A solid shell element based on relative displacements concept theory and applications

Conventional shell elements with curved two dimensions fail to investigate the mechanical variation along the thickness direction because the mechanical parameters of the conventional sh

RELATIVE DISPLACEMENTS CONCEPT:

THEORY AND APPLICATIONS

MA YONGQIAN

NATIONAL UNIVERSITY OF SINGAPORE

2006

DISPLACEMENTS CONCEPT: THEORY AND APPLICATIONS

MA YONGOIAN B.Eng (Nanjing University of Aeronautics and Astronautics, P.R.China),

M.Eng (Zhejiang University, P.R.China)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

2006

The author wishes to express a sincere appreciation to his supervisor, Associate Professor Ang Kok Keng for his invaluable advice, effective guidance and constructive criticism throughout this study The author wishes to express a profound gratitude to Professor Wang Chien Ming for his effective, efficient guidance and great encouragement Professor Wang’s quick mind and clear concepts in mechanics impress the author deeply

The author would also thank his colleagues, Dr Jin Jing, Dr Tua Puat Siong, Mr Duan Wenhui, Mr Li Zhijun, Mr Zhou Enhua and Ms Zhang Yingyan for their useful discussions and suggestions

The author would also show an appreciation to National University of Singapore for rendering the necessary support to carry out this research

Finally, Ms Li Hongxia, the wife of the author, is also appreciated for her understanding, patience and continuous support throughout the entire study

1.3 MOẨIVAfIOT Ặ.QQ SH n ST HT HT ke 8 L.A ODJeCtives cc ccccccce ee eeeeececeteeesssseeeeeeeseceeennttaaaeeeeeeees 10 V5 Outline eee cece eceeeeeeneeecneeecneeesseeesteeeeneeesteeestieeesteeenses 11 Chapter 2 Mathemafical Model , 55s ssssssss 13 2.1 Geometric definition of a shelÏ 2 + 2+2 ‡‡+2xvssseeesss 14 2.2 Differential geometry on the mid-surface - s++scc5>› 16 2.3 3D differential geometry for shells cece ceeeetteeeees 21 Chapter 3 Solid Shell TÍh€OrFY 5-5 555555SSSS% 23 3.1 Comparison of the displacement mode in existing solid shell t€OTI€S 0022 2222001111 111111111111 001111 1111k KT ng 1 1k ket 24 3.2 The proposed relative displacements concept .- - 26 3.2.1 From solid to shelÏ - L2 111222221111 1122111 1111522111 111522111 11122111 keg 26 3.2.2 Single-layered solid shelÏL - 111222221111 11222211 1115221111115 22111 e2 28

3.3 Main asSu1npfIOIS 1111 111111111551582111 111111 re 31 3.4 Coordinaf€ SVSf€INS 111111111 111111121211 11111 1tr he 32 3.5 Boundary condIfiONS - - c c1 1111121211111 1111111 re 33

°nHuaa4 35 3.7 Fimite element formulation ¿+ + + 22222 ‡++szseevesserss 36 3.7.1 — PoftentiaÏl energy c1 2222 11112222 1111222 111tr ch 36 3.7.2 Assumed natural strain method - 2221122 22222511222221xxsex 37 3.7.3 Tensor transformatfIon 2 1 2222111111321 1 1115221111 1152211 111152 k2 38 3.7.4 Shape functions 1220111 12222111112221 1111152211111 1 5211111 39 3.7.5 Stifness mafrIX 0002222211 111122211 1111522111 1112111 1111122111 ktg 41 3.8 Numerical f€SfS - 001122222111 111115821 111111581111 ky 43 3.8.1 Patch test for plateS 2222011122222 1111522 1111k eg 43 3.8.2 Cantilever linear anaÏyS1s - 112222221111 1522121 1111522211111 àg 47 3.8.3 A square plate under a concentrated load - ¿cc +22 2ccsccs>: 49 3.8.4 Scordelis-Lo roof (single-curved shel]) ¿¿ 2222222222 2zxss2 51 3.8.5 Twisted cantilever beam - L2 1222221111112 1 111522111112 kg 53 3.8.6 Spherical roof (double-curved shell) .0000000ccccecceccccceeteeeetetees 54 S999: a 55 Chapter 4 Composife PÏAÍ€S 3555555555556 S7 4.1 IntroduCfIOTI 21 1222222111111 155 1111111158 1111111 xe 58 4.2 Simply supported (0/90/0) laminated plate under uniform load 6] 4.3 Simply supported (45/-45/45/-45) laminated plate under uniform

