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Presentation made by Prof. Luis Celorrio-Barragué @ University of Porto during the OpenSees Days Portugal 2014 workshop
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1 | Universidad de La Rioja | 11/07/2014
APPLICATION OF OPENSEES IN RBDO OF STRUCTURES
OPENSEES DAYS PORTUGAL 2014Luis Celorrio Barragué
Deparment of Mechanical Engineering – Universidad de La Rioja - Spain
2 | Universidad de La Rioja | 11/07/2014
SUMMARY
APPLICATION OF OPENSEES IN RELIABILITY
BASED DESIGN OPTIMIZATION OF STRUCTURES
RELIABILITY / SENSITIVITY ANALYSIS
RBDO PROBLEM
RBDO METHODS
RBDO WITH OPENSEES
ANALITICAL EXAMPLE
10 BARS TRUSS EXAMPLE
STEELFRAME EXAMPLE
CONCLUSIONS
3 | Universidad de La Rioja | 11/07/2014
RELIABILITY / SENSTIVITY ANALYSIS• Recently, changes in Reliability Modules of OpenSees have been carried out.
Also some examples, presentations and videos are available in the OpenSees
Internet site.
• New commands provide sensitivity of response with respect to parameters.
Also, parameters can be used to map probability distributions to uncertain
properties.
• A script-level mechanism for identifying and updating parameters has been
added
• Methods to quantify uncertainty are available in OpenSees.
FOSM, FORM, SORM, etc.
Response Sensitivity
Monte Carlo Simulation (Importance Sampling MCS)
System Reliability
4 | Universidad de La Rioja | 11/07/2014
RBDO PROBLEM
ULUL
t
fiifi niPgPPts
f
XXX
PXμd,
μμμddd
PXd
μμdx
,
,...,1 ,0,, ..
,,min
mRX : vector of random design variables
kRd : vector of deterministic design variables
qRP : vector of random parameters
Single objective function
Component level probabilistic constraints
0,, PXdig Indicates Failure
PX, Correlated random input variables
where:
The most used formulation of a Reliability Based Design Optimization problem is:
5 | Universidad de La Rioja | 11/07/2014
RBDO PROBLEM
6 | Universidad de La Rioja | 11/07/2014
RBDO PROBLEM
7 | Universidad de La Rioja | 11/07/2014
RBDO METHODS
Double loop formulations:
Reliability Index Approach (RIA)-based double loop RBDO
Performance Measure Approach (PMA)-based double loop RBDO
Several PMA algorithms: AMV, HMV, HMV+, PMA+ (B.D.Youn et al
2003, 2005).
Single loop approaches:
SLSV (Single Loop Single Vector)
To Collapse KKT conditions of inner loop as constraints of the outer
design loop.
Decoupled (or sequential) approaches:
SORA. (Du and Chen, 2004)
8 | Universidad de La Rioja | 11/07/2014
RBDO WITH OPENSEES• Structural Reliability applications are useful when large structures supporting
extreme actions are considered. These extreme actions are wind loads,
seismic ground motions or wave loads.
• Then, nonlinear structural behavior must be considered. Also dynamic
analysis is necessary when load are time variant. Because that an advanced
finite element analysis software is needed.
• OpenSees is a powerful software with advanced structural analysis
capabilities. Also reliability and sensibility functions have been recently
modified. Because that OpenSees becomes a powerful FEA tool.
• Here some RBDO problems are solved combining some MATLAB functions
with the power of OpenSees. These MATLAB functions were originally
integrated with FERUM and forming the RBDO – FERUM toolbox. [1]
[1] L. Celorrio-Barragué, “Development of a Reliability-Based Design Optimization Toolbox for the FERUM Software”,
SUM 2012, LNAI 7520, pp. 273–286, 2012. Springer-Verlag Berlin Heidelberg 2012
9 | Universidad de La Rioja | 11/07/2014
RBDO WITH OPENSEES• RBDO RIA-based double loop method
• Outer loop or Design Optimization loop is carried out in Matlab using RBDO-
FERUM functions. Reliability analysis is carried out in OpenSees using
FORM. Writing/reading of files is used.
