X - FGG-KM

JCSS Workshop on Reliability Based Code Calibration
Zurich, March 2002
Reliability-Based Code
Calibration for Earthquake-
Resistant Design
Christiana Dymiotis
Department of Civil & Offshore Engineering

Presentation overview
Basic EC8 requirements and limit states
Uncertainties in structural behaviour
Code calibration at member level
Structural vulnerability and seismic risk (SLS
and ULS considerations)
Calibration of behaviour factors
Conclusions

Eurocode 8 (EC8) design
No collapse requirement: the structure must be able to
withstand the design seismic action without partial or total
collapse so that structural integrity and a residual loadbearing
capacity are retained following the earthquake.
Damage limitation requirement: the structure must be able
to withstand a seismic action that has a shorter return
period than the design seismic action (i.e. higher
probability of occurrence), so that the damage sustained
does not impose any limitations on structural usage.

Eurocode 8 (EC8) design
3 limit states
• Serviceability limit state (SLS): no structural damage
due earthquakes of low intensity.
• Ulitmate limit state (ULS): only light and repairable
damage due to earthquakes of moderate intensity.
• Collapse limit state: no collapse due to earthquakes of
high intensity.
Capacity design approach
• energy dissipation through large, inelastic deformations
• strong column - weak beam design

Uncertainty at different levels
Material
behaviour
f cc
σ
Section
behaviour
µ ϕ
ν
ε cu
ε
Structural
behaviour
h
Member
behaviour
Μ
∆x i
/h i
θ
STRUCTURAL RELIABILITY

Investigated structural systems
10×3m
6m
4m
6m
Bare frame
(EC8-designed,
DC“M”, A d =0.25g)
Fully infilled
frame
“Pilotis” frame
(soft ground storey)

Investigated cross-sections
sections
6Ø8
162
9Ø16
6Ø8
114
8Ø12
B250×850
850 762
Ø5/110
250
700 614
Ø4/85
200
B200×700
12Ø20
Ø10/110
8Ø16
Ø6/80
500
398
300
212
C500×500
500
300
C300×300

Random variables
Variable Mean COV (%) Distribution
f c 28MPa 18 Normal
X m,εcu 1.0 39 Normal
f y 440MPa 6 Normal
f u 506MPa 6 Normal
ε su 0.09 9 Normal
ULS: δ cr 6.6% 31 Lognormal
(+ 2 additional lognormal variables for infilled frames)

Uncertainty in member capacity
N
Nc
Strength
response
µ ϕ
µ ϕ,1
Ductility
response
Nb
Nt
M0
Mb
M
µ ϕ,2
ν 1 ν 2
ν
Fibre model approach - unit mean ratio of experimental to analytical
values, COV≈7%
Enhancement in strength and ductility due to confinement considered
Ductility estimated based on various criteria for estimation of ε cu

Response Surface Method
Identification of random variables
Simple expressions for estimation of strength and ductility
parameters (based on 2 nd order regression models)
Very good agreement with fibre model values for sections
not considered in derivation
e.g. for column curvature ductility,
ν=N/N ult =0.2
( )
2
+ 0.02 fˆ
− 0.06 fˆ
0.01fˆ
µ ϕ µ X +
= ϕ
m 1 c y c
RSM
15
10
5
where X m =model uncertainty factor
µ ϕ =fibre model prediction corresponding
to mean values
0
0 5 10 15
Fibre Model

Column flexural strength
N (MN)
4
ε
=
3 ε
2
300×300mm Column
cu
sy
⎛ d − x
⎜
⎝ xu
y
⎞
⎟
⎠
12
9
6
500×500mm Column
EC2
Fibre model -
characteristic values
Fibre model -
mean values
µ ϕ
⎟
⎠
⎞
1
µ 0 ε cu ⎛ 0.60 − 0.50ν
= ⎜
ϕ ε ⎝ 1. 35 ν
-1
0 0.06 0.12 0.18
sy
3
0
-3
M (MNm)
0 0.2 0.4 0.6 0.8
EC2 models lead to conservative strength estimates
Higher safety margins above balance point

