A critical look into Rayleigh damping forces for seismic performance assessment of inelastic structures

TL;DR: In this article, discrepancy forces are introduced in the framework of computational dynamics and damping forces are presented as a model of these so-called discrepancy forces to represent internal energy dissipation.

Abstract: Rayleigh damping forces are commonly introduced in the numerical simulations of nonlinear structures run to assess structural performance in case of an earthquake. Their purpose is to account for energy dissipative mechanisms not otherwise explicitly represented in the model. When caused by interactions between the structure and its surrounding environment, energy dissipation is external to the structure, whereas it is internal when resulting from energy absorption mechanisms activated in the structure. In this paper, the concept of discrepancy forces is introduced in the framework of computational dynamics. Then, damping forces are presented as a model of these so-called discrepancy forces to represent internal energy dissipation. On the other hand, the discrepancy forces are identified from a set of experimental data recorded during shaking-table test of a ductile moment-resisting frame, which provides the rationale for a critical look into Rayleigh damping forces. It is in particular observed that, for the structure tested, the Rayleigh damping model used is inaccurate as a representation of the discrepancy forces. Besides, while the knowledge of the discrepancy forces allows for rationally discussing the capabilities of the inelastic structural model to represent the actual behavior of the structure, this is only possible to a limited extent with the Rayleigh damping model used.

Summary

1. Introduction

• Whether they are pertaining to the ground motion signal or to the structural response, uncertainties are numerous and can dramatically impact the conclusions of seismic risk analyses.
• Then, two inelastic structural models of the tested momentresisting reinforced concrete frame are presented.
• The discrepancy forces are calculated for both structural models in sections 5 and 6.

2. Damping forces revisited – Discrepancy forces

• The authors also assume that displacement , velocity , and accelerationproportional forces contribute to the structural response (left-hand side of the equation): (2) Mü(tn) +C(tn)u̇(tn) + F hys(u; tn) = F ext(tn) where M and C are the mass and damping matrices, Fhys is the structural hysteretic restoring force vector, and Fext is the external loading vector.
• The external forces along with the displacements y(t) and accelerations ÿ(t) recorded during shaking table test are imposed to the system, which yields R̃ = 0 because y(t) and ÿ(t) are the “true” displacements and accelerations; then Fdis can be directly computed with equation (10), also known as In other words.
• Whereas, on the other hand, the direct problem R̃(u, ü) = 0 is solved dynamically.
• G being a valid model does not imply the plastic deformations in steel rebars are accurately simulated.

3. Shaking table tests

• Experimental data recorded during the shaking-table test of a ductile moment-resisting reinforced concrete frame is used [7].
• These latter additional masses induced service cracks.
• Mode 1 is preponderant in the sense that it accounts for more than 90% of the total mass.
• The structure is subjected to two types of loading: vertical static loading due to the dead load of the frame along with the additional masses, and horizontal dynamic forces induced by the seismic acceleration time history imposed on its base.

4. Two inelastic structural models: H1 and H2

• All the numerical simulations are performed with the finite element computer program FEAP [22] where the various elements and material behavior laws have been implemented.
• For model H2, the additional nodes are considered as massless.
• Uniaxial material behavior law is then assigned to every concrete layer and steel fiber.
• All the material models considered in this work have been developed using some of the ingredients of the more general model presented in [12].
• As illustrated in figure 6, beam-column joint elements momentrotation response is modeled by upper and lower bars.

5. Model G1 = H1⋆D1

• Only an approximation of the discrepancy forces can be provided because experimental data are missing to calculate the complete discrepancy forces vector.
• The authors use this information to check whether model H1 accurately represents structural mass and stiffness distribution in the initial state before seismic loading.
• To account for the contribution of static loading to the discrepancy forces, these displacements are approximated performing quasi-static numerical analysis and storing the displacements pertaining to the N e DOFs monitored during shaking table test, hereafter referred to as usta,e.
• Discrepancy and hysteretic forces looks like they are symmetric with respect to the x-axis.
• Considering the accelerations time histories, the numerical response is in good accordance with the experimental data for the 2nd level, especially between 8 and 20 s, as for the displacements.

6. Model G2 = H2⋆D2

• The joint elements are parameterized by the elastic modulus Ej and the limit stress σj (post-yielding slope is set to less than 0.1Ej).
• Their purpose here is only to develop a structural modelH2 that is different from previous structural model H1, and not to rigorously identify the beam-tocolumn joint parameters that would lead to the best representation of the frame.
• This would require much more advanced identification procedures that are out of the scope of this work.
• Displacements at both levels are fairly good simulated, while larger errors are observed for the accelerations.
• It only means that the discrepancy forces for model G2 are not as accurately computed as for model G1.

