Static and free vibration analyses of corrugated panels using homogenization models and a smoothed finite element method

In this thesis, the cell-based smoothed three-node Mindlin plate element CS-MIN3 based on the first-order shear deformation theory FSDT is integrated with typical homogenization models t

NGUYỄN MINH NHÂN

STATIC AND FREE VIBRATION ANALYSES OF

CORRUGATED PANELS USING HOMOGENIZATION MODELS AND A SMOOTHED FINITE ELEMENT METHOD

(PHÂN TÍCH ỨNG XỬ TĨNH VÀ DAO ĐỘNG TỰ DO CỦA TẤM LƯỢN SÓNG SỬ DỤNG CÁC MÔ HÌNH ĐỒNG NHẤT HÓA VÀ

PHƯƠNG PHÁP PHẦN TỬ HỮU HẠN TRƠN)

CHUYÊN NGÀNH: CƠ HỌC TÍNH TOÁN

MÃ SỐ CHUYÊN NGÀNH: 60460136

LUẬN VĂN THẠC SĨ

TP HỒ CHÍ MINH – 2017

Cán bộ hướng dẫn khoa học : PGS TS Nguyễn Thời Trung

Cán bộ chấm nhận xét 1 : PGS TS Vũ Công Hòa

Cán bộ chấm nhận xét 2 : TS Lê Minh Hưng

Luận văn thạc sĩ được bảo vệ tại Hội Đồng Chấm Bảo Vệ Luận Văn Thạc Sĩ Trường Đại học Bách Khoa, ĐHQG Tp HCM ngày 13 tháng 07 năm 2017

Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm:

1 Chủ tịch hội đồng : PGS TS Trương Tích Thiện

2 Phản Biện 1 : PGS TS Vũ Công Hòa

3 Phản Biện 2 : TS Lê Minh Hưng

5 Thư Ký : TS Trần Thị Thu Hạnh

Xác nhận của Chủ tịch Hội đồng đánh giá LV và Trưởng Khoa quản lý chuyên ngành sau khi luận văn đã được sửa chữa (nếu có)

CHỦ TỊCH HỘI ĐỒNG TRƯỞNG KHOA KHOA HỌC ỨNG DỤNG

PGS TS Trương Tích Thiện PGS TS Huỳnh Quang Linh

NHIỆM VỤ LUẬN VĂN THẠC SĨ

Ngày, tháng, năm sinh: 24/01/1991

Nơi sinh: Đăk-Lăk

Mã số: 60 46 01 36

I TÊN ĐỀ TÀI: “STATIC AND FREE VIBRATION ANALYSES OF CORRUGATED

PANELS USING HOMOGENIZATION MODELS AND A SMOOTHED FINITE ELEMENT METHOD” hay “PHÂN TÍCH ỨNG XỬ TĨNH VÀ DAO ĐỘNG TỰ DO CỦA TẤM LƯỢN SÓNG SỬ DỤNG CÁC MÔ HÌNH ĐỒNG NHẤT HÓA VÀ PHƯƠNG PHÁP PHẦN TỬ HỮU HẠN TRƠN”

NHIỆM VỤ VÀ NỘI DUNG:

1 Tổng quan các phương pháp đồng nhất hóa điển hình trong phân tích tấm gợn sóng

2 Tổng quan về phương pháp phần tử hữu hạn trơn CS-MIN3 cho phân tích tấm trực hướng

3 Lập trình bằng ngôn ngữ Matlab phân tích ứng xử tĩnh và dao động tự do của tấm gợn sóng hình thang và hình sin sử dụng phương pháp đồng nhất hóa và phần tử tấm CS-MIN3

4 Đánh giá tính chính xác và tin cậy của các mô hình đồng nhất hóa cho tấm gợn sóng và phần tử tấm CS-MIN3

III NGÀY HOÀN THÀNH NHIỆM VỤ: 13/07/2017

IV CÁN BỘ HƯỚNG DẪN PGS TS NGUYỄN THỜI TRUNG

CÁN BỘ HƯỚNG DẪN

Tp HCM, ngày tháng năm 20

CHỦ NHIỆM BỘ MÔN ĐÀO TẠO

PGS TS Nguyễn Thời Trung TS Đỗ Ngọc Sơn

TRƯỞNG KHOA KHOA HỌC ỨNG DỤNG

PGS TS Huỳnh Quang Linh

Abstract

The behavior analysis of complex structures like corrugated panels usually requires high-cost shell modeling which is an obstacle in subsequent studies such as parametric study and optimization To overcome the difficulty, many homogenization models in which equivalent orthotropic plates replaces the original shells have been proposed However, there is a lack of investigations for verifying their accuracy and reliability In this thesis, the cell-based smoothed three-node Mindlin plate element (CS-MIN3) based on the first-order shear deformation theory (FSDT) is integrated with typical homogenization models to give some homogenization methods for static and free vibration analyses of trapezoidally and sinusoidally corrugated panels Using the results of ANSYS and ABAQUS shell simulations as references, the accuracy and reliability of the homogenization methods are assessed through several numerical examples The influences of geometric parameters such as the number of corrugation units, corrugation amplitude, trough angle, and panel thickness are studied and some important remarks about the performances of these methods are derived

