Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm: Extensions and Applications 3031255720, 9783031255724

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Studies in Systems, Decision and Control 463

Ali Kaveh Ataollah Zaerreza

Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm Extensions and Applications

Studies in Systems, Decision and Control Volume 463

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Ali Kaveh · Ataollah Zaerreza

Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm Extensions and Applications

Ali Kaveh Department of Civil Engineering Iran University of Science and Technology Tehran, Iran

Ataollah Zaerreza Department of Civil Engineering Iran University of Science and Technology Tehran, Iran

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-031-25572-4 ISBN 978-3-031-25573-1 (eBook) https://doi.org/10.1007/978-3-031-25573-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Due to restrictions in available resources and the increase in human population, the significance of optimization continues to grow in the modern world. Engineers always seek to create structural systems that are both cost effective and robust enough to handle the most demanding functional requirements that may arise throughout their service life. The typical trial-and-error method to structural design is insufficient to provide solutions that fulfill both economic and safety criteria. Therefore, the computational method is developed for optimization. Metaheuristic algorithms are the most popular computational optimization tools. This book introduces a new metaheuristic algorithm named Shuffled Shepherd Optimization Algorithms and different versions of it. New methods for structural damage detection and reliability-based design optimization are also introduced. The concepts presented in this book are not only applicable to the design of skeletal structures, but can equally be used in different optimization techniques in civil engineering. These concepts are also applicable in the optimal design of other systems such as hydraulic and electrical networks. The authors and colleagues have been involved in various developments and applications of various metaheuristic algorithms to structural optimization in the last two decades. This book contains part of this research suitable for various aspects of optimization in civil engineering. The book is likely to be of interest to civil, mechanical, industrial, and electrical engineers who use optimization methods for design, as well as to those students and researchers in structural optimization who will find it to be necessary professional reading. Chapter 1 explains the purpose of the book and provides an overview of the remaining chapters. In Chap. 2, a newly developed multi-community metaheuristic optimization algorithm known as Shuffled Shepherd Optimization Algorithm (SSOA) is introduced, in which the agents are first decomposed into multicommunities, and the optimization process is then performed, mimicking the behavior of a shepherd in nature, operating on each community. In Chap. 3, the SSOA is modified to make it less dependent on parameter tuning. The new version is called parameter reduced SSOA requiring less parameters to be tuned. Chapter 4, v

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the Enhanced Shuffled Shepherd Optimization Algorithm, is presented. Shuffled Shepherd Optimization Algorithm is a swarm intelligence-based optimizer inspired by the herding behavior of shepherds in nature. In Chap 5, a new strategy, namely Boundary Strategy (BS), for the process of optimization-based damage detection, is presented. This strategy gradually neutralizes the effects of structural elements that are healthy in the optimization process. In Chap. 6, the discrete optimum design of two types of portal frames, including planar steel Curved Roof Frame and Pitched Roof Frame with tapered I-section members, is provided. Chapter 7 presents application of the SSOA in optimal design of castillated beams.The use of additional three algorithms is discussed in this chapter. In Chap. 8, an efficient graph-theoretical force method is presented. This graph-theoretical force method is utilized in the analysis of the frame structures to decrease the time required for optimization. Chapter 9 presents a new framework for reliability-based design optimization using metaheuristic algorithms based on decoupled methods. This framework is named sequential optimization and reliability assessment-double metaheuristic. Finally, in Chap. 10, the reliability-based design optimization of the frame structures using the force method and sequential optimization and reliability assessment-double metaheuristic framework is presented. We would like to take this opportunity to acknowledge a deep sense of gratitude to a number of colleagues and friends who have helped us in different ways in the process of writing this book. Our special thanks are due to Dr. Thomas Ditzinger, Editorial Director of Interdisciplinary and Applied Sciences and Engineering from Springer, and Series Editor of Studies in Systems, Decision and Control, Prof. Janusz Kacprzyk for the publication of our new book within Springer and the series, for their comments and suggestions during the preparation of this book. Our sincere appreciation is extended to our Springer colleagues who prepared the layout design of this book. We would also like to thank our colleagues, Mr. P. Almasi, Mr. A. Khodagholi, Mr. M. I. Karmi Dashtestani, Mr. S. M. Hosseini, and Mr. K. Biabani Hammedani, for their contribution to our shared knowledge. We would like to thank the publishers who permitted some of our papers to be utilized in the preparation of this book, consisting of Springer, Elsevier, Emerald, and Budapest University of Technology and Economics. Our warmest gratitude is due to our families for their continued support in the course of preparing this book. Every effort has been made to render this book error free. However, the authors would appreciate any remaining errors being brought to their attention through the email addresses: [email protected] (Ali Kaveh) and a_zaerreza@ civileng.iust.ac.ir (Ataollah Zaerreza). Tehran, Iran September 2022

Ali Kaveh Ataollah Zaerreza

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Main Phase of the Metaheuristic Algorithms . . . . . . . . . . . . . 1.2.1 The Problem Definition Phase . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The Algorithm Parameter Definition Phase . . . . . . . . . . . 1.2.3 The Initialization Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The Main Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Structural Optimum Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Goals and Organization of the Present Book . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 3 4 4 4 6 8

2

Shuffled Shepherd Optimization Method: A New Meta-Heuristic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Shuffled Shepherd Optimization Algorithm . . . . . . . . . . . . . . . . . . 2.2.1 Inspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Steps of the Optimization Algorithm . . . . . . . . . . . . . . . . . 2.3 Validation of the SSOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Mathematical Optimization Problems . . . . . . . . . . . . . . . . 2.3.2 Engineering Optimization Problems . . . . . . . . . . . . . . . . . 2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The 25-Bar Spatial Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The 47-Bar Planer Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 The 72-Bar Spatial Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 The 120-Bar Dome Truss . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 A 272-Bar Transmission Tower . . . . . . . . . . . . . . . . . . . . . 2.4.6 A 1016-Bar Double-Layer Grid . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 12 12 14 16 17 17 27 28 29 30 32 34 39 43 50

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Shuffled Shepherd Optimization Method Simplified for Reducing the Parameter Dependency . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Shuffled Shepherd Optimization Algorithm (SSOA) . . . 3.2.2 Parameters Reduced Shuffled Shepherd Optimization Algorithm (PRSSOA) . . . . . . . . . . . . . . . . . 3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The 160-Bar Spatial Truss . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The 272-Bar Transmission Tower . . . . . . . . . . . . . . . . . . . 3.3.3 The 1016-Bar Double-Layer Grid . . . . . . . . . . . . . . . . . . . 3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 54 57 59 59 61 65 73 75

4

An Enhanced Shuffled Shepherd Optimization Algorithm and Application to Space Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Shuffled Shepherd Optimization Algorithm (SSOA) . . . . . . . . . . . 78 4.3 Enhanced Shuffled Shepherd Optimization Algorithm . . . . . . . . . 82 4.3.1 Enhancement on the Initialization Phase . . . . . . . . . . . . . 83 4.3.2 Enhancement on the Stepsize Part . . . . . . . . . . . . . . . . . . . 84 4.4 Statement of the Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5.1 A 693-Bar Double-Layer Barrel Vault . . . . . . . . . . . . . . . 87 4.5.2 A 1016-Bar Double-Layer Grid . . . . . . . . . . . . . . . . . . . . . 90 4.5.3 A 1410-Bar Dome Structure . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5

A New Strategy Added to the SSAO for Structural Damage Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Shuffled Shepherd Optimization Algorithm . . . . . . . . . . . . . . . . . . 5.2.1 Steps of SSOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Structural Damage Detection Approach . . . . . . . . . . . . . . . . . . . . . 5.3.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Proposed Objective Function . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 The Boundary Strategy (BS) in Metaheuristic-Based Damage Detection . . . . . . . . . . . 5.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 25-Bar Planar Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 40-Element Continuous Beam . . . . . . . . . . . . . . . . . . . . . . 5.4.3 A 23-Element Asymmetrical Planar Frame . . . . . . . . . . . 5.4.4 A 72-Bar Spatial Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 108 109 113 113 114 115 116 118 121 123 126 128 131

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Optimum Design of Curve Roof Frames by SSOA and Comparison with TLBO, ECBO, and WSA . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Metaheuristic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Teaching–Learning-Based Optimization (TLBO) . . . . . . 6.2.2 Enhanced Colliding Bodies Optimization (ECBO) . . . . . 6.2.3 Shuffled Shepherd Optimization Algorithm (SSOA) . . . 6.2.4 Water Strider Algorithm (WSA) . . . . . . . . . . . . . . . . . . . . 6.3 Statement of the Discrete Optimization Problem . . . . . . . . . . . . . . 6.3.1 Checking the Design Constraints of the Problem . . . . . . 6.3.2 Optimum Design of the Structures Using the SAP2000-OAPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Structural Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Load Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Vertical Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Lateral Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Discussion and Results for the Frames with L = 16.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Discussion and Results for the Frames with L = 32.0 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum Design of Castellated Beams Using SSOA and the Other Four Meta-Heuristic Algorithms . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Geometry of the Castellated Beams . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Design of Castellated Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Overall Beam Flexural Capacity . . . . . . . . . . . . . . . . . . . . 7.3.2 Beam Shear Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Flexural and Buckling Strength of Web Post . . . . . . . . . . 7.3.4 Vierendeel Bending of Upper and Lower Tees . . . . . . . . 7.3.5 Geometric Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Deflection of Castellated Beams . . . . . . . . . . . . . . . . . . . . 7.4 Castellated Beams Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Design of Castellated Beams with Circular Holes . . . . . . 7.4.2 Design of Castellated Beams with Hexagonal Opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Recently Developed Meta-Heuristic Algorithms . . . . . . . . . . . . . . 7.5.1 Shuffled Shepherd Optimization Algorithm (SSOA) . . . 7.5.2 Improved Shuffled Based JAYA Algorithm (IS-JAYA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Plasma Generation Optimization (PGO) . . . . . . . . . . . . . . 7.5.4 Set-Theoretical-Based Jaya Algorithm (ST-JA) . . . . . . . .

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133 133 134 134 136 138 140 143 144 147 149 149 149 150 153 156 156 161 167 169 169 170 173 173 174 175 175 177 177 178 179 180 182 182 182 183 183

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8

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Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Castellated Beam with 4-M Span . . . . . . . . . . . . . . . . . . . . 7.6.2 Castellated Beam with 8-m Span . . . . . . . . . . . . . . . . . . . . 7.6.3 Castellated Beam with 9-m Span . . . . . . . . . . . . . . . . . . . . 7.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 184 184 187 190 191

An Improved PSO Using the SRM of the ESSOA for Optimum Design of the Frame Structures via the Force Method . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Force Method of Frame Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Graph-Theoretical Force Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Optimization Problems with Discrete Design Variables . . . . . . . . 8.5 PSO-SRM Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Statistical Regeneration Mechanism (SRM) . . . . . . . . . . . 8.5.3 PSO-SRM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 The 1-Bay 10-Story Steel Frame . . . . . . . . . . . . . . . . . . . . 8.6.2 The 3-Bay 15-Story Steel Frame . . . . . . . . . . . . . . . . . . . . 8.6.3 The 3-Bay 24-Story Steel Frame . . . . . . . . . . . . . . . . . . . . 8.7 Discussion and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 194 195 197 197 198 198 199 199 201 201 203 208 217

An Efficient ESSOA for the Reliability Based Design Optimization Using the New Framework . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Formulation of Optimization and Reliability . . . . . . . . . . . . . . . . . 9.2.1 Formulation of RBDO Problem . . . . . . . . . . . . . . . . . . . . . 9.2.2 Sequential Optimization Together with Reliability Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 New Reliability-Based Design Optimization Framework . . . . . . . 9.3.1 Reliability Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Termination Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 SORA-Double-Metaheuristic . . . . . . . . . . . . . . . . . . . . . . . 9.4 SORA-DESSOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Enhanced Shuffled Shepherd Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Benchmark Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Structural Benchmark Examples . . . . . . . . . . . . . . . . . . . . 9.5.3 New Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

10 Reliability-Based Design Optimization of the Frame Structures Using the ESSOA and ERao . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Force Method of Structural Analysis . . . . . . . . . . . . . . . . . . . . 10.3 RBDO Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 No Constraint Most Probable Point Finder . . . . . . . . . . . . 10.3.2 Termination Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 SORA-DM Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Shuffled Shepherd Optimization Algorithms . . . . . . . . . . 10.4.2 Rao Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Enhanced Shuffled Shepherd Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Enhanced Rao Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 The 1-Bay 10-Story Steel Frame . . . . . . . . . . . . . . . . . . . . 10.5.2 The 3-Bay 15-Story Steel Frame . . . . . . . . . . . . . . . . . . . . 10.5.3 The 3-Bay 24-Story Steel Frame . . . . . . . . . . . . . . . . . . . . 10.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

1.1 Introduction The world we live in today is very competitive, and people are constantly striving to maximize their output or profit using a limited amount of resources that they have access to. As an example, in engineering design, it is essential to select design variables that satisfy all design criteria and have the lowest feasible cost. The primary purpose of engineering design is to adhere to fundamental standards while simultaneously achieving the best possible economic results. The optimization methods provide a solution for these kinds of problems [1]. Optimizing is the process of making the most of a situation or resource to the greatest extent possible. The optimization mathematically is defined as follows. minimi ze f i (x) i = 1, 2, 3, . . . n subject to μ j (x) = 0 j = 1, 2, 3, . . . , m ϕh (x) ≤ 0 h = 1, 2, 3, . . . , b ωg (x) ≥ 0 g = 1, 2, 3, . . . , v where the f i (x) called the objective function. μ j (x), ϕh (x), and ωg (x) are the constraint functions. Here, x is the design vector. If there is only one objective function for an optimization problem, it is called a single objective problem. In contrast, it is known as the multi-objective problem. The objective and constraint functions can have a simple formulation, allowing the optimal solution to be found using a simple program or manual method. The other type of the function is the one that does not have a simple formulation or not have © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_1

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1 Introduction

formulation (called the black box problem). To solve this kind of optimization problems, stochastic optimization techniques are created. Therefore, the optimization method can be divided into the gradient-based method and metaheuristic algorithms. As compared to stochastic methods, gradient-based methods converge faster and produce more accurate results. However, gradient-based methods are susceptible to local optima trapping and are dependent on the process’s initiation point, and they cannot be applied to the black box optimization problems. Metaheuristic algorithms do not have these shortage; in addition, they can be utilized for a variety of optimization issues and are simple to implement [2]. Therefore, metaheuristic algorithms have grown in favor during the past decades. The single metaheuristic algorithm is not capable of finding the optimum solution for all types of optimization problems [3]. Therefore, new metaheuristic algorithms have been developed by researchers. In addition, the performance of current approaches is improved by modifying their various components in response to researchers’ issues. The hybridization of current techniques is a strategy for enhancing the performance of the metaheuristic algorithms [4]. The four primary categories of metaheuristic algorithms are evolution-based, physics-based, swarm-based, and human-based, depending on their inspiration. Evolutionary algorithms are inspired by the characteristics of the biological evolution, including crossover, mutation, and selection. Inspiration of swarm intelligence algorithms is based on the social behavior of creatures living in a group, which might be a swarm, herd, or flock. The human-based algorithms consist of optimizers that simulate certain human behaviors. As the fourth class of metaheuristic algorithms, physics-based algorithms are motivated by physical rules [1]. Examples for each category are provided in Fig. 1.1. Metaheuristic algorithms have two phases of exploration (diversification) and exploitation (intensification) [5]. Exploring the optimization search space is performed to acquire a better solution, while the exploitation phase consists of searching close to the best answer thus far discovered. If the algorithm exhibits more exploration than exploitation, the metaheuristic algorithm cannot converge to the optimal solution. Alternatively, if the exploitation phase of the algorithm is greater

Fig. 1.1 Classification of nature-inspired metaheuristic algorithms

1.2 The Main Phase of the Metaheuristic Algorithms

3

than its exploration phase, the algorithm becomes trapped in local optima. Consequently, there should always be a constant equilibrium between the exploration and exploitation of the algorithms [6].

1.2 The Main Phase of the Metaheuristic Algorithms There are four main phases that should be considered when the metaheuristic algorithms are utilized to solve the optimization algorithms. These phases include problem definition, algorithm parameters definition, initialization phase, and the main loop of the optimization algorithms. Each phases are explained in the following sections.

1.2.1 The Problem Definition Phase The problem definition (also known as objective function definition) phase is the most crucial stage of optimization using the metaheuristic algorithms. To achieve the desired outcome, the optimization problem must be precisely coded. In addition, the problem is continuously analyzed determining the value of the objective function; and the majority run time of the algorithm is utilized during this phase. There are two types of optimization problems. The first type is the non-constraint objective function. Handling this type of objective function is straightforward, and there is no need for further explanation. Another type of optimization problem is constrained problems. Most of the real problems are of this type. To handle the constraint, different methods are developed. Two of these are described here. One of the simplest ways to consider the constraint is to add the penalty to the objective function value when the constraint function is not satisfied. The penalty value can be a constant value or a variable according to the constraint function. For example, in structural optimization problems with the constraint on the stress ratio, the value of the stress ratio is added to the objective function when it is more than one. The other way is using multi-objective optimization algorithms, where the constraint functions are considered as the objective functions, and the optimization process is performed. At the end of the optimization process, the solution which satisfies the constraint function is selected from the Pareto front.

1.2.2 The Algorithm Parameter Definition Phase All of the metaheuristic algorithms have at least some basic optimization parameters, which are defined by the user. The basic parameters of the metaheuristic algorithms is the population size and a maximum number of iterations (or a maximum number

4

1 Introduction

of function evaluations). Any metaheuristic algorithms that have only these two parameters are called the parameters less. The algorithms may contain additional parameters beyond the fundamental parameters. These parameters influence the main step size and have advantages and disadvantages. Adjustment of these parameters can help the algorithms to have good performance in a variety of optimization problems. On the other hand, parameter adjustment can be time-consuming.

1.2.3 The Initialization Phase The initialization phase of most of the optimization algorithms is the same. In this phase, the population (solutions) are randomly produced in the search space, and then they are evaluated. This randomly generated population is transformed for the next phase. After the random generation of the population in some optimization algorithms, further strategies are employed to improve the quality of the initialized population. For example, different opposition-based learning (OBL) techniques are utilized in the enhanced shuffled shepherd optimization algorithm (ESSOA).

1.2.4 The Main Loop In the main loop of the algorithms, one tries to find a new and better solution for the considered optimization problem. The main loop has at least one step size, and the algorithms use step size to search the optimization space. Also, the other mechanism can exist in the main loop. One of them is the replacement strategy. In the replacement strategy, the new solution is compared to its old solution, and the best of them are selected. The other mechanism is memory. The memory used in the algorithms that do not have the replacement strategy to hold the best solution found. The other type of mechanism that can be used in this phase is to build the algorithm multi-population (multi-communities). For example, the shuffling technique is utilized in the SSOA to have the multi-population algorithm. The shuffling process for the population size of NP is provided in Fig. 1.2.

1.3 Structural Optimum Design In this book, the field of structural optimization using metaheuristic algorithms is taken into consideration. Structural optimization refers to the process of finding the best or most efficient of (a) the size of structural elements (size optimization), (b) the shape of a structure (shape optimization), and (c) the connectivity between structural elements (topology optimization) [7]. In size optimization, design variables can be either continuous or discrete. Each design variable represents a cross-section

1.3 Structural Optimum Design

5

Fig. 1.2 A schematic of the shuffling technique

of either a member or a member element group. If design variables are discrete, they are selected from a list of discrete cross-sections. Nevertheless, when design variables are continuous, they can fluctuate constantly within an allowable range. In simultaneous size and layout optimization, design variables are divided into two overall sets. The first group of variables is related to discrete sizing variables, whereas the second group is related to continuous geometry variables. Using continuous and discrete design variables simultaneously can provide a different convergence rate and induce the optimization problem involved with the ill-conditioning. Being involved with ill-conditioned characteristics may result in a non-unique solution for the size and layout optimization of the structure. In simultaneous size and topology optimization, the optimization methods involve unacceptable and singular topologies resulting in an increase in the complexity and difficulty of the problem. The most common structural analysis tools are displacement and force methods. The displacement method utilizes the displacement of node structures as unknowns. Using the equilibrium and stress–strain equations, each member’s force is then calculated. In the force method, some members’ forces are selected as unknowns. Using the stress–strain and compatibility equations, the forces of each member are then calculated. The number of equations solved in the displacement method corresponds to the degree of kinematical indeterminacy (DKI), while the number of equations

6

1 Introduction

Fig. 1.3 Flowchart for the analysis and optimum design of structures

needed in the force approach corresponds to the degree of statical indeterminacy (DSI). In structures with low DSI, it is anticipated that the force method will be faster. On the other hand, it is anticipated that the displacement method will be faster for structures with a lower DKI. For further clarification, the optimum design of the structure using the force or displacement methods is provided in Fig. 1.3.

1.4 Goals and Organization of the Present Book The contribution of this book is concerned with the sizing optimization and simultaneous size/layout optimization of the benchmark and real-size structures using the shuffled shepherd optimization algorithm and its variant. The chapters of the book are organized into two parts. The first part is entitled ‘Extension’. In this part, the shuffled shepherd optimization algorithm is introduced first. Then, the two other versions of

1.4 Goals and Organization of the Present Book

7

SSOA, named parameter less SSOA (PR-SSOA) and enhanced SSOA (ESSOA), are introduced. The second part is entitled ‘Application’. In this part, the efficiency of the SSOA and its variant in the different problems investigated inducing the structural damage detection, optimum design of the portal frame, optimum design of castellated beams, and reliability-based design optimization. Additionally, the statistically regeneration mechanism (SRM), which is used to improve the SSOA, is added to the Rao algorithms and particle swarm optimization algorithms to enhance their performance. The remaining chapters of this book are organized as follows: Chapter 2 introduces a newly developed multi-community metaheuristic optimization algorithm. This algorithm is called the shuffled shepherd optimization algorithm (SSOA), in which the agents are first separated into multi-communities, and the optimization process is then performed mimicking the behavior of a shepherd in nature, operating on each community. The SSOA is tested with 17 mathematical benchmark optimization problems, 2 classic engineering problems, 5 truss design problems, and one double-layer grid design problem. The results show that SSOA is competitive with other considered metaheuristic algorithms [8]. Chapter 3 introduces the modified version of the shuffled shepherd optimization algorithm. Shuffled shepherd optimization algorithm is modified to make it less dependent of parameter tuning. The new version is called parameter reduced SSOA (PRSSOA) requiring less parameters to be tuned [9]. Chapter 4 presents the Enhanced Shuffled Shepherd Optimization Algorithm (ESSOA). Shuffled Shepherd Optimization Algorithm (SSAO) is a swarm intelligence-based optimizer inspired by the herding behavior of shepherds in nature. SSOA may suffer from some shortcomings, including being trapped in a local optimum and starting from a random population without prior knowledge. In order to solve these issues, SSOA is modified by two efficient mechanisms in this chapter. The first mechanism is the opposition-based learning (OBL) concept, which was first presented by Tizhoosh [10]. The OBL is used for improving the initialization phase of the SSOA. This is because it improves the convergence rate of the algorithm by giving prior knowledge about the search space. The second mechanism is introduced a new solution generator based on the statistical results of the solutions. The presented mechanism is called statistically regenerated step size. This mechanism provides a good exploration in the early iterations of the algorithm and causes the algorithm to escape from local optima in the last iterations [11]. Chapter 5 contains new strategy, namely Boundary Strategy (BS), for the process of optimization-based damage detection. This strategy gradually neutralizes the effects of structural elements that are healthy in the optimization process. BS causes the optimization method to find the optimum solution better than conventional methods that do not use the proposed BS. This technique improves both aspects of the accuracy and convergence speed of the algorithms in identifying and quantifying the damage [12]. Chapter 6 describes the discrete optimum design of two types of portal frames, including planar steel Curved Roof Frame (CRF) and Pitched Roof Frame (PRF) with tapered I-section members. The optimal design aims to minimize the weight of

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1 Introduction

these frame structures while satisfying some design constraints based on the requirements of ANSI/AISC 360-16 and ASCE 7-10. Four population-based metaheuristic optimization algorithms are applied to the optimal design of these frames. These algorithms consist of Teaching–Learning-Based Optimization (TLBO), Enhanced Colliding Bodies Optimization (ECBO), Shuffled Shepherd Optimization Algorithm (SSOA), and Water Strider Algorithm (WSA) [13]. Chapter 7 presents the optimum design of castellated beams utilizing the SSOA. The use of castellated beams has received much attention in recent decades, because these beams have holes in their webs, and the bending moment of the cross-section increases without increasing the weight of the beam. These beams are also more practical from an architectural point of view since installations, and plumbing can be passed through the holes of these beams used in the roofs [14]. Chapter 8 investigates an efficient graph-theoretical force method. A graphtheoretical force method is utilized in the analysis of the frame structures to decrease the time required for optimization. The performance and speed of the graphtheoretical force method are compared to those of the displacement method in the optimal design of frame structures. Additionally, the standard particle swarm optimization algorithm (PSO) is improved to enhance its performance in the optimal design of the steel frames [15]. Chapter 9 present a new framework for reliability-based design optimization (RBDO) using metaheuristic algorithms based on decoupled methods. This framework is named sequential optimization and reliability assessment-double metaheuristic (SORA-DM). The efficiency of the SOAR-DM is investigated using the enhanced shuffled shepherd optimization algorithm (ESSOA). The efficiency of the proposed framework is evaluated by six RBDO problems. The results show that the SORA-DM can have better performance than the gradient-based method in the RBDO and can easily be utilized in a wide range of RBDO problems [16]. Chapter 10 provides the reliability-based design optimization (RBDO) of the frame structures using the force method and sequential optimization and reliability assessment-double meta-heuristic framework (SORA-DM). In the SORA-DM, the meta-heuristic algorithm is utilized in both the optimization process and reliability assessment. The considered frames have a lower degree of statical indeterminacy than the degree of kinematical indeterminacy. The force method is used for the first time in the structural analysis of the RBDO problems [17].

References 1. Kaveh, A.: Advances in Metaheuristic Algorithms for Optimal Design of Structures, 3rd edn. Springer (2021) 2. Mirjalili, S., Mirjalili, S.M., Lewis, A.: Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014) 3. Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1(1), 67–82 (1997) 4. Blum, C., Raidl, G.R.: Hybrid Metaheuristics: Powerful Tools for Optimization. Springer (2016)

References

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5. Kaveh, A., Bakhshpoori, T.: Metaheuristics: Outlines, MATLAB Codes and Examples, Springer (2019) 6. Yang, X.-S.: Nature-Inspired Metaheuristic Algorithms. Luniver Press (2010) 7. Kaveh, A., Ilchi Ghazaan, M.: Meta-Heuristic Algorithms for Optimal Design of Real-Size Structures. Springer (2018) 8. Kaveh, A., Zaerreza, A.: Shuffled shepherd optimization method: a new meta-heuristic algorithm. Eng. Comput. 37(7), 2357–2389 (2020) 9. Kaveh, A., Zaerreza, A., Hosseini, S.M.: Shuffled shepherd optimization method simplified for reducing the parameter dependency. Iranian J. Sci. Technol. Trans. Civ. Eng. 45(3), 1397–1411 (2021) 10. Tizhoosh, H.R.: Opposition-based learning: a new scheme for machine intelligence. In: International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA-IAWTIC’06). IEEE (2005) 11. Kaveh, A., Zaerreza, A.,Hosseini, S.M.: An enhanced shuffled Shepherd optimization algorithm for optimal design of large-scale space structures. Engineering with Computers (2021) 12. Kaveh, A., Hosseini, S.M., Zaerreza, A.: Boundary strategy for optimization-based structural damage detection problem using metaheuristic algorithms. Periodica Polytech. Civ. Eng. 65(1), 150–167 (2021) 13. Kaveh, A., Karimi Dastjerdi, M.I., Zaerreza, A., Hosseini, M.: Discrete optimum design of planar steel curved roof and pitched roof portal frames using metaheuristic algorithms. Periodica Polytech. Civ. Eng. 65(4), 1092–1113 (2021) 14. Kaveh, A., Almasi, P., Khodagholi, A.: Optimum design of castellated beams using four recently developed meta-heuristic algorithms. Iranian J. Sci. Technol. Trans. Civ. Eng. (2022) 15. Kaveh, A., Zaerreza, A.: Comparison of the graph-theoretical force method and displacement method for optimal design of frame structures. Structures 43, 1145–1159 (2022) 16. Kaveh, A., Zaerreza, A.: A new framework for reliability-based design optimization using metaheuristic algorithms. Structures 38, 1210–1225 (2022) 17. Kaveh, A., Zaerreza, A.: Reliability-based design optimization of the frame structures using the force method and SORA-DM framework. Structures 45, 814–827 (2022)

Chapter 2

Shuffled Shepherd Optimization Method: A New Meta-Heuristic Algorithm

2.1 Introduction This chapter introduces a recently established multi-community metaheuristic optimization algorithm introduced by Kaveh and Zaerreza [1]. This algorithm is known as shuffled shepherd optimization algorithm (SSOA), in which the agents are first divided into multi-communities. Then the optimization procedure inspired from the behavior of a shepherd in nature is performed, on each community. The SSOA is tested on 17 mathematical benchmark optimization problems, 2 classic engineering problems, 5 truss design problems, and one double-layer grid design problem. The results demonstrate that SSOA is competitive with other considered metaheuristic algorithms. The term “optimization” refers to the study of problems in which one seeks to minimize or maximize a function by systematically selecting the values of variables from/within a permissible set. There are two categories of optimization techniques: meta-heuristic optimization algorithms and gradient-based optimization algorithms. Meta-heuristic optimization algorithms are becoming more and more popular in engineering applications because they: (i) rely on rather simple concepts and are easy to implement; (ii) do not require gradient information; (iii) can bypass local optima; (iv) can be applied in a wide range of problems covering different disciplines [2]. The objective of the majority of meta-heuristic algorithms is to transfer the worst agents into better positions using equations that prioritize good agents while disregarding the worst agents. Additional attention to worse agents may improve the overall performance of algorithms resulting in a better solution and/or using a fewer number of function evaluations. This chapter’s objective is to present a new multi-community and straightforward meta-heuristic algorithm that will be referred to as the Shuffled Shepherd Optimization Algorithm. This algorithm draws its motivation from the behavior of a shepherd, which makes use of the natural instincts of the animals under his care. In the initial

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_2

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2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

stage of this algorithm, the agents are partitioned into communities, and the optimization progress is inspired by the behavior of a shepherd in nature operating in each community. During the process of optimization, consideration is given to both good and bad agents; this ultimately results in an improvement in the algorithm’s overall performance. This chapter is organized as follows: In Sect. 2.2 inspiration, mathematical model and the steps for Shuffled Shepherd Optimization Algorithm are described. In Sect. 2.3, some benchmark functions and 2 classic engineering problems are investigated using the SSOA. In Sect. 2.4, five truss design problems and a largescale double-layer gird design problem are optimized utilizing the SSOA, and finally conclusions are derived in Sect. 2.5.

2.2 Shuffled Shepherd Optimization Algorithm 2.2.1 Inspiration In nature, animals utilize instinct to determine the best way to live. Human beings learn how to use animals’ instincts for their own goals. Shepherd utilizes animal instinct to determine the best route to the pasture. Always in a herd, a shepherd puts one horse to find the best stiff and fast way to pasture. Horses have an instinct to find the best stiff and fast way. In nature, we can see the trail of animals’ movement (ways). Horses or other animals always follow these quick and rigid routes. In addition to the use of this trail by shepherds, road engineers in the past have used this trail to build new roads. Shepherd put one or more horses in the herd to move their tools and find the way. Shepherd tries to guide sheep behind horses to pasture and bring them back because this trail is the best trail they can ever find. Shepherd’s behavior has been an inspiration in this chapter, and it is utilized for mathematical modeling of optimization.

2.2.2 Mathematical Model In this section, mathematical models of herd and shepherd are illustrated.

2.2.2.1

Herd

In nature, a district contains a large number of herds. The sheep in the herd are not the same, and each herd has both nice and bad sheep. These characteristics of the herd are identical to those of the community in a multi-communities method. Each community contains the worst and best agents compared to the others, similar to a

2.2 Shuffled Shepherd Optimization Algorithm

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Fig. 2.1 A schematic of the shuffling technique

herd. There are various approaches for constructing multi-community algorithms. In this chapter, the shuffling method is applied to have a multi-community algorithm. In the shuffling method, the entire population is divided into the communities at the start iteration of the optimization process, and then it merges into one population at the end of each iteration. Each community in the process of shuffling evolves individually, and their data is shared at the end of each iteration. After the initialization of the algorithm, the population is evaluated and sorted based on the fitness function. Then, the entire population is partitioned into the m communities (herd). To do this, the best m solutions (x1 , x2 , . . . , xm ) are selected and assigned to communities randomly. In the following, the next best solutions (xm+1 , xm+2 , . . . , x2m ) are randomly assigned to the communities. This process is repeated for other solutions until all of them are assigned to the communities. Figure 2.1 shows how to partition NP agents into m communities.

2.2.2.2

Shepherd

In the course of time, humans have learned how to use animal abilities for their own benefit. As an example, shepherds have used fast-riding horses to herd domesticated

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2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

sheep and cows. To do this, the shepherd tries to lead the animals toward the horses. This behavior is used as the step size of the algorithm introduced in this chapter. Each community member is selected in order, and the step size is calculated for each member. The selected member is named shepherd. A better and a worse member are chosen from the same community in which the shepherd belongs, based on their objective functions. The selected members are called the horse and the sheep, respectively. According to the herding behavior of the shepherd, first, the shepherd moves toward the sheep. The shepherd then leads the sheep toward the horse. This movement is the step size of the SSOA, and mathematically it is described as follows: ) ( Stepsizei = β × rand ◦ (X d −X i ) + α × rand ◦ X j −X i

(2.1)

in which X i, X d , and X j are solution vectors of shepherd, selected horse, and selected worse sheep in an m-dimensional search space, respectively; rand is a random vector in which each component is in range [0, 1]; α is a parameter set to α0 in the start of the algorithm, then decreases to zero by increasing iteration number of the algorithm and can be computed by Eq. (2.2); β is a parameter equal to β 0 in the start of the algorithm then increases to β max and β can be computed by using Eq. (2.3) and sign “z” represents element-by-element multiplication. The first sheep selected in the herd does not have better than itself, so the first term of the step size is set to zero; and for the last sheep selected in herd which does not have worse than itself, the second term of the step size is zero. Decreasing α and increasing β gradually reduce the exploration and increase the algorithm’s exploitation. ∝= α0 −

α0 × iteration maxiteration

(2.2)

β = β0 +

βmax − β0 × iteration maxiteration

(2.3)

After computing the step size for all sheep in a herd, the temporary solution vector is computed for each sheep by the following equation. temporary

xi

= xiold + stepsi zei

(2.4)

If temporary objective function is not worse than the old objective function, then temporary , otherwise xinew = the position of the sheep is updated, so we have xinew = xi old xi .

2.2.3 Steps of the Optimization Algorithm The flowchart of the SSOA is illustrated in Fig. 2.2, and the steps are as follows:

2.2 Shuffled Shepherd Optimization Algorithm Fig. 2.2 Flowchart of the SSOA

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Step 1: Initialization The SSOA parameters are defined, and the initial position of the ith agent is obtained randomly in an n-dimensional search space by the following equation: xi0 = xmin + rand ◦ (xmax − xmin ) i = 1, 2, . . . , N P

(2.5)

in which xi0 represents the initial solution vector of the ith sheep, xmax andxmin represent the bound of design variables, rand is a random vector with each component being in the rang [0, 1]; NP represents the total number of sheep. Step 2: Evaluations The value of the objective function for each sheep is evaluated. Step 3: Shuffling process The agents are divided into the communities based on the shuffling process illustrated in Sect. 2.2.2.1. Step 4: Calculate the step size The step size is computed for each agent as described in Sect. 2.2.2.2 employing Eq. (2.1). Step 5: Calculate the temple solution vector The temporary solution vector is computed utilizing Eq. (2.4), and the objective function is evaluated and referred to as temporary solution vector. Step 6: Update the agent and merge If the temporary objective function is not worse than the old objective function, then the position of the sheep is updated and combined the herds. Step 7: Update the parameters The values of ∝ and β are updated utilizing Eqs. (2.2) and (2.3). Step 8: Termination condition Steps 3 to 7 are repeated until the specified maximum number of iterations is reached.

2.3 Validation of the SSOA In order to verify the efficiency of the new algorithm, seventeen mathematical benchmark problems and two classic engineering design problems, five truss structures, and a double-layer grid are optimized employing the SSOA and compared to other algorithms. Section 2.3.1 examines mathematical problems, and Sect. 2.3.2 examines engineering challenges. Section 2.4 optimizes the double-layer grid and trusses.

2.3 Validation of the SSOA

17

2.3.1 Mathematical Optimization Problems The mathematical problems picked from Ref. [3] are given in Table 2.1. Like any other meta-heuristics algorithm, to have good performance and increase converge speed of the algorithm in the least computational cost, the number of agents and maximum iteration number should have balance to find the best computational cost for increasing converge speed of the algorithm. Here the number of herd and size of each herd is set to 4, and the maximum number of the permitted iterations is considered as 200. The value of α0 is considered as 0.5, β0 is set to 2.4, and βmax is taken as 2.8. However, for Rosenbrock, the values of β0 and βmax are set to 2 and 3.5, respectively. The result of the optimization of GA [3], CPA [4], CSS [5], and present work are compared in Table 2.2. Each mathematical function is optimized 50 times independently utilizing SSOA, and the mean numbers of function evaluations are reported in Table 2.2. The numbers in the parentheses indicate the ratio of the successful runs in which the algorithm has located the global minimum with predefined accuracy, which is taken as ε = f min − f max = 10−4 . The absence of the parentheses means that the algorithm has been successful in all independent runs. Table 2.2 demonstrates that SSOA has generally performed better than variations of GA, CPS, and CSS. Moreover, SSOA has superior performance than CPS and variants of GA in each function, except for BL and Rosenbrock.

2.3.2 Engineering Optimization Problems In this section, performance of the SSOA is investigated by two classic engineering problems, and the penalty method is employed for constraint handling. The maximum number of iterations is set to 400 for the welded beam design and set to 1000 for the pressure vessel design. Sensitivity analysis is performed to identify the best combination of the parameters of the SSOA. The values of α0 is taken as 0.5, 1, and 1.5; β0 is varied between 1.5 and 3.5 and βmax is altered between 2 and 4 with increments of 0.5. Consequently, 45 possible combinations of parameters are generated. For statistical compression mean and standard deviation are calculated for 30 independent runs of the SSOA for each problem.

2.3.2.1

The Welded Beam Design

The first classic engineering problem considers is the design optimization of the welded beam, as given in Fig. 2.3. The aim of this problem is to identify the minimum constructing cost of the welded beam subjected to constraints on shear stress (s), bending stress (r), buckling load (Pc), deflection (d) and side constraints. The design variables are the thickness of the weld h(=x 1 ), length of attached part of the bar l(=x 2 ), the height of the bar t(=x 3 ) and thickness of the bar b(=x 4 ).

)2

−1.0316 0.0

f (X ) = 2x12 − 1.05x15 + 16 x16 + x1 x2 + x22 ∑n 1 ∑n f (X ) = i=1 xi2 − 10 i=1 cos(5π x i )

X ∈ [−5, 5]

n = 4, X ∈ [−1, 1]n

X ∈ [−5.12, 5.12]3

n = 2, 4, 8, X ∈ [−1, 1]n

X ∈ [−2, 2]2

X ∈ [−100, 100]2

Cosine mixture

Dejoung

Exponential

Goldstein and price

Griewank

f (X ) = 1 + 1 200 2 i=1 x i

∑2 − i=1

2

cos

(

xi √ i

)

f (X ) = x12 + x22 + x32 ) ( ∑n f (X ) = −ex p −0.5 i=1 xi2 ( )] [ f (X ) = 1 + (x1 + x2 + 1)2 19 − 14x1 + 3x12 − 14x2 + 16x1 x2 + 3x22 30 ( ) + (2x1 − 3x2 )2 18 − 32x1 − 12x12 + 48x2 − 36x1 x2 + 27x22

cos(x1 ) + 10

Cb3

)

f (X ) = 4x12 − 2.1x14 + 13 x16 + x1 x2 − 4x22 + 4x24

2

1 8π

X ∈ [−5, 5]2

( + 10 1 −

Camel

5 π x1

−5 ≤ x1 ≤ 10 0 ≤ x5 ≤ 15

Branin

+

( f (X ) = x2 −

X ∈ [−10, 10]2

Becker and Lago 5.1 2 x 4π 2 1

f (X ) = f (X ) =

X ∈ [−50, 50]2

Bohachevsky 2

−0.352386

0.0

3

−1

0.0

−0.4

(continued)

0.397887

0.0

0.0

0.0

f (X ) = f (X ) =

X∈

Global minimum

X ∈ [−100, 100]2

1 2 1 1 2 1 4 4 x 1 − 2 x 1 + 10 x 1 + 2 x 2 4 7 3 2 2 x1 − 2x2 − 10 cos(3π x1 ) − 10 cos(4π x2 ) + 10 3 3 x12 − 2x22 − 10 cos(3π x1 ) cos(4π x2 ) + 10 (|x1 | − 5)2 + (|x2 | − 5)2

Aluffi-Pentiny

Function

Bohachevsky 1

Interval

[−10, 10]2

Function name

Table 2.1 Specification of the mathematical optimization problems

18 2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

X∈

X ∈ [−1, 1]2

X ∈ [−30, 30]n , n = 2

Hartman 3

Rastrigin

Rosenbrock

[0, 1]3

Interval

Function name

Table 2.1 (continued)

3 10 30

1

⎤

0.3689 0.117

0.2673

⎤

f (X ) =

i=1

∑2 (

) xi2 − cos(18xi ) ( )2 ∑n−1 f (x) = i=1 100 xi+1 − xi2 + (xi − 1)2

⎢ ⎥ ⎢ 0.4699 0.4387 0.747 ⎥ ⎥ p=⎢ ⎢ ⎥ ⎣ 0.1091 0.8732 0.5547 ⎦ 0.03815 0.5743 0.8828

⎡

⎡

⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥, c = ⎢ 1.2 ⎥and ⎥ ⎢ ⎥ ⎦ ⎣ 3 ⎦ 3.2 0.1 10 35

⎤

( ∑ ( )2 ) 3 i=1 ci ex p − j=1 ai j x j − pi j

∑4

⎢ ⎢ 0.1 10 35 a=⎢ ⎢ 3 10 30 ⎣

⎡

f (X ) = −

Function

0.0

−2.0

−3.862782

Global minimum

2.3 Validation of the SSOA 19

20

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.2 Performance comparison for the mathematical optimization problems Function

GEN–S–M [3]

GEN–S–M–LS [3]

CPA [4]

CSS [5]

Present work (SSOA [1])

AP

1277

1253

560

804

295

Bf1

1640

1615

1173

1187

472

Bf2

1676

1636

1376

742

451

BL

2439

1436

424

423

510

Branin

1404

1257

708

852

459

Camel

1336

1300

482

575

334

Cb3

1163

1118

548

436

266

CM

1743

1539

1612

1563

507

Dejoung

1462

1281

670

630

353

Exp2

817

807

435

132

123

Exp4

2054

1496

781

867

312

Exp8

2054

1496

1105

1426

673

Goldstein and price

1408

1325

805

682

430

Griewank

1764

1652 (0.99)

1572

1551

867

Hartman3

1332

1274

1128

860

326

Rastrigin

1392

1381

n/a

1402

960

Rosenbrock

1675

1462

n/a

1452

1810

Total

26,636

23,328(0.999)

12,869

15,584

9148

N/A not available Fig. 2.3 Schematic of the welded beam

2.3 Validation of the SSOA

21

The mathematical formulation of the optimization problem is defined as following: f cost (x) = 1.10471x12 x2 + 0.04811x3 x4 (14.0 + x2 ) To be minimized and constraints are g1 (x) = τ (x) − τmax ≤ 0 g2 (x) = σ (x) − σmax ≤ 0 g3 (x) = x1 − x4 ≤ 0 g4 (x) = 1.10471x12 x2 + 0.04811x3 x4 (14.0 + x2 ) − 5.0 ≤ 0 g5 (x) = 0.125 − x1 ≤ 0 g6 (x) = δ(x) − δmax ≤ 0 g7 (x) = P − Pc ≤ 0 In which √ τ (x) =

(τ ' )2 + 2τ ' τ ''

x2 + (τ '' )2 2R

P MR τ' = √ , τ '' = J 2x1 x2 √ ( ) ( x1 + x3 2 x22 x2 ) ,R = + M=P L+ 2 4 2 { [ ( ) ]} √ x22 x1 + x3 2 + 2x1 x2 J =2 12 2 4P L 3 6P L δ(x) = x4 x32 E x33 x4 √ ) √ x32 x46 ( 4.013E 36 E x3 Pc (x) = 1− L2 2L 4G σ (x) =

22

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

P = 6000 lb L = 14 in E = 30 × 106 psi G = 12 × 106 psi Variable boundaries are 0.1 ≤ x1 ≤ 2.0 0.1 ≤ x2 ≤ 10 0.1 ≤ x3 ≤ 10 0.1 ≤ x4 ≤ 2.0 Table 2.3 demonstrates that in almost all parameter combinations, the SSOA identifies near optimal solution. However, parameter adjustment is required for obtaining the best solution. In 7 parameters combination, SSOA identifies an optimal solution but the worst solution is different. This shows which parameters combination is appropriate for this problem. In parameter combination number 36 (α0 = 1.5, β0 = 2, βmax = 2.5) the worst solution is 1.724871, which is extremely near to the optimum value (1.724852). As a result, this parameter combination is appropriate for this problem. It can be seen from Table 2.4 that SSOA found minimum weight and constraints are g1 (x) = −9.048E − 07, g2 (x) = −6.979E − 04, g3 (x) = −1.779E − 08, g4 (x) = −3.433, g5 (x) = −0.801, g6 (x) = −0.236 and g7 (x) = −0.001. Therefore, g1 (x), g2 (x) and g3 (x) have controllers’ role. Table 2.5 demonstrates that the SSOA has a minimum average. The average of SSOA is less than the best solution of other methods except for MCSS [6] and IGMM [7].

2.3.2.2

Pressure Vessel Design

The second engineering problem investigated is the optimization of the pressure vessel, as illustrated in Fig. 2.4. The aim of this problem is to determine the minimum constructing cost of the pressure vessel. The design variables are the thickness of the shell T s (= x 1 ), thickness of the head T h (= x 2 ), the inner radius R (= x 3 ) and length of cylindrical section of vessel L (= x 4 ). The mathematical formulation of the optimization problem is defined as follows. f cost (x) = 0.6224x1 x3 x4 + 1.7781x2 x32 + 3.1661x12 x4 + 19.84x12 x3 To be minimized and the constraints are: g1 (x) = −x1 + 0.0193x3 ≤ 0

2.3 Validation of the SSOA

23

Table 2.3 Results of the sensitivity analysis for the welded beam problem No.

∝0

β0

βmax

Best

Worst

Mean

Std

1

0.5

1.5

2

1.960431

3.825033

2.667163

0.502

2

0.5

1.5

2.5

1.831520

3.005574

2.306228

0.308

3

0.5

1.5

3

1.724857

3.450027

2.073674

0.384

4

0.5

1.5

3.5

1.727421

2.220498

1.823992

0.124

5

0.5

1.5

4

1.724855

1.890547

1.771640

0.044

6

0.5

2

2.5

1.724852

1.994225

1.775230

0.083

7

0.5

2

3

1.724853

1.741982

1.726642

0.004

8

0.5

2

3.5

1.724856

1.753322

1.727385

0.005

9

0.5

2

4

1.724862

1.746769

1.728080

0.005

10

0.5

2.5

3

1.724854

1.726242

1.724955

2.68E-04

11

0.5

2.5

3.5

1.724872

1.726221

1.725263

3.85E-04

12

0.5

2.5

4

1.724870

1.732302

1.725973

0.002

13

0.5

3

3.5

1.724960

1.728071

1.725438

6.06E-04

14

0.5

3

4

1.725013

1.731427

1.726302

0.001

15

0.5

3.5

4

1.725214

1.744705

1.728456

0.004

16

1

1.5

2

1.724853

2.781531

2.178187

0.285

17

1

1.5

2.5

1.724852

2.570316

2.043011

0.240

18

1

1.5

3

1.724852

2.269081

1.872679

0.172

19

1

1.5

3.5

1.724853

1.907858

1.754319

0.043

20

1

1.5

4

1.724856

1.762913

1.729981

0.009

21

1

2

2.5

1.724852

1.729270

1.725003

8.06E-04

22

1

2

3

1.724853

1.725898

1.724937

2.55E-04

23

1

2

3.5

1.724861

1.725345

1.724928

9.98E-05

24

1

2

4

1.724888

1.727619

1.725423

7.03E-04

25

1

2.5

3

1.724856

1.725370

1.724949

1.01E-04

26

1

2.5

3.5

1.724891

1.727444

1.725278

5.49E-04

27

1

2.5

4

1.724890

1.728569

1.725809

8.20E-04

28

1

3

3.5

1.724954

1.727341

1.725683

6.79E-04

29

1

3

4

1.725115

1.730349

1.726995

0.001

30

1

3.5

4

1.725605

1.735186

1.728416

0.002

31

1.5

1.5

2

1.724852

2.307840

1.840313

0.162

32

1.5

1.5

2.5

1.724852

2.015799

1.750629

0.074

33

1.5

1.5

3

1.724853

1.865128

1.729554

0.026

34

1.5

1.5

3.5

1.724858

1.726011

1.724985

2.56E-04

35

1.5

1.5

4

1.724870

1.729607

1.725228

8.72E-04

36

1.5

2

2.5

1.724852

1.724871

1.724855

4.32E-06 (continued)

24

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.3 (continued) No.

