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Development of an acoustic computational software to analyse composite and sandwich panels

Carlos António de Francisco Machado

Thesis submitted to

Faculdade de Engenharia da Universidade do Porto

as a requirement to obtain the MSc degree in Mechanical Engineering.

Under supervision of

Professor Jorge Américo Oliveira Pinto Belinha

Professor Aurélio Lima Araújo

Professor Renato Manuel Natal Jorge

Professor Lúcia Maria de Jesus Simas Dinis

Porto, July 2017

Development of an acoustic computational software to analyse composite and sandwich panels.

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À minha família.


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Agradecimentos

Ao professor Jorge Belinha, um grande obrigado, não só por todo o apoio, tempo, orientação

e ajuda prestadas durante todo o semestre, mas também por me ter dado a oportunidade de

realizar esta tese.

Ao professor Renato Natal Jorge, professor Aurélio Araújo e à professora Lúcia Dinis,

agradeço pelo esclarecimento de dúvidas assim como a sua disponibilidade em auxiliar em

qualquer altura do trabalho. Agradeço também a todos os professores que tive a oportunidade

de travar conhecimento e pelos preceitos transmitidos ao longo dos anos que estudei na

FEUP.

Obrigado a todos os meus amigos que me acompanharam durante este percurso e que

influenciaram de forma positiva a minha formação pessoal e académica. Um obrigado

especial aos que estiveram lá nos momentos mais secantes durante as épocas de exames, e que

graças a eles todos esses momentos agora dão saudades.

Em especial, gostava de agradecer à minha família, não só por todo o apoio, carinho e alegria

que me deram durante todos estes anos, mas sim por serem a principal razão porque todo este

meu percurso foi possível.


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Funding

The author truly acknowledge the funding provided by Ministério da Educação e Ciência –

Fundação para a Ciência e a Tecnologia (Portugal), by project funding UID/EMS/50022/2013

– “Advanced materials for noise reduction: modeling, optimization and experimental

validation” (funding provided by the inter-institutional projects from LAETA and INEGI).

The author truly acknowledges the work conditions provided by the Applied Mechanics

Division (SMAp) of the department of mechanical engineering (DEMec) of FEUP and by the

project NORTE-01-0145-FEDER-000022 – SciTech – Science and Technology for

Competitive and Sustainable Industries, co-financed by Programa Operacional Regional do

Norte (NORTE2020), through Fundo Europeu de Desenvolvimento Regional (FEDER).

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by


Thesis submitted in fulfilment of the degree of Master of Science in

Mechanical Engineering in the Faculdade de Engenharia da Universidade do Porto, under the supervision of:


Professor at Faculdade de Engenharia da Universidade do Porto

and


Associated Professor at Instituto Superior Técnico


Associated Professor at Faculdade de Engenharia da Universidade do Porto

Professor Lúcia Maria de Jesus Simas Dinis

Associated Professor at Faculdade de Engenharia da Universidade do Porto

Abstract

When numerical techniques are used to study structures subjected to dynamic loads, the Finite

Element Method (FEM) is usually the method adopted. However, other accurate and efficient

numerical methods, such as meshless methods, have been gaining appeal over the last few

years. Meshless methods avoid the need to construct elements to assure nodal connectivity,

instead it relies on the overlap of “influence-domains”, allowing more freedom on the

placement of the nodes and providing a more smooth stress field. In this work, the dynamic

and acoustical analysis of structures is extended to two different meshless methods, the RPIM

and NNRPIM. For this, algorithms are created respecting both formulations. In the end,

several examples, focusing on laminated and sandwich plates and beams, are analysed to

verify the accuracy of both methods, and the results compared with those produced by the

FEM and documented in the literature.

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por


Tese apresentada à Faculdade de Engenharia da Universidade do Porto,

para obtenção do grau de Mestre em Engenharia Mecânica,

sob a orientação de:


Professor da Faculdade de Engenharia da Universidade do Porto

e


Professor Associado do Instituto Superior Técnico


Professor Associado da Faculdade de Engenharia da Universidade do Porto

Professora Lúcia Maria de Jesus Simas Dinis

Professor Associado da Faculdade de Engenharia da Universidade do Porto

Sumário

Quando métodos numéricos são aplicados no estudo do comportamento de estruturas sujeitas

a solicitações dinámicas, o Método dos Elementos Finitos (MEF) é geralmente o processo

aplicado. No entanto, outras técnicas númericas, também precisas e eficientes, como os

Métodos Sem Malha, tem vindo a gerar grande interesse junto da comunidade cientifica nos

ultimos anos. Métodos Sem Malha distinguem-se por não necessitarem de utilizar elementos

para garantir a ligação entre os vários nós, para tal, esta conexão é assegurada pela

sobreposição dos vários “dominios de influencia”, conferindo a estes métodos mais liberdade

na colocação dos nós e a obtenção de campos de tensão mais suaves, dado as tensões não

terem que ser interpoladas entre os elementos. Neste trabalho, a analise dinamica e acustica de

estruturas é extendida a dois métodos sem malha, o RPIM e o NNRPIM. Para tal, são criados

algoritmos para ambas formulações. No final, são resolvidos vários exemplos, focados

também na análise de placas e vigas laminadas e em “sandwich”, para demostrar a

performance destes dois métodos, sendo os resultados comparados com aqueles produzidos

com o MEF e documentados na literatura.

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Table of Contents

1. Introduction .......................................................................................................................................... 1 1.1 Acoustic ................................................................................................................................................ 2 1.2 Meshless Methods ............................................................................................................................... 4 1.3 Thesis Objectives ................................................................................................................................. 6 1.4 Thesis Arrangement ............................................................................................................................. 7

2. Meshless Methods .............................................................................................................................. 9 2.1 Radial Point Interpolation Method (RPIM) .......................................................................................... 11 2.2 Natural Neighbour Radial Point Interpolation Method (NNRPIM) ....................................................... 14

3. Solid Mechanics ................................................................................................................................ 17 3.1 Stress and Strain Fields ..................................................................................................................... 18 3.2 Strong and Weak Form Formulations ................................................................................................ 19

3.2.1 Weak Form of Galerkin .................................................................................................... 20

4. Linear Deformation Theory ............................................................................................................... 23 4.1 Plane Elasticity ................................................................................................................................... 24

4.1.1 Plain Strain ...................................................................................................................... 25

4.1.2 Plane Stress .................................................................................................................... 26

4.1.3 Virtual Work Principle....................................................................................................... 27 4.2 Three Dimensional Solids .................................................................................................................. 29

5. Vibrations .......................................................................................................................................... 31 5.1 Free Vibration ..................................................................................................................................... 32 5.2 Forced Vibrations ............................................................................................................................... 33

5.2.1 Modal Superposition ........................................................................................................ 34

5.2.2 Duhamel Integral ............................................................................................................. 35

5.2.3 Frequency Analysis Response ........................................................................................ 37

5.2.4 Direct Integration ............................................................................................................. 39

Central Difference Method ............................................................................................................... 40

Houbolt Method ............................................................................................................................... 41

Newmark Method ............................................................................................................................ 42

Wilson-θ Method ............................................................................................................................. 43

6. Vibro-Acoustics ................................................................................................................................. 45 6.1 Calculation of local matrices ............................................................................................................... 47 6.2 Uncoupled Acoustic Problem ............................................................................................................. 50 6.3 Interior Acoustic Problems ................................................................................................................. 51 6.4 Vibroacoustic Indicators ..................................................................................................................... 54

7. Routines ............................................................................................................................................ 55 7.1 Structural Analysis ............................................................................................................................. 56 7.2 Acoustic Analysis ............................................................................................................................... 63

8. Numerical Examples ......................................................................................................................... 67 8.1 Elastostatic Analysis of a Cantilever Beam ........................................................................................ 68 8.2 Free Vibration of a Cantilever Beam .................................................................................................. 73 8.3 Forced Vibrations of a Cantilever Beam ............................................................................................. 79 8.4 Free Vibration of a Sandwich Beam ................................................................................................... 82 8.5 Analysis of Sandwich Cantilever Beam with Corrugated Core ........................................................... 85

8.5.1 Static Analysis ................................................................................................................. 86

8.5.2 Free Vibrations Analysis .................................................................................................. 91

8.5.3 Forced Vibrations Analysis .............................................................................................. 95 8.6 Forced Vibrations of a Sandwich Plate with Corrugated Core............................................................ 98 8.7 Free and Forced Vibrations of a Laminated Plate ............................................................................ 100 8.8 Free Vibrations of an Acoustic Tube ................................................................................................ 105 8.9 Free Vibrations of a 2D Car .............................................................................................................. 106 8.10 Free Vibrations of a Coupled Fluid-Structure Cavity ........................................................................ 107

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9. Conclusions ..................................................................................................................................... 111 9.1 Conclusions and Final Remarks ....................................................................................................... 112 9.2 Future Works .................................................................................................................................... 114

References ........................................................................................................................................... 115

Appendix A: Solutions for the static analysis of Sinusoidal Core Sandwich Beams ....................... 121

Sinusoidal Core Sandwich Beam With One Corrugated Layer. ........................................................... 122

Sinusoidal Core Sandwich Beam With Two Corrugated Layers .......................................................... 137

Appendix B: Free and Forced Vibrations for Various Sinusoidal Core Sandwich Beams ................... 153

Sinusoidal Core Sandwich Beam with One Corrugated Layer ............................................................ 154

Appendix C: Forced Vibrations of a Sandwich Plate with Corrugated Core ........................................ 171

Sinusoidal Core Sandwich Plate with One Corrugated Layer .............................................................. 172

Sinusoidal Core Sandwich Plate with Two Corrugated Layers ............................................................ 175

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List of Figures

Figure 1: Typical FRF (Frequency Response Function) of a vibroacoustic system [3]. ............ 2

Figure 2: a) Irregular mesh; b) regular mesh [28]. ..................................................................... 9

Figure 3: Background integration mesh a) nodal independent; b) nodal depedent (Voronӧi diagram) [28]. ............................................................................................................................. 9

Figure 4: Interpolation function 𝑢ℎ(𝑥) and the discrete nodal parameters (𝑥𝑖) [28].............. 11

Figure 5: Triangle of Pascal [28]. ............................................................................................. 12

Figure 6: a) Initial set of nodes of the structure. b) Normal to straight line that connects the

closest node to 𝑛0. c) Closed geometry and neighbor nodes. d) Voronoї cell 𝑉0for node 𝑛0 [28]. ........................................................................................................................................... 14

Figure 7: a) First degree influence-cell b) second degree influence cell [28]. ......................... 15

Figure 8: a) Irregular mesh generating quadrangular areas b) regular mesh generating

triangular areas [28]. ................................................................................................................. 15

Figure 9: Example of a structure under: a) plane stress; b) plane strain [54]. .......................... 26

Figure 10: Solicitation as a set of impulsive forces [59]. ......................................................... 35

Figure 11: Linear acceleration approximation [57]. ................................................................. 42

Figure 12: Wilson-θ method. [59] ............................................................................................ 43

Figure 13: Fluid domain and boundary conditions [3]. ............................................................ 45

Figure 14: Fluid-structure interior coupled problem [3]. ......................................................... 51

Figure 15: Model of the cantilever beam on study [28]. .......................................................... 68

Figure 16: Medium displacement errors for a regular mesh (left side) and irregular mesh (right

side): a)displacement u b) displacement v. Logarithmic scales. .............................................. 69

Figure 17: Medium stress errors for a regular mesh (left side) and irregular mesh (right side):

a) 𝜎𝑥𝑥b) 𝜎𝑥𝑦. Logarithmic scales. ........................................................................................... 70

Figure 18: Position on the beam of the analyzed points. .......................................................... 70

Figure 19: Model of the cantilever beam on study [28]. .......................................................... 73

Figure 20: Robutsness study of the cantilever beam for the different formulations. ............... 76

Figure 21: Natural vibration frequencies: regular mesh (left-side) and irregular mesh (right-

side).

a) first mode b) second mode c) third mode. 77

Figure 22: First three vibration modes of the cantilever beam. ................................................ 78

Figure 23: Vertical displacement for the cantilever beam measured on point 𝐴(𝐿, 𝐷2).......... 79

Figure 24: Vertical displacement on point A using different methods (FEM). ........................ 80

Figure 25: Vertical displacement on point A using different methods (RPIM). ...................... 80

Figure 26: Vertical displacement on point A using different methods (NNRPIM). ................ 80

Figure 27: Vertical displacement of point A for the cantilever beam subjected to a harmonic

distributed load. ........................................................................................................................ 81

Figure 28: Vertical displacement of the clamped-free beam presented on the literature [63]. 81

xviii

Figure 29: First two vibration modes for the simply supported sandwich beam. .................... 84

Figure 30: Sandwich beam with sinusoidal corrugated core along its length [66]................... 86

Figure 31: Scheme of the three point bending for the sandwich beam with corrugated core

[66]. .......................................................................................................................................... 86

Figure 32: Sandwich beam with three layers of sinusoidal corrugated core. ........................... 88

Figure 33: First vibrations modes for the corrugated beam under different boundary

conditions. ................................................................................................................................ 94

Figure 34: Vertical deflection for the corrugated core beam with one layer and C-C boundary

conditions. ................................................................................................................................ 96

Figure 35: Scheme of the laminated plate, with two sides clamped and two sides free. ........ 100

Figure 36: Representation of the harmonic solicitation. ........................................................ 101

Figure 37: Vertical displacement /m, for the laminated plate on point A. ............................. 102

Figure 38: Vertical displacement /m, for the laminated plate on point B. ............................. 102

Figure 39: Geometry of the 2D interior acoustic car [69]. ..................................................... 106

Figure 40: Geometric properties of the coupled system [70]. ................................................ 107

Figure 41: First natural vibration mode for the uncoupled rigid fluid cavity (𝜔 =1068 𝑟𝑎𝑑/𝑠) filled with air. .................................................................................................. 109

Figure 42: Second natural vibration mode for the uncoupled rigid fluid cavity (𝜔 =1068 𝑟𝑎𝑑/𝑠) filled with air. .................................................................................................. 109

Figure 43: Third natural vibration mode for the uncoupled rigid fluid cavity (𝜔 =1511 𝑟𝑎𝑑/𝑠) filled with air. .................................................................................................. 109 .

Figure A. 1: Location of the points where the stresses were measured.................................. 125

Figure A. 2: Location of the points where the stresses were measured for the beam with two

layers. ...................................................................................................................................... 137 .

Figure B. 1: Vertical deflection for the corrugated core beam with one layer and C-F

boundary conditions. .............................................................................................................. 161

Figure B. 2: Vertical deflection for the corrugated core beam with one layer and C-H


Figure B. 3: Vertical deflection for the corrugated core beam with one layer and H-H


Figure B. 4: Vertical deflection for the corrugated core beam with two layers and C-C


Figure B. 5: Vertical deflection for the corrugated core beam with two layers and C-F


Figure B. 6: Vertical deflection for the corrugated core beam with two layers and C-H


Figure B. 7: Vertical deflection for the corrugated core beam with two layers and H-H

boundary conditions. .............................................................................................................. 170 .

Figure C. 1: Vertical deflection for the corrugated core plate with one layer and C-C boundary

conditions. .............................................................................................................................. 173

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Figure C. 2: Vertical deflection for the corrugated core plate with one layer and C-F boundary

conditions. .............................................................................................................................. 174

Figure C. 3: Vertical deflection for the corrugated core plate with one layer and C-H boundary

conditions. .............................................................................................................................. 174

Figure C. 4: Vertical deflection for the corrugated core plate with one layer and H-H


Figure C. 5: Vertical deflection for the corrugated core plate with two layers and C-C


Figure C. 6: Vertical deflection for the corrugated core plate with two layers and C-F


Figure C. 7: Vertical deflection for the corrugated core plate with two layers and C-H


Figure C. 8: Vertical deflection for the corrugated core plate with two layers and H-H


xxi

List of Tables

Table 1: Punctual displacement errors using a regular mesh and FEM formulation. .............. 71

Table 2: Punctual displacement errors using a regular mesh and RPIM formulation. ............. 71

Table 3: Punctual displacement errors using a regular mesh and NNRPIM formulation. ....... 71

Table 4: Relative frequency error: polynomial basis with one monomial and 𝑐 = 1.42 and

𝑝 = 1.03. .................................................................................................................................. 74

Table 5: Relative frequency error: polynomial basis with one monomial and 𝑐 = 0.0001 and

𝑝 = 0.999. ................................................................................................................................ 74

Table 6: Relative frequency error: polynomial basis with three monomials and 𝑐 = 1.42 and

𝑝 = 1.03. .................................................................................................................................. 75

Table 7: Relative frequency error: polynomial basis with three monomials and 𝑐 = 0.0001

and 𝑝 = 0.9999. ...................................................................................................................... 75

Table 8: Relative frequency error for the NNRPIM formulation for 𝑐 = 1.42 and 𝑝 = 1.03.

.................................................................................................................................................. 75

Table 9: Relative frequency error for the NNRPIM formulation for 𝑐 = 0.0001 and 𝑝 =0.9999. ..................................................................................................................................... 76

Table 10: Geometric and material properties of the sandwich beam subjected to different

boundary conditions. ................................................................................................................ 82

Table 11: First four natural frequencies (Hz) for the sandwich beam with different boundary

conditions. ................................................................................................................................ 82

Table 12: Geometric and material properties of the sandwich beam subjected to different

length-thickness ratios. ............................................................................................................. 83

Table 13: Non-dimensional natural frequencies for the sandwich beam with varying 𝜆 = 𝐿𝐻

(FEM). ...................................................................................................................................... 83


(RPIM). ..................................................................................................................................... 83


(NNRPIM). ............................................................................................................................... 84

Table 16: Geometric and material properties of the sandwich beam with sinusoidal corrugated

core. .......................................................................................................................................... 86

Table 17: Maximum relative deflection, 𝑤𝑚𝑎𝑥, for the sandwich beam under three point

bending. .................................................................................................................................... 87

Table 18: Relative deflection, 𝑤 , for the corrugated core beam with 2 layers under punctual

load (FEM). .............................................................................................................................. 88

Table 19: Relative deflection, 𝑤 , for the corrugated core beam with 2 layers under punctual

load (RPIM). ............................................................................................................................. 89

Table 20: Relative deflection, 𝑤 , for the corrugated core beam with 2 layers under distributed

load (FEM). .............................................................................................................................. 89

Table 21: Relative deflection, 𝑤 , for the corrugated core beam with 2 layers under distributed

load (RPIM). ............................................................................................................................. 90

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Table 22: Fundamental natural frequency (Hz) for the beam with one sinusoidal core (FEM).

.................................................................................................................................................. 91

Table 23: Fundamental natural frequency (Hz) for the beam with one sinusoidal core (RPIM).

.................................................................................................................................................. 92

Table 24: First natural frequency (Hz) for the beam with one sinusoidal core (NNRPIM). .... 93

Table 25: Relative deflection, 𝑤 , for the corrugated core beam under a harmonic distributed

load (FEM). .............................................................................................................................. 95


load (RPIM). ............................................................................................................................. 96


load (NNRPIM). ....................................................................................................................... 96

Table 28: Relative deflection, 𝑤 , for the corrugated core plate under a harmonic distributed

load (FEM). .............................................................................................................................. 98


load (RPIM). ............................................................................................................................. 98


load (NNRPIM). ....................................................................................................................... 99

Table 31: Geometric and material properties of the laminated plate. .................................... 100

Table 32: First 10 natural frequencies /Hz for the laminated plate (mesh-1). ........................ 100

Table 33: First 10 natural frequencies /Hz for the laminated plate (mesh-2). ........................ 101

Table 34: Stresses /MPa for the laminated plate, point A. ..................................................... 102

Table 35: Stresses /MPa for the laminated plate, point A1. ................................................... 103

Table 36: Stresses /MPa for the laminated plate, point B. ..................................................... 103

Table 37: Stresses /MPa for the laminated plate, point B1. ................................................... 103

Table 38: First 10 natural frequencies /Hz for the acoustic tube. ........................................... 105

Table 39: First 8 natural frequencies /Hz for the 2D Car. ...................................................... 106

Table 40: Physical properties of the structure and fluid of the coupled vibro-acoustic system.

................................................................................................................................................ 107

Table 41: Ten first eigenvalues /𝑟𝑎𝑑𝑠 − 1 of the coupled fluid-structure cavity filled with air.

................................................................................................................................................ 108

Table 42: Ten first eigenvalues /𝑟𝑎𝑑𝑠 − 1 of the coupled fluid-structure cavity filled with

water. ...................................................................................................................................... 108 .

Table A. 1: Relative deflection, 𝑤 , for the corrugated core beam under punctual load (FEM).

................................................................................................................................................ 122

Table A. 2: Relative deflection, 𝑤 , for the corrugated core beam under punctual load (RPIM).

................................................................................................................................................ 122

Table A. 3: Relative deflection, 𝑤 , for the corrugated core beam under a punctual load

(NNRPIM). ............................................................................................................................. 123

Table A. 4: Relative deflection, 𝑤 , for the corrugated core beam under a distributed load

(FEM). .................................................................................................................................... 123

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(RPIM). ................................................................................................................................... 124


(NNRPIM). ............................................................................................................................. 124

Table A. 7: Stress values \MPa, for the corrugated core beam under a punctual load and C-C

boundary conditions (FEM).................................................................................................... 125

Table A. 8: Stress values \MPa, for the corrugated core beam under a punctual load and C-F


Table A. 9: Stress values \MPa, for the corrugated core beam under a punctual load and C-H


Table A. 10: Stress values \MPa, for the corrugated core beam under a punctual load and H-H



boundary conditions (RPIM). ................................................................................................. 127








boundary conditions (NNRPIM). ........................................................................................... 129







Table A. 19: Stress values \MPa, for the corrugated core beam under a distributed load and C-

C boundary conditions (FEM). ............................................................................................... 131


F boundary conditions (FEM). ............................................................................................... 131


H boundary conditions (FEM). ............................................................................................... 132

Table A. 22: Stress values \MPa, for the corrugated core beam under a distributed load and H-

H boundary conditions (FEM). ............................................................................................... 132


C boundary conditions (RPIM). ............................................................................................. 133


F boundary conditions (RPIM). .............................................................................................. 133


H boundary conditions (RPIM). ............................................................................................. 134

xxiv


H boundary conditions (RPIM). ............................................................................................. 134


C boundary conditions (NNRPIM)......................................................................................... 135


F boundary conditions (NNRPIM). ........................................................................................ 135


H boundary conditions (NNRPIM). ....................................................................................... 135


H boundary conditions (NNRPIM). ....................................................................................... 136

Table A. 31: Stress values \MPa, for the 2 layers beam under a punctual load and C-C


Table A. 32: Stress values \MPa, for the 2 layers beam under a punctual load and C-F


Table A. 33: Stress values \MPa, for the 2 layers beam under a punctual load and C-H


Table A. 34: Stress values \MPa, for the 2 layers beam under a punctual load and H-H


Table A. 35: Stress values \MPa, for the 2 layers beam under a punctual load and C-C


Table A. 36: Stress values \MPa, for the 2 layers beam under a punctual load and C-F


Table A. 37: Stress values \MPa, for the 2 layers beam under a punctual load and C-H


Table A. 38: Stress values \MPa, for the 2 layers beam under a punctual load and H-H


Table A. 39: Stress values \MPa, for the 2 layers beam under a distributed load and C-C


Table A. 40: Stress values \MPa, for the 2 layers beam under a distributed load and C-F


Table A. 41: Stress values \MPa, for the 2 layers beam under a distributed load and C-H


Table A. 42: Stress values \MPa, for the 2 layers beam under a distributed load and H-H


Table A. 43: Stress values \MPa, for the 2 layers beam under a distributed load and C-C






Table A. 46: Stress values \MPa, for the 2 layers beam under a distributed load and H-H

boundary conditions (RPIM). ................................................................................................. 152 ..

Table B. 1: Second natural frequency (Hz) for the beam with one sinusoidal core (FEM). .. 154

xxv

Table B. 2: Third natural frequency (Hz) for the beam with one sinusoidal core (FEM). ..... 155

Table B. 3: Second natural frequency (Hz) for the beam with one sinusoidal core (RPIM). 156

Table B. 4: Third natural frequency (Hz) for the beam with one sinusoidal core (RPIM). ... 157

Table B. 5: Second natural frequency (Hz) for the beam with one sinusoidal core (NNRPIM).

................................................................................................................................................ 158

Table B. 6: Third natural frequency (Hz) for the beam with one sinusoidal core (NNRPIM).

................................................................................................................................................ 159

Table B. 7: Maximum absolute stresses for the beam with 1 layer (FEM). ........................... 159

Table B. 8: Maximum absolute stresses for the beam with 1 layer (RPIM). ......................... 160

Table B. 9: Maximum absolute stresses for the beam with 1 layer (NNRPIM)..................... 160

Table B. 10: Fundamental natural frequency (Hz) for the beam with two sinusoidal cores

(FEM). .................................................................................................................................... 162

Table B. 11: Second natural frequency (Hz) for the beam with two sinusoidal cores (FEM).

................................................................................................................................................ 163

Table B. 12: Third natural frequency (Hz) for the beam with two sinusoidal cores (FEM). . 164

Table B. 13: Fundamental natural frequency (Hz) for the beam with two sinusoidal cores

(RPIM). ................................................................................................................................... 165

Table B. 14: Second natural frequency (Hz) for the beam with two sinusoidal cores (RPIM).

................................................................................................................................................ 166

Table B. 15: Third natural frequency (Hz) for the beam with two sinusoidal cores (RPIM). 167

Table B. 16: Maximum absolute stresses for the beam with 2 layers (FEM). ....................... 167

Table B. 17: Maximum absolute stresses for the beam with 2 layers (RPIM). ...................... 168

Table B. 18: Maximum absolute stresses for the beam with 2 layers (NNRPIM). ................ 168 ..

Table C. 1: Maximum absolute stresses for the plate with 1 layer (FEM). ............................ 172

Table C. 2: Maximum absolute stresses for the plate with 1 layer (RPIM). .......................... 172

Table C. 3: Maximum absolute stresses for the plate with 1 layer (NNRPIM). .................... 173

Table C. 4: Maximum absolute stresses for the plate with 2 layers (FEM). .......................... 175

Table C. 5: Maximum absolute stresses for the plate with 2 layers (RPIM). ......................... 175

Table C. 6: Maximum absolute stresses for the plate with 2 layers (NNRPIM). ................... 176


1

1. Introduction

The resources and time spent on the experimental analysis of the dynamic behavior of

structures and other physical phenomenon can have an exponential increase when dealing

with complex geometries and materials. In recent decades, a significant effort has been

applied on the development of numerical computational tools that simulate the behavior of

systems under different solicitations. Although, frequently, this methods do not present a

good correlation when compared to experimental results [1]. One scientific field that has been

attracting recent attention is the field of vibro-acoustics. The need to optimize the vibratory

response of structures on areas like aeronautics and aerospace can be a major factor for

competitiveness in the industry.

Discrete numerical methods, such as the FEM, have been extended to the acoustical

formulation. Although, despite being the most popular method, it can prove to be inefficient

some times for vibro-acoustic problems, showcasing difficulties in correctly simulating the

wave propagations [2]. Other numerical techniques, like the meshless methods, have gained

popularity over the last few years, mainly for being able to solve some difficulties associated

with FEM when denser nodal zones are required.

In this work, the weak formulation of acoustical problems is extended to two different

meshless methods, the RPIM (Radial Point Interpolation Method) and NNRPIM (Natural

Neighbor Radial Point Interpolation Method). Several dynamic analysis (free, forced

vibrations and vibro-acoustics) are performed and compared with the results displayed by the

finite element method and also compared with the available results documented in literature.


2

1.1 Acoustic

The study of an acoustic problem can be breakdown to the interaction between a fluid

inducing pressure over a vibrating structure and vice-versa. To characterize the behavior of

the structure, the coupling with the surrounding fluid and the boundary conditions must be

taken into account, which will vary with both geometry and properties of the fluid and

structure.

The analysis of the coupled system can be divided in three particular cases: interior problems,

where the fluid is encased inside the structure; exterior problems, where the fluid domain

surrounds the structures; interior/exterior problems, which is a combination of both problems.

Besides this division, in acoustic problems the frequency domain must also be taken into

consideration, since it will define the approach that must be considered when solving the

coupled problem. The system can be working in low, mid (MF) and high frequencies (HF) -

Figure 1.

Figure 1: Typical FRF (Frequency Response Function) of a vibroacoustic system [3].

In the low frequencies domain, the resonances are well spaced, corresponding to a modal-

controlled behavior. For this case, FEM or BEM solutions [3] usually present accurate results

to describe the response of the coupled system.

In the high frequencies domain, the response no longer has visible resonance of local strong

variations, implying that the modal density of the system is uniform, thus the number of

modes present is high. For acoustic problems under this domain, generally statistical energy

analysis (SEA) [4] is the best approach.

On the intermediate domain (mid-frequencies) both low and high frequency behaviors can be

found, for which hybrid deterministic-statistical methods are usually applied in solving the

system [5].

For an acoustic pressure disturbance 𝑝(𝒙, 𝑡) in a perfect fluid of volume Ωf, being 𝜌0 the

density and 𝑐0 the sound speed on the fluid, the inhomogeneous wave equation, also known as

Helmhotlz equation, is given as [6],

∇2𝑝(𝒙, 𝑡) −1

𝑐02 ��(𝒙, 𝑡) = −𝑄(𝒙, 𝑡) (1)

where 𝑄(𝒙, 𝑡) is the acoustic source distribution.

The resolution of the Helmhotlz equation through numerical computational methods is still a

challenging process and many times produces inaccurate results, which increase in error as


3

the frequency range increases. For instance, methods like the FE still cannot be fully applied

to solve this systems due to problems from the pollution error [7] that arises from the

dispersive nature of the numerical wave.

To try and eliminate this dispersion, many numerical methods have been proposed. The first

idea was to stabilize the finite element method [8, 9], but at the end it was shown that such

technique did not enhanced significantly the FEM. Next, higher order approximations based

on the hp formulation [10] and meshless approaches [11] were proposed.

Bouillar et al. [12] applied the EFGM to the acoustic field, but concluded that this meshless

method is still affected by pollution errors, even though in a much lesser magnitude compared

to the FEM. An improved method for the EFGM was proposed [13]. Interpolator methods

were later tested, with the RPIM being used to study the dispersion effect in 2D acoustic

problems [14], showing that such method can significantly reduce the dispersion error.

Adaptations to the RPIM were proposed, like the linearly conformed Radial Point

Interpolation Method (LC-RPIM) [15], the edge-based smoothed Point Interpolation Method

(ES-PIM) [16], the cell-based smoothed Radial Point Interpolation Method (CS-RPIM) [17],

and others.

In both cases, there were still documented problems arising from the pollution error, and that

may be the reason for the inexistence of efficient discrete numerical methods for the medium

frequency range on acoustical problems [18].

Thus, the principal interest of this work lies in the analysis of the behavior and accuracy of

two different meshless methods on the acoustical study of coupled fluid-structure problems.


4

1.2 Meshless Methods

In past decades, the finite element method (FEM) has been used in a wide range of

engineering applications. It is a numerical method best used to solve problems with complex

geometries and/or boundary conditions, since it is based on the subdivision of the initial

geometry into small domains called elements. The association of these elements forms a mesh

that assures nodal connectivity for all the structure. [19]

Despite the popularity and competence of the FEM, meshless methods have been seek with

increase interest over the last few years as an efficient technique for numerically solving

partial differential equations. Since it does not relies on a mesh, such as the case of the FEM,

this formulation becomes more suitable for solving problems with complex geometries where

mesh construction could represent a computational burdensome task [20].

Comparing both methods, meshless formulations presents some major advantages, being

more appropriated for solving problems with moving discontinuities like crack propagation,

more capable of handling large deformations (it is simpler to incorporate h-adaptivity), having

higher-order continuous shape functions, non-local interpolation features and no mesh

alignment sensitivity. As drawbacks, besides the higher computational costs, the shape

functions require high-order integration schemes in order to be correctly computed, also for

some formulations imposing essential boundary conditions can be troublesome [21].

In meshless methods, the geometry is discretized in nodes arbitrarily distributed along the

domain, and instead of using elements, the field functions are approximated within

“influence-domains” set for each interest point [22]. The first works on this method date back

to the seventies with the Smooth Particle Hydrodynamics (SPH) methods [23], using kernel

estimates to solve problems involving fluid masses moving arbitrarily in the absence of

boundaries [24]. This method gave origin to the Reproducing Kernel Particle Method

(RKPM) [25].

Later on, new approaches provided significant progress to the meshless method. Starting with

the Diffuse Element Method (DEM), that uses moving least-square approximations in the

Galerkin formulation, this way replacing the FEM interpolation, valid on an element, with a

local weighted least squares fitting, valid in a small neighborhood of an arbitrary point “x”

[26]. Belytschko gave significant improvements on the DEM, creating its own method called

the Element-Free Galerkin Method (EFGM) [27].

Meshless methods can be divided in two categories depending on the used formulation, which

can be either the strong-form or the weak-form. The first seeks the direct resolution of the

partial differential equations ruling the studied problem. The weak-form uses a variational

principle in order to minimize the residual weight of the differential equations, to do that the

method replaces the exact solution with an approximated function affected by a test function

[28].

Girault, Kao & Perrone and Liszka [29-31] gave significant contributions in the development

of the global strong-form formulation for meshless methods, such as the General Finite

Difference Method (GFDM). Later works on this formulation involve the Finite Point Method

(FPM) [32], that uses a stabilization technique in the collocation point method and the Radial

Basis Function Method (RBFM) [33], which uses radial basis functions, respecting an

Euclidean norm, in order to approximate the variable fields within the entire global or partial

local domain. Another simple local meshless approach is the Diffuse Approximate Method

[34] [35].

Other methods based on the global weak formulation, such as the Partition of Unity FEM

(PUFEM) [36] and the hp-cloud method [37], demonstrated that the methods based on the


5

MSL are instances of partitions of unity, which lead to significant improvements on these

methods. Another class of meshless methods are based on the local weak form, that are

generated on overlapping subdomains, instead of using a global weak form [21]. For example,

the Meshless Free Petrov-Galerkin (MLPG) [38], initially used for solving linear and

nonlinear potential problems, later evolved into the Method of Finite Spheres (MFS) [39].

Due to the complexity on computing the shape function and their derivatives, and the lack of

the Kronecker delta property, which arises problems when essential and natural boundary

conditions are to be imposed, several interpolant meshless methods were developed. Liu

initially proposed the Point Assembly Method (PAM) [40] and the Point Interpolation Method

(PIM) [41], the later demonstrating some singularity problems for regular nodal distributions.

The Radial Point Interpolation Method (RPIM) [42] was then introduced to avoid such

problems, relying on the use of radial basis functions, mainly the multi-quadrics (MQ)

functions [43]. Other relevant interpolant meshless methods are the Meshless Finite Element

Method (MFEM) [44] and the Natural Element Method (NEM) [45]. Recently, Belinha

combined both the NEM and RPIM formulations, giving origin to the Natural Neighbour

Radial Point Interpolation Method (NNRPIM) [46]. This last method uses the concepts of

influence-cell, such as the Voronoї diagrams [47] and the Delaunay tessellation to construct

the integration mesh and influence domains of each node.

Both the RPIM and NNRPIM use radial interpolation functions, possessing the delta

Kronecker and compact support properties. These methods differ in the way the nodal

connectivity is imposed. On the RPIM, the nodal connectivity is established by the

overlapping of the “influence-domains”, and thus, requires the use of a background

integration mesh to allocate the integrations points, being nodal independent.

On the other hand, the recently developed NNRPIM uses the concept of “influence-cells”,

relying on mathematical methods such as the Voronoї diagrams and Delaunay tessellation.

Thus, for the distribution of the integration points, only the spatial position of the nodes is

necessary, making the NNRPIM a truly meshless method, whose integration scheme is nodal

dependent.

Despite being a recent method, the NNRPIM has already been applied in many different

fields: in the static analysis of isotropic and orthotropic plates [48], nonlinear problems [49],

crack opening problems [50], bone tissue remodeling [28] and others.

In the present work, the FEM, RPIM and NNRPIM formulations are used to solve the diverse

numerical examples of static, free and forced vibrations and vibro-acoustic problems. For the

NNRPIM, it is the first time this method is applied to the analysis of acoustical problems.


6

1.3 Thesis Objectives

This thesis aims to analyze of the performance of two meshless methods, the RPIM and

NNRPIM, on the prediction of the behavior of structures and coupled fluid-structures

problems under dynamic loads.

For this matter, a sequence of MATAB® algorithms will be created. These consists on

algorithms that permit the study of free and forced vibrations on 2D and 3D problems as well

as the acoustical study of uncoupled and interior coupled fluid-structure problems.

As a final result, for the free and forced vibrations studies, these algorithms must be able to:

Solve eigenvalue problems and plot the normalized eigenvectors of the problem;

Calculate the nodal dynamic displacements for any given load and damping present in

the system;

Calculate the dynamic stresses and strains at each integration point;

Plot the dynamic displacement at any given point in time;

Plot the displacement of any node along the time domain;

For harmonic loads, plot the displacement of any node along the frequency domain;

For the acoustical algorithms, they must be able to:

Solve eigenvalue problems and plot the normalized eigenvectors of the Helmholtz

equation;

Calculate the nodal pressure for uncoupled acoustical problems;

Solve eigenvalue problems and plot the normalized eigenvectors of the interior

coupled problem;

Calculate the nodal pressure for the fluid and nodal displacements for the structure for

an interior coupled problem;

Plot of the displacements (for the structure) and pressure (for the fluid) of any node

along the frequency domain;

Calculation of acoustic parameters;

For the developed software, both task must be able to be performed on both the FEM, RPIM

and NNRPIM formulations.

After the creation of the algorithms, examples consisting mainly in laminated and sandwich

beams and plates will be solved and their results compared for both formulations and with the

literature whenever possible.


7

1.4 Thesis Arrangement

This thesis is divided into 9 chapters: Introduction, Meshless Methods, Solid Mechanics,

Plane Elasticity, Vibrations, Vibro-Acoustics, Routines, Numerical Examples and

Conclusions.

In the first chapter, an overview of the topics presented in the thesis is given. Explaining the

development of the meshless methods, forced vibrations and the acoustical formulation

In the next five chapters is provided a detailed explanation of the theoretical formulations

used on the thesis.

In chapter 7. Routines, an explanation of the computed algorithms is given. In this chapter its

possible to understand the code that the MATLAB® software runs in order to obtain the

desired results.

In chapter 8, are described several numerical examples run with both formulations using the

created algorithms. When possible, the results are compared with the theoretical solutions and

other results present in the literature.

In the last chapter, the main conclusions about this work are discussed, retaining also some

possibilities and recommendations for future works on the subject.


8


9

2. Meshless Methods

Meshless methods follow a general procedure: define the nodal mesh that discretizes the

geometry; construct the background mesh constituted by the integration points; establish the

influence domains for each integration point; creat the interpolation functions; create the local

stiffness and mass matrix; assemble the local matrices into the global matrices of the problem;

impose the essential and natural boundary conditions and solve the system of equations.

The distribution of the nodal mesh has a direct influence in the quality and accuracy of the

results. Regular meshes usually perform better than irregular meshes - Figure 2, and for

problems involving stress concentrations (crack propagations, clamped boundaries, etc.) it

may be necessary to have a higher nodal density around those areas to assure good results.

As such, there is not an exact nodal mesh to better discretize a geometry. The choice must

also take in consideration the computational costs involved in analyzing the amount of

degrees of freedom chosen, and so an equilibrium between accuracy and efficiency has to be

attained.

Figure 2: a) Irregular mesh; b) regular mesh [28].

It is after creating the nodal mesh that the meshless methods differ from the FEM, since no

elements connecting the nodes are established. Instead, a background integration mesh is

constructed. Depending on the method adopted, this background mesh can be nodal

dependent (like in the case of the NNRPIM), or nodal independent (RPIM for example) -

Figure 3. On truly meshless methods, the nodes are used as the integration points or used to

directly define the integration points, removing the need to create a background mesh.

In most cases, this integration mesh is distributed having in account the Gauss-Legendre

quadrature rule

Figure 3: Background integration mesh a) nodal independent; b) nodal depedent (Voronӧi diagram) [28].


10

After the integration mesh has been assembled the “influence-domain” for each integration

point is established by generating concentric areas (or volumes for 3D cases) around this

points, and every node inside belongs to the “influence-domain” of that point. Unlike the

FEM, on meshless methods the nodal connectivity is imposed with the overlapping of the

various influence domains.

With the “influence-domain” defined for each interest point, the shape functions and its

derivatives can be calculated. This process heavily depends on the formulation used, as they

can be either interpolation or approximation functions. The RPIM and NNRPIM methods are

interpolator methods, meaning that the shape functions pass at every node, thus possessing the

delta Kronecker property.

Next the local matrices for each integration point can be constructed and assembled into the

global matrices. This process is in everything similar to the FEM, where the global matrices

are obtained by allocating the local matrix in their respective degrees of freedom, that is,

𝑲 =∑𝑹𝑗𝑇

𝑁

𝑗=1

· 𝑲𝑗∗ · 𝑹𝑗 (2)

Where 𝑹𝒋 is the allocation matrix connecting the local degrees of freedom from the local

matrix with the global degrees of freedom. The same process is applied for the mass matrix

and force vector.

To impose the essential boundary conditions, the most three common methods are: the

Lagrange multipliers, the direct imposition method and the penalty method [28]. Since the

RPIM and NNRPIM are interpolations methods, the essential boundary conditions can be

imposed using either method.

