Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L...
Transcript of Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L...
![Page 1: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/1.jpg)
Resistance vs. Load Reliability Analysis
Let L be the load acting on a system (e.g. footing) andlet R be the resistance (e.g. soil)
Then we are interested in controlling R such that the probabilitythat R > L (i.e. the reliability) is acceptably high or, equivalently,that R < L (i.e. the failure probability) is acceptably low, where
[ ]P ( , )RLr l
R L f r l drdl∞ ∞
−∞ <
< = ∫ ∫and where is the joint (bivariate) distribution of R and L.( , )RLf r l
[ ]( , ) PRLf r l dr dl r R r dr l L l dl= < ≤ + < ≤ +∩
![Page 2: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/2.jpg)
Bivariate Distributions
[ ]2 2
1 1
1 2 1 2P ( , )l r
RLl r
l L l r R r f r l drdl< ≤ < ≤ = ∫ ∫∩
![Page 3: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/3.jpg)
Resistance vs. Load Reliability Analysis
The estimation of typically requires vast amountsof data, which is generally impractical.
Simplifications:1. assume R and L are independent so that
and
( , )RLf r l
( , ) ( ) ( )RL R Lf r l f r f l=
[ ]P ( ) ( ) ( ) ( )R L L Rr l r l
R L f r f l drdl f l f r drdl∞ ∞ ∞ ∞
−∞ < −∞ <
< = =∫ ∫ ∫ ∫
![Page 4: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/4.jpg)
Resistance vs. Load Reliability Analysis
2. Assume R and L are either normally or lognormallydistributed.The event {R < L} is the same as the events
i) {R – L < 0}ii) {R/L < 1}
If both R and L are normally distributed, thenX R L= −
is also normally distributed with parameters
(assuming R and L are independent).
2 2 2X R L
X R L
μ μ μ
σ σ σ
= −
= +
![Page 5: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/5.jpg)
Reliability Index
• the reliability index, β, is the number of standard deviations the mean is from failure.
• superior to the Factor-of-Safety approach because it depends on both the mean and the standard deviation.
• failure occurs if X < 0 (normal) or ln X < 0 (lognormal).Defining
ln
ln
, (normal)
, (lognormal)
X
X
X
X
μβσμβσ
=
=
[ ] [ ] ( )
[ ] [ ] ( )ln
ln
P failure P 0 , (normal)
P failure P ln 0 , (lognormal)
X
X
X
X
X
X
μ βσ
μ βσ
⎛ ⎞= < = Φ − = Φ −⎜ ⎟
⎝ ⎠⎛ ⎞
= < = Φ − = Φ −⎜ ⎟⎝ ⎠
![Page 6: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/6.jpg)
Resistance vs. Load Reliability Analysis
• Suppose load is normally distributed with mean 10 and standard deviation 3
• Suppose resistance is normally distributed with mean 20 and standard deviation 4.
• Then X = R – L has mean and variance
• Mean FS = μR /μL = 20/10 = 2• Reliability index = β = μX /σX = 10/5 = 2• Probability of failure =
2 2 2 2 2
20 10 10
4 3 25X R L
X R L
μ μ μ
σ σ σ
= − = − =
= + = + =
[ ] [ ] 0 10P P 05
( 2) 0.023
R L X −⎛ ⎞< = < = Φ⎜ ⎟⎝ ⎠
= Φ − =
![Page 7: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/7.jpg)
Resistance vs. Load Reliability Analysis
Now [ ] [ ] [ ]P P 0 = P 0 X
X
R L R L X μσ
⎛ ⎞< = − < < = Φ −⎜ ⎟
⎝ ⎠where is the standard normal cumulative distributionfunction.
( )xΦ
![Page 8: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/8.jpg)
Resistance vs. Load Reliability Analysis
Alternatively, if R and L are lognormally distributed, thenRXL
=
is also lognormally distributed with parameters
ln ln ln2 2 2ln ln ln
X R L
X R L
μ μ μ
σ σ σ
= −
= + (assuming independence)
so that [ ] [ ] [ ] [ ]ln
ln
P P / 1 P 1 P ln 0
X
X
R L R L X X
μσ
< = < = < = <
⎛ ⎞= Φ −⎜ ⎟
⎝ ⎠
![Page 9: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/9.jpg)
Resistance vs. Load Reliability Analysis
• Suppose load is lognormally distributed with mean 10 and standard deviation 3
• Suppose resistance is lognormally distributed with mean 20 and standard deviation 4.