∝0

β0

βmax

Best

Worst

Mean

Std

37

1.5

2

3

1.724857

1.724999

1.724894

3.83E-05

38

1.5

2

3.5

1.724871

1.725779

1.725027

2.17E-04

39

1.5

2

4

1.724999

1.728106

1.725488

6.41E-04

40

1.5

2.5

3

1.724868

1.725358

1.724989

1.21E-04

41

1.5

2.5

3.5

1.724885

1.726522

1.725429

3.76E-04

42

1.5

2.5

4

1.724991

1.730796

1.726354

0.001

43

1.5

3

3.5

1.725099

1.730144

1.726135

0.001

44

1.5

3

4

1.726169

1.732557

1.728059

0.002

45

1.5

3.5

4

1.726031

1.734503

1.729984

0.003

Table 2.4 Optimization results for the welded beam design problem Method

h(= x 1 )

l(= x 2 )

t(= x 3 )

b(= x 4 )

fcost

GA2 [8]

0.208800

3.420500

8.997500

0.210000

1.748310

ESs [9]

0.199742

3.61206

9.0375

0.206082

1.7373

RO [10]

0.203687

3.528467

9.00423

0.20724

1.735344

CDE [11]

0.203137

3.542998

9.033498

0.206179

1.733462

WOA [12]

0.205396

3.484293

9.037426

0.206276

1.730499

GA3 [13]

0.205986

3.471328

9.020224

0.206480

1.728226

CPSO [14]

0.202369

3.544214

9.04821

0.205723

1.728024

CE-CBA[15]

0.205726

3.47056

9.036630

0.20573

1.724858

IAFOA [16]

0.205726

3.470562

9.036630

0.20573

1.724856

IGMM [7]

0.205729

3.470496

9.306625

0.205730

1.724853

MCSS [6]

0.205729

3.470493

9.03662

0.20572

1.724853

SSOA [1]

0.2057296

3.4704888

9.0366236

0.2057297

1.7248524

g2 (x) = −x2 + 0.00954x3 ≤ 0 4 g3 (x) = −π x32 x4 − π x33 + 1,296,000 ≤ 0 3 g4 (x) = x4 − 240 ≤ 0 Variable boundaries are 0. ≤ x1 ≤ 99 0 ≤ x2 ≤ 99

2.3 Validation of the SSOA

25

Table 2.5 Statistical results of different methods for the welded beam problem Methods

Best

Mean

Worst

Std Dev

GA2 [8]

1.748309

1.771973

1.785835

0.011220

ESs [9]

1.737300

1.813290

1.994651

0.070500

RO [10]

1.735344

1.9083

N/A

0.173744

CDE [11]

1.733461

1.768158

1.824105

0.022194

GA3 [13]

1.728226

1.792654

1.993408

0.074713

CPSO [14]

1.728024

1.748831

1.782143

0.012926

CE-CBA [15]

1.724858

1.724858

1.724858

3.5641E-15

IAFOA [16]

1.724856

1.724856

1.424856

8.991E-0.7

IGMM [7]

1.724853

1.732152

1.74769

7.14E-03

MCSS [6]

1.724853

1.735438

1.753681

0.009527

SSOA [1]

1.724852

1.724855

1.724871

4.32E-06

Fig. 2.4 Schematic of the pressure vessel

10 ≤ x3 ≤ 200 10 ≤ x1 ≤ 200 Table 2.6 demonstrates that in almost all parameter combinations the SSOA can identify solutions that are close to optimum. In 7 combinations, SSOA able to determine the optimum solution, but in combination number 26 (α0 = 1, β0 = 2.5, βmax = 3.5) the difference between the worst and the best solution is equal to 0.611. To this end, this combination is the best parameters for this problem. Reducing α0 and increasing β0 value compared to the welded beam problem indicate that the pressure vessel problem needs fewer exploration and more exploitation compared to the welded beam problem. Table 2.7 compares the outcomes, whereas Table 2.8 compares the statistical outcomes of the current study with those of previous optimization techniques. Table

26

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.6 Results of the sensitivity analysis for the pressure vessel problem No

∝0

β0

βmax

Best

1

0.5

1.5

2

5937.9221

Worst 9678.2863

Mean 6784.5424

Std 937.052

2

0.5

1.5

2.5

6105.8884

7318.993

6642.038

392.250

3

0.5

1.5

3

5960.321

7407.2313

6589.6663

459.588

4

0.5

1.5

3.5

5957.5773

7167.1373

6272.9405

340.698

5

0.5

1.5

4

5887.6915

6737.8591

6074.1398

212.064

6

0.5

2

2.5

5885.5534

7110.4809

6058.9547

279.313

7

0.5

2

3

5885.4021

6538.2835

5942.1205

136.0747 25.609

8

0.5

2

3.5

5885.3516

6000.4418

5898.0198

9

0.5

2

4

5885.4596

5916.5175

5890.7515

8.903

10

0.5

2.5

3

5885.3417

5893.1437

5886.7253

1.869

11

0.5

2.5

3.5

5885.3491

5902.4701

5887.0622

3.170

12

0.5

2.5

4

5885.3328

8834.4409

5985.0822

538.170

13

0.5

3

3.5

5885.3318

8834.4091

5984.3788

538.288

14

0.5

3

4

5885.3374

8620.3227

6250.4736

9.453

15

0.5

3.5

4

5885.345

8834.4091

6264.8624

983.313

16

1

1.5

2

5987.6779

7305.9870

6403.9379

347.046

17

1

1.5

2.5

5885.8300

7283.3699

6304.6366

373.328

18

1

1.5

3

5885.6348

6940.9005

6113.3912

283.922

19

1

1.5

3.5

5885.3666

6299.4091

5940.4521

89.152

20

1

1.5

4

5885.5543

6291.8489

5920.6347

74.624

21

1

2

2.5

5885.3290

6043.9412

5894.7704

34.052

22

1

2

3

5885.3278

5890.2633

5885.7295

0.957

23

1

2

3.5

5885.3290

5896.7648

5886.4192

2.412

24

1

2

4

5885.3262

5890.5371

5885.7512

1.010

25

1

2.5

3

5885.3265

8893.5559

6076.9052

729.349

26

1

2.5

3.5

5885.3258

5885.9368

5885.4415

0.178

27

1

2.5

4

5885.3285

5889.6133

5885.5488

0.774

28

1

3

3.5

5885.3268

8627.6884

5976.8349

500.666

29

1

3

4

5885.3364

8893.5559

6359.7345

107.963

30

1

3.5

4

5885.3285

8893.5559

6359.5763

107.957

31

1.5

1.5

2

5885.3279

7282.0770

6114.4404

340.909

32

1.5

1.5

2.5

5885.3269

6730.1070

5966.3242

170.228

33

1.5

1.5

3

5885.3261

5986.5065

5899.7867

27.082

34

1.5

1.5

3.5

5885.3265

5900.9876

5886.7982

3.374

35

1.5

1.5

4

5885.3258

5929.4165

5887.9769

36

1.5

2

2.5

5885.3258

6103.6344

5893.6254

8.8108 39.922 (continued)

2.4 Numerical Examples

27

Table 2.6 (continued) No

∝0

β0

βmax

Best

37

1.5

2

3

5885.3258

Worst 5886.1050

Mean 5885.3677

Std 0.146

38

1.5

2

3.5

5885.3258

8834.4091

5983.6813

538.417

39

1.5

2

4

5885.3258

8619.9955

5976.5543

499.259

40

1.5

2.5

3

5885.3258

8634.9585

6250.5333

947.002

41

1.5

2.5

3.5

5885.3259

8834.4091

6364.6986

109.087

42

1.5

2.5

4

5885.3261

8893.5556

6266.4057

988.673

43

1.5

3

3.5

5885.3260

11,628.780 11,628.178

44

1.5

3

4

5885.3276

45

1.5

3.5

4

5885.3281

8893.5559

6624.1909

1456.87

6740.3234

1704.06

7012.4183

1405.91

Table 2.7 Optimization results for the pressure vessel problem Methods

Ts(= × 1)

Th(= × 2)

R(= × 3)

L(= × 4)

fcost

GA [17]

0.812500

0.437500

42.097398

176.654050

6059.9463

CPSO [14]

0.812500

0.437500

42.091266

176.746500

6061.0777

ESs [9]

0.812500

0.437500

42.098087

176.640518

6059.7456

CSS [5]

0.812500

0.437500

42.103624

176.572656

6059.0888

CDE [11]

0.812500

0.437500

42.0984

176.6376

6059.7340

IGMM [7]

0.812500

0.437500

42.098445

176.63659

6059.7143

GA3 [13]

0.812500

0.437500

42.0974

176.6540

6059.9463

IACO [18]

0.812500

0.437500

42.098353

176.637751

6059.7258

CE-CBA[15]

0.812500

0.437500

42.09984456

176.6365958

6059.7143

MCSS [6]

0.812500

0.437500

42.107406

176.525589

6058.6233

LWOA [19]

0.778858

0.385321

40.32609

200

5893.339

SSOA [1]

0.778179

0.384660

40.320140

199.999927

5885.3258

2.7 shows that the SSOA finds best results than other considered algorithms, and constraints are g1 (x) = −2.65E − 07, g2 (x) = −5.87E − 06, g3 (x) = −33.33 and g4 (x) = −40.00. This indicates that g1 (x) and g2 (x) have controlled optimization progress more than other constraints. Table 2.8 shows that the SSOA has the least average and standard deviation in comparison to the other algorithms.

2.4 Numerical Examples In this section, six numerical examples are provided to examine the performance of the SSOA on the optimum design of the structures. These examples are divided into three categories. The first two examples are the size and shape optimization

28

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.8 Statistical results of different methods for the pressure vessel problem Methods

Best

Mean

Worst

Std Dev

GA [17]

6059.95

6177.25

6469.32

130.9297

CPSO [14]

6061.08

6147.13

6363.8

86.4545

ESs [9]

6059.75

6850

7332.88

426

CSS [5]

6059.09

6067.91

6085.48

10.2564

CDE [11]

6059.73

6085.23

6371.05

43

IACO [18]

6059.7258

6081.7812

6150.1289

67.2418

CE-CBA [15]

6059.7143

6099.9218

6336.3404

104.25721

IGMM [7]

6059.7143

6060.1598

6061.2868

0.5421

MCSS [6]

6058.623

6073.5931

6108.4579

24.6712

LWOA [19]

5893.339

6223.765

7070.343

418.7902

SSOA [1]

5885.3258

5885.4415

5885.9368

0.178

Table 2.9 Parameters setting for the SSOA for truss optimization problems Problem

∝0

β0

βmax

Number of herds

Size of herds

Maximum iteration number

25-bar spatial truss

0.5

2.4

2.6

4

4

300

47-bar planer truss

0.5

2

2.3

4

5

1100

72-bar spatial truss

0.5

2.3

2.6

4

5

1000

120-bar dome truss 0.5

2.3

2.6

4

5

1000

272-bar transmission tower

0.5

2.0

2.3

4

5

700

1016-bar double-layer grid

0.5

2.3

2.7

4

5

600

of the truss structures. The third and fourth example is the size optimization of the truss structures with the frequency constraints, and the last two examples are the size optimization of the truss structure with the stress and displacement constraints. The results obtained for each example are compared to other optimization techniques. All numerical examples are run 30 times independently to provide statistically meaningful results. Parameter settings of the SSOA and the number of iteration limits on numeric examples are listed in Table 2.9.

2.4.1 The 25-Bar Spatial Truss The first example is layout optimization of the 25-bar spatial truss, as shown in Fig. 2.5. The optimization problem includes 13 design variables containing 8 discrete sizing variables for the cross-section areas and 5 continuous layout variables for nodal

2.4 Numerical Examples

29

Fig. 2.5 Schematic of the 25-bar spatial truss

coordinate. All members are subjected to a stress limitation of ± 40 ksi, and all nodal displacement in all directions is limited to ± 0.35 in. Optimization variables and input data of this truss are provided in Table 2.10. Table 2.11 compare the result obtained by the SSOA with the other methods. According to this table, SSOA has found the solution with the least number of analyses among the other algorithms. It shows that SSOA can easily escape from local optima and coverage to the optimum solution easily. The average weight and standard deviation for 30 independent runs of the SSOA are 122.4073 and 6.3443 lb, respectively. The optimum layout found by SSOA is shown in Fig. 2.6. Convergence curves for the best result and the mean performance of 30 independent runs for the 25-bar spatial truss are shown in Fig. 2.7.

2.4.2 The 47-Bar Planer Truss The 47-bar planer truss shown in Fig. 2.8 is optimized by different researchers for three load cases given in Table 2.12. The optimization problem includes 44 design variables containing 27 discrete sizing variables for the cross-section areas and 17 continuous layout variables for nodal coordinates. All members are subjected to stress limitation in tension and compression of 20 ksi and 15 ksi, respectively. Euler buckling stresses for compression members (the buckling strength of the ith element) are set to 3.96EA/L2 , and there is no limitation for node displacement. Optimization variables and input data of the truss are given in Table 2.12.

30

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.10 Simulation data for the 25-bar spatial truss Sizing variables A1 ; A2 = A3 = A4 = A5 ; A6 = A7 = A8 = A9 ; A10 = A11 ; A12 = A13 ; A14 = A15 = A16 = A17 ; A18 = A19 = A20 = A21 ; A22 = A23 = A24 = A25 Layout variables x4 = x5 = −x3 = −x6 ; x8 = x9 = −x7 = −x10 ; y3 = y4 = −y5 = −y6 ; y7 = y8 = −y9 = −y10 ; z3 = z4 = z5 = z6 Possible sizing variables Ai ∈ S = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, ( ) 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 3.0, 3.2, 3.4} in2 Layout variables bounds 20 in. ≤ x4 ≤ 60 in.; 40 in. ≤ x8 ≤ 80 in.; 40 in. ≤ y4 ≤ 80 in.; 100 in. ≤ y8 ≤ 140 in.; 90 in. ≤ z 4 ≤ 130 in.; Loads Nodes

Fx (kips)

Fy (kips)

Fz (kips)

1

1.0

−10

−10

2

0.0

−10

−10

3

0.5

0.0

0.0

6

0.6

0.0

0.0

Young modulus E = 104 (ksi) ( ) Material density ρ = 0.1 lb/in3

The comparison of the optimal design found by this work with optimum designs obtained by Salajegheh and Vanderplaats [26], Hasançebi and Erbatur [27, 28], and Panagant and Bureerat [29] is provided in Table 2.13. It can be seen that SSOA found the lightest weight (1869.876 lb) in less number of analyses (20,020), with average and standard deviation being 1929.91 lb and 29.55 lb, respectively. The optimum layout found by SSOA is shown in Fig. 2.9. Figure 2.10 shows the convergence curves for the best result and the mean performance of 30 independent runs for the 47-bar planar truss.

2.4.3 The 72-Bar Spatial Truss The third example is the 72-bar spatial truss with the frequency constraint. The structural members are divided into 16 groups, and their cross-sectional areas are selected from the range of [0.645, 4] cm2 . The material density and elastic modulus

2.4 Numerical Examples

31

Table 2.11 Optimum result for the 25-bar spatial truss Design variables

A1

Wu and Chow [20]

0.1

Kaveh and Kalatjari [21]

Tang et al. [22]

0.1

0.1

Rahami et al. [23]

0.1

Ho-Huu et al. [24]

Present work

R-ICDE

D-ICDE

SSOA [25]

0.2

0.1

0.1

A2

0.2

0.1

0.1

0.1

0.2

0.1

0.1

A6

1.1

1.1

1.1

1.1

0.9

0.9

1.0

A10

0.2

0.1

0.1

0.1

0.2

0.1

0.1

A12

0.3

0.1

0.1

0.1

0.2

0.1

0.1

A14

0.1

0.1

0.2

0.1

0.2

0.1

0.1

A18

0.2

0.1

0.2

0.2

0.2

0.1

0.1

A22

0.9

1.0

0.7

0.8

1.0

1.0

0.9

x4

41.07

36.23

35.47

33.0487

36.380

36.83

37.6762

y4

53.47

58.56

60.37

53.5663

57.080

58.53

54.4273

z4

124.6

115.59

129.07

129.9092

126.62

122.67

129.9991

x8

50.80

46.46

45.06

43.7826

48.200

49.21

51.9006

y8

131.48

127.95

137.06

136.8381

139.90

136.74

139.5535

Weight (lb)

136.20

124.0

124.943

120.115

145.275

118.76

117.2591

No. of analyses

N/A

N/A

6000

10,000

6000

6000

4816

Fig. 2.6 Comparison of optimized layout for the 25-bar spatial truss

32

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Fig. 2.7 Convergence histories of the optimization for the 25-bar spatial truss

are 2767.99 kg/m3 and 68.95 GPa. As shown in Fig. 2.11, the nonstructural masses are added to the last story nodes. There are two frequency constraints. The first frequency must be 4 Hz, and the minimum value of the third frequency is 6 Hz. The optimized designs found by Standard CSS [30], Enhanced CSS [30], HS [31], CBO [32], CS [33], WEO [33], CPA [33], and SSOA are compared in Table 2.14. SSOA has found better results compared to the other methods. Additionally, the statical result obtained using the SSOA is better than other considered methods. The first five natural frequencies of optimum design are given in Table 2.15. According to this table, frequency constraint is satisfied in all of the methods. The convergence history of the SSOA is given in Fig. 2.12.

2.4.4 The 120-Bar Dome Truss The 120-bar dome truss is the second example with the frequency constraint considered in this chapter. The members are divided into the 7 groups, as shown in Fig. 2.13, and their cross-sectional areas are varied between 1 and 129.3 cm2 . The material density is 7971.810 kg/m3 , and the modulus of elasticity is 210 GPa. The 3000 kg nonstructural masses are added at node 1. The 500 nonstructural masses are added at nodes 2–13, and 100 kg nonstructural masses are added to the remaining nodes. The frequency constraints are as f 1 ≥ 9Hz and f 2 ≥ 11H z

2.4 Numerical Examples

Fig. 2.8 Schematic of the 47-bar planer truss

33

34

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.12 Simulation data for the 47-bar planar truss

Sizing variables A3 = A1 ; A4 = A2 ; A5 = A6 ; A7 ; A8 = A9 ; A10 ; A12 = A11 ; A14 = A13 ; A15 = A16 ; A18 = A17 ; A20 = A19 ; A22 = A21 ; A24 = A23 ; A26 = A25 ; A27 ; A28 ; A30 = A29 ; A31 = A32 ; A33 ; A35 = A34 ; A36 = A37 ; A38 ; A40 = A39 ; A41 = A42 ; A43 ; A45 = A44 ; A46 = A47 Layout variables x2 = −x1 ; x4 = −x3 ; y4 = y3 ; x6 = −x5 ; y6 = y5 ; x8 = −x7 ; y8 = y7 ; x10 = −x9 ; y10 = y9 ; x12 = −x11 ; y12 = y11 ; x14 = −x13 ; y14 = y13 ; x20 = −x19 ; y20 = y19 ; x21 = −x18 ; y21 = y18 Possible sizing variables

( ) Ai ∈ S = {0.1, 0.2, 0.3, 0.4, . . . , 4.8, 4.9, 5.0} in2

Loads case

Nodes

Fx (kips)

Fy (kips)

1

17

6.0

−14.0

22

6.0

−14.0

2

17

6.0

−14.0

3

22

6.0

−14.0

Young modulus E = 3 × 104 (ksi) ) ( Material density ρ = 0.3 lb/in3

The comparison of the results of the SSOA with the other optimization method is provided in Table 2.16. The best weight is 8707.32 kg which is found by the SSOA. The SSOA also finds the minimum average weight. In the term standard deviation, the best result obtained by the OMGSA and the SSOA is second place. According to Table 2.17, all of the constraints is satisfied in the optimum weight found by the SSOA. The convergence history of the SSOA is given in Fig. 2.14.

2.4.5 A 272-Bar Transmission Tower The 272-bar transmission tower presented by Kaveh and Massoudi [37],as shown in Fig. 2.15. All nodal coordinates and end nodes of the member are presented in Ref. [37]. Members are divided into 28 groups because of symmetry, as shown in Fig. 2.15. In this chapter, we imposed 11 load cases for the basic load case, as shown in Table 2.18. The displacement of nodes 1, 2, 11, 20, 29 in Z-direction is limited to 20 mm, and in X- and Y-directions is limited to 100 mm. The modulus of elasticity is 2 × 108 kN/m2 , and the maximum available stresses for all member is ± 275,000 kN/m2 .

2.4 Numerical Examples

35

Table 2.13 Optimum result for the 47-bar planar truss Design variables

Salajegheh and Vanderplaats [26]

Hasançebi and Erbatur [27]

Hasançebi and Erbatur [28]

Panagant and Bureerat [29]

Present work SSOA [25]

A3

2.61

2.5

2.5

2.7

2.8

A4

2.56

2.2

2.5

2.6

2.5

A5

0.69

0.7

0.8

0.7

0.7

A7

0.47

0.1

0.1

0.1

0.1

A8

0.80

1.3

0.7

0.8

1.0

A10

1.13

1.3

1.3

1.2

1.1

A12

1.71

1.8

1.8

1.7

1.8

A14

0.77

0.5

0.7

0.8

0.7

A15

1.09

0.8

0.9

0.9

0.8

A18

1.34

1.2

1.2

1.3

1.5

A20

0.36

0.4

0.4

0.3

0.4

A22

0.97

1.2

1.3

1.0

1.0

A24

1.00

0.9

0.9

1.0

1.1

A26

1.03

1.0

0.9

1.0

1.0

A27

0.88

3.6

0.7

0.9

5.0

A28

0.55

0.1

0.1

0.1

0.1

A30

2.59

2.4

2.5

2.6

2.7

A31

0.84

1.1

1.0

0.9

0.9

A33

0.25

0.1

0.1

0.1

0.1

A35

2.86

2.7

2.9

2.8

3.0

A36

0.92

0.8

0.8

1.1

0.8

A38

0.67

0.1

0.1

0.1

0.1

A40

3.06

2.8

3.0

3.0

3.2

A41

1.04

1.3

1.2

1.1

1.1

A43

0.10

0.2

0.1

0.1

0.1

A45

3.13

3.0

3.2

3.1

3.3

A46

1.12

1.2

1.1

1.1

1.1

x2

107.76

114

104

109.61

100.5396

x4

89.15

97

87

93.078

81.0279

y4

137.98

125

128

126.65

137.2003

x6

66.75

76

70

70.752

63.8334

y6

254.47

261

259

246.32

254.1838

x8

57.38

69

62

56.172

56.1445

y8

342.16

316

326

356.26

327.9040

x10

49.85

56

53

48.498

48.2708 (continued)

36

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.13 (continued) Design variables

Salajegheh and Vanderplaats [26]

Hasançebi and Erbatur [27]

Hasançebi and Erbatur [28]

Panagant and Bureerat [29]

Present work

y10

417.17

414

412

436.37

407.5132

x12

44.66

50

47

42.37

42.4458

y12

475.35

463

486

490.66

468.8267

x14

41.09

54

45

41.61

45.8692

y14

513.15

524

504

521.04

515.2907

x20

17.90

1.0

2.0

1.4026

0.0010

y20

597.92

587

584

597.36

586.9443

x21

93.54

99

89

95.312

80.7351

y21

623.94

631

637

625.99

621.5769

Weight (lb)

1900

1925.79

1871.7

1871.7

1869.876

100,000

N/A

187,488

22,020

No. of analyses Fig. 2.9 Comparison of optimized layout for the 47-bar planar truss

SSOA [25]

2.4 Numerical Examples

37

Fig. 2.10 Convergence histories of the optimization for the 47-bar planar truss

Fig. 2.11 Schematic of the 72-bar spatial truss

The optimum volume found by the SSOA is presented in Table 2.19. SSOA found optimum volume after 14,020 analyses. The maximum stress ratio is 0.7639, which has happened in load case 10 in element 169, and the maximum displacement is − 20 mm for node 11 in Z-direction in load case 3. Displacement for nodes 1, 2, 11, 20, 29 are shown in Fig. 2.16. Figure 2.17 shows the convergence curves for the best result and the mean performance of 30 independent runs for the 272-bar transmission tower.

38

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.14 Optimum result for the 72-bar spatial truss Members Standard Enhanced HS [31] CSS CSS [30] in the group [30]

CBO [32]

CS [33]

WEO [33]

CPA [33] Present work SSOA

1–4

2.528

2.522

3.6803

3.7336

3.2273

3.4301

3.7438

3.4900

5–12

8.704

9.109

7.6808

7.9355

7.7472

7.8474

7.8904

8.0313

13–16

0.645

0.648

0.6450

0.6450

0.6450

0.6508

0.6450

0.6455

17–18

0.645

0.645

0.6450

0.6450

0.6450

0.6527

0.6450

0.6452

19–22

8.283

7.946

9.4955

8.3765

8.4921

8.0996

8.6836

8.1095

23–30

7.888

7.703

8.2870

8.0889

8.0895

7.9830

8.0373

7.8763

31–34

0.645

0.647

0.6450

0.6450

0.6450

0.6543

0.6450

0.6454

35–36

0.645

0.6456

0.6461

0.6450

0.6749

0.6502

0.6457

0.6452

37–40

14.666

13.465

11.4510

12.9491

12.9831

12.6499

12.2459

12.5289

41–48

6.793

8.250

7.8990

8.0524

7.9272

8.0932

8.1989

8.0794

49–52

0.645

0.645

0.6473

0.6450

0.6450

0.6587

0.6450

0.6450

53–54

0.645

0.646

0.6450

0.6450

0.6476

0.6555

0.6450

0.6490

55–58

16.464

18.368

17.4060

16.6629

17.0308

17.5009

17.0873

17.5636

59–66

8.809

7.053

8.2736

8.0557

8.3732

8.1990

7.9965

8.1336

67–70

0.645

0.645

0.6450

0.645

0.6450

0.6489

0.6450

0.6466

71–72

0.645

0.646

0.6450

0.645

0.6450

0.6878

0.6450

0.6456

Best weight (kg)

328.814

328.393

328.334

327.740

327.87

327.86

327.74

327.62

Average weight (kg)

337.70

335.77

332.64

328.20

342.98

352.80

338.93

327.71

5.42

7.20

2.39

0.54

44.31

65.25

34.60

Standard deviation

0.0697

Table 2.15 Natural frequencies (Hz) evaluated at the optimum designs of the 72-bar spatial truss Frequency number

Standard CSS [30]

Enhanced CSS [30]

HS [31]

CBO [32]

CS [33]

WEO [33]

CPA [33]

Present work SSOA

1

4.000

4.000

4.0000

4.000

4.0003

4.0000

4.0000

4.0000

2

4.000

4.000

4.0000

4.000

4.0003

4.0000

4.0000

4.0000

3

6.006

6.004

6.0000

6.000

6.0001

6.0002

6.0000

6.0000

4

6.210

6.155

6.2723

6.267

6.2502

6.2614

6.2696

6.2452

5

6.684

8.390

9.0749

9.101

9.0143

9.0780

9.0981

9.0761

2.4 Numerical Examples

39

Fig. 2.12 Convergence histories of the optimization for the 72-bar spatial truss

2.4.6 A 1016-Bar Double-Layer Grid Design optimization of double-layer grid with the configuration of a square on diagonal grid investigated in this section, as shown in Fig. 2.18. A span is 40 × 40 m, and height is equal to 3 m. All connections assumed to be ball jointed. This grid has 1016 members and 320 nodes, and simple support conditions are employed for the bottom layer at the nodes demonstrated in Fig. 2.18a. Each top layer joint is subjected to a concentrated vertical load of 30 kN. The elements are divided into 25 groups which are selected from the list of steel pipe sections in the manual of steel construction, as given in Table 2.20. The modulus of elasticity is 205 GPa, the yield stress is 248.2 MPA, and the material density is 7833.413 kg/m3 . Displacement limitation is 20/3 cm were imposed on all nodes in the vertical direction, and limitation on stress and stability of truss elements are imposed according to the provisions of AISC 360-10 [38] as follows. Tension member constraint { pu ≤ pr ; pr = min

∅t Fy A g ; ∅t = 0.9 ∅t Fu Ae ; ∅t = 0.75

where pu is the required strength; pr is the nominal axial strength; A g is gross area of member; Ae effective net area; Fy specified minimum yield stress and Fu is specified minimum tensile strength. Compression member constraint pu ≤ pr ; pr = ∅c Fcr A g ; ∅c = 0.9

40

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Fig. 2.13 Schematic of the 120-bar dome truss

Fcr =

⎧( ) Fy ⎨ 0.658 Fe Fy ; ⎩ 0.877∗Fe ;

√ ≤ 4.71 FEy √ > 4.71 FEy

KL r

KL r

π2E Fe = ( )2 KL r

where Fe is elastic bulking stress; Fcr is critical stress; E is the modulus of elasticity; L laterally unbraced length of the member; r is radius of gyration and K is effective length factor taken equal to 1.

4.26

15.414 8890.48

89.38

14.249

7

12.758

Standard deviation

12.039

6

10.060

21.025

8895.99

9.964

5

9251.84

24.839

4

11.136

Average weight (kg)

11.492

3

41.290

19.607

9171.93

32.976

Democratic PSO [34]

Best weight (kg)

23.494

2

Standard PSO [34]

1

Design variable

Table 2.16 Optimum result for the 120-bar dome truss

38.33

8945.64

8890.69

15.1417

12.2866

9.8104

21.4601

11.6056

40.6757

19.7738

CBO [32]

7.195

8798.55

8727.28

14.960

12.190

10.795

20.664

10.218

41.418

19.043

IGSA [35]

1.183

8745.58

8724.97

15.877

11.738

9.603

20.563

9.989

39.294

20.263

OMGSA [35]

128.63

8916.9

8776.3

14.8123

13.4254

10.9003

20.4079

9.8161

44.4985

18.2999

BBO [36]

7.15

8718.5

8711.95

15.1282

11.6648

9.4245

21.0929

10.5496

39.8248

19.8878

EBBO [36]

1.977

8709.99

8707.32

14.7977

11.7851

9.8937

21.1206

10.6173

40.4707

19.4507

SSOA

Present work

2.4 Numerical Examples 41

42

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.17 Natural frequencies (Hz) evaluated at the optimum designs of the 120-bar dome truss Frequency Standard Democratic CBO number PSO [34] PSO [34] [32]

IGSA [35]

OMGSA BBO [35] [36]

EBBO [36]

Present work SSOA

1

9.0000

9.0001

2

11.0000

11.0007

9.000

9.001

9.002

11.000 11.003 11.003

9.0001

9.0000

9.0000

11.0007 11.0000 11.0000

3

11.0052

11.0053

11.000 11.003 11.003

11.0007 11.0002 11.0000

4

11.0134

11.0129

11.010 11.017 11.007

11.0015 11.0002 11.0000

5

11.0428

11.0471

11.049 11.089 11.076

11.0735 11.0657 11.0671

Fig. 2.14 Convergence histories of the optimization for the 120-bar dome truss

Slenderness ratio constraints KL ≤ 200; r KL ≤ 300; r

f or compr ession member f or tension member

Table 2.21 presents the optimum designs obtained by CBO [39], ECBO [39], and present work. Table 2.21 shows that SSOA has found the solution after 12,020 analyses that is 0.82% is higher than the best solution found by ECBO, but has least average and standard deviation than other solution obtained by other considered algorithms. Figure 2.19 shows the convergence histories of the best result and the mean performance of 30 independent runs for the 1016-bar double-layer grid.

2.5 Conclusions

43

Fig. 2.15 Schematic of the 272-bar transmission tower

2.5 Conclusions This chapter presents a new multi-community meta-heuristic algorithm called Shuffled Shepherd Optimization Algorithm (SSOA). This algorithm is inspired by the behavior of the shepherd. For the analysis of the exploration and exploitation ability of the algorithm 17 mathematical benchmark functions are analyzed and shown that the present algorithm works better than other considered algorithms. In order to show the efficiency and robustness of the SSOA, two classic engineering problems (i.e., design of welded beam and design of pressure vessel), five truss design problems (i.e., design of 25-bar spatial truss, 47-bar planer truss, 72-bar spatial truss, 120-bar dome truss, 272-bar transmission tower) and double-layer grid design problem (i.e., design of 1016-bar double-layer grid) are solved by the SSOA. Results indicate that SSOA can find the optimal solution with less number of analyses for some problems compared to other considered algorithms.

44

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.18 Loading condition for the 272-bar transmission tower Case 1

2

3

4

5

6

7

8

9

Force direction

nodes 1

2

11

20

29

Other free nodes

Fx (kN)

20

20

20

20

20

5

Fy (kN)

20

20

20

20

20

5

Fz (kN)

−40

−40

−40

−40

−40

0

Fx (kN)

0

20

20

20

20

5

Fy (kN)

0

20

20

20

20

5

Fz (kN)

0

−40

−40

−40

−40

0

Fx (kN)

20

0

20

20

20

5

Fy (kN)

20

0

20

20

20

5

Fz (kN)

−40

0

−40

−40

−40

0

Fx (kN)

20

20

20

0

20

5

Fy (kN)

20

20

20

0

20

5

Fz (kN)

−40

−40

−40

0

−40

0

Fx (kN)

20

0

0

0

0

5

Fy (kN)

20

0

0

0

0

5

Fz (kN)

−40

0

0

0

0

0

Fx (kN)

0

20

0

0

0

5

Fy (kN)

0

20

0

0

0

5

Fz (kN)

0

−40

0

0

0

0

Fx (kN)

0

0

0

20

0

5

Fy (kN)

0

0

0

20

0

5

Fz (kN)

0

0

0

−40

0

0

Fx (kN)

0

0

20

20

20

5

Fy (kN)

0

0

20

20

20

5

Fz (kN)

0

0

−40

−40

−40

0

Fx (kN)

0

20

20

0

20

5

Fy (kN)

0

20

20

0

20

5

Fz (kN)

0

−40

−40

0

−40

0 (continued)

2.5 Conclusions

45

Table 2.18 (continued) Case 10

11

12

Force direction

nodes 1

2

11

20

29

Other free nodes

Fx (kN)

0

0

20

0

20

5

Fy (kN)

0

0

20

0

20

5

Fz (kN)

0

0

−40

0

−40

0

Fx (kN)

0

0

0

20

20

5

Fy (kN)

0

0

0

20

20

5

Fz (kN)

0

0

0

−40

−40

0

Fx (kN)

0

0

20

20

0

5

Fy (kN)

0

0

20

20

0

5

Fz (kN)

0

0

−40

−40

0

0

Table 2.19 Optimum results for the 272-bar transmission tower Group number

Cross section area (mm2 ) Group number Cross section area (mm2 )

1

1000.551

15

9320.549

2

1240.013

16

1000.028

3

2491.871

17

1000.307

4

1017.829

18

1002.518

5

9618.809

19

8389.809

6

1000.000

20

1000.814

7

12,063.816

21

1000.004

8

1001.777

22

1003.288

9

1000.188

23

7982.259

10

1000.457

24

1000.445

11

10,217.022

25

1000.591

12

1000.064

26

1000.053

13

1000.015

27

7504.298

14

1000.005

28

1000.076

Volume

(cm3 )

1,168,200.624

Average volume (cm3 ) 1,168,668.715 Std (cm3 )

310.7557

46

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Fig. 2.16 Compression of allowable and existing displacements for the 272-bar transmission tower

Fig. 2.17 Convergence histories of the optimization for the 272-bar transmisson tower

2.5 Conclusions

47

(a)

(c )

(b)

(d)

Fig. 2.18 Schematic of the1016-bar double layer grid; a 3D view, b top layer members, c bottom layer members, and d web members

48

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.20 The steel pipe sections Area (cm2 )

No

Type

Nominal diameter (in.)

1

a ST

½

1.6129

Gyration radius (cm) 0.662432

2

b EST

½

2.064512

0.635

3

ST

¾

2.129028

0.846582

4

EST

¾

2.774188

0.818896

5

ST

1

3.161284

1.066038

6

EST

1

4.129024

1.034542

7

ST

1¼

4.322572

1.371346

8

ST

1½

5.16128

1.582166

9

EST

1¼

5.677408

1.331214

10

EST

1½

6.903212

2.003806

11

ST

2

6.903212

1.53543

12

EST

2

9.548368

1.945132

13

ST

2½

10.96772

2.41681

14

ST

3

14.387068

2.955798

15

EST

2½

14.5161

2.346452

16

c DEST

2

17.161256

1.782572

17

ST

3½

17.290288

3.395726

18

EST

3

19.483832

2.882646

19

ST

4

20.451572

3.835908

20

EST

3½

23.741888

3.318002

21

DEST

2½

25.999948

2.143506

22

ST

5

27.74188

4.775454

23

EST

4

28.451556

3.749548

24

DEST

3

35.290252

2.65811

25

ST

6

35.999928

5.700014

26

EST

5

39.419276

4.675124

27

DEST

4

52.25796

3.490976

28

ST

8

54.19344

7.462012

29

EST

6

54.19344

5.577332

30

DEST

5

72.90308

4.379976

31

ST

10

76.77404

9.342628

32

EST

8

82.58048

33

ST

12

94.19336

34

DEST

6

100.64496

5.236464

35

EST

10

103.87076

9.216898

7.309358 11.10361

(continued)

2.5 Conclusions

49

Table 2.20 (continued) No

Type

Nominal diameter (in.)

Area (cm2 )

Gyration radius (cm)

36

EST

12

123.87072

11.028934

37

DEST

8

137.41908

7.004812

a

ST standard weight b EST extra strong c DEST double-extra strong

Table 2.21 Results of optimization for the 1016-bar double-layer grid Element group

Sections CBO [39]

ECBO [39]

Present work (SSOA [1])

1

EST 5

EST 5

EST 5

2

DEST 3

EST 5

ST 5

3

ST 3 ½

ST 3

ST 4

4

ST 2 ½

ST 3 ½

EST 2 ½

5

ST 2 ½

ST 2 ½

ST 3 ½

6

ST 2

ST 2

EST 1 ½

7

ST 2

DEST 2

EST 1 ½

8

ST 2 ½

DEST 2

EST 1 ½

9

DEST 2 ½

EST 2

ST 4

10

DEST 2 ½

ST 6

DEST 2 ½

11

ST 1 ½

ST 2

ST 2 ½

12

DEST 5

EST 8

ST 10

13

EST 3 ½

EST 3 ½

EST 4

14

EST 3 ½

ST 5

ST 4

15

EST 4

ST 4

EST 4

16

ST 6

EST 5

ST 6

17

ST 5

ST 5

ST 5

18

EST 4

EST 5

EST 5

19

EST 5

EST 5

DEST 4

20

ST 8

ST 8

DEST 4

21

ST 6

ST 5

ST 6

22

ST 3

ST 3

ST 3 ½ (continued)

50

2 Shuffled Shepherd Optimization Method: A New Meta-Heuristic …

Table 2.21 (continued) Element group

Sections CBO [39]

ECBO [39]

Present work (SSOA [1])

23

EST 6

EST 2 ½

ST 3 ½

24

ST 3 ½

ST 5

ST 2 ½

25

EST 1 ½

ST 4

ST 3 ½

Weight (kg)

74,849

67,839

68,398

Average optimized weight (kg)

79,422

73,042

72,084

Standard deviation on average weight (kg)

8154

9158

1802

No. of structural analyses

9760

15,760

12,020

Fig. 2.19 Convergence histories of the optimization for the 1016-bar double-layer grid

References 1. Kaveh, A., Zaerreza, A.: Shuffled shepherd optimization method: a new meta-heuristic algorithm. Eng. Comput. 37(7), 2357–2389 (2020) 2. Kaveh, A.: Advances in Metaheuristic Algorithms for Optimal Design of Structures. 3rd Edn, Springer (2021) 3. Tsoulos, I.G.: Modifications of real code genetic algorithm for global optimization. Appl. Math. Comput. 203(2), 598–607 (2008) 4. Kaveh A., Zolghadr A.: Cyclical parthenogenesis algorithm: a new meta-heuristic algorithm (2017) 5. Kaveh, A., Talatahari, S.: A novel heuristic optimization method: charged system search. Acta Mech. 213(3–4), 267–289 (2010) 6. Kaveh, A., Share, M.A.M., Moslehi, M.: Magnetic charged system search: a new meta-heuristic algorithm for optimization. Acta Mech. 224(1), 85–107 (2013) 7. Varaee, H., Ghasemi, M.R.: Engineering optimization based on ideal gas molecular movement algorithm. Eng. Comput. 33(1), 71–93 (2017)

References

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8. Coello, C.A.C.: Use of a self-adaptive penalty approach for engineering optimization problems. Comput. Ind. 41(2), 113–127 (2000) 9. Mezura-Montes, E., Coello, C.A.C.: An empirical study about the usefulness of evolution strategies to solve constrained optimization problems. Int. J. Gen. Syst. 37(4), 443–473 (2008) 10. Kaveh, A., Khayatazad, M.: A new meta-heuristic method: ray optimization. Comput. Struct. 112, 283–294 (2012) 11. Huang, F.-Z., Wang, L., He, Q.: An effective co-evolutionary differential evolution for constrained optimization. Appl. Math. Comput. 186(1), 340–356 (2007) 12. Mirjalili, S., Lewis, A.: The whale optimization algorithm. Adv. Eng. Softw. 95, 51–67 (2016) 13. Coello, C.A.C., Montes, E.M.: Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv. Eng. Inform. 16(3), 193–203 (2002) 14. He, Q., Wang, L.: An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng. Appl. Artif. Intell. 20(1), 89–99 (2007) 15. Xiao, W., Liu, Q., Zhang, L., Li, K., Wu, L.: A novel chaotic bat algorithm based on catfish effect for engineering optimization problems. Eng. Comput. (2019) 16. Wu, L., Liu, Q., Tian, X., Zhang, J., Xiao, W.: A new improved fruit fly optimization algorithm IAFOA and its application to solve engineering optimization problems. Knowl.-Based Syst. 144, 153–173 (2018) 17. Deb, K.: Optimal design of a welded beam via genetic algorithms. AIAA J. 29(11), 2013–2015 (1991) 18. Kaveh, A., Talatahari, S.: An improved ant colony optimization for constrained engineering design problems. Eng. Comput. 27(1), 155–182 (2010) 19. Zhou, Y., Ling, Y., Luo, Q.: Lévy flight trajectory-based whale optimization algorithm for engineering optimization. Eng. Comput. 35(7), 2406–2428 (2018) 20. Wu, S.-J., Chow, P.-T.: Integrated discrete and configuration optimization of trusses using genetic algorithms. Comput. Struct. 55(4), 695–702 (1995) 21. Kaveh, A., Kalatjari, V.: Size/geometry optimization of trusses by the force method and genetic algorithm. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik: Appl. Math. Mech. 84(5), 347–357 (2004) 22. Tang, W., Tong, L., Gu, Y.: Improved genetic algorithm for design optimization of truss structures with sizing, shape and topology variables. Int. J. Numer. Meth. Eng. 62(13), 1737–1762 (2005) 23. Rahami, H., Kaveh, A., Gholipour, Y.: Sizing, geometry and topology optimization of trusses via force method and genetic algorithm. Eng. Struct. 30(9), 2360–2369 (2008) 24. Ho-Huu, V., Nguyen-Thoi, T., Nguyen-Thoi, M., Le-Anh, L.: An improved constrained differential evolution using discrete variables (D-ICDE) for layout optimization of truss structures. Expert Syst. Appl. 42(20), 7057–7069 (2015) 25. Kaveh, A., Zaerreza, A.: Size/layout optimization of truss structures using shuffled shepherd optimization method. Periodica Polytechnica Civ. Eng. 64(2), 408–421 (2020) 26. Salajegheh, E., Vanderplaats, G.N.: Optimum design of trusses with discrete sizing and shape variables. Struct. Optim. 6(2), 79–85 (1993) 27. Hasançebi, O., Erbatur, F.: Layout optimization of trusses using improved GA methodologies. Acta Mech. 146(1–2), 87–107 (2001) 28. Hasançebi, O., Erbatur, F.: On efficient use of simulated annealing in complex structural optimization problems. Acta Mech. 157(1–4), 27–50 (2002) 29. Panagant, N., Bureerat, S.: Truss topology, shape and sizing optimization by fully stressed design based on hybrid grey wolf optimization and adaptive differential evolution. Eng. Optim. 50(10), 1645–1661 (2018) 30. Kaveh A., Zolghadr A.: Shape and size optimization of truss structures with frequency constraints using enhanced charged system search algorithm. Asian J. Civil. Eng. 12(4), 487–509 (2011) 31. Miguel, L.F.F., Fadel Miguel, L.F.: Shape and size optimization of truss structures considering dynamic constraints through modern metaheuristic algorithms. Expert Syst. Appl. 39(10), 9458–9467 (2012)

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32. Kaveh, A., Ilchi Ghazaan, M.: Enhanced colliding bodies algorithm for truss optimization with frequency constraints. J. Comput. Civ. Eng. 29(6), 04014104 (2015) 33. Kaveh, A., Biabani, H.K., Barzinpour, F.: Optimal size and geometry design of truss structures utilizing seven meta-heuristic algorithms: a comparative study. Iran Univer. Sci. Technol. 10(2), 231–260 (2020) 34. Kaveh, A., Zolghadr, A.: Democratic PSO for truss layout and size optimization with frequency constraints. Comput. Struct. 130, 10–21 (2014) 35. Khatibinia, M., Sadegh, N.S.: Truss optimization on shape and sizing with frequency constraints based on orthogonal multi-gravitational search algorithm. J. Sound Vib. 333(24), 6349–6369 (2014) 36. Taheri, S.H.S., Jalili, S.: Enhanced biogeography-based optimization: a new method for size and shape optimization of truss structures with natural frequency constraints. Latin Am. J. Solids Struct. 13, 1406–1430 (2016) 37. Kaveh, A., Massoudi, M.S.: Multi-objective optimization of structures using charged system search. Scientia. Iranica. Trans. A. Civ. Eng. 21(6), 1845 (2014) 38. Committee, A.: Specification for structural steel buildings (ANSI/AISC 360-10). American Institute of Steel Construction, Chicago-Illinois (2010) 39. Kaveh, A., Ilchi Ghazaan, M.: Meta-Heuristic Algorithms for Optimal Design of Real-Size Structures. Springer (2018)

Chapter 3

Shuffled Shepherd Optimization Method Simplified for Reducing the Parameter Dependency

3.1 Introduction Shuffled shepherd optimization algorithm (SSOA) is a newly developed multicommunity algorithm inspired by shepherd behavior. Since the SSOA is a parameterdependent method, the optimum performance requires using the right parameters for each problem. In this chapter, SSOA is modified to become less reliant on parameter adjustment. The new version is called parameter reduced SSOA (PRSSOA), developed by Kaveh et al. [1], being less parameter dependent. In today’s extremely competitive world, one tries to get the most output or profit out of the few resources that are available. Optimization offers a suitable technique for this purpose. That is why popularity of optimization approaches is growing. Among different optimization methods, meta-heuristic algorithms have gained higher popularity due to not requiring gradient information. Engineers are constantly searching for the optimum possible solution and have devoted much attention to applications of optimization methods, especially metaheuristic algorithm. For example: Gandomi et al. [2] employed evolutionary strategy to optimum design of retaining wall. Du et al. [3] utilized Jaya algorithm for damage identification of structure. Upper bound strategy for metaheuristic based design optimization of steel frames is developed by Kazemzadeh Azad [4]. SSOA has one parameter for maximum number of iterations, two parameters for population and three parameters for computing the step size of the agents. These parameters help the algorithm in enhancing the performance of the SSOA when solving various issues. Nevertheless, fine-tuning parameters related to step size can take some time. Due of this, a new version of SSOA known as the parameters reduced shuffled shepherd optimization method (PRSSOA) is introduced in this chapter. This version of SSOA does not require tuning the step size parameters (PRSSOA). The rest of the chapter is organized as follows: In Sect. 3.2, gives an introduction of the SSOA and introduces the PRSSOA. Comparison of the performances of the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_3

53

54

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

SSOA, PRSSOA, and other methods for truss optimization is provided in Sect. 3.3. Finally, conclusions are derived in Sect. 3.4.

3.2 Optimization Algorithms This section introduces the parameter-reduced version of the SSOA after giving a quick description of the SSOA.

3.2.1 Shuffled Shepherd Optimization Algorithm (SSOA) Kaveh and Zaerreza [5] have introduced the Shuffled Shepherd Optimization Algorithms (SSOA), as a novel multi-community population-based metaheuristic. This optimization algorithm mimics from the behavior of shepherds in nature. Over time, humans have learned how to use animal instincts for their own purposes. Shepherd put horses or other animals in a herd utilizing the instinct of these animals to find the best way to pasture. For this purpose, the shepherd directs sheep toward horses. The SSOA’s fundamental inspiration is this behavior. In the SSOA, each candidate solution is considered as a sheep. SSOA begins with a randomly created sheep. To separate the sheep into herds, first the sheep are ordered according to the values of their objective functions and chosen with the size equal to “m” (number of herd) and randomly placed in each herd. Following that, the next m arranged sheep among the remaining sheep are randomly allocated in the next herd based on their quality values like the previously generated herd. Until every herd has formed, this process is repeated. This process is known as shuffling method which enhances survivability by the exchanging of information in the search process among herds. The step size for each sheep is determined in the following step so that each chosen sheep is referred to as a shepherd, and sheep with a better objective function than the shepherd is called horses. Therefore, each shepherd has both horses and sheep. Then for each shepherd, one worse sheep than shepherd and one horse are chosen randomly. The shepherd tries to guide sheep toward horse, and then step size is determined based on their movement (i.e., movement of shepherd to sheep and horse). The shepherd’s position is updated if the new one is not worse than the prior one (replacement strategy). This process is repeated for the sheep of all herds. Finally, the herds combine with each other. Repeatedly, the sheep are separated into herds, and aforementioned process is repeated until stopping criterion is met. For simplicity, according to the shuffled complex evolution (SCE) proposed by Duan et al. [6], herd and sheep can represent a community (complex) and a member of each community (complex), respectively. In the following, the steps of the SSOA are given briefly as follows: Step 1: Initialization

3.2 Optimization Algorithms

55

The initial member of community (MOC) in the search space is produced randomly by SSOA using the equation: M OCi,0 j = M OCmin + rand × (M OCmax − M OCmin ); i = 1, 2, . . . , m and j = 1, 2, . . . , n

(3.1)

in which rand is a random vector with each component being generated between 0 and 1; M OCmin and M OCmax represent respectively the lower and upper bounds of design variables; m is the number of communities, and n specifies number of members that belong to each community. In this regard, it can be said that the total number of member of communities are determined as: n MC = m × n

(3.2)

Step 2: Shuffling process In this procedure, first m members of the communities are chosen based on their objective function values and randomly assigned to the first column of multicommunity (MC) matrix (see Eq. (3.3)) as the first member of each community. Next, in order to form the second column of MC, the next m members are selected similarly to the previous phase and are randomly assigned to the column. Until the MC matrix is generated as follows, this process is carried out n times separately. ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ MC = ⎢ ⎢ ⎢ ⎢ ⎣

M OC1,1 M OC1,2 M OC2,1 M OC2,2 .. .. . . M OCi,1 M OCi,2 .. .. . .

. . . M OC1, j . . . M OC2, j .. .. . . . . . M OCi, j .. .. . .

. . . M OC1,n . . . M OC2,n .. .. . . . . . M OCi,n .. .. . .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(3.3)

M OCm,1 M OCm,2 . . . M OCm, j . . . M OCm,n It is important to note that each row of MC denotes the members of each community so that first column of MC contains the best members in each community. Moreover, The worst members of each community are those who are listed in the last column. Step 3: Movement of community members A unique stepsize of movement for each community member is computed based on two vectors. The first vector (i.e. stepsi zei,Wjor se ) shows the ability to explore new regions of the search space (diversification strategy). In contrast, the ) shows the capability to explore the area around second vector (i.e. stepsi zei,Better j of the previously visited promising search space regions (intensification strategy).

56

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

The stepsize can be mathematically stated in the following ways: stepsi zei, j = stepsi zei,Wjor se + stepsi zei,Better j i = 1, 2, . . . , m and j = 1, 2, . . . , n

(3.4)

in which stepsi zei,Wjor se and stepsi zei,Better are defined as follows: j ) ( stepsi zei,Wjor se = α × rand1 × M OCi,w − M OCi, j

(3.5)

) ( stepsi zei,Better = β × rand2 × M OCi,b − M OCi, j j

(3.6)

where rand1 and rand2 are random vector with each component being generated between 0 and 1; M OCi,b (chosen horse) and M OCi,w (chosen sheep) are the better and worse members in terms of objective function value compared to M OCi, j (shepherd). It is important to note that the first member of the ith commuBetter nity (M OCi,1 ) does not have a member better than itself. Therefore, stepsi zei,1 is set to zero. On the other hand, M OCi,n does not have a member worse than W or se is itself because of the last member of the ith community so that stepsi zei,n set to zero. Additionally, α and β are the parameters that control exploration and exploitation, respectively. Following is a definition of these parameters: α = α0 − α0 × t; t =

iteration Max iteration

β = β0 + (βmax − β0 ) × t

(3.7) (3.8)

It is clear that when iteration number t increases, β and α increases and decreases, respectively. As a result, exploration rate declines while exploitation rate raises. Step 4: Updating position of each community member According to the previous step, the new position of the M OCi, j is computed utilizing Eq. (3.9). After that, the position of M OCi, j will be modified if it is not worse than its old objective function value. newM OCi, j = M OCi, j + stepsi zei, j

(3.9)

Step 5: Checking termination conditions After a fixed number of iterations as stopping criterion (Maxiteration), the optimization process will be ended. Otherwise, it goes to step 2 for a new round of optimization. From this description, it can be realized that SSOA has six parameters, one of which is the maximum number of iterations, two parameters for population, and

3.2 Optimization Algorithms

57

three parameters for calculating the step size. For further details, the reader may refer to [5].

3.2.2 Parameters Reduced Shuffled Shepherd Optimization Algorithm (PRSSOA) This section introduces the parameters for the reduced version of SSOA, which does not need step size parameters. According to the previous section, SSOA contains three parameters for computing the step size of agents, two parameters for population, and one parameter for the maximum number of iterations. These factors help the SSOA algorithm to enhance its performance in dealing with different optimization problems. Nevertheless, fine tuning the parameters associated to step size can take some time. To this end, a new version of SSOA that doesn’t need the adjustment of step size parameters is presented. The maximum number of iterations as a stopping condition and the number of populations are the main parameters in the parameterreduced SSOA (PRSSOA). These parameters do not require time for tuning and do not depend on the considered problem. Since PRSSOA and SSOA have similar steps that have been quick explained in the previous section, here the differences between these two are provided. This difference arises when the stepsize of each community member is computed. Unlike the previously created stepsize in the Step 3 of the SSOA, a unique parameter-less stepsize of PRSSOA is defined as follows: ) ( stepsi zei,Wjor se = randα × M OCi,w − M OCi, j

(3.10)

) ( stepsi zei,Better = randβ × M OCi,b − M OCi, j j

(3.11)

in which randα and randβ are random vectors with the same size as M OCi, j in [0, αmax ] and [β0 , βmax ] intervals, respectively; β0 , βmax and αmax are calculated as: β0 = 0.5 × t

(3.12)

βmax = 2 − β0

(3.13)

αmax = 1 − t

(3.14)

For further clarity, the flowchart of the PRSSOA is given in Fig. 3.1.

58

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

Fig. 3.1 Flowchart of the PRSSOA

3.3 Numerical Examples

59

3.3 Numerical Examples In this section, PRSSOA is employed to three truss optimization problems in order to confirm its capability and efficiency in comparison to SSOA and some other stateof-art optimization methods published in the literature. The first example is the size optimization of a 160-bar spatial tower with 38 design variables. The second one is the layout optimization of a 272-bar transmission tower with 72 design variables. The last problem is a size optimization of a 1016-bar double-layer grid with 25 design variables. The number of communities (m) and the number of members belonging to each community (n) are respectively set to 4 and 5 for the first and last problems, and both of them (i.e., m and n) are set to 5 for the second problem. In order to compare the results acquired by PRSSOA with SSOA, its parameters including α0 , β0 andβmax are respectively set to be 0.5, 2.3, and 2.8. It should be noted that the results of the other examined case studies determined by SSOA are taken from literature. Additionally, the maximum number of iterations as a stopping criterion are set to 600 for the first and last problems and 1700 for the second problem. Each design example is separately run 30 times to provide statistically meaningful results.