8. Conclusions

• The general concept of discrepancy forces has been introduced in the framework of computational dynamics.
• Better knowledge of how additional damping forces should be computed thus is highly desirable.
• The purpose of this proposed shift of perspective is to provide a critical look into Rayleigh damping forces.
• While discrepancy forces provide insight into the capability of the structural model for representing the actual structural behavior, the modeled Rayleigh damping forces only shed light on the structural model to a more limited extent; .
• As a final remark, this work illustrates how experimental data can be effectively used to identify the forces that are usually expected to be represented by Rayleigh damping in seismic structural analyses.

Figures

Figure 4. Finite element meshes for structural models H1 (black nodes only) and H2 (black and white nodes connected through inelastic joints).

Figure 15. [top] 2nd-level relative displacement [left] and relative acceleration [right] time histories; [bottom] 1st-level relative displacement [left] and relative acceleration [right] time histories. Experimental data (plain line / black) and numerical simulation with model G2 (dashed line with markers).

Figure 6. Beam-to-column element.

Figure 16. Experimental (black) and simulated (grey) 2nd-level relative displacement [top] and relative acceleration [bottom] time histories. Simulations are run with model H1 along with massproportional Rayleigh damping (ξ̂1 = 3%).

Figure 17. Experimental (black) and simulated (grey) 2nd-level relative displacement [top] and relative acceleration [bottom] time histories. Simulations are run with model H2 along with massproportional Rayleigh damping (ξ̂1 = 3%).

Figure 3. Elastic response spectrum to the seismic motion with a critical viscous damping ratio of 5% (pseudo-acceleration with respect to period).

Figure 2. Acceleration time history recorded on the base of the frame during the test.

Figure 8. Concrete behavior laws: unconfined (plain line) and confined (dashed line) concrete.

Figure 9. Beam-to-column joint behavior law. Illustration with Ej = 300 GPa, σj = 600 MPa.

Figure 12. [top] 2nd-level relative displacement [left] and relative acceleration [right] time histories; [bottom] 1st-level relative displacement [left] and relative acceleration [right] time histories. Experimental data (black) and numerical simulation with model G1 (grey).

Figure 18. Damping (black) and hysteretic (grey) forces time histories simulated using model H1 [left] and H2 [right] along with mass-proportional Rayleigh damping (ξ̂1 = 3%) at nodes 4 [top], 5 [center], and 7 [bottom]. At node 5, horizontal velocity being almost identical all along the 1st-level beam, damping forces are almost null.

Figure 7. Steel longitudinal rebar constitutive law.

Figure 13. Resisting bending moment time history at node 7 in element 9 simulated running model G1. Bending moment is not null initially because of preceding static loading.

Figure 11. Model D1 [left] and model D2 [right]. Forces at nodes 4 [top], 5 [center], and 7 [bottom]: discrepancy forces (black), resisting forces developed by inelastic structural models (grey).

Figure 5. Frame element cross section. Sections are divided into confined and unconfined concrete layers according to whether it is inside or outside the steel reinforcement stirrups (in the y direction). Longitudinal steel rebars cross sections are explicitly introduced in the model description, not the stirrups. Dashed lines represent frontiers between layers.

Figure 19. Rayleigh damping-like discrepancy forces consistent with the experimental data. Forces are simulated with massproportional Rayleigh damping (ξ̂1 = 3%) at nodes 4 [top] and 7 [bottom]. Mass-proportional damping forces are independent of the structural model because stiffness matrix is not considered. Forces at node 5 are null because same horizontal velocity is assumed all over 1st-level beam.

Figure 1. RC frame structure tested on the shaking table at École Polytechnique Montreal. The frame is 5-meter wide (2×2.5 m) and 3-meter high (2 × 1.5 m). Every beam supports additional masses to account for service static loads. The system is symmetric.

Figure 14. Resisting bending moment time history at node 7L (element 9) simulated running model G2. Bending moment is not null initially because of preceding static loading.

Figure 10. Seismic (black) and inertia forces (grey) at node 5. The mass distribution for model H1 being the same as for model H2, seismic and inertia forces are identical for both models. Seismic and inertia forces at any other node are less or equal to those shown here at node 5.