ELASTO-PLASTIC ANALYSIS OF CORRUGATED SANDWICH STEEL PANELS

Transcription

1 The Pennslvania State Universit The Graduate School College of Engineering ELASTO-PLASTIC ANALYSIS OF CORRUGATED SANDWICH STEEL PANELS A Thesis in Civil Engineering b Wan-Shu Chang 004 Wan-Shu Chang Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosoh August 004

2 The thesis of Wan-Shu Chang was reviewed and aroved* b the following: Theodor Krauthammer Professor of Civil Engineering Thesis Advisor Chair of Committee Eduard S. Ventsel Professor of Engineering Science and Mechanics Thesis Co-Advisor Sabih I. Haek Distinguished Professor Emeritus of Engineering Mechanics Vija K. Varadan Universit Distinguished Professor of Engineering Science and Mechanics Andrew Scanlon Professor of Civil Engineering Head of the Deartment of Civil Engineering *Signatures are on file in the Graduate School

3 ABSTRACT iii In this stud, the elasto-lastic analsis of a corrugated-core sandwich late has been develoed. The develoed stud based on generalized Galerkin method (or Navior method for siml suorted edges) and incremental theor of lasticit with the initial incremental lastic moments calculated b an iterative rocedure can be alied in both elastic erfectl-lastic and strain-hardening behaviors. However, due to currentl absent eerimental data of the ost ield behavior of corrugated-core sandwich lates, in these numerical studies, onl the behaviors of elastic erfectl-lastic cases were analzed in this work. Also, a comrehensive stress analsis for corrugated-core lates is develoed. The effects of geometric arameters and various lengths and widths of corrugated-core sandwich lates with all siml suorted boundar conditions on the late behavior and strength were studied. Some new henomena observed in eerimental investigations of corrugated-core sandwich lates, but not found in revious numerical analsis, have been confirmed and reorted. In articular, our investigation has strengthened some eerimental results obtained in (Tan et al. 1989) which lacked suorted from 3D FEM, and the other analtical solutions.

4 TABLE OF CONTENTS iv LIST OF FIGURES...vii LIST OF TABLES...i ACKNOWLEDGEMENTS... Chater 1 Introduction Background and Problem Statement Imortance Objectives Scoe Thesis Laout...7 Chater Literature Review Plate Bending Theories...8. Sandwich Plates Develoment of Sandwich Constructions Corrugate-Core Sandwich Plates Previous Works on Corrugate-Core Sandwich Panels Elasto-Plastic Analsis Finite Element Method Boundar Element Method Yielding Criteria Tresca Criterion (Maimum Shear Theor) von Mises Criterion (Distortion Energ Theor)....5 Plastic Stress-Strain Relations Incremental Theor of Plasticit (Plastic Flow Rule) Deformation Theor Methods of Analsis The Galerkin method Initial stiffness aroach...30 Chater 3 Methodolog Introduction Elastic Analsis Equilibrium Equations Curvature-Dislacement Relations Constitutive Equations Governing Equations Theor of Plasticit...40

5 3.3.1 The Yield Criterion Elasto-Plastic Analsis Incremental Form of Elastic Constitutive Equations The Incremental Form of the Governing Equations Boundar Conditions Research Aroach...63 v Chater 4 Numerical Analsis and Results Introduction Linear Elastic Analsis Numerical Algorithm Elastic Analsis Comarison with Eerimental Results Convergence of Elastic Analsis Conclusion Elasto-Plastic Analsis Numerical Algorithm Numerical Procedure Convergence Criterion Numerical Eamles for Elastic Perfectl-Plastic Material Conclusions Geometric Parameters Analsis...96 h c Ratios of Core Deth to Thickness,...10 tc 4.4. Corrugation Angle, α t c Ratios of Core to Facing Plate Thickness, t Ratios of Pitch to Core Deth, h c f Effects of length, a, and width, b Discussion of negative moments in the direction normal to the corrugation Conclusions Chater 5 Conclusions Introduction Conclusions Recommendations for Future Work...10 Bibliograh...13 Aendi A ELASTIC CONSTANTS...131

6 Aendi B ORTHOTROPIC PARAMETERS OF PLASTICITY vi

7 LIST OF FIGURES vii Fig. 1-1: A sandwich late... Fig. 1-: Smmetrical te of a corrugated-core sandwich late...3 Fig. -1: Corrugated-Core sandwich anel and a anel unit...13 Fig. -: Projections of the von Mises and Tresca ield surfaces...0 Fig. 3-1: Sign convention used in the late analsis...34 Fig. 3-: Yielding of non-laered cross-section...4 Fig. 3-3: Material behavior...45 Fig. 3-4: the cross section and stress distribution in direction...46 Fig. 3-5: the cross section and stress distribution in direction...48 Fig. 3-6: Equivalent Stress-Strain Curve...5 Fig. 3-7: Definition of boundar conditions (for =0 or b)...59 Fig. 4-1: Rectangular siml...67 Fig. 4-: Flow chart for elastic analsis...70 Fig. 4-3: Deflection of the corrugated-core sandwich late...7 Fig. 4-4: Central deflection comarison...73 Fig. 4-5: Convergence of w at central oint...76 Fig. 4-6: Convergence of M at central oint...76 Fig. 4-7: Convergence of M at central oint...77 Fig. 4-8: Initial stiffness solution algorithm...8 Fig. 4-9: Program organization for elasto-lastic alication...85 Fig. 4-10: Elastic Perfectl-Plastic Behavior...87 Fig. 4-11: Central deflection vs. Loading...91 Fig. 4-1: Moment vs. Curvature in -direction...93

8 viii Fig. 4-13: Moment vs. Curvature in -direction...94 Fig. 4-14: Corrugation unit...97 Fig. A-1: Dimensions of the corrugation unit Fig. A-: One corrugation leg...13 Fig. B-1: Stri beam in?direction for 0 A Fig. B-: Moment vs. lastic curvature curve in and directions...138

9 LIST OF TABLES i Table 4-1: Geometric Parameters and Elastic Constants...71 Table 4-: Comarison of central deflection...71 Table 4-3: Moments at central oint...74 Table 4-4: Comarison of convergence of w, M, and M...75 Table 4-5: Comarison of stiffness constants, ielding loadings, and deflections...89 Table 4-6: Comarison of various geometric arameters for siml suort, h c = t c Table 4-7: Comarison of various geometric arameters for siml suort, h c = t c Table 4-8: Comarison of various geometric arameters for siml suort, h c = t c Table 4-9: Comarison of various ratios α for h c /t c =10, t c /t f =0.6, and /h c = Table 4-10: Comarison of various ratios t c /t f for h c /t c =10, α=60, and /h c = Table 4-11: Comarison of various ratios /h c for h c /t c =10, α=60, and t c /t f = Table 4-1: Comarison of various ratios /h c for h c /t c =10, α=80, and t c /t f = Table 4-13: Comarison of moments of inertia on various /h c and α Table 4-14: Comarison of central deflections and moments with various length Table 4-15: Comarison of elastic constant for negative-moment cases...115

10 ACKNOWLEDGEMENTS I would like to eress m sincere gratitude and rofound thanks to m advisors, Professor Ted Krauthammer and Professor Edward S. Ventsel, for their remarkable atience, valuable encouragement and advice throughout all the stages of this work. The great care with which the have read the manuscrit, their numerous suggestions for imrovements of the English stle, as well as helful technical comments, are greatl areciated. I would also like to thank the members of m doctoral committee, Dr. Sabih I Haek, Dr. Andrew Scanlon, and Dr. Vija K. Varadan, for their valuable advice and review during the course of research. I also want to eress m gratefulness to all the members of the Protective Technolog Center at the Penn State Universit for their hel and suort. Finall, I wish to thank m wife, Ching-Yi, for the full suort she gave to me. This work would not have been ossible without the affection and sacrifice of m famil; m arents in Taiwan and lovel son, Ethan. This stud is dedicated to them.

11 Chater 1 Introduction 1.1 Background and Problem Statement A close-to-analtic solution is roosed for the elasto-lastic analsis of corrugated-core sandwich late bending. The 3-dimensional (3D) sandwich anel will be reduced to an equivalent D structurall orthotroic thick late continuum. In this analsis, the Galerkin method and the incremental method with initial lastic moments are emloed to solve the elasto-lastic roblem. For man ears, the analsis and design of structures have been based on the linear theor of elasticit with the assumtion of isotroic material roerties, but it is well known that this aroach is undul conservative because it fails to take advantage of the abilit of man materials to carr stresses above their ield stresses. The main difficult of lastic analsis is the mathematical comleit. However, that difficult has been overcome b the advent of high-seed comuters and the efficient use of comutational methods. A sandwich late shown in Figure 1-1 is a three-laer structure, comrised of a thick core between two thin, flat face sheets. Generall, the face lates are made of highstrength materials and the core is made of a low strength and densit. Due to the high-

12 strength faces and low-weight core, the significant feature of this structure is its high strength to weight ratio. Face sheet Core z Fig. 1-1: A sandwich late The advantages of sandwich lates are summarized as follows: 1. High stiffness and strength to weight ratios.. The good surface finish reduces an aerodnamic resistance. 3. Good thermal and acoustic insulation. 4. High energ absortion caabilit with different fillers and cores. 5. Increase the interior sace and ease of equiment installation.

