Higher Order Shear Deformation Plate Theory

Transcription

1 Higher Order Shear Deformation Plate Theory by Gjermund Mæsel Kolvik THESIS for the degree of MASTER OF SCIENCE Master i Anvendt matematikk og mekanikk Faculty of Mathematics and Natural Sciences University of Oslo May 01 Det matematisk- naturvitenskapelige fakultet Universitetet i Oslo

2

3 Preface This thesis has been written to fulll the degree of Master of Science at the University of Oslo, Department of Mathematics, Mechanics Division. Professors Noël Challamel and Jostein Hellesland at the University of Oslo have been my supervisors during this project. Professor Challamel's expertise on higher order theories has been of immense help. I would like to thank him for his motivating enthusiasm and welcoming presence. I feel fortunate to have made his acquaintance during his year at the University of Oslo. I would like to express my gratitude to professor Jostein Hellesland for his excellent guidance and advice throughout the project. His vast knowledge in solid mechanics and encouraging support is highly appreciated. i

4

5 Abstract Several plate theories have been developed to describe the static and dynamic behaviour of plates. This thesis is predominantly a study of plate theories including shear eects, with emphasis on higher order shear deformation theories. The plate theories of Reddy and Shi are specically analysed. An eort towards the development of a unied higher order shear deformation plate theory is presented in this thesis. The buckling behaviour of some generic higher order shear plate models is investigated in a unied framework. The governing equations of the buckling problem are obtained from a variational approach, leading to generic partial dierential equations and associated boundary conditions. Buckling problems are analytically solved using the Navier method on isotropic simply supported plates under uniform in-plane loads. Buckling load relationships between classical plate theory and the plate theories including shear eects are also investigated. The accuracy of the unied shear deformation theory is demonstrated through these buckling results. The numerical results of the buckling problems, indicate that the theories of Reddy and Shi yield exactly the same buckling loads for the problems in question, whereas the buckling loads estimated from some other higher order theories vary slightly. Due to the simple nature of the solved buckling problems, in terms of geometrical and material properties, all the higher order theories yield almost the same buckling loads as the rst order shear deformation theory. This coincides with the fact that higher order plate theories have their advantages when being used for laminated composite plates. It is the author's belief that the unied higher order shear deformation plate theory presented in this thesis, can contribute to gathering many of the higher order theories presented in the literature in a common framework. iii

6

7 Contents 1 Introduction Background and motivation Objectives and Scope Outline of the report Development of plate theories 4.1 Short background Theoretical preliminaries The principle of minimum potential energy PMPE The divergence theorem Navier method Mindlin plate theory Strain energy Potential energy of external forces System of partial dierential equations Boundary conditions Solving the Mindlin buckling problem HSDT; a gradient elasticity approach 6 v

8 4.1 Gradient elasticity FSDT model Stiness parameters HSDT Strain energy System of partial dierential equations Boundary conditions Describing Reddy's and Shi's theories by unied dierential equations Solving the HSDT buckling problem Relationship between HSDT and CPT Introducing respective dimensionless stiness parameters into relationship Buckling results Case A Case B Case C Conclusion Conclusion of results Suggestions for further work References 6 Appendix 64 A Finding the relationship between HSDT and CPT 64 B Matlab scripts 69 vi

9

10 Chapter 1 Introduction 1.1 Background and motivation Plate structures are major load carrying elements in structural mechanics, both in aeronautics, on land and in naval engineering. Such plates are often subjected to signicant in plane compression forces and/or shear loading. Various plate theories are available to describe the static and dynamic behaviour of such plates. Depending on the plate geometry and material properties, it is of interest to utilize one plate theory over another. Understanding the dierences between the theories, and the application of them, is of interest both to engineers working in the eld of plate structures, as well as researchers working with the development of new knowledge on plates. Since the middle of the 19th century there has been ongoing research and development of plate theories. This research has resulted in three main categories in the eld of plate theories: Approximately 1850: Kirchho plate theory, classical plate theory CPT. Suitable for thin plates with thickness to width ratio less than 1/10. Neglects shear eects. Approximately 1950: Mindlin plate theory, rst order shear deformation plate theory FSDT. Suitable for thick plates with thickness to width ratio more than 1/10. Includes shear eects. Approximately 1980: Higher order shear deformation plate theories HSDT. Can represent the kinematics better than FSDT and are especially suitable for composite plates. Includes shear eects. 1

11 A few years prior to Mindlin, Reissner developed a plate theory including shear eects and therefor suitable for thick plates. The theories of Mindlin and Reissner are similar, and they are often referred to as one Reissner-Mindlin plate theory. According to Wang et al. 1] this is a misleading description as the two theories are based on dierent assumptions. In this thesis we do not treat the theory of Reissner, and when reference is made to FSDT we refer to Mindlin plate theory only. Many higher order shear deformation theories have been developed in the last 30 years. Without further comment we mention the theories of Touratier transverse strain distribution as a sine function, Soldatos hyperbolic shear deformation theory, Mechab hyperbolic shear deformation theory, Karama et al. exponential variation for the transverse strain in addition to Reddy, and Shi. The latter two proposed a parabolic variation of the transverse shear strain ]. Among the higher order plate theories, the one of J. N. Reddy is considered to be the most popular theory used for analysis of laminated composite plates 10]. In 007 Guangyu Shi presented a new shear deformation theory of plates, similar to Reddy's in the sense that they both proposed a parabolic variation of the transverse shear strain. Both Reddy's and Shi's theories are third order plate theories, meaning that the displacement eld is assumed to be described by a function of third order. The motivation behind this thesis is to describe several of the above mentioned higher order theories in a unied framework, emphasizing the theories of Reddy and Shi. In 011 Challamel 4] presented an article, in which he analytically studied and treated the buckling problem of a third order shear beam-column in the framework of gradient elasticity Timoshenko beam theory. This thesis is a similar study in the eld of plate theories. 1. Objectives and Scope The main objective of the thesis is to investigate the buckling behaviour of some generic higher order shear plate models in a unied framework. In order to achieve that, eorts will be made to show that most higher order shear plate models developed in the literature, whatever shear strain distribution assumptions over the cross section, can be classied in a common gradient elasticity rst order shear plate theory. A short review of CPT and FSDT will be given, but the thesis will primarily focus on the two plate theories of Reddy and Shi. Based on the displacement eld

