Proceedings of the 4th International Conference on Numerical Modelling in Engineering: Volume 1: Numerical modelling in Civil Engineering, NME 2021, ... (Lecture Notes in Civil Engineering, 217) 9811681848, 9789811681844

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Lecture Notes in Civil Engineering

Magd Abdel Wahab   Editor

Proceedings of the 4th International Conference on Numerical Modelling in Engineering Volume 1: Numerical modelling in Civil Engineering, NME 2021, 24–25 August, Ghent University, Belgium

Lecture Notes in Civil Engineering Volume 217

Series Editors Marco di Prisco, Politecnico di Milano, Milano, Italy Sheng-Hong Chen, School of Water Resources and Hydropower Engineering, Wuhan University, Wuhan, China Ioannis Vayas, Institute of Steel Structures, National Technical University of Athens, Athens, Greece Sanjay Kumar Shukla, School of Engineering, Edith Cowan University, Joondalup, WA, Australia Anuj Sharma, Iowa State University, Ames, IA, USA Nagesh Kumar, Department of Civil Engineering, Indian Institute of Science Bangalore, Bengaluru, Karnataka, India Chien Ming Wang, School of Civil Engineering, The University of Queensland, Brisbane, QLD, Australia

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Magd Abdel Wahab Editor

Proceedings of the 4th International Conference on Numerical Modelling in Engineering Volume 1: Numerical modelling in Civil Engineering, NME 2021, 24–25 August, Ghent University, Belgium

Editor Magd Abdel Wahab Soete Laboratory, Faculty of Engineering and Architecture Ghent University Zwijnaarde, Belgium

ISSN 2366-2557 ISSN 2366-2565 (electronic) Lecture Notes in Civil Engineering ISBN 978-981-16-8184-4 ISBN 978-981-16-8185-1 (eBook) https://doi.org/10.1007/978-981-16-8185-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Organising Committee

Chairman Prof. dr. ir. Magd Abdel Wahab, Laboratory Soete, Ghent University, Belgium

International Scientific Committee Prof. D. Ribeiro, School of Engineering, Polytechnic of Porto (ISEP-IPP), Portugal Prof. J. Santos, University of Madeira, Portugal Prof. J. Toribio, University of Salamanca, Spain Prof. B. B. Zhang, Glasgow Caledonian University, UK Prof. V. Silberschmidt, Loughborough University, UK Prof. T. Rabczuk, Bauhaus University Weimar, Germany Prof. L. Vanegas Useche, Universidad Tecnológica de Pereira, Colombia Prof. N. S. Mahjoub, Institut Préparatoire aux Etudes d’Ingénieurs de Monastir, Tunisia Prof. A. Cheknane, Amar Telidji University of Laghouat, Algeria Prof. E. N. Farsangi, Kerman Graduate University of Advanced Technology (KGUT), Iran Prof. N. A. Noda, Kyushu Institute of Technology, Japan Prof. K. Oda, Oita University, Japan Prof. S. Abdullah, Universiti Kebangsaan Malaysia, Malaysia Prof. C. Zhou, Nanjing University of Aeronautics and Astronautics, China Prof. B. Bhusan Das, National Institute of Technology Karnataka, India Prof. R. V. Prakash, Indian Institute of Technology, India Prof. H. N. Xuan, Hutech University, Vietnam Prof. Giuseppe Carbone, University of Calabria, Italy Prof. Fadi Hage Chehade, Lebanese University, Lebanon Prof. Sohail Nadeem, Quaid-i-Azam University, Pakistan

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Organising Committee

Dr. A. San-Blas, Miguel Hernández University of Elche, Spain Dr. G. Minafo, University of Palermo, Italy Dr. A. Caggiano, Technische Universität Darmstadt, Germany Dr. S .Khatir, Ghent University, Belgium Dr. T. Yue, Ghent University, Belgium Dr. A. Rudawska, Lublin University of Technology, Poland Dr. L. V. Tran, Sejong University, South Korea Dr. X. Zhuang, Leibniz Universität Hannover, Germany Dr. I. Hilmy, International Islamic University Malaysia, Malaysia Dr. C. Wang, Liaocheng University, China Dr. M. Mirrashid, Semnan University, Iran Prof. A. G. Correia, University of Minho, Portugal Dr. M. Wang, Los Alamos National Laboratory, USA Dr. Filippo Genco, Adolfo Ibáez University, USA Dr. Denis Benasciutti, University of Ferrara, Italy Dr. Y. L. Zhou, Xi’an Jiaotong University, China

Preface

This volume contains the proceedings of the 4th International Conference on Numerical Modelling in Engineering: Volume 1 Numerical Modelling in Civil Engineering. Numerical Modelling in Engineering NME 2021 is the 4th NME conference and is held online via MS Teams, during the period 24–25 August 2021. Previous NME conferences were celebrated in Ghent, Belgium (2018), Beijing, China (2019) and online (2020). The overall objective of the conference is to bring together international scientists and engineers in academia and industry in fields related to advanced numerical techniques, such as FEM, BEM and IGA, and their applications to a wide range of engineering disciplines. The conference covers industrial engineering applications of numerical simulations to civil engineering, aerospace engineering, materials engineering, mechanical engineering, biomedical engineering, etc. The presentations of NME 2021 are divided into 2 main sessions, namely (1) civil engineering and (2) mechanical and materials engineering. This volume is concerned with the applications to civil engineering. The organising committee is grateful to keynote speaker, Prof. Yaroslav D. Sergeyev, University of Calabria, Rende, Italy, and Lobachevsky State University, Nizhni Novgorod, Russia, for his keynote speech entitled ‘Computations with numerical infinities and infinitesimals’. Special thanks go to members of the Scientific Committee of NME 2021 for reviewing the articles published in this volume and for judging their scientific merits. Based on the comments of reviewers and the scientific merits of the submitted manuscripts, the articles were accepted for publication in the conference proceedings and for presentation at the conference venue. The accepted papers are of a very high scientific quality and contribute to advancement of knowledge in all research topics relevant to NME conference.

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Preface

Finally, the organising committee would like to thank all authors, who have contributed to this volume and to those who have presented their research work at the conference in MS Teams. Zwijnaarde, Belgium

Prof. Magd Abdel Wahab Chairman of NME 2021

Contents

Static and Dynamic Analysis of Framed Buildings by Means of an Equivalent Multi-stepped Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ilaria Fiore, Annalisa Greco, Salvatore Caddemi, and Ivo Caliò

1

Collapse of Shear Wall—Experimental and Numerical Analysis . . . . . . . . 17 Adrian Bekö and Peter Rosko Numerical Simulation of Water Impoundment at a High Arch Dam Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Christine Detournay, Guotao Meng, Jing Hou, Jianrong Xu, Zhao Cheng, Ryan Peterson, and Peter Cundall 3-D Forward and Inverse Scattering Analyses for Cavity in Viscoelastic Media Using Convolution Quadrature Time-Domain Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Haruhiko Takeda, Takahiro Saitoh, and Sohichi Hirose Importance of the Geotechnical Variability in the Bearing Capacity of Shallow Foundations Through Random Fields . . . . . . . . . . . . . . . . . . . . . . 59 Cristhian C. Mendoza, Jorge E. Hurtado, and Jairo A. Paredes Prediction of Wave Overtopping Discharge on Coastal Protection Structure Using SPH-Based and Neural Networks Method . . . . . . . . . . . . . 71 Bao-Loi Dang, Quoc Viet Dang, Magd Abdel Wahab, and H. Nguyen-Xuan Structural Robustness of RC Frames Under Blast Events . . . . . . . . . . . . . . 81 Marco Mennonna, Mattia Francioli, Francesco Petrini, and Franco Bontempi Influence of Pretensioned Rods on Structural Optimization of Grid Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Valentina Tomei, Ernesto Grande, and Maura Imbimbo

ix

Static and Dynamic Analysis of Framed Buildings by Means of an Equivalent Multi-stepped Beam Ilaria Fiore, Annalisa Greco, Salvatore Caddemi, and Ivo Caliò

Abstract In order to evaluate the static and dynamic response of multi-storey buildings many researchers have focused, in the last thirty years, their attention on simplified models which allow to drastically reduce the computational effort without losing the required accuracy. In this context, with the aim of analyzing buildings characterised by a decreasing size of the column cross sections along the height of the structure, the present study refers to a simplified model represented by a multistepped beam. Therefore, in the proposed approach, each inter-storey of the building is modelled by means of a beam segment with uniform stiffness and mass properties. The model is adopted for the evaluation of the elastic response of multi-storey buildings either to horizontal static forces or to seismic accelerations at the base. The equations of motion of the considered multi-stepped beam, which exhibits only shear and torsional deformability, have been derived by means of the Hamilton’s principle and its deformed shapes have been evaluated through a Rayleigh–Ritz approach based on a certain number of mode shapes of the uniform shear beam. Numerical applications to a reinforced concrete multi-storey building have been performed by comparing the static and dynamic responses, evaluated by means of the proposed approach, to the results obtained according to 3D FEM models. Furthermore, the static response of the considered multi-stepped beam has been also validated by means of an original closed form solution for step-wise shear-torsional elastic beams. Keywords Beam-like model · Static analysis · Dynamic analysis · Rayleigh–Ritz · Shear beam · Multi-stepped beam · FEM models

1 Introduction The evaluation of the static and dynamic behaviour of multi-storey buildings has represented one of the most studied topics for structural engineers in the last century. I. Fiore (B) · A. Greco · S. Caddemi · I. Caliò Department of Civil Engineering and Architecture, University of Catania, via Santa Sofia 64, 95125 Catania, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. A. Wahab (eds.), Proceedings of the 4th International Conference on Numerical Modelling in Engineering, Lecture Notes in Civil Engineering 217, https://doi.org/10.1007/978-981-16-8185-1_1

1

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I. Fiore et al.

Significant improvements have been reached in the last years thanks to the development of sophisticated computational procedures and parallel computing. With the aim of performing large scale simulations of the seismic response of entire urban areas simplified numerical models of multi-storey buildings have been presented in the scientific literature [1, 2]. Among these models the beam-like one, which is based on the equivalence of structures to continuum beams, aims at simulating the dynamic behaviour of multi-storey buildings with a great reduction in the computational burden. The presented models of equivalent beam are based either on Timoshenko or Euler–Bernoulli theories. Many studies refer to homogeneous elements in which the stiffness of the beam is uniformly distributed [3]. Few studies consider uniform beams having eccentricity between the centres of stiffness and mass [4–7]. In order to model buildings whose mass and stiffness are variably distributed along the height and that are characterized by asymmetrical plans, a non-uniform beam-like model has been presented by the present authors [8]. In the latter paper the effectiveness in evaluating the dynamic response of multi-storey buildings by means of the proposed equivalent beam has been shown. In the present paper the multi-stepped model proposed in [8] has been applied both to the evaluation of the static and seismic response of multi-storey buildings. The beam has shear and torsional deformability only and provides a simple mathematical model. Numerical applications show that, in spite of its simplicity, the proposed multi-stepped beam is able to provide excellent results for the evaluation of the static and dynamic responses of low- and mid-rise buildings reducing the computational effort with respect to that required by more general models. Hamilton’s principle has been used for the derivation of the governing equations of the static and dynamic behaviour of the proposed multi-stepped beam. The linear static and dynamic behaviour of the multi-stepped beam is then evaluated by discretizing the continuous model with a Rayleigh–Ritz approach based on a certain number of mode shapes of the uniform shear torsional beam model. The static response of multi-storey buildings has been also studied by means of a closed form solution obtained for the considered beam following the formulation presented by Caddemi et al. [9]. The evaluated static and dynamic responses, with reference to a four-storey building assumed as case study, have been compared to those obtained through a 3D FEM modeling approach. The results have shown a very good agreement thus validating the efficacy of the proposed simplified model for investigating both the static and seismic behaviour of multi-storey buildings with a strong reduction of the computational demand.

2 The Equivalent Multi-stepped Beam In this paper multi-storey asymmetrical buildings are modeled by means of equivalent 3D multi-stepped shear-torsional beams as described in Fig. 1. Each k-th building

Static and Dynamic Analysis of Framed Buildings …

3

Fig. 1 The beam-like model

inter-storey is modeled by means of equivalent beam segments with the same length and having constant cross section. The multi-stepped beam is assumed to be a cantilever beam with axis z, clamped at the base. At each k-th storey (in the x, y plane) the different positions of the centres of mass and stiffness are appropriately calculated, generating torsional effects during either the static or the seismic response of the considered building. The beam shear and torsional stiffness, at each floor, are here appropriately evaluated by means of an equivalence to the corresponding inter-storey values of the considered building following the procedure proposed by Piccardo et al. [6]. With reference to the equivalent multi-stepped beam, distributed masses mxk , myk and second order moment I ok are assumed for each inter-storey beam segment. Furthermore, the masses M xk and M yk and second order moment I ok of the k-th storey are assumed to be concentrated at the floor levels. Continuous displacement functions in the x and y directions ux (z,t), uy (z,t) and torsional rotations ϑ z (z,t) of the beam have been considered. A Rayleigh–Ritz discretization is performed with the aim of reducing the computational effort related to the evaluation of the response of the equivalent non-uniform beam. The latter is based on the use of N mode shapes of a uniform shear cantilever beam of total length h; these can be expressed as: ψm (ζ ) = sin

π 2

(2m − 1)ζ



m = 1, 2, ..., ∞

(1)

where ζ = z/ h is the dimensionless abscissa of the beam. The displacement components, u x (z, t), u y (z, t), ϑz (z, t), can be expressed as the sum of each shape function contribution as follows: u x (ζ, t) =

N  i=1

ψi (ζ )qi x (t) =

N  i=1

ψi qi x

(2)

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I. Fiore et al.

u y (ζ, t) =

N 

ψi (ζ )qi y (t) =

i=1

ϑz (ζ, t) =

N  i=1

N 

ψi qi y

(3)

ψi qiϑ

(4)

i=1

ψi (ζ )qiϑ (t) =

N  i=1

qi x (t), qi y (t), qiϑ (t) being the generalized i-th coordinates which represent the contribution of the single shape function to the total response. For the evaluation of the seismic response of the building, displacements at the base of the beam have been introduced. However, the static response of the building has been calculated considering transversal loads applied along the height with assumed distributions. The governing equations of the static and dynamic problem of the multi-stepped beam in the generalized space are derived through the application of Hamilton’s principle. In the case of the static problem, the contribution of the elastic energy and the work associated with the non-conservative forces have been considered only, while in the dynamic problem the base displacements have been considered in the formulation of the kinetic energy in order to take into account seismic excitations; in the latter case non-conservative forces have been neglected and the damping has been subsequently introduced as modal damping in the reduced space. It is worth pointing out that the torsional rotation of the beam around the centre of stiffness CS, determines transversal displacements that must be added to the ones due to shear deformability (Fig. 2). Fig. 2 Displacements in x (ux ) and y (uy ) direction of a corner node due to shear deformability (v) and torsional rotation (θ)

Static and Dynamic Analysis of Framed Buildings …

5

2.1 Static Analysis In this section the static response of multi-storey buildings subjected to horizontal forces has been evaluated according to two different approaches. The first one considers the described Rayleigh–Ritz discretization of the multi-stepped beam and evaluates the static displacements by means of the construction of the stiffness matrix. The second approach is based on a closed form solution of the considered multi-stepped beam appropriately derived according to the procedure proposed in [9]. The results obtained by means of both the approaches have been compared in the applicative section to those obtained through a 3D FEM modeling of the building. Rayleigh–Ritz discretization. The governing equations of the static problem of the proposed equivalent multi-stepped beam in the generalized space in matrix notation are as follows: Kq = F

(5)

where K is the generalized Stiffness Matrix, q is a vector collecting the generalized coordinates and F the corresponding load vector. Equation (5) can be partitioned as follows: ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ Kxϑ Fx qx Kx F = ⎣ Fy ⎦ q = ⎣ qy ⎦ (6) K=⎣ K y K yϑ ⎦ Kϑ x Kϑ y Kϑ Mϑ qϑ All the sub-matrices of K have NxN dimension (being N the number of used mode shapes for each direction as well as the torsional response), vectors qx , qy , qϑ collect the corresponding generalized coordinates qi x (t), qi y (t), qiϑ (t)(i = 1…N). The expressions of the stiffness matrix and the load vector components are shown below. Nf ζk 1 G A x,k ψi ψ j dζ h k=1 ζk−1

(7)

Nf ζk 1 = G A y,k ψi ψ j dζ h k=1 ζk−1

(8)

Nf  ζk   1 

2 2 G Jz,k + e y,k G A x,k + ex,k G A y,k = ψi ψ j dζ h k=1 ζk−1

(9)

K jx i x =

K jy i y

K jϑ iϑ

K jx iϑ = K jϑ i x

Nf ζk 1 = e y,k G A x,k ψi ψ j dζ h k=1 ζk−1

(10)

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I. Fiore et al.

K jy iϑ = K jϑ i y

F jx =

Nf 

Nf ζk 1 =− ex,k G A y,k ψi ψ j dζ h k=1 ζk−1

Fx,k ψ j,k

F jy =

k=1

M jϑ =

Nf 

Nf 

(11)

Fy,k ψ j,k

k=1

 Fx,k e¯ y − Fy,k e¯x + Mϑ,k ψ j,k

(12)

k=1

In Eqs. (7)–(12) differentiation with respect to the dimensionless abscissa has been denoted with the primes, GAx, and GAy are the shear stiffnesses in the x and y directions, respectively, and GJ z is the torsional stiffness (assumed constant within each inter-storey length of the equivalent beam). Furthermore, ex and ey are the coordinates of the centre of stiffness, N f is the number of floors and e x and e y are the eccentricities of the applied forces with respect to the centre of mass. Closed-form solution. The equivalent multi-stepped beam is characterized by an arbitrary number m of abrupt variations of the shear G(x)A(x) and torsional G(x)JT (x) stiffness with respect to the reference values G0 A0 and G0 J T 0 , respectively, modelled by means of the adoption of the Heaviside unit-step generalized function as follows: ⎤ ⎡ m 



 G(x)A(x) = G 0 A0 ⎣1 − β j − β j−1 U x − xβ j ⎦ (13) j=1

⎤ m 



 G(x)JT (x) = G 0 JT 0 ⎣1 − α j − α j−1 U x − xα j ⎦ ⎡

(14)

j=1

G A

G J

where β j = 1 − G 0j A0j and α j = 1 − G 0j JTT 0j and G j A j and G j JT j are the constant values of the shear and torsional stiffness, respectively, of the j-th beam segment. The considered multi-stepped beam is assumed to be subjected to transversal loads p(x) and distributed torsional moments t(x) as shown in Fig. 3. Taking into account the equilibrium and compatibility equations and the constitutive law, the following uncoupled differential equations governing the shear and torsional problems are obtained: ⎞ ⎤ m 



  ⎣G 0 A0 ⎝1 − β j − β j−1 U x − xβ j ⎠v (x)⎦ = − p(x) ⎡

⎛

j=1

(15)

Static and Dynamic Analysis of Framed Buildings …

7

Fig. 3 Equivalent multi-stepped Beam-Like model

⎤ ⎞ m 



  ⎣G 0 JT 0 ⎝1 − α j − α j−1 U x − xα j ⎠θ (x)⎦ = −t(x) ⎡

⎛

(16)

j=1

or in dimensionless form: ⎞ ⎤ m 



  ⎣⎝1 − β j − β j−1 U ξ − ξβ j ⎠u (ξ )⎦ = − p(ξ ) ⎡⎛

(17)

j=1

⎤ ⎞ m 



  ⎣⎝1 − α j − α j−1 U ξ − ξα j ⎠θ (ξ )⎦ = −t (ξ ) ⎡⎛

j=1

where ξ = x/L; u(ξ ) =

v(ξ ) ; L

p(ξ ) =

p(ξ )L ; G 0 A0

t (ξ ) =

t(ξ )L 2 . G 0 JT 0

(18)

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I. Fiore et al.