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5.] IntroducfHOT c c2 1111111221111 1 1155211111111 k vn ky 68 5.2 Mindlin theory with relative displacements concept 69 5.2.1 Displacement componenfs - c1 2222111111222 1 11152211 trse 69 5.2.2 Strain-displacement reÏafions - 2 1 2222221111221 11152211 xe2 69 5.243 Stress resultant-displacement reÏations -::c 2222222 ssssc2 70 5.2.4 Energy functionals§ - 22.11122222 11112222 111tr rree 79 5.2.5 Governing equations OŸ mofiOï - 22 2222211222221 E222k2 71 5.2.6 The rotary Inertia efÍfect - -.- L1 2221111 112212111 115221111115 2211 1111 àg 71 5.3 Natural frequencies of a circular plate 0 - -<<<5: 72 5.3.1 Frequency parameters A= oR? | ph/D of circular plates with soft simply supported edge (Case Ï) 20000222211 11112221211 1115221111121 x tre 74 5.3.2 Frequency parameters 4= oR? | ph/D of circular plates with clamped edge (Case 2) cece eee eee eee cn reece tee eb cbbeteeeecttseeeeetttiteeeetties 77 5.3.3 Frequency parameters A = wR’ /ph/D of circular plates with free edge

5.4 Discussion and COncluSIONS .0 ccccceeceeeeeeeeeeteeeeeteeeeetteeeeens 82

6.] IntroduCfION ccecececeeeceeeeeeeeeeetseeeeetseeeetieeeenieeeeens 85 6.2 Modeling 0 2222222111111 11H HT HT TT 011111 1k khe he S8 6.3 Results and điSCuSSIOT - 2 1 222011112211 11 1511111158111 rg 9]

6.3.1 Comparison with MD results .cccccccccccsssesseessesseessessessvesvessessesseee 91 6.3.2 Parametric Study c.ccccscsccscsssecsessvessessesssessessvessessesssessessvssvessessvesveee 95

6.4 Con€ÏUSIOTS 001112201 11112211 1112111115111 1 11181111192 khe 106 Chapter7 Buckling ofSuper Flipsoidal Shells 108

7.2 Problem Definifion - -c c1 2222111111232 11111511 crsee 111 7.3 Finite Element Mesh Design 22222222222 1c csssssea 113 7.4 Results and Discussion .0 00cccccccceeececeeeeeeeeeneeeetneeeeeteeees 113 7.5 COnCÏUSIOTS 0001122111112 11111211111 511 11198111119 k tre 125 Chapfer8 Conclusions and Recommendafions - 126

Appendix A The deflection ofa Mindlin plate - 141 AppendxB Critical buckling strain formulas - 141

Conventional shell elements with curved two dimensions fail to investigate the mechanical variation along the thickness direction because the mechanical parameters

of the conventional shell elements are based on the reference surface Therefore, a general three-dimensional solid shell element based on relative displacements concept was proposed The proposed solid shell element takes the strain and stress of the thickness direction into consideration and has two real surfaces, top and bottom, replacing the middle surface in conventional shell elements The integration of the solid shell element is conducted in three dimensions simultaneously, which makes the variations of mechanical behavior in each direction clearly demonstrated These new features bring new applications beyond the conventional concept of shell structures It accurately simulates all types of shell structures, i.e thin, moderately thick and thick shell structures It is much closer to the real shell structures than conventional ones, especially in some drastic mechanical process

This thesis includes two parts The first is to develop a solid shell element based on relative displacement concept, incorporating the Assumed Natural Strain method and Enhanced Assumed Strain method in the formulation in order to avoid some locking phenomena and numerical difficulty The proposed solid shell element was verified against a group of standard benchmark problems The second part is to apply the proposed solid shell element to real engineering cases, including the Composite Laminated Shell (CLS) structures, vibrations of the Very Large Floating Structures (VLSF), infinitesimal buckling of the Carbon NanoTube (CNT) structures and

VI

the applicability and suitability of the proposed solid shell element In the case of CLS structures the thickness integration accuracy of the proposed element was verified From the study of VLSF, the proposed element was examined on its ability to obtain accurate natural frequencies of the structures The study of infinitesimal buckling of CNTs was aimed at checking the capability of the eigenvalue extraction of structural bifurcations The results show that the proposed shell element can pass the standard benchmark problems and the solutions obtained from the proposed solid shell elements are convergent to the analytical solutions The results also show that the proposed solid shell element is versatile and successful when applied in the abovementioned engineering fields This thesis comes to conclusion that the solid shell element with new characteristics has a much wider range of applicability and versatility in the engineering applications

VI

The external load matrix The metric tensor The determinant of a tensor The shear modulus

The thickness of a shell The independent covariant basis in a coordinate The independent contravariant basis in a coordinate The Gaussian curvature

The length of a cylinder of DWNT The moments in the shell edges The shape functions

The origin in coordinates The resultant shear forces in a shell element The external pressure exerted on the shell surface The rotational matrix

3D mathematical space The reference curved plane space of a shell The radius of a cylindrical shell

The natural coordinates in the shell element The time parameter in Chapter 5

The middle surface of a shell The space between walls of DWNT nth-order tensor, superscript omitted when n=1 Contravariant components of a tensor