13 3 A corrugated-core sandwich late is shown in Figure 1-. To analze the bending behavior of the corrugated-core sandwich late, the following assumtions are made: 1. The late is smmetrical te corrugated-core sandwich late shown in Figure 1-. Therefore, both face lates are identical in material and thickness.. The material of the corrugated-core sandwich late is isotroic. 3. The deformation of this late is small. 4. Facing lates are thin comared with the corrugation deth. Consequentl, the local bending stiffness of the face sheets is negligible. 5. The core contributes to the anel fleural stiffness in the -direction but not in the -direction. 6. The elastic modulus of the late in the z-direction is assumed to be infinite. 7. The core can resist the transverse shear stresses and also contribute to the fleural and etensional stiffness. 8. The transverse shortening of the core is ignored. z Fig. 1-: Smmetrical te of a corrugated-core sandwich late

14 4 The basis for this corrugated-core sandwich late bending roblem is the Reissner-Mindlin late theor. In this theor, straight material lines normal to the middle surface are assumed to remain straight, but not necessaril normal to the middle surface during distortion of the late. Namel, the three indeendent degrees of freedom corresond to the deflection w and the sloes θ and θ. w is the deflection in z-direction (i.e. the direction of thickness). θ and θ are the sloes of the z- and z-lanes, resectivel. Due to several elastic constants derived b Libove and Hubka (1951), the 3D corrugated-core flat sandwich late here can be regarded as a D equivalent continuum. The governing equations for the elasto-lastic analsis of corrugated-core sandwich lates are made b aroimating them with continuous structurall orthotroic structures. The Galerkin method and the incremental theor of lasticit aroach with initial lastic moments are emloed to solve this roblem. An iterative method is then used to evaluate the incremental lastic moments in each stage. In this elasto-lastic analsis, the material maintains a linear elastic behavior until ield. Then, ield condition, flow rule, and strain-hardening rule are used to calculate stress and lastic strain increments. It can be seen in literature reviews that recentl the finite element method (FEM) is widel used in the elasto-lastic analsis. FEM is now firml acceted as the most owerful numerical technique in engineering. Tan et al. (1989) showed eerimentall that for the 3D corrugated-core sandwich late, the 3D FEM model agreed well with

15 5 measured deflections in linear elastic range. However, in the moment analsis, there are discreancies between the 3D FEM and eerimental results (Tan et al. 1989). In the eerimental results, the signs of moments in - and -directions are oosite, but the 3D FEM showed the same signs in the studied case. In addition, it is ver difficult to modif the boundar conditions where using the 3D FEM. 1. Imortance The theor of elasticit cannot be used to address the behavior after material ielding occurs. Furthermore, an elasto-lastic analsis of corrugated-core sandwich lates is currentl unavailable. Therefore, the rocedure for the elasto-lastic analsis of sandwich lates was develoed in this stud. Also, the ield criteria for naturall isotroic and anisotroic materials have been develoed and alied reviousl. However, because the corrugated-core sandwich late is a structurall orthotroic sstem, there is no ield criterion for such a structure. Therefore, a ield criterion for structurall orthotroic constructions was develoed in this stud. Hill s and Ilushin s ield criteria were modified and etended to include structurall orthotroic constructions.

16 1.3 Objectives 6 The main goal of this stud is to enable the develoment of lastic analsis of corrugated-core sandwich lates. The three ke issues that need to be considered in this elasto-lastic analsis are reliable elastic analsis, ield criterion, and aroriate constitutive descritions. Hill s ield criterion develoed for the naturall anisotroic material will be etended to include structurall orthotroic constructions. The results obtained from linear elastic analsis b the above-develoed solution of the corrugated-core late with all siml suorted boundaries are comared with the results rovided b other authors (e.g. Tan et al. 1989) before roceeding to analze the elasto-lastic roblem. The elasto-lastic solution begins on a linear elastic analsis. The linear analsis is also develoed and is mainl to check the results. The behavior of a late is related to its stiffness. Several geometric arameters might contribute to the stiffness of the corrugated sandwich anel. The effects of geometric arameters such as ratio of face to core thickness, ratio of itch to core deth, the core deth to core thickness ratio, and corrugation angle and the length in corrugation direction and the width in the direction normal to corrugation are analzed. 1.4 Scoe This research is limited to onl the elastic erfectl-lastic analsis of rectangular corrugated-core sandwich lates with all siml suorted edges. The theoretical

17 formulation can be adated to elastic erfectl-lastic and strain-hardening roblems; however, in this stud, onl elastic erfectl-lastic roblem will be considered. 7 Furthermore, it is also ver imortant to know what geometric arameters in the corrugated-core sandwich late will affect the behaviors of that late significantl. Different geometric arameters such as the corrugation angle (α), the ratio of core to facing sheet thickness (t c /t f ), the ratio of corrugation itch to core deth (/h c ), and the core deth to core thickness ratio (h c /t c ) and various lengths and widths are investigated. 1.5 Thesis Laout This thesis has five chaters. Chater One is a brief introduction to the concet, and also lists the scoe and objectives. Chater Two is the review of revious work for sandwich lates, elasto-lastic analsis, and numerical methods. The theoretical methodolog is resented in Chater Three. Chater Four contains analsis and numerical eamles to comare and investigate the effects of different geometric arameters. Chater Five contains the conclusion and recommendations for future work in this work.

18 8 Chater Literature Review.1 Plate Bending Theories Several late bending theories have been develoed in the ast decades. Two widel used theories of these late bending theories are Kirchhoff s theor also well known as the classical theor (or thin late theor) and Reissner theor. The linear Kirchhoff theor assumed that there is no deformation in the middle lane of the late and the straight line normal to the middle lane remains the normal to the middle surface after deformation. According to the Kirchhoff assumtion, the transverse shear strains and stresses, as well as stress normal to the late midsurface are neglected in the classical late theor. Reissner (1945) modified the classical theor (or Kirchhoff s theor). He roosed that the rotations of normal to the late midsurface in the z- and z- lane could be introduced as indeendent variable in the late theor. Mindlin (1951) simlified Reissner s assumtion that normal to the late midsurface before deformation remains straight but not necessar normal to the late after deformation and the stress normal to the late midsurface is disregarded as in the Kirchhoff theor. This late bending theor modified b Reissner and Mindlin is well known as the Reissner-Mindlin late theor. The Reissner-Mindlin late theor is suitable to analze both thin and

19 moderatel thick lates. The analsis of the corrugated-core sandwich lates in this thesis is based on this theor. 9. Sandwich Plates..1 Develoment of Sandwich Constructions Sandwich anels are etensivel and increasingl used due to its lightweight advantage in civil engineering, aerosace engineering, and shibuilding industr where weight is an imortant design issue. With the variet of faces and cores, the sandwich anels have wide alications in man fields such as acoustic and thermal insulation and fire safet. Therefore, the sandwich anels have been used commerciall in eterior walls and internal artition walls in architectural engineering. The detail introduction of aling sandwich anels in engineering is directl in (Davies 001; Karbhari 1997) With the advent of high-seed flight, sandwich lates were widel used in airlane design because of the romised high strength and the well-finish surface maintaining aerodnamicall smooth surface. Also, the sandwich construction can be a ossible substitute for sheet-stringer to reduce the weight of an aircraft. Recentl, the US Nav has been studing alication of the laser-welded corrugated-core sandwich constructions (Marsico et al. 1993; Wiernicki et al. 1991). These alications cover bulkheads and decks on accommodation areas, deckhouses, deck edge elevator doors, and hangar b division doors.