The solutions of the differential Eqs. (17) and (18) can be written in the form: u(ξ ) = C1 f 1 (ξ ) + C2 f 2 (ξ ) + f 3 (ξ )

(19)

θ (ξ ) = D1 g1 (ξ ) + D2 g2 (ξ ) + g3 (ξ )

(20)

where C 1 , C 2 , D1 , D2 are integration constants and: f 1 (ξ ) = 1 f 2 (ξ ) = ξ +

m 



 β ∗j ξ − ξβ j U ξ − ξβ j

(21)

(22)

j=1

f 3 (ξ ) = − p [2] (ξ ) −

m 





 β ∗j p [2] (ξ ) − p [2] ξβ j U ξ − ξβ j

(23)

g1 (ξ ) = 1

(24)

j=1

g2 (ξ ) = ξ +

m 



 α ∗j ξ − ξα j U ξ − ξα j

(25)

j=1

g3 (ξ ) = −t [2] (ξ ) −

m 





 α ∗j t [2] (ξ ) − t [2] ξα j U ξ − ξα j

(26)

j=1

where the apex [j] indicates a primitive of order j of the relevant function and β ∗j = βj 1−β j

β

α

α

− 1−βj−1j−1 ; α ∗j = 1−αj j − 1−αj−1j−1 . In case the multi-stepped beam is subjected only to n F concentrated forces F r , r = 1, . . . , n F , applied at the abscissae ξ Fr with eccentricity er , the load terms become: p(ξ ) =

nF 



Fr δ ξ − ξ Fr

(27)



Tr δ ξ − ξ Fr

(28)

r =1

t (ξ ) =

nF  r =1

where Fr =

Fr G 0 A0

and Tr =

F r er G 0 JT 0

L.

Static and Dynamic Analysis of Framed Buildings …

9

The integration constants for a cantilever beam clamped at the abscissa ξ = 0 turn out to be: C1 = 0; C2 =

nF 

Fr ;

D1 = 0;

D2 =

r =1

nF 

Tr

(29)

r =1

It is worth pointing out that, differently from the Rayleigh–Ritz approach, the solution provided by the presented closed form expression does not introduce approximations in the static solution of the inhomogeneous beam problem.

2.2 Dynamic Analysis In order to evaluate the seismic response of the multi-storey building, the equations of motion of the undamped multi-stepped beam in the generalized space turn out to be: MR q + Kq = P

(30)

being K the generalized stiffness matrix already introduced in Sect. 2.1, M and P the generalized mass matrix and the load vector equivalent to the seismic excitation, respectively, as follows: ⎡ M=⎣

⎤

Mx

⎦

My Mϑ

⎡

⎤ Px P = ⎣ Py ⎦ Pϑ

(31)

where: M jx i x = h

Nf 

m x,k

ζk−1

k=1

M jy i y = h

Nf 

m y,k

k=1

M jϑ iϑ = h

Nf 

ζk ζk−1

I0,k

k=1

P jx = −u¨ gx h

ζk

Nf  k=1

ζk ζk−1

m x,k

ψi ψ j dζ +

Nf 

Mx,k ψi,k ψ j,k

(32)

M y,k ψi,k ψ j,k

(33)

I o,k ψi,k ψ j,k

(34)

k=1

ψi ψ j dζ +

Nf  k=1

ψi ψ j dζ +

Nf  k=1

ζk

ζk−1

ψ j dζ − u¨ gx

Nf  k=1

Mx,k ψ j,k

(35)

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I. Fiore et al.

P jy = −u¨ gy h

Nf 

m y,k

ζk

ψ j dζ − u¨ gy

ζk−1

k=1

Nf 

M y,k ψ j,k

(36)

k=1

P jϑ = 0

(37)

where ψi,k is the shape function evaluated at the floor level with abscissa z k . The undamped vibration modes in the generalized space collected in the vector ψˆ and the vibration frequencies ω of the equivalent beam are obtained from the solution of the following eigen-problem:   K − ω2 M ψˆ = 0

(38)

The dynamic response in the generalized space can be expressed combining S modes of vibration as follows: q(t) =

S 

ψˆ j · z j (t)

(39)

j=1

Substituting Eq. (39) in the equations of motion (30) and adopting the orthogonality properties of the vibration modes, the equations of motion are simplified as follows: Mmod, j · z¨ j (t) + Cmod, j · z˙ j (t) + K mod, j · z j (t) = Pmod, j

(40)

T T T Mmod,n = ψˆ j Mψˆ j K mod,n = ψˆ j Kψˆ j Pmod,n = ψˆ j P

(41)

where:

It is worth noting that generalized modal damping Cmod, j are related to the C j = 2ξ j ω j . corresponding modal damping ratios as follows: Mmod, mod, j Finally, the dynamic response in geometric coordinates is obtained as follows: u x (z, t) =

S  N 

ψi (z)ψˆ i x j · z j (t)

(42)

ψi (z)ψˆ i y j · z j (t)

(43)

ψi (z)ψˆ iϑ j · z j (t)

(44)

j=1 i=1

u y (z, t) =

S  N  j=1 i=1

ϑ(z, t) =

S  N  j=1 i=1

Static and Dynamic Analysis of Framed Buildings …

11

where N is the number of displacement shape functions adopted in the discretization and S is the number of vibration modes adopted to compute the dynamic response.

3 Numerical Applications In this section the proposed multi-stepped beam is adopted for simulating the static and dynamic behaviour of a four-storey building with asymmetric plan and vertical mass and stiffness variation. The seismic response to the x and y component of the L’Aquila’s earthquake accelerogram has been evaluated in terms of the displacements of the control point P located at the top of the building (whose position in plan is shown in Fig. 4). The size of the cross section of the perimeter beams are assumed to be 30 × 50 cm2 , while all the remaining beams have a 110 × 23 cm2 size. An equivalent floor thickness of 9.32 cm at each level is considered. The span length is 4.5 m, the inter-storey height is 3.3 m and the cross sections of all the columns are reported in Table 1. Fig. 4 Plan of the case study building asymmetric with eccentricity between centres of mass and stiffness. The control point P is highlighted on the plan

Table 1 Assignment of the concrete column sections

Floor

Section A

Section B

4

30 × 30

50 × 50

3

35 × 35

50 × 50

2

40 × 40

50 × 50

1

45 × 45

50 × 50

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The columns are assumed fully fixed at the base. Dead and live loadings are considered uniformly distributed on each floor with a total value equal to 5.654 kN/m2 . The plan of the building is reported in Fig. 4. The material is assumed to be linear elastic and characterized by a Young’s modulus equal to 29,962 MPa, a Poisson ratio equal to 0.2 and a specific weight of 25 kN/m3 . The analyses on FEM models have been developed with SAP2000 [10].

3.1 Static Response The static displacements of the nodes at each floor have been evaluated by considering a uniform and an inverse triangular force distribution. The forces in the FEM model are applied to the four corner nodes of each floor, separately in x and y direction. The values of the applied forces from the bottom to the top floor in the two load conditions are assumed as follows: (a) (b)

FEM: [25 25 25 25] kN; FEM: [6.25 12.5 18.75 25] kN;

The static analysis of the equivalent beam model takes into account a load distribution which is the sum of the forces applied at the four corner nodes of the building. Therefore, the above-described force distributions adopted for the FEM model turn out to be the following for the equivalent beam: (a) (b)

BEAM LIKE: [100 100 100 100] kN; BEAM LIKE: [25 50 75 100] kN;

For the sake of brevity, only the static displacements at each floor corresponding to the control point P shown in Fig. 4, have been reported in the following Fig. 5. The figure shows the displacements in x and y directions for the beam-like model obtained by means of the Rayleigh–Ritz approach (red line), the closed form solution (blue line) and the ones obtained on the FEM model of the building (black line). The results plotted in Fig. 5 clearly show the accuracy of both the proposed discretization and closed form solution for the evaluation of the static response of the considered building.

3.2 Seismic Response A linear dynamic analysis of the above four storey building, excited by a real seismic record, has been also performed. In particular, the time histories of the ground accelerations in x and y directions that occurred in 2009 in L’Aquila (Italy), have been considered.

Static and Dynamic Analysis of Framed Buildings …

13

Fig. 5 Static displacements in x and y direction for a Uniform force distribution. b Inverse triangular force distribution. Red line: Multi-stepped equivalent beam with Rayleigh–Ritz discretization. Blue line: Closed form solution for the equivalent beam multi-stepped beam. Black line: FEM model of the building

The comparisons of the time histories of the x and y displacements of the control point P shown in Fig. 4 have been used for the assessment of the reliability of the beam-like model. For the sake of brevity only the time histories of the displacements at the highest floor, evaluated by means of the FEM and the proposed beam-like model, are reported in Fig. 6. The observation of Fig. 6 clearly proves the accuracy of the equivalent beam model compared to the displacements of the FEM model. Furthermore, in Fig. 7 the maximum displacements of the control point P located at each floor and the related inter-storey drifts are reported, showing once again the reliability of the proposed model.

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Fig. 6 Time histories of the displacements in x and y direction of the equivalent multi-stepped beam (BL) and FEM model subjected to L’Aquila earthquake

Fig. 7 L’Aquila earthquake. Maximum floor displacements (left) and inter-storey drifts (right)

4 Conclusions The paper presents and discuss a simplified model for the static and dynamic behaviour of multi-storey buildings based on the adoption of an equivalent multistepped beam. The proposed simplified model allows to consider both non-uniform mass and stiffness distribution along the height of the building and asymmetrical plans. A Rayleigh–Ritz discretization of the continuous beam model has been performed considering the mode shapes of the uniform shear-torsional beam. The static response to applied transversal loads has been evaluated not only by means of

Static and Dynamic Analysis of Framed Buildings …

15

the described discretization but also with an original closed form solution equation developed ad hoc for the proposed beam-like model. The dynamic response in the generalized space has been evaluated by considering seismic excitations at the base of the building. The accuracy of the proposed procedure has been tested comparing the static and seismic response of a four-storey building modeled by means of both the beam-like approach and 3D FEM models. The results showed a satisfactory correspondence thus confirming the capability of the considered multi-stepped beam model to provide reliable results with a small computational burden.

References 1. Rahgozar R, Ahmadi A, Sharifi Y (2010) A simple mathematical model for approximate analysis of tall buildings. Appl Math Model 34:2437–2451 2. Cluni F, Gioffrè M, Gusella V (2013) Dynamic response of tall buildings to wind loads by reduced order equivalent shear-beam models. J Wind Eng Ind Aerodyn 123:339–348 3. Chajes M, Romstad K, McCallen D (1993) Analysis of multiple-bay frames using continuum model. J Struct Eng ASCE 119(2):522–546 4. Zalka K (2014) Maximum deflection of asymmetric wall-frame buildings under horizontal load. Periodica Polytech: Civil Eng 58(4):387–396 5. Potza G, Kollar L (2003) Analysis of building structures by replacement sandwich beams. Int J Solids Struct 535–553 6. Piccardo G, Tubino F, Luongo A (2019) Equivalent Timoshenko linear beam model for the static and dynamic analysis of tower buildings. Appl Math Model 71:77–95 7. Piccardo G, Tubino F, Luongo A (2015) Equivalent nonlinear beam model for the 3-D analysis of shear-type buildings: application to aeroelastic instability. Int J Non-Linear Mech 80:52–65 8. Greco A, Fiore I, Occhipinti G, Caddemi S, Spina D, Caliò I (2020) An equivalent non-uniform beam-like model for dynamic analysis of multi-storey irregular buildings. Appl Sci 10:3212 9. Caddemi S, Caliò I, Cannizzaro F (2013) Closed form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports. Arch Appl Mech 83:559–577 10. CSI (2007) CSI analysis reference manual for SAP2000. Computers and Structures Inc.

Collapse of Shear Wall—Experimental and Numerical Analysis Adrian Bekö and Peter Rosko

Abstract The paper deals with the analysis of a reinforced concrete shear wall under varying cyclic loading with focus on its global behavior until collapse. The structural shear walls of the Kashiwazaki-Kariwa nuclear power plant were reported to sustain no significant structural damage during the 2007 Chuetsu earthquake. It was the initial motive to analyze the resistance of the applied shear walls. The IRIS research project on industrial safety assessment and management included the cyclic testing of full-scale reinforced concrete walls. During testing each wall was fixed at the bottom and subjected to a combination of constant vertical load and variably increasing horizontal load applied at the top of the shear wall. The test data have been analyzed. The most important results of testing are the load–displacement curve and diagram of shear displacement vs. cumulative vertical displacement. With help of both diagrams it is possible to identify the steel yielding and the collapse of the wall. The numerical model of the wall was derived, validated and applied in seismic analysis. The results may point towards unaccounted overstrength of RC shear walls. Keywords Reinforced concrete shear wall · Collapse · Experimental analysis · Numerical analysis

1 Introduction The initial incentive for this research came from the behavior of bearing structures of the Kashiwazaki-Kariwa nuclear power plant in the 2007 Niigata-Chuetsu-Oki earthquake after which the structural damages observed were less than might be expected for an earthquake of such magnitude [1]. A. Bekö (B) Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia e-mail: [email protected] P. Rosko Department of Civil Engineering, Vienna University of Technology, Vienna, Austria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. A. Wahab (eds.), Proceedings of the 4th International Conference on Numerical Modelling in Engineering, Lecture Notes in Civil Engineering 217, https://doi.org/10.1007/978-981-16-8185-1_2

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Horizontal displacements due to ground shaking are most commonly kept within desirable limits by the stiffness of shear walls, especially so in the context of nuclear power plants. As shear walls are built of reinforced concrete the considerable variability of parameters is given and specific laboratory experiments have a limited applicability based on the type of shear wall and the particular set of variables. Depending on element parameters such as the wall geometry, its cross-section, concrete compressive strength, detailing and quantity of reinforcement, steel grade [2, 3], or loading [4, 5] and boundary conditions, different failure modes may occur. In this research we focus on low-rise shear walls which, in contrast to high-rise walls dominated by flexural behavior, are governed by shear. To circumvent scaling issues a full scale shear wall was studied.

2 Testing The intention of the tests was to reach the limit capacity of the wall in pure shear failure mode. To achieve this aim the wall was chosen to be massive with a thickness of 400 mm not to let the wall enter buckling. Secondly, the top surface of the wall was forced to remain horizontal during testing to provoke the shearing behavior. This was achieved by four vertical actuators which were coupled by equilibrium conditions. The main horizontal loading was applied by an elaborate frame (see Fig. 1) in which the actuators acted at the centerline of the specimen. An additional vertical load was applied to simulate structural continuity. Detailed description of the specimen including reinforcement, testing apparatus and applied load histories can be found in [6]. The specimens were tested under various cyclic loading regimes. Due to the applied actuators, which were selected based on the maximum required force, the loading was of a quasi-static nature. At each load level two cycles were performed and the load was increased by 500 or 1000 kN increments. One load cycle was defined

Fig. 1 A wall specimen placed in the shearing device, centered before testing (left). Wall specimen exhibiting diagonal cracks after failure (right)

Collapse of Shear Wall—Experimental …

19

as reaching the same load level (target) in two opposite directions and subsequent unloading. In case of all four tested specimens clear shear failure was reached.

2.1 Results We present the response of the wall by plotting the select principal parameters which are horizontal load, shear displacement and vertical displacement in Fig. 2. The shear displacement was defined as an average of the shear distortions at the two ends of the wall, which, in turn, were taken as the difference between the top and bottom displacements. The plots are for a specimen with gradually increasing load history. In the data points of interest have been marked. Interpreting the response may not be straightforward as only global behavior parameters are available and it is noted that the reinforcement itself was not instrumented. In case of concrete, linearity is a convenient engineering simplification but actually the material exhibits nonlinear features early on after load is applied to the element. Once the first cracks form in the specimen nonlinear behavior governs its

Fig. 2 Response of wall specimen over time samples (load, shear and vertical displacement)

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response. As load cycles are applied in two opposite directions the wall develops two distinct sets of cracks. The integrity of the element is but already compromised in the opposite direction half of the cycle by the damages suffered under loading in the initial direction, which is shown by the fact that the specimen reaches the target load level (the control variable) at a larger horizontal displacement (see Fig. 3 top). As expected the specimen exhibits lower stiffness in the second or opposite load direction. As the direction of the shear force changes the first set of cracks closes and its existence has a reduced effect in the behavior of wall while the second group of cracks starts to open and becomes the dominating factor reducing the stiffness. Progressively degradation occurs, starting with concrete cracking, trough reinforcement yielding which accumulate over time until the wall fails due to concrete crushing in its central region. The identification of points at which the behavior of the wall changes significantly was based on considerations of its global response as shown in Fig. 3. Cracks in concrete behave differently based on their width. While micro-cracks close easily wider cracks may remain open due to aggregate interlocking on shifted adjacent surfaces and can require additional force to be closed. As the wall undergoes increasing shear distortion this translates into increasing vertical displacements which cause existing cracks to open and grow. Based on this we can consider the first major crack to have formed when vertical displacement start to ramp up (see Fig. 4). From this point on the vertical displacements also start to accumulate with each load cycle. When crack width extends beyond the yield strain of the reinforcement this goes into yielding. Once reinforcement yielding occurs, crack closure becomes difficult. We may postulate that the reinforcement first yields in a cycle at the end of which upon unloading there remain non-recovered vertical displacements. Plastic deformation of the reinforcement prevents closing of the crack to a higher degree and this manifests itself in larger residual vertical displacements. It can be seen in Figs. 3 and 4 that residual vertical displacements at the end of the cycle exhibit a stepwise increase. The onset of this phenomenon is considered to be the initiation of reinforcement yielding. For the data shown it corresponds to about 3.75 ‰ vertical elongation and 2.3 ‰ shear distortion. The so-called pinching effect of the load–displacement curve around the origin (Fig. 3 top) can be explained by the existence of partially open cracks after unloading. Upon load reversal first the open cracks are being closed. During this phase the wall provides lower stiffness. Friction along the crack surface is low initially but increases as the crack closes and surfaces become locked. Resistance to sliding increases and so does the overall stiffness of the element.

3 Numerical Model and Analysis Using the work diagrams gained from laboratory testing a numerical model has been developed which is based on hysteresis rules introduced originally by Takeda [7].

