# Influence of the constitutive model in the damage distribution of buildings designed with an energy-based method

^{1}Department of Mechanical and Mining Engineering, University of Jaén, Jaén, Spain^{2}Department of Mechanical Engineering, Universidad Politécnica de Madrid, Madrid, Spain^{3}Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy

It is widely accepted in the seismic design of buildings a certain level of damage under moderate or severe seismic actions but preventing the damage concentration in them. On the other hand, the energy-based design methodology proposes an optimum strength distribution for designing the structure of the building aimed at achieving an approximated even distribution of the damage—energy dissipated by plastic deformations—under seismic actions. Different approaches for the optimum strength distribution have been proposed in both existing literature and standards. Most of them were formulated from the results obtained in non-linear numeric evaluations of elastic-perfectly plastic (EPP) structures, such as the findings proposed recently by the authors of this study. However, studies on the optimum strength distributions of reinforced concrete (RC) structures are scarce. The present study sheds light on this issue. Accordingly, the structures of four prototype buildings with 3, 6, 9, and 12 stories were designed through an energy-based method by using five approaches for the optimum strength distribution: those proposed by the authors and two others from the literature and standards. Then, different prototypes of the structures arose considering the different approaches for the optimum strength distribution, two soil classes (dense and medium dense), and two ductility levels (low and high). Such prototype structures were subjected to two sets of far-field ground motion records by using three different constitutive models for the shear force-interstory drift relationship: EPP, Clough model, and Modified Clough model. The first characterizes the steel structures and the rest are typical for RC structures. A complete analysis was carried out to obtain the distribution of damage for EPP and RC structures, their deviations with respect to the “ideal” even distribution of damage, and the possible damage concentration on specific stories. RC structures showed a higher dispersion for the distribution of damage than EPP structures although those designed with the optimum strength distributions proposed by the authors showed the lowest values in the order of those obtained with EPP structures designed with optimum strength distributions proposed in the literature.

## 1 Introduction

One of the most important issues in the seismic design of structures is related to the form in which the energy of an earthquake is introduced and dissipated in buildings. Up to the 50s of the 20th century, the seismic design of structures was based, overall, on elastic criteria with the absence of damage. Later, the seismic design evolved to enable plastic deformations in the structure under severe earthquakes for economic reasons, and this was reflected in the use of reduction factors for the lateral strength based on the ductility coefficient, *i* story. Since that time, the use of equivalent lateral forces allowing inelastic deformations in seismic designs has been a practice accepted worldwide and incorporated into most of the standards. Nevertheless, the force-based design methodology has not yet solved the problems derived from the damage concentration of the seismic actions that lead to severe damage and even the collapse of structures when moderate or strong earthquakes occur. Most of the standards put forward heuristic recommendations or criteria for the safe seismic design of structures, like symmetry, absence of irregularities in the geometry of the buildings, or the hierarchy of damage in structural elements based on the strong column-weak beam, that show deficiencies in practice (Kuntz and Browning, 2003). Furthermore, different studies have shown that the simplified lateral load patterns proposed by standards do not lead to a uniform ductility demand distribution in building structures (Mohammadi et al., 2004; Chopra, 2011). By contrast, there are some approaches in the literature aimed to address the problem of damage concentration. They propose optimization procedures and formulations of lateral load patterns or their equivalent strength distributions for the seismic design of structures to achieve a uniform distribution of the damage expressed in terms of deformations or dissipated energies. The lateral load pattern is defined by the relationship *F*_{i} and *Q*_{y,1} are the lateral force of the *i*-th story and the yield shear force at the first story—base shear force, respectively. The most important drawback is that the optimum strength or lateral load pattern distribution is different for each ground motion. Then, differences between the actual and the “ideal” even distribution of the damage are expected in structures when they are subjected to different ground motions.

The approaches based on the control of the lateral deformations of the structure are aimed at achieving a uniform distribution of the interstory drift, *T*, the number of stories¸ *N*, and *T* (between 0.1 and 3 s) and formulated an optimum load pattern to achieve the uniform ductility distribution that depended on *T* and *T*, *N,* *,* and

On the other hand, there are approaches aimed at achieving a uniform distribution of the dissipated energy which is the main goal of the energy-based methods. Akiyama (2000), Akiyama (2003) proposed and demonstrated that the lateral strength of the stories in a structure should be “optimal” to achieve a uniform distribution of the damage. For this purpose, the damage of the *i* story was expressed through the dimensionless parameter *W*_{p,i}, and divided by the product of *Q*_{y,i} is the yield shear force of the story. In turn, *Q*_{y,i} can be dimensionless and expressed through the yield shear-force coefficient defined as *M* is the total mass of the structure and *m*_{j} is the mass of the *j*-th story, *g* is the gravity acceleration. Then Akiyama (2000), Akiyama (2003), defined the optimum yield shear-force coefficient distribution as the ratio *T* such as that formulated by the Japanese Building Code (Building Research Institute, 2009b). Benavent-Climent (2011) put forward an expression for EPP models that depends on the stiffness distribution, *T,* and the predominant period of the ground motion, *T*_{G}. Donaire-Ávila (2013) and Donaire-Ávila and Benavent-Climent (2020) formulated an equation of