12 that the two theories propose, a unied system of partial dierential equations that describe both theories will be developed. Variational consistent boundary conditions will also be presented as they appear when using the principle of minimum potential energy, but they will not be thoroughly investigated. The dierential equations will be used to solve the buckling problem for isotropic simply supported plates under uniform in-plane compressive load on all edges. Even though this study primarily considers the two theories of Reddy and Shi, and only solves fairly basic buckling problems simply supported, isotropic plates of homogenous material, it is the author's belief that the equations developed in this thesis can be transferred to solve more complex buckling problems orthotropic, laminated composite plates, and describe several higher order shear plate theories. In that respect, this study represents an eort towards the development of a unied higher order shear deformation plate theory. 1.3 Outline of the report Chapter gives a background frame. A review of dierent plate theories is presented in Section.1, and central methods for deriving the dierential equations and boundary conditions, as well as a description of the Navier method for solving buckling problems are presented in Section.. Chapter 3 deals with FSDT. The methodology presented in this chapter will be used also when dealing with HSDT, and the equations derived for FSDT give a basis for comparison with HSDT. Chapters 4 and 5 contain the theory and relationships linked with HSDT. Chapter 4 presents HSDT in the light of gradient elasticity, whereas the dierential equations for HSDT are derived and used to solve the buckling problem in Chapter 5. In Chapter 6 the buckling results are presented for CPT, FSDT and HSDT in tables. Chapter 7 contains conclusion and suggestions for further work. 3

13 Chapter Development of plate theories.1 Short background Several plate theories have been developed to describe the static and dynamic behaviour of plates, and many of them are based on displacement approximations. According to Altenbach 1] engineers prefer theories which are based on hypotheses. The rst theory of plates based on displacement assumptions, was presented by Kirchho in Kirchho's assumptions read as follows 13]: Straight lines perpendicular to the mid-surface i.e., transverse normals before deformation remain straight after deformation. The transverse normals do not experience elongation i.e., they are inextensible. The transverse normals rotate such that they remain perpendicular to the mid-surface after deformation. The consequence of the Kirchho hypothesis is that the transverse shear strains are zero, and consequently, the transverse stresses do not enter the theory. The theory is known as the Kirchho plate theory and it is an extension of Euler- Bernoulli beam theory. Often Kirchho plate theory is referred to as the classical plate theory. It does not include shear eects and is therefore applicable to thin plates only. The classical plate theory will give erroneous results when being used for thick plates, especially plates made of advanced composites 6]. To account for the transverse shear strains, shear deformation plate theories have been developed. Mindlin proposed his theory in 1951, which is an extension of Kirchho plate theory. In Mindlin plate theory the basic equations are derived 4

14 by assumption that the in-plane displacements are linearly distributed across the plate thickness. This leads to the transverse shear stresses being constant across the plate thickness, so the zero shear stress condition on the plate face is not satised. This forces the use of shear correction factors, comparable to the need for shear correction factors in the Timoshenko beam theory. Mindlin plate theory is often referred to as rst order shear deformation plate theory, and it has been extensively used in the analysis of shear exible plates and shells. But when Mindlin plate theory is applied to composite plates, the diculty in accurately evaluating the shear correction factors presents the shortcommings of FSDT 10]. To properly approximate the nonlinear distribution of transverse shear strains along the plate thickness, quite a number of higher order shear deformation plate theories were developed. Such HSDTs have proven to be highly applicable to laminated composite plates. Levinson and Murthy developed plate theories that employ three-order polynomials to expand the in plane displacement across the plate thickness, which in turn excludes the need for shear correction factors. However, in Levinson's and Murthy's plate theories they used the equilibrium equations of the classical plate theory, which is variationally inconsistent with the kinematics of displacements. In order to rectify this defect, Reddy presented his plate theory in 1984, which developes variational consistent equilibrium equations for plates 6]. The two-dimensional plate theories with higher order in plane displacements but a constant deection through the plate thickness are the so called simple higher order shear deformation theories 10]. A two dimensional structure is here deined as 1]: A two dimensional load bearing structural element is a model for analysis in Engineering/Structural Mechanics, having two geometrical dimensions which are of the same order and which are signicantly larger in comparison with the third thickness direction. Reddy's plate theory is known for being the most popular simple HSDT used for composite plate analysis 10]. According to Liu 6], another HSDT was developed by Ambartsumian in which he proposed another transverse shear stress function in order to explain deformation of layered anisotropic plates. Reddy's and Ambartsumian's theories formed a solid benchmark for the development of a new simple HSDT. In 007 Shi presented a new HSDT which is developed on the basis of Murthy's and Reddy's theories. Shi derived a new set of variational consistent governing equations and associated proper boundary conditions. Both Reddy and Shi employed third order polynomials in the expansion of the displacement components through the thickness of the plate, leading to 5

15 a parabolic variation of the transverse shear stresses. They are therefore sometimes referred to as Third order shear deformation theories or Parabolic shear deformation theories. It should be remarked that it is not preferable to always use HSDT in order to get as accurate numerical results as possible. On the contrary, Reddy 9] expresses that: Higher-order theories can represent the kinematics better, may not require shear correction factors, and can yield more accurate interlaminar stress distributions. However, they involve higher-order stress resultants that are dicult to interpret physically and require considerably more computational eort. Therefore, such theories should be used only when necessary. Figure.1 shows an illustration of how CPT, FSDT and HSDT dier from eachother in terms of in-plane displacements. Wang et al. presented this gure in 11]. Aydogdu ] presented in 006 a study in which he compared various HSDTs with available three-dimensional analysis. He showed that while the transverse displacement and the stresses are best predicted by the exponential shear deformation theory Karama et al., the parabolic shear deformation Reddy and the hyperbolic shear deformation Soldatos theories yield more accurate predictions for the natural frequencies and the buckling loads. 6

16 Undeformed z x,u z,w w w CPT u,w u 0,w 0 φ x w FSDT u,w u 0,w 0 φ x w HSDT u,w u 0,w 0 Figure.1: Undeformed and deformed geometries of an edge of a plate in various plate theories. u 0 denotes the in-plane displacement. 7

17 . Theoretical preliminaries..1 The principle of minimum potential energy PMPE Variational principles can be used to obtain governing dierential equations and associated boundary conditions. PMPE is a special case of the principle of virtual displacements that deals with linear as well as nonlinear elastic bodies 9]. For elastic bodies in the absence of temperature variations there exists a strain energy U and a potential of external forces V. The sum U + V = Π is called the total potential energy of the elastic body. In general the potential energy expresses the potential or the ability of a body to perform work 3]. Let us consider an arbitrary body under the impact of traction and volume forces, with given boundary conditions, see Figure.. T x,u Γ y,v F Ω z,w Figure.: Arbitrary body in equilibrium. In the cartesian coordinate system the volume force at a point is given by the components F = {F x F y F z } T Force per unit volume.1 The traction force at a surface point is given by the componets T = {T x T y T z } T Force per unit area. The displacement of an arbitrary point is a vector that can be decomposed along the three coordinate axes 8