3.3.1 The 160-Bar Spatial Truss The first example investigated in this chapter is the 160-bar spatial truss, as depict in Fig. 3.2. The material density and modulus of elasticity are 0.00785 kg/cm3 and 2.047 × 106 kgf/cm2 , respectively. The nodal coordinates and end nodes of the members are presented in Ref. [7]. Furthermore, load conditions for design of this truss is provided in Table 3.1. Cross-sectional areas are chosen form the set S = {1.84, 2.26, 2.66, 3.07, 3.47, 3.88, 4.79, 5.27, 5.75, 6.25, 6.84, 7.44, 8.06, 8.66, 9.40, 10.47, 11.38, 12.21, 13.79, 15.39, 17.03, 19.03, 21.12, 23.20, 25.12, 27.50, 29.88, 32.76, 33.90, 34.77, 39.16, 43.00, 45.65, 46.94, 51.00, 52.10, 61.82, 61.90, 68.30, 76.38, 90.60, 94.13 cm2 } and the corresponding radius of gyration are r = {0.47, 0.57, 0.67, 0.77, 0.87, 0.97, 0.97, 1.06, 1.16, 1.26, 1.15, 1.26, 1.36, 1.46, 1.35, 1.36, 1.45, 1.55, 1.75, 1.95, 1.74, 1.94, 2.16, 2.36, 2.57, 2.35, 2.56, 2.14, 2.33, 2.97, 2.54, 2.93, 2.94, 2.94, 2.92, 3.54, 3.96, 3.52, 3.51, 3.93, 3.92, 3.92 cm}. The stress range for all members is limited to ± 1500 kg/cm2 , and the compression members are subjected to buckling stress constraints as follows: { 2 1300 − (λ24i ) i f λi ≤ 120 − σi = 107 (3.15) i f λi > 120 (λ )2 i

in which σi− is the allowable compressive stress of member i; λi represents the slender ratio for member i (λi = k L i /ri ); L i represents the length of member i; ri denotes the corresponding radius of gyration, and k is the member effective length factor fixed as 1 for all members.

60

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

Fig. 3.2 Schematic of the 160-bar spatial truss

The optimization result obtained by SSOA, PRSSOA and some other methods in the literature are given in Table 3.2. The best solution determined is by PRSSOA which is lighter than those obtained by other methods. Number of required analyses and all statistic results of the SSOA and PRSSOA are superior to other methods. This demonstrates that SSOA and PRSSOA converged more efficiently than other methods. The difference between SSOA and PRSSOA is only 0.763 lb, demonstrating that PRSSOA behaves similarly to SSOA. Also it shows that SSOA’s parameters are

3.3 Numerical Examples

61

Table 3.1 Eight cases of load distribution on nodes for the 160-bar space truss structure [7] Load case

Node

1

52

2

3

4

Px

Pz

Load case

Node

−868

Py 0

−491

5

52

37

−996

0

−546

25

−1091

0

−546

28

−1091

0

−546

52

−493

1245

−363

37

−996

0

−546

25

−1091

0

−546

28

−1091

0

−546

52

−917

0

−491

37

−951

0

25

−1015

28

−1015

52

6

Px

Py

Pz

−917

0

−491

37

−951

0

−546

25

−1015

0

−546

28

−636

1259

−428

52

−917

0

−491

37

−572

1303

−428

25

−1015

0

−546

28

−1015

0

−546

52

−917

0

−491

−546

37

−951

0

−546

0

−546

25

−1015

0

−546

0

−546

28

−636

1303

−428

−917

0

−546

52

−498

1460

−363

37

−572

1259

−428

37

−951

0

−546

25

−1015

0

−546

25

−1015

0

−546

28

−1015

0

−546

28

−1015

0

−546

7

8

appropriately adjusted. Figure 3.3 displays the convergence histories of the best run and the average performance of 30 independent runs of SSOA and PRSSOA for the 160-bar spatial truss.

3.3.2 The 272-Bar Transmission Tower In this section, layout optimization of the 272-bar transmission tower is investigated as second optimization problem. The 272-bar transmission tower, as depicted in Fig. 3.4, for the first is presented by Kaveh and Massoudi [11], and Kaveh and Zaerreza [12] applied configuration variables for this problem. Except for nodes 1, 2, 11, 20, and 29, which are fixed in all three directions, and nodes 62, 63, 64, and 65, which are fixed in the z-direction, other nodes are free to move in all directions. Nodal coordinates for the 272-bar transmission tower and node connectivity are available in Ref. [11]. Members are organized into 28 groups, as stated in Ref [5]. As a result, the problem has 28 sizing variables and 44 configuration variables. The truss is under 12 load conditions, as given in Table 3.3. Stress for each member is limited to the range of ± 275,000 kN/m2 . Euler buckling stress for compression members is limited to 4EA/L 2 . Displacement of nodes 1, 2, 11, 20, and 29 is limited to 20 mm in Z-direction and to 100 mm in X and Y directions. Table 3.4 lists the truss optimization variables. The results obtained by PRSSOA and SSOA [12] are presented in Table 3.5. PRSSOA is identified better solution than SSOA, and the mean of 30 independent

19.03

15.39 5.75

5.75 13.79 6.25 5.75 2.66 7.44

2.66 3.07

19.03

5.27

19.03

5.75

17.03

6.25

13.79

6.25

5.75

12.21

6.84

5.75

2.66

7.44

1.84

8.66

2.66

3.07

A3

A4

A5

A6

A7

A8

A9

A10

A11

A12

A13

A14

A15

A16

A17

A18

A19

A20

8.66

1.84

5.75

13.79

5.75

19.03

5.27

5.27

5.27

A2

RGA 19.03

SDR

19.03

Groenwold et al. [9]

Groenwold and Stander [8]

A1

Design variables (area cm2 )

Table 3.2 Optimum and statistical results for the 160-bar spatial truss

3.88

3.88

12.21

2.26

8.66

6.84

5.75

7.44

12.21

6.84

5.75

13.79

5.75

17.03

5.75

19.03

5.27

19.03

5.27

19.03

DE

3.07

2.66

8.66

1.84

7.44

3.88

5.75

6.25

12.21

5.75

5.75

13.79

5.75

15.39

5.75

19.03

5.27

19.03

5.27

19.03

aeDE

Ho-Huu, Nguyen-Thoi [7]

3.07

2.66

8.66

1.84

7.44

4.79

5.75

6.84

13.79

5.75

12.21

13.79

5.75

15.39

5.75

19.03

5.27

19.03

5.75

19.03

EM

3.07

2.66

8.66

1.84

7.44

3.88

5.75

6.25

12.21

5.75

5.75

13.79

5.75

15.39

5.75

19.03

5.27

19.03

5.27

19.03

EFA

Le et al. [10]

3.07

2.66

8.66

1.84

7.44

3.88

5.75

6.25

12.21

5.75

5.75

13.79

5.75

15.39

5.75

19.03

5.27

19.03

5.27

19.03

SSOA [1]

(continued)

3.88

2.66

8.66

1.84

7.44

3.47

5.75

6.25

12.21

5.75

5.75

13.79

5.75

15.39

5.75

19.03

5.27

19.03

5.27

19.03

PRSSOA [1]

Present work

62 3 Shuffled Shepherd Optimization Method Simplified for Reducing …

4.79 2.66 3.47 1.84 2.26 3.88

3.88 1337.442

4.79

2.66

3.47

1.84

2.26

3.88

1.84

1.84

3.88

1.84

1.84

3.88

1359.781

A27

A28

A29

A30

A31

A32

A33

A34

A35

A36

A37

A38

Weight (lb)

1.84

1.84

3.88

1.84

1.84

5.75 1.84

6.25

1.84

A24

A25

6.25

7.44

A23

A26

8.06 5.27

8.06

5.27

A22

1448.306

4.79

3.88

2.26

4.79

2.66

2.26

3.88

2.66

1.84

4.79

4.79

6.25

4.79

9.4

7.44

6.25

8.66

3.88

1336.634

3.88

1.84

1.84

3.88

1.84

1.84

3.88

2.26

1.84

3.47

2.66

4.79

2.26

5.75

6.25

5.75

8.06

2.66

aeDE

DE

2.66

RGA

SDR

2.66

Ho-Huu, Nguyen-Thoi [7]

Groenwold et al. [9]

Groenwold and Stander [8]

A21

Design variables (area cm2 )

Table 3.2 (continued)

1429.406

4.79

3.07

2.66

3.88

1.84

3.07

3.88

2.66

2.66

3.88

3.07

4.79

2.26

7.44

7.44

6.84

8.06

2.66

EM

1336.704

3.88

1.84

1.84

3.88

1.84

1.84

3.88

2.26

1.84

3.47

2.66

4.79

1.84

6.25

6.25

5.75

8.06

2.66

EFA

Le et al. [10]

1336.794

3.88

1.84

1.84

3.88

1.84

1.84

3.88

2.26

1.84

3.47

3.47

4.79

1.84

5.75

6.25

5.75

8.06

2.66

SSOA [1]

(continued)

1336.031

3.88

1.84

1.84

3.88

1.84

1.84

3.88

2.26

1.84

3.47

2.66

4.79

1.84

5.75

6.25

5.27

8.06

3.47

PRSSOA [1]

Present work

3.3 Numerical Examples 63

–

Mean weight (lb)

Standard deviation (lb) –

– –

–

–

Worst weight (lb) 81.930

1417.346

1743.596

50,025

18.805

1355.875

1410.611

23,925

aeDE

DE

–

RGA

SDR

–

Ho-Huu, Nguyen-Thoi [7]

Groenwold et al. [9]

Groenwold and Stander [8]

No of analyses

Design variables (area cm2 )

Table 3.2 (continued)

101.848

1574.834

1773.773

31,640

EM

34.706

1372.551

1429.253

16,870

EFA

Le et al. [10]

12.938

1352.023

1406.570

12,020

SSOA [1]

12.272

1350.089

1384.653

12,020

PRSSOA [1]

Present work

64 3 Shuffled Shepherd Optimization Method Simplified for Reducing …

3.3 Numerical Examples

65

Fig. 3.3 Convergence histories of the SSOA and PRSSOA for the 160-bar spatial truss

runs of PRSSOA is superior to SSOA. Nevertheless, standard deviation of the SSOA is better than PRSSOA. This indicates that better parameters can be found for SSOA and also shows the problem of parameter tuning for SSOA. The maximum stresses ratio of the best result of SSOA is 0.97, which occurred in load Case 1 in member 245. The displacements for the nodes whose constraint are considered taken into account displayed in Fig. 3.5. Optimum layout determined by PRSSOA is displayed in Fig. 3.6. The converge histories of the SSOA and PRSSOA are given in Fig. 3.7.

3.3.3 The 1016-Bar Double-Layer Grid The last problem examined in this chapter is the double-layer grid with configuration of a square on diagonal as illustrated in Fig. 3.8 having a span and height of 40 m and 3 m, respectively. Simple support conditions are utilized for 12 nodes in the bottom layer, as shown in Fig. 3.8a. Vertical load of 30 kN is applied on each joint of the top layer. The 1016 members of the grid are separated into 25 groups, as illustrated in Fig. 3.8, and sections are chosen from steel pipes presented in the manual of steel construction [13]. The modulus of elasticity is 205 GPa, yield stress is 248.2 MPA, and the material density is 7833.413 kg/m3 . All nodes in the vertical direction have a displacement constraint of 6.667 cm, and the allowed stress and stability of the truss components are defined by AISC 360-10 [14] as follows:

66

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

Fig. 3.4 Schematic of the 272-bar transmission tower

Tension member constraint } pu ≤ pr ;

pr = min

∅t Fy A g ; ∅t = 0.9 ∅t Fu Ae ; ∅t = 0.75

(3.16)

3.3 Numerical Examples

67

Table 3.3 Loading condition for the 272-bar transmission tower Case 1

2

3

4

5

6

7

8

9

10

11

12

Force direction

Nodes 1

2

11

20

29

Other free nodes

Fx (kN)

20

20

20

20

20

5

Fy (kN)

20

20

20

20

20

5

Fz (kN)

−40

−40

−40

−40

−40

0

Fx (kN)

0

20

20

20

20

5

Fy (kN)

0

20

20

20

20

5

Fz (kN)

0

−40

−40

−40

−40

0

Fx (kN)

20

0

20

20

20

5

Fy (kN)

20

0

20

20

20

5

Fz (kN)

−40

0

−40

−40

−40

0

Fx (kN)

20

20

20

0

20

5

Fy (kN)

20

20

20

0

20

5

Fz (kN)

−40

−40

−40

0

−40

0

Fx (kN)

20

0

0

0

0

5

Fy (kN)

20

0

0

0

0

5

Fz (kN)

−40

0

0

0

0

0

Fx (kN)

0

20

0

0

0

5

Fy (kN)

0

20

0

0

0

5

Fz (kN)

0

−40

0

0

0

0

Fx (kN)

0

0

0

20

0

5

Fy (kN)

0

0

0

20

0

5

Fz (kN)

0

0

0

−40

0

0

Fx (kN)

0

0

20

20

20

5

Fy (kN)

0

0

20

20

20

5

Fz (kN)

0

0

−40

−40

−40

0

Fx (kN)

0

20

20

0

20

5

Fy (kN)

0

20

20

0

20

5

Fz (kN)

0

−40

−40

0

−40

0

Fx (kN)

0

0

20

0

20

5

Fy (kN)

0

0

20

0

20

5

Fz (kN)

0

0

−40

0

−40

0

Fx (kN)

0

0

0

20

20

5

Fy (kN)

0

0

0

20

20

5

Fz (kN)

0

0

0

−40

−40

0

Fx (kN)

0

0

20

20

0

5

Fy (kN)

0

0

20

20

0

5

Fz (kN)

0

0

−40

−40

0

0

68

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

Table 3.4 Simulation data for the 272-bar transmission tower Sizing variables Group 1, Group 2, Group 3, …, Group 28 Layout variables x10 = x9 = −x4 = −x3 ; y10 = y4 = −y3 = −y9 ; z 10 = z 9 = z 4 = z 3 ; x8 = x7 = −x5 = −x6 ; y8 = y6 = −y5 = −y7 ; z 8 = z 7 = z 6 = z 5 ; x19 = x18 = −x12 = −x13 ; y19 = y13 = −y12 = −y18 ; z 19 = z 18 = z 13 x17 = x16 = −x14 = −x15 ; y17 = y15 = −y14 = −y16 ; z 17 = z 16 = z 15 x28 = x27 = −x21 = −x22 ; y28 = y22 = −y21 = −y27 ; z 28 = z 27 = z 22 x26 = x25 = −x23 = −x24 ; y26 = y24 = −y23 = −y25 ; z 26 = z 25 = z 24 x37 = x36 = −x30 = −x31 ; y37 = y31 = −y30 = −y36 ; z 37 = z 36 = z 31 x35 = x34 = −x32 = −x33 ; y35 = y33 = −y32 = −y34 ; z 35 = z 34 = z 33 x41 = x40 = −x38 = −x39 ; y41 = y39 = −y38 = −y40 ; z 41 = z 40 = z 39 x45 = x44 = −x42 = −x43 ; y45 = y43 = −y42 = −y44 ; z 45 = z 44 = z 43 x49 = x48 = −x46 = −x47 ; y49 = y47 = −y46 = −y48 ; z 49 = z 48 = z 47 x53 = x52 = −x50 = −x51 ; y53 = y51 = −y50 = −y52 ; z 53 = z 52 = z 51 x57 = x56 = −x54 = −x55 ; y57 = y55 = −y54 = −y56 ; z 57 = z 56 = z 55 x61 = x60 = −x58 = −x59 ; y61 = y59 = −y58 = −y60 ; z 61 = z 60 = z 58 x65 = x64 = −x62 = −x63 ; y65 = y63 = −y62 = −y64 Layout variables bounds 1 ≤ x10 , x19 , x28 , x37 ≤ 2.25 0.1 ≤ x8 , x17 , x26 , x35 ≤ 0.9 0.1 ≤ x45 , x41 ≤ 1.5 0.1 ≤ x49 , x53 ≤ 2 0.1 ≤ x57 , x61 ≤ 2.5 0.1 ≤ x65 ≤ 3 0.1 ≤ y10 , y8 , y19 , , y17 , , y28 , y26 , y37 , y35 , y41 ≤ 1 0.1 ≤ y45 , y49 , y53 , y57 , y61 , y65 ≤ 1 17.3 ≤ z 10 , z 8 ≤ 19 15.5 ≤ z 19 , z 17 ≤ 16.6 14.9 ≤ z 28 , z 26 ≤ 15.4 13.8 ≤ z 37 , z 35 ≤ 14.8 11 ≤ z 41 ≤ 13.7 9.6 ≤ z 45 ≤ 10.9 7.5 ≤ z 49 ≤ 9.5 5.6 ≤ z 53 ≤ 7.4 3.6 ≤ z 57 ≤ 5.5 1 ≤ z 61 ≤ 3.5 Possible sizing variables 1000 mm2 ≤ Group 1, Group 2, Group 3, . . . , Group 28 ≤ 16, 000 mm2 Young modulus E = 2 × 108 (kN/m2 )

= z 12 ; = z 14 ; = z 21 ; = z 23 ; = z 30 ; = z 32 ; = z 38 ; = z 42 ; = z 46 ; = z 50 ; = z 54 ; = z 59 ;

3.3 Numerical Examples

69

Table 3.5 Optimum and statistical results for the 272-bar transmission tower Design variables

Kaveh and Zaerreza [12]

Present work (PRSSOA [1])

Design variables

Kaveh and Zaerreza [12]

Present work (PRSSOA [1])

Group 1

1000.2

1041.4

z 19

16.1015

16.3892

Group 2

1000.0

1000.0

x17

0.7327

0.7030

Group 3

1000.2

1000.2

y17

0.4548

0.4354

Group 4

1000.0

1000.0

z 17

15.5185

15.8748

Group 5

3412.4

2569.5

x28

1.2562

1.5786

Group 6

1000.4

1000.4

y28

0.1003

0.1009

Group 7

3786.5

3609.0

z 28

15.1170

15.1559

Group 8

1003.0

1000.2

x26

0.7572

0.7248

Group 9

1008.1

1000.5

y26

0.4915

0.5037

Group 10

1003.2

1000.9

z 26

15.1625

15.3917

Group 11

4498.3

3398.6

x37

1.1038

1.3188

Group 12

1001.0

1000.5

y37

0.1902

0.1214

Group 13

1000.1

1000.1

z 37

14.3021

14.6397

Group 14

1000.4

1001.1

x35

0.8821

0.8622

Group 15

4615.9

3725.2

y35

0.6002

0.6455

Group 16

1000.3

1000.2

z 35

13.9793

14.2282

Group 17

1000.0

1000.0

x41

0.9384

0.9918

Group 18

1005.6

1001.6

y41

0.7835

0.8549

Group 19

4826.0

4245.3

z 41

12.3261

12.3787

Group 20

1000.4

1.2195

x45

1.0161

1.1378

Group 21

1001.7

1000.5

y45

0.9049

1.0333

Group 22

1001.4

1000.1

z 45

10.7022

10.5050

Group 23

5092.8

4665.6

x49

1.1052

1.3040

Group 24

1000.8

1002.4

y49

0.9440

1.1668

Group 25

1008.1

1000.7

z 49

8.9221

8.6061

Group 26

1007.6

1000.1

x53

1.2439

1.4541

Group 27

5072.8

5003.6

y53

1.0765

1.2337

Group 28

1000.8

1122.5

z 53

7.1762

7.0111

x10

1.0311

1.0126

x57

1.5098

1.7542

y10

0.1003

0.1002

y57

1.2954

1.3653

z 10

17.3610

17.4132

z 57

4.9926

4.8853

x8

0.2283

0.1856

x61

1.9012

2.1184

y8

0.4300

0.4495

y61

1.6430

1.6269

z8

17.30005

17.3013

z 61

2.4470

2.6018

x19

1.0000

1.1445

x65

2.3101

2.6246 (continued)

70

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

Table 3.5 (continued) Design variables

Kaveh and Zaerreza [12]

Present work (PRSSOA [1])

Design variables

Kaveh and Zaerreza [12]

Present work (PRSSOA [1])

y19

0.2649

0.1380

y65

1.9996

1.9671

Volume (cm3 )

736,814.944

734,948.633

Number of analyses

51,030

42,525

Worst volume (cm3 )

800,631.147

Mean volume (cm3 )

764,061.589

763,458.079

Standard deviation (cm3 )

15,485.12

16,383.16

Fig. 3.5 Compression of allowable and existing displacements for the 272-bar transmission tower

3.3 Numerical Examples

71

Fig. 3.6 Comparison of optimized layout for the 272-bar transmission tower

in which pu indicates the required strength; pr represents the nominal axial strength; A g denotes gross area of member; Ae effective net area; Fy represents the minimum yield stress and Fu is the specified minimum tensile strength. Compression member constraint pu ≤ pr ; pr = ∅c Fcr A g ; ∅c = 0.9

Fcr =

⎧( ) Fy ⎨ 0.658 Fe Fy ;

KL r

⎩ 0.877∗Fe ;

KL r

π2E Fe = ( )2 KL

√ ≤ 4.71 FEy √ > 4.71 FEy

(3.17)

(3.18)

(3.19)

r

in which Fe indicates the elastic bulking stress; Fcr represents the critical stress; E denotes the modulus of elasticity; L represents the laterally unbraced length of the

72

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

Fig. 3.7 Convergence histories of the SSOA and PRSSOA for the 272-bar transmission tower

member; r denotes the radius of gyration and K is the effective length factor taken equal to 1. Slenderness ratio constraints KL ≤ 200; f or compr ession member r KL ≤ 300; f or tension member r The achieved solution of the PRSSOA, SSOA [5] and ECBO [15] are given in Table 3.6. Comparison of the answers show that PRSSOA similar to SSOA can obtain optimal solution. Statistic result of the PRSSOA and SSOA are superior to other method. The PRSSOA and SSOA convergence histories are shown in Fig. 3.9.

3.4 Concluding Remarks

73

(a)

(c )

(b)

(d)

Fig. 3.8 Schematic of the1016-bar double layer grid; a 3D view, b top layer members, c bottom layer members, and d web members

3.4 Concluding Remarks Shuffled shepherd optimization algorithm is a meta-heuristic algorithm mimicking the behavior of shepherd in nature, and its performance has been explored in several studies. SSOA is a parameters-dependent algorithm (it contains six parameters), and the step size computation parameters have a significant impact on the algorithm’s performance. This study introduces a parameter-less variant of SSOA that does not need adjustment for the step size parameters, making PRSSOA independent of each particular issue, comparable to SSOA. The PRSSOA is tested on three structural optimization problems. For the 160-bar and 1016-bar truss structures the results obtained by PRSSOA are close to those of

74

3 Shuffled Shepherd Optimization Method Simplified for Reducing …

Table 3.6 Optimum and statistical results for the 1016-bar double-layer grid

Element group Kaveh and Ghazaan [15] ECBO

Kaveh and Zaerreza [5]

Present work

SSOA

PRSSOA [1]

1

EST 5

EST 5

EST 5

2

EST 5

ST 5

EST 4

3

ST 3

ST 4

EST 3

4

ST 3 ½

EST 2 ½

ST 2 ½

5

ST 2 ½

ST 3 ½

ST 3

6

ST 2

EST 1 ½

EST 1 ½

7

DEST 2

EST 1 ½

EST 1

8

DEST 2

EST 1 ½

ST 2 ½

9

EST 2

ST 4

EST 2

10

ST 6

DEST 2 ½

ST 3 ½

11

ST 2

ST 2 ½

ST 4

12

EST 8

ST 10

ST 10

13

EST 3 ½

EST 4

ST 6

14

ST 5

ST 4

ST 5

15

ST 4

EST 4

ST 5

16

EST 5

ST 6

ST 5

17

ST 5

ST 5

ST 6

18

EST 5

EST 5

EST 5

19

EST 5

DEST 4

EST 5

20

ST 8

DEST 4

DEST 4

21

ST 5

ST 6

ST 6

22

ST 3

ST 3 ½

ST 3 ½

23

EST 2 ½

ST 3 ½

ST 3 ½

24

ST 5

ST 2 ½

ST 2 ½

25

ST 4

ST 3 ½

EST 1 ½

Weight (kg)

67,839

68,398

67,407

Mean weight (kg)

73,042

72,084

70,054

Standard deviation (kg)

9,158

1,802

1,864

Number of analyses

15,760

12,020

12,020

References

75

Fig. 3.9 Convergence histories of the SSOA and PRSSOA for the 1016-bar double-layer grid

SSOA. However, For the 272-bar structure result obtained by PRSSOA is better than SSOA. This demonstrate that SSOA is more parameter dependent and can identified better parameters for SSOA. As the results show, PRSSOA performs like SSOA and need less parameter adjusting than SSOA, which makes it easy to employ the PRSSOA to other optimization problems.

References 1. Kaveh, A., Zaerreza, A., Hosseini, S.M.: Shuffled shepherd optimization method simplified for reducing the parameter dependency. Iran. J. Sci. Technol. Trans. Civ. Eng. 45(3), 1397–1411 (2021) 2. Gandomi, A.H., Kashani, A.R., Roke, D.A., Mousavi, M.: Optimization of retaining wall design using evolutionary algorithms. Struct. Multidiscip. Optim. 55(3), 809–825 (2017) 3. Du, D.-C., Vinh, H.-H., Trung, V.-D., Hong Quyen, N.-T., Trung, N.-T.: Efficiency of Jaya algorithm for solving the optimization-based structural damage identification problem based on a hybrid objective function. Eng. Optim. 50(8), 1233–1251 (2018) 4. Azad, S.K., Hasançebi, O., Azad, S.K., Erol, O.: Upper bound strategy in optimum design of truss structures: a big bang-big crunch algorithm based application. Adv. Struct. Eng. 16(6), 1035–1046 (2013) 5. Kaveh, A., Zaerreza A.: Shuffled shepherd optimization method: a new meta-heuristic algorithm. Eng. Comput. 37(7), 2389–2357 (2020)

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3 Shuffled Shepherd Optimization Method Simplified for Reducing …

6. Duan, Q., Gupta, V.K., Sorooshian, S.: Shuffled complex evolution approach for effective and efficient global minimization. J. Optim. Theory Appl. 76(3), 501–521 (1993) 7. Ho-Huu, V., Nguyen-Thoi, T., Vo-Duy, T., Nguyen-Trang, T.: An adaptive elitist differential evolution for optimization of truss structures with discrete design variables. Comput. Struct. 165, 59–75 (2016) 8. Groenwold, A., Stander, N.: Optimal discrete sizing of truss structures subject to buckling constraints. Struct. Optim. 14(2–3), 71–80 (1997) 9. Groenwold, A., Stander, N., Snyman, J.: A regional genetic algorithm for the discrete optimal design of truss structures. Int. J. Numer. Meth. Eng. 44(6), 749–766 (1999) 10. Le, D.T., Bui, D.-K., Ngo, T.D., Nguyen, Q.-H., Nguyen-Xuan, H.: A novel hybrid method combining electromagnetism-like mechanism and firefly algorithms for constrained design optimization of discrete truss structures. Comput. Struct. 212, 20–42 (2019) 11. Kaveh, A., Massoudi, M.: Multi-objective optimization of structures using charged system search. Sci. Iranica Trans. A, Civ. Eng. 21(6), 1845 (2014) 12. Kaveh, A., Zaerreza, A.: Size/layout optimization of truss structures using shuffled shepherd optimization method. Periodica. Polytech. Civ. Eng. 64(2) 408–421 (2020) 13. (AISC) A.I.o.S.C.: Manual for Steel Construction, Load and Resistance Factor Design, 3rd Edn. American Institute of Steel Construction-AISC, Chicago (2001) 14. Committee A.: Specification for structural steel buildings (ANSI/AISC 360-10). American Institute of Steel Construction, Chicago-Illinois (2010) 15. Kaveh, A., Ilchi Ghazaan, M.: Meta-Heuristic Algorithms for Optimal Design of Real-Size Structures. Springer (2018)

Chapter 4

An Enhanced Shuffled Shepherd Optimization Algorithm and Application to Space Structures

4.1 Introduction In this chapter, the Enhanced Shuffled Shepherd Optimization Algorithm (ESSOA) is presented and utilized for optimal design of space structures, Kaveh et al. [1]. Shuffled Shepherd Optimization Algorithm (SSAO) is an optimizer inspired by the herding behavior of shepherds in nature. SSOA may suffer from some disadvantages, including being caught in a local optimum and starting from a random population without previous information. This chapter aims to improve the performance of the SSOA by incorporating two efficient devices. Gradient-based approaches and metaheuristic methods are two independent classifications of optimization methods. Gradient-based methods utilize gradient information of the involved functions to explore the solution in close an initial starting point. Despite the that fact these methods can converge more quickly and identify the global optimum solution with high accuracy than metaheuristic methods, the gaining of gradient information based on these methods can either be costly or impossible to find the global optimum. In addition, a good beginning point that influences the search for the optimal solution can be considered as another drawback of gradient-based methods. In majority of optimization problems, the objective function is often complex, and getting its gradient information can be difficult or even impossible. In contrast gradient-based optimization methods, metaheuristic algorithms do not require any gradient information and are independent on the starting position. Moreover, they are favorable options when dealing with discontinuous, complex, non-convex, and non-smooth search spaces at an acceptable computational time. Shuffled Shepherd Optimization Algorithm (SSOA) is a swarm intelligence-based optimization method inspired by the herding behavior of shepherds in nature. SSOA is an effective and simple optimizer, so it has been recently employed to solve different optimization problems. However, SSOA suffers from some shortcomings. The first problem is associated with beginning with a random population without previous knowledge about the randomly created solutions. The second problem is being © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_4

77

78

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

trapped in a local optimum when it near to an optimum solution. This chapter modifies SSOA with two efficient techniques to address these difficulties. The improved version is named as Enhanced Shuffled Shepherd optimization algorithm (ESSOA). The first mechanism is the opposition-based learning (OBL) concept, which was introduced by Tizhoosh [2]. The OBL is utilized to enhance the initialization phase of the SSOA. Because it enhances the convergence rate of the algorithm by providing prior knowledge about the search space. Starting with a random population in the initialization step of the algorithm may result the algorithm being trapped into a local optimum since the optimization algorithm lacks previous information of the search space. In this sense, OBL provides a strategy in which the optimization technique effectively explores the full search space. The second mechanism is developed a novel solution generator based on the statistical results of the solutions. The presented mechanism is called statistically regenerated stepsize. This mechanism provides a good exploration in the early iterations of the algorithm and leads the algorithm to escape from local optimums in the last iterations. The robustness and capability of the suggested ESSOA are demonstrated in three large-scale design examples consisting of a 693-bar double layer barrel vault, a 1016-bar double layer grid, and a 1410-bar dome structure. Comparing the optimization results obtained by ESSOA to those of other metaheuristics, namely Colliding Bodies Optimization (CBO) and its Enhanced version (ECBO), Vibrating Particles System (VPS), and Multi Design Variable Configurations-Upper bound Vibrating Particles System (MDVC-UVPS) demonstrates the competence and robustness of the proposed ESSOA in obtaining optimal design in all design examples. This paper’s remaining sections are organized as follows: In Sect. 4.2, the SSOA is concisely reviewed. Section 4.3 is devoted to presenting the improved variant of SSOA. The statement of the optimization problem is presented in Sect. 4.4. In Sect. 4.5, 3 distinct large-scale structures are examined to demonstrate the competence and robustness of the ESSOA in comparison to other metaheuristics. The concluding remarks are finally driven at the end.

4.2 Shuffled Shepherd Optimization Algorithm (SSOA) The shuffled shepherd optimization algorithm (SSOA) is a multi-population metaheuristic algorithm imitates the herding behavior of shepherds [3]. Humans have increasingly learned that they can employ animal abilities to attain their objectives. For example, shepherds have learned how to tend and rear sheep. They usually utilize fast-ridden horses for collecting sheep and guide their herd of sheep to remain them in the proper direction. In this regard, shepherds place animals such as horses or herding dogs into their herd of sheep and use the instinct of these animals to guide the herd. This method is the basis for determining the stepsize of the sheep in the SSOA. SSOA is started with a set of randomly initialized solutions represented by a herd of sheep as:

4.2 Shuffled Shepherd Optimization Algorithm (SSOA)

79

) ( xi,0 j,q = xq,min + r × xq,max − xq,min ; i = 1, 2, . . . , nh, j = 1, 2, . . . , nS/nh, and q = 1, 2, . . . , nV ar

(4.1)

in which xi,0 j,q indicates the initial value of qth variable of the jth sheep in the ith herd; r is a random number generators from a uniform distribution in the interval [0, 1]; xq,min and xq,max represent the lower and upper bounds of the qth design variable, respectively; nS, nh, and nV ar are the number of sheep, herds, and design variables, respectively. This optimization method (SSOA) is based on multi-population techniques. There is a fixed number of herds (nh) that contain a fixed number of sheep (nS/nh). Therefore, the total population of candidate solutions (nS) is divided into nh herds so that each herd has an same number of sheep (nS/nh). For dividing the entire population, first of all, all sheep are evaluated and arranged depending on the quality of solutions. Then, SSOA picks the first nh sheep from the whole sorted population to be allocated into the nh herds randomly. Up to this step, each herd consist of one sheep. It can be claimed that the random assignment of sheep to herds guarantees the population diversity of the algorithm. Next, the nh + 1, nh + 2, . . . , 2nh sheep from the remaining arranged population (nS − nh) are chosen and allocated again to nh herds randomly. It is necessary to note that up to this step, each herd consist of two sheep. This population dividing method is performed based on the shuffling method until the whole population of sheep is allocated randomly to the nh herds. Dividing the entire population into smaller sub-populations makes the SSOA exploring different areas of search space concurrently. For more clarity, Fig. 4.1 shows how the whole population of sheep is classified into h herds in the cyclic body of the SSOA. After partitioning the entire population of sheep, a unique stepsize is computed for each considered sheep in each herd. To do this, sheep with better and worse objective function values than the considered sheep are chosen randomly from the same herd. The considered, better, and worse sheep are referred as the shepherd (xi, j ), horse (xi,h ), and sheep (xi,s ), respectively. To direct the sheep toward the horse, the shepherd first goes toward the sheep and then into the horse’s position. Therefore, the new position of the shepherd is acquired and evaluated. This behavior can be mathematically expressed as follows: shepher d

stepsi zei, j sheep

in which stepsi zei, j

sheep

= stepsi zei, j

+ stepsi zei,horj se

(4.2)

and stepsi zei,horj se are obtained as: ) ( = α × r1 × xi,s − xi, j

(4.3)

( ) stepsi zei,horj se = β × r2 × xi,h − xi, j

(4.4)

sheep

stepsi zei, j

80

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

Fig. 4.1. Population partitioning technique used in the SSOA sheep

where stepsi zei, j and stepsi zei,horj se represent the stepsize vectors of the SSOA with d design variables. These vectors demonstrate global search (diversification) and local search (intensification) capabilities of the SSOA, respectively; r1 and r2 are random vectors generators in which each component is in the range [0, 1]. It is worth mentioning that the last sheep of each herd (x1,nS/nh , x2,nS/nh , x3,nS/nh , . . . , xnh,nS/nh ) sheep does not have a sheep worse than itself. Thus, stepsi zei,nS/nh is set to zero. Similarly, when the first sheep of the herds which are x1,1 , x2,1 , x3,1 , . . . , xn1 are examined, they do not have a sheep superior than themselves in the respective herd. Therefore, hor se stepsi zei,1 is set to zero. α and β are among the most important parameters of the SSOA that control exploration and exploitation rates, respectively.

4.2 Shuffled Shepherd Optimization Algorithm (SSOA)

81

α = α0 − α0 × t β = β0 + (βmax − β0 ) × t; t =

(4.5) it Maxit

(4.6)

where it and Maxit represent the current iteration and the maximum number of iterations, respectively. According to Eqs. (4.5) and (4.6), as t increases,α and β decreases and increases, respectively. The alterations of α and β in the cyclic body of the algorithm in a manner that makes a counterbalance between the exploitation and exploration when seeking promising solutions in the search space. After computing the stepsize for the shepherds, the their new positions are found using Eq. (4.7). Also, Fig. 4.2 is an illustration how the new position of the jth shepherd in the ith herd is determined. new shepher d

xi, j

shepher d

= xi, j

shepher d

+ stepsi zei, j

(4.7)

In the following, the replacement strategy is implemented by comparing and replacing the old shepherds with the newly produced shepherds. Therefore, the shepherd with a better objective function value is selected. After conducting the aforementioned procedure for the sheep of all herds, the herds are combined to exchange information among themselves. Again, all sheep

Fig. 4.2 Schematic of the position updating in the SSOA

82

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

are sorted depending on the quality of the solutions and divided into h herds using population partitioning technique. This optimization procedure is repeated in the cyclic body of the SSOA until the termination criterion of the algorithm is reached. Like other well-known metaheuristic algorithms, the maximum number of iterations (Maxit) is considered as the stopping criterion of the SSOA. Hence, if the current iteration (t) becomes more than Maxit, the process will end. The pseudo-code of SSOA is given in Algorithm 4.1.

Algorithm 4.1: Pseudo-code of the SSOA Set the algorithm parameters; α0 ,β0 , βmax , nS, nh, and Maxit Create random initial sheep using Eq. (4.1) Evaluate initial sheep While it < Maxit Arrange the population according to the quality of their solutions For j:1 to nS/nh Choose the first nh members from the remaining population Place randomly the nh selected sheep as the jth sheep of each herd End For For i:1 to nh For j:1 to nS/nh Choose randomly xi,h and xi,s for xi, j shepher d Compute stepsi zei, j using Eq. (4.2) Calculate the position of each shepherd using Eq. (4.7) newshepher d Evaluate xi, j and apply the replacement strategy between old and new shepherds End For End For it = it + 1 End While Return the best solution

4.3 Enhanced Shuffled Shepherd Optimization Algorithm This section proposes an improved version of the SSOA. This modified version is named as Enhanced Shuffled Shepherd Optimization Algorithm (ESSOA). The upgraded algorithm has two effective features. According to the new area of study

4.3 Enhanced Shuffled Shepherd Optimization Algorithm

83

that has gained significant interest over the last decade, the initialization phase of the SSOA is improved by employing the opposition-based learning (OBL) concept, first developed by Tizhoosh [2]. This efficient mechanism provides a strategy for enhancing the convergence behavior of metaheuristic algorithms to get the global optimum solution of the optimization problem. Due to the fact that metaheuristic methods begin with a random population, they cannot converge to an optimal solution. In addition, when metaheuristic methods randomly explore the search space, they are faced with a time-consuming process due to a lack of prior knowledge about the search space. In some cases, a randomly generated population causes the algorithm to get trapped in a local optimum as well. As a solution to solve this problem, the OBL approach provides a strategy in which the optimization algorithm enables to search the entire search space effectively. The last feature utilized for improving the performance of the SSOA is called statistically regenerated stepsize. This feature is integrated into the cyclic body of the algorithm based on the statistical results of the solutions found in each herd. This mechanism provides a good exploration in the early iterations of the algorithm and leads the algorithm to escape from local optima in the last iterations.

4.3.1 Enhancement on the Initialization Phase In order to enhance the initialization of SSOA, the OBL technique is employed. For this purpose, in the initialization phase, the proposed algorithm not only employs the basic version of OBL (opposite) but also three different version of OBL, namely Quasi-OBL (QOBL), Quasi-Reflection OBL (QROBL), and super-opposite-based learning (SOBL). The opposite of the considered solution x is stated as follows: O X q = xq,max + xq,min − xq q = 1, 2, . . . , nV ar

(4.8)

Thus, O X is the opposite of the considered solution x. The considered solution and opposite of it are utilized to generate the other variant of OBL as follows: (1)

The Quasi-Reflection solution Q R X of the considered solution x is defined as ) which is generated between the x and middle point M I Dq = (a random solution xq,max + xq,min /2. Q R X for the qth design variable is calculated as follows: ) ( Q R X q = M I Dq + M I Dq − xq × rand

(2)

(4.9)

The Quasi-opposite solution QX of the considered solution x is described as a random solution that is created between the OX and middle point and computed utilizing the following equation ) ( Q O X q = M I Dq + M I Dq − O X q × rand

(4.10)

84

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

(3)

The super-opposite solution SUX of the considered solution x is determined using the following equation. { SU X q =

) ( O X q + (xq,max − O X q ) × rand O X q > M I Dq xq,min + O X q − xq,min × rand other wise

(4.11)

All the randomly created solutions utilizing Eq. (4.1) and those computed by Eqs. (4.8–4.11) are considered so that the whole population reaches 5 × nS members. In other words, four solutions are calculated by Eqs. (4.8–4.11) for each randomly created solution utilizing Eq. (4.1). After that, the whole population is ordered depending on the quality of solutions, and the first nS members are picked from the entire population as the initial population of the ESSOA.

4.3.2 Enhancement on the Stepsize Part SSOA is a simple population-based metaheuristic. However, the algorithm suffers from some disadvantages; for example, the possibility of being stuck in local minima when it is near to an optimal solution. This issue is caused by insufficient population diversity. As an efficient solution to alleviate this shortcoming, the stepsize component for creating a new solution is modified. To do this, we defined a flow control by if statement so that if the randomly generated number is less than 0.8, the stepsize is calculated according to Eq. (4.2). If not, 20% of design variables in the choosen agent regenerate based on the statistically regenerated stepsize as follows: new shepher d

xi, j,q

( ) = U Mean j,q − Std j,q − sigma j,q , Mean j,q + Std j,q + sigma j,q (4.12)

in which U represents the operator that returns a random number generated from the continuous uniform distribution with lower and upper endpoints specified by Mean j,q − Std j,q − sigma j,q and Mean j,q + Std j,q + sigma j,q ; Mean j,q and Std j,q are the average and standard deviation of the qth variable in the jth herd; sigma j,q is a parameter that helps the statistically regenerated stepsize (Eq. 4.12) to work efficiently when the whole population converges to the specified value. { sigma j,q =

) ) ( ( 0.01 × xq,max − xq,min i f Std j,q < 0.01 × xq,max − xq,min 0 other wise (4.13)

The pseudo-code of the ESSOA is given in the following:

4.3 Enhanced Shuffled Shepherd Optimization Algorithm

Algorithm 4.2: Pseudo-code of the ESSOA Set the algorithm parameters; α0 , β0 , βmax , nS, nh, and Maxit Create random initial sheep using Eq. (4.1) Calculate O X , Q R X , Q O X , and SU X for each initial sheep utilizing Eqs. (4.8–4.11) Evaluate initial sheep and those calculated by O X , Q R X , Q O X , and SU X Sort the entire population based on the quality of solutions Choose the first nS members from the whole population as the initial population of ESSOA While it < Maxit Arrange the population according to the quality of their solutions For j: 1 to nS/nh Choose the first nh members from the remaining population Place randomly the nh selected sheep as the jth sheep of each herd End For For i:1 to nh For j:1 to nS/nh If rand < 0.8 Choose randomly xi,h and xi,s for xi, j shepher d Compute stepsi zei, j utilizing Eq. (4.2) Calculate the position of each shepherd utilizing Eq. (4.7) Else Employ statistically regenerated stepsize utilizing Eq. (4.12) End If newshepher d Evaluate xi, j and apply the replacement strategy between old and new shepherds End For End For it = it + 1 End While Return the best solution

85

86

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

4.4 Statement of the Optimization Problem This section presents the formulation of the sizing optimization of the space structures. The optimization aims to minimize the weight of the structures by the following equation: [ ] Find {X } = x1 , x2 , . . . , xntg T o minimi ze : W ({X }) = { Subject to :

nte ∑

ρi · Ai · L i

i=1

g j ({X }) ≤ 0; j = 1, 2, . . . , ntc xi,min ≤ xi ≤ xi,max

(4.14)

where {X } represents the vector of design variables; ntg denotes the number of the element groups (number of the design variables); W ({X }) indicates the weight of the entire space structures; nte represents the total number of structural elements; ρi , Ai , and L i are the material density, cross-sectional area, and length of the ith member, respectively; g j ({X }) denotes the jth constraint of the optimization problem; ntc indicates the number of the constraints; xi,min and xi,max are the lower and upper bounds of the design variable xi . The well-known penalty technique is used to manage the constraints of the sizing optimization of space structures. The mathematical formulation of the penalty function is stated as follows: f penalt y ({X }) = (1 + ε1 · ν)ε2 × W ({X }); ν=

ntc ∑

{ } max 0, g j ({X })

(4.15)

j=1

where ν represents the sum of the violations of the optimization problem constraints; ε1 is set equal to 1, whereas ε2 starts from 1.5 and then increases linearly to 3 at the last iteration of the optimization method.

4.5 Design Examples Three large-scale structures are investigated in this section in order to determine the efficiency of the proposed ESSOA. These examples are a 693-bar double-layer barrel vault, a 1016-bar double-layer grid, and a 1410-bar dome structure. In all design examples, the results achieved by ESSOA are compared to those obtained by SSOA and other existing optimization methods published in the literature. The algorithm-specific parameters of all test examples are α0 = 1, β0 = 2, βmax = 3, nh = 4, and nS = 20. The maximum number of structural analyses is set to

4.5 Design Examples

87

20,000 in the first and third design examples, whereas it is set to 12,000 in the second design example. Due to the stochastic nature of the metaheuristic optimization algorithms, 30 independent runs are performed to provide statistically meaningful results. The optimization algorithms are programmed in MATLAB, and the structures are analyzed using the direct stiffness method.

4.5.1 A 693-Bar Double-Layer Barrel Vault The first design example deals with a 693-bar double-layer barrel vault consisting of 259 nodes and 693 elements, as shown in Fig. 4.3. The top layer configuration consist of an orthogonal grid with a single bracing of the Pratt truss. As seen in Fig. 4.3b, the barrel vault’s free span is equal to 19.03 m, and its height and length are respectively equal to 5.75 and 22.9 m. There is a ball-jointed connection between structural elements. Two load cases in which top layer joints are exposed to concentrated vertical loads of 1114.44 lb (4.97 kN) and –1168.06 lb (–5.19 kN) are considered. The material density, modulus of elasticity, and yield stress of this steel structure are ρ = 0.283 lb/in3 (7833.413 kg/m3 ), E = 29,000 ksi (203,893.6 MPa), and Fy = 36 ksi (253.1 MPa), respectively. The 693 members of the structure are organized into 23 element groups due to structural symmetry. The design variables are the cross-sectional area of the structure members chosen from the steel pipe sections provided in Table 4.1. Optimization constraints dealing with stress limitation on truss members are imposed based on the requirements of ASD-AISC [4]. These requirements are as follows: { + σi i f σi ≥ 0 (4.16) σi− i f σi < 0 The allowable stress for tension members is computed as follows: σi+ = 0.6Fy

(4.17)

where σi+ represents allowable tensile stress of member i; Fy yield stress of steel, and σi− indicates the allowable compressive stress of member i computed as: σi−

=

⎧ [( ⎨ 1− ⎩ 12π 22E 23λi

λi2 2Cc2

)

]√ [ Fy

5 3

+

3λi 8Cc

−

λi3 8Cc3

]

i f λi < Cc other wise

(4.18)

where λi represents the slenderness ratio for member i (λi = k L i /ri ); L i represents the length of the member i; ri is the corresponding radius of gyration for member I; k denotes the member effective length factor assumed to be 1 for all truss elements; Cc is the slenderness ratio separating the elastic and inelastic buckling

88

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

(a)

(b)

(c) Fig. 4.3 a 3D view, b plan view with group numbers of the top layer, and c flatten cross-sectional view with the group numbers of bracing and the bottom layer elements of the 693-bar double-layer barrel vault

) ( √ regions Cc = 2π 2 E/Fy , and E is the modulus of elasticity. According to the provisions of AISC-ASD, the slenderness ratio of tension and compression elements must not exceed 300 and 200, respectively. The obtained displacement for each node in any direction must not be greater than ± 0.1 in (0.254 cm).

4.5 Design Examples

89

Table 4.1 The steel pipe sections Area (cm2 )

No.

Type

Nominal diameter (in.)

1

a ST

½

2

b EST

½

2.064512

0.635

3

ST

¾

2.129028

0.846582

4

EST

¾

2.774188

0.818896

1.6129

Gyration radius (cm) 0.662432

5

ST

1

3.161284

1.066038

6

EST

1

4.129024

1.034542

7

ST

1¼

4.322572

1.371346

8

ST

1½

5.16128

1.582166

9

EST

1¼

5.677408

1.331214

10

EST

1½

6.903212

2.003806

11

ST

2

6.903212

1.53543

12

EST

2

9.548368

1.945132

13

ST

2½

10.96772

2.41681

14

ST

3

14.387068

2.955798

15

EST

2½

14.5161

2.346452

16

c DEST

2

17.161256

1.782572

17

ST

3½

17.290288

3.395726

18

EST

3

19.483832

2.882646

19

ST

4

20.451572

3.835908

20

EST

3½

23.741888

3.318002

21

DEST

2½

25.999948

2.143506

22

ST

5

27.74188

4.775454

23

EST

4

28.451556

3.749548

24

DEST

3

35.290252

2.65811

25

ST

6

35.999928

5.700014

26

EST

5

39.419276

4.675124

27

DEST

4

52.25796

3.490976

28

ST

8

54.19344

7.462012

29

EST

6

54.19344

5.577332

30

DEST

5

72.90308

4.379976

31

ST

10

76.77404

9.342628

32

EST

8

82.58048

33

ST

12

94.19336

34

DEST

6

100.64496

7.309358 11.10361 5.236464

35

EST

10

103.87076

9.216898

36

EST

12

123.87072

11.028934

DEST

8

137.41908

7.004812

37 a ST=

Standard weight; b EST= Extra Strong; c DEST= Double-Extra Strong

90

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

Table 4.2 presents the comparison results obtained by the ESSOA and other optimization methods. As can be seen, ESSOA identifies the best optimal weight (9053.4 lb) among all other optimization methods shown in this table, namely CBO (10,221.8 lb), ECBO (9240.5 lb), VPS (9201.4 lb), MDVC-UVPS (9091.1 lb), and SSOA (9165.7 lb). In addition, the lowest mean weight (9265.6 lb) is acquired by ESSOA. This table indicates that the number of needed structural analyses (NSAs) obtained by the ESSOA is equal to 19,100, which is less than that acquired by SSOA (i.e., 19,780). Convergence histories of the best and mean of runs recorded for the SSOA and ESSOA are provided in Fig. 4.4. This figure demonstrates that the proposed ESSOA performs much better than the SSOA in terms of both computing cost and convergence speed. The optimum weight obtained by ESSOA in each independent run is shown in Fig. 4.5. As apparent, ESSOA has reached the optimal weight smaller than the average weight in 16 execution out of 30 independent runs, and the best optimal weight is related to the third run. Figure 4.6 through Fig. 4.8 demonstrates that there is no violation of displacement and stress on the best run obtained by the ESSOA. In addition, the maximum displacement and stress ratio are, respectively, equal to 0.0997 in and 96.652% (Fig. 4.7).