Covariant components of a tensor

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The displacement components in Cartesian space The displacement components of the shell reference plane in Cartesian space The relative displacement of a points in the shell domain with respect to the reference plane of the shell

The potential energy of external loads The arc-length in the shell reference surface The position vector of a point in continuum before deformation The position vector difference between the shell surface and the reference plane in the deformed configuration

The position vector of a point in continuum after deformation The position vector difference between the shell surface and the reference plane in the undeformed configuration

The global Cartesian coordinates The local Cartesian coordinates attached on the Gaussian integration point The strain tensor

The critical buckling strain The stress tensor

The shear correction factor The Kronecher symbol The Poisson ratio The rotational angles of a vector The coordinates of a point in a certain coordinate The closure of middle surface domain of a shell The mapping from @ to Š

The Euclidean space The closed reference domain The mapping from 2 intod’

The surface Christoffel symbols The total potential energy

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Table 3-1 Global coordinates of patch test nodes 2 1 0222211111222 1111522 11111221 x xe 44 Table 3-2 Nodal forces of membrane patch test - 2 1 2222211111122 1111522111111 xe 45 Table 3-3 Strain and stress components of each element - - -.- c2 22222111 122222111 1152211 x£2 45 Table 3-4 Nodal displacements ofbending patch test 2 2 22222111 222222111 11522213 x°2 46 Table 3-5 Comparison of the results of (a) and (b) 2 1 2222221111122 1111522111111 xe 48 Table 3-6 Comparison of the results of (c) and (d) c1 2222211111222 1111522111111 xe 48

Table 3-7 The Vertical displacement at B (wg) in the Scordelis-Lo roof case 52

Table 3-8 The normalized end displacement in load direction 0.00 00000ccccecccceeteeeentees 53 Table 3-9 The displacement at load DOITĂ 2 22222211 11112221111 115222111 11522111 111522111 xe2 55 Table 4-1 The material properties of the three-layered square plate . -::-: 61

Table 5-1 Coarse mesh of Case Ì - 01022211 121211 111211111011 111011 1110111110111 1 111111111 k tre 74 Table 5-2 Fine mesh of Case Ì 00 22201122211 111211 111211111011 11101 1111011111111 1 1111k key 75 Table 5-3 The finest mesh of Case Ì - L2 1122211111211 111211 111211 111011111 2111111111111 1 key 76 Table 5-4 Coarse mesh of Case 2 - Q0 20002211 111211 111011111011 11101 1110111110111 1 111111111 k tre 77 Table 5-5 Fine mesh of Case 2 2 0200002211 112211 111211111011 111011 1110111110111 1 111111111 k tre 78 Table 5-6 The finest mesh of Case 2 - Q1 2010221111121 111211111011 111011 1110111111111 1 111k key 79 Table 5-7 Coarse mesh of Case 3 - Q0 0000221111121 1 111211111011 111011 1110111110111 1 11 11kg S0 Table 5-8 Fine mesh of Case 3 0000220112221 1111211 111211111011 11101 1111011111111 1111k tre 81 Table 5-9 The finest mesh of Case 3 Q20 102211111211 11121 1111011111011 1 1101111121111 1k tre 82 Table 6-1 Comparison of buckling strains obtained using present solid shell model and MD simulation for DWNT with length 6 nm 2012222211111 122221 111152211111 92 Table 6-2 Critical buckling strains for various radii of SS-SS DWNT with 7⁄8, =10 98

Table 6-3 Critical buckling strains for various lengths of SS-SS DWNT with #; = 0.4 nm 98

Table 6-4 Critical buckling strains for various radii of SS-C DWNT with /⁄R=10 100

Table 6-5 Critical buckling strains for various lengths of SS-C DWNT with &; = 0.4 nm 101

Table 6-6 Critical buckling strains for various radii of C-C DWNT with L/ R; =10 103

Table 6-7 Critical buckling strains for various lengths of C-C DWNT with &; = 0.4nm 103 Table 7-1 Convergence study of finite element results with respective number of elements

Table 7-2 Normalized buckling pressures of spherical shells with various thickness-to-radius

r9 2 115

Table 7-3 Buckling pressures of a super spheroid with various values oŸ?ø 118

Table 7-4 Normalized buckling pressures p/p of ellipsoidal shell (#7 = 1) 121

Table 7-5 Normalized buckling pressures p/p of super ellipsoidal shell (7 = 2) 122

Table 7-6 Normalized buckling pressures p./po of super ellipsoidal shell (7 = Š5) 122

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2-1 Geometric description Oƒq ShelÏ cv 1 1111 11111111111 111111111111 x key 15 2-2 Intersection of the surface with a plane CONTAINING C3 0 cece cette tet 18

3-1 A solid shell element based on relative displacement COHC€JÍ 28

3-2 Relative displacements concept in a single layer solid shell element 29

3-3 Kinematics oƒrelative displacemermfs solid shell eÏeI€HI cà cv 29 3-4, Relative displacements concept in a multi-layered solid shell element 30

