15.12.2010 daniel honfi - konstruktionsteknik considering the vertical deflections of structural...
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Reliability based serviceability design15.12.201015.12.2010
Daniel [email protected]
Introduction
Such approaches commonly focus on the ultimate limit state (ULS)
Design codesCode calibration Relatively consistent
probability of failure
Reliability based serviceability design
Such approaches commonly focus on the ultimate limit state (ULS)
The present study:
Investigation the reliability of Eurocode specifications for serviceability, considering the vertical deflections of structural members subjected to bending made of different materials using Second Order Reliability Method (SORM).
Special focus on imposed loads
Introduction
Surveys:
• most defects are related to serviceability conditions rather than strength (Woodcock, 1986)
• 28% of building damages are due to building cracking and deflections (Loss and Kennett, 1987)
• 50% of damage cases, the damage cost exceeds $50,000
Reliability based serviceability design
• 50% of damage cases, the damage cost exceeds $50,000
• 15% of cases damage costs exceed $1 million
Influence of tributary area on βs, for Australian serviceability loading provision ΨsQ.
Stewart, 1996
Steel beams
Reliability based serviceability design
Influence of Ψs on βs, for Australian serviceability loading provision G+ΨsQ.
target βs= 1.65
Range of Reliability Indices for Total Deflection Criteria
Stewart, 1996
Concrete beams
Reliability based serviceability design
Range of Reliability Indices for Incremental Deflection Criteria
Li et al., 2005
Probability of serviceability failure due to deflection for different limits
Concrete beams
Reliability based serviceability design
Li et al., 1994
Failure probability for different COV of creep
Timber beams
Philpot et al. 1993
Reliability based serviceability design
Β-Φc Plots for Selecting Resistance Factor
Timber beams
Leichti and Tang, 1989
Reliability based serviceability design
Reliability with respect to serviceability decreases with time for sawn 2 x 10 lumber andwood-composite I-beams under constant load.
SAKO, NKB Report, 1999
Eurocode
Reliability based serviceability design
Expert review, BRE, 2003
EC load combinations for serviceability
Characteristic combination (irreversible limit states):
∑∑>≥
++1
,,0
1
1,,
i
iki
j
kjk QQG ψ
Frequent combination (reversible limit states):
Reliability based serviceability design
∑∑≥≥
+1
,,2
1
,
i
iki
j
jk QG ψ
Quasi-permanent combination (long-term effects and appearance):
∑∑>≥
++1
,,2
1
1,1,1,
i
iki
j
kjk QQG ψψ
Deflections
• wc is the precamber (unloaded);
(EN1990:2002 A1.4.3)
Reliability based serviceability design
• wc is the precamber (unloaded);
• w1 is the initial deflection (permanent loads);
• w2 is the long-term part deflection (permanent loads);
• w3 is the additional deflection (variable actions);
• wtot is the total deflection (w1+w2+w3);
• wmax is the remaining total deflection.
Deflection limits - steel
wmax w3
Denmark - L/400
Finland L/400 -
National Annexes:
different prescriptions based on function, importance, type of the carried material etc.
Reliability based serviceability design
Finland L/400 -
France L/200 L/300
Greece L/250 L/300
Hungary L/250 L/300
Spain - L/300
UK - L/200
Deflection limit values for a general steel beam not carrying brittle finish (characteristic combination)
Deflection limits - steel
1.5
2
2.5
3
IPE 100
IPE 200
IPE 300
QSLS/QULS
Reliability based serviceability design
0
0.5
1
1.5
0 5 10 15 20 25
IPE 400
IPE 500
IPE 600
L [m]
Serviceability vs. ultimate load for different IPE beams (wmax=L/250, χ=Qk/(Gk+Qk)=0.5, S235).
Deflection limits - concrete
The appearance and general utility of the structure:
quasi-permanent loads: L/250
precamber: max. L/250
Deflections that could damage adjacent parts of the structure:
deflection after construction, quasi-permanent loads: L/500.