In this work an extension of the penalty method was adopted to impose the boundary

conditions, where the rows and columns of the constrained degrees of freedom are removed

from the global matrices, resulting in a smaller system of equations to solve and avoiding the

appearance of rigid body modes when using modal superposition.


11

2.1 Radial Point Interpolation Method (RPIM)

As already stated, the Point Interpolation Method [41] is an efficient method to create

interpolation shape functions for meshless methods possessing the Kronecker delta property.

It uses polynomial basis functions to construct shape functions, forcing the interpolation

function to pass through all the scattered nodes in an influence domain. Despite those

advantages, the PIM may lead to singular moment matrices when creating the shape functions

if the nodes are perfectly aligned.

To bypass this problem, a radial basis function was added to the PIM generic shape functions,

creating the RPI shape functions [42]. However, the inclusion of these functions increases the

computational costs.

Figure 4: Interpolation function 𝑢ℎ(𝑥) and the discrete nodal parameters (𝑥𝑖) [28].

It has been proved that the absence of a polynomial basis on the RPI method would lead to

failure of the standard patch test, since a 𝐶1 continuity needs to be assured. Adding this

polynomial basis ensures the stability of this method.

Considering an approximation function 𝑢ℎ(𝒙) in an influence domain, being 𝑛 the number of

nodes inside the influence domain of 𝒙𝐼. The RPIM forces the approximation function to pass

through all nodal data within the influence domain, using a radial basis function 𝑟𝑖(𝒙) and a

polynomial basis function 𝑝𝑗(𝒙). Thus, the interpolated value for an interest point 𝒙𝐼 can be

obtained with:

𝑢ℎ(𝒙𝐼) =∑𝑟𝑖(𝒙𝐼)𝑎𝑖

𝒏

𝒊=𝟏

+∑𝑝𝑗(𝒙𝐼)𝑏𝑗

𝒎

𝒋=𝟏

= 𝑹𝑇(𝒙𝐼)𝒂 + 𝑷𝑇(𝒙𝐼)𝒃 (3)

where 𝑎𝑖 and 𝑏𝑗 are the coefficients of 𝑟𝑖(𝒙) and 𝑝𝑗(𝒙). The vectors can be defined as:

𝒂𝑻 = {𝑎1 𝑎2 ⋯ 𝑎𝑛}

𝒃𝑻 = {𝑏1 𝑏2 ⋯ 𝑏𝑛}

𝑹𝑻(𝒙) = [𝑟1(𝒙) 𝑟2(𝒙) ⋯ 𝑟𝑛(𝒙)]

𝑷𝑻(𝒙) = [𝑝1(𝒙) 𝑝2(𝒙) ⋯ 𝑝𝑚(𝒙)]

(4)

The polynomial basis function 𝑝𝑗(𝒙) is defined according to triangle of Pascal - Figure 5 -

being 𝑚 the number of monomials of the complete polynomial.


12

𝑷𝑻(𝒙) = [1 𝑥 𝑦 𝑥2 𝑥𝑦 …] (5)

For the radial basis functions 𝑟𝑖(𝒙𝐼) the only variable present is the Euclidian norm between

the field nodes and the considerer point 𝒙. For a three-dimensional space, this distance

between an interest point 𝒙𝐼 and another point 𝒙𝑖 can be expressed as,

𝑑𝐼𝑖 = ‖𝒙𝑖 − 𝒙𝐼‖ = √(𝑥𝑖 − 𝑥𝐼)2 + (𝑦𝑖 − 𝑦𝐼)2 + (𝑧𝑖 − 𝑧𝐼)2 (6)

There are several different RBF that can be applied on the construction of the shape functions

[43], being the multi-quadratics (MQ) functions the most used: 𝑟𝑖(𝒙𝐼) = (𝑑𝐼𝑖2 + 𝑐2)𝑝.

Comprehensively, the MQ-RBF is dependent on two shape parameters, 𝑐 and 𝑝, that must be

determined and optimized in order to obtain accurate results [51].

Figure 5: Triangle of Pascal [28].

In order to calculate the values of the coefficients 𝑎𝑖 and 𝑏𝑗 a new set of equations must be

taken into consideration [52], which can be presented as:

∑𝑝𝑗(𝒙)

𝑛

𝑖=1

𝑎𝑖 = 0, 𝑗 = 1,2, … ,𝑚 (7)

Now the system with 𝑛 +𝑚 unknowns can be solved by forcing the interpolant functions to

pass through all nodes in the influence domain, expressed in matrix form as:

[𝑹 𝑷𝑷𝑻 𝟎

] · {𝒂𝒃} = 𝑴𝑇 {

𝒂𝒃} = {

𝒖𝑠𝟎} (8)

Where 𝑴𝑇 is the total moment matrix, and 𝒖𝑠 the vector that represents the function values at

the nodes inside the influence domain. The coefficient matrix 𝑹 is defined as,


13

𝑹 = [

𝑟1(𝒙1) 𝑟2(𝒙1)𝑟1(𝒙2) 𝑟2(𝒙2)

⋯ 𝑟𝑛(𝒙𝑛)⋯ 𝑟𝑛(𝒙𝑛)

⋮ ⋮𝑟1(𝒙𝑛) 𝑟2(𝒙𝑛)

⋱ ⋮⋯ 𝑟𝑛(𝒙𝑛)

] (9)

The coefficient matrix 𝑷 is,

𝑷 = [

𝑝1(𝒙1) 𝑝2(𝒙1)𝑝1(𝒙2) 𝑝2(𝒙2)

⋯ 𝑝𝑚(𝒙1)⋯ 𝑝𝑚(𝒙2)

⋮ ⋮𝑝1(𝒙𝑛) 𝑝2(𝒙𝑛)

⋱ ⋮⋯ 𝑝𝑚(𝒙𝑛)

] (10)

Both matrixes are symmetrical, which means 𝑴𝑇 is also symmetric. By inverting the total

moment matrix 𝑴𝑇, it is possible to obtain the values of the coefficients 𝑎𝑖 and 𝑏𝑗,

{𝒂𝒃} = 𝑴𝑇

−1 {𝒖𝑠𝟎} (11)

Substituting back on equation (3),

𝑢ℎ(𝒙) = [𝑹𝑇(𝒙) 𝑷𝑇(𝒙)]𝑴𝑇−1 {

𝒖𝑠𝟎} = 𝝋𝑇(𝒙)𝒖𝑠 (12)

The matrix of shape functions is defined as,

𝝋(𝒙) = [𝜑1(𝒙) 𝜑2(𝒙) ⋯ 𝜑𝑛(𝒙)] (13)

The approximation function 𝑢ℎ(𝒙) is now given by,

𝑢ℎ(𝒙) =∑𝜑𝑖(𝒙)

𝑛

𝑖=1

𝑢𝑖 (14)

For two distinct points, sharing the same 𝑛 nodes inside their influence-domain, the obtained

coefficients 𝑎𝑖 and 𝑏𝑗 will be the same for both points, therefore the total moment matrix is

not directly dependent on the spatial position of the considered point. The derivatives of the

shape functions can be easily obtained since only the radial and polynomial basis needs to be

derivate, that is,

𝜕𝑢ℎ(𝒙)

𝜕𝜉=∑

𝜕𝜑𝑖(𝒙)

𝜕𝜉𝑢𝑖

𝑛

𝑖=1

(15)

where 𝜉 is a generic variable. The RPI shape functions enjoy a range of properties, such as

consistency, reproducibility, partition of unity, compact support and the Kronecker delta

property. A detailed description of RPI shape functions can be found in the literature [28].


14

2.2 Natural Neighbour Radial Point Interpolation Method (NNRPIM)

The NNRPIM shares many resemblances with the RPIM formulation, in which both are

interpolator methods possessing the Kronecker Delta property, and use a radial basis functions

to construct the shape functions. As such, after the construction of the “influence-domain” for

each interest point both formulations work the same way.

Where they differ is in the imposition of the nodal connectivity and in the construction of the

background integration mesh. Here the NNRPIM uses the concept of “influence-cell” to draw

the “influence-domain” for each interest point. The method relies on the use of geometrical

concepts such as the Voronoї diagrams and the Delaunay tessellations.

To obtain the Voronoї diagram for each node the mathematical concept of natural neighbor is

used [53]. This method consists in finding the closest subset of nodes for a given point and

giving to each node a weight based on its proportional area. As a result, the Voronoї cell for

each node consist in the aggregate of all points that are more close to the considered node than

to any other node on the structure.

In Figure 6 is illustrated the construction of a Voronoї cell, region 𝑉0. For the considered

point, 𝑛0, a straight line is drawn from this node to the closest node. Next, a line normal to

this one is drawn on the closest node, and every point on the other side of this line is rejected.

This process is then repeated until all normal lines form a closed geometry, from this point all

neighbor nodes have been found, which correspond to the nodes on the lines.

After the neighbor nodes are found, the Voronoї cell for 𝑛0 is obtained dividing the distance

to each node by half, that is, the Voronoї cell is the homothetic form of the closed geometry,

being

𝑑0𝑖∗ =

‖𝑥0 − 𝑥𝑖‖

2 (16)

Figure 6: a) Initial set of nodes of the structure. b) Normal to straight line that connects the closest node to 𝑛0.

c) Closed geometry and neighbor nodes. d) Voronoї cell 𝑉0for node 𝑛0 [28].


15

After applying this methodology to every node on the structure, the influence cell concept can

be used to construct the “influence-domain” for each interest point. There are mainly two

degrees of influence cell used: “first degree influence-cell” - where the neighbor nodes of a

point of interest are considered - and “second degree influence-cell” - where the natural

neighbor nodes of the neighbor nodes of a point of interest are considered on the influence-

cell - Figure 7.

Figure 7: a) First degree influence-cell b) second degree influence cell [28].

To impose the background integration mesh, a nodal based integration scheme is used. Here,

the Voronoї cells are divided into small areas, either quandrangular (for irregular meshes) or

triangular (for regular meshes). Each of these areas is then populated with Gauss Points,

depending on the desired quadrature. The integration weight of each integration point is given

by

𝜔𝐼 = 𝜔𝜉 · 𝜔𝜂 · (𝐴

4) (17)

Where 𝐴 is the area of the respective sub-quadrilateral and 𝜔𝜉 and 𝜔𝜂 are the integration

weights given by the Gauss-Legendre Quadrature. In Figure 8 is illustrated how to divide the

cells into small areas, and the small shapes obtained from the division with one gauss point.

Figure 8: a) Irregular mesh generating quadrangular areas b) regular mesh generating triangular areas [28].


16


17

3. Solid Mechanics


18

3.1 Stress and Strain Fields

Consider a 3D solid structure. At any given point of the structure the stress and strain fields

are defined as:

𝝈 = {𝜎𝑥 𝜎𝑦 𝜎𝑧 𝜏𝑥𝑦 𝜏𝑦𝑧 𝜏𝑧𝑥}𝑇 (18)

𝜺 = {휀𝑥 휀𝑦 휀𝑧 𝛾𝑥𝑦 𝛾𝑦𝑧 𝛾𝑧𝑥}𝑇 (19)

The strain components can be directly obtained from the displacement field. Denoting the

components 𝑢, 𝑣 and 𝑤 as the displacements along the 𝑥, 𝑦 and 𝑧 axis, the strain field can be

calculated using the following expressions

휀𝑥 =𝜕𝑢

𝜕𝑥 ; 휀𝑦 =

𝜕𝑣

𝜕𝑦 ; 휀𝑧 =

𝜕𝑤

𝜕𝑧

𝛾𝑥𝑦 =𝜕𝑢

𝜕𝑦+𝑑𝑣

𝑑𝑥 ; 𝛾𝑥𝑧 =

𝜕𝑢

𝜕𝑧+𝑑𝑤

𝑑𝑥 ; 𝛾𝑦𝑧 =

𝜕𝑣

𝜕𝑧+𝑑𝑤

𝑑𝑦

(20)

And the stress tensor is calculated multiplying the constitutive matrix, 𝑫, defined by material

properties of the structure, with the strain tensor, resulting

𝝈 = 𝑫𝜺 (21)

If initial stresses or strains are to be considered, then, expression (21) becomes,

𝝈 = 𝑫(𝜺 − 𝜺0) + 𝝈0 (22)

The equilibrium equations for the static analysis of a general 3D problem are

𝜕𝜎𝑥𝜕𝑥

+𝜕𝜏𝑥𝑦

𝜕𝑦+𝜕𝜏𝑥𝑧𝜕𝑥

+ 𝑓𝑥 = 0

𝜕𝜏𝑥𝑦

𝜕𝑥+𝜕𝜎𝑦

𝜕𝑦+𝜕𝜏𝑦𝑧

𝜕𝑧+ 𝑓𝑦 = 0

𝜕𝜏𝑥𝑧𝜕𝑥

+𝜕𝜏𝑦𝑧

𝜕𝑦+𝜕𝜎𝑧𝜕𝑧

+ 𝑓𝑧 = 0

(23)

Or in matrix form

𝑳𝝈 + 𝒇 = 0 (24)


19

3.2 Strong and Weak Form Formulations

In the strong form formulations, a solution is seek at every single point of the structure, which

means that the equations governing the strong form must be satisfied for any point in the

considered global domain. For complex geometries and boundary conditions, a solution to the

differential system equations governing the studied phenomenon may not always be possible

to attain.

In those cases, a weak form formulation is often sought. Here, the differential system

equations no longer needs to be valid at every point on the structure, but instead is established

at discrete points spread across the domain. The implementation of boundary conditions is

also simple, since they can be directly applied on an arbitrary node. As a downside, this

approximation means the accuracy of the solution is dependent on the number of points

discretizing the problem domain.


20

3.2.1 Weak Form of Galerkin

The Galerkin form is a variational principle that is based on the energy principle - the

Hamilton’s principle.

The Hamilton’s principle states that: “of all admissible displacements configurations

satisfying the compatibility conditions, the essential boundary conditions and the initial and

final time conditions, the real solution correspondent configuration is the one which

minimizes the Lagrangian functional L” [28].

𝐿 = 𝑇 − 𝑈 +𝑊𝑓 (25)

where 𝑇 is the kinetic energy, 𝑈 is the strain energy and 𝑊𝑓 is the work produced by the

external forces. For a solid domain with volume Ω, being 𝒖 and �� the displacement and

velocity fields, the kinetic energy is define by,

𝑇 =1

2∫𝜌��𝑇�� 𝑑Ω

Ω

(26)

being 𝜌 the solid’s mass density. For elastic materials, the strain energy is defined as,

𝑈 =1

2∫𝜺𝑇

Ω

𝝈 𝑑Ω (27)

being 𝜺 and 𝝈 the strain and stress vectors respectively. The work produced by the external

forces can be given by,

𝑊𝑓 = ∫ 𝒖𝑇𝒃 𝑑Ω + ∫𝒖𝑇𝒕 𝑑Γ

Γ

Ω

(28)

being 𝒃 body forces and Γ the traction boundary where the external forces 𝒕 are applied. Thus,

the Hamilton’s principle can be written as,

𝛿 ∫ [1

2∫𝜌��𝑇�� 𝑑Ω

Ω

−1

2∫𝜺𝑇

Ω

𝝈 𝑑Ω + ∫ 𝒖𝑇𝒃 𝑑Ω + ∫𝒖𝑇𝒕 𝑑Γ

Γ

Ω

]𝑡2

𝑡1

𝑑𝑡 = 0 (29)

Moving the variation operator 𝛿 to inside the integral results,

∫ [1

2∫𝛿 (𝜌��𝑇��) 𝑑Ω

Ω

−1

2∫𝛿(𝜺𝑇

Ω

𝝈) 𝑑Ω + ∫ 𝛿𝒖𝑇𝒃 𝑑Ω + ∫𝛿𝒖𝑇𝒕 𝑑Γ

Γ

Ω

]𝑡2

𝑡1

𝑑𝑡 = 0 (30)

Using the chain rule variation and the scalar property, that is

∫ 𝛿(𝒖 𝑇𝒖 ) 𝑑𝑡

𝑡2

𝑡1

= ∫ (𝛿𝒖 𝑇

𝑡2

𝑡1

𝒖 + 𝒖 𝑇𝛿𝒖 ) 𝑑𝑡 = 2∫ (𝛿𝒖𝑇𝒖) 𝑑𝑡

𝑡2

𝑡1

(31)

and integrating by parts, the first term of equation (29) becomes,

∫ [1

2∫𝛿 (𝜌��𝑇��) 𝑑Ω

Ω

]𝑡2

𝑡1

𝑑𝑡 = −∫ [𝜌∫𝛿 (𝜌𝒖𝑇��) 𝑑Ω

Ω

]𝑡2

𝑡1

(32)

For the second term of the equation, the next simplifications can be made,

𝛿(𝜺𝑇𝝈) = 𝛿𝜺𝑇𝝈 + 𝜺𝑇𝛿𝝈

𝜺𝑇𝛿𝝈 = (𝜺𝑇𝛿𝝈)𝑇 = 𝛿𝝈𝑇𝜺 = 𝛿(𝑫𝜺)𝑇𝜺 = 𝛿𝜺𝑇𝑫𝑇𝜺 = 𝛿𝜺𝑇𝝈

𝛿(𝜺𝑇𝝈) = 2𝛿𝜺𝑇𝝈

(33)


21

Therefore, the second term becomes,

∫ [1

2∫𝛿 (𝜺𝑇𝝈) 𝑑Ω

Ω

]𝑡2

𝑡1

𝑑𝑡 = ∫ [∫𝛿 𝜺𝑇𝝈 𝑑Ω

Ω

] 𝑑𝑡𝑡2

𝑡1

(34)

The Hamilton’s principle now becomes,

∫ [−𝜌∫𝛿 (𝒖𝑇��) 𝑑Ω

Ω

−∫𝛿 𝜺𝑇𝝈 𝑑Ω

Ω

+∫ 𝛿𝒖𝑇𝒃 𝑑Ω + ∫𝛿𝒖𝑇𝒕 𝑑Γ

Γ

Ω

] 𝑑𝑡 = 0𝑡2

𝑡1

(35)

In order for the equation to be zero at any given time, the integrand must be null, leading to

what is known as the ‘Galerkin weak form’, or the principle of virtual work,

−𝜌∫𝛿 (𝒖𝑇��) 𝑑Ω

Ω


Ω


Γ

Ω

= 0 (36)


22


23

4. Linear Deformation Theory


24

4.1 Plane Elasticity

The plane elasticity theory tries to simplify the 3D analyses of solid structures by assuming

that sections transversal to the prismatic axis z deform in the same manner, therefore, the

displacement along this axis can be neglected. This way, the study of a given structure

consists in the analyses of a generic transversal section on the xy plane.

Therefore, for a static analysis, the equilibrium equations for plane elasticity are,

𝜕𝜎𝑥𝜕𝑥

+𝜕𝜏𝑥𝑦

𝜕𝑦+ 𝑓𝑥 = 0

𝜕𝜏𝑥𝑦

𝜕𝑥+𝜕𝜎𝑦

𝜕𝑦+ 𝑓𝑦 = 0

(37)

Depending on the geometry and boundary conditions applied, the plane elasticity analysis can

be divided into two different types: plain strain and plain stress.


25

4.1.1 Plain Strain

For a structure whose thickness is considerably larger than the other dimensions is considered

to be in plane strain deformation. On this type of analysis, the deformations along the z axis

are considered null, and the displacement field is defined as,

𝑢 = 𝑢(𝑥, 𝑦); 𝑣 = 𝑣(𝑥, 𝑦); 𝑤 = 0 (38)

And the strain field yields,

𝛾𝑥𝑧 = 𝛾𝑦𝑧 = 휀𝑧 = 0

휀𝑥 =𝜕𝑢

𝜕𝑥; 휀𝑦 =

𝜕𝑣

𝜕𝑦; 𝛾𝑥𝑦 =

𝜕𝑢

𝜕𝑦+𝑑𝑣

𝑑𝑥

(39)

The constitutive matrix, for an isotropic material under plain strain is defined as,

𝑫 =𝐸

(1 + 𝜈)(1 − 2𝜈)[

1 − 𝜈 𝜈 0𝜈 1 − 𝜈 0

0 01 − 2𝜈

2 ] (40)

where 𝐸 is the Young’s modulus of the material and 𝜈 the Poisson ratio.


26

4.1.2 Plane Stress

For cases where the thickness is relatively small when compared to the other dimensions of a

given geometry, the structure is said to be under plane stress. For these conditions, the stress

field is defined as,

𝜏𝑥𝑧 = 𝜏𝑦𝑧 = 𝜎𝑧 = 0

𝜎𝑥 = 𝜎𝑥(𝑥, 𝑦); 𝜎𝑦 = 𝜎𝑦(𝑥, 𝑦); 𝜏𝑥𝑦 = 𝜏𝑥𝑦(𝑥, 𝑦) (41)

The constitutive matrix for a plane stress analysis is defined as,

𝑫 =𝐸

1 − 𝜈2[

1 𝜈 0𝜈 1 0

0 01 − 𝜈

2 ] (42)

Figure 9: Example of a structure under: a) plane stress; b) plane strain [54].


27

4.1.3 Virtual Work Principle

The virtual work principle states that the sum of all work forces applied on an element must

be null, for any given virtual displacement. This means,

𝛿𝑊𝑒 + 𝛿𝑊𝑖 + 𝛿𝑊𝑗 = 0 (43)

where 𝛿𝑊𝑒is the external forces virtual work, 𝛿𝑊𝑖 the internal forces virtual work and 𝛿𝑊𝑗

the virtual work of the kinetic forces. From Hamilton’s principle, equation (36), it can be

written

−𝜌∫𝛿 (𝒖𝑇��) 𝑑Ω

Ω


Ω


Γ

Ω

= 0 (44)

Since for this theory the deformations along the z axis are constant, the past equation can be

easily integrated along the thickness, yielding

𝜌∬𝛿 (𝒖𝑇��)𝑡 𝑑A

𝐴

+∬(𝛿 𝜺𝑇𝝈

𝐴

)𝑡 𝑑𝐴 = ∬𝛿𝒖𝑇𝒃

𝐴

𝑡 𝑑𝐴 + ∫𝛿𝒖𝑇𝒕 𝑡 𝑑L

L

(45)

being 𝑡 the thickness of the structure. For plane elasticity, the relation between the

displacements and the strains is given as,

𝜺 = 𝑳𝒖

𝑳 =

[ 𝜕

𝜕𝑥0

0𝜕

𝜕𝑦𝜕

𝜕𝑦

𝜕

𝜕𝑥]

(46)

From Hook’s law, equation (21), and substituting back on equation (45) the virtual work

principle writes,

𝜌∬𝛿 (𝒖𝑇��)𝑡 𝑑A

𝐴

+∬(𝛿 (𝑳𝒖)𝑇𝑫𝑳𝒖

𝐴

)𝑡 𝑑𝐴 =∬𝛿𝒖𝑇𝒃

𝐴

𝑡 𝑑𝐴 + ∫𝛿𝒖𝑇𝒕 𝑡 𝑑L

L

(47)

Which is the global equilibrium equation for a plane elasticity problem. As stated before, in

more general cases we seek the weak form formulation, where equation (47) is valid at

interest points. As such, the relation between the global and nodal displacements at a point 𝒙𝐼 is defined as,

𝒖(𝒙𝐼) =∑𝜑𝑗(𝒙𝐼) 𝒖𝑗∗

𝑛

𝑗=1

(48)

Therefore, the virtual displacements will also be,

𝛿𝒖(𝒙𝐼) =∑𝜑𝑗(𝒙𝐼) · 𝛿𝒖𝑗∗

𝑛

𝑗=1

(49)

For plane elasticity, 𝝋, the shape functions matrix, is defined as,

𝝋 = [𝜑1 0 𝜑2 0 … 𝜑𝑛 00 𝜑1 0 𝜑2 0 𝜑𝑛

] (50)


28

Finally, relating the deformations with the nodal displacements yields,

𝜺 = 𝑳 𝝋 𝒖∗ = 𝑩 𝒖∗

𝑩 =

[ 𝜕𝜑1𝜕𝑥

0

0𝜕𝜑1𝜕𝑦

𝜕𝜑1𝜕𝑦

𝜕𝜑1𝜕𝑥

𝜕𝜑2𝜕𝑥

0

0𝜕𝜑2𝜕𝑦

𝜕𝜑2𝜕𝑦

𝜕𝜑2𝜕𝑥

…

𝜕𝜑𝑛𝜕𝑥

0

0𝜕𝜑𝑛𝜕𝑦

𝜕𝜑𝑛𝜕𝑦

𝜕𝜑𝑛𝜕𝑥 ]

(51)

where 𝑩 is the deformation matrix. Substituting back on the virtual work principle equation

results in

𝛿𝒖·𝑇 ([∬ 𝜌𝑡 𝝋𝑇𝝋 𝑑A·

𝐴·] ��· +∬ 𝑩𝑇𝑫 𝑩

𝐴·𝑡 𝑑𝐴· 𝒖·)

= 𝛿𝒖·𝑇 (∬ 𝝋𝑇𝒃

𝐴·𝑡 𝑑𝐴· +∫ 𝝋𝑇𝒕 𝑡 𝑑L·

L·)

[∬ 𝜌𝑡 𝝋𝑇𝝋 𝑑A·

𝐴·] ��· +∬ 𝑩𝑇𝑫 𝑩

𝐴·𝑡 𝑑𝐴· 𝒖· =∬ 𝝋𝑇𝒃

𝐴·𝑡 𝑑𝐴· +∫ 𝝋𝑇𝒕 𝑡 𝑑L·

L·

(52)

Which is the equilibrium equation that must be valid at every element of the domain for the

FEM and for every influence-domain for the meshless formulations. Analyzing each term of

the equations one can construct both the stiffness and mass matrix for the given domain, that

are

𝑲 =∬ 𝑩𝑇𝑫 𝑩

𝐴∗𝑡 𝑑𝐴∗ (53)

𝑴 =∬ 𝜌𝑡 𝝋𝑇𝝋 𝑑A·

𝐴· (54)

Which are matrices with size (2𝑁, 2𝑁), where N is the total number of nodes that discretizes

the geometry domain. The force vector, with size (2𝑁, 1) is defined as

𝑭 =∬ 𝝋𝑇𝒃

𝐴∗𝑡 𝑑𝐴∗ +∫ 𝝋𝑇𝒕 𝑡 𝑑L∗

L∗ (55)

Being 𝐴∗ the area in which the body force 𝒃 is assumed and L∗ the curve in which the external

force 𝒕 is applied. Both matrices must be evaluated numerically, and as already mentioned, a

Gauss-Legendre quadrature scheme will be used. Therefore, the integrals are to be

transformed into sums analyzed at every integration point, being:

𝑲∗ =∑𝜔𝐼 𝑩(𝒙𝐼)𝒋𝑇𝑫𝒋𝑩(𝒙𝐼)𝒋

𝒏

𝑗=1

𝑡 (56)

𝑴∗ =∑𝜔𝐼 𝜌 𝑡 𝝋(𝒙𝐼)𝒋𝑇𝝋(𝒙𝐼)𝒋

𝒏

𝑗=1

(57)

where 𝜔𝐼 is the weight of each integration point. After the local matrices have been

constructed, the global matrix are assembled using the procedure described in chapter 2.


29

4.2 Three Dimensional Solids

Many structures, due to their geometric properties or boundary conditions, are impossible to

simulate using plane elasticity. For this scenarios, three dimensional theories must be applied

to study the structure.

The relations used in chapter 2.2. describes the formulation behind the three dimensional

analysis of linear solids. This time, the displacement field is defined as,

𝒖 = (𝑢, 𝑣, 𝑤) (58)

Being 𝑢, 𝑣 and 𝑤 the displacements along the x, y and z axis, respectively. This means that

each node has three degrees of freedom. Thus, the stiffness and mass matrices are of size

(3𝑁, 3𝑁), being 𝑁 the total number of nodes that discretize the solid domain. These matrices

are now defined as

𝑲∗ =∑𝜔𝐼 𝑩(𝑥𝐼)𝒋𝑇𝑫𝒋𝑩(𝑥𝐼)𝒋

𝒏

𝑗=1

(59)

𝑴∗ =∑𝜔𝐼 𝜌 𝝋(𝑥𝐼)𝒋𝑇𝝋(𝑥𝐼)𝒋

𝒏

𝑗=1

(60)

With the shape function matrix now being,

𝝋 = [

𝜑1 0 00 𝜑1 00 0 𝜑1

𝜑2 0 00 𝜑2 00 0 𝜑2

⋯

𝜑𝑛 0 00 𝜑𝑛 00 0 𝜑𝑛

] (61)

And the deformation matrix now being,

𝜺 = 𝑳 𝝋 𝒖∗ = 𝑩 𝒖∗

𝑩 =

[ 𝜕𝜑1𝜕𝑥

0 0

0𝜕𝜑1𝜕𝑦

0

0 0𝜕𝜑1𝜕𝑧

𝜕𝜑1𝜕𝑦

𝜕𝜑1𝜕𝑥

0

𝜕𝜑1𝜕𝑧

0𝜕𝜑1𝜕𝑥

0𝜕𝜑1𝜕𝑧

𝜕𝜑1𝜕𝑦

…


0 0

0𝜕𝜑𝑛𝜕𝑦

0

0 0𝜕𝜑𝑛𝜕𝑧

𝜕𝜑𝑛𝜕𝑦


0

𝜕𝜑𝑛𝜕𝑧

0𝜕𝜑𝑛𝜕𝑥

0𝜕𝜑𝑛𝜕𝑧

𝜕𝜑𝑛𝜕𝑦 ]

(62)


30

And the constitutive matrix, for orthotropic materials is

𝑺 =

[ 1

𝐸1−𝜈21𝐸2

−𝜈31𝐸3

0 0 0

−𝜈12𝐸1

1

𝐸2−𝜈32𝐸3

0 0 0

−𝜈13𝐸1

−𝜈23𝐸2

1

𝐸30 0 0

0 0 01

𝐺120 0

0 0 0 01

𝐺230

0 0 0 0 01

𝐺31]

𝑫 = 𝑺−𝟏

(63)


31

5. Vibrations

The system of equilibrium equations describes the dynamic behavior of a structure. In order to

obtain the response of the system under certain boundary conditions the system of equations

must be solved. Generally, these equations can be solved using a step-by-step integration of

the equations in the time domain (direct integration) or using the modal superposition method

[55]. In the last method, the displacements are given as:

𝒖(𝑡) = 𝚽𝒖𝒎(𝑡) (64)

Where 𝚽 is a 𝑚 ×𝑚 matrix (𝑚 = 2𝑁 for two-dimensional problems and 𝑚 = 3𝑁 for three-

dimensional problems, being 𝑁 the total number of nodes in the domain). The following

considerations can also be established:

��(𝑡) = 𝚽��𝒎(𝑡) (65)

��(𝑡) = 𝚽��𝒎(𝑡) (66)


32

5.1 Free Vibration

For a system vibrating under no load and no damping, the equilibrium equation is:

𝑴�� + 𝑲𝒖 = 𝟎 (67)

The solution for the nodal displacements 𝒖(𝑡) can be represented as the following

𝒖(𝑡) = 𝛟 sin(𝜔(𝑡 − 𝑡0)) (68)

Substituting back on equation (67) the generalized eigenproblem is obtained, for which the

natural frequencies, 𝜔 represents the eigenvalues and the modal amplitudes, 𝛟 are the

eigenvectors of the problem,

𝑲𝛟− 𝜔2𝑴𝛟 = 𝟎 (69)

The eigensolutions are given by:

{

𝑲𝛟𝟏 = 𝜔1

2𝑴𝛟𝟏

𝑲𝛟𝟐 = 𝜔22𝑴𝛟𝟐

⋮𝑲𝛟𝐦 = 𝜔𝑚

2𝑴𝛟𝐦

(70)

Defining a matrix containing all the eigenvectors,

𝚽 = [𝛟𝟏 𝛟𝟐 ⋯ 𝛟𝐦] (71)

And a matrix containing all the eigenvalues,

𝛀𝟐 =

[ 𝜔12 0

0 𝜔22

⋯ 0⋯ 0

⋮ ⋮0 0

⋱ ⋮⋯ 𝜔𝑚

2 ] (72)

Since the eigenvectors are orthogonal to both mass and stiffness matrix, if the modal

amplitudes are normalized respecting modal masses, that is

∭𝒖2𝑑𝑉

𝑉

= 1 (73)

The eigenvectors will be orthonormalized with respect to both matrices, which yields

𝚽𝐓𝑴𝚽 = 𝐈

𝚽𝐓𝑲𝚽 = 𝛀𝟐 (74)


33

5.2 Forced Vibrations

The general equilibrium equation for a dynamic system takes into account the damping and

loads in the system, and is given by

𝑴�� + 𝑪�� + 𝑲𝒖 = 𝑭 (75)

where 𝑪 is a matrix denoting the damping of the system. As stated earlier, two approaches can

be taken to solve this equation, by direct integration or by modal superposition method. The

second approach usually produces more accurate results, but is limited to excitations under

low frequencies, which means only the first modes of the system will be excited. For high-

frequency excitations, a direct integration must be applied, a method that usually has stability

problems and present difficulties when choosing the time step for the integration [56].

If the system is subjected to harmonic loads, then the response of the system will also be

harmonic, in which case a frequency analysis response can be applied. The equilibrium

equation can be simplified as presented in [56],

(−𝜔2𝑴+ 𝑖𝜔𝑪 + 𝑲) · 𝒖 = 𝑭 (76)

In general, finite element and meshless methods deal with a high number of degrees of

freedom, which turns the direct resolution of the equation into a costly process. Usually, a

modal truncated analysis is applied in those cases.


34

5.2.1 Modal Superposition

In this method, the size of the matrices are reduced assuming that only the first eigenmodes

contribute to the dynamic response of the system. The modal analysis has to take in account

the damping on the system, since in the case of an undamped structure the eigenmodes are

real and for a damped structure are complex. Thus, the later can cause problem when

performing a modal analysis. For weakly damped systems, the 𝑪 matrix can be ignored when

calculating the eigenmodes.

To solve a problem of forced vibrations using a truncated modal analysis, all modes with

natural frequencies up to one-half or two times the highest frequency of the excitation should

be kept, truncating the remaining modes. Considering an undamped system, and normalizing

the eigenvectors according to equation (73) results in

��𝒎(𝑡) + 𝛀𝟐𝒖𝒎(𝑡) = 𝚽

T𝑭(𝑡) (77)

Which is a decoupled system of equations. Based on the time-dependent loading conditions

applied on the problem the modal displacements 𝒖𝒎(𝑡) can be obtained [3].

The initial conditions of 𝒖𝒎(𝑡) are given by

{𝒖𝒎𝟎 = 𝚽𝑇𝑴𝒖𝟎��𝒎𝟎 = 𝚽𝑴��𝟎

(78)

Finally, the generalized displacements of the system can be obtained by modal superposition

of the response in each mode, being 𝑚𝑒𝑞 the number of considered modes after truncating the

system,

𝒖(𝑡) = ∑𝛟𝑖𝒖𝒎𝑖(𝑡)

𝑚𝑒𝑞

𝑖=1

(79)

The use of truncated modal analysis is sometimes associated with errors in the final results,

which is to be expected since it is an approximation technique. To improve the convergence

and accuracy of the solution, the modal acceleration method can be applied [57]. In this

method, the contribution of the truncated modes is taken into account assuming that they

respond in a quasi-static manner. Thus, the corrected response of the system is given by

𝒖(𝑡) = 𝚽 𝒖𝒎 +𝑲

−1𝑭 −𝚽 𝛀−1 𝚽𝐓 𝑭 (80)

The added terms represent the static contributions from all the modes minus the static

contribution of the truncated modes.


35

5.2.2 Duhamel Integral

For a decoupled system, such as those that result from the application of modal superposition,

the solution for a differential equation can be obtained by assuming that the solicitation is a

set of impulsive forces [55]. Therefore, under these circumstances, the Duhamel integral can

be used to obtain the solution, and for an undamped system is given by the following equation

𝑥𝑖(𝑡) =1

𝑚·𝜔𝑛∫ 𝑓(𝜏) · sin(𝜔𝑛 · (𝑡 − 𝜏)) 𝑑𝜏𝑡

0+ 𝑥0 · cos(𝜔𝑛 · 𝑡) +

𝑣0·sin(𝜔𝑛·𝑡)

𝜔𝑛 (81)

Where 𝑥0 and 𝑣0 denote the initial position and velocity of the system, 𝜔𝑛 is the natural

frequency of the undamped system and 𝑚 the equivalent mass. Due to the nature of the

integral, it is usually solved numerically using either the Trapezoidal or Simpsons method

[58]. Therefore, the Duhamel integral can be used to solve all the decoupled equations

obtained from modal superposition.

Figure 10: Solicitation as a set of impulsive forces [59].

In some applications the structural damping must be taken into consideration. Although,

usually, there is not a realistic approach to create a damping matrix for a structure, and even if

there was, it would be impossible to decouple a system of equations through modal

superposition. To avoid such problems, there are generally two ways to implement the

damping effect:

Structural modal damping: it consists in including the damping effect in Young’s

modulus for a structural problem, �� = 𝐸 · (1 + 𝑖 · 𝜂𝑠), being 𝜂𝑠 the loss factor of the

structure. As a result, the stiffness matrix for a system after modal superposition is

𝛀𝑠2 = [

𝜔12(1 + 𝑖𝜂1)

𝜔22(1 + 𝑖𝜂2)

⋱ 𝜔𝑛

2(1 + 𝑖𝜂𝑛)

] (82)

Proportional damping model: in this case, the damping matrix is assumed to be

somewhat proportional to the mass and stiffness matrix multiplied by some

coefficients. The general law to obtain the damping matrix can be given by using the

Caughey series:

𝐂 = 𝐌 ·∑𝑎𝑘[𝑴−1𝑲]𝑘

𝑟−1

𝑘=0

(83)


36

From this, the damping ratios are given by

𝜉𝑖 =1

2(𝑎0𝜔𝑖+ 𝑎1𝑖 + 𝑎2𝜔𝑖

3 +⋯+ 𝑎𝑟−1𝜔𝑖2𝑟−3) (84)

If 𝑟 = 2, equation (83) reduces to Rayleigh damping, that is

𝑪 = 𝛼𝑴+ 𝛽𝑲 (85)

Where 𝛼 and 𝛽 are constants to be determined from two given damping ratios that

correspond to two unequal frequencies of vibration [55]. After the damping matrix is

obtained, applying the modal superposition yields:

𝚽𝒊𝐓 𝑪 𝚽𝑗

= 2𝜔𝑖𝜉𝑖𝛿𝑖𝑗 (86)

If proportional damping is present in the system differential equations, the Duhamel integral

is now given by

𝑥𝑖(𝑡) =1

𝑚 · 𝜔𝑑∫ 𝑓(𝜏) · 𝑒𝜉𝜔𝑛(𝜏−𝑡) sin(𝜔𝑑 · (𝑡 − 𝜏)) 𝑑𝜏𝑡

0

+ 𝑒−𝜉𝜔𝑛𝑡

· (𝑥0 · cos(𝜔𝑑 · 𝑡) +(𝑣0 + 𝜉𝜔𝑛𝑢0) · sin(𝜔𝑑 · 𝑡)

𝑤𝑑)

(87)

Where the natural damped frequency is 𝜔𝑑 = 𝜔𝑛 · √1 − 𝜉2 for underdamped systems.


37

5.2.3 Frequency Analysis Response

For problems involving harmonic loads, the dynamic equilibrium equation can be written as

follows.

(−𝜔2𝑴+ 𝑖𝜔𝑪 +𝑲)𝒖 = 𝑭 (88)

Where the left part of the equation is called the impedance matrix, that is,

𝒁(𝜔) = (−𝜔2𝑴+ 𝑖𝜔𝑪 + 𝑲) (89)

Instead of looking for a solution in the time domain, for harmonic problems it may be more

interesting to obtain a solution in the frequency domain, since it gives an easier view of at

which frequencies should and should not the system operate [3].

A simple solution, although costly, to equation (88) is to invert the impedance matrix,

resulting

𝒖 = 𝒁(𝜔)−1 ∙ 𝑭 (90)

But as pointed out, this process is not very efficient and should only be used when no other

approach is applicable. In most situations, a modal analysis can be performed, making the

impedance matrix diagonal and much easier to invert. Thus, equation (89) becomes

𝒁(𝜔) = [

(𝜔12 − 𝜔2) + 𝑖2𝜉1𝜔1𝜔 0 0

0 ⋱ 00 0 (𝜔𝑛

2 − 𝜔2) + 𝑖2𝜉𝑛𝜔𝑛𝜔 ]

𝑍(𝜔) = 𝛀𝟐 − 𝜔2𝑰 + 𝑖2𝜔 · 𝝃Ω

(91)

The force vector on the modal base is

𝑭𝒎 = 𝚽𝑇𝑭 (92)

Thus, the displacements on the modal base become

𝒖𝒎 =

{

𝐹1𝑚(𝜔1

2 − 𝜔2) + 𝑖2𝜉1𝜔1𝜔 𝐹2𝑚

(𝜔22 − 𝜔2) + 𝑖2𝜉2𝜔2𝜔

⋮𝐹𝑛𝑚

(𝜔𝑛2 − 𝜔2) + 𝑖2𝜉𝑛𝜔𝑛𝜔 }

(93)

Where in this case 𝑛 represents the number of modes kept from the truncated modal analysis.

