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