20 10 The advantage of using two faces searated b a distance is thought to have been first discussed b Duleau, French, in 180. This concet was first alied commerciall after 100 ears. After World War Two, due to the blossoming of aviation, there was an increased interest in sandwich construction. In the late 1940s, Gough et al. (1940) and Williams et al. (1941) ublished the ioneer theoretical works on structural sandwich constructions. The core of the sandwich construction kees the faces aart and stabilizes them b resisting vertical deformations, and also enables the whole structure to act as a single thick late as a virtue of its shear strength. The second feature that characterizes the sandwich construction is its outstanding strength. In addition, the core carries a ortion of the bending load. Therefore, sandwich anels are commonl used in aviation, aerosace, civil engineering and other alications where weight is an imortant design issue due to its ecetionall high fleural stiffness-to-weight ratio. Develoment of core materials has continued from 1940s through to toda in an effort to reduce the weight of sandwich anels. Man sandwich-te configurations emloing different roduction techniques have been suggested. In general, sandwich cores could be classified into four tes: (a) foam or solid core, (b) honecomb core, (c) web core, and (d) corrugated or truss core. Most oular materials of sandwich cores are soft and light. In most foam and honecomb core sandwich anels all the in-lane and bending loads are assumed to be carried b the faces onl. However, in web and

21 corrugated-core constructions, a ortion of the in-lane and bending loads are also carried b the core. 11 Plate theories alicable to general sandwich lates for small deflection have been develoed b Libove and Batdorf (1948) for flat anels and b Stein and Maers (1950) for curved anels. In these two aers, deflections due to transverse shear are taken into account and sandwich lates are regarded as structurall orthotroic anels because the face sheets or core (or both) ma have orthotroic stretching roerties in general. To reduce a 3D sandwich construction to a D equivalent continuum, seven hsical constants are introduced in flat lates. The include the transverse shear stiffness D Q and D Q, the bending stiffness D and D, a twisting stiffness D, and the Poisson ratios υ and υ. In 1960s, studies in sandwich construction had sread widel. Plantema (1966) ublished the first book on sandwich construction in In 1969, the book b Allen (1969) followed. In the earl work u through the 1960s, including these two books written b Plantema and Allen, a nomenclature and analtical resentation of sandwich anels was develoed. A book b Zenkert (1995) sulements with more eamles and solved roblems contained in the Plantema and Allen books. A review b Noor et al. (1996) rovided over 800 relevant references that were all discussed in that review and another 559 references as a sulemental bibliograh. In this review recent develoments in the comutational

22 1 modeling of sandwich lates and shells are introduced. Vinson (1999) ublished a tetbook for sandwich structural analsis. Later, Vinson (001) had given an etensive list of 179 references to rovide a review of several asects of sandwich structures. Almost all imortant literatures on the subject of sandwich structures can be found in that review... Corrugate-Core Sandwich Plates Most oular materials of sandwich cores are soft and light such as balsa wood and foam. Conventional forms of sandwich cores are honecomb-shaed. Honecomb cores currentl rovide the greatest shear strength and stiffness to weight ratios but require secial care in ensuring adequate bonding to the facing sheets. In the earl literature on the instabilit of sandwich constructions, much attention was devoted to wrinkling henomena. This te of failure is made ossible b the finite resistance of the core to deformations erendicular to the faces and is of secial imortance when the core is made of an eanded material. However, modern sandwich cores are made of high strength materials that have a high resistance in this resect, such as a corrugated steel sheet. For the uroses of an analsis it can be assumed that the transverse normal stiffness of the core is infinitel large. The innovative corrugated-core sandwich late consists of a corrugated sheet laser welded between two face sheets. Unlike soft honecomb-shaed core, a corrugated core resists not onl vertical shear but also bending and twisting. In this thesis, a corrugated-core sandwich late is analzed. A

23 13 corrugated-core sandwich late is consisted of two facing lates and a corrugation core shown in Figure -1. In general, both facing lates and corrugation are made of the same material. The corrugated core is often stiff enough to contribute to the fleural rigidit. z h h t f α t c h c Fig. -1: Corrugated-Core sandwich anel and a anel unit..3 Previous Works on Corrugate-Core Sandwich Panels In 1951, Libove and Hubka (1951) derived formulas for evaluating elastic constants of a corrugated-core sandwich late. These elastic constants are necessar for the analsis of flat sandwich late theor derived b Libove and Batdorf (1948). Also, those formulas for evaluating elastic constants are used in this elasto-lastic analsis in this thesis.

24 14 In the 1980s, through the earl 1990s, several eeriments of sandwich anels have been conducted in UK. Montague and Norris (1987) and Norris et al. (1989) studied the structural behavior of sot-welded and corrugated-core steel anels. Tan et al. (1989) ublished a aer on eerimental testing of corrugated-core sandwich anels and comared the results with finite element method and the closed solution develoed from the work of Libove and Bandorf (1948). In these aers, various dimensional anels were analzed. In the all around siml suorted late, both finite element analsis and the closed solution redicted the deflection well; however, in the redictions of bending stresses and moments, both solutions redicted the same sign on the facing lates in both - and -directions. These results conflicted with the eerimental investigations that bending moment M and M have the oosite sign in some cases. It is believed that the roduction difficulties revent the wide use of steel corrugated-core sandwich anels. The main manufactured difficult is firml joining face sheet to the corrugated core. Laser welding enables construction of steel corrugated-core sandwich constructions b welding from one side onl. In 1989, configurations of laserwelded corrugated-core steel sandwich anels were tested b Forbes (1989) to illustrate behavior of the structure under comression, bending and shear. In that eeriment, before etreme ost-buckling deformation, no samle failed due to the failure of the laser-welded connection. Therefore, the laser weld shows the ecellent caacit of joining faces to the core.

25 15 Recentl, the corrugated-core sandwich anels are widel emloed in the shibuilding industr. The US Nav has studied alication of the laser-welded corrugated-core sandwich anels since the 1990 s (Marsico, 1993; Eiernicki, 1991). Also, in UK in 1991, Bird (1991) conducted several eeriments to stud static strength, fatigue, and blast trials on laser-welded steel sandwich anels. In these tests the roerties of laser-welded steel sandwich anels have been shown to be adequate for shi construction. Also the corrugated-core anels have shown good blast characteristics and could be ecellent for resisting fragmentation damage with incororation of aroriate absorbent materials. The third international conference on sandwich construction was held at Southamton, UK, in Coule aers were resented in that conference about the alication and analsis of corrugated-core sandwich anels. Kattan (1995) and Kujala and Tuhkuri (1995) resented the caabilit of using laser-welded corrugated-core sandwich anels in shibuilding. Those structures were found to be u to 40-50% lighter than the conventional steel grillages. More recentl, several different tes of core have been investigated. Fung et al. studied the C-core (1993; 1996) and Z-core (Fung et al., 1994) sandwich anels. In 001, Lok et al. (000) introduced a truss-core sandwich anel and studied its elastic stiffness roerties and behavior. A truss core is ver similar to a corrugated-core in the outlook. The difference between those two structures is that the corrugation is a continuous structure but the truss core is assembled b two inclined lates in a anel unit.

26 .3 Elasto-Plastic Analsis 16 Hill (1950) introduced the further concet of lastic anisotro into the theor of lasticit b Mises. He also roosed a ield criterion constituting a generalization of the Huber-Mises criterion. Olszak and Urbanowski (1956) introduced the anisotroic arameters, the tensor moduli of lasticit, into the lastic otential and ield criterion for erfectl lastic materials. Hu (1956) etended the theor for strain-hardening materials. These anisotroic arameters var as the material strain-hardens as was first ointed out b Hill (1950) and was later used b Jensen et al. (1966) for a shear lag roblem. The nonlinear difficult created b mathematical comleit has been overcome b the advent of high-seed comuters and the efficient use of comutational methods. The formulation and solution of the incremental roblem in elastic-lastic solids is a fundamental roblem in lasticit. Finite element method and boundar element method are well-known for the solution of this te of roblems and solutions can be carried out in standard finite element and boundar element codes. Beginning with the 1970s, the advent of structural mechanics dealing with laminated structures shifted attention of researchers to orthotroic lates and much work has been done in characterizing elasto-lastic analsis of these lates. All sandwich anels can be regard as laminated structures. One face is the lamina 1, the core is lamina

27 , and the other face is lamina 3. The theor of laminate can be alied in all sandwich structures. Those idea and advantages have been discussed b Vinson (1999). 17 Man aers in elasto-lastic analsis of comosite lates were ublished. In these studies, two aroimate aroaches, finite element method (Marcal and King 1967; Owen and Figueiras 1983a; Owen and Figueiras 1983b; Valliaan et al. 1976) and boundar element method (Karam and Telles 199; Telles and Brebbia 1979; Telles 1983; Telles and Carrer 1991), were emloed..3.1 Finite Element Method The finite element solution of elastic-lastic roblems has been intensel develoed since the 1960s and a number of different aroaches have been roosed. A considerable literature eists on the elasto-lastic analsis using finite element method. Oden (1969) had given an etensive list of references that include the various investigations in the fields of material, geometric and combined nonlinear behavior of structures in In the same ear, Whang (1969) ublished a aer on elasto-lastic orthotroic lates and shells using finite element method. In that aer, a finite element method was develoed for the elasto-lastic analsis for bi-linear strain-hardening orthotroic lates and shells. Two aroaches, the initial stiffness aroach and tangent stiffness aroach, were resented and comared. Both alied the Huber-Mises ield function and the

28 Prandtl-Reuss flow rule. A generalized ield function for anisotroic material with the anisotroic arameters was used. 18 In 1980, Owen and Hinton (1980) ublished the book of finite elements in lasticit. In that book, the two-dimensional laer and non-laer structures were analzed. The FEM rocedures and rogramming dealing with the elastic-lastic materials were introduced..3. Boundar Element Method The boundar element method has been develoed for several kinds of non-linear roblems such as elasto-lasticit, viscolasticit and cree. A book on boundar elements b Banerjee and Butterfield (1981) rovided solutions for elasto-lastic roblems of two or three dimensions b means of initial strain or stress. In 1986, Moshaiov and Vorus (1986) ublished a aer on late bending roblem b boundar element method and incremental theor with initial lastic moments. Whether using finite element or boundar element methods to solve elasto-lastic roblems, an incremental aroach must be alied in which the loading is divided into increments after the ielding occurs.