Tóm lược

Phân tích ứng xử của các kết cấu phức tạp như tấm gợn sóng thường yêu cầu

mô phỏng vỏ với chi phí cao và đây là một trở ngại trong các nghiên cứu tiếp sau như phân tích tham số và tối ưu hóa Để khắc phục khó khăn này, nhiều mô hình đồng nhất hóa trong đó các tấm trực hướng tương đương thay thế vỏ ban đầu được

đề xuất Tuy nhiên, những khảo sát nhằm xác minh sự chính xác và tin cậy của các

mô hình này còn thiếu Trong luận văn này, phần tử tấm Mindlin được làm trơn dựa trên phần tử (CS-MIN3), dựa trên lý thuyết biến dạng cắt bậc nhất (FSDT), sẽ được tích hợp vào các mô hình đồng nhất hóa điển hình để tạo ra các phương pháp đồng nhất hóa cho phân tích tĩnh vào dao động tự do của tấm gợn sóng hình thang và hình sin Sử dụng kết quả của các mô phỏng vỏ bằng ANSYS và ABAQUS, sự chính xác và tin cậy của các phương pháp đồng nhất hóa được đánh giá thông qua vài ví dụ số Ảnh hưởng của các tham số hình học như số lượng đơn vị gợn sóng, biên độ gợn sóng, góc nghiêng, và độ dày tấm sẽ được nghiên cứu và một vài nhận xét quan trọng về hiệu năng các phương pháp đồng nhất hóa sẽ được đưa ra

List of keywords

Homogenization, Corrugated panel, Asymptotic analysis, Smoothed finite element method (S-FEM), First-order shear deformation theory (FSDT), Cell-based smoothed three-node Mindlin plate element (CS-MIN3)

Acknowledgments

First of all, I would like to send my deep gratitude to my family including my parents and my younger brother Although, many difficulties have come and gone, their concern and belief devoted to me have never wavered

I would also like to express my greatest appreciation to my supervisor, Dr Trung Nguyen-Thoi, for his precious guidance throughout my master study My research cannot reach its final stage without his advice and feedback

Special thanks go to Ph.D student Thang Bui-Xuan who has directed me from the beginning of my research I have strengthened my skill as well as come over many obstacles through discussion with him

Many thanks to my colleagues and friends at Ton Duc Thang University, University of Science, and Bach Khoa University Their opinion, encourage, and support help me a lot to find out the best approach to my problems

Finally, the thesis was partly funded by the University of Science, Vietnam National University Hochiminh City (VNU-HCM) under the grant number T2015-

3 I am very grateful for this support

Lời cam đoan

Tôi xin cam đoan rằng luận văn này là kết quả nghiên cứu của bản thân, được thực hiện sau khi đăng ký chương trình Cao học ngành Khoa học Tính toán thuộc khoa Khoa học Ứng dụng, trường Đại học Bách Khoa, Việt Nam, và chưa được công bố trong bất kỳ luận văn nào khác đã nộp cho khoa hay bất kỳ đơn vị học thuật khác nhằm đạt một bằng cấp hay chứng chỉ

