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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

12

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

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Solution – 2. Iteration

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pU []

q [kN/m]N [kN]

1

0

35

30 10

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Solution – 2. Iteration

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pS []

q [kN/m]N [kN]

1

0

35

30 10

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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

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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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