If modal damping is to be considered instead, the displacements become

𝒖𝒎 =

{

𝐹1𝑚𝜔12(1 + 𝑖𝜂1) − 𝜔2

𝐹2𝑚𝜔22(1 + 𝑖𝜂2) − 𝜔2

⋮𝐹𝑛𝑚

𝜔𝑛2(1 + 𝑖𝜂𝑛) − 𝜔2 }

(94)


38

If no damping is present on the system, 𝜉 = 0, equation (93) simply becomes

𝒖𝒎 =

{

𝐹1𝑚(𝜔1

2 −𝜔2)

𝐹2𝑚(𝜔2

2 −𝜔2) ⋮𝐹𝑛𝑚

(𝜔𝑛2 − 𝜔2) }

(95)

The displacements can be transformed back to the spatial base using equation (64).


39

5.2.4 Direct Integration

Another method used to solve the equilibrium equations is through direct integration. Here,

the system is numerically solved using a step-by-step procedure, resulting in a solution at

discrete intervals of time, named time-step, ∆𝑡, instead of a continuous solution in the time

spectrum.

To solve the equilibrium equations this approach assumes variational formulations for the

displacements, velocities and accelerations, and is the different types of formulations that

determinate the stability and accuracy of the procedure.

Any direct integration approach is more efficient when the matrices are either diagonal or

symmetric and banded. Therefore, the combination of a modal superposition and direct

integration is often used to solve systems of equations.

Since this resolution does not rely on any prior transformation of the system of equations, it is

mostly used when proportional damping cannot be applied on the damping matrix. Also, for

dynamic systems with higher solicitation frequencies where a modal superposition would

become a highly expensive computational option, a direct integral approach can prove to be a

reliant alternative for solving the problem.

Next, a brief description of some different direct integration methods is presented. On Chapter

7 is described a routine for each method [55, 57].


40

Central Difference Method

This method is based on the approximation of the velocity and acceleration fields in terms of

the discrete values of the displacement, 𝒙(𝑡). Thus, it is possible to assume that the

displacement on the instance 𝑡𝑖 + ∆𝑡 can be given as a Taylor expansion around the instant 𝑡𝑖. The velocity and acceleration at a given instance 𝑡 are

��𝑡 =1

(∆𝑡)2(𝒙𝑡+∆𝑡 − 2𝒙𝑡 + 𝒙𝑡−∆𝑡)

��𝑡 =1

2∆𝑡(𝒙𝑡+∆𝑡 − 𝒙𝑡−∆𝑡)

(96)

Considering the equilibrium equation at the same instant

𝑴��𝒕 + 𝑪��𝒕 +𝑲𝒙𝒕 = 𝒇𝒕 (97)

Substituting back the velocity and accelerations the equation yields

𝑴1

(∆𝑡)2(𝒙𝒕+∆𝒕 − 2𝒙𝒕 + 𝒙𝒕−∆𝒕) + 𝑪

1

2∆𝑡(𝒙𝒕+∆𝒕 − 𝒙𝒕−∆𝒕) + 𝑲𝒙𝒕 = 𝒇𝒕 (98)

Rearranging the equation in order of instances a system of algebraic equations is obtained:

(1

(∆𝑡)2𝑴+

1

2∆𝑡𝑪) 𝒙𝒕+∆𝒕 = 𝒇𝒕 − (𝑲 −

2

(∆𝑡)2𝑴) 𝒙𝒕 − (

1

(∆𝑡)2𝑴−

1

2∆𝑡𝑪) 𝒙𝒕−∆𝒕 (99)

This is the recurrent expression from which the discrete displacements are calculated for this

method. To be noted, the solution 𝒙𝑡+∆𝑡 is obtained using the equilibrium equation at the

instant 𝑡, making the central difference method an explicit integration method, and it does not

require a factorization of the stiffness matrix to become more effective.

As a drawback, since it requires the displacement at the instant 𝑡 − ∆𝑡 to obtain the

displacement at the instant 𝑡 = 𝑡 + ∆𝑡, the method needs a special starting procedure, which

often is given by

𝒙−∆𝒕 = 𝒙𝟎 − ∆𝑡��𝟎 +(∆𝑡)2

2𝑴−𝟏 · (𝒇𝟎 − 𝑪��𝟎 −𝑲𝒙𝟎

) (100)

Also, to assure the numerical stability of the integration procedure, the time step for the

integration, ∆𝑡, must be inferior to a critical limit,

∆𝑡 ≤𝑇𝑛𝜋=2

𝜔𝑛 (101)

Where 𝑇𝑛 is the smallest undamped period of the system, and 𝜔𝑛 the biggest natural

undamped frequency. As such, using a time step larger than ∆𝑡𝑐𝑟𝑡 results in an unstable

integration, wielding the calculations worthless in most cases. This integration is said to be

conditionally stable.


41

Houbolt Method

Similar to the previous method, this integration scheme also uses standard finite difference

expressions to approximate the velocity and accelerations fields from the discrete

displacements. The equations of this formulation are

��𝑡+∆𝑡 =1

(∆𝑡)2· (2 · 𝒙𝑡+∆𝑡 − 5 · 𝒙𝑡 + 4 · 𝒙𝑡−∆𝑡 − 𝒙𝑡−2∆𝑡)

��𝑡+∆𝑡 =1

6∆𝑡· (11 · 𝒙𝑡+∆𝑡 − 18 · 𝒙𝑡 + 9 · 𝒙𝑡−∆𝑡 − 2 · 𝒙𝑡−2∆𝑡)

(102)

Considering the equilibrium equation at the instant 𝑡 + ∆𝑡

𝑴��𝒕+∆𝒕 + 𝑪��𝒕+∆𝒕 +𝑲𝒙𝒕+∆𝒕 = 𝒇𝒕+∆𝒕 (103)

And substituting back the acceleration and velocity at the same instance gives

(2

(∆𝑡)2𝑴+

11

6∆𝑡𝑪 + 𝑲)𝒙𝒕+∆𝒕 = 𝒇𝒕+∆𝒕 + (

5

(∆𝑡)2𝑴+

3

∆𝑡𝑪)𝒙𝒕

−(4

(∆𝑡)2𝑴+

3

2∆𝑡𝑪) 𝒙𝒕−∆𝒕 + (

1

(∆𝑡)2𝑴+

1

3∆𝑡𝑪) 𝒙𝒕−𝟐∆𝒕

(104)

Which is the recurrent expression from which the discrete displacements are calculated for

this method. To be noted, the solution 𝒙𝑡+∆𝑡 is obtained using the displacement at instants 𝑡 −∆𝑡 and 𝑡 − 2∆𝑡, which renders to the method the need for a special starting procedure.

Usually, another method such as the central difference method can be used to evaluate the

displacements at the first two time steps.

As opposed to the previous method, the Houbolt method is an implicit integration scheme,

and has no critical time step limit. What also should be noted is that if the mass and damping

matrix are neglected, applying the Houbolt method yields the static solution for the time-

dependent load, as opposed to the central difference method that cannot be applied under such

conditions.


42

Newmark Method

The Newmark method is solely based on the linear acceleration method, and uses the

following assumptions

��𝑡+∆𝑡 = ��𝑡 + [(1 − 𝛿)��𝑡 + 𝛿��𝑡+∆𝑡]∆𝑡

𝒙𝑡+∆𝑡 = 𝒙𝑡 + ��𝑡∆𝑡 + [(1

2− 𝛼) ��𝑡 + 𝛼��𝑡+∆𝑡] (∆𝑡)

2 (105)

Where 𝛼 and 𝛿 are parameters that assure the precision and stability of the integration. If 𝛿 =1

2 and 𝛼 =

1

6 are chosen as the parameters, the Newmark method becomes the linear

acceleration method.

Figure 11: Linear acceleration approximation [57].

If the Newmark parameters chosen are 𝛿 =1

2 and 𝛼 =

1

4 the method becomes unconditionally

stable. From the displacement and velocity equations at the instant 𝑡 + ∆𝑡, the acceleration at

the same instant is

��𝑡+∆𝑡 =1

𝛼(∆𝑡)2(𝒙𝑡+∆𝑡 − 𝒙𝑡) −

1

𝛼∆𝑡��𝑡 − (

1

2𝛼− 1) ��𝑡 (106)

Substituting on the equilibrium equation at the instant 𝑡 + ∆𝑡 gives

(

4

(∆𝑡)2𝑴+

2

∆𝑡𝑪 + 𝑲)𝒙𝒕+∆𝒕 = 𝒇𝒕+∆𝒕 +𝑴(

4

(∆𝑡)2𝒙𝒕 +

4

∆𝑡��𝒕 + ��𝒕) +

𝑪(2

∆𝑡𝒙𝒕 + ��𝒕)

(107)

Which is the recurrent expression from which the discrete displacements are calculated for

this method. This method is an implicit integration scheme, and if the parameters chosen

respect the following expressions

𝛿 ≥1

2 𝑎𝑛𝑑 𝛼 ≥

1

4(𝛿 +

1

2)2

(108)

The method is unconditionally stable. Note that if 𝛿 <1

2 the method introduces artificial

damping in the system.


43

Wilson-θ Method

This method is an extension of the linear acceleration method, and adopts a linear acceleration

variation over the interval [𝑡, 𝑡 + 𝜃∆𝑡], with 𝜃 ≥ 1. Thus, the acceleration field is given by

��𝑡+𝜏 = ��𝑡 +𝜏

𝜃∆𝑡(��𝑡+𝜃∆𝑡 − ��𝑡) (109)

Integration over 𝜏 yields

��𝑡+𝜏 = ��𝑡 + ��𝑡𝜏 +𝜏2

2𝜃∆𝑡(��𝑡+𝜃∆𝑡 − ��𝑡)

𝒙𝑡+𝜏 = 𝒙𝑡 + ��𝑡𝜏 +1

2��𝑡𝜏

2 +𝜏3

6𝜃∆𝑡(��𝑡+𝜃∆𝑡 − ��𝑡)

(110)

Note if 𝜃 = 1 the Wilson-θ method becomes the linear acceleration method.

Figure 12: Wilson-θ method. [59]

For 𝜏 = 𝜃∆𝑡, and substituting back on the equilibrium equations, the recurring equation can

be obtained, which is

[𝑲 +6

(𝜃∆𝑡)2𝑴+

3

𝜃∆𝑡𝑪] 𝒙𝒕+𝜽∆𝒕 = 𝒇𝒕 + 𝜃(𝒇𝒕+∆𝒕 + 𝒇𝒕)

+𝑴(6

(𝜃∆𝑡)2𝒙𝒕 +

6

(𝜃∆𝑡)2��𝒕 + 2��𝒕) + 𝑪(

3

(𝜃∆𝑡)2𝒙𝒕 + 2��𝒕 +

𝜃∆𝑡

2��𝒕)

(111)

Thus, this method is an implicit integration scheme. Note that for 𝜃 ≥ 1.37 the Wilson-θ

method is unconditionally stable. Usually, 𝜃 = 1.4 is a recurrent value for this parameter.


44


45

6. Vibro-Acoustics

Recalling the inhomogeneous wave equation, equation (1),

∇2𝑝(𝒙, 𝑡) −1

𝑐02 ��(𝒙, 𝑡) = −𝑄(𝒙, 𝑡) (112)

it describes the dynamic changes in pressure inside a fluid due to the propagation of waves

generated from a sound source. This equation must me be completed with initial conditions

𝑝(𝒙, 𝑡)|𝑡=0 and 𝜕𝑝(𝒙,𝑡)

𝜕𝑡|𝑡=0

at all points of the domain.

In most acoustical problems, the sound source is a harmonic solicitation, meaning that the

temporal dependency of the pressure is sinusoidal, with circular frequency 𝜔 [3]. Therefore,

for harmonic temporal dependence, equation (112) can be rewritten as,

∇2𝑝(𝒙) −𝜔2

𝑐02 𝑝(𝒙) = −𝑄(𝒙) (113)

For acoustic problems, there are generally three different boundary conditions that can be

imposed, a prescribed acoustic pressure, a normal acoustic displacement and an impedance

boundary condition. In Figure 13 is described a fluid domain bounded by various boundary

conditions. To note that for exterior acoustical problems, a condition must be specified to

ensure that the wave amplitude vanishes at infinity, given by Sommerfeld radiation condition

[60],

lim𝑟→∞

𝑟 (𝜕𝑝

𝜕𝑟+1

𝑐0

𝜕𝑝

𝜕𝑡) = 0 (114)

Figure 13: Fluid domain and boundary conditions [3].


46

On boundary 𝛿Ω𝑓,𝑁 is prescribed a normal acoustic displacement, given by,

𝜕𝑝

𝜕𝒏= 𝜌0𝜔

2𝒖 · 𝒏 (115)

And on boundary 𝛿Ω𝑓,𝑅 an impedance boundary condition,

𝜕𝑝

𝜕𝒏+𝑖𝜔

𝑐0��𝑝 = 0 (116)

being �� =𝜌0𝑐0

𝑍𝑛, where 𝑍𝑛 is the specific acoustic impedance applied on the boundary.

The weak integral Galerkin’s formulation associated with an acoustic problem, recalling the

concepts presented on Chapter 3.2 yields,

∫ [∇𝑝 · ∇𝛿𝑝 −𝜔2

𝑐02 𝑝𝛿𝑝] 𝑑𝑉 − ∫ 𝜌0𝜔

2𝒖𝑛𝛿𝑝 𝑑𝑆

𝛿Ω𝑓,𝑁

+∫𝑖𝜔

𝑐0��𝑝𝛿𝑝

𝛿Ω𝑓,𝑅

𝑑𝑆 = 0

Ω𝑓

(117)

Making the following simplifications,

𝑯(𝑝, 𝛿𝑝) = ∫1

𝜌0(∇𝑝 · ∇𝛿𝑝)

Ω𝑓

𝑑𝑉 (118)

𝑸(𝑝, 𝛿𝑝) = ∫1

𝜌0𝑐02 𝑝𝛿𝑝 𝑑𝑉

Ω𝑓

(119)

𝑨(𝑝, 𝛿𝑝) = ∫��

𝜌0𝑐0𝑝𝛿𝑝

𝛿Ω𝑓,𝑅

𝑑𝑆 (120)

𝑹(𝛿𝑝) = 𝜔2∫ 𝒖𝑛𝛿𝑝 𝑑𝑆

𝛿Ω𝑓,𝑁

(121)

Equation (117) can be rewritten as,

𝑯(𝑝, 𝛿𝑝) − 𝜔2𝑸(𝑝, 𝛿𝑝) + 𝑖𝜔𝑨(𝑝, 𝛿𝑝) − 𝑹(𝛿𝑝) = 0 (122)


47

6.1 Calculation of local matrices

The matrices from equation (122) must be evaluated at each element for the FEM and at each

integration point for the meshless methods adopted on this work. On these sections is

represented the construction of these four matrices.

Kinetic energy matrix, 𝑸(𝑝, 𝛿𝑝)

Evaluating equation (117) over each domain (either an element or the influence domain)

results in,

𝑸∗(𝑝, 𝛿𝑝) = ∫1

𝜌0𝑐02 𝑝𝛿𝑝 𝑑𝑉

Ω ∗ (123)

𝑸∗(𝑝, 𝛿𝑝) = {𝛿𝑝∗}𝑸∗{𝑝∗} (124)

where,

𝑸∗ = ∫1

𝜌0𝑐02𝝋

𝑇𝝋 𝑑𝑉

Ω ∗ (125)

which is a positive semi-defined matrix of size (𝑛 , 𝑛 ), where 𝑛 is the number of nodes inside

each domain. This integral can be transformed into a sum, being,

𝑸∗ =∑��𝐼

1

𝜌0𝑐02

𝝋(𝒙𝐼)𝒋𝑇

𝒏

𝑗=1

𝝋(𝒙𝐼)𝒋

(126)

Where ��𝐼 is the weight integration point 𝒙𝐼. For an acoustic problem, the shape functions

matrix, 𝝋 is,

𝝋 = [𝜑1 𝜑2 ⋯ 𝜑𝑛] (127)

The assemblage process has been described in Chapter 2. After this process, results matrix 𝑸

of size (𝑁,𝑁), being 𝑁, the total number of nodes that discretize the fluid.


48

Compression energy matrix, 𝑯(𝑝, 𝛿𝑝)

From equation (118), applying over each domain results,

𝑯∗(𝑝, 𝛿𝑝) = ∫1

𝜌0∇δ𝑝∗ ∇𝑝∗ 𝑑𝑉

Ω ∗ (128)

∇𝑝 = ∇𝑥 𝜑 {𝑝∗} = 𝑩{𝑝∗} (129)

𝑯(𝑝, 𝛿𝑝) = {𝛿𝑝∗} 𝑯∗ {𝑝∗} (130)

For a 3D acoustic problem, the deformation matrix is,

𝑩 =

[ 𝜕𝜑1𝜕𝑥

𝜕𝜑2𝜕𝑥

𝜕𝜑1𝜕𝑦

𝜕𝜑2𝜕𝑦

𝜕𝜑1𝜕𝑧

𝜕𝜑2𝜕𝑧

…

𝜕𝜑𝑛𝜕𝑥𝜕𝜑𝑛𝜕𝑦𝜕𝜑𝑛𝜕𝑧 ]

(131)

And results,

𝑯∗ = ∫1

𝜌0𝑩𝑇𝑩 𝑑𝑉

Ω ∗ (132)

which is a positive semi-defined matrix of size (𝑛 , 𝑛 ). This integral can be discretized as,

𝑯∗ =∑��𝐼 ·1

𝜌0

𝑩(𝒙𝐼)𝒋𝑇

𝒏

𝑗=1

𝑩(𝒙𝐼)𝒋 (133)

The assemblage process has been described in Chapter 2. After this process, results matrix 𝑯

of size (𝑁,𝑁), being 𝑁, the total number of nodes that discretize the fluid.


49

Impedance Boundary Matrix, 𝑨(𝑝, 𝛿𝑝)

From equation (119), applying over each boundary domain results,

𝑨∗(𝑝, 𝛿𝑝) = ∫��

𝜌0𝑐0δ𝑝∗ 𝑝∗ 𝑑𝑆

δΩ ∗𝑓,𝑅

(134)

𝑨∗(𝑝, 𝛿𝑝) = {𝛿𝑝∗}𝑨∗{𝑝∗} (135)

with,

𝑨∗ = ∫��

𝜌0𝑐0𝝋𝑇𝝋 𝑑𝑆

δΩ ∗𝑓,𝑅

(136)

which is a positive semi-defined matrix of size (𝑛 𝑓 , 𝑛𝑓 ), where 𝑛𝑓 is the number of nodes on

surface domain 𝛿Ω𝑓,𝑅. This integral can be transformed into a sum, being,

𝑨∗ =∑��𝐼

��

𝜌0𝑐0

𝝋(𝒙𝐼)𝒋𝑇

𝒏

𝑗=1

𝝋(𝒙𝐼)𝒋

(137)

Force Vector

From equation (120), applying over each boundary domain results,

𝑹∗(𝛿𝑝) = 𝜔2∫ {𝛿𝑝∗} 𝒖𝑛 𝑑𝑆

δΩ ∗𝑓,𝑁

(138)

𝑹∗(𝑝, 𝛿𝑝) = {𝛿𝑝∗} 𝒇∗ (139)

with,

𝒇∗ = 𝜔2 𝑪𝒖𝒑 𝒖𝑛 (140)

and

𝑪𝒖𝒑 = ∫ 𝝋𝑇𝝋 𝑑𝑆

δΩ ∗𝑓,𝑁

(141)

This matrix is square, symmetric of size (𝑛 𝑓 , 𝑛𝑓 ), In the case where the nodal displacement

vector is given (and not the normal displacements), a nodal approximation must be performed,

𝒖𝑛 =1

|𝑗∗|𝒏 · 𝒖 (142)

Being 𝑗∗ = (𝜕𝒙 𝜕𝜉1⁄ ) × (𝜕𝒙 𝜕𝜉2⁄ ) , where 𝜉1 and 𝜉2 are parametric coordinates of the

surface element [3]. The force vector can be transformed into a sum, given by

𝒇𝒂∗ = 𝜔2∑��𝐼 𝝋(𝒙𝐼)𝒋

𝑇

𝒏

𝑗=1

𝒖𝑛 𝝋(𝑥𝐼)𝒋

(143)

After the assemblage process results a vector of size (𝑁, 1).


50

6.2 Uncoupled Acoustic Problem

For an uncoupled acoustic problem, the interaction between the fluid and the structure can be

neglected. This means that the fluid gradient pressure is not capable of interfering with the

behavior of the structure, as such, this type of problems usually resumes in finding the

displacements of a structure due to an external excitation and then the wave propagations on

the fluid due to those displacements.

Recalling the weak form Galerkin’s integral, equation (122), now simplified

𝛿𝒑𝑇[𝑯 − 𝜔2𝑸+ 𝑖𝜔𝑨 − 𝒇] = 0 (144)

Since this equation must be satisfied for every 𝛿𝒑, this means that,

−𝜔2𝑸+ 𝑖𝜔𝑨 + 𝑯 = 𝒇 (145)

Which is the equation that governs the dynamic behavior of the fluid. This expression

resembles the equilibrium equations of a damped system, where 𝑸 replaces the mass matrix,

𝑨 the damping matrix and 𝑯 the stiffness matrix.

Since most acoustic problems involve harmonic loads, usually a frequency analysis response

is used to solve the system equations. Thus, as described in Chapter 5.2.3, after the modal

analysis, equation (145) results

𝒁(𝝎) · 𝒑𝒎 = 𝒇𝒎 (146)

where,

𝒁(𝝎) = 𝛀𝟐 − 𝜔2𝑰 + 𝑖2𝜔 · 𝝃Ω (147)

Thus,

𝒑𝒎 = 𝒁(𝝎)−1 ∙ 𝒇𝒎

𝒑𝒎 =

{

𝑓1𝑚(𝜔1

2 − 𝜔2) + 𝑖2𝜉1𝜔1𝜔

𝑓2𝑚(𝜔2

2 − 𝜔2) + 𝑖2𝜉2𝜔2𝜔 ⋮𝑓𝑛𝑚

(𝜔𝑛2 − 𝜔2) + 𝑖2𝜉𝑛𝜔𝑛𝜔 }

(148)

If modal damping is to be considered, then equation (148) becomes,

𝒑𝒎 =

{

𝑓1𝑚𝜔12(1 + 𝑖𝜂1) − 𝜔2

𝑓2𝑚𝜔22(1 + 𝑖𝜂2) − 𝜔2

⋮𝑓𝑛𝑚

𝜔𝑛2(1 + 𝑖𝜂𝑛) − 𝜔2 }

(149)


51

6.3 Interior Acoustic Problems

Depending on the nature of the fluid and the structure, the effects of the fluid dynamic

behavior on the structure may not be neglected. A simple criteria to analyse the amount of

coupling in a fluid-structure system is given by [61],

𝛽𝑐 =𝜌0𝑐0𝜌𝑠ℎ𝑠𝜔1

(150)

being 𝜌0 and 𝑐0 the density and sound speed of the fluid, 𝜌𝑠 and ℎ𝑠 the density and thickness

of the structure and 𝜔1 the critical frequency of the structure.

For 𝛽𝑐 ≪ 1 the coupling is considered weak. On this case the system can be analyzed as an

uncoupled fluid-structure problem. For 𝛽𝑐 ≥ 1, the coupling is considered strong. Here, the

dynamic pressures on the fluid modify the behavior of the structure, and the problem must be

solved simultaneously for the structure and the fluid. On this work, a solution for the forced

response using coupled modes is adopted [3]

In interior problems, the fluid domain is bounded by a structure. Recalling the equilibrium

equations for both of them, the coupled system equations for bounded fluids is,

[𝑲 − 𝜔2𝑴 −𝑪𝒖𝒑

−𝜔2𝑪𝒖𝒑𝑇 𝑯−𝜔2𝑸

] {𝒖𝒑} = {

𝑭𝒇} (151)

The previous equation is not valid for 𝜔 = 0. For this case the static pressure is given by,

1

𝜌0𝑐02∫ 𝒑

Ω𝑓

𝑑𝑉 + ∫ 𝒖 · 𝒏

Ω𝑓

𝑑𝑆 = 0 (152)

Dividing equation (151) by 𝜔2 results in,

([𝑲 −𝑪𝒖𝒑𝟎 𝑯

] − 𝜔2 [𝑴 𝟎𝑪𝒖𝒑𝑇 𝑸]) {

𝒖𝒑} = {

𝑭𝒇} (153)

Figure 14: Fluid-structure interior coupled problem [3].

To solve equation (153) a method using coupled modes was adopted [3]. It consists in

obtaining the forced response using coupled modes on a system of equations resulting from

the uncoupled real modes of the fluid and structure.

First, the eingenvalues for the uncoupled fluid are calculated. Since the fluid is enclosed in a

rigid cavity, there exists a rigid body mode associated with the static pressure that must be

removed. Thus, it was retained a matrix containing the elastic eigenvalues of the fluid, Ω𝒇,𝒆𝟐

and eigenvectors 𝚽𝒇,𝒆.

The pressure at any given node will be given by the sum of the static pressure, obtained from

equation (152), with the elastic pressure (obtained from the elastic cavity modes).


52

𝒑 = 𝒑𝒔 + 𝒑𝒆 (154)

The acoustic source can also be separated, resulting,

𝒇 = 𝒇𝒔 + 𝒇𝒆 (155)

After the modal analysis of the fluid cavity, the following simplifications can be done,

Ω𝒇,𝒆𝟐 = 𝚽𝒇,𝒆

𝑇 𝑯 𝚽𝒇,𝒆

𝑰𝒇,𝒆 = 𝚽𝒇,𝒆

𝑇 𝑸 𝚽𝒇,𝒆

𝒇𝒆,𝒎 = 𝚽𝒇,𝒆𝑇 𝒇𝒆

𝒑𝒆,𝒎 = 𝚽𝒇,𝒆𝑇 𝒑𝒆

(156)

The coupling term from equation (153), becomes,

∫ 𝑝𝒏 · 𝛿𝑢

𝛿Ω𝑤

𝑑𝑆

= −𝜌0𝑐0

2

Ω𝑓(∫ 𝒏 · 𝒖

𝛿Ω𝑤

𝑑𝑆)(∫ 𝒏 · 𝛿𝒖

𝛿Ω𝑤

𝑑𝑆) + ∫ 𝑝𝑒𝒏 · 𝛿𝒖 𝑑𝑆

𝛿Ω𝑤

(157)

Which leads to two different matrices,

𝛿𝒖𝑇 𝑲𝒔 𝒖 =𝜌0𝑐0

2

Ω𝑓(∫ 𝒏 · 𝒖

𝛿Ω𝑤

𝑑𝑆)(∫ 𝒏 · 𝛿𝒖

𝛿Ω𝑤

𝑑𝑆) (158)

𝛿𝒖𝑇 𝑪𝒖𝒑 𝒑𝒆 = ∫ 𝑝𝑒𝒏𝛿𝒖 𝑑𝑆

𝛿Ω𝑤

(159)

Both matrices can be evaluated numerically for each integration point, resulting in a sum. For

𝑲𝒔 results,

𝑲𝒔∗ =

1

Ω𝑓∑𝜔𝐼 𝜌0𝑐0

2 (𝝋𝒔(𝒙𝐼)𝒋

𝑇 𝒏)

𝒏

𝑖=1

∙∑𝜔𝐼 ∙ (𝒏 𝑇 𝝋

𝒔(𝒙𝐼)𝒋

)

𝑛

𝑗=1

(160)

Which is a square matrix, of size (2𝑛, 2𝑛). For 𝑪𝒖𝒑 results,

𝑪𝒖𝒑∗ =∑𝜔𝐼 (𝝋𝒔(𝒙𝐼)𝒋

𝑇 𝒏

𝒏

𝑗=1

𝝋𝒇(𝒙𝐼)𝒋

) (161)

Where 𝝋𝒔 is the shape function matrix for the structure and 𝝋𝒇 the shape function matrix for

the fluid. 𝑪𝒖𝒑 is a matrix of size (2𝑛, 𝑛). Both matrices are evaluated on the boundary where

the fluid interacts with the structure. The equation for the coupled problem now yields.

([𝑲 + 𝑲𝒔 𝟎𝟎 𝑰

] − 𝜔2 [𝑴 +𝑴𝒆 𝑪𝒖𝒑,𝒎

Ω𝒇,𝒆−𝟐

Ω𝒇,𝒆−𝟐 𝑪𝒖𝒑,𝒎

𝑇 Ω𝒇,𝒆−𝟐 ]) {

𝒖𝒑𝒆,𝒎

} =

{𝑭 + 𝑪𝒖𝒑,𝒎

Ω𝒇,𝒆−𝟐 𝒇𝒆,𝒎


}

(162)

with

𝑴𝒆 = 𝑪𝒖𝒑,𝒆 Ω𝒇,𝒆−𝟐 𝑪𝒖𝒑,𝒆

𝑇 (163)


53

𝑪𝒖𝒑,𝒎 = 𝑪𝒖𝒑 𝚽𝒇,𝒆 (164)

The eigenvalue problem is now symmetric and can be efficiently used to solve the coupled

modes problem. Although, to further simplify the system equations, the following eigenvalue

problem can be solved,

[(𝑲 + 𝑲𝒔) − 𝜔2(𝑴 +𝑴𝒆)]𝒖 = 0 (165)

Resulting the elastic eigenvalues matrix Ω𝑠𝟐 and associated eigenvectors 𝚽𝒔. The reduced

coupled system is,

([Ω𝑠𝟐 𝟎𝟎 𝑰

] − 𝜔2 [𝑰 𝑪𝒖𝒑

𝑪𝒖𝒑�� Ω𝒇,𝒆−𝟐]) {

𝒖𝒎𝒑𝒆,𝒎

} = {𝚽𝒔𝑇𝑭 +𝚽𝒔

𝑇 𝑪𝒖𝒑,𝒎 Ω𝒇,𝒆

−𝟐 𝒇𝒆,𝒎


} (166)

Where

𝒖𝒎 = 𝚽𝒔 𝒖 (167)

𝑪𝒖𝒑 = 𝚽𝒔

𝑇 𝑪𝒖𝒑,𝒎 𝛀𝒇,𝒆

−𝟐 (168)

The resulting eigenvalue problem is again symmetric and therefore can be used to calculate

the coupled modes. From equation (166), denoting the obtained eigenvalues 𝛀𝒇𝒔𝟐 and

associated eigenvectors 𝚽𝒇𝒔 the forced response is,

{𝒖𝒎𝒑𝒆,𝒎

} = [Ω𝑓𝑠𝟐 − 𝜔2𝑰]

−1 𝚽𝒇𝒔

𝑇 {𝚽𝒔𝑇𝑭 +𝚽𝒔

𝑇 𝑪𝒖𝒑,𝒎 Ω𝒇,𝒆

−𝟐 𝒇𝒆,𝒎


} (169)

It is a system of size (𝑛𝑠𝑓 − 1, 𝑛𝑠𝑓 − 1), where 𝑛𝑠𝑓 is the number of eigenvalues kept for the

structure and fluid during the modal truncation analysis. The displacements and pressures are

returned to the spatial base using equation (79). As previously stated, to the calculated elastic

pressure a static contribution must be added, which can be simplified as,

𝑝𝑠 = −𝜌0𝑐0

2

Ω𝑓∫ 𝑢 ∙ 𝑛

𝜕Ω𝑤

𝑑𝑆 (170)


54

6.4 Vibroacoustic Indicators

Once the displacements of the structure and pressure on the fluid have been determined,

vibroacoustic indicators can be calculated [3].

Kinetic Energy of the Structure

Ec𝑠 =1

4𝜔2𝒖𝒎

𝑇𝒖𝒎 (171)

Kinetic Energy of the Fluid

𝐸𝑐𝑓 =1

4𝜔2𝒑𝑇 𝑯 𝒑 (172)

Mean Square Pressure in the Fluid

(𝑝2) =1

2Ω𝑓𝜌0𝑐0

2𝒑𝒎𝑇𝒑𝒎 (173)

Radiated Power

Π𝑒𝑥𝑐 =𝜔

2ℑ(𝒖𝒎

𝑇 𝑪𝒖𝒑,𝒎 𝒑𝒎) (174)

Where ℑ denotes the imaginary part of a complex number.

Sound Transmission Loss

𝑆𝑇𝐿 = log10 (1

𝜏)

(175)

where 𝜏 is the transmission factor, defined as

𝜏 =ΠtΠinc

(176)

With Π𝑡, the transmitted sound power and Π𝑖𝑛𝑐 the incident sound power, defined as:

Π𝑡 =1

2ℛ(−𝑖𝜔 𝒑 𝑪𝒖𝒑

𝑇 𝒖) (177)

Π𝑖𝑛𝑐 = ℛ(|𝐴𝑖|

2 ∙ 𝑆𝜋

2𝜌0𝑐0) (178)

Where ℛ denotes the real part of a complex number. The incident sound power

describes the intensity that a wave flows through the surface of a structure as if it was

not there, being 𝑆 the surface of the structure and |𝐴𝑖| the incident plane wave.


55

7. Routines

On this chapter a brief description of the created software is given. The routines that each

computed method uses to solve the dynamic system of equation are described to give a better

understanding of how the algorithms used solve the various examples that were tested in this

work.

The created software can perform the analysis of two different physical phenomena: structural

analysis and acoustical analysis. In the first, the software simulates the behavior of linear

structures subjected to either static or dynamic forces, obtaining in both cases the

displacements of every node on the structure and the stresses and strains on the integration

points.

On the acoustical analysis, the software simulates the propagation of waves on a fluid

subjected to harmonic loads, induced either by an acoustic source or from the vibratory

response of the structure. In this problems the pressure in every node of the fluid is

determined.


56

7.1 Structural Analysis

Introduction of all the geometric and material properties, integration parameters and

analysis type;

Nodal discretization of the geometric domain;

Finite Element Method

Nodes are connected with each other using elements;

Distribution of the Gauss Points in each element using a Gauss-Legendre quadrature rule;

Construction of the shape functions and its derivatives;

Radial Point Interpolation Method

Nodal independent distribution of the Gauss Points;

Construction of the influence-domain for each node;

Nodal connectivity is insured by the overlapping of the influence-domains;


Natural Neighbor Radial Interpolation Method

Determination of the natural neighbors of each node and construction of the Voronӧi cells;

Distribution of the integration points using nodal based integrations;

Nodes are connected to other nodes present on their first or second neighbor cells;


Assignment to each Gauss Point of a variable indicating its material;

Construction of the Constitutive Material Matrix, 𝑫;

Construction of the local stiffness matrix, 𝑲∗;

𝑲∗ =∑𝜔𝐼 · 𝑩(𝒙𝐼)𝒋𝑇𝑫𝒋𝑩(𝒙𝐼)𝒋 ·

𝒏

𝑗=1

𝑡

Assemblage of the local matrices onto the global stiffness matrix, 𝑲;


57

As pointed out before, the created software permits three different types of analysis: static

analysis, free vibrations analysis and force vibrations analysis. Next are presented the routines

used for each analysis.

Static Analysis

Construction of the force vector, may contain body forces, distributed and concentrated

loads;

Imposition of the essential boundary conditions using direct imposition method;

Determination of the static displacement by inverting the stiffness matrix;

𝒖 = 𝑲−1𝑭

Calculation of the stress and strain components;

𝜺 = 𝑩 𝒖 𝝈 = 𝑫 𝜺

Free Vibrations Analysis

Construction of the local mass matrix, 𝑴∗;


𝒏

𝑗=1

Assemblage of the local matrices onto the global mass matrix, 𝑴;

Imposition of the essential boundary conditions by removing the columns and rows from

the stiffness and mass matrices with restrained degrees of freedom;

Specify the number of modes to be kept on the modal truncated analysis;

Resolution of the eigenvalue problem;

𝑲− 𝜔2𝑴 = 𝟎

Normalization of the eigenvectors respecting modal masses;

𝚽 =1

√𝝓𝑇 𝑴 𝝓· 𝝓


58

Forced Vibrations

Construction of the local mass matrix, 𝑴∗;


𝒏

𝑗=1

Assemblage of the local matrices onto the global mass matrix, 𝑴;

Creation of a time variable to indicate for how long the displacements will be evaluated,

another for how long the force vector will be applied and other indicating the time step for

the integration;

Construction of the dynamic force vector, may contain body forces, distributed and

concentrated loads;


the stiffness and mass matrices with restrained degrees of freedom;

Construction of the damping matrix, C, using proportional damping;

The dynamic equilibrium equations of a system,

𝑴�� + 𝑪�� + 𝑲𝒖 = 𝒇 (179)

can be solved in three different ways on the created software: modal superposition, direct

integration and frequency analysis response. Each different method has its advantages and

disadvantages and are better applied on specific situations.

Modal superposition should be applied on undamped or weakly damped systems when the

damping matrix is obtained through proportional damping, since for strong damped systems

the damping may affect the natural vibrations modes. Also, it should be taken into

consideration that the highest kept natural vibration frequency for a system should be, in most

cases, at least double that of the excitation frequency. Thus for very high excitation

frequencies modal superposition may require a high number of nodes to discretize the system,

becoming computational inefficient (due to the high computational cost associated).

To culminate these problems, direct integration can be used, although the results may not be

as accurate as those obtained from modal superposition, the computational cost can

compensate the lack of accuracy. Also, for strong damped systems it is the preferred method

to be used.

Lastly, if the solicitation is harmonic and there is interest in obtaining a response on the

frequency domain, a frequency analysis response is used. Although, since this method, for

being efficient requires the use of modal superposition, it carries its disadvantages.