• Then X = R/L is lognormally distributed
20, 4R Rμ σ= = →( )
22ln 2
2ln ln
ln 1 0.0392
ln 0.5 2.976
RR
R
R R R
σσμ
μ μ σ
⎛ ⎞= + =⎜ ⎟
⎝ ⎠= − =
10, 3L Lμ σ= = →( )
22ln 2
2ln ln
ln 1 0.0862
ln 0.5 2.259
LL
L
L L L
σσμ
μ μ σ
⎛ ⎞= + =⎜ ⎟
⎝ ⎠= − =
![Page 10: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/10.jpg)
Resistance vs. Load Reliability Analysis/ ln ln lnX R L X R L= → = −
ln ln ln2 2 2ln ln ln ln
0.717
0.125 0.354X R L
X R L X
μ μ μ
σ σ σ σ
= − =
= + = → =
[ ]ln
ln
0.717 2.02 P / 1 ( 2.02) 0.0220.354
X
X
R Lμβσ
= = = → < = Φ − =
![Page 11: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/11.jpg)
Reliability Index
![Page 12: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/12.jpg)
Reliability Index
More generally, system failure can be defined in terms ofa failure or limit state function. Also called the safety margin
1 2( , , )M g Z Z= …
Failure occurs when M = g(Z1, Z2, …) < 0. In this case, thereliability index is defined as
M
M
μβσ
=
Problem: different choices of the function M lead to differentreliability indices (e.g. M = R – L or M = ln(R/L) both implyfailure when M < 0, but lead to different values of β in firstorder approximations).
![Page 13: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/13.jpg)
Reliability IndexExample 1: suppose that M = R – Land R and L are normally distributed. Then
2 2 (assuming independence)M R L
M R L
μ μ μ
σ σ σ
= −
= +and
so that2 2
R L
R L
μ μβσ σ
−=
+
and [ ] [ ] ( )P failure P 0M β= < = Φ −
This is exact, so long as R and L are normally distributedand independent (if not independent then must involvecovariances in computation of σM).
![Page 14: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/14.jpg)
Reliability IndexExample 2: suppose that M = ln( R / L ) = lnR – lnLand R and L are lognormally distributed. Then
ln ln
2 2ln ln (assuming independence)
M R L
M R L
μ μ μ
σ σ σ
= −
= +and
so that ln ln2 2ln ln
R L
R L
μ μβσ σ
−=
+
and [ ] [ ] ( )P failure P 0M β= < = Φ −
This is exact, so long as R and L are lognormally distributedand independent (if not independent then must involvecovariances in computation of σM).
![Page 15: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/15.jpg)
Reliability IndexExample 3: suppose that M = ln( R / L ) = lnR – lnLand R and L are normally distributed. Then the distributionof M is complex and we must approximate its moments;
2 22 2ln ln
2 22 2
2 2
ln ln (to first order)
=
R L
M R L
M R L
R LR L
R L
M MR L
V V
μ μ
μ μ μ
σ σ σ
σ σμ μ
−
∂ ∂⎛ ⎞ ⎛ ⎞+⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠
+ = +
2 2R L
R L
μ μβσ σ
−=
+now
2 2
ln lnR L
R LV Vμ μβ −
=+
It was
These are clearly different.
in Example 1
![Page 16: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/16.jpg)
Hasover-Lind Reliability Index
Hasover and Lind (1974) solved this ambiguity by mappingthe set of system variables, Z, onto a set of standardized (meanzero, unit variance) and uncorrelated variables, X
[ ]( )E= −X A Z Z
where the transformation matrix A is the solution ofT =ZAC A I
where CZ is the matrix of covariances between the systemvariables, Z, and I is the identity matrix. In terms of Z,
[ ]( ) [ ]( )1min E EZ
T
Lβ −
∈= − −Zz
z Z C z Z
where LZ is the failure surface. The value of z which minimizesthis is called the design point, z*.
