by the Fixed-Strut-Angle-Model (Gullu et al. 2019), which represents a complex plane-stress relationship that relates the strain field imposed on a RC panel to the resultant of stresses developing in concrete and reinforcing steel, converted into smeared stresses in concrete. In this way shear–flexural interaction is considered.

Both models captured well load–displacement response for very different types of walls and loading, including lateral load capacity, lateral stiffness and its cyclic degradation, pinching, as well as shear deformations and their coupling with flexural deformations. In some cases, it was even possible to describe (Isakovic and Fischinger 2019) 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. On the other hand, it is clear that macro elements can provide only average values for the local deformation quantities and that the applied Bernoulli–Euler hypothesis imposes some limitations.

(B) Nonlinear Truss Model is another popular type of a macro model, which is capable to capture axial–shear–flexural interaction in RC members on the basis of the strut-and-tie approach. It uses overlapping vertical, horizontal and diagonal elements to represent the discontinuous strain field of reinforced concrete in its cracked state. This model is used in papers contributed by Alvarez et al. (2019) and Arteta et al. (2019).

In the first paper (Alvarez et al. 2019), the Truss Model proposed by Panagiotou et al. (2012) is validated by the pioneering seismic testing of two 1:4 scale seven-story reinforced concrete coupled structural walls with different detailing of coupling beams reported in Santhakumar (1974) and in Paulay and Santhakumar (1976). In the applied version of the 2D Truss Model, the vertical and horizontal elements are nonlinear trusses except the beam elements in the base of boundaries and ends of coupling beams used to model dowel action. In addition to the simulation of the experimental results of the complex response of coupled walls with conventionally and diagonally reinforced coupling beams, the paper also presents valuable information about the distribution of shear demand between the leading and trailing wall, as well as the distribution of shear demand in coupling beams along the height of the wall.

A potential disadvantage of the Truss Model is that it may be time consuming in its construction and is computationally expensive as compared to nonlinear beam-column elements. Therefore, Arteta et al. (2019) proposed and tested a hybrid solution by applying Truss Model only at the critical part of the wall and fiber beam-column elements for other sections. The connection between both parts was provided by an elastic rigid beam. The hybrid alternative was validated using the experimental results for walls with different aspect ratios, which had been chosen as benchmark specimens within the Wall Institute. The part of the wall modelled by Truss elements was varied from 25 to 100% of the total height of the wall.

C) Interested reader may find additional information about some conceptually different macro models in Kolozvari et al. (2018), in particular about the shear wall element implemented in the widely used commercial software Perform 3D (Perform 3D 2019). This model was also used (see continuation of this Section) by Ugalde et al. (2019). In this research the model was additionally validated and verified by the experimental results of two slender wall specimens chosen within the Wall Institute.

(D) The next three papers are analysing the efficiency of micro models, based on more refined FE representations. By definition such models can provide more detailed information of local response and complex response mechanisms if enough data in line with their concepts are available.

Among the models in this group, the Fixed Strut Angle Finite Element (FSAFE) model has been qualified by the authors (Gullu et al. 2019) as a mesoscopic finite element retaining some features of macro elements. The model is an assembly of membrane elements (with zero out-of-plane stiffness) with a smeared stress–strain formulation used to describe the plane stress behaviour of RC. The constitutive relation between the average strain field and the smeared stress field follows the Fixed Strut Angle Formulation developed by Orakcal et al. (2012). The same model is used to define the behaviour of the vertical macro fibres in the MVLEM described in Kolozvari et al. (2019b).

Although the working principles of the FSAFE (mesoscopic) model are relatively simple, the model provided accurate prediction of the experimentally-measured key results of the complex nonlinear response of very different RC walls except for the abrupt degradation in lateral resistance related to buckling of reinforcement or out-of-plane instability of the wall boundary regions.

(E) The in-plane and out-of-plane behaviour of RC walls can be well documented by an adequate shell element. An efficient quadrilateral thin flat layered shell (QTFLS) element has been proposed by Rojas et al. (2019) by combining the robust quadrilateral layered membrane element with drilling DOF developed by Rojas et al. (2016) for the modelling of RC walls and the well-known Discrete Kirchhof Quadrilateral Element formulated by Batoz and Tahar (1982). Such element can be applied in complex 3D RC wall systems to simulate the interaction between elements at different direction of planes, such as wall and slabs, and to study the local (i.e. variations of strains) and global behaviour as well as the axial-shear-flexural interaction. The proposed element was verified by the comparison with the test results for two T-shaped RC walls available in the literature.

