20100315104024541

8/20/2019 20100315104024541

1/44

Chapter 7Chapter 7

Reliability-Based Design Methods of

Structures

8/20/2019 20100315104024541

2/44

Chapter 7: Reliability-Based Design Methods of StructuresChapter 7: Reliability-Based Design Methods of Structures

7.2 Reliability-Based Design Formulas

7.5 Practical RFD Formulas in !urrent !odes

7." Reliability-Based Design !odes

Contents

7.# !alibration for Deterministic !odes

7.$ %arget Reliability &nde' in !hinese !odes

8/20/2019 20100315104024541

3/44

7." Reliability-Based Design !odes

Chapter 7 Chapter 7

Reliability-Based Design Methods ofReliability-Based Design Methods of

StructuresStructures

8/20/2019 20100315104024541

4/44

7.1 Reliability-Based Design Codes …1

7."." Role of a !ode in the Building Process

– The building process includes planning, design, manufacturing ofmaterials, transportation, construction, operation/use, and demolition.

– The role of a design code is to establish the requirements needed to

ensure an acceptable level of reliability for a structure.

–

The central role of a code is diagrammed in the following figure:

()ner

Designer

!ontractor

*ser!ode

8/20/2019 20100315104024541

5/44

7.1 Reliability-Based Design Codes …

7.".2 !ode e+els

– Level Codes Ⅰ : Use deterministic formulas

( )k k G Q k

K S S R+ ≤

– Level Codes Ⅱ : Use approimate probability limit state design formula

– Level Codes Ⅲ : Use full probability analysis and design formula

– Level Codes Ⅳ : Use the total epected life cycle cost of the design as

the optimi!ation criterion

8/20/2019 20100315104024541

6/44

7.1 Reliability-Based Design Codes …

7.".# Reliability-Based Design !odes

,eneral Princiles on Reliability for Structures

&S(2#/$0 "//1

". &nternational Standard

2. !hinese !odes

*nified Standard for Reliability Design of3ngineering Structures

,B54"5# /2

8/20/2019 20100315104024541

7/44

7.1 Reliability-Based Design Codes …!

". *nified Standard for Reliability Design of

Building Structures ,B54461 244"

2. *nified Standard for Reliability Design ofigh)ay 3ngineering Structures ,B8%5421# "///

#. *nified Standard for Reliability Design ofRail)ay 3ngineering Structures ,B542"6 /$

$. *nified Standard for Reliability Design ofydraulic 3ngineering Structures ,B54""/ /$

5. *nified Standard for Reliability Design ofarbor 3ngineering Structures ,B54"51 /2

8/20/2019 20100315104024541

8/44

7.2 Reliability-Based Design Formulas

Chapter 7 Chapter 7



8/20/2019 20100315104024541

9/44

7. Reliability-Based Design "or#ulas …1

7.2." Formulas of Reliability !hec9ing

– There are three "inds of reliability chec"ing formulas:

[ ] s P where ,target failure probability , or target reliability inde .

[ ] f P [ ]β , , are called target safety probability ,

s s P P ≥[ ]

f f P P ≤[ ]

β β ≥[ ]

: : : : :"

: : : : :2

: : : : :#

– The third formula is generally used in practical engineering.

Given: the probability distribution and digital characteristic

of the loads and resistance

Find: design vector

Subjected to: ( )

x

xβ β

≥[ ]

8/20/2019 20100315104024541

10/44

7. Reliability-Based Design "or#ulas …

S µ where

,

is the mean value of load effect

is the mean value of resistance R µ

is the central safety factor 0 K

7.2.2 Single Factor Design Formulas

– The single factor formula based on mean values is as following:

– # $ % are normal distributions

0 S R K µ µ ≤

2 2 2 2

0 2 2 ( )