Write RVDATA.tcl
Design Variables,
𝑑𝑖 𝑖 = 1, … , 𝑛Optimization
Loop
RBDO-FERUM
Call !OpenSees file.tcl
Read betas.out
Read gradbetas.out
Read LSFE.out
OPENSEES
Reliability
Loop
10 | Universidad de La Rioja | 11/07/2014
RBDO WITH OPENSEES• RBDO PMA-based double loop method
• Now, Values of Random Variables are passed to OpenSees to compute the
response and the gradients of the response wrt random variables.
Optimization loop and the search of MPPIR are computed using RBDO-
FERUM. Also files are used as interfaces.
Write VECTORDATA.tcl
Random Variables,
𝑋𝑖 𝑖 = 1, … , 𝑁Optimization
Loop
Sensitivity
AnalysisCall !OpenSees filegrad.tcl
Read RES.out
Read GRADRES.out
Reliability
Loop
RBDO-FERUM OPENSEES
11 | Universidad de La Rioja | 11/07/2014
ANALYTICAL EXAMPLE
.3 ,2 ,1 ,0.2 iti
To minimize 𝐶𝑜𝑠𝑡 𝛍𝐗 = 𝜇𝑋1 + 𝜇𝑋2
Subject to 𝑃 𝑔𝑖 𝑋 ≤ 0 ≤ Φ −𝛽𝑖𝑡, 𝑖 = 1,2,3
0 ≤ 𝜇𝑋1 ≤ 10 ; 0 ≤ 𝜇𝑋2 ≤ 10
Where the Limit State Functions are
𝑔1 𝐗 = 𝑋12𝑋2 20 − 1
𝑔2 𝐗 = 𝑋1 + 𝑋2 − 5 2 30 + 𝑋1 + 𝑋2 − 12 2 120 − 1
𝑔3 𝐗 = 80 𝑋12 + 8𝑋2 + 5 − 1
The distribution of the random variables are:
Initial design: 𝛍𝐗𝟎 = 5.0, 5.0 𝑇
Convergence Tolerance of the optimization loop: 10−4
𝑋1~𝑁 𝜇𝑋1 , 𝐶𝑜𝑉 = 0.12
𝑋2~𝑁 𝜇𝑋2 , 𝐶𝑜𝑉 = 0.12
12 | Universidad de La Rioja | 11/07/2014
ANALYTICAL EXAMPLE
Results obtained using RIA based RBDO.
Design Values at the probabilistic optimum: 𝜇𝑋1 = 3.4163 𝜇𝑋2 = 3.1335
Cost Function at the probabilistic optimum: 𝐶𝑜𝑠𝑡 𝛍𝐗 =6.5497
Reliability Indexes at the optimum: 𝛽1 = 2.0171 , 𝛽2 = 2.0109 , 𝛽3 = 7.7892
Number of Optimization Iterations: 15
Number of LSFEs: 1032. It’s very high. We use very small convergence
tolerance (10−4 in the external loop). Also, no technique to reduce
computational effort has been considered.
Gradients are computed using Direct Differentiation Method (Implicit in
OpenSees).
13 | Universidad de La Rioja | 11/07/2014
ANALYTICAL EXAMPLE
14 | Universidad de La Rioja | 11/07/2014
10 BARS TRUSS EXAMPLE
Classic Example in Structural Optimization.
RBDO Problem: To minimize the weight or volume of the truss subject to
reliability constraints in terms of displacements or stresses.
15 | Universidad de La Rioja | 11/07/2014
10 BARS TRUSS EXAMPLE
CASE 1.- Linear Elastic Material, Linear Analysis.
RBDO Problem: To minimize the volume of the truss subject to reliability
constraints in terms of the vertical displacement of node 2.