18 u = {u v w} T.3 The potential of external forces acting on the body is given by V = F x u + F y v + F z w dv Ω T x u + T y v + T z w ds Γ σ = u T Fdv u T Tds Γ σ Ω.4 where Ω is the volume of the body and Γ σ is the part of the surface where the traction force is located. dv and ds denote the volume and surface elements of Ω. The energy that is stored in the body due to deformation is called the strain energy. To nd a proper expression for the strain energy it is necessary to know the material's stress-strain relationship. For a linear elastic material the strain energy is simply found from the stress strain diagram, see Figure.3. σ U 0 = 1 σɛ ɛ Figure.3: Strain energy density in the linear elastic case. The strain energy stored in a volume unit dv, is called the strain energy density and it is given by U 0 = ɛ 0 σ T dɛ.5 9

19 For a linear elastic material the strain energy density is U 0 = 1 σ xɛ x + σ y ɛ y + σ z ɛ z + τ xy γ xy + τ yz γ yz + τ xz γ xz + = 1 ɛt σ = 1 ɛt Cɛ.6 where C expresses the generalized Hooke's law: σ = Cɛ.7 The strain energy is now found by integrating the strain energy density over the volume U = Ω U 0 dv.8 PMPE states that the deections in a body in static equilibrium, will settle in such a way that the total potential energy is a minimum. This results in the derivative of Π with respect to the displacements, must equal zero. U + V = U + V Π = 0.9 In other words, it means that of all admisssible displacements, those which satisfy static equlibrium make the total potential energy a minimum: Πu Πū.10 where u is the true solution and ū is any admissible displacement eld. The equality holds only if u = ū... The divergence theorem When nding the total potential energy of a body we need to integrate the expressions for the strain energy and the potential of external forces. In that operation we use the divergence theorem to obtain the dierential equations and associated boundary conditions. The divergence theorem is explained as follows 9]. Let Ω denote a region in space sorrounded by the surface Γ, and let ds be a dierential element of the surface whose unit outward normal is denoted by n. 10

20 Let dv be a dierential volume element and A a vector function dened over the region Γ. Then the following integral identity holds: A i dv = n i A i ds.11 Ω i Γ where a circle on the integral sign signies integration over the total boundary...3 Navier method There are several approaches to solving a buckling problem. In this study we use the Navier method. According to Reddy 7] the Navier solutions can be developed for a rectangular plate or laminate when all four edges are simply supported. Other methods frequently used for solving plate buckling problems are the ones of Levy and Rayleigh-Ritz. Levy's solutions can be developed for plates with two opposite edges simply supported and the remaining two edges having any possible combination of boundary conditions: free, simple support, or xed support. The Rayleigh-Ritz method can be used to determine approximate solutions for more general bounday conditions. In Navier's method the generalized diplacements are expanded in a double trigonometric series in terms of unknown parameters. The choice of the functions in the series is restricted to those which satisfy the boundary conditions of the problem. Substitution of the displacement expansions into the governing equations should result in a unique, invertible, set of algebraic equations among the parameters of the expansion. Otherwise, the Navier solution can not be developed for the problem 7]. 11

21 Chapter 3 Mindlin plate theory Before treating HSDT we show in this chapter how to obtain the dierential equations and boundary conditions in FSDT. The methodology will be the same in both cases and can be summarized as: Start from the displacement eld assumptions. Consider the stress strain relationship for a linear elastic material. Find the expressions for U and V. Use the principle of minimum potential energy to obtain the governing equations and associated boundary conditions. Solve the buckling problem by using the Navier method. The objective is to obtain a system of three partial dierential equations by investigating the minimum potential energy of the plate subjected to uniform in-plane loads. We want to nd the rst variation of the strain energy and the potential energy of external forces, and use the principle of minimum potential energy to obtain the dierential equations and associated boundary conditions. That is to say we need to solve δπ = δu + δv = Strain energy The Mindlin plate theory is based on the displacement eld 1

22 u = zφ x v = zφ y w = w 3. where u,v,w are the displacement components along the x,y,z coordinate directions, respectivly. φ x and φ y denote rotations about the y and x axes, respectivly. In view of that displacement eld the strains are given by ɛ x = u = z φ x ɛ y = v = z φ y γ xy = u + v = z φx + φ y γ xz = u z + w = φ x + w γ yz = v z + w = φ y + w 3.3 Assuming the plate material is isotropic and obeys Hooke's law the stress strain relationship is given by σ x = E 1 ν ɛ x + νɛ y σ y = E 1 ν ɛ y + νɛ x σ xy = Gγ xy = σ xz = Gγ xz = σ yz = Gγ yz = This can be written in matrix format as E 1 + ν γ xy E 1 + ν γ xz E 1 + ν γ yz

23 σ x σ y τ xy τ xz τ yz = E 1 ν νe 1 ν νe 1 ν E 1 ν G G G ɛ x ɛ y γ xy γ xz γ yz 3.5 where E is the Young's modulus, ν is Poisson's ratio and the shear modulus G is related to E and ν by G = E 1+ν. In general the strain energy for a linear elastic material can be written, see Eqs..6 and.8 U = 1 σ x ɛ x + σ y ɛ y + σ z ɛ z + τ xy γ xy + τ yz γ yz + τ zx γ zx dxdydz 3.6 Since we have plane stress σ z ɛ z is excluded. Introducing the stress strain relationship into Eq. 3.6 gives U = E 1 ν ɛ x + ɛ 1 ν y + νɛ x ɛ y + γ xy + γxz + γyz dxdydz 3.7 Introducing the strain expressions from Eq. 3.3 into Eq. 3.7 gives E U = 1 ν 1 ν + z φx z φx + φ y Now integrating Eq. 3.8 with respect to z gives + z φy + νz φx φy + φ x + w + φ y + w dxdydz 3.8 U = D φx + + Eh ν φy φx + ν φ x + w + φ y + w φ y 1 ν + ] φx + φ ] y dxdy dxdy

24 where the plate's bending stiness D = and h is the plate thickness. E h/ 1 ν z dz = h/ Eq. 3.9 can be written in matrix format as Eh ν U = 1 { } φ x,x φ y,y φ x,y + φ y,x w,x + φ x w,y + φ y D 11 D D 1 D D A A 55 φ x,x φ y,y φ x,y + φ y,x w,x + φ x w,y + φ y dxdy 3.10 where in the isotropic case D 11 = D 1 = νd D = D Eh 3 11 ν = D 1 ν D 66 = D Eh A 44 = A 55 = 1 + ν 3.11 Using variational calculus gives the rst variation of the strain energy 15

25 φx δu = D + Eh φ x + w 1 + ν + ν φ y φx δ φy + + ν φ x φy δ 1 ν φ x 1 ν ν φ x 1 ν + + δφ x + + φ y + w φ y δ φ y δ φ x + w δφ y + φ y + w φx ] φy dxdy w ] w dxdy 3.1 δ δ The plate constitutive equations are given by M xx = M yy = M xy = Q x = κ Q y = κ h h h h h h h h h φx σ x zdz = D + ν φ y σ y zdz = D ν φ x + φ y φx h σ xy zdz = σ xz dz = σ yz dz = D1 ν κeh 1 + ν κeh 1 + ν + φ y φ x + w φ y + w 3.13a 3.13b 3.13c 3.13d 3.13e where κ is the shear correction factor. Since the transverse shear stresses are represented as constant through the plate thickness in FSDT, we introduce κ to modify the transverse shear stresses. It is a well known fact that the transverse shear stresses are parabolic through the plate thickness 11]. It is normal to introduce κ = 5 6. Introducing the plate constitutive equations into Eq. 3.1 gives 16

