Artificial Neural Network-based Optimized Design of Reinforced Concrete Structures

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Author
1 Introduction to Lagrange optimization for engineering applications
1.1 Significance of this chapter
1.2 An optimality formulation based on equality constraints
1.2.1 Formulation of Lagrange functions
1.2.2 Formulation of gradient vectors
1.2.3 Optimality conditions of objective functions constrained by equality functions based on gradient vectors; finding stationary points based on gradients vectors
1.2.4 Optimizations of an objective function constrained by equality functions
1.3 An optimality formulation based on inequality constraints
1.3.1 KKT (Karush-Kuhn-Tucker Conditions) optimality conditions
1.3.2 Formulation of KKT optimality conditions (active and inactive), their implications on economy and structural engineering
1.3.3 Optimality examples with inequality conditions
1.3.3.1 Example #1
Summary
1.3.3.2 Example # 2
1.3.3.3 Example # 3
1.4 How many KKT conditions (Kuhn and Tucker, 1951; Kuhn and Tucker, 2014) must be considered?
1.5 Conclusions
References
2 Lagrange optimization using artificial neural network-based generalized functions
2.1 Importance of an optimization for engineering designs
2.1.1 Significance of ANN-based optimization
2.1.2 Why ANN-based generalized functions?
2.2 ANN-based Lagrange formulation constrained by inequality functions
2.3 ANN-based generalizable objective and constraining functions
2.3.1 A limitation of an analytical function-based objective and inequality functions
2.3.2 Formulation of ANN-based Lagrange functions and KKT condition
2.3.3 Formulation of ANN-based objective and inequality functions
2.3.4 Linear approximation of a first derivative (Jacobi) of Lagrange functions
2.3.4.1 Optimization based on linearized Lagrange functions based on first-order (Jacobian matrix ∇ℒ(x[sup((k))], λ[sub(c)sup((k))], λ[sub(v)sup((k))] using Newton–Raphson iteration
2.3.4.2 Formulation of generalized Jacobian and Hessian matrices
2.3.4.3 Formulation of KKT non-linear equations based on Newton–Raphson iteration
2.3.5 Stationary points of Lagrange functions based on gradient vectors
2.3.6 ANN-based generalized functions replacing analytical functions
2.3.6.1 Formulation of Jacobian and Hessian matrices
2.3.6.2 Formulation of Jacobian matrix based on ANN
2.3.6.3 Formulation of universally generalizable Hessian matrix based on ANN
2.3.6.4 Flow chart for Lagrange-based optimization
2.3.6.5 Summary
2.4 Examples of optimizing Lagrange functions using ANN-based objective and constraining functions with KKT conditions
2.4.1 Purpose of examples
2.4.2 Optimization of a fourth-order polynomial with KKT conditions
2.4.2.1 Optimization of a fourth-order polynomial considering inequality constraints based on analytical objective and constraining functions
2.4.2.2 ANN-based optimization of a fourth-order polynomial constrained by inequality functions
2.4.2.3 Conclusions
2.4.3 A design of a truss frame based on Lagrange optimization
2.4.3.1 Lagrange optimization of a truss frame based on analytical objective and constraining functions
2.4.3.2 Lagrange optimization of a truss frame based on ANN-based object and constraining functions
2.4.3.3 Conclusions
2.4.4 Maximizing flying distance of a projectile based on Lagrange optimization
2.4.4.1 Analytical function-based Lagrange optimization
2.4.4.2 ANN-based Lagrange optimization
References
3 Design of reinforced concrete columns using ANN-based Lagrange algorithm
3.1 Introduction
3.1.1 Overview of Lagrange multiplier method-based KKT conditions
3.1.2 Optimization implemented in structural engineering
3.1.3 Significance of the chapter
3.2 ANN-based on Lagrange networks
3.2.1 Obtaining minimum design parameters for reinforced concrete columns based on ACI 318-19 and ACI 318-19
3.2.2 ANN-based functions including objective functions of RC columns
3.2.2.1 Weight and bias matrices based on forward ANNs to derive objective functions
3.2.2.2 Weight and bias matrices based on reverse ANNs to derive objective functions