Chữ ký

Nguyễn Minh Nhân

Contents

Abstract i

List of keywords iii

Acknowledgments iv

Declaration v

Contents vi

List of figures viii

List of tables xi

Publications xiv

Chapter 1 Introduction 1

1.1 Motivation 1

1.2 Review of Previous Work 1

1.3 Present Work and Outline 4

Chapter 2 Homogenization Methods for Corrugated Plates 5

2.1 Introduction to homogenization of corrugated structures 5

2.2 Equivalent stiffness terms of some typical homogenization models 8

Chapter 3 Smoothed Finite Element Methods for Analyses of Equivalent Plates 11

3.1 Galerkin weak form for static and free vibration analyses of equivalent Reissner-Mindlin orthotropic plate 11

3.2 A brief introduction to CS-MIN3 12

Chapter 4 Numerical Examples for Free Vibration Analysis 15

4.1 Validation example 16

4.2 Free vibration analysis of a trapezoidally corrugated panel 18

4.3 Effect of some geometric parameters 22

4.3.1 Number of corrugation unit n 22 c

4.3.2 Corrugation Amplitude 23

4.3.3 Trough angle 26

4.3.4 Thickness 27

4.4 Free vibration analysis of a sinusoidally corrugated panel 28

4.5 Effect of some geometric parameters 32

4.5.1 Number of corrugation units 33

4.5.2 Corrugation Amplitude 34

4.5.3 Thickness 36

Chapter 5 Numerical Examples for Static Analysis 39

5.1 Validation example 39

5.2 Static analyses of a trapezoidally corrugated panel 42

5.2.1 Effect of the number of corrugation units 44

5.2.2 Effect of corrugation amplitude 47

5.2.3 Effect of trough angle 49

5.3 Static analysis of a sinusoidally corrugated panel 50

5.3.1 Effect of the number of corrugation units 51

5.3.2 Effect of thickness 54

Chapter 6 Conclusions and Future Works 56

6.1 Conclusions 56

6.2 Future Work 57

Appendix 59

References 61

List of figures

Fig 1-1 a) A corrugated panel and its representative volume element (RVE), b) A unit of trapezoidal corrugation, c) A unit of sinusoidal corrugation 1Fig 2-1 a) A representative volume element (RVE) of a corrugated panel and its equivalent plate element, b) A Reissner-Mindlin plate and its field variables, c) CS-MIN3 element with three sub-triangles 6Fig 4-1 Typical constructive meshes for a) CS-MIN3 plate model, b) ABAQUS plate model, c) shell model of trapezoidally corrugated panel, and d) shell model of sinusoidally corrugated panel 16Fig 4-2 Relative errors between non-dimensional frequencies of a square orthotropic plate computed by different methods and those computed by ABAQUS 17Fig 4-3 Mode 10 and Mode 11 of a simply supported orthotropic plate which are computed by ABAQUS and three different smoothed finite element methods 18Fig 4-4 The 1st, 3rd, 4th, 10th, and 16th mode shapes (ordered by increasing frequencies) of a trapezoidally corrugated panel and its equivalent plates in SSSS boundary condition 20Fig 4-5 Relative errors between frequencies of homogenization methods and ANSYS Workbench shell simulation in modal analysis of a trapezoidally corrugated panel in SSSS and CCCC boundary conditions 22Fig 4-6 Relative errors between fundamental frequencies of homogenization methods and ANSYS APDL shell simulation of a trapezoidally corrugated panel when the number of corrugation units (n ) is changed 23 c

Fig 4-7 Relative errors between fundamental frequencies of homogenization models and ANSYS APDL shell simulation of a trapezoidally corrugated panel when the corrugation amplitude is changed 25

Fig 4-8 Relative errors between fundamental frequencies of homogenization methods and ANSYS APDL shell simulation of a trapezoidally corrugated panel when the trough angle is modified 27Fig 4-9 Relative errors between fundamental frequencies of homogenization methods and ANSYS APDL shell simulation of a trapezoidally corrugated panel when the thickness is changed 28Fig 4-10 The 13th, 14th, 15th, 16th, and 17th mode shapes (ordered by increasing frequencies) of a sinusoidally corrugated panel and its equivalent plates in CCCC boundary condition 30Fig 4-11 Relative errors between frequencies of homogenization methods and ANSYS Workbench shell simulation in modal analysis of a sinusoidally corrugated panel in SSSS and CCCC boundary conditions 32Fig 4-12 Relative errors between fundamental frequencies of homogenization methods and ANSYS APDL shell simulation of a sinusoidally corrugated panel when the number of corrugation units (n ) is changed 34 c

Fig 4-13 The fundamental mode shapes of the sinusoidally corrugated panel whose corrugation amplitude is high (r f  4) 36Fig 4-14 Relative errors between homogenization methods and ANSYS APDL shell simulation of a sinusoidally corrugated panel when the corrugation amplitude

is changed 36Fig 4-15 Relative errors between fundamental frequencies of homogenization methods and ANSYS APDL shell simulation of a sinusoidally corrugated panel when the thickness is changed 38Fig 5-1 Relative errors between non-dimensional deflections by CS-MIN3 and other FEMs and those by ABAQUS plate simulation 42Fig 5-2 Deflection along central lines of simply supported trapezoidally corrugated panels under a uniform pressure (ABAQUS-VAPAS is the analysis of the homogenization model VAPAS using ABAQUS plate simulation) 45