26 δu = M xx + M yy + Q x + M xy + Q x δφ x δφ y + M xy + Q y ] + Q y δw dxdy 3.14 Now integrating Eq by parts and using the divergence theorem we obtain From δφ x : M xx + M xy + Q x δφ x dxdy = M xx n x + M xy n y δφ x ds Γ M xx,x + M xy,y Q x δφ x dxdy 3.15 From δφ y : M yy + M xy + Q y = δφ y dxdy M yy n y + M xy n x δφ y ds Γ M yy,y + M xy,x Q y δφ y dxdy 3.16 From δw: Q x + Q y = δw dxdy Q x n x + Q y n y δw ds Γ Q x,x + Q y,y δw dxdy 3.17 A comma followed by subscripts denotes dierentiation with respect to the subscripts. For example M xx,x = M xx. n x and n y denote the direction cosines of the unit normal n on the boundary Γ. The same convention is used in 11]. That completes the rst variation of the strain energy. We now turn to the potential energy of external forces and repeat the procedure. 17

27 3. Potential energy of external forces The potential energy of external forces V is given by the forces acting on the boundary of the plate and the curvature, assuming small rotations. Note that V depends on the variable δw only V = 1 w N xx + N yy w w + N xy w dxdy 3.18 Variational calculus leads to w δv = N xx δ w + N w yy δ w w +N xy δ w + w δ w dxdy 3.19 When integrating Eq. obtain 3.19 by parts and using the divergence theorem we = Γ N xx w δ w N xx w n x dxdy δwds + N xx δwdxdy 3.0 = Γ N yy w δ w N yy w n y dxdy δwds + N yy δwdxdy 3.1 w N xy δ w w = N xy Γ n x + + w δ w w N xy n y dxdy ] δwds + N xy δwdxdy 3. Summarizing the results in Eqs. 3.0, 3.1 and 3. gives 18

28 δv = N xx Γ N xx w n x + N yy + N w xy δwdxdy w + N yy n w y + N xy n x + ] w N xy n y δwds System of partial dierential equations We have now found the rst variation of the strain energy and the potential energy of external forces. Inserting that into gives δπ = δu + δv = δπ = M xx n x + M xy n y δφ x ds Γ M xx,x + M xy,y Q x δφ x dxdy + M yy n y + M xy n x δφ y ds Γ M yy,y + M xy,x Q y δφ y dxdy w + Q x n x + Q y n y N xx Γ n x + N xx + N yy + N xy w + N yy ] w + N xy n w x + N xy n y δw ds n y Q x,x + Q y,y δw dxdy = For this expression to be kinematically admissible each seperate section must equal zero. From the eld integrals we get the three partial dierential equations, and from the boundary integrals we get the boundary conditions. We present the system of partial dierential equations rst. From the section coupled with δφ x we get 1. Mxx + M xy + Q x =

29 From the section coupled with δφ y we get. Myy + M xy + Q y = From the section coupled with δw we get 3. Qx + Q y + N x + N xy + N y = The above equations can be expressed in terms of displacements φ x, φ y and w by substituting for the force and moment resultants D1 ν φ x + φ x D1 ν φ y + φ y κeh 1 + ν D1 + ν D1 + ν + w + φ x + φ y + + N xx φx + φ y + κeh 1 + ν φx + φ y + κeh 1 + ν + N xy + N yy w + φ x = w + φ y = = 0 It can be usefull to express the dierential equations by the moment sum, also called the Marcus moment 11]. Introducing the moment sum M s M xx + M yy 1 + ν and using the Laplace operator φx = D + φ y 3.3 = , the dierential equations can be expressed in the form 0

30 D1 ν φ x 1 + ν M s + κeh w 1 + ν + φ x = D1 ν φ y 1 + ν M s + κeh w 1 + ν + φ y = κeh w + M s = N xx 1 + ν D + N xy + N yy Boundary conditions The boundary conditions are found from the boundary integrals in Eq. 3.5, and involve specifying one element of the following three pairs. On an edge parallel to the x or y coordinate either δφ x = 0 or M xx n x + M xy n y = either δφ y = 0 or M yy n y + M xy n x = either δw = 0 or Q x n x + Q y n y w N xx + N w xy n x w N yy + N w xy n y =

31 3.5 Solving the Mindlin buckling problem We consider a plate under uniform in-plane compressive load on all edges, thus N xx = N yy = N and N xy = 0. y x z N yy x b N xx a y Figure 3.1: Simply supported plate under hydrostatic in-plane load. The boundary conditions for the simply supported plate are x = 0 : w0, y = 0 M xx 0, y = 0 x = a : wa, y = 0 M xx a, y = 0 y = 0 : wx, 0 = 0 M xx x, 0 = 0 y = b : wx, b = 0 M xx x, b = We have previously found three equilibrium equations in Eqs. 3.6, 3.7, and 3.8. Note that N xy does not appear in the nal equilibrium equation. From the section coupled with δφ x we have 1. Mxx + M xy + Q x = From the section coupled with δφ y we have

32 . Myy + M xy + Q y = From the section coupled with δw we have 3. Qx + Q y + N xx + N yy = The plate constitutive equations are given by M xx = M yy = M xy = Q x = κ Q y = κ h h h h h h h h h φx σ x zdz = D + ν φ y σ y zdz = D ν φ x + φ y φx h σ xy zdz = σ xz dz = σ yz dz = D1 ν κeh 1 + ν κeh 1 + ν + φ y φ x + w φ y + w 3.44a 3.44b 3.44c 3.44d 3.44e Introducing the constitutive equations into the equations of equlibrium gives φ x D + ν φ y D1 ν φ x + φ y + κeh φ x + w = ν φ y D + ν φ x D1 ν φ y + φ x + κeh φ y + w = ν κeh 1 + ν + w + φ x + φ y + N xx + N yy = 0 3

33 For an isotropic plate we have D 11 = D D 1 = νd D1 ν D 66 = K = κeh 1 + ν the equation set that solves the buckling problem is φ x D 11 D φ y 1 D 66 φ y D 11 D φ x 1 D 66 φ x φ y + φ y + K φ x + w = 0 + φ x + K φ y + w = 0 3. K + w + φ x + φ y + N xx + N yy = 0 We assume that w, φ x and φ y can be represented by the following double Fourier series wx, y = φ x x, y = φ y x, y = m=1 n=1 m=1 n=1 m=1 n=1 mπx W mn sin a mπx X mn cos a mπx Y mn sin a sin sin cos nπy b nπy b nπy b 3.46 a 3.46 b 3.46 c where W mn, X mn and Y mn are series coecients, m and n are positive integers. Introducing the Fourier series into the equation set gives the following matrix system D 11 α D 66 β K D 1 αβ D 66 αβ Kα D 1 αβ D 66 αβ D 11 β D 66 α K Kβ Kα Kβ N α + β Kα Kβ 3.47 X mn Y mn W mn =

