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

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 (2 min read)

### 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.

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##### Citations

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### Cites methods from "A critical look into Rayleigh dampi..."

...Terming from the age when the seismic analyses were first developed for elastic structures, considering its computational efficiency and its convenient implementation, adopting a viscous damping model is, most of time, reasonable.(13) However, the nonlinear time-history analysis, which is based on the nonlinear constitutive model, is the most appropriate way to account for the hysteretic behaviors of structures....

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5 citations

##### References

1,984 citations

### "A critical look into Rayleigh dampi..." refers methods in this paper

...The HHT-α method [11] implemented in FEAP with α = 0....

[...]

479 citations

### "A critical look into Rayleigh dampi..." refers methods in this paper

...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....

[...]

...The HHT-α method [11] implemented in FEAP with α = 0.65, β = 0.5 and γ = 1 is used and a time step of 5 ms; the value for α has been found to be adequate for accurately simulating the maximum displacements....

[...]

...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....

[...]

319 citations

### "A critical look into Rayleigh dampi..." refers methods in this paper

...For instance, in the case of moment-resisting frame structures, maximum interstory drift is often used as the EDP of interest (see [6, 16, 5] among others) because it is possible to map it to meaningful PL such as “immediate occupancy”, “structural damage” or “collapse prevention” [6]....

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246 citations

245 citations

### "A critical look into Rayleigh dampi..." refers background in this paper

...Although there presently is a consensus in the earthquake engineering community that using Rayleigh damping forces is far from reaching this ultimate goal [3, 10], common implementation of inelastic time history seismic analyses adds Rayleigh damping forces to the hysteretic structural forces....

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...as a requirement for a good modeling of damping in other works [10, 3]....

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...On the other hand, it has been shown that using Rayleigh damping forces along with an inelastic structural model can be problematic and lead to unintended consequences that can compromise the validity of the analyses outputs [10, 3]....

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