“S8 6) nnốốốốốốốeee 33

3-6 Simply-supported bouHdlqFV COHdÌÏÍÍOHS e cece eet e cette tt ttettteees 34 3-7 Clamped boundary COHÌIEÏOHS ccc ccc cece cette eet e te etetettteeeetteteeniaes 34 3-8 Patch test membrane and bending dẰ€ƒOFMAfiOH .Ă à cành xi 44 3-9 Linear anq]JsiS Qƒ CaHÍÍÏ@V€F, ST HS ST HS 1S KTS k TK ket 47 3-10 One quarter of a plate with simply-supported edges and a concentrated load .49

3-11 Resulis oƒ1moderately thick pÌafe (C$€ Ì) à S112 vS SE E19 x11 xxx, 49 3-12 Results oƒ thin pÏafe (CS 2) Ặ- TQ HS S 1 HE S1 v11 v11 1k 11111111111 ky 50 3-13 Scordelis-LO VOOf hốằẦốẦốeẦốeẦeẢ 51

3-14 Pre-twisted Ded 8n raẨỐdẦ1À šẼÈẼ.ẽ 53

K ?) ⁄ 0 nem dete bbtittteeettteeees 54 4-1 Through-the-thickness đisiribufion oƒ the in-pÌlane nor1mdl sÍf€SS Øy 63

4-2 Through-the-thickness distribution of the out-of-plane sheqF' siIeSS Øyz 63

4-3 Through-the-thickness distribution of the out-of-plane shear stress Oyz 0 64

4-4 Through-the-thickness distribulion oƒ the iH-DÌqHe HOFIMdlÏ SÍf€SS Øxx 64

4-5 Through-the-thickness distribufion oƒ the in-plane sheqF Si€SS Øay 65

4-6 Through-the-thickness distribution of the out-of-plane shear stress Oyz 0 65

3-1 A set oƒ typical modal shapes oƒq circHỈqF DÌqf€ 73

6-1 Modeling scheme oƒ DWNT by solid shell eÏeHI€HfS cà cha 89 6-2 Axial linear springs between shell elements of inner and outer walls 90

6-3 Buckling modes of DWNT with ƒour types oƒ clamped eHdi$ à c2 93 6-4 Typical local buckling mode (a) and global buckling mode (b) of DWNT 96

6-6 Comparison of thin shell FEM model and thick shell FEM model with

J////2/8701//272⁄9/8/2282/- E08 t tee bttnttteeeteaes 99 6-7 Comparison of thin shell FEM model and thick shell FEM model with

}////2/1101/12/9/A12282//209/2///J52282/7AEANYYYŒ—ặũạAẠII|)Aa 101 6-8 Comparison of thin shell FEM model and thick shell FEM model with clamped

6-9 Effect of boundary conditions OH CFHICdÏ SÍFYQÏHS 0Q Q ST nhà 105 6-10 Effect of length-to-radius ratio on critical StrQINS à cài 105 7-1 A typical super ellipSOid Ư“nMaaaaŸÂẲỶÝ 111 7-2 Cross sections of a family of super ellipsoidal sheÏÏS cà chi 112 7-3 A prolate ellipsoid and a super ellipsoid constrained at their poles 113 7-4 Curve fitting of normalized buckling pressure of spheroidal shells with various 7/84 A55//20422/// 9/2/72 nh ố.- 117 7-5 Normalized buckling pressures of cubic spheroidal shells with various n 119 7-6 Two typical buckling mode shapes of a cubic spheroidal shell 0 000.0 119 7-7 Comparison of buckling pressures 0ƒ prolafe spheroidal shells 120 7-8 Normalized buckling pressures of ellipsoidal shells with various c/a and b/a rafios and various HN F@ÍÏOS à à HS ST ST kg kg tk tk key 124 7-9 A typical buckling mode of a long general ellipsoidal shell 000.0.0000000.- 124

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1.1 Background

Most shell structures are beautiful externally and also have a very effective load carrying capacity internally A shell structure is a three-dimensional structure, thin in one direction and long in the other two directions The ancient people already knew this fact and constructed many shell structures based on empirical method With the development of calculus and geometry, people established various shell mechanical theories based on various assumptions and simplifications However, only regular shell structures (e.g cylinder, cone and sphere) have analytical solutions[1] and their applications are very limited In the 1960s, with the presence of digital computers and numerical methods, aerospace engineers tried to use these new tools to solve the response of airplane structures which demanded more accurate computation that could not be accomplished by traditional methods Finite Difference Method (FDM)[2] and Finite Element Method (FEM)[3,4] are the main tools of computational mechanics, which brought scientists and engineers into a new era FDM is based on Partial Differential Equations which are not available for general shell structures In contrast, the FEM is based on Energy Functionals which are available for general structure problems So the FEM is more powerful in the computation of shell structures Moreover, once the Energy Functionals or equivalent Energy Functionals exist in a certain physics problem and the solution concerns the minimum of Energy Functionals, FEM can be readily applied Therefore FEM can be applied to various disciplines and inter-disciplines (e.g solid mechanics, fluid mechanics, seepage and magnetism etc.) Furthermore, FEM has the characteristics of relatively simple

theories and strong capabilities in large-scale computations and tedious iterations, which cater for digital computers So with the rapid development of digital computers nowadays, FEM becomes the most efficient tool to analyze shell structures