3.2.2.3 Jacobian and Hessian matrices derived based on ANNs
3.2.2.4 Stationary points of Lagrange functions ℒ(x, λ[sub(c), λ[sub(v)]) subject to constraining conditions based on Newton–Raphson iteration
3.3 Optimization of column designs based on an Ann-based Lagrange algorithm
3.3.1 Column design scenario minimizing CI[sub(c)]
3.3.1.1 Formulation of Lagrange optimization based on forward network
3.3.1.2 Formulation of Lagrange optimization based on a reverse network
3.3.1.3 Verifications
3.3.1.4 P–M diagram
3.3.2 Column design scenario minimizing CO[sub(2)]
3.3.2.1 Formulation of forward network vs. reverse network
3.3.2.2 Solving KKT nonlinear equations based on Newton–Raphson iteration
3.3.2.3 Verifications
3.3.2.4 P–M diagram
3.3.3 Column design scenario minimizing weight
3.3.3.1 Formulation of forward network vs. reverse network
3.3.3.2 Solving KKT nonlinear equations based on Newton–Raphson method
3.3.3.3 Verifications
3.3.3.4 P–M diagram
3.3.3.5 Influence of optimization on P-M diagrams
3.4 Noticeable updates with ACI 318-19 compared with 318-14
3.5 Conclusions
References
4 Optimization of a reinforced concrete beam design using ANN-based Lagrange algorithm
4.1 Significance of the Chapter
4.1.1 Current research
4.1.2 Motivations and objective
4.1.3 Significance of the proposed methodology
4.2 Optimization of a reinforced concrete beam designs based on ANNs
4.2.1 Beam design scenarios
4.2.2 Formulation of a Lagrange function for optimizing a reinforced concrete beam based on ANNs
4.2.2.1 Derivation of ANN-based objective functions
4.2.2.2 Derivation of ANN-based Lagrange functions
4.2.2.3 Formulation of KKT conditions based on equality and inequality constraints
4.3 Generation of Large Structural Datasets
4.3.1 Input and output parameters selected for large datasets
4.3.2 Random design ranges
4.3.3 Network training based on parallel training method (PTM) training
4.3.4 Training for forward Lagrange networks
4.3.5 Training for rebar placements with multiple layers
4.4 Network Verification
4.4.1 Verification of design parameters based on a forward Lagrange network
4.4.2 Verification of Selected parameters based on large datasets
4.4.3 Cost savings based on Lagrange algorithm
4.5 Design Charts Based on ANN-Based Lagrange Optimizations Minimizing CI[sub(b)]
4.5.1 Optimization of the cost (CI[sub(b)]) for material and manufacture for design ductile beam sections based on design charts
4.5.2 Use of design charts to design ductile beam sections
4.5.3 Verification of optimization
4.6 Use of ANN-Based Lagrange Networks to Investigate Changes between ACI 318-14 and ACI 318-19
4.6.1 ACI 318-19
4.6.1.1 Revised limit of tension-controlled sections
4.6.1.2 Reduction in effective moment of inertia for ACI 318-19
4.6.2 The Comparisons between ACI 318-14 and ACI 318-19 Based on Conventional Structural Calculations
4.6.3 Changes of Optimized Results between ACI 318-14 and ACI 318-19 Using ANNs
4.7 Results and Discussions
4.7.1 ANN-based formulation of objective functions
4.7.2 Design charts obtained based on Lagrange networks optimizing cost (material and manufacture) of ductile doubly reinforced concrete beams
4.7.3 Verifying optimized objective functions
4.7.4 ANN-based structural designs beyond human efficiency
4.8 Conclusions
References
5 ANN-based structural designs using Lagrange multipliers optimizing multiple objective functions
5.1 Introduction
5.1.1 Significance of optimizing multiple objective functions
5.1.1.1 Previous studies
5.1.1.2 Problem Descriptions and Motivations of the Chapter
5.1.1.3 Significance of optimizing UFOs
5.1.1.4 Contents of Chapter 5
5.1.2 Review of Pareto frontier
5.1.3 Criterion space and Pareto frontier
5.1.4 Weighted sum method
5.1.4.1 The first method - minimization of bi-objective functions based on a definition of nondominated points
5.1.4.2 The second method - minimizing bi-objective functions (UFO) based on weighted sum method
5.1.5 Normalized unified function of objectives implementing weighted sum method