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Fig. 3 Load–displacement curve of the wall (top), cumulative vertical displacements over shear displacements (bottom)

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Fig. 4 Cumulative vertical displacements over shear displacements—detail

The constitutive model was adopted within a single degree of freedom mass-spring system representing the shear wall as a single element. The equation of motion was taken in the dimensionless form of μ¨ + 2ζ ωn μ˙ + ωn2 f s (μ, μ) ˙ = −ωn2

u¨ g (t) ay

(1)

where μ is the ductility, ωn is the angular natural frequency of the inelastic system vibrating within its elastic range, ζ is the viscous damping ratio, f s is the dimensionless nonlinear force–deformation function, ay is the acceleration of the mass required to produce the yield force [8] and üg is the ground acceleration. The constitutive relation is defined by a discrete piecewise linear model, which was initially developed with a trilinear backbone as shown in Fig. 5, where YI stand for yielding, LO for loading, UN for unloading, PI for pinching, QE for quasi-elastic. A detailed description of the model can be found in [6]. Here we describe in short its basic features. The model features four steering parameters separately for the positive and the negative quadrants of the work diagram. These are linked to the unloading stiffness k u , reloading point Y, pinching stiffness k s and pinching point P. In the advanced model the steering parameters were extended to a function form as it was observed that they do not remain constant during progressive deterioration of the wall.

Collapse of Shear Wall—Experimental …

23

Fig. 5 Nonlinear hysteretic constitutive model

The relation governing the steering parameters is based on the hyperbolic tangent function and has the feature that it gives a smooth transition within the transition interval (defined in terms of ductility μ) and a near constant value outside this interval. Finally, the model was extended to quadrilinear backbone to better represent test data. The performance of the tuned model which was validated on different test data can be seen in Fig. 6. The calibrated numerical model was used to run an earthquake simulation of a theoretical shear wall which behaves analogously to the tested walls. The strong motion record for the 1985 Mexico earthquake was selected which had a maximum acceleration of 0.103 g and a maximum velocity of 0.318 m/s and was of an unusually long duration. The response of the developed nonlinear model is compared to a linear response of single degree of freedom (sDOF) system. The response of the models in terms of displacements, velocities and accelerations are plotted in Fig. 7. To analyze the response of the wall to ground accelerations the harmonic wavelet transform has been applied [9]. This analysis method yields a time frequency plot which can be very useful to follow the change in frequency response over time. It can be observed in Fig. 8 that until the excitation remains of lower amplitudes the wall responds within its elastic domain. Once the wall is pushed beyond the yield point and its stiffness decreases significantly the response period of the wall changes dramatically. In the present case it is from 4.0 Hz to about 1.5 Hz. Such significant

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Fig. 6 Numerical model output compared to test data in terms of force over ductility (left). Response of the numerical model to the 1985 Mexico earthquake within its “elastic” limit, i.e. only the part before going into yielding (right)

Fig. 7 Response to the 1985 Mexico earthquake of a linear sDOF system (left) and the nonlinear hysteretic model (right) in terms of displacements (top, [m]), velocities (center, [m/s]) and accelerations (bottom, [m/s2 ])

Fig. 8 Response to the 1985 Mexico earthquake of a linear sDOF system (left) and the nonlinear hysteretic model (right) in terms of frequency over time

Collapse of Shear Wall—Experimental …

25

shift of behavior may be perceived as a change of the action of the load on the structure. Meaning that if one uses a simplified normative classification for designing the structure, for example classified as having “load” or “displacement” controlled behavior, this changes during the earthquake excitation. This may indicate possible design safety margins that may be exploited by the wall during severe excitation.

4 Conclusions Massive reinforced concrete shear walls, classified as short walls, were tested in a way that forced pure shear response up until failure. Horizontal loading was applied through a self-equilibrating frame at the center line of the specimen. In addition vertical loads were applied and constraints on the horizontality of the top of the wall. The data was cleared and analyzed to gain understanding of the global behavior of the wall and to shed light into the governing mechanisms of reinforced concrete as applied to shear walls. An assessment of the vertical displacements which showed to be significant was used to determine the onset of cracking and reinforcement yielding. Based on the test data a numerical model has been developed. It is a piece-wise linear hysteretic constitutive model which was implemented within the frame of a single degree of freedom system for which shear displacement, in terms of ductility, is the governing parameter. Comparing linear and nonlinear time–frequency responses to strong ground motion showed that during the duration of an earthquake the behavior of the shear wall may change drastically and by this additional design margins may be exercised.

References 1. IAEA Homepage. https://www.iaea.org/newscenter/news/iaea-issues-report-kashiwazaki-kar iwa-nuclear-plant, last accessed 2021/4/1 2. Barda F (1972) Shear strength of low-rise walls with boundary elements. PhD dissertation, Lehigh University 3. Iliya R, Bertero V (1980) Effects of amount and arrangement of wall-panel reinforcement on hysteretic behavior of reinforced concrete walls. Rep No. UCB/EERC 80/04, Univ of California, Berkeley CA 4. Greifenhagen C, Lestuzzi P (2005) Static cyclic test on lightly reinforced concrete shear walls. Eng Struct 27(11):1703–1712 5. Rothe D (1992) Untersuchungen zum nichtlinearen Verhalten von Stahlbetonwandscheiben unter Erdbebenbeanspruchung, VDI-Forschrittsberichte, Reihe 4(117). VDI-Verlag, Düsseldorf 6. Beko A, Rosko P, Wenzel H, Pegon P, Markovic D, Molina FJ (2015) RC shear walls: full-scale cyclic test, insights and derived analytical model. Eng Struct 102:120–131 7. Takeda T, Sozen MA, Nielsen NN (1970) Reinforced concrete response to simulated earthquakes. J Struct Div ASCE 96(12):2557–2573 8. Chopra AK (2007) Dynamics of structures, 3rd edn. Prentice Hall, Englewood Cliffs NJ 9. Newland DE (1994) Harmonic and Musical Wavelets. Proc R Soc A 444(1922):605–620

Numerical Simulation of Water Impoundment at a High Arch Dam Site Christine Detournay, Guotao Meng, Jing Hou, Jianrong Xu, Zhao Cheng, Ryan Peterson, and Peter Cundall

Abstract In a former study, water impoundment at a high arch dam (located in southwest China) was simulated using fluid-mechanical elasto-plastic analyses to predict deformation mechanisms taking place at the scale of the valley (Hou et al., IOP Conf Ser: Earth Environ Sci 570:022033, 2020). In this continuation work, we review the findings, clarify the mechanism of valley deformation caused by changes in impoundment level, and present the results of a sensibility study that accounts for the presence of drainage volumes at the site. With no account for drainage volumes, the model predicts valley expansion along measuring lines upstream from the dam, and valley contraction downstream. Also, valley contraction increases with measuring line elevation. Valley deformation occurs during changes of impoundment; this reflects the mechanical response to the change in dead weight of standing impoundment water. Deformation also takes place at constant impoundment level; this is caused by large-scale water seepage under and around the dam in the valley banks. A decrease in impoundment level generates a relative increase in valley contraction in the model, both upstream and downstream of the dam. This behavior is explained by the mechanical relaxation of lateral pressure that acts on the valley banks when the impoundment level is reduced. With drainage domains (associated with tunnels and powerhouse caverns) accounted for in the model, it was expected that the associated drop in fluid pressure would reduce the predicted valley contraction in the dam vicinity. Indeed, in this case, a mechanism of valley expansion is predicted to develop along all monitoring lines in the model. Keywords Civil Engineering · Dam impoundment · Valley contraction/Expansion mechanisms · Numerical study · Fluid-mechanical analysis C. Detournay (B) · Z. Cheng · R. Peterson · P. Cundall Itasca Consulting Group, Inc, Minneapolis, MN 55401, USA e-mail: [email protected] G. Meng · J. Hou · J. Xu PowerChina Huadong Engineering Corporation Limited, Hangzhou 311122, Zhejiang, China G. Meng Hydro-China Itasca Research and Development, Hangzhou 311122, Zhejiang, China © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. A. Wahab (eds.), Proceedings of the 4th International Conference on Numerical Modelling in Engineering, Lecture Notes in Civil Engineering 217, https://doi.org/10.1007/978-981-16-8185-1_3

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1 Introduction The Jinsha River (a tributary of the Yangtze river), in the Southwest of China, is the site of several large arch dams, with heights greater than 200 m, designed and successfully constructed within the last 20 years. These dams are part of large-scale hydro-power plan projects located in sites characterized by high steep slopes, the presence of large underground caverns at great depth, and associated tunnels. Line monitoring of valley width during dam impoundment has indicated that valleys experience contraction at the site of some of those projects. The observed valley contraction during filling of the dam is a phenomenon quite specific to high dams, and several potential mechanisms have been proposed to explain it, including rock mass creep, coupled thermal creep effects, and subsidence caused by a break in confined aquifer at depth; however, there is little consensus on its root cause [1–3]. The focus of this ongoing numerical work [4–6] is to explore if large-scale seepage after dam impoundment could cause valley contraction and if a similar mechanism could develop at the Baihetan site. Recent findings from coupled large-scale seepage and deformation simulations performed with FLAC3D [7] are reported here.

2 Study of Basic Mechanisms Generic impoundment scenarios were analyzed in 2D as part of this project. The poro-elastic study, conducted using coupled fluid-mechanical simulations in FLAC [8], highlighted two competing mechanisms associated with dam impoundment: a “mattress effect” caused by the mechanical pressure applied by the impoundment water load, and a “swelling effect” caused by the large-scale seepage and associated changes in fluid pressure that would occur in the valley after impoundment [9]. These competing behaviors, associated with undrained and drained responses, are illustrated by the results of 2D simulations in Figs. 1 and 2.

Fig. 1 Illustration of 2D mattress effect

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Fig. 2 Illustration of 2D swelling effect

The short-term “mattress” effect (caused by the added water load) is conceptualized by pushing on a bedspring, thereby causing instantaneous nearby motion downwards and inwards. On the other hand, the “swelling” effect, caused by pore pressure increase associated with seepage under the valley, generates time-dependent upward and outward movement in the 2D simulations. The impoundment water exerts pressure on both the river bottom and the riverbanks. As a result, the mattress effect manifests itself in a combination of valley contraction at high elevations (as pointed out earlier), and a valley expansion at low depth, below the impoundment water level. Both mechanisms are represented in Fig. 1. The swelling effect generates displacements that are opposite, and significantly smaller compared to those of the mattress effect in the simulations. The results of the 2D simulations showed that (with the model and properties used for the runs) valley contraction was larger at a higher height and larger in the short term after impoundment.

3 Numerical Model The FLAC3D poro-elasto-plastic model is 4 km long by 3 km wide. The model stratigraphy, the dam, and the discrete features (faults and bedding planes) are shown in Fig. 3. The dam body is not discretized in detail, however, both dam self-weight and the grout curtain are included in the simulations. The system of reference axes is centered at the base of the dam in the model, and positive y-values denote downstream distance. The rock mass includes Basalt, Columnar Basalt, Sandstone, and Weathered layers as well as fractures and bedding planes. The constitutive behavior is simulated using a Mohr–Coulomb model. The properties for the simulations are listed in Tables 1, 2 and 3.

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Fig. 3 FLAC3D model with stratigraphy, dam, and structures for the 3D analyses Table 1 Rock mass mechanical and hydraulic properties Density (kN/m3 )

Young’s modulus (GPa)

Poisson’s ratio

f

c (MPa)

Permeability (cm/s)

Strong unloading

25

3.5

0.31

0.75

0.55

3 × 10–3

Weakly weathering zone

26

8

0.27

0.95

0.95

8 × 10–4

Fresh zone

27

15

0.24

1.20

1.30

5 × 10–5

Strong unloading

22

1.5

0.35

0.40

0.10

5 × 10–3

Weakly weathering zone

24

3

0.32

0.60

0.50

8 × 10–4

Fresh zone

25

7

0.30

0.75

0.70

1 × 10–4

Strong unloading

22

1

0.34

0.45

0.25

1 × 10–3

Weakly weathering zone

24

3

0.31

0.65

0.55

2 × 10–4

Fresh zone

26

6

0.28

0.75

0.70

5 × 10–5

Rock stratum Basalt

Tuff

Sandstone

Table 2 Mechanical properties of discrete features Rock stratum

Thickness (cm)

Deformation modulus (GPa)

Poisson’s ratio

f

c (MPa)

Bedding planes

10–40

0.04–0.25

0.25–0.32

0.25–0.45

0.04–0.15

Faults

15–60

0.1–0.25

0.25–0.28

0.35–0.40

0.04–0.1

Numerical Simulation of Water Impoundment … Table 3 Hydraulic properties of discrete features

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Rock stratum

Permeability coefficient (cm/s) Parallel

Vertical

Bedding planes

5.0 × 10–3 –1.0 × 10–2

5.0 × 10–4 –1.0 × 10–3

Faults

1.0 × 10–2 –1.5 × 10–2

1.0 × 10–3 –1.5 × 10–3

The permeability of the grout curtain is 10−5 cm/s. The Biot coefficient of Basalt, Sandstone, and Tuff are 0.5, 0.75, and 1, respectively. The simulated impoundment schedule is plotted in Fig. 5 of Sect. 4, where the red dot indicates the time corresponding to the contour plots.

4 Simulation Results 4.1 Seepage Domain The in-situ phreatic surface level on a vertical plane located at y = 250 m in the model is compared to the measured field value, shown by the yellow trace in Fig. 4. Contours of saturation in situ and at the (maximum) impoundment level of 825 m in the simulation are plotted for comparison to the left and right in the figure, respectively. (The dam is not present at the in-situ state in the model, it is shown there for reference only.) As seen on the left plot in the figure, the match between simulated and measured phreatic surface level in situ is good. Comparison of left and right plots shows that the impact of impoundment on saturation is apparent upstream of the dam and more subtle downstream, as expected.

Fig. 4 Saturation contours and trace of measured in-situ water table at y = 250 m

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Fig. 5 Excess pore pressure contours at times shown by the red dot on the impoundment graph

4.2 Excess Water Pressure on Cut Plane Along the Valley Axis Excess water pressure contours on a vertical plane aligned with the valley axis are plotted in Fig. 5 at four times, indicated by the red dot on the graph of impoundment schedule. The excess pore pressure ‘front’ advances downstream (to the right on the Figure) as time progresses. The distribution of pore pressure has reached a steady state by the end of the 5-year simulation.

4.3 Predicted Overall x-displacements at the End of the 5-Year Simulation Figure 6 shows contours of x-displacements in the whole model 5 years after the start of impoundment. The model predicts valley contraction downstream of the dam and valley expansion upstream below the dam crest. Downstream of the dam, a sliding mechanism is observed towards the valley on the left bank, and superficial toppling is apparent at high elevations on the right bank. The development of two different deformation mechanisms on the right and left banks is attributed to the fact that the bedding planes are sloping toward the valley on the left bank and into the slope on the right bank.

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Fig. 6 Contours of x-displacement and displacement vectors at 5 years

A close-up view of displacements in the vicinity of the dam shows that the predicted displacement mechanism is controlled in part by the presence of the main faults in the model, see Fig. 7.

4.4 Predicted Valley Contraction Versus Time at Different Elevations The valley contraction predicted by the model along measuring lines is recorded versus time at different elevations. Representative measurement results at y-location –350 m upstream from the dam and 225 m downstream are plotted in Figs. 8 and 9. Overall, the model predicts valley expansion at measuring lines upstream from the dam and valley contraction at the downstream measuring lines. Valley contraction increases with elevation along most measuring lines in the model. Valley deformation occurs mainly during the changes of impoundment upstream and up to about 400 m downstream of the dam; this behavior is attributed to the mattress effect, caused by changes in dead weight of standing impoundment water. However, deformation also occurs during the period at a constant impoundment level; this behavior reflects the impact of the swelling effect caused by water seepage under and around the dam in the valley banks.

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Fig. 7 Contours of x-displacement and displacement vectors 5 years after the start of impoundment—close-up view downstream of the dam

Fig. 8 Impoundment level and valley contraction versus time—upstream monitoring line 4–4

A somewhat counter-intuitive behavior is observed whereby a decrease in impoundment level generates a relative increase in valley contraction. This (elastic rebound) mechanism is explained by the mechanical relaxation of lateral pressure that acts on the valley banks upstream of the dam when the impoundment level is

Numerical Simulation of Water Impoundment …

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Fig. 9 Impoundment level and contraction versus time—downstream monitoring line 7–7

reduced. Although the pressure relaxation (and its increase) originates upstream of the dam, it also affects the valley behavior downstream.

5 Addition of Drainage Volumes The impact of the presence of drainage volumes on the magnitude of valley contraction at the site is analyzed in this section. The results of a recent analysis are presented below. The drainage volumes associated with the tunnels behind the dam and with the powerhouse caverns at the site are pictured to the left in Fig. 10. They are represented in the FLAC3D model by the blue domains to the right in the figure. Drainage is activated at the start of impoundment in the model, which is achieved by fixing water pressure at zero in the drainage volumes represented in blue in Fig. 10. Thus, drainage in the model is supplied by a combination of the drop in phreatic surface and water supplied by the impoundment domain. The decrease in

Fig. 10 Drainage volumes at the site (left) and in the model (right)—in blue on both images

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Fig. 11 Map view of x-displacement contours and iso-surface of zero x-displacement shown in red at about 2.5 years—without (left) and with (right) drainage volumes

water pressure around the drainage volumes is responsible for rock mass ‘shrinkage’ there; this effect is expected to cause an increase in valley expansion. A map view with the iso-surface of x-displacement 2.5 years after the start of impoundment is compared without and with drainage volumes in Fig. 11. The red arrows on Fig. 11 indicate regions of x-displacements away from the river valley, and the blue arrows indicate domains where valley contraction is predicted. The model with drainage volumes predicts valley expansion both upstream and downstream of the dam, while convergence is only predicted at high elevations along the riverbanks. The sliding mechanism towards the valley observed downstream on the left bank and the superficial toppling at high elevations on the right bank identified in Fig. 7 are no longer there when the drainage volumes are considered in the FLAC3D model. A close-up view of the displacement mechanisms in the vicinity of the dam is shown in Fig. 12. The plots in Figs. 11 and 12 show that with drainage volumes accounted for, the predicted displacement mechanisms are only weakly controlled by the presence of the main faults in the model.

6 Conclusions Coupled fluid-mechanical elasto-plastic simulations were carried out to simulate the effect of dam impoundment at the site of a high dam in Southwest China.

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Fig. 12 Close-up view of x-displacement contours and displacement vectors downstream of the dam about 2.5 years after the start of impoundment—without (left) and with (right) drainage volumes

A 2D generic study has highlighted two competing mechanisms associated with dam impoundment: the “mattress effect” and the “swelling effect”. The first is a short-term effect caused by the mechanical pressure applied by the impoundment water load upstream of the dam. The second is a time-dependent effect caused by the large-scale seepage and associated changes in fluid pressure that would occur in the valley after dam impoundment. These basic mechanisms help explain the valley behavior observed after impoundment in high dams.