The quadrilateral four-node curved shell element (Q20SH) available in the commercial finite element analysis software DIANA (2011) is used in the study contributed by Dashti et al. (2019). The efficiency of the element is first verified by the experimental results obtained for the benchmark specimens chosen within the Wall Institute. The following complex failure patterns were successfully predicted: (a) concrete crushing in slender walls under high axial load and with low confinement, (b) global out-of-plane instability of slender walls under in-plane cyclic loading, (c) diagonal tension in shear dominant walls (squat walls) with light horizontal reinforcement, (d) sliding shear proceeded by development of concrete crushing along the web and boundary regions. The model was not able to represent (a) bar buckling and subsequent progressive concrete crushing, (b) bar fracture and the subsequent stress redistribution, and (c) out-of-plane instability as the secondary failure mode triggered by bar buckling, asymmetric cover spalling and concrete crushing. A special value of this paper is provided by an additional parametric study, which helps to understand how failure modes of RC walls change with alternation of key damage parameters.

(F) In the next three papers the existing models are applied to solve some important engineering problems, which highly depend on adequate modelling of nonlinear seismic response of RC structural walls. It is of particular importance that the results of the first two papers in this group are supported and verified by the empirical observations after the 2010 Chile earthquake.

In the paper contributed by Ugalde et al. (2019) the authors discuss the observation that two RC wall buildings of 17 and 26 stories survived the 2010 Chile earthquake practically undamaged although the standard elastic analysis would suggest that the demand of this strong earthquake was well above the code-specified demand.

The wall models in PERFORM-3D, which is now extensively used for non-linear seismic analysis in design offices, were first validated and verified by the experimental results of two slender wall specimens chosen within the Wall Institute. Then both buildings were analysed by push-over and response history analysis using one of the records of the 2010 Chile earthquake.

Analysis explained the main factors influencing the good behaviour of the buildings. These were somehow expected. But it is important that the study provided quantitative measure of their relevance, which turned out to be much larger than expected. The main factor was the large over-strength factor in the span of two to even six. This over-strength was primarily due to two reasons: (a) gross cross-section stiffness (which is very much higher than the actual stiffness of cracked sections during nonlinear seismic response) is typically used in the Chilean design practice and (b) slab elements are typically removed from elastic models in design, leaving only a rigid in-plane diaphragm constrained. The coupling of walls through slabs was found to be far larger than expected.

The reduction of the length of RC walls in the floors designated for parking places in comparison to the upper floors impose another important challenge in seismic design of buildings, which is addressed in the paper contributed by Massone et al. (2019). All over the world, and in particular in Chile, the resulting walls with setback discontinuities (flag walls) frequently suffered considerable damage at the location of discontinuity. In the paper, the experimental study of three flag walls and additional benchmark rectangular wall is supported by numerical modelling and parametric study of walls with different heights and setback characteristics. Two numerical models were used in the analyses: quadrilateral layered membrane finite element developed by Rojas et al. (2016) and strut-and-tie model. The analyses confirmed the observed behaviour characterized with concentration of plastic demand at the location of setbacks and reduction of drift capacity up to 40%.

Another important challenge in earthquake engineering related to older non-ductile concrete buildings is addressed in the paper contributed by Parra et al. (2019). The ATC-78 methodology provides simplified nonlinear procedure for relatively rapid assessment of potentially hazardous frame-wall buildings. It requires determining the controlling plastic collapse mechanism, the base shear strength, and the ratio between the story drift ratio and the roof drift ratio at collapse level. In the paper this procedure was calibrated by extensive nonlinear static and dynamic analyses for a set of 4-story and 8-story structures with different wall arrangements and considering a large set of ground motions. Three major conclusions were made: (a) the walls of modest length may prevent soft story mechanism, (b) walls that are discontinuous in upper stories may cause partial collapse at the location of discontinuity, and (c) for the walls supported on columns in the first story, failure could occur at low lateral drift ratio. It should be noted, however, that these results were obtained by adopting several simplifications in the model to accelerate the extensive computational work.

(G) Even the »best« numerical model cannot provide adequate results if the data needed within the range of its applicability are not available. The next two papers in this Special Issue provide some information needed for nonlinear modelling of RC walls.

Kusunoki et al. (2019) contributed an impressive and valuable verification of the empirical formulas for RC walls based on the comprehensive analysis of the test data for 507 RC walls tested in the period between 1975 and 2013. Pushover analysis has become the procedure of choice in many advanced design offices. However the results of these analyses highly depend on the input data, such as initial and post-cracking stiffness, crack/yield/ultimate strength, shear strength, stiffness and stress degradation, as well as ultimate deformation capacity of the elements used in the model. This is particularly true for RC structural walls. These data are typically estimated by empirical formulas. However, the accuracy and applicability of these formulas have been not always comprehensively verified. Some of them were also derived in the past, when different materials and structural details were used.