5.1.5.1 Normalized UFOs implementing weighted sum method
5.1.5.2 Discussion on normalized objective and nonnormalized functions
5.2 ANN-based Lagrange functions optimizing multiple objective functions
5.2.1 Significance of considering UFO
5.2.2 Unified function of objectives
5.2.3 ANN-based Lagrange optimization algorithm of five steps based on UFO
5.3 ANN-based Lagrange optimization design of RC circular columns having multiple objective
5.3.1 Forward design of circular RC columns
5.3.2 Optimization design scenarios
5.3.3 Five steps to optimize circular RC column based on three-objective functions
5.3.4 Discussions on an optimization based on three objective functions
5.3.5 Verification to large datasets
5.3.6 Generation of evenly spaced fractions
5.3.7 Interpretation of data trend
5.3.7.1 Relationships among three objective functions
5.3.7.2 Exploring trend of large datasets
5.3.8 Examples of optimal designs based on Pareto frontier
5.3.8.1 Identifying design parameters for a designated fraction
5.3.8.2 Optimized P-M diagram
5.3.9 Decision-making based on Pareto frontier
5.4 An ANN-based optimization of UFO for circular RC columns sustaining multiple loads
5.4.1 Reusing components of weight matrices subject to one biaxial load pair (ANN-1LP) to derive weight matrices subject to multiple biaxial load pairs (ANN-nLP) load pairs
5.4.1.1 Generalized ANN (Model-LPs) used to derive n load pairs (P[sub(u,i)], M[sub(u,i)])
5.4.1.2 Formulation of the Network subject to multi-load pairs
5.4.2 An optimization of a circular RC column sustaining five load pairs based on three-objective functions
5.4.2.1 Optimization design scenario
5.4.2.2 Five steps to optimize a circular RC column sustaining five load pairs based on three-objective functions
5.4.3 Verification of Pareto frontier based on large datasets
5.5 ANN-based Lagrange optimization for UFO to design uniaxial rectangular RC columns sustaining multiple loads
5.5.1 Optimization scenario based on a forward design
5.5.2 Five-step optimization based on multiple objective functions
5.5.3 Verification of Pareto frontier to large dataset
5.5.4 Design parameters corresponding to three fractions of Pareto frontier
5.6 ANN-based Lagrange optimization for UFO to design biaxial rectangular RC columns sustaining multiple loads
5.6.1 Optimization scenario based on a forward design subject to multi-loads with small magnitude
5.6.2 Five steps optimization based on multiple objective functions
5.6.3 Verification of Pareto frontier based on large datasets
5.6.4 Design parameters corresponding to three points of Pareto frontier
5.6.5 An example of ANN-based Lagrange optimization design based on multi-objective functions for biaxial rectangular RC columns sustaining multiple loads with big magnitude
5.6.5.1 Identifying design parameters for designated fractions based on two neural networks based on Tables 5.6.2.3 and 5.6.5.6
5.6.5.2 Design accuracies based on the two neural networks based on Tables 5.6.2.3 and 5.6.5.6 for the two Pareto curves
5.7 ANN-based Lagrange multi-objective optimization design of RC beams
5.7.1 Design scenarios of doubly reinforced concrete beams
5.7.1.1 Selection of design parameters based on design criteria of doubly reinforced concrete beams
5.7.1.2 Selection of objective functions
5.7.1.3 An optimization scenario
5.7.2 Five steps to optimize a design of RC beams with which CI[sub(b)], CO[sub(2)] and W[sub(b)] are minimized
5.7.2.1 Step 1–Deriving ANNs
5.7.2.2 Step 2–Defining MOO problems
5.7.2.3 Step 3–Optimization based on a single-objective function
5.7.2.4 Step 4–Formulating UFO
5.7.2.5 Step 5–Optimizing UFO
5.7.3 Design parameters corresponding to various fractions on Pareto frontier
5.7.4 Verification of Pareto frontier
5.7.5 Decision-making based on the Pareto frontier
5.7.6 Interpretation of data trend based on relationships among three objective functions
5.8 Design recommendations and conclusions
5.8.1 Design recommendations
5.8.2 Conclusions
References
Acknowledgments
Appendix A
Appendix B
Appendix C
Appendix D
Index