Fig 5-3 Deflection along central lines of clamped trapezoidally corrugated panels under a uniform pressure when the number of corrugation units is changed (ABAQUS-VAPAS is the analysis of the homogenization model VAPAS using ABAQUS plate simulation) 46Fig 5-4 Contour plots of the deflections of a) a simply supported trapezoidally corrugated panel (by ANSYS shell simulation) and b) its equivalent plate (by CS-MIN3-YeHM) 47Fig 5-5 Relative errors between central deflections by homogenization methods and those by ANSYS shell simulation when the corrugation amplitude of a trapezoidally corrugated panel is changed 49Fig 5-6 Relative errors between central deflections by homogenization methods and those by ANSYS shell simulations when the trough angle of a trapezoidally corrugated panel is changed 50Fig 5-7 Deflection along central lines of simply supported sinusoidally corrugated panels under a uniform pressure (ABAQUS-VAPAS is the analysis of the homogenization model VAPAS using ABAQUS plate simulation) 52Fig 5-8 Deflection along central lines of clamped sinusoidally corrugated panels under a uniform pressure (ABAQUS-VAPAS is the analysis of the homogenization model VAPAS using ABAQUS plate simulation) 53Fig 5-9 Contour plots of the deflections of a) a clamped sinusoidally corrugated panel (by ANSYS shell simulation) and b) its equivalent plate (by CS-MIN3-YeHM) 54Fig 5-10 Relative errors between central deflections by homogenization methods and those by ANSYS shell simulation when the thickness of a sinusoidally corrugated panel is changed 55

List of tables

Table 2-1 Equivalent extensional, bending, and transverse shear stiffness terms for corrugated panels of trapezoidal, sinusoidal, and general profiles 8Table 4-1 The first thirteen non-dimensional frequencies of a square orthotropic plate in simply supported (SSSS) and clamped (CCCC) boundary conditions 17Table 4-2 Equivalent stiffness terms for a trapezoidally corrugated panel with ten corrugation units 19Table 4-3 The first fifteen natural frequencies of a simply supported (SSSS) trapezoidally corrugated panel with ten corrugation units 20Table 4-4 The first fifteen natural frequencies of a clamped (CCCC) trapezoidally corrugated panel with ten corrugation units 21Table 4-5 Fundamental frequencies of a trapezoidally corrugated panel corresponding to different values of number of corrugation units 23Table 4-6 Ratios of stiffness terms between homogenization models and the flat plate f 0 when the amplitude of corrugation is reduced 16 times 24Table 4-7 Fundamental frequencies of a trapezoidally corrugated panel corresponding to different levels of corrugation amplitude 25Table 4-8 Fundamental frequencies of a trapezoidally corrugated plate corresponding to different values of trough angle 26Table 4-9 Fundamental frequencies of a trapezoidally corrugated panel corresponding to different levels of panel thickness 27Table 4-10 Equivalent stiffness terms for a sinusoidally corrugated panel with nine corrugation units 29Table 4-11 The first nineteen natural frequencies of the simply supported (SSSS) sinusoidally corrugated panel with nine corrugation units 31Table 4-12 The first nineteen natural frequencies of the clamped (CCCC) sinusoidally corrugated panel with nine corrugation units 31

Table 4-13 Fundamental frequencies of a sinusoidally corrugated panel corresponding to different values of number of corrugation units 33Table 4-14 Fundamental frequencies of a sinusoidally corrugated panel corresponding to different levels of corrugation amplitude 35Table 4-15 Fundamental frequencies of a sinusoidally corrugated panel corresponding to different levels of panel thickness 37Table 5-1 Fifteen cases for static analysis of an orthotropic plate 40Table 5-2 Non-dimensional central deflection of a simply supported (SSSS) orthotropic plate under uniform pressure 41Table 5-3 Non-dimensional central deflection of a clamped (CCCC) orthotropic plate under uniform pressure 42Table 5-4 Equivalent stiffness terms for a trapezoidally corrugated panel with 9 corrugation units 43Table 5-5 Relative errors between central deflections by homogenization methods and those by ANSYS shell simulation when the number of corrugation units of a trapezoidally corrugated panel is changed 46Table 5-6 Comparison of central deflection of a simply supported (SSSS) trapezoidally corrugated panel under a uniform pressure when the corrugation amplitude is changed 47Table 5-7 Comparison of central deflection of a clamped (CCCC) trapezoidally corrugated panel under a uniform pressure when the corrugation amplitude is changed 48Table 5-8 Central deflections of a trapezoidally corrugated panel under a uniform pressure when the trough angle is changed 49Table 5-9 Equivalent stiffness terms for a sinusoidally corrugated panel with 11 corrugation units 51

Table 5-10 Relative errors between central deflections by homogenization methods and those by ANSYS shell simulation when the number of corrugation units is changed 53Table 5-11 Comparison of central deflections of a simply supported (SSSS) sinusoidally corrugated panel under a uniform pressure when the thickness is changed 54Table 5-12 Comparison of central deflection of a clamped (CCCC) sinusoidally corrugated panel under a uniform pressure when the thickness is changed 55