59

Modal Superposition



𝑲− 𝜔2𝑴 = 𝟎


𝚽 =1

√𝝓𝑇 · 𝑴 · 𝝓· 𝝓

For the kept modes, calculate the damped natural frequencies, 𝜔𝑑 = 𝜔𝑛√1 − 𝜉2;

Define the initial displacements, 𝒖𝟎 , and velocity,��𝟎;

Transformation of the various variables on the modal base;

𝛀𝟐 = 𝚽𝐓𝑲𝚽 𝐈 = 𝚽𝐓𝑴𝚽 2𝜔𝑖𝜉𝑖𝛿𝑖𝑗 = 𝚽𝒊𝐓 𝑪 𝚽𝑗

𝒇𝒎 = 𝚽𝐓 𝒇

𝒖𝒎 = 𝚽𝑇𝒖 𝒖𝒎𝟎 = 𝚽

𝑇𝑴𝒖𝟎 ��𝒎𝟎 = 𝚽𝑴��𝟎

Separation of the Duhamel integral;

1

𝑚 · 𝜔𝑑∫ 𝑓𝑚(𝜏) · 𝑒

𝜉𝜔𝑛(𝜏−𝑡) sin(𝜔𝑑 · (𝑡 − 𝜏))𝑑𝜏𝑡

0=

=1

𝑚 · 𝜔𝑑((∫ 𝑓𝑚(𝜏) · 𝑒

𝜉𝜔𝑛𝜏 cos(𝜔𝑑𝜏) 𝑑𝜏𝑡

0

) 𝑒−𝜉𝜔𝑛𝑡 sin(𝜔𝑡)

− (∫ 𝑓𝑚(𝜏) · 𝑒𝜉𝜔𝑛𝜏 sin(𝜔𝑑𝜏) 𝑑𝜏

𝑡

0

) 𝑒−𝜉𝜔𝑛𝑡 cos(𝜔𝑡))

Loop for every kept mode;

Numerically solve both integrals using a trapezoids integration rule;

𝐴 = ∫ 𝑓𝑚(𝜏) · 𝑒𝜉𝜔𝑛𝜏 cos(𝜔𝑑𝜏) 𝑑𝜏

𝑡

0

𝐵 = ∫ 𝑓𝑚(𝜏) · 𝑒𝜉𝜔𝑛𝜏 sin(𝜔𝑑𝜏) 𝑑𝜏

𝑡

0

Add the initial displacement and velocity;

𝑢𝑚0 · cos(𝜔𝑑 · 𝑡)+(��𝑚0 + 𝜉𝜔𝑛𝑢𝑚0) · sin(𝜔𝑑 · 𝑡)

𝜔𝑑

Transforming back the displacement on the spatial coordinates;

𝒖 = 𝚽 𝒖𝒎

Add the static contribution of the truncated modes;

𝒖 = 𝚽 𝒖𝒎 +𝑲−1𝒇 −𝚽 𝛀−1 𝚽𝐓𝒇

Calculate the dynamic stress and strains;

𝜺 = 𝑩 𝒖 𝝈 = 𝑫 𝜺


60

Direct Integration

Central Difference Method

Define the initial displacements, 𝒖𝟎 , velocity,��𝟎 and acceleration, ��𝟎;

��𝟎 = 𝑴−1 · (𝒇𝟎 − 𝑪 · ��𝟎 −𝑲 · 𝒖𝟎)

Assure time step is smaller than critical time step;

Calculate integration constants;

𝑎0 =1

Δ𝑡2 𝑎1 =

1

2Δ𝑡 𝑎2 = 2𝑎0 𝑎3 =

1

𝑎2

Fictional displacement at 𝑡 = −Δ𝑡;

𝒖−𝚫𝐭 = 𝒖𝟎 − Δ𝑡��𝟎 + 𝑎3��𝟎

Effective mass matrix

�� = 𝑎0𝑴+ 𝑎1𝑪

Loop for every time step;

Effective loads at time 𝑡;

��𝒕 = 𝒇𝒕 − (𝑲− 𝑎2𝑴) · 𝒖𝒕 − (𝑎0𝑴− 𝑎1𝑪) · 𝒖𝒕−𝚫𝒕

Displacement at instant 𝑡 + Δ𝑡

𝒖𝒕+𝚫𝒕 = ��−1 · ��𝒕

Acceleration and velocity at the instant 𝑡;

��𝒕 = 𝑎0(𝒖𝒕−𝚫𝒕 − 2 · 𝒖𝒕 + 𝒖𝒕+𝚫𝒕)

��𝒕 = 𝑎1(𝒖𝒕+𝚫𝒕 − 𝒖𝒕−𝚫𝒕)


61

Houbolt Method


��𝟎 = 𝑴−1 · (𝒇𝟎 − 𝑪 · ��𝟎 −𝑲 · 𝒖𝟎)


𝑎0 =2

Δ𝑡2 𝑎1 =

11

6Δ𝑡 𝑎2 =

5

Δt2 𝑎3 =

3

Δt 𝑎4 = −2𝑎0

𝑎5 = −𝑎32 𝑎6 =

𝑎02 𝑎7 =

𝑎39

Use Central Difference Method to calculate 𝒖𝚫𝒕 and 𝒖𝟐𝚫𝒕;

Loop for each time step;

Effective loads at time 𝑡 + Δ𝑡;

��𝒕+𝚫𝒕 = 𝒇𝒕+𝚫𝒕 +𝑴(𝑎2𝒖𝒕 + 𝑎4𝒖𝒕−𝚫𝒕 + 𝑎6𝒖𝒕−𝟐𝚫𝒕) + 𝑪(𝑎3𝒖𝒕 + 𝑎5𝒖𝒕−𝚫𝒕 + 𝑎7𝒖𝒕−𝟐𝚫𝒕)


𝒖𝒕+𝚫𝒕 = 𝑲−1 · ��𝒕+𝚫𝒕

Newmark Method


��𝟎 = 𝑴−1 · (𝒇𝟎 − 𝑪 · ��𝟎 −𝑲 · 𝒖𝟎)

Input the Newmark parameters 𝛿 and 𝛼;


𝑎0 =1

𝛼(Δ𝑡)2 𝑎1 =

𝛿

𝛼Δ𝑡 𝑎2 =

1

𝛼Δ𝑡 𝑎3 =

1

2𝛼− 1

𝑎4 =𝛿

𝛼− 1 𝑎5 =

Δ𝑡

2(𝛿

𝛼− 2) 𝑎6 = Δ𝑡(1 − 𝛿) 𝑎7 = 𝛿Δ𝑡

Effective stiffness matrix

�� = 𝑲 + 𝑎0𝑴+ 𝑎1𝑪


Effective loads at time 𝑡 + Δ𝑡;

��𝒕+𝚫𝒕 = 𝒇𝒕+𝚫𝒕 +𝑴(𝑎0𝒖𝒕 + 𝑎2��𝒕 + 𝑎3��𝒕) + 𝑪(𝑎1𝒖𝒕 + 𝑎4��𝒕 + 𝑎5��𝒕)


𝒖𝒕+𝚫𝒕 = ��−1 · ��𝒕+𝚫𝒕

Acceleration and velocity at instant 𝑡 + Δ𝑡

��𝒕+𝚫𝒕 = 𝑎0(𝒖𝒕+𝚫𝒕 − 𝒖𝒕) − 𝑎2 · ��𝒕 − 𝑎3 · ��𝒕

��𝒕+𝚫𝒕 = ��𝒕 + 𝑎6 · ��𝒕 + 𝑎7 · ��𝒕+𝚫𝒕


62

Wilson-θ Method


��𝟎 = 𝑴−1 · (𝒇𝟎 − 𝑪 · ��𝟎 −𝑲 · 𝒖𝟎)

Input the parameter 𝜃;


𝑎0 =6

(𝜃Δ𝑡)2 𝑎1 =

3

𝜃Δ𝑡 𝑎2 = 2𝑎1 𝑎3 =

𝜃Δ𝑡

2 𝑎4 =

𝑎0𝜃

𝑎5 = −𝑎2𝜃 𝑎6 = 1 −

3

𝜃 𝑎7 =

Δ𝑡

2 𝑎8 =

(Δ𝑡)2

6

Effective stiffness matrix

�� = 𝑲 + 𝑎0𝑴+ 𝑎1𝑪


Effective loads at time 𝑡 + 𝜃Δ𝑡;

��𝒕+𝜽𝚫𝒕 = 𝒇𝒕 + 𝜃(𝒇𝒕+𝚫𝒕 − 𝒇𝒕) +𝑴(𝑎0𝒖𝒕 + 𝑎2��𝒕 + 2��𝒕) + 𝑪(𝑎1𝒖𝒕 + 2��𝒕 + 𝑎3��𝒕)

Displacement at instant 𝑡 + 𝜃Δ𝑡

𝒖𝒕+𝜽𝚫𝒕 = ��−1 · ��𝒕+𝜽𝚫𝒕

Acceleration, velocity and displacement at instant 𝑡 + Δ𝑡

��𝒕+𝚫𝒕 = 𝑎4(𝒖𝒕+𝜽𝚫𝒕 − 𝒖𝒕) + 𝑎5��𝒕 + 𝑎6��𝒕

��𝒕+𝚫𝒕 = ��𝒕 + 𝑎7(��𝒕+𝚫𝐭 + ��𝒕)

𝒖𝒕+𝚫𝒕 = 𝒖𝒕 + Δ𝑡��𝒕 + 𝑎8(��𝒕+𝚫𝐭 + 2��𝒕)

Frequency Analysis Response



𝑲− 𝜔2𝑴 = 𝟎


𝚽 =1

√𝝓𝑇 · 𝑴 · 𝝓· 𝝓

Loop for every kept mode

𝑢𝑚𝑖=

𝐹𝑖𝑚(𝜔𝑖

2(1 + 𝑖𝜂1) − 𝜔2) + 𝑖2𝜉1𝜔1𝜔

Transforming back the displacement on the spatial coordinates;

𝒖 = 𝚽 · 𝒖𝒎

Calculate the dynamic stress and strains;

𝜺 = 𝑩 · 𝒖 𝝈 = 𝑫 · 𝜺


63

7.2 Acoustic Analysis

Assemble of the structure matrices, 𝑲,𝑴 and 𝑪;

Introduction of all the geometric and fluid properties, integration parameters and analysis

type;

Nodal discretization of the fluid domain;

Finite Element Method

Nodes are connected with each other using elements;

Distribution of the Gauss Points in each element using a Gauss-Legendre quadrature rule;


Radial Point Interpolation Method

Nodal independent distribution of the Gauss Points;

Construction of the influence-domain for each node;

Nodal connectivity is insured by the overlapping of the influence-domains;


Natural Neighbor Radial Interpolation Method

Determination of the natural neighbors of each node and construction of the Voronӧi cells;

Distribution of the integration points using nodal based integrations;

Nodes are connected to other nodes present on their first or second neighbor cells;


Construction of the local kinetic energy matrix, 𝑸∗;

𝑸∗ =∑𝜔𝐼 ·1

𝜌0𝑐02

𝝋(𝒙𝐼)𝒋𝑇 ·

𝒏

𝑗=1

𝝋(𝒙𝐼)𝒋

Construction of the local compression energy matrix, 𝑯∗;

𝑯∗ =∑𝜔𝐼 ·1

𝜌0

𝑩(𝒙𝐼)𝒋𝑇 ·

𝒏

𝑗=1

𝑩(𝒙𝐼)𝒋

Construction of the local impedance boundary matrix, 𝑨∗;

𝑨∗ =∑𝜔𝐼 ·��

𝜌0𝑐0

𝝋(𝒙𝐼)𝒋𝑇 ·

𝒏

𝑗=1

𝝋(𝒙𝐼)𝒋

Assemblage of the local matrices onto the global matrices, 𝑸, 𝑯 and 𝑨;


64

Depending on the strength of the coupled fluid-structure problem, the resolution of system

equilibrium equations is performed differently. If the system is weakly coupled, the

equilibrium equations for the structure and the fluid can be solved separately, thus the

structural problem is solved using the routines presented on chapter 7.1 and the displacements

are inserted in the force vector. From here the equilibrium equations for the fluid are solved.

If the system is strongly coupled, then the equilibrium equations must be solved

simultaneously for the fluid and the structure.

In either cases, since acoustic problems are in general harmonic problems, a response in the

frequency domain is usually seek. Thus, the algorithms used to solve this problems focus

mainly on the frequency domain analysis.


65

Uncoupled Acoustic Problem

Construction of the local force vector, 𝒇𝒂∗ from the displacements of the structure;

𝒇𝒂∗ = 𝜔2∑𝜔𝐼 𝝋(𝒙𝐼)𝒋

𝑇

𝒏

𝑗=1

𝒖𝑛 𝝋(𝒙𝐼)𝒋

Assemblage of the local matrices onto the global matrices, 𝑸, 𝑯 and 𝑨;


the various matrices with restrained degrees of freedom;

Construction the equilibrium equations of the system;

−𝜔2𝑸+ 𝑖𝜔𝑨 + 𝑯 = 𝒇


Solve the eigenvalue problem;

𝑯−𝜔2𝑸 = 0

Normalization of the eigenvectors;

𝚽 =1

√𝝓𝑇 𝑸 𝝓· 𝝓

Transformation of the various variables into the modal base;

𝝎𝒏𝟐𝒊= 𝚽𝐓𝑯𝚽 2𝜔𝑖𝜉𝑖𝛿𝑖𝑗 = 𝚽

𝐓 𝑨 𝚽 𝒇𝒎 = 𝚽𝑇𝒇 𝒑𝒎 = 𝚽𝑇𝒑

Solve for every kept mode;

If no damping is present;

𝑝𝑖𝑚 =𝑓𝑖𝑚

𝜔𝑖2 −𝜔2

If modal damping is considered;


𝜔𝑖2(1 + 𝑖𝜂𝑖) − 𝜔

2

If proportional damping is considered;


(𝜔𝑖2 −𝜔2)+ 𝑖2𝜉𝑖𝜔𝑖𝜔

For other damping situations, either solve the equilibrium equations using direct integration

or invert the impedance matrix;

𝒁 = Ω𝟐 + 𝑖𝜔𝚽𝐓 𝑨 𝚽 −𝝎2𝑰

𝒑𝒎 = 𝒁−1 𝒇𝒎

Transforming back the pressure on the spatial coordinates;

𝒑 = 𝚽 𝒑𝒎

Add the static contribution of the truncated modes;

𝒑 = 𝚽 𝒑 + 𝑯−1𝒇 −𝚽 𝛀−1 𝚽𝐓𝒇


66

Interior Coupled Acoustic Problem

Imposition of the essential boundary conditions on the structure by removing the columns

and rows from the various matrices with restrained degrees of freedom;

Determine the nodes belonging to both the structure and the fluid;

Construction of the local coupling matrices, 𝑪𝒖𝒑∗ and 𝑲𝒔

∗;

𝑲𝒔∗ =

1

Ω𝑓∑𝜔𝐼 𝜌0𝑐0

2 (𝝋𝒔(𝒙𝐼)𝒋

𝑇 𝒏)

𝒏

𝑗=1

∙∑𝜔𝐼(𝒏 𝑇𝝋

𝒔(𝒙𝐼)𝒋

)

𝑛

𝑗=1

𝑪𝒖𝒑∗ =∑𝜔𝐼(𝝋𝒔(𝒙𝐼)𝒋

𝑇𝒏

𝒏

𝑗=1

𝝋𝒇(𝒙𝐼)𝒋 )

Assemblage of the local matrices onto the global matrices, 𝑲𝒔 and 𝑪𝒖𝒑;

If damping is present, solve the equilibrium equation using direct integration;

([𝑲 −𝑪𝒖𝒑𝟎 𝑯

]+ 𝑖𝜔 [𝑪 𝟎𝟎 𝑨

]−𝜔2 [𝑴 𝟎𝑪𝒖𝒑𝑇 𝑸

]) {𝒖𝒑} = {

𝑭𝒇}

Else, solve the eigenvalue problem for the fluid and normalize the eigenvectors;

𝑯−𝜔2𝑸 = 0

Remove the rigid body mode (the first mode) and separate the static pressure and elastic

pressure;

𝒑 = 𝒑𝒔 + 𝒑𝒆


𝛀𝒇𝟐 = 𝚽𝒇

𝐓𝑯𝚽𝒇 𝑪𝒖𝒑𝒎= 𝑪𝒖𝒑𝚽𝒇 𝒇𝒆𝒎

= 𝚽𝑇𝒇𝒆 𝒑𝒆𝒎 = 𝚽𝒇

𝑇𝒑𝒆

Solve the structural coupled eigenvalue problem and normalize the eigenvectors;

𝑴𝒆 = 𝑪𝒖𝒑𝒎 𝛀𝒇𝟐 𝑪𝒖𝒑𝒎

(𝑲−𝑲𝒔)−𝜔2(𝑴+𝑴𝒆) = 0


𝛀𝒔𝟐 = 𝚽𝒔

𝐓(𝑲 − 𝑲𝒔)𝚽𝒔 𝑪𝒖𝒑 = 𝚽𝒔𝑇 𝑪𝒖𝒑,𝒎

𝛀𝒇−𝟐 𝒖𝒎 = 𝚽𝒇

𝑇𝒖

Solve the coupled structure-fluid eigenvalue problem and normalize the eigenvectors;

[Ω𝑠𝟐 𝟎𝟎 𝑰

]−𝜔2 [𝑰 𝑪𝒖𝒑

𝑪𝒖𝒑�� Ω𝒇,𝒆

−𝟐] = 0

𝛀𝒇𝒔𝟐 = 𝚽𝒔

𝐓 [Ω𝑠𝟐 𝟎𝟎 𝑰

]𝚽𝒔

Calculate the modal displacements and elastic pressure;

{𝒖𝒎𝒑𝒆𝒎

} = [𝛀𝒇𝒔𝟐 −𝜔2𝑰]

−1 𝚽𝒇𝒔

𝑇 {𝚽𝒔𝑇𝑭+𝚽𝒔

𝑇 𝑪𝒖𝒑𝒎

Ω𝒇,𝒆−𝟐 𝒇𝒆𝒎

Ω𝒇−𝟐 𝒇𝒆𝒎

}

Transforming back the pressure and displacements on the spatial coordinates;

𝒑𝒆 = 𝚽𝒇𝒔 · 𝒑𝒆𝒎 𝒖 = 𝚽𝒇𝒔 · 𝒖𝒎 𝒑 = 𝒑𝒔 + 𝒑𝒆


67

8. Numerical Examples

In this chapter firstly some benchmark examples are solved to verify the convergence and

validate the created software by comparing the results obtained with those presented on the

literature. Afterwards some complex problems, involving laminated and sandwich plates and

beams, are analysed with both the FEM, RPIM and NNRPIM and their results and effieciency

are compared with the literature when possible.


68

8.1 Elastostatic Analysis of a Cantilever Beam

Consider the cantilever beam represented in Figure 15 with the prescribed degrees of

freedom, being the beam length 𝐿 = 2 𝑚, and the beam height 𝐷 = 1 𝑚. The material

properties are 𝐸 = 1 𝑘𝑃𝑎 and 𝑣 = 0.3. The following essential boundary conditions are

applied:

𝑝𝑥𝑥(𝑥 = 0, 𝑦) =𝑃 · 𝐿 · 𝑦

𝐼

𝑝𝑦𝑦(𝑥 = 0, 𝑦) = −𝑃·𝐷2

8·𝐼· (1 −

4·𝑦2

𝐷2)

𝑝𝑦𝑦(𝑥 = 𝐿, 𝑦) = −𝑃·𝐷2

8·𝐼· (1 −

4·𝑦2

𝐷2)

(180)

where 𝐼 =𝐷3

12 and 𝑃 = 10 𝑁. The analytical stress field for the described beam is:

𝜎𝑥𝑥(𝑥, 𝑦) = −𝑃 · (𝐿 − 𝑥) · 𝑦

𝐼

𝜎𝑦𝑦(𝑥, 𝑦) = 0

𝜎𝑥𝑦(𝑥, 𝑦) = −𝑃 · 𝐷2

8 · 𝐼· (1 −

4 · 𝑦2

𝐷2)

(181)

Therefore, the following displacement field is obtained

𝑢(𝑥, 𝑦) = −2 · 𝑃

𝐸 · 𝐷3[3𝑥 · (2𝐿 − 𝑥) · 𝑦 + (2 + 𝑣) · (𝑦2 −

𝐷2

4) · 𝑦]

𝑣(𝑥, 𝑦) =2 · 𝑃

𝐸 · 𝐷3[𝑥2 · (3𝐿 − 𝑥) + 3 · 𝑣 · (𝐿 − 𝑥) · 𝑦2 + 𝑥4+5·𝑣 ·

𝐷2

4]

(182)

Figure 15: Model of the cantilever beam on study [28].

The problem was evaluated considering a two-dimensional analysis, with increasing denser

nodal meshes applied. Both regular and irregular mesh discretization were considered for the

FEM, RPIM and NNRPIM formulations, the results pertaining the displacement and stress

medium errors are presented in Figure 16 and Figure 17 respectively. In both cases, the

medium errors are given by

𝐸𝑚𝑒𝑑 =1

𝑁∑

√((𝑢𝑖)𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 − (𝑢𝑖)𝑒𝑥𝑎𝑐𝑡)2

√(𝑢𝑖)𝑒𝑥𝑎𝑐𝑡2

𝑁

𝑖=1

(183)


69

Figure 16: Medium displacement errors for a regular mesh (left side) and irregular mesh (right side):

a)displacement u b) displacement v. Logarithmic scales.

a)

b)


70

Figure 17: Medium stress errors for a regular mesh (left side) and irregular mesh (right side): a) 𝜎𝑥𝑥b) 𝜎𝑥𝑦.

Logarithmic scales.

A punctual displacement error was conducted on four different points for a regular mesh,

Figure 18. For comparison the analytical results were calculated using the coordinates of the

nodes more close to the chosen points. The results can be found in Table 1 to Table 3.

Figure 18: Position on the beam of the analyzed points.

a)

b)

1

2

3 4


71

Table 1: Punctual displacement errors using a regular mesh and FEM formulation.

Mesh Discretization 4x2 8x4 16x8 32x16 64x32 128x64

Point 1

Displacement u 0 6.06E-02 1.95E-02 6.65E-03 2.11E-03 6.40E-04

Displacement v 9.20E-03 2.96E-02 1.14E-02 4.08E-03 1.33E-03 4.10E-04

Point 2



Point 3

Displacement u 0 0 0 0 0 0


Point 4



Table 2: Punctual displacement errors using a regular mesh and RPIM formulation.


Point 1



Point 2



Point 3



Point 4



Table 3: Punctual displacement errors using a regular mesh and NNRPIM formulation.


Point 1



Point 2



Point 3



Point 4




72

As shown, both the regular and irregular meshes present a good convergence for the stresses.

For the displacements, the methods converge towards a solution and then oscillate around this

value. Overall, the use of a regular mesh attains better accuracy than using an irregular one.


73

8.2 Free Vibration of a Cantilever Beam

To validate the resolution of free vibration problems, a simple example of a cantilever beam

was considered, Figure 19. The beam has length 𝐿 = 100 𝑚𝑚 and height 𝐷 = 10 𝑚𝑚. The

material properties are 𝐸 = 2.1 · 105 𝑀𝑃𝑎, 𝑣 = 0.3 and 𝜌 = 8.0 · 10−9 𝑘𝑔 𝑠2/𝑚𝑚4.

Figure 19: Model of the cantilever beam on study [28].

For the convergence study, only the first three vibrations modes were taken into consideration

and compared to the results obtained with another FEM program, ABAQUS, using a regular

mesh distribution of 4000 nodes, which were the following: 𝑓1 = 830 𝐻𝑧, 𝑓2 = 4,979 𝐻𝑧 and

𝑓3 = 12,826 𝐻𝑧.

The relative frequency error is given in percentage by the expression:

𝐸𝑓𝑖 =√((𝑓𝑖)𝐴𝐵𝐴𝑄𝑈𝑆 − (𝑓𝑖)𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑)

2

(𝑓𝑖)𝐴𝐵𝐴𝑄𝑈𝑆· 100 (184)

In order to obtain the optimal integration scheme for the RPI method, some preliminary

studies were conducted where the effect of the radial shape function parameters, the Gauss

quadrature (GQ) and the influence domain size were observed. For these studies an irregular

mesh was used, where the coordinates of the nodes are

𝑥𝑖𝑟𝑟 = 𝑥𝑟𝑒𝑔 + 2 · (𝑟𝑎𝑛𝑑 − 0.5) · (𝐿

𝐷𝑖𝑣𝐿 · Υ)

𝑦𝑖𝑟𝑟 = 𝑦𝑟𝑒𝑔 + 2 · (𝑟𝑎𝑛𝑑 − 0.5) · (𝐷

𝐷𝑖𝑣𝐷 · Υ)

(185)

Where 𝑟𝑎𝑛𝑑 is a random number generated for each mesh, 𝐷𝑖𝑣𝐿 and 𝐷𝑖𝑣𝐷 the number of

divisions along the length and width of the beam, respectively. The parameter Υ defines the

irregularity of the mesh, the higher it is, the more regular the nodal mesh becomes. To note

that the nodes belonging to the boundaries of the beam were not affected by equation (185),

being in the same position as those of the regular mesh, as represented in Figure 20.

From Table 4 to Table 7 are presented the frequency errors for the an irregular mesh with Υ =5, varying some of the described parameters and using two different polynomial basis, with

one and three monomials . As can be seen, in order to attain relatively accurate results (𝐸𝑓𝑖 <

10%) a Gauss quadrature rule with at least 5x5 integration points is necessary. Using more

integration points has no justification since the improvement on the accuracy is almost

insignificant.

Figure 20: (a) Regular mesh; (b) Irregular mesh.


74

The other parameters did not considerably affect the obtained results, as such adopting an

influence domain with 16 nodes is the better option since it requires less computational effort.

For the shape parameters, using 𝑐 = 0.0001 and 𝑝 = 0.9999 provides slightly better accuracy

for the considered influence domain.

Under those conditions, a robustness test was performed where the relative frequency error

was, for the first natural vibration mode, obtained while varying the irregularity parameter Υ.

The results are represented in Figure 21.

Table 4: Relative frequency error: polynomial basis with one monomial and 𝑐 = 1.42 and 𝑝 = 1.03.

Influence

Domain

Natural

Frequency

GQ = 2 GQ = 3 GQ = 4 GQ = 5 GQ = 6

16 nodes 1 91.6529 41.6236 24.2630 2.2880 2.3753

2 92.1512 42.5627 25.0343 2.3388 2.3648

3 94.6889 52.7456 26.6357 2.1126 1.4685

21 nodes 1 91.2066 35.1148 25.3981 2.0278 2.8582

2 92.1007 35.3537 26.1197 2.0126 2.7498

3 94.7166 43.3402 30.0396 1.5534 2.1802

27 nodes 1 91.7317 32.6295 22.9363 1.9117 2.7300

2 92.3133 33.0738 23.4836 1.8690 2.5823

3 94.8952 38.9528 26.1227 1.3397 2.2860

Table 5: Relative frequency error: polynomial basis with one monomial and 𝑐 = 0.0001 and 𝑝 = 0.999.

Influence

Domain

Natural

Frequency

GQ = 2 GQ = 3 GQ = 4 GQ = 5 GQ = 6

16 nodes 1 91.5930 40.3059 24.2496 2.2171 2.2958

2 91.9910 41.1493 24.8336 2.1550 2.1591

3 94.5188 51.2431 26.2868 1.9368 1.2620

21 nodes 1 91.1400 33.6481 25.4750 1.9824 2.8867

2 92.0090 33.8177 26.0859 1.9180 2.7213

3 94.6202 41.7715 29.7674 1.4904 2.1726

27 nodes 1 91.6247 31.1689 23.2451 1.9493 2.8362

2 92.2237 31.5340 23.7433 1.8823 2.6582

3 94.7980 37.3315 26.2952 1.3679 2.4029


75

Table 6: Relative frequency error: polynomial basis with three monomials and 𝑐 = 1.42 and 𝑝 = 1.03.

Influence

Domain

Natural

Frequency

GQ = 2 GQ = 3 GQ = 4 GQ = 5 GQ = 6

16 nodes 1 91.7717 41.4829 24.0860 2.1743 2.2319

2 92.1485 42.3920 24.7258 2.1169 2.1028

3 94.6572 52.6283 26.2748 1.9024 1.2241

21 nodes 1 91.2482 34.9563 25.3293 1.9324 2.7660

2 92.1188 35.2139 25.9731 1.8719 2.6029

3 94.7170 43.2707 29.8831 1.4424 2.0583

27 nodes 1 91.7518 32.5670 22.9413 1.8986 2.6979

2 92.3338 33.0167 23.4485 1.8377 2.5206

3 94.9000 38.9303 26.0939 1.3098 2.2376

Table 7: Relative frequency error: polynomial basis with three monomials and 𝑐 = 0.0001 and 𝑝 = 0.9999.

Influence

Domain

Natural

Frequency

GQ = 2 GQ = 3 GQ = 4 GQ = 5 GQ = 6

16 nodes 1 91.5932 40.3069 24.2506 2.2179 2.2967

2 91.9911 41.1500 24.8347 2.1559 2.1601

3 94.5189 51.2436 26.2880 1.9375 1.2628

21 nodes 1 91.1400 33.6488 25.4754 1.9829 2.8873

2 92.0090 33.8183 26.0865 1.9185 2.7218

3 94.6202 41.7718 29.7679 1.4908 2.1731

27 nodes 1 91.6247 31.1692 23.2452 1.9495 2.8365

2 92.2236 31.5343 23.7435 1.8824 2.6585

3 94.7980 37.3316 26.2953 1.3680 2.4031

In order to obtain the optimal integration scheme for the NNRPI method, some preliminary

studies were conducted where the effect of the radial shape function parameters, the degree of

the polynomial used (P) and the influence domain size (using first and second degree

influence cells) were observed. For these studies an irregular mesh with Υ = 5 was used. The

results are available in Table 8 and Table 9.


Influence

Domain

Natural

Frequency

P = 1 P = 2 P = 3

First Degree

Influence Cells

1 10.8258 15.5650 8.6764

2 3.8229 2.4305 3.6250

3 6.2716 6.9091 5.7573

Second Degree

Influence Cells

1 5.7472 3.3015 4.2125

2 5.1802 4.4074 4.3840

3 6.3846 4.2465 4.1092


76


Influence

Domain

Natural

Frequency

P = 1 P = 2 P = 3

First Degree

Influence Cells

1 4.5741 4.1241 5.7874

2 4.2360 4.5531 5.1146

3 4.9605 5.2264 5.7290

Second Degree

Influence Cells

1 4.5754 4.9899 4.0871

2 4.6370 4.9839 4.3878

3 4.0094 4.5198 4.0853

Based on these results, a robustness test was performed for the NNRPIM, using polinomials

of the first degree since they require less computational cost and present good results with the

shape parameters being 𝑐 = 0.0001 and 𝑝 = 0.9999. For the influence cell domain, a second

degree influence cell was used since it provided errors below 10% for all the cases tested. The

results of the robustness test are also displayed in Figure 21.

Figure 21: Robutsness study of the cantilever beam for the different formulations.

For the RPIM, from Figure 21 it can be noted that using the integration scheme previously

defined (that is, applying a 5x5 integration mesh) the values for the fundamental natural

vibration always stay with a relative error below 10% for Υ ≥ 1.

For the NNRPIM, the relative error is kept below 5% with the chosen parameters. It can be

seen that as the mesh becomes less irregular the relative error stabilizes around a value,

instead of converging to one.

Therefore, those will be the integration parameters adopted for the remaining studies of free

and forced vibrations where RPIM and NNRPIM formulations are applied.

The results for the first three frequencies for the different applied methods with increase

denser meshes can be seen in Figure 22.


77

Figure 22: Natural vibration frequencies: regular mesh (left-side) and irregular mesh (right-side).

a) first mode b) second mode c) third mode.

As can be seen, both solutions converge towards the result obtained with the ABAQUS

software. Particularly, the use of an irregular mesh seems to decrease the natural frequencies

values, which can mean that the use of such meshes decreases the level of rigidity of the

system in analysis.

a)

b)

c)


78

In Figure 23 are represented the first three vibration modes of the studied cantilever beam.

Figure 23: First three vibration modes of the cantilever beam.

Mode 3

Mode 1

Mode 2


79

8.3 Forced Vibrations of a Cantilever Beam

For the cantilever beam of Figure 19 a harmonic load, 𝑓(𝑡) = sin (0.04 · 𝑡), was applied on

the right side of the beam, in the y-direction. The geometry of the square section beam is:

𝐿 = 4 𝑚 and 𝐷 = 1 𝑚. The material properties are: 𝐸 = 1 𝑃𝑎, 𝑣 = 0.3 and 𝜌 = 1 𝑘𝑔/𝑚3

and the Rayleigh damping coefficients are 𝛼 = 0.005 and 𝛽 = 0.272.

An initial analysis was performed with FEM using direct integration with the Newmark

Method, and the results were compared with those presented on [62]. For such case, the

Newmark parameters used were 𝛼 = 0.5 and 𝛿 = 1.0 and the time step for the integration

∆𝑡 = 1.27 𝑠. Five simulations were performed with an increasing denser mesh to verify the

convergence of the method, the results are displayed in Figure 24.

Comparing the results, they converge towards a solution since both meshes with 64x16 and

128x32 nodes apparent the same behavior. Intriguingly, a mesh with 16x4 nodes provides the

most similar results with those presented on the literature.

Figure 24: Vertical displacement for the cantilever beam measured on point 𝐴(𝐿, 𝐷 2⁄ ).

Under the same conditions, other simulations were conducted using different direct

integration methods and even a modal superposition analysis using the Duhamel integral. For

the Wilson-𝜃 method a value of 𝜃 = 1.4 was used, and for the modal superposition a

truncated analysis with only the first five modes was performed. This time, besides the FEM,

the results with RPIM and NNRPIM formulations were also obtained, they can be found in

Figure 25 to Figure 27.


80

Figure 25: Vertical displacement on point A using different methods (FEM).

Figure 26: Vertical displacement on point A using different methods (RPIM).

Figure 27: Vertical displacement on point A using different methods (NNRPIM).


81

Another study was conducted on a clamped-free cantilever beam, this time with dimensions

𝐿 = 1 𝑚, 𝐷 = 0.04 𝑚 and width of 𝐻 = 0.06 𝑚. The material properties are: 𝐸 = 70 𝐺𝑃𝑎,

𝑣 = 0.32 and 𝜌 = 2730 𝑘𝑔/𝑚3. On the upper side of the beam a dynamic distributed load,

𝑓𝑥(𝑡) = 𝑓𝑦(𝑡) = 2000 · sin(𝜋 · 𝑥) · sin(8 · 207.0236 · 𝑡) , 𝑦 = 𝐷 (186)

is applied, and no damping is considered on this simulation. The simulation was performed

with both FEM and RPIM and NNRPIM formulations, using modal superposition with a

truncated analysis considering the first five modes. The results are displayed in Figure 28.

Figure 28: Vertical displacement of point A for the cantilever beam subjected to a harmonic distributed load.

To verify the validity of the presented software the results for the dynamic displacement of

the free edge were compared with those on literature [63], Figure 29. As can be seen, the

displacements are almost identical to one another, showing the applicability of the software to

predict the response of structures to forced vibration problems.

Figure 29: Vertical displacement of the clamped-free beam presented on the literature [63].


82

8.4 Free Vibration of a Sandwich Beam

To verify the applicability of the developed software on the analysis of the behavior of

sandwich beams with soft-core some benchmark examples of free vibrations were conducted

using the FEM, RPIM and NNRPIM formulations. Firstly, different essential boundary

conditions were enforced on the beam, the results obtained were compared with those

presented on literature [64].

Table 10: Geometric and material properties of the sandwich beam subjected to different boundary conditions.

Geometry Material Properties

Length 𝐿 = 260 𝑚𝑚 Face Sheets

Width 𝐷 = 59.9 𝑚𝑚 𝐸 = 210 𝐺𝑃𝑎 𝑣 = 0.3 𝜌 = 7900 𝑘𝑔/𝑚3

Face sheet thickness 𝑡𝑓 = 1.9 𝑚𝑚 Soft Core

Core thickness 𝑡𝑐 = 34.8 𝑚𝑚 𝐸 = 56 𝑀𝑃𝑎 𝑣 = 0.27 𝜌 = 60 𝑘𝑔/𝑚3

The sandwich beam in study is constituted as follows (aluminum face sheet/PE open cell

foam core/aluminum face sheet). In Table 10 are presented the geometric and material

properties of the beam. A 260x40 regular discretization mesh was used in all the examples. In

Table 11 the first four natural frequencies of the beam for the different boundary conditions

are shown, compared with those presented on the literature.

Table 11: First four natural frequencies (Hz) for the sandwich beam with different boundary conditions.

Mode nr. Theory C-F C-C C-H H-H F-F

1 FEM 164.2 367.1 342.3 322.1 649.4

RPIM 165.1 376.8 354.1 335.8 644.0

NNRPIM 167.0 372.8 347.6 327.0 659.7

Literature 164.6 366.5 339.0 316.6 658.6

2 FEM 511.7 778.3 730.7 687.2 953.1

RPIM 513.7 792.7 750.0 710.2 946.2

NNRPIM 518.0 781.6 733.8 690.7 960.7

Literature 513.5 774.2 721.9 675.0 962.3

3 FEM 915.8 1268.9 1192.2 1122.3 1463.9

RPIM 917.1 1280.2 1210.7 1147.3 1479.2

NNRPIM 921.1 1256.5 1180.8 1111.7 1501.4

Literature 920.6 1256.2 1174.4 1099.8 1530.0

4 FEM 1390.0 1859.9 1751.4 1642.4 2075.4

RPIM 1383.7 1859.8 1760.2 1672.4 2048.5

NNRPIM 1382.9 1817.2 1710.7 1612.6 2068.9

Literature 1396.1 1831.5 1718.6 1613.7 2123.7

On the boundary conditions, C denotes “clamped”, H is “hinged” and F is a “free edge”. As

can be seen from the results, the software performs well under the varying boundary

conditions applied, showing perhaps higher discrepancies under free-free conditions.


83

For the simply supported sandwich beam of Table 12 various tests were conducted in which

the ratio length-thickness was changed. Only the first two natural vibration modes were

considered in this study.

Table 12: Geometric and material properties of the sandwich beam subjected to different length-thickness ratios.


Length 𝑣𝑎𝑟𝑖𝑎𝑣𝑒𝑙 Face Sheets

Width 𝐷 = 20 𝑚𝑚 𝐸 = 200 𝐺𝑃𝑎 𝑣 = 0.3 𝜌 = 7800 𝑘𝑔/𝑚3

Face sheet thickness 𝑡𝑓 = 3 𝑚𝑚 Soft Core

Core thickness 𝑡𝑐 = 14 𝑚𝑚 𝐸 = 660 𝑀𝑃𝑎 𝑣 = 0.27 𝜌 = 60 𝐾𝑔/𝑚3

In Table 13 to Table 15 the first two natural frequencies for three different ratios are

compared with those present on [65]. The frequencies are in their non-dimensional form,

given by the expression

�� = 𝜔 𝐿2

𝐷√𝜌𝑓

𝐸𝑓 (187)

Table 13: Non-dimensional natural frequencies for the sandwich beam with varying 𝜆 = 𝐿 𝐻⁄ (FEM).

Mode nr. Theory 𝜆 = 5 𝜆 = 10 𝜆 = 100

1 FEM 1.103 1.932 4.199

Literature 1.097 1.915 4.103

Error 0.55 % 0.89 % 2.34 %

2 FEM 2.659 4.436 15.99

Literature 2.660 4.386 15.55

Error 0.04% 1.14 % 2.83 %

Table 14: Non-dimensional natural frequencies for the sandwich beam with varying 𝜆 = 𝐿 𝐻⁄ (RPIM).


1 RPIM 1.135 1.932 4.102

Literature 1.097 1.915 4.103

Error 3.46 % 0.89 % 0.02 %

2 RPIM 2.787 4.474 15.65

Literature 2.660 4.386 15.55

Error 4.77 % 2.01 % 0.64 %


84

Table 15: Non-dimensional natural frequencies for the sandwich beam with varying 𝜆 = 𝐿 𝐻⁄ (NNRPIM).


1 NNRPIM 1.159 2.005 4.083

Literature 1.097 1.915 4.103

Error 5.65 % 4.70 % 0.49 %

2 NNRPIM 2.746 4.599 15.61

Literature 2.660 4.386 15.55

Error 3.23 % 4.86 % 0.39 %

The results displayed show that the software, when using the FEM formulation performs very

well on mimicking the behavior of thick sandwich plates, showcasing very small percentages

for the error. On the other hand, as the beam becomes thinner with the increase in 𝜆 the

bigger the discrepancies between the two solutions for the FEM. The opposite seems to

happen for meshless methods. Nevertheless both formulations give very accurate results on

free vibrations analysis of laminated beams.

In Figure 30 are represented the natural vibration modes for the first two modes of the simply

supported sandwich beam.

Figure 30: First two vibration modes for the simply supported sandwich beam.

Mode 1

Mode 2


85

8.5 Analysis of Sandwich Cantilever Beam with Corrugated Core


86

8.5.1 Static Analysis

For the sandwich beam of Figure 31, with a sinusoidal corrugated core a static analysis was

performed. The beam is under three-point bending, obeying the boundary conditions in Figure

32.

Figure 31: Sandwich beam with sinusoidal corrugated core along its length [66].

In Table 16 is described the geometric and material properties of the considered beam. Two

theories for analytical expressions for the maximum deflection are given in [66], the “Euler-

Bernoulli” and “Broken Line” theories.. For an applied bending force of 𝐹 = 10,000 𝑁, the

comparison between the results obtained with the software using FEM and Meshless

formulations and those presented in the literature can be found in Table 17 varying different

parameters for each example given. The results are given as relative displacements, that is

�� =𝑤

𝐿.

Table 16: Geometric and material properties of the sandwich beam with sinusoidal corrugated core.


Length 𝐿 = 𝑣𝑎𝑟𝑦𝑖𝑛𝑔 𝐸 = 7.2 𝐺𝑃𝑎

Width 𝐷 = 90 𝑚𝑚

𝑡𝑓 = 1.0 𝑚𝑚 𝑡𝑐 = 10 𝑚𝑚 𝑣 = 0.3

𝑡0 = 0.6 𝑚𝑚 𝑏0 = 𝑣𝑎𝑟𝑦𝑖𝑛𝑔

Further examples were conducted for the same beam with different boundary conditions

applied. In Appendix A, from Table A. 1 to Table A. 3 are presented the results for a static

punctual force applied in the middle of the beam and in Table A. 4 to Table A. 6 are presented

the displacements for a constant pressure of 𝑝 = 10 𝑁/𝑚𝑚2 applied on the top of the beam

along its length. In both cases the vertical displacements were obtained on a point located at

the middle span of the beam, with coordinates (𝐿 2⁄ , 𝐷).

Figure 32: Scheme of the three point bending for the sandwich beam with corrugated core [66].


87

The principal stresses, both for the face sheets and for the corrugated core, measured at the

points described in Figure A. 1 are represented from Table A. 7 to Table A. 30, all in

Appendix A.

Table 17: Maximum relative deflection, ��𝑚𝑎𝑥, for the sandwich beam under three point bending.

b0 \mm Theory 𝜆 = 5 𝜆 = 10 𝜆 = 20 𝜆 = 50 𝜆 = 100

10 “Euler-Bernoulli” 0.0191 0.0763 0.3052 1.9076 7.6302

“Broken Line” 0.0349 0.0924 0.3214 1.9238 7.6465

Present FEM 0.0716 0.1232 0.3439 1.9330 7.6330

Present RPIM 0.0749 0.1362 0.3661 1.9727 7.7881

Present NNRPIM 0.0487 0.1036 0.3228 - -

15 “Euler-Bernoulli” 0.0191 0.0763 0.3052 1.9075 7.6299

“Broken Line” 0.0455 0.1033 0.3324 1.9348 7.6573

Present FEM 0.0669 0.1209 0.3416 1.9292 7.6254

Present RPIM 0.0682 0.1258 0.3536 1.9808 7.8340

Present NNRPIM 0.0547 0.1079 0.3216 - -

20 “Euler-Bernoulli” 0.0191 0.0763 0.3052 1.9074 7.6297

“Broken Line” 0.0587 0.1168 0.3462 1.9487 7.6710

Present FEM 0.0770 0.1268 0.3468 1.9289 7.6252

Present RPIM 0.0783 0.1270 0.3524 1.9524 7.7717

Present NNRPIM 0.0751 0.1220 0.3401 - -

30 “Euler-Bernoulli” 0.0191 0.0763 0.3052 1.9073 7.6294

“Broken Line” 0.0934 0.1530 0.3831 1.9860 7.7083

Present FEM 0.0984 0.1487 0.3669 1.9376 7.6242

Present RPIM 0.0915 0.1463 0.3646 1.9587 7.7339

Present NNRPIM 0.1171 0.1669 0.3972 - -

As should be noted, the increase of the beam length results in the need to use more nodes in

order to attain accurate values, which brings substantial computational costs.

The same conclusions can be taken when varying the pitch length of the sinusoidal core.

Since the smaller the pitch is the more nodes are needed to correctly represent the sinusoidal

geometry of the corrugated core.