4.5.2 A 1016-Bar Double-Layer Grid The 1016-bar double layer grid with 1016 members and 320 nodes is considered the second design example, as shown in Fig. 4.9. The span length and height of the structure are 40 m and 3 m, respectively. There is a ball-jointed connection between structural members so that they can only withstand tension or compression. Each top layer joint is exposed to a concentrated vertical load, which is equal to 30 kN. The material density, modulus of elasticity, and yield stress of this steel structure are 7833.413 kg/m3 , 205 GPa, and 248.2 MPa, respectively. Due to structural symmetry, the 1016 members of the structure are divided into 25 element groups. Similar to the previous design example, the design variables are the cross-sectional area of the structure members picked from the steel pipe sections mentioned in Table 4.1. Optimization constraints dealing with stress limitation on truss members are applied according to AISC-LRFD provisions [6]. These provisions are as follows: ⎧ ⎨

{

∅t Fy A g ∅t = 0.9 if σi ≥ 0 ∅t Fu Ae ∅t = 0.75 ⎩ pu ≤ pr ; pr = ∅c Fcr A g ∅c = 0.9 if σi < 0 pu ≤ pr ; pr = min

(4.19)

where pu and pr represent the needed strength and nominal axial strength; A g and Ae denote the gross cross-sectional area and the effective net cross-sectional area of member i; Fy and Fu represent the yield and ultimate tensile stress of the steel, and Fcr is determined as follows:

4.5 Design Examples

91

Table 4.2 Comparison of optimization results obtained by ESSOA and other considered metaheuristic algorithms for the 693-bar double-layer barrel vault Element group

Kaveh and Ilchi Ghazaan [5] CBO

ECBO

VPS

Present work MDVC-UVPS

SSOA [1]

ESSOA [1]

1

ST 4

ST 4

EST 3

ST 4

EST 3

EST 3

2

ST 1

ST 1

ST 1

ST 1

ST 1

ST 1

3

ST 1 1/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

4

ST 1 1/4

ST 1

ST 1

ST 1

ST 1

ST 1

5

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

6

EST 3

ST 3

ST 3 1/2

ST 3 1/2

DEST 2

EST 3

7

ST 1

ST 1

ST 1

ST 1

ST 1

ST 1

8

ST 3/4

ST 1

ST 3/4

ST 1

ST 3/4

ST 3/4

9

ST 1 1/2

ST 1

ST 1

ST 1

ST 1

ST 1

10

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

11

ST 3

EST 2

ST 3

EST 2 1/2

DEST 2

EST 2 1/2

12

ST 1

ST 1 1/4

EST 1 1/4

ST 1

ST 1 1/4

EST 1

13

ST 1 1/4

EST 2

EST 1

ST 1 1/2

EST 1

EST 1

14

ST 1 1/4

ST 1

ST 1

ST 1

ST 1

ST 1

15

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

16

ST 2

ST 1

EST 1 1/2

EST 1 1/4

EST 1 1/2

EST 1 1/2

17

ST 1 1/2

ST 1

ST 1

ST 1

EST 1

ST 1

18

EST 1 1/2

ST 3

EST 1 1/2

EST 2

EST 1 1/4

ST 2

19

ST 1 1/2

ST 1

ST 1

ST 1

ST 1

ST 1

20

ST 3/4

ST 3/4

EST 3/4

ST 3/4

ST 3/4

ST 3/4

21

ST 2 1/2

ST 3/4

ST 1

ST 1

ST 1

ST 1

22

ST 1

ST 3/4

ST 1

ST 1

ST 1

ST 1

23

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

ST 3/4

Best weight (lb)

10,221.8

9240.5

9201.4

9091.1

9165.7

9053.4

Average weight (lb)

15,563

9577

9823

9475

9418.7

9265.6

Worst weight (lb)

N/A

N/A

N/A

N/A

9722.5

9548.2

Standard deviation (lb)

3976

505

598

765

135.7

111.5

NSAs

4400

16,720

9800

4120

19,780

19,100

92

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

Fig. 4.4 The best and average convergence curves for the 693-bar double-layer barrel vault

Fig. 4.5 The obtained structural weight in each independent run for the 693-bar double-layer barrel vault

Fcr =

⎧( ) Fy ⎨ 0.658 Fe Fy ; ⎩ 0.877∗Fe ;

√ ≤ 4.71 FEy √ > 4.71 FEy

KL r

KL r

(4.20)

where Fcr is the critical buckling stress; K represents the effective length factor assumed to be 1 for all truss members; L represents the length of the member; r is the corresponding radius of gyration;E denotes the modulus of elasticity, and Fe is computed using the following equation:

4.5 Design Examples

93

Fig. 4.6 Displacement values in two different load conditions found by the ESSOA for the 693-bar double-layer barrel vault

Fig. 4.7 Stress ratio values in the first load condition found by the ESSOA for the 693-bar doublelayer barrel vault

π2E Fe = ( )2 KL

(4.21)

r

In addition, AISC-LRFD recommends that the tension and compression members’ maximum slenderness ratio must not be greater than 300 and 200, respectively. For displacement constraint, the limitation of 40/600 m was imposed on all nodes in the vertical direction.

94

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

Fig. 4.8 Stress ratio values in the second load condition found by the ESSOA for the 693-bar double-layer barrel vault

Table 4.3 displays the comparative findings between ESSOA and other optimization techniques. From this table, it can be observed that ESSOA finds the best weight (67,079 kg) among the different optimization methods, including CBO with a weight of 74,849 kg, ECBO with a weight of 67,839 kg, VPS with a weight of 67,229 kg, and SSOA with a weight of 68,398 kg. Moreover, the ESSOA has the lowest mean weight, which is 70,408 kg. According to this table, ESSOA needs 11,680 structural analyses (i.e., NSAs = 11,680) to determine the best weight (67,079 kg). The NSAs acquired by ESSOA are better than those acquired by ECBO (15,760), VPS (15,220), and SSOA (12,020) and slightly inferior to the NSAs acquired by CBO (9760). The diagram of convergence histories for the best and mean runs achieved by SSOA and proposed ESSOA is compared in Fig. 4.10. As can be seen, ESSOA has superior performance than SSOA in both aspects of accuracy and convergence speed. The optimum weight identified by the proposed ESSOA in each independent run is shown in Fig. 4.11. According to this figure, ESSOA discovers the optimized weights in 22 separate runs, which are smaller than the mean weight. This outcome demonstrates that ESSOA is a reliable optimization method. The displacement and stress ration values found by ESSOA in the best run are respectively shown in Figs. 4.12 and 4.13. These figures indicate that there is no violation of displacement and stress ratio values. The maximum values of the stress ratio and displacement obtained by the ESSOA are equal to 97.212% and 6.648 cm, respectively.

4.5 Design Examples

95

(a)

(c)

(b)

(d)

Fig. 4.9 Schematic of the1016-bar double layer grid; a 3D view, b top layer members, c bottom layer members, and d web members

4.5.3 A 1410-Bar Dome Structure As the last design example, a 1410-dome structure comprised of 1410 members and 390 nodes is studied. Figure 4.14 shows the schematic configuration of this largescale structure in both 3D and top views. A sub-structure of this structure with further details for nodal numbering is given in Fig. 4.15. Table 4.4 provides nodal coordinates of the sub-structure of the 1410-bar dome structure. There is a ball-jointed connection between structural members. A single loading condition acting on the sub-structure of the 1410-bar dome structure is given in Table 4.4. The material density, modulus of elasticity, and yield stress of this steel structure are ρ = 7850 kg/m3 , E = 200 GPa, and Fy = 400 MPa, respectively. The 1410 members of the structure are organized into 47 element groups due to structural symmetry. The design variables are the

96

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

Table 4.3 Comparison of the different optimization methods for the 1016-bar double layer grid structure Element group

Kaveh and Ilchi Ghazaan [5]

Present work

CBO

SSOA [1]

ECBO

VPS

ESSOA [1]

1

EST 5

EST 5

ST 6

EST 5

ST 6

2

DEST 3

EST 5

ST 5

ST 5

ST 5

3

ST 3 ½

ST 3

ST 3½

ST 4

EST 3

4

ST 2 ½

ST 3 ½

ST 2½

EST 2 ½

EST 2 ½

5

ST 2 ½

ST 2 ½

ST 4

ST 3 ½

ST 3

6

ST 2

ST 2

EST 1

EST 1 ½

EST 1 ½

7

ST 2

DEST 2

EST 2

EST 1 ½

EST 1 ½

8

ST 2 ½

DEST 2

DEST 2

EST 1 ½

ST 2 ½

9

DEST 2 ½

EST 2

EST 3

ST 4

EST 3

10

DEST 2 ½

ST 6

DEST 2½

DEST 2 ½

EST 2 ½

11

ST 1 ½

ST 2

EST 12

ST 2 ½

EST 4

12

DEST 5

EST 8

DEST 5

ST 10

ST 10

13

EST 3 ½

EST 3 ½

ST 4

EST 4

ST 4

14

EST 3 ½

ST 5

ST 5

ST 4

ST 5

15

EST 4

ST 4

ST 5

EST 4

EST 4

16

ST 6

EST 5

DEST 4

ST 6

ST 6

17

ST 5

ST 5

EST 4

ST 5

EST 4

18

EST 4

EST 5

EST 4

EST 5

ST 5

19

EST 5

EST 5

EST 4

DEST 4

EST 6

20

ST 8

ST 8

DEST 4

DEST 4

EST 6

21

ST 6

ST 5

ST 6

ST 6

ST 6

22

ST 3

ST 3

ST 3½

ST 3 ½

ST 3 ½

23

EST 6

EST 2 ½

EST 2½

ST 3 ½

ST 3 ½

24

ST 3 ½

ST 5

ST 2½

ST 2 ½

EST 2 ½

25

EST 1 ½

ST 4

EST 1½

ST 3 ½

EST 1 ½

Best weight (kg)

74,849

67,839

67,229

68,398

67,079

Average weight (kg)

79,422

73,042

72,366

72,084

70,408

Worst weight (kg)

N/A

N/A

N/A

75,626

80,828

Standard deviation (kg)

8154

9158

5545

1802

2703

NSAs

9760

15,760

15,220

12,020

11,680

cross-sectional area of the structure members selected from continuous ranges with the minimum and maximum allowable values equal to 1 × 10− 4 and 100 × 10− 4 m2 , respectively. Stress limitation on structural elements and stability of truss members are imposed according to the requirements of ASD-AISC [4], as presented in the

4.5 Design Examples

97

Fig. 4.10 The best and average convergence curves for the 1016-bar double-layer grid

Fig. 4.11 The obtained structural weight in each independent run for the 1016-bar double-layer grid

first design example. Furthermore, the optimization constraint dealing with nodal displacement for each node in every direction must be less than ± 8 cm. Table 4.5 compares the optimized results obtained by the presented ESSOA and other existing optimization methods. As observed in this table, ESSOA identifies the lowest weight (7331.6 kg) after 19,400 structural analyses. The optimal weights determined by the CBO, ECBO, VPS, MDVC-UVPS, and SSOA are respectively equal to 8413.46, 7860.01, 7848.68, 7661.64, and 7689.8 kg. These values are respectively acquired after 18,940 structural analyses, 19,840 structural analyses, 19,860

98

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

Fig. 4.12 Displacement values found by the ESSOA for the 1016-bar double-layer grid

Fig. 4.13 Stress ratio values found by the ESSOA for the 1016-bar double-layer grid

structural analyses, 16,308 structural analyses, and 19,820 structural analyses. The lowest mean weight acquired by ESSOA is equal to 7602.3 kg, which is the better weight among the reported optimization methods. Convergence histories of the best and mean of 30 separate runs recorded for the SSOA and proposed ESSOA are presented in Fig. 4.16. As can be seen from this figure, ESSOA has converged to the optimum solution better than SSOA. Consequently, it can be concluded that ESSOA has superior performance than SSOA in both aspects of computational cost and convergence speed. All results dealing with the final weights in each of 30 independent runs are shown in Fig. 4.17. Displacement and stress ratio values determined

4.5 Design Examples

99

Fig. 4.14 Schematic of the 1410-bar dome truss; a 3D view, b Top view

(a)

(b)

Fig. 4.15 Details of a substructure of the 1410-bar dome truss

by the ESSOA in the best run are respectively given in Figs. 4.18 and 4.19. These figures demonstrate that the optimization constraints defined for this design example have not been violated. The maximum values of the displacement and stress ratio determined by the ESSOA are equal to 0.5131 cm and 99.53%, respectively.

Coordinates (x, y, z)

(1.0, 0.0, 4.0)

(3.0, 0.0, 3.75)

(5.0, 0.0, 3.25)

(7.0, 0.0, 2.75)

(9.0, 0.0, 2.0)

(11.0, 0.0, 1.25)

(13.0, 0.0, 0.0)

Node number

1

2

3

4

5

6

7

0

0

0

0

0

0

0

Fx kN

0

0

0

0

0

0

0

Fy kN 8 9 10 11 12 13

−200 −600 −1000 −1500 −2000 −2500 0

Node number

Fz kN

(11.934, 1.2543, −0.5)

(9.945, 1.0453, 1.0)

(7.956, 0.836, 1.75)

(5.967, 0.627, 2.25)

(3.978, 0.418, 2.75)

(1.989, 0.209, 3.0)

Coordinates (x, y, z)

Table 4.4 Nodal coordinates and a single loading condition imposed on the sub-structure of the 1410-bar dome structure

0

0

0

0

0

0

Fx kN

0

0

0

0

0

0

Fy kN

−1000

−2000

−1500

−1200

−1000

−400

Fz kN

100 4 An Enhanced Shuffled Shepherd Optimization Algorithm …

4.5 Design Examples

101

Fig. 4.16 The best and average convergence curves for the 1410-bar dome truss structure Table 4.5 Comparison of the different optimization methods for the 1410-bar dome truss structure Element number (nodes)

Kaveh and Ilchi Ghazaan [5] CBO

ECBO

VPS

Present work MDVC-UVPS

SSOA [1]

ESSOA [1]

1 (1–2)

5.1214

5.217

4.6048

4.8489

4.7985

4.8298

2 (1–8)

2.2479

2.213

1.5208

1.5104

2.7592

2.0165

3 (1–14)

1

4.0413

1.4229

4.3939

4.5517

7.5790

4 (2–3)

5.6721

5.3523

4.785

4.8489

5.1241

4.8331

5 (2–8)

2.5777

2.8635

2.3714

2.3413

2.9311

3.8613

6 (2–9)

1.6817

1.8832

2.2803

1.6246

2.1854

1.8555

7 (2–15)

1.4126

1.0007

6.0836

4.3939

2.0614

1.5531

8 (3–4)

6.8558

6.4681

5.037

4.8489

5.6518

5.3270

9 (3–9)

2.1922

1.2068

2.1952

2.1707

6.4816

2.8348

10 (3–10)

2.0673

1.738

1.6864

1.6765

2.6323

2.3982

11 (3–16)

8.9218

12.5144

2.9786

4.3939

2.8623

2.8265

12 (4–5)

6.4513

6.3101

5.8296

7.6688

6.1446

6.1447

13 (4–10)

2.5147

1.7218

2.4275

2.4287

3.5949

1.4803

14 (4–11)

2.3745

2.4362

4.4668

1.8282

2.4946

2.3323

15 (4–17)

4.273

3.5615

3.0016

5.5832

4.3819

4.0043

16 (5–6)

6.5994

6.1832

6.1684

7.6688

6.2206

6.5057

17 (5–11)

3.3831

2.7977

2.5737

2.5749

2.6964

3.0395

18 (5–12)

2.7308

4.1412

4.5709

3.6629

4.1729

4.0052

19 (5–18)

8.5163

4.1542

4.2362

5.5832

4.6500

4.2089

20 (6–7)

7.834

7.9148

8.7333

7.6688

8.1462

7.8692 (continued)

102

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

Table 4.5 (continued) Element number (nodes)

Kaveh and Ilchi Ghazaan [5] CBO

ECBO

VPS

Present work MDVC-UVPS

SSOA [1]

ESSOA [1] 3.0147

21 (6–12)

3.6101

5.894

3.3266

3.7234

3.3501

22 (6–13)

5.0307

3.3083

5.439

3.1638

3.2705

3.5449

23 (6–19)

6.127

6.6223

5.8551

5.5832

5.7243

5.7876

24 (7–13)

3.8352

3.6804

3.7713

3.64

3.8860

4.1487

25 (8–9)

5.3726

4.8207

4.6028

6.1741

4.9043

5.0264

26 (8–14)

2.0258

1.5864

1.5129

1.5104

2.3993

2.7103

27 (8–15)

5.5215

2.5913

2.3505

2.3413

3.7742

3.5774

28 (8–21)

3.6576

1.0843

4.334

4.0242

1.6565

1.1653

29 (9–10)

5.638

5.9325

8.0424

6.1741

5.0077

5.1417

30 (9–15)

1.7705

3.0351

1.5699

1.6246

1.9835

1.8443

31 (9–16)

2.3381

1.2356

2.5573

2.1707

2.4325

2.4992

32 (9–22)

3.316

1.708

7.4354

4.0242

3.9596

2.3271

33 (10–11)

6.4184

4.8743

4.8246

6.3156

5.6558

5.2449

34 (10–16)

5.0152

3.429

1.6796

1.6765

3.5557

2.2090

35 (10–17)

2.9268

1.9623

3.3532

2.4287

2.8379

1.7165

36 (10–23)

5.7701

2.7079

2.4308

4.8511

4.4655

3.9555

37 (11–12)

8.4621

5.0557

5.1426

6.3156

5.1250

5.3902

38 (11–17)

1.925

4.1289

1.9981

1.8282

2.3498

2.4328 3.0903

39 (11–18)

3.0442

3.4292

2.5741

2.5749

2.8006

40 (11–24)

4.4108

4.9348

3.491

4.8511

3.9112

3.8433

41 (12–13)

8.4293

7.3564

6.3216

6.3156

6.6407

6.4713

42 (12–18)

2.295

4.4329

3.7521

3.6629

4.1691

3.8202

43 (12–19)

4.1246

3.3212

7.627

3.7234

3.5060

3.3478

44 (12–25)

5.3458

4.9391

4.8609

4.8511

5.0970

5.2009

45 (13–19)

3.199

3.7342

7.1805

3.1638

3.5661

3.4564

46 (13–20)

4.0629

4.1154

3.7848

3.64

3.9221

3.9634

47 (13–26)

3.6865

5.0799

3.7592

4.8511

3.6169

3.8686

Best weight (kg)

8413.46

7860.01

7848.68

7661.64

7689.8

7331.6

Average weight (kg)

9932.11

8250.20

8959.27

8106.52

8468.5

7602.3

Worst weight (kg)

N/A

N/A

N/A

N/A

9380.7

8049.4

Standard deviation (kg)

1726.69

409.09

1277.34

244.08

395.0

185.5

NSAs

18,940

19,840

19,860

16,308

19,820

19,400

4.6 Concluding Remarks

103

Fig. 4.17 The obtained structural weight in each independent run for the 1410-bar dome truss structure

Fig. 4.18 Displacement values found by the ESSOA for the 1410-bar dome truss structure

4.6 Concluding Remarks In this chapter, an ESSOA is proposed by incorporating two efficient features: The Opposition-Based Learning (OBL) method and a solution generator based on the statistical results of the solutions. To enhance the convergence rate of the algorithm, the OBL technique was only applied to the initialization phase. A solution generator is established based on the statistical results of the solutions was incorporated into the cyclic body of the algorithm. The proposed feature provides a good balance between

104

4 An Enhanced Shuffled Shepherd Optimization Algorithm …

Fig. 4.19 Stress ratio values found by the ESSOA for the 1410-bar dome truss structure

the exploration and exploitation capability of the algorithm such that it decreases the possibility of becoming trapped in a local optimum. Optimization aims to minimize the weight of the entire structure while satisfying some constraints on displacements and stresses. Three large-scale design examples, including a 693-bar double layer barrel vault, a 1016-bar double layer grid, and a 1410-bar dome structure, were examined, and the optimization results were provided. The optimization results obtained by the ESSOA were compared to those of the standard SSOA. In all investigated design examples, ESSOA outperformed SSOA in terms of the best weight, mean weight, worst weight, standard deviation, number of needed analyses, and convergence rate. These outcomes reveal that the ESSOA is a better and more robust algorithm than its standard version. Moreover, comparing the results found by ESSOA with those of some other state-of-art metaheuristics, namely CBO, ECBO, VPS, and MDVC-UVPS, demonstrates the superiority of ESSOA for size optimization of these large-scale design examples.

References 1. Kaveh, A., Zaerreza, A., Hosseini, S.M.: An enhanced shuffled shepherd optimization algorithm for optimal design of large-scale space structures. Eng. Comput. 38(2), 1505–1526 (2021) 2. Tizhoosh, H.R.: Opposition-based learning: a new scheme for machine intelligence. In: International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA-IAWTIC’06). IEEE (2005) 3. Kaveh, A., Zaerreza, A.: Shuffled shepherd optimization method: a new Meta-heuristic algorithm. Eng. Comput. 37(7), 2357–2389 (2020)

References

105

4. AISC, A.: Manual of steel construction—allowable stress design. American Institute of Steel Construction (AISC), Chicago (1989) 5. Kaveh, A., Ilchi Ghazaan, M.: Meta-Heuristic Algorithms for Optimal Design of Real-Size Structures. Springer (2018) 6. Construction AIoS: Load and resistance factor design. Amer Inst of Steel Construction (2001)

Chapter 5

A New Strategy Added to the SSAO for Structural Damage Detection

5.1 Introduction In order to provide a good performance for Shuffled Shepherd Optimization Algorithm (SSOA), in the damage detection problems, Kaveh et al. [1] applied the a boundary strategy (BS) to SSOA. This strategy gradually neutralizes the impact of healthy structural elements progressively throughout the optimization procedure. BS improves the performance of the optimization approach compared to traditional methods that do not use the suggested BS. This strategy increases the accuracy and convergence time of the SSOA and comparable algorithms for detecting and estimating the damages. Every engineering structure is susceptible to the occurrence of damage, which causes the structure’s performance to deteriorate over time. On the other hand, structural damage is likely to spread owing to changing mechanical characteristics of the structures which are occurred due to crack, creep, corrosion, and so on. Methods for recognizing and measuring the location and amount of damage in the components of engineering structures have received considerable attention. Among these methods, vibration-based damage detection methods, due to simplicity and being independence from external excitation, can be used as an efficient indicator of structural behavior before and after damage incidence. In a general, vibration-based damage detection approaches can be divided into two types. The first category comprises with non-model (data-driven) techniques. Although these methods can easily identify the site of the damages without utilizing structural analytical programs, they are unable of determining the amount of damages with a high level of accuracy [2]. The second category consists of model updating techniques in which the damage identification problem is defined as an inverse problem. This chapter focuses on finite element model updating methods using evolutionary algorithms. Evolutionary algorithm-based finite element model updating methods, have been more widely used than indirect methods in recent decades. Alternatively, they can © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_5

107

108

5 A New Strategy Added to the SSAO for Structural Damage Detection

efficiently identify the locations and intensity of the damages. When the metaheuristic optimization methods are used for damage identification of large-scale structures, the algorithms begin searching in high-dimensional search space. This is because the number of design variables is equal to the number of structural members. Consequently, the optimization method may unable to precisely predict the location and extent of damage. On the other hand, there is a high possibility for the metaheuristic algorithms to get trapped in local optima when they are applied to solve the damage detection problem. To this end, a novel strategy, namely Boundary Strategy (BS), is presented for structural damage detection problems utilizing metaheuristic optimization algorithms. This strategy reduces the complexity of the search space and enhances the effectiveness of the used optimization method for finding damaged members by reducing the search space’s complexity. In contrast, in typical damage detection methods using metaheuristic algorithms, the effects of healthy members are not neutralized in the optimization process. In this chapter, these conventional techniques are considered Without Boundary Strategy (WBS). In this chapter, a penalty function is introduced by integrating into a damagesensitive cost function based on vibration data. The Shuffled Shepherd Optimization Algorithm (SSOA) proposed by Kaveh and Zaerreza [3] is used to solve the damage detection problem. The selection of SSOA is based on its poor degree of accuracy in recognizing damaged components when WBS is used to solve the problem. However, when the SSOA incorporates the BS in the optimization procedure, it accurately identifies the location and severity of damage. To evaluate the capability of the proposed cost function using BS, four test examples, including a 25-bar planar truss, a 40-element continuous beam, a 23-element asymmetrical planar frame, and a largescale 72-bar spatial truss, are examined. The obtained results are compared with the three well-known parameter-less optimizers: Teaching–learning based optimization (TLBO) [4], Grey Wolf Optimizer (GWO) [5], and Moth-flame Optimization Algorithm (MFO) [6]. In addition, the robustness of the BS in comparison to WBS is investigated in different optimization methods and cost functions. The remainder of this chapter is structured as follows: Sect. 5.2 provides an overview of the SSOA. Section 5.3 presents the formulation of the problem under consideration and introduces BS. Numerical examples are provided in Sect. 5.4, and the obtained results are discussed. Concluding remarks are finally driven in Sect. 5.5.

5.2 Shuffled Shepherd Optimization Algorithm Shuffled Shepherd Optimization algorithms (SSOA) is a novel multi-community population-based metaheuristic presented by Kaveh and Zaerreza [3]. This optimization method mimics the behavior of shepherds in nature. In this method, firstly, members of each community are randomly created. Next, the shuffling procedure is carried out to improve survivability by exchange information in the search process. This procedure may result in the enhancement of the community via sharing of its knowledge with other communities. In the SSOA, to determine the new position of

5.2 Shuffled Shepherd Optimization Algorithm

109

each member in each community, the better and worse members are randomly chosen from the community of which the member under consideration belongs. Then, if the objective function value of the newly determined position is better than the previously created one, the newly determined position will be replaced by the previous one. Finally, the optimization process will be ended if the termination condition is satisfied.

5.2.1 Steps of SSOA The steps of SSOA are characterized by five main steps. These steps are as follows: 1. 2. 3. 4. 5.

Forming the initial community members. Shuffling process. Movement of community member. Updating the position of each community member. Termination condition of SSOA.

In the following, the steps mentioned above are described, and their mathematical interpretations are discussed.

5.2.1.1

Forming the Initial Community Members

In the SSOA, the initial positions of members of communities (MOC) are determined with a randomly generated population in a d-dimensional search space: M OCi,0 j = xmin + r × (xmax − xmin ); i = 1, 2, . . . , cand j = 1, 2, . . . , nm

(5.1)

in which r represents a random vector in which each component is produced between 0 and 1; xmin and xmax indicate the minimum and maximum permissible values, respectively; c and nm refer to the number of communities and number of members that belong to each community, respectively. Since each community has nm members and the total numbers of communities are equal to c, the population size is determined as follows: n Pop = c × nm

(5.2)

110

5.2.1.2

5 A New Strategy Added to the SSAO for Structural Damage Detection

Shuffling Process

The shuffling process denotes to merging communities into one community and formation of new communities. Consequently, initially, the whole population (nPop) is arranged based on the quality of solutions. Next, in order to generate the MC matrix (see Eq. 5.3), nPop are divided into c communities in which each community has the nm members. To this end, in the first step, the first c members are chosen from the arranged population and are randomly allocated to c communities so that each community has one member so far. Consequently, the first column of the MC matrix was created. In the subsequent step, the next c members are chosen from the remaining population and are again assigned to the c communities randomly. Thus, the second column of the matrix was formed as well. This procedure is performed nm times until each c communities has nm members. Therefore, the MC matrix is obtained as follows: ⎡ ⎤ M OC1.1 M OC1,2 · · · M OC1, j · · · M OC1,nm ⎢ M OC M OC · · · M OC · · · M OC ⎥ ⎢ 2,1 2,2 2, j 2,nm ⎥ ⎢ ⎥ .. .. .. .. .. .. ⎢ ⎥ . . . . . . ⎢ ⎥ (5.3) MC = ⎢ ⎥ ⎢ M OCi,1 M OCi,2 · · · M OCi, j · · · M OCi,nm ⎥ ⎢ ⎥ .. .. .. .. .. .. ⎢ ⎥ ⎣ ⎦ . . . . . . M OCc,1 M OCc,2 · · · M OCc, j · · · M OCc,nm where M OCi, j represents to the jth member of the ith community. According to the MC matrix (Eq. 5.3), each row denotes the members belonging to each community. In addition, the first column represents the members of each community with the highest quality, and members placed in the last column are the worst members of each community.

5.2.1.3

Movement of Community Member

After forming MC matrix, the stepsize of each M OCi, j is computed using two vectors. For this reason, for each M OCi, j , the members that have the better and worse objective function values than M OCi, j are chosen randomly. These members named M OCi,b and M OCi,w , respectively. In SSOA, M OCi, j not only goes toward the M OCi,b but it tends go the M OCi,w . Moving M OCi, j toward the M OCi,b illustrates its intensification tendency. In contrast, tending M OCi, j toward the M OCi,w illustrates its diversification tendency. This concept is represented graphically in Fig. 5.1 and is mathematically stated as follows: stepsi zei, j = stepsi zei,Wjor se + stepsi zei,Better j i = 1, 2, . . . , c and j = 1, 2, . . . , nm

(5.4)

5.2 Shuffled Shepherd Optimization Algorithm

111

Fig. 5.1 A schematic of position updating in SSOA

where stepsi zei,Wjor se and stepsi zei,Better are defined as follows: j ) ( stepsi zei,Wjor se = α × r1 × M OCi,w − M OCi, j

(5.5)

) ( stepsi zei,Better = β × r2 × M OCi,b − M OCi, j j

(5.6)

where stepsi zei,Wjor se and stepsi zei,Better are the stepsize vectors with d design varij ables. These vectors respectively illustrate the diversification and intensification tendencies of the algorithm; r1 and r2 represent random vectors whose components are created between 0 and 1. Since M OCi,nm is located in the last column of the W or se MC matrix (Eq. 5.3), it has no members worse than itself. Therefore, stepsi zei,nm Better (Eq. 5.5) is equal to zero. Similarly, stepsi zei,1 will be equal to zero because M OCi,1 does not have a member better than itself. α and β are the parameters that control exploration and exploitation, respectively. These parameters are specifications as follows: α = α0 − α0 × t β = β0 + (βmax − β0 ) × t; t =

(5.7) it Maxit

(5.8)

112

5 A New Strategy Added to the SSAO for Structural Damage Detection

where it and Maxit represent the iteration and the maximum number of iterations. According to Eqs. (5.7) and (5.8), α and β are among the most crucial parameters of the SSOA because they control the balance between exploration and exploitation. In this regard, here, decreasing α and increasing β respectively result to explore the search space more efficiently in the early iterations and search around the better solutions in the last iterations.

5.2.1.4

Updating the Position of Each Community Member

In this step, first, the new position of each community member is calculated as: newM OCi, j = M OCi, j + stepsi zei, j

(5.9)

in which newM OCi, j represents the new position of jth member of the ith community. Then, associated objective function is evaluated. In order to decide which positions (newM OCi, j or M OCi, j ) return to the population, the replacement strategy is used. Consequently, the objective function value of new M OCi, j and M OCi, j are compared, and the superior one is returned to the population.

5.2.1.5

Termination Condition of SSOA

In SSOA, the maximum number of iterations (Maxit) is assumed as a termination condition. Therefore, if the current iteration is less than Maxit, SSOA returns to Step 2 for a new round of iteration. If not, the algorithm ends, and the best community member is reported. The pseudo-code of SSOA is shown in Algorithm 5.1.

Algorithm 5.1 Framework of SSOA The procedure of Shuffled Shepherd Optimization Algorithm (SSOA) Begin Set the algorithm parameters; α0 , αmax , and β0 Initialize number of members belong to each community (nm) and number of communities (c), and termination criterion (Maxit) Create the initial candidate solutions and evaluate them While (termination criterion not satisfied) do For j:1 to nm Choose the c members from the remaining population based on the quality of solutions Put c chosen members randomly in the jth column of the MC matrix End For

5.3 Structural Damage Detection Approach

113

For i:1 to c For j:1 to nm Chose M OCi,b and M OCi,w randomly for M OCi, j Compute movement of M OCi, j utilizing Eq. (5.4) Modify the position of each community member based on step 4 End For End For End While End

5.3 Structural Damage Detection Approach In this section, first, the theoretical background of the inverse damage detection problem, including damage modeling and obtaining vibration data, are discussed. In second part, the suggested damage-sensitive cost function is described. In the last part, an effective strategy for resolving the problem is presented.

5.3.1 Theoretical Background In the vibration-based damage detection method, the main modal parameters, including natural frequency and mode shapes vector of a vibrating structure, can be determined using the following equation: (

) K − ωi2 M ϕi = 0; i = 1, 2, . . . , ndo f

(5.10)

in which ωi and ϕi represents natural frequency and mode shape vector in ith mode, respectively; M and K are respectively the mass and stiffness matrices with the dimension of ndo f × ndo f . An approach that is extensively used to model damage considers damage as the reduction of stiffness parameters such as modulus of elasticity (E), cross-sectional area (A), and moment of inertia (I ). In this method, it is considered that mass variations before and after damage are negligible. Likewise, here, the damage is modeled as a relative decrease of E in each structural member so that: R Fe = (1 − xe ); 0 ≤ xe ≤ 1

(5.11)

114

5 A New Strategy Added to the SSAO for Structural Damage Detection

E ed = R Fe × E eh ; e = 1, 2, . . . , nte

(5.12)

where xe represents the damage ratio of the eth member; nte denotes the total number of structural elements; E eh and E ed indicate modulus of elasticity of the eth healthy and damaged members, respectively;xe = 0 represents that the element is healthy, while xe = 1 shows that the member is fully damaged. Taking this into account, the overall stiffness matrix of structures is equal to the summation of the stiffness matrices of damaged and healthy members: K =

nte Σ

R Fe × ke

(5.13)

e=1

where ke represents the stiffness matrix of the eth member.

5.3.2 Proposed Objective Function In this section, the suggested cost function is discussed in detail. Damage occurrence results to changes in natural frequencies and related mode shapes of the structure before and after the damage. Although natural frequencies variations as a result of damage occurrence can be easily determined, its changes are low-sensitive to damage. Consequently, minor structural damage cannot be only determined by the natural frequency changes. Unlike natural frequencies, mode shapes include local information, which makes them more sensitive to local damage. Therefore, considering mode shapes make them be used directly in multiple damage detection. Additionally, mode shapes are less susceptible to external influences (e.g. temperature) than natural frequencies. Nevertheless, detecting mode shapes needs a large number sensors, which are measured with less accuracy than natural frequencies. Consequently, they are more susceptible to noise contamination than natural frequencies. In order to tackle the disadvantages of each main modal characteristics (natural frequencies and mode shapes), in this chapter, the combination of them is considered. Accordingly, an optimization problem’s damage-sensitive cost is composed of two functions. The first function is a penalty function that weights against an increasing number of damaged members. The second function is considered the combination of natural frequency and mode shapes. Since the effect of measurement noise causes optimization algorithms to predict many structural members as damaged ones, a penalty function is suggested against the increase of damaged members. The damage-sensitive cost function used for damage identification in this paper is defined as follows: Find X = [x1 , x2 , . . . , xnte ]T ; 0 ≤ xe ≤ 1 in which vector X denotes the ratio of structural damage.

(5.14)

5.3 Structural Damage Detection Approach

Minimi ze F(X );

115

F(X ) = (1 + γ · P(X )) × G(X )

(5.15)

in which F(X ) represents the suggested cost function; γ indicates the penalty factor (equal to 0.5 in this work), and P(X ) and G(X ) are respectively penalty function and the cost function without penalty: P(X ) = G(X ) =

nmod Σ

m d (X ) nte

(5.16)

(Ri × (1 − M AC(i, i )))

(5.17)

i=1

in which m d (X ) represents the number of damaged members identified by the metaheuristic algorithm in the solution X ; nmod number of utilized modes and Ri and M AC(i, i ) are computed as follows: ( d )2 ω Ri = ( i )2 , ωia

M AC(i, i ) = ( dT ϕi

)2 ( dT ϕi × ϕia )( ) × ϕid ϕiaT × ϕia

(5.18)

where ωi and ϕi represent the ith natural frequency and its corresponding mode shape, respectively. The superscript d and a indicate, respectively, for the damaged model and the analytical one, and M AC denotes the modal assurance criteria (MAC).

5.3.3 The Boundary Strategy (BS) in Metaheuristic-Based Damage Detection Structural damage detection problem utilizing finite element model updating is a very complex problem with many of local optimum. Despite a lot of work has been done to the damage detection methods applying metaheuristic algorithms, it is noted several drawbacks in damage detection outcomes. Some of them are listed below: (1) when too many design variables are considered in the problem, some metaheuristic algorithms cannot discover the position and amount of damage properly or may not predict it with a high degree of accuracy. (2) there is a high possibility of becoming caught in local optima when some metaheuristic algorithms are implemented. Thus, the algorithms fail to find the global optimum solution as the damage detection results. In order to alleviate these disadvantages, here, a simple strategy is presented for damage detection problems employing metaheuristic algorithms. This tactic is called Boundary Strategy (BS). In this technique, the optimization process progressively neutralizes the impacts of structural components that are associated to the healthy ones.

116

5 A New Strategy Added to the SSAO for Structural Damage Detection

In BS, first of all, the lower and upper limits of design variables are respectively set to be −1 and 1 instead of 0 and 1 in WBS. Then, the metaheuristic algorithm is implemented. Before evaluating the objective function of each agent in each phase, the vibration properties of the analytical model (like modal data) must be determined first according to Sect. 5.3.1. Therefore, in order to compute R Fe in Eq. (5.11), the value of each negative solution component must be set to zero. This is because the extent of damage in each structural member according to this equation is placed in [0, 1] interval. In other words, when the vibration characteristic of the model structure is computed, each solution component less than zero is converted to zero. It is worth mentioning that this change from [−1, 1] to [0, 1] is only performed to compute the R Fe in Eq. (5.11) and is not returned to the optimization process. This indicates that the values of the design variables do not change throughout the optimization process, with each variable remaining inside the [−1, 1] range. The BS causes the design variables associated to healthy members to be in [−1, 0] interval. When any design variable among all solutions of the population is placed in this interval, it traps in this range. Consequently, the influence of the relevant design variable is neutralized throughout the optimization procedure. SSOA as a population-based metaheuristic algorithm is utilized to evaluate the capability of BS in comparison to WBS. Figure 5.2 depicts the flowchart of SSOA for damage detection using BS.

5.4 Numerical Examples This section examines four numerical examples to illustrate the capabilities and effectiveness of the suggested technique. These examples are as follows: a 25-bar planar truss, a 40-element continuous beam, a 23-element asymmetrical planar frame, and a large-scale 72-bar spatial truss. All numerical case studies are investigated in two states. The first is the ideal situation in which input data are not contaminated by measurement noise. The second deals with the noisy situation in which each component of eigenvalue and eigenvector are contaminated with measurement noise as follows: input noise = input × (1 + rand × σ )

(5.19)

in which input noise and input represent natural frequencies value or mode shape vector in the noisy and ideal state, respectively. rand is a random number between −1 and 1, and σ represents the intensity of the applied noise. This chapter includes 1 and 3% noise contamination of natural frequencies and mode shape vectors, respectively. In order to compare the ratio of the detected and actual damage, an error index is determined as follows:

5.4 Numerical Examples

Fig. 5.2 Flowchart of the SSOA using BS for damage detection

117

118

5 A New Strategy Added to the SSAO for Structural Damage Detection

100(%) Σ |ADe − I De | × nte e=1 nte

Err or =

(5.20)

where ADe represents the actual damage ratio and I De denotes the identified damage ratio. SSOA is used to solve the optimization-based damage detection problem. The algorithm parameters in all test examples are assumed to be as follows:m = 4, n = 5, Maxit = 1000, α0 = 0.5, β0 = 2, and βmax = 2.5. Three more wellknown metaheuristics including Teaching–learning-based optimization (TLBO) [4], Grey Wolf Optimizer (GWO) [5], and Moth-Flame Optimization (MFO) [6] are run with the identical maxNSAs, and their finding results are compared to those discovered by the SSOA. In all cases, the number of required structural analyses (NSAs) is determined. For this purpose, an iteration in which differences between its corresponding cost function value and cost function value of the Maxit is less than 10−6 is found. Next, the discovered iteration is multiplied to the population size, which gives NSAs. All examined structures are modeled numerically in the MATLAB environment and are analyzed using the direct stiffness method. In order to get statistically meaningful findings, ten separate runs are performed on each test case. The mean values of the discovered results are reported in the figures. The healthy members which have negative values in the vector of the optimal solution using BS are considered equal to zero. Therefore, zero values in all bar graphs show that the respective member is healthy. It should be noted that in the first three test examples, the first five vibration modes are employed for determining damage, however in the last large-scale example this value is considered 12.

5.4.1 25-Bar Planar Truss The first example is a 25-bar planar truss, as shown in Fig. 5.3. This example has 12 nodes and 21 degrees of freedom (DOFs). For each member, the modulus of elasticity, material density, and cross-sectional area are respectively as follows: E = 200 GPa and ρ = 7780 kg/m3 , and A = 10 cm2 . Table 5.1 gives two different damage scenarios. In this example the capacity of BS as opposed to WBS is examined. For this purpose, the mean values of damage detection finding discovered by SSOA employing BS and WBS in various scenarios are presented in Figs. 5.4 and 5.5, respectively. A detailed inspection of these figures demonstrates that the results produced by using BS are much superior than those found by employing WBS in all cases. In other words, utilizing BS localized and quantified damaged members precisely even when the input data are contaminated by measurement noise. In contrast, employing WBS reveals that all members have damage even in the ideal state. Table 5.2 displays the statistical findings obtained employing SSOA utilizing

5.4 Numerical Examples

119

Fig. 5.3 Finite element model of the 25-bar planar truss

Table 5.1 Two different damage scenarios in the 25-bar planar truss Scenario Element no Damage ratio (%)

I

II 2

21

3

7

15

20

25

10

20

25

20

25

BS and WBS for two distinct damage scenarios in the presence and absence of noise. In all damage situations, the statistical findings achieved with BS are considerably superior than those obtained with WBS. For further investigation, it can be seen that employing BS reduces NSAs by more than 50% compared to the use of WBS in both damage scenarios. In addition, the error index [computed according to Eq. (5.20)] found from employing BS compared to WBS is very low and near to zero in all cases. It can be concluded that unlike utilizing WBS for damage identification, employing BS has a high level of acceptable accuracy.

(a)

(b)

Fig. 5.4 Average value of damage detection results for the scenario I of the 25-bar planar truss a using BS, and b WBS

120

5 A New Strategy Added to the SSAO for Structural Damage Detection

(a)

(b) Fig. 5.5 Average value of damage detection results for scenario II of the 25-bar planar truss a using BS, and b WBS Table 5.2 Statistical damage identification results in the 25-bar planar truss for both damage scenarios in the case with noise and without noise Scenario I

Noise level Actual location

Actual ratio

BS Avg. value

Std. value

Avg. value

Std. value

Noise-free

2

25

25

9.72E–7

42.42

6.0536

21

10

Noisy

10

2.89E–6

27.16

5.6981

Error (%)

1.34E–7

9.09E–8

19.7787

7.1536

NSAs

7126

2

25

25

0.4195

41.68

5.9545

10

10.24

1.7127

28.84

5.6336

0.1532

0.1011

21.1405

5.9146

4.72E–7

33.57

NSAs Noise-free

Noisy

19,800

21 Error (%) II

WBS

3

9986 20

20

19,822 3.5304

7

25

25

1.50E–6

39.51

13.2814

15

20

20

5.17E–6

21.27

14.8681

20

25

25

2.00E–6

36.21

3.1153

Error (%)

3.33E–7

6.04E–8

14.4617

3.9270

NSAs

4808

3

20

20.29

19,802 0.1928

36.51

8.0478

7

25

27.34

0.0516

47.31

11.5700

15

20

21.04

0.2180

33.06

17.3246

20

25

24.07

0.0520

37.77

4.2017

Error (%)

0.2081

0.0456

18.7160

4.9669

NSAs

7120

19,896

5.4 Numerical Examples

121

5.4.2 40-Element Continuous Beam A 40-element continuous beam is considered as the second test example to verify the capability of the suggested method. The finite element model of the beam is illustrated in Fig. 5.6. Each node of this beam has two degrees of freedom, and only the vertical components of the supports have been restricted. This results in a total of 79 DOFs. Both the width and height of each member are set to 15 cm. The modulus of elasticity and material density for all elements are E = 210 GPa and ρ = 7860 kg/m3 , respectively. Two distinct damage scenarios, as given in Table 5.3, are considered. In this example, three other well-known optimization algorithms, including TLBO, GWO, and MFO, are compared to SSOA in order to evaluate its performance. In order to do this, the average results achieved by these algorithms using BS are compared to those obtained by SSOA. These comparisons for the damage scenarios I and II are shown in Figs. 5.7 and 5.8, respectively. A detailed analysis of these figures demonstrates that by employing BS, SSOA could gain much better results than other optimization methods, whether the input data are contaminated with noise or not. For further examination to demonstrate the efficiency of the suggested method in the case when TLBO, GWO, and MFO are utilized for identified damaged elements, Fig. 5.9 presents the results employing WBS for the second damage scenario in the noisy condition. As indicated in this figure and Fig. 5.8b, it can be concluded that when BS in comparison to WBS is used, the results achieved by BS are much superior than those obtained by employing WBS. Table 5.4 provides statistical results for explored algorithms in addition to the results discovered by SSOA when BS is utilized. These statistical results include mean values for damage ratios, Errors, and NSAs. The standard deviations of damage ratios and Errors are also included in this table. All of these data indicate that SSOA achieves the better results among all other algorithms, both in ideal and noisy conditions. The most important finding is that the mean NSAs acquired by SSOA is much less than those achieved by other methods in all cases. Likewise, from the inspecting

Fig. 5.6 Finite element model of the 40-element continuous beam

Table 5.3 Two different damage scenarios in the 40-element continuous beam Scenario Element no Damage ratio (%)

I

II 7

20

37

2

6

8

26

32

35

10

60

45

55

20

55

60

122

5 A New Strategy Added to the SSAO for Structural Damage Detection

(a)

(b)

Fig. 5.7 Comparison of the average value of damage detection results from different metaheuristics for the scenario I of the 40-element continuous beam in: a noise-free state, b noisy state

(a)

(b) Fig. 5.8 Comparison of the average value of damage detection results from different metaheuristics for the scenario II of the 40-element continuous beam in: a noise-free state, b noisy state

Fig. 5.9 Comparison of the average value of damage detection results using WBS found by MFO, GWO, and TLBO for scenario II of the 40-element continuous beam in the noisy state

5.4 Numerical Examples

123

of Tables 5.4 and 5.5 in damage scenario II of the beam, it can be seen that when BS is employed, TLBO, MFO, and GWO algorithms gain better results than when WBS is used.

5.4.3 A 23-Element Asymmetrical Planar Frame The third test example is considered a 23-element asymmetrical planar frame. As shown in Fig. 5.10, the finite element model of this frame is consisting of 23 members, which include 14 columns and 9 beams. The frame includes 14 free nodes, and each node has three degrees of freedom, leading to 42 total DOFs. The cross-sectional area, mass per unit length, and moment of inertia for beam elements are equal to Abeam = 1.6 × 10−2 m2 , m = 1300 kg/m, and Ibeam = 3.5 × 10−4 m4 , whereas these values for column elements are respectively equal to Acolumn = 1.62 × 10−2 m2 , m = 125.6 kg/m, and Icolumn = 3.85×10−4 m4 . In addition, the modulus of elasticity and material density for all elements are the identical and respectively equal to E = 200 GPa and ρ = 7850 kg/m3 . Two distinct damage scenarios are suggested as given in Table 5.6. In order to demonstrate the superiority of the suggested cost function, a comparison is made with a second cost function in this case. There are three components to this cost function. The first and second portions are concerned with the difference between natural frequencies and mode shapes of the measured structure and analytical model. The third portion is a penalty against too many damaged sites, so that it weights against an increasing number of damaged locations. The following describes the cost function [7]: ( ) E(X ) = 1 + γ · m d (X ) ×

|| ( (nmod || ) ||) || d a 2 || Σ || φ d − φ a || nmod Σ || || ωi − ωi || i || || i || ( )2 || || φ d + φ a || + || || ωid i i i=1 i=1

(5.21)

Similar the previous examples, it is supposed that the first five mode’s data are available for comparison. The mean of damage identification results by utilizing the F(X ) and E(X ),and using BS for both damage scenarios are given in Fig. 5.11. Although in both damage scenarios both investigated cost functions can detect damaged members, there are several false predictions for the results obtained by E(X ). In addition, F(X ) employing BS can precisely locate the real location of both damage scenarios in this frame, even in a noisy condition. To compare the results achieved by BS with those found by WBS utilizing E(X ), scenario II of this frame in the case with noise and without noise is chosen for comparison as indicated in Fig. 5.12. As can be seen, when WBS is used, the results significantly get worse, and the error index is rises. When E(X ) employing WBS is incorporated by BS,

II

I

59.97

60

Noisy

55

45

55

20

2

6

8

22.09

54.88

44.44

9920

15

NSAs

55

15

26

32

20

3.06E-6

20

8

45 55

Error (%)

45

55

5400

NSAs

6

0.0194

Error (%)

37

10.70

10

20

3964 34.96

35

NSAs

7

Noise-free 2

Noisy

10 60 1.17E-7

10

60

20

37

TLBO

MFO

GWO

0.2129

0.0990

0.0843

8.96E-6

1.11E-4

5.52E-5

9.79E-6

3.08E-5

1.36E-4

1.77E-6

7.53E-6

6.87E-5

3.11E-5

4.79E-8

3.98E-7

2.88E-6

1.66E-6

14.85

49.44

42.15

16,632

1.2417

13.77

49.47

16.29

49.34

44.73

14,764

0.3624

59.62

7.66

33.81

13,828

0.0997

60.04

10.05

34.97

7.4640

16.4957

14.1305

1.7322

5.0890

16.5006

8.2478

16.4525

1.4099

0.2221

0.1730

3.8591

0.6507

0.0882

0.0393

0.1135

0.1922

9.39

54.66

33.49

11,338

2.4310

9.48

55.23

11.83

44.58

40.91

8066

2.1939

42.11

3.67

24.31

13,714

1.4111

60.70

6.92

28.96

9.5739

1.9565

21.9301

3.4466

7.9751

2.9115

9.7590

22.4081

13.9595

1.5369

27.5657

5.6206

15.9213

1.7126

1.0909

6.3282

14.6466

3.50

50.03

29.82

19,696

3.2446

1.80

44.34

0.40

49.26

31.15

18,702

0.8746

60

3.72

30.88

18,972

0.8636

60.17

4.27

30.78

(continued)

5.3420

16.6915

19.3838

2.5940

3.6374

22.1782

1.1870

16.4396

20.4255

0.7454

0.1774

4.5616

10.3096

0.6306

0.1289

5.1799

10.2790

Avg. value Std. value Avg. value Std. value Avg. value Std. value Avg. value Std. value 35

Error (%)

35

Actual location Actual ratio SSOA

Noise-free 7

Scenario State

Table 5.4 Comparison of statistical damage identification results using BS in the 40-element continuous beam for both damage scenarios in the case with noise and without noise

124 5 A New Strategy Added to the SSAO for Structural Damage Detection

Scenario State

Table 5.4 (continued)

54.74 0.1597 8422

Error (%)

15.77

NSAs

55

15

26

32

TLBO

MFO

GWO

0.0855

0.7954

0.0687

16,120

1.4316

13.90

49.39 1.6151

2.2986

16.4998

10,534

1.9666

6.23

49.45 1.5646

6.3742

16.5186

19,760

2.7639

0.34

55.11

2.0630

1.0284

1.3397

Avg. value Std. value Avg. value Std. value Avg. value Std. value Avg. value Std. value

Actual location Actual ratio SSOA

5.4 Numerical Examples 125

126

5 A New Strategy Added to the SSAO for Structural Damage Detection

Table 5.5 Comparison of statistical damage identification results obtained by different algorithms using WBS for scenario II of the 40-element continuous beam in the case with noise and without noise Noise level Actual location Noise-free

Noisy

Actual ratio

TLBO Avg value

MFO Std value

Avg. value

GWO Std. value

Avg. value

Std. value

2

45

36.12

18.0612

35.78

30.7190

35.75

17.8866

6

55

54.72

1.1301

60.99

21.8961

49.24

15.0559

8

20

20.76

1.0692

37.69

19.6206

8.73

9.6139

26

55

54.93

0.6419

65.84

10.2809

49.78

16.6101

32

15

13.07

4.7426

25.40

25.7395

7.96

6.8583

Error (%)

0.8986

0.9816

22.3001

15.7348

2.8388

2.1340

NSA

16,144

2

45

44.34

18,596 3.3786

35.57

19,884 23.8740

35.29

16.3648

6

55

49.13

16.3893

49.20

25.3673

49.95

16.6521

8

20

18.98

6.8690

26.67

20.8168

6.50

8.3067

26

55

48.91

16.3765

55.83

19.8007

54.92

1.2156

32

15

9.88

8.3634

26.53

17.0036

6.34

5.6541

Error (%)

1.4825

1.7455

13.6565

12.0996

3.1814

1.8289

NSA

16,256

16,122

19,890

the achieved results improve significantly even for noise-contaminated data. Consequently, it can be concluded that the BS is also capable of enhancing the performance of E(X ). For further examination, Table 5.7 shows the statistical results consisting of the mean and standard deviation values of damage ratios and errors. In addition, the mean of NSAs in both cost functions for both damage scenarios are shown in this table. A close examination of this table indicates the following: (1) the mean of detected damage is much close to real damage in all cases when F(X ) is used with BS. (2) the mean and standard deviation of errors estimated by F(X ) are superior than those calculated by E(X ). (3) the mean NSA in the case F(X ) is utilizing for damage identification is much superior than those achieved by E(X ). Consequently, in a general view employing F(X ) has better performance than utilizing E(X ).

