EGGN 598 – Probabilistic Biomechanics
Ch.7 – First Order Reliability Methods
Anthony J Petrella, PhD
Review: Reliability Index
• To address limitations of risk-based reliability with greater efficiency
than MC, we introduce the safety index or reliability index, b
• Consider the familiar limit state, Z = R – S, where R and S are
independent normal variables
2
2
• Then we can write, Z   R  S ;  Z   R   S
and POF = P(Z ≤ 0), which can be found as follows…
0.45
Z
R  S
Z


Z
 R2   S2
b
0.30
fZ(z)
b 
0  Z
0.15
POF  ( b )  1  ( b )
0.00
-4
-3
-2
-1
0
1
Z Value (Limit State)
2
3
4
Review: MPP
Safe
Safe
Failure
Failure
Geometry of MPP
• Recognize that the point on any curve or n-dimensional surface that is
closest to the origin is the point at which the function gradient passes
through the origin
• Distance from the origin is the radius of a circle tangent to the
curve/surface at that point (tangent and gradient are perpendicular)
*
MPP → closest to the origin
→ highest likelihood on joint PDF
in the reduced coord. space
gradient
direction
gradient → perpendicular
to tangent direction
Review: AMV Example
• For example, consider the non-linear limit state,
 hip
where,
Mc ( Fap l fem )  12 h hip  6 Fap  l fem



3
1
 b  h2
I
b
h
12 hip hip
 hip  hip
l fem  0.66  0.05
( m)
hhip  0.06  0.012
( m)
bhip  0.08
( m)
Fap  836.65
(N )
Review: AMV Geometry
h_hip' = reduced variate for xsection height
• Recall the plot depicts (1) joint PDF of l_fem and h_hip, and (2) limit
state curves in the reduced variate space (l_fem’,h_hip’),
• To find g(X) at a certain prob level, we wish to find the g(X) curve that
is tangent to a certain prob contour of the joint PDF – in other words,
2
the curve that is tangent to a circle
of certain radius b
g
• We start with the linearization of
1
g(X) and compute its gradient
g
• We look outward along the gradient
0
until we reach the desired prob level
b   (0.9)  1.28
• This is the MPP because the linear
-1
g
g(X) is guaranteed to be tangent to
the prob contour at that point
linear_10%
linear_50%
1
linear_90%
-2
-2
-1
0
1
l_fem' = reduced variate for femur length
2
Review: AMV Geometry
h_hip' = reduced variate for xsection height
• The red dot is (l_fem’*,h_hip’*), the tangency point for glinear_90%
• When we recalculate g90% at (l_fem’*,h_hip’*) we obtain an updated
value of g(X) and the curve naturally passes through (l_fem’*,h_hip’*)
• Note however that the updated
2
curve may not be exactly tangent
Linear
MV
AMV
to the 90% prob circle, so there
1
may still be a small bit of error
(see figure below)
a
10%
10%
10%
0
MV90%
-1
AMV90%
b   1(0.9)  1.28
Linear90%
-2
-2
-1
0
1
l_fem' = reduced variate for femur length
2
AMV+ Method
• The purpose of AMV+ is to reduce the error exhibited by AMV
• AMV+ simply translates to…“AMV plus iterations”
• Recall Step 3 of AMV: assume an initial value for the MPP, usually at
the means of the inputs
• Recall Step 5 of AMV: compute the new value of MPP
• AMV+ simply involves reapplying the AMV method again at the new
MPP from Step 5
• AMV+ iterations may be continued until the change in g(X) falls below
some convergence threshold
AMV+ Method (NESSUS)
Assuming one is seeking values of the performance function (limit state)
at various P-levels, the steps in the AMV+ method are:
1. Define the limit state equation
2. Complete the MV method to estimate g(X) at each P-level of interest, if
the limit state is non-linear these estimates will be poor
3. Assume an initial value of the MPP, usually the means
*
 g 