To minimize 𝑉 𝐝, 𝛍𝐗, 𝛍𝐏
Subject to 𝑃 𝑔𝑖 𝑋 ≤ 0 ≤ Φ −𝛽𝑖𝑡, 𝑖 = 1
5𝑐𝑚2 ≤ 𝜇𝑋𝑗 ≤ 75𝑐𝑚2; 𝑗 = 1,2,3
Displacement constraint: Vertical displacement at node 2 is limited
ntdisplacemeallowedcmua 2 𝑔1 𝐝, 𝐗, 𝐏 = 1 −𝑢𝑦2 𝐝, 𝐗, 𝐏
𝑢𝑎
Convergence Tolerance of the optimization algorithm: 10−3
16 | Universidad de La Rioja | 11/07/2014
10 BARS TRUSS EXAMPLECASE 1.- Linear Elastic Material, Linear Analysis
RANDOM VARIABLES OF THE PROBLEM
Random
Variable Description
Distribution
type
Mean Value
(initial)
CoV or
Standard
Desviation
Design
Variable
1X 1A LN 20.0 cm2 CoV = 0.05
1X
2X 2A LN 20.0 cm2 CoV = 0.05
2X
3X 3A LN 20.0 cm2 CoV = 0.05
3X
4X E LN 21000.0 kN/cm2 1050 kN/cm
2 -
5X 1P LN 100.0 kN 20 kN -
6X 2P LN 50.0 kN 2.5 kN -
1X
2X
3X
Mean value of the cross section area in horizontal bars.
Mean value of the cross section area in vertical bars.Mean value of the cross section area in diagonal bars.
17 | Universidad de La Rioja | 11/07/2014
10 BARS TRUSS EXAMPLE
Results obtained using RIA based RBDO.
Design Values at the probabilistic optimum:
𝜇𝑋1 = 24.1668 𝑐𝑚2 𝜇𝑋2 = 18.2887 𝑐𝑚2 𝜇𝑋3 = 10.2211 𝑐𝑚2
Volume of Steel at the probabilistic optimum: 𝐶𝑜𝑠𝑡 𝛍𝐗 = 68783.08 𝑐𝑚3
Reliability Index at the optimum: 𝛽1 = 3.7000,
Number of Optimization Iterations: 61 (very high)
Number of LSFEs: 602. Note that the convergence tolerance is small (10−3).
Also, no strategy to reduce computational effort has been considered.
Gradients are computed using DDM (Implicit in OpenSees).
CASE 1.- Linear Elastic Material, Linear Analysis
18 | Universidad de La Rioja | 11/07/2014
10 BARS TRUSS EXAMPLE
#######################################################################
# FORM ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Limit-state function value at start point: ......... 0.80548 #
# Limit-state function value at end point: ........... -1.6552e-006 #
# Number of steps: ................................... 4 #
# Number of g-function evaluations: .................. 10 #
# Reliability index beta: ............................ 3.7 #
# FO approx. probability of failure, pf1: ............ 1.07801e-004 #
# #
# rvtag x* u* alpha gamma delta eta #
# 1 2.342e+001 -5.994e-001 -0.16211 -0.16211 0.16746 -0.10514 #
# 2 1.806e+001 -2.309e-001 -0.06246 -0.06246 0.06337 -0.01752 #
# 3 1.002e+001 -3.629e-001 -0.09809 -0.09809 0.10017 -0.04044 #
# 4 1.993e+004 -1.017e+000 -0.27517 -0.27517 0.29001 -0.29337 #
# 5 1.948e+002 3.465e+000 0.93649 0.93649 -0.34563 -3.00061 #
# 6 5.074e+001 3.188e-001 0.08637 0.08637 -0.08526 -0.02319 #
# #
#######################################################################
CASE 1.- Linear Elastic Material, Linear Analysis
FORM Results for the last iteration.
