1. ARISTOTLE UNIVERSITY OF THESSALONIKI CIVIL ENGINEERING DEPARTMENT LABORATORY OF REINFORCED CONCRETE COMPILED GREGORY G. PENELIS ANDREAS J. KAPPOS 3D PUSHOVER ANALYSIS: THE ISSUE OF TORSION 12 th European Conference on Earthquake Engineering LONDON – SEPTEMBER 2002

2. INTRODUCTION  Torsional strain is often observed on damaged buildings after earthquakes  This effect is more transparent in the nonlinear response of stuctures (I.e. severe damage)  The nonlinear analysis of buildings is gradually being introduced in codes and guidelines (ATC- 40, FEMA 273 & 356, HAZUS, RISK-UE etc)- mainly by utilising the more perceptible by the practicing engineer PUSHOVER ANALYSIS.

3. INELASTIC TORSION TO DATE: STATE OF THE ART  Two “categories” of reports: (Α) The theoretical study of inelastic torsion (Β) The design of torsionally restrained new buildings From these: The static eccentricity is modified as the elastic center CR shifts towards the center of shear CS. (PAULAY). The limit surface BST (BASE SHEAR TORSION) defined by triads of points corresponding to different failure mechanisms (Chopra).

4. From the state of the art the issue of nonconvergence between static nonlinear analysis and dynamic nonlinear analysis is obvious. - All approaches seem to be case sensitive to the excitation - The modal loads (elastic) seem to be the load vector approximating the dynamic nonlinear analysis better

5. SCOPE OF WORK  The primary results of a 3D static nonlinear analysis methodology for the assessment of the vulnerability of structures which converges with the results of 3D dynamic nonlinear analysis. Α) Definition of an appropriate load vector for the static nonlinear analysis Β) Definition of the equivalent single dof oscillator for the spectral assessment of the vulnerability under a specific excitation. C) The introduction of the excitation.

6. PRINCIPLES OF THE METHODOLOGY Α) LOAD VECTOR: One that causes the same displacement and torque on a structure using static linear analysis as the ones calculated by elastic spectral dynamic analysis (icluding all important modes). A kind of modal loads… Β) EQUIVALEN SDOF OSCILATOR: ( For translation & torque) The methodology of Saidi& Sozen (1981) which defined the sdof oscillator for translation was modified to take into account the torsional effect. C) SPECTRA: Mean normalised inelastic acceleration- displacement spectra (ADRS)

7. ONE STOREY BUILDING (1) 1) Selection of accelerograms (3-5) which are normalised (acc. Pga or Ι) 2) Calculation of the mean elastic spectra of the selected accelerograms and execution of spectral dynamic analysis in order to define the elastic translation and rotation of the center of mass. 3) The displacement vector of step 2 is used as a constraint in order to calculate the corresponding load vector. 4) Calculation of the modification factors for the sdof oscillator..

8. ONE STOREY BUILDING (2) ψ δ = P 1 /M 1 (1) ψ Μ = -1 (2) c 1 = (m  u y2 2 + J m  θ z2 2 ) / m  u y2 (3) c 2 = (u y2  ψ δ + ψ M  θ z2 )/ ψ δ (4) m * = m  u y2 (5) Where ψ δ, ψ Μ : parameters related to the modal loads, P 1, M 1 : the load vector defined by step 3 c 1, c 2 : parameters for the tranformation of a mdof to a sdof system, In general parameter c 1 corresponds to displacements and parameter c 2 to loading.

10. ONE STOREY BUILDING (3) 5) Pushover analysis with the load vector at Center of Mass (P 1, M 1 ). The P-δ curve of the multi dof -> single dof using c 1, c 2 Ρ * = c 2  p/m* δ * = c 1  u y (6) 6) For the selected accelerograms the mean inelastic normalised spectra (A-D) are calculated. The demand is defined for several ductilities (I.e. Fajfar-Dolsek, 2000) 7) The P-δ curve of the sdof is plotted on the demand spectraand the performace point is defined. This is the target displacement of the sdof -> u * targ. 8) The target displacement of the mdof is calculated u targ = u * targ / c 1 (7) u targ = u * targ / c 1 (7) and the target rotation (R targ ) as it is defined by the pushover analysis (P-θ curve) of the mdof for the target dispacement u targ and the target rotation (R targ ) as it is defined by the pushover analysis (P-θ curve) of the mdof for the target dispacement u targ

11. RESULTS - COMPARISON Α) Comparison of the P-δ and Ρ-θ curves of the pushover analysis (steps 1-3 &5) with the corresponding dynamic envelope Β) Calculation of the target displacement and rotation using pushover analysis with inelastic spectra and comparison with the results of nonlinear time history analysis. Torsionally Unrestrained Torsionally Restrained

12. Α) P-δ and Ρ-θ curves  The dynamic envelope is calculated for the 1st set of 4 accellerograms using: T.UR :40 time history nonlinear analysis T.UR :40 time history nonlinear analysis T.R. : 80 time history nonlinear analysis T.R. : 80 time history nonlinear analysis

13. TORSIONALLY RESTRAINED TORSIONALLY UNRESTRAINED

14. Β) TARGET DISPLACEMENT & ROTATION The 4 selected accelerograms scaled to pga = 0.4g  6% deviation in displacement and 2% in rotation for the torsionally unrestrained building.  3.7% deviation in displacement and 6.8% in rotation for the torsionally restrained building.

15. CONCLUSIONS - COMMENTS  The Ρ-δ and Ρ-θ curves of the pushover analysis approximate the dynamic envelope  The target displacement and rotation are accurately calculated for the one storey building  The implementation for multi storey buildings is yet to come Problems - Observations Α) Adaptive pushover analysis Change in Κ -> [Φ] -> [V, T]

16. Mean inelastic normalised spectra Β) Mean inelastic normalised spectra / Highly damped spectra Mean inelastic spectra without normalisation

17. C) Inconsistency of t-h inelastic analysis?

18. Dynamic Envelope MaxV -> disp & rot Maxdisp -> V & rot