29 .4 Yielding Criteria 19 An accurate ielding function is the ke factor for the accurac of the lastic analsis. The ield criterion determines the stress level at which lastic deformation begins. Numerous criteria have been roosed for the ielding of materials. The most widel used criteria are the Tresca maimum shear criterion and the von Mises ield criterion. Those two criteria have the significant agreement with eerimental redictions and closel aroimate metal lastic behavior. In this section, those two criteria are briefl introduced. The detail of those ielding surfaces can be found in several references (e.g. Lubliner 1990; Mendelson 1983; Shames and Cozzarelli 199). The rojections of Tresca and von Mises ield surfaces in the π-lane are regular heagon and circle, resectivel, as shown in Figure -(a). In (σ 1 -σ 3 ) and (σ -σ 3 )-lane those are irregular heagon and ellise, resectivel, as shown in Figure -(b). Generall, the Tresca ield surface is a regular heagon inscribed within the von Mises ield surface.

30 0 σ -σ 3 σ 3 von Mises Tresca von Mises Tresca σ 1 -σ 3 σ 1 σ (a) -lane (b) (σ 1 -σ 3 ) (σ -σ 3 )-lane Fig. -: Projections of the von Mises and Tresca ield surfaces In Figure -, it can be seen that the Tresca ield criterion is linear between segments. The singularities on the rincial aes make the lasticit difficult. Some difficulties regarding the direction of lastic flow arise at the corners of the heagon. Singular ield surface with corners has been discussed b Koiter (1953, 1960). Generall, the lastic analsis as based on the smooth von Mises ield surface seems to be more aroriate for the numerical discussion. Also, the von Mises ield criterion usuall fits the eerimental data better than other criteria..4.1 Tresca Criterion (Maimum Shear Theor) (Doltsinis 000; Kachanov 1971) The Tresca criterion is the historicall oldest. This theor assumes that the lastic deformation occurs when the magnitude of the maimum shear stress over all lanes reaches a critical value. The limit value can be determined b simle tension eeriment. This criterion ma be written in the following:

31 1 τ 1 = σ σ 3 k (.1) τ τ 3 = σ = σ 3 1 σ σ 1 k k Where f k some function a material arameter to be determined eerimentall σ 1, σ, σ 3 τ 1, τ, τ 3 rincial stresses shear stresses The forms Eq..1 are not unique and not an analtic function. It can be rewritten as: f [( σ ) k ] ( σ σ 3 ) [ 4k ][( σ1 3 ) ] = σ 4 σ k (.) 1 4 The above form has the advantage of being analtic and, moreover, it can be eressed in terms of the rincial stress-deviator invariants J and J 3. f = 4J 7 J3 36k J + 96k J 64k (.3) According to Eq..3, to al the Tresca criterion, one major difficult of is that it is necessar to know in advance the maimum and minimum rincial stresses. Or it will suffer from the great comleit of the Tresca criterion in its general form, Eq..3. Onl in the case that the maimum and minimum rincial stresses are known a riori can the Tresca criterion be reduced into a simle form.

32 .4. von Mises Criterion (Distortion Energ Theor) von Mises suggested that ielding occurs when the second deviatoric stress invariant, J, reaches a critical value. Or it is called the distortion energ theor. This theor assumes that ielding begins when the distortion energ equals the distortion energ at ielding in simle tension. An analtic form of the Mises ield function (Calcote, 1968) is f = J k (.4) Or [( σ ) + ( σ σ ) + ( σ ) ] 1 f = σ σ1 k (.5) 6 Where f J some function the second deviatoric stress invariant σ 1, σ, σ 3 rincial stresses k a material arameter to be determined eerimentall A hsical meaning of the constant k can be obtained b considering the ielding of materials under simle stress states. In terms of the stress comonent referred to the -, -, and z- aes, the von Mises ield criterion ma be written as: Where ( σ ) + ( σ σ ) + ( σ σ ) + 6( τ + τ + τ ) 6k σ (.6) z z z z = σ, σ, σ z stresses in -, -, and z- directions, resectivel

33 3 τ, τ z, τ z shear stresses Hill (1950) generalized the von Mises ield criterion for isotroic materials to include anisotroic materials. He assumed the ield criterion to be a quadratic in the stress comonents as given b Where F, G, H, L, M, N material anisotroic arameters For most metals von Mises criterion fits the eerimental result more closel that Tresca criterion. Throughout the elasto-lastic analsis in this thesis, the von Mises ield criterion is emloed. It should be noted that there is no ield criterion for structurall orthotroic lates. In the revious research, both von Mises and Tresca ield criteria deal with the naturall isotroic and aniosotroic materials. However, in this stud, the structure is made of the isotroic material but the behavior of the structure is orthotroic. The Ilushin ield surface, shown in Eq..8 and eressed in terms of dimensionless stress resultant, is considered and modified to the structurall orthotroic late. ( ) ( ) ( ) 1 ) ( = = z z z z ij N M L H G F f τ τ τ σ σ σ σ σ σ σ (.7) = + + M M M M M M M M M M (.8)

34 4 Because of the structural te, the behaviors of the structurall orthotroic late are different in both the corrugation direction and the direction normal to the corrugation. The ield criteria develoed for the naturall anisotroic material are not aroriate in this analsis. However, it seems that the emirical data of structurall orthotroic lates for the ost ield behaviors are not available. Therefore, several hsical assumtions are reasonabl made in this stud and introduced in Chater three and these assumtions coincide with the naturall anisotroic material..5 Plastic Stress-Strain Relations A more comlicated distinction between elastic and lastic stress-strain relations arises from the fact that in the elastic range, the strains are unique and can be determined b the stress; therefore, for a given set of stresses we can calculate the strains b using Hooke s law without an regard as to how the stress state was reached; however, in general, in the lastic range, the strains are not uniquel determined b the stresses but deend on the histor of loading or the stresses state. Because of the deendence of the lastic strains on the loading ath, it is necessar to calculate the increments of lastic strain throughout the loading histor and then obtain the total strains b summation. However, there is at least one imortant class of loading aths for which the lastic strains are indeendent of the loading ath and deend onl on the final state of stress. These are called roortional loading aths, in which all the stresses increase in the same ratio. These will be discussed subsequentl.

35 .5.1 Incremental Theor of Plasticit (Plastic Flow Rule) (Mendelson 1983) 5 In general, the incremental theor is considered as the correct henomenological theor of lasticit. The rocess of deformation in lastic stage is irreversible. The equations of the theor of lastic flow establish a connection between infinitesimal increments of strain and stress, the stresses themselves and certain arameters of lastic state. The total strain increments are combined the increments in elastic strains and lastic strains and shows as follow: dε + e = dε dε (.9) Where dε e is the increment in elastic strain comonents and is connected with the increment in elastic stress according to Hooke s law. dε is the increment in lastic strain comonents and can be eressed b the flow rule: dε ij g = dλ (.10) σ Where dλ is a ositive scalar that deends on the stress increment. The function g, defining the ratios of the comonents of the lastic strain increment, is known as the lastic otential. In the view of the similarit of the roerties of the ield function and the lastic otential, it is generall assumed that the are actuall identical. ij The flow rule obtained on the basis of identit of lastic otential and ield function is known as the associated flow rule for the given ield criterion. There are two articular stress-strain relations associated with von Mises ield criterion and known as: (a) Lev-Mises equations and (b) Prandtl-Reuss equations.

36 (a) Lev-Mises Equations 6 Lev first introduced the general three-dimensional equations relating the increments of total strain to the stress deviations in Also, in 1913, these equations were given indeendentl b von Mises. These are known as the Lev-Mises equations. These equations are shown as follows: dε S dε dε dε ε dε z z d z = = = = = = dλ (.11) S S τ τ τ z z z or dε ij = Sij dλ (.1) Where S ij dλ the stress deviator tensor nonnegative constant which ma var through out the loading histor In these equations the elastic strains are assumed to be ignored and the total strain increments are assumed to be equal to the lastic strain increments. Therefore, these equations can be alied to large lastic flow so that the elastic strains can be neglected and cannot be used in the elasto-lastic transitional range. (b)prandtl-reuss-equations Instead of the Lev-Mises equations, Prandtl and Ruess included both elastic and lastic comonents of strains in the generalization of Eq..11. Reuss assumed that the lastic strain increment is roortional to the instantaneous stress deviation at an instant of loading. The equations are as follows:

37 7 or Eq..11 can be regarded as a secial case of Eq..13 where the elastic strains are neglected. The theor in which the lastic strains are governed b Eq..1 or Eq..14 is known as the incremental or flow theor of lasticit. We can al von Mises ield criterion, the equivalent or effective stress σ e, and the equivalent or effective lastic strain increment dε into the Prandtl-Ruess equations. Then, the constant dλ can be written as: and the stress-strain relations become: λ τ ε τ ε τ ε ε ε ε d d d d S d S d S d z z z z z z = = = = = = (.13) ε dλ S d ij ij = (.14) e d d σ ε λ 3 = (.15) ( ) ( ) ( ) + = + = + = z e z z e z e d d d d d d σ σ σ σ ε ε σ σ σ σ ε ε σ σ σ σ ε ε (.16)

