Published by Elsevier Science Ltd. All rights reserved 12th European Conference on Earthquake Engineering
Paper Reference 015 (quote when citing this paper)
3D PUSHOVER ANALYSIS: THE ISSUE OF TORSION
Gr. G. Penelis1, A.J. Kappos11
Department of Civil Engineering, Aristotle University of Thessaloniki, 54006, Greece
ABSTRACT
A methodology is presented for modelling the inelastic torsional response of buildings in nonlinear static (pushover) analysis, aiming to reproduce to the highest possible degree the results of inelastic dynamic time history analysis. The load vectors are defined using dynamic elastic spectral analysis while the dynamic characteristics of an equivalent single mass system, which incorporates both translational and torsional modes, are derived using an extension of earlier methods based on the SDOF approach. The proposed method is verified for the case of single-storey monosymmetric buildings.
Keywords: Inelastic torsion; Pushover analysis; Nonlinear response; SDOF systems
INTRODUCTION
A common observation in buildings damaged by earthquakes is the torsional deformation they have been subjected to. This has long been recognised, and most modern codes include special provisions to account for this torsional deformation and, whenever feasible, to control its magnitude. However these provisions are typically related to the elastic properties of the building and are strictly applicable to cases of elastic analyses, whether static or dynamic. On the other hand these same codes permit and encourage the design of structures with allowance for inelastic behaviour, by introducing the ‘behaviour’ or ‘force reduction’ factor (q, or R), but generally failing to provide clear guidance for the treatment of inelastic torsional response of buildings.
Recently, nonlinear analysis has been introduced into codes or guidelines for the design and/or assessment of structures, the most reliable version of which is considered to be the inelastic time history analysis. This analysis, albeit useful and rewarding for research purposes, tends to be cumbersome and expensive whenever used by practising engineers, since it requires rather sophisticated input data (sets of accelerograms, damping coefficients, constitutive cyclic laws for inelastic members) and provides output, such as variation of internal forces and displacements with time, absorbed energy, etc., which is difficult to interpret for design purposes. So the method that is currently favoured [3,4,5] for inelastic analysis and design of structures is the nonlinear static (pushover) analysis that combines the advantage of explicit treatment of yielding and inelastic deformations with the simplicity of static loading patterns. Properly accounting for torsional effects in such an analysis is an issue at the forefront of current research.
REVIEW OF PREVIOUS STUDIES
The issue of inelastic torsion has been tackled in recent works by De La Llera & Chopra [2] and Paulay [10,11]. They mainly focused on the design of torsionally “insensitive” structures, i.e. structures with an arrangement of stiffness elements adequate for the control of torsional deformations. With regard to the translation-torque pair and the position of the lateral load vector, Paulay has demonstrated that the static eccentricity (defined as the distance between the elastic centre or centre of rigidity, CR and the centre of mass, CM), which is constant for an elastic system, is indeed varying in inelastic systems, since the elastic centre is moving towards the centre of shear CS as structural elements enter the post-yield range. Hence, the static eccentricity, in the case of inelastic analysis, is varying between two extreme values, the distance of the centre of mass CM from the elastic centre CR, when all the structural elements are elastic, and the distance of the mass centre CM from the shear centre CS when all the structural elements have yielded. Therefore the calculation of some dynamic amplification of the static eccentricity (as is the case in elastic analysis) should take into account that it is not a constant value. With regard to the force capacity of structures subject to torsion, De La Llera & Chopra introduced the Base Shear - Torsion surface (BST) which is defined by several sets of shears (Vx, Vy), and torques (T) which correspond to several different failure mechanisms of the structural system. As shown both theoretically and analytically, all triads of points from different nonlinear dynamic analyses fall within this BST surface.
