Chapter 7 Reliability-Based Design Methods of Structures.
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Transcript of Chapter 7 Reliability-Based Design Methods of Structures.
Chapter 7Chapter 7
Reliability-Based Design Methods of
Structures
Chapter 7: Reliability-Based Design Methods of StructuresChapter 7: Reliability-Based Design Methods of Structures
7.2 Reliability-Based Design Formulas
7.5 Practical LRFD Formulas in Current Codes
7.1 Reliability-Based Design Codes
Contents
7.3 Calibration for Deterministic Codes
7.4 Target Reliability Index in Chinese Codes
7.1 Reliability-Based Design Codes
Chapter 7Chapter 7 Reliability-Based Design Methods of Reliability-Based Design Methods of StructuresStructures
7.1 Reliability-Based Design Codes …1
7.1.1 Role of a Code in the Building Process
– The building process includes planning, design, manufacturing of materials, 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:
Owner
Designer
Contractor
UserCode
7.1 Reliability-Based Design Codes …2
7.1.2 Code Levels
– Level CodesⅠ : Use deterministic formulas
( )k kG Q kK S S R ≤
– Level CodesⅡ : Use approximate probability limit state design formula
– Level CodesⅢ : Use full probability analysis and design formula
– Level CodesⅣ : Use the total expected life cycle cost of the design as the optimization criterion
7.1 Reliability-Based Design Codes …2
7.1.3 Reliability-Based Design Codes
General Principles on Reliability for Structures (ISO2394: 1998)
1. International Standard
2. Chinese Codes
Unified Standard for Reliability Design of Engineering Structures
(GB50153 — 92)
7.1 Reliability-Based Design Codes …3
1. Unified Standard for Reliability Design of Building Structures (GB50068 — 2001)
2. Unified Standard for Reliability Design of Highway Engineering Structures (GB/T50283 — 1999)
3. Unified Standard for Reliability Design of Railway Engineering Structures (GB50216 — 94)
4. Unified Standard for Reliability Design of Hydraulic Engineering Structures (GB50119 — 94)
5. Unified Standard for Reliability Design of Harbor Engineering Structures (GB50158 — 92)
7.2 Reliability-Based Design Formulas
Chapter 7Chapter 7 Reliability-Based Design Methods of Reliability-Based Design Methods of StructuresStructures
7.2 Reliability-Based Design Formulas …1
7.2.1 Formulas of Reliability Checking
– There are three kinds of reliability checking formulas:
[ ]sPwhere,target failure probability, or target reliability index.
[ ]fP [ ], , are called target safety probability,
s sP P≥[ ]
f fP P≤[ ]
≥[ ]
… … … … …(1)
… … … … …(2)
… … … … …(3)
– 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
x
≥[ ]
7.2 Reliability-Based Design Formulas …2
Swhere,
is the mean value of load effect
is the mean value of resistanceRis the central safety factor0K
7.2.2 Single Factor Design Formulas
– The single factor formula based on mean values is as following:
– R & S are normal distributions
0 S RK
2 2 2 2
0 2 2
1 (1 )
1R S R
R
K
2 20 exp( )R SK – R & S are lognormal distributions
7.2 Reliability-Based Design Formulas …2
kSwhere,
is the characteristic value of load effect
is the characteristic value of resistancekR
is the characteristic safety factorK
7.2.2 Single Factor Design Formulas
– The single factor formula based on characteristic values is as following:
k kKS R
0
1
1R R
S S
kK K
k
(1 )k R R RR k
(1 )k S S SS k
7.2 Reliability-Based Design Formulas …3
0S
Frequency
S , load effect
S nS nSMean load
Nominal load
Factored load
Relationships among nominal load, mean load, and factored load
7.2 Reliability-Based Design Formulas …4
Relationships among nominal resistance, mean resistance, and factored resistance
0R
Frequency
R , Resistance
RnRnRMean resistance
Nominal resistance
Factored resistance
7.2 Reliability-Based Design Formulas …2
niSwhere,
is the nominal (design) value of load effect component,
is the load partial factor for load component,Siis the nominal (design) value of resistance or capacity,nR
is the resistance partial factor.R
7.2.3 Multiple Factor Design Formulas
(Load and Resistance Factor Design, LRFD)
– The LRFD formula is as following:
1ni n
R
S R
Si
Factored nominal resistance Total factored nominal load effect
7.2 Reliability-Based Design Formulas …2
7.2.3 Multiple Factor Design Formulas
(Load and Resistance Factor Design, LRFD)
* * *1 2( , , , ) 0ng X X X
– The partial safety factors and must be calibrated based on the target index adopted by the code.
