Nonlinear seismic analysis of masonry buildings · 2013-10-01 · for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, 1787-1799) Pushover analysis

A. Penna – Software Forum - Erlenbach Sept. 12, 2013

Nonlinear seismic analysis of

masonry buildings

Erlenbach, September 12th, 2013

Department of Civil Engineering and Architecture

University of Pavia, Italy

Andrea [email protected]

EUCENTRE Foundation

• Highly nonlinear behaviour

• Need for nonlinear analysis recognized sincelate 1970s (Tomazevic, 1978; Braga and Dolce, 1982)

• Pushover analysis

• Equivalent frame modelling

Seismic response of masonry buildings


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

T

N


Modelling of the mechanical behaviour

Flexure-rocking

Shear-sliding (friction)

Seismc response

-100

-80

-60

-40

-20

0

20

40

60

80

100

-8 -6 -4 -2 0 2 4 6 8

displacement (mm)

forc

e (k

N)

Cyclic behaviour: stiffnessdegradation and strength

deterioration

0.4 0.8-0.4-0.8

40

20

-20

-40

0

First-Story Drift, %

Sto

ry S

hea

r, k

N

0

Dynamic response


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


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

TTb Tc Td

SD

TTb Tc Td


Pushover analysis

Spectral coordinatesSA

TTb Tc TdSD

TTb Tc Td

SD

Tb Tc

Td

SA

2

2 2

)(

)(

TTS

TS

D

A

Pushover analysis

Seismic response of nonlinear systems

F

Ddy dmax = dy

“Rigid” structures

Fe

Fy

F

Ddy dmax = dy

“Flexible” structures

Fe

Fy


Pushover analysis

Ductility demand and spectral reduction factors

cy

eR

cc

y

eR

TTseF

F

TTseT

T

F

F

11

(Fajfar, 1999)

cD

cc

D

TTseq

TTseT

Tq

11

y

e

y

e

F

TmS

F

Fq

)(

Spectral reductioncoefficient or “behaviour

factor”

Pushover analysis

Displacement demand for a «rigid» system

SD

SA

max,

max,

max11

e

Ce dT

Tq

q

dd

m

Fy

dmax

de,max

y

A

y

e

F

TmS

F

Fq

)(


Pushover analysis

Structural capacity

• 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 itsnonlinear (static) response

• Hypothesis: the lateral response of the structure under the effect of a properly incremented vector of horizontalforces can be assumed as the envelope of the possibleresponse obtained by nonlinear time-history analysis


PUSHOVER ANALYSIS

Base shear

F1

Fi

Fi+1

Fn

n

ib FT1

dtop

PUSHOVER ANALYSIS

Capacity curve

DTOP

TB

SA

SD

SA

SD

SA

SD


Pushover analysis

Performance Displacement limit states

• Performance limit states

• Damage limit states for structural members

• Interstorey drift ratio

• Damage limitation

• Ultimate limit states

Pushover analysis

Analysis results:

• Capacity curve

• Limit states: from local element damage to global limit states

• Safety assessment in terms of global displacements


Simple model for masonry structural members

Du = 0.004-0.008 hDy

VR

VR

N

22

FLEXURAL STRENGTH

In-plane bending failure ↔ toe-crushing

For relatively low compression values (N)the wall tends to overturn similarly to arigid body

The analysis of the wall bending responsecan be based on an appropriate definitionof a “stress-block” for the compressedpart of the masonry cross section

tf

Na

u

u

mm

uu f

tl

ltf

NNlalNM

1

21

22

2

Vertical translation:

Rotation : = 0.85-1


23

Flexural strength

esup

einf

PV

VP

H

H

0

D

eP

a

D/2D/2

x

M=Pe=VH0

fu

Dt

Pp

f

pDPMePHV

uu

;

1

2inf0max

24

Cyclic shear response

-100

-80

-60

-40

-20

0

20

40

60

80

100

-8 -6 -4 -2 0 2 4 6 8

displacement (mm)

forc

e (k

N)


25

SHEAR STRENGTH

The definition of “shearfailure” usually includesdifferent cracking modesassociated with thecombined effect of shearand compression stress

Two main shear failuremodes can be identified:

a) diagonal-cracking

b) shear-sliding

Diagonal crackig: weak joints

Diagonal cracking: strong joints

26

Shear strength (1)

Dt

Pp

f

p

b

DtfV

tu

tuu

; 1

P

V

ftu = tensile strength

(Turnsek & Sheppard, 1980)


27

Shear strength (2)

pc

pcDt

pcDt

Dt

PcDtV

V

u

31

5.1

Strength of the cracked section:

Sliding on bed-joints:

c

P

V

28

Shear-compression interaction diagram

0100200300400500600700800900

100011001200

0 10 20 30 40 50 60 70 80 90 100

N/Nu [%]

Vre

s [k

N]


(Lagomarsino S, Penna A, Galasco A, Cattari S [2013] TREMURI program: An equivalent frame model for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, 1787-1799)

Pushover analysis

Analysis control

A pushover analysis consists of applying to the structure gravity loads and asystem of of distributed horizontal forces in the considered analysis direction,at each building level, proportionally to the inertial masses (sum of thehorizontal forces = base shear).

