Modern Trends in Structural and Solid Mechanics 3

6 Chapter 5Figure 5.1. CDF of the stochastic safety factor Figure 5.2. Effect of COV on the probability of failure Figure 5.3. Flowchart of structural design to reliability process

7 Chapter 6Figure 6.1. Upper and lower bounds of spatial uncertainty on the thickness t of ...

8 Chapter 7Figure 7.1. Three typical two-dimensional convex models. For a color version of ...Figure 7.2. Two-step bar Figure 7.3. Convex polytopic models of the stiffness terms and the external load...Figure 7.4. Static displacements u 1 and u 2of the vertex solution theorem, based...Figure 7.5. Ten-bar truss Figure 7.6. Convex polytopic models of the external loads and the cross-sectiona...Figure 7.7. Bounds of the vertical displacements at nodes 3, 4, 5 and 6 of the t...Figure 7.8. Comparison of the upper and lower bounds on the vertical displacemen...Figure 7.9. Plane frame Figure 7.10. Convex polytopic models of Young’s moduli and the external loads of...Figure 7.11. Bounds of the horizontal displacements at nodes 188, 192, 198 and 2...Figure 7.12. Comparison of the upper and lower bounds on the horizontal displace...

9 Chapter 8Figure 8.1. Damaged beam: finite element model and continuous model. For a color...Figure 8.2. Function | versus the dimensionless crack depth |. For a color versi...Figure 8.3. Compliance function | versus the dimensionless crack depth |. For a ...Figure 8.4. (a) Lower and (b) upper bounds of the function | for different level...Figure 8.5. (a) Lower and (b) upper bounds of the function | for different level...Figure 8.6. (a) Central value | and (b) deviation amplitude | for different leve...Figure 8.7. Lower and upper bounds of the deflection function (in m) for a finit...Figure 8.8. (a) Sensitivity functions and (b) lower and upper bounds of the defl...Figure 8.9. Lower and upper bounds of the deflection function (in m) for a finit...Figure 8.10. (a) Sensitivity functions and (b) lower and upper bounds of the def...

10 Chapter 9Figure 9.1. Classic bit and qubit represented by two electronic levels in a sphe...Figure 9.2. A typical checkerboard pattern of a continuum structure Figure 9.3. Polar plot of the quantum gate for a qubit Figure 9.4. Finite element encoding scheme and topological form Figure 9.5. Flowchart for the presented quantum-inspired methodology Figure 9.6. Schematic indication of the advantage of quantum annealing over clas...Figure 9.7. The diagram of element density updating Figure 9.8. Design domain of a short cantilever Figure 9.9. Result of different initial distribution of material schemes Figure 9.10. Result of different methods Figure 9.11. A wing rib structure under multiple loads Figure 9.12. Topological configuration of wing rib under different methods

11 Chapter 10Figure 10.1. Quarter-car model rolling at constant speed on an uneven road with ...Figure 10.2. Amplitude ratio (red) and rms ratio (blue) for two damping values a...Figure 10.3. Half-car model rolling on a random road profile with two degrees of...Figure 10.4. Root mean square ratios of the response and excitation of the half-...Figure 10.5. Resonance and absorption speeds for k=1,2,3,… calculated from equat...Figure 10.6. Extended half-car model with four degrees of freedom and two viscoe...Figure 10.7. Related rms ratios of all displacements versus the speed frequency ...Figure 10.8. Mean amplitudes and densities of the displacements At and Bt versus...Figure 10.9. Resonance of the resultant mean amplitude for growing noise that in...Figure 10.10. Resonance reduction of the mean amplitude Vm for growing noise int...Figure 10.11. Resonance induction from blue to red resonance peaks for growing n...Figure 10.12. Limit cycles and transient projections of angles Δτ, Ψt and their ...Figure 10.13. Boundaries for almost sure stability and stability in mean. Critic...

12 Chapter 11Figure 11.1. Cumulative distribution function and robustness level. For a color ...Figure 11.2. Deign acceleration response spectra for damping factors 0.02, 0.05 ...Figure 11.3. Building model Figure 11.4. Acceleration response spectra of seismic motions at bedrock and sur...Figure 11.5. Histogram of observed maximum inter-story drift angles Figure 11.6. Observed maximum inter-story drift angles of each story Figure 11.7. A 20-story shear frame model (Ohsaki et al . 2019) Figure 11.8. Relation between the maximum inter-story drift and order k for the ...Figure 11.9. Distributions of additional damping coefficients: (a) Pareto optima...

1 Chapter 2 Table 2.1. Some parametric classes of correlation functions for a Gaussian proce...Table 2.2. COV values of the stochastic displacement statically determinate stoc...Table 2.3. Example 1: COV values of the redundant force X for L = 10 and for L =...Table 2.4. Example 1: COV values of the redundant force X for L = 10 and for L =...

2 Chapter 3Table 3.1. Mean values and standard deviations of the buckling load of the von M...Table 3.2. Results of optimizations of the three-bar truss example with random c...Table 3.3. Results of optimizations of the three-bar truss example with random h...Table 3.4. Mean value and standard deviation of the compliance of the design obt...Table 3.5. Results of the cantilever beam example with random Young’s modulus

3 Chapter 4Table 4.1. Knowledge entity of the variable representation Table 4.2. Definition of a variable knowledge entity

4 Chapter 5Table 5.1. Reliability and engineering confidence matrix

5 Chapter 7Table 7.1. Bounds of the vertical displacements at nodes 3, 4, 5 and 6 of the te...Table 7.2. Physical parameters of the plane frame (m) Table 7.3. Bounds of the horizontal displacements at nodes 188, 192, 198 and 202...

6 Chapter 8Table 8.1. Deflection bounds (in m) and “error in bound” εb % evaluated at x = 2...

7 Chapter 9Table 9.1. The number of iterations and final compliance of different methods

8 Chapter 11Table 11.1. Sample size requirements for α = γ = 0.9 Table 11.2. Content ratio for order statistics of α = 0.9, n = 150 Table 11.3. Optimal solutions (cross-sectional areas of braces, unit: 102 mm2) Table 11.4. Summary of observed maximum inter-story drift angles Table 11.5. Summary of observed maximum inter-story drifts

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2 Table of Contents

3 Title Page Series Editor Noël Challamel

4 Copyright First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2021 The rights of Noël Challamel, Julius Kaplunov and Izuru Takewaki to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2020952868 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-718-7