Unlike the approaches for which the damage is based on the maximum lateral deformations, the use of optimization algorithms to obtain *N* and

This study sheds light on the influence of the constitutive model in the damage distribution of low and mid-rise shear-type buildings subjected to seismic actions along one of their principal directions—planar analysis. The structures were designed with energy-based methods by using different approaches for optimum strength distributions. Accordingly, the constitutive models Clough and Modified Clough that characterize the reinforced concrete (RC) structures are used and the results are compared with those obtained in structures with EPP constitutive model. Results in terms of energy dissipation and its deviation with respect to the even distribution are shown and discussed. The main limitation of the study is that it addresses only low and mid-rise shear-type buildings. In future studies, investigating the influence of the global flexural deformation that can be relevant in high-rise buildings and extending the analysis to spatial models subjected to more than one component of the seismic action is recommended.

## 2 Methodology

This study focuses on the distribution of damage among stories in building structures designed with different optimum strength distributions

1. Selection of realistic distributions of mass and lateral stiffness for the benchmark prototype structures.

2. Calculation of the base shear-force coefficient *E*_{I} (expressed in terms of equivalent velocity

3. Calculation of the shear-force coefficient of the upper stories

4. Development of a numerical model for each prototype structure using different constitutive models for the shear force-interstory drift relationships.

5. Selection of ground motion records and scaling to the *V*_{E} adopted in the design.

6. Non-linear time history analyses of the numerical models and evaluation of the damage distributions.

## 3 Benchmark prototype structures

Prototype structures—benchmark—of 3, 6, 9, and 12 stories used by the authors (Donaire-Avila et al., 2020) in previous studies are used here, modifying only the constitutive models for the shear force-interstory drift relationship. Figure 1 illustrates the geometry of them as well as the structural elements—beams and columns. They have a fundamental period of 0.40 s, 0.80 s, 1.2 s, and 1.6 s, respectively, with a uniform mass distribution (*k*_{1} to the top story *k*_{T}, with *k*_{1}/*k*_{T} equal to 1, 1.5, 2.0, and 2.5.

**FIGURE 1**. The geometry of the prototype structures: **(A)** 3 stories; **(B)** 6 stories; **(C)** 9 stories; **(D)** 12 stories, where *b x h* in beams and columns refer to the width x depth of the transverse section. Dimensions in cm.

The base shear-force coefficient *E*_{I} and the expected level of damage *E*_{I}, expressed in terms of the equivalent velocity *V*_{E}, was obtained from the study carried out by Donaire-Avila et al. (2020) by applying the ground motion prediction equation developed by Cheng et al. (2014), whose functional form is defined by a set of variables which were applied in this case as follows: i) a moment magnitude for the earthquake, *M*_{w} = 6.5; ii) a reverse or oblique-reverse fault type; iii) the closest distance to a fault of 10 km; iv) set of coefficients that depend on *T;* iv) the average shear-wave velocity in the top 30 m, *V*_{s30}. The moment magnitude and the fault type were established to ensure later in the non-linear evaluations a minimum number of records that achieve the target input energy in a wide range of periods, preventing the use of large-scale factors. The closest distance to the fault was determined to obtain a significant input energy *V*_{E} for each benchmark prototype structure that leads to reasonable values for the base shear coefficient *V*_{E} were obtained for each benchmark structure; one per soil class, which are reported in Table 1.

The expected level of damage

## 4 Design of prototype structures with different optimum shear coefficient distributions

According to the energy-based design methodology, once

### 4.1 Optimum distributions for systems with EPP restoring force characteristics

Five optimum distributions

The optimum distribution proposed by Donaire-Avila et al. (2020) was obtained from an optimization procedure based on non-linear time history analysis (NLTHA) applied to 3, 6, 9, and 12-story prototype building structures designed as stated above for two ductility levels and two soil classes. The numerical model of each prototype structure, a lumped-mass model with EPP restoring force characteristics, was subjected to a seismic record in an iterative process. In each iteration, the strength distribution of the model, except

Then, a predictive equation for

The *z*_{i} coefficients were obtained for each soil class and ductility level and can be found elsewhere (Donaire-Avila et al., 2020). Eventually, a simplified expression of

where *N* is the number of stories of the building; *i* is the story level (*i* = 1 for the first story and *i* = *N* for the top story).