Such forces are scaled to monotonically increase, in both positive andnegative directions up to local/global collapse conditions, the horizontaldisplacement of a control point (usually coincident with the barycenter of thetop storey, excluding turrets)


Pushover analysis

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.5 1 1.5 2 2.5 3d [cm]

Fb

[kN

]

Curva modello SLD

Curva modello SLU

Stati limite

Capacity curve and limit states

DLSULS


Pushover analysis

Loading pattern Fi: force at the i-th storey

N

jj

ibasei

m

mFF

1

j

N

jj

iibasei

m

mFF

1

b):

a):

(“inverse triangular” or «modal»)

(“uniform”)

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


Modelling Strategies (in-plane response)

Limit analysis POR Method Finite Elements

Macro-elementsspandrel beam

pier

joint

F1

F2

Como & Grimaldi Tomaževič, Braga & DolceGambarotta & Lagomarsino,

Anthoine, Papa & Nappi, Lourenço

Pagano et al. SAM - Magenes et al.D’Asdia & ViskovicBraga & Liberatore

(Podestà, 2002)

PIER

RIGID NODE

SPANDREL

Equivalent frame macro-element modelling

PIER

RIGID NODE

SPANDREL


Macro-element wall models

Earthquake Damage Observation

FEM Non-linear Continuum Model

RigidNode

Lintel

Pier

2012 Emilia earthquake

Damage to masonry piers Damage to spandrels


Emilia 2012 earthquake

Damage to spandrels

Damage to piers




BENDING -ROCKING SHEAR


BENDING -ROCKING SHEAR


TREMURI ProgramNon-linear analysis procedures implemented

STATIC INCREMENTAL (FORCE / DISPLACEMENT)DYNAMIC (Newmark integration, Rayleigh viscous damping)

PUSHOVER (with fixed and adaptive load pattern)

mm m mk x f

FF Fm FC F FT TFm Cm

CF Cm CC C C

K k K x f

k k

K k K x r

1 1 1 ... ... 0i i ii m im mm m in mn n

m m m

f f fk k x k k x k k x

f f f

mm m mk x f

FF Fm FC FT TFm Cm

CF Cm CC C C

K k K x 0

k k

K k K x r

(Lagomarsino S., Penna A., Galasco A., Cattari S., 2013, TREMURI program: an equivalent frame model for the nonlinear seismic analysis of masonry buildings, Engineering Structures)

EQUIVALENT FRAME IDEALIZATION


Steps for the identification of the equivalent-frame mesh


EQUIVALENT FRAME IDEALIZATION


Irregular wall geometry

PIERSPANDRELRIGID NODE

THREE-DIMENSIONAL MODELING

node

rigid offset

node

hinge

rigid offset

PLAN

ELEVATION

Pier EL.

Spandrel EL.

Beam EL.

Rigid offset


3D MODEL

2D NODE

(3 d.o.f.)

3D NODE

(5 d.o.f.)

Z

z

xlo

c

X X

Y

φI J

K

Z

X

Y



In-plane floor stiffness – Membrane elements

12

21

21

21

21

00

011

011

ˆ

Gm

mE

m

Emm

Em

m

E

D

k

ji

x

y

...

... ...

e e eii ij ik

e ejj jk

ekk

e

k k k

K k k

k

Ase Tij i jk B DB

01

02

j k

k j

k j j k

y y

x xA

x x y y

iB

i

y

x

j

kl

ji

lk

j

lk

i

= ½ +

DRRD T

Model validation: numerical simulation of experimental testing on a full-scale URM building

(University of Pavia – Magenes, Calvi & Kingsley, 1995)

25 20 15 10 5 0 5 10 15 20 25

Numerical results


Damage pattern

n1 n2 n3

n4

n5

n6 n7

n8 n9

1 2 3

4 5 6

7 8

9 10

1 2

3 4

• 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

Requirements for computational models for pushoveranalysis of masonry structures


The user should never forget he/she isperforming a nonlinear analysis!

To check analysis convergence, he/she should perform one analysis for each direction starting from default values…

Example using the 3muri program

Convergence in an iterative process!

•Check the results (X and Y)

It is very “irregular”

Example in x direction

Look at the curves!

Repeat the analysis decreasing the allowed tolerance!

Anyway a new analysis is needed



Example in y direction

Look for specific suggestion!

The program suggests changing the control node in order to follow better the deformation of the building


It’s “smooth”!!!

Example in y direction

No specific need for new analysis for the curve but the sensitivity to the tolerance has to be checked!!




Decrease the tolerance up to convergence: 0.005‐> 0.001‐>…..



Compare the analyses after the

tolerance is decreased

0.005 > 0.001>0.00075>0.0005>0.0004

αuX αu

Y εrrX εrrY tolerance

3.117 2.243 0.005

2.907 2.155 -7.22% -4.08% 0.001

2.935 1.993 0.95% -8.13% 0.00075

2.921 2.122 -0.48% 6.08% 0.0005

2.934 2.098 0.44% -1.14% 0.0004

Convergence procedure

•Compare the curves after the tolerance is decreased

0.005 > 0.001>0.00075>.0005>0.0004


Convergence procedure is completed when decreasing the tolerance (more accurate analyses) there is no change in

•Pushover curve (especially bilinear idealization)

•Assessed safety factors

When the convergence procedure is completed, all the analyses can be evaluated!

CONVERGENCE

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3

0.00%0.10%0.20%0.30%0.40%0.50%0.60%

tollerance

Saf

ety

val

ue

.

x directiony direction

THANK YOU FOR YOUR ATTENTION!

Documents

Nonlinear seismic analysis of masonry buildings · 2013-10-01 · for the nonlinear seismic analysis of masonry buildings, Engineering Structures, 56, 1787-1799) Pushover analysis