R S R

R

K β δ δ β δ β δ

+ + −=−

2 2

0 e!p( ) R S K β δ δ = + – # $ % are lognormal distributions

8/20/2019 20100315104024541

11/44


k S where

,

is the characteristic value of load effect

is the characteristic value of resistancek R

is the characteristic safety factor K

7.2.2 Single Factor Design Formulas

– The single factor formula based on characteristic values is as following:

k k KS R≤

0

R R

S S

k K K

k

δ

δ −=+

( )k R R R R k µ δ = −( )k S S S S k µ δ = +

8/20/2019 20100315104024541

12/44

7. Reliability-Based Design "or#ulas …!

0 S

Fre"uency

S , load effect

S µ nS nS γ

&ean load

'ominal load

(actored load

Relationshis among nominal load; mean load; and factored load

8/20/2019 20100315104024541

13/44

7. Reliability-Based Design "or#ulas …$

Relationshis among nominal resistance; mean resistance; and

factored resistance

0 R

Fre"uency

R , #esistance

R µ n Rn Rγ

&ean resistance

'ominal resistance

(actored resistance

8/20/2019 20100315104024541

14/44


niS where

,

is the nominal )design* value of load effect component,

is the load partial factor for load component,Siγ

is the nominal )design* value of resistance or capacity,n R

is the resistance partial factor . Rγ

7.2.# Multile Factor Design Formulas

oad and Resistance Factor Design; RFD

– The L#(+ formula is as following:

ni n

RS Rγ γ ≤∑

Si

(actored nominal resistance≤Total factored nominal load effect

8/20/2019 20100315104024541

15/44


7.2.# Multile Factor Design Formulas

oad and Resistance Factor Design; RFD

# # #

2( $ $ $ ) 0n g X X X =L

– The partial safety factors and must be calibrated based on the

target inde adopted by the code. Rγ Siγ

#

S S

S k S S

S

S k

α βδ γ

δ

+= =

+

# ( )S S S S S S S µ α βσ µ α βδ = + = +

( )k S S S

S k µ δ = +

#

k R R

R R R

R k

R

δ γ

α βδ

−= =

+

# ( ) R R R R R R R µ α βσ µ α βδ = + = +

( )k R R R

R k µ δ = −

8/20/2019 20100315104024541

16/44

7.# !alibration for Deterministic !odes

Chapter 7 Chapter 7



8/20/2019 20100315104024541

17/44

7.! Calibration for Deter#inistic Codes …1

7.#." !alibration of %arget Reliability &nde'

". Basic Princiles

( ) 0k Gk Qk R K S S − + =

where, safety factor, K

Consider a structural member which carry a dead load and a variant load.

-ccording to the original deterministic structural design code , the design

formula of ultimate limit state design for this member can be stated as

follows:

k R characteristic value of member resistance ,

Gk S , characteristic value of permanent load effect and variant load effect designed according to the

deterministic code .

Qk S

8/20/2019 20100315104024541

18/44

7.! Calibration for Deter#inistic Codes …

0G Q

R S S − − =

'ow, the problem can be reformulated as follows:

ow much is the reliability implicit in the original deterministic structural

design code )Level Code*0 Ⅰ

– 1hen the calibration method is used, the limit state equation in simple

load combination condition can be formulated as:

where, structural member resistance, R

GS dead load effect,

QS live load effect.

– 2t is assumed that the parameters and the probability distribution types of

the three basic random variables are "nown.

– The calibration method can be implemented by the (3#& method, for

eample, 4C &ethod.

8/20/2019 20100315104024541

19/44

7.! Calibration for Deter#inistic Codes …!

– 2t is assumed that the following parameters of the basic random

variables are "nown:

$ $ QG

G Q

S S R R S S

k Gk Qk R S S µ µ µ λ λ λ = = =bias factor:

$ $ QGG Q

G Q

S S R R S S

R S S

V V V σ σ σ

µ µ µ

= = =variation factor:

– 2t is assumed that is linearly related with and .k R Gk S Qk S

LetQk

Gk

S

S

ρ =

then ( ) ( )

( )

k Gk Qk Gk Gk

Gk

R K S S K S S

K S

ρ

ρ

= + = +

= +

, is called load effect ratio, ρ

8/20/2019 20100315104024541

20/44

7.! Calibration for Deter#inistic Codes …$

" -ssume one value of the load effect ratio 5 ρ

k R

2. !alculation Procedure

2 +etermine the characteristic value of member resistance :

# +etermine the mean values and standard deviations of the

basic variables :$ $

G G Q Q R R k S S Gk S S Qk R S S µ λ µ λ µ λ = = =mean values:

$ $G G G Q Q Q R R R S S S S S S

V V V σ µ σ µ σ µ = = =standard deviations:

( )k Gk R K S ρ = +

$ +etermine the limit state equation:

0G Q R S S − − =

5 %olve the reliability inde by the 4C method.β

6 -d6ust the load effect ratio, calculate the mean value of different

reliability indees.