19 | Universidad de La Rioja | 11/07/2014
10 BARS TRUSS EXAMPLE
#OPENSEES CODE
probabilityTransformation Nataf -print 0
randomNumberGenerator CStdLib
runImportanceSamplingAnalysis truss10MCSa.out -type responseStatistics -maxNum 250000 -targetCOV 0.01 -print 0
runImportanceSamplingAnalysis truss10MCSb.out -type failureProbability -maxNum 250000 -targetCOV 0.01 -print 0
#######################################################################
# SAMPLING ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Estimated mean: .................................... 0.77538 #
# Estimated standard deviation: ...................... 0.16102 #
# #
#######################################################################
#######################################################################
# SAMPLING ANALYSIS RESULTS, LIMIT-STATE FUNCTION NUMBER 1 #
# #
# Reliability index beta: ............................ 3.7151 #
# Estimated probability of failure pf_sim: ........... 0.00010155 #
# Number of simulations: ............................. 250000 #
# Coefficient of variation (of pf): .................. 0.17007 #
#######################################################################
CASE 1.- Linear Elastic Material, Linear Analysis
Sampling Analysis Results, using 250000 simulations.
20 | Universidad de La Rioja | 11/07/2014
10 BARS TRUSS EXAMPLE
21 | Universidad de La Rioja | 11/07/2014
RBDO 10 BARS TRUSS EXAMPLE
uniaxialMaterial Hardening 1 $E $fy 0.0 [expr $b/(1-$b)*$E]
A random variable is added: fy (elastic limit) ~ 𝐿𝑁 𝜇 = 15.5 𝑘𝑁 𝑐𝑚2 , 𝐶𝑜𝑉 = 0.05 .
$b is the hardening ratio and is considered determinist: set b 0.02
CASE 2.- Nonlinear Material, Nonlinear Analysis
22 | Universidad de La Rioja | 11/07/2014
RBDO 10 BARS TRUSS EXAMPLE
RANDOM VARIABLES OF THE PROBLEM
Random
Variable Description
Distribution
type
Mean Value
(initial)
CoV or
Standard
Desviation
Design
Variable
1X 1A LN 20.0 cm2 0.05
1X
2X 2A LN 20.0 cm2 0.05
2X
3X 3A LN 20.0 cm2 0.05
3X
4X E LN 21000.0 kN/cm2 1050 kN/cm
2 -
5X fy LN 15.5 kN/cm2 0.775 kN/cm
2 -
6X 1P LN 100.0 kN 20 kN -
7X 2P LN 50.0 kN 2.5 kN -
1X
2X
3X
Mean value of the cross section area in horizontal bars.
Mean value of the cross section area in vertical bars.Mean value of the cross section area in diagonal bars.
CASE 2.- Nonlinear Material, Nonlinear Analysis
23 | Universidad de La Rioja | 11/07/2014
RBDO 10 BARS TRUSS EXAMPLECASE 2.- Nonlinear Material, Nonlinear Analysis
Results obtained using RIA based RBDO.
Design Values at the probabilistic optimum:
𝜇𝑋1 = 27.4826 𝑐𝑚2 𝜇𝑋2 = 14.5461 𝑐𝑚2 𝜇𝑋3 = 11.7636 𝑐𝑚2
Volume of Steel at the probabilistic optimum: 𝐶𝑜𝑠𝑡 𝛍𝐗 = 74004.32 𝑐𝑚3
Reliability Index at the optimum: 𝛽1 = 3.7002,
Number of Optimization Iterations: 100 (very high)
Number of LSFEs: 1360.
Gradients are computed using DDM (Implicit in OpenSees).
Note that areas of cross sections are larger than in the case of elastic material.
24 | Universidad de La Rioja | 11/07/2014
STEELFRAME EXAMPLE3 Stories and 3 Bays Steel Frame
Modified version of the structural model in the file steelframe.tcl [2] downloaded
from OpenSees forum.