34 where α = mπ a and β = nπ b. By dening we simplify the matrix. C 1 = D 11 α D 66 β K C = D 1 αβ D 66 αβ C 3 = Kα C 4 = D 11 β D 66 α K C 5 = Kβ C 6 = N α + β + αc 3 + βc C 1 C C 3 C C 4 C 5 C 3 C 5 C 6 X mn Y mn W mn = The non trivial solution we are seeking is found by setting the determinant of the matrix in Eq equal to zero 13], 14]. C 1 C C 3 C C 4 C 5 C 3 C 5 C 6 = C 1 C 4 C 6 C 1 C 5 C C 6 + C C 3 C 5 + C C 3 C 5 C 3C 4 = 0 C 6 C 1 C 4 C = C 1 C 5 + C 3C 4 C C 3 C Inserting for C 6 and solving for N gives N = C1 C5 + C 3 C 4 C C 3 C 5 1 C 1 C 4 C αc 3 βc 5 α + β 3.5 The critical buckling load N cr occurs at n = 1 while m can vary. 5

35 Chapter 4 HSDT; a gradient elasticity approach 4.1 Gradient elasticity FSDT model The objective is to describe the energy functional Π = U + V of higher order shear plate theories in a gradient elasticity rst order shear plate model, meaning that one or more variables in HSDT are gradients of the variables in FSDT. The potential energy of external forces V is the same in HSDT and FSDT. The challenge is therefore describing the strain energy U. We will focus on the theory of Reddy and the one of Shi, which are based on dierent kinematics functions fz. We want to show that both can be classied in a common gradient elasticity rst order shear plate theory. The two theories are based on the following kinematics functions: Reddy: fz = z 1 4z 3h Shi: fz = 5 4 z 1 4z 3h Note that the polynomials are of third order, meaning that the displacements are described by a function of third order. We want to ceep these functions xed and derive the energy equations without inserting for fz. To describe the theories in a unied way we introduce the parameter ζ which takes dierent value in the various HSDTs, depending on the respective kinematics function. The unied kinematics function can then be presented as fz = ζz 1 4z 3h 4.1 6

36 where ζ = 1 in Reddy's theory ζ = 5 4 in Shi's theory The displacement eld for a plate in HSDT can be described by: ux, y, z = u 0 x, y zw,x + fz φ x + w,x vx, y, z = v 0 x, y zw,y + fz φ y + w,y wx, y, z = wx, y 4. The in-plane displacement of the middle surface u 0, and v 0 will in the following be omitted without aecting the basic bending features of shear exible plates. In view of that displacement eld the strains are given by ɛ x = u = zw,xx + fz φ x,x + w,xx = fzφ x,x + fz z] w,xx ɛ y = v = zw,yy + fz φ y,y + w,yy = fzφ y,y + fz z] w,yy γ xz = u z + w = f z φ x + w,x ] γ yz = v z + w = f z φ y + w,y ] γ xy = u + v = zw,xy + fz φ x,y + w,xy zw,yx + fz φ y,x + w,yx = fzφ x,y + fz z] w,xy + fzφ y,x + fz z] w,yx 4.3 Organizing the above strain expressions to release the gradients of the deection w and rotations φ x and φ y : ɛ x = zφ x,x + fz z w,xx + φ x,x ɛ y = zφ y,y + fz z w,yy + φ y,y γ xz = f z w,x + φ x ] γ yz = f z w,y + φ y ] γ xy = zφ x,y + fz z w,xy + φ x,y + zφ y,x + fz z w,yx + φ y,x Considering an orthotropic elastic constitutive law: 4.4 7

37 σ x = Q 11 ɛ x + Q 1 ɛ y σ y = Q 1 ɛ x + Q ɛ y τ xz = Q 44 γ xz τ yz = Q 55 γ yz τ xy = Q 66 γ yz 4.5 In matrix form: σ x σ y τ xz τ yz τ xy = Q 11 Q Q 1 Q Q Q Q 66 ɛ x ɛ y γ xz γ yz γ xy 4.6 where Q ij are the plane stress reduced elastic constants in the material axes of the plate. In general the strain energy under the plain stress assumption can be written: U = 1 σ x ɛ x + σ y ɛ y + τ xz γ xz + τ yz γ yz + τ xy γ xy dxdydz 4.7 See Eqs..6 and.8. Introducing stresses from Eq. 4.6 gives U = 1 Q11ɛ x + Q 1 ɛ x ɛ y + Q ɛ y + Q 44 γ xz + Q 55 γ yz + Q 66 γ xy dxdydz 4.8 Itroducing strain expressions from Eq. 4.4 gives 8

38 Q 11 zφ x,x + fz z w,xx + φ x,x] U = 1 +Q 1 zφ x,x + fz z w,xx + φ x,x ] zφ y,y + fz z w,yy + φ y,y ] +Q zφ y,y + fz z w,yy + φ y,y ] +Q 44 f z w,x + φ x ] ] +Q 55 f z w,y + φ y ] ] +Q 66 φ x,y + fz z w,xy + φ x,y + zφ y,x + fz z w,yx + φ y,x ] ] dxdydz 4.9 We need to identify the variables we are delaing with in Eq If we enumerate the six lines in Eq. 4.9 from 1-6 and treat them separately we have 1.. U = 1 U = Q 11 1 Q 11 zφ x,x + fz z w,xx + φ x,x ] dv z φ x,x + zφ x,x fz z w,xx + φ x,x + fz z w,xx + φ x,x + ] dv From 1. we recognize the two variables: φ x,x and w,xx + φ x,x. U = 1 Q 1 zφ x,x + fz z w,xx + φ x,x ] zφ y,y + fz z w,yy + φ y,y ] dv 1 U = Q 1 z φ x,xφ y,y + z fz z φ x,x w,yy + φ y,y + z fz z φ y,y w,xx + φ x,x ] + fz z w,xx + φ x,x w,yy + φ y,y dv From. we recognize the four variables: φ x,x, φ y,y, w,xx + φ x,x and w,yy + φ y,y, which are the same as in 1. and U = 1 U = Q 1 Q zφ y,y + fz z w,yy + φ y,y ] dv z φ y,y + zφ y,y fz z w,yy + φ y,y + fz z w,yy + φ y,y + ] dv From 3. we recognize the two variables: φ y,y and w,yy + φ y,y. U = 1 Q 44 f z w,x + φ x ] dv 1 U = Q 44 f z w,x + φ x ] dv From 4. we recognize the one variable: w,x + φ x. 9