Shear strength
Overall member shear capacity: V R3 =V c +V w
Assessment: V c =V c1 +V c2
• RSM function for V c2 , empirical functions for V c1 , V w
Design: V c =V R1 (EC2)
Simulated / Code values
V R3, 0.05 / V R3, k V R3, 0.5 / V R3, m
C300×300 1.74 1.97
C500×500 1.55 1.82

Column curvature ductility
Exact EC8 equation:
ε ⎛ d − x ⎞
cu y
µ ϕ =
⎜
⎟
ε sy ⎝ xu
⎠
µ ε cu ⎛ 0.60 − 0.50ν
⎞
= ⎜
⎟ ϕ ε ⎝ 1. 35 ν ⎠
Approximate EC8
equation:
sy
300×300mm Column 500×500mm Column
frequency
30
20
N/N ult =0.1 30
20
10
10
0
0
0 15 30 45 0 15 30 45
30
N/N ult =0.2 30
20
20
10
10
0
0
0 15 30 45 0 15 30 45
µ ϕ
µ ϕ
Distributions corresponding to buckling or 0.85fc
15
Distributions 10 corresponding to buckling or 0.5fcc
5
Exact EC8 equation, characteristic material properties
0
1 2 3
Approximate EC8 equation, characteristic material properties

Column curvature ductility
Exact EC8 equation:
ε ⎛ d − x ⎞
cu y
µ ϕ =
⎜
⎟
ε sy ⎝ xu
⎠
µ ε cu ⎛ 0.60 − 0.50ν
⎞
= ⎜
⎟ ϕ ε ⎝ 1. 35 ν ⎠
Approximate EC8
equation:
sy
300×300mm Column 500×500mm Column
frequency
30
20
N/N ult =0.1 30
20
10
10
0
0
0 15 30 45 0 15 30 45
30
N/N ult =0.2 30
20
20
10
10
0
0
0 15 30 45 0 15 30 45
µ ϕ
µ ϕ
15
10
5
Distributions corresponding to buckling or 0.85fc
Distributions corresponding to buckling or 0.5fcc
Exact EC8 equation, mean material properties
0
1 2 3
Approximate EC8 equation, mean material properties

Column curvature ductility
Distributions
corresponding to 0.85fc
Distributions
corresponding to 0.5fcc
ν d ν µ m µ k,0.05 P(µ≤9) µ m µ k,0.05 P(µ≤9)
C300×300 0.107 0.0465 15.12 5.71 0.14 24.86 8.97 0.05
C500×500 0.526 0.215 9.21 3.67 0.48 13.77 5.65 0.17

Methodology for probabilistic assessment
START
Uncertainty
modelling
Optimum
sample size
Simulation
of r.v.’s
Seismic
modelling
Structural
analysis
Convolution of
vulnerability
with seismic
hazard
NO
More
scenarios?
Comparison
with other
scenarios
Mean
vulnerability
Structural
vulnerability
YES
YES
More return
periods?
More
variables?
YES
Simulation
of r. v.’s
NO
NO
SEISMIC
RELIABILITY

Structural analysis
Inelastic time-history dynamic analysis using
DRAIN-2000
Lumped plasticity approach
Either moderately or fully cracked sections
assumed
Member capacity estimated using derived
response surface functions
Beam failures accounted for during analysis

Modelling of uncertainty in seismic input
Natural accelerograms
chosen to cover a wide
range of seismic
parameters.
Seven input motions
initially considered
Three carefully chosen
motions adequate for ULS
vulnerability
mean P f (%)
100
80
60
40
20
0
0 1 2 3 4 5 6 7
A′/A d
All 7 Records
5 Records
3 Records

Structural vulnerability - SLS
3 SLS criteria considered significant differences in
vulnerability predictions
X m,εcu
not critical
mean P f (%)
100
80
60
40
20
0
0 1 2
A′/A d
SLS criteria:
θ>θ y
θ>2θ y
δ max >0.5%