5.4.4 A 72-Bar Spatial Truss As the last large-scale test case, a 72-bar spatial truss is explored. Four nonstructural masses are applied to the fourth story nodes in which each mass has a weight equal to 2270 kg, as illustrated in Fig. 5.13. The truss contains 16 free nodes, resulting to 48 active DOFs. For each element, the modulus of elasticity, material density,

5.4 Numerical Examples

127

Fig. 5.10 Finite element model of the 23-element asymmetrical planar frame

Table 5.6 Two different damage scenarios in the asymmetrical 23-element planar frame Scenario Element no Damage ratio (%)

I

II 4

10

4

18

21

15

25

15

35

20

and cross-sectional area are respectively E = 69.8 GPa and ρ = 2770 kg/m3 , and A = 25 cm2 . Two distinct damage scenarios are investigated as given in Table 5.8. Figure 5.14 shows the mean value of damage identification results in the case with noise and without noise for both damage scenarios. From this figure, although the structure has many members, it is clear that even in noisy conditions, all damaged elements are detected with high accuracy. Similar to previous examples, the statistical results in both noise-free and noisy conditions for damage scenarios I and II are given in Table 5.9. A close investigation of this table indicates that the number of successful

128

5 A New Strategy Added to the SSAO for Structural Damage Detection

(a)

(b)

Fig. 5.11 Comparison of damage detection results for the 23-element asymmetrical planar frame obtained from the SSOA using two different cost functions for a scenario I and b scenario II

Fig. 5.12 Obtained damage detection results for Scenario II of the 23-element asymmetrical planar frame using E(x) and WBS in the noisy condition

runs is equal to 100%, and the suggested method can identify damaged members with a maximum mean error equal to 0.1626%.

5.5 Concluding Remarks This study introduces a novel strategy called Boundary Strategy (BS) for damage identification issues based on optimization. In this strategy, despite the typical damage identification approaches that only zero values in the vector of design variables indicate healthy members, the range between −1 and 0 represents healthy members. BS neutralizes progressively the impacts of structural members that are healthy throughout the optimization process. This strategy reduces the complexity of the search space. Shuffled Shepherd Optimization Algorithm (SSOA) as a new multi-community metaheuristic is considered to solve the problem. Utilizing vibration data together with a penalty function, the damage-sensitive cost function is defined. Several instances are examined to evaluate the capabilities of the suggested

II

Noise-free

I

Noisy

Noise-free

Noisy

Noise level

Scenario

3282

21.60

NSA

20

21

35.65 0.1166

35

18

14.58

Error (%)

15

4

4002

20

35

NSA

20

21 3.88E-7

35

18

15

Error (%)

15

5824

4

0.0754

NSA

24.19

Error (%)

25

10

14.56

3538 15

4

NSA

25

15

2.42E-7

1.49E-6

5.9E-6

8.94E-7

2.19E-7

2.17E-6

7.34E-6

7.39E-7

0.0171

0.0090

0.0823

5.11E-8

5.55E-7

1.26E-6

7544

4.3392

13.18

5.92

13.64

10,066

2.2448

18.09

27.33

11.94

4326

1.3798

17.28

9.47

7004

1.7533

20.01

6.00

Avg. value

9.84E-8

25

BS, using E(X)

Avg. value

Std. value

BS, using F (X)

Error (%)

15

10

Actual ratio

4

Actual location

3.3973

12.4078

17.7746

12.5562

4.1413

6.0329

13.7262

5.9714

1.6445

11.3157

7.7532

1.7458

10.01

7.35

Std. value

19,388

18.5783

14.91

34.59

16.78

19,840

16.8918

18.24

29.90

25.58

19,282

12.3229

36.37

20.67

19,320

11.9748

30.46

20.71

Avg. value

9.2930

16.9992

37.1335

16.1385

7.1453

20.0152

30.4757

13.4528

6.4175

10.8711

13.6939

10.1250

15.1078

10.8528

Std. value

WBS, using E(X)

Table 5.7 Comparison of statistical damage identification results obtained by different cost functions for both scenarios of the 23-element asymmetrical planar frame in the case with noise and without noise

5.5 Concluding Remarks 129

130

5 A New Strategy Added to the SSAO for Structural Damage Detection

Fig. 5.13 Finite element model of the 72-bar spatial truss Table 5.8 Two different damage scenarios in the 72-bar spatial truss Scenario Element no Damage ratio (%)

I

II 5

1

21

37

30

25

20

30

(a)

(b)

Fig. 5.14 Damage detection results obtained by SSOA for the 72-bar spatial truss: a scenario I, b scenario II

References

131

Table 5.9 Comparison of statistical damage identification results for both scenarios of the 72-bar spatial truss in the case with noise and without noise Scenario

Noise level

Actual location

Actual ratio

I

Noise-free

5

30

Noisy

II

Noise-free

Noisy

Avg. value

Std. value

30

0.0088

Error (%)

2.89E-4

9.14E-4

NSA

11,704

5

30.85

0.0606

Error (%)

30

0.0407

0.0433

NSA

9978

1

25

25.03

0.1069

21

20

20.21

0.5871

37

30

29.82

0.4998

Error (%)

0.0323

0.1020

NSA

11,809

1

25

25.31

0.2523

21

20

19.76

1.4704

37

30

29.38

0.3828

Error (%)

0.1626

0.1673

NSA

16,436

technique. They consist of a 25-bar planar truss, a 40-element continuous beam, a 23-element asymmetrical planar frame, and a large-scale 72-bar spatial truss. In the first numerical examples, the performance of the BS in comparison to WBS in identifying and quantifying damage was examined. In the second case, the SSOA was compared to three well-known metaheuristics, namely TLBO, GWO, and MFO. In the third example, the suggested cost function was compared to another cost function. In the last example, a large-scale truss with 72 design variables is examined with the suggested BS. All gained results demonstrate that SSOA considering BS and the proposed cost function have appropriately functioning for both noisy state and large-scale issues.

References 1. Kaveh, A., Hosseini, S.M., Zaerreza, A.: Boundary strategy for optimization-based structural damage detection problem using metaheuristic algorithms. Period Polytech Civ. Eng. 65(1), 150–167 (2021) 2. Dinh, D., Nguyen-, T., Nguyen, D.T.: A FE model updating technique based on SAP2000-OAPI and enhanced SOS algorithm for damage assessment of full-scale structures. Appl. Soft Comput. 89, 106100 (2020) 3. Kaveh, A., Zaerreza, A.: Shuffled shepherd optimization method: a new meta-heuristic algorithm. Eng. Comput. 37(7), 2357–2389 (2020)

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4. Rao, R.V., Savsani, V.J., Vakharia, D.P.: Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput.-Aided Des. 43(3), 303–315 (2011) 5. Mirjalili, S., Mirjalili, S.M., Lewis, A.: Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014) 6. Mirjalili, S.: Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm. Know.-Based Syst. 89, 228–249 (2015) 7. Kaveh, A., Maniat, M.: Damage detection based on MCSS and PSO using modal data. Smart Struct. Syst. 15(5), 1253–1270 (2015)

Chapter 6

Optimum Design of Curve Roof Frames by SSOA and Comparison with TLBO, ECBO, and WSA

6.1 Introduction In this chapter, the discrete optimum design of two types of portal frames, including planar steel Curved Roof Frame (CRF) structures and Pitched Roof Frame (PRF) structures with tapered I-section members are presented, which is investigated by Kaveh et al. [1]. The optimal design aims to minimize the weight of these frame structures while satisfying some design constraints based on the requirements of ANSI/AISC 360-16 and ASCE 7-10. Four population-based metaheuristic optimization algorithms are applied to the optimal design of these frames. These algorithms consist of Shuffled Shepherd Optimization Algorithm (SSOA) [2], Teaching– Learning-Based Optimization (TLBO) [3], Enhanced Colliding Bodies Optimization (ECBO) [4], and Water Strider Algorithm (WSA) [5]. Portal frames are a type of structural frame in which its elements comprise columns and curved or pitched rafters. In these frames, the connections between columns and rafters are considered as moment-resisting. However, the connections between columns and base plates can be either pin-jointed or fixed-jointed, and the pinbased are more economical than the fixed-based. In the construction of industrial buildings, warehouses, gyms, fire stations, agricultural structures, hangars, etc., portal frames are commonly utilized. These can be constructed in different forms. This chapter investigates two types of them, namely Pitched Roof Frame (PRF) and Curved Roof Frame (CRF). These frame’s components can be prismatic or non-prismatic (tapered). The prismatic members have a constant cross-section over their whole length, whereas tapered members have a varied cross-section along their entire length. Economically, the incorporation of tapered members into the portal frames results to non-uniform distribution of bending moments and reduction in material consumption over a wide range of spans [6]. Optimization of structures is one of the most researched areas in engineering, and it has garnered several research articles. Since there is a limited number of

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_6

133

134

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

existing resources in engineering design, designers attempt to identify most costeffective solution that meets all requirements. Obtaining this optimal solution is a challenging task. It cannot be found with an exact method within a reasonable amount of time. Using approximate algorithms such as metaheuristic algorithms is the main alternative to find this solution. Focusing on a relatively simple concept and easy implementation, not requiring the gradient information, and bypassing most local optima indicate why metaheuristic optimization algorithms are more effective than other optimization methods and have grown more prominent in engineering applications in recent years [2]. Two main objectives are followed in this chapter. The first one deals with comparing the optimized weight of the CRF and PRF structures with the same dimensions for height and span in two different span lengths (16.0 and 32.0 m), and the second compares the performance of the proposed metaheuristic algorithms in the optimal design of these portal frames. For the fair comparison, it is assumed that the geometrical properties (height and span) of the frames are same, and they are compared in two different span lengths: 16.0 and 32.0 m. The optimal design of the frame structures is performed based on linking the existing commercial software SAP2000 and MATLAB via Open Application Programming Interface (OAPI) feature. Since the frame members are non-prismatic, the dimensions of the crosssections at the beginning and end of the frame members are considered as design variables. The rest of this chapter is organized as follows: In Sect. 6.2, the employed metaheuristic optimization techniques are briefly described. Section 6.3 presents the mathematical formulation for the discrete structural optimization of the portal frame problem and related design constraints. A brief description of structural loading is presented in Sect. 6.4. In Sect. 6.5, design examples and explanations of the outcomes gained are provided. This chapter’s concluding remarks are presented in Sect. 6.6.

6.2 Metaheuristic Algorithms In this chapter, four population-based metaheuristic algorithms are utilized for the optimum design of planar steel CRFs and PRFs. These algorithms are Teaching– Learning- Based optimization (TLBO), Enhanced Colliding Bodies Optimization (ECBO), Shuffled Shepherd Optimization Algorithm (SSOA), and Water Strider Algorithm (WSA). These optimization techniques are briefly detailed in the following subsections.

6.2.1 Teaching–Learning-Based Optimization (TLBO) Rao et al. [3] presented the Teaching–learning-based optimization (TLBO) method, which is based on the learning process occurring in the school. Like majority of the

6.2 Metaheuristic Algorithms

135

population-based optimization techniques established in the literature, TLBO begins with random solutions, each of which is referred as student or Learner (L). In each iteration of the TLBO, the best student whose solution is of the greatest quality is designated as the teacher. The stages of this algorithm are the teacher phase and the learner phase. These steps are repeated repeatedly inside the algorithm’s iterative body to find the optimal student. TLBO contains two parameters: number of learners (nL) as population size and maximum number of function evaluations (MaxNFEs) as a stopping criterion. Since these two parameters exist in every other population-based technique, TLBO may be referred to as a parameter-free optimizer. During the teacher phase, learners are updated depending on the teacher’s knowledge transfer. Therefore, the performance of the class as measured by a normal distribution of marks is enhanced by shifting the average position of the students toward the top student (teacher). The teacher phase is expressed mathematically as follows: Lnew,i = Lold ,i + randi,j · (T − Fi × ML); i = 1, 2, . . . , nL , j = 1, 2, . . . , nd (6.1) where Lnew,i and Lold ,i are respectively the new and old positions of the student; randi,j is a random number created inside the [0,1] interval; T represents the best learner who is considered as the teacher; Fi is a teaching factor that can be either 1 or 2; ML is the mean position of the learners in the search space, and nd is the number of design variables. This phase denotes intensification or global search capability of the TLBO algorithm by moving the ML toward the teacher. In the leaner phase, learners are upgraded cyclically depending on the transfer of information through contact with a randomly chosen individual. This phase can be mathematically expressed as follows: { Lnew,i = Lold ,i + randi,j ·

Li − Lrs iff (Li ) < f (Lrs ) Lrs − Li iff (Li ) > f (Lrs )

(6.2)

in which randi,j is a random number generators which generate the in the [0,1] interval; Li denotes the ith learner; Lrs represents to a randomly chosen learner (rs /= i); f (Li ) and f (Lrs ) indicate the objective function values of ith and randomly chosen learners, respectively. This step demonstrates the diversification or local search capability of the TLBO algorithm. Because each learner attempts to discover a better position by searching around its neighborhood and exchanging information with a learner chosen randomly. After each searching stage, the replacement technique is used to keep the old students or replace them with the newly created ones. In this sense, the learner with the lowest objective function value or highest quality is preferred to the old learner. Algorithm 1 provides the pseudo-code of the TLBO algorithm for more explanation.

136

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Algorithm 1: pseudo-code of the TLBO Set the algorithm parameters: nL and MaxNFEs Generate the initial students randomly in the search space Evaluate the initial students While NFEs ≤ MaxNFEs Determine the teacher for the students Determine the average position of the students Create the new students utilizing Eq. (6.1) Evaluate the new students Employ the replacement technique between the new and old students Create the new students utilizing Eq. (6.2) Evaluate the new students Employ the replacement technique between the new and old students End While Report the best students discover by the TLBO

6.2.2 Enhanced Colliding Bodies Optimization (ECBO) Colliding Bodies Optimization (CBO) is a population-based metaheuristic approach that is simple and effective. This optimization technique is inspired by the collision of two bodies in a single dimension. CBO, like TLBO, lacks algorithm-specific parameters, hence it is considered a parameter less optimization method. Although CBO has these benefits, it also has certain drawbacks. In order to alleviate these disadvantages, Kaveh and Ilchi Ghazan [4] created an improved form of the method, namely ECBO. In their suggested method, a memory that stores the number of the best solutions so far acquired and a mechanism that modifies certain components of colliding bodies (CBs) are employed to enhance the algorithm’s performance. The memory can boost the convergence speed of ECBO relative to CBO, and the method allows CBs to escape from local optima and avoids unintended premature convergence. Similar to previous population-based metaheuristics, ECBO begins with a set of candidate solutions, each referred to Colliding Body (CB). These CBs are randomly created within the search space. Thereafter, the objective function values of the CBs are evaluated. For each CB, a certain mass is assigned according to the following equation: 1/f (CBi ) ; i = 1, 2, . . . , nCB mi = ∑nCB i=1 1/f (CBi )

(6.3)

6.2 Metaheuristic Algorithms

137

where f (CBi ) denotes the objective function value for the ith CB, and nCB represents the number of colliding bodies. To store a number of the best solutions obtained so far, a Colliding Memory (CM) is employed in the cyclical body of the ECBO algorithm. To do this, the vector of solutions stored in CM is added to the current population, and the equal number of the existing worst CBs are removed. Next, the CBs are sorted according in ascending order by their related masses. CBs are separated into two different categories next: stationary and moving. The first half of the objects (i = 1, 2, . . . , nCB ) are supposed to be stationary 2 objects, while the second half of them are assumed to be moving objects (i = nCB + 1, nCB + 2, . . . , nCB). For collision, moving objects move toward to the corre2 2 sponding stationary objects. The velocities of stationary and moving objects before ' collision (vi ) and after collision (vi ) can be determined respectively by the following equations: vi = 0; i = 1, 2, . . . , vi = CBi− nCB − CBi ; i = 2

(

nCB 2

(6.4)

nCB nCB + 1, + 2, . . . , nCB 2 2

) mi+ nCB + εmi+ nCB vi+ nCB

nCB 2 2 ; i = 1, 2, . . . , mi + mi− nCB 2 2 ) ( nCB vi m − εm i i− nCB nCB ' 2 + 1, + 2, . . . , nCB vi = ; i= mi + mi− nCB 2 2 '

vi =

2

(6.5)

(6.6)

(6.7)

2

ε =1−

it MaxNITs

(6.8)

in which ε is the coefficient of restitution (COR) decreasing linearly from one to zero; it represents the current iteration number of the algorithm; MaxNITs represents the maximum number of algorithm iterations. The ECBO method assumes that the present position of the stationary objects is the origin of both moving and stationary objects. Therefore, the new position of both stationary and moving objects are determined by adding their new velocities to their present positions according to the following equations: '

CBnew,i = CBold ,i + randi ◦ vi ; i = 1, 2, . . . , '

CBnew,i = CBold ,i− nCB + randi ◦ vi ; i = 2

nCB 2

nCB nCB + 1, + 2, . . . , nCB 2 2

(6.9) (6.10)

138

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

where randi generates a uniformly distributed random vector in which each component is inside the range of [–1,1] and the sign "◦" denotes the element-by-element multiplication between two vectors. In the following phase of the ECBO algorithm, a method is added to escape from local optima. To do this, a uniformly distributed random number like rni is produced in the range of [0,1] for each CBi . This randomly generated value is then compared to a parameter inside [0,1] such as pro. If rni < pro is true, one randomly chosen component of ith CB is regenerated, and its value is modified as follows: ( ) CBij = CBj,min + randij × CBj,max − CBj,min

(6.11)

where CBij represents the jth design variable of the ith CB; CBj,min and CBj,max denote the minimum and maximum values for the jth design variable, respectively. As a stopping condition of the ECBO method, the optimization procedure will end when the maximum number of function evaluations (MaxNFEs) is achieved. Algorithm 2 provides the ECBO algorithm’s pseudo-code.

Algorithm 2: Pseudo-code of the ECBO Set the algorithm parameters: nCB, MaxNFEs, size of CM , and pro Randomly generate the initial CBs in the search space Evaluate the initial CBs While NFEs ≤ MaxNFEs Compute the mass of each CB utilizing Eq. (6.3) Modify the population Sort the population from lowest to highest Generate the groups Determine the velocity of each CBs utilizing Eq. (6.6) and (6.7) Create the new CBs utilizing Eq. (6.9) and (6.10) Employ escape from local optima mechanism utilizing Eq. (6.11) Evaluate the new CBs End While Report the best CB identified by the ECBO algorithm.

6.2.3 Shuffled Shepherd Optimization Algorithm (SSOA) In 2020, Kaveh and Zaerreza [2] established a novel multi-population metaheuristic algorithm, namely the Shuffled Shepherd Optimization Algorithm (SSOA). This algorithm is based on the herding behavior of shepherds in nature. Similar to the

6.2 Metaheuristic Algorithms

139

TLBO and ECBO algorithms, SSOA begins with randomly generated solutions called Sheep (S). Sheep are separated into the distinct nh herds utilizing the shuffling procedure. To do this, initially all sheep are evaluated and arranged in ascending order according to their objective function values. Next, nh of the first sorted sheep are chosen and allocated randomly to each herd. Hence, each herd has one sheep at the first step of forming herds. After allocating the first sheep for each herd, the sorted sheep from nh+1 to 2nh are chosen and again allocated randomly to the herds. Each herd has two sheep at the end of forming herds in this step. This procedure is repeated until all sheep are distributed to the herds. There are an equal number of sheep in each herd, and the best and worst sheep are the first and last members of each herd. After establishing the herds and allocating the sheep to them, the step size can be determined for each sheep. To do this, sheep with better and worse objective function values than the considered sheep are chosen randomly from the same herd. In the SSOA, the considered, superior, and inferior sheep of the corresponding herd are referred as the shepherd (Si,j ), horse (Si,h ), and sheep (Si,s ). In order to guide the sheep toward the horse, the shepherd changes his position to the sheep and then moves toward the horse. This concept can be expressed mathematically as follows: sheep

stepsizei,j = stepsizei,j

horse + stepsizei,j

i = 1, 2, . . . , nh j = 1, 2, . . . , nS/nh

(6.12)

where nS and nh are the number of sheep and herds, respectively. Moreover, sheep horse stepsizei,j and stepsizei,j are calculated as: ) ( = α × rand1 ◦ Si,s − Si,j

(6.13)

) ( horse stepsizei,j = β × rand2 ◦ Si,h − Si,j

(6.14)

sheep

stepsizei,j

where rand1 and rand2 are random vectors generators in the range of [0,1]; α and β are control parameters. They are employed to control the exploration and exploitation rates of the SSOA, respectively. These controlling parameters are described as follows: ) ( it (6.15) α = αmax × 1 − MaxNITs β = βmin + (βmax − βmin ) ×

it MaxNITs

(6.16)

where it and MaxNITs represent respectively the current number of iteration and the maximum number of iterations; αmax , βmax , and βmin are user-defined algorithm parameters. Equations (6.15) and (6.16) demonstrate that if the number of algorithm iterations is increased, α decreases linearly from αmax to zero, and β grows linearly from βmin to βmax . Decreasing α and increasing β provide a good counterbalance

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6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

between exploration and exploitation capabilities during the course of the optimization process. After computing the stepsize for the sheep of all herds, the new location of each sheep is as follows: new old Si,j = Si,j + stepsizei,j

(6.17)

new In the subsequent phase, the replacement technique is implemented between Si,j old and Si,j , and the best position of sheep is going to the subsequent iteration. This procedure is done for all of the sheep in all herd. Thereafter, the generated herds are merged together, and the sheep are arranged in ascending order according to the objective function values. Again, the sheep are separated into nh herds utilizing the shuffling process. The aforementioned process is repeated in the cyclic body of the algorithm until the algorithm is ended. Similar to the TLBO and ECBO algorithms, SSOA uses the MaxNFEs as its ending criteria. The pseudo-code of the SSOA algorithm is provided in Algorithm 3.

Algorithm 3: Pseudo-code of the SSOA Set the algorithm parameters: nS, nh, αmax , βmin , βmax , and MaxNFEs Randomly generate sheep in the search space Evaluate the initial sheep While NFEs ≤ MaxNFEs Employ shuffling prosses. Compute the stepsize utilizing Eq. (6.12) Create the new sheep utilizing Eq. (6.17) Evaluate the new sheep Employ the replacement technique between the new and old sheep Combine all herds into the single population End While Report the best S identified by the SSOA algorithm

6.2.4 Water Strider Algorithm (WSA) Water Strider Algorithm (WSA), proposed by Kaveh and Dadras Eslamlou [5], is a novel nature-inspired metaheuristic algorithm. This method simulates water striders’ territorial behavior, intelligent ripple communication, mating style, feeding processes, and succession of water striders. WSA is initiated with a population generated randomly in the search space. In this algorithm, each candidate solution is termed

6.2 Metaheuristic Algorithms

141

a Water Strider (WS). After the algorithm’s startup step, the territories are generated. Each territory has at least one adult male (keystone) and few female insects. The following approach is used to allocate nws number of WSs to nt number of territory. In the initial phase of generating territories, WSs are evaluated and arranged in ascending order of their objective function values. Then, the population of WSs is orderly separated into nws/nt groups. Next, the first WS from each group is picked and allocated orderly to the first territory. At the end of the first step, the first territory has nws/nt WSs. In the subsequent stage, the second WS of each group is selected and allocated orderly in the second territories. At the end of this step, the second territory has the same number of WSs as the first territory. The process of generating territory continues until each WSs has been allocated to the territories. At the end of last step, each territory has an equal number of WSs. It is clear that the first and last WSs are respectively the best and worst agents within a specified territory, and they are respectively considered as female and male (keystone). WSs are updated cyclically to search the optimal within three consecutive steps: mating, feeding, and finally succession of keystone. Each of these phases is described in brief as follows:

6.2.4.1

Mating

The keystone transmits a ripple to an objective female in order to mate. She reacts to him by emitting messages of attraction or repulsion. Since this answer is uncertain, the probability of mating has been determined. This chance is considered to be 50% for simplicity. The following equation gives an equal chance of mating and repelling and updates the position of the keystone: { WSic+1

=

if rand < p WSic + R ◦ rand WSic + R ◦ (1 + rand ) otherwise

(6.18)

where WSic represents the location of the i th WS in the c th cycle; rand is a random vector generator whose components are created between 0 and 1, and R is a vector defined as follows: R = WSFc−1 − WSic−1

(6.19)

in which WSFc−1 and WSic−1 denote the location of female and male WSs in the (c − 1) th cycle, respectively.

6.2.4.2

Feeding

The keystone expends a significant amount of energy whether or not the mating procedure is successful. In the subsequent phase, he must recuperate by looking for food sources. For finding the food supplies, the objective function is evaluated.

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6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

If the objective function value is smaller than that gained in the mating process, he has already gotten sufficient food. Alternatively, if the objective function of the keystone is worse than that of in the previous state, the keystone should migrate toward the best WS of the lake (WSBL ) to search for food availability according to the equation below: ) ( c WSic+1 = WSic − 2 × randi ◦ WSBL − WSic

6.2.4.3

(6.20)

Succession of Keystone

In this stage, if the keystone’s quality is inferior than its prior stage, it will perish since it cannot find food. Consequently, a new keystone is randomly formed in the lake as follows: ) ( c c c (6.21) WSic+1 = WSj,min + rand ◦ WSj,max − WSj,min In the cyclical body of the WSA, the aforementioned processes for constructing territories are repeated until the algorithm’s ending requirement is fulfilled. Similar to the other discussed algorithms, the MaxNFEs is considered as the ending condition of the WSA. Algorithm 4 provides the pseudo-code for the WSA algorithm for more understanding.

Algorithm 4: Pseudo-code of the WSA Set the algorithm parameters: nws, nt, p, and MaxNFEs Randomly generate WSs in the search space Evaluate the initial MaxNFEs While NFEs ≤ MaxNFEs Create territories Determine the new location of keystone utilizing Eq. (6.18) Evaluate him If the old keystone is superior than the new one Determine the other new keystone utilizing Eq. (6.20) Evaluate him If the old keystone is superior than the new one again Regenerate the keystone randomly in the search space Evaluate him Substitute the new keystone with the old one

6.3 Statement of the Discrete Optimization Problem

143

Else Substitute the new keystone with the old one End If Else Substitute the new keystone with the old one End If End While Report the best WS identified by the WSA algorithm

6.3 Statement of the Discrete Optimization Problem The definition of the optimization problem for the optimum design of the frame structures is as follows: Find {X } = [x1 , x2 , x3 , . . . , xn ] To minimize : W ({X })

(6.22)

in which {X } represents the vector of design variable; n represents the number of design variables, and W ({X }) is the total weight of the steel frame structures. We are aware that the described algorithms (i.e., TLBO, ECBO, SSOA, and WSA) were created for continuous search space. However, such methods are readily applicable to discrete optimization issues. In this chapter, the vector of continuous design variables is converted to discrete design variables using the rounding function. In other words, the vector of design variables (i.e., {X } = [x1 , x2 , x3 , . . . , xn ] in Eq. (6.22)) created by the algorithm is rounded to the closest possible value utilizing the rounding function. The objective function stated in Eq. (6.22) is minimized subjected to the following design constraints: G1 : Check the stability constraint G2 : Check the buckling constraints G3 : Check the strength constraints G4 : Check the maximum vertical displacement G5 : Check the maximum horizontal displacement

(6.23)

To handle all the aforementioned restrictions, the penalty technique is utilized. Consequently, if these design constraints are not violated, the penalty will have value of to zero. Alternatively, if any of the design restrictions is violated, objective function is penalized as follows:

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6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

( fpenalty ({X }) = W ({X }) × 1 +

nte ∑

) pi

(6.24)

i=1

where nte is the number of structural members, and pi is the penalty for the i th member, which can be computed as follows: ⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨ r pi = ⎪ 1 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ 0

if G1 is violated if G2 is violated if G3 is violated if G4 is violated if G5 is violated otherwise

(6.25)

in which r > 1 is the stress ratio.

6.3.1 Checking the Design Constraints of the Problem In the following subsections, the approach of checking design constraints provided by Eq. (6.23) is examined.

6.3.1.1

Checking Constraint G1

To check the stability of the structure, constraint G1 is used. The definition of this limitation based on ANSI/AISC 360-16 standards [7]. For evaluating the stability of a structure, the stability index under P-delta effects that produce additional forces in the members is determined using the following equation: θ=

Px ΔIe Vx hsx Cd

(6.26)

in which θ denotes the coefficient of stability; Px represents to the total vertical design load above level x with a maximum load factor of 1.0 (kip or kN); Δ indicates the design story drift happening concurrently with Vx ; Ie represents the importance factor; Vx denotes the seismic shear force acting between levels x and x − 1; hsx indicates the story height below level x, and Cd represents the deflection amplification factor. The upper bound of θ is defined as follows: θmax =

0.5 ≤ 0.25 βCd

(6.27)

6.3 Statement of the Discrete Optimization Problem

145

where β represents the ratio of shear demand to shear capacity for the story between levels x and x−1. It is important to note that β is allowed to be conservatively taken as 1.0. If θ ≤ 1.0, P-delta effects are not required to be considered. If 1.0 < θ ≤ θmax , the incremental factor associated with the P-delta effects on member forces and displacements must be determined by the rational analysis. Alternately, it is allowed to multiply member forces and displacements by 1.0/(1 − θ ). Otherwise, if θ > θmax , the structure is potentially unstable and should be redesigned [7, 8].

6.3.1.2

Checking Constraint G2

This restriction checks the buckling of the structural members. According to the ANSI/AISC 341-16 [8] requirements for designing slender compression members, the logical and practical width-to-thickness ratios (i.e., Eqs. (6.28) and (6.29)) must be met so that the buckling does not occur.

6.3.1.3

bf /tf ≤ 18

(6.28)

h/tw ≤ 0.4E/Fy ≤ 260

(6.29)

Checking Constraint G3

This restriction checks the strength of each section according to the requirements of ANSI/AISC 360-16 [8]: Pu Mu Pu + − 1 ≤ 0; if < 0.2 2φc P n φb M n φc P n ) ( Pu Pu 8 Mu − 1 ≤ 0; if + ≥ 0.2 9 φc P n φb M n φc P n

(6.30)

where Pu represents the needed strength (tension or compression); P n indicates the nominal axial strength (tension or compression); φc denotes the resistance factor (φc = 0.9 for tension, φc = 0.85 for compression); Mu represents the required flexural strength; M n indicates the nominal flexural strength, and φb represents the flexural resistance reduction factor (φb = 0.90). The following equation computes the nominal tensile strength: Pn = Ag × Fy

(6.31)

where Ag represents the gross cross-sectional area of the member and Fy indicates specified minimum yield stress. The nominal compressive strength of a member is calculated as:

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6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Pn = Ag × Fcr √ ) ( Fy E KL ≤ 4.71 Fcr = 0.658 Fe Fy ; for r Fy √ E KL Fcr = 0.877 × Fe ; for > 4.71 r Fy π 2E Fe = ( )2 KL

(6.32)

(6.33)

(6.34)

(6.35)

r

in which Fcr represents the critical stress of the member, Fe indicates elastic buckling stress; E denotes the modulus of elasticity, r denotes the radius of gyration, L indicates the laterally unbraced length of the member, and k represents the effective length factor.

6.3.1.4

Checking Constraint G4

The maximum vertical displacement is checked by this restriction as follows: ΔV − RV ≤ 0 L

(6.36)

where ΔV indicates the maximum vertical displacement of apex in CRF or roof in the PRF; L represents the span length of the CRF or PRF, and RV denotes the allowable vertical displacement and equal to 1/360 and 1/240 under the dead and live loadings, respectively.

6.3.1.5

Checking Constraint G5

This constraint checks the maximum horizontal displacement by the following equation: ΔH − RH ≤ 0 H

(6.37)

in which ΔH indicates the maximum horizontal displacement of the eaves point in the CRF or PRF; H denotes the height of the column, and RH indicates the allowable horizontal displacement and equal to H /200 under all loading conditions.

6.3 Statement of the Discrete Optimization Problem

147

6.3.2 Optimum Design of the Structures Using the SAP2000-OAPI The optimum design of the structure is an optimization problem where the solution can be the optimal size for the structural members (sizing optimization), optimal coordinate for the structure nodes (shape optimization), or optimal size and connectivity between structural members (topology optimization). In this chapter, the optimal sizes for structural members of the PRF and CRF is determined. This problem can be solved as an optimization task using Eq. (6.24). For the purpose of determining the value of the vector {X } in Eq. (6.24), metaheuristic methods as a powerful and reliable optimization technique is chosen to minimize fpenalty ({X }). Using the Open Application Programming Interface (OAPI) feature, we connect the SAP2000 and MATLAB programs in order to analyze the structural model and collect the required data to validate design constraints. The flowchart of obtaining the optimal size for structural elements based on SAP2000-OAPI and the metaheuristic algorithm is provided in Fig. 6.1. In accordance with this diagram, the parameters of the metaheuristic algorithm, such as population size and the maximum number of function evaluations as a termination criterion, are first determined. After that, the population of candidate solutions (algorithm agents) is initialized randomly. The randomly initialized solutions are created in the continuous form. However, as indicated previously, here, we deal with discrete search space in which the design variables of the optimization problem are picked from the discrete set. Therefore, the rounding function is utilized to transform the produced solutions from continuous to discrete, and the solutions round to the closest discrete value. Consequently, the acquired values are discrete ones that compose our initial sections. At the same time, the model of the structure is simultaneously initialized in SAP2000. Utilizing the OAPI feature, we contact the SAP2000 software via MATLAB to modify the sections of the initial model. By considering this, the initial sections generated from MATLAB are assigned to elements modeled in SAP2000. The model is then analyzed to determine the member forces. In the design phase, the necessary information needed to verify the problem’s design constraints (discussed in Sect. 6.3.1) is obtained. By determining the needed information, these limitations are checked in MATLAB. If each design constraint is violated, it will be penalized according to Eq. (6.25). It mentions that the value∑of W ({X }) is acquired from the SAP 2000. By obtaining the value of W ({X }) and nte i=1 pi in Eq. (6.24), the value of the objective function, which reflects the entire weight of the structure, is determined. After that, we will go to the main loop of the metaheuristic algorithm, which is implemented iteratively until the stopping criterion is met. The metaheuristic method updates the vector of the design variable {X } at each iteration. In other words, since we deal with the population-based metaheuristic algorithm, each algorithm agent creates new values for the components of the vector {X } in each iteration. Then, each value of this vector is transformed to the discrete value by using the rounding function, and the preceding steps are repeated until the new value for the objective function is determined. This procedure is repeated in the

148

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Fig. 6.1 The flowchart of finding the optimal size of structural members based on SAP2000-OAPI and the metaheuristic algorithm

cyclic body of the algorithm for all agents until stopping criterion of the algorithm is met. Finally, the optimal values for the components of the vector {X } is reported as the optimal solution discovered by the metaheuristic algorithm.

6.4 Structural Loading

149

6.4 Structural Loading 6.4.1 Load Combinations In designing steel portal frames, certain load combinations should be taken into account. According to the ASCE/SEI 7-10 criteria [9], the following equation is assumed for designing members of the frames. These combinations are specified for vertical loads (i.e., dead, live, and snow) and lateral loads (i.e., seismic and wind). 1. 1.4D 2. 1.2D + 1.6L + 0.5(S or R) 3. 1.2D + 1.6(S or R) + (L or 0.8W ) 4. 1.2D + 1.0 W + L + 0.5(S or R) 5. (1.2 + 0.2SDS )D + E + L + 0.2S 6. 0.9D + 1.0 W 7. (0.9−0.2SDS )D + E

(6.38)

6.4.2 Vertical Loads 6.4.2.1

Dead Loads

The dead and collateral loads (D) consist of the self-weight of the structure and the weight of the roof purlins and panels with a mass equal to 14.65 kg/m2 , as given in Table 6.1. In this case, the impact of collateral loads are ignored, therefore it is assumed to be zero.

6.4.2.2

Live Loads

According to the ASCE/SEI 7-10, the live loads acting on the roof beams are almost 100 kg/m2 . Table 6.2 is a summary of the live loads. Table 6.1 The dead load parameters

Dead load (kg/m2 ) Loading width (per m) Frame distributed dead load (kg/m)

14.65 6.0 87.85

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6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Table 6.2 The live load parameters

Table 6.3 The snow load parameters

Live load (kg/m2 )

100

Loading width (per m)

6.0

Frame distributed live load (kg/m)

590

Ce

1.0

Ct

1.0 1.0

Is (kg/m2 )

98

Pf (kg/m2 )

69

Cs

1.0

Pg

6.4.2.3

Snow Load

These structures have two distinct kinds of snow load. The balanced and unbalanced snow loads. The flat snow load (Pf ) and the balanced snow load (Ps0 ) are computed as follows: Pf = 0.7Ce Ct Is Pg

(6.39)

Ps0 = Cs Pf

(6.40)

in which Ce indicates the exposure factor, Ct denotes the thermal factor, Is represents the importance factor, Pg denotes the ground snow load, and Cs denotes the roof slope factor. Table 6.3 provides the details of the necessary parameters for the snow load calculation. In our design problems, the roof slope is taken a value less than 30 degrees. Thus, the value of Cs will be determined equal to 1.0 according to the code. Figure 6.2 illustrates how the distribution of balanced and unbalanced snow loads is computed.

6.4.3 Lateral Loads 6.4.3.1

Seismic Load

In order to compute the seismic load, first, the seismic base shear is computed: V = Cs W

(6.41)

where W indicates the effective seismic weight, and Cs represents the seismic response coefficient determined as follows:

6.4 Structural Loading

151

Fig. 6.2 Calculation of balanced and unbalanced snow load

Table 6.4 The summarized calculation of Cs

SDS

0.2768

R

4.5 1

Ie Cs =

SDS RIe

Cs =

0.041

SDS RSIe

(6.42)

where SDS denotes the design spectral response acceleration parameter in short period range, R represents the response modification factor, and Ie represents the importance factor. Since the researched design examples are located in Clay County of Kansas in the United States of America, Table 6.4 provides a summary of the Cs computation.

6.4.3.2

Wind Load

In order to determine the wind load for low-rise buildings, first, the wind pressure should be acquired: qz = 0.613KZ KZt Kd V 2

( ) N/m2

(6.43)

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6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

where KZ denotes the velocity pressure exposure coefficient,KZt indicates to the topographic factor, Kd represents the wind directionality factor, and the V denotes the basic wind speed. qh represents the velocity pressure at height h (average height of roof) determined as follows: qh = 0.613Kh KZt Kd V 2

( ) N/m2

(6.44)

According to Table 27.3-1 in ASCE 7-10, Kh is determined as follows: ( Kh = 2.01

h 274.32

) 9.52 in metric

(6.45)

in which h represents the average height of the roof, and Kh denotes a function of the average height of the roof so that its value is changed by altering the slope of the roof. The design wind pressures for the frame system of an enclosed and partially enclosed rigid building at all heights are computed as below: P = qGCP − qi (GCPi )

(6.46)

where q represents velocity pressure (kg/m2 ); G denotes the gust-effect factor; CP represents the external pressure coefficient computed according to Table 6.5 for PRFs and Table 6.6 for CRFs; qi represents velocity pressure for internal pressure determination, and GCPi denotes the internal pressure coefficient. For q and qi , we have: • q = qz for windward walls evaluated at height z above the ground. • q = qh for leeward walls, side walls, and roofs evaluated at height h. • qi = qz for the positive internal pressure evaluation in partially enclosed buildings in which the height z is defined as the level of the highest opening in the building that could affect the positive internal pressure. For positive internal pressure evaluation (qi ) may conservatively be evaluated at height h. Table 6.5 The coefficient of CP in two orthogonal directions of wind for PRFs

No. case

The directions of wind

Transverse wind direction (Case 1)

Windward wall

0.8

Windward roof

−0.7

Leeward roof

−0.5

Leeward wall

−0.5

Transverse wind direction (Case 2)

CP

Windward wall

0.8

Windward roof

−0.18

Leeward roof

−0.5

Leeward wall

−0.5

6.5 Design Examples

153

Table 6.6 The coefficient of CP in two orthogonal directions of wind for CRFs Rise-to-Span Ratio, r

Windward quarter

Center half

Leeward quarter

0 < r < 0.2

−0.9

−0.7 − r

−0.5

0.2 ≤ r < 0.3

1.5r − 0.3

−0.7 − r

−0.5

0.3 ≤ r ≤ 0.6

2.75r − 0.7

−0.7 − r

−0.5

a

r is the rise-to-span ratio

Table 6.7 Details of wind load parameters for each design example Wind load parameters

The first design example

The second design example

KZ

0.89

0.89

KZt

1.0

1.0

Kd

0.85

0.85

V

90 mph (40.234 m/s)

90 mph (40.234 m/s)

h

26.57 ft

53.14 ft

G

0.85

0.85

GCPi

± 0.18

± 0.18

qz

15.834 psf

15.834 psf

qh

16.232 psf

18.782 psf

qi

16.232 psf

18.782 psf

r

0.0093

0.0093

• qi = qh for the windward wall, side walls, leeward wall, and roof of enclosed buildings and negative internal pressure evaluation in partially enclosed buildings. Table 6.7 summarizes the details of wind load parameters that need for computing wind load parameters of the examined design examples.

6.5 Design Examples In this section, two design examples of symmetric portal frames including steel CRF and PRF with varying span lengths are examined. Due to the repetition of a series of longitudinally braced transverse frames, the design of portal frames is mostly performed for a two-dimensional frame. For the first and second design examples, the span length of the portal frames is supposed to be 16.0 m (L = 16.0 m) and 32.0 m (L = 32.0 m), respectively. Figures 6.3 and 6.4 depict the geometrical form of the portal frames for the first and second design examples, respectively. The columns and rafters of the design examples are web-tapered I-section with the identical flange width, and the inside and outside of the flange thickness also have an equal value. In the analysis and design of the portal frames, the connections between columns

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6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Fig. 6.3 Geometrical shape of the first design example with L = 16.0 m

Fig. 6.4 Geometrical shape of the second design example with L = 32.0 m

and base plates are considered as pin-jointed. However, the connections between columns and rafters are assumed as moment-resisting. The nodal geometry of the members is provided based on the neutral axis of the members. The design examples are located in Clay County from Kansas in the United States of America. TLBO, ECBO, SSOA, and WSA algorithms are considered for determining the best, average, and worst optimal weights for the design examples, and the acquired results of the algorithms are compared. For all algorithms, population size and the maximum number of function evaluations (MaxNFEs) are set to 20 and 4000, respectively. The internal parameters of ECBO, SSOA, and WSA are according to the literature as follows: ECBO, size of CM = 2 and pro = 0.3; SSOA, nh = 4, αmax = 1, βmin = 2, and βmax = 3; and WSA, nt = 10, and p = 0.5. According to the Fig. 6.5, 13 and 12 design variables are used for the optimal design of steel CRFs and PRFs, respectively. The properties of the design variables are provided in this figure. For both investigated examples, the values for the thickness of web and flange, the web height, and flange width should be chosen from the discrete set as provided in Table 6.8. The characteristic and number of design variables are fixed when the span length of the portal frames is increased from 16.0 to 32.0 m. In this chapter, the material density, modulus of elasticity, yield stress, and poison ratio of the all examined

6.5 Design Examples

155

Fig. 6.5 The considered design variables for the optimal design of steel a CRF and b PRF

Table 6.8 The values of design variables for investigated portal frames Design examples

The first design example

The second design example

CRF and PRF with L = 16.0

CRF and PRF with L = 32.0

Thickness of web and flange

T = {8, 10, 12, 15, 20, 22, 25}

T= {10, 12, 15, 20, 22, 25, 30, 35, 40}

Web height

WH = {200, 210, 220, . . . , 690, 700}

WH = {200, 210, 220, . . . , 1490, 1500}

Flange width

FW = {200, 210, 220, . . . , 490, 500}

FW = {200, 210, 220, . . . , 590, 600}

design examples are ρ = 7850 kg/m3 , E = 2.1 × 106 kg/cm2 , Fy = 2520 kg/cm2 , and ν = 0.3, respectively. The final results of the metaheuristic algorithm in each independent run are not the same with those found in other executions. Because they are stochastic solvers. In this regard, 20 independent runs are carried out for each case study of this chapter. The investigated algorithms are coded in MATLAB environment, and the analysis and design of the examples are performed based on the existing commercial software SAP2000-OAPI.

156

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

6.5.1 Discussion and Results for the Frames with L = 16.0 m Table 6.9 compares the optimization results obtained for the first design example utilizing TLBO, ECBO, SSOA, and WSA algorithms. The outputs of optimization include optimal sections and statistical measures including best, worst, mean, and standard deviation. Each algorithm was executed twenty times to provide statistically meaningful results. From Table 6.9, the average weights obtained by TLBO, ECBO, SSOA, and WSA for CRF are 1789.07 kg, 1798.14 kg, 1773.54 kg, and 1803.21 kg, respectively. The average weight obtained by SSOA lighter than the average weight determined by other algorithms. The relative weights for the PRF are 2064.78 kg, 2012.28 kg, 1993.09 kg, and 2037.9 kg, respectively. Again, this result demonstrates that SSOA obtained the average weight better than other utilized methods. In terms of determining the best optimum weight for the CRF, ECBO is ranked first, and the optimal weight determined by SSOA is slightly inferior to the results obtained by ECBO. However, in PRF, ECBO, SSOA, and WSA found an equal value (i.e., 1882.39 kg) for the best optimum weight. The final structural weights determined by the explored algorithms in 20 separate runs for CRF and PRF are shown in Figs. 6.6 and 6.7, respectively. The convergence histories of TLBO, ECBO, SSOA, and WSA for the CRF and PRF are illustrated in Figs. 6.8 and 6.9, A zoomed segment is added to the convergence histories to simplify the comparison between the used methods. A close examination of these figures demonstrates that the convergence speed of WSA and ECBO algorithms is much higher than TLBO and SSOA in the early iterations. However, SSOA achieved the lowest mean weight in both frames at the completion of the optimization process. The structural weight of CRF and PRF are compared in Fig. 6.10. With the same height and span, the CRF has a less structural weight than the PRF. Consequently, the optimized outcomes demonstrate that CRF is more economical than PRF. The obtained stress ratios of members of both frames for the best optimal design utilizing optimization algorithms are given in Fig. 6.11.

6.5.2 Discussion and Results for the Frames with L = 32.0 m The optimized weight of the second design example determined by TLBO, ECBO, SSOA, and WSA algorithms are compared in Table 6.10. As can be seen from this table, the mean weights of CRF determined by TLBO, ECBO, SSOA, and WSA algorithms are respectively 7783.62 kg, 7731.3 kg, 7530.4 kg, and 7675.39 kg, whereas the mean weights of PRF determined by these algorithms are 12,280.9 kg, 12,182.9 kg, 11,507.5 kg, and 12,057.5 kg, respectively. Consequently, it can be concluded that the mean weight of both frames determined by the SSOA better than other investigated methods. In addition, the TLBO determines the optimal CRF weight, whereas the SSOA determines the optimal PRF weight. Figures 6.12 and 6.13 show the final structural weights of steel CRF and PRF derived by TLBO, ECBO, SSOA, and WSA algorithms in 20 separate runs. Convergence curves of the used

69.627

Standard deviation

76.2658

1798.14

1789.07

Mean weight (kg)

1669.71

1913.52

1672.14

1910.92

Best weight (kg)

350

390

WH7

Worst weight (kg)

230

380

250

360

210

490

200

200

WH5

200

WH4

220

200

WH6

200

490

WH2

200

WH1

WH3

220

200

FW1

FW2

8

8

FT2

8

15

8

15

WT2

8

FT1

8

66.727

1773.54

1868.29

1670.99

400

350

240

210

490

200

200

200

220

8

15

8

8

57.9758

1803.21

1866.49

1707.76

500

480

440

200

420

200

200

200

210

8

15

8

8

70.6406

2064.78

2282.42

1901.81

–

480

400

210

440

220

200

310

210

8

15

8

8

TLBO [1]

WSA [1]

PRF SSOA [1]

TLBO [1]

ECBO [1]

CRF

WT1

Design variables

Table 6.9 Comparison of results of different optimization methods for the first design example

87.7254

2012.28

2281.88

1882.39

–

430

360

200

480

200

200

310

210

8

15

8

8

ECBO [1]

92.999

1993.09

2281.34

1882.39

–

430

360

200

480

200

200

310

210

8

15

8

8

SSOA [1]

127.52

2037.9

2281.85

1882.39

–

250

200

200

600

200

210

290

200

8

20

8

10

WSA [1]

6.5 Design Examples 157

158

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Fig. 6.6 The final structural weight in each independent run for the steel CRF of the first design example obtained by a TLBO, b ECBO, c SSOA, and d WSA

Fig. 6.7 The final structural weight in each independent run for the steel PRF of the first design example obtained by a TLBO, b ECBO, c SSOA, and d WSA

6.5 Design Examples

159

Fig. 6.8 Comparison of the convergence histories from TLBO, ECBO, SSOA, and WSA metaheuristic algorithms for the steel CRF of the first design example

Fig. 6.9 Comparison of the convergence histories from TLBO, ECBO, SSOA, and WSA metaheuristic algorithms for the steel PRF of the first design example

160

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Fig. 6.10 Comparison of the portal frames of the first design example in terms of a the best structural weight and b the average structural weight

algorithms for the CRF and PRF are compared in Figs. 6.14 and 6.15, respectively. Similar to the preceding design example, although the convergence rate of ECBO and WSA are greater than TLBO and SSOA in the early iterations, SSOA surpasses other algorithms in terms of determining the mean weight in both frames. A comparison between the structural weight of CRF and PRF is depicted in Fig. 6.16. Again, the outcomes demonstrate that the CRF is economically superior than the PRF with the same height and span. Figure 6.17 reports the obtained stress ratio of members of both portal frames for the best execution of the considered algorithms.

6.6 Concluding Remarks

161

Fig. 6.11 The obtained stress ratios from different optimization algorithms for the best optimum design of the first design example: a CRF and b PRF

6.6 Concluding Remarks This chapter examines the optimal design of two kinds of portal frames consisting of planar steel Curved Roof Frame (CRF) and Pitched Roof Frame (PRF) with tapered I-section members. The optimal design seeks to decrease the weight of these frame structures while fulfilling certain design limitations based on the requirements of

15

12

290

580

8441.08

7731.3

15

12

480

250

200

460

1160

610

220

470

560

7310.62

8979.15

7783.62

376.416

FT1

FT2

FW1

FW2

WH1

WH2

WH3

WH4

WH5

WH6

WH7

Best weight (kg)

Worst weight (kg)

Mean weight (kg)

Standard deviation

300.34

7343.64

670

550

1100

440

200

250

480

10

10

WT2

10

10

122.213

7530.4

7918.12

7366.46

580

490

230

620

1180

460

210

250

480

12

15

10

10

332.611

7675.39

8650.9

7375.23

600

520

230

660

1290

500

220

230

470

12

15

10

10

559.753

12,280.9

13,766.7

334.92

12,182.9

12,748.7

11,616.9

−

− 11,343.9

340

350

300

1120

330

200

510

350

15

30

10

20

ECBO [1]

940

730

200

700

390

200

540

360

15

30

10

12

PRF WSA [1]

TLBO [1]

SSOA [1]

TLBO [1]

ECBO [1]

CRF

WT1

Design variables

Table 6.10 Comparison of results of different optimization methods for the second design example

222.721

11,507.5

12,011.7

11,287.6

−

1210

990

310

540

300

240

560

340

15

30

10

10

SSOA [1]

434.038

12,057.5

13,225.2

11,362.9

−

590

420

200

870

290

200

560

360

15

30

10

15

WSA [1]

162 6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

6.6 Concluding Remarks

163

Fig. 6.12 The final structural weight in each independent run for the steel CRF of the second design example obtained by a TLBO, b ECBO, c SSOA, and d WSA

Fig. 6.13 The final structural weight in each independent run for the steel PRF of the second design example obtained by a TLBO, b ECBO, c SSOA, and d WSA

164

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Fig. 6.14 Comparison of the convergence histories from TLBO, ECBO, SSOA, and WSA metaheuristic algorithms for the steel CRF of the second design example

Fig. 6.15 Comparison of the convergence histories from TLBO, ECBO, SSOA, and WSA metaheuristic algorithms for the steel PRF of the second design example

6.6 Concluding Remarks

165

Fig. 6.16 Comparison of the portal frames of the second design example in terms of a the best structural weight and b the average structural weight

ANSI/AISC 360-16 and ASCE 7-10. Existing commercial software SAP2000 and MATLAB are integrated through Open Application Programming Interface (OAPI) to optimize the design of these types of portal frames. In this chapter, two primary goals are discussed. The first involves comparing the optimization outcomes of CRF and PRF structures with identical height and span dimensions. In this case, the frame structures are compared in two different span lengths, which are 16.0 and 32.0 m. The second purpose of this study is to examine the performance of four populationbased metaheuristic optimization methods for the optimum design of CRF and PRF

166

6 Optimum Design of Curve Roof Frames by SSOA and Comparison …

Fig. 6.17 The obtained stress ratios from different optimization algorithms for the best optimum design of the second design example: a CRF and b PRF

structures. The investigated metaheuristics are Teaching–learning-based optimization (TLBO), Enhanced Colliding Bodies Optimization (ECBO), Shuffled Shepherd Optimization Algorithm (SSOA), and Water Strider Algorithm (WSA). The data gathered allow us to conclude the following conclusions: • CRF structures are considerably much economical than PRF structures in both investigated design examples, and the optimized weights obtained in CRF structures are lower than those found in PRF structures.

