Nonlinear seismic analysis of masonry buildings

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1 Erlenbach, September 12 th, 2013 Nonlinear seismic analysis of masonry buildings Andrea Penna Department of Civil Engineering and Architecture University of Pavia, Italy EUCENRE Foundation Seismic response of masonry buildings Highly nonlinear behaviour Need for nonlinear analysis recognized since late 1970s (omazevic, 1978; Braga and Dolce, 1982) Pushover analysis Equivalent frame modelling

2 Global seismic analysis of masonry buildings Modelling of the mechanical behaviour (Nonlinear static) pushover analysis Models for pushover analysis Mixed masonry-r.c. buildings Modelling of the mechanical behaviour N

3 Modelling of the mechanical behaviour Shear-sliding (friction) Flexure-rocking Seismc response force (kn) displacement (mm) Cyclic behaviour: stiffness degradation and strength deterioration Story Shear, kn Dynamic response First-Story Drift, %

4 Analysis of the seismic response Earthquake-resistant structure: walls + floor diaphragms Walls resisting elements (both vertical and horizontal loads) Floor diaphragms share vertical loads on walls and are in-plane stiffening elements Out-of-plane behavior of walls and flexural response of floors negligible with respect to the global behavior (under certain conditions) Highly nonlinear behaviour Computational approaches Pushover analysis Seismic demand (seismic action) Structural capacity (capacity curve) Performance Displacement limit states Definition of an equivalent nonlinear SDOF system Choice of the horizontal loading pattern Global assessment

5 Pushover analysis Representation of seismic action Acceleration and displacement response spectra Spectral coordinates Seismic response of nonlinear systems Inelastic spectra Reduction factors and ductility demand Pushover analysis Acceleration and displacement response spectra Se SD b c d b c d

6 Pushover analysis Spectral coordinates SA SA b c SD b c d d SD b c d S ( ) A S ( ) D Pushover analysis Seismic response of nonlinear systems F Fe F Fe Fy Fy dy dmax = dy D dy dmax = dy D Rigid structures Flexible structures

7 Pushover analysis Ductility demand and spectral reduction factors c y e R c c y e R se F F se F F 1 1 (Fajfar, 1999) c D c c D se q se q 1 1 y e y e F ms F F q ) ( Spectral reduction coefficient or behaviour factor Pushover analysis Displacement demand for a «rigid» system SD SA max, max, max 1 1 e C e d q q d d m F y d max d e,max y A y e F ms F F q ) (

8 Structural capacity Pushover analysis Base shear reference displacement Capacity curve Spectral coordinates PUSHOVER ANALYSIS Basic idea of the method: apply an horizontal force distribution to the structural model to directly evaluate its nonlinear (static) response Hypothesis: the lateral response of the structure under the effect of a properly incremented vector of horizontal forces can be assumed as the envelope of the possible response obtained by nonlinear time-history analysis

9 Base shear PUSHOVER ANALYSIS Fn dtop Fi+1 Fi F1 n b F i 1 PUSHOVER ANALYSIS Capacity curve SA B DOP SD SA SA SD SD

10 Pushover analysis Performance Displacement limit states Performance limit states Damage limit states for structural members Interstorey drift ratio Damage limitation Ultimate limit states Analysis results: Pushover analysis Capacity curve Limit states: from local element damage to global limit states Safety assessment in terms of global displacements

11 Simple model for masonry structural members VR N VR Dy Du = h FLEXURAL SRENGH In-plane bending failure toe-crushing For relatively low compression values (N) the wall tends to overturn similarly to a rigid body he analysis of the wall bending response can be based on an appropriate definition of a stress-block for the compressed part of the masonry cross section Vertical translation: N a fut Rotation : 2 l a Nl N ml t m M u N f ult 2 f u =

12 Flexural strength V P e sup P e inf D V H0 H M=Pe=VH 0 D/2 e P a x D/2 f u V max H 0 P e inf M u P D 2 p 1 f u ; p P Dt 23 Cyclic shear response force (kn) displacement (mm) 24

13 SHEAR SRENGH he definition of shear failure usually includes different cracking modes associated with the combined effect of shear and compression stress wo main shear failure modes can be identified: a) diagonal-cracking b) shear-sliding Diagonal crackig: weak joints Diagonal cracking: strong joints 25 Shear strength (1) V P V u p f tu Dt b P Dt 1 p f tu ; f tu = tensile strength (urnsek & Sheppard, 1980) 26

14 Shear strength (2) V P Sliding on bed-joints: c Strength of the cracked section: V u Dt c P Dt p 1.5c p Dt c Dt 1 3cV p 27 Shear-compression interaction diagram Vres [kn] N/Nu [%] 28

15 (Lagomarsino S, Penna A, Galasco A, Cattari S [2013] REMURI program: An equivalent frame model for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, ) Pushover analysis Analysis control A pushover analysis consists of applying to the structure gravity loads and a system of of distributed horizontal forces in the considered analysis direction, at each building level, proportionally to the inertial masses (sum of the horizontal forces = base shear). Such forces are scaled to monotonically increase, in both positive and negative directions up to local/global collapse conditions, the horizontal displacement of a control point (usually coincident with the barycenter of the top storey, excluding turrets)

16 Pushover analysis Capacity curve and limit states Fb [kn] DLS ULS Curva modello SLD 600 Curva modello SLU 400 Stati limite d [cm]