Publications

The research is related to the following publications

1 Nguyen-Minh N, Tran-Van N, Bui-Xuan T, Nguyen-Thoi T Static analysis

of corrugated panels using homogenization models and a cell-based smoothed Mindlin plate element (CS-MIN3) Frontiers of Structural and Civil Engineering (FSCE), 2017 (Submitted)

2 Nguyen-Minh N, Tran-Van N, Bui-Xuan T, Nguyen-Thoi T Free vibration

analysis of corrugated panels using homogenization models and a cell-based smoothed Mindlin plate element (CS-MIN3) Thin-Walled Structures, 2017 (Submitted)

Chapter 1 Introduction

1.1 Motivation

With high stiffness-to-weight ratio, corrugated panels are incorporated in many structural components in civil, mechanical, marine, and aerospace engineering [1] Furthermore, the anisotropic property due to different flexure characteristics in two perpendicular directions makes them become promising structures in morphing designing [1] In practical applications, the overall behavior of corrugated panels is usually taken into consideration [2] However, the subsequent precise analyses using finite element method (FEM) involve high computational cost which makes the parametric studies and optimization of the structures expensive To overcome the obstacle, many homogenization methods in which a flat orthotropic plate with equivalent stiffness replaces the original panel have been proposed (see Fig 1-1a)

Fig 1-1 a) A corrugated panel and its representative volume element (RVE), b) A unit of

trapezoidal corrugation, c) A unit of sinusoidal corrugation

1.2 Review of Previous Work

There is a rich literature about homogenization techniques for an extensive range of corrugated structures from single-layer sheets of isotropic [2–12] or composite

material [2, 11] to multi-layer structures like corrugated core sandwich plates [13–19] and corrugated laminates [20] Some of these homogenization methods are employed in parameter studies and optimization processes of corrugated structures [12, 21, 22] Most of the existing homogenization methods belong to engineering approaches in which different boundary conditions and assumptions of strain/stress distribution are used to derive the formulas of equivalent stiffness terms

Shimansky and Lele [4] derived an analytical model for initial transverse stiffness of a sinusoidally corrugated panel and then highlighted the impact of plate thickness and degree of corrugation to this stiffness component Also working with

a sinusoidal profile, Briassoulis [3] reviewed existing classical equivalent models and modified expressions of some extensional and flexural rigidities Samanta and Mukhopadhyay [5] used a similar approach to determine the equivalent extensional rigidities for trapezoidally corrugated sheets Combining them with the equivalent flexural rigidities derived by McFarland [23], the authors then qualified the resulting model in buckling, linear static, geometric nonlinear, and free vibration analyses Liew et al [6–9] used a meshless Galerkin method to investigate many mechanical behaviors of stiffened corrugated panels of sinusoidal and trapezoidal profiles In their studies, besides modifying formulas of some equivalent flexural rigidities, the authors also employed the derivation of equivalent transverse shear terms [24] in their FSDT plate model Xia et al [2] formulated generalized expressions to estimate equivalent stiffness terms for thin corrugated laminates of any shape This generalized model was then extended to cover the transverse shear stiffness by Park et al [11]

An approach using asymptotic method for homogenization of thin corrugated panels was proposed by Ye et al [10, 25] The method was based on governing equations of a thin shell theory in which the variable fields were represented in asymptotic expansions and then substituted back into the equations The consequently derived systems of governing differential equations were used to find the relationship between the equivalent plate and the original structure The method

could handle corrugations of any shape as well as provide a set of formulas to recover the local fields in the corrugated panels Although there are many proposal treatments for corrugated panels, a deeper study of their effectiveness and accuracy

On another front of the development of numerical methods, Liu and Thoi [26] have integrated the strain smoothing technique [27] into the FEMs to create a series of smoothed finite element methods (S-FEMs), such as the cell/element-based smoothed FEM (CS-FEM) [28], the node-based smoothed FEM (NS-FEM) [29, 30], the edge-based smoothed FEM (ES-FEM) [31], and the face-based smoothed FEM (FS-FEM) [32] These S-FEMs with different properties have been applied to improve the solution for a wide class of benchmark and practical mechanics problems, especially two-dimensional (2-D) and three-dimensional (3-D) linear elastic mechanics problems Related to analyses of plate and shell structures, the smoothing techniques have improved the performance of traditional plate elements such as DSG3 [33], MITC4 [34] and MIN3 [35] to produce smoothed counterparts such as ES-DSG3 [36], NS-DSG3 [37], CS-DSG3 [38], ES-MIN3 [39], CS-MIN3 [40], and MISCk [41]

