Poisson’s Theory for Analysis of Bending of Isotropic and Anisotropic Plates

Abstract

Sixteen-decade-old problem of Poisson-Kirchhoff’s boundary conditions paradox is resolved in the case of isotropic plates through a theory designated as “Poisson’s theory of plates in bending.” It is based on “assuming” zero transverse shear stresses instead of strains. Reactive (statically equivalent) transverse shear stresses are gradients of a function (in place of in-plane displacements as gradients of vertical deflection) so that reactive transverse stresses are independent of material constants in the preliminary solution. Equations governing in-plane displacements are independent of the vertical (transverse) deflection 𝑤₀(𝑥,𝑦). Coupling of these equations with 𝑤₀ is the root cause for the boundary conditions paradox. Edge support condition on 𝑤₀ does not play any role in obtaining in-plane displacements. Normally, solutions to the displacements are obtained from governing equations based on the stationary property of relevant total potential and reactive transverse shear stresses are expressed in terms of these displacements. In the present study, a reverse process in obtaining preliminary solution is adapted in which reactive transverse stresses are determined first and displacements are obtained in terms of these stresses. Equations governing second-order corrections to preliminary solutions of bending of anisotropic plates are derived through application of an iterative method used earlier for the analysis of bending of isotropic plates.

1. Introduction

Kirchhoff’s theory [1] and first-order shear deformation theory based on Hencky’s work [2] abbreviated as FSDT of plates in bending are simple theories and continuously used to obtain design information. Kirchhoff’s theory consists of a single variable model in which in-plane displacements are expressed in terms of gradients of vertical deflection 𝑤₀(𝑥,𝑦) so that zero face shear conditions are satisfied. 𝑤₀ is governed by a fourth-order equation associated with two edge conditions instead of three edge conditions required in a 3D problem. Consequence of this lacuna is the well-known Poisson-Kirchhoff boundary conditions paradox (see Reissner’s article [3]).

Assumption of zero transverse shear strains is discarded in FSDT forming a three-variable model. Vertical deflection 𝑤₀(𝑥,𝑦) and in-plane displacements [𝑢,𝑣]=𝑧[𝑢₁(𝑥,𝑦),𝑣₁(𝑥,𝑦)] are coupled in the governing differential equations and boundary conditions. Reactive (statically equivalent) transverse shears are combined with in-plane shear resulting in approximation of associated torsion problem instead of flexure problem. In Kirchhoff's theory, in-plane shear is combined with transverse shear implied in Kelvin and Tait’s physical interpretation of contracted boundary condition [4]. In fact, torsion problem is associated with flexure problem whereas flexure problem (unlike directly or indirectly implied in energy methods) is independent of torsion problem. Reissner [3] in his article felt that inclusion of transverse shear deformation effects was presumed to be the key in resolving Poisson-Kirchhoff’s boundary conditions paradox. Recently, it is shown that the second-order correction to 𝑤₀(𝑥,𝑦) by either Reissner’s theory or FSDT corresponds to approximate solution of a torsion problem [5]. Coupling between bending and associated torsion problems is eliminated earlier [6] by using zero rotation 𝜔_𝑧=(𝑣_,𝑥−𝑢_,𝑦) about the vertical axis. The proposal of zero 𝜔_𝑧 does not, however, resolve the paradox in a satisfactory manner. In the reference [6], face deflection 𝑤₀ in (8) is from zero face shear conditions whereas neutral plane deflectio n 𝑤₀ from edge support condition is associated with self-equilibrating transverse shear stresses.