![Page 17: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/17.jpg)
Hasover-Lind Reliability Index
Hasover-Lind’s reliability index is the minimum distance fromthe mean to the failure surface in standardized space(figure from Madsen, Krenk, and Lind, 1986)
![Page 18: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/18.jpg)
Going Beyond Calibration• Must move beyond calibration for real benefits of LRFD• Simple probability-based methods take load and
resistance distributions into account- nominal or characteristic resistance: Rn = kRμR- nominal or characteristic load: Ln = kLμL
• Design: ϕ Rn = γ Ln
P[failure] P[ ] P[ / 1]R L R L= < = <
![Page 19: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/19.jpg)
• let M = ln(R/L)• then P[failure] P[ 0]M= <
• β is the reliability index• typically β ranges from 2.0 to 3.0
Going Beyond Calibration
![Page 20: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/20.jpg)
We want to produce a design such that the mean and standard deviation ofresistance satisfies
[ ]P 1 P ln ln 0 ( )R R LL
β⎡ ⎤< = − < = Φ −⎢ ⎥⎣ ⎦
In detailln ln
2 2ln ln
0 E[ln ln ]P[ln ln 0] P ( )SD[ln ln ]
R L
R L
R LR L ZR L
μ μ βσ σ
⎛ ⎞⎡ ⎤ −− − ⎜ ⎟− < = < = Φ − = Φ −⎢ ⎥ ⎜ ⎟−⎣ ⎦ +⎝ ⎠so that2ln ln
ln ln ln ln2 2ln ln
R LR L R L
R L
μ μ β μ μ β σ σσ σ
−= ⇒ = + +
+ln ln ln0.75 ( )L R Lμ β σ σ+ +
Now, since 2ln lnln( ) 0.5L L Lμ μ σ= − and ( )2
ln lnexp 0.5R Rμ μ σ= + Rwe get
To determine both load and resistance factors:
{ } { }2 2ln ln ln lnexp 0.5 0.75 exp 0.5 0.75R L R R L Rμ μ σ βσ σ βσ⎡ ⎤= + − +⎣ ⎦
or
{ } { }2 2ln ln ln lnexp 0.5 0.75 exp 0.5 0.75R R R L R Lσ βσ μ σ βσ μ+ = − +
![Page 21: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/21.jpg)
Writing this in terms of the nominal load and resistance ( 1) ( 1)
n R R R
n L L L
R k kL k k
μμ
= <
= >
gives us{ } { }2 2
ln ln ln lnexp 0.5 0.75 exp 0.5 0.75R R L Ln n
R L
R Lk k
σ βσ σ βσ⎡ ⎤ ⎡ ⎤− − − +⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
Recalling that our LRFD has the form n nR Lϕ γ= implies
Resistance factor:{ }
{ }
2ln ln
2ln ln
exp 0.5 0.75
exp 0.5 0.75
R R
R
L L
L
k
k
σ βσϕ
σ βσγ
− −=
− +=Load factor:
![Page 22: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/22.jpg)
If load factors are known (e.g. from structural codes) then the resistancefactor becomes dependent on both the resistance variability and theload variability. In this case, our LRFD can be written
i i
i
i n i ni i
n i ni n R R
L LR L
R k
γ γϕ γ ϕ
μ= ⇒ = =
∑ ∑∑
{ }2
2 2ln ln2
1exp
1
ii nLi
R LR L R
Q Vk V
γβ σ σ
μ
⎛ ⎞+⎜ ⎟= − +⎜ ⎟+⎝ ⎠
∑
wherei i in L LL k μ=
i
i
i
nL L
i i L
Lk
μ μ= =∑ ∑2 2i iL LiL
LL L
VV
μσμ μ
= =∑ (assuming loads are independent)
![Page 23: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/23.jpg)
Going Beyond Calibration
• thus, for given target reliability index and variances, the resistance factor can be computed as
which depends on- coefficient of variation of load ( )- coefficient of variation of resistance ( )- load factors γι- characteristic coefficients,
LV
RV
and iR Lk k
{ }2
2 2ln ln2
1exp
1
ii nLi
R LR L R
L Vk V
γϕ β σ σ
μ
⎛ ⎞+⎜ ⎟= − +⎜ ⎟+⎝ ⎠
∑
![Page 24: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/24.jpg)
Problems Implementing LRFD
• the coefficient of variation of resistance depends on;- variability in material properties- error in design models- measurement and correlation errors- construction variability
• with steel and concrete, the material property variability does not change significantly with location (quality controlled materials)
• with soils, the material property mean and variability change within a site and from site to site
![Page 25: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/25.jpg)
Problems Implementing LRFD
• there is a dependence between resistance and load which is generally absent (or small) in structural engineering, e.g. shear strength is dependent on stress;
tanf cτ σ φ= +
![Page 26: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/26.jpg)
Problems Implementing LRFD
• No common definition of “characteristic value”- often defined as a “cautious estimate of the
mean”, but sometimes as a low percentile- we badly need a standard definition (median?)
• VR changes with intensity of site investigation- resistance factor should approach 1.0 as the
site is more thoroughly investigated- this would lead to a complex table of resistance
factors (however, see, e.g., AS 5100, AS 2159,AS4678, Eurocode 7, NCHRP507)
![Page 27: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/27.jpg)
Future Directions
• probabilistic methods generally limited to “single random variable” models (e.g. R vs. L)
• to consider the effect of spatial variability, random field simulation combined with finite element analysis is necessary (RFEM)
• the random field simulation allows the representation of a soil’s spatial variability
• the finite element analysis allows the soil to fail along “weakest paths” (decreased model error)
![Page 28: Resistance vs. Load Reliability AnalysisResistance vs. Load Reliability Analysis 2. Assume R and L are either normally or lognormally distributed. The event {R < L} is the same as](https://reader033.fdocuments.in/reader033/viewer/2022060910/60a4d166a7a63c1599584a0e/html5/thumbnails/28.jpg)
Conclusions
• geotechnical engineers led the way with Limit States concepts (1940’s) but have been slow to migrate to reliability-based design methods.
• the most advanced LRFD codes currently are AS 5100 and Eurocode 7.
• all current LRFD geotechnical codes have load and resistance factors calibrated from older WSD codes, with some adjustments based on engineering judgement and simple probability methods.