In general, both the FEM and meshless methods are in good agreement with the theoretical

results for thin beams. On the other hand, more specifically, the results for 𝜆 = 5 are very

different. This may be due to the fact that the “Euler-Bernoulli” and “Broken Line” theories

present shear locking, that increases when the beam becomes thinner. Thus, for such small

length-to-thickness ratio these theories are not very reliable.

On the overall, the NNRPIM seems to present the most accurate results for the measured

cases.


88

Figure 33: Sandwich beam with three layers of sinusoidal corrugated core.

More than one corrugated core can be considered on the analysis of sandwich beams, as can

be seen in Figure 33. Studies were performed under the previous conditions for sandwich

beams with two sinusoidal cores, using the same thickness for the face sheets and sinusoidal

core (𝑡𝑓 = 1.0 𝑚𝑚 and 𝑡𝑐 = 1.0 𝑚𝑚) and keeping the same thickness (𝑡 = 12 𝑚𝑚) for the

beam. Thus, the amplitude for the core was adjusted accordingly.

From Table 18 to Table 21 are summarized the relative deflections for a punctual and

distributed load. In Appendix A, from Table A. 31 to Table A. 46 are described the stresses

on the points of Figure A. 2.

Table 18: Relative deflection, �� , for the corrugated core beam with 2 layers under punctual load (FEM).

b0 \mm Boundary 𝜆 = 5 𝜆 = 10 𝜆 = 20 𝜆 = 50 𝜆 = 100

10 C-F 0.0648 0.1743 0.6243 3.8529 15.113

C-C 0.0205 0.0321 0.0873 0.5196 1.9629

C-H 0.0253 0.0476 0.1454 0.8801 3.3833

H-H 0.0343 0.0885 0.3140 1.9378 7.6074

15 C-F 0.0775 0.1839 0.6307 3.7731 15.050

C-C 0.0295 0.0384 0.0917 0.4832 1.9348

C-H 0.0345 0.0542 0.1499 0.8378 3.3506

H-H 0.0429 0.0943 0.3174 1.8977 7.5754

20 C-F 0.0976 0.1992 0.6418 3.7673 15.108

C-C 0.0413 0.0485 0.0987 0.4862 1.9764

C-H 0.0469 0.0647 0.1547 0.8399 3.3952

H-H 0.0547 0.1035 0.3233 1.8949 7.6041

30 C-F 0.1495 0.2537 0.6848 3.8005 14.902

C-C 0.0744 0.0846 0.1248 0.5051 1.8905

C-H 0.0842 0.1023 0.1851 0.8604 3.2933

H-H 0.0926 0.1382 0.3474 1.9113 7.4997


89

Table 19: Relative deflection, �� , for the corrugated core beam with 2 layers under punctual load (RPIM).


10 C-F 0.0557 0.1620 0.6001 3.7068 14.179

C-C 0.0167 0.0286 0.0827 0.4704 1.7813

C-H 0.0212 0.0438 0.1397 0.8233 3.1269

H-H 0.0307 0.0851 0.3080 1.8912 7.2059

15 C-F 0.0619 0.1658 0.5990 3.6589 14.142

C-C 0.0211 0.0315 0.0843 0.4656 1.7753

C-H 0.0257 0.0467 0.1409 0.8134 3.1167

H-H 0.0346 0.0872 0.3072 1.8642 7.1471

20 C-F 0.0743 0.1748 0.6036 3.6179 13.901

C-C 0.0286 0.0374 00884 0.4628 1.7470

C-H 0.0334 0.0527 0.1450 0.8065 3.0656

H-H 0.0415 0.0921 0.3094 1.8427 7.0368

30 C-F 0.1002 0.2051 0.6254 3.5600 13.519

C-C 0.0454 0.0564 0.1030 0.4641 1.7060

C-H 0.0512 0.0727 0.1603 0.8016 2.9873

H-H 0.0586 0.1093 0.3205 1.8126 6.8629

Table 20: Relative deflection, �� , for the corrugated core beam with 2 layers under distributed load (FEM).


10 C-F 3.47E-03 2.11E-02 1.57E-01 2.44E+00 1.92E+01

C-C 5.31E-04 1.81E-03 1.03E-02 1.55E-01 1.17E+00

C-H 7.20E-04 3.05E-03 1.96E-02 2.99E-01 2.31E+00

H-H 1.08E-03 6.31E-03 4.66E-02 7.22E-01 5.69E+00

15 C-F 3.98E-03 2.18E-02 1.58E-01 2.40E+00 1.91E+01

C-C 8.02E-04 2.09E-03 1.07E-02 1.44E-01 1.16E+00

C-H 1.00E-03 3.34E-03 2.00E-02 2.86E-01 2.29E+00

H-H 1.33E-03 6.53E-03 4.68E-02 7.10E-01 5.67E+00

20 C-F 4.90E-03 2.30E-02 1.60E-01 2.39E+00 1.92E+01

C-C 1.24E-03 2.60E-03 1.14E-02 1.45E-01 1.17E+00

C-H 1.45E-03 3.89E-03 2.07E-02 2.87E-01 2.32E+00

H-H 1.74E-03 6.96E-03 4.72E-02 7.08E-01 5.68E+00

30 C-F 6.96E-03 2.73E-02 1.66E-01 2.41E+00 1.90E+01

C-C 2.22E-03 4.70E-03 1.38E-02 1.50E-01 1.13E+00

C-H 2.62E-03 6.12E-03 2.34E-02 2.92E-01 2.25E+00

H-H 2.95E-03 8.94E-03 4.92E-02 7.12E-01 5.62E+00


90

Table 21: Relative deflection, �� , for the corrugated core beam with 2 layers under distributed load (RPIM).


10 C-F 3.04E-03 1.98E-02 1.52E-01 2.6E+00 1.90E+01

C-C 4.25E-04 1.61E-03 9.77E-03 1.40E-01 1.12E+00

C-H 6.06E-04 2.82E-03 1.89E-02 2.81E-01 2.25E+00

H-H 9.78E-04 6.11E-03 4.58E-02 7.08E-01 5.68E+00

15 C-F 3.28E-03 2.01E-02 1.51E-01 2.33E+00 1.89E+01

C-C 5.54E-04 1.74E-03 9.90E-03 1.39E-01 1.11E+00

C-H 7.34E-04 2.94E-03 1.89E-02 2.78E-01 2.23E+00

H-H 1.08E-03 6.15E-03 4.55E-02 6.98E-01 5.61E+00

20 C-F 3.89E-03 2.07E-02 1.52E-01 2.30E+00 1.88E+01

C-C 8.51E-04 2.02E-03 1.03E-02 1.38E-01 1.11E+00

C-H 1.03E-03 3.24E-03 1.93E-02 2.75E-01 2.23E+00

H-H 1.33E-03 6.34E-03 4.55E-02 6.90E-01 5.60E+00

30 C-F 4.83E-03 2.31E-02 1.55E-01 2.26E+00 1.81E+01

C-C 1.23E-03 3.05E-03 1.17E-02 1.38E-01 1.07E+00

C-H 1.45E-03 4.32E-03 2.08E-02 2.73E-01 2.15E+00

H-H 1.72E-03 7.17E-03 4.62E-02 6.77E-01 5.40E+00

Comparing the results of the beam with one sinusoidal core, both three formulations show

overall good agreement, which is to be expected since they both performed well when

compared with the documented results for the three-point bending studies, repreented in Table

17.

For the beam with two sinusoidal cores, the results for the FEM and RPIM show good

agreement with each other, showing only a few punctual cases where the error between the

two increases a little bit more than the medium error.


91

8.5.2 Free Vibrations Analysis

A free vibration study was conducted on the same beam, only this time aluminium was

considered as the material, with properties: 𝐸 = 30 𝐺𝑃𝑎, 𝑣 = 0.32 and 𝜌 = 3,690 𝑘𝑔/𝑚3.

The first natural frequencies for the beam with one sinusoidal core are displayed from Table

22 to Table 24. For a beam with one core, the second and third natural frequencies can be

seen in Appendix B, from Table B. 1 to Table B. 6. For a beam with two sinusoidal cores, the

natural frequencies are displayed in Table B. 10 to Table B. 15.

Table 22: Fundamental natural frequency (Hz) for the beam with one sinusoidal core (FEM).


10 C-F 1517.8 466.56 126.13 20.496 5.2010

C-C 4470.4 2010.8 690.46 120.10 32.471

C-H 3945.1 1633.7 512.76 86.578 22.660

H-H 1325.2 1258.1 351.30 57.575 14.651

F-F 7012.7 2561.9 761.35 126.25 32.699

15 C-F 1545.6 471.48 126.42 20.676 5.2141

C-C 4479.3 2099.0 699.57 130.28 33.215

C-H 3976.5 168..1 517.20 90.797 22.953

H-H 1281.9 1279.3 352.67 58.640 14.724

F-F 6170.7 259.5.6 764.54 129.04 32.893

20 C-F 1465.7 468.09 126.20 20.693 5.2202

C-C 3662.4 2007.4 688.40 130.06 33.229

C-H 3388.1 1636.9 511.99 90.758 22.970

H-H 1238.8 1258.8 351.30 58.677 14.739

F-F 3957.0 2491.5 759.42 129.09 32.924

30 C-F 1255.9 454.14 124.96 20.687 5.2203

C-C 2327.6 1650.2 635.17 129.39 33.191

C-H 2081.7 1424.4 486.64 90.509 22.958

H-H 1193.4 1169.2 344.11 58.629 14.739

F-F 2228.7 1959.7 731.68 128.96 32.924


92

Table 23: Fundamental natural frequency (Hz) for the beam with one sinusoidal core (RPIM).


10 C-F 1326.8 443.39 124.34 20.650 5.1541

C-C 3669.0 1612.3 614.37 123.03 32.719

C-H 3550.2 1373.1 474.87 87.972 22.672

H-H 3073.9 1142.3 340.36 58.069 14.572

F-F 5641.9 2287.5 729.19 127.45 32.345

15 C-F 1387.2 453.90 125.38 20.552 5.1723

C-C 3717.9 1715.6 642.70 124.29 32.651

C-H 3406.2 1446.3 489.96 88.264 22.688

H-H 2209.0 1176.7 345.23 57.899 14.615

F-F 5081.9 2350.0 741.67 127.20 32.426

20 C-F 1384.0 455.89 125.62 20.627 5.1534

C-C 3316.2 1716.0 648.90 127.47 32.624

C-H 3121.8 1451.7 493.47 89.609 22.636

H-H 2114.9 1180.1 346.50 58.276 14.564

F-F 3674.0 2275.0 742.62 128.16 32.316

30 C-F 1252.2 446.22 125.03 20.633 5.1964

C-C 2374.3 1490.2 623.47 126.47 32.985

C-H 2191.9 1311.4 481.41 89.272 22.860

H-H 1793.5 1110.9 343.28 58.256 14.695

F-F 2356.4 1762.2 723.28 127.99 32.608


93

Table 24: First natural frequency (Hz) for the beam with one sinusoidal core (NNRPIM).

b0 \mm Boundary 𝜆 = 5 𝜆 = 10 𝜆 = 20

10 C-F 1432.0 434.34 116.30

C-C 4393.5 1851.3 636.71

C-H 3878.4 1505.1 472.44

H-H 2351.2 1158.0 323.54

F-F 6640.3 2423.0 708.10

15 C-F 1451.5 455.10 123.34

C-C 4070.5 1834.2 657.07

C-H 3670.8 1516.9 493.25

H-H 2106.8 1195.7 341.95

F-F 5481.9 2455.7 745.20

20 C-F 1382.3 452.39 124.29

C-C 3339.3 1697.5 640.27

C-H 3105.4 1430.4 486.79

H-H 1933.8 1157.3 342.11

F-F 3503.1 2276.0 741.81

30 C-F 1097.1 422.06 121.95

C-C 2013.8 1273.0 555.49

C-H 1885.2 1132.0 440.39

H-H 1610.9 992.28 325.53

F-F 1906.9 1575.5 688.52


94

In Figure 34 are represented the first vibration modes for different boundary conditions for the

corrugated core beam with two layers.

Figure 34: First vibrations modes for the corrugated beam under different boundary conditions.

Again, for the beam with one sinusoidal core, the natural frequencies obtained for the three

methods are in very good agreement.

On the other hand, for the beam with two sinusoidal cores, even though the first natural

frequencies show overall good agreement, for higher frequencies the differences between the

FEM and RPIM start to increase. The difference in these values may be a result from not

using the most correct nodal mesh for this domain. This happened due to hardware

limitations, since free vibrations analysis require more computational cost than a static

analysis due to the need to solve the eigenvalue problem, equation (69), thus fewer nodes had

to be used on this simulations.


95

8.5.3 Forced Vibrations Analysis

Considering as material properties 𝐸 = 30 𝐺𝑃𝑎, 𝑣 = 0.32 and 𝜌 = 3,690 𝑡𝑜𝑛𝑛/𝑚3, a forced

vibration analysis was performed, with a distributed harmonic load, 𝑓(𝑡) = 5 · sin(80 · 𝑡) +5, applied on the upper side of the beam. Modal superposition was used to solve the

equilibrium equations, with a truncated analysis considering only the first five natural

vibration modes.

The maximum relative vertical displacements for a point with coordinates (𝐿 2⁄ , 𝐷) are

displayed in Table 25 to Table 27, for a beam of length 𝐿 = 120 𝑚𝑚, varying the pitch

distance and number of sinusoidal layers. In Appendix B, from Table B. 7 to Table B. 9 are

shown the maximum stresses for the same point for a beam with one sinusoidal core and in

Table B. 16 to Table B. 18 the maximum stress for a beam with two sinusoidal cores. To be

noted that for the forced vibrations analysis the thickness of the beam was kept (𝑡 = 12 𝑚𝑚)

and the other thickness were adjust proportionally.

Figure 35 is a temporal representation of the vertical displacement of the same point,

considering a beam with one layer and a harmonic force applied only during 0.14 seconds, for

a pitch length of 𝑏0 = 15 𝑚𝑚 and both edges clamped. From Figure B. 1 to Figure B. 7 are

represented the vertical displacement for the same point varying the boundary conditions and

number of layers.

Table 25: Relative deflection, �� , for the corrugated core beam under a harmonic distributed load (FEM).

Layers Boundary 𝑏0 = 10 𝑏0 = 15 𝑏0 = 20 𝑏0 = 30

1 C-F 1.91E-02 1.82E-02 1.88E-02 2.14E-02

C-C 1.16E-03 1.04E-03 1.12E-03 1.39E-03

C-H 1.65E-03 1.52E-03 1.63E-03 2.02E-03

H-H 2.93E-03 2.75E-03 2.90E-03 3.35E-03

2 C-F 3.02E-02 2.95E-02 3.08E-02 3.28E-02

C-C 1.59E-03 1.38E-03 1.61E-03 2.42E-03

C-H 2.60E-03 2.29E-03 2.60E-03 3.60E-03

H-H 4.86E-03 4.62E-03 5.04E-03 5.98E-03


96

Table 26: Relative deflection, �� , for the corrugated core beam under a harmonic distributed load (RPIM).


1 C-F 1.87E-02 1.88E-02 2.02E-02 2.48E-02

C-C 9.96E-04 1.07E-03 1.23E-03 1.89E-03

C-H 1.55E-03 1.62E-03 1.88E-03 2.65E-03

H-H 2.82E-03 2.86E-03 3.10E-03 3.87E-03

2 C-F 2.27E-02 2.51E-02 2.73E-02 3.09E-02

C-C 1.03E-03 1.27E-03 1.79E-03 3.95E-03

C-H 1.76E-03 2.09E-03 2.78E-03 5.44E-03

H-H 4.20E-03 4.54E-03 5.20E-03 7.63E-03

Table 27: Relative deflection, �� , for the corrugated core beam under a harmonic distributed load (NNRPIM).


1 C-F 1.89E-02 1.88E-02 2.08E-02 2.65E-02

C-C 1.06E-03 1.19E-03 1.37E-03 2.26E-03

C-H 1.62E-03 1.74E-03 2.01E-03 3.11E-03

H-H 2.84E-03 2.85E-03 3.21E-03 4.28E-03

2 C-F 2.21E-02 2.41E-02 2.62E-02 3.24E-02

C-C 8.94E-04 1.18E-03 1.74E-03 4.24E-03

C-H 1.60E-03 1.89E-03 2.48E-03 5.53E-03

H-H 3.48E-03 3.78E-03 4.42E-03 7.17E-03

Figure 35: Vertical deflection for the corrugated core beam with one layer and C-C boundary conditions.


97

Comparing both methods, we can verify that both the RPIM and NNRPIM display very

similar results with each other. On the other hand, the results obtained from the FEM, despite

having the same behavior, displays a different frequency of vibration, and thus the vertical

displacement does not accompany the other two methods.

This may be due to the different way the FEM and Meshless methods assure the nodal

connectivity. Since the meshless methods rely on the overlapping of the “influence-domains”,

this results in more strength bond between the corrugated core and the face sheets, thus

increasing the stiffness of the structure.

For the beam with two layers, the maximum relative displacements obtained with the finite

element method are generally higher than the results from the meshless. Despite this,

analysing the figures showing the displacements along the time domain shows they have very

similar frequencies. Thus, the difference in the maximum relative displacements may be due

to the FEM not having full converged.


98

8.6 Forced Vibrations of a Sandwich Plate with Corrugated Core

A forced vibrations study was conducted for a structure with the same geometric and material

properties of the beam used on example 8.5.3. But this time, a plain strain analysis was

considered to simulate a sandwich plate.

The same distributed harmonic load, 𝑓(𝑡) = 5 · sin(80 · 𝑡) + 5, was applied on the upper side

of the plate and a modal truncated analysis considering only the first five natural vibration

modes was adopted.

In Table 28 to Table 30 are presented the maximum relative vertical displacements for a point

with coordinates (𝐿 2⁄ , 𝐷). In Appendix C, from Table C. 1 to Table C. 6 are shown the

maximum stresses for the same point for a plate with one and two sinusoidal cores. In Figure

C. 1 to Figure C. 8 are shown temporal representation of the vertical displacement of the plate

on the same point for the various cases and both formulations applied.

Table 28: Relative deflection, �� , for the corrugated core plate under a harmonic distributed load (FEM).


1 C-F 1.46E+00 1.36E+00 1.44E+00 1.64E+00

C-C 9.56E-02 8.28E-02 9.12E-02 1.11E-01

C-H 1.39E-01 1.28E-01 1.33E-01 1.69E-01

H-H 2.32E-01 2.17E-01 2.29E-01 2.61E-01

2 C-F 2.55E+00 2.47E+00 2.58E+00 2.81E+00

C-C 1.22E-01 1.09E-01 1.38E-01 1.92E-01

C-H 1.99E-01 1.81E-01 2.11E-01 2.90E-01

H-H 3.84E-01 3.66E-01 4.00E-01 4.75E-01

Table 29: Relative deflection, �� , for the corrugated core plate under a harmonic distributed load (RPIM).


1 C-F 1.65E+00 1.64E+00 1.73E+00 2.19E+00

C-C 9.06E-02 9.65E-02 1.14E-01 1.76E-01

C-H 1.30E-01 1.38E-01 1.62E-01 2.34E-01

H-H 2.45E-01 2.49E-01 2.69E-01 3.42E-01

2 C-F 2.22E+00 2.45E+00 2.68E+00 3.15E+00

C-C 9.01E-02 1.16E-01 1.59E-01 3.50E-01

C-H 1.65E-01 1.88E-01 2.41E-01 4.79E-01

H-H 3.72E-01 4.02E-01 4.59E-01 6.78E-01


99

Table 30: Relative deflection, �� , for the corrugated core plate under a harmonic distributed load (NNRPIM).


1 C-F 1.50E+00 1.45E+00 1.60E+00 2.12E+00

C-C 8.38E-02 9.14E-02 1.12E-01 1.78E-01

C-H 1.22E-01 1.30E-01 1.59E-01 2.44E-01

H-H 2.21E-01 2.23E-01 2.48E-01 3.39E-01

2 C-F 1.87E+00 2.03E+00 2.20E+00 2.86E+00

C-C 7.39E-02 9.24E-02 1.36E-01 3.38E-01

C-H 1.25E-01 1.53E-01 1.97E-01 4.42E-01

H-H 2.77E-01 3.01E-01 3.52E-01 5.74E-01

For the plate with one corrugated core, the RPIM and NNRPIM formulations show very

similar displacements. For some boundary conditions, such as C-C an C-F, analyzing the

variation of the vertical displacement along the time domain one can see very good agreement

between the three formulations. In general, both meshless methods can predict the behavior of

a structure under plain strain with good accuracy.

For the plate with two corrugated cores, despite the two layers increasing the number of nodes

and the complexity of the geometry, the NNRPIM and the FEM seem to be in very good

agreement, with the RPIM showing, in some cases, a larger difference from the other two

methods.


100

8.7 Free and Forced Vibrations of a Laminated Plate

For the laminated plate of Figure 36, clamped on two edges, a free and forced vibrations study

was conducted. The plate is constituted by two carbon/epoxy T800/3900 laminas (0/45) [67],

with geometry and material properties described in Table 31,

The plate is excited by a harmonic force on the z axis applied in the middle, point

𝐴(𝐿 2⁄ , 𝐷 2⁄ ,𝐻), where 𝐻 = 5 𝑚𝑚 is the thickness of the plate. The force is represented in

Figure 37, and is given by the expression:

𝑓𝑧(𝑡) = sin(800𝜋 ∙ 𝑡) + sin(1100𝜋 ∙ 𝑡) + sin (1320𝜋 ∙ 𝑡) (188)

Table 31: Geometric and material properties of the laminated plate.


Side Length 𝐿 = 100 𝑚𝑚 𝐸1 = 142 𝐺𝑃𝑎 𝐸2 = 𝐸3 = 7.79 𝐺𝑃𝑎 𝜌 = 1550 𝑘𝑔 ∙ 𝑚−3

Thickness of

each lamina 𝑡𝑙 = 2.5 𝑚𝑚 𝐺12 = 4.00 𝐺𝑃𝑎 𝐺13 = 4.00 𝐺𝑃𝑎 𝐺23 = 2.55 𝐺𝑃𝑎

𝜈12 = 0.34 𝜈13 = 0.34 𝜈23 = 0.59

Figure 36: Scheme of the laminated plate, with two sides clamped and two sides free.

Here, a 3D analysis was performed. Thus, two 3D mesh discretization were used on the

analysis, one discretized in 25x25x5 nodes, used for both formulations, FEM, RPIM and

NNRPIM (mesh-1) and another discretized in 41x41x5 nodes, used on the FEM and RPIM

(mesh-2). A modal superposition with a truncated analysis was performed, where the first 100

vibration modes were kept. In Table 32 are represented the ten first natural vibration

frequencies for the plate for the first mesh and in Table 33 for the second mesh.

Table 32: First 10 natural frequencies /Hz for the laminated plate (mesh-1).

FEM RPIM NNRPIM

3548.96 3445.49 3122.88

3780.60 3548.99 3211.23

4760.37 4096.67 3664.64

7037.08 5657.79 4999.85

7852.92 7811.74 7502.73

8779.94 8439.19 7818.70

9074.66 8464.85 7851.40

10145.50 8568.16 7955.15

10787.73 9065.71 8342.72

12254.52 10220.5 9279.29


101

Table 33: First 10 natural frequencies /Hz for the laminated plate (mesh-2).

FEM RPIM

3426.93 3271.79

3608.13 3368.90

4420.20 3872.34

6395.61 5310.29

7815.13 7795.87

8480.06 7916.04

8704.14 8068.46

9543.47 8188.97

9724.92 8647.72

11237.94 9709.58

Even though there are no clear evidences that the values have already converged towards the

exact results, from comparing the frequencies obtained from the different meshes it can be

seen that the RPIM methods converge faster than the FEM. In fact, with the increase in the

number of nodes, the results produced by the FEM got more close to those given by the RPIM

with the less denser mesh, with the higher vibrations modes showing to converge slower.

Figure 37: Representation of the harmonic solicitation.

The forced vibrations analysis was run using the first mesh for the NNRPIM second mesh for

the FEM and RPIM. The displacements along the z axis were measured at two points of the

plate, point 𝐴(𝐿 2⁄ , 𝐷 2⁄ ,𝐻) and 𝐵(𝐿 2⁄ , 0, 𝐻) and their temporal response is represented in

Figure 38 and Figure 39. The stresses were measured at those points and also on points

𝐴1(𝐿2⁄ , 𝐷 2⁄ , 0) and 𝐵1(

𝐿2⁄ , 0,0), the results are described in Table 34 to Table 37, showing

the maximum positives stresses (traction) and the maximum negative stresses (compression).


102

Figure 38: Vertical displacement /m, for the laminated plate on point A.

Figure 39: Vertical displacement /m, for the laminated plate on point B.

Table 34: Stresses /MPa for the laminated plate, point A.

𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑧𝑧 𝜏𝑥𝑦 𝜏𝑥𝑧 𝜏𝑦𝑧

Traction

FEM 2.87E-01 1.19E-01 1.49E-01 3.38E-03 6.33E-02 4.04E-02

RPIM 5.41E-01 2.65E-01 3.60E-01 2.16E-03 9.29E-02 5.90E-02

NNRPIM 1.18E-01 1.47E-03 2.60E-03 2.16E-03 3.74E-03 4.31E-03

Compression

FEM -2.84E-01 -1.19E-01 -1.49E-01 -3.46E-03 -6.32E-02 -4.04E-02

RPIM -5.28E-01 -2.64E-01 -3.59E-01 -2.15E-03 -9.27E-02 -5.89E-02

NNRPIM -1.37E-01 -1.34E-03 -2.30E-03 -2.38E-03 -4.40E-03 -4.93E-03

-0,001

-0,0008

-0,0006

-0,0004

-0,0002

0

0,0002

0,0004

0,0006

0,0008

0,001

0 0,005 0,01 0,015 0,02

Verti

cal

Dis

pla

cem

en

t \

mm

Time \s

FEM

RPIM

NNRPIM

-0,0002

-0,00015

-0,0001

-0,00005

0

0,00005

0,0001

0,00015

0,0002

0 0,001 0,002 0,003 0,004 0,005

Verti

cal

Dis

pla

cem

en

t \

mm

Time \s

FEM

RPIM

NNRPIM


103

Table 35: Stresses /MPa for the laminated plate, point A1.


Traction

FEM 2.04E-01 1.03E-01 7.68E-02 2.28E-03 4.32E-02 3.55E-02

RPIM 2.06E-01 8.29E-02 1.58E-02 6.72E-04 3.97E-02 3.05E-02

NNRPIM 1.55E-01 1.46E-01 1.38E-01 2.02E-03 1.06E-02 7.90E-03

Compression

FEM -2.07E-01 -1.03E-01 -7.63E-02 -2.28E-03 -4.33E-02 -3.55E-02

RPIM -2.13E-01 -8.37E-02 -1.58E-02 -6.77E-04 -4.00E-02 -3.06E-02

NNRPIM -1.33E-01 -1.25E-01 -1.60E-01 -1.76E-03 -9.15E-03 -7.02E-03

Table 36: Stresses /MPa for the laminated plate, point B.


Traction

FEM 5.39E-03 6.68E-04 3.34E-04 3.55E-04 2.45E-03 1.90E-04

RPIM 2.19E-02 6.31E-04 2.07E-04 2.83E-04 6.99E-04 1.79E-04

NNRPIM 9.2E-02 2.25E-02 3.81E-02 1.35E-03 1.08E-02 6.51E-03

Compression

FEM -5.09E-03 -6.88E-04 -3.48E-04 -3.73E-04 -2.61E-03 -1.82E-04

RPIM -1.96E-02 -6.56E-04 -2.04E-04 -3.18E-04 -7.81E-04 -1.59E-04

NNRPIM -1.05E-01 -2.55E-02 -4.32E-02 -1.48E-03 -1.21E-02 -6.01E-03

Table 37: Stresses /MPa for the laminated plate, point B1.


Traction

FEM 1.81E-02 2.78E-03 3.45E-03 2.64E-04 7.05E-04 1.87E-04

RPIM 1.68E-02 1.14E-03 4.56E-04 1.59E-04 5.16E-04 1.47E-04

NNRPIM 2.43E-02 5.39E-02 6.60E-02 1.78E-03 1.22E-02 3.65E-03

Compression

FEM -1.94E-02 -2.81E-03 -3.18E-03 -2.98E-04 -7.14E-04 -1.85E-04

RPIM -1.86E-02 -1.12E-03 -4.55E-04 -1.54E-04 -5.81E-04 -1.33E-04

NNRPIM -2.65E-02 -6.06E-02 -5.86E-02 -1.69E-03 -1.07E-02 -3.13E-03


104

The differences for both the displacements and the stress values obtained with the FEM,

RPIM and NNRPIM can be explained with the low density of the discretization mesh. As

already pointed out, since no convergence test was performed on this simulation, there is no

evidence that any of the methods has converged to the exact solution. However, as the

literature and the results show, meshless methods converge faster than the FEM, specially the

tetrahedral element which was the one used for 3D FEM.

The fact that not the best mesh was applied is also due to hardware limitation, since the

maximum number of nodes that the computer can simulate is about 8000 nodes for 3D

analysis, hindering further analysis using denser nodal discretizations.


105

8.8 Free Vibrations of an Acoustic Tube

Considering a one-dimensional tube of length 𝐿 = 1 𝑚, filled with air ( 𝜌 = 1.225 𝑘𝑔/𝑚3

and 𝑐 = 340 𝑚/𝑠 ). The exact natural frequencies of the acoustic tube, assuming both ends

are perfectly rigid, are given by the following expression [68]:

𝑓𝑚 =(𝑚 − 1) · 𝑐

2 · 𝐿 /𝐻𝑧 (189)

The same example was computed using the created software, and in Table 38 are represented

the first 25 natural frequencies for different formulations using a nodal discretization of 100

linear elements.

Table 38: First 10 natural frequencies /Hz for the acoustic tube.

Mode nr. FEM RPIM NNRPIM Exact

1 0.000 0.000 0.000 0

2 170.007 170.000 170.000 170

3 340.056 340.003 339.958 340

4 510.189 510.009 509.837 510

5 680.448 680.021 679.597 680

6 850.874 850.041 849.198 850

7 1021.512 1020.070 1018.603 1020

8 1192.399 1190.110 1187.771 1190

9 1363.582 1360.161 1356.667 1360

10 1535.101 1530.226 1525.253 1530

11 1706.999 1700.303 1693.494 1700

12 1879.318 1870.395 1861.354 1870

13 2052.101 2040.501 2028.799 2040

14 2225.389 2210.621 2195.798 2210

15 2399.227 2380.756 2362.318 2380

16 2573.655 2550.905 2528.33 2550

17 2748.718 2721.069 2693.804 2720

18 2924.458 2891.248 2858.713 2890

19 3100.917 3061.443 3023.032 3060

20 3278.140 3231.654 3186.735 3230

21 3456.168 3401.883 3349.801 3400

22 3635.045 3572.132 3512.207 3570

23 3814.814 3742.403 3673.935 3740

24 3995.518 3912.701 3834.966 3910

25 4177.199 4083.030 3995.284 4080

As can be seen, both formulations calculate with good accuracy the natural frequencies for the

first few modes. Despite this, with the increase in the natural frequencies, the relative errors

also increase. For the higher modes the RPIM still attains very accurate results, showing that

this methodology does not suffer much dispersion error. The same can’t be said for the FEM

and NNRPIM based on this results.


106

8.9 Free Vibrations of the Interior of a 2D Car

To verify the performance of both meshless methods in simulating more complex problems, a

2D geometry that resembles the interior of a car - Figure 40 - was use for this study.

Considering that the acoustic domain is fully bounded and that the fluid inside is air, with 𝜌 =1.25 𝑘𝑔/𝑚3 and 𝑐 = 340 𝑚𝑠−1, a free vibrations analysis was performed using both

methods, with a mesh discretization using 6859 nodes. The results for the natural frequencies

were compared with those presented on [69] and are displayed in Table 39.

Figure 40: Geometry of the 2D interior acoustic car [69].

Table 39: First 8 natural frequencies /Hz for the 2D Car.

Literature FEM RPIM NNRPIM

0 0.000 0.000 0.000

86 86.212 86.194 86.504

153 153.067 153.148 153.972

164 163.830 163.959 165.842

208 207.424 207.539 208.258

240 240.234 240.435 240.693

273 273.016 273.305 274.494

297 296.651 296.468 296.319

As can be seen again, even for more complex geometries both meshless methods perform

with very good accuracy, being able to simulate the dynamic behavior of the fluid.


107

8.10 Free Vibrations of a Coupled Fluid-Structure Cavity

An analysis of the free vibrations of a linear elastic structure coupled with an interior inviscid

compressible fluid was performed for a rectangular steel cavity. Two analysis were performed

using two different fluids: air, which represents a weakly couple structure, and water which

represents a strong coupled structure. The physical properties for the structure and fluid are

found in Table 40 and the geometrical properties of the system are displayed in Figure 41.

Table 40: Physical properties of the structure and fluid of the coupled vibro-acoustic system.

Steel Air Water

𝐸 = 1.44 · 1011 𝐺𝑃𝑎 𝜌 = 1 𝑘𝑔/𝑚3 𝜌 = 1000 𝑘𝑔/𝑚3

𝑣 = 0.35 𝑐 = 340 𝑚𝑠−1 𝑐 = 1430 𝑚𝑠−1

𝜌 = 7700 𝑘𝑔/𝑚3

Figure 41: Geometric properties of the coupled system [70].

As should be expected, for the weak coupled system (cavity filled with air), the natural

vibration modes should be very similar to those of an uncoupled structure with the same

properties. On the other hand, increasing the coupling of the system, such as using a more

heavy fluid (like water for example) should translate in a change of the eigenvalues of the

problem compared with those of the uncoupled one.

For this analysis the first ten eigenvalues were calculated for the coupled structure using,

again, three different formulations: FEM, RPIM and NNRPIM. To show the validity of the

created software the results were compared with those presented on the literature [70] and are

shown in Table 41 and Table 42. Since the reference used to compare the calculated

frequencies shows two different results obtained from different formulations, both were used

for comparison purposes, named “Ref. 1” and “Ref. 2”.


108

Table 41: Ten first eigenvalues /𝑟𝑎𝑑𝑠−1 of the coupled fluid-structure cavity filled with air.

Mode Air

“Ref. 1” “Ref. 2” FEM RPIM NNRPIM

1 664.121 676.926 669.891 673.691 672.068

2 1068.129 1068.562 1068.248 1068.215 1068.216

3 1068.152 1068.607 1068.266 1068.266 1068.220

4 1510.589 1511.191 1510.734 1510.640 1510.544

5 2136.102 2139.448 2137.061 2136.975 2136.893

6 2136.240 2139.707 2137.137 2137.056 2136.916

7 2258.686 2304.012 2290.450 2293.244 2290.413

8 2388.418 2391.688 2389.250 2388.982 2388.671

9 2388.539 2391.734 2389.348 2389.020 2388.682

10 - 3026.000 3022.365 3021.717 3020.904

Table 42: Ten first eigenvalues /𝑟𝑎𝑑𝑠−1 of the coupled fluid-structure cavity filled with water.

Mode Water

“Ref. 1” “Ref. 2” FEM RPIM NNRPIM

1 641.837 654.159 657.103 648.474 666.948

2 2116.398 2159.301 2253.299 2265.401 2265.412

3 3201.475 3445.498 3433.365 3416.174 3531.527

4 3804.124 3907.321 3792.468 3855.430 3878.114

5 4211.620 4221.192 4345.587 4394.918 4446.555

6 4687.927 4710.677 4558.097 4493.534 4485.225

7 5155.246 5168.735 4898.962 4978.288 4828.081

8 5385.805 5454.176 5426.502 5435.455 5485.183

9 6239.332 6280.978 6251.436 6184.674 6193.533

10 - 7597.43 7575.224 7558.473 7613.561

In both analyses, a regular discretization mesh with 5508 nodes for the structure and 4225

nodes for the fluid was adopted. As a result of that, some eigenvalues may have not fully

converged to the reference value on the literature, which results in a larger error.

In general, the results for weak coupled system are in very good agreement with those

presented in both references. For the fluid-structure cavity filled with water, the error in the

eigenvalues increases, but never surpassing the 10 % for both references compared.


109

In Figure 42 to Figure 44 are represented the first three natural vibrations modes for an

uncoupled rigid fluid cavity filled with air.

Figure 42: First natural vibration mode for the uncoupled rigid fluid cavity (𝜔 = 1068 𝑟𝑎𝑑/𝑠) filled with air.

Figure 43: Second natural vibration mode for the uncoupled rigid fluid cavity (𝜔 = 1068 𝑟𝑎𝑑/𝑠) filled with air.

Figure 44: Third natural vibration mode for the uncoupled rigid fluid cavity (𝜔 = 1511 𝑟𝑎𝑑/𝑠) filled with air.


110


111

9. Conclusions


112

9.1 Conclusions and Final Remarks

In this work two meshless methods, the RPIM and the NNRPIM, were used to solve problems

involving free and forced vibrations and acoustical analysis. When compared to exact results

found on the literature, both formulations proved to be reliable methods, showcasing accurate

results, constituting thus, a good and efficient alternative to the FEM.

The convergence tests analyzed backup the claims from the theory, in which meshless

methods convergence faster than the FEM, this can be said both for the static and free

vibrations analysis. Additionally, for denser meshes the NNRPIM provided least medium

displacement errors from all formulations tested.

In the other benchmarks tests performed involving free and forced vibrations, both methods

were able to deliver accurate solutione when compared with theoretical results, never

presenting errors larger than 10%. And in many cases for the NNRPIM the errors were bellow

0.1%.

When analyzing more complex geometries, like the beams and plates with corrugated

sinusoidal core the results between formulations show some discrepancies, with both

meshless methods having very similar results but the FEM not following the same values.

Nonetheless, the results were very accurate with the theory when under three point bending,

and the discrepancies mainly appear when considering two layers of sinusoidal core. This can

mean that for that geometry, the FEM may not have fully converged.

On the acoustical analysis performed, the RPIM showed to be less affected by dispersion

errors than the remaining methods. The FEM performed the worst. Still, both formulations

presented results very similar with those found on the literature.

From the accuracy of the results compared with the literature, it can be concluded that the

developed algorithms (using the MATLAB® software) to perform free and forced vibrations

and acoustical (uncoupled and interior coupled fluid-structure) analysis were successfully and

correctly built. Moreover, the implementation of the FEM and the extension of two meshless

methods, the RPIM and the NNRPIM to these types of analysis was achieved with success.

With the experience gained through the creating of the algorithms, some comparisons can be

made. Despite the meshless methods requiring more programmable knowledge and being

harder to implement than the FEM, after the algorithms to establish the “influence-domain”

and the shape functions are implemented, they become much easier to work with, since it

brings some added freedom to distribute the nodal mesh than the FEM.

Analyzing the computational time, as the mesh becomes increasingly denser, meshless

methods’ codes become very expensive to run. This is also the case for the NNRPIM, whose

integration scheme is nodal dependent, and this aggravates when three-dimensional solids are

studied. Still, if not many nodes are considered, meshless methods’ codes run faster than the

FEM, and this allied with the fact that mesh free methods converge faster to the solution make

them in general less computational demanding.

It must not be forgotten that the algorithms created run on the MATLAB® software and thus

are very dependent on the hardware components of the computer used to run the simulations.

This, in return creates some barriers for the amount of nodes that can be used, which

translates in some limitations when running examples, mainly on very thin beams and 3D

analysis. Another issue to point out on MATLAB® is the biggest and smallest numbers that

the software can work with. For the first, when this limit is trespassed the result comes out as

infinite, which gave serious trouble when trying to implement the damped effect on the

Duhamel integral and forced to the alteration of the Trapezoidal rule code. The second may

lead to incoherent results due to approximations when rounding the algorisms.


113

Overall, with the creation of the software and the validity of its application to laminated and

sandwich panels was completed with success. From the agreement of the various results

obtained with the literature, it can be concluded the success of this research work.


114

9.2 Future Works

It would be interesting to expand the meshless methods to the analysis of coupled exterior

acoustical problems. Although, some major barriers shall be found in the way since the BEM

is currently the most popular method to simulate the fluid for problems of this type. For this

method, only nodes placed on the boundary of the structure are necessary to discretize the

fluid domain, which makes satisfying the Sommerfield radiation equation much less

expensive.

It would also be interesting to combine the forced vibrations and acoustical algorithms created

with plate theories, such as the Reissner-Mindlin Theory and other High-Order Shear

Deformation Theories. They present some major advantages when compared to the

conventional 3D elasticity, since they require less nodes to obtain accurate results.


115

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70. Claude, B., et al., Eigensolutions to a vibroacoustic interior coupled problem with a

perturbation method. Comptes Rendus Mécanique, 2017. 345(2): p. 130-136.


119


120


121

Appendix A: Solutions for the static analysis of Sinusoidal Core Sandwich Beams


122

Sinusoidal Core Sandwich Beam With One Corrugated Layer.

Table A. 1: Relative deflection, �� , for the corrugated core beam under punctual load (FEM).