References

167

• The SSOA metaheuristic is more efficient than other considered algorithms due to finding the lightest weight in the average of runs. Thus, it is highly recommended for the optimal design of these types of portal frames.

References 1. Kaveh, A., Karimi Dastjerdi, M.I., Zaerreza, A., Hosseini, M.: Discrete optimum design of planar steel curved roof and pitched roof portal frames using metaheuristic algorithms. Periodica Polytechnica Civil Eng. 65(4), 1092–1113 (2021) 2. Kaveh, A., Zaerreza, A.: Shuffled shepherd optimization method: a new Meta-heuristic algorithm. Eng. Comput. 37(7), 2357–2389 (2020) 3. Rao, R.V., Savsani, V.J., Vakharia, D.P.: Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput. Aided Des. 43(3), 303–315 (2011) 4. Kaveh, A., Ilchi Ghazaan, M.: Enhanced colliding bodies optimization for design problems with continuous and discrete variables. Adv. Eng. Softw. 77, 66–75 (2014) 5. Kaveh, A., Dadras Eslamlou, A.: Water strider algorithm: a new metaheuristic and applications. Structures 25, 520–541 (2020) 6. Fraser, D.J.: Design of tapered member portal frames. J. Constr. Steel Res. 3(3), 20–26 (1983) 7. AISC.: In: Specification for Structural Steel Buildings. ANSI/AISC 360-16 (2016) 8. AISC.: In: Seismic Provisions For Structural Steel Buildings. American Institute of Steel Construction (AISC): Chicago, Illinois, USA (2016) 9. ASCE/SEI.: In: Minimum Design Loads for Buildings and Other Structures ASCE/SEI 7-10 (2010)

Chapter 7

Optimum Design of Castellated Beams Using SSOA and the Other Four Meta-Heuristic Algorithms

7.1 Introduction This chapter presents the optimum design of the castellated beams introduced by Kaveh et al. [1]. In recent decades, castellated beams have gained much attention. Due to the perforations in the webs of these beams, the bending moment capacity of the cross-section rises without increasing the beam’s weight. These beams are also more practical from an architectural point of view and installations, and plumbing can be passed through the holes of these beams that are used in the roof. The increasing usage of castellated beams in a variety of structures, such as Parking lots, industrial buildings and warehouses, office buildings, schools, and hospitals, makes the optimization of these beams crucial. In this chapter, the optimization of castellated beams with circular and hexagonal holes with cost objective function has been done using the shuffled shepherd optimization algorithm (SSOA). In addition, the performance of four different meta-heuristic algorithms known as particle swarm optimization (PSO), improved shuffled based Jaya (IS-Jaya), plasma generation optimization (PGO), and set theoretical based Jaya algorithm (ST-JA) is compared. Moreover, the results demonstrate the good performance of these meta-algorithms in optimizing the castellated beams problems. In recent years, due to the increase in the dimensions of structures, their weight has also increased due to the volume of construction materials employed. Thus, engineers are more inclined to utilize structural components with high strength and at the same time with better economic and architectural features. Since the 1940s, the generation of structural beams with higher strength and lower expense has been a demand to engineers in their endeavors to design more effective steel structures. The castellated I-shaped steel beam is one of these endeavors that have a wide range of applications in steel structures, particularly within the shape of simply supported main gravity girders. Due to the extensive application of these beams in structures, the optimization of these beams in terms of the structure’s economic efficiency is of special relevance. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_7

169

170

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

Due to the nonlinear nature of the problems involved in optimizing these beams, meta-heuristic techniques are utilized to determine optimal solutions for the design variables of these beams. The main idea of meta-heuristic algorithms is based on the simulation of natural phenomena and, unlike the traditional optimization methods, do not require extensive mathematical computations during the optimization process. The main purpose of this chapter is to investigate the cost differences between the castellated beams with hexagonal openings and cellular beams with the optimum design under equal conditions of loading and boundary conditions. Also, another aim is to compare the optimum design results of castellated beams using meta-heuristic algorithms. These algorithms are used for optimization, and the cost of the beam is considered as the objective function. The design approach for castellated beams is the final strength method in this chapter. The rest of the chapter is structured as follows: In Sect. 7.1 of this chapter, the geometry of castellated beams is introduced. In Sect. 7.3, the design of castellated beams is introduced. In Sect. 7.4, the optimum design problem of these beams is formulated. In Sect. 7.5, four recently developed meta-heuristic algorithms (SSOA, IS-Jaya, PGO, and ST-JA) are briefly introduced. In Sect. 7.6, design examples and comparisons are shown with considering the minimization of the cost of the castellated beams as the design objective function. Finally, concluding remarks are given in Sect. 7.7.

7.2 Geometry of the Castellated Beams The main advantage of these beams is their structural properties. These beams are manufactured by cutting the web of an I-shaped rolled beam, following a certain pattern along the beam. Due to the increase in the height of the castellated beam compared to the main beam, the flexural strength and stiffness of the beam are enhanced. In the design of steel structures, beams with web-opening are generally used to pass the under-floor services pipes, such as water pipes and air ducts. A castellated beam is produced by cutting a standard wide-flange beam longitudinally in a zigzag or semicircular pattern, segregation and offsetting the two halves, and welding them back together. The resultant openings in the webs allow mechanical ducts, plumbing, and electrical lines to pass through the beam rather than under the beam. The geometric properties of castellated beams with hexagonal holes are illustrated in Fig. 7.1. Based on the cutting pattern in castellated beams with hexagonal holes, their geometry is determined by the following parameters: HS = h + d

(7.1)

where HS represents the overall depth, d represents the cutting depth, and h is the depth of UB section. The distance between the hole centers is determined by the following equation:

7.2 Geometry of the Castellated Beams

171

Fig. 7.1 The geometric characteristics of castellated beams with hexagonal hol

S = 2e + 2 d cot θ

(7.2)

where e represents the length of horizontal cutting and θ represents the cutting angle. The value of d can geometrically have a maximum value equal to h − 2t f . In contrast, in terms of strength limitations, its value is considered a certain limit. The angle θ can also vary between 45 and 64 degrees. The horizontal length of the cut, e, can take various values based on the value of d and θ . If the weld length is too short, the weld will break under horizontal shear force, whereas the weld too long will increase the length of the tee-section, which may cause the vierendeel bending mechanism. Therefore, in order to achieve a balance between these two states, the welding length is limited to a specific length. The depth of the tee-section is equal to: dT =

(HS − 2d) 2

(7.3)

Figure 7.2 illustrates the geometric properties of castellated beams with round perforations. The geometric characteristics of honeycomb beams with circular holes are as follows:

172

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

Fig. 7.2 The geometric characteristics of castellated beams with circular holes

√ HS = h +

D 2 − e2 2

S = D+e dT =

HS − D 2

where D represents the diameter of holes.

(7.4) (7.5) (7.6)

7.3 Design of Castellated Beams

173

7.3 Design of Castellated Beams Beams must be robust enough to withstand the bending moments and shear forces generated by the applied loads. The effectiveness of a beam is determined by its size, cross-section geometry and shape. As a result of the holes in the web, the structural behavior of a castellated steel beam differs from that of solid web beams. Due to the complexity of the behavior of castellated beams, there is no generally accepted method up to now. Castellated beams design criteria is assumed similar to usual beam limit states, however web holes and welds can lead to other forms of failure. Under different applied loads on the beam, failure in castellated beams happens in one of the following situations and must be controlled: 1. 2. 3. 4. 5. 6.

Vierendeel bending mechanism; Lateral-torsional buckling; Rupture of the welded joint; Web post buckling due to shear force; Compression web post buckling; Flexural failure mechanism;

Lateral-torsional buckling is possible in an unrestrained beam. A beam is assumed to be unrestrained when its compression flange is free to displace laterally and rotate. In this chapter, it is assumed that the compression flange of the castellated beam is restrained by the floor system. Therefore, the total buckling strength of the castellated is be eliminated from the design requirements. These modes are closely associated with beam geometry, shape parameters, type of loading, and provision of lateral supports. Each of these failure modes is explained in detail below. The first step in designing castellated beams is to compute the total flexural moment and shear force due to external loads at each opening and web post. These forces are referred to as general forces. General forces are used to calculate local forces at the top and bottom of T-shaped sections, web posts, and gross cross-sections. These beam components are then tested for rupture under these local loads.

7.3.1 Overall Beam Flexural Capacity The maximum bending moment under external loading must be smaller than the plastic moment capacity of the castellated beam cross-section, which is expressed by the following equation: MU < M P = A L T PY HU

(7.7)

where A L T represents the area of lower tee, PY represents the design strength of steel, and HU represents distance between center of gravities of upper and lower tees.

174

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

7.3.2 Beam Shear Capacity In designing castellated beams, it is necessary to control two types of shear failure mechanism. The first mode of shear failure is related to the vertical shear that the upper and lower tee-sections of the castellated beam must withstand. The total shear capacity of the upper and lower tees, should be determined using the following equations, and the shear force should be smaller than PV Y . For circular opening PV Y = 0.6PY (0.9A W U L ) For hexagonal opening PV Y

√ 3 PY (A W U L ) = 3

(7.8)

(7.9)

The second mode of shear failure that should be considered in the design of castellated beams is related to horizontal shear. This failure mode occurs due to changes in axial force in the tee-section (Fig. 7.3). The horizontal shear capacity is controlled by the following equations: For circular opening PV H = 0.6PY (0.9A W P ) For hexagonal opening PV H

√ 3 PY (A W P ) = 3

(7.10)

(7.11)

where A W U L represents the total area of the webs of the tees and A W P represents the minimum area of web post.

Fig. 7.3 Horizontal shear in the web post of castellated beams, a hexagonal opening, b circular opening

7.3 Design of Castellated Beams

175

7.3.3 Flexural and Buckling Strength of Web Post The overall buckling of the castellated beam is omitted from the design consideration due to the assumption of this study. It is assumed that the compression flange of the castellated beam is restrained by the floor system. Web post buckling is caused by a combination of horizontal shear force and bending moment in the middle of the height of the area. The web post flexural and buckling of capacity in the castellated beam is given by: ] [ M M AX = C1 · α − C2 · α 2 − C3 ME

(7.12)

where M M AX represents the maximum allowable web post moment and M E represents the web post capacity at critical section A-A shown in Fig. 7.2. C1 , C2 and C3 are constants obtained by following expressions: C1 = 5.097 + 0.1464(β) − 0.00174(β)2

(7.13)

C2 = 1.441 + 0.0625(β) − 0.000683(β)2

(7.14)

C3 = 3.645 + 0.0853(β) − 0.00108(β)2

(7.15)

Also α and β are obtained by following equations: For circular opening α =

D0 S ,β= D0 tW

For hexagonal opening α =

2d S ,β= 2d tW

(7.16) (7.17)

where S represents the spacing between the centers of holes, d represents the cutting depth of hexagonal opening, D0 represents the holes diameter and tW represents the web thickness.

7.3.4 Vierendeel Bending of Upper and Lower Tees This failure mode is related to the high shear on the cross-section in the area of the teesections. Vierendeel bending results in the formation of four plastic hinges above and below the web opening. The Olander technique is used to calculate the vierendeel flexural stresses in castellated beams having round perforations. According to the Olander method, the critical point of stress occurs at a section at an angle θ from the line perpendicular to the center of the hole (Fig. 7.4). This angle varies according

176

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

Fig. 7.4 Olander’s curved beam approach

to the amount of secondary moment in the section. For zero secondary moment, the critical section occurs at a zero-degree angle just above the center of the hole, and this angle increases as the secondary bending increases. For ease of calculating stresses, this angle is commonly regarded as 45 degrees. The interaction between the vierendeel bending moment and the axial force in the critical area of the tee-section should be controlled by the following equation: P0 M + ≤ 1.0 PU MP

(7.18)

where P0 and M are the force and the bending moment on the section, respectively. PU is equal to area of critical section ×PY , M P is calculated as the plastic modulus of critical section ×PY in plastic section or elastic section modulus of critical section ×PY for other sections. For castellated beams with hexagonal holes, the total plastic moment of the teesection area at the top and bottom of the hole is known as the vierendeel resistance. The interaction between the vierendeel moment and the shear force in the beam cross section is controlled by the following equation: VO M AX .e − 4MT P

(7.19)

where VO M AX and MT P are the maximum shear force and the moment capacity of tee section, respectively.

7.3 Design of Castellated Beams

177

7.3.5 Geometric Criteria Criteria in references [2] and [3] have been used to apply geometric constraints to castellated beams with circular holes, whereas criteria from reference [4] have been used for castellated beams with hexagonal holes. In castellated beams with circular holes, the following equations are controlled: 1.08 ≤ S/D0 ≤ 1.60

(7.20)

1.25 ≤ HS /D0 ≤ 1.75

(7.21)

Additionally, in castellated beams with hexagonal holes, the following equations should be controlled: ) 3( d − . HS − 2t f ≤ 0 8 ( ) ( ) HS − 2t f − 10 × dT − t f ≤ 0

(7.22) (7.23)

2 · d · cot θ ≤ e ≤ 2d · cot θ 3

(7.24)

2d · cot θ + e − 2d ≤ 0

(7.25)

45◦ ≤ θ ≤ 64◦

(7.26)

where t f represents the thickness of beam’s web.

7.3.6 Deflection of Castellated Beams Serviceability control is crucial factor in the design of structures. In castellated beams, due to the high shear deformations, the importance of this issue becomes more apparent. Hence, the analysis of displacement is more difficult compared to full web beams. The deflection of a castellated beam under applied load combinations should not exceed span/360. In castellated beams with circular openings, the deflection at each point is calculated by the following formula: YT O T = Y M T + YW P + Y AT + Y ST + Y SW P

(7.27)

where Y M T , YW P , Y AT , Y ST and Y SW P are deflection caused by the bending moment in tee, deflection caused by the bending moment in web post of beam, deflection

178

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

caused by the axial force in tee, deflection caused by the shear in tee and deflection caused by the shear in web post, respectively. These equations are provided in Ref. [2]. For a castellated beam with hexagonal opening and length L subjected to transverse loading, the total deflection is composed by two terms: the first term corresponds to pure moment action f b , and the second one corresponds to shear action f s . Consequently, the total deflection can be computed by the following expression: f = f b + f s = C1 L 3 + C2 L

(7.28)

Here, C1 and C2 are determined by means of a curve fitting technique.

7.4 Castellated Beams Optimization In the majority of engineering works and especially in issues related to structural optimization, the weight of the structure is considered as the objective function. Hence, the optimization process is carried out to reduce the weight. In matters related to the design of castellated beams, due to the inclusion of parameters related to the cutting of the beam and the horizontal welding of the cutting area, the beam with the minimum weight is not necessarily the most economical beam. For this reason, a cost of the beam is considered as objective function in the optimization process. One of the main goals of making and using castellated beams is to reduce the cost of materials. There are many factors to consider when estimating the cost of making castellated beams, including the cost of working hours, the cost of materials, the cost of cutting, and the cost of welding. It is assumed that the cost of construction hours for castellated beams with circular and hexagonal holes is the same. Therefore, in this chapter, the objective function related to the cost of construction of these beams consists of three components: 1. The first part is related to the cost of the selected section or the weight of the structure. 2. The second part is the cost of the cutting process of these beams. 3. The third part is the cost related to welding. The objective function in the design of these beams is expressed by combining the above three sentences and taking into account various coefficients, as follows: ) ( S + P2 · L cut + P3 · L weld Fcost (x) = P1 · ρ · Ainitial · L + 2

(7.29)

where P1 ,P2 and P3 are the price of the weight of the beam per unit weight, length of cutting and welding for per unit length, L 0 is the initial length of the beam before castellation process, ρ is the density of steel, Ainitial is the area of the selected universal beam section, L cut and L weld are the cutting length and welding

7.4 Castellated Beams Optimization

179

length, respectively. The length of cutting is different for hexagonal and circular web-openings. The dimension of the cutting length is described by the following equations: For circular opening L cut = π D.N H + 2e(N H + 1) + (

For hexagonal opening L cut

d = 2N H e + sin θ

πD +e 2

(7.30)

d sinθ

(7.31)

) + 2e +

where NH is the total number of holes, e is the length of horizontal cutting of web, D0 is the diameter of holes, d is the cutting depth, and θ is the cutting angle. The welding length for both of circular and hexagonal openings is determined by: L weld = e(N H + 1)

(7.32)

The coefficients P2 and P3 are considered per unit of length and the coefficient P1 per unit of weight, which in this study P1 = 0.85, P2 = 0.3 and P3 = 1.0.

7.4.1 Design of Castellated Beams with Circular Holes The design process of a cellular beam consists of three parts: The selection of a rolled beam, the selection of a diameter, and the total number of holes in the beam [2]. Hence, the sequence number of the rolled beam section in the standard steel sections tables, the circular holes diameter, and the total number of holes are taken as design variables in the optimization process. The optimum design problem formulated by considering the constraints explained in the previous sections can be expressed as the following: Find an integer design vector {X } = {x1 , x2 , x3 }T where x1 is the sequence number of the rolled steel profile in the standard steel section list, x2 is the sequence number for the hole diameter which contains various diameter values, and x3 is the total number of holes for the cellular beam [2]. Hence the design problem can be expressed as: Minimize Eq. (7.29) Subjected to g1 = 1.08D0 − S ≤ 0

(7.33)

g2 = S − 1.6D0 ≤ 0

(7.34)

g3 = 1.25D0 − HS ≤ 0

(7.35)

180

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

g4 = HS − 1.75D0 ≤ 0

(7.36)

g5 = MU − M P ≤ 0

(7.37)

g6 = VM AX SU P − PV ≤ 0

(7.38)

g7 = VO M AX − PV Y ≤ 0

(7.39)

g8 = VH M AX − PV H ≤ 0

(7.40)

g9 = M A−AM AX − MW M AX ≤ 0

(7.41)

g10 = VT E E − 0.5 × PV Y ≤ 0

(7.42)

g11 =

M P0 + − 1.0 ≤ 0 PU MP

g12 = Y M AX − L/360 ≤ 0

(7.43) (7.44)

where tW is the web thickness, HS and L are the overall depth and the span of the cellular beam, and S is the distance between centers of holes. MU is the maximum moment under the applied loading, M P is the plastic moment capacity of the cellular beam, VM AX SU P is the maximum shear at support, VO M AX is the maximum shear at the opening, VH M AX is the maximum horizontal shear, M A−AM AX is the maximum moment at A-A section shown in Fig. 7.3. MW M AX is the maximum allowable web post moment. VT E E represents the vertical shear on the tee at θ = 0 of web opening. P0 and M are the internal forces on the web section, and Y M AX denotes the maximum deflection of the cellular beam [2].

7.4.2 Design of Castellated Beams with Hexagonal Opening In the design of castellated beams with hexagonal openings, the design vector includes four design variables: The selection of a rolled beam, the selection of a cutting depth, the spacing between the center of holes or total number of holes in the beam, and the cutting angle as shown in Fig. 7.1. Consequently, the optimum design problem formulated is expressed as follows: Find an integer design vector {X } = {x1 , x2 , x3 , x4 }T where x1 is the sequence number of the rolled steel profile in the standard steel section list, x2 is the sequence number for the cutting depth which contains various values, x3 is the total number of

7.4 Castellated Beams Optimization

181

holes for the castellated beam and x4 is the cutting angle. Hence, the design problem can be expressed as: Minimize Eq. (7.29) Subjected to ) 3( g1 = d − . HS − 2t f ≤ 0 8 ) ( ) ( g2 = HS − 2t f − 10 × dT − t f ≤ 0

(7.45) (7.46)

2 · d · cot ∅ − e ≤ 0 3

(7.47)

g4 = e − 2 · d · cot ∅ ≤ 0

(7.48)

g5 = 2d · cot ∅ + e − 2d

(7.49)

g6 = 45◦ − ∅ ≤ 0

(7.50)

g7 = ∅ − 64◦ ≤ 0

(7.51)

g8 = MU − M P ≤ 0

(7.52)

g9 = VM AX SU P − PV ≤ 0

(7.53)

g10 = VO M AX − PV Y ≤ 0

(7.54)

g11 = VH M AX − PV H ≤ 0

(7.55)

g12 = M A−AM AX − MW M AX ≤ 0

(7.56)

g13 = VT E E − 0.5 × PV Y ≤ 0

(7.57)

g14 = VO M AX .e − 4MT P

(7.58)

g15 = Y M AX − L/360 ≤ 0

(7.59)

g3 =

182

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

where t f is the flange thickness, dT is the depth of the tee-section, M P is the plastic moment capacity of the castellated beam, M A−AM AX is the maximum moment at AA section shown in Fig. 7.3. MW M AX is the maximum allowable web post moment. VT E E represents the vertical shear on the tee, MT P is the moment capacity of teesection, and Y M AX denotes the maximum deflection of the cellular beam.

7.5 Recently Developed Meta-Heuristic Algorithms In recent years, meta-heuristic optimization algorithms have become very popular and widely applied to solve many problems in different fields. The meta-heuristics are inspired by nature, typically related to physical phenomena, animal’s behaviors, or evolutionary concepts. Four newly developed meta-heuristic algorithms are utilized to optimum design of castellated beams. The four meta-heuristic algorithms described below are SSOA, IS-JAYA, PGO, and ST-JA.

7.5.1 Shuffled Shepherd Optimization Algorithm (SSOA) In this algorithm, each solution candidate, X i , which contains a number of variables, is considered a sheep. Each sheep is sorted according to its objective function and then separated into herds using the shuffling method. In each herd, the sheep are selected in order, the selected sheep are called shepherds, and the sheep that perform better in the herd are called horses. So there are a number of sheep and horses for each shepherd. A shepherd tries to guide the sheep to the horse, and the shepherd’s new position is achieved by moving towards one of the sheep and horses. This is done for two purposes: first, moving to a worse agent leads to exploration, and second, moving to a better member leads to exploitation. The new shepherd position is updated when the new objective function is not worse than the old goal function, which leads to elitism in the algorithm. The steps of this algorithm are explained in the references [5].

7.5.2 Improved Shuffled Based JAYA Algorithm (IS-JAYA) The Jaya algorithm is a simple and efficient population-based meta-heuristic algorithm. In addition to simplicity, the algorithm has no specified parameters. Despite these benefits, this method has certain drawbacks, such as unwanted early convergence and the possibility of being caught in the local minimum due to insufficient population diversity. The IS-JAYA algorithm is proposed to reduce these barriers. In the IS-JAYA, the shuffling process is added to the JAYA algorithms to increase the population diversity. In addition, the escaping from the local optima mechanism is

7.6 Examples

183

employed in the IS-JAYA algorithms. These two mechanisms improve the performance of the JAYA algorithms in structural optimization problems. The steps of the IS-JAYA algorithm are described in the reference [6].

7.5.3 Plasma Generation Optimization (PGO) The basics of PGO are inspired by the plasma generation process. During plasma generation, some chemical reactions, including excitation, de-excitation, and ionization, occur in the presence of electrons (algorithmic agents). The simulation of these mentioned processes, which lead to the production of plasma, is based on a special mechanism developed in quantum physics. Here, these processes are explained using some assumptions about electron motion. The steps of the PGO algorithm are described in the references [7].

7.5.4 Set-Theoretical-Based Jaya Algorithm (ST-JA) JA is based on the idea of dividing solutions into a number of pre-defined wellorganized subpopulations [8]. In addition to the common control parameters, population size (nP) and maximum number of iterations (MaxIt), ST-JA requires an additional parameter, the number of subpopulations (nS). ST-JA steps are described in the reference [8].

7.6 Examples In this section, in order to investigate the method of this study in castellated beams with hexagonal and circular holes and to compare the obtained results, 2 examples have been used. The meta-heuristic algorithms i.e. SSOA, PGO, IS-Jaya and St-JA have been applied to them for optimization. From the list of standard British steel profiles, 64 UB sections from 254 × 102 × 28 U B to 914 × 419 × 388 U B for castellated beams have been selected and inspected. For the diameter of circular holes, 421 values from 180 to 600 mm with an increment of one millimeter have been selected. Also, for cutting depth in castellated beams with hexagonal holes, 351 values from 50 to 400 mm with incremental step of 1 mm have been considered. The cutting angle for this type of hole also varies from 45 to 64 degrees. A discrete set is also considered for the number of holes. In addition, in all examples, the modulus of elasticity of steel is assumed to be 205 KN/mm2 and the design strength is 355 MPa.

184

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

Fig. 7.5 Simply supported beam with 4-m span

7.6.1 Castellated Beam with 4-M Span The first example is a beam with a span length of 4 m, as shown in Fig. 7.5, which has a concentrated load of 50 kN in the center of the beam as well as a wide load of 5 kN/m considering the weight of the beam itself. The maximum number of iterations for all meta-heuristic algorithms is 200, and the number of function evaluation is 10000. The optimization results are shown in the tables by 4 meta-heuristic algorithms for castellated beams with circular and hexagonal holes, including cost, selected section, number of holes, diameter or cutting depth of the hole, as well as cutting angle. Also, to comparing the results between castellated beams with circular and hexagonal holes, the optimization results of the algorithms are compared with the PSO [9] optimization algorithm. The optimum result for the castellated beams with circular and hexagonal holes is given in Tables 7.1 and 7.2, respectively. As shown in the tables, a castellated beam with the hexagonal holes costs less than a castellated beam with the circular holes with the same condition. In castellated beams with circular holes, the large cutting length and the effect of secondary bending in the cross-section are costly to increase. Also, PSO, SSOA, and PGO algorithms result in the minimum cost in both castellated beams with circular and hexagonal holes. In addition, it can be seen that IS-Jaya and ST-JA algorithms result near-optimal cost which shows that they are very competitive algorithms. The CPU time for all algorithms is approximately about 6 s just with differences of about hundredth of a second. The convergence history of the algorithms is provided in Figs. 7.6 and 7.7.

7.6.2 Castellated Beam with 8-m Span The beam with a span length of 8 m is considered as a second example, as illustrated in Fig. 7.8, which has two concentrated dead and live loads of 70 kN in two points of the beam as well as a wide load of 0.4 kN/m considering the weight of the beam itself. The maximum number of iterations in all meta-heuristic algorithms is set to 200, and the number of function evaluations is 10,000. The optimization results of the 4 new meta-heuristic algorithms and the PSO for castellated beams with circular

7.6 Examples

185

Table 7.1 Optimum design of the castellated beams with circular holes with 4-m span Optimization algorithm

Optimum UB section

Hole diameter or cutting depth

Total number of holes

Cutting angle

Minimum cost ($)

PSO [1]

UB 305 × 102 × 25

245

15

−

91. 2215

SSOA [1]

UB 305 × 102 × 25

245

15

−

91.2215

IS-Jaya [1]

UB 305 × 102 × 25

263

14

−

91.4248

PGO [1]

UB 305 × 102 × 25

245

15

−

91.2215

ST-JA [1]

UB 305 × 102 × 25

263

14

−

91.4248

Table 7.2 Optimum design of the castellated beams with hexagonal holes with 4-m span Optimization algorithm

Optimum UB section

Hole Diameter Total number or cutting of holes depth

Cutting angle

Minimum cost($)

PSO [1]

UB 305 × 102 116 × 25

14

54

89.8937

SSOA [1]

UB 305 × 102 116 × 25

14

54

89.8937

IS-Jaya [1]

UB 305 × 102 120 × 25

14

55

89.9264

PGO [1]

UB 305 × 102 116 × 25

14

54

89.8937

ST-JA [1]

UB 305 × 102 112 × 25

14

54

89.9159

and hexagonal holes are given in the tables, including cost, selected section, number of holes, diameter or cutting depth of the hole, as well as cutting angle. The optimum result for the castellated beams with circular and hexagonal holes are presented in Tables 7.3 and 7.4, respectively. According to the tables, a castellated beam with the hexagonal holes costs less than a castellated beam with the circular holes under the identical condition. In castellated beams with circular holes, due to the large cutting length and the effect of secondary bending in the cross-section, the cost of the beam is increase. Also, PSO, SSOA, and PGO algorithms result in the lowest cost for both castellated beams with circular and hexagonal holes and ST-JA algorithm results the minimum cost in castellated beams with hexagonal holes. In addition, it can be seen that IS-Jaya and ST-JA algorithms result near-optimal cost in castellated beams with circular holes. Moreover, the CPU time for all algorithms is approximately about 7 s just with differences of about hundredth of a second. The convergence history of the algorithms is provided in Figs. 7.9 and 7.10.

186

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

Fig. 7.6 Convergence curves of the castellated beam with circular holes and 4-m span

Fig. 7.7 Convergence curves of the castellated beam with hexagonal holes and 4-m span

7.6 Examples

187

Fig. 7.8 Simply supported beam with 8-m span

Table 7.3 Optimum design of the castellated beams with circular holes with 8-m span Optimization Optimum UB algorithm section

Hole diameter Total number Cutting angle Minimum cost or cutting of holes ($) depth

PSO [1]

UB 610 × 229 526 × 101

14

−

720.8255

SSOA [1]

UB 610 × 229 526 × 101

14

−

720.8255

IS-Jaya [1]

UB 610 × 229 525 × 101

14

−

720.8316

PGO [1]

UB 610 × 229 526 × 101

14

−

720.8255

ST-JA [1]

UB 610 × 229 525 × 101

14

−

720.8316

Table 7.4 Optimum design of the castellated beams with hexagonal holes with 8-m span Optimization Optimum UB section algorithm

Hole diameter Total number Cutting angle Minimum cost or cutting of holes ($) depth

PSO [1]

UB 610 × 229 232 × 101

14

54

717.5993

SSOA [1]

UB 610 × 229 232 × 101

14

54

717.5993

IS-Jaya [1]

UB 610 × 229 231 × 101

14

54

717.6032

PGO [1]

UB 610 × 229 232 × 101

14

54

717.5993

ST-JA [1]

UB 610 × 229 232 × 101

14

54

717.5993

7.6.3 Castellated Beam with 9-m Span The last example is the beam with a span length of 9 m, as depicted in Fig. 7.11. The two concentrated dead load of 50 kN is applied in the two point of the beam. Also, the wide load of the 40 kN/m is imposed considering the weight of the beam itself. The maximum number of iterations and the number of function evaluations

188

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

Fig. 7.9 Convergence curves of the castellated beam with circular holes and 8-m span

Fig. 7.10 Convergence curves of the castellated beam with hexagonal holes and 8-m span

7.6 Examples

189

Fig. 7.11 Simply supported beam with 9-m span

for algorithms are the same as the previous examples and set to 200 and 1000, respectively. The optimization results of the 4 new meta-heuristic algorithms and the PSO for castellated beams with circular and hexagonal holes are given in the tables, including cost, selected section, number of holes, diameter or cutting depth of the hole, as well as cutting angle. Tables 7.5 and 7.6 give the optimal results for castellated beams with circular and hexagonal holes, respectively. Under equal conditions, a castellated beam with hexagonal holes costs less than one with circular holes. SSOA, IS-Jaya, PGO, and ST-JA algorithms result the minimum cost in castellated beams with hexagonal holes compared with PSO. Moreover, due to the length of the span and complexity in this example compared with prior examples, in general convergence to the optimum cost is happened a little later. Also, the CPU time for all algorithms is approximately about 9 s, with variations of approximately one hundredth of a second. The algorithms’ convergence history is depicted in Figs. 7.12 and 7.13. Table 7.5 Optimum design of the castellated beams with circular holes with 9-m span Optimization algorithm

Optimum UB section

Hole diameter or cutting depth

Total number of holes

Cutting angle

Minimum cost($)

PSO [1]

UB 762 × 267 × 134

600

13

−

1033.9148

SSOA [1]

UB 762 × 267 × 134

600

13

−

1033.9148

IS-Jaya [1]

UB 762 × 267 × 134

600

12

−

1034.3251

PGO [1]

UB 762 × 267 × 134

600

13

−

1033.9148

ST-JA [1]

UB 762 × 267 × 134

600

13

−

1033.9148

190

7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

Table 7.6 Optimum design of the castellated beams with hexagonal holes with 9-m span Optimization algorithm

Optimum UB section

Hole diameter or cutting depth

Total number of holes

Cutting angle

Minimum cost($)

PSO [1]

UB 762 × 267 × 134

258

14

54

1030.2141

SSOA [1]

UB 762 × 267 × 134

229

16

54

1030.1671

IS-Jaya [1]

UB 762 × 267 × 134

229

16

54

1030.1671

PGO [1]

UB 762 × 267 × 134

229

16

54

1030.1671

ST-JA [1]

UB 762 × 267 × 134

229

16

54

1030.1671

Fig. 7.12 Convergence curves of the castellated beam with circular holes and 9-m span

7.7 Concluding Remarks In this chapter, four new meta-heuristic algorithms, namely SSOA, IS-Jaya, PGO and ST-JA, are employed to determine the optimum design of castellated beams with circular and hexagonal holes with cost objective function. The three beams with span of the 4, 8, and 9 m is investigated in this study. In each example, cost of a castellated beam with a circular and hexagonal hole is examined. A comparison made between

References

191

Fig. 7.13 Convergence curves of the castellated beam with hexagonal holes and 9-m span

the results of 4 new meta-heuristic algorithms and the PSO meta-heuristic algorithm illustrating the efficiency, potency, and applicability of the new algorithms. Moreover, it is noted that, due to the rapid convergence to the optimal answer in less iteration, all 4 algorithms are suitable in terms of time-saving. According to the results obtained for the optimal design of castellated beams under different loads, castellated beams with hexagonal holes in terms of construction has better performance and is more economical in compared to castellated beams with circular holes. The reason for this difference in the answers is the effect of crosssectional cutting and welding parameters. Due to the considerable cutting length and the impact of secondary bending in the cross-section, the cost of castellated beams with round holes is increased.

References 1. Kaveh, A., Almasi, P., Khodagholi, A.: Optimum design of castellated beams using four recently developed meta-heuristic algorithms. Iranian J. Sci. Technol. Tran. Civil Eng. (2022). https:// doi.org/10.1007/s40996-022-00884-z 2. Erdal, F., Do˘gan, E., Saka, M.P.: Optimum design of cellular beams using harmony search and particle swarm optimizers. J. Construct. Steel. Res. 67(2), 237–247 (2011) 3. Ward, J.: In: Design of Composite and Non-Composite Cellular Beams. Steel Construction Institute Ascot, UK (1990) 4. Tkalˇcevi´c, V., Džeba, I., Androi´c, B.: Analysis of castellated beams according to Eurocode 3. Gradevinar 58(9), 709–716 (2006)

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7 Optimum Design of Castellated Beams Using SSOA and the Other Four …

5. Kaveh, A., Zaerreza, A., Hosseini, S.M.: Shuffled shepherd optimization method simplified for reducing the parameter dependency. Iranian J. Sci. Technol. Tran. Civil Eng. 45(3), 1397–1411 (2021) 6. Kaveh, A., Hosseini, S.M., Zaerreza, A.: Improved Shuffled Jaya algorithm for sizing optimization of skeletal structures with discrete variables. Structures 29, 107–128 (2021) 7. Kaveh, A., Hosseini, S.M., Zaerreza, A.: Size, layout, and topology optimization of skeletal structures using plasma generation optimization. Iranian J. Sci. Technol. Tran. Civil Eng. 45(2), 513–543 (2021) 8. Kaveh, A., Biabani Hamedani, K., Joudaki, A., Kamalinejad, M.: Optimal Analysis for Optimal Design of Cyclic Symmetric Structures Subject to Frequency Constraints. Elsevier, Structures (2021) 9. Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: MHS’95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science. IEEE (1995)

Chapter 8

An Improved PSO Using the SRM of the ESSOA for Optimum Design of the Frame Structures via the Force Method

8.1 Introduction The efficient graph-theoretical force method, investigated by Kaveh and Zaerreza [1], is presented in this chapter. A graph-theoretical force method is used in the analysis of the frame structures to decrease the time required for optimization. The performance and speed of the graph-theoretical force method are compared to those of the displacement method in the optimal design of frame structures. In addition, the standard particle swarm optimization algorithm (PSO) is improved to enhance its performance in the optimal design of the steel frames. In recent decades, structural optimization has been a popular research topic among civil engineers [2]. Metaheuristic algorithms are mostly simple and easily programmed methods, and computational effort for the different algorithms are close to each other when considering the same number of objective function evaluations. In these methods, the time required to finish the optimization process is dependent on the time needed to calculate the objective function. In structural optimization problems, structural analyses consume the majority of time. Therefore, utilizing an efficient method for structural analysis helps accelerate the optimization process. The two widely used methods for structural analysis are displacement and force methods. The displacement method regards the displacement of structures in the nodes as unknowns. The force of each member is then calculated using the equilibrium and stress–strain equations. In the force method, certain member’s forces are selected as unknowns. The forces of each member is then calculated utilizing the stress–strain and compatibility equations [3]. The number of the equations solved in the displacement technique corresponds to the degree of kinematical indeterminacy (DKI), whereas the number of the equations needed in the force method corresponds to the degree of statical indeterminacy (DSI). Hence, in the structure with less DSI, it is expected the force method to be faster. On the contrary, in the structure with less DKI, the displacement method is expected to be faster.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_8

193

194

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

The force method can be divided into five groups, including the topological force method, integrated force method, algebraic force method, mixed algebraiccombinatorial force method, and combinatorial force method [4]. The flexibility matrix generated using the combinatorial force method is sparser than those generated by other force methods. Thus, the combinatorial force method requires less time for structural studies than other force methods. Hence, the efficiency of the graphtheoretical force method, which is the type of the combinatorial force method, is investigated in this chapter. This chapter examines the effectiveness of the graph-theoretical force method in the optimal design of frame structures. In addition, the new improved version of particle swarm optimization (PSO) is presented. A statistical regeneration mechanism (SRM) is applied to the PSO to enhance its performance. The improved PSO algorithm is named PSO-SRM, and its good performance is proven in the steel frame design problems. The remainder of this chapter is structured as follows. Section 8.2 describes the force method. Section 8.3 presents the graph-theoretical force method. In Sect. 8.4, the formulation of structural optimization with discrete design variables is provides. The PSO-SRM optimization algorithm is introduced in Sect. 8.5. The efficiency of the proposed algorithm in optimizing steel frames is evaluated in Sect. 8.6. The conclusion is finally presented in Sect. 8.7.

8.2 Force Method of Frame Analysis The structure S with the M(S) members, N(S) nodes, and γ(S) times statically indeterminate is considered. γ (S) independent unknowns are chosen as redundants, and their constraints are removed. Then, the stress resultants in members are computed using the Eq. (8.1) [3]. r = B0 p + B1 q

(8.1)

where r denotes the stress of the members, p represents the joint loads, q represents the forces of redundants, B 0 and B 1 are rectangular matrices with m rows, and n and γ columns, respectively, n denotes the number of the components for joint loads, and m is the number of independent components for the member. B 0 p and B 1 q are known as a particular and complementary solution, respectively [3]. By applying the load–displacement relationship and utilizing the principle of the virtual work, the displacement and stresses of the members are computed using the following equations. ] [ ( )−1 v 0 = B t0 F m B 0 − B 0 F m B 1 B t1 F m B 1 B t1 F m B 0 p

(8.2)

] [ ( )−1 r = B 0 − B 1 B t1 F m B 1 B t1 F m B 0 p

(8.3)

8.3 Graph-Theoretical Force Method

195

where the v 0 represents the displacement corresponding to the force components of p, F m is the unassembled flexibility matrix, G = B t1 F m B 1 is known as the flexibility matrix of the structure.

8.3 Graph-Theoretical Force Method In this chapter, the force method is utilized to analyze the structure. The force method can be divided into five categories, as mentioned in the introduction. Kaveh [3] demonstrated the graph-theoretical force method is more effective than the other type of force method. To this end, the graph-theoretical force method is used as structural analysis in this chapter. Graph theory is used to generate the B 0 and B 1 matrices in the graph-theoretical force method. The B 0 matrix is defined as the 6 M × 6NL matrix, where M represents the number of structural elements and NL represents the number of nodes loaded in the considered structure. To generate the B 0 matrix, a spanning forest from the structure’s support is grown using graph theory. Then, each element is given an orientation based on the direction in which the spanning forest grows from its support node. The matrix B 0 consists of the sub-matrices, where each element of these submatrices is calculated by transferring each joint load to a support node. The sub-matrix [B 0 ]i j for the ith member and the jth node is computed as follows: ⎡

1 0 0 ⎢ 0 1 0 ⎢ ⎢ 0 1 ⎢ 0 [B 0 ]i j =∝i j ⎢ ⎢ 0 −Δz B0 Δy B0 ⎢ ⎣ Δz B0 0 −Δx B0 0 −Δy B0 Δx B0

0 0 0 1 0 0

0 0 0 0 1 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1

(8.4)

where Δx B0 = x j − xk

(8.5)

Δy B0 = y j − yk

(8.6)

Δz B0 = z j − z k

(8.7)

⎧ ⎨ +1 i f member is positi vely orriented in the tree containing node j ∝i j = −1 i f member is negativly orriented in the tr ee containing node j ⎩ 0 i f member is not in the tr ee containing node j (8.8)

196

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

where, x j , y j , and z j are coordinate of the jth node; xk , yk , and z k are the coordinate of the lower numbered node of the ith member. More detail is accessible in Ref. [3]. B 1 matrix consists of 6 M rows and 6 b1 (S) columns, where the b1 (S) is the first Betti number and is determined by Eq. (8.9). b1 (S) = M(S) − N (S) + b0 (S)

(8.9)

In the graph-theoretical force method, first, the cycle basis of the structure is formed, then B 1 matrix is calculated using the elements of the selected cycle basis. One of the graph-theoretical algorithms of the Kaveh [3] is employed to generate the cycle basis. An element of structure is selected randomly to form the cycle on this memeber, and then the smallest cycle on this element is created. Each element of the cycle basis is chosen, and the smallest cycles using elements are formed. After generating the new cycle, their admissibility condition (increase of the Betti number by unity) of cycle is checked. If the formed cycle is admissible, the cycle is added to the set of the cycles so-far formed. Then the new cycles on the unused members are formed. This procedure is repeated until the b1 (S) the cycle basis is selected. After the generation of the cycles, the generator of each cycle is cut in the neighborhood of its beginning node, and six bi-actions are applied. The term "generator" refers to one of the members of a specified cycle. In sub-matrix of [B 1 ]i j , the columns show the internal forces at the lower-numbered end of the ith member under the application of six bi-actions at the cut of the jth generator, as given in Eq. (8.10). ⎡

1 0 0 ⎢ 0 1 0 ⎢ ⎢ 0 1 ⎢ 0 [B 1 ]i j = βi j ⎢ ⎢ 0 −Δz B1 Δy B1 ⎢ ⎣ Δz B1 0 −Δx B1 0 −Δy B1 Δx B1

0 0 0 1 0 0

0 0 0 0 1 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1

(8.10)

where Δx B1 = x j − xk

(8.11)

Δy B1 = y j − yk

(8.12)

Δz B1 = z j − z k

(8.13)

⎧ ⎨ +1 i f member has same orriented o f the cycle generated on j βi j = −1 i f member has r ever se orriented o f the cycle generated on j ⎩ 0 i f member is not in the tcycle whose generator is j (8.14)

8.5 PSO-SRM Optimization

197

where, x j , y j , and z j represent the coordinate of the beginning node of the generator j, xk , yk , and z k are the coordinate of lower number of the ith member. More detail is accessible in Ref. [3]. The B 0 and B 1 matrices are dependent on the shape of the structures but not on their cross-sections. Consequently, it is unnecessary to calculate B 0 and B 1 in each iteration of the iterative optimization approach. To this purpose, these matrices are calculated at the start of the optimization process in this study, and they are not required to be updated for each solution. B 0 and B 1 are both stored in memory, and their data is retrieved when required.

8.4 Optimization Problems with Discrete Design Variables Optimization problems can be categories into two types based on their design variables: discrete and continuous. While manipulating continuous design variables is straightforward, manipulating discrete design variables is harder. Different method is developed to handle the discrete design variables using continuous optimization algorithms. In this study, the round operator is used to handle the discrete design variables. The minimum value for design variables is one with this method, and the maximum value for each design variable is limited to the number of members in the set that can be used to represent this design variable. Each design variable is rounded to the nearest integer value in the cost function. This integer value indicates which of the value in the set of the allowable design variables is picked. To further clarify, the formulation of optimization using discrete design variables is given in Eq. (8.15). f ind {X } = {x1 , x2 , x3 , . . . , xn } 1 ≤ xi ≤ number o f element Di { } Di = d1 , d2 , d3 , . . . , dneDi to minimi ze : f (D(r ound(x1 ))1 , D(r ound(x1 ))2 , . . . , D(r ound(xn ))n ) (8.15) Subjec to : dcl < 0, dce = 0, dcg > 0 where {X } represents the set of design variables. n is the number of design variables. Di is the set of the allowable variables for ith design variable. f(.) is the cost function. D(r ound(x1 ))1 is the r ound(x1 ) th member of a set D1 . dc is the constraint function of the optimization problem.

8.5 PSO-SRM Optimization This section provides an improved version of the PSO algorithm for discrete optimization problems. To begin, the particle swarm optimization (PSO) algorithm is described. The statistical regeneration mechanism (SRM) is then defined, followed

198

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

by a minor modification to this mechanism. The PSO-SRM method is described lastly.

8.5.1 Particle Swarm Optimization PSO is a well-known and well-established metaheuristic algorithm inspired by the social behavior of animals in nature [5]. Similar to the other metaheuristics, the initial population of the PSO is generated at randomly in the search space. Then optimization’s main loop is started to generate the new solution. Each member’s step size is determined by the previous step size, the best position discovered by the considered population member, and the best position found by the entire population, so the step size of each population member is defined as follows. ( ) ( ) stepsi zeit+1 = w × stepsi zeit + c1r1 popibest − popit + c2 r2 pop Gbest − popit (8.16) where stepsi zeit represents the step size for the ith member when the iteration number is t. popit i is the position vector of the ith member in the tth iteration. popibest and pop Gbest are the best position found by the ith member and the best position found by the entire population, respectively. r1 and r2 are the random vector generators between 0 and 1. w, c1 , and c2 are the user-defined parameters of the PSO. It should be noted that the value of the w at the end of each algorithm iteration is Multiplied by 0.99. Therefore, the value of the w is decreased by one percent in comparison to the previous value in each iteration to increase the exploitation ability of the PSO. Then, the position of each population member is updated using Eq. (8.17), and this process is repeated until the termination condition of the algorithm is reached. popit+1 = popit + stepsi zeit+1

(8.17)

8.5.2 Statistical Regeneration Mechanism (SRM) Kaveh et al. [6] introduced SRM to improve the shuffling shepherd optimization algorithm (SSOA) for solving large-scale optimization problems. In the SRM, first, the mean and standard deviation of the entire population positions in the search space are computed then the new solution is generated using Eq. (8.18). new xi,s = U N F I R AN D(Mean s − stds − sigmas , Mean s + stds + sigmas ) (8.18)

8.6 Design Examples

199

where U N F I R AN D is the mechanism that generates the new solution randomly in the range of [Mean s −stds −sigmas , Mean s +stds +sigmas ]. Mean s and stds are the average and standard deviation of the sth design variable of candidate solutions, and sigmas is the mechanism that helps the U N F I R AN D work perfectly and is defined as follows [6]. ( sigmas =

) ) ( ( 0.01 × X smax − X smin i f stds < 0.01 × X smax − X smin 0 other wise

(8.19)

where X smax and X smin are the upper and lower bound of the search space. In this study, SRM is used with minor modifications in the value of the sigmas to achieve high performance in discrete size optimization problems. The various values and formulations for the sigmas is tested. The test result reveals that the ideal value for sigmas is the fixed value of 3. As stated previously, the continuous optimization algorithm is connected to the discrete cost function using the operator that rounds each continuous value to the nearest integer, and this integer value specifies which section is picked. Therefore, using the fixed value of 3 for sigmas help the U N F I R AN D mechanism to select at least three smaller or larger sections than Mean s .

8.5.3 PSO-SRM Algorithm In this section, the PSO-SRM algorithm is introduced. In order to improve the exploration and exploitation of the PSO, the statistical regeneration mechanism (SRM) with minor modification, which is described in the previous section, is added to the PSO. Different manners are tested to find the best way to add the SRM. The optimal way we reached is defined as follows. To add the SRM into the PSO, first, the mean and standard deviation of the best position found by each population is computed. Then fifty percent of the population regenerated via SRM in each iteration. In the SRM, each population position is replaced with the best position found by considered population, and then its position is regenerated. If the current iteration number (CIN) of the optimization algorithm is less than half of the maximum number of iterations (MNI), twenty percent of the design variables are chosen and regenerated using SRM. Otherwise, only one of the design variables is selected and regenerated using SRM. Figure 8.1 provides a flowchart of the PSO-SRM for more explanation.

8.6 Design Examples Three benchmark steel frame examples first time are investigated utilizing the force method. Also, the efficiency and capability of PSO-SRM are tested using these

200

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

Fig. 8.1 Flowchart of the PSO-SRM algorithm

examples. These examples include the 1-bay 10-story steel frame, 3-bay 15-story steel frame, and 3-bay 24-story steel frame. The outcomes of the PSO-SRM in the optimization of these examples are compared to the PSO and other existing methods. The maximum number of function evaluations of PSO and PSO-SRM in all examples is set to 20,000. C1 and C2 in both of the algorithms are set as the same value of the 2. It should be noted that in comparing the required time for each structure, the total time is reported. In particular, the time required to calculate the B 0 and B 1 matrices is considered in the total time.

8.6 Design Examples

201

8.6.1 The 1-Bay 10-Story Steel Frame The first problem is investigated in this chapter a 1-bay 10-story steel frame, as shown in Fig. 8.2. This frame’s elements are separated into nine main categories. The design variables for the beam element are picked from 267 W-section, whereas design variables for the column elements are chosen from W 12 and W 14 sections. Members’ yield stress and elasticity modulus are set to 36 ksi and 29,000 ksi, respectively. The strength and displacement constraints are considered according to the AISC-LRFD requirements. The degree of statical indeterminacy (DSI) and degree of kinematical indeterminacy (DKI) of this structure are 30 and 60, respectively. Therefore, it is predicted that the method of forces is faster than the displacement method. The B 0 , B 1 ,and G matrix pattern for this structure is given in Fig. 8.3. The comparison of the result obtained by PSO and PSO-SRM using the displacement and force method with other metaheuristic algorithms employing the displacement method are summarized in Table 8.1. The results found by PSO-SRM and PSO are superior to ECBO and GSU-PSO. The statistical result obtained by PSO-SRM is better than PSO in both of the analysis methods. PSO-SRM required 2 s more than PSO to complete the optimization. However, according to Fig. 8.4, the displacement method needs 45.38% longer time to complete the 20,000 structural analyses than the force method, as expected according to the DSI and DKI. In addition, there is no significant difference between the optimal and mean weights determined using the force and displacement method. According to Fig. 8.5, the convergence history of the best run and average runs of the PSO-SRM is under the PSO in both analyzing methods. This demonstrates that the PSO-SRM can easily escape local optima and converge to the optimal solution. The section for each group of the members chosen by utilizing the force and displacement methods in the best run of the PSO-SRM are the same. Therefore, as shown in Figs. 8.6 and 8.7, both methods’ stress ratio and inter-story drift are very close, with no discernible difference. This indicates that there is no noticeable difference in the accuracy of the force method and displacement approach. According to Figs. 8.6 and 8.7, the optimization constraint is satisfied, and none of them are violated.

8.6.2 The 3-Bay 15-Story Steel Frame The 3-bay 15-story steel frame is utilized as the second problem to study the PSOSRM’s performance via the force method. This structure consists of 105 members organized into 11 groups, as illustrated in Fig. 8.8. Variables for these groups are picked from a database of 267 W-section variables. Members’ yield stress and elasticity modulus are set to 36 ksi and 29,000 ksi, respectively. Constraints on stress and displacement are taken into account in accordance with the AISC-LRFD standard. Additionally, the top story’s sway is limited to 8.25 in. the DSI and DKI of this

202

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

Fig. 8.2 The schematic of the 1-bay 10-story steel frame

example are 135 and 180, respectively. Figure 8.9 shows the B 0 ,B 1 , and G matrix pattern for the 3-bay 15-story steel frame. Table 8.2 compares results of PSO and PSO-SRM algorithms with other available results acquired by hybrid Eagle Strategy with Differential Evolution (ES-DE) [9] and Plasma Generation Optimization (PGO) [10]. Compared to the other methods considered, PSO-SRMs’ best solutions are significantly better. In addition, the average optimal weight determined by PSO-SRM is significantly better than the average

8.6 Design Examples

203

Fig. 8.3 Sparsity pattern of the B 0 ,B 1 , and G for 1-bay 10-story steel frame

weight determined by PSO. According to Fig. 8.10, the PSO-SRM algorithm has a quicker convergence rate than the PSO algorithm in both the best and average runs. Due to the stochastic properties of the metaheuristic algorithms, the best result found in the displacement method is better than the result found using the force method, and there is no significant difference, as shown in Fig. 8.11. The number of the equations solved in the displacement method is 180, and the number of equations solved in the force method is 135. Therefore, according to Table 8.2 force method is faster than the displacement method and requires 100 s less to complete the optimization process. The maximum inter-story drift in the PSO-SRM’s best run utilizing the force and displacement methods are 0.4447 and 0.4539, respectively, which is less than the allowable value (0.46), as shown in Fig. 8.12. Additionally, according to Fig. 8.13, the maximum stress ratios are less than one, indicating that all of the limitations have been satisfied.