Nguyen-Among these smoothed plate elements, CS-MIN3 [40], which is a combination of the cell-based smoothed technique and the Mindlin plate element, MIN3 [35], possess many significant computational advantages In this method, each triangular element is divided into three sub-triangles on which the MIN3 is

employed locally to compute the strains Afterward, the strain smoothing technique

on the entire element is applied to smooth the strains on these sub-triangles The CS-MIN3 is free of shear locking and achieves high accuracy compared to exact solutions and some other existing elements in the literature [40] Compared to other S-FEMs, the CS-FEM is much simpler in implementation since it is conducted within a target element and does not need any extra information from adjacent elements With the mentioned advantages, the CS-MIN3 has been applied in different analyses of plate structures For examples, isotropic plates [40, 42], laminated composite plates [43, 44], functional graded plates [45], and recently cracked FGM plates [46]

1.3 Present Work and Outline

This thesis hence further extends the CS-MIN3 by integrating itself with homogenization models to give equivalent homogenization methods for free vibration and static analyses of corrugated panels of some common shapes

The remainder of the present work is organized as follows The next chapter provides a summary of some most cited homogenization models with equivalent stiffness terms given explicitly Chapter 3 represents the Galerkin weak forms for free vibration and static analyses of equivalent orthotropic plates and a brief formulation of the CS-MIN3 Some numerical examples to evaluate the reliability and accuracy of the homogenization methods are described in Chapter 4 and Chapter 5 Last but not least, the thesis ends up with some concluding remarks in Chapter 6

Chapter 2 Homogenization Methods for Corrugated Plates

2.1 Introduction to homogenization of corrugated structures

In practical applications, depending on the usage and functions, corrugated panel structures possess numerous kinds of shape In this thesis, we only consider shell panels of periodic and symmetric corrugation profiles in one direction Notably, the focus of this work is on trapezoidally and sinusoidally corrugated panels (Fig 1-1b and Fig 1-1c) The geometry of such structures can be defined by several parameters such as the half-period c , half-length l , half-amplitude f , and the

trough angle  In several formulas, the period of a corrugation unit  is used instead of the half one c

The mid-surface of corrugated panels can be represented analytically using

two coordinate systems including a global Cartesian coordinate system xyz and a local coordinate system syn (see Fig 2-1a) The corrugation profile is designed in

the xz -plane and then extruded in y -direction The local coordinate s represents

the arch length at a position on the profile curve The location of a point on the surface in the global coordinate system is defined as

mid-  mid- s y, x s  y z s 

where i , j , and k are unit vectors of x -, y - and z - axes respectively Then we

have d / dx scos and d / dz ssin where  is the tangential angle at that point For instance, the trapezoidal and sinusoidal corrugation profiles in Fig 1-1b and Fig 1-1c are in turn represented by following equations

where x0  c, x1  c f / tan, x2  f / tan, x3 0, and

 1

1

d d 2

where  is the mid-surface domain, N N s N y N syT,

the extensional, bending, and transverse shear stiffness terms in A , D , and b D , s

respectively

2.2 Equivalent stiffness terms of some typical homogenization models

As mentioned in the introduction section, there are two main approaches for homogenization of corrugated panels In engineering approaches, the RVE is constrained in some specific boundary and loading conditions Equivalent force and energy methods are then applied to derive the equivalent stiffness terms as functions

of material and geometric parameters of the original structure (more details in [2]) This approach is highlighted in many studies about trapezoidally and sinusoidally corrugated panels by Samanta et al [5], Peng et al [6–9] and Xia et al [2] Using thin shell governing equations, the asymptotic approach proposed by Ye et al [10]

is also applicable for generalized corrugation profiles Especially, their study finds out that the equivalent orthotropic model with no extension-bending coupling is only suitable for symmetric corrugation profiles For convenience, from this point onwards, we label the above four homogenization models as SamantaHM, PengHM, XiaHM, and YeHM, respectively

Consider a corrugated panel containing n corrugation units with half-period c c

and uniform thickness h We assume that the panel is made from an isotropic material with Young’s modulus E, Poisson’s ratio  , and uniform density  The equivalent models for the panel when its profile is a trapezoidal, sinusoidal, or general one are respectively given in Table 2-1

Table 2-1 Equivalent extensional, bending, and transverse shear stiffness terms for corrugated panels of trapezoidal, sinusoidal, and general profiles

Term

s

Trapezoidal profile Sinusoidal

profile General profile (symmetric) SamantaH

M [5] PengHM [8] PengHM [8] XiaHM [2] YeHM [10]