8/20/2019 20100315104024541

21/44

7.! Calibration for Deter#inistic Codes …%

3'amle 7."

Consider a #C aial compression short column carrying a dead load and an

office live load, the column was designed according to the old 7+esign Code

of Concrete %tructures )T489*;.

-ssume that the following parameters are "nown:

-ssume that the ratio of live load to dead load ,

Try to calibrate the reliability inde of the ultimate limit state in T489 code.

% &0Qk Gk S S ρ = =

&'' Rλ = R is lognormal 0& RV =

&0GS λ =GS is normal 0&0GS V =

0&0 LS

λ = LS is

8/20/2019 20100315104024541

22/44

7.! Calibration for Deter#inistic Codes …&

" +etermine

&0 ρ =

%olution

k R

% 0 % 0k k G L

S S ρ = = =

( ) &++ (0 0) 'k Gk Qk R K S S = + = × + =2 +etermine the means and standard deviations

0&,2G G GS S S

V σ µ = =

&'' ' ,&2' R R k R µ λ = = × =

0& ,&2' &00* R R R

V σ λ = = × =

0&G GS S Gk

S µ λ = =

2&0' L L LS S S

V σ µ = =

&0 L LS S Lk

S µ λ = =

8/20/2019 20100315104024541

23/44


# +etermine the ultimate limit state equation

0G L R S S − − =$ +etermine the reliability inde by the 4C method

%he solution rocess of

%he solution result is 0 '&-0-2β =&f the load effect ratio ; then2&0 ρ = '&+-2-β =

=lease refer to the reference boo" 7 #eliability of

%tructures; by =rofessors 3u and +uan.

Turn to =age 89, loo" at the table >.? carefully@

8/20/2019 20100315104024541

24/44

7.! Calibration for Deter#inistic Codes …'

7.#.2 !alibration of Partial Factors

". Basic Princiles– The partial factors in the L#(+ format must be calibrated based on the

target reliability inde adopted by the code.

– 2n determining partial factors, the problem is reversed compared with

reliability analysis contet introduced in Chapter?.

#eliability analysis

i X µ

i X V Anown: ,

(ind: ,β

=artial factor calibration

i X µ

Anown: ,

(ind: ,

[ ]β β =i X

V

#

i

di i

X ri ri

X x

X X γ = =

#

i x#

i x

8/20/2019 20100315104024541

25/44

7.! Calibration for Deter#inistic Codes …(

2. &teration =lgorithm

" (ormulate the limit state function and design equation.

+etermine the probability distributions and appropriate parameters for basic variables.

There can be at most only two un"nown mean values needed tosolve. 3ne is , the other corresponds a variant load effect .

Load effect ratios are used to relate the means of the load effects.

R µ iS

µ

#

i x2 3btain an initial design point by assuming mean values.

(or the first iteration, we can use the limit state equationevaluated at the mean values to get a relationship between the twoun"nown means.

0 Z =

i

e

X µ # (or each of the design point values corresponding to a non

normal distribution, determine the equivalent normal mean

and standard deviation by using equivalent normali!ation.

#i x

i

e

X σ

i i

e

X X µ µ =

i i

e

X X σ σ =

8/20/2019 20100315104024541

26/44

7.! Calibration for Deter#inistic Codes …1)

#

i x5 Calculate the n values of design point# [ ]

i ii X i X x µ α β σ = + ( $ 2$ $ )i n= L

6 Update the relationship between the two un"nown mean values by

solving the limit state function.# # #

2

( $ $ $ ) 0n

g x x x =L

7 #epeat %teps ?B until converge.. /iα

iα $ Calculate the n values of direction cosine

#

#

2

i

i

X i P

i

n

X

i i P

g

X

g

X

σ

α

σ

=

∂− ×

∂= ∂

× ÷ ÷∂ ∑

( $2$ $ )i n= L

1 3nce convergence is achieved, calculate the partial factors.# %

i X i ri x X γ =

8/20/2019 20100315104024541

27/44


3'amle 7.2

=lease refer to the tetboo"

7 #eliability of %tructures;

by =rofessor -. %. 'owa".