When analyzed by the FEM, shell structures can be modeled theoretically by using the general three-dimensional solid elements However, modeling in this way will result in very low efficiency, because a large number of elements will be needed to analyze even a simple shell structure Therefore various shell elements have been developed to reduce the computational cost by degenerating a 3-D problem into a 2-D curved plane problem Historically, Exact Shell Elements (2-D prone) and Degenerated Shell Elements, or so-called Continuum-based Shell Elements (3-D prone) are two main branches They simulate the classical shell theory in different directions Specifically, the reduction to resultant form is typically carried out analytically in the Exact Shell Theory and numerically in the Degenerated Shell Theory, although the hypotheses underlying the two theories are essentially the same Recently, Solid Shell Elements were developed and studied from a totally new point

of view These new elements alter some basic hypotheses underlying the aforementioned shell elements and are not consistent with classical shell theory any more These solid shell elements concerned more in the solid behaviour of shell elements, where the normal strain and stress cannot be neglected

The basic assumption of shell theories is straight normals: the normals to the reference plane remain straight, but not be necessarily normal to the reference plane The significance of this basic assumption is that it is possible to analyze a real 3-D

solid structure by a curved 2-D shell theory Also, this assumption leads to the omission of normal strain of shell structures The next section will examine how shell assumptions evolved historically

1.2 Literature review

The degenerated shell element was proposed by Ahmad ef a/.[5] The most important advantage of this kind of element is its simplicity in formulations and applications over the conventional shell element Since the degenerated shell element

is derived directly from fundamental equations of continuum mechanics, some parasitic problems arise due to the differences between shells and continua, which can

be observed in the shell assumptions, as mentioned in the dissertation of Stanley[6]: a) Straight normals: The normals to the reference plane remain straight, but not necessarily normal to the reference plane

b) Incrementally-Rigid normals: The lengths of the normals to the reference plane are unchanged during deformation

Mathematical interpretation of these kinematical hypotheses is

where X is the normal vector of the original reference plane, xis the “normal vector”

of the deformed reference plane and R is the rotational matrix which is given by

0 +6; -9, R=l-9 0 +3 (1.2) +0, -@ 0

After deformation, the normals remain straight and their length unchanged

Another two important assumptions are static hypotheses mentioned by Ahmad ez

al.[5]:

c) The normal stress along the normal is zero

d) The normal strain along the normal is zero too

With these assumptions, shell model has been constructed from three-dimensional continuum However difficulties associated numerical computation will appear

e Low efficiency in computation;

e Locking phenomena: shear locking and membrane locking;

As the degenerated shells come from the three-dimensional continuum, the certain integrations should be implemented in all three dimensions Therefore, the computational cost of this kind of degenerated shell is larger than that of conventional two-dimensional one, especially in multi-layered shell structures To follow the traditional stress and stiffness concepts in classical shell theories, stress-resultant degenerated shell theory and corresponding element was developed by Stanley[6] and Liu et al.[7] It should be emphasized that one additional assumption is made to fulfill this reduction: in each fixed configuration, the transverse material fibers are approximated as normal to the reference This pre-integration technique will also reduce the computational cost as the traditional shell element does Another technique

to reduce the computational effort is the simplification of Jacobian matrix, as detailed

in the dissertation of Stanley[6] and the appendix of the paper of Zienkiewicz ef al.[8] This simplification involves two steps, namely, treating the Jacobian matrix as constant along the normals, and then uncoupling the in-plane part and out-of-plane part in the Jacobian matrix The purpose is to avoid the difficulties when dealing with

displacement derivatives and reduce the three-dimensional problem into two-dimensional one completely

Soon after the paper of Ahmad ef a/.[5], researchers found the plate bending element suffered from the fact that it is much too stiff when thickness is reduced, which is a result of the so-called shear locking Reduced or selective integration by Zienkiewicz et al.[8] and Hughes ef a/.[9] was soon found to be an efficient technique

to circumvent the poor results from shear locking Then, Malkus ef a/.[10] presented the mechanical proof of the reduced integration and selective integration Meanwhile,

it was pointed by Cook[11] and Hughes e¢ a/.[9] that spurious zero-energy or mode rank deficiency would emerge with these reduced or selective integration Substantial research effort was made by Belytschko ef a/.[12,13] and Reese[14] On the other hand, the problem of membrane locking was found in curved element (quadratic or cubic element) by Stolarski et a/.[15,16], The reduced and selective integrations were also found to be efficient in circumventing this kind of membrane locking Furthermore, Cook[17] found membrane locking also occurs in flat element (linear, e.g four-node shell element) and non-conforming displacement mode is regarded as a good tool to solve it by Choi ef a/.[ 18,19]

