Dissipations in reinforced concrete components: Static and dynamic experimental identification strategy
Introduction
The numerous structural constitutive laws which have been developed since the second part of the 20th century laws now allow to provide realistic and reliable results on the nonlinear behavior of reinforced concrete (RC) structures. The more complex is the model, the more precise is the required knowledge of the material properties – a knowledge which is not obvious for engineers when the studied structure is still at the design state. Moreover, the variability of these parameters may lead to a necessary extensive numerical study to assess its influence on the structural behavior and the numerical cost of the associated nonlinear simulations is a strong counterpart that designers and engineers are not always prone to pay for. In addition, no model is precise enough to account for every single dissipation phenomenon occurring in a RC structure during a seismic loading.
For these reasons, the common practice is to consider a simpler structural model associated to an additional viscous damping to account for the dissipations not explicitly modeled. Especially, energy dissipation appears even in the linear domain of the material behaviors [12]. The origins of these dissipations may be multiple: soil-structure interaction, nonstructural elements dissipation, friction in joints, friction, etc. Rayleigh-based damping models – including Caughey’s series [8] – are convenient and popular in the earthquake engineering community since they allow a fuzzy description of these sources through a viscous force field. Classical Rayleigh-damping models come with now well-known drawbacks [29], [30], depending on which version of the model is chosen (mass proportional, initial stiffness proportional, tangent stiffness proportional, or Caughey’s series). Additional viscous damping should be considered carefully when used in combination with a hysteretic model as emphasized in [9], [21]. Indeed, the viscous contribution should be reduced progressively once in the nonlinear domain [12], otherwise the total dissipated energy may be overestimated thus leading to a non-conservative result.
For this reason, the amount of dissipated energy is a strong concern to calibrate whatever the chosen model is. Let us consider a (nonlinear) single degree of freedom (SDOF) oscillator of constant mass m and angular eigenfrequency excited by a sinusoidal displacement of angular frequency . Jacobsen [27], [28] has shown that a linear viscous damping force of the form of Eq. (1), where c is the viscous damping coefficient and is the oscillator velocity, is able to represent with an acceptable accuracy the dissipations of a more general nonlinear viscous damping force.His method can be graphically summarized on Fig. 1 for a linear viscous SDOF oscillator response. The restoring force versus displacement plot allows for a quick estimation of both the energy dissipated during one oscillation , corresponding to the area enclosed in the red curve, and the maximum energy stored energy during this cycle corresponding to the area under the straight line between the origin and the point of maximum displacement (since the oscillator is linear). Then, the equivalent viscous damping ratio (EVDR) defined as the ratio of the actual damping coefficient c over the so-called critical damping corresponding to the damping coefficient below which oscillations exist if the SDOF is relieved from an out-of-equilibrium state. Then the following equations arise:andEq. (3) being the one proposed by Jacobsen [28] and further discussed for nonlinear cases in Section 4.3. Basically, the EVDR can be seen as proportional to the ratio of energy dissipated during one cycle over the energy storage capacity of the SDOF.
This method stays reliable up to a certain extent whether viscous [1] or nonviscous phenomena are involved [2]. Consequently, a N-degrees of freedom (N-DOF) oscillator would require N equivalent viscous damping coefficients. From this point arises challenging problems regarding the equivalent viscous damping coefficients values associated to each eigenmode, their evolution throughout the inelastic time history analysis, and the possible existing couplings between modal dampings.
Two goals have driven the development of the experimental campaign in order to address the aforementioned issues:
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it should allow for a mode-per-mode as well as mode-coupled dissipations identification;
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the tests must be driven by the degradation level in order to identify the influence of this parameter on energy dissipations. The sensitivity studies regarding other parameters such as material properties should not be corrupted by an uncontrolled evolution of the structural state.
This paper will firstly give an overview of existing experimental campaigns. This will then help to introduce the experimental campaign design for this work. Finally, the relevance of the design is supported by the presentation of noticeable post-treated results.
Section snippets
Quasi-static tests
Quasi-static tests are generally easier to setup, and allow for cancelling inertial effects that are inherent to seismic loadings. This characteristic makes them more convenient to identify dissipations which are independent on the velocity or on the acceleration, since both are negligible. However, there is an information loss regarding the dependency of the damping on the excitation frequency. According to Jacobsen [28], the approximation of structural damping by an equivalent viscous damping
Framework
In the framework of SINAPS@ project [25], the IDEFIX experimental campaign took place in the TAMARIS experimental facility at the French Alternative Energies and Atomic Energy Comission (CEA) from May to November 2016. Twenty RC beams have been casted and their dimensions have been designed in order to make their first two eigenmodes being in the optimally-controlled frequency range of the shaking table, Azalée, i.e. under 30 Hz according to IDEFIX boarded mass. The experimental protocol
Experimental protocol
The QSC1 tests (Fig. 10a) performed on the different beam designs allow for computing the capacity curves by considering only the envelope of the force-displacement curves obtained such as pushover analysis. The force considered is the sum of the force measured on each beam support in X-axis, whereas the displacement is measured at midspan (only the positive displacements are considered, i.e. when the actuators push on the beam, by convention). These results are plotted in Fig. 13. For each
Conclusions
A proper evaluation of the energy dissipated by RC structures during seismic events is required to ensure their capability to resist, but this remains a challenging task. The necessity of experimental data to study the damping phenomena has motivated numerous experimental campaign. The present experimental campaign is an attempt to provide further information regarding the evolution of the dissipations throughout time-history analysis, and interactions between modes. Thanks to an innovative
Acknowledgments
The authors wish to express their most grateful thanks to CEA/DEN for its financial support. The work carried out under the SINAPS@ project has benefited from French funding managed by the National Research Agency under the program Future Investments (SINAPS@ reference No. ANR-11-RSNR-0022). The work reported in this paper has also been supported by the SEISM Institute (http://www.institut-seism.fr).
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