From the review of the literature, it is clear that there is a weakness of the 3D pushover analysis to model the torsional effects as reproduced by nonlinear dynamic (time history) analysis. More specifically, the work of Moghadam & Tso [9] produced clear evidence of serious discrepancies between dynamic and static nonlinear analysis. The work of Rutenberg and De Stefano [12,13], indicates that existing methods for modelling an asymmetric building using pushover analysis generally do not reproduce the results of dynamic nonlinear analysis. Serious differences were found when different nonlinear constitutive laws were used, varying from simple bilinear to complex hysteretic, and a very high sensitivity to the input motion (accelerogram) used was also observed, making all the proposed modelling techniques extremely sensitive to that parameter. The work of Humar & Kumar [6] in which the effect of the transverse structural elements on the torsional response of a building is assessed, re-evaluates the findings of Correnza et al. [1] and Paulay [10,11] which were different from those of Humar & Kumar, who challenged the conclusions theoretically derived by Paulay, using several inelastic dynamic analyses. Based on the results reported in [6], it appears that the introduction of accidental eccentricity in that study, a parameter not included in the work of Paulay, is the source of the observed differences.
Another important observation, resulting from the work of Moghadam & Tso [9,15], is the effect of the distribution of the lateral loads along the height on the torsional response of the building. It was shown that using the modal load pattern produces results more accurate than a triangular or uniform distribution. In the same work [15] the estimation of the target displacement is done using the results of a spectral dynamic analysis, which along with the previous observation suggest that a spectral dynamic elastic analysis provides very useful information relevant to the inelastic torsional response of a building.
Based on the above observations (also supported by other studies not cited here due to space limitations) it seems that today static nonlinear analysis is, with respect to dynamic nonlinear analysis, at the same level as static elastic analysis was with regard to elastic modal spectral analysis in the late fifties.
PROPOSED METHODOLOGY
Principles of the method
Based on the review of the literature presented in the previous section, the objective of the study presented herein is the development of a method that will allow the modelling of the torsional response of buildings using a type of 3D pushover analysis that will produce results which do not deviate significantly from the ones produced by inelastic time history analysis. More specifically the aim is to account for the effects of dynamically amplified torque on the
available nonlinear capacity of a building, as represented by its pushover (P- ) curve.
The results of the 3D pushover analysis will be compared with the corresponding ones from the time history inelastic analysis of the same models, with common assumptions regarding member rigidity, strength, and moment-curvature sceleton curve at plastic hinges. It is understood that the use of inelastic dynamic analysis introduces sensitivity to the input motion, since the time-history method requires the use of input accelerograms, while the pushover analysis involves acceleration or displacement spectra (for estimating the target displacements). Hence, the first important observation is that the proposed method should be based on mean values resulting from sets of input motions appropriately selected according to specific seismological characteristics (magnitude, focal depth, distance), as required by all modern codes when dealing with inelastic dynamic analysis.
Critical points in the procedure are the determination of the force vector (lateral forces and torques) for the pushover analysis of the building and the determination of the equivalent generalised single-DOF oscillator, which will account for both the translational and torsional response of the building.
With regard to the force vector (loading pattern) of the building, based on existing literature and several parametric analyses, the load vector causing translations and rotations of the structure, calculated from elastic static analysis, that reproduce those predicted from modal spectral analysis, taking into account all significant modes, has been selected.
For the definition of the dynamic properties of the generalised equivalent SDOF system with both translational and torsional response, the methodology adopted in the early work of Saiidi & Sozen [14] wherein only the translational characteristics were accounted for, has been appropriately extended. For the case of single-storey monosymmetric buildings these characteristics are defined in step 4 of next section. It is understood that the general equations can be applied, in principle, into more complex cases, which are currently being studied and verified by the writers.
Steps of the methodology
1. The accelerograms to be used as input motion for the inelastic analysis are selected. A minimum of 3 to 5 accelerograms is required, and they are scaled according to peak ground acceleration (PGA) or spectrum intensity.
2. The mean elastic spectrum for the selected accelerograms is calculated and a dynamic response spectrum analysis of the building is performed, from which the translation and torque (uy1, z1) at the mass centre (CM) are calculated (for each floor diaphragm in multistorey structures).
uy1 P1 uy2 = 1
⇒
⇒
z1 M1 z2 = ]X\ C.M ExcitationDisplacement – Tosional Angle: uy1, z1
0 2 4 6 8 10 12 0 1 2 3 sec ) Ac c (m /se c 2) Elastic
Mean acceleration spectra
}
}
}
Static elastic analysis with constraint uy1, z1
Scale displacement vector to uy1 Spectral Dynamic Analysis
Figure 1: Summary of procedure for calculating the load vector for pushover analysis
3. Carrying out an elastic static analysis of the building the load vector (lateral force P1, torque M1) at the CM due to the translation and rotation pair (uy1, z1) calculated from the response spectrum analysis of the previous step, is calculated. In practical terms, the displacements calculated in step 2 are introduced as a constraint at the CM and the resulting reactions constitute the load vector (P1, M1).