The Assumed Strain Method proposed by MacNeal ef a/.[20], Dvorkin et al.[21] and Park ef a/[22] is a kind of mixed formulations which can overcome the aforementioned locking phenomena without introducing any rank deficiency The core

of this method lies in the accurate description of pure bending deformation mode without coupling the membrane deformation via the independent shear strains

assumptions instead of the derivatives of the assumed displacements Later, Huang er al.[23] found the assumed strain method could be applied to circumvent the membrane locking problem in quadratic elements

Non-conforming element or incompatible modes proposed by Taylor ef al/.[24], Ibrahimbegovic ef al.[25] and Buchter ef a/.[26] can improve the performance of continuum elements including two-dimensional and three-dimensional cases Thus, with non-conforming elements, good results can be obtained with few elements In shell elements, the membrane behavior actually is a two-dimensional plane stress problem So the incompatible mode can be borrowed to improve the accuracy of membrane part of shell elements Moreover, using this technique to overcome membrane locking in linear element is essentially improving the accuracy of this element Based on the non-conforming concept, Simo ef al.[27] proposed the Enhanced Assumed Strain method, which generalized the incompatible modes and also gave out the conditions to prescribe the application of this method

The degenerated shell elements are elements that are derived from the solid element Historically, most researchers focused on various approaches of degenerating continua to shells However, some researchers were more concerned with the shell’s solid behavior, Kanok-Nukulchai ef a/.[28], Andelfinger ef a/.[29], Buchter et a/.[26], Parish[30], Hauptmann ef a/.[31], Smith ef a/.[32], and Harnau ef a/.[33] were the pioneers in this field The main general features of these solid-shell elements are: (1) using translational degrees-of-freedom instead of rotational ones; (2) considering general three-dimensional material law; (3) considering the strains and stresses along

the thickness direction for the purpose of connection with solid element, and (4) adding quadratic or higher terms into the thickness direction interpolation

From another point of view, researches on the shell element are always classified

by groups Zienkiewicz and Irons initiated the concept of “degenerated shells” and proposed reduced integration [5,8,34,35] Hughes investigated on reduced integration and selective integration techniques and extended this concept into nonlinear finite element of shells [9,10,36,37,38] Belytschko and Liu mainly focused on the reduced integration with stabilization matrix and general nonlinear analysis [12,39,40,41,42] Stanley in his Ph.D thesis [6] conducted a comprehensive research regarding the degenerated shells Bathe constructed a set of shell elements, called MITC elements

(21,43,44,45,46,47,48]

On the other hand, Simo ef a/.[49,50,51,52,53,54,] contributed in the classical shell element, theory and applications The hypothesis underlying the degenerated shell and the classical shell theory are the same, while the reduction to resultant form is typically carried out numerically in the former and analytically in the latter which is the essential difference Also, it should be noted Simo ef al [27] proposed another important concept, Enhanced Assumed Strain method, in finite element method history Another important group from Germany studied the “Solid-Shell” element

[29,30,31,32,33,55]

1.3 Motivation

As mentioned above, according to Ahmad ef a/.[5], the motion of the normal of the shell can be described by Eq (1.1) and Eq (1.2), however, these two formulae are

only applicable to the infinitesimal displacements Furthermore, when the finite rotation is considered, it was pointed out by Kanok-Nukulchai ef a/.[28] that these descriptions will involve the trigonometrical functions and the coordinate transformation often requires a complex and unorthodox treatment, and the element formulation end up with highly complicated expressions Moreover, the displacement field involving rotational degrees of freedom is not truly iso-parametric, although in some literature, it was called so On the other hand, the main limitation of the degenerated shell is the exclusion of the normal strain and the normal stress This exclusion is fulfilled not only in the omission of the corresponding components in material tensor, but also in shell kinematics, which is assumed in the Straight Normals and Incrementally-Rigid Normals assumption In the application aspect, modern industries are much more concerned about the normal strain and normal stress, e.g sheet metal forming, car crash analysis and the multi-layered composite shells with their low transverse shear stiffness For example, in a metal forming process, a sheet might be held on its top and bottom surfaces which can lead to significant non-zero transverse stress and strain that are fully coupled to the other stress components So, the interpolation function in thickness direction should be flexible according to various circumstances Based on this consideration, a solid shell element based on relative displacements was developed in the thesis, which involves displacement degree-of-freedom only, which is a true iso-parametric element; which considers normal strains and stresses and which can be applied to the engineering applications (composite shell structures, shell vibration and buckling).

