The objective of code-compliant pushover analysis is to reduce the structure to an inelastic SDOF system properly de ned, and usually related to the fundamental vibration mode.
F.1 Eurocode 8 - Part 2: Bridges
Following Eurocode 8 (Part 2) [EC8 2005a], pushover is performed separately in both longitudinal and transverse directions. The investigation of two pattern loads is recommended;
• Triangular: The displacement at the level of the deck is assumed constant and piers are subjected to linearly increasing loads from the foundation to their top:
s∗i = mi· zi
Htot (F.1)
Where s∗i, mi and zi are respectively the load, mass and height associated with node i, whilst Htot is the height of the pier.
• Principal Mode: The load pattern is proportional to the principal mode in each direction, which is considered the one with the highest corresponding participation factor Γ:
s∗i = mi· φi,p (F.2)
Where φi,p is the modal displacement at node i in the principal mode.
The target inelastic displacement demand of the equivalent SDOF (um) is con-sidered equal to the elastic displacement of the centre of mass of the deformed deck (control point) when linear multi-mode spectrum analysis is performed1, tacitly
1According to Eurocode 8 (Part 2) [EC8 2005a], the target displacement is considered equal to the maximum displacement obtained between two seismic elastic combinations; (1) longitudinal earthquake spectrum combined with 30 % of the transverse spectrum and (2) transverse earthquake spectrum combined with 30 % of the longitudinal spectrum.
398 Appendix F. Pushover analysis in seismic codes and guidelines
considering that bridges are exible enough to apply the `equal deformation' rule (um ≈ uo, see gureF.1), which is reasonable for vibration periods in the velocity-and displacement-sensitive regions of the spectrum, i.e T > TC s 2. This rule may be deemed appropriate in cable-stayed bridges due to their important exibility, nevertheless, it has been veri ed in the present work that several in uencing lon-gitudinal or transverse modes could be located in the acceleration-sensitive region of the spectrum, specially if the foundation soil is soft3, and therefore the `equal deformation' rule should be questioned.
Eurocode 8 (Part 1) [EC8 2004] is prepared for this situation (when relatively sti modes in uence the response) and provides a simpli ed expression to estimate the inelastic target displacement of such periods (where T <= TC):
um ≈ uo qu
1 + (qu− 1)TC T
≥ uo; if T ≤ TC
um ≈ uo;if T > TC
(F.3)
uo = Sd(T, ξ) being the peak displacement obtained by considering the initial elastic SDOF properties (see gure F.1) and T the vibration period of this elastic response, whereas qu is the ratio between the acceleration in the elastic equivalent problem (Sa(T, ξ)) and the inelastic SDOF system (Fy/Mef f, where Mef f is the mass of the SDOF4 and Fy its elastic limit); hence qu = Sa(T, ξ)Mef f/Fy. Finally, TC is the aforementioned corner frequency.
Figure F.1. Elastoplastic system and its corresponding linearization.
2Following Eurocode 8 [EC8 2005a], in rocky soil class (TA) TC = 0.4s, whereas in soft soil class (TD) TC= 0.8s.
3Table 6.1includes the governing frequencies of several cable-stayed bridges on soft soil, and there may be appreciated that models with 200 m main span could be dominated by periods lower than TC= 0.8s (TD), i.e. frequencies higher than f = 1.25 Hz.
4Mef f is the eective mass of the studied mode, obtained with eq. (4.3).
F.2. ATC-40 399
Anyhow, Eurocode 8 (Part 2) [EC8 2005a] discourages pushover analysis if the mass of some piers has a signi cant eect on its dynamic behaviour, which is clearly the case in cable-stayed bridges due to the large mass of the towers, recommending nonlinear dynamics (NL-RHA) in such cases.
F.2 ATC-40
ATC-40 [ATC 1996] was one of the rst guidelines proposing pushover as a method-ology to study the nonlinear seismic behaviour of buildings. The load pattern pro-posed in ATC-40 is the `Principal Mode' described above.
The inelastic target displacement is estimated by means of an iterative pro-cedure called `capacity-spectrum' method, where a sequence of elastic SDOF are analyzed by updating the period and equivalent damping ratio5. This procedure does not always obtain convergence, and when it does converge, the solution could be misleading [Chopra 2007]. The pushover analysis leads to the capacity curve, which is expressed in ADRS coordinates6. The reduced spectra, taking into account hysteretic damping by means of simpli ed rules focused on building structures, is represented along with this plot. The reduction of the spectrum due to such equiv-alent viscous damping (ξ) may be done by using the reduction factor proposed by seismic codes (for example η = p10/(5 + ξ) ≥ 0.55 is proposed by Eurocode 8 [EC8 2004]), which is known to present lack of physical basis.
Shortcomings of ATC-40 methodology in the estimation of the inelastic displace-ment demand appear to have been recti ed in FEMA-440 report [fem 2005], but the bene ts in the equivalent linearization detour is unclear when there are available ex-pressions for the inelastic deformation ratio, like the one proposed by Eurocode 8 and presented in equation (F.3) [Chopra 2007].
ATC-40 suggests the possibility of including higher mode eect for high-rise buildings with fundamental period larger than 1 s, which are observed to be the most sensitive to such modes, by repeating the `capacity-spectrum' method for important modes dierent than the rst one.
F.3 FEMA-356
FEMA-356 [fem 2000] is a seismic guideline focused on building structures. At least two load distribution patterns must be considered separately and the most demanding results are selected. The rst load distribution is to be selected from among the following:
• Principal Mode
• Equivalent Lateral Force (ELF), analogous to triangular distribution.
5The `capacity spectrum' procedure has been described by the author elsewhere [Camara 2008].
6ADRS coordinates represent a plot of spectral displacements (horizontal axis) versus spectral accelerations (vertical axis).
400 Appendix F. Pushover analysis in seismic codes and guidelines
• MRSA pattern: s∗ is de ned with the storey shears obtained from the elastic spectrum analysis (MRSA) of the structure.
The second pattern is either the `Uniform' distribution (where the load is equal to the mass associated with each point, s∗i = mi) or an adaptive one which varies as the structure yields. There are many research works available in literature about the applicability of these load patterns to building structures, several concluding that `Uniform' distribution is not appropriate since it grossly underestimates the drifts in upper stories, and overestimates them in lower stories [Chopra 2007]. In the present work `Principal Mode' and `Uniform' patterns were studied in several cable-stayed bridges and `Uniform' distribution is also found to yield wrong results (see section6.4.4.2).
The target displacement of the SDOF system is estimated by means of the
`coe cient method':
um = C1C2C3 T 2π
2
Sa
| {z }
Sd
(F.4)
Where the elastic spectral displacement of the considered vibration mode (Sd) is increased by three coe cients; C1 represents the inelastic deformation ratio um/uo (again, umis the maximum inelastic displacement and uo is the maximum displace-ment of the equivalent elastic SDOF, see gure F.1) for SDOF systems with stable hysteresis loops (i.e. without stiness degradation, pinching, etc.), the coe cient C2 considers the increase in the deformation due to the damage eects which are not covered in C1, whereas C3 takes into account the increase in the deformation because of the negative post-yielding stiness arising from P-∆ eects. The princi-pal problem is that values provided for such coe cients are not prepared to be used in bridges, and some numerical results are not supported by research [Chopra 2007].
Improvements in coe cients C1 and C2 were established in FEMA-440 [fem 2005]
report.
If higher modes are important in the response of the structure, pushover is supplemented by the linear dynamic analysis according to FEMA-356.