17 Loading pattern Pushover analysis F i : force at the i-th storey a): F F i base N m j 1 i m j ( uniform ) b): F F i base N m j1 i j i m j ( inverse triangular or «modal») Global response of existing URM buildings: wall in-plane behaviour Local devices resist to out-of-plane mechanisms and favour a global behaviour governed by wall in-plane response Piers and spandrels

18 Modelling Strategies (in-plane response) Limit analysis POR Method Finite Elements Como & Grimaldi omaževič, Braga & Dolce (Podestà, 2002) Gambarotta & Lagomarsino, Anthoine, Papa & Nappi, Lourenço Macro-elements F 2 spandrel beam pier F 1 joint Pagano et al. Braga & Liberatore D Asdia & Viskovic SAM - Magenes et al. Equivalent frame macro-element modelling PIER PIER RIGID NODE RIGID NODE SPANDREL SPANDREL

19 Macro-element wall models Rigid Node Lintel Pier Earthquake Damage Observation FEM Non-linear Continuum Model 2012 Emilia earthquake Damage to masonry piers Damage to spandrels

20 Emilia 2012 earthquake Damage to spandrels Damage to piers 2012 Emilia earthquake

21 2012 Emilia earthquake BENDING - ROCKING SHEAR 2012 Emilia earthquake BENDING - ROCKING SHEAR

22 REMURI Program Non-linear analysis procedures implemented SAIC INCREMENAL (FORCE / DISPLACEMEN) DYNAMIC (Newmark integration, Rayleigh viscous damping) PUSHOVER (with fixed and adaptive load pattern) KFF kfm KFC xf ff k kfm mm kcm xm fm KCF kcm KCC xc rc K k K x 0 FF Fm FC F kfm kmm kcm xm fm K CF kcm KCC xc rc f i f i f i ki1 km 1x1... kim kmmxm... kin kmnxn 0 fm fm fm (Lagomarsino S., Penna A., Galasco A., Cattari S., 2013, REMURI program: an equivalent frame model for the nonlinear seismic analysis of masonry buildings, Engineering Structures) EQUIVALEN FRAME IDEALIZAION Steps for the identification of the equivalent-frame mesh (Lagomarsino S, Penna A, Galasco A, Cattari S [2013] REMURI program: An equivalent frame model for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, )

23 EQUIVALEN FRAME IDEALIZAION PIER SPANDREL RIGID NODE Irregular wall geometry (Lagomarsino S, Penna A, Galasco A, Cattari S [2013] REMURI program: An equivalent frame model for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, ) HREE-DIMENSIONAL MODELING nod e rigid offset nod e rigid offset Pier EL. hinge Spandrel EL. Beam EL. Rigid offset ELEVAION PLAN

24 3D MODEL 2D NODE (3 d.o.f.) 3D NODE (5 d.o.f.) z K Z Y x lo c φ I J X X Z Y X (Lagomarsino S, Penna A, Galasco A, Cattari S [2013] REMURI program: An equivalent frame model for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, )

25 In-plane floor stiffness Membrane elements ˆ G m me m E m m E m m E D k j i x y e e e ii ij ik e e jj jk e kk e k k k K k k k As e ij i j k B DB j k k j k j j k y y x x A x x y y B i i y x j k l j i l k j l k i = ½ + DR R D Model validation: numerical simulation of experimental testing on a full-scale URM building (University of Pavia Magenes, Calvi & Kingsley, 1995) Numerical results

26 Damage pattern n8 9 3 n n n6 7 1 n4 8 2 n n1 n2 n3 Requirements for computational models for pushover analysis of masonry structures Failure modes for all structural members (not only masonry) Local and global equilibrium. Reasonable compromise between accuracy, simplicity of use and interpretation of the results Possibility of identification of meaningful damage thresholds in the structural members

27 he user should never forget he/she is performing a nonlinear analysis! o check analysis convergence, he/she should perform one analysis for each direction starting from default values Example using the 3muri program he user should never forget he/she is performing a nonlinear analysis! Convergence in an iterative process! Check the results (X and Y) Look at the curves! Example in x direction It is very irregular Anyway a new analysis is needed Repeat the analysis decreasing the allowed tolerance!

28 he user should never forget he/she is performing a nonlinear analysis! Example in y direction Look for specific suggestion! he program suggests changing the control node in order to follow better the deformation of the building he user should never forget he/she is performing a nonlinear analysis! Example in y direction It s smooth!!! No specific need for new analysis for the curve but the sensitivity to the tolerance has to be checked!!

29 he user should never forget he/she is performing a nonlinear analysis! he user should never forget he/she is performing a nonlinear analysis! Decrease the tolerance up to convergence: > >..

30 0.005 > 0.001> > > Compare the analyses after the tolerance is decreased α X u α Y u εrr X εrr Y tolerance % -4.08% % -8.13% % 6.08% % -1.14% Convergence procedure Compare the curves after the tolerance is decreased > 0.001> >.0005>0.0004

31 Convergence procedure is completed when decreasing the tolerance (more accurate analyses) there is no change in Pushover curve (especially bilinear idealization) Assessed safety factors CONVERGENCE x direction y direction Safety value % 0.50% 0.40% 0.30% 0.20% 0.10% 0.00% tollerance When the convergence procedure is completed, all the analyses can be evaluated! HANK YOU FOR YOUR AENION!

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