Application of OpenSees in Reliability-based Design Optimization of Structures

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


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


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


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:


RBDO PROBLEM


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)


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


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


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


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


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).


ANALYTICAL 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.



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


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.




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



#######################################################################

# 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 #

# #

#######################################################################


FORM Results for the last iteration.



#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 #

#######################################################################


Sampling Analysis Results, using 250000 simulations.


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


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


RBDO 10 BARS TRUSS EXAMPLECASE 2.- Nonlinear Material, Nonlinear Analysis



𝜇𝑋1 = 27.4826 𝑐𝑚2 𝜇𝑋2 = 14.5461 𝑐𝑚2 𝜇𝑋3 = 11.7636 𝑐𝑚2




Number of LSFEs: 1360.


Note that areas of cross sections are larger than in the case of elastic material.


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



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



1P Vertical Load LN 50 kN 0.05

2P Vertical Load LN 100 kN 0.05



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


STEELFRAME EXAMPLE

Results obtained using PMA – HMV+ based RBDO.


𝜇𝑑1 = 29.5624 𝑐𝑚 𝜇𝑑2 = 49.4783 𝑐𝑚 𝜇𝑋3 = 35.2246 𝑐𝑚




Number of LSFEs: 336. Convergence tolerance is small (10−2).


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


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.


𝜇𝑑1 = 20.8792 𝑐𝑚 𝜇𝑑2 = 34.9506 𝑐𝑚 𝜇𝑋3 = 26.1249 𝑐𝑚



Number of Optimization Iterations: 256 (very high). Time: 1 hour.

Number of LSFEs: 1221. Convergence tolerance is small (10−3).




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



1P Vertical Load LN 2.5797303295e-5

2P Vertical Load LN 1.6692914381e-5

REMARK: Units used

are: 𝑘𝑁, 𝑘𝑁 𝑐𝑚2 𝑦 𝑐𝑚


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.


𝜇𝑑1 = 20.8704 𝑐𝑚 𝜇𝑑2 = 34.9277 𝑐𝑚 𝜇𝑋3 = 26.1391 𝑐𝑚



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 .



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


QUESTIONS – COMENTS

THANK YOU

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Application of OpenSees in Reliability-based Design Optimization of Structures

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