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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 Penna [email protected]

EUCENTRE Foundation

• Highly nonlinear behaviour

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

• Pushover analysis

• Equivalent frame modelling

Seismic response of masonry buildings

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

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

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

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)

fo rc

e (k

N )

Cyclic behaviour: stiffness degradation and strength

deterioration

0.4 0.8-0.4-0.8

40

20

-20

-40

0

First-Story Drift, %

St or

y Sh

ea r,

kN

0

Dynamic response

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

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

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

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

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

Pushover analysis

Spectral coordinates SA

TTb Tc Td SD

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

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

Pushover analysis

Ductility demand and spectral reduction factors

c y

e R

c c

y

e R

TTse F F

TTse T T

F F

11 (Fajfar, 1999)

cD

c c

D

TTseq

TTse T Tq

11

y

e

y

e

F TmS

F Fq )(

Spectral reduction coefficient or “behaviour

factor”

Pushover analysis

Displacement demand for a «rigid» system

SD

SA

max,max,max 11 eCe dT Tq

q d

d

m Fy

dmax de,max

y

A

y

e

F TmS

F Fq )(

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

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 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

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

PUSHOVER ANALYSIS

Base shear

F1

Fi

Fi+1

Fn

n

ib FT 1

dtop

PUSHOVER ANALYSIS

Capacity curve

DTOP

TB

SA SD

SA

SD

SA

SD

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

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

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

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 a rigid body

The 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

tf Na u

u

mm

u u f

tl ltf

NNlalNM

1

2 1

22

2

Vertical translation:

Rotation : = 0.85-1

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

23

Flexural strength

esup

einf

P V

V P

H H

0

D

e P

a

D/2D/2

x

M=Pe=VH0

fu

Dt Pp

f pDPMePHV

u u

;

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)

fo rc

e (k

N )

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

25

SHEAR STRENGTH

The definition of “shear failure” usually includes different cracking modes associated with the combined effect of shear and compression stress

Two main shear failure modes 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

tu u

; 1 P

V

ftu = tensile strength

(Turnsek & Sheppard, 1980)

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

27

Shear strength (2)

pc pcDtpcDt

Dt PcDtV

V

u

31 5.1

Strength of the cracked section:

Sliding on bed-joints:

c P

V

28

Shear-compression interaction diagram

0 100 200 300 400 500 600 700 800 900

1000 1100 1200

0 10 20 30 40 50 60 70 80 90 100

N/Nu [%]

V re

s [ kN

]

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

(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 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 t