Turn to =age ?D, loo" at the eample E.D carefully@

0& R

V = 0&2Q

V = [ ] '&0β =

Z R Q= −

R R Q Qγ µ γ µ ≥

%he limit state function:

%he design equation:

>no)n arameters:

Probability information: # and F are all normal and uncorrelated.

8/20/2019 20100315104024541

28/44


%olution

2teration cycle D

" -ssume iteration initial values

#

Rr µ =#

Qq µ =

# # 0r q− = R Q µ µ =

2 Calculate direction cosine

#

0& R R R R R Q P

Z G V

Rσ σ µ µ

∂= − × = − = − = −

∂

#

0&2Q Q Q Q Q Q P

Z G V Q σ σ µ µ

∂= − × = + = =∂

2 20&,02 R R

R S

G

G Gα = = −

+ 2 20&-2

Q

Q

R S

G

G Gα = =

+

8/20/2019 20100315104024541

29/44

7.! Calibration for Deter#inistic Codes …1!

# Calculate design points

# # 0r q− = &+-0 R Q µ µ =

" Calculate direction cosine

2 20&*, R R

R S

G

G Gα = = −

+ 2 20&0,-

Q

Q

R S

G

G Gα = =

+

# [ ] 0&,02 '&0 0& 0&-0* R R R R R R

r µ α β σ µ µ µ = + = − × × =# [ ] 0&-2 '&0 0&2 &2Q Q Q Q Q Qq µ α β σ µ µ µ = + = + × × =

$ Update the relationship between the two un"nown means

2teration cycle

0& 0&+-0 R R QG µ µ = − = −

0&2Q QG µ =

8/20/2019 20100315104024541

30/44

7.! Calibration for Deter#inistic Codes …1$

2 Calculate design points

# # 0r q− = &+*** R Q µ µ =


2 20&-000 R R

R S

G

G Gα = = −

+ 2 20&000

Q

Q

R S

G

G Gα = =

+

# [ ] 0&*, '&0 0& 0& R R R R R R

r µ α β σ µ µ µ = + = − × × =# [ ] 0&0,- '&0 0&2 &2Q Q Q Q Q Qq µ α β σ µ µ µ = + = + × × =

# Update the relationship between the two un"nown means

2teration cycle ?

0& 0&+*** R R QG µ µ = − = −

0&2Q QG µ =

8/20/2019 20100315104024541

31/44

7.! Calibration for Deter#inistic Codes …1%

2 Calculate design points

# # 0r q− = &000 R Q µ µ =


2 20&-000 R R

R S

G

G Gα = = −

+ 2 20&000

Q

Q

R S

G

G Gα = =

+

# [ ] 0&- '&0 0& 0&00 R R R R R R

r µ α β σ µ µ µ = + = − × × =# [ ] 0& '&0 0&2 &20Q Q Q Q Q Qq µ α β σ µ µ µ = + = + × × =

# Update the relationship between the two un"nown means

2teration cycle

0& 0&00 R R QG µ µ = − = − 0&2Q QG µ =

have converge. The iteration stop.. /iα

8/20/2019 20100315104024541

32/44

7.! Calibration for Deter#inistic Codes …1&

-ssuming the mean values are the nominal design values, then the partial factors are :

Rα

?umbers of&teration

" 2 # $

-4.6$42 -4.7/6$ -4.1444 -4.1444

4.7612 4.64$1 4.6444 4.6444Qα

Table 9.D Convergence process for

8/20/2019 20100315104024541

33/44

7.$ %arget Reliability &nde' in !hinese !odes

Chapter 7 Chapter 7



8/20/2019 20100315104024541

34/44

7.$ *arget Reliability +nde, in Chinese Codes …1

7.$." Safety !lass of Building Structures

– -ccording to the importance and the consequences of structural damage,the safety class of buildings in Unified %tandard for #eliability +esign of

Guilding %tructures )HG>IIBE IID* is divided into three categories.

Safety

!lass

!onse@uences ofDamage

%yes of

Buildings

&mortancefactor

!lass one Aery se+ere &mortant buildings "."