(EN1992-1-1:2002 7.4)
Reliability based serviceability design
12 )1( δζζδδ −+=
2
1
−=
sr
s
σ
σβζ
),(1 0
,t
EE cm
effc∞+
=ϕ
Members which are expected to crack should behave a manner intermediate between the uncracked and fully cracked conditions.
Effect of creep:
Deflection limits - concrete
1
1.2
1.4
1.6
1.8
10
15
hSLS/hULS
qtot [kN/m]
Reliability based serviceability design
0
0.2
0.4
0.6
0.8
2 3 4 5 6 7 8 9 10
20
25
30
L [m]
SLS vs. ULS beam height for different spans and load levels(wmax=L/250, χ=Qk/(Gk+Qk)=0.5, ρ=0.01, L/b=20, d/h=0.92, C25/30, B500).
Deflection limits - timber
• u0 is the precamber (if applied);
(EN1995-1-1:2002 7.2)
Reliability based serviceability design
uinst unet,fin ufin
L/300 to L/500 L/250 to L/350 L/150 to L/300
• u0 is the precamber (if applied);
• uinst is the instantaneous deflection;
• ucreep is the creep deflection;
• ufin is the final deflection;
• unet,fin is the net final deflection.
Deflection limits - timber
For the instantaneous deflection:
400)(
384
5
,0
4L
QGIE
cLu kk
mean
inst ≤+=
[ ]200
)1()1(384
52
,0
4 LkQkG
IE
cLu defQkdefk
mean
fin ≤+++= ψ
For the final deflection:
Reliability based serviceability design
200384 ,0 IE mean
[ ]300
)1()1(384
502
,0
4
,
LukQkG
IE
cLu defQkdefk
mean
finnet ≤−+++= ψ
[ ]400
)1(384
52
,0
4
,2
LkQkG
IE
cLu defQkdefk
mean
fin ≤++= ψ
For the net final deflection:
For the deflection from the time dependent part of the permanent loads and the variable loads (w2+w3):
Deflection limits - timber
1.5
2
uinst,lim
ufin,lim
hSLS/hULS
Reliability based serviceability design
0
0.5
1
10 20 30 40
ufin,lim
unet,fin,lim
u2,inst,lim
L/hSLS
Serviceability vs. ultimate height of the beam for a glulam beam(χ=Qk/(Gk+Qk)=0.5, GL37).
Reliability
Safety Serviceability Durability
Reliability Robustness
Reliability based serviceability design
Reliability Robustness
[ ] )(0)( β−Φ=<= XgPPfβ: reliability index
Target reliabilities
Target reliability index β for Class RC2
(EN1990:2002 C6)
Reliability based serviceability design
Class RC2:
medium consequence for loss of human life, economic, social or environmental consequences considerable
Reliability analysis
For probabilistic calculations this failure can be described by a limit state function:
)(XgZ =
In case of deflections of the beam the expression can be generally formulated as:
( )limit
limitgXgδ
δδδ max
max 1,)( −==
Reliability based serviceability design
limit
max1)(δθ
δθ
R
EXg −=
Limit state function
limitδ
[ ]0)( <= XgPp f
Probability of failure
Second Order Reliability Method (SORM) with the structural reliability software COMREL 8.10
Material models
Description X Dist. µX σX
Young’s modulus E Normal En 0.04µX
Moment of Inertia I Normal In 0.03µX
Resistance factor θR LN 1 0.05µX
Steel
Reliability based serviceability design
JCSS Probabilistic Model Code (JCSS 2001)
Description X Dist. µX σX
Young’s modulus E Normal En 0.13µX
Width of the beam b Normal bn 0.005µX
Height of the beam h Normal hn 0.02µX
Resistance factor θR LN 1 0.10µX
Timber
Material model
Description X Dist. µX σX
Young’s modulus Es Normal Es,n 0.04µX
Width of the beam b Normal bn-0.003bn 4mm+0.006bn
Height of the beam h Normal hn-0.003hn 4mm+0.006hn
Concrete
Reliability based serviceability design
Effective depth d Normal dn 0.02 µX
Reinforcement area As Normal As,n 0.02 µX
Concrete comp. strength fck LN fck+2σX 0.17µX
Resistance factor θR LN 1 0.10µX
JCSS Probabilistic Model Code (JCSS 2001)
Load model
SAKO; Joint Committee of NKB and INSTA-B. NKB Report:
1999
Reliability based serviceability design
Expert review, BRE, 2003
Load model
Reference µX COV
Hendrickson et al.: 0.30Qk 0.60
Ellingwood: 0.45Qk 0.40
Reliability based serviceability design
Gulvanessian and Holicky 0.60Qk 0.35
Melchers, 1999 0.39Qk 0.60
The load intensity of the sustained live load may be represented by a stochastic process in two dimensions (random field) W(x, y) defined:
m is the overall mean for a particular user category V is a zero mean random variable and U(x, y) is a zero mean random field.
