15.12.2010 Daniel Honfi - Konstruktionsteknik considering the vertical deflections of structural members subjected to bending made of different materials using ... Ellingwood: 0.45Q

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


• 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


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


Range of Reliability Indices for Incremental Deflection Criteria

Li et al., 2005

Probability of serviceability failure due to deflection for different limits

Concrete beams


Li et al., 1994

Failure probability for different COV of creep

Timber beams

Philpot et al. 1993


Β-Φc Plots for Selecting Resistance Factor

Timber beams

Leichti and Tang, 1989


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


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):


∑∑≥≥

+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)


• 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.


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


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)


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]


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)


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.


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:


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):


1.5

2

uinst,lim

ufin,lim

hSLS/hULS


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 Robustness

[ ] )(0)( β−Φ=<= XgPPfβ: reliability index

Target reliabilities

Target reliability index β for Class RC2

(EN1990:2002 C6)


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,)( −==


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


JCSS Probabilistic Model Code (JCSS 2001)


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


Timber

Material model


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


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



Load model

SAKO; Joint Committee of NKB and INSTA-B. NKB Report:

1999


Expert review, BRE, 2003

Load model

Reference µX COV

Hendrickson et al.: 0.30Qk 0.60

Ellingwood: 0.45Qk 0.40


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


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


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


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


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


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



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:


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


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


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


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


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


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



Dead load G Normal Gk 0.10µX

Live load Q Gumbel 0.44Qk 0.50µX

Action effect θE LN 1 0.10µX


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


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



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


2.50

3.00

3.50

β


-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


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


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


• 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!


Thank you for your attention!

Documents

15.12.2010 Daniel Honfi - Konstruktionsteknik considering the vertical deflections of structural members subjected to bending made of different materials using ... Ellingwood: 0.45Q