R Si
* 1
1S S
Sk S S
S
S k
* (1 )S S S S S SS
(1 )k S S SS k
*
1
1k R R
RR R
R k
R
* (1 )R R R R R RR
(1 )k R R RR k
7.3 Calibration for Deterministic Codes
Chapter 7Chapter 7 Reliability-Based Design Methods of Reliability-Based Design Methods of StructuresStructures
7.3 Calibration for Deterministic Codes …1
7.3.1 Calibration of Target Reliability Index
1. Basic Principles
( ) 0k Gk QkR K S S
where, — safety factor,K
Consider a structural member which carry a dead load and a variant load.
According to the original deterministic structural design code, the design formula of ultimate limit state design for this member can be stated as follows:
kR — characteristic value of member resistance ,
GkS , — characteristic value of permanent load effect and
variant load effect designed according to the
deterministic code .
QkS
7.3 Calibration for Deterministic Codes …2
0G QR S S
Now, the problem can be re-formulated as follows:
How much is the reliability implicit in the original deterministic structural design code (Level Code)?Ⅰ
– When 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.
– It is assumed that the parameters and the probability distribution types of the three basic random variables are known.
– The calibration method can be implemented by the FORM method, for example, JC Method.
7.3 Calibration for Deterministic Codes …3
– It is assumed that the following parameters of the basic random variables are known:
, , QG
G Q
SSRR S S
k Gk QkR S S
bias factor:
, , QG
G Q
G Q
SSRR S S
R S S
V V V
variation factor:
– It is assumed that is linearly related with and .kR GkS QkS
Let Qk
Gk
S
S
then ( ) ( )
(1 )
k Gk Qk Gk Gk
Gk
R K S S K S S
K S
, is called load effect ratio,
7.3 Calibration for Deterministic Codes …4
(1) Assume one value of the load effect ratio ;
kR
2. Calculation Procedure
(2) Determine the characteristic value of member resistance :
(3) Determine the mean values and standard deviations of the basic variables :
, ,G G Q QR R k S S Gk S S QkR S S mean values:
, ,G G G Q Q QR R R S S S S S SV V V standard deviations:
(1 )k GkR K S
(4) Determine the limit state equation:
0G QR S S (5) Solve the reliability index by the JC method. (6) Adjust the load effect ratio, calculate the mean value of different reliability indexes.
7.3 Calibration for Deterministic Codes …5
Example 7.1
Consider a RC axial compression short column carrying a dead load and an office live load, the column was designed according to the old “Design Code of Concrete Structures (TJ9-74)”.
Assume that the following parameters are known:
Assume that the ratio of live load to dead load ,
Try to calibrate the reliability index of the ultimate limit state in TJ9-74 code.
/ 1.0Qk GkS S
1.33R R is lognormal 0.17RV
1.06GS
GS is normal 0.07GS
V
0.70LS
LS is Extreme Ⅰ 0.29LS
V
1.55K 10LkS kN m
7.3 Calibration for Deterministic Codes …6
(1) Determine
1.0
Solution
kR
/ 10 /1 10k kG LS S
( ) 1.55 (10 10) 31k Gk QkR K S S
(2) Determine the means and standard deviations
0.742G G GS S SV
1.33 31 41.23R R kR
0.17 41.23 7.009R R RV
10.6G GS S GkS
2.03L L LS S SV
7.0L LS S LkS
7.3 Calibration for Deterministic Codes …7
(3) Determine the ultimate limit state equation
0G LR S S
(4) Determine the reliability index by the JC method
The solution process of JC method is omitted.
The solution result is : 3.8082
If the load effect ratio , then2.0 3.5828
Please refer to the reference book “Reliability of Structures” by Professors Ou and Duan.
Turn to Page 97, look at the table 5.3 carefully!
