Advances in Materials Science and Engineering

Volume 2016, Article ID 5750672, 21 pages

http://dx.doi.org/10.1155/2016/5750672

## Stiffness-Displacement Correlation from the RC Shear Wall Tests of the SAFE Program: Derivation of a Capacity Line Model

^{1}European Commission, Joint Research Centre (JRC), Institute for the Protection and Security of the Citizen (IPSC), 21027 Ispra, Italy^{2}Université Paris-Est, RENON (IRC-ESTP, IFSTTAR), IRC-ESTP, 28 avenue du Président Wilson, 94234 Cachan, France

Received 24 September 2015; Accepted 21 March 2016

Academic Editor: Santiago Garcia-Granda

Copyright © 2016 Francisco J. Molina et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The response of 13 reinforced concrete shear walls submitted to successive seismic tests has been postprocessed to produce time histories of secant stiffness and displacement oscillation amplitude. For every wall an envelope curve of displacement amplitude versus stiffness is identified which is fairly modelled by a straight line in double logarithmic scale. This relatively simple model, when used as a capacity line in combination with the demand response spectrum, is able to predict in an approximate manner the maximum response to the applied earthquakes. Moreover, the graphic representation of the demand spectrum and a unique model capacity line for a group of equal walls with different assumed design frequencies on them gives a visual interpretation of the different safety margins observed in the experiments for the respective walls. The same method allows as well constructing vulnerability curves for any design frequency or spectrum. Finally, the comparison of the different identified line models for the different walls allows us to assess the qualitative effect on the behaviour of parameters such as the reinforcement density or the added normal load.

#### 1. Introduction

The seismic tests performed on reinforced concrete (RC) shear walls within the SAFE program were aimed at observing the effect of several parameters in the response. Particularly, as discussed by Labbé et al. [1], the study of the effect of the design frequency on the safety margin was one of the objectives. The current work takes advantage of those experimental results in order to derive a structural degradation model that potentially may be used for predicting in an approximate manner the maximum response of those walls to any given seismic excitation.

The description of the degradation in the structural behaviour as a function of the damage, and particularly by means of a reduction in the stiffness (or eigenfrequency) that is a function of the maximum performed displacement, is present in many studies and we will refer here just to a few of them that have especial common points with the current work.

In the study of Benedetti and Limongelli [2], shaking-table tests results on masonry buildings were used to plot “effective modal stiffness” values as a function of the displacement regarding single cycles in the response to successive earthquakes on the same specimen. They observed how for those structures the stiffness was dependent only on the maximum previously reached displacement at every cycle. By keeping in this representation only the envelope points of growing displacement, they constructed a model that reproduced the experimental envelope curve. The model was based on the combination of brittle-elastic elements and elastoplastic elements and was defined by four parameters that were identified from the experimental results.

In the work of Brun et al. [3], results from the seismic tests on the RC shear walls of the SAFE program were reproduced by a finite element model based on the fixed smeared crack concept. The dynamic results of such finite element model to a series of excitations were used for representing the fundamental frequency associated with every level of maximum displacement reached. There, the fundamental frequency was derived from the secant stiffness of single cycles. After doing similar observations to the ones of Benedetti and Limongelli [2], this envelope curve from the numerical simulations was substituted by a formula based on a combination of continuous functions and defined by eight parameters that were identified. Then, a simplified dynamic model was used to produce the time response of a wall by just step-by-step integration of the linear 1-DoF equation of motion, but with variable natural frequency at every step, decided from the recorded maximum displacement according to the previously obtained curve . The damping ratio for this simplified dynamic model was chosen fixed at 7% and considered as an average of the experimental values.

On the other hand, in the more recent work of Brun et al. [4], the finite element model based on fixed smeared crack that had been used to produce dynamic response was now used to produce static pushover curves from which to extract the secant stiffness and derive the fundamental frequency at every level of displacement. Moreover, the derived ) curve for every wall was successfully compared with the equivalent envelope curve derived from the experimental results of the SAFE program. This time the simplified dynamic model was implemented by using directly the function, either observed from the numerical pushover simulations (first case) or from the PsD tests for every wall (second case), without additional simplification by a formula and with the use of damping values derived from the experimental ones. The computed responses of the simplified model were similar between the two cases and also with the experimental ones for most of the studied walls.

In the current work, as in the mentioned studies, the variation of the stiffness with respect to the displacement level is observed from the experimental results. The special methods used for deriving an instant secant stiffness and displacement amplitude, as well as the rule for defining the envelope curve or , are introduced and applied for all the tested walls of the SAFE program. Then, as a step forward with respect to other works, such curve is used as a capacity line that can be directly checked with the demand spectrum [5] in order to predict the maximum response to a given earthquake. Moreover, for every single tested wall, or for groups of them in some cases, the envelope curve is adjusted to a very simple straight line in double logarithmic scale, which is also used as the capacity line instead of the experimental curves. Interestingly, this simple model (characterised by only two parameters) is able to reproduce the order of magnitude of the maximum response to the tested inputs in most of the cases. It is also effective for producing vulnerability lines that give justification to the observations made by Labbé et al. [1] regarding the influence of the design frequency in the safety margin, for example. Moreover, this study gives one more graphical illustration of the influence of two main factors on the safety margin, which are, according to Labbé [6], the type of structural stiffness degradation and the shape of the demand spectrum at the performed frequencies.

#### 2. SAFE Program Data Processing

##### 2.1. Testing Campaign

As described in other publications [1, 3, 4, 7], the wall specimens of the SAFE program were 13 with names T01⋯T13 and they were all seismically tested in pure shear at the ELSA laboratory by means of the pseudodynamic (PsD) method, which is a hybrid testing technique by which the inertial forces are modelled numerically [8].

For all the walls, the length wasand the height wasThe main parameters with different value among the walls are displayed in Tables 1 and 2. The thickness of the walls wasfor the first four specimens T01⋯T04 andfor the remaining ones T05⋯T13. Consequently, the section of every wall wasEven though the concrete used was of two types with different actual capacities, the design of the walls assumed a conventional capacity value ofcorresponding to a design shear modulus offor a Poisson’s ratio 0.2 (according to Labbé et al. [1]), and the design stiffnessIn the SAFE program, the design frequency was made a key parameter that was imposed with several values among the different walls in order to observe the effect of it in the response. As shown in Table 1, its value waswhereas the design damping ratio was adopted constantfor all the walls.