The condition 𝜔_𝑧=0 decoupling the bending and torsion problems is satisfied in Kirchhoff’s theory. If this condition is imposed in FSDT, sum of the strains (𝜀_𝑥+𝜀_𝑦) in the isotropic plate is governed by a second-order equation due to applied transverse loads. Reactive transverse shear stresses are in terms of gradients of (𝜀_𝑥+𝜀_𝑦) uncoupled from 𝑤₀. Thickness-wise linear distribution of 𝜎_𝑧 is zero from the equilibrium equation governing transverse stresses. Normal strain 𝜖_𝑧 from constitutive relation is linear in 𝑧 in terms of (𝜀_𝑥+𝜀_𝑦). Reactive transverse shear stresses and thickness-wise linear strain 𝜀_𝑧 form the basis for resolving the paradox and for obtaining higher order corrections to the displacements. The theory thus developed is designated as “Poisson’s theory of plates in bending”.

The previously mentioned Poisson’s theory is applied to the analysis of anisotropic plates using reactive transverse shear stresses as gradients of a function. Normally, solutions to the displacements are obtained from governing equations based on stationary property of relevant total potential and the reactive transverse shear stresses are expressed in terms of these displacements. In the present work, reverse process is adapted in which reactive transverse stresses are determined first and the displacements are obtained in terms of these stresses. Equations governing second-order corrections to the preliminary solutions are derived through the application of an iterative method used earlier [7] for the analysis of bending of isotropic plates.

2. Equations of Equilibrium and Edge Conditions

For simplicity in presentation, a rectilinear domain bounded by 0≤𝑋≤𝑎, 0≤𝑌≤𝑏, 𝑍=±ℎ, with reference to Cartesian coordinate system (𝑋,𝑌,𝑍) is considered. For convenience, coordinates 𝑋,𝑌,𝑍 and displacements 𝑈,𝑉,𝑊 in nondimensional form 𝑥=𝑋/𝐿, 𝑦=𝑌/𝐿, 𝑧=𝑍/ℎ,𝑢=𝑈/ℎ, 𝑣=𝑉/ℎ, 𝑤=𝑊/ℎ and half-thickness ratio 𝛼=ℎ/𝐿 with reference to a characteristic length 𝐿 in 𝑋-𝑌 plane are used (𝐿 is defined such that mod of 𝑥 and 𝑦 are equal to or less than 1). With the previous notation and ⇔ indicating interchange, equilibrium equations in terms of stress components are 𝛼𝜎_𝑥,𝑥+𝜏_{𝑥𝑦,𝑦}+𝜏_{𝑥𝑧,𝑧}𝛼𝜏=0⟺𝑥,𝑦,_{𝑥𝑧,𝑥}+𝜏_{𝑦𝑧,𝑦}+𝜎_𝑧,𝑧=0, (1)(2) in which suffix after “,” denotes partial derivative operator.

Edge conditions are prescribed such that (𝑤,𝜏_𝑥𝑧,𝜏_𝑦𝑧,𝛾_𝑥𝑧,𝛾_𝑦𝑧) are even in 𝑧, and (𝑢,𝑣,𝜎_𝑥,𝜎_𝑦,𝜏_𝑥𝑦,𝜀_𝑥,𝜀_𝑦,𝛾_𝑥𝑦,𝜎_𝑧,𝜀_𝑧) are odd in 𝑧. In the primary flexure problem, the plate is subjected to asymmetric load 𝜎_𝑧=±𝑞(𝑥,𝑦)/2 and zero shear stresses along 𝑧=±1 faces. Three conditions to be satisfied along 𝑥 (and 𝑦) constant edges are prescribed in the form 𝑢=0or𝜎_𝑥=𝑧𝑇_𝑥𝑦⟺𝑥,𝑦,(𝑢,𝑣),𝑣=0or𝜏_𝑥𝑦=𝑧𝑇_𝑥𝑦𝑦⟺𝑥,𝑦,(𝑢,𝑣),𝑤=0or𝜏_𝑥𝑧=121−𝑧²𝑇_𝑥𝑧𝑦⟺𝑥,𝑦. (3a)(3b)(3c)

3. Stress-Strain and Strain-Displacement Relations

In displacement based models, stress components are expressed in terms of displacements, via six stress-strain constitutive relations and six strain-displacement relations. In the present study, these relations are confined to the classical small deformation theory of elasticity.