6.1 Results Without Drainage Domains Downstream of the dam, the valley deformation is controlled by large structural units (faults and bedding planes); the model predicts displacement towards the valley along a major fault and bedding plane on the left bank and large, superficial toppling on steep slopes on the right bank. The maximum amount of valley contraction along the monitoring lines does not exceed 5 cm in the simulations. Valley expansion is predicted by the model along most measuring lines upstream from the dam, while the estimate is for valley contraction at the downstream measuring lines. Also, valley contraction is generally predicted to increase with measuring line elevation. From upstream to a distance of about 400 m downstream of the dam, the valley deformation occurs mainly during the changes of impoundment; this behavior is attributed to the mattress effect caused by the change in dead weight of standing impoundment water. However, deformation also occurs during the period at a constant impoundment level, and this behavior is caused by the increase of fluid pressure associated with water seepage under the dam and around in the valley banks. An interesting effect is predicted whereby a decrease in impoundment level generates a relative increase in valley contraction both upstream (where the effect is stronger) and downstream of the dam. This behavior (a manifestation of the mattress

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effect) is explained by the mechanical relaxation of lateral pressure that acts on the valley banks and valley bottom upstream of the dam when the impoundment level is reduced in the model.

6.2 Results with Drainage Domains The large drainage domain associated with tunnels present at the site and with the powerhouse caverns has been represented in the FLAC3D model. It was expected that the associated drop in fluid pressure would reduce the predicted valley contraction in the dam vicinity, and this is indeed the case. In fact, so much so that valley expansion is predicted by the model along all monitoring lines. The simulation results show the effectiveness of the drainage in reducing both the valley contraction and the dominance of structural features (faults and bedding planes) on the deformation mechanism in the model. The effect of drainage domains on model results needs further analysis. In a continuation study, we propose separating the effect of drainage on the initial water table (which may have already occurred during dam construction and well before the start of impoundment) from the drainage of water supplied by the impoundment. In addition, a more refined representation of drainage tunnels will be implemented. By accounting for a more realistic impact of drainage in the model, it is expected that the trends observed in the current study will be reduced to more realistic magnitudes. However, it is still expected that drainage will reduce valley contraction. Acknowledgements This work was conducted with the financial support of Mr. Hou, PowerChina Huadong Engineering Corporation Limited, Hangzhou 311122, Zhejiang, China. The help and collaboration of our friend and colleague Weijiang Chu is gratefully acknowledged. Without his enduring support, this work would not have been possible.

References 1. Liang G, Hu Y, Fan Q, Li Q (2016) Analysis on valley deformation of Xiluodu high arch dam during impoundment and its influencing factors. J Hydroelectr Eng 35(9):101–110 2. Zhou Z, Li M, Zhuang C, Guo Q (2018) (2018) Impact factors and forming conditions of valley deformation of Xiluodu Hydropower Station. J Hohai Univ (Natural Sciences) 46(6):497–505 3. Zhang G, Cheng H, Zhou Q, Liu Y (2019) Analysis of mechanism of valley creep deformation of high arch dam during impoundment. China Sci Pap 14(1):77–84 4. Detournay C, Cheng Z, Peterson R, Cundall PA (2020) Baihetan Dam—stress and seepage analysis final report, phase 1”, Itasca Consulting Group, Inc., Report to PowerChina Huadong Engineering Corporation Limited, 2-6179-01:20R01. Minneapolis, Minnesota 5. Hou J, Xu J, Meng G, Chu W, Cheng Z, Detournay C, Cundall PA (2020) Analysis of the effect of dam impoundment at the Baihetan site using coupled fluid-mechanical elasto-plastic simulations. IOP Conf Ser: Earth Environ Sci 570:022033

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6. Meng G, Detournay C, Cundall P (2020) Formulation and application of a constitutive model for multijointed material to rock mass engineering. Int J Geomech 20(6) 7. Itasca Consulting Group, Inc. (2018) FLAC3D—Fast Lagrangian analysis of Continua in three dimensions, Ver. 7.0. Minneapolis: Itasca 8. Itasca Consulting Group, Inc. (2015) FLAC—Fast Lagrangian analysis of Continua, Ver. 8.0. Minneapolis: Itasca 9. Cundall P (2020) The art of numerical modeling in geomechanics. Geo-Congress 2020 GSP 321. ASCE

3-D Forward and Inverse Scattering Analyses for Cavity in Viscoelastic Media Using Convolution Quadrature Time-Domain Boundary Element Method Haruhiko Takeda, Takahiro Saitoh, and Sohichi Hirose Abstract In this research, forward and inverse scattering analyses for a cavity in 3-D viscoelastic media are presented. After the formulation for a forward analysis using the convolution quadrature time-domain boundary element method (CQBEM) for 3-D viscoelastodynamics is discussed, a linearized inverse scattering analysis based on the Born approximation for a cavity in 3-D viscoelastic media is introduced. The scattered wave forms in time-domain obtained by the CQBEM are transformed into those in frequency-domain, and their transformed data are utilized to reconstruct a cavity in 3-D viscoelastic media. Some numerical results for a cavity obtained by these forward and inverse scattering analyses are shown to validate the proposed methods. Keywords Convolution quadrature time-domain boundary element method · Viscoelastodynamics · Inverse scattering analysis · Born approximation

1 Introduction In this paper, forward and inverse scattering techniques are developed for a cavity in 3-D viscoelastic media. The boundary element method (BEM) has been developed as one of the effective numerical tools for wave propagation [1–3]. The classical time-domain boundary element method, in which the differential approximation is applied to time-discretization of the convolutions of a time-domain boundary integral equation, has a problem for the stability of numerical calculations [4]. To improve H. Takeda · T. Saitoh (B) Department of Civil and Environmental Engineering, Gunma University, 1-5-1, Tenjin, Kiryu, Gunma 376-8515, Japan e-mail: [email protected] URL: https://civil.ees.st.gunma-u.ac.jp/~applmech/ S. Hirose Department of Civil and Environmental Engineering, Tokyo Institute of Technology, 2-12-1, Ookayama, Meguro, Tokyo 152-8552, Japan © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. A. Wahab (eds.), Proceedings of the 4th International Conference on Numerical Modelling in Engineering, Lecture Notes in Civil Engineering 217, https://doi.org/10.1007/978-981-16-8185-1_4

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this numerical stability problem, the convolution quadrature time-domain boundary element method (CQBEM) is developed by applying the convolution quadrature method (CQM) proposed by Lubich [5], which is a method to improve the stability of the numerical calculations of the convolution integrals, to the classical time-domain BEM. In the past few years, the CQBEM has been applied to various problems for wave propagation [6, 7]. In addition, the boundary nonlinear problem [8], which is difficult to solve using the classical time-domain BEM due to the numerical instability, was solved by the CQBEM. In general, Laplace-domain fundamental solutions are required for the CQBEM formulation. The explicit fundamental solutions cannot be derived mathematically in time-domain for the problems where the dispersion property of wave propagation, such as viscoelastic and poroelastic wave propagation, exists [9]. However, in Laplace-domain, the closed-form of fundamental solutions can be obtained for viscoelastic and poroelastic wave propagation [10]. Thus, the advantages of the CQBEM, namely, the improvement of the numerical stability and the use of the Laplace-domain fundamental solutions, have expanded the application range of the time-domain BEM. Moreover, the CQBEM has been accelerated by the fast multipole method (FMM) [11, 12] to improve the computational efficiency by several researchers in the past decade [13–15]. On the other hand, the BEM has been utilized for some inverse problems. Especially, the inverse scattering analysis, proposed by Kitahara et al. [16], based on a boundary integral equation with the Born and Kirchhoff approximation [17], is used for the reconstruction of defect shape and position in a 3-D isotropic elastic material. In recent years, the inverse scattering technique was applied to more practical problems. For instance, Nakahata et al. proposed a 3-D flaw imaging method for ultrasonic testing with a matrix array transducer and the inverse scattering technique [18]. Tsunoda et al. tried to visualize the state of a steel-concrete interface using the inverse scattering technique [19]. After the development of a general fundamental solution for anisotropic elastodynamics [20], the inverse scattering technique was extended to the reconstruction of a cavity in anisotropic materials [21]. However, no numerical results for the viscoelastodynamics can be found in the literature. This is probably because it was difficult to solve viscoelastodynamic problems using the conventional time-domain BEM, as mentioned before. Moreover, from the point of view of new materials, various materials which possesses viscoelastic properties like a Carbon Fiber Reinforced Plastic (CFRP) have been developed in recent years, and attracted in many engineering fields. Therefore, it is significant to expand the application range of the inverse scattering technique to the reconstruction of defects in viscoelastic materials. In this paper, the forward and inverse scattering technique formulations for viscoelastic wave propagation are proposed. In the following section of this paper, a forward analysis using CQBEM for 3-D viscoelastic wave scattering is firstly discussed. Then, the inverse scattering formulation for the reconstruction of a cavity in a 3-D infinite viscoelastic medium is presented. In this research, the scattered wave data used for the inverse scattering are obtained by the forward analysis using CQBEM. As numerical examples, an incident wave, and the scattering by a cavity in a viscoelastic medium are demonstrated to validate the CQBEM. In addition, the

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reconstruction of a cavity in a 3-D infinite viscoelastic medium using the proposed inverse scattering technique is implemented. Finally, the perspective of our future research works is discussed.

2 Formulation for Forward Analysis for a Cavity in 3-D Viscoelastic Media Using CQBEM In this section, the formulation for a forward analysis based on CQBEM for 3-D viscoelastic wave scattering is discussed. The small index throughout this paper, such as ( )i , ranges 1–3, unless otherwise stated. Moreover, well known summation over repeated subscripts is used in this paper.

2.1 Problem to Be Solved Let us consider the viscoelastic wave scattering by a cavity Dc with boundary surface S in a 3-D infinite viscoelastic solid D, as depicted in Fig. 1. When the incident wave u iin (x, t) arrives at the surface S of a cavity Dc , scattered waves are generated by the interaction between the cavity Dc and the incident wave. Assuming the zero initial condition, u i (x, t = 0) = 0 and ∂u i (x, t = 0)/ t = 0, the governing equations for the displacement u i (x, t) at the position x and time t in time-domain, and the boundary condition are written as   1 μ(t) ∗ u˙ i, j j (x, t) + K (t) + μ(t) ∗ u˙ j,i j (x, t) = ρ u¨ i (x, t) 3 ti = 0 on S

in D

(1) (2)

where ρ is the density, the symbol * shows the convolution. ( ),i and (˙) show the partial derivative with respect to the space ∂/∂ xi and time ∂/∂t, respectively. In addition, μ(t) and K (t) are the relaxation functions for the shear modulus and the

Fig. 1 Viscoelastic wave scattering model

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bulk modulus, respectively, which satisfy the axiom of nonretroactivity μ(t) = 0 and K (t) = 0 for −∞ < t < 0. The traction component ti satisfies ti = 0 on the boundary surface of a cavity S due to the traction free condition, as shown in Eq. (2).

2.2 CQBEM for 3-D Viscoelastic Wave Scattering In general, the closed form of time-domain fundamental solutions (or Green’s function) for 3-D viscoelastodynamics cannot be obtained, as mentioned in Sect. 1. However, if we can obtain the fundamental solutions for this problem, the time-domain boundary integral equation for the displacement can be written as follows:  Ci j (x)u j (x, t) = u iin (x, t) −

Ti j (x, y, t) ∗ u j ( y, t)d S y

(3)

S

where Ci j is a free term which depends on the boundary surface of the position x and Ti j (x, y, t) is the double layer kernel for the 3-D viscoelastodynamics in the time-domain. As mentioned before, the fundamental solutions for viscoelastic wave propagation cannot be obtained by an explicit form in time-domain. Therefore, in this research, the CQM, proposed by Lubich [5], is applied to the time-convolutions of the boundary integral equation (3) for the time discretization. Discretizing the smooth boundary surface S into M piecewise constant boundary elements and considering the limit as x ∈ D to x ∈ S, discretized time-domain boundary integral equation with the time increment t at nth time step can be written as follows: M  1 u i (x, nt) + Bi0j (x, yα )u j ( yα , nt) 2 α=1

= u iin (x, nt) −

M  n−1 

α α Bin−k j (x, y )u j (kt)

(4)

α=1 k=1

where yα is the representative of each boundary element and u αj (kt) is kth timestep displacement at the position yα . Moreover, Bimj (x, y) is the influence function, which is defined as follows: Bimj (x, y) =

L−1  R−m  2πiml Tˆi j (x, y, sl )e− L d S y L l=0

(5)

S

where sl is the Laplace-parameter defined as sl = δ(zl )/t, i is the imaginary unit. R, L, and δ(zl ) are the CQM parameters [5]. Since Eq. (5) is expressed in the form of Fourier transform, the FFT can be applied to Eq. (5) to accelerate the computational time when the total time-step number N is equal to the CQM parameter L. Moreover,

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Tˆi j (x, y, s) is the double layer kernel for 3-D viscoelastodynamics in the Laplacedomain, which is obtained by the Laplace-transform of Ti j (x, y, t) in time-domain, as follows: ∗2 ˆ Tˆi j (x, y, s) = n j ( y)ρ(c∗2 L − 2cT )Uik,k (x, y, s)   + ρcT∗2 Uˆ i j,k (x, y, s) + Uˆ ik, j (x, y, s) n k ( y)

(6)

where n j ( y) is the outward unit normal vector component at the point y on the surthe complex phase velocities for longitudinal and face S. In addition, c∗L and cT∗ are √ √ transverse waves, given by c∗L = (K ∗ (s) + (4/3)μ∗ (s))/ρ and cT∗ = μ∗ (s)/ρ, respectively. The complex bulk modulus K ∗ (s) and the complex shear modulus μ∗ (s) ˆ respectively, where Kˆ (s) and are calculated by K ∗ (s) = s Kˆ (s) and μ∗ (s) = s μ(s), μ(s) ˆ are Laplace transforms of relaxation functions K (t) and μ(t), respectively. In addition, Uˆ ik (x, y, s) is the displacement fundamental solution for 3-D viscoelastodynamics in the Laplace-domain as follows: Uˆ ik (x, y, s) =

1 4π μ∗ (s)



e

− cs∗ r T

r

c∗2 ∂ 2 δik − T2 s ∂ yi ∂ yk



e

− cs∗ r T

r

−

e

− cs∗ r



L

r

(7)

where δik is a Kronecker delta and r is calculated by r = |x − y|. As shown in Eqs. (6) and (7), the closed form of the Laplace-domain fundamental solutions for 3-D viscoelastodynamics can be obtained analytically. At the nth time step in Eq. (4), the incident wave term u iin (x, nt) and the remaining term in the right hand side of Eq. (4) have been already known in advance. Therefore, the displacements u i (x, t) on the boundary surface S and in the domain D can be obtained by solving Eqs. (3) and (4), according to the initial and given boundary conditions, step-by-step from the first time step at n = 0.

3 Formulation for 3-D Inverse Scattering Analysis for a Cavity in Viscoelastic Media In this section, the formulation for an inverse scattering analysis based on the Born approximation [17] is presented to estimate the location and the shape of a cavity Dc in a 3-D infinite viscoelastic medium D.

3.1 Problem Setting It is assumed that the incident wave u iin (x, t) is transmitted from sufficiently far from the point o, and the scattered waves by a cavity Dc are obtained at the observation

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Fig. 2 Inverse scattering analysis model

point x as shown in Fig. 2. This problem setting is called the pulse-echo method [22] in the field of the ultrasonic nondestructive evaluation. In this paper, only the longitudinal scattered wave component is utilized to reconstruct a cavity Dc .

3.2 Formulation for 3-D Inverse Scattering Analysis Taking the Fourier transform of Eq. (2) with respect to time t, the governing equation in frequency-domain for viscoelastic wave propagation is obtained as follows:   1 μ˜ ∗ (ω)u˜ i, j j (x, ω) + K˜ ∗ (ω) + μ˜ ∗ (ω) u˜ j,i j (x, ω) 3 = −ρω2 u˜ i (x, ω)

in D

(8)

where ω is the angular frequency. In addition, μ˜ ∗ (ω) and K˜ ∗ (ω) are complex relaxation function for the shear modulus and the bulk modulus, respectively, in frequency-domain. Assuming the traction free boundary condition on the boundary surface S, as shown in Eq. (2), and the total displacement u˜ i (x, ω), which is the sum of the incident wave field u˜ iin (x, ω) and the scattered wave field u˜ isc (x, ω) (u˜ i (x, ω) = u˜ iin (x, ω) + u˜ isc (x, ω)), the boundary integral equation for the scattered wave u˜ isc (x, ω) in 3-D viscoelastodynamics in frequency-domain can be obtained as follows:  sc u˜ i (x, ω) = − T˜i j (x, y, ω)u˜ j ( y, ω)d S y (9) S

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where T˜i j (x, y, ω) is the double layer kernel for 3-D viscoelastodynamics in frequency-domain as follows: ∂ ˜ Ui p (x, y, ω) T˜i j (x, y, ω) = n k ( y)G˜ ∗jkpq (ω) ∂ yq

(10)

where G˜ ∗jkpq (ω) is a complex relaxation function for isotropic and homogeneous viscoelastic media as follows:   2 (11) G˜ ∗jkpq (ω) = K˜ ∗ (ω) − μ˜ ∗ (ω) δ jk δ pq + μ˜ ∗ (ω)(δ j p δkq + δ jq δkp ) 3 In addition, U˜ i p (x, y, ω) is the fundamental solution for displacement for the 3-D viscoelastodynamics in frequency-domain as follows: 1 U˜ i p (x, y, ω) = 4π μ˜ ∗ (ω)



˜

1 ∂2 e ik T r δi p + 2 r k˜ T ∂ yi ∂ y p



˜

˜

e ik T r e ik L r − r r

 (12)

where k˜ T and k˜ L are the complex wave number for transverse and longitudinal waves in viscoelastic wave propagation, respectively. Normally, scattered waves are observed far away from a defect, and the propagation distance between a defect and an observation point is much longer than a defect length in the field of the ultrasonic nondestructive testing. Therefore, the longitudinal and shear wave components of the scattered wave can be distinguished in the far field [23]. Applying the far-field approximation (r ≈ |x| − xˆ · y) to Eq. (12), the boundary integral equation for the scattered waves defined in Eq. (9) is rearranged as follows: u˜ isc (x, ω) = −

ik˜ L BiLjk 4π |x|

˜

eik L |x|



˜

n k ( y)e−ik L xˆ · y u˜ j ( y, ω)d S y

(13)

S

where xˆ is a unit vector of x. BiLjk is given by BiLjk = − (1 − 2κ 2 )δ jk + 2κ 2 xˆ j xˆ k xˆ i , where κ = k˜ L /k˜ T . In Eq. (13), the displacement u˜ j ( y, ω) at the point y on the boundary surface S of a cavity is unknown. Therefore, the Born approximation is applied to the unknown displacement u˜ j ( y, ω), and the unknown displacement u˜ j ( y, ω) can be replaced by the incident wave u inj ( y, ω). In this research, the incident wave u inj ( y, ω) is given as follows: ˜

u˜ inj ( y, ω) = F v (k˜ L )dˆ j e−ik L xˆ · y

(14)

where dˆ j is the polarization vector component. In addition, the function F v (k˜ L ) is given by

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u 0 ω0 2 (1 − eiωT0 ) F v (k˜ L ) = 2iω(ω2 − ω0 2 )

(15)

where u 0 is the incident wave amplitude, ω0 is the central angular frequency, and T0 is the corresponding period of the incident wave defined by T0 = 2π/ω0 . Next, applying Gauss’s divergence theorem and the characteristic function ( y) defined as  1 ( y ∈ Dc ) (16)

( y) = 0 for otherwise to Eq. (13), and extending the integration range of Eq. (13) into all space A, we can obtain the following equation for the scattered wave: u˜ isc (x, ω)

 k˜ L2 dˆi F v (k˜ L ) ik˜ L |x| ˜ e =−

( y)e−2ik L xˆ · y d Vy 2π |x|

(17)

A

ˆ the characteristic function In Eq. (17), considering the K space defined by K = 2k˜ L x,

( y) in Eq. (17) can be derived using the inverse Fourier transform as follows: 2π π ∞

( y) = 0

0

0

2|x|u˜ isc (x, ω) sin θ −ik˜ L (|x|−2 xˆ · y) ˜ e d k L dθ dϕ π 2 F v (k˜ L )dˆi

(18)

where ϕ and θ are the azimuth and zenith angles, respectively, as shown in Fig. 2. Recalling Eq. (16), we note that the characteristic function ( y) takes the value

( y) = 1 inside a cavity. Therefore, we can reconstruct the shape and position of a cavity if we calculate the right-hand side of Eq. (18). Equation (18) includes the frequency-domain scattered wave u˜ isc (x, ω). The scattered wave u˜ isc (x, ω) in frequency-domain is obtained by using the Fourier transform of the corresponding scattered wave u isc (x, t) in time-domain obtained by the CQBEM as shown in the previous section.

