A critical look into Rayleigh damping forces for seismic performance assessment of inelastic structures
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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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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References
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"A critical look into Rayleigh dampi..." refers background in this paper
...For assessing structural performance of reinforced concrete frames, beam-column joint elements have been developed and implemented in finite element procedures for modeling reinforced concrete moment-resisting frames, whether they are ductile or not [1, 17]....
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141 citations
"A critical look into Rayleigh dampi..." refers background or methods in this paper
...In [15], the authors use FOSM method to investigate the sensitivity of a series of EDPs to uncertain parameters among which Rayleigh damping ratio....
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...In particular, the studies in [18, 15, 2] explicitly account for Rayleigh damping as a potential contributor to the uncertainty in the EDP of interest....
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...On the one hand, it has been observed that Rayleigh damping can be a significant contributor to the overall uncertainty in the EDPs of interest for seismic performance assessment of inelastic structures [15, 2]....
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115 citations
"A critical look into Rayleigh dampi..." refers background in this paper
...For assessing structural performance of reinforced concrete frames, beam-column joint elements have been developed and implemented in finite element procedures for modeling reinforced concrete moment-resisting frames, whether they are ductile or not [1, 17]....
[...]