38 8 dε 3 dε = τ σ e dε z 3 dε = τ σ e z (.17) dε z = 3 dε σ e τ z or dε ij 3 Sij = dε (.18) σ e The Prandtl-Ruess stress-strain relation is well known as the most satisfactor basis for dealing with lastic analsis when anisotro and Bauschinger effect are secondar imortance (Chakrabart, 1987); however, the incremental theor generall leads to mathematical comleities..5. Deformation Theor In contrast to the incremental or flow theories, Henck roosed total stress-strain relations in 194. Considerable simlifications are often achieved b using the deformation theor. The comonents of the total lastic strain are taken to be roortional to the corresonding deviatoric stresses. The lastic strain relation roosed b Henck can be written as: ε = λs (.19) ij ij

39 9 Where λ is ositive during loading and zero during unloading. From Eq..19, the lastic strain is a function onl of the current state of stress and is indeendent of the loading ath. It can easil be roved that for the case of roortional loading, i.e., if all the stresses are increasing in a ratio, the incremental theor described reviousl can reduce to the deformation theor. The verification can be found in (Kachanov 1971; Mendelson 1983). Furthermore, Budiansk (1959) roosed that there are a range of loading aths other than roortional loading for which the general requirements of lasticit theor are satisfied b deformation theories..6 Methods of Analsis The main difficult of lastic analsis is the mathematic comleit. However, that difficult has been overcome b advent of high-seed comuters and the efficient use of comutational methods. In this numerical analsis, the general Galerkin method, direct iteration, and initial stiffness aroach are emloed..6.1 The Galerkin method The Galerkin method can be emloed successfull to diverse tes of late bending roblems. The main idea of this method is minimizing errors b orthogonalizing with resect to the assigned set of trial functions and reducing the sstem of differential

40 30 equations into a sstem of algebraic equations (Giri and Smitses 1980; Niogi 1973). B aling the Galerkin method, Bennett (1971) studied the nonlinear vibration of unsmmetrical angle-l lates. This method can ield a ver eas and fast solution. The detail treatment can be found in (Ventsel and Krauthammer 001) or an late bending theor monograh..6. Initial stiffness aroach The mathematical rogramming concets to incremental elastic-lastic analsis have been discussed b Martin et al. (1987). There are two aroaches, initial stress and initial strain aroaches. The initial stiffness aroach (Owen and Figueiras 1983a; Owen and Figueiras 1983b) has been develoed and etensivel used for elasto-lastic roblem. Zienkiewicz et al. (1969) develoed the finite element rogramming using the initial stress method. The initial lastic moment method develoed b Moshaiov et al. (1986) has been adoted as the comutational rocess in the roosed elasto-lastic analsis.

41 31 Chater 3 Methodolog 3.1 Introduction In this chater the elasto-lastic analsis of a corrugated-core sandwich late is addressed. In general, the corrugated-core sandwich late is made of an isotroic material. To develo a mathematical theor of lasticit, several idealizations of ield criterion are taken into account. First, it is assumed that the conditions of loading are such that all strain rate and thermal effects can be neglected. Secondl, the Bauschinger effect is disregarded. Finall, the material is assumed to be isotroic. As mentioned reviousl, the 3D corrugated-core sandwich late can be reduced to a D structurall orthotroic continuum using several elastic constants. The basic elasto-lastic material behavior in the two-dimensional structurall orthotroic late must be resented before the numerical asects. The Reissner-Mindlin late theor is alied to this analsis; namel, the transverse shear deformation effects are taken into account for a sandwich late subjected to static loads. Also, the von Mises and Ilushin s ield surfaces are considered and modified in this analsis.

42 The analsis is based on the Reissner-Mindlin late theor. Therefore, three hotheses of this theor are as follows: 3 1. The straight line normal to the middle lane of the late remains straight but not necessaril normal to the middle lane when the late is subjected to bending.. The transverse normal stress, σ z, is small when comared with the stresses σ and σ, i.e. the σ z can be neglected. 3. The middle lane remains unstrained subsequent to bending. An elastic-lastic analsis based on the incremental theor of lasticit can be carried out into two stages, elastic and elasto-lastic stes. The elastic ste is that the elastic solution where internal stresses and strains follow Hook s law and the elastolastic ste is that this elastic sstem is modified to combine lastic strains. The lastic strain is defined as the difference between the total strain and the elastic strain. Unlike elastic solids, in which the state of strain deends onl on the final state of stress, the lastic deformation is determined b the comlete histor of the loading. Therefore, the lastic roblems are essentiall incremental in nature. The detail will be discussed in the following sections. 3. Elastic Analsis Before the onset of lasticit the relationshi between stress and strain remains in the linear elastic stage. In elastic range, the strains are linearl related to the stresses b

43 33 Hook s law. We consider the corrugated-core sandwich lates with thin faces under the assumtions stated in Section 1.1. It is assumed that the sheet faces work as membranes; namel, onl the direct stresses that are uniforml distributed over the thickness act in the faces. The core contributes to the fleural stiffness in the corrugation direction, - direction, but not in the direction normal to the corrugation, -direction. The shear strain deformations of this late are taken into account. Due to these elastic constants introduced b Libove and Batdorf (1948), the 3D corrugated-core sandwich late can be reduced to a D structurall orthotroic late Equilibrium Equations A ositive bending moment results in tensile stresses forming in the bottom fibers. Accordingl, all the moments and shear forces acting on the element in Figure 3-1 are ositive. Consider the equilibrium of an infinitesimal element with length d and width d cut from the late subjected to vertical distributing load of intensit q(, ) alied to an uer surface of the late shown in Figure 3-1. Since the stress resultants and the stress coules are assumed to be alied to the middle lane of this element, a distributed load q(,) is transferred to the middle lane.

44 34 z q d d M h M M M Q Q Fig. 3-1: Sign convention used in the late analsis We can take the equilibriums of moments about ais and ais and the sum of forces in z direction. Therefore, the following three indeendent conditions of equilibrium can be formed, resectivel. M M + Q = 0 (3.1) M M + Q = 0 (3.) Q Q + + q = 0 (3.3) Where M bending moment in the -direction

45 M bending moment in the -direction 35 M twisting moment and M =M Q transverse shear force in the -z lane Q transverse shear force in the -z lane q transverse load intensit 3.. Curvature-Dislacement Relations According to the first Mindlin late hothesis in Section 3.1, three indeendent degrees of freedom corresonding to the deflection w in z-direction and the angles of rotation θ and θ, which are not related to the deflection w are acquired for the late roblem. Therefore, we can obtain the curvature-dislacement relationshi as follows: θ θ χ =, χ = (3.4) χ 1 θ = θ + (3.5) Where: θ and θ sloes of the normal to the middle lane of the sandwich late about the and aes, resectivel. The above angles are assumed to be indeendent variables with resect to w χ curvature along the ais

46 36 χ curvature along the ais χ twisting curvature with resect to the and aes 3..3 Constitutive Equations The 3D corrugated-core sandwich late can be considered as a D structurall orthotroic construction due to several elastic constants. Thus, we can obtain the following bending moments, Eq. 3.6 and Eq. 3.7, and twisting moment relation, Eq. 3.8, for orthotroic lates b substituting curvature-dislacement relations, Eq. 3.4 and Eq. 3.5, as follows: Where: ( ) ( ) + = + = + = D D D M θ υ θ χ υ χ χ υ χ υ υ 1 (3.6) ( ) ( ) + = + = + = D D D M θ υ θ χ χ υ χ χ υ υ υ 1 (3.7) + = = D D M θ θ χ (3.8)

47 D and D fleural stiffnesses of the sandwich late as a whole in the z and z 37 lanes, resectivel. D twisting stiffness of the sandwich late as a whole in the lane. υ and υ Poinson s ratios in and aes, resectivel. Here D υ =D υ and this has been derived b Libove and Batdorf (1951), and D D υ υ = and 1 D = D 1 υ υ, thus, D υ =D υ. The shear forces can be eressed as Eq. 3.9 and Eq. 3.10: w Q = DQγ z = DQ θ + (3.9) w Q = DQγ z = DQ θ + (3.10) Where w D Q and D Q deflection in z-direction. shear stiffnesses of lates in z and z lanes, resectivel. As described reviousl, a 3D sandwich construction can reduce to a D structurall equivalent orthotroic late due to the above elastic constants. These elastic constants, D, D, D, D Q, and D Q, of corrugated-core sandwich lates were derived b