4 Numerical Results Numerical results are shown in this section. First, the viscoelastic model used in the following numerical results is briefly presented. Next, the results for the forward analysis for 3-D viscoelastic wave scattering by a cavity using CQBEM are shown. For comparison, the numerical results for forward analysis for an isotropic elastic medium which possesses no viscoelastic properties are also shown. After that, numerical results for the inverse scattering analysis for the reconstruction of a cavity in a 3-D infinite medium are demonstrated.

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Fig. 3 Three-element standard linear viscoelastic model

4.1 Three-Element Standard Linear Viscoelastic Model In general, there are some modeling methods for dealing with the viscoelastic property. In this paper, the three-element standard linear model [24] with two spring constants, μ1 , μ2 , and the viscosity coefficient η, as shown in Fig. 3, is considered as viscoelastic model. In the three-element standard linear model, the complex shear modulus in the Laplace-domain is given by μ∗ (s) =

1 + sτσ μR 1 + sτ

(19)

where τ and τσ are the stress relaxation time and the strain relaxation time, respectively, and expressed as follows:  τσ = η

1 1 + μ1 μ2

 , τ =

η μ1

(20)

In Eq. (20), μ2 = μ R is the relaxation shear modulus when t → ∞. Also, μ0 is the initial relaxation shear modulus when t → 0 and is calculated by using the stress relaxation time τσ and τ as follows: μ0 = lim μ∗ (s) = μ R s→∞

τσ τ

(21)

In the following numerical results, the three-element standard linear model is considered and the viscoelastic property is given for only shear modulus.

4.2 Numerical Results for Forward Analysis The scattering of an incident plane wave by a cavity with radius a, as shown in Fig. 4, in a 3-D viscoelastic medium is solved using the CQBEM. The cavity with radius

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Fig. 4 Viscoelastic wave scattering by a cavity

a is discretized into 384 boundary elements using a piecewise constant approximation. The CQM parameters, N and L, are given by N = L = 128 to use the FFT in Eq. (5). The ratio of the relaxation shear modulus μ R to the initial shear modulus μ0 is μ R /μ0 = 0.85, and the ratio of the bulk modulus K to the initial shear modulus μ0 is K√/μ0 = 5/3. In addition, the initial longitudinal wave velocity c L0 is given by c L0 = (K + (4/3)μ0 )/ρ, the corresponding period T0 is given by T0 = 2a/c L0 , τσ = 0.5T0 and τ = 17T0 /40. The non-dimensionless time increment c L0 t/a is set as c L0 t/a = 0.08 unless otherwise noted. Incident Plane Wave. The incident plane wave u iin (x, t) in time-domain propagating to x1 direction used in the following analysis is defined by using the inverse Fourier transform F −1 for Eqs. (14) and (15) as follows: u iin (x, t)

= u 0 δi1 F

−1



ω0 2 (1 − eiωT0 ) ik˜ L x1 e 2iω(ω2 − ω0 2 )

 (22)

Figure 5 shows the time histories of the incident waveforms obtained by solving Eq. (22) at x1 /a = 0.0, 5.0, 10.0, and 15.0. For comparison, the corresponding incident waveforms propagating in the isotropic elastic media without viscoelastic properties are also shown in Fig. 5 by plotting dotted lines. As shown in Fig. 5, the incident plane wave displacements u 1 /u 0 are attenuated as the incident waves propagate to x1 direction due to the viscoelastic effect with the advancement in time. CQBEM Accuracy Confirmation. Next, the accuracy of the CQBEM is confirmed before the numerical results for scattering problems are shown. To verify the accuracy of the CQBEM, the case where the cavity Dc is filled with the same material as

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Fig. 5 Incident wave forms propagating in a three-element standard linear viscoelastic medium calculated by Eq. (22) at x1 /a = 0.0, 5.0, 10.0 and 15.0

Fig. 6 Time histories of displacements u 1 /u 0 at the points B and C in Fig. 4

surrounding viscoelastic solid D is considered. In this case, scattered waves do not propagate and it can be considered as a just transmission problem in which only incident wave propagates. Therefore, the solutions obtained by the CQBEM for this problem are equivalent to the incident plane wave. Figure 6 shows the solution obtained by the CQBEM and incident wave calculated by Eq. (22) at the points B and C in Fig. 4 using difference time increments t = 0.04 and t = 0.008. The time nt/T0 = 0.0 is set at the moment the incident plane wave in Eq. (22) arrives at x1 /a = −1.0. As shown in Fig. 6, the accuracy of the displacements u 1 /u 0 at the point B seem to be good for both time increments t = 0.04 and t = 0.008. On the other hands, the displacements u 1 /u 0 at the point C obtained by the CQBEM with smaller time increments t = 0.008 are in good agreement with the exact solutions which are the same as the incident wave displacement in Eq. (22). Therefore, it can be confirmed that the CQBEM can produce high precision solutions for the smaller time increment t = 0.008.

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Fig. 7 Time histories of displacements u 1 /u 0 at the points A, B, C and D in Fig. 4 a isotropic elastic medium and b viscoelastic medium

Viscoelastic Wave Fields Around a Cavity. Figure 7a–d show the time histories of u 1 /u 0 at the points A, B, C and D in Fig. 4, respectively. In Fig. 7, the solid and dashed lines show the time histories of u 1 /u 0 in a viscoelastic and an isotropic material without viscosity, respectively. The time nt/T0 = 0.0 is set at the moment the incident wave in Eq. (22) arrives at the point A, x1 /a = −2.0. The wave velocity in an isotropic elastic medium is set as initial longitude wave velocity c L0 for a viscoelastic medium. As seen in Fig. 7a, the displacements u 1 /u 0 at the point A in a viscoelastic medium for 0.0 ≤ nt/T0 ≤ 1.0 are almost the same as those u 1 /u 0 in an isotropic elastic medium. The peak values of displacements u 1 /u 0 at about nt/T0 = 1.5 and nt/T0 = 2.2, however, are decreased due to viscoelastic effects. In addition, it is clearly confirmed that the displacements u 1 /u 0 observed at the points B, C and D are also attenuated due to the viscoelastic effect, as shown in Fig. 7b–d. Figures 8 and 9 show the dimensionless total displacements |u|/u 0 around the cavity in an isotropic and a viscoelastic medium, respectively. Figures 8a–d and 9a–d in each denote the results at nt/T0 = 1.28, 2.56, 3.84 and 5.12, respectively.

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Fig. 8 Total wave fields |u|/u 0 around the cavity in an isotropic elastic medium a nt/T0 = 1.28, b nt/T0 = 2.56, c nt/T0 = 3.84, and d nt/T0 = 5.12

The time nt/T0 is set at the moment the incident wave in Eq. (22) arrives at x1 /a = −5.0. As seen in Fig. 8 for the isotropic elastic medium, it can be confirmed that the incident wave and scattered waves propagate without attenuation. On the other hand, as shown in Fig. 9 for the viscoelastic medium, the incident wave and scattered waves decay as time progress compared with Fig. 8 because of the viscoelastic properties.

4.3 Numerical Results for Inverse Scattering Analysis Next, numerical results for the proposed inverse scattering technique are shown in this section. The defect to be reconstructed in this inverse scattering technique is a cavity Dc with radius a in the infinite region D, as shown in Fig. 2. The scattered waveforms used for this inverse scattering analysis to reconstruct the cavity are computed by using the CQBEM. The scattered waves are observed at the several

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Fig. 9 Total wave fields |u|/u 0 around the cavity in a viscoelastic medium a nt/T0 = 1.28, b nt/T0 = 2.56, c nt/T0 = 3.84, and d nt/T0 = 5.12

observation points x in Fig. 2 where R = 12a distance away from the center of a cavity, o, and 10◦ angular increment for ϕ and θ is considered. In the calculation of scattered waves, the cavity with radius a is discretized into 384 boundary elements. The total number of time-steps N is given by N = L = 1024 and the time increment c L0 t/a is set as c L0 t/a = 0.08. The viscoelastic parameters μ R /μ0 = 0.85, K /μ0 = 5/3, τσ = 0.5T0 , and τ = 17/40T0 are given to calculate the scattered waves using the CQBEM. A thread-parallel implementation using OpenMP is applied to reduce the computational time for calculating scattered waves. Figures 10a, 11a and 12a show the reconstruction results obtained by the proposed inverse scattering technique in x2 − x3 plane for the cavity in the viscoelastic medium, as shown in Fig. 2. On the other hand, Figs. 10b, 12b and 12b show the corresponding results in 3-D view. The characteristic function ( y)/ max defined in Eq. (16) is plotted in each figure to reconstruct the cavity where max is the maximum value of

( y) in each case. In addition, the white dashed circle in Figs. 10a, 11a and 12a shows actual shape and position of the cavity. In Fig. 10, the shape and position of the cavity is reconstructed accurately under the viscoelastic effect using the scattered wave data

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Fig. 10 Shape reconstruction of the cavity in the viscoelastic material a x2 − x3 plane and b 3-D space view (0◦ ≤ ϕ ≤ 360◦ , 0◦ ≤ θ ≤ 180◦ )

Fig. 11 Shape reconstruction of the cavity in the viscoelastic material a x2 − x3 plane and b 3-D space view (0◦ ≤ ϕ ≤ 180◦ , 0◦ ≤ θ ≤ 180◦ )

obtained within 0◦ ≤ ϕ ≤ 360◦ and 0◦ ≤ θ ≤ 180◦ . On the other hand, when the observation points are limited as 0◦ ≤ ϕ ≤ 180◦ and 0◦ ≤ θ ≤ 180◦ , the upper half of the spherical shape and position of the cavity where the incident waves hit directly are reconstructed as shown in Fig. 11. When the observation points are limited as 0◦ ≤ ϕ ≤ 180◦ and 0◦ ≤ θ ≤ 90◦ , the position of the cavity is well estimated. However, the spherical shape of the cavity cannot be reconstructed accurately. Therefore, it is concluded that if scattered wave data are enough, the shape and position of a cavity in viscoelastic media can be reconstructed using the proposed inverse scattering technique.

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Fig. 12 Shape reconstruction of the cavity in the viscoelastic material a x2 − x3 plane and b 3-D space view (0◦ ≤ ϕ ≤ 180◦ , 0◦ ≤ θ ≤ 90◦ )

5 Conclusion In this paper, a forward analysis technique for 3-D viscoelastic wave scattering using the CQBEM and an inverse scattering technique for the reconstruction of a cavity in 3D viscoelastic media were presented. The effectiveness of the forward analysis using the CQBEM was confirmed by simulating scattered wave fields in 3-D viscoelastic media. Also, the proposed inverse scattering formulation for a cavity reconstruction was presented and the results are demonstrated for several cases. The proposed formulation for forward and inverse scattering analyses will be applied to the case for a crack in viscoelastic media in near future. In addition, a forward and an inverse scattering techniques are developed for a defect in materials with both anisotropic and viscoelastic properties. Moreover, the fast multipole method [11, 12] and H matrix method [25] will be used to obtain scattered wave data efficiently for the reconstruction of many cavities in 3-D viscoelastic media. Acknowledgements This work was supported by “Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures”, and “High Performance Computing Infrastructure” in Japan (Project ID: jh200052-NAH and jh210033-NAH) and the bilateral exchange research between Japan and India (project number: JPJSBP1 20207707).

References 1. Beskos D (1987) Boundary element methods in mechanics. North Holland 2. Ang WT (2007) A beginner’s course in boundary element methods. Universal Publishers, Boca Raton 3. Mansur WJ, Brebbia CA (1983) Transient elastodynamics using a time-stepping technique. In: Brebbia CA, Futagami T, TanakaM (eds) Boundary elements, pp 677–698

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4. Fukui T, Tani K (1992) Stability of time domain boundary element method in wave propagation problems. In: Kobayashi S, Nishimura N (eds) Boundary element methods, fundamentals and applications, pp 82–91 5. Lubich C (1988) Convolution quadrature and discretized operational calculus I. Numer Math 52:129–145 6. Abreu AI, Carrer JAM, Mansur WJ (2003) Scalar wave propagation in 2D: a BEM formulation based on the operational quadrature method. Eng Anal Bound Elem 27:101–105 7. Furukawa A, Saitoh T, Hirose S (2014) Convolution quadrature time-domain boundary element method for 2-D and 3-D elastodynamic analyses in general anisotropic elastic solids. Eng Anal Bound Elem 39:64–74 8. Maruyama T, Saitoh T, Hirose S (2017) Numerical study on sub-harmonic generation due to interior and surface breaking cracks with contact boundary conditions using time-domain boundary element method. Int J Solids Struct 126–127:74–89 9. Dominguez J (1992) Boundary element approach for dynamic poroelastic problems. Int J Numer Methods Eng 35:307–324 10. Schanz M (2001) Wave propagation in viscoelastic and poroelastic continua: a boundary element approach. Springer 11. Rokhlin V (1985) Rapid solution of integral equations of classical potential theory. J Comput Phys 60:187–207 12. Greengard L, Rokhlin V (1987) A fast algorithm for particle simulations. J Comput Phys 73:325–348 13. Saitoh T, Hirose S, Fukui T, Ishida T (2007) Development of a time-domain fast multipole BEM based on the operational quadrature method in a wave propagation problem. In: Minutolo V, Aliabadi MH (eds) Advances in boundary element techniques VIII. EC Ltd., Eastleigh, pp 355–360 14. Saitoh T, Hirose S, Fukui T (2009) Convolution quadrature boundary element method and acceleration by fast multipole method in 2-D viscoelastic wave propagation. Theoret Appl Mech Jpn 57:385–393 15. Schanz M (2018) Fast multipole method for poroelastodynamics. Eng Anal Bound Elem 89:50– 59 16. Kitahara M, Nakahata K, Hirose S (2002) Elastodynamic inversion for shape reconstruction and type classification of flaws. Wave Motion 36:443–455 17. Schmerr LW (1998) Fundamentals of ultrasonic nondestructive evaluation. Plenum Press 18. Nakahata K, Saitoh T, Hirose S (2006) 3-D flaw imaging by inverse scattering analysis using ultrasonic array transducer. Rev Prog Quant Nondestr Eval 34:717–724 19. Tsunoda T, Suzuki S, Saitoh T (2018) Application of linearized inverse scattering methods for the inspection in steel plates embedded in concrete structures. Rev Prog Quant Nondestr Eval 1949:040007 20. Wang C-Y, Achenbach JD (1994) Elastodynamic fundamental solutions for anisotropic solids. Geophys J Int 118:384–392 21. Saitoh T, Shimoda M, Inagaki Y, Hirose S (2016) Forward and inverse scattering analysis for defect in anisotropic plate using convolution quadrature time-domain boundary element method. J Jpn Soc Civ Eng A2 72(2):237–246 (in Japanese) 22. Rose JL (2008) Ultrasonic waves in solid media. Cambridge University Press 23. Achenbach JD (1987) Wave propagation in elastic solids. North Holland 24. Fung YC (1965) Foundation of mechanics. Prentice-Hall 25. Bebendorf M (2008) Hierarchical matrices: a means to efficiently solve elliptic boundary value problems. Springer-Verlag

Importance of the Geotechnical Variability in the Bearing Capacity of Shallow Foundations Through Random Fields Cristhian C. Mendoza , Jorge E. Hurtado , and Jairo A. Paredes

Abstract This study presents the variability influence of the geotechnical parameters using random fields for shallow foundations. Knowing the variation of bearing capacity can lead to a better foundation design. The present study was made through continuous-footing finite element models (FEM). These FEM models used a constitutive elastoplastic model that takes the geological history of the soil. The model’s parameters were generated as random fields with the decomposition matrix technique (Cholesky factorization). Later, these parameters were integrated into the FEM models. Then, more than 2400 FEM simulations were made. The simulations show the parameters that must be carefully obtained to reduce the variability of the load capacity. In addition, the results show that the soil compressibility parameters have a significant influence on the value of the bearing capacity. However, the compression parameters aren’t taken into account in the classical equations of bearing capacity. Keywords Random fields · Shallow foundations · Elastoplastic model

1 Introduction The influence of geotechnical parameters variability is the great importance in the design of foundation structures. However, understanding the importance of each parameter is not easy work. A common problem in geotechnical engineering is the bearing capacity of the shallow foundation. Nowadays, equations to bearing capacity only consider the parameters to shear resistance, but the behavior of foundations is a connected process between bearing capacity and settlements. This research used an elastoplastic model that takes the compressibility of the soil. In addition, the influence of geotechnical parameters variability in the bearing capacity of the shallow foundation was studied. C. C. Mendoza (B) · J. E. Hurtado · J. A. Paredes Facultad de Ingeniería y Arquitectura, Departamento de Ingeniería Civil, Universidad Nacional de Colombia Sede Manizales, Campus Palogrande, 170004 Manizales, Colombia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. A. Wahab (eds.), Proceedings of the 4th International Conference on Numerical Modelling in Engineering, Lecture Notes in Civil Engineering 217, https://doi.org/10.1007/978-981-16-8185-1_5

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The equation to bearing capacity was presented by Terzaghi in 1943 [1]. Then other proposes were made as those made by Hansen [2]; Meyerhof [3]; Vesic [4], and others. These equations are based on the Mohr-Columb criterion. A limitation of these equations is that they do not take into account the strains on the soil. Furthermore, the equations don’t take into account the effect of the variability of the geotechnical parameters. The variability of the geotechnical parameters was addressed in the researches by Lump [5]; Schultze [6]; Kirby [7]; Phoon and Kulhawy [8]; Griffiths and Fenton [9]; Ching et al. [10]; Zevgolis et al. [11]; Pua and Caicedo [12]; MolinaGómez et al. [13], among others. In addition, the variability had been applied in problems of geotechnical structures as foundations, slope stability, retaining walls, dams, tunnels, among others [14–22]. However, the applications of methodologies with variability in real design are low. One reason is that the safety factor covers the variability of the geotechnical parameters. However, a higher safety factor does not always have a lower probability of failure [23]. Then an interesting problem is bearing capacity with the inclusion of variability of the geotechnical parameters. The variability is currently being used using Monte Carlo methods, random fields, Bayesian analysis, among other methods [13, 21, 24– 28]. Many proposals are because a consensus has not been reached with the different techniques used and their results. In the present research, random fields were used from geotechnical parameters with the Cholesky decomposition technique. These parameters were created for the elastoplastic model with the Drucker-Prager rupture criterion and a cap. This model uses the compressibility of the soil using a cap. These fields were incorporated into FEM finite element simulations. Then 800 simulations were made with the Montecarlo technique. From the simulations were obtained the importance of the geotechnical parameters in the bearing capacity with more accuracy. In addition, the influence of the compression parameters on the bearing capacity is shown. Then this research shows which parameters should be obtained with greater care to reduce the variability of the bearing capacity in a shallow foundation.