10 C-F 0.1441 0.2460 0.6843 3.8425 15.204

C-C 0.0571 0.0661 0.1154 0.5051 1.9239

C-H 0.0652 0.0861 0.1778 0.8656 3.3526

H-H 0.0716 0.1232 0.3439 1.9329 7.6329

15 C-F 0.1348 0.2415 0.6797 3.8353 15.188

C-C 0.0525 0.0638 0.1133 0.5005 1.9193

C-H 0.0603 0.0836 0.1754 0.8605 3.3468

H-H 0.0669 0.1209 0.3416 1.9292 7.6254

20 C-F 0.1408 0.2505 0.6901 3.8345 15.189

C-C 0.0552 0.0682 0.1186 0.5025 1.9193

C-H 0.0633 0.0885 0.1814 0.8624 3.3468

H-H 0.0697 0.1252 0.3468 1.9288 7.6251

30 C-F 0.1588 0.2783 0.7241 3.8519 15.169

C-C 0.0611 0.0818 0.1355 0.5142 1.9291

C-H 0.0698 0.1035 0.2005 0.875 3.3557

H-H 0.0759 0.1388 0.3635 1.9375 7.6241

Table A. 2: Relative deflection, �� , for the corrugated core beam under punctual load (RPIM).


10 C-F 0.1486 0.2659 0.7168 3.8678 15.395

C-C 0.0603 0.0784 0.1364 0.5507 1.9996

C-H 0.0687 0.1000 0.2018 0.9151 3.4524

H-H 0.0749 0.1362 0.3661 1.9724 7.7875

15 C-F 0.1294 0.2431 0.6915 3.8845 15.482

C-C 0.0511 0.0672 0.1243 0.5399 2.0279

C-H 0.0590 0.0876 0.1879 0.9067 3.4881

H-H 0.0656 0.1248 0.3533 1.9807 7.8338

20 C-F 0.1253 0.2376 0.6858 3.8297 15.363

C-C 0.0489 0.0644 0.1215 0.5145 1.9584

C-H 0.0566 0.0845 0.1847 0.8770 3.4103

H-H 0.0633 0.1220 0.3503 1.9512 7.7713

30 C-F 0.1284 0.2512 0.7000 3.8397 15.953

C-C 0.0470 0.0712 0.1288 0.5223 1.9554

C-H 0.0550 0.0919 0.1929 0.8851 3.3998

H-H 0.0615 0.1286 0.3572 1.9548 7.7317


123

Table A. 3: Relative deflection, �� , for the corrugated core beam under a punctual load (NNRPIM).


10 C-F 0.0962 0.2090 0.6512

C-C 0.0337 0.0485 0.1041

C-H 0.0404 0.0662 0.1632

H-H 0.0487 0.1036 0.3228

15 C-F 0.1065 0.2168 0.6491

C-C 0.0391 0.0536 0.1081

C-H 0.0463 0.0717 0.1666

H-H 0.0545 0.1078 0.3215

20 C-F 0.1250 0.2382 0.6821

C-C 0.0478 0.0629 0.1189

C-H 0.0564 0.0825 0.1802

H-H 0.0655 0.1193 0.3387

30 C-F 0.1999 0.3193 0.7825

C-C 0.0825 0.1019 0.1596

C-H 0.0934 0.1261 0.2283

H-H 0.1020 0.1615 0.3902

Table A. 4: Relative deflection, �� , for the corrugated core beam under a distributed load (FEM).


10 C-F 7.20E-03 2.70E-02 1.69E-01 2.44E+00 1.94E+01

C-C 1.70E-03 3.90E-03 1.38E-02 1.51E-01 1.15E+00

C-H 2.00E-03 5.50E-03 2.38E-02 2.95E-01 2.30E+00

H-H 2.30E-03 8.50E-03 5.03E-02 7.22E-01 5.72E+00

15 C-F 6.80E-03 2.74E-02 1.68E-01 2.44E+00 1.93E+01

C-C 1.60E-03 3.80E-03 1.36E-02 1.50E-01 1.15E+00

C-H 1.90E-03 5.40E-03 2.35E-02 2.94E-01 2.29E+00

H-H 2.10E-03 8.40E-03 5.01E-02 7.21E-01 5.71E+00

20 C-F 7.30E-03 2.82E-02 1.70E-01 2.43E+00 1.94E+01

C-C 1.90E-03 4.10E-03 1.42E-02 1.51E-01 1.15E+00

C-H 2.20E-03 5.70E-03 2.42E-02 2.94E-01 2.29E+00

H-H 2.40E-03 8.60E-03 5.07E-02 7.21E-01 5.71E+00

30 C-F 8.00E-03 3.08E-02 1.76E-01 2.44E+00 1.93E+01

C-C 1.80E-03 4.90E-03 1.62E-02 1.54E-01 1.16E+00

C-H 2.20E-03 6.60E-03 2.66E-02 2.98E-01 2.30E+00

H-H 2.50E-03 9.40E-03 5.25E-02 7.23E-01 5.71E+00


124

Table A. 5: Relative deflection, �� , for the corrugated core beam under a distributed load (RPIM).


10 C-F 7.29E-03 2.93E-02 1.74E-01 2.43E+00 1.96E+01

C-C 1.71E-03 4.58E-03 1.62E-02 1.64E-01 1.19E+00

C-H 2.05E-03 6.31E-03 2.66E-02 3.10E-01 2.36E+00

H-H 2.29E-03 9.19E-03 5.29E-02 7.32E-01 5.82E+00

15 C-F 6.46E-03 2.73E-02 1.69E-01 2.45E+00 1.97E+01

C-C 1.49E-03 3.96E-03 1.48E-02 1.61E-01 1.21E+00

C-H 1.80E-03 5.58E-03 2.50E-02 3.08E-01 2.38E+00

H-H 2.06E-03 8.55E-03 5.14E-02 7.37E-01 5.86E+00

20 C-F 6.50E-03 2.68E-02 1.68E-01 2.43E+00 1.96E+01

C-C 1.63E-03 3.79E-03 1.45E-02 1.53E-01 1.17E+00

C-H 1.93E-03 5.38E-03 2.45E-02 2.98E-01 2.33E+00

H-H 2.19E-03 8.35E-03 5.10E-02 7.28E-01 5.82E+00

30 C-F 6.53E-03 2.81E-02 1.71E-01 2.43E+00 1.95E+01

C-C 1.33E-03 4.15E-03 1.53E-02 1.56E-01 1.17E+00

C-H 1.64E-03 5.77E-03 2.55E-02 3.01E-01 2.32E+00

H-H 1.88E-03 8.64E-03 5.16E-02 7.28E-01 5.79E+00

Table A. 6: Relative deflection, �� , for the corrugated core beam under a distributed load (NNRPIM).


10 C-F 5.02E-03 2.44E-02 1.61E-01

C-C 9.92E-04 2.89E-03 1.25E-02

C-H 1.27E-03 4.31E-03 2.20E-02

H-H 1.62E-03 7.33E-03 4.75E-02

15 C-F 5.47E-03 2.49E-02 1.60E-01

C-C 1.16E-03 3.21E-03 1.30E-02

C-H 1.44E-03 4.65E-03 2.23E-02

H-H 1.76E-03 7.55E-03 4.72E-02

20 C-F 6.51E-03 2.70E-02 1.67E-01

C-C 1.60E-03 3.78E-03 1.43E-02

C-H 1.97E-03 5.34E-03 2.41E-02

H-H 2.36E-03 8.28E-03 4.95E-02

30 C-F 1.00E-02 3.44E-02 1.87E-01

C-C 2.55E-03 6.17E-03 1.92E-02

C-H 2.95E-03 8.04E-03 3.03E-02

H-H 3.14E-03 1.08E-02 5.64E-02


125

Figure A. 1: Location of the points where the stresses were measured.

Table A. 7: Stress values \MPa, for the corrugated core beam under a punctual load and C-C boundary

conditions (FEM).

b0 \mm 𝜆 Stress P. 1 Stress P. 2 Stress P. 3

𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜏𝑥𝑦

10 5 -390.34 -626.58 142.79 3.75 -77.04 -214.79 64.85 10 -432.85 -530.89 206.33 3.09 -53.53 -200.60 62.85

20 -528.52 -392.92 342.29 2.28 -29.39 -187.23 62.77 50 -938.29 -214.01 812.55 3.22 -29.95 -181.95 60.93

100 -1743.83 -173.24 1619.74 6.04 -53.77 -182.37 54.95

15 5 -335.25 -619.16 148.05 4.00 -192.98 -239.61 101.91 10 -393.94 -524.69 218.81 3.75 -54.43 -176.12 112.69 20 -486.14 -386.86 360.94 3.26 -17.74 -143.04 99.81

50 -915.68 -204.49 814.37 3.27 -24.56 -125.68 78.32

100 -1690.00 -171.90 1620.89 6.05 -42.44 -126.01 67.63

20 5 -357.99 -616.47 162.25 5.14 -210.11 -228.01 186.99 10 -387.94 -521.31 229.44 4.32 -296.88 -242.44 143.86 20 -502.61 -382.94 374.13 3.98 -84.85 -153.09 166.05

50 -880.76 -202.68 824.90 3.85 -163.57 -139.63 71.28

100 -1735.48 -161.15 1628.12 6.38 -35.21 -91.88 108.01

30 5 -442.43 -619.36 159.29 4.67 158.36 15.82 -25.91 10 -450.18 -522.48 242.02 5.01 150.73 20.43 -19.83 20 -577.78 -385.14 393.42 5.05 174.41 28.09 -11.36

50 -1019.43 -206.03 847.21 5.10 136.81 0.60 -94.56 100 -1704.80 -158.51 1637.29 6.94 366.72 -16.19 -280.26


126

Table A. 8: Stress values \MPa, for the corrugated core beam under a punctual load and C-F boundary conditions

(FEM).


𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜏𝑥𝑦 10 5 -308.99 -627.27 62.49 3.45 49.65 40.47 -6.08 10 -268.37 -531.69 45.33 2.54 37.09 26.77 -2.39 20 -199.79 -392.03 19.02 1.28 26.65 11.98 1.51

50 -102.54 -207.79 10.07 1.15 -6.92 -5.24 -0.07

100 -91.37 -164.03 -6.71 0.57 -8.72 -6.58 0.11

15 5 -253.35 -620.27 67.39 3.68 64.09 53.83 -17.58 10 -228.96 -526.54 57.46 3.12 8.53 29.88 -23.11 20 -160.88 -389.92 38.34 2.09 -9.00 12.58 -15.99

50 -96.99 -206.59 5.10 0.58 -13.81 -0.67 -6.46

100 -22.39 -160.36 8.58 0.83 -18.00 -6.14 -4.27

20 5 -264.46 -617.36 88.46 4.81 63.84 61.22 -50.44 10 -222.19 -523.55 68.61 3.72 96.57 65.78 -32.18 20 -167.95 -386.59 52.18 2.79 12.74 29.07 -39.61

50 -80.27 -208.58 19.74 1.12 40.75 20.80 -0.29 100 -80.35 -164.03 7.57 0.63 -15.81 1.98 -18.05

30 5 -350.32 -619.21 79.35 4.46 120.84 4.93 -9.03 10 -269.39 -522.58 81.05 4.42 80.21 2.65 -3.69 20 -216.63 -385.11 71.87 3.88 82.64 6.23 -3.68

50 -134.72 -205.84 46.58 2.43 91.84 2.54 -17.02 100 -64.80 -167.42 24.33 1.33 102.55 8.07 -24.40

Table A. 9: Stress values \MPa, for the corrugated core beam under a punctual load and C-H boundary

conditions (FEM).



10 5 -440.12 -626.95 188.01 3.89 -48.61 -158.32 49.23 10 -496.34 -531.51 263.36 3.28 -23.10 -126.25 41.67

20 -620.80 -393.39 431.74 2.68 -6.05 -113.21 40.53 50 -1141.25 -213.84 1018.28 4.19 -28.55 -116.40 36.42

100 -2152.07 -174.00 2030.41 8.11 -59.09 -119.71 29.77

15 5 -388.56 -619.88 191.14 4.13 -134.03 -172.42 74.57 10 -463.58 -526.02 274.44 3.90 -33.80 -108.54 68.10

20 -592.71 -388.87 448.76 3.52 -15.08 -85.85 56.93 50 -1139.22 -206.84 1019.95 4.13 -22.58 -79.25 46.05

100 -2094.87 -171.96 2026.42 7.63 -50.04 -82.46 34.18

20 5 -403.92 -617.31 212.63 5.20 -148.22 -162.83 133.46 10 -462.27 -523.07 285.74 4.46 -169.02 -142.40 86.80

20 -617.87 -385.99 462.50 4.20 -49.21 -86.33 90.48 50 -1114.41 -206.75 1028.85 4.52 -81.64 -76.88 44.48

100 -2179.30 -165.76 2034.82 7.91 -28.13 -56.94 61.07

30 5 -507.06 -620.48 200.83 4.82 158.93 14.82 -24.33 10 -531.04 -524.50 301.68 5.16 143.40 17.12 -15.20

20 -699.15 -388.73 484.59 5.26 168.92 24.55 -6.65 50 -1283.22 -213.58 1050.13 5.58 146.90 -1.17 -107.05

100 -2170.66 -166.48 2041.91 8.27 384.43 -16.09 -294.09


127

Table A. 10: Stress values \MPa, for the corrugated core beam under a punctual load and H-H boundary

conditions (FEM).



10 5 -476.91 -626.86 223.45 4.01 -76.34 -214.42 64.83 10 -605.80 -531.42 368.51 3.63 -52.06 -199.89 62.87

20 -861.66 -394.16 668.28 3.48 -24.77 -184.91 62.88

50 -1763.12 -217.46 1622.05 6.01 -41.49 -182.96 58.26 100 -3392.87 -179.99 3252.35 12.64 -89.38 -187.69 47.59

15 5 -429.06 -619.79 227.54 4.27 -192.33 -239.05 101.72 10 -577.70 -525.77 379.57 4.28 -54.35 -175.73 112.37

20 -840.58 -388.27 684.35 4.34 -18.57 -142.35 98.80

50 -1764.08 -206.99 1628.04 6.26 -27.83 -126.35 77.52 100 -3341.28 -178.93 3238.70 11.73 -69.31 -128.33 57.29

20 5 -441.73 -617.27 249.28 5.29 -209.03 -227.06 186.18 10 -577.10 -522.85 388.59 4.82 -295.48 -241.52 143.44

20 -874.48 -385.65 696.32 5.00 -84.53 -152.12 164.64

50 -1733.54 -205.66 1633.62 6.55 -154.92 -135.38 71.31 100 -3441.62 -166.80 3252.72 12.29 -35.51 -92.36 108.66

30 5 -545.82 -620.71 231.76 4.92 167.01 17.26 -28.07 10 -649.84 -525.38 399.68 5.49 171.92 23.63 -20.38

20 -968.81 -390.75 714.38 6.03 217.07 34.78 -8.30 50 -1969.13 -218.20 1649.85 7.46 179.81 -3.39 -160.83

100 -3435.76 -165.92 3253.52 12.43 553.71 -30.60 -456.08


conditions (RPIM).



10 5 -242,24 -186,64 158,61 -1,82 -7,92 -168,83 75,01 10 -330,45 -186,05 239,51 -1,92 -8,05 -169,07 75,05

20 -510,85 -184,80 399,64 -2,10 -8,27 -168,89 74,79 50 -1208,75 -142,55 847,88 -8,43 -108,11 -222,31 7,72

100 -2301,85 -411,47 1702,64 -13,71 46,02 -330,69 -533,51

15 5 -200,19 -181,21 171,95 -2,07 16,12 -268,39 197,95 10 -301,10 -180,18 250,43 -2,14 15,93 -265,59 195,89

20 -487,69 -178,17 409,75 -2,30 15,77 -265,10 195,48 50 -877,17 -116,39 895,22 -13,11 49,57 48,14 136,14

100 -1647,20 -62,55 1718,45 -27,74 325,07 82,99 307,58

20 5 -239,15 -177,47 187,33 -2,52 -104,80 -251,14 196,13 10 -323,38 -176,26 254,62 -2,22 -101,36 -244,51 189,32

20 -511,74 -173,80 414,08 -2,39 -101,43 -244,71 189,46 50 -884,90 -111,29 898,87 -13,47 -119,15 -266,73 154,91

100 -1571,11 -95,15 1693,22 -21,56 30,75 -210,25 146,50

30 5 -292,57 -176,58 169,91 -2,04 141,61 -2,57 3,17 10 -340,54 -176,24 265,24 -2,44 86,31 -3,06 4,31

20 -539,03 -173,91 422,30 -2,57 98,30 -6,46 2,51 50 -980,29 -91,68 906,84 -14,25 38,01 -40,16 41,27

100 -1815,76 -41,75 1710,09 -23,73 44,13 -32,87 -14,14


128

Table A. 12: Stress values \MPa, for the corrugated core beam under a punctual load and C-F boundary

conditions (RPIM).



10 5 -173,48 -187,86 80,99 -1,65 -24,71 6,18 -15,72 10 -171,89 -187,90 81,72 -1,66 -24,51 6,28 -15,66 20 -171,91 -187,90 81,72 -1,66 -24,51 6,28 -15,66

50 -434,46 -158,95 53,33 -6,61 -81,69 -287,49 29,84

100 -653,13 -346,80 77,35 -14,54 -658,43 -27,39 -268,07

15 5 -125,46 -181,12 93,52 -1,83 -18,58 67,42 -57,63 10 -134,00 -181,09 92,61 -1,82 -20,24 71,93 -61,76 20 -134,45 -181,08 92,49 -1,82 -20,32 72,15 -61,97

50 -200,83 -157,11 88,07 -8,06 -42,66 15,28 -45,30 100 -242,27 -122,71 137,84 -16,20 1,51 -2,74 -27,75

20 5 -145,57 -176,68 115,54 -2,27 37,43 90,19 -78,09 10 -144,43 -176,59 96,90 -1,92 36,89 88,27 -77,16 20 -142,82 -176,58 97,20 -1,93 36,28 86,85 -75,87

50 -127,23 -127,94 91,56 -8,25 15,40 60,19 -41,86 100 -116,02 -100,81 78,95 -9,22 -5,54 43,98 -25,90

30 5 -205,44 -175,54 91,14 -1,89 167,26 1,67 -4,56 10 -154,19 -176,24 107,39 -2,17 109,56 2,85 -3,03 20 -160,04 -176,17 107,14 -2,17 116,20 2,71 -3,20

50 -129,68 -125,47 104,30 -9,52 131,51 9,13 12,11 100 -123,30 -112,92 104,18 -12,29 148,54 45,56 8,64


conditions (RPIM).



10 5 -299,95 -186,40 206,50 -1,85 -11,65 -130,83 55,23 10 -405,20 -185,77 300,31 -1,96 -13,44 -113,13 46,00

20 -621,50 -184,31 491,30 -2,17 -14,41 -105,43 41,81

50 -1432,04 -144,09 1056,64 -9,62 -94,59 -242,30 24,35 100 -2750,29 -438,99 2116,25 -16,95 -145,06 -201,49 -404,43

15 5 -256,52 -180,39 216,54 -2,09 7,93 -189,46 137,87 10 -373,59 -179,11 307,36 -2,15 3,95 -154,46 111,04

20 -598,18 -176,66 498,61 -2,34 2,42 -141,79 101,30

50 -1098,45 -118,66 1102,68 -14,25 66,28 55,10 97,49 100 -2049,75 -59,00 2116,13 -30,40 455,14 122,30 298,46

20 5 -293,07 -176,30 230,60 -2,51 -70,85 -169,73 130,70 10 -393,23 -174,70 310,60 -2,23 -55,52 -134,22 100,99

20 -620,39 -171,66 502,36 -2,43 -51,02 -123,41 92,38 50 -1105,84 -113,20 1103,93 -14,56 -58,34 -169,79 78,77

100 -1962,38 -99,54 2098,31 -24,44 83,10 -149,58 99,12

30 5 -348,43 -175,46 213,80 -2,04 148,96 -2,88 0,81 10 -412,28 -174,70 323,19 -2,45 97,75 -3,39 0,84

20 -649,03 -171,85 511,49 -2,60 111,79 -7,33 -1,58 50 -1225,07 -89,08 1111,04 -15,25 60,19 -40,58 45,33

100 -2274,12 -30,69 2110,38 -26,38 73,05 -36,11 -23,33


129


conditions (RPIM).



10 5 -334,43 -186,00 241,76 -1,91 -8,04 -168,79 74,89 10 -513,78 -184,78 402,68 -2,11 -8,27 -168,90 74,79 20 -875,30 -182,29 722,80 -2,46 -8,70 -168,54 74,27

50 -2030,28 -135,19 1655,79 -11,43 -102,18 -215,05 21,05

100 -4016,39 -494,69 3342,11 -18,42 164,52 -306,73 -486,70

15 5 -296,45 -180,18 254,37 -2,17 16,05 -268,11 197,74 10 -490,67 -178,14 412,55 -2,31 15,78 -265,23 195,58 20 -863,90 -174,12 731,43 -2,63 15,46 -264,40 194,87

50 -1637,64 -95,57 1711,89 -17,97 131,72 79,02 183,26 100 -3140,40 -20,61 3304,05 -38,97 729,03 198,05 495,13

20 5 -336,05 -176,22 268,79 -2,59 -104,52 -250,57 195,60 10 -516,30 -173,76 416,32 -2,38 -101,24 -244,29 189,12 20 -894,82 -168,83 735,38 -2,71 -101,20 -244,29 189,10

50 -1696,38 -104,31 1714,29 -18,37 -102,81 -308,54 151,55 100 -3078,12 -98,80 3317,57 -33,64 136,68 -266,56 174,66

30 5 -392,05 -175,42 249,03 -2,08 143,35 -4,25 2,39 10 -536,55 -173,96 426,34 -2,59 92,07 -6,38 2,70 20 -927,55 -169,38 741,47 -2,85 109,28 -13,05 -0,66

50 -1885,71 -67,14 1715,44 -18,70 17,11 -70,54 65,36 100 -3580,03 19,15 3313,80 -34,91 26,76 -85,61 -42,69


conditions (NNRPIM).



10 5 -168,648 -219,134 112,505 0,112 -15.97 -95.22 45.97 10 -251,629 -222,007 189,876 0,113 -15.70 -94.20 45.83 20 -417,461 -221,737 344,190 0,135 -15.64 -92.39 45.10

15 5 -135,155 -214,574 122,010 0,130 -58.53 -171.96 112.36 10 -223,749 -214,819 198,687 0,144 -58.23 -170.72 111.63 20 -390,357 -214,389 349,554 0,166 -57.95 -169.54 110.90

20 5 -134,546 -217,758 138,424 0,188 -134.77 -201.83 178.35 10 -210,599 -217,964 214,820 0,178 -133.07 -200.18 176.38 20 -373,644 -217,233 371,071 0,205 -132.75 -199.66 175.90

30 5 -157,133 -222,791 150,743 0,216 203.93 -2.27 -45.22 10 -175,856 -221,242 227,846 0,222 225.35 -4.41 -48.08 20 -338,735 -214,729 399,612 0,290 220.13 -3.79 -42.97


130

Table A. 16: Stress values \MPa, for the corrugated core beam under a punctual load and C-F boundary




10 5 -75,089 -220,400 35,187 0,097 2.819 24.901 -12.935 10 -74,145 -223,421 35,407 0,089 2.582 24.457 -13.045 20 -74,186 -223,596 35,412 0,076 2.544 24.520 -13.008

15 5 -27,354 -216,481 46,934 0,123 11.888 36.928 -25.380 10 -33,425 -217,070 48,226 0,123 12.624 39.322 -26.993 20 -33,673 -217,345 48,325 0,111 12.603 39.538 -27.061

20 5 -80,802 -216,465 69,987 0,183 32.576 47.581 -44.112 10 -11,198 -220,917 56,838 0,148 33.200 48.112 -44.868 20 -9,883 -221,246 56,757 0,161 32.851 47.818 -44.513

30 5 -22,275 -225,434 68,517 0,193 93.913 -15.725 37.450 10 40,142 -225,255 66,925 0,187 80.798 -18.276 50.830 20 15,946 -227,986 67,061 0,173 71.431 -19.672 41.345





10 5 -202,892 -219,322 147,349 0,117 -10.883 -62.183 29.844 10 -300,378 -222,186 239,207 0,121 -9.474 -52.910 25.446 20 -502,407 -221,840 427,515 0,147 -8.912 -47.497 22.974

15 5 -168,484 -214,772 156,718 0,136 -40.341 -117.913 76.696 10 -271,675 -215,087 247,847 0,153 -34.540 -100.287 65.174 20 -473,100 -214,551 431,439 0,179 -31.987 -92.263 59.937

20 5 -174,939 -217,106 189,953 0,205 -94.311 -141.088 124.374 10 -257,107 -218,204 266,807 0,185 -78.756 -119.024 104.071 20 -452,521 -217,313 456,234 0,220 -72.431 -109.512 95.579

30 5 -206,462 -222,760 195,631 0,226 192.012 -6.898 -26.479 10 -226,314 -221,164 291,031 0,238 203.175 -11.394 -18.147 20 -409,566 -212,006 496,488 0,321 204.722 -10.726 -12.521





10 5 -250,095 -218,925 188,725 0,124 -15.923 -94.199 45.573 10 -415,624 -221,555 343,089 0,135 -15.625 -92.329 45.068 20 -747,062 -220,828 651,679 0,179 -15.487 -88.632 43.572

15 5 -214,750 -214,167 194,341 0,140 -58.620 -172.130 112.471 10 -389,618 -214,102 346,586 0,166 -57.909 -169.418 110.827 20 -721,967 -212,961 648,244 0,210 -57.278 -166.840 109.215

20 5 -205,182 -217,344 234,536 0,225 -135.421 -202.032 178.904 10 -375,913 -216,904 367,669 0,206 -132.467 -199.178 175.504 20 -701,059 -215,096 680,287 0,262 -131.577 -197.757 174.206

30 5 -259,418 -222,282 225,148 0,232 217.460 -4.466 -43.337 10 -334,026 -219,384 384,424 0,257 253.673 -8.644 -46.366 20 -638,825 -203,375 727,548 0,392 276.550 -5.903 -41.554


131

Table A. 19: Stress values \MPa, for the corrugated core beam under a distributed load and C-C boundary

conditions (FEM).



10 5 1.87 0.01 2.51 0.05 -0.82 -2.26 0.68 10 -3.28 0.00 7.13 0.06 -0.54 -2.09 0.66

20 -22.81 -0.12 26.30 0.12 0.00 -1.77 0.65

50 -161.13 -0.79 161.19 0.57 -2.50 -2.03 0.11 100 -653.00 -2.82 652.08 2.64 -14.35 -3.92 -2.29

15 5 6.70 0.40 2.77 0.06 -3.21 -3.92 1.66 10 0.66 0.33 7.47 0.07 -0.86 -2.78 1.79

20 -19.45 0.23 26.51 0.12 -0.29 -2.18 1.51

50 -161.19 -0.23 162.15 0.60 -0.83 -2.03 1.11 100 -650.31 -2.78 645.29 2.27 -10.96 -2.77 -2.96

20 5 11.06 0.79 3.82 0.12 -4.77 -5.14 4.22 10 5.24 0.71 8.00 0.09 -6.24 -5.07 2.99

20 -16.77 0.56 26.95 0.14 -1.74 -3.15 3.43

50 -155.38 0.09 161.40 0.56 -2.22 -2.28 1.42 100 -667.46 -1.60 648.13 2.37 -0.57 -2.06 2.69

30 5 14.42 0.81 4.78 0.14 -5.40 0.61 -1.63 10 11.45 0.90 9.10 0.12 -4.74 1.30 -1.76

20 -12.40 0.56 28.17 0.18 -0.18 1.86 -1.51 50 -169.97 -1.13 161.47 0.54 -4.70 -2.06 -14.55

100 -660.63 -1.08 645.25 2.22 77.86 -7.47 -77.84

Table A. 20: Stress values \MPa, for the corrugated core beam under a distributed load and C-F boundary

conditions (FEM).



10 5 8.53 -0.03 -3.93 0.03 6.77 13.05 -3.57 10 23.43 -0.08 -18.67 -0.03 10.28 25.16 -7.17 20 82.78 0.19 -77.35 -0.22 13.08 45.85 -14.06

50 505.30 3.63 -482.22 -1.23 13.62 104.20 -35.95

100 1988.01 10.92 -1954.33 -6.56 53.95 209.16 -65.16

15 5 13.47 0.34 -3.71 0.03 12.21 13.69 -5.52 10 27.86 0.15 -18.36 -0.03 6.69 21.93 -14.50 20 87.03 -0.39 -76.79 -0.25 1.87 35.12 -26.20

50 499.66 -0.98 -486.24 -1.61 6.27 73.11 -49.59

100 2010.89 13.89 -1937.12 -6.27 29.11 142.01 -85.11

20 5 18.65 0.76 -2.26 0.10 11.65 12.20 -10.02 10 32.81 0.50 -17.73 0.00 40.92 31.88 -18.12 20 93.44 -0.10 -76.11 -0.23 21.65 40.49 -45.82

50 495.64 -2.85 -483.39 -1.62 118.65 93.14 -41.53 100 2000.85 -2.80 -1947.01 -6.90 22.86 110.78 -148.83

30 5 22.92 0.89 -1.40 0.13 -7.94 -0.04 -0.66 10 41.45 1.01 -16.65 0.03 -14.07 -0.97 0.21 20 105.98 1.02 -74.73 -0.18 -25.65 -3.92 0.09

50 551.30 1.42 -479.40 -1.54 -40.27 -0.10 45.25 100 1999.67 -8.81 -1936.96 -6.71 -313.96 27.41 299.52


132

Table A. 21: Stress values \MPa, for the corrugated core beam under a distributed load and C-H boundary

conditions (FEM).



10 5 -0.18 -0.01 4.23 0.06 0.32 -0.01 0.06 10 -8.38 -0.05 11.65 0.07 1.89 3.85 -1.03

20 -37.57 -0.19 40.61 0.18 3.73 10.07 -2.91

50 -242.31 -0.72 243.47 0.96 -1.94 24.19 -9.70 100 -979.58 -3.43 980.61 4.30 -18.61 46.21 -22.43

15 5 4.34 0.36 4.27 0.06 -0.87 -1.25 0.57 10 -5.02 0.22 11.80 0.08 0.79 2.61 -1.77

20 -36.52 -0.09 40.53 0.16 0.14 6.97 -5.34

50 -250.60 -1.17 244.37 0.95 -0.04 16.54 -11.80 100 -974.19 -2.82 969.71 3.54 -17.05 32.07 -29.72

20 5 9.41 0.76 5.88 0.12 -2.32 -2.57 2.11 10 -0.89 0.57 12.28 0.10 3.92 2.88 -1.54

20 -35.24 0.07 41.03 0.17 3.95 7.51 -8.65

50 -248.82 -1.53 242.97 0.82 30.55 22.82 -9.29 100 -1022.50 -5.29 973.48 3.59 5.10 25.89 -34.86

30 5 11.59 0.78 5.76 0.15 -5.37 0.60 -1.64 10 4.53 0.73 13.35 0.14 -5.30 1.04 -1.38

20 -31.87 -0.02 42.60 0.21 -1.05 1.30 -0.75 50 -275.43 -4.15 242.58 0.73 -0.67 -2.77 -19.55

100 -1033.29 -7.46 968.91 3.28 92.03 -7.39 -88.90

Table A. 22: Stress values \MPa, for the corrugated core beam under a distributed load and H-H boundary

conditions (FEM).



10 5 -1.65 0.00 5.63 0.06 -0.79 -2.25 0.68 10 -17.15 -0.05 20.04 0.10 -0.42 -2.04 0.66 20 -76.09 -0.32 78.45 0.31 0.74 -1.40 0.67

50 -491.04 -2.17 484.97 1.68 -7.12 -2.43 -0.96

100 -1972.21 -8.23 1958.15 7.92 -42.84 -8.18 -8.18

15 5 2.65 0.36 5.68 0.07 -3.17 -3.88 1.64 10 -14.18 0.24 20.13 0.11 -0.85 -2.75 1.76 20 -76.17 0.00 78.18 0.29 -0.42 -2.07 1.35

50 -500.52 -1.23 487.59 1.80 -2.14 -2.30 0.79

100 -1971.30 -8.40 1939.52 6.81 -32.46 -4.63 -11.23

20 5 7.94 0.76 7.32 0.13 -4.71 -5.09 4.18 10 -10.13 0.59 20.38 0.13 -6.15 -5.01 2.97 20 -76.29 0.13 78.34 0.30 -1.69 -3.00 3.21

50 -496.43 -1.10 484.83 1.64 1.24 -0.58 1.43 100 -2032.33 -6.13 1947.76 7.09 -0.80 -2.44 3.21

30 5 10.22 0.78 6.88 0.15 -5.27 0.68 -1.76 10 -5.09 0.66 20.89 0.16 -3.01 1.56 -1.79 20 -74.96 -0.34 79.14 0.33 6.63 2.93 -1.01

50 -549.64 -6.00 482.32 1.48 12.49 -3.66 -41.04 100 -2045.27 -7.01 1938.10 6.61 227.44 -19.00 -218.48


133


conditions (RPIM).



10 5 1,11 -0,22 2,55 -0,02 -0,08 -1,77 0,79 10 -4,07 -0,18 7,28 -0,03 -0,09 -1,79 0,79 20 -25,50 -0,03 26,26 -0,05 -0,11 -1,77 0,76

50 -161,08 0,90 159,54 -0,66 0,14 -1,17 2,89

100 -679,25 -35,13 650,88 -1,95 43,31 6,80 12,13

15 5 4,98 -0,14 3,06 -0,03 0,20 -4,24 3,10 10 -1,13 -0,07 7,64 -0,03 0,19 -4,04 2,95 20 -23,31 0,16 26,52 -0,05 0,17 -3,98 2,90

50 -144,23 3,63 161,54 -1,07 16,54 7,75 11,21 100 -585,08 16,26 631,02 -4,70 159,28 46,51 76,80

20 5 7,19 -0,09 4,38 -0,06 -2,10 -5,04 3,89 10 0,88 -0,02 8,09 -0,04 -1,77 -4,35 3,26 20 -21,49 0,28 27,00 -0,06 -1,78 -4,38 3,28

50 -152,50 0,66 161,84 -1,10 0,81 -12,23 2,12 100 -589,05 -2,37 644,17 -4,94 42,30 -21,69 14,04

30 5 11,11 -0,30 5,16 -0,06 -15,41 0,27 0,46 10 8,42 -0,28 9,51 -0,06 -18,89 0,25 0,53 20 -15,20 0,00 28,27 -0,08 -17,43 -0,16 0,32

50 -164,42 3,78 162,41 -1,10 -20,28 -5,16 4,94 100 -681,07 22,34 642,26 -4,70 -21,24 -20,14 -9,78


conditions (RPIM).



10 5 7,14 -0,30 -3,77 -0,01 -1,09 8,73 -4,65 10 22,35 -0,45 -18,18 0,01 -2,06 19,24 -10,08 20 85,06 -0,98 -75,90 0,08 -3,98 40,24 -20,90

50 467,38 -10,41 -479,24 1,03 14,85 -41,66 13,49 100 1983,76 75,69 -1956,80 -1,07 -849,29 361,00 311,86

15 5 11,48 -0,15 -3,30 -0,01 -1,88 15,90 -12,24 10 26,63 -0,27 -17,80 0,01 -4,14 36,45 -27,96

20 91,69 -0,86 -75,38 0,09 -8,47 76,90 -58,84

50 413,65 -24,97 -486,18 2,94 -55,22 -18,14 -107,08 100 1696,32 -72,67 -1902,03 13,63 -390,01 -102,06 -400,41

20 5 14,95 -0,07 -1,59 -0,04 6,43 15,43 -12,56 10 30,31 -0,16 -17,32 0,00 14,82 35,58 -28,72

20 97,96 -0,79 -74,77 0,08 31,25 75,17 -60,38

50 464,28 -10,73 -485,85 3,01 78,28 192,26 -115,26 100 1759,39 -7,71 -1943,26 14,70 -43,58 305,86 -204,09

30 5 18,92 -0,26 -1,09 -0,05 -13,91 0,57 0,01 10 39,27 -0,37 -15,89 -0,02 -16,33 1,10 -0,28

20 107,54 -0,91 -72,93 0,04 -14,01 2,58 -0,80 50 526,70 -21,40 -481,44 2,62 39,95 30,48 -17,36

100 2050,22 -87,29 -1931,55 13,50 110,49 94,92 28,93


134


conditions (RPIM).



10 5 -1,19 -0,21 4,45 -0,02 0,06 -3,28 -1,57 10 -10,04 -0,16 12,14 -0,03 0,33 -6,25 -3,10 20 -43,19 0,04 40,92 -0,06 0,83 -11,89 -5,99

50 -250,39 0,28 243,03 -1,14 -4,05 8,31 1,03

100 -1037,96 -57,15 981,74 -4,54 243,74 -86,93 72,33

15 5 2,74 -0,10 4,80 -0,03 0,52 -7,34 -5,45 10 -6,91 0,01 12,17 -0,03 1,14 -12,87 -9,69 20 -40,97 0,41 40,72 -0,06 2,28 -23,63 -17,90

50 -232,73 2,72 244,52 -1,52 26,59 11,25 -36,26 100 -907,11 19,10 949,15 -6,83 217,78 61,37 -159,56

20 5 5,11 -0,04 6,06 -0,06 -3,40 -8,17 -6,40 10 -4,67 0,11 12,51 -0,04 -5,41 -13,10 -10,26 20 -38,85 0,62 41,08 -0,07 -9,81 -23,71 -18,75

50 -240,87 -0,11 243,85 -1,54 -20,21 -59,44 -31,89 100 -902,06 -5,88 968,23 -7,25 42,75 -92,73 -63,20

30 5 8,98 -0,25 6,77 -0,06 -15,62 0,21 -0,51 10 2,77 -0,15 14,02 -0,06 -19,46 0,08 -0,71 20 -32,72 0,33 42,44 -0,08 -18,60 -0,61 -0,68

50 -262,29 4,82 244,05 -1,51 -33,38 -11,14 -8,20 100 -1047,71 31,18 962,45 -6,82 -51,32 -38,65 13,85


conditions (RPIM).



10 5 -2,56 -0,19 5,85 -0,03 -0,09 -1,77 0,78 10 -18,70 -0,08 20,31 -0,04 -0,11 -1,78 0,77

20 -83,77 0,37 77,94 -0,11 -0,18 -1,72 0,68

50 -489,65 3,84 482,67 -1,86 2,51 1,73 8,23 100 -2050,73 -101,71 1962,32 -5,71 138,10 25,97 49,57

15 5 1,14 -0,09 6,28 -0,03 0,20 -4,23 3,09 10 -16,24 0,09 20,54 -0,05 0,18 -4,01 2,92

20 -83,44 0,81 77,92 -0,11 0,12 -3,87 2,80

50 -448,38 11,96 488,18 -3,01 49,40 20,10 30,05 100 -1779,58 49,81 1899,44 -13,68 482,43 138,55 226,83

20 5 3,46 -0,04 7,55 -0,06 -2,09 -5,02 3,86 10 -14,46 0,18 20,89 -0,05 -1,77 -4,34 3,25

20 -82,67 1,07 78,27 -0,11 -1,75 -4,31 3,22

50 -477,04 3,45 487,95 -3,06 7,34 -28,95 0,77 100 -1794,63 -5,29 1943,62 -14,61 127,04 -66,74 36,57

30 5 7,31 -0,25 8,06 -0,06 -15,37 0,21 0,43 10 -7,00 -0,10 22,07 -0,07 -18,44 -0,02 0,40

20 -77,04 0,72 79,01 -0,12 -15,69 -1,21 -0,19 50 -526,41 13,59 485,68 -2,89 -28,63 -17,30 14,58

100 -2092,31 71,05 1925,06 -13,64 -35,13 -62,33 -32,62


135





10 5 1.3485 -0.1452 2.0585 0.0014 -0.1668 -0.9899 0.4816 10 -3.6638 -0.1317 6.7094 0.0019 -0.1626 -0.9384 0.4635 20 -23.6606 -0.0766 25.2807 0.0045 -0.1537 -0.7117 0.3716

15 5 5.4718 -0.0551 2.3213 0.0018 -0.9219 -2.7287 1.7792 10 0.2299 -0.0355 6.9376 0.0026 -0.9045 -2.6574 1.7371 20 -19.8500 0.0506 25.1006 0.0052 -0.8634 -2.4926 1.6341

20 5 13.4760 0.1438 3.0375 0.0041 -3.0343 -4.5318 4.0174 10 7.4086 0.1437 7.6897 0.0038 -2.7749 -4.1798 3.6841 20 -12.2202 0.2748 26.4968 0.0071 -2.7118 -4.0803 3.5935

30 5 31.5899 0.5782 4.5671 0.0065 4.8683 -0.0550 -2.3783 10 29.5570 0.7002 8.9322 0.0061 6.1590 -0.1761 -2.5133 20 14.1279 2.3718 29.3210 0.0133 6.8869 0.3621 -2.5620



B0 \mm 𝜆 Stress P. 1 Stress P. 2 Stress P. 3


10 5 8.6046 -0.2253 -4.1022 0.0003 0.9594 6.1958 -3.0446 10 24.2070 -0.3194 -17.9527 -0.0019 2.0281 13.2262 -6.5708 20 85.1155 -0.5948 -73.4227 -0.0139 4.1958 27.0490 -13.4500

15 5 13.2911 -0.1808 -3.6578 0.0011 3.2997 9.7928 -6.4711 10 29.6421 -0.3340 -17.0856 -0.0009 7.5837 22.4933 -14.8637 20 92.1868 -0.7719 -71.1933 -0.0122 16.0112 47.4702 -31.3382

20 5 18.1938 0.2111 -2.7079 0.0034 7.0161 10.4392 -9.3392 10 37.7986 -0.2504 -17.4625 -0.0008 17.1592 25.5827 -22.8388 20 101.0310 -0.8569 -73.8242 -0.0074 36.9277 55.1581 -49.1613

30 5 46.2255 0.5691 -1.8272 0.0048 0.0468 -0.6324 0.9182 10 60.6135 0.1263 -16.6770 0.0005 -12.6455 -1.7243 9.6047 20 121.9747 -1.6651 -76.9335 -0.0235 -33.1229 -3.2927 17.5778





10 5 -0.0052 -0.1528 3.4172 0.0016 0.0385 0.3434 -0.1696 10 -7.5609 -0.1460 10.6304 0.0025 0.3361 2.3686 -1.1690 20 -37.2496 -0.0931 38.5963 0.0064 0.9225 6.4722 -3.1690

15 5 3.8264 -0.0663 3.6725 0.0020 -0.1965 -0.5720 0.3617 10 -3.6778 -0.0567 10.7993 0.0033 0.9912 2.9791 -1.9808 20 -33.1942 0.0252 38.0908 0.0073 3.2900 9.8683 -6.5182

20 5 11.9364 0.1690 5.1995 0.0049 -1.3836 -2.0513 1.8146 10 3.5393 0.1274 11.7051 0.0044 1.5819 2.3282 -2.1158 20 -25.0759 0.2636 39.9097 0.0094 6.9415 10.3478 -9.2606

30 5 33.5611 0.7429 6.3307 0.0067 6.5311 -0.0597 -3.2608 10 24.0128 0.6894 13.7051 0.0076 4.1105 -0.7903 0.1757 20 1.8930 2.8447 44.2084 0.0183 4.4885 -0.7688 2.3797


136





10 5 -1.8963 -0.1368 5.0889 0.0019 -0.1657 -0.9555 0.4685 10 -16.7923 -0.0954 18.9498 0.0036 -0.1566 -0.7886 0.4026 20 -76.3975 0.0688 74.4635 0.0115 -0.1295 -0.1097 0.1268

15 5 1.7238 -0.0438 5.1713 0.0022 -0.8953 -2.6412 1.7237 10 -13.1307 0.0221 18.7012 0.0043 -0.8789 -2.5529 1.6724 20 -73.0259 0.2797 72.7707 0.0122 -0.7568 -2.0648 1.3669

20 5 10.6735 0.1584 7.0404 0.0057 -3.0558 -4.5302 4.0323 10 -5.9285 0.2326 19.7824 0.0060 -2.7433 -4.1254 3.6368 20 -64.9015 0.6189 75.7758 0.0161 -2.5320 -3.7863 3.3331

30 5 37.3669 0.8569 7.2985 0.0066 6.3207 -0.1729 -2.8472 10 14.3307 0.8141 21.2466 0.0093 8.3218 -0.5371 -2.2536 20 -35.2606 4.2421 81.5258 0.0297 16.1151 0.0110 -2.3180


137

Sinusoidal Core Sandwich Beam With Two Corrugated Layers

Figure A. 2: Location of the points where the stresses were measured for the beam with two layers.