Load model
Reliability based serviceability design
Statistical parameters equivalently uniformly distributed load:
U(x, y) is a zero mean random field.
A is the influence area (i.e. the loaded area from which the considered load effect isinfluenced) and A0 is the so-called correlation area
Load model
Reliability based serviceability design
The variance of the stochastic field mostly depends on theinfluence area, where as the type of covariance function and κ (A) has minor importance.
Load model
Reliability based serviceability design
Madsen at al.
Load model
If it can be assumed that load changes occur as events of a Poisson process with rate λ the probability distribution function of the maximum load within a given reference period T is given by the exponential distribution.
where F (x) is the so-called random point in time probability distribution function of
Reliability based serviceability design
where FQ (x) is the so-called random point in time probability distribution function of the load (the probability distribution function of the maximum load in a reference period equal to 1/λ ).
The sustained live load is best represented by a Gamma or Gumbel distribution.
Load model
Although, transient live load events normally occur in the form of concentrated loads transient loads are usually represented in the probabilistic modelling in the form of a stochastic fields. Therefore the following moments for an equivalent uniformly distributed load Pequ due to transient loads may be derived as:
In JCSS it is suggested to use an exponential probability distribution function to
Reliability based serviceability design
In JCSS it is suggested to use an exponential probability distribution function to describe the transient load. The transient live loads may be described by a Poisson spike process with a mean occurrence rate equal to 1/ν and mean duration of dp
days.
Hence, the probabilitydistribution function for the maximum transient live load corresponding to a reference period T is given by:
Load model
Reliability based serviceability design
JCSS Probabilistic Model Code (JCSS 2001)
Load model
The total live load is the sum of the sustained live load and the transient live load. The maximum total live load corresponding to a reference period T can be assessed as the maximum of the following two loads:
Reliability based serviceability design
where Q,max L is the maximum sustained live load, LQ is the arbitrary point in time sustained live load, LP is the arbitrary point in time live load and LP,max is the maximum transient liveload. It can be assumed that the combined total live load has a Gumbel distribution function.
Load model
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
LS
A0
[m2]
mq [kN/m
2]
σv [kN/m
2] σu [kN/m
2]
1/λ [a]
mp [kN/m
2]
σp [kN/m
2]
1/ν [a]
dp [a]
20 0.5 0.3 0.6 5 0.2 0.4 0.3 1-3
Reliability based serviceability design
0 5 10 15 20 25 30 35 40 45 50
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1
LE
Annual max. load
LS = (0.5, 0.59)
LE = (0.39, 0.26)
L = (0.89, 0.64) Qk = 2.52 kN/m2
1Empirical CDF
1Empirical CDF
Load model
AT = 20 m2
COV: 0.71
Reliability based serviceability design
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
q1
F(x
)
sustained
extroardinary
total
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
q1 total
F(x
)
Empirical
Fitted Gamma
Fitted GEV
Lifetime max. load
LS = (1.66, 0.80)
LE = (1.11, 0.25)
L = (2.36, 0.80)