The optimum distribution proposed by Akiyama,

Finally, the expression proposed by the Japanese Building code,

Among the different approaches

**FIGURE 2**. Mean deviation of **(A)**, 6 **(B)**, 9 **(C)**, and 12-story buildings **(D)**.

Next, the differences with respect to

**FIGURE 3**. Deviation of the optimum distributions over **(A,E)**, 6 **(B,F)**, 9 **(C,G)**, and 12-story buildings **(D,H)**.

## 5 Damage distribution in structures with different constitutive models

This section investigates the distribution of damage on the prototype structures designed with the optimum distributions described in Section 4 for different constitutive models to describe the restoring forces: the EPP, the Clough model, and the Modified Clough model.

### 5.1 Numerical models

The prototype structures were represented by a lumped-mass model with one translational degree of freedom per level. Three constitutive models were considered in this study for the shear force-interstory drift relationships (Figure 4): i) the EPP model, ii) the Clough model (Clough and Johnston, 1966), and iii) the Modified Clough model (Otani, 1981). The first one is used to represent the mechanical behavior of steel structures and was also the main reference used by Akiyama (2000), Akiyama (2003) to develop the energy-based design methodology applied to multistory buildings. The Clough model and the Modified Clough model are two of the most widely used models to characterize the hysteretic behavior of reinforced concrete structural elements. They are phenomenological models based on experimental investigations that have been validated by many studies for the simulation of the monotonic and cyclic behavior of reinforced concrete (RC) structures (Gupta et al., 2001; Grammatikou et al., 2019). The postyield stiffness was established to be null, *i* story, *k*_{r,i}. For the Clough model, *k*_{r,i} remains constant and equal to the initial lateral stiffness of the story *k*_{e,i}. But for the Modified Clough model, *k*_{r,i} shows a progressive degradation with the maximum interstory drift, *a* is a constant that is calibrated with tests. Akiyama 2000, Akiyama, 2003 proposed to use *L*_{s} is the shear span and *h* is the depth of the transverse section of the structural element. For the size of the members of the prototype structures used in this study (Figure 1), the expression proposed by Grammatikou et al. (2019) gives *a* between −0.50 and −0.33. The value adopted in this study (*a* =

**FIGURE 4**. Constitutive models for the shear force-interstory drift relationship: **(A)** EPP; **(B)** Clough; **(C)** Modified Clough.

To further validate the Modified Clough numerical model (with *a* = *Q-δ* curves and with bold lines the prediction. It can be seen that the numerical model captures well the shape of the hysteretic loops. Finally, Figure 5C compares the energy dissipated by the numerical model and by the test specimen up to each peak of the history of the loading plot in Figure 5A. It can be seen that both histories of dissipated energy are very close.

**FIGURE 5**. Validation of the Modified Clough model with the experimental results obtained on an RC column: **(A)** History of loading; **(B)** Lateral force vs. lateral displacement; **(C)** Histories of cumulated energy.

Accordingly, 240 (=80·3) numerical models were built for the analysis, 120 for each soil class. The Rayleigh damping model was considered for an equivalent damping ratio

### 5.2 Selection of ground motion records

The ground motions used in this study are the same employed by Donaire-Avila et al. (2020). Two sets of ground motion records were selected, fifteen for each soil class −30 in total, obtained from the database of Campbell and Bozorgnia (2007) by excluding the records with *S*_{a} and *V*_{E} of the selected ground motion records, where *S*_{a} is the response acceleration. The scale factor applied to each record for matching *V*_{E} (Table 1) was lower than four.

**FIGURE 6**. Spectra of *S*_{a} **(A,C)** and *V*_{E} **(B,D)** for the selected ground motion records in Soil Class B (upper row) and Soil Class C (lower row). Each record is identified by the following sequence: earthquake short name-station short name-component of the record (horizontal one or two) according to the criterion established in the database.

### 5.3 Damage evaluation

The numerical models were subjected to the selected set of ground motion records and the response was obtained through non-linear time history analysis (NLTHA). This entailed 3,600 dynamic evaluations (= 120 numerical models x 2 soil classes x 15 records per soil class). They were conducted with OpenSees (2022).

#### 5.3.1 Results in terms of $\mathit{\eta}$ and ${\mathit{\mu}}_{\mathit{d}}$

The energy dissipated—normalized by the product *i* story of each prototype structure subjected to the ground motion *j*,

**FIGURE 7**. **(A,C)** and **(B,D)**, using EPP and RC constitutive models (Clough model (upper row) and Modified Clough model (lower row)).

**FIGURE 8**. **(A,C)** and **(B,D)**, using EPP and RC constitutive models (Clough model (upper row) and Modified Clough model (lower row)).