2 Constitutive Model The elastoplastic model with the Mohr–Coulomb criterion is the most used in geotechnics [29, 30]. However, this model doesn’t take into account the compressibility behavior of the soil. This research used an elastoplastic model with a DruckerPrager criterion and a cap. The advantage is that this model use material compressibility. Their parameters can be obtained from triaxial tests and compression tests in an oedometer [30]. The model works in two parts. The initial part is the elastic range, where two parameters, Young’s Modulus E and Poisson’s ratio, μ are working to relate stresses and deformations. This part is valid until the stress paths reached the yielding envelope. Then, the second part starts to work with parameters that can be correlated to the friction angle of the soil φ and the cohesion c in the Mohr–Coulomb model

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Fig. 1 The yield surface in the Drucker and Prager model with cap

(see Fig. 1). The relations between the Mohr–Coulomb parameters and the DruckerPrager parameters are shown in Eqs. 1 and 2. Three parameters control the size and form of the cap: the eccentricity of the cap ellipse R, a constant K that controls the shape of the yield surface in the plane π, and pp , that controls hardening and softening of the material with the evolution of the isotropic compression law. This compression law used the virgin isotropic compressibility slope λ to the plane of the void ratio e, and the natural logarithm of p, κ is a parameter for the slope of the p unloading–reloading line of the material, e0 is the initial void ratio, and εv is the volumetric plastic strain, as shown in Eq. 3. tan β =

(1)

18c cos φ 3 − sin φ

(2)

λ−κ p ln 1 + e0 pp

(3)

d= εvp =

6 sin φ 3 − sin φ

This research uses Speswhite kaolin as the material. This material was used for the models because its mechanical behavior has been characterized in many other geotechnical studies [31, 32]. Table 1 have geotechnical parameters to kaolin divided into parameters from the shearing and compression behavior. Table 1 Parameters and variables used for kaolin Parameter

E

c

φ

μ

Unit

kPa

kPa

°

–

–

–

–

kPa

Mean

7500

15

23

0.34

0.8

0.18

0.03

250

Coefficient of variation

0.5

0.25

0.3

0.15

0.33

0.35

0.4

0.5

Standard deviation

3750

2.5

6.9

0.051

0.264

0.063

0.012

100

e0

λ

κ

pp

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Fig. 2 Boundaries conditions for analysis of shallow foundation

3 Random Analysis with Finite Elements 3.1 Finite Element Model The continuous footing was developed in a finite element model (FEM) with the ABAQUS program. The model is a function of the footing width, and their dimensions were a depth of five times the footing width (B = 1 m), a width of 10 times the footing width [29, 33]. The model geometry is shown in Fig. 2. The elements were used CPE4 elements (plane strain quadrilaterals, two dimensions, four nodes) were used to mesh the FEM model. The elements of the model have a size of dx = 0.1 m and dy = 0.1 m. The boundary conditions are a boundary condition fixed in all directions at the base of the FEM model; a boundary conditions to lateral sides fixed in the horizontal direction and free in the vertical direction; a condition to simulate the depth of the footing through the unit weight of the soil (17 kN/m3 ) per one meter of depth. The simulations were made in two steps. The initial step was a geostatic step that induces geostatic initial overburden stresses. The second step was created, a loading step with a constant strain rate up to a total vertical displacement of 0.18 m. This was made to simulate a load test at a constant strain rate.

3.2 Integration of the Random Field to the FEM Model The random fields were made by the decomposition Cholesky technique for the spatial distribution. The random numbers were obtained following a log-normal distribution. In addition, the random fields were created with Eq. 4, the values of

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the output variable Y. This equation is composed of the mean value μ, a vector ε of random values with a normal distribution, and a matrix L (defined in Eq. 5). Equation 5 shows the covariance matrix A of the field. Matrix A is the positive part of the decomposition of a lower triangular matrix with positive diagonal inputs. This concept is called Cholesky factorization. On the other hand, Matrix A (Eq. 6) has the autocorrelation distance in x and y, using L x and L y to take vertical and horizontal anisotropy. The autocorrelation distances in each direction are L x = 5.0 and L y = 1.0. Furthermore, the length in x was divided into 50 parts, and the length in y was divided into 25 parts.

Ai, j

Y = Lε + μ

(4)

A = LLT

(5)

⎤ ⎡  x 2 y 2  di, j di, j ⎥ ⎢ = σ 2 exp⎣ + ⎦ Lx Ly

(6)

The random fields with the Cholesky decomposition technique are generated with a normal distribution. A common alternative is to change the statistical parameters with log-normal parameters, as shown in Eqs. 7 and 8. Then, 200 random fields were generated for the geotechnical parameters shown in Table 1, with a log-normal distribution. Each random field matrix generated is loaded in the FEM model. This was made with a subroutine in Python language that automatically processes the data through some loops. After running the models, the important output variables (strains and stresses) are saved in an external file. The ultimate stress is obtained by the absolute limit settlement at 0.15 m from the load-deformation curves of the simulations. 

  2   σ  σln = ln 1 + (7) μ 1 μln = ln μ − σ 2ln 2

(8)

Figure 3a shows an example of a random field generated. Figure 3b shows the cumulative probability for the 200 random fields for the friction angle with statistical parameters from Table 1. The vector strains generated on the soil for the FEM model, as shown in Fig. 3c. From this figure, the influence of the random fields on the vector deformation of the FEM model has been observed where the magnitudes of the strain are not symmetric. In addition, non-symmetric failure mechanisms in the deformations are emphasized [15].

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Fig. 3 a Random fields for friction angle from Table 1. b Distributions of cumulative probability of random fields for friction angle. c Strain vectors generated from the FEM simulations with random fields

4 Results The analyses performed with the FEM model are 200 simulations varying all parameters (E, c, φ, μ, λ, κ, pp ) from Table 1, 200 simulations changing shear parameters (E, c, φ, μ), 200 simulations varying compression parameters (λ, κ, pp ), 200 simulations varying Young’s Modulus, 200 simulations varying cohesion, 200 simulations varying Poisson’s ratio, 200 simulations varying friction angle, 200 simulations varying the slope λ of the virgin isotropic compressibility line, 200 simulations varying the slope κ, 200 simulations varying the isotropic yielding stress pp , 600 simulations varying the autocorrelation distance (L x = 1.0, L y = 1.0; L x = 5.0, L y = 1.0; L x = 10.0, L y = 1.0). The full simulations for the present paper were 2400. Figure 3a shows the stress-settlement curves from FEM simulations. The simulations in Fig. 3a were performed by varying all parameters of the elastoplastic model from Table 1. From these curves, the ultimate stress was obtained with the absolute limit settlement method [34]. This method takes a limiting value of the settlement to obtain the ultimate stress. The limit settlement was taken 0.15 m. The boundary is where structures can begin to get damaged [35]. Figure 3b shows the density curve and histogram of ultimate stresses with a mean of 411.9 kPa and a standard deviation of 39.7 kPa (Fig. 4).

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Fig. 4 a Vertical stress-settlement curves from simulations with the Mohr–Coulomb rupture criterion and the Drucker-Prager rupture criterion. b Density curve and histogram of ultimate stresses from the FEM models

4.1 Shear and Compression Behaviors Figure 5 shows a higher mean value when the shearing parameters are varied. However, there is a larger standard deviation when all parameters were varying. The standard deviation when all parameters varying is 39.7 kPa, the shear parameters is 29.33 kPa, and the compression parameters are 23.4 kPa. The variation coefficient changes from 9.6% (all parameters), 6.9% (shear), and 5.9% (Compression). Then, the shear parameters have more influence on the ultimate load in comparison with compression parameters.

Fig. 5 Box-and-whisker plots to ultimate stresses for variation of all parameters, compression parameters, and shear parameters

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In conclusion, a decrease in the standard deviation of the shear parameters can decrease the variation of the ultimate load obtained. This is important because the bearing capacity equations are a function of the shear parameters. In the following analysis, a specific observation is made of the influence of each shear and compression parameter.

4.2 Influence of Geotechnical Parameters on the Bearing Capacity The influence of variations of the random fields of the parameters was made by dividing them into shearing parameters and compression parameters. The results of the variations can be seen in Fig. 6. Figure 6a shows the influence comparison of the shear parameters on the ultimate vertical. This figure shows a marked influence of the ultimate stress by Young’s modulus. In addition, the coefficients of variation of the simulated parameters are 6.1% for Young’s modulus, 1.9% for Poisson’s ratio, 2.6% for friction angle, and 1.7% for cohesion. An advantage of a model with a cap is that it can take into account the compressive behavior of the soil. Then Fig. 6b shows that the compression parameter that most influences the bearing capacity is the yield stress. The coefficients of variation of the simulated parameters are 6.5% for isotropic yield stress, 1.3% for λ, and 1.0% for κ. Also, Fig. 6 shows that the highest coefficients of variation were Young’ modulus with 6.1% and the isotropic yield stress with 6.5%. This is important because the yield stress is not taken into account when calculating the bearing capacity of the soil. Then, the foundation behavior depends on the elastic part with Young’s modulus and the compression with the yield stress. Regarding the angle of friction and cohesion, the influence is low because few trajectories reach up to the rupture criterion. This was shown by Mendoza and Caicedo [36], where the stress trajectories of the gauss points of the finite elements do not reach the failure criterion at the same time depending on the type and whether or not it has a cap.

Fig. 6 Box-and-whisker plots to ultimate stresses: a Shear parameters. b Compression parameters

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Fig. 7 Relation between ultimate stress and autocorrelation distance

4.3 Influence of the Autocorrelation Distance Two hundred random fields were made for each parameter of the model used (Table 1), varying the autocorrelation distance, as shown in Fig. 7. Later, these parameters were integrated into the finite element model, as shown above. The autocorrelated distances used are Lx = 1.0, Ly = 1.0; Lx = 5.0, Ly = 1.0; Lx = 10.0, Ly = 1.0. These variations of the autocorrelation length are for a case where the distance in x and y are equal and a case where the distance in x and y are different. Figure 7 shows the results for the different cases used in a box-and-whisker plot. The results show that the mean value of the final load varies little from 405.36 kPa (Lx = 1.0, Ly = 1.0), 411.92 kPa (Lx = 5.0, Ly = 1.0), and 407.24 kPa (Lx = 10.0, Ly = 1.0). However, the greater the length of x, the standard deviation increases from 26.72 kPa (Lx = 1.0, Ly = 1.0) to 42.46 kPa (Lx = 10.0, Ly = 1.0). Then, the value of the coefficient of variation changes from 6.6% (Lx = 1.0, Ly = 1.0) to 10.4 (Lx = 10.0, Ly = 1.0). This means that the greater the distance from x to y, there is an increase in the coefficient of variation.

5 Conclusions The simulations with the random fields were able to show the influence of the geotechnical parameters on the bearing capacity. This was made for an elastoplastic model with a cap. This model can be described the shear behavior and compressibility behavior of the soil. Then, the parameters were divided into two groups to describe the shear behavior and the compression behavior of the material. It was shown the influence of the compression in the ultimate stress with the isotropic yielding stress. So, the bearing capacity is a function of the isotropic yield stress (the pre-consolidation stress in the compression test), the modulus of elasticity, and friction angle. This is

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important because the bearing capacity equations do not consider the compressive behavior of the soil and its influence. In addition, the simulations show that the greater the autocorrelation distances, there is an increase in the coefficient of variation.

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21. Llano-Serna MA, Farias MM, Pedroso DM, Williams DJ, Sheng D (2018) An assessment of statistically based relationships between critical state parameters. Géotechnique 68(6):556–560 22. Sainea-Vargas CJ, Torres-Suárez MC (2019) Damage probability assessment for adjoining buildings to deep excavations in soft soils. Int J Geotech Eng 23. Lacasse S, Nadim F (2007) Probabilistic geotechnical analyses for offshore facilities, Georisk. Assess Manage Risk Eng Syst Geohazards 1(1):21–42 24. Lua YJ, Sues RH (1996) Probabilistic finite-element analysis of airfield pavements. Transp Res Rec 1540:29–38 25. El-Kadi AI, Williams SA (2000) Generating two-dimensional fields of autocorrelated, normally distributed parameters by the matrix decomposition technique. Ground Water 38:530–532 26. Daniel C, Caro S (2014) Probabilistic modeling of air void variability of asphalt mixtures in flexible pavements. Constr Build Mater 61:138–146 27. Pieczy´nska-Kozłowska JM, Puła W, Griffiths DV, Fenton GA (2015) Influence of embedment, self-weight and anisotropy on bearing capacity reliability using the random finite element method. Comput Geotech 67(June):229–238 28. Marcin C (2020) Soil sounding location optimisation for spatially variable soil. Géotech Lett 10(3):1–10 29. Helwany S (2007) Applied soil mechanics with ABAQUS applications, 1st edn. Wiley 30. Desai CS, Siriwardane HJ (1984) Constitutive laws for engineering materials with emphasis on geologic materials, 1 edn. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, USA. ISBN 0-13-167940-6 31. Bolton MD, Britto AM, Powrie W, White TP (1989) Finite element analysis of a centrifuge model of a retaining wall embedded in a heavily overconsolidated clay. Comput Geotech 7(4):289–318 32. Hagiwara T, Grant RJ, Calvello M, Taylor RN (1999) The effect ofoverlying strata on the distribution of ground movements induced by tunneling in clay. Soils Found 39(3):63–73 33. Mendoza C, Ruge J, Caicedo B (2018) The geological history analysis of the friction angle in transported soils and their importance in the bearing capacity of shallow foundations. Rev int métodos numér cálc diseño ing 34 (1):11 34. Velloso DA, Lopes F (2010) Fundações profundas, vol 21 edn. Oficina de textos, Brasil 35. Sowers G, Sowers G (1970) Introductory soil mechanics and foundations, 3rd edn. Macmillan, New York 36. Mendoza C, Caicedo B (2018) Elastoplastic framework of relationships between CBR and Young’s modulus for granular material. Road Mater Pavement Des 19(8):1796–1815

Prediction of Wave Overtopping Discharge on Coastal Protection Structure Using SPH-Based and Neural Networks Method Bao-Loi Dang, Quoc Viet Dang, Magd Abdel Wahab, and H. Nguyen-Xuan

Abstract This paper adopts DualSPHysics, the powerful SPH models, to investigate a large-scale 2-D numerical simulation of wave-structure interactions. As a case study, a non-conventional seawall structure built at Vietnam’s coastline is considered. The hydraulic performance of such a structure is assessed using the value of wave overtopping over structure. It is one of the most important considerations when evaluating the efficiency of proposed designs. Due to the geometrical differences, traditional methods such as empirical equations are inconvenient for analyzing such novel structure design with complicated shapes. As a supplement to the experimental study, numerical modeling and machine learning approaches are being studied for assessing such problems. The reliability and effectiveness of two approaches have been proven in several studies in literature. In this work, a large-scale computational model of wave-structure interaction under regular wave conditions is carried out. The simulation results demonstrate good agreement when compared to neural networkbased prediction approaches, and analytical solution as well. Keywords Coastal structure · Wave overtopping · SPH model · Neural networks

1 Introduction Wave overtopping at coastal protection structures is becoming more common as a result of a combination of climate change, sea level rise, and coastal landslides. B.-L. Dang (B) · M. A. Wahab Department of Electrical Energy, Metal, Mechanical Construction and Systems, Faculty of Engineering and Architecture, Ghent University, Ghent, Belgium e-mail: [email protected] B.-L. Dang · Q. V. Dang Department of Civil Engineering, Mientrung University of Civil Engineering, Tuy Hoa City, Vietnam H. Nguyen-Xuan CIRTech Institute, Ho Chi Minh University of Technology (HUTECH), Ho Chi Minh City, Vietnam © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. A. Wahab (eds.), Proceedings of the 4th International Conference on Numerical Modelling in Engineering, Lecture Notes in Civil Engineering 217, https://doi.org/10.1007/978-981-16-8185-1_6

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During extreme condition, e.g., high tide or in stormy conditions, waves overtopping can cause considerable damage to coastal structures, resulting in flooding, human loss, and damage to inland infrastructure. In practice, the mean overtopping discharge is a crucial factor in determining the cross-section of a coastal dike as well as its crest height in order to minimize the mean overtopping discharge. There are several methods for evaluating the overtopping discharge over coastal structures. Traditionally, empirical formulae have been used to calculate the mean overtopping discharge for decades. A comprehensive usage of those empirical equations can be found in literature [1, 2]. However, the empirical solution owns the drawbacks themselves since in practice the range of structure configurations and loading conditions is very wide. As a result, the use of empirical formulas in problems outside of the specified applicable range comes with uncertainty. Therefore, experimental studies are commonly conducted when analyzing the cases with complex shape and wave conditions [3–5]. With the fast development of numerical methods and computing technology, numerical modeling has served as a promising solution along with analytical and experimental approaches. Several numerical models for computational fluid dynamics (CFD) problems have been proposed in literature. They have their pro and con and need to be improved eventually. The accuracy, computational load, and robustness are the primary considerations as considering the use of numerical models. Recently, with the rapid development of computing method using graphics processing unit (GPU) brings a lot of expectation in CFD modeling thank to its parallel and robust computing which can significantly reduce the computing time for complicated CFD problems. In this work, DualSPHysics, an open-source codes based on Smoothed Particle Methods (SPH), is adopted to investigate the overtopping of sea wave on a given sea dike structure [6]. Several studies on overtopping discharge modeling using DualSPHysics codes can be found in literature. Rogers et al. analyzed the stability of a caisson breakwater using SPHysics [7]. A case study of wave attack on a realistic urban furniture has been carried out by Barreiro et al. [8]. Altomare et al. proposed a wave run-up investigation using this approach [9]. Many studies have shown the considerable application and validation of this useful tool [10–13]. Several studies have been conducted to find the suitable DualSPHysics model parameters that ensure the accuracy for corresponding problems [14, 15]. Within the scope of this work, the validation of model is not presented as well as the chosen of model parameters. The model parameters are selected based on the recommendation of previous publications and the experience as listed in Table 1. The case study investigated in this work is a novel sea dike proposed by BUSADCO, Vietnam (see Fig. 1) [16, 17], and the 2D model for simulation is simplified as shown in Fig. 2. Table 1 Selection of DualSPHysics model parameters adopting [14] for modeling Coefficients

Speed of sound

Artificial viscosity

Smoothing length

CFL

Value

15.8

0.00354

1.8

0.176

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Fig. 1 Case study in Vietnam coastal and its cross-section [16, 17]

Fig. 2 Simplified 2D large-scale numerical model

The detail of numerical set-up, simulation results, and discussion will be presented in the following sections.