Table A. 31: Stress values \MPa, for the 2 layers beam under a punctual load and C-C boundary conditions

(FEM).

b0

\mm 𝜆

Stress P. 1 Stress P. 2 Stress P. 3 Stress P. 4 Stress P. 5

𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜏𝑥𝑦 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜏𝑥𝑦

10 5 -328.86 -527.18 50.88 -0.03 42.77 12.08 44.42 8.02 -0.56 -91.74 -75.52 -21.02

10 -352.32 -387.61 131.55 0.21 44.50 8.49 42.77 2.75 8.71 -106.41 -148.24 -11.85

20 -528.13 -388.21 292.39 0.74 43.49 7.12 41.69 1.13 8.54 -107.56 -147.22 0.31

50 -778.27 -208.94 788.03 3.14 58.91 -8.76 58.91 -8.76 24.52 -208.94 -66.64 42.92

100 -1577.6 -118.98 1544.9 7.11 167.49 -12.73 167.5 -12.73 70.86 -118.98 -31.83 179.72

15 5 -342.41 -522.61 32.42 -0.34 68.09 28.77 67.59 27.11 6.39 -225.80 -267.30 1.52

10 -364.03 -383.90 114.88 -0.03 69.86 25.62 69.86 25.62 -4.97 -383.90 -190.32 21.47

20 -545.50 -385.31 275.16 0.46 67.45 25.79 67.45 25.79 -4.40 -385.31 -193.94 47.39

50 -960.11 -204.42 757.70 2.28 62.36 -1.40 62.36 -1.40 -25.52 -204.42 -117.72 198.00

100 -1729.7 -123.07 1448.5 3.04 208.50 -8.86 208.5 -8.86 49.68 -123.07 -83.77 459.48

20 5 -401.45 -522.27 17.32 -0.27 87.67 22.12 87.61 21.93 4.65 -225.49 -267.28 -90.05

10 -398.12 -381.86 100.56 -0.18 100.96 30.16 101.0 30.16 -5.71 -381.86 -194.07 -81.92

20 -580.99 -384.10 260.77 0.27 98.78 30.96 98.78 30.96 -5.13 -384.10 -197.66 -62.00

50 -937.20 -202.82 736.21 1.55 76.07 1.57 76.07 1.57 -36.31 -202.82 -113.23 237.12

100 -1499.0 -117.92 1544.8 4.91 101.80 -11.67 101.8 -11.67 65.35 -117.92 -41.59 427.70

30 5 -451.56 -525.19 11.75 0.04 120.77 7.27 120.8 7.27 -2.53 -525.19 -300.50 -77.28

10 -481.76 -385.13 67.87 -0.59 166.02 19.85 166.0 19.85 -8.56 -385.13 -197.59 -181.08

20 -661.23 -387.88 233.43 -0.04 167.87 21.51 167.9 21.51 -8.47 -387.88 -200.87 -180.25

50 -1061.5 -204.76 712.33 1.41 131.52 25.33 131.5 25.33 -20.19 -204.76 -138.43 -32.87

100 -1771.6 -116.98 1517.2 4.69 85.65 6.87 85.65 6.87 -76.65 -116.98 -197.37 414.30


138

Table A. 32: Stress values \MPa, for the 2 layers beam under a punctual load and C-F boundary conditions

(FEM).

b0

\mm 𝜆



10 5 -239.43 -527.25 -27.29 -0.19 41.78 13.81 37.62 19.84 -26.41 -47.75 -91.88 -78.87

10 -179.47 -387.70 -26.76 -0.12 38.50 12.37 41.50 22.38 -16.51 -85.45 -107.13 -153.75

20 -179.49 -387.70 -26.78 -0.12 38.53 12.37 41.53 22.38 -16.51 -85.47 -107.13 -153.75

50 66.54 -198.94 0.22 2.88 53.20 -5.20 53.20 -5.20 18.14 -2.82 -212.63 -49.59

100 41.91 -108.87 -34.30 5.93 155.69 -6.33 155.7 -6.33 32.61 4.49 -116.71 1.32

15 5 -248.16 -522.65 -45.73 -0.56 73.49 30.45 74.10 32.47 -22.99 -148.44 -226.29 -277.70

10 -186.51 -384.02 -42.19 -0.38 70.18 26.67 70.18 26.67 -36.39 -167.19 -380.89 -194.77

20 -186.66 -384.03 -42.14 -0.38 70.11 26.44 70.11 26.44 -36.15 -167.26 -380.90 -194.81

50 -94.63 -204.01 -42.80 -0.14 63.42 11.94 63.42 11.94 -48.66 -66.77 -199.39 -83.38

100 -88.30 -112.80 -123.65 3.53 194.33 -1.80 194.3 -1.80 9.44 -118.98 -119.60 -45.58

20 5 -291.67 -521.73 -53.90 -0.53 109.55 25.94 109.8 26.76 -19.08 -191.09 -224.54 -279.05

10 -215.36 -381.59 -55.47 -0.54 106.91 29.95 106.9 29.95 -37.22 -192.47 -377.37 -199.94

20 -214.57 -381.61 -55.22 -0.55 105.75 29.71 105.8 29.71 -36.67 -191.34 -377.33 -200.06

50 -76.12 -203.49 -51.76 -0.32 76.24 16.90 76.24 16.90 -69.87 -29.90 -195.66 -81.50

100 150.04 -102.86 -23.38 3.18 92.59 -4.46 92.59 -4.46 25.88 69.48 -120.39 -12.16

30 5 -370.82 -524.92 -77.17 -0.21 155.37 8.96 155.4 8.96 -24.69 -360.33 -522.84 -309.08

10 -288.37 -383.95 -87.35 -1.02 179.29 20.26 179.3 20.26 -37.04 -269.26 -380.08 -202.24

20 -284.60 -383.87 -80.01 -0.82 177.78 21.05 177.8 21.05 -37.77 -265.74 -380.06 -202.05

50 -166.93 -198.36 -77.71 -0.65 155.31 25.39 155.3 25.39 -54.69 -108.81 -187.38 -104.11

100 -112.80 -110.49 -72.81 -0.41 133.35 18.94 133.4 18.94 -94.32 -33.34 -95.35 -58.73


139

Table A. 33: Stress values \MPa, for the 2 layers beam under a punctual load and C-H boundary conditions

(FEM).

b0

\mm 𝜆



10 5 -359.83 -527.42 78.83 0.09 41.44 12.28 41.29 11.44 -8.83 -175.27 -92.01 -77.44

10 -402.70 -388.03 176.48 0.43 41.18 9.04 41.12 8.85 -0.54 -314.95 -107.41 -149.58

20 -620.72 -388.78 375.20 1.09 39.51 7.39 39.41 7.04 -1.03 -533.08 -108.84 -148.36

50 -972.00 -207.19 996.84 4.73 60.14 -9.66 60.14 -9.66 28.39 -997.91 -212.31 -65.99

100 -1975.5 -119.08 1960.46 10.65 170.97 -14.40 171.0 -14.40 79.31 -1989.5 -122.01 -40.91

15 5 -379.76 -523.19 59.65 -0.21 64.65 29.90 64.46 29.27 -2.42 -287.41 -226.55 -270.17

10 -418.59 -384.85 160.88 0.16 67.66 26.11 67.66 26.11 -15.78 -411.68 -383.74 -194.27

20 -643.91 -386.62 358.02 0.77 63.67 26.38 63.67 26.38 -15.45 -636.71 -385.45 -198.88

50 -1186.8 -205.97 960.47 3.19 52.91 -3.28 52.91 -3.28 -34.68 -1176.4 -204.24 -129.39

100 -2134.7 -123.53 1865.33 7.18 211.66 -10.64 211.7 -10.64 59.38 -2146.2 -126.08 -95.43

20 5 -449.34 -523.27 39.68 -0.20 84.89 24.00 84.92 24.08 -1.50 -356.89 -226.36 -270.12

10 -455.89 -383.26 147.89 -0.01 99.47 30.35 99.47 30.35 -16.28 -447.88 -381.79 -198.51

20 -683.10 -386.08 344.23 0.56 95.67 31.44 95.67 31.44 -15.97 -674.52 -384.51 -203.20

50 -1168.5 -205.11 935.88 2.20 63.67 0.26 63.67 0.26 -49.66 -1151.1 -202.17 -126.01

100 -1891.4 -116.23 1952.63 7.45 103.95 -13.50 104.0 -13.50 75.91 -1921.6 -122.80 -53.10

30 5 -527.59 -526.61 24.49 0.07 120.84 9.02 120.8 9.02 -6.17 -524.45 -525.99 -303.96

10 -548.40 -386.73 117.18 -0.38 161.01 20.19 161.0 20.19 -17.30 -542.10 -385.46 -201.47

20 -766.28 -390.13 319.49 0.24 165.34 21.43 165.3 21.43 -18.27 -759.42 -388.75 -205.71

50 -1305.6 -209.82 913.66 2.10 123.62 26.14 123.6 26.14 -28.44 -1283.9 -205.72 -152.69

100 -2213.9 -123.36 1917.64 6.30 66.72 5.80 66.72 5.80 -81.97 -2184.1 -117.67 -235.38


140

Table A. 34: Stress values \MPa, for the 2 layers beam under a punctual load and H-H boundary conditions

(FEM).

b0

\mm 𝜆



10 5 -415.71 -527.50 130.87 0.22 42.62 11.49 44.41 7.34 -0.40 -234.53 -92.07 -76.75

10 -528.76 -388.21 292.69 0.74 43.36 7.13 41.56 1.13 8.54 -444.56 -107.56 -147.21

20 -882.15 -389.41 614.06 1.81 40.41 4.42 38.46 -2.07 8.17 -798.28 -109.85 -145.17

50 -1591.3 -212.12 1591.50 5.83 64.47 -12.33 64.47 -12.33 34.51 -1591.3 -212.12 -75.77

100 -3186.5 -125.20 3157.78 13.49 180.05 -19.25 180.1 -19.25 107.3 -3186.5 -125.20 -66.26

15 5 -435.26 -523.40 108.53 -0.07 63.17 30.33 62.62 28.51 6.72 -346.21 -226.63 -266.83

10 -545.92 -385.31 275.08 0.45 67.52 25.79 67.52 25.79 -4.39 -545.92 -385.31 -193.95

20 -912.93 -388.14 594.85 1.44 60.47 26.34 60.47 26.34 -3.24 -912.93 -388.14 -201.26

50 -1843.6 -207.16 1562.91 5.18 46.22 -12.41 46.22 -12.41 -26.25 -1843.6 -207.16 -157.06

100 -3366.3 -129.98 3059.50 9.37 220.64 -15.95 220.6 -15.95 89.39 -3366.3 -129.98 -125.41

20 5 -507.82 -523.79 79.40 -0.10 83.40 25.08 83.33 24.84 5.29 -418.71 -227.00 -266.50

10 -582.25 -384.12 259.56 0.28 97.62 30.60 97.62 30.60 -5.07 -582.25 -384.12 -197.69

20 -955.90 -388.69 579.22 1.21 90.77 32.46 90.77 32.46 -3.88 -955.90 -388.69 -205.02

50 -1823.8 -206.08 1529.20 3.71 56.49 -9.74 56.49 -9.74 -37.55 -1823.8 -206.08 -152.71

100 -3115.3 -124.25 3137.54 10.01 110.85 -18.91 110.9 -18.91 105.9 -3115.3 -124.25 -77.65

30 5 -587.58 -527.25 50.47 0.10 120.65 10.06 120.7 10.06 -2.24 -587.40 -527.21 -301.90

10 -668.74 -387.99 222.29 -0.08 157.41 20.51 157.4 20.51 -7.87 -668.67 -387.98 -201.00

20 -1037.5 -393.65 549.84 0.84 160.71 21.71 160.7 21.71 -7.25 -1037.5 -393.65 -207.73

50 -1985.5 -216.67 1507.19 3.73 105.35 26.61 105.4 26.61 -12.78 -1985.5 -216.67 -181.70

100 -3480.7 -131.12 3114.60 10.34 25.39 -2.10 25.39 -2.10 -74.37 -3480.7 -131.12 -341.85


141

Table A. 35: Stress values \MPa, for the 2 layers beam under a punctual load and C-C boundary conditions

(RPIM).

b0

\mm 𝜆



10 5 -230.59 -177.67 54.652 -0.048 56.381 1.397 56.38 1.397 -0.893 -7.8117 -10.908 -5.0640

10 -318.40 -176.98 132.922 -0.086 56.291 1.443 56.29 1.443 -0.777 0.1755 -6.3202 2.0205

20 -498.92 -175.54 288.518 -0.182 53.885 1.528 53.89 1.528 -0.527 15.5170 2.5268 15.5938

50 -820.16 -107.78 820.940 -3.719 32.748 -0.274 32.75 -0.274 3.286 118.274 54.9670 17.3450

100 -1479.3 -1.053 1313.78 -32.52 -26.848 -1.743 -26.85 -1.743 4.431 298.698 33.8276 -0.2305

15 5 -267.18 -177.49 45.531 -0.040 78.284 -0.802 78.28 -0.802 -1.009 -94.780 -0.3867 1.4674

10 -352.31 -176.63 125.612 -0.105 79.994 -0.577 79.99 -0.577 -0.856 -89.252 -0.1890 1.9895

20 -526.90 -174.91 280.285 -0.189 75.675 -0.024 75.68 -0.024 -0.420 -65.148 0.3070 2.9896

50 -901.54 -92.285 776.189 -6.185 41.939 0.077 41.94 0.077 4.123 58.9383 7.9909 19.8829

100 -1624.0 -40.932 1622.1 -13.52 2.910 -9.112 2.910 -9.112 21.59 245.261 21.2720 67.7632

20 5 -310.81 -176.19 39.586 -0.165 87.734 -3.398 87.73 -3.398 -2.270 -117.20 -0.0332 0.7616

10 -391.80 -175.13 120.688 -0.168 101.410 -3.619 101.4 -3.619 -2.398 -118.92 -0.0094 0.9663

20 -560.34 -173.13 275.130 -0.247 95.655 -2.989 95.66 -2.989 -1.944 -89.076 0.0279 1.0285

50 -927.6 -95.764 780.085 -6.377 51.049 -0.456 51.0 -0.456 2.568 59.9256 -0.0609 1.7550

100 -1618.1 -51.433 1515.26 -10.31 -28.422 -13.29 -28.42 -13.29 27.23 294.989 16.9583 41.4701

30 5 -390.16 -174.21 37.989 -0.297 113.121 -3.410 113.1 -3.410 -2.604 -105.11 0.4634 2.8323

10 -485.23 -173.22 111.037 -0.298 139.253 -4.762 139.2 -4.762 -3.603 -158.66 0.6040 3.6862

20 -643.93 -171.71 268.227 -0.408 133.698 -4.134 133.7 -4.134 -3.141 -127.72 0.5564 3.3586

50 -972.45 -95.278 764.382 -6.411 61.710 0.017 61.71 0.017 3.124 84.4312 -0.5272 1.0054

100 -1494.6 -51.762 1565.24 -18.26 -26.998 -15.48 -27.00 -15.48 30.31 414.902 0.2302 16.8710


142

Table A. 36: Stress values \MPa, for the 2 layers beam under a punctual load and C-F boundary conditions

(RPIM).

b0

\mm 𝜆



10 5 -141.97 -178.31 -17.908 -0.036 53.707 0.353 53.71 0.353 -16.06 -134.81 -178.49 -69.583

10 -141.37 -178.32 -18.274 -0.027 53.450 0.373 53.45 0.373 -15.93 -134.15 -178.50 -69.650

20 -141.39 -178.32 -18.277 -0.027 53.469 0.373 53.47 0.373 -15.94 -134.17 -178.50 -69.648

50 -116.18 -128.18 -16.918 -0.262 51.211 4.732 51.21 4.732 -9.690 -94.453 -123.26 -53.394

100 -174.90 -55.300 -6.550 0.882 4.635 -8.297 4.635 -8.297 -14.15 -167.80 -52.908 -66.915

15 5 -176.78 -177.71 -27.134 0.005 81.970 -1.724 81.97 -1.724 -18.63 -172.70 -179.06 -70.654

10 -177.26 -177.67 -24.055 -0.028 82.750 -1.738 82.75 -1.738 -18.62 -173.10 -179.04 -70.634

20 -177.44 -177.69 -24.220 -0.028 82.764 -1.741 82.76 -1.741 -18.59 -173.27 -179.04 -70.638

50 -135.99 -115.04 -13.795 -0.637 49.355 -1.746 49.36 -1.746 -31.97 -104.85 -107.40 -50.546

100 -112.56 -85.969 -8.425 -0.395 50.311 -3.027 50.31 -3.027 -18.31 -68.824 -75.174 -43.355

20 5 -211.19 -176.66 -24.821 -0.112 98.437 -4.412 98.44 -4.412 -21.09 -206.61 -178.11 -70.727

10 -219.80 -176.47 -27.989 -0.082 108.163 -4.861 108.2 -4.861 -21.90 -215.35 -177.88 -70.381

20 -218.99 -176.48 -27.989 -0.080 107.102 -4.802 107.1 -4.802 -21.61 -214.48 -177.91 -70.472

50 -167.26 -110.99 -19.329 -0.724 65.831 -3.070 65.83 -3.070 -42.09 -130.69 -102.29 -52.853

100 -144.85 -78.739 -8.440 -1.704 71.405 6.384 71.41 6.384 -20.55 -94.906 -66.738 -48.715

30 5 -307.99 -174.22 -35.313 -0.249 121.067 -4.180 121.1 -4.180 -23.04 -302.29 -175.70 -70.051

10 -319.56 -174.03 -38.263 -0.183 153.048 -6.020 153.1 -6.020 -24.93 -314.08 -175.45 -69.549

20 -317.31 -174.03 -33.651 -0.212 151.319 -6.142 151.3 -6.142 -24.70 -311.72 -175.48 -69.689

50 -285.08 -106.70 -18.685 -1.557 88.770 -0.252 88.77 -0.252 -57.05 -243.43 -97.129 -60.978

100 -243.62 -66.327 -15.325 -1.512 87.513 10.49 87.53 10.49 -26.37 -198.14 -56.726 -58.182


143

Table A. 37: Stress values \MPa, for the 2 layers beam under a punctual load and C-H boundary conditions

(RPIM).

b0

\mm 𝜆



10 5 -261.91 -177. 80.7434 -0.074 53.4523 1.075 58.51 1.744 -4.309 -259.45 -177.47 -66.521

10 -369.02 -176.55 175.241 -0.126 52.7948 1.109 58.21 1.826 -4.825 -366.34 -176.62 -66.460

20 -592.90 -174.77 367.749 -0.245 49.5512 1.211 55.04 1.938 -5.215 -590.18 -174.83 -66.045

50 -1004.1 -104.2 1032.53 -4.762 27.9460 -1.219 28.35 -1.603 -7.154 -995.83 -102.34 -76.293

100 -1809.2 11.8430 1650.62 -40.09 -51.861 -5.313 -31.97 0.987 -9.013 -1806.6 12.7277 -150.66

15 5 -298.13 -176.97 72.4466 -0.066 75.3974 -0.890 78.63 -0.515 -4.828 -296.77 -177.42 -65.960

10 -401.60 -175.91 168.090 -0.132 76.7951 -0.630 80.38 -0.213 -5.637 -400.06 -176.40 -65.196

20 -618.02 -173.77 359.122 -0.236 71.0218 0.058 74.67 0.482 -6.302 -616.45 -174.28 -63.277

50 -1104.3 -88.956 975.070 -7.588 30.9190 -1.330 41.73 0.913 -14.87 -1092.5 -86.068 -76.305

100 -2020.4 -32.987 2028.80 -16.57 -14.155 -9.903 -11.69 -10.70 -32.71 -2003.8 -28.908 -101.69

20 5 -344.82 -175.56 66.7235 -0.175 85.0996 -3.439 87.89 -3.125 -3.510 -343.40 -176.01 -65.592

10 -441.08 -174.30 164.169 -0.191 97.5938 -3.620 101.0 -3.239 -4.614 -439.47 -174.81 -64.937

20 -650.02 -171.81 354.275 -0.290 90.1332 -2.851 93.57 -2.463 -5.313 -648.33 -172.34 -63.275

50 -1131.0 -94.666 981.485 -7.802 37.8010 -1.699 49.08 0.580 -15.84 -1117.1 -91.379 -77.504

100 -2003.7 -48.374 1897.60 -12.43 -54.422 -17.57 -53.63 -18.38 -40.54 -1984.9 -43.848 -90.775

30 5 -428.39 -173.65 69.6379 -0.304 109.807 -3.434 111.9 -3.215 -2.89 -426.78 -174.07 -63.948

10 -535.73 -172.52 158.392 -0.330 134.197 -4.659 137.0 -4.372 -3.717 -533.84 -173.01 -63.014

20 -731.85 -170.63 349.267 -0.457 127.176 -3.863 130.1 -3.560 -4.705 -729.79 -171.17 -61.064

50 -1159.8 -95.409 962.473 -7.645 44.6758 -2.035 56.64 0.504 -20.46 -1144.1 -91.795 -76.321

100 -1823.9 -51.180 1961.74 -22.32 -56.220 -21.02 -56.69 -22.22 -46.16 -1806.7 -47.563 -80.688


144

Table A. 38: Stress values \MPa, for the 2 layers beam under a punctual load and H-H boundary conditions

(RPIM).

b0

\mm 𝜆



10 5 -322.78 -176.94 134.096 -0.092 55.045 1.437 55.05 1.437 -0.759 0.227 -6.246 2.040

10 -502.64 -175.51 290.597 -0.184 53.281 1.535 53.28 1.535 -0.516 15.666 2.595 15.717

20 -865.40 -172.63 601.533 -0.376 47.578 1.713 47.58 1.713 -0.005 46.249 20.206 42.761

50 -1543.9 -89.618 1671.70 -7.500 14.229 -4.854 14.23 -4.854 11.60 267.439 108.670 38.306

100 -3096.6 -17.328 2524.42 -63.44 139.722 59.74 139.7 59.74 64.48 936.839 904.192 755.448

15 5 -355.67 -176.63 124.341 -0.098 74.679 -0.512 74.68 -0.512 -0.783 -83.368 -0.162 1.912

10 -530.87 -174.88 281.587 -0.191 74.669 -0.002 74.67 -0.002 -0.405 -65.013 0.303 2.974

20 -882.81 -171.42 590.680 -0.361 64.506 1.134 64.51 1.134 0.482 -17.525 1.274 4.930

50 -1693.1 -73.099 1575.58 -11.81 19.374 -1.067 19.37 -1.067 27.45 190.681 -2.617 42.002

100 -2673.3 -31.575 3125.24 -6.408 -64.065 38.18 -64.07 38.18 118.5 841.837 810.645 395.055

20 5 -399.61 -175.12 112.575 -0.195 84.498 -3.079 84.50 -3.079 -2.041 -102.44 -0.016 0.786

10 -564.56 -173.08 275.411 -0.250 93.605 -2.942 93.61 -2.942 -1.913 -88.378 0.023 0.998

20 -907.10 -169.02 583.982 -0.415 80.772 -1.694 80.77 -1.694 -1.016 -31.034 0.086 1.086

50 -1716.3 -84.760 1588.70 -12.10 20.501 -0.865 20.50 -0.865 31.68 233.098 -3.996 11.591

100 -2831.4 -108.02 3072.37 -9.899 -36.150 9.093 -36.15 9.093 84.39 519.434 -9.534 -42.111

30 5 -474.40 -173.43 110.551 -0.323 107.913 -3.197 107.9 -3.197 -2.442 -96.679 0.440 2.673

10 -646.40 -171.74 265.272 -0.405 127.403 -3.955 127.4 -3.955 -3.004 -122.49 0.531 3.206

20 -973.61 -168.64 575.603 -0.599 114.382 -2.530 114.4 -2.530 -1.959 -52.882 0.422 2.461

50 -1689.9 -88.577 1555.87 -11.33 17.552 -3.106 17.55 -3.106 46.87 329.482 -3.897 4.046

100 -2626.0 -104.32 3104.21 -24.24 224.699 -132.8 -80.98 15.63 87.64 789.870 -22.744 -82.317


145

Table A. 39: Stress values \MPa, for the 2 layers beam under a distributed load and C-C boundary conditions

(FEM).

b0

\mm 𝜆



10 5 1.70 0.05 1.63 0.01 0.05 0.31 0.09 0.21 0.00 0.17 0.00 0.01

10 -3.57 0.03 6.47 0.02 -0.09 0.19 -0.13 0.05 0.18 -0.36 0.00 -0.01

20 -24.74 -0.05 25.67 0.09 -0.28 0.03 -0.33 -0.14 0.15 -2.48 -0.01 0.00

50 -154.40 -0.60 160.28 0.56 1.08 -0.94 1.08 -0.94 2.63 -15.44 -0.06 -0.05

100 -635.06 -2.51 638.47 2.51 9.28 -3.06 9.28 -3.06 17.13 -63.51 -0.25 -1.14

15 5 4.47 0.33 1.13 -0.01 0.86 0.98 0.84 0.92 0.20 0.45 0.03 -0.02

10 -0.61 0.29 6.50 0.02 0.11 0.98 0.11 0.98 -0.11 -0.06 0.03 0.01

20 -22.64 0.12 25.59 0.08 -0.35 0.96 -0.35 0.96 -0.03 -2.26 0.01 -0.03

50 -168.43 -0.24 160.35 0.57 -2.75 -1.80 -2.75 -1.80 -0.89 -16.84 -0.02 -0.70

100 -645.66 -2.75 634.25 2.32 12.52 -3.15 12.52 -3.15 17.62 -64.57 -0.27 -1.40

20 5 5.88 0.45 0.62 -0.01 1.65 0.76 1.65 0.75 0.14 0.59 0.04 -0.03

10 1.65 0.48 6.34 0.01 1.27 1.54 1.27 1.54 -0.18 0.16 0.05 0.01

20 -20.53 0.20 25.64 0.07 0.20 1.49 0.20 1.49 -0.07 -2.05 0.02 -0.04

50 -162.98 0.09 157.87 0.43 -3.64 -1.57 -3.64 -1.57 -1.59 -16.30 0.01 -0.63

100 -622.96 -2.12 634.16 2.05 2.95 -3.23 2.95 -3.23 18.11 -62.30 -0.21 -0.86

30 5 11.93 0.50 0.34 0.00 3.62 0.21 3.62 0.21 -0.08 1.19 0.05 0.02

10 6.57 0.55 4.60 -0.03 5.76 1.23 5.76 1.23 -0.41 0.66 0.05 0.02

20 -13.87 0.24 26.06 0.06 2.99 1.45 2.99 1.45 -0.33 -1.39 0.02 -0.01

50 -165.50 -1.05 158.72 0.46 -3.02 1.76 -3.02 1.76 0.17 -16.55 -0.11 -0.77

100 -660.07 -4.20 636.74 2.24 -21.23 -2.52 -21.23 -2.52 -3.58 -66.01 -0.42 -5.49


146

Table A. 40: Stress values \MPa, for the 2 layers beam under a distributed load and C-F boundary conditions

(FEM).

b0

\mm 𝜆



10 5 8.84 0.05 -4.69 -0.01 -0.02 0.43 -0.33 0.93 -1.54 9.49 9.49 0.04

10 24.29 0.04 -18.99 -0.04 -0.74 0.71 -0.21 2.47 -2.84 25.45 25.45 -0.04

20 87.33 0.17 -76.69 -0.21 -1.21 1.50 -0.09 5.21 -5.83 89.68 89.68 0.15

50 515.12 6.04 -473.14 -0.14 -3.46 1.91 -3.46 1.91 -3.20 473.50 473.50 -2.18

100 1951.51 12.10 -1901.7 -1.45 -10.13 7.23 -10.13 7.23 -43.36 1906.59 1906.59 2.70

15 5 11.84 0.34 -5.26 -0.03 1.17 1.10 1.22 1.26 -1.54 12.48 12.48 0.30

10 28.09 0.33 -18.79 -0.04 0.24 1.09 0.24 1.09 -3.91 30.40 30.40 0.71

20 93.01 0.65 -76.20 -0.20 0.87 1.07 0.87 1.07 -7.75 97.67 97.67 1.40

50 527.42 0.56 -481.31 -1.46 0.93 8.41 0.93 8.41 -14.62 544.13 544.13 3.33

100 1977.27 12.34 -1897.4 0.37 -10.48 8.16 -10.48 8.16 -46.54 1940.46 1940.46 4.18

20 5 14.46 0.51 -5.44 -0.03 2.76 0.94 2.77 0.99 -1.33 15.13 15.13 0.52

10 31.10 0.61 -18.82 -0.05 2.01 1.51 2.01 1.51 -3.98 33.85 33.85 1.11

20 97.64 1.17 -75.78 -0.20 2.56 1.07 2.56 1.07 -7.75 103.21 103.21 2.20

50 531.18 0.34 -473.64 -1.13 0.40 9.89 0.40 9.89 -21.48 558.91 558.91 5.04

100 2002.22 18.48 -1885.0 -2.07 -11.91 8.31 -11.91 8.31 -45.47 1905.56 1905.56 -2.55

30 5 19.29 0.56 -6.55 -0.01 5.40 0.27 5.40 0.27 -1.47 19.89 19.89 0.68

10 37.00 0.80 -20.79 -0.10 7.37 1.30 7.37 1.30 -3.84 39.30 39.30 1.27

20 107.03 1.67 -74.76 -0.20 5.92 1.32 5.92 1.32 -7.47 111.55 111.55 2.58

50 556.39 5.18 -474.57 -1.25 16.54 1.54 16.54 1.54 -22.02 591.26 591.26 11.76

100 2012.98 9.23 -1911.7 -6.15 59.09 15.57 59.09 15.57 -25.56 2108.33 2108.33 27.39


147

Table A. 41: Stress values \MPa, for the 2 layers beam under a distributed load and C-H boundary conditions

(FEM).

b0

\mm 𝜆



10 5 0.40 0.04 2.68 0.01 -0.07 0.33 -0.10 0.35 -0.32 0.55 0.03 0.03

10 -7.64 -0.01 10.01 0.04 -0.39 0.24 -0.30 0.54 -0.56 -7.37 -0.08 -0.25

20 -39.55 -0.14 38.90 0.14 -0.92 0.07 -0.70 0.80 -1.38 -39.03 -0.34 -0.20

50 -231.87 0.10 243.80 1.19 1.59 -1.30 1.59 -1.30 4.17 -242.24 -1.95 -0.25

100 -953.32 -2.59 970.92 5.35 12.07 -4.39 12.07 -4.39 23.89 -964.55 -4.94 -18.66

15 5 2.83 0.31 2.10 0.00 0.63 1.07 0.63 1.05 -0.11 2.96 0.28 -0.35

10 -5.10 0.21 10.05 0.03 -0.20 1.01 -0.20 1.01 -0.97 -4.55 0.30 -0.21

20 -38.40 -0.09 38.79 0.13 -0.98 1.05 -0.98 1.05 -1.80 -37.25 0.09 -1.13

50 -259.09 -0.86 241.43 0.94 -6.53 -2.56 -6.53 -2.56 -4.55 -254.91 -0.16 -11.63

100 -969.66 -3.12 967.97 5.63 15.05 -4.57 15.05 -4.57 25.38 -978.88 -5.16 -23.36

20 5 4.00 0.41 1.37 -0.01 1.53 0.85 1.53 0.85 -0.06 4.13 0.40 -0.40

10 -3.16 0.37 9.92 0.03 0.86 1.57 0.86 1.57 -1.01 -2.53 0.48 -0.30

20 -36.90 -0.11 38.89 0.12 -0.35 1.57 -0.35 1.57 -1.80 -35.53 0.14 -1.26

50 -255.45 -0.82 237.69 0.69 -8.60 -2.09 -8.60 -2.09 -6.92 -248.51 0.35 -11.42

100 -936.85 -0.77 960.39 4.08 4.68 -4.69 4.68 -4.69 26.56 -961.00 -6.02 -17.80

30 5 9.08 0.45 0.59 0.00 3.65 0.29 3.65 0.29 -0.19 9.20 0.47 0.07

10 0.92 0.41 8.19 -0.01 5.11 1.30 5.11 1.30 -1.07 1.44 0.52 -0.08

20 -30.79 -0.13 39.57 0.11 2.37 1.43 2.37 1.43 -1.89 -29.71 0.09 -0.89

50 -263.15 -3.08 239.16 0.73 -6.18 2.09 -6.18 2.09 -3.12 -254.46 -1.44 -13.41

100 -1013.8 -9.31 957.01 3.52 -36.37 -3.37 -36.37 -3.37 -7.84 -989.95 -4.76 -85.31


148

Table A. 42: Stress values \MPa, for the 2 layers beam under a distributed load and H-H boundary conditions

(FEM).

b0

\mm 𝜆



10 5 -1.85 0.04 4.70 0.02 -0.06 0.30 -0.02 0.19 0.01 -0.01 -1.83 0.03

10 -17.71 -0.02 19.26 0.06 -0.23 0.08 -0.28 -0.08 0.16 -0.16 -17.73 -0.09

20 -81.35 -0.24 77.08 0.26 -0.79 -0.40 -0.86 -0.65 0.09 -0.09 -81.43 -0.51

50 -479.55 -1.87 481.65 1.64 3.33 -2.36 3.33 -2.36 6.62 -6.62 -479.55 -1.87

100 -1922.2 -7.49 1928.74 7.61 19.33 -8.27 19.33 -8.27 46.31 -46.31 -1922.2 -7.49

15 5 0.57 0.30 3.97 0.00 0.53 1.11 0.51 1.04 0.23 -0.21 0.57 0.27

10 -15.29 0.18 19.08 0.06 -0.27 0.99 -0.27 0.99 -0.06 0.06 -15.29 0.18

20 -81.39 -0.34 76.59 0.23 -1.51 1.04 -1.51 1.04 0.15 -0.15 -81.39 -0.34

50 -521.73 -1.33 482.33 1.74 -9.20 -6.21 -9.20 -6.21 -1.18 1.18 -521.73 -1.33

100 -1954.9 -8.28 1923.26 7.39 22.23 -8.82 22.23 -8.82 49.39 -49.39 -1954.9 -8.28

20 5 1.78 0.39 2.80 0.00 1.46 0.90 1.46 0.89 0.18 -0.15 1.78 0.37

10 -13.27 0.30 18.71 0.05 0.61 1.59 0.61 1.59 -0.12 0.12 -13.27 0.30

20 -80.48 -0.53 76.32 0.22 -1.17 1.73 -1.17 1.73 0.13 -0.13 -80.48 -0.53

50 -517.44 -1.21 474.91 1.29 -11.47 -6.09 -11.47 -6.09 -2.08 2.08 -517.44 -1.21

100 -1915.9 -7.18 1908.24 6.13 10.20 -9.03 10.20 -9.03 50.55 -50.55 -1915.9 -7.18

30 5 6.82 0.43 1.44 0.00 3.66 0.34 3.66 0.34 -0.06 0.12 6.82 0.43

10 -8.65 0.31 16.37 0.01 4.75 1.34 4.75 1.34 -0.35 0.40 -8.64 0.32

20 -74.12 -0.69 76.03 0.20 1.44 1.47 1.44 1.47 -0.12 0.12 -74.12 -0.69

50 -534.79 -5.81 476.31 1.38 -13.48 2.27 -13.48 2.27 3.13 -3.13 -534.79 -5.81

100 -2027.0 -15.52 1914.32 6.75 -69.43 -9.69 -69.43 -9.69 -1.76 1.76 -2027.0 -15.52


149

Table A. 43: Stress values \MPa, for the 2 layers beam under a distributed load and C-C boundary conditions

(RPIM).

b0

\mm 𝜆



10 5 0,06 -0,13 1,57 0,00 0,40 0,04 0,40 0,04 -0,01 0,06 -0,13 -0,10

10 -5,21 -0,09 6,12 -0,01 0,19 0,04 0,19 0,04 0,00 -5,21 -0,09 -0,09

20 -26,76 0,08 24,50 -0,02 -0,19 0,05 -0,19 0,05 0,03 -26,76 0,08 -0,05

50 -140,64 3,34 166,66 -0,75 -3,45 -0,81 -3,45 -0,81 1,56 -140,64 3,34 -3,87

100 -483,07 2,45 576,55 -45,5 -28,91 2,39 -28,9 2,39 9,16 -483,07 2,46 -15,47

15 5 1,17 -0,16 1,39 0,00 1,25 0,00 1,25 0,00 -0,01 1,17 -0,16 -0,10

10 -3,76 -0,12 6,29 -0,01 0,66 0,02 0,66 0,02 0,00 -3,76 -0,12 -0,06

20 -24,73 0,09 24,51 -0,02 -0,03 0,09 -0,03 0,09 0,06 -24,73 0,09 0,13

50 -153,24 3,53 157,39 -1,13 -3,82 -0,17 -3,82 -0,17 4,22 -153,24 3,53 -4,15

100 -605,96 7,28 613,68 -27,2 -10,06 5,05 -10,1 5,05 28,01 -605,96 7,29 -23,08

20 5 2,72 -0,24 1,25 0,00 1,65 -0,09 1,65 -0,09 -0,06 2,72 -0,24 -0,11

10 -2,02 -0,18 6,50 -0,01 1,55 -0,10 1,55 -0,10 -0,07 -2,02 -0,18 -0,07

20 -22,27 0,06 24,71 -0,02 0,33 -0,01 0,33 -0,01 0,00 -22,27 0,06 0,10

50 -151,60 1,81 159,46 -1,16 -4,90 -0,08 -4,90 -0,08 5,09 -151,60 1,81 -3,90

100 -567,50 -12,71 666,29 -13,1 -25,90 2,34 -25,9 2,34 38,70 -567,51 -12,71 -38,98

30 5 7,26 -0,36 1,30 -0,01 3,34 -0,11 3,34 -0,11 -0,08 7,26 -0,36 -0,23

10 2,27 -0,32 6,40 -0,01 4,21 -0,22 4,21 -0,22 -0,16 2,27 -0,32 -0,18

20 -15,97 -0,15 25,69 -0,03 1,67 -0,12 1,67 -0,12 -0,09 -15,97 -0,15 0,02

50 -135,38 0,78 157,07 -1,04 -6,53 -0,52 -6,53 -0,52 7,27 -135,38 0,78 -2,10

100 -492,46 0,56 635,56 -11,3 -22,47 6,29 -22,5 6,29 51,85 -492,46 0,56 -29,91


150


(RPIM).