[2] T. Haukaas and M. H. Scott, Shape Sensitivities in the Reliability Analysis of Nonlinear Frame Structures, Computer
and Structures, v. 84, 15-16, p964-977, 2006
1
2
3
1 1
2 2
5 5 5
4
4
4
1
1
1
1
2
2
2
2
25 | Universidad de La Rioja | 11/07/2014
STEELFRAME EXAMPLE3 Stories and 3 Bays Steel Frame
Random
Variable Description Dist.
Initial
Mean CoV
Design
Variable
1d Height LC N 0.4 m 0.02 1d
2d Height CC N 0.4 m 0.02 2d
3d Height B N 0.4 m 0.02 3d
1E Modulus LC LN 200E+6 kPa 0.05 -
1fy Yield Stress LC LN 300E+3 kPa 0.1 -
1Hkin Hard. Kin.LC LN 4.0816E+6 kPa 0.1 -
2E Modulus CC LN 200E+6 kPa 0.05 -
2fy Yield Stress CC LN 300E+3 kPa 0.1 -
2Hkin Hard. Kin.CC LN 4.0816E+6 kPa 0.1 -
3E Modulus B LN 200E+6 kPa 0.05 -
3fy Yield Stress B LN 300E+3 kPa 0.1 -
3Hkin Hard. Kin.B LN 4.0816E+6 kPa 0.1 -
1H Lateral Load LN 400 kN 0.05
2H Lateral Load LN 267 kN 0.05
3H Lateral Load LN 133 kN 0.05
1P Vertical Load LN 50 kN 0.05
2P Vertical Load LN 100 kN 0.05
26 | Universidad de La Rioja | 11/07/2014
STEELFRAME EXAMPLE3 Stories and 3 Bays Steel Frame
Member are grouped in three groups: Lateral Columns, Central Columns and
Beams. All member assigned to a group have the same rectangular cross
section, with width b = 20 cm (fixed and deterministic) and height 𝑑𝑖 (random,
design variable). 3 design variables, 𝜇𝑑𝑖 𝑤𝑖𝑡ℎ 𝑖 = 1,2,3.
.j
PgPts
VMin
jd
t
tt
f
3,2,1 cm 50cm 10
0.3 where
0,, ..
,,
PXd
μμd PX
Reliability constraint: the horizontal displacement of node 13 is limited. 𝑈𝑚𝑎𝑥 =3.6 𝑐𝑚 𝑃 𝑢𝑥13 𝐝, 𝐗, 𝐏 − 𝑈𝑚𝑎𝑥 ≤ 0 ≤ Φ −𝛽𝑡
27 | Universidad de La Rioja | 11/07/2014
STEELFRAME EXAMPLE
Results obtained using PMA – HMV+ based RBDO.
Design Values at the probabilistic optimum:
𝜇𝑑1 = 29.5624 𝑐𝑚 𝜇𝑑2 = 49.4783 𝑐𝑚 𝜇𝑋3 = 35.2246 𝑐𝑚
Volume of Steel at the probabilistic optimum: 𝐶𝑜𝑠𝑡 𝛍𝐗 = 3482083.3054 𝑐𝑚3
Reliability Index at the optimum: 𝛽1 = 3.0025,
Number of Optimization Iterations: 168 (very high)
Number of LSFEs: 336. Convergence tolerance is small (10−2).
Gradients are computed using DDM (Implicit in OpenSees).
Nonlinear Material and Beam-Column elements are considered. However,
material works in the linear elastic zone because gradients wrt parameters
𝑓𝑦𝑖 , 𝐻𝑘𝑖𝑛𝑖 are 0.
3 Stories and 3 Bays Steel Frame
28 | Universidad de La Rioja | 11/07/2014
STEELFRAME EXAMPLE
Results obtained using PMA – HMV+ based RBDO. (DDM)
CASE Nonlinear. Now, allowed horizontal displacement at node 13 is 20 cm.
Mean Values of Horizontal loads H1, H2 and H3 are the double that in the first
case. Then, large deformations occur and material works in the plastic zone.