11

A

3 2

212

11

Eh





l Eh

12 12 11 2

11 22 12 11

A A A

2 1

Eh C

1

D C

12 1

T

Eh

Eh c

2 11

[8] The constants k and 1 k depend on the number of corrugation units 2 n , the c

half-amplitude f , the thickness h , and the total length L as follows

 

2 2 1

2 2 2 2

2 2 0

Chapter 3 Smoothed Finite Element Methods for Analyses of

Equivalent Plates

The homogenization models represented in Chapter 2 are now used to approximate the static and free vibration behavior of the original corrugated panels Particularly, the first-order shear deformation theory (FSDT) is applied in the derivation of the Galerkin weak forms while the problems are then solved using the cell-based smoothed MIN3 (CS-MIN3)

3.1 Galerkin weak form for static and free vibration analyses of equivalent Reissner-Mindlin orthotropic plate

The displacements in x -, y -, and z -directions of the plate (Fig 2-1b) are

respectively given as follows

about y - and x - axes of its normal Using the small deformation theory, the

membrane, bending, and transverse shear strains of the equivalent orthotropic plate are

ε L u κ L u γ L u (3.2) where uT  u v w x y is the field vector containing displacements and rotation angles and the partial differential operators are

When the panel is subjected to a distributed load

3.2 A brief introduction to CS-MIN3

In the formulation of the CS-MIN3, each triangular element e is divided into three sub-triangles 1, 2, and 3 by connecting the central point O of the element to three elemental nodes 1 , 2 , and 3 (see Fig 2-1c) Then, in each sub-triangle, the MIN3 is used to estimate the strain fields The formulation of MIN3 on the whole element 1, 2, 3 can be described shortly as follows 

Firstly, the rotation angles x and  y are approximated linearly and the

deflection w is approximated quadratically We have

3 1

i i

i



in which d and i  F are nodal displacement vector and shape function matrix i

corresponding to the node i

i i

N N

N N

The shape functions N , i H , i L are given as in [35] As a result, the strain fields of i

a MIN3 element are

where A is the area of the entire triangular element e

Let the central point of the element e be the node 0 Respectively replacing the triangle 1, 2, 3 in the above formulation by the sub-triangles  1, 2, 0 ,  0, 2, 3 and 1, 0, 3 , we can derive the MIN3 approximation of strain fields in each sub-element

e e

The displacement vector d at the center point 0 is simply the average of the 0e

three displacement vectors d , 1e d , and 2e d , i.e 3e d0e d1e d2e d3e/ 3 As a result, the smoothed strains and the smoothed elemental stiffness matrix of the CS-MIN3 are derived as follows

Chapter 4 Numerical Examples for Free Vibration Analysis

In this section, we conduct several numerical examples to verify the accuracy and reliability of some typical homogenization models in free vibration analyses of trapezoidally and sinusoidally corrugated panels First, a validation study will clarify the computational ability of the CS-MIN3 in free vibration analysis of orthotropic plates Second, the CS-MIN3 will be integrated with four typical homogenization models (SamantaHM, PengHM, XiaHM, and YeHM) to give four homogenization methods labeled by CS-MIN3-SamantaHM, CS-MIN3-PengHM, CS-MIN3-XiaHM and CS-MIN3-YeHM, respectively Here, the equivalent transverse shear terms in CS-MIN3-PengHM are added to the other three methods

to make them compatible with the FSDT plate model The four methods are applied

to derived the free vibration behavior of two typical panels which were first analyzed in [9] Their results, including frequencies and mode shapes are compared

to those of ANSYS Workbench, ANSYS APDL, and ABAQUS shell simulations Finally, the effect of some geometric parameters such as number of corrugation units, corrugation amplitude, trough angle, and panel thickness on the accuracy of homogenization methods is figured out Consistently, in all numerical examples, orthotropic plates are analyzed using constructive meshes of 21x21 nodes (Fig 4-1a and Fig 4-1b) and original corrugated panels are simulated using constructive rectangular meshes (Fig 4-1c and Fig 4-1d) The element types used in ANSYS and ABAQUS are SHELL181 and S4R, respectively

Fig 4-1 Typical constructive meshes for a) CS-MIN3 plate model, b) ABAQUS plate model, c) shell model of trapezoidally corrugated panel, and d) shell model of sinusoidally corrugated panel

4.1 Validation example

In this example, we will evaluate the accuracy of the CS-MIN3 in the free vibration analysis of a square orthotropic plate whose material and geometric properties are given by