39 5. U = 1 Q 55 f z w,y + φ y ] dv 1 U = Q 55 f z w,y + φ y ] dv From 5. we recognize the one variable: w,y + φ y. 6. U = 1 Q 66 zφ x,y + fz z w,xy + φ x,y + zφ y,x + fz z w,xy + φ y,x ] ] dv 1 U = Q 66 z φ x,y + z fz z φ x,y w,xy + φ x,y + z φ x,y φ y,x + z fz z φ x,y w,xy + φ y,x +z fz z φ x,y w,xy + φ x,y + fz z w,xy + φ x,y +z fz z φ y,x w,xy + φ x,y + fz z w,xy + φ x,y w,xy + φ y,x +z φ y,x φ x,y + z fz z φ y,x w,xy + φ x,y + z φ y,x + z fz z φ y,x w,xy + φ y,x +z fz z φ y,x w,xy + φ y,x + fz z w,xy + φ y,x w,xy + φ x,y +z fz z φ y,x w,xy + φ y,x + fz z w,xy + φ y,x ] dv From 6. we recognize the four variables: φ x,y, φ y,x, w,xy + φ x,y and w,xy + φ y,x. We have now targeted the ten variables: { φ x,x φ y,y φ x,y φ y,x } w,x + φ x w,y + φ y w,xx + φ x,x w,yy + φ y,y w,xy + φ x,y w,xy + φ y,x The next thing we want to do is to write the strain energy including the above lines 1-6 in a 10x10 matrix system. We will show that the 10x10 stiness matrix can be reduced to an 8x8 matrix. The stiness parameters that enter the matrix system can be introduced in the following format: 30

40 h/ h/ h/ h/ h/ h/ h/ h/ Q ij z dz = D ij Q ii f z ] dz = κaii Q ij fz z] dz = b 0κA ij Q ij z fz z] dz = c 0κA ij 4.10a 4.10b 4.10c 4.10d where i, j = 1,, 6 and i, i = 4, 5. κ in HSDT is not a correction factor that we assume to be for example κ = 5 6 as in FSDT. In HSDT κ is a constant computed from the kinematics. We dene T ij = κa ij and T ii = κa ii 4.11 so that D ij is the exural stiness and T ij and T ii is the transverse shear stiness of an orthotropic plate. We will return to the stiness parameters in Section 4. where they are further discussed. 31

41 The strain energy is now written in matrix form as: U = 1 { } φ x,x φ y,y φ x,y φ y,x w,x + φ x w,y + φ y w,xx + φ x,x w,yy + φ y,y w,xy + φ x,y w,xy + φ y,x D 11 D c 0 T 11 c 0 T D 1 D c 0 T 1 c 0 T D 66 D c 0 T 66 c 0 T D 66 D c 0 T 66 c 0 T T T c 0 T 11 c 0 T b 0 T 11 b 0 T c 0 T 1 c 0 T b 0 T 1 b 0 T c 0 T 66 c 0 T b 0 T 66 b 0 T c 0 T 66 c 0 T b 0 T 66 b 0 T 66 φ x,x φ y,y φ x,y φ y,x w,x + φ x w,y + φ y dxdy w,xx + φ x,x w,yy + φ y,y w,xy + φ x,y w,xy + φ y,x 4.1 We observe in the 10x10 stiness matrix that rows and columns 3 and 4 connected to the variables φ x,y and φ y,x are identical. Hence, the two variables can be combined to the one variable φ x,y + φ y,x, and the rows and columns are merged. The same goes for rows and columns 9 and 10 connected to the variables w,xy + φ x,y and w,xy + φ y,x. The two variables are summed to one variable, namely φ x,y + w,xy + φ y,x, and the rows and columns are merged. 3

42 The matrix in Eq. 4.1 is now reduced to the 8x8 stiness matrix: U = 1 { } φ x,x φ y,y φ x,y + φ y,x w,x + φ x w,y + φ y w,xx + φ x,x w,yy + φ y,y φ x,y + w,xy + φ y,x D 11 D c 0 T 11 c 0 T 1 0 D 1 D c 0 T 1 c 0 T D c 0 T T T c 0 T 11 c 0 T b 0 T 11 b 0 T 1 0 c 0 T 1 c 0 T b 0 T 1 b 0 T c 0 T b 0 T 66 φ x,x φ y,y φ x,y + φ y,x w,x + φ x w,y + φ y w,xx + φ x,x w,yy + φ y,y φ x,y + w,xy + φ y,x dxdy 4.13 From this matrix system we make several important observations. First the matrix is symmetric, as it should be. Second we notice that rows and columns 6, 7 and 8 include two additional length scales, namely b 0 and c 0. If we omit rows and columns 6, 7 and 8 we get the same matrix system as in FSDT, see Eq We also notice that the three variables connected to rows and columns 6, 7 and 8 are the gradients of the variables connected to rows and columns 4 and 5. This leads to the conclusion that we are dealing with a gradient elasticity FSDT model. 33

43 4. Stiness parameters The stiness parameters in Eq. 4.10, repeated here for ready reference, are calculated for each HSDT as follows. We will utilize the unied kinematics function in Eq. 4.1 to obtain dimensionless values for the stiness parameters. We will see that the length scale c 0 will vanish in the case of Shi. The dimensionless value of the stiness parameters play a key role in describing the two HSDTs in a unied framework, and they are presnted for Shi's and Reddy's theories in Table 4.1. h/ h/ h/ h/ h/ h/ h/ h/ Q ij z dz = D ij Q ii f z ] dz = κaii Q ij fz z] dz = b 0κA ij Q ij z fz z] dz = c 0κA ij 4.14a 4.14b 4.14c 4.14d where i, j = 1,, 6, i, i = 4, 5, κa ij = T ij and κa ii = T ii. The unied kinematics function is fz = ζz 1 4z 3h 4.15 The derivative of fz is f z = ζ ζ 4z h 4.16 Recall that ζ takes the value 1 for Reddy 5 4 for Shi h/ h/ Q ij z dz = Q ij h 3 1 = D ij