48 38 Libove and Hubka (1951). D and D are the fleural stiffness of the sandwich late as a whole in the z and z coordinate lanes, resectivel, and D is the twisting stiffness of the sandwich late as a whole in the lane. The can be defined as Eq. 3.11: D Q and D Q are the shear stiffness of the sandwich late as a whole in the z and z lanes, resectivel. The are defined as Eq. 3.1: The detailed eressions of these elastic constants can be found in Aendi A Governing Equations Substituting for the stress resultants and stress coules from Eq. 3.4 to Eq. 3.8 into Eq. 3.1 to Eq. 3.3, we obtain the following sstem of governing differential equations for an orthotroic sandwich late: w M D w M D w M D = = = ; ; (3.11) z Q z Q Q D Q D γ γ = = ; (3.1) 0 = w D D D D D D Q Q θ υ θ (3.13) 0 = w D D D D D D Q Q θ θ υ (3.14) 0 = q w D D D D Q Q Q Q θ θ (3.15)

49 The above sstem can be reresented in the following oerator form: 39 ({ }) { F} L ϑ = (3.16) Where: L11 L1 L13 L = L1 L L3 (3.17) L 31 L3 L33 θ ϑ = θ (3.18) w { } 0 F = 0 q { } (3.19) L L L = D D = = D Q ( ) D ( ) ( ) + + D υ ( ) D Q L L 1 3 = L 1 = D, L Q ( ) = D ( ) D ( ) + D Q (3.0) L L = L = D 13 Q, L 3 ( ) ( ) = L + D 3 Q

50 q intensit of lateral loading Theor of Plasticit The objective of the theor of lasticit is to obtain a theoretical relationshi between stress and strain for an elastic-lastic behavior of materials. There are three requirements in order to formulate a theor that models elasto-lastic material behavior: 1. An elicit relationshi between stress and strain must be formulated to describe material behavior under elastic conditions, i.e., before the onset of lastic deformation.. A ield criterion, which secifies the state of the multiaial stresses corresonding to the occurrence of lastic flow. 3. A flow rule, which determines the relations between lastic strain increments and stresses increments after ielding. The second requirement the Huber-Mises ield criterion, with isotroic hardening etended for naturall anisotroic materials b Hill, and Ilushin s dimensionless ield surface, will be modified to a structurall orthotroic late for this analsis. To achieve the elasto-lastic analsis, the ield criterion should be determined first. As mentioned reviousl, this corrugated-core sandwich late is aroimated b a D structurall orthotroic late. The material of that late is isotroic material; however, the structural behavior of that late is orthotroic. No ield criterion was used in revious studies for

51 structurall orthotroic behavior of lates. Therefore, in this stud, the ield criterion will be modified for a structurall orthotroic. 41 To derive the governing differential equations for the elastic-lastic analsis of corrugated-core sandwich lates, the following assumtions will be made: 1. The ield function f is a function of the direct stresses associated with fleure of the late onl, but not of the transverse shear stresses.. All assumtions regarding the behavior and mechanical resonse of the corrugated-core sandwich late adated in mechanics of sandwich lates and discussed reviousl are valid here ecet for the constitutive equations. In what follows, it is also assumed that dislacements are small, and the material of the late is isotroic. 3. When the bending moment reaches the ield moment, the whole cross section of the reduced late lastifies simultaneousl, namel, the whole cross section of the late is assumed to become lastic instantaneousl. The third assumtion is however a convenient fiction as in realit there is alwas a gradual lastification of the whole cross-section of the late with the outer fibers becoming initiall lastic. The lastic zone then sreads inward until the whole section ultimatel ields shown in Figure 3-. Moreover, this assumtion can reduce a significant difficult in calculation.

52 4 < σ 0 σ 0 > σ 0 σ 0 < σ > σ 0 0 σ 0 σ 0 Fig. 3-: Yielding of non-laered cross-section According to the above conditions, the develoment of the elasto-lastic analsis is introduced in the following sections. Essentiall, lastic behavior is characterized b an irreversible strain, which can onl be obtained once a certain level of stress has been reached The Yield Criterion It is suosed that a solid continuum is subjected to graduall increasing stresses. The initial deformation of the solid is entirel elastic and recoverable on comletel unloading. For certain critical combinations of alied stresses, lastic deformation first occurs in the element. A rule that defines the limit of elastic behavior under an ossible stress combinations is called ield criterion.

53 43 An accurate ield function is the ke factor for the accurac of the elasto-lastic analsis. Because of the assumtion that the whole cross section ields simultaneousl in Section 3.3, the ield function in terms of stress coules is considered here. For Mindlin lates we ma assume that the ield function F is eressed as a function of the bending moments {M}, but not of the shear forces {Q} (Karam 199). During ielding it is assumed that the bending and twisting moments must remain on the ield surface so that: ({ M} ) k( ξ ) f = (3.1) Where f is some function and k is a material arameter that is determined eerimentall. The term k ma be a function of a hardening arameter, ξ. Rearrange Eq. 3.1: ({ M}, ξ ) = f ({ M }) k( ξ ) = 0 F (3.) The von Mises ield surface for the bending of structurall orthotroic sandwich lates can be obtained b modifing Ilushin s ield surface eressed in terms of dimensionless stress resultants for isotroic lates and shells (Shi and Voiadjis 199) or the late ield criteria for isotroic late in (Lubliner 1990) as follows: m m m + m + m 1 (3.3) = Where: m = αβ M αβ M U (3.4)

54 M U = σ h 0 4 (3.5) 44 α,β =1, h = the thickness of late σ 0 = ield stress Using the same dimensionless idea as Eq. 3.3 and introducing the anisotroic arameters into the von Mises ield function, we can obtain: A 11 m A + m A1mm + 3A33m 1= 0 (3.6) The terms m, m, and m are dimensionless bending and twisting moments, resectivel defined as: m = M M 0, m = M M 0, m = M M 0 (3.7) Where : M and M bending moments in the (the corrugation direction) and (erendicular to the corrugation) directions, resectivel M twisting moment for the structurall orthotroic sandwich late M 0 and M 0 uniaial ield bending moments in the (the corrugation direction) and (erendicular to the corrugation) directions, resectivel M 0 ield twisting moment for the structurall orthotroic sandwich late

55 45 A 11, A 1, A, and A 33 Anisotroic arameters deend on the structural orthotro of the sandwich late. The formulas of these anisotroic arameters are given in Aendi B. Accordingl, the ield criterion for structurall orthotroic sandwich lates can be formulated. Since the corrugated-core sandwich late is regarded as a structurall orthotroic late, the ield moments in each - and -direction should be different. In an elastic erfectl-lastic material, the effects of strain hardening are disregarded. That means once the ield moment M 0 is reached the material could not take more strength. The strain hardening rovides an increase in the strength of the material. Figure 3-3 is the behaviors of elastic erfectl-lastic and strain-hardening structures. Moment Strain Hardening M 0 Perfectl Plastic Curvature Fig. 3-3: Material behavior

56 46 Net, let us define the uniaial ield moment. According to the third assumtion in Section 3.3, the whole cross section ields simultaneousl. Geometric arameters of a late unit are defined in Figure 3-4. Therefore, referring to Figure 3-4, this moment in the direction of corrugation, M 0, can be calculated as follows: M 0 = σ 0 t h + f h c c σ 0 h zda c = σ 0 t f ( h + t + t ) c c f h + c 0 σ 0 zda c (3.8) = σ 0 [ t ( h + t + t ) + Z ] f c c f Where: Z first moment of area of one-half of the corrugation about the central ais of the late er unit length σ 0 the ield stress of the material in a tension test σ 0 z h h f t f α t c ` f h c f σ 0 Fig. 3-4: the cross section and stress distribution in direction

57 Where: 47 t c, t f thicknesses of core and face sheets, resectivel. α corrugation angle h c deth of corrugation measured from the center line at crest to center line at trough h f distance between middle surfaces of face sheets corrugation itch, the distance between two crowns length of corrugation flat segment The fleural contribution of core in the direction normal to the corrugation is significantl smaller than the face sheets. Furthermore, the osition of the core varies in different cross section selected in the -direction. Therefore, according to the basic assumtion of the corrugated-core sandwich late in Section 1.1, the corrugation gives no contribution into the fleural resistance in the -direction, i.e. the direction normal to the corrugation. Hence, referring to Figure 3-5, the ield moment M 0 can be defined as: h t c + t f c M ( ) = t f 0 σ 0 + = 0t f hc + t c + t f σ (3.9)

58 48 1 σ 0 h z h t f h c σ 0 Fig. 3-5: the cross section and stress distribution in direction The same reason of moment resistance for twisting moment as that in -direction, the corrugation also gives no contribution into the twisting resistance to fleure. Thus: ( h + t t ) M + 0 = τ 0 t f c c f (3.30) Since τ 0 = σ 0 / 3, we have: Eq f ( h + t t ) σ 0 M = t + 3 c c f (3.31) 3.3. Elasto-Plastic Analsis The rocess of lastic deformation is irreversible. Most of the work of deformation is transformed into heat. A more comlicated distinction between elastic and lastic stress-strain relations is that in the elastic stage the stains are uniquel determined b the stresses, whereas in lastic range the strains are generall not uniquel determined b the stresses, but deend on the whole histor of loading. To analze the elasto-lastic