Table 4-1 represents the first thirteen non-dimensional frequencies of the plate The ABAQUS results are taken as references beside those of the exact 3D analysis [47] (in SSSS case only) The relative errors between these S-FEMs and ABAQUS are depicted in Fig 4-2 We see that all S-FEMs agree well with ABAQUS and exact 3D analysis The agreement in mode shape is also attained Specially, as

shown in Fig 4-3, only CS-MIN3 agrees with ABAQUS at the 10th and 11th mode shapes of the plate in SSSS boundary condition

Table 4-1 The first thirteen non-dimensional frequencies of a square orthotropic plate in simply supported (SSSS) and clamped (CCCC) boundary conditions

CS- DSG3

ES-ABAQU

S

Exact [47]

DSG3

MIN3

CS- DSG3

Fig 4-2 Relative errors between non-dimensional frequencies of a square orthotropic plate

computed by different methods and those computed by ABAQUS

Fig 4-3 Mode 10 and Mode 11 of a simply supported orthotropic plate which are computed by

ABAQUS and three different smoothed finite element methods

4.2 Free vibration analysis of a trapezoidally corrugated panel

We now conduct a modal analysis of a square trapezoidally corrugated panel using four homogenization methods CS-MIN3-SamantaHM, CS-MIN3-PengHM, CS-MIN3-XiaHM, and CS-MIN3-YeHM The material and geometric parameters of the panel are given as follows

The equivalent stiffness terms and densities are listed in Table 4-2 We see that the four equivalent models have similar extensional terms The flexural terms

of SamantaHM are significantly different from other models Fig 4-4 visualizes the

1st, 3rd, 4th, 10th, and 16th mode shapes of the panel and equivalent plates in SSSS boundary condition Here, the order is defined by increasing frequencies We see that all four equivalent methods agree with ANSYS about the 1st mode shape while disagree with it about the 16th one Compared to ANSYS result, the first wrong mode of CS-MIN3-SamantaHM is 3 while that of CS-MIN3-XiaHM is 10 Both CS-MIN3-PengHM and CS-MIN3-YeHM begin their failure at the 16th mode We recognize that the wrong mode shape likely comes from a swap between two close modes For example, the mode 3 and mode 4 of CS-MIN3-SamantaHM (Fig 4-4)

To make the comparison between homogenization models more reasonable,

we rearrange the first fifteen modes of SamantaHM and XiaHM to attain a consistency with the arrangement in ANSYS result In this new arrangement, the first fifteen natural frequencies of the four homogenization models

CS-MIN3-in SSSS and CCCC boundary conditions are listed CS-MIN3-in Table 4-3 and Table 4-4, respectively With the mean absolute percentage error (MAPE) of 13.68% in SSSS case and of 8.94% in CCCC case, CS-MIN3-SamantaHM cannot produce good prediction for all modes Fig 4-5 shows the relative errors between natural frequencies of CS-MIN3-PengHM, CS-MIN3-XiaHM, and CS-MIN3-YeHM and those of ANSYS The three methods give well-agreed results compared to ANSYS, especially at low frequency modes It should be noticed that the frequencies of CS-MIN3-YeHM are bounded by those of the other two methods

Table 4-2 Equivalent stiffness terms for a trapezoidally corrugated panel with ten corrugation units

M

PengHM

XiaHM

YeHM

CS-MIN3-PengHM [9]

Meshfree-Shell

SamantaHM

PengHM

XiaHM

YeHM

CS-MIN3-PengHM [9]

Meshfree-Shell

MAPE 8.94 % 1.67 % 2.81 % 1.43 % 5.14 % -

Fig 4-5 Relative errors between frequencies of homogenization methods and ANSYS Workbench shell simulation in modal analysis of a trapezoidally corrugated panel in SSSS and CCCC boundary

conditions

4.3 Effect of some geometric parameters

Using the previous trapezoidally corrugated panel, we evaluate the accuracy of the four homogenization models for various values of the number of corrugation units, corrugation amplitude, trough angle and panel thickness In these parametric studies, we take the results of ANSYS APDL shell simulations as references

4.3.1 Number of corrugation unit n c

The fundamental frequencies of the panel when the number of corrugation units n c

is modified are listed in Table 4-5 The relative errors (compared to ANSYS APDL results) of four homogenization methods and other benchmark software are represented in Fig 4-6 Except CS-MIN3-SamantaHM, other homogenization methods produce acceptable results at many levels of corrugation degree We also noticed the results of YeHM are softer than XiaHM while stiffer than PengHM

Table 4-5 Fundamental frequencies of a trapezoidally corrugated panel corresponding to different values of number of corrugation units

c

Cond

Fundamental frequency (Hz) CS-MIN3-

SamantaH

M

PengHM

XiaHM

YeHM

CS-MIN3- Shell

ABAQUS-Workbench -Shell

Shell

of stiffness terms between homogenization models and the flat plate ( f 0) in the