44 h/ h/ h/ h/ h/ h/ Q ii f z ] dz = Tii ] Q ii ζ ζ 4z h dz = T ii Q ii ζ ζ 8z h Q ii ζ z ζ 8z3 16z5 + ζ 3h 5h 4 Q ii ζ h 16 h 3 ζ 3h 8 Q ii ζ h ] = T ii 5 + ζ 16z4 h 4 ] h/ h/ + ζ 3 5h 4 h 5 3 dz = T ii = T ii ] = T ii T ii Q ii h = 8 ζ h/ h/ h/ h/ h/ h/ Q ij fz z] dz = b 0T ij Q ij ζz ζ 4z3 3h z dz = b 0T ij Q ij ζ z + ζ 16z6 9h 4 + z ζ 8z4 3h ζz + ζ 8z4 3h dz = b 0T ij ] h/ Q ij ζ z3 16 z 7 + ζ 3 9h z3 3 8 z 5 ζ z3 3h ζ ζ 8 z 5 3h = b 5 0T ij h/ Q ij ζ h 3 3 h 7 + ζ h h 3 16 h 5 ζ h 3 ζ 4 h ζ 16 h 5 ] 15h = b 3 0T ij Q ij h 3 ζ ζ ζ ζ ζ ] = b 30 0T ij Q ij h 3 ζ ζ ζ ] = b 15 0T ij b 0 T ij D ij = ζ ζ , since Q ij h 3 = h/ h/ Q ijz 1dz = 1D ij, see Eqs

45 h/ h/ h/ h/ h/ h/ Q ij z fz z] dz = c 0T ij Q ij z ζz ζ 4z3 3h z dz = c 0T ij Q ij ζz ζ 4z4 3h z Q ij ζ z3 3 ζ 4 z 5 3h 5 z3 3 Q ij ζ h ζ 8 h 5 15h Q ij h 3 ζ 1 1 ζ ] h/ dz = c 0T ij h/ ] = c 0T ij 3 h 3 = c 3 8 0T ij ] = c 0T ij c 0 T ij D ij = 1 ζ By introducing ζ = 1 for Reddy and ζ = 5 4 in Table 4.1. for Shi, we obtain the values given 36

46 Table 4.1: Dimensionless stiness parameters of Shi's and Reddy's HSDT. HSDT fz Shi fz = 5 4 z 1 4z 3h Reddy fz = z 1 4z 3h T ii Q ii h b 0 T ij D ij c 0 T ij D ij In a similar manner Challamel et al. 5] found the values of the stiness parameters in the HSDTs of Touratier, Karama et al. and Mechab, shown in Table 4.. Table 4.: Dimensionless stiness parameters of Touratier's, Karama et al.'s and Mechab's HSDT. HSDT Touratier fz fz = h π sin π z h T ii Q ii h b 0 T ij D ij c 0 T ij D ij 1 = 0, , , 6 Karama et al fz = ze h z , , , 53 Mechab fz = zcosh π h π sinhπ z h cosh π , , ,

47 Chapter 5 HSDT For a plate with transverse shear deformations the principle of minimum potential energy states δπ = δu φ x, φ y, w + V w] = We want to use the matrix system in Eq to nd an expression for δu and substitute that expression into Eq We also need an expression for δv, but as previously mentioned δv is the same in HSDT as in Mindlin plate theory. In Chapter 4. we found δv = N xx Γ N xx w n x + N yy + N w xy δwdxdy w + N yy n w y + N xy n x + ] w N xy n y δwds 5. δv is added to δu in Eq. 5.9 below. 5.1 Strain energy When multiplying out the matrix system in Eq we obtain the following expression for the strain energy: 38

48 U = 1 D 11 φ x,x + D 1 φ x,x φ y,y + D φ y,y + D 66 φ x,y + φ x,y φ y,x + φ y,x +T 44 w,x + w,x φ x + φ x + T55 w,y + w,y φ y + φ y +b 0T 11 w,xx + w,xx φ x,x + φ x,x + b 0 T w,yy + w,yy φ y,y + φ yy +b 0T 1 w,xx w,yy + w,xx φ y,y + w,yy φ x,x + φ x,x φ y,y +b 0T 66 φ x,y + 4w,xy + φ y,x + 4w,xy φ x,y + φ x,y φ y,x + 4w,xy φ y,x c 0T 11 φ x,x w,xx + φ x,x c 0T 1 φ x,x w,yy + φ y,y c 0T 1 φ y,y w,xx + φ x,x c 0T φ y,y w,yy + φ y,y ] c 0T 66 φ x,y + φ y,x φ x,y + w,xy + φ y,x dxdy 5.3 When using the fundamental lemma of variational calculus we obtain 39

49 { δu = D 11 φ x,x + D 1 φ y,y + b 0T 11 w,xx + b 0T 11 φ x,x + b 0T 1 w,yy + b 0T 1 φ y,y ] c 0T 11 w,xx c 0T 11 φ x,x c 0T 1 w,yy c 0T 1 φ y,y δ φ x + D 66 φ x,y + D 66 φ y,x + b 0T 66 φ x,y + b 0T 66 w,xy + b 0T 66 φ y,x ] c 0T 66 φ x,y c 0T 66 w,xy c 0T 66 φ y,x δ φ x + T 44 w,x + φ x ] δφ x + D φ y,y + D 1 φ x,x + b 0T w,yy + b 0T φ y,y + b 0T 1 w,xx + b 0T 1 φ x,x ] c 0T w,yy c 0T φ y,y c 0T 1 w,xx c 0T 1 φ x,x δ φ y + D 66 φ x,y + D 66 φ y,x + b 0T 66 φ y,x + b 0T 66 w,xy + b 0T 66 φ x,y ] c 0T 66 φ y,x c 0T 66 w,xy c 0T 66 φ x,y δ φ y + T 55 w,y + φ y ] δφ y ] + T 44 w,x + φ x δ w ] + T 55 w,y + φ y δ w ] + b 0T 11 w,xx + b 0T 11 φ x,x + b 0T 1 w,yy + b 0T 1 φ y,y c 0T 11 φ x,x c 0T 1 φ y,y δ w ] + b 0T w,yy + b 0T φ y,y + b 0T 1 w,xx + b 0T 1 φ x,x c 0T φ y,y c 0T 1 φ x,x δ w } ] + 4b 0T 66 w,xy + b 0T 66 φ x,y + b 0T 66 φ y,x c 0T 66 φ x,y c 0T 66 φ y,x δ w dxdy 5.4 We need to integrate Eq. 5.4 by parts and include δv to obtain the dierential 40

50 equations and boundary conditions. The integration process, using the divergence theorem, is executed in the following way, taking the rst part of Eq. 5.4 as an example: φ x D 11 δ φ x φ x = D 11 Γ n x dxdy δφ x ds D 11 φ x δφ x dxdy 5.5 When doing this procedure on all the parts of Eq. 5.4 we obtain a large expression including a eld integral and a boundary integral. As we can see from the example in Eq. 5.5 the eld integral, and hence the dierential equations, is easy to spot from Eq The boundary integral is more complicated, and to abbreviate the terms including b 0 and c 0 we introduce the dimensionless stiness parameters from Table 4.1 as: so that for example b 0 T ij D ij = S 1 5.6a c 0 T ij = S D ij 5.6b T ii Q ii h = S 3 5.6c and b φ x 0T 11 = b 0 T 11 φ x D 11 D 11 = S φ x 1D 11 w T 44 + φ x = T 44 w w Q 44 h Q 44h + φ x = S 3 Gh + φ x ,since Q 44 = Q 55 = G. When integrating Eq. 5.4 and including δv we obtain the expression shown on the next two pages: 41