It is convenient to denote displacement and stress components of anisotropic plates as [𝑢,𝑣,𝑤]=𝑢_𝑖,𝜎_𝑥,𝜎_𝑦,𝜏_𝑥𝑦=𝜎_𝑖),𝜏(𝑖=1,2,3_𝑥𝑧,𝜏_𝑦𝑧,𝜎_𝑧=𝜎_𝑖(𝑖=4,5,6). (4a)(4b)

Strain-stress relations in terms of compliances [𝑆_𝑖𝑗] with the usual summation convention are 𝜀_𝑖=𝑆_𝑖𝑗𝜎_𝑗,𝜀_𝑟=𝑆_𝑟𝑠𝜎_𝑠,𝑖,𝑗=1,2,3,6(𝑟,𝑠=4,5). (5) We have from semi-inverted strain-stress relations with [𝑄_𝑖𝑗] denoting inverse of [𝑆_𝑖𝑗]: 𝜎_𝑖=𝑄_𝑖𝑗𝜀_𝑗−𝑆_6𝑗𝜎_𝑧𝑖,𝑗=1,2,3. (6) Normal strain 𝜀_𝑧 is given by 𝜀_𝑧=𝑆_6𝑗𝜎_𝑗,𝑗=1,2,3,6. (7) Transverse shear strains with [𝑄_𝑟𝑠] denoting inverse of [𝑆_𝑟𝑠] are 𝜀_𝑟=𝑄_𝑟𝑠𝜎_𝑠,(𝑟,𝑠=4,5). (8) Strains 𝜀_𝑖 from strain-displacement relations are 𝜀₁,𝜀₂,𝜀₃=𝛼𝑢_,𝑥,𝑣_,𝑦,𝑢_,𝑦+𝑣_,𝑥,𝜀₄,𝜀₅,𝜀₆=𝑢+𝛼𝑤_,𝑥,𝑣+𝛼𝑤_,𝑦,𝑤_,𝑧. (9)

4. 𝑓_𝑛(𝑧) Functions and Their Use

We use thickness-wise distribution functions𝑓_𝑛(𝑧) generated from recurrence relations [7] with 𝑓₀=1, 𝑓_2𝑛+1,𝑧=𝑓_2𝑛, 𝑓_2𝑛+2,𝑧=−𝑓_2𝑛+1 such that 𝑓_2𝑛+2(±1)=0. They are (up to 𝑛=5) 𝑓₁,𝑓₂,𝑓₃=𝑧,1−𝑧²2,𝑧−𝑧³/32,𝑓₄,𝑓₅=5−6𝑧²+𝑧⁴,2425𝑧−10𝑧³+𝑧⁵.120 (10) Displacements, strains, and stresses are expressed in the form (sum 𝑛=0,1,2,3,…) [𝑤,𝑢,𝑣]=𝑓_2𝑛𝑤_2𝑛,𝑓_2𝑛+1𝑢_2𝑛+1,𝑓_2𝑛+1𝑣_2𝑛+1,𝜀_𝑥,𝜀_𝑦,𝛾_𝑥𝑦,𝜀_𝑧=𝑓_2𝑛+1𝜀_𝑥,𝜀_𝑦,𝛾_𝑥𝑦,𝜀_𝑧2𝑛+1. (11) (Note that 𝑤_2𝑛=−𝜀_{𝑧2𝑛−1},  𝑛≥1) 𝜎_𝑥,𝜎_𝑦,𝜏_𝑥𝑦,𝜎_𝑧=𝑓_2𝑛+1𝜎_𝑥,𝜎_𝑦,𝜏_𝑥𝑦,𝜎_𝑧2𝑛+1,𝛾_𝑥𝑧,𝛾_𝑦𝑧,𝜏_𝑥𝑧,𝜏_𝑦𝑧=𝑓_2𝑛𝛾_𝑥𝑧,𝛾_𝑦𝑧,𝜏_𝑥𝑧,𝜏_𝑦𝑧. (12) In the previous equations, variables associated with 𝑓(𝑧) functions are functions of (𝑥,𝑦) only.