AT = 20 m2
Load model
1Empirical CDF
1Empirical CDF
COV: 0.34
Reliability based serviceability design
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
q50 total
F(x
)
Empirical
Fitted Gamma
Fitted GEV
0 1 2 3 4 5 6 70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
q50 total
F(x
)
sustained
extroardinary
total
0
0,2
0,4
0,6
0,8
1
1,2
0,1
0,6
1,1
1,6
2,1
2,6
3,1
3,6
4,1
Eq9
Eq8
Eq16
simul
ation
LE1
[ ]))(1)((exp)()(max
xFxFTxFxF LLLL −⋅−= ν
[ ]))(1(exp)()(max
xFTxFxF LLL −−= ν
[ ]))(1(exp)(max
xFTxF LL −−= ν
Load model
Reliability based serviceability design
LS50
0
0,2
0,4
0,6
0,8
1
1,2
0
0,5 1
1,5 2
2,5 3
3,5 4
4,5 5
5,5 6
6,5 7
7,5 8
8,5 9
9,5 10
Eq9
Eq8
Eq16
simul
ation
0
0,2
0,4
0,6
0,8
1
1,2
0,1
0,6
1,1
1,6
2,1
2,6
3,1
3,6
4,1
4,6
5,1
Eq9
Eq8
Eq16
simul
ation
LE50
0,6
0,8
1
1,2
Q1.0.98/Qk
Q50mean/Qk
αA (EC)
Office
Reliability based serviceability design
0
0,2
0,4
0 50 100 150 200 250
2
00
0 10;0.17
5mA
A
AA =≤+= ψα
AT [m2]
Annual max. load COV=0.92-0.46
Load model
Annual max. load COV=0.50
( )
−−−+= 5772.0))98.0ln(ln(
61
πµ XXk VQ kX Q44.0=µ
Hendrickson et al.: kX Q30.0=µ 6.0=COV
Ellingwood: kX Q45.0=µ 4.0=COV
Gulvanessian and Holicky: Q60.0=µ 35.0=COV
Reliability based serviceability design
Description X Dist. µX σX
Dead load G Normal Gk 0.10µX
Live load Q Gumbel 0.44Qk 0.50µX
Action effect θE LN 1 0.10µX
JCSS Probabilistic Model Code (JCSS 2001)
Gulvanessian and Holicky: kX Q60.0=µ 35.0=COV
Melchers:kX Q39.0=µ 60.0=COV
2,00
2,50
3,00
3,50
0,7
β
ResultsSteel - remaining total deflection, wmax
Reliability based serviceability design
0,00
0,50
1,00
1,50
2,00
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0,7
0,5
0,3
kk
k
QG
Q
+=χ
2,00
2,50
3,00
3,50
β
ResultsSteel - remaining total deflection, wmax
Reliability based serviceability design
0,00
0,50
1,00
1,50
2,00
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
code
Ltot
Ls+Le
kk
k
QG
Q
+=χ
ResultsSteel - remaining total deflection, wmax
2.50
3.00
3.50
β
Reliability based serviceability design
-0.50
0.00
0.50
1.00
1.50
2.00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
char
freq
qp
χ
kk
k
QG
Q
+=χ
ResultsConcrete - remaining total deflection, wmax
2.5
3.0
3.5
β
Reliability based serviceability design
0.0
0.5
1.0
1.5
2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
char
freq (st)
freq (lt)
qp
χkk
k
QG
Q
+=χ
ResultsTimber
2.50
3.00
3.50
β
Reliability based serviceability design
0.00
0.50
1.00
1.50
2.00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
uinst
ufin
unet,fin
u2,fin
χkk
k
QG
Q
+=χ
Conclusions
• Modelling of live loads in not an easy task.
• The reliability in SLS is not consistent and is below the target value (β=2.9) given by the code.
• The characteristics of the curves are similar for the 3 different materials.
• Low values of variable loads: very low reliability.
• The reliability index for different load combinations is different but
Reliability based serviceability design
• The reliability index for different load combinations is different but the target value is only given for irreversible limit states.
• The deflection limit itself does not influence the reliability of the SLS. But influences the design!
• The origin of these values is not clear and they are not harmonized among the different countries, therefore an intensive investigation on this topic is essential.
Thank you for your attention!
Reliability based serviceability design
Thank you for your attention!