4. Compute the partial derivatives  X i  and find alpha (unit vector in
direction of the function gradient)
5. Now, if you are seeking to find the performance (value of limit state) at
various P-levels, then there will be a different value of the reliability
p
  1 ( p)
index bHL at each P-level. It will be some known value b HL
and you can estimate the MPP for each P-level as…
p
p
X i*  a i b HL
AMV+ Method (NESSUS)
The steps in the AMV+ method (continued):
6. Convert the MPP from reduced coordinates back to original
coordinates
7. Obtain an updated estimate of g(X) for each P-level using the relevant
MPP’s computed in step 6
8. Check for convergence by comparing g(X) from Step 7 to g(X) from
Step 2. If difference is greater than convergence criterion, return to
Step 3 and use the new MPP found in Step 5.
AMV+ Example
• We will continue with the AMV example already started and extend it
with the AMV+ method
AMV Method – Iteration 1
Mean Value Method
Forward Difference Trials
Trial
Variable Perturbation
g
dg/dX
ref = mean
--1.1504E+07
-1
l_fem
0.05
1.2375E+07 1.7430E+07
2
h_hip
0.012
7.9889E+06 -2.9293E+08
Mean Value (MV) Method
mean
SD
1.1504E+07 3621533.28
p
0.05
0.1
0.5
0.9
0.95
g (hand)
5547090
6862800
11503982
16145164
17460874
NESSUS
g
5547074
6862856
11504017
16145012
17460908
g(X)
6
10 MC
g
6349887
7155118
11488743
20975184
25804032
X = MPP-1
AMV+ Example
• We will continue with the AMV example already started and extend it
with the AMV+ method
AMV Method – Iteration 1
Advanced Mean Value (MV) Method - Iteration 1
dg/dX'
alpha
p
g (hand)
8.7151E+05 2.4065E-01
0.05
--3.5151E+06 -9.7061E-01
0.1
-0.5
-0.9
-0.95
-MPP Table: AMV Iteration 1, Level 4 (p = 0.90)
0.05
0.10
0.50
0.90
-0.3958
-0.3084
0.0000
0.3084
1.5965
1.2439
0.0000
-1.2439
0.6402
0.6446
0.6600
0.675420
0.0792
0.0749
0.0600
0.045073
AMV Method – Iteration 2
NESSUS
g
6411184
7204743
11503982
20860680
25572242
Advanced Mean Value (MV) Method - Iteration 2
dg/dX'
alpha
p
g (hand)
1.5453E+06 1.9316E-01
0.05
--7.8494E+06 -9.8117E-01
0.1
-0.5
-0.9
-0.95
--
0.95
0.3958
-1.5965
0.6798
0.0408
MPP Table: AMV Iteration 2, Level 4 (p = 0.90)
0.05
0.10
0.50
0.90
-0.3177
-0.2475
0.0000
0.2475
1.6139
1.2574
0.0000
-1.2574
0.6441
0.6476
0.6600
0.672377
0.0794
0.0751
0.0600
0.044911
Forward Difference Trials: AMV Iteration 1, Level 4 (p = 0.90)
Trial
Variable Perturbation
g
dg/dX
ref = MPP
--2.0861E+07
-1
l_fem
0.05
2.2406E+07 3.0905E+07
2
h_hip
0.012
1.3011E+07 -6.5412E+08
l_fem'
h_hip'
l_fem
h_hip
NESSUS
g
20917711
0.95
0.3177
-1.6139
0.6759
0.0406
X = MPP-2
g(X)
l_fem'
h_hip'
l_fem
h_hip
AMV+ Example
AMV Method – Iteration 1
AMV Method – Iteration 2
2
MV10%
AMV10%
Linear10%
1
a
0
MV90%
-1
AMV90%
b   1(0.9)  1.28
Linear90%
-2
-2
-1
h_hip' = reduced variate for xsection height
h_hip' = reduced variate for xsection height
2
MV10%
1
l_fem' = reduced variate for femur length
2
Linear10%
1
0
MV90%
-1
AMV90%
Linear90%
-2
0
AMV10%
-2
b   1(0.9)  1.28
-1
0
1
l_fem' = reduced variate for femur length
2
AMV+ Example
AMV Method – Iteration 2
AMV Method – Iteration 2
-0.9
MV10%
AMV10%
Linear10%
1
0
MV90%
-1
AMV90%
Linear90%
-2
-2
b   1(0.9)  1.28
-1
0
1
l_fem' = reduced variate for femur length
h_hip' = reduced variate for xsection height
h_hip' = reduced variate for xsection height
2
90% Probability Contour
Linear-1 (about means)
-1.0
Linear-2 (about MPP-1)
AMV-1 (g = 20860680)
-1.1
AMV-2 (g = 20917711)
-1.2
MPP-2
-1.3
MPP-1
-1.4
2
0.0
0.1
0.2
0.3
0.4
l_fem' = reduced variate for femur length
0.5
AMV+ Example
0.91
1.0
8-Trial AMV Method
0.8
11-Trial AMV+ (90% only)
10e6 Trials MC
0.6
3-Trial MV Method
FG(g)
FG(g)
10e6 Trials MC
8-Trial AMV Method
0.90
11-Trial AMV+ (90% only)
0.4
0.2
0.89
0.0
0.0E+0
2.5E+7
5.0E+7
Bending Stress in Hip (Pa)
7.5E+7
2.0E+7
2.1E+7
Bending Stress in Hip (Pa)
2.2E+7