1.4 Objectives

In this thesis, corresponding to the limitations of the exact shell elements and the degenerated shell elements, a solid shell element based on the relative displacement concept will be proposed This thesis will

1 Introduce the relative displacement concept and replace the traditional rotational degree-of-freedom with this new relative displacement degree-of-freedom, so that

a characteristic of isoparametrization can be fulfilled;

Interpolate the thickness shape of the solid shell elements by different functions; Include the strain and stress components in the thickness direction which are absent in the conventional shell elements;

Derive the general relative displacement shell element in conventional engineering notation without distinguishing between the thin and thick shell cases;

Conduct comprehensive benchmark tests to show the versatility and accuracy of the proposed element in various cases of loading conditions and boundary conditions;

Adopt the existing successful strategies (Assumed Strain Method, Enhanced Assumed Strain Method and Reduced Integration) to overcome the various locking phenomena, shear locking and membrane locking, etc

Apply the proposed element to the practical field: composite structures, dynamics problem, buckling of shells and nano-mechanics

The proposed element in this thesis is believed to enrich the growing branch, the Solid Shell Elements, in the shell element library The main feature of the proposed

10

element introduces the normal interpolation and normal strain & stress which are the dominated mechanical parameters in certain circumstances The reason is that the new materials, the new manufacturing procedures and the extremely intensive load conditions make the normal mechanical characteristics significant while they were ignored before Another feature of the proposed element introduces the relative displacement concept This concept retains the compatibility of the various boundary conditions (e.g simply-supported, clamped) in traditional shell elements involving rotational degree-of-freedom Moreover, a characteristic of iso-parametrization which

is ignored in traditional shell elements is fulfilled This iso-parametrization of interpolation functions is the resolution of the existing dilemma that the shell elements with rotational degrees-of-freedom are not real isoparametric elements, whose shape functions in the geometric interpolations and the displacement interpolations are different

This thesis focuses on the development of a solid shell based on the relative displacement concept Static analysis of regular shaped solid structures is the main work here The next chapter will focus on the construction of the mathematical model

of the shell element

1.5 Outline

The outline of the thesis is as below:

Chapter I presents the introduction and background of the shell structures, shell theories and shell elements

Chapter II provides some basic mathematical knowledge on tensor theories which

II

will be used in the proposed shell derivations

Chapter III is core of the whole thesis, which derives the solid shell element based

on the relative displacements concept

Chapter IV applies the proposed shell element to composite shell structures to verify the thickness integration accuracy of the proposed element

Chapter V computes the natural frequencies of the circular plates to test the dynamic computational capability of the proposed element

Chapter VI and Chapter VII use the buckling of double walled carbon nanotubes and buckling of super ellipsoidal shells to check the capability of the eigenvalue extraction of structural bifurcations

Chapter VIII draws some useful conclusions and provides some heuristic

recommendations

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The purpose of this section is to introduce the geometric concepts (notations, definitions and basic properties) needed in the analysis of mathematical shell models

2.1 Geometric definition of a shell

A shell is considered to be a solid medium geometrically defined by a mid-surface immersed in the physical space x, and a parameter representing the thickness of the medium around this surface In general, the mid-surface of a given shell is defined by

a collection of two-dimensional chart, i.e smooth injective mappings from the reference domain of R° into © Note that the mid-surface of a general shell may consist of the collection of several smooth surfaces assembled along folds, and that even a smooth surface cannot always be defined by a single two dimensional (2D) chart (e.g a sphere) However, in complex configurations, the analysis will be decomposed according to each chart and each reference domain Therefore, without loss of generality, we now focus on shells defined using a single chart

It is then considered that a shell with a mid-surface (denoted by 5) defined by a 2D chart @ which is an injective mapping from the closure of a bounded open subset of R’, denoted bya, into ©, hence S=/(@) It is assumed that is such that the vectors

In this definition, A(E' £’), represents the thickness of the shell at the point of coordinates (E' &’)

The 3D chart ® and the reference domain © provide a natural parameterization of the shell body, i.e describe the shell with a natural (since it is based on the mid-surface) curvilinear coordinate system In order to be able to easily express and manipulate tensors in this specific coordinate system, some basic concepts of surface differential geometry are introduced

SS

ẹ Fig 2-1 Geometric description of a shell

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2.2 Differential geometry on the mid-surface

It is useful to introduce surface tensors on the mid-surface of the shell in the similar manner mentioned in the above section At each point, recalling that (a1, a2) is

a basis of the tangent plane, it is called the covariant basis and the contravariant basis (a', a’) of this tangent plane is defined by

First-order surface tensors are vectors of the tangent plane; hence they are uniquely determined by their components in either one of the above-defined bases To distinguish the surface tensors from 3D tensors, we denote the former by an under-bar for the first-order tensor or a parenthesized number as left subscript for higher ones Note that we use Greek indices for the components of surface tensors, in order to distinguish them form components of 3D tensors denoted with Latin indices, hence Greek indices will henceforth implicitly vary in (1, 2)