b0

\mm 𝜆



10 5 7,260 -0,183 -4,436 -0,001 0,252 -0,024 0,252 -0,024 -0,921 7,689 -0,193 -0,370

10 23,486 -0,309 -18,378 0,006 -0,020 -0,092 -0,020 -0,092 -1,828 24,353 -0,330 -0,668

20 88,449 -0,817 -74,226 0,036 0,224 -0,241 0,224 -0,241 -3,708 90,182 -0,860 -1,320

50 426,609 -12,534 -506,16 2,084 11,399 3,111 11,40 3,111 -7,892 439,643 -9,581 10,123

100 1454,50 -12,442 -1741,7 138,1 82,578 -10,30 82,59 -10,30 -21,51 1473,30 -4,951 40,371

15 5 8,387 -0,195 -4,700 0,003 1,476 -0,061 1,476 -0,061 -1,075 8,632 -0,276 -0,493

10 24,532 -0,316 -18,016 0,006 1,208 -0,138 1,208 -0,138 -2,148 25,031 -0,477 -0,962

20 87,776 -0,859 -73,525 0,036 2,583 -0,413 2,583 -0,413 -4,378 88,776 -1,182 -2,185

50 464,194 -13,956 -476,88 3,327 4,878 -1,025 4,878 -1,025 -22,09 482,876 -9,372 11,920

100 1817,26 -29,052 -1855,9 82,16 -6,221 -24,93 -6,221 -24,93 -97,59 1863,06 -16,031 75,687

20 5 10,603 -0,295 -4,418 -0,001 2,248 -0,155 2,248 -0,155 -1,195 10,877 -0,382 -0,529

10 25,689 -0,425 -17,743 0,005 2,637 -0,273 2,637 -0,273 -2,427 26,223 -0,593 -0,975

20 87,656 -1,079 -72,940 0,035 4,300 -0,547 4,300 -0,547 -4,798 88,739 -1,421 -2,195

50 462,139 -9,522 -482,37 3,382 9,810 -1,568 9,810 -1,568 -27,52 484,078 -4,302 11,371

100 1691,42 32,800 -2017,4 39,32 56,530 -12,19 56,53 -12,19 -133,4 1760,57 44,260 124,794

30 5 14,087 -0,381 -4,889 -0,007 3,927 -0,163 3,927 -0,163 -1,318 14,428 -0,469 -0,705

10 28,768 -0,475 -18,166 0,005 6,201 -0,401 6,201 -0,401 -2,744 29,425 -0,646 -1,234

20 89,241 -0,962 -71,766 0,033 7,468 -0,731 7,468 -0,731 -5,357 90,581 -1,309 -2,701

50 420,319 -7,418 -471,78 2,858 18,178 -0,051 18,18 -0,051 -37,59 445,312 -1,675 7,409

100 1467,96 -13,433 -1919,3 33,62 50,031 -27,31 50,03 -27,31 -174,0 1534,73 11,197 97,627


151


(RPIM).

b0

\mm 𝜆



10 5 -1,18 -0,12 2,59 0,00 0,28 0,03 0,28 0,03 -0,21 -1,08 -0,12 -0,15

10 -9,24 -0,05 9,49 -0,01 -0,09 0,01 -0,09 0,01 -0,43 -9,03 -0,06 -0,21

20 -41,78 0,21 37,16 -0,03 -0,88 0,00 -0,88 0,00 -0,84 -41,34 0,19 -0,28

50 -214,19 4,77 251,29 -1,16 -5,37 -1,18 -5,37 -1,18 1,66 -210,90 5,52 -6,40

100 -729,63 2,89 868,23 -68,4 -45,35 2,73 -45,4 2,73 15,37 -724,90 4,76 -24,76

15 5 -0,05 -0,14 2,42 0,00 1,12 0,00 1,12 0,00 -0,23 0,01 -0,16 -0,16

10 -7,68 -0,06 9,65 -0,01 0,39 0,02 0,39 0,02 -0,49 -7,56 -0,10 -0,18

20 -39,28 0,27 37,08 -0,02 -0,78 0,11 -0,78 0,11 -0,94 -39,03 0,19 -0,06

50 -234,32 4,86 236,92 -1,69 -8,23 -0,73 -8,23 -0,73 4,56 -229,62 6,02 -6,66

100 -920,72 9,39 920,10 -40,7 -27,29 5,17 -27,3 5,17 39,60 -909,21 12,65 -33,31

20 5 1,46 -0,22 2,25 -0,01 1,55 -0,09 1,55 -0,09 -0,27 1,51 -0,24 -0,17

10 -5,92 -0,11 9,91 -0,01 1,22 -0,10 1,22 -0,10 -0,60 -5,80 -0,15 -0,20

20 -36,56 0,27 37,31 -0,03 -0,55 0,01 -0,55 0,01 -1,08 -36,29 0,18 -0,12

50 -232,92 2,25 239,99 -1,73 -10,20 -0,58 -10,2 -0,58 5,58 -227,39 3,57 -6,41

100 -863,28 -20,57 1001,9 -19,7 -45,02 2,33 -45,0 2,33 55,07 -845,89 -17,69 -57,46

30 5 5,85 -0,34 2,43 -0,01 3,22 -0,11 3,22 -0,11 -0,28 5,91 -0,36 -0,28

10 -1,69 -0,26 10,07 -0,02 3,79 -0,21 3,79 -0,21 -0,70 -1,54 -0,30 -0,31

20 -29,94 0,02 38,52 -0,04 0,61 -0,08 0,61 -0,08 -1,23 -29,61 -0,07 -0,22

50 -210,29 0,73 236,25 -1,53 -13,34 -1,34 -13,3 -1,34 9,09 -204,01 2,18 -3,66

100 -752,21 -2,27 953,22 -16,9 -41,83 7,60 -41,8 7,60 75,54 -735,42 3,93 -44,15


152

Table A. 46: Stress values \MPa, for the 2 layers beam under a distributed load and H-H boundary conditions

(RPIM).

b0

\mm 𝜆



10 5 -3,59 -0,10 4,68 0,00 0,33 0,04 0,33 0,04 0,00 -3,59 -0,10 -0,09

10 -19,89 0,03 18,67 -0,01 -0,06 0,05 -0,06 0,05 0,02 -19,89 0,03 -0,06

20 -85,32 0,55 74,52 -0,05 -1,20 0,08 -1,20 0,08 0,12 -85,32 0,55 0,07

50 -430,11 10,61 506,93 -2,26 -10,86 -2,64 -10,9 -2,64 4,88 -430,11 10,61 -11,97

100 -1465,6 7,94 1744,2 -138 -90,10 7,27 -90,1 7,27 27,76 -1465,6 7,97 -46,69

15 5 -2,30 -0,13 4,43 0,00 1,08 0,01 1,08 0,01 -0,01 -2,30 -0,13 -0,07

10 -17,93 0,02 18,63 -0,01 0,21 0,07 0,21 0,07 0,04 -17,93 0,02 0,07

20 -81,54 0,65 74,03 -0,04 -1,82 0,28 -1,82 0,28 0,20 -81,54 0,65 0,65

50 -469,79 11,21 477,06 -3,38 -12,85 -0,63 -12,9 -0,63 13,55 -469,79 11,21 -12,93

100 -1848,6 22,28 1844,1 -81,6 -42,06 15,22 -42,1 15,22 86,07 -1848,6 22,31 -70,11

20 5 -0,58 -0,20 3,94 -0,01 1,53 -0,08 1,53 -0,08 -0,05 -0,58 -0,20 -0,09

10 -15,68 -0,02 18,67 -0,02 0,88 -0,04 0,88 -0,04 -0,03 -15,68 -0,02 0,05

20 -77,53 0,71 73,89 -0,05 -2,05 0,20 -2,05 0,20 0,15 -77,53 0,71 0,57

50 -466,99 6,22 482,80 -3,44 -17,12 -0,25 -17,1 -0,25 16,73 -466,99 6,22 -12,40

100 -1722,0 -38,52 2016,0 -39,5 -79,49 7,22 -79,5 7,22 118,4 -1722,0 -38,52 -118,71

30 5 4,16 -0,33 3,91 -0,01 3,16 -0,10 3,16 -0,10 -0,08 4,16 -0,33 -0,20

10 -10,34 -0,20 18,39 -0,02 3,23 -0,15 3,23 -0,15 -0,11 -10,34 -0,20 -0,04

20 -68,31 0,33 74,41 -0,06 -1,44 0,14 -1,44 0,14 0,10 -68,31 0,33 0,58

50 -422,17 3,47 473,47 -3,01 -24,18 -1,75 -24,2 -1,75 24,75 -422,18 3,46 -7,37

100 -1501,9 1,64 1910,6 -33,7 -77,81 19,37 -77,8 19,37 159,2 -1501,9 1,62 -91,71


153

Appendix B: Free and Forced Vibrations for Various Sinusoidal Core Sandwich Beams


154

Sinusoidal Core Sandwich Beam with One Corrugated Layer

Table B. 1: Second natural frequency (Hz) for the beam with one sinusoidal core (FEM).


10 C-F 5271.3 2164.5 706.91 120.76 32.061

C-C 8813.6 4222.1 1622.0 297.96 86.579

C-H 8473.6 3997.5 1443.1 257.28 71.589

H-H 3518.4 2397.1 1248.1 215.75 57.574

F-F 10929 5233.2 1851.4 324.09 88.365

15 C-F 5110.4 2234.9 713.66 127.59 32.546

C-C 7076.6 4353.6 1651.2 348.11 90.795

C-H 7001.5 4112.6 1463.6 287.27 73.915

H-H 3538.0 2040.5 1093.4 230.62 58.639

F-F 6170.7 2595.6 764.54 329.04 90.228

20 C-F 3865.1 2144.5 705.08 127.49 32.568

C-C 4501.1 3868.1 1596.9 346.83 90.752

C-H 4320.4 3710.9 1425.4 286.47 73.923

H-H 3132.3 1869.8 1033.6 230.39 58.674

F-F 3957.0 2491.5 759.42 329.09 90.267

30 C-F 2313.5 1761.6 662.33 127.08 32.548

C-C 2621.1 2198.1 1354.7 342.29 90.509

C-H 2329.4 2185.0 1250.8 284.19 73.800

H-H 2030.2 1676.5 973.93 229.38 58.629

F-F 2228.7 1959.7 731.68 328.96 90.188


155

Table B. 2: Third natural frequency (Hz) for the beam with one sinusoidal core (FEM).


10 C-F 9586.3 4719.9 1729.9 310.65 87.504

C-C 12930 6740.7 2746.1 523.85 163.02

C-H 10117 5629.2 2593.2 484.28 144.27

H-H 8144.0 3749.8 1252.0 442.61 125.94

F-F 11060 7904.1 3139.5 578.30 168.20

15 C-F 7063.5 4795.6 1753.0 348.79 90.550

C-C 7592.0 6622.3 2789.0 657.04 176.08

C-H 7430.0 5457.6 2628.6 580.03 152.84

H-H 6917.3 3839.2 1263.7 505.15 131.00

F-F 7653.7 7336.1 3157.7 665.57 175.56

20 C-F 4509.6 4131.0 1699.5 347.67 90.549

C-C 4950.6 4765.6 2620.4 651.62 175.80

C-H 4514.3 4722.1 2485.9 576.57 152.71

H-H 4191.8 3523.0 1241.0 503.26 130.98

F-F 5570.2 4744.4 2942.0 662.32 175.49

30 C-F 2756.7 2182.0 1450.0 344.61 90.388

C-C 3456.2 2280.4 1925.6 636.76 174.95

C-H 3140.3 2271.5 1885.5 566.71 152.19

H-H 2101.7 2180.3 1131.5 497.33 130.70

F-F 3310.2 2311.7 1994.2 652.72 175.07


156

Table B. 3: Second natural frequency (Hz) for the beam with one sinusoidal core (RPIM).


10 C-F 4390.3 1817.7 648.11 122.92 32.023

C-C 7141.1 3284.8 1365.2 309.97 88.569

C-H 7031.0 3189.2 1256.8 265.00 72.475

H-H 3153.8 2159.1 1133.4 220.00 57.743

F-F 8264.0 4171.3 1647.9 330.80 88.235

15 C-F 4289.8 1907.0 670.68 123.51 32.024

C-C 5697.7 3381.3 1444.0 317.29 87.776

C-H 5663.7 3277.6 1316.9 269.07 72.141

H-H 3180.8 1570.0 1102.5 221.55 57.698

F-F 5794.6 4162.0 1709.9 333.77 88.106

20 C-F 3530.6 1890.2 675.09 125.67 31.960

C-C 4174.1 3119.0 1448.5 332.57 87.993

C-H 4078.4 3046.0 1321.0 278.21 72.160

H-H 2962.8 1511.4 1065.2 226.10 57.599

F-F 4308.2 3493.0 1694.0 341.36 87.972

30 C-F 2430.0 1600.2 652.73 125.06 32.284

C-C 2747.7 1947.1 1302.3 326.58 89.277

C-H 2457.9 1935.0 1211.0 274.87 73.067

H-H 2161.2 1406.2 1005.7 224.65 58.225

F-F 2614.8 1925.3 1482.9 337.99 88.938


157

Table B. 4: Third natural frequency (Hz) for the beam with one sinusoidal core (RPIM).


10 C-F 7952.5 3755.8 1500.2 320.22 88.443

C-C 9983.6 5106.6 2239.5 552.40 169.72

C-H 9875.0 5026.8 2160.7 505.89 148.34

H-H 6883.4 3079.6 1505.4 457.66 127.95

F-F 10547 6004.9 2635.3 597.90 167.81

15 C-F 5774.1 3799.4 1570.6 325.48 87.983

C-C 6318.1 4856.8 2357.4 571.58 166.83

C-H 6217.7 4797.3 2265.5 519.86 146.62

H-H 5266.6 3154.7 1177.3 466.58 127.12

F-F 6563.0 5331.0 2737.6 611.03 165.66

20 C-F 4249.8 3348.5 1565.1 337.07 88.024

C-C 4843.4 3713.7 2302.6 610.89 167.86

C-H 4413.4 3703.6 2218.1 548.08 147.13

H-H 4064.7 2956.0 1180.0 484.90 127.24

F-F 5443.5 3694.2 2611.4 636.18 171.48

30 C-F 2856.6 1935.4 1393.2 332.35 89.149

C-C 3836.0 2048.5 1801.3 590.53 170.96

C-H 3473.0 2021.5 1770.1 533.77 149.48

H-H 2513.1 1925.7 1112.5 475.82 128.97

F-F 3571.8 2113.3 1862.8 619.52 171.26


158

Table B. 5: Second natural frequency (Hz) for the beam with one sinusoidal core (NNRPIM).


10 C-F 5112.8 2024.1 655.64

C-C 8404.0 3883.2 1505.9

C-H 8185.4 3681.9 1338.0

H-H 3444.5 1647.9 1148.1

F-F 9907.2 4897.4 1728.1

15 C-F 4649.7 2024.0 681.94

C-C 6296.3 3682.8 1519.5

C-H 6184.9 3532.4 1365.6

H-H 3340.0 1478.5 1033.8

F-F 6124.2 4531.7 1772.5

20 C-F 3466.4 1881.0 670.10

C-C 4136.7 3127.6 1435.1

C-H 3967.8 3037.4 1305.9

H-H 2898.7 1361.2 954.13

F-F 4242.7 3452.9 1693.4

30 C-F 1988.3 1403.4 596.55

C-C 2267.7 1175.8 1126.5

C-H 1997.1 1732.2 1060.3

H-H 1903.4 1264.1 871.72

F-F 2216.8 1696.1 1321.4


159

Table B. 6: Third natural frequency (Hz) for the beam with one sinusoidal core (NNRPIM).


10 C-F 8306.1 4386.2 1611.8

C-C 11259 6119.2 2556.3

C-H 8965.7 4896.5 2412.8

H-H 7946.2 3450.7 1158.5

F-F 10649 7242.7 2933.5

15 C-F 6206.4 4128.3 1639.7

C-C 6675.8 5373.2 2533.3

C-H 6598.6 5092.7 2401.3

H-H 6088.1 3357.6 1199.2

F-F 6744.8 5823.1 2912.8

20 C-F 4135.5 3344.6 1558.4

C-C 4481.9 3819.8 2294.3

C-H 4136.1 3750.1 2203.8

H-H 3843.5 2931.2 1163.8

F-F 4918.2 3761.9 2616.8

30 C-F 2433.3 1728.9 1228.6

C-C 2892.0 1833.0 1589.1

C-H 2702.1 1826.6 1558.2

H-H 1988.6 1708.5 986.70

F-F 2856.3 1845.0 1623.8

Table B. 7: Maximum absolute stresses for the beam with 1 layer (FEM).

Boundary Stress 𝑏0 = 10 𝑏0 = 15 𝑏0 = 20 𝑏0 = 30

C-F 𝜎𝑥𝑥 8,50E+01 7,40E+01 8,08E+01 9,50E+01

𝜎𝑦𝑦 1,36E+00 1,34E+00 2,36E+00 2,78E+00

𝜏𝑥𝑦 6,24E-01 7,27E+00 1,11E+01 1,48E+01

C-C 𝜎𝑥𝑥 6,73E+00 3,52E+00 1,00E+01 1,98E+01

𝜎𝑦𝑦 1,40E-01 3,94E-01 8,92E-01 2,19E+00

𝜏𝑥𝑦 5,97E-01 1,15E+00 2,11E+00 9,68E-01

C-H 𝜎𝑥𝑥 1,21E+01 7,87E+00 6,48E+00 1,00E+01

𝜎𝑦𝑦 6,49E-02 1,87E-01 4,97E-01 1,60E+00

𝜏𝑥𝑦 4,29E-01 1,41E+00 2,71E+00 1,64E+00

H-H 𝜎𝑥𝑥 2,71E+01 2,12E+01 1,19E+01 6,47E+00

𝜎𝑦𝑦 2,29E-01 3,14E-01 8,34E-01 1,87E+00

𝜏𝑥𝑦 4,32E-01 7,28E-01 1,77E+00 2,42E-01


160

Table B. 8: Maximum absolute stresses for the beam with 1 layer (RPIM).


C-F 𝜎𝑥𝑥 8,30E+01 1,17E+02 8,21E+01 1,50E+02

𝜎𝑦𝑦 2,81E+00 2,74E+00 6,40E+00 2,87E+00

𝜏𝑥𝑦 5,36E+00 6,97E+00 9,19E+00 1,57E+01

C-C 𝜎𝑥𝑥 7,57E+00 2,79E+00 6,75E+00 7,57E+00

𝜎𝑦𝑦 1,63E-01 4,82E-01 9,81E-01 1,63E-01

𝜏𝑥𝑦 5,08E-02 2,90E-01 8,11E-01 5,08E-02

C-H 𝜎𝑥𝑥 1,52E+01 8,60E+00 4,30E+00 1,86E+01

𝜎𝑦𝑦 1,65E-01 2,99E-01 1,21E+00 1,44E+00

𝜏𝑥𝑦 4,93E-01 1,59E+00 8,16E-01 2,62E+00

H-H 𝜎𝑥𝑥 3,06E+01 2,70E+01 1,69E+01 6,78E+00

𝜎𝑦𝑦 2,57E-01 6,73E-01 6,13E-01 1,75E+00

𝜏𝑥𝑦 1,06E-01 1,10E+00 3,58E-01 1,38E+00

Table B. 9: Maximum absolute stresses for the beam with 1 layer (NNRPIM).


C-F 𝜎𝑥𝑥 8,93E+01 9,87E+01 1,10E+02 1,44E+02

𝜎𝑦𝑦 7,15E-01 9,37E-01 9,54E-01 1,14E+00

𝜏𝑥𝑦 3,31E+00 7,81E+00 1,19E+01 1,90E+01

C-C 𝜎𝑥𝑥 7,04E+00 2,29E+00 7,98E+00 3,92E+01

𝜎𝑦𝑦 1,33E-01 5,49E-02 2,09E-01 1,01E+00

𝜏𝑥𝑦 6,47E-02 3,23E-01 8,99E-01 2,98E+00

C-H 𝜎𝑥𝑥 1,34E+01 7,99E+00 5,54E+00 3,31E+01

𝜎𝑦𝑦 1,55E-01 6,04E-02 2,08E-01 1,06E+00

𝜏𝑥𝑦 5,32E-01 1,23E+00 2,30E+00 4,97E+00

H-H 𝜎𝑥𝑥 2,87E+01 2,33E+01 1,40E+01 2,06E+01

𝜎𝑦𝑦 1,05E-01 7,89E-02 3,68E-01 1,32E+00

𝜏𝑥𝑦 2,08E-01 4,80E-01 1,33E+00 3,75E+00


161

Figure B. 1: Vertical deflection for the corrugated core beam with one layer and C-F boundary conditions.

Figure B. 2: Vertical deflection for the corrugated core beam with one layer and C-H boundary conditions.

Figure B. 3: Vertical deflection for the corrugated core beam with one layer and H-H boundary conditions.


162

Sinusoidal Core Sandwich Beam with Two Corrugated Layer

Table B. 10: Fundamental natural frequency (Hz) for the beam with two sinusoidal cores (FEM).

b0 \mm Boundary 𝜆 = 5 𝜆 = 10 𝜆 = 20 𝜆 = 50

10 C-F 1448.1 362.21 101.39 16.285

C-C 5709.0 1473.9 610.58 95.883

C-H 4709.7 1210.0 433.79 68.763

H-H 3722.6 960.20 284.78 45.519

F-F 7608.4 1927.5 629.50 101.09

15 C-F 1404.4 388.29 101.32 16.331

C-C 5044.5 1947.4 599.30 98.491

C-H 4287.1 1500.1 429.35 69.852

H-H 3533.6 1070.6 283.96 45.808

F-F 6730.8 2261.4 626.90 101.76

20 C-F 1350.3 384.99 101.07 16.274

C-C 4287.8 1830.7 582.31 94.733

C-H 3808.8 1439.6 422.29 68.265

H-H 3306.1 1052.8 282.38 45.425

F-F 4337.9 2200.8 622.24 100.64

30 C-F 1198.1 370.65 100.20 16.430

C-C 2626.6 1495.4 538.46 101.83

C-H 2362.9 1245.3 402.71 713.18

H-H 2232.9 981.82 277.46 462.39

F-F 2304.4 1931.5 607.25 103.05


163

Table B. 11: Second natural frequency (Hz) for the beam with two sinusoidal cores (FEM).


10 C-F 6239.4 1614.1 608.19 96.595

C-C 11529 3116.5 1558.4 240.78

C-H 10929 2921.4 1320.8 206.50

H-H 10216 2698.2 1087.4 172.03

F-F 13985 3590.2 1645.7 261.01

15 C-F 5581.2 2031.3 600.75 98.348

C-C 7707.8 4302.3 1503.2 253.99

C-H 7537.5 3953.3 1287.9 214.40

H-H 7481.6 3556.0 1071.8 175.95

F-F 7025.0 5084.8 1617.6 267.03

20 C-F 4501.5 1936.1 589.06 95.744

C-C 4994.3 3857.6 1424.8 236.44

C-H 4801.7 3616.5 1239.4 203.43

H-H 4682.1 3326.7 1047.3 170.21

F-F 4604.2 4359.3 1572.2 256.72

30 C-F 2416.0 1634.8 557.14 100.76

C-C 2765.3 2202.4 1240.8 269.02

C-H 2571.0 2167.8 1116.6 223.73

H-H 2444.4 2154.0 979.04 180.84

F-F 2384.9 2024.7 1436.4 276.82


164

Table B. 12: Third natural frequency (Hz) for the beam with two sinusoidal cores (FEM).


10 C-F 9644.1 3378.3 1600.1 250.65

C-C 15191 474.0.2 2806.3 428.02

C-H 13703 4547.6 2549.3 393.02

H-H 11306 4401.3 2285.3 356.56

F-F 14382 5036.4 3003.4 469.05

15 C-F 7186.3 4673.6 1557.8 260.53

C-C 8046.0 6894.6 2664.9 463.84

C-H 7874.1 6664.9 2445.5 418.52

H-H 7653.6 6204.9 2216.8 373.01

F-F 7738.0 7167.4 2906.9 490.29

20 C-F 4650.2 4187.4 1494.3 246.41

C-C 5098.4 4508.3 2463.6 418.82

C-H 4921.9 4497.7 2294.7 384.51

H-H 4834.3 4268.2 2113.7 349.32

F-F 4982.4 4435.0 2736.4 455.28

30 C-F 2703.7 2081.3 1333.1 273.02

C-C 3187.6 2317.3 1991.9 501.20

C-H 2945.8 2297.3 1914.0 446.72

H-H 2536.7 2254.0 1824.9 392.56

F-F 3146.6 2263.1 1989.1 522.26


165

Table B. 13: Fundamental natural frequency (Hz) for the beam with two sinusoidal cores (RPIM).


10 C-F 1456.7 397.22 102.10 16.668

C-C 5947.9 2130.7 620.44 106.23

C-H 4882.0 1593.2 438.59 73.397

H-H 3820.6 1099.4 286.60 47.055

F-F 7569.2 2338.6 632.10 104.57

15 C-F 1437.8 396.89 102.35 16.716

C-C 5517.6 2071.2 616.34 106.41

C-H 4636.1 1567.9 437.67 73.580

H-H 3730.7 1095.5 287.08 47.200

F-F 7045.0 2320.4 632.75 104.87

20 C-F 1391.3 3945.8 102.47 16.791

C-C 4654.1 1951.3 605.26 106.55

C-H 4098.0 1510.5 433.83 73.798

H-H 3491.4 1081.5 286.87 47.394

F-F 5578.7 2266.3 631.31 105.27

30 C-F 1270.0 3852.6 102.26 16.945

C-C 3152.9 1650.6 570.32 106.33

C-H 2988.2 1348.2 419.77 74.057

H-H 2802.3 1030.2 284.45 47.778

F-F 3165.0 2044.7 621.90 106.05

.


166

Table B. 14: Second natural frequency (Hz) for the beam with two sinusoidal cores (RPIM).


10 C-F 6380.6 2175.3 615.27 103.76

C-C 11988 4897.0 1594.4 288.82

C-H 11419 4386.5 1343.8 235.39

H-H 10738 3831.8 1100.0 186.86

F-F 14310 5555.0 1661.7 285.90

15 C-F 5967.4 2133.0 613.32 103.99

C-C 9203.7 4639.3 1568.6 288.91

C-H 9106.0. 4211.7 1330.6 235.73

H-H 8969.9 3734.1 1095.9 187.32

F-F 9155.0 5328.6 1652.2 286.49

20 C-F 5038.3 2039.9 606.55 104.26

C-C 6125.4 4141.5 150..9 288.19

C-H 6041.9 3846.2 1296.8 235.73

H-H 5959.3 3499.4 1081.2 187.72

F-F 5919.1 4759.4 1622.0 286.94

30 C-F 3195.0 1776.4 582.60 104.49

C-C 3674.8 2865.6 1342.7 283.71

C-H 3347.3 2803.4 1190.6 234.11

H-H 3236.7 2709.9 1026.5 187.88

F-F 3507.2 2968.3 1506.2 286.61


167

Table B. 15: Third natural frequency (Hz) for the beam with two sinusoidal cores (RPIM).


10 C-F 9710.7 4936.2 1628.2 287.49

C-C 17162 8143.3 2889.8 556.33

C-H 14711 7724.1 2610.9 483.91

H-H 12118 7277.7 2327.4 415.49

F-F 16632 9197.9 3054.9 553.49

15 C-F 9007.4 4925.3 1610.2 287.83

C-C 9462.3 7366.1 2811.7 555.51

C-H 9444.6 7087.3 2558.6 483.91

H-H 9136.2 6777.9 2297.5 416.05

F-F 9442.1 8142.8 3004.0 553.97

20 C-F 6031.8 4449.1 1564.8 287.71

C-C 6217.4 5748.2 2644.1 551.48

C-H 6096.9 5675.9 2437.3 481.94

H-H 6004.0 5510.3 2218.8 415.58

F-F 6128.4 5804.3 2871.7 552.88

30 C-F 3695.2 2936.4 1421.3 285.30

C-C 4144.6 3039.6 2177.2 534.08

C-H 3903.9 3035.2 2069.3 471.81

H-H 3469.4 3028.8 1948.5 2683.3

F-F 4300.8 3050.6 2393.0 545.07

Table B. 16: Maximum absolute stresses for the beam with 2 layers (FEM).


C-F 𝜎𝑥𝑥 1,54E+02 1,34E+02 1,34E+02 1,65E+02

𝜎𝑦𝑦 2,54E+00 5,41E+00 6,36E+00 5,57E+00

𝜏𝑥𝑦 3,83E+00 1,71E+01 1,96E+01 2,68E+01

C-C 𝜎𝑥𝑥 6,33E+00 1,18E+01 2,62E+01 9,13E+01

𝜎𝑦𝑦 2,05E-01 1,88E+00 3,39E+00 4,77E+00

𝜏𝑥𝑦 3,47E+00 1,60E+00 3,20E+00 2,02E+01

C-H 𝜎𝑥𝑥 1,58E+01 1,20E+01 2,10E+01 7,69E+01

𝜎𝑦𝑦 4,05E-01 1,27E+00 2,71E+00 3,77E+00

𝜏𝑥𝑦 3,30E+00 1,44E+00 3,15E+00 2,05E+01

H-H 𝜎𝑥𝑥 4,51E+01 3,82E+01 1,84E+01 5,86E+01

𝜎𝑦𝑦 1,59E-01 1,85E+00 3,62E+00 4,35E+00

𝜏𝑥𝑦 3,11E+00 1,73E+00 2,18E+00 1,87E+01


168

Table B. 17: Maximum absolute stresses for the beam with 2 layers (RPIM).


C-F 𝜎𝑥𝑥 1,33E+02 1,86E+02 1,43E+02 1,83E+02

𝜎𝑦𝑦 4,40E+00 5,73E+00 9,68E+00 6,27E+00

𝜏𝑥𝑦 2,73E+00 6,59E+00 6,44E-01 5,11E+00

C-C 𝜎𝑥𝑥 1,24E+01 9,71E+00 1,52E+01 4,66E+01

𝜎𝑦𝑦 5,50E-02 3,40E-02 3,03E-01 1,03E+00

𝜏𝑥𝑦 2,42E-01 3,61E-01 6,21E-01 2,56E+00

C-H 𝜎𝑥𝑥 2,53E+01 1,05E+01 1,11E+01 2,90E+01

𝜎𝑦𝑦 3,71E-01 9,55E-01 9,21E-01 3,53E-01

𝜏𝑥𝑦 7,78E-01 8,47E-01 4,85E-01 1,23E+00

H-H 𝜎𝑥𝑥 5,93E+01 4,90E+01 4,17E+01 2,12E+01

𝜎𝑦𝑦 3,30E-01 3,45E-01 5,64E-01 9,84E-01

𝜏𝑥𝑦 1,60E+00 2,40E+00 4,50E-01 1,08E+00

Table B. 18: Maximum absolute stresses for the beam with 2 layers (NNRPIM).


C-F 𝜎𝑥𝑥 8,88E+00 7,20E+00 1,51E+01 5,23E+01

𝜎𝑦𝑦 2,08E-02 1,64E-01 3,13E-01 2,04E+00

𝜏𝑥𝑦 1,29E+00 4,15E+00 6,98E+00 2,27E+01

C-C 𝜎𝑥𝑥 1,21E+02 1,22E+02 1,27E+02 1,17E+02

𝜎𝑦𝑦 6,10E-01 3,26E-01 5,24E-01 1,86E+00

𝜏𝑥𝑦 6,05E+00 9,65E+00 9,46E+00 2,00E+01

C-H 𝜎𝑥𝑥 1,95E+01 1,09E+01 1,07E+01 3,40E+01

𝜎𝑦𝑦 4,75E-02 2,74E-01 4,44E-01 2,25E+00

𝜏𝑥𝑦 1,17E+00 3,83E+00 6,60E+00 2,36E+01

H-H 𝜎𝑥𝑥 4,38E+01 3,39E+01 2,20E+01 2,94E+01

𝜎𝑦𝑦 4,05E-02 2,82E-01 4,63E-01 2,21E+00

𝜏𝑥𝑦 2,13E+00 5,55E+00 9,06E+00 2,47E+01


169

Figure B. 4: Vertical deflection for the corrugated core beam with two layers and C-C boundary conditions.

Figure B. 5: Vertical deflection for the corrugated core beam with two layers and C-F boundary conditions.

Figure B. 6: Vertical deflection for the corrugated core beam with two layers and C-H boundary conditions.


170

Figure B. 7: Vertical deflection for the corrugated core beam with two layers and H-H boundary conditions.


171

Appendix C: Forced Vibrations of a Sandwich Plate with Corrugated Core


172

Sinusoidal Core Sandwich Plate with One Corrugated Layer

Table C. 1: Maximum absolute stresses for the plate with 1 layer (FEM).


C-F 𝜎𝑥𝑥 7,05E+03 6,30E+03 6,78E+03 8,25E+03

𝜎𝑦𝑦 2,20E+02 1,82E+02 3,21E+02 3,93E+02

𝜏𝑥𝑦 5,76E+01 5,90E+02 9,38E+02 1,22E+03

C-C 𝜎𝑥𝑥 6,14E+02 3,28E+02 9,44E+02 1,78E+03

𝜎𝑦𝑦 6,16E+00 6,75E+01 1,44E+02 2,72E+02

𝜏𝑥𝑦 5,24E+01 8,84E+01 1,67E+02 7,32E+01

C-H 𝜎𝑥𝑥 1,13E+03 7,07E+02 6,03E+02 9,62E+02

𝜎𝑦𝑦 9,36E+00 3,60E+01 8,13E+01 1,89E+02

𝜏𝑥𝑦 3,62E+01 1,21E+02 2,26E+02 1,36E+02

H-H 𝜎𝑥𝑥 2,40E+03 1,86E+03 1,02E+03 5,07E+02

𝜎𝑦𝑦 1,92E+01 5,22E+01 1,29E+02 2,25E+02

𝜏𝑥𝑦 3,75E+01 5,46E+01 1,35E+02 1,87E+01

Table C. 2: Maximum absolute stresses for the plate with 1 layer (RPIM).


C-F 𝜎𝑥𝑥 7,66E+02 2,76E+02 6,69E+02 2,59E+03

𝜎𝑦𝑦 1,51E+01 4,62E+01 9,48E+01 1,62E+02

𝜏𝑥𝑦 5,14E+00 2,81E+01 7,32E+01 2,51E+01

C-C 𝜎𝑥𝑥 7,75E+03 1,13E+04 7,87E+03 1,52E+04

𝜎𝑦𝑦 3,67E+02 6,02E+02 8,19E+02 6,79E+02

𝜏𝑥𝑦 5,34E+02 6,97E+02 8,63E+02 1,51E+03

C-H 𝜎𝑥𝑥 1,43E+03 8,16E+02 4,47E+02 1,82E+03

𝜎𝑦𝑦 1,87E+01 1,83E+01 1,38E+02 1,22E+02

𝜏𝑥𝑦 4,81E+01 1,46E+02 8,60E+01 2,14E+02

H-H 𝜎𝑥𝑥 2,98E+03 2,68E+03 1,64E+03 6,44E+02

𝜎𝑦𝑦 2,97E+01 9,55E+01 5,94E+01 2,04E+02

𝜏𝑥𝑦 9,54E+00 9,73E+01 3,31E+01 1,02E+02


173

Table C. 3: Maximum absolute stresses for the plate with 1 layer (NNRPIM).


C-F 𝜎𝑥𝑥 7,64E+03 8,47E+03 9,58E+03 1,33E+04

𝜎𝑦𝑦 4,11E+01 2,19E+01 2,08E+01 1,07E+02

𝜏𝑥𝑦 2,98E+02 6,77E+02 1,07E+03 1,79E+03

C-C 𝜎𝑥𝑥 6,32E+02 2,05E+02 7,16E+02 3,50E+03

𝜎𝑦𝑦 1,15E+01 3,52E+00 2,27E+01 9,85E+01

𝜏𝑥𝑦 6,32E+00 2,73E+01 8,24E+01 2,55E+02

C-H 𝜎𝑥𝑥 1,13E+03 6,73E+02 5,03E+02 2,97E+03

𝜎𝑦𝑦 1,13E+01 4,78E+00 3,05E+01 1,17E+02

𝜏𝑥𝑦 4,53E+01 1,03E+02 2,05E+02 4,35E+02

H-H 𝜎𝑥𝑥 2,49E+03 2,03E+03 1,23E+03 1,35E+03

𝜎𝑦𝑦 9,45E+00 8,63E+00 3,37E+01 1,15E+02

𝜏𝑥𝑦 1,87E+01 4,16E+01 1,11E+02 2,95E+02

Figure C. 1: Vertical deflection for the corrugated core plate with one layer and C-C boundary conditions.


174

Figure C. 2: Vertical deflection for the corrugated core plate with one layer and C-F boundary conditions.

Figure C. 3: Vertical deflection for the corrugated core plate with one layer and C-H boundary conditions.

Figure C. 4: Vertical deflection for the corrugated core plate with one layer and H-H boundary conditions.


175

Sinusoidal Core Sandwich Plate with Two Corrugated Layers

Table C. 4: Maximum absolute stresses for the plate with 2 layers (FEM).


C-F 𝜎𝑥𝑥 1,42E+04 1,21E+04 1,29E+04 1,64E+04

𝜎𝑦𝑦 4,49E+02 8,23E+02 1,04E+03 8,39E+02

𝜏𝑥𝑦 3,17E+02 1,52E+03 1,78E+03 2,47E+03

C-C 𝜎𝑥𝑥 5,23E+02 1,13E+03 2,49E+03 8,44E+03

𝜎𝑦𝑦 5,28E+01 2,43E+02 4,75E+02 7,41E+02

𝜏𝑥𝑦 2,80E+02 1,08E+02 2,29E+02 1,63E+03

C-H 𝜎𝑥𝑥 1,37E+03 1,04E+03 1,95E+03 7,18E+03

𝜎𝑦𝑦 8,88E+01 1,64E+02 3,28E+02 5,96E+02

𝜏𝑥𝑦 2,75E+02 9,82E+01 2,22E+02 1,68E+03

H-H 𝜎𝑥𝑥 3,94E+03 3,33E+03 1,58E+03 5,69E+03

𝜎𝑦𝑦 4,29E+01 2,16E+02 4,49E+02 6,89E+02

𝜏𝑥𝑦 2,60E+02 1,71E+02 1,34E+02 1,53E+03

Table C. 5: Maximum absolute stresses for the plate with 2 layers (RPIM).


C-F 𝜎𝑥𝑥 1.42E+04 2.03E+04 1.50E+04 2.14E+04

𝜎𝑦𝑦 7.85E+02 1.25E+03 1.52E+03 1.40E+03

𝜏𝑥𝑦 2.71E+02 5.33E+02 1.03E+02 4.35E+02

C-C 𝜎𝑥𝑥 9.56E+02 9.56E+02 1.52E+03 4.66E+03

𝜎𝑦𝑦 1.52E+01 1.52E+01 1.31E+01 9.49E+00

𝜏𝑥𝑦 3.28E+01 3.28E+01 4.33E+01 2.87E+02

C-H 𝜎𝑥𝑥 2.65E+03 1.03E+03 1.06E+03 3.05E+03

𝜎𝑦𝑦 6.03E+01 1.59E+02 1.23E+02 1.41E+02

𝜏𝑥𝑦 6.99E+01 7.51E+01 2.82E+01 1.78E+02

H-H 𝜎𝑥𝑥 5.85E+03 4.87E+03 4.07E+03 2.11E+03

𝜎𝑦𝑦 5.23E+01 1.26E+01 7.95E+01 4.11E+01

𝜏𝑥𝑦 1.39E+02 1.93E+02 5.47E+01 1.31E+02


176

Table C. 6: Maximum absolute stresses for the plate with 2 layers (NNRPIM).


C-F 𝜎𝑥𝑥 1,14E+04 1,11E+04 1,18E+04 1,17E+04

𝜎𝑦𝑦 1,51E+02 1,64E+02 2,21E+02 3,75E+02

𝜏𝑥𝑦 5,74E+02 8,83E+02 9,32E+02 1,66E+03

C-C 𝜎𝑥𝑥 8,05E+02 6,41E+02 1,40E+03 4,90E+03

𝜎𝑦𝑦 9,06E+00 3,67E+00 5,34E+00 3,28E+02

𝜏𝑥𝑦 1,13E+02 3,42E+02 5,62E+02 1,85E+03

C-H 𝜎𝑥𝑥 1,71E+03 9,79E+02 1,03E+03 3,24E+03

𝜎𝑦𝑦 4,40E+00 2,81E+01 3,82E+01 3,64E+02

𝜏𝑥𝑦 9,33E+01 3,19E+02 5,28E+02 1,89E+03

H-H 𝜎𝑥𝑥 3,89E+03 2,99E+03 1,87E+03 2,74E+03

𝜎𝑦𝑦 1,64E+01 1,18E+01 1,25E+01 3,68E+02

𝜏𝑥𝑦 1,83E+02 4,66E+02 7,52E+02 2,05E+03

Figure C. 5: Vertical deflection for the corrugated core plate with two layers and C-C boundary conditions.


177

Figure C. 6: Vertical deflection for the corrugated core plate with two layers and C-F boundary conditions.

Figure C. 7: Vertical deflection for the corrugated core plate with two layers and C-H boundary conditions.

Figure C. 8: Vertical deflection for the corrugated core plate with two layers and H-H boundary conditions.

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