Transcript Predicting masonry properties from component properties

Using Monte Carlo and Directional Sampling
combined with an Adaptive Response Surface
for system reliability evaluation
L. Schueremans, D. Van Gemert
[email protected],
[email protected]
Department of Civil Engineering
KULeuven, Belgium
Praha
Euro-Sibram, June 24 tot 26, 2002, Czech Republic
Introduction
Framework:
– Ph. D. “Probabilistic evaluation of structural unreinforced masonry”,
– Ongoing Research: “Use of Splines and Neural Networks in structural
reliability - new issues in the applicability of probabilistic techniques for
construction technology”.
Target:
– obtain an accurate value for the global pf, accounting for the exact PDF of
the random variables;
– minimize the number of LSFE, which is of increased importance for
complex structures;
– remain workable for a large number of random variables (n). In practice,
the number of LSFE should remain proportional with the number of
random variables (n).
Introduction
Level II and Level III methods:
p f  P g X  0   ...  f X x dx
g  X  0
Level
Definition
III
Le vel III met ho ds s uc h as Mo nte C arlo ( M C) sam pling and Nu meric al
Integr atio n (NI) are co nsi dere d most accurate. The y co m pute t he e xact
pro bability of failure o f the w hole struct ur al syste m, or o f str uctural
ele me nts, usi ng t he exact pro babilit y de nsit y fu nctio n o f all r ando m vari ables.
II
Level II methods such as FORM and SORM compute the probability of failure by
means of an idealization of the limit state function where the probability density
functions of all random variables are approximated by equivalent normal
distribution functions.
I
Level I methods verify whether or not the reliability of the structure is sufficient
instead of computing the probability of failure explicitly. In practice this is often
carried out by means of partial safety factors.
Table 1: Levels for the calculation of structural safety values (EC1, 1994; JCSS, 1982)
Introduction - reliability methods
Reliability met ho ds - Le vel ( I, II, III) - Direct/In direct ( D, ID)
Integration
methods
Analytical or Numerical Integration (AI/NI, III, D)
#LSFE~9n
Directional Integration (DI, III, D)
#LSFE~3/pf VI
Sampling
methods
(Importance Sampling) Monte Carlo (ISMC, III, D)
FORM/SORM
methods
First Order Second Moment reliability method (FOSM, II, D)
Combined
methods using
Adaptive
Response
Surface
techniques
Directional Adaptive Response surface Sampling (DARS, III, D-ID)
(Importance) Directional Sampling (IDS, III, D)
#LSFE~cte.n
First Order and Second Order Reliability Method (FORM/SORM, II, D)
in combination with a system analysis (FORM/SORM-SA, III, D)
Monte Carlo Adaptive Response surface Sampling (MCARS, III, D-ID)
FORM with an Adaptive Response Surface (FORMARS, II, D-ID) in
combination with a system analysis (FORMARS-SA, III, D-ID)
Table 2: Overview of reliability methods for a level III reliability analysis
Methods for System Reliability using an
Adaptive Response Surface
Real structure: high degree of
Reliability analysis
mechanical complexity,
numerical algorithms, nonlinear FEM
Response Surface: low order
polynomial, Splines, Neural
Network,...
Optimal scheme: DARS or MCARS+VI
DARS: Matlab 6.1 [Schueremans, 2001], Diana 7.1 [Waarts,2000]
MCARS+VI: Matlab 6.1 [Schueremans, 2001]
DARS-Directional Adaptive Response surface Sampling
u2
Component
reliability:
fU1,U2(u1,u
2)
fU1,U2(u1,u2)
unsafe
g3(u1,u2)<0
unsafe
g1(u1,u2)<0
g3>0
safe
g1>0
u2
g2>0
u2
g4>0
unsafe u1
g1(u1,u2)<0
g3(u1,u2)<0
g2(u1,u2)<0
unsafeunsafe
pf,g1= 0.0161, g1=2.14
pf,g2= 0.0161, g2=2.14
pf,g3= 0.0062, g3=2.50
pf,g4= 0.0062,
g1(u1,u2)<0
g4=2.50
unsafe
safe
System reliability:
pf= 0.0446
u1 =1.70
u1
g4(u1,u2)<0
unsafe
unsafe g4(u1,u2)<0
u1  u 2 
2
g1u1, u 2   2.0  0.1u1  u 2  
2
g2(u1,u2)<0