Response gradients wrt material parameters 𝑓𝑦𝑖 , 𝐻𝑘𝑖𝑛𝑖 are ≠ 0.
Design Values at the probabilistic optimum:
𝜇𝑑1 = 20.8792 𝑐𝑚 𝜇𝑑2 = 34.9506 𝑐𝑚 𝜇𝑋3 = 26.1249 𝑐𝑚
Volume of Steel at the probabilistic optimum: 𝐶𝑜𝑠𝑡 𝛍𝐗 = 2515535.2701 𝑐𝑚3
Reliability Index at the optimum: 𝛽1 = 3.0025,
Number of Optimization Iterations: 256 (very high). Time: 1 hour.
Number of LSFEs: 1221. Convergence tolerance is small (10−3).
3 Stories and 3 Bays Steel Frame
29 | Universidad de La Rioja | 11/07/2014
STEELFRAME EXAMPLE3 Stories and 3 Bays Steel Frame
Random
Variable Description Dist.
Gradient of Response
wrt Random Variable
1d Height LC N -0.726905589
2d Height CC N -0.796264509
3d Height B N -2.066431991
1E Modulus LC LN -0.000235612
1fy Yield Stress LC LN -0.021961307
1Hkin Hard. Kin.LC LN -5.731584490e-6
2E Modulus CC LN -0.000233264
2fy Yield Stress CC LN -0.240438494
2Hkin Hard. Kin.CC LN -0.000403937
3E Modulus V LN -0.000544820
3fy Yield Stress V LN -0.393476132
3Hkin Hard. Kin.V LN -0.000298610
1H Lateral Load LN 0.0271410404
2H Lateral Load LN 0.0204777759
3H Lateral Load LN 0.0103512600
1P Vertical Load LN 2.5797303295e-5
2P Vertical Load LN 1.6692914381e-5
REMARK: Units used
are: 𝑘𝑁, 𝑘𝑁 𝑐𝑚2 𝑦 𝑐𝑚
30 | Universidad de La Rioja | 11/07/2014
STEELFRAME EXAMPLE
Results obtained using PMA – HMV+ based RBDO.(DDM) WARM-UP = yes
CASE Nonlinear. Same case than last slide: 𝑢𝑥13 𝑎𝑑𝑚 = 20 𝑐𝑚
Loads H1, H2 and H3 are the double that in the linear case.
Design Values at the probabilistic optimum:
𝜇𝑑1 = 20.8704 𝑐𝑚 𝜇𝑑2 = 34.9277 𝑐𝑚 𝜇𝑋3 = 26.1391 𝑐𝑚
Volume of Steel at the probabilistic optimum: 𝐶𝑜𝑠𝑡 𝛍𝐗 = 2515413.2267 𝑐𝑚3
Reliability Index at the optimum: 𝛽1 = 3.0025,
Number of Optimization Iterations: 244 (very high).
Number of LSFEs: 560 ≪ 1221. This reduction is motivated by Warm-Up strategy
Convergence tolerance is small (10−3),
Warm-Up Tolerance = 10−2 .
3 Stories and 3 Bays Steel Frame
31 | Universidad de La Rioja | 11/07/2014
CONCLUSIONS
Sensitivity and Reliability capabilities of OpenSees can be combined with an
optimization tool, such as Optimization Toolbox of Matlab to carry out RBDO.
Double loop RBDO methods have been implemented using OpenSees and
Matlab.
An analytical and two structural examples have been studied.
Complex problems can be solved thanks to advanced structural analysis
algorithms implemented in OpenSees.
Computational cost is very high and convergence problems can occur, specially
when an increased number of random design variables are considered.
Some special techniques to reduce the computational cost must be added:
Warm up: to start the MPP search in the MPP of the last Iteration.
To use deterministic optimum as initial design
32 | Universidad de La Rioja | 11/07/2014
QUESTIONS – COMENTS
THANK YOU
luis.celorrio@unirioja.es
luis.celorrio@gmail.com
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