2 Analysis Methods 2.1 Numerical Modeling The numerical tank is 105 m long, and 11 m height for large-scale simulation. The structure is located at 83 m from the wavemaker. The geometry of structure is extracted from the drawings of project which was implemented in practice. The structure is made of concrete; hence it is impermeable. A wave probe is defined at a distance 30 m from wavemaker to measure the surface elevation for comparison purpose. The effect of reflected waves can be vanished by imposing the active wave absorption system (AWAS) to wave generation boundary. The overtopping discharge is measured by defining a domain which is started from the crest of structure. The FlowTool included in DualSPHysics package is used to calculate the value of overtopping discharge. The detail of numerical mode set-up is shown in Fig. 2. In this paper, five regular wave conditions with different wave heights, but same wave period (T s = 7.2 s) are considered for investigation as listed in the first column of Table 3. The water depth is 4.81 m for all cases as well.

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It is recommended that the inter-particle distance d p should be approximately 1/10 of wave height (H s ) to ensure proper wave propagation, hence d p = 0.18 m is chosen for domain discretization, resulting more than 12,000 particles for each case. Within this paper, only second-order regular wave is reproduced. The simulation run for 500 s to obtain reliable results of mean overtopping discharges. Using NVIDIA GeForce RTX 3060 Laptop GPU, it takes approximately 1 h for each simulation which is very robust computing in large-scale problem.

2.2 Empirical Formula For comparison purpose, the analytical analysis using empirical formula is also presented in this work. The EurOtop (2018) [1], a manual on wave overtopping assessment, is used to perform the hand calculation of mean overtopping discharge. The expression of mean overtopping discharge, q is shown as below. The detail of definition of involved parameters in Eqs. (1) & (2) can be found in EurOtop (2018) manual.   0.067 Rc q  (1) =√ γb ξm−1,0 exp −4.75 ξm−1,0 Hm0 γb γ f γβ γv 3 tanα g Hm0 with a maximum of: 

q 3 g Hm0

 = 0.2exp −2.6

Rc Hm0 γ f γβ γv

 (2)

2.3 Neural Network Approach Along with empirical prediction, an approach based on neural network (NN) algorithms has also been considered. This NN tool has been proposed by van Gent (2007) using the CLASH-database [18]. To be able to adopt this NN tool, the geometry of structure must be converted into corresponding structure configuration suggested in guidance of tool. For instance, based on the shape of investigated structure, the conversion with the instruction as shown in Fig. 3 is adopted. As a result, 12 inputs of structure combined with 2 input of wave conditions are used for NN prediction as presented in Table 2.

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Fig. 3 Instruction for converting of structure configuration adopting [18]

Table 2 Inputs of structure configuration for NN prediction model Input

h (m)

ht (m)

Bt (m)

γf (−)

cotα d (−)

cotα u (−)

Rc (m)

B (m)

hb (m)

tanα B (−)

Ac (m)

Gc (m)

Value

4.81

4.81

0.0

1.0

0.625

0.0

1.89

4.49

1.31

0.0

1.89

0

Table 3 Results of mean overtopping discharge on structure for various methods

Wave height

Empirical formula

NN prediction

Numerical model

m

m3 /m/s

1.8 1.9

0.028490

0.019940

0.024000

0.038690

0.025750

0.043280

2.0

0.050991

0.032610

0.051040

2.1

0.065530

0.040620

0.063280

2.2

0.082390

0.049900

0.063440

3 Result and Discussion This study presents a numerical investigation modeling for predicting the wave overtopping discharge on a specified sea dike under regular wave conditions. Figure 4 is the snapshot during simulation process. Overtopping volume is obtained and

Fig. 4 Snapshot of simulation result

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Free surface elevation, m

2 Th Du

1.5 1 0.5 0 -0.5 -1

360

380

400

420

440

Time, s

Fig. 5 Comparison of free surface elevation at x = 30 m (with H s = 2.0 m, T s = 7.2 m)

measured in the reservoir behind the sea dike. For model verification, the proper wave propagation is investigated to estimate the wave reproducing capability of numerical model. Figure 5 shows the free surface elevation in model compared with theoretical one under regular wave condition with H s = 2 m, and T s = 7.2 s. The acceptable agreement can be found visibly which proves the capability of DualSPHysics model. The result of all five simulations of overtopping discharge is plotted in Fig. 6 and presented in Table 3. The cumulative overtopping discharge increases corresponding to the increase of wave height. The step of increase is consistent to wave period which indicates the proper productivity of the model. Figure 7 shows the comparison between simulated and calculated mean overtopping discharge using empirical equations, while Fig. 8 shows the comparison to predictions using NN method. In order to estimate the performance of the numerical model, the Index of Agreement (d) is used as criterion [19]. This index represents the ratio of the mean square error and the potential error, which is expressed in Eq. 3. d equals to 1 that indicates the Fig. 6 Cumulative overtopping discharges for all cases (m3 /m) Cumulative overtopping volume, m

3

35 Hs=1.8m Hs=1.9m Hs=2.0m Hs=2.1m Hs=2.2m

30 25 20 15 10 5 0 0

100

200 300 Time, s

400

500

Prediction of Wave Overtopping Discharge … Fig. 7 Comparison of mean overtopping discharge between numerical results and empirical formula (m3 /m/s)

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0.1

SPH simulation

0.08

+10% error 0.06

0.04 -10% error 0.02

0 0

0.02

0.04

0.06

0.08

0.1

Empirical formula

Fig. 8 Comparison of mean overtopping discharge between numerical data and neural network prediction (m3 /m/s)

0.1

0.08

SPH simulation

+10% error 0.06

0.04

-10% error

0.02

0 0

0.02

0.04 0.06 0.08 Neural network prediction

0.1

perfect agreement between numerical and analytical results. The d value for regression analysis shown in Figs. 7 and 8 are 0.927 and 0.643, respectively. It can be seen that the numerical results are better agreed with analytical ones than predicted values by NN. It is worth noting that the Index of Agreement is extremely sensitive with outliers values due to the squared differences. The indicative ±10% error band w.r.t a perfect match (diagonal line) is also performed in those plots to describe differences

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between those approaches. n

2 i=1 (Oi − Pi )    2     i=1 ( Pi − O + Oi − O )

d =1−  n

(3)

where Oi is the target value, Pi is the predicted value, and O is the mean target values. The numerical and analytical results seem to be in better agreement than predicted values by NN model. It is conceivable that the numerical model and analytical approach adequately considered the influence of all components of sea dike, while NN model could not include suitable inputs to accurately describe the structure configuration. This is the reason why the mean overtopping discharge predicted by NN model is smaller than other models and is the model that gives the most divergent results.

4 Conclusions In this work, we numerically investigated the wave overtopping discharge on a specified sea dike using DualSPHysics model. A simplified large-scale two-dimension model has been conducted for various regular wave conditions. The good agreements are shown between numerical results and empirical formulae. The Index of Agreement when comparing results of these approaches is 0.927, which is better than other method, i.e., neural network prediction. The average deviation is approximately 10%, which is acceptable in coastal engineering in general. For further research, the numerical mode needs to be improved such as boundary particles problem, optimal model parameters to guarantee the accuracy and numerical stability. Moreover, various experiment tests should be conducted to validate the numerical results. Acknowledgements The authors acknowledge the financial support of VLIR-OUS TEAM Project, VLIRUOS2017-2021-75900, funded by the Flemish Government.

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References 1. van der Meer J et al (2016) EurOtop: manual on wave overtopping of sea defences and related structures—an overtopping manual largely based on European research, but for worldwide application, 2nd edn 2. van der Meer J (2002) Technical report wave run-up and wave overtopping at Dikes. Advisory Committee on Flood Defence, Delft 3. Saville T (1955) Laboratory data on wave run-up and overtopping on shore structures 4. Garcia N, Lara JL, Losada IJ (2004) 2-D numerical analysis of near-field flow at low-crested permeable breakwaters. Coast Eng 51(10):991–1020 5. Lara JL et al (2016) Large-scale 3-D experiments of wave and current interaction with real vegetation. Part 1: guidelines for physical modeling. Coastal Eng 107:70–83 6. Crespo AJC et al (2015) DualSPHysics: open-source parallel CFD solver based on smoothed particle hydrodynamics (SPH). Comput Phys Commun 187:204–216 7. Rogers B (2009) Simulation of caisson breakwater movement using 2-D SPH. J Hydraul Res 48 8. Barreiro A et al (2013) Smoothed particle hydrodynamics for coastal engineering problems. Comput Struct 120:96–106 9. Altomare C et al (2014) Numerical modelling of armour block sea breakwater with smoothed particle hydrodynamics. Comput Struct 130:34–45 10. Wang S, Garlock M, Glisic B (2021) Parametric modeling of depth-limited wave spectra under hurricane conditions with applications to kinetic umbrellas against storm surge inundation. Water 13:251 11. Dang B-L, Nguyen-Xuan H, Abdel Wahab M (2021) Numerical study on wave forces and overtopping over various seawall structures using advanced SPH-based method. Eng Struct 226:111349 12. Altomare C et al (2015) Applicability of smoothed particle hydrodynamics for estimation of sea wave impact on coastal structures. Coast Eng 96:1–12 13. Altomare C et al (2018) Improved relaxation zone method in SPH-based model for coastal engineering applications. Appl Ocean Res 81:15–33 14. Rota Roselli RA et al (2018) Ensuring numerical stability of wave propagation by tuning model parameters using genetic algorithms and response surface methods. Environ Model Softw 103:62–73 15. Altomare C et al (2017) Long-crested wave generation and absorption for SPH-based DualSPHysics model. Coast Eng 127:37–54 16. Ba Ria – Vung Tau Urban sewerage and development one member limited company (BUSADCO). Available from: www.busadco.com.vn 17. Dang B-L et al (2019) Numerical investigation of novel prefabricated hollow concrete blocks for stepped-type seawall structures. Eng Struct 198:109558 18. van Gent MRA et al (2007) Neural network modelling of wave overtopping at coastal structures. Coast Eng 54(8):586–593 19. AgriMetSoft (2019) Online calculators. Available on: https://agrimetsoft.com/calculators/ Index%20of%20Agreement

Structural Robustness of RC Frames Under Blast Events Marco Mennonna, Mattia Francioli, Francesco Petrini, and Franco Bontempi

Abstract This paper presents a numerical procedure for the robustness quantification of RC frames under blast-induced damage scenarios. The procedure is supported by a non-linear numerical analysis, by quantifying the structural response at the global level (i.e., response of the structural system/frame to the blast-induced damage) and by obtaining the so-called “robustness curves”, representing the residual strength of the structure under increasing damage levels. The procedure is then applied to a 2D RC frame structure. The sensitivity of the robustness curves with respect to a set of analysis parameters is discussed. Keywords RC structures · Structural robustness · Blast · Non-linear dynamics

1 Introduction Although the events of progressive collapse have a very low probability of occurring, the consequences have usually a very high impact [1]. Progressive collapse can be triggered by many factors such as blast loading from explosives or gas leakage, design faults, vehicle impact, construction errors, debris impact, and other extreme loadings such as fire and high-magnitude earthquake. In many instances, a significant propagation of direct damage to key structural components throughout the structure have produced a progressive collapse of residential, iconic and public buildings, M. Mennonna · M. Francioli · F. Petrini (B) · F. Bontempi Department of Structural and Geotechnical Engineering, Sapienza University of Rome, Rome, Italy e-mail: [email protected] M. Mennonna e-mail: [email protected] M. Francioli e-mail: [email protected] F. Bontempi e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. A. Wahab (eds.), Proceedings of the 4th International Conference on Numerical Modelling in Engineering, Lecture Notes in Civil Engineering 217, https://doi.org/10.1007/978-981-16-8185-1_7

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resulting in huge losses of life and property. The interest in the explosion and their potential of causing progressive collapse of structures began after an important event, which was the partial collapse of the Ronan Point tower in the UK in 1968 [2]. A comprehensive definition of structural robustness is reported in EN 1991-1-7 (2010) as “the ability of a structure to resist events such as fires, explosions, impacts or the consequences of human error, without being damaged in a disproportionate way compared to the original cause”, that explicitly refers to the kind of actions that are relevant to the robustness. One of the most established procedures for robustness analyses in research is based on the so-called “damage-presumption approach”, that is a non-linear analysis where a certain damage level is assumed for the structure (typically for a framed structure consisting in the removal of a column), and the residual strength of the structure is then evaluated. As for any structural collapse analysis, the results of the damage-presumption approach are very sensitive with respect to the various analysis parameters who play a certain role in the numerical solution: the two most important parameters are the time interval for column removal (which is representative of the blast explosion typology) and the value assumed by the structural damping. In this paper, a sensitivity analysis is carried out on these two parameters in order to quantify the scattering of the numerical results coming out from the uncertainty/variability affecting the phenomena.

2 Global Robustness of Reinforced Concrete Frames 2.1 Structural Behavior Aspects As reported in Starrosek [1], redundancy or compartmentation are the two main conceptual design strategies at global structural scale that can be pursued to satisfy robustness requirements, together with local ductility requirements. In this view, the good seismic design practice which has been established for modern reinforced concrete (RC) framed structures matches well with robustness requirements: they are highly hyper static structures; therefore, they allow to have alternative paths of different loads and to be able to suffer many local damages before a global collapse occurs, the sections of the elements are designed to be ductile and to have a bending behavior at failure. In general then, the criteria of seismic engineering have positive impact regarding the robustness of the building. Additional specific structural behaviors, like the membrane or catenary effects allow an over strength which can be well exploited in case of damage such as the removal of a key element like a column [3].

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2.2 Numerical Analysis for Structural Robustness Regarding the load resistance corresponding to the column removal scenario, a number of experimental studies have been conducted on the reduced scale specimens mostly in order to study the progressive collapse resistance in such scenario [4, 5]. The main drawback of the experimental studies is that the column removal and the successive load is usually applied quasi-statically, then not capturing the dynamic amplification effects, which can play a prominent role in this kind of problems. From the numerical analysis point of view, to capture the kind of effects listed in previous section, a material and geometric nonlinear Finite Element (FE) analysis must be put in place during the design and assessment phases of RC structures for robustness. The nonlinear dynamic procedure for progressive collapse is the most thorough method of analysis in which a primary load-bearing structural element (e.g., column) is removed dynamically and the structural material is allowed to experiment nonlinear behavior. This allows larger deformations and energy dissipation through material yielding, cracking, and fracture. The nonlinear dynamic procedure for RC frames consists in analyzing the frame dynamic response under the sudden removal of a number “n” of columns for the frame starting from the static equilibrium configuration reached by the structure under vertical loads (due to the seismic “permanent+0.3*variable” mass combination). The outcome of the nonlinear dynamic analysis can be of two typologies [6]: (a) after an initial damped transitory phase (fast dynamics), the structure reaches a static equilibrium condition; (b) the collapse occur. Regarding the collapse, it can be defined to occur when: (i) there is “run-away” behavior observed in the vertical displacement time history of the nodes around the removed column, or; (ii) the vertical relative drift (DV ) between the beam-column nodes located around the removed column reaches the value of 20–15% [7]. The latter is calculated starting from the vertical displacement of the node δV and the length of the beam to which the node belongs Lb DV = tan−1 (δV /Lb )

(1)

If the outcome of the non-linear dynamic analysis is not the collapse, an incremental static nonlinear analysis of the structure is carried out under horizontal forces (pushover) in order to evaluate the residual capacity of the damaged structure. In this way, each number of simultaneously removed column “n” is associated to a residual lateral force capacity (λu ) as evaluated by the pushover, and expressed as percentage (λu /λ %) of the force capacity (λ) of the non-damaged structure, evaluated by a pushover analysis on the undamaged structure. It is worth noting, that the response to the initial dynamic analysis (typically represented by the time history of the vertical displacement of the node) is strongly influenced by some parameters regarding the analysis procedure or the structural model. One of these parameters is the removal time interval (td ) for the column: the less is td for the complete removal of the columns, the more impulsive is the response, the larger is the structural response, the larger is the damage (e.g., formation

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of plastic hinges) suffered by the rest of the structure. In this view, the identification or setting of the column removal time interval td for a certain “n” would be of value.

2.3 Robustness Curves As a result of the above-mentioned numerical outcome, the robustness of the structure can be quantified and efficiently represented by the so-called “robustness curves” (RCs) as introduced by Olmati et al. [6]. RC are represented on a Cartesian plane in which the x-axis reposts the damage level suffered by the structure (“n” in the previous section), while the y-axis reports the corresponding residual force capacity percentage (λu /λ % in the previous section), see for example the qualitative representation of robustness curves represented in Fig. 1. The less is the steepness of the robustness curve, the more is the robustness of the structure under the considered damage. The procedure for the evaluation of the robustness curves in Fig. 1, is depicted in the flowchart reported in Fig. 2: the non-linear dynamic analysis (NDA in the flowchart) consisting in the column sudden removal is conducted for a number of different locations (NL ) in the structure, and for an increasing damage level “Dscenario (i, j)” (where “i” indicates the location and “j” the presumed damage level). After each NDA, if the collapse does not occur, the pushover non-linear analysis under lateral load is carried out to determine one point (λu /λ %) of the robustness curve. The procedure is repeated for different locations and damage levels in order to obtain a set of robustness curves under blast presumed damage. Fig. 1 Typical robustness curves

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Fig. 2 Flowchart of the procedure to evaluate the structural robustness against blast damage [6]

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3 Application to an Existing Structure 3.1 Case Study Structure and FEM Model In order to develop the full procedure for the robustness quantification of RC frames under blast load scenarios, a 2D RC frame structure is considered (Fig. 3). The building to which the 2D frame belongs is a part of a very complex hospital system completed in the early 2000s. It is a RC structure made of concrete C28/35 and steel B450C. The structure is modeled using SAP 2000® commercial structural code, with a Finite Element modeling that allows the possibility to define the nonlinear properties of the materials: the nonlinear behavior is implemented using the approximation of plastic hinges, which are obtained from the Moment-rotation relationship (Mθ) evaluated from the equations provided by the Italian Standards NTC2018 [8]. Moreover, geometric nonlinearity is taken into account with large displacement and P- options. As stated before, the 2D frame is extrapolated form a complex structure: in order to simulate the membrane effect that occurs in a 3D structure due to the connection of the different beams with the floor slab, a dedicated non-linear beam finite element is added. In particular, this latter element has been modeled on the basis of the behavior of the floor slab: a fiber-based model with diffusive plasticity has been used in order to identify its exact axial and bending behavior; thus, a simplified model with the approximation of plastic hinges has been created and connected to the columns of the 2D frame in such a way that the so-called membrane (catenary effect in 2D) effect can be taken into account in the 2D plane. Considering the 2D frame structure, a nonlinear static analysis has been made in order to evaluate the effectiveness of these new elements. After the removal of a certain column, vertical loads have been amplified using a λ multiplier. A pushdown analysis has been made on two models, which difference was the presence of the

Fig. 3 2D RC frame structure and different locations assumed

Structural Robustness of RC Frames Under Blast Events Fig. 4 Membrane forces in RC beam-column substructures under pushdown analysis

87

180 150

Force [kN]

120 90 60 no membrane

30 0

membrane

0

100

200 300 Displacement [mm]

400

afore mentioned elements. During the analysis, the vertical displacement of the node at the top of the removed element was monitored, together with the resultant vertical forces. The outcome is shown in Fig. 4: the presence of the membrane/catenary effect allows an increase in strength without modifying the initial stiffness.