!lass t)o Se+ere !ommon buildings ".4

!lass three not se+ere *nimortant buildings 4./

– The safety class is considered through the importance factor 0γ

0γ

Table 9. %afety class of building structures

8/20/2019 20100315104024541

35/44

7.$ *arget Reliability +nde, in Chinese Codes …

7.$.2 %arget Reliability &nde' for *ltimate imit State

%yes of damageSafety class

!lass one !lass t)o !lass three

Ductile #.7 #.2 2.7

Brittle $.2 #.7 #.2

Table 9.? Target reliability inde for UL% of structural member [ ]β

8/20/2019 20100315104024541

36/44

7.$ *arget Reliability +nde, in Chinese Codes …!

7.$.# %arget Reliability &nde' for Ser+iceability imit State

2rreversible Limit %tate

#eversible Limit %tate

Table 9. Target reliability inde for %L% of structural member [ ]β

&+≥

0≥

?. ow are these target reliability indees determined 0

. 1hy are the target reliability indees for ultimate limitstate and serviceability limit state different 0

D. 1hat are the rules of target reliability indees 0

8/20/2019 20100315104024541

37/44

7.5 Practical RFD Formulas in !urrent !odes

Chapter 7 Chapter 7



7 % ti l R"D " l i C t C d 1

8/20/2019 20100315104024541

38/44

7.% ractical R"D "or#ulas in Current Codes …1

7.5." *ltimate imit State Design Formulas

where, structural importance factor,0

γ

Gγ partial factor for dead load,

Q

γ , partial factors for the Dst and i th variant load,iQ

γ

0

2

( ) ( $ $ )i i

n

G Gk Q Q k Q ci Q k k k k

i

S S S R f aγ γ γ γ ϕ γ =

+ + ∑ LR

1≤

0

( ) ( $ $ )i i

n

G Gk Q ci Q k k k k

i

S S R f aγ γ γ ϕ

γ =

+ ∑ LR

1≤

Gk S effect of permanent load characteristic value

Q k S effect of variant load characteristic value which

dominates the load effect combination.

7 % ti l R"D " l i C t C d

8/20/2019 20100315104024541

39/44

7.% ractical R"D "or#ulas in Current Codes …

iQ k S effect of the i th variant load characteristic value

icϕ combination factor of the i th variant load

( ) R × function of structural member

Rγ partial factor for structural member resistance,

k f characteristic value of material behavior,

k a characteristic value of geometric parameter.

– The second formula is mainly used in the structures, which is dominated

by permanent load. The most unfavorable one of the above two formulas

should be used in practical design situations.

– The partial factors in the above two formulas are determined by the

principles introduced in this course and optimi!ation method. Jou can

refer to the =.8EDID in the reference boo".

7 % ti l R"D " l i C t C d !

8/20/2019 20100315104024541

40/44

7.% ractical R"D "or#ulas in Current Codes …!

7.5.2 Ser+iceability imit State Design Formulas

2

[ ]i

n

Gk Q k ci Q k

i

S S S f ϕ =

+ + ∑ ≤

". Design Formula for !haracteristic Aalues

2

2

[ ]i

n

Gk f Q k qi Q k

i

S S S f ϕ ϕ =

+ + ∑ ≤2. Design Formula for Fre@uent Aalues

'

[ ]i

n

Gk qi Q k

i

S S f ϕ =

+ ∑ ≤#. Design Formula for uasi-Permanent Aalues

7 % ti l R"D " l i C t C d $

8/20/2019 20100315104024541

41/44

7.% ractical R"D "or#ulas in Current Codes …$

where, f Q k S ϕ effect of a variant load frequent value which

dominates the frequent load combination.

iqi Q k S ϕ effect of quasipermanent value of a variant load.

[ ] f the deformation or crac" limit value corresponding

to characteristic value combination.

2[ ] f the deformation or crac" limit value corresponding

to frequent value combination.

'[ ] f the deformation or crac" limit value correspondingto quasipermanent value combination.

Ch t 7 / 7

8/20/2019 20100315104024541

42/44

ome)or9 7

=rogramming the above algorithms in &-TL-G

environment according to the iteration algorithm proposed

by this course.

)D* Gy using your own subroutine, rechec" the eample 9.

in this course.

)* Gy using your own subroutine, recalculate the eample

E.? in the tet boo" on =.?D

Chapter 7: /o#e0or 7

8/20/2019 20100315104024541

43/44

8/20/2019 20100315104024541

44/44

Documents

20100315104024541