8.6.3 The 3-Bay 24-Story Steel Frame The final case study in this study is a 3-bay 24-story steel frame with 168 members, as illustrated in Fig. 8.14. The structural members are divided into twenty groups, where the beam elements are chosen from 262 W-sections, and the column elements are selected from the W14 sections. Members have a yield stress of 33.4 ksi and an elasticity modulus of 29,732 ksi. In a manner similar to the preceding example, the stress and displacement limits are taken into consideration in accordance with the AISC-LRFD standard. The B 0 ,B 1 , and G matrix patterns for this example are shown in Fig. 8.15. Additionally, the DSI and DKI are 216 and 288.

N/A

N/A

Average time (second)

68027.172 N/A

N/A

N/A

Average weight (lb)

Standard deviation (lb)

72736.918

N/A

Worst weight (lb)

W21 × 44

W21 × 55 64475.2

W18 × 40

64561.068

9

Best weight (lb)

W27 × 84

W27 × 84

W27 × 84

8

W33 × 118

31.17

1023.06

65533.35

67403.68

64133.98

W30 × 90

W33 × 118 W30 × 99

W33 × 118

W30 × 99

W12 × 65

6

W12 × 79

W12 × 65

5

W14 × 99

W14 × 159

W14 × 176

W14 × 233

PSO [1]

Present work

7

W14 × 145 W12 × 106

W14 × 145

W12 × 106

3

4

W14 × 211 W14 × 176

W14 × 233

W14 × 176

1

ECBO [7]

GSU-PSO [8]

Displacement method

2

Element group

33.08

925.06

64666.00

68968.99

64001.98

W18 × 46

W27 × 84

W30 × 90

W33 × 118

W14 × 61

W14 × 99

W14 × 159

W14 × 176

W14 × 233

PSO-SRM [1]

21.44

1804.32

66045.15

71532.68

64001.98

W18 × 46

W27 × 84

W30 × 90

W33 × 118

W14 × 61

W14 × 99

W14 × 159

W14 × 176

W14 × 233

PSO [1]

Force method

23.42

640.86

64607.08

66150.02

64001.98

W18 × 46

W24 × 84

W30 × 99

W33 × 118

W14 × 61

W14 × 99

W14 × 159

W14 × 176

W14 × 233

PSO-SRM [1]

Table 8.1 Comparative results of the PSO-SRM and PSO algorithms with some other methods for the 1-bay 10-story frame structure (DSI = 30 and DKI = 60)

204 8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

8.6 Design Examples

205

Fig. 8.4 Percentages of variation of the best weight, average weight, and average run time between the force method and displacement method in the 1-bay 10-story steel frame

Fig. 8.5 Convergence histories of the PSO and PSO-SRM for the 1-bay 10-story frame structure

Table 8.3 makes a comparison between the results obtained by PSO and PSOSRM and those obtained by other optimization methods. The best weight for PSOSRM is 201402.05 lb, which is the least weight among the other results, including those of the ES-DE at 212,478.17 lb, PGO at 202,194 lb, PSO (force method) at 204,066.03 lb, and PSO (displacement method) at 204,738.00 lb. Although the time required for PSO-SRM is very close to the PSO, PSO-SRMs obtain an average weight of 203,400.11 and 204,050.13 lb using the force and displacement method, respectively, which is significantly lighter than the average weight achieved by other optimization algorithms (see Table 8.3). DSI is less than DKI in this example, so

206

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

Fig. 8.6 Stress ratio for the 1-bay 10-story frame using the displacement and force method

Fig. 8.7 Inter-story drifts for the 1-bay 10-story frame using the displacement and force method

as expected, the force method is faster than the displacement method. The force method’s average time is 126.72 s, which is 71 s less than that of the displacement method. Additionally, the maximum percentage of variation of the average and best weight of the force and displacement method is 2.39%, which is very less than the maximum percentage of variation of the average time, as shown in Fig. 8.16. The inter-story drift and stress ratios, calculated in accordance with the AISC-LRFD, are shown in Figs. 8.17 and 8.18. These figures demonstrate that displacement and

8.6 Design Examples Fig. 8.8 The schematic of the 3-bay 15-story steel frame

207

208

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

Fig. 8.9 Sparsity pattern of the B 0 , B 1 , and G matrices for the 3-bay 15-story steel frame

stress limitations are satisfied. Similar to the first example, the section selected for each member is the same in both methods of analysis. As a result, the stress ratio and inter-story drift of these methods are very close, as shown in Figs. 8.17 and 8.18. Figure 8.19 contains the convergence history of the PSO and PSO-SRM. According to the convergence history figure, the PSO-SRM can readily escape local optima, whereas the PSO has difficulty escaping local optima.

8.7 Discussion and Concluding Remarks This chapter has two main purposes. Firstly, a graph-theoretical analysis is incorporated into the optimal design of frame structures. Three benchmark frame design examples are optimized using the graph-theoretical force method and are compared to the displacement method. Secondly, an enhanced version of the particle swarm optimization algorithm named PSO-SRM is developed. The suggested method incorporates a statistical regeneration mechanism (SRM) into the PSO to boost its exploration capability in the early iterations and its exploitation capability in the final iterations. In the SRM, the position of each particle is replaced by the best position found. Then, twenty present or one of its positions are regenerated using the statistical information. The performance of the PSO-SRM is tested by utilizing the force and displacement methods on the three benchmark examples. These examples include a 1-bay 10-story steel frame, 3-bay 15-story steel frame, and 3-bay 24-story steel frame. In all of these examples, the results obtained by the PSO-SRM are better than those of the standard PSO and other considered methods. In the first and last example, the best results are found by the PSO-SRM algorithm using the force and displacement methods are the

N/A

N/A

Average time (second)

90682 2103

N/A

N/A

Average weight (lb)

N/A

N/A

Worst weight (lb)

Standard deviation (lb)

W21 × 44 87399

W21 × 48

93309

11

W10 × 39

W12 × 40

10

Best weight (lb)

W12 × 65 W8 × 28

W18 × 65

W8 × 28

8

W18 × 50

W18 × 71

7

9

W21 × 68 W18 × 86

W30 × 90

W10 × 88

5

W24 × 104

W27 × 114

4

6

W27 × 161 W27 × 84

W36 × 150

W12 × 79

2

W14 × 99

W18 × 106

1

3

PGO [10]

Displacement method

ES–DE [9]

Element group

251.87

2328.99

91436.88

98442.96

87823.53

W21 × 44

W8 × 40

W6 × 25

W21 × 68

W14 × 48

W30 × 90

W21 × 68

W24 × 104

W27 × 84

W27 × 146

W24 × 117

PSO [1]

Present work

256.04

722.49

87705.73

91258.5

86950.79

W21 × 44

W10 × 39

W14 × 30

W14 × 61

W18 × 50

W18 × 86

W12 × 65

W24 × 104

W14 × 82

W36 × 170

W14 × 90

PSO-SRM [1]

154.56

2588.55

91324.54

96805.93

87735.36

W21 × 44

W8 × 40

W6 × 25

W21 × 68

W14 × 48

W30 × 90

W14 × 61

W27 × 114

W18 × 86

W27 × 161

W12 × 96

PSO [1]

Force method

126.72

318.36

87606.54

88861.77

87183.39

W21 × 44

W8 × 40

W6 × 25

W12 × 65

W8 × 48

W30 × 90

W14 × 61

W21 × 111

W27× 84

W27× 161

W12 × 96

PSO-SRM [1]

Table 8.2 Comparative results of the PSO-SRM and PSO algorithms with other methods for the 3-bay 15-story frame structure (DSI = 135 and DKI = 180)

8.7 Discussion and Concluding Remarks 209

210

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

Fig. 8.10 Convergence histories of the PSO and PSO-SRM for the 3-bay 15-story frame structure

Fig. 8.11 Percentages of variation of the best weight, average weight, and average run time between the force method and displacement method in the 3-bay 15-story steel frame

same. In the second example, the result found utilizing the displacement method is better than those of the force method. There is no significant difference between the time required to complete analyses using the PSO and PSO-SRM utilizing the same analyzing method in all examples. On the other hand, these examples have less degree of static indeterminacy (DSI) than the degree of kinematical indeterminacy. Thus, the time required for structural analyses using the force method is less than those of the displacement method. Also, when the difference between the DSI and DKI is greater, the difference in required time increases. Therefore, it can be concluded that in a structure with less DSI than DKI it is better to utilize the force method as a

8.7 Discussion and Concluding Remarks

211

Fig. 8.12 Inter-story drifts for the 3-bay 15-story frame using the displacement and force method

Fig. 8.13 Stress ratio for the 3-bay 15-story frame using the displacement and force methods

structural analysis tool. One should do this, especially when the difference between DSI and DKI is significant.

212

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

Fig. 8.14 The schematic of the 3-bay 24-story steel frame

8.7 Discussion and Concluding Remarks

213

Fig. 8.15 Sparsity pattern of the B 0 , B 1 , and G for the 3-bay 24-story steel frame

Table 8.3 Comparative results of the PSO-SRM and PSO algorithms with some other methods for the 3-bay 24-story frame structure (DSI = 216 and DKI = 288) Element group

Displacement method ES–DE [9]

PGO [10]

Force method Present work PSO [1]

PSO-SRM [1]

PSO [1]

PSO-SRM [1]

1

W14 × 145

W14 × 159

W14 × 193

W14 × 159 W14 × 211

W14 × 159

2

W14 × 99

W14 × 120

W14 × 120

W14 × 132 W14 × 109

W14 × 132

3

W14 × 109

W14 × 132

W14 × 90

W14 × 109 W14 × 99

W14 × 109

4

W14 × 132

W14 × 74

W14 × 68

W14 × 74

W14 × 74

W14 × 74

5

W14 × 99

W14 × 61

W14 × 61

W14 × 82

W14 × 68

W14 × 82

6

W14 × 109

W14 × 48

W14 × 38

W14 × 48

W14 × 53

W14 × 48

7

W14 × 145

W14 × 38

W14 × 61

W14 × 30

W14 × 43

W14 × 30

8

W14 × 68

W14 × 22

W14 × 22

W14 × 22

W14 × 22

W14 × 22

9

W14 × 109

W14 × 90

W14 × 90

W14 × 90

W14 × 90

W14 × 82

10

W14 × 68

W14 × 109

W14 × 99

W14 × 99

W14 × 109

W14 × 99 (continued)

214

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

Table 8.3 (continued) Element group

Displacement method ES–DE [9]

PGO [10]

Force method Present work PSO [1]

PSO-SRM [1]

PSO [1]

PSO-SRM [1]

11

W14 × 48

W14 × 82

W14 × 99

W14 × 90

W14 × 99

W14 × 90

12

W14 × 68

W14 × 90

W14 × 99

W14 × 90

W14 × 90

W14 × 90

13

W14 × 38

W14 × 74

W14 × 82

W14 × 61

W14 × 68

W14 × 61

14

W14 × 61

W14 × 53

W14 × 68

W14 × 53

W14 × 53

W14 × 53

15

W14 × 30

W14 × 30

W14 × 22

W14 × 34

W14 × 30

W14 × 34

16

W14 × 22

W14 × 22

W14 × 22

W14 × 22

W14 × 22

W14 × 22

17

W30 × 90

W30 × 90

W30 × 90

W30 × 90

W30 × 90

W30 × 90

18

W21 × 55

W8 × 18

W14 × 26

W6 × 15

W6 × 15

W6 × 15

19

W21 × 48

W24 × 55

W24 × 55

W24 × 55

W24 × 55

W24 × 55

20

W10 × 45

W6 × 8.5

W6 × 8.5

W6 × 8.5

W6 × 8.5

W6 × 8.5

Best weight (lb)

212478.17

202194

204738

201402.05

204066.03

201402.05

Worst weight (lb)

N/A

N/A

228143.92

208253.93

271433.87

207372.11

Average weight (lb)

N/A

218596

213814.53

204050.13

219050.38

203400.11

Standard deviation (lb)

N/A

12721

7021.10

2158.68

16395.64

1538.31

Average time (second)

N/A

N/A

483.21

474.25

408.34

403.55

8.7 Discussion and Concluding Remarks

215

Fig. 8.16 Percentages of variation of the best weight, average weight, and average run time between the force method and displacement method in the 3-bay 24-story steel frame

Fig. 8.17 Inter-story drifts for the 3-bay 24-story frame using the displacement and force method

216

8 An Improved PSO Using the SRM of the ESSOA for Optimum Design …

Fig. 8.18 Stress ratio for the 3-bay 24-story frame using the displacement and force method

Fig. 8.19 Convergence histories of the PSO and PSO-SRM for the 3-bay 24-story frame structure

References

217

References 1. Kaveh, A., Zaerreza, A.: Comparison of the graph-theoretical force method and displacement method for optimal design of frame structures. Structures 43, 1145–1159 (2022) 2. Kaveh, A.: Advances in Metaheuristic Algorithms for Optimal Design of Structures. 3rd Edn, Springer (2021) 3. Kaveh, A.: Structural Mechanics: Graph and Matrix Methods. vol. 6, Macmillan International Higher Education, UK (1992) 4. Henderson J C de, C., Maunder, E.W.A.: A problem in applied topology: on the selection of cycles for the flexibility analysis of skeletal structures. IMA J. Appl. Math. 5(2), 254−269 (1969) 5. Jain, N.K., Nangia, U., Jain, J.: A review of particle swarm optimization. J. Instit. Eng. (India): Series B 99(4), 407–411 (2018) 6. Kaveh, A., Zaerreza, A., Hosseini, S.M.: An enhanced shuffled Shepherd optimization algorithm for optimal design of large-scale space structures. Eng. With Comput. 38(2), 1505–1526 (2021) 7. Kaveh, A., Hoseini Vaez, S.R., Hosseini, P.: Modified dolphin monitoring operator for weight optimization of frame structures. Period Polytech Civil Eng. 61(4), 770−779 (2017) 8. Khajeh, A., Ghasemi, M.R., Ghohani Arab, H.: Hybrid particle swarm optimization, grid search method and univariate method to optimally design steel frame structures. IUST 7(2), 173-191 (2017) 9. Talatahari, S., Gandomi, A.H., Yang, X.-S., Deb, S.: Optimum design of frame structures using the eagle strategy with differential evolution. Eng. Struct. 91, 16–25 (2015) 10. Kaveh, A., Hosseini, S.M., Zaerreza, A.: Size, layout, and topology optimization of skeletal structures using plasma generation optimization. Iranian J. Sci. Technol. Trans. Civil Eng. 45(2), 513–543 (2021)

Chapter 9

An Efficient ESSOA for the Reliability Based Design Optimization Using the New Framework

9.1 Introduction In this chapter, a novel framework is presented for reliability-based design optimization (RBDO) employing decoupled approaches and metaheuristic algorithms, developed by Kaveh and Zaerreza [1]. This framework is referred to as sequential optimization and reliability assessment-double metaheuristic (SORA-DM). Utilizing enhanced shuffled shepherd optimization technique (ESSOA), the effectiveness of the SOAR-DM is examined. Six RBDO problems are used to assess the efficiency of the suggested framework. The findings demonstrate that the SORA-DM can outperform the gradient-based technique in the RBDO and is applicable to a broad variety of RBDO problems. In recent decades, deterministic optimization approaches have been effectively applied to a wide variety of structural optimization problems [2], and this trend is anticipated to continue. Nevertheless, it is generally accepted that there is always some uncertainty associated with any structural system owing to differences in material characteristics, erroneous characterization of the loading environment, and manufacturing tolerances, all of which add to the overall uncertainty [3]. In light of the uncertainty, reliability-based design optimization (RBDO) is more important than deterministic optimization. The RBDO techniques can be categorized into three categories consisting of double-loop, single-loop, and decoupled methods. The double-loop approaches necessitated high processing costs [4], and the singleloop techniques are unsuitable for the complex nonlinear constraint functions [5]. In the decoupled approaches, the optimization and reliability analysis components are separated from each other and do not have the shortage of the other methods. Thus, the SORA-DM framework uses the sequential optimization and reliability assessment strategy (SORA) [6]. There are two types of reliability analyses. The first type is computing the reliability index. In double-loop approaches, this form of reliability analyses is utilized to verify the solution’s viability in each cycle. Moreover, it is utilized to validate © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_9

219

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9 An Efficient ESSOA for the Reliability Based Design Optimization …

the solutions identified by other methods. Techniques for calculating the reliability index can be categorized into three types: moment methods, sampling methods, and optimization methods. The second method of reliability analysis involves calculating the most probable point (MPP). In recent decades, many approaches, such as Chaos Control (CC) [3], have been developed for calculating the MPP. These techniques depend on the starting position. In addition, they cannot be exploited in big systems since gradient information is required. Hence, metaheuristic methods that do not require gradient information and are independent of the initial position, can be applied for reliability analysis. To reduce the unneeded function evaluations in metaheuristic optimization methods, a novel termination condition inspired by gradient-based approaches is provided. This chapter’s objective is to present a framework in which metaheuristics are used for both reliability and optimization. To do this, a novel approach for reliability analysis based on metaheuristics is first devised. In order to reduce the number of unnecessary function evaluations, a novel termination condition inspired by gradientbased approaches is introduced to present the general form of RBDO problems. The remainder of the chapter is structured as follows: in Sect. 9.2 presents the general form of the RBDO problems and the sequential optimization and with reliability method are presented. The new RBDO framework for is described in Sect. 9.3. ESSOA is added to the framework to test the capability of the method in Sect. 9.4. In Sect. 9.5, Six numeral examples containing, including two new instances are provided. Section 9.6 concludes the study with a conclusion.

9.2 Formulation of Optimization and Reliability First, a general definition of the RBDO problem is given in this section. Then the sequential optimization and reliability analysis method, which is one of the wellknown techniques for RBDO problems, is next discussed.

9.2.1 Formulation of RBDO Problem The RBDO problem is defined in general form by the following equation. Find {D, μ X } Minimi ze f (D, X, P)

) ( Subject to : P R O B(gi (D, X, P) ≤ 0) ≤ Φ −βit i = 1, 2, 3, . . . , m Where D max ≤ D ≤ D min μmax ≤ μ X ≤ μmin X X

(9.1)

where D is a vector of deterministic design variables with D max and D min as upper and lower limits, respectively; X is the vector of random design variables; P is the

9.2 Formulation of Optimization and Reliability

221

vector of random design parameters; μ X is the mean vector of the X with μmax and X μmin as upper and lower limits, respectively; g is ith constraint function (limit (.) i X state function); PROB(.) is the probability of the failure; Φ(.) is the standard normal cumulative distribution function; βit is the target reliability index for ith constraint function; m is the number of constraint functions.

9.2.2 Sequential Optimization Together with Reliability Assessment Du and Chen [6] presented sequential optimization and reliability analysis (SORA) as one of the effective strategies for probabilistic design. In the first cycle of SORA a deterministic optimization is carried out, where the optimization approach is utilized to minimize the f (D, μ X , P) subjected to the gi (D, μ X , P) ≤ 0, disregarding the probability functions in order to discover {D, μ X } . It should be emphasized that at the beginning of the SORA, the random design parameters are assumed to be equal to their mean. Next, the reliability analysis is used to identify the most probable point (MPP). Finding the MPP is the most critical part of the SORA, which is utilized to verify the feasibility of the obtained result and to compute the next cycle’s shifting vector. For determining the feasibility of the outcomes, the constraint functions’ values are computed in the MPP. The positive value of all constraint functions shows that the solution is feasible. In other words, the positive value of all constraint functions indicates that the constraint functions are fulfilled. If the solution is not practicable, the subsequent cycle will start. The shifting vector based on MPP is computed by Eq. (9.2) for updating the constraint functions in the second loop. SVicn = μcn−1 − X M P Picn−1 i = 1, 2, 3, . . . , m X

(9.2)

where the cn represents the cycle number; SVicn is the shifting vector for the ith represents the average value of random design constraint in the cnth cycle; μcn−1 X variables obtained in the previous cycle; X M P Picn−1 represents the MPP point for the ith constraint function acquired in the previous cycle. Similar to the first loop, deterministic optimization is applied in the second loop. However, the constraint functions are shifted, and the random design parameter’s values are the value obtained in the reliability assessment. Therefore, the objective function of the optimization process in the second cycle is described as follows: Find {D, μ X } Minimi ze f (D, μ X , μ P ) ( ) Subject to : gi D, μ X − SVicn , P M P Picn−1 ≤ 0 i = 1, 2, 3, . . . , m

(9.3)

where the μ P is the average value of the random design parameters; P M P Picn−1 is the most probable point of the random design parameters in the preceding cycle. After

222

9 An Efficient ESSOA for the Reliability Based Design Optimization …

the optimization processes, the reliability analysis is applied like the first cycle, and the viability of the solution is examined. If the solution is not realizable, the shifting vectors are computed. The whole procedure is repeated until a suitable solution is reached.

9.3 New Reliability-Based Design Optimization Framework This section presents the novel framework for reliability-based optimization utilizing metaheuristic algorithms and SORA. Unlike previous studies, this framework utilizes the metaheuristic method for both optimization and reliability analysis. To do this, a novel objective function for the reliability analysis utilizing metaheuristic methods is created. Next, a new termination condition for the metaheuristic technique is introduced. Finally, the new framework for RBDO is represented.

9.3.1 Reliability Assessment In the SORA, a reliability analysis is conducted to determine the most probable point (MPP). For finding the MPP, gradient-based approaches such as advanced mean value (AMV) and chaos control (CC) are available. To acquire the gradient information, it is necessary to compute a symbolic stiffness matrix for the structure. Although computing the symbolic stiffness matrix is not time-consuming, however determining the structure’s displacement utilizing the symbolic stiffness matrix is time-consuming. This is particularly true of structures with high degrees of freedoms. Alternatively, there are effective numerical techniques for computing the structural displacements when the numerical stiffness matrix is used. Due to the use of numerical approaches, the metaheuristic method for locating the MPP is quicker than gradient-based methods. Additionally, they can be applied in large structural systems. To this end, a novel approach for reliability analysis is provided in this chapter. The traditional objective function for finding the MPP utilizing optimization techniques is stated as follows [6]: f ind{X, P} minimi ze g(U ) )1/2 ( =β subject to U T U

(9.4)

where U represents the vector of the transformed of the random variables (X, P) into an independent and standardized normal space; β is called the reliability index; g(.) is the limit state function. Although Eq. (9.4) can be employed in metaheuristic algorithms to discover MPP, it is hard to find a solution that precisely fulfills the constraint function owing to the

9.3 New Reliability-Based Design Optimization Framework

223

equality restriction. In light of this, a suitable range is chosen for this kind of constraint function. The allowed range decreases the accuracy of the traditional approach for identifying the MPP. The inaccuracy of the MPP impacts the whole RBDO and causes an inaccurate solution to be found. Therefore, a new objective function is proposed in this chapter to improve the precision of the reliability analysis utilizing metaheuristic algorithms. This method is named no constraint most probable point finder (NCMPPF). In NCMPPF, as opposed to the conventional reliability analysis technique based on metaheuristic algorithms, the constraint function is excluded from the optimization procedure. Due to the omission of the acceptable range, the accuracy of the MPP calculated with the NCMPPF is improved. In the NCMPPF, one of the design variables is omitted from optimization procedures of reliability analysis and the objective function is used to determine the omitted variable. To do this, the other variables changed into independent and standardized normal space. Then, the value of the missing design variable in the standardized normal space is determined utilizing Eq. (9.5). U omit 2 = β 2 − U other s T × U other s

(9.5)

where the U other s denotes the vector of the transformation of the random variables into an independent and standardized normal space, except one which is removed; U omit 2 is the power of two of the removed design variable in the standardized normal space. Although the U omit can be calculated using Eq. (9.5), the value of Uomit 2 may be negative, rendering the calculation of the U omit impossible. In order to overcome this problem, if the value of U omit 2 goes negative, U other s T × U other s × 109 is regarded as the objective value. This leads the optimization process to seek the least value for U other s T × U other s, causing U omit 2 to approach zero, and at last, becomes a positive value. When the value of the U omit 2 becomes positive, it is possible to determine the value of the limit state function. However, due to the radical of U omit 2 is calculated, there are two values, one of which is positive and the other of which is negative, as given in Eq. (9.6). √ U omit 2 √ U omit2 = − U omit 2

U omit1 =

(9.6)

where both U omit1 and U omit2 reflect the missing design variable in the standardized normal space. Thus, each of these numbers should be added to U other s separately to calculate the limit state function value, and whichever one results in a lower limit state function value should be considered as the actual value. In addition, the lower limit state function value is considered as an objective value. The NCMPPF flowchart is shown in Fig. 9.1 for better clarification.

224

9 An Efficient ESSOA for the Reliability Based Design Optimization …

Fig. 9.1 Flowchart of the NCMPPF

9.3 New Reliability-Based Design Optimization Framework

225

9.3.2 Termination Condition In this chapter, a novel termination condition for reliability analysis is presented that is inspired by the gradient-based approach termination condition. The gradient-based technique to prevent excessive function evaluation ends when there is no substantial difference between the current and prior cycle’s values. To determine if a change is substantial or not, Eq. (9.7) is utilized. O k − O k−1 ≤ε Ok

(9.7)

where O is the variable that the gradient-based technique tries to find; k represents the cycle number and ε is tolerance for determining the stopping criterion. In the metaheuristic algorithms, a similar manner like Eq. (9.7) is utilized to convergence of the algorithm. When the algorithm converges, it cannot discover a new solution, thus, the process must be ended to avoid excessive function evaluation. To determine whether or not the algorithm has converged, the variations of the best and worst agents are examined using Eq. (9.8). O B k − O B k−1 ≤ε O Bk O W k − O W k−1 ≤ε OWk

(9.8)

where OB and OW denote the objective function of the best and worst agent, respectively. If Eq. (9.8) is satisfied, the convergence of the whole population is checked utilizing the following equation. Otherwise, the algorithm advances to the next cycle of optimization. O M k − O M k−1 ≤ε O Mk

(9.9)

where OM is the average value of the objective function of the population. When Eq. (9.9) is satisfied for fifty cycles in a row without interruption, it indicates that the algorithm’s whole population has converged, so it can no longer discover a new solution. Hence, the procedure must be ended in order to avoid unnecessary function evaluations.

9.3.3 SORA-Double-Metaheuristic This section represents the new framework of the RBDO. In the SORA-DoubleMetaheuristic (SORA-DM) technique, the metaheuristic algorithm is employed for both optimization and reliability analysis. In the optimization phase, the objective

226

9 An Efficient ESSOA for the Reliability Based Design Optimization …

function described in Sect. 9.2.2 is used, and the NCMPPF is employed to conduct a reliability analysis. Except for the maximum number of iterations (Maxiter), the parameters of the optimization technique for both the reliability analysis and optimization sections are regarded identical in SORA-DM. In SORA, the accuracy of the MPP identified influences the whole optimization procedure; hence, the Maxiter considered for the reliability analysis section is three times the Maxiter considered for optimization. In addition, the Maxiter for both the reliability analysis and optimization portions is raised by 10% every cycle to enhance the precision of the identified solution without increasing the number of cycles. In addition, the termination condition introduced in Sect. 9.3.2 is applied to the optimization algorithm’s primary termination condition as the second termination condition. Consequently, the optimization procedure ends when one of the termination conditions is met. Figure 9.2 illustrates a flowchart of the SORA-DM to give more clarification.

9.4 SORA-DESSOA In order to analyze the performance of the SORA-DM methodology, the Enhanced Shuffled Shepherd Optimization Algorithm (ESSOA) is utilized as an optimization strategy in SORA-DM, thus the method is named SORA-DESSOA. The SORA-DM method has been covered in preceding sections. Consequently, the components of the ESSOA are detailed in this section.

9.4.1 Enhanced Shuffled Shepherd Optimization Algorithm Kaveh et al. [7] develop ESSOA as an improved variant of the shuffling shepherd optimization technique (SSOA). SSOA lacks a strategy for escaping from the local optimum and beginning from a random population without previous information. Consequently, to improve the functioning of the SSOA. Opposition-Based Learning (OBL) approaches have been included into the initialization phase of the algorithm. Statistically Regeneration Mechanism (SRM), a new method for escaping from local optimum, has been applied to the algorithm’s main loop. The ESSOA’s main steps are outlined: Step 1: Initialization During the startup step of the ESSOA, the search space population is created at randomly using the following formulae: ) ( 0 xi,n = xnmax + xnmax − xnmin × rand

i = 1, 2, 3, . . . ,

Fig. 9.2 Detailed flowchart of the SORA-DM

9.4 SORA-DESSOA 227

228

9 An Efficient ESSOA for the Reliability Based Design Optimization …

N pop n = 1, 2, 3, . . . , N var

(9.10)

0 where the xi,n is the randomly generated value for nth variable of ith population; xnmax and xnmin are the upper and lower bound of the nth design variables, respectively; Nvar is the number of the design variables; Npop is the population size of the algorithm; rand is a random number generator that generates values between 0 and 1. After generating solutions at randomly, four OBL approaches are implemented. First of all, the opposite of the solutions (OS) is determined utilizing the following equation.

O Sn = xnmax + xnmin − xn n = 1, 2, 3, . . . , N var

(9.11)

The second OBL strategy is the Quasi-Reflection. The Quasi-Reflection of solution (x) is defined as a solution that is randomly produced between the x and center point, which is determined as follows: (

x max + xnmin M I Dn = n 2

)

Q RSn = M I Dn + (M I Dn − xn ) × rand

(9.12) (9.13)

where M I D is the center point of the search space, the QRS represent the QuasiReflection of the solution x under consideration. The third OBL approach is referred to Quasi-Opposite. A quasi-opposite solution (QOS) is defined as a solution randomly generated between the solution’s opposite and center point, utilizing Eq. (9.14). Q O Sn = M I Dn + (M I Dn − O Sn ) × rand

(9.14)

In the initialization phase, the last OBL strategy is the Super-Opposite solution (SOS), which is defined as follows. ( S O Sn =

) ( O Sn + ( xnmax − O Sn) × rand O Sn ≥ M I Dn other wise xnmin + O Sn − xnmin

(9.15)

All the aforementioned solutions create and combine into a single population. Then, Npop of the best solution is chosen based on their objective function as the initialization population of the algorithm. Step 2: Partitioning the population This phase divides the algorithm’s overall population into m subpopulations using the shuffling technique. To do this, the population is sorted based on their objective function. Then, the first m solutions from the whole population (i.e. NP) are chosen and randomly allocated to each subpopulation. Following this, the second m members

9.4 SORA-DESSOA

229

are selected from the remainder of NP and randomly assigned to each subpopulation as the second members [7]. This process is repeated until all solutions have been assigned to subpopulations. Step 3: Generating a new solution There are two distinct techniques for generating new solutions in the ESSOA with different probabilities as follows: ( xinew

=

the basic method rand < 0.8 (9.16) Statistically Regeneration Mechanism(S R M) other wise

To generate new solutions in each subpopulation using the basic technique, first, a better and worse solution based on objective function from the same subpopulation of the considered solution is randomly selected. Then the step size for each agent is computed as follows: ( ( ) ) stepsi zei,n = α × xs,n − xi,n × rand1 + β × x h,n − xi,n × rand2 α = αmax × (1 − t); t =

iter Maxiter

β = βmin + (βmax − βmin ) × t

(9.17) (9.18) (9.19)

where xi,n represents the nth variable of the considered agent; xs,n and x h,n indicate the nth variables of the worst and best agents, respectively, which are randomly selected for each xi in the same subpopulation; rand1 and rand2 are random number generators that generate values between 0 and 1; iter and Maxiter represent the current iteration and the maximum number of iterations, respectively; αmax , βmax , andβmin are the algorithm’s user-defined parameters. It should be noted that if no better agent exists than those examined, the second part of the step size is eliminated. Additionally, if there is no agent that is worse than the agents that are considered, the first part of the step size is ignored. Following the step size calculation, the following equation is used to determine the new position of each solution in the basic method: xinew = stepsi zei + xi

(9.20)

In the SRM, to begin, the average and standard deviation of population objective values in each subpopulation are determined. Then the twenty percent of the chosen agent position is alternated as the following manner: new xi,n = U N F I R AN D(Mean n − stdn − sigman , Mean n + stdn + sigman ) (9.21)

230

9 An Efficient ESSOA for the Reliability Based Design Optimization …

where U N F I R AN D is the operator that produces a random number from a continuous uniform distribution with lower and upper bounds are defined by Mean n − stdn − sigman and Mean n + stdn + sigman ; Mean n and stdn are the average and standard deviation of the nth variable of subpopulation which xi is a member. sigman is a parameter that facilitates the Statistically Regeneration mechanism operation when the whole population converges to the specified value and has a different value for each subpopulation. ( sigman =

) ) ( ( 0.01 × xnmax − xnmin i f stdn < 0.01 × xnmax − xnmin 0 other wise

(9.22)

Step 4: Evaluating and implementing replacement strategy In this step, the newly developed solutions are evaluated. If the new solution is better to the old one, the existing solution will be abandoned. If not, the current solution will be maintained. Step 5: Merge all subpopulations In order to share information among subpopulations, they are combined. Consequently, a single population is created. Shuffling the population and creating new subpopulations ensures that the knowledge acquired individually in one subpopulation is shared with other subpopulations. Step 6: Checking termination condition When the number of iterations approaches the maximum value, the algorithm terminates. Otherwise, the process returns to Step 2 for the subsequent repetition.

9.5 Numerical Examples First, two benchmark examples investigated in this section include the mathematical example, and the design of the speed reducer, then two structural benchmark examples include the design of the 10-bar truss and the design of the 72-bar truss, are examined. These examples are based on previous studies using gradient-based methods, so they help to illustrate the proposed method’s capabilities. Moreover, to demonstrate the superiority of the proposed method, two examples are presented in which the gradient-based method cannot be used owing to the higher degree of freedom. The parameters used in the ESSOA are the same as the main article of this method. However, the maximum number of iterations differs across examples; in the first benchmark example, the Maxiter is set to 50, while in the other benchmark examples, it is set to 150. Finally, the Maxiter is set to 400 in the new examples. The ε is set to

9.5 Numerical Examples

231

10–5 as the second termination condition. 30 independent runs are performed to get statistically meaningful results. In the last two examples, there is no result by using the other algorithms. Hence, to have comparable results, the capability of the Teaching Learning-Based Optimization algorithm (TLBO) [8], one of the well-known optimization methods in structural optimization, is investigated in the RBDO using the SORA-DM framework. In TLBO, the maximum number of function evaluations and population size are considered identical values as used in the ESSOA.

9.5.1 Benchmark Examples This section examines two benchmark examples, including the mathematical example and the design of the speed reducer.

9.5.1.1

The Mathematical Examples

The first example is a mathematical problem whose objective and constraint functions are nonlinear. This example includes two random design variables x = {X 1 , X 2 } with a normal distribution and}a standard deviation of 0.3. The average value of design { variables μx = μx1 , μx2 are considered as optimization variables. The problem is characterized as follows: } { Find : μx = μx1 , μx2

)2 )2 ( ( μx1 − μx2 + 10 μx1 + μx2 − 10 − Minimi ze : f (μx ) = − 30 120 ( t) Subjected to : P R O B(gi (x) ≤ 0) ≤ Φ βi i = 1, 2, 3 X 12 X 2 −1 20 g2 (x) = 1 − (Y − 6)2 − (Y − 6)3 + 0.6 × (Y − 6)4 − Z 80 ) −1 g3 (x) = ( 2 X 1 + 8X 2 + 5 W her e : g1 (x) =

Y = 0.9063X 1 + 0.4226X 2 Z = 0.4226X 1 − 0.9063X 2 β1t = β2t = β3t = 3.0 1 ≤ μx1 , μx2 ≤ 10

(9.23)

Table 9.1 compares the outcomes obtained by SORA-DESSOA and different RBDO techniques. As can be seen, the result found by SORA-ESSOA is the same as the other RBDO techniques, which need gradient information at least for reliability

232

9 An Efficient ESSOA for the Reliability Based Design Optimization …

Table 9.1 Comparison of the RBDO methods in the mathematical examples Reliability analysis needing gradient information

Reliability analysis using the global optimization method

Design variable

PMA [9] SORA [9]

AH-SLM NDL-IDE ESORA-IDE Present work [10] [11] [11] (SORA-DESSOA) [1]

d1

4.5581

4.5581

4.5581

4.5581

4.5581

4.5581

d2

1.9645

1.9645

1.9645

1.9645

1.9645

1.9645

Objective

−1.7247

−1.7247 −1.7247

−1.7247

−1.7247

−1.7247

NFE in the optimization part

−

−

−

−

−

6260

NFE in reliability analysis part

−

−

−

−

−

21,600

Total NFE

2160

853

254

256,967

7104

27,860

Average of objective

−

−

−

−

−

−1.7247

The standard − deviation of objective

−

−

−

−

3.269E-05

assessment. This demonstrates that the present method has the capability to find the optimum solution in the RBDO problems. Due to using the metaheuristic as a global optimizer, the total number of function evaluations (NFE) is greater than other RBO methods except for NDL-IDE. NDL-IDE found the best result in the 256,967 function evaluation, but the function evaluation of the SORA-DESSOA is 27860, which is much less than that of the NDL-IDE. As seen in Fig. 9.3, the SORA-DESSOA requires five cycles to find the optimum solution in this example.

9.5.1.2

The Speed Reducer Design

The second example is the design of a speed reducer by considering the probability constraints. This example has 7 design variables x = {X 1 , X 2 , . . . , X 7 } with a normal distribution and a standard deviation of 0.005. The objective function is the total weight of the speed reducer and include the 11 probability constraint. Like the previous example, the average value of the design variables is selected as optimization variables. The mathematical expression for the speed reducer design is as follows: Find : μx =

} { μx1 , μx2 , μx3 , μx4 , μx5 , μx6 , μx7

9.5 Numerical Examples

233

Fig. 9.3 Convergence history of the optimization part in the best run for mathematical example

( ) Minimi ze : f (μx ) = 0.7854μx1 μ2x2 3.3333μ2x3 + 14.9334μx3 − 43.0934 ( ) ( ) ( ) − 1.508μx1 μ2x6 + μ2x7 + 7.477 μ3x6 + μ3x7 + 0.7854 μx4 μ2x6 + μx5 μ2x7 ; ( ) Subjected to : P R O B(gi (x) ≤ 0) ≤ Φ βit i = 1, 2, 3, . . . , 11 27 W her e : g1 (x) = 1 − X 1 X 22 X 3 397.5 g2 (x) = 1 − X 1 X 22 X 32 g3 (x) = 1 − g4 (x) = 1 −

1.93X 43 X 2 X 3 X 64 1.93X 53 X 2 X 3 X 74 /(

754X 4 X2 X3

g5 (x) = 1100 − /( g6 (x) = 850 −

)2

+ 16.9 × 106

0.1X 63 )2 754X 5 + 157.5 × 106 X2 X3

g7 (x) = 40 − X 2 X 3 X1 −5 g8 (x) = X2 X1 g9 (x) = 12 − X2

0.1X 73

234

9 An Efficient ESSOA for the Reliability Based Design Optimization …

1.5X 6 + 1.9 X4 1.1X 7 + 1.9 g11 (x) = 1 − X5 t βi = 3.0 g10 (x) = 1 −

2.6 ≤ μx1 ≤ 3.6, 0.7 ≤ μx2 ≤ 0.8, 17 ≤ μx3 ≤ 28, 7.3 ≤ μx4 , μx5 ≤ 8.3, 2.9 ≤ μx6 ≤ 3.9, 5.0 ≤ μx7 ≤ 5.5

(9.24)

Table 9.2 provides a summary of the SORA-ESSOA and other RBDO technique outcomes. The objective of the best optimum result found by SORA-DESSOA is 3038.52, which is superior than other methods. In addition, the average value of the 30 independent runs of the present method is better than the best result of the other RBDO techniques. However, the function evaluation of the SORA-DESSOA is more than other methods due to using the metaheuristic in both the reliability assessment and optimization parts except NDL-IDE. Monte Carlo simulation (MCS) is performed to demonstrate that the reliability criteria are met. MCS with 107 samples is used to calculate the reliability index. The reliability index for the RBDO methods is present in Table 9.3. As can be seen, the SORA-DESSOA is satisfied all reliability constraints. Figure 9.4 indicates that SORA-DESSOA only needs two cycles to obtain the optimum solution.

9.5.2 Structural Benchmark Examples In this section, two of the well-known structural benchmarks in the area of the RBDO are investigated.

9.5.2.1

The 10-Bar Truss Design Problem

The ten-bar truss design problem is one of the well-known benchmarks in deterministic optimization. In addition, in the recent decade, researchers investigated this problem using probabilistic constraints. To this end, the RBDO of the 10-bar truss with the probabilistic constraint is investigated in this chapter. In this example, as shown in Fig. 9.5, the external forces of P are applied in the bottom nodes in the negative y-direction. The elements’ cross-sectional areas are selected as random design variables, and their average value is regarded as optimization variables. The applied forces and modulus of the elasticity (E) are assumed as random design parameters. The required information for the 10-bar truss is provided in Table 9.4. The objective function is the total weight of the truss, and the reliability constraint function is the deflection of node 2 in the y-direction. The mathematically expressed as follows: f ind : μ A =

{

μ A1 , μ A2 , . . . , μ A10

}

9.5 Numerical Examples

235

Table 9.2 Comparison of the RBDO methods in the speed reducer design Reliability analysis needing gradient information

Design variable

RIA + envelope function [12]

PMA + NDL-GA envelope [13] function [12]

Reliability analysis using the global optimization method

NDL-IDE ESORA-IDE Present study [11] [11] (SORA-DESSOA) [1]

d1

3.60

3.60

3.5767 3.5765

3.5765

3.5765

d2

0.70

0.70

0.7

0.7000

0.7000

0.7000

d3

17.0

17.2

17.0

17.0000

17.0001

17.000

d4

7.61

8.30

7.3

7.3013

7.3015

7.3000

d5

8.15

8.30

7.7550 7.7543

7.7546

7.7542

d6

3.43

3.58

3.366

3.3654

3.3654

3.3652

d7

5.50

5.45

5.3018 5.3017

5.3017

5.3015

Objective value

3207

3100

3038.73

3038.76

3038.52

3038.99

NFE in the − optimization part

−

−

−

−

6300

NFE in − reliability analysis part

−

−

−

−

101,900

Total NFE

5304

10,917

−

146,517

11,255

108,200

Average of objective

−

−

−

−

−

3038.55

The standard − deviation of objective

−

−

−

−

0.0219

minimi ze : f (μ A ) = ρ

10 Σ

Ai L i (lb)

i=1

( ) Subjected to : P R O B(g(x) ≤ 0) ≤ Φ β t | y | W her e : g(μ A , P, E) = 2 in − |u (μ A , P, E)| 2

0.1 in 2 ≤ μ A ≤ 35 in 2 β t = 3.0

(9.25)

Table 9.5 compares results of SORA-DESSOA with other available methods. The optimum result found by the SORA-DESSOA is significantly smaller than other techniques, demonstrating that its performance is superior than gradient-based methods.

Infinite

Infinite

SORA-DESSOA [1]

Infinite

Infinite

Infinite

Infinite

NDL-IDE [11]

Infinite

Infinite

NDL-GA [13]

ESORA-IDE [11]

−

−

−

−

RIA + envelope function

β2MC S

β1MC S

PMA + envelope function

RBDO method

Infinite

Infinite

Infinite

Infinite

−

−

β3MC S

Infinite

Infinite

Infinite

Infinite

−

−

β4MC S

3.00

3.00

3.00

3.15

−

−

β5MC S

3.00

3.00

3.00

3.02

−

−

β6MC S

Table 9.3 Reliability index of constraint function of different RBDO methods in Example 1

Infinite

Infinite

Infinite

Infinite

−

−

β7MC S

3.00

3.00

3.00

3.00

−

−

β8MC S

Infinite

Infinite

Infinite

Infinite

−

−

β9MC S

Infinite

Infinite

Infinite

Infinite

−

−

MC S β10

3.03

3.05

3.01

3.09

−

−

MC S β11

236 9 An Efficient ESSOA for the Reliability Based Design Optimization …

9.5 Numerical Examples

237

Fig. 9.4 Convergence history of the optimization part in the best run for speed reducer design Fig. 9.5 Schematic and loads of the 10-bar truss structure

Table 9.4 Simulation data for 10-bar truss structure Variable

Description

Type

Mean or deterministic value

Distribution

Coefficient of variation

Ai

Cross-sectional area

Random design variables

Founded by the optimization process

Normal

0.05

E

Elasticity modulus

Random design parameter

107 psi

Normal

0.05

P

Applied forces

Random design parameter

105 lb

Normal

0.05

ρ

Material density Deterministic

0.1 lb/in3

−

−

238

9 An Efficient ESSOA for the Reliability Based Design Optimization …

In addition, the average value of the 30 independent runs of the SORA-DESSOA is less than the best results of the other RBDO methods considered in this chapter. Like the previous example, to verify the result obtained, the MCS is applied. As shown in Table 9.5, the reliability index of the obtained result is satisfied the reliability requirement. To obtain the optimum solution, the present method needed seven cycles, as shown in Fig. 9.6. Table 9.5 Comparison of the RBDO methods in the 10-bar truss

Design variable A1

Reliability analysis needing gradient information

Reliability analysis using the global optimization method

NDL-GA [13]

34.352

NDL-IDE [11]

ESORA-IDE [11]

Present study (SORA-DESSOA) [1]

34.999

34.999

34.9905

A2

0.1

0.1

0.1

0.1047

A3

29.683

27.8086

27.9481

27.2906

A4

26.275

19.8943

19.8276

19.5793

A5

0.1

0.1

0.1

0.1001

A6

0.1

0.1

0.1

0.1005

A7

3.337

3.4353

3.3929

3.3577

A8

28.354

29.2083

29.1896

29.3749

A9

26.138

28.4209

28.4286

28.3492

A10 Objective value

0.1 6211.30

0.1

0.1

0.1

6102.04

6102.02

6072.83

NFE in the optimization part

−

−

−

28,700

NFE in reliability analysis part

−

−

−

85,720

Total NFE

−

2,943,447

81,944

114,420

β MC S

3.01

3.1464

3.1464

3.02

Average of objective

−

−

−

6085.44

The standard deviation of objective

−

−

−

10.41

9.5 Numerical Examples

239

Fig. 9.6 Convergence history of the optimization part in the best run for the 10-bar truss design problem

9.5.2.2

The 72-Bar Truss Design Problem

The fourth example is the 72-bar truss design problem. This structure includes 20 nodes and 72 members, as shown in Fig. 9.7. Like the previous example, the objective function is the overall weight of the truss, but there are two probability constraints in this example. The probability constraints are the displacement of node 1 in the x and y-direction. The structure is subjected to the single load condition that is applied in all directions of node 1. The cross-sectional area of the members is selected as random design variables, and the applied forces and elasticity modulus are considered as random design parameters. Due to the symmetries of the structure, the members are separated into 16 groups, hence there are 16 random design variables. The average value of the random design variables is assumed as optimization variables. The mean value of the cross-sectional area is selected from the discrete set of S = {0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 1.7, 1.9, 2.1, 2.3, 2.5, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7, 3.9, 4.1, 4.3, 4.5}. The required information is given in Table 9.6 and mathematically expressed in Eq. 9.26. } { f ind : μ A = μ A1 , μ A2 , . . . , μ A16 minimi ze : f (μ A ) = ρ

72 Σ i=1

Ai L i (lb)

( ) ( ) Subjected to : P R O B g j (x) ≤ 0 ≤ Φ β tj j = 1, 2 | y( ( ) )| y y W her e : g1 μ A , P1x , P1 , P1z , E = 0.3in − |u 1 μ A , P1x , P1 , P1z , E | | ( ( ) )| y y g2 μ A , P1x , P1 , P1z , E = 0.3in − |u 1x μ A , P1x , P1 , P1z , E |

240

9 An Efficient ESSOA for the Reliability Based Design Optimization …

β1t = β2t = 3.0

(9.26)

Fig. 9.7 Schematic of the 72-bar truss structure

Table 9.6 Simulation data for the 72-bar truss structure Variable

Description

Type

Mean or deterministic value

Distribution

Coefficient of variation

Ai

Cross-sectional area

Random design variables

Founded by the optimization process

Normal

0.05

E

Elasticity modulus

Random design parameter

107 psi

Lognormal

0.05

P1x

Applied forces in the x-direction of first node

Random design parameter

5 kip

Lognormal

0.1

P1

y

Applied forces in the y-direction of first node

Random design parameter

5 kip

Lognormal

0.1

P1z

Applied forces in the z-direction of the first node

Random design parameter

-5 kip

Lognormal

0.1

ρ

Material density deterministic

0.1 lb/in3

−

−

9.5 Numerical Examples

241

Table 9.7 provides the comparison results of the SORA-DESSOA with other methods which need gradient information in reliability analysis. The results indicated that the SORA-DESSOA could identify the best solution, similar to NDL-IDE and ESORA-IDE, and superior than NDL-GA. However, the total number of function evaluations is fewer than the gradient-based method. This superiority is indicated that the present method has better performance than the gradient-based method when the number of structural components rises. To verify the found results, the MCS with 107 samples is applied. The MCS results show that the reliability index in limit state functions is more than 3. As shown in Fig. 9.8, SORA-DESSOA is needed the two cycles to found the best solution.

9.5.3 New Examples In this section two new examples are investigated that can be utilized as new benchmark examples for future research.

9.5.3.1

The 120-Bar Dome Truss Design Problem

The 120-bar dome truss is one of the well-known benchmark structures in the deterministic optimization investigated by many researchers [2]. In this chapter, for the first time, the probabilistic constraint function based on the displacement of the nodes is used in this structure. These constraint functions are imposed on the displacement of nodes 1, 2, 14, and 15. The elements of the truss are organized into seven classes, as shown in Fig. 9.9. Therefore, there are seven random design variables that their mean value is considered as optimization variables. The only random design parameter is the elasticity modulus, and the loading of the truss is assumed as a deterministic value. Table 9.8 provides the loading conditions and needed data, and the design problem is stated as follows: { } f ind : μ A = μ A1 , μ A2 , . . . , μ A7 minimi ze : f (μ A ) = ρ

120 Σ i=1

Ai L i (lb)

( ) ( ) Subjected to : P R O B g j (x) ≤ 0 ≤ Φ β tj j = 1, 2, 3, 4 | | W her e : g1 (μ A , E) = 0.2in − |u 1Z (μ A , E)| | | g2 (μ A , E) = 0.2in − |u 2Z (μ A , E)| | Z | g3 (μ A , E) = 0.2in − |u 14 (μ A , E)| | | g4 (μ A , E) = 0.2in − |u Z (μ A , E)| 15

0.775in 2 ≤ μ A ≤ 20in 2

242

9 An Efficient ESSOA for the Reliability Based Design Optimization …

Table 9.7 Comparison of the RBDO methods in the 72-bar truss Design variable

A1

Reliability analysis needing gradient information

Reliability analysis using the global optimization method

NDL-GA [13]

0.1

NDL-IDE [11]

ESORA-IDE [11]

Present study (SORA-DESSOA) [1]

0.1

0.1

0.1

A2

0.9

0.7

0.7

0.7

A3

0.3

0.5

0.5

0.5

A4

0.5

0.7

0.7

0.7

A5

1.1

0.9

0.9

0.9

A6

0.9

0.7

0.7

0.7

A7

0.1

0.1

0.1

0.1

A8

0.1

0.1

0.1

0.1

A9

1.5

1.7

1.5

1.7

A10

0.5

0.7

0.7

0.7

A11

0.3

0.1

0.1

0.1

A12

0.3

0.1

0.1

0.1

A13

2.3

2.3

2.5

2.3

A14

0.9

0.7

0.7

0.7

A15

0.1

0.1

0.1

0.1

A16 Objective value

0.1 535.79

0.1

0.1

0.1

492.8687

492.8687

492.8687

NFE in the optimization part

−

−

−

6300

NFE in reliability analysis part

−

−

−

37,840

11,462,892

174,266

44,140

Total NFE β1MC S β2MC S

2.95

3.0397

3.0368

3.01

2.96

3.0397

3.0368

3.00

Average of objective

−

−

−

495.4266

The standard deviation of objective

−

−

−

8.8847

9.5 Numerical Examples

243

Fig. 9.8 Convergence history of the optimization part in the best run for the 72-bar truss design problem

β1t = β2t = β3t = β4t = 3.0

(9.27)

The result of the SORA-DESSOA and SORA-DTLBO are presented in Table 9.9. The best, mean, and standard deviation of the solutions discovered by SORADESSOA are better than the latter method. The SORA-DESSOA is required 49,020 function evaluation in the optimization part and 72,780 function evaluation in the reliability analysis part to find the optimum solution. The SORA-DTLBO requires the same NFE as the SORA-DESSOA in the optimization part. However, the required NFE for reliability analysis is significantly more than that of SORA-DESSOA. The Monte Carlo simulation verifies the results by computing the reliability index. The reliability index for each constraint function is more than 3; this shows that all of the constraint functions are satisfied. The convergence history of the best solution in the optimization part of SORA-DESSOA is provided in Fig. 9.10.

