Post on 02-Jan-2020
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 Pennaandrea.penna@unipv.it
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
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)
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
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 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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
)(
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 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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
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 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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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)
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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)
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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]
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 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)
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
Emilia 2012 earthquake
Damage to spandrels
Damage to piers
2012 Emilia earthquake
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
2012 Emilia earthquake
BENDING -ROCKING SHEAR
2012 Emilia earthquake
BENDING -ROCKING SHEAR
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
(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)
Steps for the identification of the equivalent-frame mesh
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
EQUIVALENT FRAME IDEALIZATION
(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)
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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
(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)
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
The user should never forget he/she isperforming a nonlinear analysis!
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
The user should never forget he/she isperforming a nonlinear analysis!
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!!
The user should never forget he/she isperforming a nonlinear analysis!
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
The user should never forget he/she isperforming a nonlinear analysis!
Decrease the tolerance up to convergence: 0.005‐> 0.001‐>…..
The user should never forget he/she isperforming a nonlinear analysis!
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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
A. Penna – Software Forum - Erlenbach Sept. 12, 2013
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!