Discontinuities and irregularities at the location of the of lap splices has been ignored in the majority of nonlinear studies, which could be a gross and dangerous simplification driven primarily due to the lack of experimentally supported knowledge about the behaviour of the splices. In the paper contributed by Tarquini et al. (2019) the results of the experimental investigation on the deformation capacity (rather than strength capacity alone) of lap splices under cyclic loading is presented. A total of 24 specimens were tested varying in length of the splice, confinement, location of casting (top or bottom), and type of loading (monotonic or cyclic). The key result of the study is an expression for predicting the deformation capacity of lap splices, which is defined as the average strain at the onset of splice failure.

4 Concluding remarks

The 12 papers in this Special Issue of the Bulletin of Earthquake Engineering provide a systematic overview of the state-of-the-art macroscopic and microscopic modelling approaches for RC walls available in commercial and research-oriented software.

The models were validated with the experimental results obtained from cyclic and shake-table tests that cover a wide spectrum configurational and behavioural characteristics of RC walls. Some of them exhibited extremely complex behaviour at the near collapse stage.

The guest editors do not want to provide any discussion about which element is “better”. This would be pointless and misplaced. The mission of this volume is to provide detailed information about a large variety of conceptually different advanced models and to highlight their strengths and weaknesses and, first of all, their range of applicability related to the basic assumptions, which were used in their development.

It has been demonstrated that the presented models include advanced capabilities to monitor complex response mechanisms, which should be considered in up-to-date research and design. Strength and deformation properties can be now successfully modelled on global and local level well into the inelastic range and some complex features of behaviour like nonlinear shear–flexure interaction are now much better understood and modelled as one decade ago.

Although much progress has been achieved, there are still several issues which should be further investigated. In particular, the post-critical near-collapse mechanisms such as bar buckling, lateral instability, shear sliding as well as combined 3D mechanisms are on average still not fully understood and adequately modelled.

After all these new developments a question arises if the elastic FE elements and simplified beam-column models widely used in design practice and research should be totally abandoned and replaced by more advanced models? The answer is no. The traditional models predominantly work fine in the case of the moderate flexural inelastic behaviour of slender walls in the buildings with regular architectural concept. However, designers and researchers should be aware of the possible and in fact quite common problems, listed in the Introduction that can lead to dangerous complex mechanisms, which should be addressed with more advanced models presented in this Special Issue.

For example, the more complex models are needed for nowadays popular hazard studies. These studies inherently include near collapse behaviour involving very complex responses. Yet the authors of these studies have frequently used over-simplified models to reduce the huge computational effort associated with such research, which is contradictive and may lead to deceiving results of the seemingly comprehensive and “exact” studies.

Code development is an integrated process considering a whole span of different parameters and not just an isolated improvement of structural modelling only. Typical issue is the common over-strength provided by the use of traditional (elastic) models in design, which is discussed for example in Ugalde et al. (2019) or Fischinger et al. (2011). There has been an aggressive tendency in the new developments of codes (e.g., Eurocodes) to eliminate theoretically “wrong” assumptions leading to over-strength and related “un-economical” design. However, at the same time, the seismic force reduction factors have remained practically unchanged. In this way it has been ignored that the seismic force reduction factors were empirically calibrated based on the common design practice including the way of modelling. It is clear that such solution changes the level of seismic safety and may lead to increased damage during strong and even moderate earthquakes.

Any numerical model, either macroscopic or microscopic, cannot provide adequate results if the experimental data within the range of the model applicability are not available. Unfortunately this crucial need is frequently neglected in setting the research priorities.

Given the crucial role of nonlinear simulation of RC walls and wall systems in improving structural design and community resilience, the guest editors believe that this Special Issue in Bulletin of Earthquake Engineering will effectively disseminate new knowledge on this topic to the audience of interest.



Guest Editors would like to convey their thanks to the authors who enthusiastically responded to their invitation by contributing high-quality and interesting papers to this Special Issue and to the reviewers whose meticulous critical assessment and advice further improved the original submissions. Special thanks go to the Editor-in-Chief of the Bulletin of Earthquake Engineering Professor Atilla Ansal and Mrs. Petra van Steenbergen, the Springer Executive Editor in Earth Sciences, Geography and Environment for their encouragement and support, as well as to Mr. Madan Ellappan for his technical support during the publishing process.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • M. Fischinger
    • 1
    Email author
  • T. Isaković
    • 1
  • K. Kolozvari
    • 2
  • J. Wallace
    • 3
  1. 1.Faculty of Civil and Geodetic EngineeringUniversity of LjubljanaLjubljanaSlovenia
  2. 2.CSU FullertonLos AngelesUSA
  3. 3.University of California, Los AngelesLos AngelesUSA

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