9.5.3.2

The 272-Bar Transmission Tower Design Problem

The last example investigated in this study is a 272-bar transmission tower, as depicted in Fig. 9.11. For reliability-based optimization of this problem, five probabilistic constraints are considered. These limitations are put in the displacement of nodes 1, 2, 11, 20, and 29 at the Z-direction. The single deterministic load condition is assumed in all directions at nodes 1, 2, 11, 20, and 29. This structure consist of 28 element groups, which the details of element group and nodal coordinate are available in Ref. [2]. Similar to the previous example, the element groups’ cross-sectional is considered the random design variable, and elasticity modulus is assumed as a random design parameter. The necessary information is provided in Table 9.10, and

244

9 An Efficient ESSOA for the Reliability Based Design Optimization …

Fig. 9.9 Schematic of the 120-bar dome truss structure

the problem is stated as follows: } { f ind : μ A = μ A1 , μ A2 , . . . , μ A28 minimi ze : f (μ A ) =

272 Σ

( ) Ai L i m 3

i=1

( ) ( ) Subjected to : P R O B g j (x) ≤ 0 ≤ Φ β tj j = 1, 2, 3, 4, 5 | | W her e : g1 (μ A , E) = 0.02m − |u 1Z (μ A , E)| | | g2 (μ A , E) = 0.02m − |u 2Z (μ A , E)| | Z | g3 (μ A , E) = 0.02m − |u 11 (μ A , E)| | | g4 (μ A , E) = 0.02m − |u Z (μ A , E)| 20

9.5 Numerical Examples

245

Table 9.8 Simulation data for the 120-bar dome truss structure Variable

Description

Type

Mean or deterministic value

Distribution

Coefficient of variation

Ai

Cross-sectional area

Random design variables

Founded by the optimization process

Normal

0.05

E

elasticity modulus

Random design parameter

30,450 ksi

Normal

0.05

P1z

Applied forces in deterministic the z-direction of the first node

−13.49 kip

−

−

z P2−13

Applied forces in deterministic the z-direction of node 2 through 13

−6.744 kip

−

−

z P14−37

Applied forces in deterministic the z-direction of node 14 through 37

−2.248 kip

−

−

ρ

Material density

0.288 lb/in3

−

−

deterministic

Fig. 9.10 Convergence history of the optimization part in the best run for the 120-bar dome truss design problem

246

9 An Efficient ESSOA for the Reliability Based Design Optimization …

Table 9.9 Result of the SORA-DESSOA in the 120-bar dome truss design problem

Design variable

SORA-DTLBO [1]

SORA-DESSOA [1]

A1

2.2692

2.2942

A2

16.9854

17.1781

A3

6.6318

6.4836

A4

2.9090

2.9592

A5

11.5758

11.4084

A6

4.0656

4.0365

A7

2.2492

2.3002

objective

37,278.11

37,276.82

NFE in the optimization part

49,020

49,020

NFE in reliability analysis part

114,720

72,780

Total NFE

163,920

121,800

β1MC S β2MC S β3MC S β4MC S

3.01

3.01

3.01

3.01

Infinite

Infinite

Infinite

Infinite

Average of objective

37,285.75

37,282.28

The standard deviation of objective

9.3667

2.8442

| Z | g5 (μ A , E) = 0.02m − |u 29 (μ A , E)| 1000mm 2 ≤ μ A ≤ 16000mm 2 β1t = β2t = β3t = β4t = 3.0

(9.28)

The best total volume of the elements discovered by SORA-DESSOA is 126.3772 m2 , which is less than that of the SORA-DTLBO, as shown in Table 9.11. In SORADESSOA, the total number of function evaluations is 343700, which reliability analysis needs 85 percent of the total number of function evaluations. In contrast, due to a large number of degrees of freedom, the gradient-based method cannot be utilized. The absence of significant differences in the mean result of the 30 independent runs and the best solution demonstrates SORA-DESSOA’s ability to locate the optimal solution. Figure 9.12 shows that the best run of the SORA-DESSOA requires five cycles to find the best solution.

9.6 Concluding Remarks

247

Fig. 9.11 Schematic of the 272-bar transmission tower structure

9.6 Concluding Remarks This chapter presents a novel reliability-based design optimization framework utilizing a metaheuristic algorithm called SORA-DM based on sequential optimization and reliability assessment (SORA). SORA-DM employs a metaheuristic algorithm for both reliability analysis and optimization parts. For this purpose, first, to improve the ability of the metaheuristic in the reliability analysis, the new reliability analysis method named no constraint most probable point finder (NCMPPF) is developed. Then, a new termination condition based on the variation of the agents in the metaheuristic is given. Enhanced shuffled shepherd optimization algorithm is utilized as an optimization algorithm in SORA-DM to show the effeminacy of the SORA-DM and is called SORA-DESSOA.

248

9 An Efficient ESSOA for the Reliability Based Design Optimization …

Table 9.10 Simulation data for the 272-bar transmission tower structure Variable

Description

Ai

Cross-sectional Random design Founded by the Normal area variables optimization process

E

Elasticity modulus

Type

Mean or deterministic value

Random design 2 × 108 kn/m2 parameter

Distribution Coefficient of variation 0.05

Normal

0.05

Deterministic

−13.49 kip

−

−

P1,2,11,20,29 Applied forces in the y-direction at nodes 1, 2, 11, 20, and 29

Deterministic

−6.744 kip

−

−

z P1,2,11,20,29 Applied forces in the z-direction at nodes 1, 2, 11, 20, and 29

Deterministic

−2.248 kip

−

−

x P1,2,11,20,29 Applied forces in the x-direction at nodes 1, 2, 11, 20, and 29 y

To evaluate the efficacy of the proposed method, first, four examples which investigated in the previous studies using the gradient-based methods are considered. These examples include the mathematical example, the speed reducer design, the 10-bar truss design, and the 72-bar truss design. The results indicate that the SORADESSOA can find the optimal solution like the other RBDO method in mathematical examples and 72-bar truss design problems, but with less function evaluation than other methods in the 72-bar design problem. In addition, SORA-DESSOA can find a better solution in speed reducer design and 10-bar truss design problems. This shows the ability of SORA-DESSOA to find the optimum solution of different problems. Second, two new larger examples considered the probability constraints are introduced in this study. These examples include the 120-bar dome truss design and 272bar transmission tower design problems. Results indicate that, unlike the gradientbased methods, the SORA-DESSOA can find the optimum solution in structures with large degrees of freedom.

9.6 Concluding Remarks Table 9.11 Result of the SORA-DESSOA in transmission tower design problem

249 Design variable

SORA-DTLBO [1]

SORA-DESSOA [1]

A1

1002.2015

1004.0831

A2

1358.4475

1393.6120

A3

2688.5624

2765.4734

A4

1000.1794

1000.0193

A5

1039.8604

10,083.9829

A6

1002.3150

1004.2609

A7

13,608.5665

14,064.4696

A8

1000.4096

1004.5068

A9

1048.6049

1004.1627

A10

1005.1194

1016.4251

A11

12,186.9179

12,092.4269

A12

1000.0000

1001.0605

A13

1025.8815

1002.7102

A14

1026.9502

1019.0136

A15

10,363.6135

10,613.8956

A16

1000.0000

1002.4913

A17

1000.7452

1009.3588

A18

1029.7371

1000.6828

A19

9817.2968

9666.6551

A20

1003.2549

1000.0204

A21

1005.6673

1000.1768

A22

1000.0000

1006.5893

A23

9162.4131

9089.6045

A24

1000.1964

1000.8428

A25

1000.0056

1015.6872

A26

1021.4054

1000.4362

A27

8802.4064

8373.7283

A28

1000.000

1000.7069

Objective value

126.4966

126.3772

NFE in the optimization part

49,020

49,020

NFE in reliability analysis part

388,960

294,680

Total NFE

438,160

343,700

β1MC S

Infinite

Infinite (continued)

250

9 An Efficient ESSOA for the Reliability Based Design Optimization …

Table 9.11 (continued)

Design variable

SORA-DTLBO [1]

SORA-DESSOA [1]

β2MC S

Infinite

Infinite

β3MC S β4MC S β5MC S

3.01

3.01

Infinite

Infinite

4.12

4.11

Average of objective

127.5904

126.4660

The standard deviation of objective

1.5956

0.0509

Fig. 9.12 Convergence history of the optimization part in the best run for the 272-bar transmission tower design problem

References 1. Kaveh, A., Zaerreza, A.: A new framework for reliability-based design optimization using metaheuristic algorithms. Structures 38, 1210–1225 (2022) 2. Kaveh, A.: Advances in Metaheuristic Algorithms for Optimal Design of Structures. 3rd Edn, Springer (2021) 3. Yang, D., Yi, P.: Chaos control of performance measure approach for evaluation of probabilistic constraints. Struct. Multidiscip. Optim. 38(1), 83 (2008) 4. Ting Lin, P., Chang Gea, H., Jaluria, Y.: A modified reliability index approach for reliabilitybased design optimization. J. Mech. Design 133(4) (2011) 5. Jiang, C., Qiu, H., Gao, L., Cai, X., Li, P.: An adaptive hybrid single-loop method for reliabilitybased design optimization using iterative control strategy. Struct. Multidiscip. Optim. 56(6), 1271–1286 (2017) 6. Du, X., Chen, W.: Sequential optimization and reliability assessment method for efficient probabilistic design. J. Mech. Des. 126(2), 225–233 (2004)

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7. Kaveh, A., Zaerreza, A., Hosseini, S.M.: An enhanced shuffled Shepherd optimization algorithm for optimal design of large-scale space structures. Eng. With Comput. 38(2), 1505–1526 (2021) 8. Rao, R.V., Savsani, V.J., Vakharia, D.P.: Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput. Aided Des. 43(3), 303–315 (2011) 9. Meng, Z., Yang, D., Zhou, H., Wang, B.P.: Convergence control of single loop approach for reliability-based design optimization. Struct. Multidiscip. Optim. 57(3), 1079–1091 (2018) 10. Xiong, F., Wang, D., Ma, Z., Chen, S., Lv, T., Lu, F.: Structure-material integrated multiobjective lightweight design of the front end structure of automobile body. Struct. Multidiscip. Optim. 57(2), 829–847 (2018) 11. Khodam, A., Mesbahi, P., Shayanfar, M., Ayyub, B.M.: Global decoupling for structural reliability-based optimal design using improved differential evolution and chaos control. ASCE-ASME J. Risk and Uncertainty in Eng. Syst. Part A: Civil Eng. 7(1), 04020052 (2021) 12. Lee, J.J., Lee, B.C.: Efficient evaluation of probabilistic constraints using an envelope function. Eng. Optim. 37(2), 185–200 (2005) 13. Shayanfar, M., Abbasnia, R., Khodam, A.: Development of a GA-based method for reliabilitybased optimization of structures with discrete and continuous design variables using OpenSees and Tcl. Finite Elem. Anal. Des. 90, 61–73 (2014)

Chapter 10

Reliability-Based Design Optimization of the Frame Structures Using the ESSOA and ERao

10.1 Introduction The reliability-based design optimization (RBDO) of the frame structures using the force method and sequential optimization and reliability assessment-double metaheuristic framework (SORA-DM), instigated by Kaveh and Zaerreza [1], is presented in this chapter. In the SORA-DM, the meta-heuristic method is used for both the optimization process and analysis of reliability. The statical indeterminacy of the examined frames is lower than their kinematic indeterminacy, so the force method is used for structural analysis. The force method is used for the first time in the structural analysis of the RBDO problems. In most of structural optimization problems, constraints are defined as deterministic values. However, it is generally accepted that structural systems always include some uncertainty owing to variances in material properties, imprecise characterization of the loading environment, and manufacturing tolerances, all of which contribute to the overall uncertainty. The Reliability-Based Design Optimization (RBDO) is implemented to address this uncertainty. Three categories comprise the RBDO: double-loop, single-loop, and decoupled approach. Techniques using a double loop incur larger computing costs than other methods. Single-loop approaches are inappropriate for complex nonlinear issues. Hence, this chapter employs the decoupled strategy, which has none of the disadvantages of the other methods. There are several ways to handle RBDO problems. In the majority of the RBDO techniques, the gradient-based approach or other iterative techniques are used to perform reliability assessment. Kaveh and Zaerreza [2] developed the sequential optimization and reliability assessment-double meta-heuristic (SORA-DM). In this framework, both optimization and reliability assessment is carried out utilizing metaheuristic methods. To this end, this framework is used in this chapter to handle the RBDO problems. The displacement and force methods are the two well-known structural analyzing techniques. The computing time needed by these approaches is proportional to the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Kaveh and A. Zaerreza, Structural Optimization Using Shuffled Shepherd Meta-Heuristic Algorithm, Studies in Systems, Decision and Control 463, https://doi.org/10.1007/978-3-031-25573-1_10

253

254

10 Reliability-Based Design Optimization of the Frame Structures Using …

number of equations that must be solved in order to calculate the stress or displacement of the nodes. The number of equations depends on the degrees of kinematical indeterminacy (DKI) and statical indeterminacy (DSI). The DKI and DSI values reflected the number of the equations to be solved by the displacement and force methods, respectively. Although the time difference is negligible for a single analysis, it increases over the optimization process as a result of several structural analyses. The structures considered in this study have lower DSI than DKI. Hence the force method is faster than the displacement method. Moreover, Kaveh and Zaerreza [3] indicate the efficacy of the force approach on the structures investigated in this chapter. For this purpose, the force method is utilized as the structural analysis method. In this chapter, the effectiveness of the SORA-DM in the RBDO of the frame structure is explored for the first time. Additionally, the first-time force method is used in the RBDO of the frame structures. This chapter examines three standard algorithms and their enhanced variants: Shuffled Shepherd Optimization Algorithms (SSOA), Rao-1, and Rao-2 algorithms. These standard and advanced algorithms are used for the first time in the RBDO of frame structures. Moreover, the considered structures are previously only used in deterministic optimization problems. However, the probabilistic constraints are considered here. The results demonstrated that the SORA-DM framework is appropriate for the RBDO of the frame, and the improved algorithms have high performance in the RBDO problems. The rest of the chapter is organized as follows: the force method is described briefly in Sect. 10.2. In Sect. 10.3, the SORA-DM framework is presented. Three standard optimization algorithms and their enhanced version is provided in Sect. 10.4. The results of the three frame structures are presented in Sect. 10.5. Finally, the conclusion is given in Sect. 10.6.

10.2 The Force Method of Structural Analysis There are several types of force methods, including the topological force method, integrated force method, algebraic force method, and graph theoretical force method. The graph-theoretical force method is simpler to implement than other force methods, and its resulting flexibility matrix is sparser. As a consequence, this chapter applies the graph-theoretical force method. Considered the structure with γ time statically independent. In order to obtain the stress of the member using Eq. (10.1), the γ independents unknown are eliminated from the structure. r = B0 p + B1 q

(10.1)

where r represents the stress of the members, p represents the joint loads; q represents the forces of redundants; B 0 and B 1 are rectangular matrices with m rows and n and

10.3 RBDO Framework

255

γ columns, respectively; n represents the number of joint load components, and m represents the number of independent member components. Unknown is the force of redundants in Eq. (10.1). Therefore, the link between load and displacement and the virtual work concept are used to exclude q from Eq. (10.1). The Eq. (10.1) is rewritten as shown below: ] [ ( )−1 v 0 = B t0 F m B 0 − B 0 F m B 1 B t1 F m B 1 B t1 F m B 0 p

(10.2)

] [ ( )−1 r = B 0 − B 1 B t1 F m B 1 B t1 F m B 0 p

(10.3)

where the v 0 represents the displacement associated with the force components of p, F m is the unassembled flexibility matrix, G = B t1 F m B 1 is known as the flexibility matrix of the structure. In different variants of the force method, the B 0 and B 1 matrices are generated in various ways. B 0 matrix is constructed by generating the spanning forest from structural supports using the graph-theoretical force method. Calculating each submatrix of the B 0 by transferring each joint load to a support node. More information is available in the references [3, 4]. For the form of the B 1 , the set of the cycle basis is necessary. There are several algorithms for finding the cycle basis. However, the Kaveh’s methods generate a sparser matrix than other techniques. After generating the cycle basis using the Kaveh methods, one element of each cycle is cut at its initial node, and six bi-actions are applied. In the B 1 sub-matrix, the columns indicate the internal forces at the lower-numbered end of the ith member when six bi-actions are applied at the jth cut. Additional information is provided in Ref. [4].

10.3 RBDO Framework Sequential optimization and reliability assessment-double meta-heuristic (SORADM) is the new framework developed by Kaveh and Zaerreza [2]. In this framework, the no constraint most probable point finder (NCMPPF) objective function is presented for the reliability assessments using the metaheuristic algorithms. The NCMPPF improves the accuracy of the most probable point (MPP) found compared to other objective functions for reliability assessments. In addition, the termination condition is introduced to both the reliability and optimization portions of the framework to prohibit the evaluation of the algorithm’s functions unnecessarily.

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10 Reliability-Based Design Optimization of the Frame Structures Using …

10.3.1 No Constraint Most Probable Point Finder All of the MPPs are not found by meta-heuristic algorithms when the NCMPPF is used as the objective function. One of the MPPs is found in the objective function. Consequently, one of the random variables is eliminated from the optimization algorithm’s variable. The procedures of the NCMPPF are outlined as follows. Step 1: Find the power two of the removed variable Using Eq. (10.4), the power two of the eliminated random variable in the standardized normal space is determined. V r 2 = β 2 − V oT × V o

(10.4)

where the V r 2 is the power two of the removed random variable in the standardized normal space; β 2 is the power two of the reliability index; V o is the vector of the transformation of the random variables into the independent standardized normal space. Step 2: Check the value of the V r 2 If the value of the V r 2 is the negative, the value of the V oT × V o × 109 is considered as an objective function value. The process of the NCMPPF is stopped, and the considered value is a return to the optimization algorithm. Otherwise, the value of the V r is computed using the following equations. V r1 =

√

Vr2

√ V r2 = − V r 2

(10.5) (10.6)

where V r1 and V r2 represent the removed random variable’s value in the standardized normal space. One of these values represents the real value, which will be confirmed in the next step. Step 3: Calculate the objective function The value of the limit state function using both random variables found is calculated. The value of the limit state function is computed using V r1 and V o together, as well as the V r2 and V o. From these limit state functions, the one which has a lower value is regarded as an objective function value, and related random variables in the standardized normal space are considered as the MPP. Algorithm 10.1 provides the pseudo-code of the NCMPPF for form detail.

Algorithm 10.1. Pseud-code of the NCMPPF

10.3 RBDO Framework

257

Input V o Output V r and objective function value Calculate the V r 2 using Eq. (10.4) If V r 2 < 0 then V r = in f init y objective function value = V oT × V o × 109 return End If Calculate the V r1 and V r2 Calculate the value of the limit state function using V r1 and V o Calculate the value of the limit state function using V r2 and V o If the limit state function value using V r1 lower than using V r2 then V r = V r1 objective function value = the limit state function value using V r1 Return Else V r = V r2 objective function value = the limit state function value using V r2 Return End If

10.3.2 Termination Condition The termination condition is used in this research based on the convergence of the best design variable, worst design variable, and the mean value of the design variables. When this condition is met, optimization procedures are terminated since there is no prospect of identifying a new solution. The following are the stages of the termination condition. Step 1: Calculate the convergence of the best and worst variable In order to calculate the convergence of the best and worst variables, Eqs. (10.7) and Eq. (10.8) are used. O B k − O B k−1 ≤ε O Bk

(10.7)

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10 Reliability-Based Design Optimization of the Frame Structures Using …

O W k − O W k−1 ≤ε OWk

(10.8)

where O W and O B represent the objective function of the worst and best variable, respectively; k is the current iteration number of the optimization algorithm; ε is the tolerance for defining convergence, which is equal to 10–5 in this chapter. Step 2: Modifying the convergence counter If either of Eqs. (10.7) and (10.8) are not satisfied, the convergence counter (CC) is reset to zero and the convergence checking procedure is ended. If not, the method proceeds to the subsequent step. Step 3: Calculate the convergence of the population To calculate the convergence of the whole population, the mean value of the objective function of the entire population is considered and calculated using Eq. (10.9). O M k − O M k−1 ≤ε O Mk

(10.9)

where the O M is the mean value of the objective function of the whole population. Step 4: Modifying the convergence counter and checking the termination If Eq. (10.9) is not met, then the convergence counter (CC) is set to zero and the convergence checking procedure is ended. If not, the one is added to the CC. If the value of the CC is fifty, there is no substantial change in the objective function value for the whole population. Consequently, the optimization method is terminated.

10.3.3 SORA-DM Framework In the SORA-DM framework, the reliability analysis and the optimization part are separated from each other. As a result, the shifting vector is used to handle the probabilistic restriction. Also, the NCMPPF and the termination condition described in the preceding sections are utilized. The SORA-DM framework includes the following steps: Step 1: Define the parameters The parameters of the optimization method are specified, such as the maximum number of iterations (MNI) and the population size. The MNI used for reliability analysis is three times that used for optimization in the SORA-DM framework. Step 2: Define the design variables The three design variables in the RBDO problems include the deterministic design variable (D), random design variable (X ), and random design parameters (P). The value of the D and mean value of the X (μx ) are determined in the optimization part.

10.3 RBDO Framework

259

Their values are initialized to zero at the beginning of the framework. The mean value of the P (μ p ) is defined by the user at the start of the framework. In addition, the MPP of the X and P is set to equal their value. Step 3: Calculate the shifting vector The shifting vector is only calculated for the random design variables using Eq. (10.10). SV C N = μCX N −1 − M P PXC N −1

(10.10)

where the SV C N is the shifting vector in the counter of the CN; μCX N −1 is the mean value of the random design variable in the counter of the CN; M P PXC N −1 is the most probable point found for the random design variables in counter of the CN. Step 4: Optimization part In the optimization phase, the metaheuristic algorithm is used to determine the D and μx values. The termination condition described in Sect. 10.3.2 is also added to the optimization method. In order to calculate the objective function, the MPP of the P (M P Pp ) and the value determined by the optimization techniques are used. However, in order to check the constraint function, the value of the random design variables is shifted. The constraint function’s first input is the D determined by the algorithms. The second input is the μx − SV C N (shifted random design variables), and the last input is the M P Pp . Step 5: Reliability assessment part After finishing the optimization phase, the reliability assessment is conducted. The metaheuristic algorithm employs the NCMPPF objective function to determine the most probable point for the random design variables and parameters. The termination condition is added to the metaheuristic algorithm, as in the previous phase. Step 6: Checking the termination condition The objective function value obtained during the last iteration of the reliability analysis is used to validate the solution. If the objective function’s value is negative, the function’s identified solution is not valid. In contrast, it suggests that the discovered solution for this function is valid. If all of the values of the objective function are valid, the process is ended. Nevertheless, just one of the functions is not met. The procedure advances to the subsequent counter, and the process from Step 3 is repeated. Before the start of the subsequent counter, the MNI for both the reliability analysis and optimization sections is raised by 10%. Figure 10.1 illustrates SORA-DM flowchart.

260

10 Reliability-Based Design Optimization of the Frame Structures Using …

Fig. 10.1 Flowchart of the SOR-DM framework

10.4 Optimization Algorithms

261

10.4 Optimization Algorithms In this chapter, six optimization methods are examined. Three of these algorithms are standard algorithms, while the other three are enhanced versions of considered basic algorithms. These methods include the Shuffled Shepherd Optimization Algorithms (SSOA), Rao-1, Rao-2, Enhanced Shuffled Shepherd Optimization Algorithms (ESSOA), ERao-1, and ERao-2. The next section describes these algorithms.

10.4.1 Shuffled Shepherd Optimization Algorithms SSOA is the simple and reliable method introduced by Kaveh and Zaerreza [5]. SSOA begins with the randomly initialized population. Then the main loop of the SSOA starts. Initially, the whole population is separated into subpopulations in the main loop using the shuffling method. Then, the new position of each member in each subpopulation is computed using the Eq. (10.11). ( ( ) ) new Si,n = Si,n + α × Sw,n − Si,n × rand1 + β × Sb,n − Si,n × rand2

(10.11)

new where Si,n is the new position of the ith member of the nth subpopulation; Si,n is the current position of the ith member of the nth subpopulation; Sw,n , and Sb,n are the worst and best members of the nth subpopulation; rand1 and rand2 are random vector generators that produce values between 0 and 1; α and β are calculated using the following equations.

NI MNI

(10.12)

β = βmin + (βmax − βmin ) × t

(10.13)

α = αmax × (1 − t); t =

where αmax , βmax , and βmin are the user-defined parameters; NI is the current number of the iteration. If the newly created solutions are not inside the search space, their values are fixed to the search space’s border. Then, the new and old positions are compared based on their objective value, and the best positions are chosen for the subsequent optimization cycle. If the maximum number of iterations has not been achieved, the algorithm advances to the subsequent optimization cycle. Subpopulations are created, and the whole process is repeated. The pseudo-code of SSOA is supplied in Algorithm 10.2.

262

10 Reliability-Based Design Optimization of the Frame Structures Using …

Algorithm 10.2. Pseudo-code of SSOA Set the algorithm parameters: population size, M N I , αmax , βmax , and βmin Generate the initial population randomly in the search space NI = 0 While MNI < NI then Create the subpopulation using the shuffling method Calculate the new solution using Eq. (10.11) If new solutions generated are not in the search space then Stick the variable that is not in the search space to the closest boundary End If Select the best solution in comparison of the old and new solutions NI = NI + 1 End While

10.4.2 Rao Algorithms In this chapter, the two algorithms devised Rao [6] are explored. Rao algorithms are simple, and there is a minor difference in their main step size of them. These algorithms, like the SSOA, begin by producing a random population in the search space. In the Rao-1 algorithm, the new solution is calculated as follows. Sinew = Si + rand × (Sb − Sw )

(10.14)

where the Sinew , Si are the new and current position of the solution in the search space; Sb is the best solution of the population, and Sw is the worst solution of the population. After generating a new solution similarly to the SSOA, if the new solutions created are not inside the search space, their values are set to the nearest search space boundary, and the best of the new and old solutions is selected. If the MNI is not fulfilled, the algorithm advances to the next optimization cycle. Step size is the sole difference between the Rao-1 and Rao-2 methods. Using the Eq. (10.15) the new solution for the Rao-2 method is computed. { Sinew = Si + rand × (Sb − Sw ) +

rand × (|X i | − |X k |) i f X i better than X k rand × (|X k | − |X i |) i f X k better than X i (10.15)

10.4 Optimization Algorithms

263

where the X k is the solution randomly selected for the X i . Algorithm 10.3 provides the pseudo-code of the Rao algorithms for better clarification.

Algorithm 10.3 Pseudo-code of Rao algorithms Set the algorithm parameters: population size,M N I Generate the initial population randomly in the search space NI = 0 While MNI < NI then Calculate the new solution using Eq. (10.14). For Rao-1 algorithm Calculate the new solution using Eq. (10.15). For Rao-2 algorithm If new solutions generated are not in the search space then Stick the variable that is not in the search space to the closest boundary End If Select the best solution in comparison of the old and new solutions NI = NI + 1 End While

10.4.3 Enhanced Shuffled Shepherd Optimization Algorithms Statistically regenerated mechanism (SRM) and Opposition-Based Learning (OBL) are applied to the SSOA to improve its performance [7]. At the beginning of ESSOA, after randomly generating solutions, OBL methods are employed to generate new solutions. Opposite of the solutions, quasi-reflection, quasi-opposite, and superopposite of solutions are produced. The opposite of the solution is derived as follows. O Si = Smax + Smin − Si

(10.16)

where the O Si is the opposite of the Si ; Smax and Smin are the upper and lower bound of the search space, respectively. In order to obtain the quasi-reflection of the solution, first, the center of the search space is calculated using Eq. (10.17). Then the quasi-reflection of the solution is produced as follows. MP =

Smax + Smin 2

Q RSi = M P + (M P − Si ) × rand

(10.17) (10.18)

264

10 Reliability-Based Design Optimization of the Frame Structures Using …

where the M P represents the center of the search space, and Q RSi represents the quasi-reflection of the ith member. The solution randomly created between the O S and the center of the search space is the quasi-opposite solution (QOS) and computed using Eq. (10.19). Q O Si = M P + (M P − O Si ) × rand

(10.19)

The last OBL technique in the initialization phase is the Super-Opposite solution (SOS) which is defined as follows. { S O Si =

O Si + (Smax − O Si ) × rand O Si ≥ M P Smin + (O Si − Smin ) × rand other wise

(10.20)

After creating the solution using the OBL methods, all the solutions are mixed, and the best of them is chosen as the initialization population of the algorithms. Then the main loop of the ESSOA begins. The sole difference between the ESSOA and the SSOA is the main step size. The main step size of the ESSOA is divided into two parts. Eighty percent of the solutions are created using the same method in the SSOA, while the SRM is applied to the other 20% of the solution. In the SRM, 20% of variables of the considered solution are regenerated using Eq. (10.21). Sinew = U N F I R AN D(Mean n − stdn − sigman , Mean n + stdn + sigman ) (10.21) where U N F I R AN D is the operator that produces a random number from a continuous uniform distribution with lower and upper bounds limits given by Mean n − stdn − sigman and Mean n + stdn + sigman .; where Mean n is the average position of the member in the nth subpopulation; stdn is the standard deviation of the member in the nth subpopulation. sigman is the parameter that helps the SRM to perform perfectly and is specified by Eq. (10.22). The pseudo-code of the ESSOA is given in Algorithm 10.4. { sigman =

0.01 × (Smax − Smin ) i f stdn < 0.01 × (Smax − Smin ) 0 other wise

(10.22)

Algorithm 10.4 Pseudo-code of ESSOA Set the algorithm parameters: population size, M N I , αmax , βmax , and βmin Generate the initial population randomly in the search space Generate the opposite of the solutions Generate the quasi-reflection of the solutions

10.4 Optimization Algorithms

265

Generate the quasi-opposite of the solutions Generate the super-opposite of solutions Merge all the solutions that generate Select the best solution same size as the population size as the initial population NI = 0 While MNI < NI then Create the subpopulation using the shuffling method If rand < 0.8 then Calculate the new solution using Eq. (10.11) Else Calculate the new solution using Eqs. (10.21) and (10.22) End If If new solutions generated are not in the search space then Stick the variable that is not in the search space to the closest boundary End If Select the best solution in comparison of the old and new solutions NI = NI + 1 End While

10.4.4 Enhanced Rao Algorithms In order to improve the Rao algorithms, the modified statistically regenerated mechanism (MSRM) is applied to them. Additionally, the strategy that keeps the solutions in the search space to have acceptable results is enhanced [8]. Similar to the SRM, the 20% of the solution is chosen in the MSRM, and their 20% variables are updated using Eq. (10.21). However, the sole difference between the SRM and MSRM is in the sigman . In order for the SRM to work well in the Rao algorithms the value of the sigman is computed as follows. { sigman =

) ) ( ( ) ( ) ( NI it 0.05 × Smax − Smin × 1 − Max I t i f stdn < 0.05 × Smax − Smin × 1 − M N I 0 other wise

(10.23)

In the basic Rao algorithms, if any variables of the solution violate the search space, their value is fixed into the nearest the search space border. This mechanism causes the solution to get trapped at the search space boundary in the Rao algorithms. Hence, the change is applied to this mechanism. In the new strategy for keeping the solution in search space, the variables recreated in the search space with probability

266

10 Reliability-Based Design Optimization of the Frame Structures Using …

of the 50%. Otherwise, their variables stick to the search space boundary. Algorithm 10.5 provides the pseudo-code for the Rao algorithms for more clarity.

Algorithm 10.5 Pseudo-code of ERao algorithms Set the algorithm parameters: population size,M N I Generate the initial population randomly in the search space NI = 0 While MNI < NI then If rand < 0.8 then Calculate the new solution using Eq. (10.14). For Rao-1 algorithm Calculate the new solution using Eq. (10.15). For Rao-2 algorithm Else Calculate the new solution using Eqs. (10.21) and (10.23) End If If new solutions generated are not in the search space then If rand < 0.5 then Stick the variable that is not in the search space to the closest boundary Else Regenerate the variable that is not in the search space End If End If Select the best solution in comparison of the old and new solutions NI = NI + 1 End While

10.5 Numerical Examples In this section, the RBDO of the three well-known frame structures is explored. furthermore, the efficiency of the SORA-DM framework is investigated using the force method as the frame structural analysis. Therefore, the SORA-D prefix is appended to the names of the optimization algorithms. These examples include the 1-bay 10-story steel frame, 3-bay 15-story steel frame, and 3-bay 24-story steel frame. The maximum number of function evaluations and population size for both basic and enhanced algorithms is set to 1000 and 20, respectively.

10.5 Numerical Examples

267

10.5.1 The 1-Bay 10-Story Steel Frame The 1-bay 10-story steel frame is the first structure examined in this chapter. There are 30 members in the frame, which are divided into nine member groups. The modulus of elasticity is defined as random design parameter with a mean of 29,000 ksi, a coefficient of variation of 0.05, and a normal distribution. Figure 10.2 depicts the loading of the frame, which is regarded as a deterministic value. The beams’ cross-section is picked from 267 W-sections, while the cross-sections of the column are chosen from W 14 and W12 sections. The cross-sectional area and the members’ second moment are regarded as the random design variables. Algorithms found their average value by choosing the sections with the coefficient of variation of 0.05 and the normal distribution. Therefore, there are 9 variables in the optimization part and 19 variables in the reliability assessment section. The probabilistic constraint is the lateral displacement of the top story, which has a value less than 4.92 inches and a reliability index of 3. The results of the considered algorithms are presented in Table 10.1. In comparing the basic optimization algorithms, the SROA-DSSOA finds a superior solution than SORA-DRao-1 and SORA-DRao-2. In addition, the mean and standard deviation of SORA-DSSOA is better than those of other conventional methods. According to Fig. 10.3, SROA-DSSOA and SORA-DRao-1 are capable to conduct the reliability assessment correctly in each of thirty runs. In contrast, the SORA-DRao-2 operates successfully in just three out of thirty trails. SORA-DESSOA identifies the optimal solution in compared to the upgraded methods. In contrast, SORA-DERao-1 had a superior mean and standard deviation. Naturally, all the enhanced algorithms perform better than the corresponding standard ones, as expected. In addition, SORA-DERao-2 overcomes the limitations of SORA-DRao-2, which is able to do accurate reliability assessment in all the runs. All the enhanced algorithms perform the reliability assessment successfully in every run, as demonstrated in Fig. 10.4. SORA-DRao-1 and SORA-DERao-2 need two cycles of optimization to found the optimal weight, as demonstrated in Fig. 10.5. SORA-SSOA, SORA-ERAO-1, and SORA-DESSOA each need three optimization cycles; whereas, SORA-DRao-2 requires four cycles. The solutions are confirmed by calculating the reliability index using Monte Carlo Simulation (MCS) with 107 samples. The reliability index is more than 3 in all the six algorithms, indicating that the constraint function is fulfilled. In addition, it indicates that the SORA-DM framework can be applied to the RBDO of the frames.

10.5.2 The 3-Bay 15-Story Steel Frame As depicted in Fig. 10.6, he 3-bay, 15-story steel frame is the second problem evaluated by RBDO utilizing the SORA-DM. This frame is made up of 105 members and

268

10 Reliability-Based Design Optimization of the Frame Structures Using …

Fig. 10.2 The schematic of the 1-bay 10-story steel frame

65 joints, which are organized into 11 groups. The modulus of elasticity is considered as random design parameter with a mean of 29,000 ksi, a coefficient of variation of 0.05, and a normal distribution. The structural loading is considered as a deterministic value. The beam and column sections are picked from a pool of 267 W sections. Similar to the previous example, the cross-sectional area and the members’ second moment are defined as random design variables with a normal distribution and a coefficient of variation of 0.05. This example has 11 design variables and

2278.23

Standard deviation (lb) 1668.71 (3 of 30)

62,618.82 (3 of 30) 58,377.58

61,570.47

82,280

994.70

3.1569

Mean weight (lb)

89,340 3.0613

23,120

52,460

48,040

NFE in reliability analysis 22,780 part

59,160

55,904.56

W 6 × 8.5

W 27 × 94

W 33 × 118

W 40 × 149

3.3673

61,067.40 41,300

58,573.53

29,680

Best weight (lb)

NFE in the optimization part

W 14 × 48 W 14 × 48

Total NFE

W 12 × 14

W 8 × 10

9

W 14 × 120 W 14 × 90

βMCS

W 36 × 150 W 33 × 118

W 33 × 118

W 30 × 99

7

W 30 × 116

W 36 × 135

6

8

W 14 × 99 W 14 × 61

W 14 × 48

W 12 × 50

4

5

W 14 × 159 W 14 × 48

W 14 × 109

W 14 × 132

2

3

609.03

56,157.28

3.0186

111,560

45,360

66,200

54,605.34

W 14 × 30

W 30 × 90

W 33 × 118

W 36 × 135

W 14 × 48

W 14 × 48

W 14 × 99

W 14 × 120

W 14 × 145

904.23

56,223.77

3.0084

59,980

17,980

42,000

54,515.30

W 14 × 30

W 27 × 84

W 36 × 135

W 36 × 150

W 14 × 48

W 14 × 48

W 14 × 48

W 14 × 120

W 14 × 145

827.11

56,246.65

3.1562

93,000

26,800

66,200

54,382.72

W 16 × 31

W 27 × 94

W 33 × 118

W 40 × 149

W 14 × 48

W 14 × 48

W 14 × 48

W 14 × 132

W 14 × 145

W 14 × 159

W 14 × 193

W 14 × 159

SORA-DRao-1 [1] SORA-DRao-2 [1] SORA-DSSOA [1] SORA-DERao-1 [1] SORA-DERao2 [1] SORA-DESSOA [1]

Element group

1

Table 10.1 Comparative results of the standard and enhanced algorithms in the 1-bay 10-story steel frame

10.5 Numerical Examples 269

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10 Reliability-Based Design Optimization of the Frame Structures Using …

Fig. 10.3 The structural weight of each independent run of the standard algorithms for the 1-bay 10-story steel frame design problem

Fig. 10.4 The structural weight of each independent run of the enhanced algorithms for the 1-bay 10-story steel frame design problem

23 random design variables. The probabilistic limitation is the maximum allowable lateral displacement of the top floor, which is 6.94 inches. Table 10.2 shows the outcome of the basic and enhanced algorithms. Standard algorithm findings reveal that SORA-DSSOA needs more NFE than SORA-DRao-1 in order to found the best solution. However, the optimal solution found by SORADSSOA is much better than that of the SORA-DRao-1. In addition, the statistical results obtained by SORA-DSSOA are superior to those of the SORA-DRao-1,

10.5 Numerical Examples

271

Fig. 10.5 Convergence histories of the best run of the normal and enhanced algorithms for the 1-bay 10-story steel frame design problem

demonstrating that SORA-DSSOA is more reliable than SORA-DRao-1. SORADRAo-2 is unable of performing reliability assessment in any of the 30 separate runs; hence, there is no result for SORA-DRAo-2 in Table 10.2. According to Fig. 10.7, SORA-DRao-1 unable conduct the reliability assessment in one of the runs. Additionally, SORA-DRao-1 obtains superior results than SORA-DSSOA in every single runs. According to the findings of the enhanced algorithms, despite the fact that SORADERao-1 discovered a better result than other enhanced algorithms, it needs much more NFE than other enhanced algorithms. In term of the statistical results, the results obtained by SORA-DESSOA is superior to those obtained by other methods. Additionally, it needs less NFE than the other optimization methods. As indicated in Fig. 10.8, the enhanced algorithms are able to successfully execute the reliability assessments in each of the thirty distinct runs. According to Fig. 10.9, the SORADSSOA and SORA-DERao-1 require three optimization cycles to get the optimal outcome. The other optimization methods require just two cycles of optimization to reach the optimum result. The obtained results are confirmed by calculating the reliability index using the MCS. All of them have a higher reliability index greater than 3, indicating that the constraint function is met.

10.5.3 The 3-Bay 24-Story Steel Frame The final example, considers the 24-story 3-bay steel frame. This frame contains 100 joints and 168 elements, as seen in Fig. 10.10. The structural components are organized into four groups for beams and sixteen groups for columns. Similar to the

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10 Reliability-Based Design Optimization of the Frame Structures Using …

Fig. 10.6 The schematic of the 3-bay 15-story steel frame

N/A N/A N/A

31,080

NFE in reliability analysis 37,240 part

68,320

3.4052

NFE in the optimization part

Total NFE

βMCS

N/A

N/A

75220.37 (29 of 30) N/A

2182.63 (29 of 30)

Mean weight (lb)

Standard deviation (lb)

N/A

N/A

W 24 × 55

70,903.56

11

N/A

W 14 × 48

10

Best weight (lb)

N/A N/A

W 21 × 44

W 6 × 8.5

8

N/A

W 12 × 22

7

9

N/A N/A

W 14 × 34

W 21 × 50

5

6

N/A N/A

W 24 × 62

W 21 × 55

3

W 24 × 62

2

4

N/A

W 18 × 50

873.57

70058.15

3.0113

94,820

28,620

66,200

68842.43

W 24 × 55

W 12 × 35

W 10 × 12

W 24 × 55

W 10 × 12

W 24 × 55

W 14 × 34

W 24 × 55

W 14 × 38

W 24 × 55

W 24 × 55

806.35

69326.66

3.1758

120,300

57,300

63,000

68009.13

W 14 × 48

W 12 × 35

W 12 × 14

W 21 × 50

W 14 × 26

W 21 × 50

W 18 × 40

W 24 × 55

W 21 × 50

W 14 × 48

W 30 × 99

667.53

69,387.14

3.0225

65,500

23,500

42,000

68112.17

W 14 × 48

W 21 × 44

W 5 × 16

W 21 × 50

W 14 × 26

W 24 × 55

W 18 × 46

W 21 × 44

W 21 × 57

W 24 × 55

W 24 × 76

556.65

69,273.46

3.1957

58,880

22,740

36,140

68017.47

W 14 × 48

W 12 × 35

W 12 × 19

W 21 × 44

W 14 × 26

W 18 × 40

W 14 × 48

W 24 × 55

W 21 × 55

W 21 × 55

W 30 × 90

SORA-DRao-2 [1] SORA-DSSOA [1] SORA-DERao-1 [1] SORA-DERao2 [1] SORA-DESSOA [1] N/A

SORA-DRao-1 [1]

Element group

1

Table 10.2 Comparative results of the standard and enhanced algorithms in the 3-bay 15-story steel frame

10.5 Numerical Examples 273

274

10 Reliability-Based Design Optimization of the Frame Structures Using …

Fig. 10.7 The structural weight of each independent run of the standard algorithms for the 3-bay 15-story steel frame design problem

Fig. 10.8 The structural weight of each independent run of the enhanced algorithms for the 3-bay 15-story steel frame design problem

previous instances, the structural loading is a deterministic value. The beam member is picked from the 267 W section, whereas the column member is taken from the W14 section. The cross-sectional area and the members’ second moment are defined as random design variables with a normal distribution and a coefficient of variation of 0.05. The modulus of elasticity is considered as a random design parameter with an mean of 29 732 ksi, a coefficient of variation of 0.05, and a normal distribution. There are 20 design variables and 41 random design variables in this example. The

10.5 Numerical Examples

275

Fig. 10.9 Convergence histories of the best run of the normal and enhanced algorithms for the 3-bay 15-story steel frame design problem

probability restriction is the maximum permissible lateral displacement of the top floor, which is 11.52 inches with a reliability index of 3. The outcomes of the considered algorithms are given in Table 10.3. According to the results of the standard algorithms, SORA-DSSOA discovered a superior solution than SORA-DRao-1. Moreover, SORA-DSSOA needs 72% fewer NFE than the SORA-DRao-1, indicating that SORA-DSSOA can rapidly converge to the best solution. In terms of statistical outcomes, SORA-DSSOA produces better results over SORA-DRao-1. Similar to the previous instance, SORA-DRao-2 cannot perform reliability assessment in any of the runs; hence, there are no results for SORADRao-2 in Table 10.3. As shown in Fig. 10.11, SORA-DRao-1 is able to complete the reliability analyses in 19 of the 30 trials. In addition, the results of each run of the SORA-DSSOA are much better than those of the SORA-DRao-1. Enhanced algorithm results reveal that the SORA-DERao-2 identified the best solution compared to all other available outcomes. SORA-DESSOA achieves a better average of the 30 independent runs than SORA-DERao-1 and SORA-DERao-2. In the term of the standard deviation, SORA-DERao-2 has better results than other methods. According to Fig. 10.12, the improved algorithms performed the reliability assessment successfully in all the runs. Similar to the other instance, MCS is utilized to confirm the solution. The MCS reliability index reveals that the constraint function is satisfied in all the investigated techniques. SORA-DERao-2 needs two optimization cycles. Nevertheless, the other optimization methods needed three optimization cycles, as shown in Fig. 10.13.

276

10 Reliability-Based Design Optimization of the Frame Structures Using …

Fig. 10.10 The schematic of the 3-bay 24-story steel frame

211126.89

Best weight (lb) N/A

N/A N/A

W 24 × 55

W 21 × 44

19

N/A

W 8 × 10

18

20

N/A N/A

W 14 × 22

W 30 × 90

16

N/A

W 14 × 61

15

17

N/A N/A

W 14 × 74

W 14 × 43

13

N/A

W 14 × 82

12

14

N/A N/A

W 14 × 90

W 14 × 90

10

N/A

W 14 × 109

9

11

N/A N/A

W 14 × 30

W 14 × 38

7

N/A

W 14 × 68

6

8

N/A N/A

W 14 × 90

W 14 × 82

4

5

N/A N/A

W 14 × 145

W 14 × 132

2

W 14 × 176

3

SORA-DRao-2 [1] N/A

SORA-DRao-1 [1]

Element group

1

209942.07

W 12 × 35

W 24 × 55

W 6 × 8.5

W 30 × 90

W 14 × 34

W 14 × 48

W 14 × 61

W 14 × 74

W 14 × 74

W 14 × 90

W 14 × 99

W 14 × 99

W 14 × 26

W 14 × 38

W 14 × 61

W 14 × 74

W 14 × 99

W 14 × 120

W 14 × 145

W 14 × 176

SORA-DSSOA [1]

210014.00

W 12 × 35

W 24 × 55

W 6 × 8.5

W 30 × 90

W 14 × 34

W 14 × 53

W 14 × 61

W 14 × 74

W 14 × 74

W 14 × 90

W 14 × 99

W 14 × 99

W 14 × 26

W 14 × 34

W 14 × 61

W 14 × 74

W 14 × 99

W 14 × 120

W 14 × 145

W 14 × 176

SORA-DERao-1 [1]

Table 10.3 Comparative results of the standard and enhanced algorithms in the 3-bay 24-story steel frame

209726.76

W 16 × 26

W 24 × 55

W 6 × 8.5

W 30 × 90

W 14 × 34

W 14 × 48

W 14 × 61

W 14 × 68

W 14 × 90

W 14 × 90

W 14 × 99

W 14 × 99

W 14 × 26

W 14 × 34

W 14 × 53

W 14 × 74

W 14 × 99

W 14 × 120

W 14 × 145

W 14 × 176

SORA-DERao2 [1]

209905.92

W 16 × 26

W 24 × 55

W 6 × 8.5

W 30 × 90

W 14 × 34

W 14 × 48

W 14 × 61

W 14 × 68

W 14 × 90

W 14 × 90

W 14 × 99

W 14 × 99

W 14 × 26

W 14 × 38

W 14 × 61

W 14 × 74

W 14 × 90

(continued)

W 14 × 120

W 14 × 145

W 14 × 176

SORA-DESSOA [1]

10.5 Numerical Examples 277

58,540

143,400

201,940

3.0041

223090.62 (19 of 30)

NFE in the optimization part

NFE in reliability analysis part

Total NFE

βMCS

Mean weight (lb)

Standard deviation 8930.15 (19 of 30) (lb)

SORA-DRao-1 [1]

Element group

Table 10.3 (continued)

N/A

N/A

N/A

N/A

N/A

N/A

SORA-DRao-2 [1]

3330.75

210877.95

3.0088

116,960

52,720

64,240

SORA-DSSOA [1]

357.18

210455.25

3.0076

184,320

118,160

66,160

SORA-DERao-1 [1]

271.07

210158.32

3.0011

131,440

90,580

40,860

SORA-DERao2 [1]

383.18

210101.77

3.0132

129,480

66,780

62,700

SORA-DESSOA [1]

278 10 Reliability-Based Design Optimization of the Frame Structures Using …

10.6 Concluding Remarks

279

Fig. 10.11 The structural weight of each independent run of the standard algorithms for the 3-bay 24-story steel frame design problem

Fig. 10.12 The structural weight of each independent run of the enhanced algorithms for the 3-bay 24-story steel frame design problem

10.6 Concluding Remarks In this chapter, reliability-based design optimization of the frame structures is investigated. These structures include the 1-bay 10-story steel frame, 3-bay 15-story steel frame, and the 3-bay 24-story steel frame. The investigated structures have lower degrees of statical indeterminacy than the degrees of kinematical indeterminacy,

280

10 Reliability-Based Design Optimization of the Frame Structures Using …

Fig. 10.13 Convergence histories of the best run of the normal and enhanced algorithms for the 3-bay 24-story steel frame design problem

hence, the force method is faster than the displacement method for structural analysis. The probabilistic constraint is considered as the top story’s lateral displacement in all the examples. Three simple optimization techniques named Shuffled Shepherd optimization Algorithm (SSOA), Rao-1, and Rao-2 are considered. Also, the efficiency of their improved versions such as the Enhanced Shuffled Shepherd optimization Algorithm (ESSOA), ERao-1, and ERao-2 are investigated in this chapter. The Monte Carlo Simulation (MCS) is used to validate the optimization technique’s outcomes. According to the acquired outcomes, The SSOA discovered a better solution than the standard algorithms in all examples in terms of the best and average solution and performed the reliability analysis in all the runs in all three examples. Rao-2 unable to conduct the reliability analysis in any of the thirty independents runs in the last two examples and performed the reliability analysis in the three runs in the first example. Rao-1 performed the reliability assessment in the first instance. However, it unable perform the reliability assessment in every run in the other examples. In the enhanced algorithms, none of them has superiority over the other algorithms in the considered examples. ESSOA performed better in the first example, ERao-1 performed better in the second example, and ERao-2 performed better in the third example. The MCS demonstrated constraint function is valid in every example. It shown that the SORA-DM framework is applicable to the RBDO of the frame structures.

References 1. Kaveh, A., Zaerreza, A.: Reliability-Based Design Optimization of the Frame Structures Using

References

281

the Force Method and SORA-DM Framework. Structures 45, 814–827 (2022) 2. Kaveh, A., Zaerreza, A.: A new framework for reliability-based design optimization using metaheuristic algorithms. Structures 38, 1210–1225 (2022) 3. Kaveh, A., Zaerreza, A.: Comparison of the graph-theoretical force method and displacement method for optimal design of frame structures. Structures 43, 1145–1159 (2022) 4. Kaveh, A.: Structural Mechanics: Graph and Matrix Methods. vol. 6, Macmillan International Higher Education, UK (1992) 5. Kaveh, A., Zaerreza, A.: Shuffled shepherd optimization method: a new meta-heuristic algorithm. Eng. Comput. 37(7), 2357–2389 (2020) 6. Rao, R.: Rao algorithms: three metaphor-less simple algorithms for solving optimization problems. Int. J. Indust. Eng. Comput. 11(1), 107–130 (2020) 7. Kaveh, A., Zaerreza, A., Hosseini, S.M.: An enhanced shuffled Shepherd optimization algorithm for optimal design of large-scale space structures. Eng. With Comput. 38(2), 1505–1526 (2021) 8. Kaveh, A., Zaerreza, A.: Enhanced Rao algorithms for optimization of the structures considering the deterministic and probabilistic constraints. Period Polytech. Civil Eng. 66(3), 694–709 (2022)