7.3 Calibration for Deterministic Codes …8
7.3.2 Calibration of Partial Factors
1. Basic Principles– The partial factors in the LRFD format must be calibrated based on the
target reliability index adopted by the code.
– In determining partial factors, the problem is reversed compared with reliability analysis context introduced in Chapter3.
Reliability analysis
iX
iXVKnown: ,
Find: ,
Partial factor calibration
iX
Known: ,
Find: ,
[ ] iX
V
*
i
di iX
ri ri
X x
X X
*ix
*ix
7.3 Calibration for Deterministic Codes …9
2. Iteration Algorithm
(1) Formulate the limit state function and design equation. Determine the probability distributions and appropriate parameters
for basic variables.
There can be at most only two unknown mean values needed to solve. One is , the other corresponds a variant load effect . Load effect ratios are used to relate the means of the load effects.
RiS
*ix(2) Obtain an initial design point by assuming mean values.
For the first iteration, we can use the limit state equation evaluated at the mean values to get a relationship between the two unknown means.
0Z
i
eX
(3) For each of the design point values corresponding to a non- normal distribution, determine the equivalent normal mean and standard deviation by using equivalent normalization.
*ix
i
eX
i i
eX X
i i
eX X
7.3 Calibration for Deterministic Codes …10
*ix(5) Calculate the n values of design point
* [ ]i ii X i Xx ( 1,2, , )i n
(6) Update the relationship between the two unknown mean values by solving the limit state function.
* * *1 2( , , , ) 0ng x x x
(7) Repeat Steps 3-6 until converge.{ }i
i(4) Calculate the n values of direction cosine
*
*
2
1
i
i
Xi P
in
Xi i P
gX
gX
( 1,2, , )i n
(8) Once convergence is achieved, calculate the partial factors.* /
iX i rix X
7.3 Calibration for Deterministic Codes …11
Example 7.2
Please refer to the textbook “Reliability of Structures” by Professor A. S. Nowak.
Turn to Page 231, look at the example 8.1 carefully!
0.1RV 0.12QV [ ] 3.0
Z R Q
R R Q Q ≥The limit state function:
The design equation:
Known parameters:
Probability information: R and Q are all normal and uncorrelated.
7.3 Calibration for Deterministic Codes …12
SolutionIteration cycle 1
(1) Assume iteration initial values*
Rr *Qq
* * 0r q R Q
(2) Calculate direction cosine
*
0.1R R R R R QP
ZG V
R
*
0.12Q Q Q Q Q Q
P
ZG V
Q
2 20.6402R
R
R S
G
G G
2 20.7682Q
Q
R S
G
G G
7.3 Calibration for Deterministic Codes …13
(3) Calculate design points
* * 0r q 1.5801R Q
(1) Calculate direction cosine
2 20.7964R
R
R S
G
G G
2 20.6048Q
Q
R S
G
G G
* [ ] 0.6402 3.0 0.1 0.8079R R R R R Rr * [ ] 0.7682 3.0 0.12 1.2766Q Q Q Q Q Qq
(4) Update the relationship between the two unknown means
Iteration cycle 2
0.1 0.15801R R QG
0.12Q QG
7.3 Calibration for Deterministic Codes …14
(2) Calculate design points
* * 0r q 1.5999R Q
(1) Calculate direction cosine
2 20.8000R
R
R S
G
G G
2 20.6000Q
Q
R S
G
G G
* [ ] 0.7964 3.0 0.1 0.7611R R R R R Rr * [ ] 0.6048 3.0 0.12 1.2177Q Q Q Q Q Qq
(3) Update the relationship between the two unknown means
Iteration cycle 3
0.1 0.15999R R QG
0.12Q QG
7.3 Calibration for Deterministic Codes …15
(2) Calculate design points
* * 0r q 1.6000R Q
(1) Calculate direction cosine
2 20.8000R
R
R S
G
G G
2 20.6000Q
Q
R S
G
G G
* [ ] 0.8 3.0 0.1 0.7600R R R R R Rr * [ ] 0.6 3.0 0.12 1.2160Q Q Q Q Q Qq
(3) Update the relationship between the two unknown means
Iteration cycle 4
0.1 0.1600R R QG 0.12Q QG
have converge. The iteration stop.{ }i
7.3 Calibration for Deterministic Codes …16
Assuming the mean values are the nominal design values, then the partial factors are :
R
Numbers of Iteration
1 2 3 4
-0.6402 -0.7964 -0.8000 -0.8000
0.7682 0.6048 0.6000 0.6000Q
Table 7.1 Convergence process for Example 7.2
*
0.7600RR
r
*
1.22QQ
q
7.4 Target Reliability Index in Chinese Codes
Chapter 7Chapter 7 Reliability-Based Design Methods of Reliability-Based Design Methods of StructuresStructures
7.4 Target Reliability Index in Chinese Codes …1
7.4.1 Safety Class of Building Structures
– According to the importance and the consequences of structural damage, the safety class of buildings in Unified Standard for Reliability Design of Building Structures (GB50068 — 2001) is divided into three categories.