u1  u 2 
2

g u1, u 2   ming 2 u1, u 2   2.0  0.1u1  u 2  
unsafe
2

 g 3 u1, u 2   u1  u 2  2.5 2
 g u , u   u  u  2.5 2
2
1
 4 1 2
DARS -Directional Adaptive Response surface Sampling
Step 1:
u2
[0,0]
[- 0,0]
=min
[0,0]
[0,-0]
u1
- Evaluate the LSF for
the origin in the uspace;
- Search the roots  of
the limit state function
for the principal
directions in the uspace (n=2):
- [1,0];[0,1];[-1,0];[0,-1]
With the root-finding
algorithm, this requires
approximately 3 to 4
LSFE
N=5=2n+1, #LSFE=21
min = 3.5, =2.85
DARS -Directional Adaptive Response surface Sampling
u2
Step 2:
Fit a response surface
through these data in
the x-space and the
resulting outcome Y,
using a least squares
algorithm.
u1
add = 3.0 min = 3.5
gRS,1 = 0
gRS,1= 1.65
2
-0.13u1
2
-0.13u2
DARS -Directional Adaptive Response surface Sampling
Step 3: iter. procedure
required accuracy
Perform DS on the
Response Surface:
u2
u1
If i,RS < min+add
Calculate
2
pi(LSF)= (i,LSF,n)
Update the response
surface with new data
Else
Calculate
2
pi(RS)=  (i,RS,n)
gRS,2 = 0
add = 3.0 min = 2.05
gRS,2= 0.92+0.046u1
-0.023u2-0.074u1u2
2
2
-0.097u1 -0.084u2
DARS -Directional Adaptive Response surface Sampling
p f  0.028
u2
  191
.
# LSFE  51
N  14
 min  2.05
u1
gRS,3 = 0
add=3 min=2
pf
N = 14
MCARS+VI
Monte Carlo Adaptive Response surface
Sampling+Variance Increase
Step 3:
g
LSF
RS
Monte Carlo Variance
Increase on the Response
Surface (vi).
add
gRS,i
eg,i
gLSF,i
g,add
g,add
RS
Sampling function: h=n-0.4
IF |gRS(v i)|<|g,add|
calculate gLSF(vi)
u
fu v
pi  I gLSF vi   0
hv v

i

update RS
update g,add
fu v
pi  I gRS vi   0
hv v
Else
.


DARS and MCARS+VI
The number of direct LSFE remains proportional to the number of
random variables (n),
There is no preference for a certain failure mode. All contributing
failure modes are accounted for, resulting in a safety value that
includes the system behavior, thus on level III.
Safety of masonry Arch
Random
variables
Probability
density
function
Mean value
µ
Standard
deviation
σ
Coefficient of
variation V [%]
x1 = r0 [m]
Normal
2.5
0.02
0.8
x2 = t [m]
Normal
0.16
0.02
12
x3= dr [m]
Normal
0
0.02
/
x4 = F [N]
Lognormal
750
150
20
Table: Random variables and their parameters
Safety of masonry Arch
To evaluate the stability of the arch, the thrust line method is used
(Heyman, 1982), which is a Limit Analysis. Following
assumptions are made:
–
–
–
–
blocs are infinitely resistant,
joints resist infinitely to compression
joints do not resist to traction
joints resist infinitely to shear
An external program Calipous is used for the Limit State Function
Evaluations [Smars, 2000]
Safety of masonry Arch
Failure modes - limit states - limit analysis based on thrust lines
 g  X  1
 g  X  1
 S  X  1
 S  X  1
 g  X   1
g  min 
 S  X   1
Safety of masonry Arch

procedure
pf
#LSFE (time)
Accuracy
reliability analysis – initial random values: d~N(0.16,(0.02)²)
()=0.15
1.26
0.11
43 (17 min)
DARS
()=0.15
1.25
0.11
23 (12 min)
MCARS+VI
reliability analysis – increased accuracy on thickness: d~N(0.16,(0.005)²)
-4
3.55
1.9 10
34 (13 min)
DARS
V()=0.05
-4
3.46
2.7 10
56 (21 min)
MCARS+VI
V()=0.05
reliability analysis – increased mean value for thickness: d~N(0.21,(0.02)²)
-4
3.69
10
45 (17 min)
DARS
V()=0.05
-4
3.67
1.2 10
23 (12 min)
MCARS+VI
V()=0.05
Table 2: Outcome of reliability analysis for masonry arch – initial parameters and update
Safety of masonry Arch
T=3.7
Increased thickness:
t: =0.21 m; =0.02 m
-4
N=192: =3.72, pf=1.0 10
Increased accuracy:
t: =0.16 m; =0.005 m
-4
N=273: =3.44, pf=2.9 10

95% Confidence interval 
Initial survey:
t: =0.16 m; =0.02 m
-1
N=371: =1.26, pf=1.0 10
Number of Samples N
Figure: DARS-outcome - reliability vernus number of samples N
Conclusions
• Focus was on the use of combined reliability methods to
obtain an accurate estimate of the global failre probability
of a complete structure, within an minimum number of
LSFE.
• A level III method is presented and illustrated:
(DARS/MCARS+VI
• Ongoing research: Splines and Neural Network instead of
low order polynomial for Adaptive Response Surface
(ARS).
• Acknowlegment: IWT-VL (Institute for the encouragement of
Innovation by Science and Technology in Flanders).