4. The factors for the reduction of the multi-DOF to a single-DOF oscillator are calculated:
= P1/M1 (1) = -1 (2) c1= (m ⋅uy22+ Jm⋅ z22) / m ⋅uy2 (3) c2= (uy2 ⋅ + M⋅ z2 )/ (4) m*= m⋅uy2 (5) where
, : parameters referring to the modal load vectors P1, M1: the load vector calculated from step 3
m*: the mass of the single-DOF oscillator
c1, c2: factors of translation from the multi-DOF to the single-DOF oscillator. In general c1 refers to displacements and c2 to loads, as explained in the following steps.
5. A 3D pushover analysis of the building is performed applying the calculated load vector (P1, M1) at the CM. The P- curve of the building is calculated and, using the factors c1 and c2(equations 3, 4), it is reduced to the capacity curve of the equivalent single-DOF:
P*= c2⋅P/m*, *= c1⋅ (6) 6. The mean inelastic acceleration-displacement spectra are calculated for the selected
accelerograms, from which the inelastic demand spectra for several ductility factors are calculated, e.g. using the procedure described by Fajfar & Dolsek (2000).
7. From the overlay of the capacity curve over the demand spectra the target displacement of the single-DOF oscillator is calculated as the intersection of the capacity curve with the
demand spectra for ductility , where is calculated using the displacement at the point
of intersection of the capacity curve and the previous spectrum.
8. The target displacement for the top of the multi-DOF is calculated using the factor c1:
utarg = u*targ/c1 (7)
while the target torsional rotation (θtarg) results from the pushover analysis for the target displacement utarg.
W1 W2 W1 W2
Torsionally Unrestrained Building (T.U) Torsionally Restrained Building (T.R)
Figure 2: Single storey buildings studied
Case study: Single-storey monosymmetric building
For the verification of the method, at this stage, two single-storey buildings are analysed, one torsionally restrained (T.R) and one torsionally unrestrained (T.U.), shown in Figure 2. As already mentioned the results will be compared with those from inelastic time history analysis using the same modelling assumptions. For the latter, four different recorded accelerograms were selected (two from the Loma Prieta 1989 earthquake, one from the Northridge 1994 earthquake and one from the 1995 Kobe earthquake) which are shown in Figure 3.
/ 3 / L F N O D E W U V F D O H G 1 R U W K U L G J H 1 H Z K D O O )L U H 6 W D W L R Q / V F D O H G / S 7 U H V X U H ,V O 7 U V F D OH G .R E H +< 2 * 2 .(1 O V F D O H G
Firstly the reliability of the procedure for the definition of the load vector is established, using as reference the envelope of the base shear – displacement (or torsion angle) calculated from the inelastic dynamic analysis of the buildings under 40 different excitations (the 4 accelerograms scaled to PGA’s varying from 0.04g to 0.4g), for the torsionally unrestrained building and 80 different excitations (the 4 accelerograms scaled to PGA’s varying from 0.04g to 1.6g) for the torsionally restrained one. The corresponding curves using the results of steps 1 to 3 and 5 of the methodology are compared with the aforementioned envelopes, and it can be observed from Figures 4 and 5 that the results are satisfactory. This indicates that the model selected for the definition of the load vector of the building in the 3D pushover analysis, using the proposed procedure, is capable of predicting the inelastic torsional response of the single-storey building (accounting for dynamic effects).
Secondly the ability of the method to accurately predict the target displacement and rotation of the building using the results of the 3D pushover analysis, the factors for reducing it to a single-DOF oscillator, and the inelastic demand spectra, is assessed. To this effect, the accelerograms are scaled to 0.4g and from the analyses performed a deviation of 2% in the torsion angle and 6% in the displacement were observed between the 3D pushover analysis and the nonlinear time history analysis for the torsionally unrestrained building (Figure 6) and 3.7% and 6.8% correspondingly for the torsionally restrained one.