On the other hand, structures designed with

It is important to note that the differences observed for

The same procedure described above to obtain *i* story and ground motion *j*,

**FIGURE 9**. **(A,C)** and **(B,D)**, using EPP and RC constitutive models (Clough model (upper row) and Modified Clough model (lower row)).

**FIGURE 10**. **(A,C)** and **(B,D)**, using EPP and RC constitutive models (Clough model (upper row) and Modified Clough model (lower row)).

Moreover, it can be observed that, overall, *n*_{eq} (Khashaee et al., 2003). The design of the prototypes was made considering *j*. Next, the mean among the stories was calculated,

#### 5.3.2 Influence of the constitutive models in the damage distribution

The energy-based design methodology proposes to design the structures using an optimum shear strength distribution such that—ideally—the energy dissipated by plastic deformations is evenly distributed (*j*,

Figures 11, 12 illustrate

**FIGURE 11**. Mean values of **(A,C)** and **(B,D)**, using EPP and RC constitutive models (Clough model (upper row) and Modified Clough model (lower row)).

**FIGURE 12**. Mean values of **(A,C)** and **(B,D)**, using EPP and RC constitutive models (Clough model (upper row) and Modified Clough model (lower row)).

Next, the differences for

Moreover, Figures 11, 12 show that there are no appreciable differences for

#### 5.3.3 Damage concentration

Another important issue is the level of damage concentration in the structures. It was estimated through the ratio *j*. Then, the average of this ratio among the set of records for each story *i*,

**FIGURE 13**. **(A,E)**, 6 stories **(B,F)**, 9 stories **(C,G)**, and 12 stories **(D,H)**.

**FIGURE 14**. **(A,E)**, 6 stories **(B,F)**, 9 stories **(C,G)**, and 12 stories **(D,H)**.

Overall, RC models offer a worse agreement of

Structures designed with

In the case of structures designed with

## 6 Conclusion

This study is relevant for the following reasons. First, because it involves a large number of parameters and cases: four prototype buildings designed for five different approaches for the optimum strength distributions in two different soil types with two different levels of ductility and three different constitutive models for the shear force-interstory drift relationships and subjected to 30 different ground motions—15 for each soil type— (total 3,600 cases). Second, the distribution of damage is investigated using realistic constitutive models for RC structures instead of the elastic perfectly plastic model used in most past studies. Third, because, as far as the authors know, a study that quantifies the goodness of different lateral strength distributions to achieve an even distribution of damage using realistic constitutive models of RC structures has never been done. Finally, this study provides valuable information on the most appropriate lateral strength distribution to design RC structures applying the energy-based approach. The optimum strength distribution

• RC structures designed with the optimum distributions considered in this study showed higher deviations in the damage distribution (

• Structures with Clough and Modified Clough restoring force characteristics showed, on average, similar distributions of dissipated energy without significant differences for

• Structures that differ only in the ductility level show similar values for

• Structures designed with

• When

•

• By contrast, the structures designed with

Eventually, it is worth noting that future studies will be needed to evaluate the influence of different aspects not developed here in the damage distribution of structures designed with different approaches for the optimum strength distribution: i) structures subjected to different seismic actions such as the near-field—pulse-like—ground motions; ii) analysis employing the two components of the seismic actions—spatial analysis; iii) type of structures where the contribution to the lateral deformations of the bending moment component is not secondary (e.g., tall buildings equipped with RC frames and shear walls).

## Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

## Author contributions

JD-A formulated the study, conducted the non-linear time history analysis, carried out the data curation, and wrote the paper. AB-C supervised the research, discussed the results, and reviewed the paper. FM proposed the seismic levels used in the study, selected the ground motions database, and reviewed the paper. All authors contributed to the article and approved the submitted version.

## Funding

This work has received financial support from the Sapienza University of Rome (Grant no. RG11916B4C241B32). The support from the Spanish Government under Project MEC_PID 2020-120135RB-I00 and received funds from the European Union (Fonds Européen de Dévelopment Régional) is also gratefully acknowledged.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

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Keywords: constitutive model, energy-based method, optimum strength distribution, damage distribution, shear-type buildings

Citation: Donaire-Avila J, Benavent-Climent A and Mollaioli F (2023) Influence of the constitutive model in the damage distribution of buildings designed with an energy-based method. *Front. Built Environ.* 9:1190923. doi: 10.3389/fbuil.2023.1190923

Received: 21 March 2023; Accepted: 02 May 2023;

Published: 25 May 2023.

Edited by:

Nicola Tarque, Universidad Politécnica de Madrid, SpainReviewed by:

Tadesse Gemeda Wakjira, University of British Columbia, Okanagan Campus, CanadaDaniel Ruiz, University of Los Andes, Colombia

Copyright © 2023 Donaire-Avila, Benavent-Climent and Mollaioli. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Jesus Donaire-Avila, jdonaire@ujaen.es