Fully probabilistic optimization of reinforced E concrete ...½_PPT.pdfFully probabilistic...
Transcript of Fully probabilistic optimization of reinforced E concrete ...½_PPT.pdfFully probabilistic...
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XVI INTERNATIONAL CONFERENCE ON
STRUCTURAL REPAIR AND REHABILITATIONPORTO | PORTUGAL 13-15 JULY 2020
ONLINE CONFERENCE
Fully probabilistic optimization of reinforced concrete elements using heuristic algorithm and neural network
Venclovský J., Štěpánek P., Laníková I.
Brno University of Technology,Faculty of Civil Engineering
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Introduction
Many mathematical programming applications in civil engineering, e.g.
• structural design
• water management
• reconstruction of transportation networks
• reliability related computations
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Structural Design
Two ways of structural designing
• partial reliability factor method• all uncertainties are replaced with their characteristic values
• calculation is simple
• obtaining input data is easy
• fully probabilistic design• complex time-consuming computation
• lack of input data
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Design Optimization
Two ways to optimize structural design
• DBSO (Deterministic-Based Structural Optimization)• based on partial reliability factor method
• provides suboptimal solution
• quick
• RBSO (Reliability-Based Structural Optimization)• based on probabilistic design
• „better“ solution
• complex and time-consuming
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DBSO Definition – Objective Function
Objective is to minimize beam’s price
min(𝐶C𝑉C(𝒙) + 𝐶S𝑊S(𝒙))
𝐶C , 𝐶S – cost of concrete per m3, cost of steel per kg
𝑉C , 𝑊S – volume of concrete [m3], weight of steel [kg]
𝒙 – vector of design variables
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DBSO Definition – Design Variables
The work focuses on beams• e.g. simply supported beam with this scheme and cross section
• here then 𝒙 = (𝑏1, 𝑏2, 𝑏3, ℎ1, ℎ2, ℎ3, 𝐴𝑠1, 𝐴𝑠3)
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q
Nb2b2
b3
h1
h2
h3
h
As1
As3
b1
5 m
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DBSO Definition – ULS
First constraint is ULS – ultimate limit state
• beam has to bear the prescribed load strains in concrete and steel cannot exceed prescribed values
𝜀 ≥ 𝜀𝑐,min in cross section vertices
𝜀 ≤ 𝜀𝑠,max in reinforcing steel positions
𝜀 – actual strain value
𝜀𝑐,min – minimal allowed strain of concrete
𝜀𝑠,max – maximal allowed strain of steel
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DBSO Definition – SLS
Second constraint is SLS – serviceability limit state
• more possibilities, here it is limitation of deflection
• beam’s deflection cannot exceed prescribed maximum value
𝑤 ≤ 𝑤𝑚𝑎𝑥
𝑤 – beam’s deflection
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DBSO Definition – FEM
• Mathematical model is rather complex
• To be able to assess ULS and SLS it is necessary to proceed with matrix calculations
𝑲𝒖 = 𝑭
𝑲 – stiffness matrix; depends on 𝒙
𝑭 – vector of load; depends on load
𝒖 – vector of node deflections; strain 𝜀 and
deflection 𝑤 depend on it
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RBSO Definition
• same as DBSO, except• some variables become random variables (such as load, material characteristics of concrete and
steel, etc.)
• ULS and SLS are redefined as probabilistic constraints
• ULS
𝑃 𝜀(vertices) ≥ 𝜀𝑐,min ∧ 𝜀(steel) ≤ 𝜀𝑠,max ≥ 𝑝ULS
• SLS𝑃 𝑤 ≤ 𝑤𝑚𝑎𝑥 ≥ 𝑝SLS
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Solution
• the probability 𝑝ULS should reach the level from 1-10-4 to 1-10-6 (depending on chosenreliability class) , which is obviously troublesome to achieve
• large number of scenarios makes the problem insolvable in a reasonable time horizon
• therefore a different approach is proposed• this is based on algortihm divided into inner deterministic optimization (based on Reduced Gradiant
Method) and outer stochastic optimization (based on Regression Analysis)
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Example
• shown on the mentioned simply supported beam
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q
Nb2b2
b3
h1
h2
h3
h
As1
As3
b1
5 m
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Example
• normal force N and distributed load q are random variables with gamma distribution
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q [kN/m]
0
0.7
p
30 350
0.7
5 10N [kN]
p
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Solution – 1. Initialization
• The calculation is initialized by choosing 12 initialization scenarios
𝝃𝑘init = 𝑞𝑘
init, 𝑁𝑘init , 𝑘 = 1,… , 12
• For these scenarios, the deterministic optimization is performed, giving design variables’ values 𝒙𝑘
init and values of objective function 𝑓𝑘init
• Probabilities 𝑝U𝑘init, 𝑝S𝑘
init that configuration 𝒙𝑘init satisfies the ULS and SLS, are assessed
using Neural Network on Monte Carlo Simulation
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Solution – 2. Iteration
• Regression analysis is applied on known scenarios to approximate behavior ofobjective function values and probabilities with regards to values of q and N
• Next scenario 𝝃𝑙iter is selected based on these approximations (so that it satisfies set
probabilities and has minimal objective function value)
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Solution – 2. Iteration
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f [CZK]
q [kN/m]N [kN]
4400
380035
30 10
5
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Solution – 2. Iteration
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pU []
q [kN/m]N [kN]
1
0
35
30 10
5
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Solution – 2. Iteration
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pS []
q [kN/m]N [kN]
1
0
35
30 10
5
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Solution – 2. Iteration
• The deterministic optimization is performed for the new scenario 𝝃𝑙iter, giving design
variables’ values 𝒙𝑙iter and values of objective function 𝑓𝑙
iter
• Also, probabilities 𝑝U𝑙iter, 𝑝S𝑙
iter that configuration 𝒙𝑙iter satisfies the ULS and SLS, are
assessed using Neural Network on Monte Carlo Simulation
• If 𝑝U𝑙iter ≥ 𝑝ULS and 𝑝S𝑙
iter ≥ 𝑝SLS• than 𝒙𝑙
iter is solution, end
• else, 𝝃𝑙iter is added to regression and solution continues with next iteration
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Solution – 2. Iteration
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f [CZK]
q [kN/m]N [kN]
4400
380035
30 10
5
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Solution – 2. Iteration
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pU []
q [kN/m]N [kN]
1
0
35
30 10
5
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Solution – 2. Iteration
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pS []
q [kN/m]N [kN]
1
0
35
30 10
5
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Results
• Results• desired probabilities 𝑝ULS = 99.99277 %, 𝑝SLS = 93.31928 %
• obtained probabilities 𝑝𝑈 = 99.9928 %, 𝑝𝑆 = 99.9324 %
• solution found for scenario 𝑞 = 34.4103 kN/m, 𝑁 = 5.3090 kN
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The End
Thank you for your attention
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