51 { δπ = { c 0 + D 11 φ x T 11 + D φ y 1 + T 1 D 66 φx T 44 w + φ x { + c 0 + D φ y T 1 T 55 w + φ y { + + φ y } δφ x + D φ x 1 + T φx D 66 + φ y } δφ y + b φ x 0 T 11 c 0 + T φ y 1 ] φ y T 11 φ x + T 1 + b 0T 66 φx + φ y + w + b φ y 0 T c 0 + T φ x 1 ] φ y T 1 φ x + T + b 0T 66 φx + φ y + w w T 44 + φ x + w T 55 + φ y b φ x 0 T 11 + T φ y 1 + b 0 b φ x 0 T 1 φ y T 11 + b 0 T 1 + T b φx 0T 66 + φ y + w N xx N xy N yy } δw + T 1 + T + b 0 T 11 + T 1 w c φx 0T 66 + φ ] y + w + b 0 T c φx 0T 66 + φ y } dxdy + T 1 w c φx 0T 66 + φ ] y + w c φ x 0 T 11 c φ y 0 T ] + T 1 + T 1 ] φ y ] φ x 4

52 { { φ x S 1 S D 11 Γ + D φ y 1 φx S 1 S D 66 + φ y { φ y S 1 S D + D φ x 1 φx S 1 S D 66 + φ y { w + S 3 Gh + φ x S 1 + S φ x D 11 + D φ y 1 w + S 3 Gh + φ y S 1 + S φ y D + D φ x 1 ] n x ] n y + S 1 S D 11 + S 1 S + D 1 D 66 + S 1 S D + S 1 S D 11 + φ x D + φ y φx S 1 D 66 + φ y + w φx + S D 66 + φ y φx S 1 D 66 + φ y + w φx + S D 66 + φ y w N xx n w x + N yy n w y + N xy n x + { + S 1 D 11 + S 1 D 1 + φ x φx + S 1 D 66 + φ y + w { + S 1 D + φ y φx + S 1 D 66 + φ y + w ] n x ]n y } δφ x + D 1 D 66 ] n y ]n x } δφ y + D 1 + φ y + D 1 + φ x ] n y ] n x N xy w n y S D 11 φ x + φ y φx S D 66 + φ y + S 1 D 1 + φ x ] } δw + D 1 ]n y } δ w S D φ y φx S D 66 + φ y + D 1 } ]n x } δ w 5.9 For the above expression to be kinematically admissible, either the integral expression or the variable must be zero. The variable being zero gives a trivial solution, while the eld integral expression being zero gives the dierential 43 ] φ y n x ] φ x n y ds = 0

53 equations. 5. System of partial dierential equations Focusing now on the eld integral in Eq By introducing the dimensionless stiness parameters from Table 4.1 as shown in Eqs. 5.6, we obtain a system of three partial dierential equations that describe both Shi's and Reddy's HSDTs. Collecting the terms corresponding to the variations of δφ x, δφ y and δw in the eld integral we obtain the following dierential equations: From δφ x φ x 1 + S 1 S D 11 + D φ y 1 φx 1 + S 1 S D 66 + w S 3 Gh + φ x = 0 + φ y + S 1 S D 11 + S 1 S + D 1 D 66 ] ] w 5.10 From δφ y φ y 1 + S 1 S D + D φ x 1 + φx 1 + S 1 S D 66 + φ y S 3 Gh w + φ y = 0 + S 1 S D + S 1 S + D 1 D 66 ] ] w

54 From δw ] w S 3 Gh + φ x + ] w S 3 Gh + φ y φ x S 1 S D 11 + D 1 φ y S 1 S D + D φ x 1 φx S 1 S D 66 + φ y φ y + S 1 D 11 + S 1 D + 4S 1 D 66 + D 1 + D 1 ] ] w ] w N xx N xy N yy = This system of dierential equations constitute the core of the unied framework that can describe several of the higher order shear deformation theories by introducing the related stiness parameters. 5.3 Boundary conditions The boundary conditions in HSDT will not be thoroughly discussed in this study. However it is worth noting that the entities in the boundary integral in Eq. 5.9 can be grouped into ve sections: δφ x, δφ y, δw, δ w and δ w. This implies that there are ve pairs of boundary conditions on each edge, as opposed to the three pairs we obtained in the Mindlin plate theory. According to Shi 10] the total dierential order of the three dierential equations in Reddy's and Shi's HSDTs is ten. Therefore, ve boundary conditions for each edge of plates are expected. By setting each of the ve sections equal to zero we obtain that either the integral expression or the variable in the following ve pairs needs to be specied on boundaries Γ of plates 10]. 45

55 Γ { { φ x 1 + S 1 S D 11 + D 1 φ y φx S 1 S D 66 + φ y + S 1 S D 11 + S 1 S + D ] w 1 n x } ]n y }δφ x D ds = 0 Γ { { φ y 1 + S 1 S D + D 1 φ x φx S 1 S D 66 + φ y + S 1 S D + S 1 S + D ] w 1 n y } ]n x }δφ y D ds = 0 Γ { { w S 3 Gh + φ x + S φ x D 11 + D φ y 1 w + S 3 Gh + φ y S 1 + S φ y D + D φ x 1 S 1 D 66 φx S 1 D 66 φx w N xx n x S w 1 D 11 ] n x ] n y + φ y + w + φ y + w w + N yy n y + φ x D + φ y + S D 66 φx + S D 66 φx + N xy w n x + D 1 + φ y + D 1 + φ x + φ y + φ y + ] n y ] n x N xy w n y ] } } δw ds = Γ { { S 1 D 11 + φ x + S 1 D 1 φx + S 1 D 66 + φ y + w + φ y φx S D 66 + φ y S D 11 φ x + D 1 ]n y } δ w } ] φ y n x ds =

56 Γ { { S 1 D + φ y + S 1 D 1 φx + S 1 D 66 + φ y + w + φ x φx S D 66 + φ y S D φ y + D 1 ]n x } δ w } ] φ x n y ds = Describing Reddy's and Shi's theories by unied dierential equations In this section we will show that the system of dierential equations in Eqs. 5.10, 5.11 and 5.1 can describe both Reddy's and Shi's theories by introducing the respective stiness parameters from Table 4.1. We will prove the correspondence by investigating the dierential equation coupled with δφ x Eq Reddy The rst dierential equation of Reddy's theory can be presented as: see 8] From δφ x 4 φ x D D 1 4 4h φ x 3h D D h 3h 35 φx D 66 + φ y D 66 φ x 3 Gh w + φ x φ y 1 D 11 5 φ y h 8 D 66 5 w + D 1 D 11 + D 1 ] ] w + D φ y 66 h 8 D ] ] w h h w 30 Gh + φ x = When multiplying out Eq we get 47