59 behavior, the modified ield criterion based on von Mises and Ilushin criteria and associated flow rule (Prandtl-Reuss equations) are used Incremental Form of Elastic Constitutive Equations The definitions of those smbols in the incremental form are the same as revious sections. The smbol, d, in front of the term is defined as the increment of that term. The total curvature increments are considered to be the comonents of elastic and lastic curvature increments and can be written as: { dχ } { dχ e } + { dχ } = (3.3) Where the suerscrits e and are the elastic and lastic curvature comonents, resectivel. According to the kinematic assumtions, the increments of the total bending strain are given in terms of the deflection and sloe increments as follows: ( dθ ) ( dθ ) dχ =, dχ = (3.33) dχ 1 = ( dθ ) ( dθ ) + (3.34) ( dw) ( dw) e e dψ = dθ +, dψ = dθ + (3.35)

60 The elastic strain increments are related to the moments through the orthotroic Hooke s law as follows: 50 dm dm dm = D = D = D e e ( dχ ) + D υ ( dχ ) υ e e ( dχ ) + D ( dχ ) e ( dχ ) (3.36) dq dq = D = D Q Q e ( dψ ) e ( dψ ) Or in matri form: { dm } [ C e ]{ dχ e } = (3.37) { dq} [ C s ]{ dψ e } = (3.38) e e 1 { d } = [ C ] { dm} χ (3.39) Where: e [ C ] D = Dυ 0 D D 0 υ 0 0 D DQ 0 s [ C ] = 0 D Q

61 51 dχ e e e { dχ } dχ, { dψ } dχ e e dψ = = e e dψ dm dm { dm } = dm, { dq} dq = dq According to the Prandtl-Ruess theor (Valliaan 1976), the lastic strain incremental comonents (curvatures) can be eressed as: { dχ } dχ = dχ dχ 1 1 = ( A M A M ) 11 ( A M + A M ) 1 3A 33 M 1 d χ M dχ M dχ M (3.40) A A = 3A33M (3.41) 11 M M A1M M + M + Where the equivalent moment, M, is defined b the Huber-Mises ield function. d χ is the incremental equivalent lastic strain associated with d M, which is obtained b imlicit differentiation of Eq as shown: dm dm dm d M = ( A11M A1M ) + ( A1M + AM ) + 3A33M (3.4) M M M

62 If H is defined as the sloe of equivalent moment vs. equivalent lastic strain curve as shown in Figure 3-6, then d χ is related to d M as follows: 5 dm d χ = (3.43) H M M M 0 H = dm d χ M 0 H = dm d χ χ χ Fig. 3-6: Equivalent Stress-Strain Curve For eamle, consider a stri-beam cut from the late in the direction. For this articular case, since the onl moment occurs in -direction, it can be shown that the equivalent moment and equivalent lastic strain increment reduce to: d M = dm d χ = dχ (3.44)

63 Therefore, Eq for the above case can be considered as the following form: 53 H d M = d χ dm = dχ dm = dχ dχ = dχ dm = 1 EI e 1 dχ dm 1 e 1 EI = 1 ( EI ) ( EI ) EI (3.45) Where: EI, (EI ) E the elastic and elasto-lastic resonse sloes, resectivel the modulus of elasticit of the late material. It can be shown that H is the sloe of the uniaial stress versus the uniaial lastic strain as obtained in a uniaial ield test. Obviousl, for structurall orthotroic sandwich lates, the values of H will be different in the and directions. Let us write Eq in the matri form as: { d } { a} χ = d χ (3.46) Where: { a} 1 = 1 ( A M A M ) 11 ( A M + A M ) 1 3A 33 M M M M 1 (3.47)

64 54 Then the incremental equivalent moment, d M, given b Eq. 3.4, can be also written in the matri form as: d M T { a} { dm } = (3.48) Rewriting the elastic relations i.e. Eq in the form: e e e { dm } [ C ]{ dχ } = [ C ]{ ( dχ} { dχ }) = (3.49) and substituting Eq into Eq. 3.49, we can get: ( dχ ) e { dm } [ C ]{ dχ} { a} = (3.50) But because of Eq. 3.43: d M ( χ ) = H d (3.51) Substituting Eq into Eq and using Eq. 3.51, ields: d M ( χ dχ ) = Hd χ T e { a} [ C ]{ d } { a} = (3.5) Now the incremental equivalent lastic strain d χ can be eressed as: T e { a} [ C ]{ dχ} T e { a} [ C ]{ a} + H d χ = (3.53) Substituting Eq into Eq. 3.49, we obtain: e { dm } = [ C ] { d } T e { a}{ a} [ C ]{ dχ} T e { a} [ C ]{ a} + H χ (3.54)

65 We can reresent the above equation in the following form: 55 e { dm } [ C ]{ dχ} = (3.55) e T e e e [ ] [ ] [ C ]{ a}{ a} [ C ] C = C T e { a} [ C ]{ a} + H (3.56) The elasto-lasticit matri [C e ] takes the lace of the elasticit matri [C e ] in the incremental analsis. It is evident that the elasto-lastic matri is smmetric and ositive definite. Furthermore, the Eq is valid whether or not H takes on a zero value The Incremental Form of the Governing Equations The incremental form of the equilibrium equations is given b: [ T ]{ d } = { dr} Ξ (3.57) Where: { dξ} = { dm, dm, dm, dq dq }, T = { dm } { dq} 0 dr = 0 dq { }

66 56 [ T ] = Using Eq the moments ma be eressed as follows: { dm } { dm e } { dm } = (3.58) e dm e dm χ = dm (3.59) e dm e e { } = [ C ]{ d } e { dm } = [ C ] dm T e { a}{ a} [ C ]{ dχ} = dm T e { } [ a C ]{ a} + H dm (3.60) Substituting Eq into Eq. 3.57, we can get: e e ( dm dm ) ( dm dm ) e e ( dm dm ) ( dm dm ) ( dq ) ( dq ) = dq dq dq = 0 = 0 (3.61)

67 Rearrange the above Eq. 3.61, we can obtain: 57 e e ( dm ) ( dm ) ( dm ) ( dm ) + dq = + e e ( dm ) ( dm ) ( dm ) ( dm ) + dq = + (3.6) ( dq ) ( dq ) + = dq Substituting Eq and Eq to Eq into Eq. 3.6, after some transformation the following equation is obtained: [ L]{ d } = { dr} + { dr} ϑ (3.63) Where: { dr} = ( dm ) ( dm ) ( dm ) ( dm ) (3.64) { dϑ } { } dθ = dθ dw 0 dr = 0 dq

68 58 [ L ] L = L L L L L 1 3 L L L The differential oerators L ij (i, j =1,,3) are identical to Eq It is evident that the governing equations of sandwich late bending, Eq. 3.63, include lateral loading and lastic moment effects. The lastic moment vector, {dm }, which is unknown at an increment, siml aears in the equations as an additional lateral load. 3.4 Boundar Conditions For most common tes of suorts, in ractice, the boundar conditions are siml suorted, clamed, and free. Let us consider the aroriate boundar conditions rescribed on an edge of a sandwich, rectangular, orthotroic late. Unlike the classical theor, the governing differential equations, Eq. 3.63, have the sith order not fourth order; therefore, three boundar conditions should be rescribed at an oint of the sandwich late side. The discussions for the different boundar conditionsis resented net. (a) Siml suorted edge The rincial boundar conditions for a siml suorted edge are that the deflections in z-direction and bending moments, M or M, are zero. There are two different tes of conditions that ma be regarded as the third boundar condition (Folie 1971), illustrated in Figure 3-7. One ma set the twisting moment, M, to zero thus

69 59 ermitting the edge to shear, e.g. on this edge arallel to the -ais γ z?0. In most ractical cases, there will be an edge stiffener or some smmetric constraint to revent such shear deformation. Therefore, the boundar condition should rather allow the eistence of a twisting moment along the edge but restrict the shear deformation of the edge. The former boundar condition, where the twisting moment along the edge is zero, is called the soft boundar condition. The latter, reventing shear deformations, is called the hard boundar condition. M =0 γ z =0 z Soft te γ z M Hard te Fig. 3-7: Definition of boundar conditions (for =0 or b) Two boundar conditions for a siml suorted edge, sa =constant are: w 0, M = 0 (3.65) = (1) Hard te boundar: The third condition will be reresented for the hard te of siml suorted boundar condition. Then the following boundar condition takes lace: w γ z = θ + = 0 (3.66)

70 From Eq. 3.10, because D Q is the elastic constant of corrugated-core sandwich lates, therefore, we can obtain: 60 w Q = DQγ z = DQ θ + = 0 (3.67) From Eq. 3.6, we can obtain: M = D θ + υ θ = 0 (3.68) Since D is the material constant, hence: θ + υ θ = 0 (3.69) Thus, the boundar conditions at = 0, a in this case can be formulated in terms of the rincial unknowns of the roblem, as follows: w = 0 θ + υ θ = 0 (3.70) θ w + = 0 Since w = 0 on the edge = constant, w = 0. Therefore, the third equation of Eq is of the form θ = 0 and thus θ = 0. Finall, we can reresent the above boundar conditions in the form:

71 For = 0, a 61 w θ 0, = 0, θ = 0 = (3.71) Meanwhile, the boundar conditions for hard te siml suort on the edges = constant can be defined as follows: For = 0, b w θ 0, = 0, θ = 0 = (3.7) () Soft te boundar: For = 0, a w 0, M = 0, M = 0 (3.73) = For = 0, b w 0, M = 0, M = 0 (3.74) = From Eq. 3.4 to Eq. 3.8, the boundar conditions, Eq and Eq. 3.74, can be eressed in terms of the deflection and sloes, therefore, For = 0, a θ θ θ θ w = 0, + υ = 0, + = 0 (3.75) For = 0, b

72 6 θ θ θ θ w = 0, + υ = 0, + = 0 (3.76) (b) Clamed edge The boundar conditions characterizing a clamed edge are no dislacements of the neutral surface and no rotation of the cross-section in the boundar. These boundar conditions are shown as follows: Eq θ = θ = w = 0 (3.77) (c) Free edge There are no force and stress coules on a free edge. Therefore, the boundar conditions for a free edge can be eressed as: For = 0, a M = M = Q = 0 (3.78) For = 0, b M = M = Q = 0 (3.79) as followings: Or the boundar conditions can be eressed all in terms of deflection and sloes For =0, a θ + υ θ θ θ w = 0, + = 0, θ + = 0 (3.80)

73 For =0, b 63 θ + υ θ θ θ w = 0, + = 0, θ + = 0 (3.81) 3.5 Research Aroach For elastic analsis, the general Galerkin method and double Fourier s eressions are alied. The trial functions satisfing the boundar conditions are selected for this analsis. We substitute the selected trial functions into the governing Eq and solve simultaneous equation sstem for the unknown coefficients in double Fourier s series of these unknown dislacements until the ielding occurs. Thus, the deflection, w, and rotated angles, θ and θ, can be found b substituting these coefficients into the Fourier s eressions. Furthermore, the corresonding moments and curvatures in - and -direction can be calculated. After the ielding occurs, the elasto-lastic analsis begins. Ecet the methods used in elastic analsis, in addition, the iteration rocedure and incremental theor of lasticit are emloed to solve the governing equations, Eq. 3.63, in the elasto-lastic analsis. Because in the elasto-lastic analsis the incremental theor of lasticit is emloed, the trial functions are emloed in an incremental stle. The most difficult of the elasto-lastic analsis is that in Eq the {dr} is unknown and relates to the incremental lastic moments {dm }. In each increment the incremental lastic moments

74 are unknown. Therefore, it is necessar to emlo iterative to find the lastic moments and {dr}. The rocedure is discussed in Chater Four. 64

75 Chater 4 Numerical Analsis and Results 4.1 Introduction The mathematical model of the elasto-lastic analsis of corrugated-core sandwich lates was roosed in revious Chater. The numerical rocedure for the elasto-lastic analsis is resented in this Chater. In this analsis, Double Fourier eansions are emloed to analze late bending roblems. The linear elastic analsis is fairl straightforward. Each load corresonds to dislacements. Thus, according to the constitutive equations, the stress coules can be obtained. The elasto-lastic analsis begins on a reliable elastic linear analsis. Onl a significant result of elastic linear analsis can induce an accurate elasto-lastic analsis. Therefore, the resent analtical solution with siml suorted edges is corroborated b eerimental results, the analtical solution, and 3D finite element analsis of corrugated-core sandwich late reorted in (Tan et al. 1989). The difficult of the elasto-lastic analsis is the mathematical comleit. The general Galerkin and incremental methods are emloed to solve the elasto-lastic bending roblem. The double iterations are alied to the load increment and the lastic

76 66 incremental moment determination. The outer loo is the load increment and the inner loo is to obtain the lastic incremental moments. Numerical simulations are done b using MATLAB (000). 4. Linear Elastic Analsis 4..1 Numerical Algorithm Double Fourier series solutions have been widel alied in late roblems eseciall for siml suorted boundar conditions (e.g., Whitne 1969; Whitne and Leissa 1970; Holston 1971; Pagano 1970). Hence, in this rectangular late, as shown in Figure 4-1 with all siml-suorted edges, the solutions of the governing differential equations, Eq. 3.16, can be sought in the form of an infinite double Fourier eansion, as follows: w = θ θ = = m= 1 n= 1 m= 1 n= 1 m= 1 n= 1 w A B mn mn mn mπ nπ sin sin a b mπ nπ cos sin a b mπ nπ sin cos a b (4.1) q = q mπ nπ sin sin a mn m= 1 n= 1 b (4.) Where w mn, A mn, and B mn are reresent coefficients to be determined. q mn can be determined b the eternal load using double Fourier eansion. a is the length in the

77 direction of corrugation and b is the width in the direction erendicular to the corrugation. 67 b a Fig. 4-1: Rectangular siml suorted late The eternal load over a rectangular late can be an te of functions. In the subsequent eamle the uniforml distributed load is discussed. Let q be the intensit of the uniforml distributed load. Then, the coefficients of the double Fourier eansion, q mn in Eq. 4., can be determined as: q mn 4 ab a b = 0 0 qsin mπ sin a nπ dd b 4q = π ab ( 1 cos( mπ ))( 1 cos( nπ ) (4.3) 4q = π ab m [ 1 ( 1) ] 1 ( 1) n [ ]

78 68 In Eq. 4.3, onl when the terms, m and n, are equal to odd numbers the q mn is a non-zero value. Otherwise, it equals to zero. Furthermore, it can be easil verified that the eressions for the deflection and two rotations, Eq. 4.1, automaticall satisf the rescribed boundar conditions Eq and Eq The general Galerkin method is not necessar for solving the linear elastic analsis of a rectangular late with all siml suorted boundar edges. The Navior method is emloed, instead. Substituting Eq. 4.1 and Eq. 4. into Eq to Eq. 3.15, the sstem of equations in terms of the above unknown coefficients is in the following form: [ ]{ U} { R} K = (4.4) Where: [ K ] K = K K K K K 1 3 K K K K K = D = D mπ a Q mπ a D + nπ b + D Q ; K 1 D = + D υ mnπ ab (4.5) nπ D mπ nπ K1 = K1 ; K = D + + DQ ; K = DQ (4.6) 3 b a b

79 69 K31 = K13; K3 = K3 ; K 33 = D Q mπ a + D Q nπ b (4.7) wmn 0 U mn = 0 Bmn q { } = A ; { R} mn Solving Eq. 4.4, the coefficients of the deflection, w, and the sloes, θ and θ, in Eq. 4.1 can be obtained. Therefore, substituting the known solutions in Eq. 4.1 into the curvature-dislacement relations, Eq. 3.4 and Eq. 3.5, and the constitutive relations, Eq. 3.6 to Eq. 3.8, the moments of this late can be obtained. Figure 4- is the rogramming flow chart for the elastic analsis.

80 70 Start The alied loads q Calculate the corresonding {ϑ }, {χ} and moments {M} Yielding Yes No q=q+dq Start elasto-lastic analsis Fig. 4-: Flow chart for elastic analsis 4.. Elastic Analsis Comarison with Eerimental Results In this section, the behavior of the 6m b.1m corrugated-core sandwich late studied b Tan et al. (1989) is analzed. The material roerties of the corrugated-core sandwich late are the same as the late in (Tan et al. 1989). In this case, the uniform distributed loading intensit is taken to be 550 N/m. The modulus of elasticit is 08 GPa and the Poisson s ratio is assumed to be 0.3. These stiffness values shown in

81 Table 4-1 which are obtained from the formulas b (Libove and Hubka 1951) are from (Lok and Cheng 000). 71 Table 4-1: Geometric Parameters and Elastic Constants (mm) h c (mm) f (mm) t f (mm) t c (mm) D D D D Q D Q (N-m) (N-m) (N-m) (N/m) (N/m) Table 4- shows the comarisons of the maimum deflection at the central oint with classical analsis develoed b Libove and Batdorf (1948), 3D corrugated-core sandwich late finite element analsis (3D FEM) both reorted in (Tan et al. 1989), and the resent solution based on Reissner-Mindlin late theor. As can be seen, the difference between the 3D FEM and the resent solution in this thesis is 0.37% and between the solution b Tan et al. and the 3D FEM is 1.19%. Figure 4-3 is the deflection diagram of the whole late, with a comlete deflects downward. solution B Tan Table 4-: Comarison of central deflection Maimum Deflection, w ma (mm) Tan et al D FEM analsis Present solution Difference (%) between the resent solution and 3D FEM Difference (%) between solution b Tan and 3D FEM

82 7 w (m) b (m) a (m) Fig. 4-3: Deflection of the corrugated-core sandwich late Figure 4-4 shows the central deflection of the anel from this solution, the 3D FEM, and the eerimental investigations. It is noted that although the resent solution is closed to the eerimentall obtained deflections, and it agrees closel with that given b the 3D FEM and rovides adequate redictions.