Safety
Class
Consequences of Damage
Types of
Buildings
Importance factor
Class one Very severe Important buildings 1.1
Class two Severe Common buildings 1.0
Class three not severe Unimportant buildings 0.9
– The safety class is considered through the importance factor 0
0
Table 7.2 Safety class of building structures
7.4 Target Reliability Index in Chinese Codes …2
7.4.2 Target Reliability Index for Ultimate Limit State
Types of
damage
Safety class
Class one Class two Class three
Ductile 3.7 3.2 2.7
Brittle 4.2 3.7 3.2
Table 7.3 Target reliability index for ULS of structural member[ ]
7.4 Target Reliability Index in Chinese Codes …3
7.4.3 Target Reliability Index for Serviceability Limit State
Irreversible Limit State
Reversible Limit State
Table 7.4 Target reliability index for SLS of structural member[ ]
1.5≥
0≥
3. How are these target reliability indexes determined ?
2. Why are the target reliability indexes for ultimate limit state and serviceability limit state different ?
1. What are the rules of target reliability indexes ?
7.5 Practical LRFD Formulas in Current Codes
Chapter 7Chapter 7 Reliability-Based Design Methods of Reliability-Based Design Methods of StructuresStructures
7.5 Practical LRFD Formulas in Current Codes …1
7.5.1 Ultimate Limit State Design Formulas
where, — structural importance factor,0
G — partial factor for dead load,
1Q , — partial factors for the 1st and ith variant load,
iQ
1 102
( ) ( , , )i i
n
G Gk Q Q k Q ci Q k k k ki
S S S R f a
R
1≤
01
( ) ( , , )i i
n
G Gk Q ci Q k k k ki
S S R f a
R
1≤
GkS — effect of permanent load characteristic value
1Q kS — effect of variant load characteristic value which
dominates the load effect combination.
7.5 Practical LRFD Formulas in Current Codes …2
iQ kS — effect of the ith variant load characteristic value
ic — combination factor of the ith variant load
( )R — function of structural member
R — partial factor for structural member resistance,
kf — characteristic value of material behavior,
ka — 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 optimization method. You can refer to the P.98-101 in the reference book.
7.5 Practical LRFD Formulas in Current Codes …3
7.5.2 Serviceability Limit State Design Formulas
1 12
[ ]i
n
Gk Q k ci Q ki
S S S f
≤
1. Design Formula for Characteristic Values
1 1 22
[ ]i
n
Gk f Q k qi Q ki
S S S f
≤
2. Design Formula for Frequent Values
31
[ ]i
n
Gk qi Q ki
S S f
≤
3. Design Formula for Quasi-Permanent Values
7.5 Practical LRFD Formulas in Current Codes …4
where, 1 1f Q kS — effect of a variant load frequent value which
dominates the frequent load combination.
iqi Q kS — effect of quasi-permanent value of a variant load.
1[ ]f — the deformation or crack limit value corresponding
to characteristic value combination.
2[ ]f — the deformation or crack limit value corresponding
to frequent value combination.
3[ ]f — the deformation or crack limit value corresponding
to quasi-permanent value combination.
Homework 7
Programming the above algorithms in MATLAB environment according to the iteration algorithm proposed by this course.
(1) By using your own subroutine, re-check the example 7.2 in this course.
(2) By using your own subroutine, re-calculate the example 8.3 in the text book on P.231
Chapter 7: Homework 7
End of
Chapter 7Chapter 7
End of
This CourseThis Course
Thank you
Very Much!