The restriction of the metric tensor to the tangent plane, also called the first fundamental form of the surface, is given by its components

Based on these definitions, the Euclidean norm of surface tensors can be written in the following manner

with

q =ang„; — (a, y (2.16) Another crucial second-order tensor is the second fundamental form of the surface, the components of which are

big = As" Ay p (2.17)

The second fundamental form is also called the curvature tensor, because it contains all the information on the curvature of the surface Consider indeed n, a unit vector in the tangent plane, and the curve obtained by intersecting the mid-surface with the plane defined by n and a3, see Fig 2-2, A parameterization of this curve by its arc-length will be of the form

x > (2'(x),€?(x)) (2.18)

with

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Fig 2-2 Intersection of the surface with a plane containing a;

Recalling x represents the arc-length and noting b,g=-a,-a3z due to a,-a3=0, it is

due to n-a3=0 along the intersection curve

Hence, noting that a3 the unit normal vector to this (planar) curve at the considered point, this imply n-@b-n is the curvature of the curve, counted positively when a3 points towards the center of curvature Therefore, by considering the quantity

point, since the denominator normalizes the numerator by the square of the norm of v and varying v in the tangent plane thus amounts to rotating the intersecting plane

around a3 in Fig 2-2, the tensors gya and yb are both symmetric, hence the quantity

Eq (2.22) can be seen as a Rayleigh quotient Therefore there exist two directions corresponding to its minimum and its maximum, and these directions are

gya-orthogonal, i.e they are orthogonal in the usual sense since ¿ya is the surface

metric tensor The values of the curvature along these directions are called the principal curvatures The half-sum and the product of the principal curvatures are classically called the mean curvature and Gaussian curvature, respectively

The mean and Gaussian curvatures of the surface can be respectively obtained by

= (i +U a, -a,}a +(u,a7, -a;)a,

then defining the surface Christoffel symbols

My = Aggy A =—Aa,-a’, (2.29)

so, (2.29) can be expressed in

aE = (ty UVa) a +(u,b2 Ja, (2.30)

Finally, calling yv tangent vector of the curve (¢' &’), namely v"= (E")’, it is defined of the surface gradient of u, denoted by @)Vu, as the second-order surface tensor Thus denoting by wp), the covariant-covariant components of @yVu

U py =U pg — Wilt; (2.31)

and wgj_, is called a surface covariant derivatives of ug Note that the expressions of

surface Christoffel symbols and of surface covariant derivatives are similar to their 3D counterparts, with Greek indices instead of Latin indices

Since it is primarily concerned with shells (hence with surfaces), we will from now

on omit the term “surface” when referring the quantities which pertain to surface differential geometry, and instead specify “3D” when referring to 3D differential geometry

The curvature tensor enjoys an additional symmetry property, which involves its covariant derivatives, called the Codazzi Equation

It is called that a surface elliptic, parabolic or hyperbolic according to whether its Gaussian curvature K is positive, zero, or negative, respectively Of course, general

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surfaces need not be of uniform nature in this respect, but this distinction can always

be made point-wise

In this section, we focus on the natural 3D curvilinear coordinate system based on

a parameterization of the mid-surface of the shell Using the definition of the 3D chart 2

it can be derived of the 3D covariant base vectors We have

Then the components of 3D metric tensor are:

Sop — 8a Sp = gy ~2E day +(F Jey

833 = 23°28, =1 Note that the expression of g., suggests an interpretation of the “fundamental form” terminology, since the components of the three fundamental forms compose the coefficients of the polynomial expansion of the tangential components of the 3D metric tensor

Next it can be shown that the g quantity appearing in volume measure is given by

2 2

It is noted from (2.37) that the mapping $ 1s well defined (hence so is the system of

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curvilinear coordinates) provided that the expression of 1-2/7¿?+ K(€) is strictly positive This is equivalent to requiring that

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The fundamental principle of shell theory is the description of the shell kinematics Specifically speaking, it is the motion and deformation of shell normals As mentioned in Chapter one, the classical Reissner-Mindlin theory restricts the shell normals according to Eq (1.1) and Eq (1.2) which eliminate the normal strain of the shell To exactly describe the deformation of shell normals, researchers have done a lot of work on this topic: Reddy et a/.[56,57] proposed the high order theory, namely, the generalized laminated plate theory (GLPT) and the layer-wise laminated theory (LWLT) Parisch[30], Hauptmann ef a/.[31] also proposed a new displacement interpolation in shell thickness direction

shell theories

@ Reddy: (GLPT)

The displacements of a certain point in the shell domain are given by

u, (x,y,z) = u(x, y)+U (x,y,z)

is associated Since the displacement vector uw’ may not violate the shell kinematics, namely, the straight normals assumptions, it must be forced to be collinear with the deformed shell director d so it is defined