3RO\QRP LDOILWWRLQH ODV WLFWLP H KLV WRU \G\QDP LFDQDO\V H V DQG FRP SDU LV RQZ LWKWKH SXV KRYH UFXU YH
3 FXUYH 75 P P 3 8 P D[ N 1 /37UHDV,VO /3/LFN/DE 1R1HZ K .RE+\ RJ.HQ 3XVKRY HU SRO\ QRPLDOWK
3RO\QRP LDOILWWRLQH ODV WLFWLP H KLV WRU \G\QDP LFDQDO\V H V DQG FRP SDU LV RQZ LWKWKH SXV KRYH U FXU YH
3 FXUYH 75 ( ( ( ( ( ( U DG 3 8 P D[ N 1 /37UHDV,VO /3/LFN/DE 1R1HZ K .RE+\ RJ.HQ 3XVKRY HU 3RO\ QRPLDOWK
Figure 4b: Comparison of results, P- curve, Torsionally Restrained
3RO\QRP LDOILWWRLQH ODV WLFWLP H KLV WRU \G\QDP LFDQDO\V H V DQG FRP SDU LV RQZ LWKWKH SXV KRYH UFXU YH
3 FXUYH P P 3 8 P D[ N 1 /37UHV,VO /3/LFN/DE 1R1HZ K .RE+\ RJ.HQ 3XVKRY HU 3RO\ QRPLDOUG
3RO\QRP LDOILWWRLQH ODV WLFWLP H KLV WRU \G\QDP LFDQDO\V H V DQG FRP SDU LV RQZ LWKWKH SXV KRYH U FXU YH
3 FXUYH ( ( ( ( ( ( ( U DG 3 8 P D[ N 1 /37UHV,VO /3/LFN/DE 1R1HZ K .RE+\ RJ.HQ 3XVKRY HU 3RO\ QRPLDOUG
Figure 5b: Comparison of results, P- curve, Torsionally Unrestrained
: :
3XVKRYHU 1RQOLQHDU'\QDPLF
Figure 6: Calculated displacements and rotations of the torsionally unrestrained building
As mentioned previously, for the definition of the target displacement the inelastic demand spectra approach was utilised; the procedure is shown in Figure 7. Alternative procedures, e.g. based on modified elastic spectra could also have been used.
0HDQ,QHODVWLF&DSDFLW\'HPDQGVSHFWUD 'LVS P $ FF P V HF (ODVWLF 3XVKRYHU SRO\QRPLDO 3RO\QRPLDO
Figure 7: The capacity curve – demand spectrum procedure
COMMENTS AND CONCLUSIONS
During the development of the method several different approaches and parameters were explored, some of which significantly affect the calculated results. The most important of these are discussed in the following.
Inelastic Spectra
The calculation of the inelastic spectra as the mean of the inelastic spectra of each scaled accelerogram introduces the problem of irregularity in the shape of the final demand curve (an issue that has already been raised in the literature). This caused a problem in the graphic method for calculating the target displacement, as illustrated in Figure 8, since this value is very sensitive to the irregularities in the demand spectra. It is widely accepted, though, that all code-defined elastic and design spectra are smoothed spectra corresponding to the seismological parameters of each region. Hence it was decided that the same principle should be applied in the proposed methodology, and the normalised average curve of the inelastic demand spectra (using 6th order polynomial fit) was used. It is clear that another approach would be to increase the damping of the elastic spectra by taking into account the equivalent damping (for the appropriate ductility level), which would however not eliminate these irregularities and is generally a less accurate approach than that based on inelastic spectra.
Adaptive 3D pushover
Since the load vector and the characteristics of the equivalent SDOF oscillator are defined based on the results of elastic response spectrum analysis, it would be reasonable to expect that yielding of one of the resisting elements of the structure would alter its modal characteristics and therefore affect both the load vector and the equivalent SDOF oscillator characteristics. However the introduction of this parameter by means of adaptive pushover analysis, in the case of the studied buildings, as well as the case of an irregular two storey 2D frame, have shown no significant change, an observation which is in agreement with observations in the literature [3] that adaptive pushover analysis produces differences only in highly irregular multistorey buildings.
Average Demand spectra not normalised 0 2 4 6 8 10 12 0 0.01 0.02 0.03 0.04 Disp (m) $ FF P V HF Elastic duct.= 1.5 duct.= 2 duct.= 4
Figure 8: Inelastic demand spectra with irregularities (without smoothing)
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