Structural vulnerability - ULS
Several spatial scenarios considered
Failure due to one of two criteria:
• large interstorey drift
• first column failure
Pf (%)
100
I
d
( θ p
)
( θ )
( VR2
)
( V )
( VR3
)
( V )
⎪⎧
⎪⎫
req req req
, el = max⎨
, , ⎬ ≤1.0
⎪⎩ p avail R2
avail R3
avail ⎪⎭
Scenario A
80
60
40
20
0
Scenario B
Scenario C
Scenario D
Scenario E
0 2 4 6 A'/A d

ULS comparisons with infilled frames
mean P f (%)
100
75
50
25
0
0 1 2 3 4 5 6
A'/A d
⎝
S a
1.5
1.0
0.5
0.0
ful. inf.
pilotis
bare
AEGL
ELC
KALW
LPRL
SFERT
0 1 2 3 T(s)
Bare Fully infilled Pilotis
mean P f (%)
100
75
50
25
0
0 1 2 3 4 5 6
A'/A d
⎝
S a
1.5
1.0
0.5
0.0
ful. inf.
pilotis
bare
NORL
0 1 2 3 T(s)

Seismic risk
Unconditional probability of failure in t d years,
P
f
∞
∫
− ∞
= f ( A′
) F ( A′
) d A′
S
R
where F R (A’) and F S (A’) are given by the structural
vulnerability and seismic hazard curves, respectively.
P(PGA>A)
1
0.8
0.6
0.4
0.2
0
Structural
lifetimes:
5.2 years
20 years
50 years
200 years
350 years
500 years
0 0.2 0.4 0.6
A(g)

Seismic risk
Results for bare frame:
12
P f (%)
9
6
3
ULS
SLS: θ>θ y
SLS: θ>2θ y
SLS: δ max >0.5%
0
0 100 200 300 400 500
t d (years)

Reliability indices
SLS
θ>θ y θ>2θ y δ max >0.5% ULS
Structural lifetime (years) 5.2 5.2 5.2 50
Earthquake return period (years) 50 50 50 475
Bare frame
Fully infilled frame
Pilotis frame
P f 0.1139 0.0716 0.0069 0.0082
β 1.21 1.46 2.46 2.40
P f 0.0936 0.0281 0.0110 0.0222
β 1.32 1.91 2.29 2.01
P f 0.1175 0.0362 0.0163 0.0278
β 1.19 1.80 2.14 1.91
β EC1 2.45 2.45 2.45 3.80
β US - - 1.5-1.8 1.55-1.75

Behaviour (q) factors
T1
S a
m 1 m n
k n , ξ
Elastic acceleration spectrum
Inelastic design spectrum
(= Elastic modified by q)
S d
T s
u1(t)
Tn
un(t)
T
..
x(t)
k 1 , ξ
………………
..
x(t)

Probabilistic evaluation of EC8 q-factorq
Actual q-factor q
developed,
q=q d (A /A d )
(A cr /A
P(q≤q d |failure) evaluated
from statistics of maximum
accelerations sustained by
structures that fail
P(q≤q d )= P(q≤q d |failure) P f
=0.75×0.0082
0.0082
≈0.6%
P(q|fail.)
1.00
0.75
0.50
0.25
0.00
q d =3.75
0 4 8 12
q

Conclusions
At member level:
Eurocode provisions give conservative estimates of strength parameters,
with higher safety margins at regions where brittle failure is expected.
EC8 equations with characteristic material properties do not lead to
adequately conservative estimates of curvature ductility.
At structure level:
Uncertainty in seismic input, member capacity and failure criteria ia taken
into account.
Better definition of target performance criteria required, especially for
the SLS.
Acceptable ULS safety margins have been observed for EC8 frames,
whereas SLS safety margins vary significantly, depending on the
adopted failure criterion.
The design behaviour (q) factor has been found to be exceeded in only
0.6% of the entire population of simulated frames.