3.2 Global Robustness Results The procedure for the robustness quantification of RC frames under blast load scenarios is conducted. The global analysis of the structure, using both nonlinear dynamic and nonlinear static analyses as depicted in the flowchart reported in Fig. 2, and its outcome is the evaluation of the global robustness. After having defined a certain number of location NL (2, column 3 and column 5 of the 2D RC frame, see Fig. 3), each of which represents the position of the first key element removed -blast location, different analyses regarding each D(i, j)-scenarios (i = location, j = damage level) are developed; a particular location is considered and starting from a certain damage level (which corresponds to a number of elements removed), the structural response is evaluated by the non-linear dynamic analysis. At this point, if failure doesn’t occur spontaneously to another key element, first the residual strength of the structure is identified using a nonlinear static analysis, then the damage level is increased (i.e., another element is removed) and again the structural response is evaluated. During the analysis that provides the evaluation of the residual strength of the structure (i.e., pushover analysis), the residual lateral force capacity (λu ) considered is the one that corresponds to the event that occurs first between the “run-away” behavior observed in the vertical displacement time history of the nodes around the removed column or the presence of a vertical drift ratio (DV, MAX ) bigger than 15% (see paragraph 2.2). Obviously, if NL is different from 1, all is repeated NL times. Before starting to apply the procedure in order to identify the robustness of the structure under blast loads, various locations of hypothetical damage were assumed (Fig. 3). Depending on which column is removed, the results obtained are different

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Fig. 5 Push-over curves of the 2D frame structure due to different locations of damage

1600 1400

Force [kN]

1200 1000 800 600 400 200 0

push-over dx 1

push-over dx 2

push-over dx 3

push-over dx 4

push-over dx 5 0

200

400 600 Displacement [mm]

800

1000

Fig. 6 Displacement of the node at the top of column 3 (location 1): effect of variation of damping ratio

Vertical Displcement [mm]

(Fig. 5): the internal columns (n°3, n°4 and n°5) are characterized by a bigger area of influence and the hypothetical damage to one of them could cause a bigger reduction of the capacity. The curves obtained, for simplicity of representation, are then bi-linearized: the real curve is replaced a with simplified curve which has at first a linear part and then a perfectly plastic plateau at the FY value for the force. The slope of the linear part is identified imposing the passage for the point 0.6FMAX of the original capacity curve, while the value of FY is obtained by imposing the equality of the areas underpinned by the bilinear curve and by the capacity curve for the maximum displacement dU . Once established the locations, for each of them, some sensitivity analyses have been carried out varying different parameters: the damping ratio (ζ) and the removal time interval of column (td ). Considering the first parameter, Fig. 6, shows the effects of the variation of the damping ratio ζ for location 1 when the removal interval td is set equal to 0.03 s. As the damping index increases, the maximum vertical displacement and the time necessary to dampen the free oscillations of the removed column node decreases. For successive analyses, the damping ratio is set equal to 4% since smallest values (e.g., 1% in the figure) do not determine a significative 0 -5 0

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difference in terms of “damping” time. For sake of completeness, it has to say that a very large damping index, such as 0.5, leads to an almost absence of oscillation. The bi-linear pushover curves in Fig. 7 show the case of damage 1, with instantaneous removal of column 3 and depict the influence of the damping index parameter at the same removal time interval td , set equal to 0.03 s. As the damping index decreases, the above-described dynamic amplification effect leads to a decreasing of both the stiffness and strength of the damaged frame (i.e., after the removal of the column) under lateral load. Figures 8 and 9 show the effect of the variation of td for location 3. As it appears in Fig. 8, which shows the results of a nonlinear dynamic analysis that captures the effects of amplification in terms of displacement and in terms of geometric and material non-linearities, the displacement of the node at the top of the removed column increases as td decreases; there is a small difference between the cases with td = 0.5–0.3 and td = 0.01–0.03. It is also possible to note that the value of td has also a certain influence on the amplitude of the oscillations around the residual displacement and on the damping shown in the time histories. Figure 9 shows the results of pushover analyses for location 1 with damage level equal to 1 (column 3 removed) and how the variation of td affects the capacity of the damaged structure: although there are just slight differences between the curves for the cases considered, as the td value decreases also the overall capacity decreases. 0 -5

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It is important to understand the effect that different values of td have in terms of decreasing the capacity of the structure because this parameter can be used to simulate the damage induced by different blast scenarios: the time interval of column removal td can be considered as the time during which the damage propagates and affects the element structural element under blast-load effects, and it is possible to assume that it depends on the different properties/intensity measures of the explosion, like stand-off distance from the ignition and equivalent kilograms of TNT.

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Figures 10, 11, 12 and 13 shows the results of the discussed analyses in terms of robustness curves, obtained as described in paragraph 2.3. Using the terminology presented in the flowchart of Fig. 2, two locations are considered (NL = 2): location 1 implies that the first element removed is column 3, while for location 2 the first element removed is column 5. In both cases, damage level 2 corresponds to the complete failure of the column adjacent to the first element removed (column 2 for L1 and column 4 for L2). Damage level 0.5, instead, corresponds to the loss of the 50% of the transversal section; this means that the explosion results in a loss of element stiffness and capacity, but not in a collapse. The damaged element continues to carry the axial load but there are no dynamic effects due to the loss of the column. It should be noted that for the damage level 1.5, considerations are fully similar to those reported for the damage level 0.5: in this case, one column is completely removed, and the loss of 50% of the second column’s section is considered. For example, in Figs. 10 and 11 it is possible to notice the effect of the variation of td : coherently to Fig. 9, where the pushover curves for damage level 1 are reported, the decrease of td determines a decrease of the residual capacity for both locations. Damage level 2 always determines the progressive collapse of the structure, while damage level 1 causes a drop in initial capacity of about 20% for L1 and about 30% for L2. Similarly, Figs. 12 and 13 shows the effect of the variation of the other parameter investigated, the damping ratio, with the same removal time of the column (td = 0.03 s). The issue related to capacity losses remains the same as previously discussed:

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if the damping index increases, there is an increase in the capacity of the RC frame compared to cases with a smaller damping value. Even in this case, damage level 2 causes the collapse of the structure and damage level 1 determines a reduction of the structure’s capacity.

4 Conclusions In this paper, a procedure for the robustness quantification of RC frames under blast induced damage has been presented, connected to a global assessment of the structure and based on nonlinear dynamic analysis in order to evaluate the behavior of the damaged structure and a nonlinear static analysis in order to find a characteristic value of the residual capacity. The influence of parameters regarding the structural aspects (damping index) and explosions typology (removal time interval of the element) has been evaluated and described. As a general conclusion, it has to be highlighted that, due to the high sensitivity of the analyses to the removal time interval of the element, the connection of the explosion typology (e.g., stand-off distance and intensity) to this parameter it is necessary for an exhaustive association of the robustness performances to the blast load. Further researches are under development on this side by the authors.

References 1. Starossek U, Haberland M (2011) Approaches to measures of structural robustness. Struct Infrastruct Eng 7(7–8):625–631 2. Agarwal J, Haberland M, Holick`y M, Sykora M, Thelandersson S (2012) Robustness of structures: lessons from failures. Struct Eng Int 22:105–111 3. Kiakojouri F, De Biagi V, Chiaia B, Sheidaii MR(2020) Progressive collapse of framed building structures: current knowledge and future prospects. Eng Struct 206:110061 4. Azim I, Yang J, Bhatta S, Wang F, Liu Q (2020) Factors influencing the progressive collapse resistance of RC frame structures. J Build Eng 27:100986 5. Izzuddin BA, Vlassis AG, Elghazouli AY, Nethercot DA (2008) Progressive collapse of multistorey buildings due to sudden column loss —part I: simplified assessment frame-work. Eng Struct 30:1308–1318 6. Olmati P, Petrini F, Bontempi F (2013) Numerical analyses for the structural assessment of steel buildings under explosions. Struct Eng Mech 45(6):803–819 7. Parisi F, Scalvenzi M (2020) Progressive collapse assessment of gravity-load designed European RC buildings under multi-column loss scenarios. Eng Struct 209:110001 8. Itailan Ministry for Trasportations and Infrastructures (2018) Norme Tecniche per le Costruzioni (NTC2018)—Technical Stardards for Strctures

Influence of Pretensioned Rods on Structural Optimization of Grid Shells Valentina Tomei , Ernesto Grande, and Maura Imbimbo

Abstract In the last decades, grid shell structures are widely spreading in many architectural works because they combine the beauty of their shape with the efficiency of the structural performance. They represent a fascinating example where the architectural design merges with the structural one since the “form” is itself the “structure”. Thanks to this, grid shells are able to cover large spans with light solutions relying on the inherent rigidity provided by the double curvature shell. In many cases grid shells are also equipped with pretensioned members aimed at enhancing some structural properties such as stability, lateral stiffening, etc. The design of grid shells is often based on optimization strategies, which guarantee structurally efficient solutions by considering the role of each structural element composing the grid shell. In this context, the paper aims at investigating the influence of pretensioned members on the solution obtained from the structural optimization process. In particular, considering the case study of the Smithsonian Museum canopy in Washington equipped with pretensioned rods, the role of these elements is explored within the optimization design procedure. Keywords Grid shell · Structural optimization · Pretension

V. Tomei (B) Department of Civil and Industrial Engineering, University Niccolò Cusano, Via Don Carlo Gnocchi, 3, 00166 Rome, Italy e-mail: [email protected] E. Grande Department of Engineering Sciences, University Guglielmo Marconi, via Plinio 44, 00193 Rome, Italy e-mail: [email protected] M. Imbimbo Department of Civil and Mechanical Engineering, University of Cassino and Southern Lazio, via G. Di Biasio 43, 00143 Cassino, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 M. A. Wahab (eds.), Proceedings of the 4th International Conference on Numerical Modelling in Engineering, Lecture Notes in Civil Engineering 217, https://doi.org/10.1007/978-981-16-8185-1_8

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1 Introduction Grid shells are fascinating examples of structures able to combine aesthetic qualities and optimal structural performances, since the “shape” is completely merged with the “structure”. To make grid shells structurally efficient is their capacity to cover large span with light systems, exploiting the inherent strength of a double curvature shell. Generally, grid shells are designed with a triangular mesh, taking advance of its inherent rigidity; examples of this are the British Museum canopy [1], the Palacio de Comunicaciones canopy [2] and the canopy of the lobby of DZ Bank. In some applications, to limit the use of material or to favor natural lighting, a quadrilateral mesh is preferred; in this case, in order to provide an adequate stiffness to the structure, the mesh is braced with pretensioned members. Some actual examples are the Smithsonian Museum grid shell canopy [3], the Hippo House at the Berlin Zoo [4], the Orangery grid shell at Chiddingstone Castle, Multihalle in Mannheim [5] or the Weald and Download Museum [6]. In many cases grid shells are also equipped with pretensioned members aimed at providing adequate lateral stiffening and stability to grid shell structures, as illustrated in [7–10]. Several studies analyze the structural optimization of grid shell structures [11– 19]; however few attention is devoted to the role of pretensioned members in the structural optimization process. In this context, the object of the paper is to investigate the effect of the pretensioned members on the optimization process. In particular, considering that pretensioned members become part of the entire structure and interact with the other elements composing the grid shell, the stress state and the stability condition of the structure will be certainly altered by the introduction of these elements. Therefore, design strategies based on structural optimization processes should consider the pretension imposed to the members as a design parameter. This analysis represents the core of the present paper which is carried out with reference to the case study of Smithsonian Museum grid shell canopy [3].

2 Case Study The case study analyzed in this paper is the grid shell canopy of the Smithsonian Museum (Fig. 1), built in Washington in 2007 and designed by Foster and Partners and Buro Happold. The grid shell covers a 39 × 84 m surface, that rests on eight columns and it is characterized by a shape composed of three “domes” connected each other (Fig. 2), reaching a maximum height of 6.3 m. It is made by a quadrangular mesh of welded steel beams (in the following called grid beams) and pretensioned rods that link the columns and surrounds the three “domes” (Fig. 2). Some of the data concerning the structure of the grid shell have been directly deduced from literature; other ones have been supposed by the authors. In particular,

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Fig. 1 Smithsonian Museum grid shell canopy [3] DOME 2

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Table 1 Cross-sections and materials employed for the Smithsonian case study

Cross-section

Material

Grid beams

Steel fy = 355 Mpa E = 210 GPa

Rods

Steel fy = 950 Mpa E = 205 GPa

the cross-sections of the grid beams are schematized as a hollow rectangular crosssection made of steel S355; more in details, the base and the thickness are defined as functions of the height hG of the section, in order to keep the proportions of the actual cross-sections (Table 1). The rods are supposed as steel full circular sections of diameter  R , made of steel with a yielding stress equal to 950 MPa (Table 1). Permanent vertical loads have been considered, including the structural weight and the glass panels weight (200 kg for each panel). Considering the shape of the canopy shown in Fig. 2, the effect of the pretension levels of rods on the solution obtained from the structural optimization process has been investigated.

3 The Effect of the Pretension in the Rods The optimization process considered herein aims to minimize the structural weight W by varying the cross-section dimensions of the grid beams (hG ) and the rods (R ), and imposing constraint conditions on the maximum utilization ratio of the grid members, Umax (the allowable value Ulim is assumed equal to 1), and on the maximum vertical displacements of the nodes, Dmax (the allowable value Dlim is assumed equal to 0.112, i.e. maximum span divided by 250). More in details, the utilization ratio of members is evaluated as in Eurocode 3 [20], and it can be described as the demand to capacity ratio, where the capacity also includes local buckling phenomena. The levels of pretension are conferred to the rods, in the numerical model, by imposing an initial value of deformation εin , ranging from 0 (lower bound: εin,lb ) to 4.5 mm/m (upper bound: εin,ub ), the latter corresponding to the attainment of the yield stress level of material. The tridimensional model is built in the Grasshopper environment [21, 22], while the numerical analyses have been developed in Karamba [23, 24], a Grasshopper plug-in. The optimization algorithms employed in the analyses are genetic [25] and they are implemented by means of the component Galapagos of Grasshopper. In Fig. 3 are reported, in function of the values of the imposed initial deformation of rods εin , the following quantities: the weight of the structure (W) normalized with respect to the maximum value of the weight of the obtained solutions (Wmax ), the

Influence of Pretensioned Rods … Fig. 3 Results of optimization process by considering different values of εin

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maximum value of displacement of joints (Dmax ) normalized with respect to the fixed limit value (Dlim ), and the maximum utilization ratio of the members (Umax ). The plot clearly shows the effect of the level of pretention of rods: values of εin within the range between 1.5 and 2.5 lead to optimized solutions characterized by lower structural weight and higher values of the stress utilization ratio. It is evident how the structural design is generally dominated by the stiffness, since the value of Dmax reaches its limit value in most of the cases. Nevertheless, the values of Umax obtained by solutions with an initial pretension εin in the rods, increases with respect the solution obtained with no pretension, leading to a better use of the structural elements from the strength point of view. In some cases (εin = 1.5, 2), the design is even dominated by strength since the utilization ratios Umax reaches one and the maximum displacements Dmax approach its limit value. This result is due to a decrease of the cross-section dimensions with respect to the solution with no pretension in the rods, that also leads to a considerable weight reduction, which varies from 12% to 75%. The greatest benefits in terms of weight reduction are obtained by imposing a pretension value εin between 1.5 and 2.5 mm/m, as also evident from Fig. 3. Since the level of pretension of rods is a parameter influencing the optimized solution, it could be useful to introduce this variable in the optimization process; to this aim, an optimization strategy has been proposed with the purpose to include also the pretension levels εin as additional variable together with the cross-section dimensions. The proposed optimization process is stated as: Minimize Subjected to

W Umax ≤ Ulim Dmax ≤ Dlim hG,lb ≤ hG ≤ hG,ub R , lb ≤ R ≤ R , ub

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where hlb,G (10 cm) and hub,G (100 cm) are the lower and upper bounds of hG , respectively; lb,R (1 cm) and ub,R (15 cm) are the lower and upper bounds of R , respectively. Introducing the pretension levels εin as variable, as expected, the optimal solution which minimizes the weight, respecting the constraint conditions, is the one corresponding to a strain level of rods equal to 2 mm/m. This result confirms the validity of the proposed approach. In particular, a value of Umax equal to 1 and a value of Dmax equal to 0.1, close to the limit one, are obtained, meaning that the structural solutions is optimal for both the strength and the stiffness requirements, allowing the structural elements to be fully exploited.

4 Conclusions In this paper, the effect of pretensioned rods on the structural optimization of grids shell structures has been investigated by considering a reference case study inspired to the Smithsonian Museum canopy in Washington. In particular, a strategy based on a structural optimization approach has been developed by the authors, which specifically takes into account the level of pretension provided to the rods. Firstly, several simple sizing optimization approaches are applied to the case study by considering different levels of the initial pretension in the rods; then, an optimization strategy including also the pretension as a design variable in addition to the cross-section dimensions is proposed and applied. Downstream of the obtained results, it is possible to state that the application of an initial pretension in the rods reduces the deformability of the grid shell canopy and, hence, allows the use of smaller cross-sections that leads to a strong reduction of the structural weight. The numerical analysis and the obtained results demonstrate the effectiveness of the proposed approaches, and highlights the important role of pretensioned members in the phase of structural optimization of grid shell structures. An important aspect in the design of grid shells is certainly related to their shape, because of the strong synergy between “form” and “structure”: generally, the hanging model shape is among the best candidates. However, given the strong influence of the pretensioned members on the structural performances of grid shell structures, a shape that is “optimal” for a grid shell not equipped with pretensioned members, it may not be “optimal” in the presence of these. Thus, a further development of this study should be related to the conception of the optimal shape by taking into account the presence and position of pretensioned members.

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22. Saremi S, Mirjalili S, Lewis A (2017) Grasshopper optimisation algorithm: theory and application. Adv Eng Softw. https://doi.org/10.1016/j.advengsoft.2017.01.004 23. Preisinger C Karamba: parametric structural modeling 24. Preisinger C (2013) Linking structure and parametric geometry. Archit Des. https://doi.org/10. 1002/ad.1564 25. Goldberg DE (1989) Genetic algorithms in search. Optim Mach