5. Sequence of Trivially Known Steps without Any Assumptions in Preliminary Analysis

(a) Due to prescribed zero (𝜏_𝑥𝑧,𝜏_𝑦𝑧) along 𝑧=±1 faces of the plate, (𝜏_𝑥𝑧0,𝜏_𝑦𝑧0)≡0 in the plate.

(b) (𝛾_𝑥𝑧0,𝛾_𝑦𝑧0)≡0 from constitutive relations (5).

(c) Static equilibrium equation (2) gives 𝜎_𝑧1≡0.

(d) In the absence of (𝑢₁,𝑣₁), (𝜀_𝑥1,𝜀_𝑦1,𝛾_𝑥𝑦1)≡0 from strain-displacement relations (8).

(e) (𝜎_𝑥1,𝜎_𝑦1,𝜏_𝑥𝑦1)≡0 from constitutive relations (6).

(f) 𝜀_𝑧1≡0 from relation (7).

(g) 𝑤=𝑤₀(𝑥,𝑦) from thickness-wise integration of 𝜀_𝑧=0.

(h) From steps (b) and (g), integration of [𝑢_,𝑧+𝛼𝑤_,𝑥,𝑣_,𝑧+𝛼𝑤_,𝑦]=[0,0] in the thickness direction gives [𝑢,𝑣]=−𝑧𝛼[𝑤_0,𝑥,𝑤_0,𝑦].

In the present study as in the earlier work [6], [𝑢,𝑣] linear in 𝑧 are unknown functions as in FSDT. From integration of [𝑢_,𝑧+𝛼𝑤_,𝑥,𝑣_,𝑧+𝛼𝑤_,𝑦]=[0,0] in the face plane instead of thickness direction, one gets vertical deflection 𝑤₀(𝑥,𝑦) in the form 𝛼𝑤₀=−𝑢₁𝑑𝑥+𝑣₁𝑑𝑦. (13) Since 𝑤₀ is from satisfaction of zero face shear conditions, it can be considered as face deflection 𝑤_0𝐹 though it is same for all face parallel planes (note that prescribed zero 𝑤₀ along a segment of the neutral plane implies zero 𝑤₀ along the corresponding segment of the intersection of face plane with wall of the plate since 𝑤₀ is independent of 𝑧 and vice versa). It is analytic in the domain of the plate if 𝜔_𝑧 is zero. In such a case, we note from strain-displacement relations (8) that 𝜀_1,𝑦,𝜀_2,𝑥,𝜀_3,𝑦,𝜀_3,𝑥=𝛼²𝑣_,𝑥𝑥,𝑢_,𝑦𝑦,2𝑢_,𝑦𝑦,2𝑣_,𝑥𝑥. (14)

(i) In the absence of applied transverse loads, integrations of (1), (2) using Kirchhoff’s displacements give a homogeneous fourth-order equation governing 𝑤₀. In the present analysis as in FSDT, static equations governing [𝑢₁,𝑣₁] are, however, given by 𝑄_1𝑖𝜀_𝑖,𝑥+𝑄_3𝑖𝜀_𝑖,𝑦=0,𝑄_2𝑖𝜀_𝑖,𝑦+𝑄_3𝑖𝜀_3𝑖,𝑥=0(𝑖=1,2,3). (15) They are subjected to the edge conditions (3a), (3b) along 𝑥- (and 𝑦-) constant edges.

Equations (15) governing [𝑢₁,𝑣₁] are uncoupled due to relations (14) in the case of isotropic plates whereas they are coupled without cross-derivatives [𝑢_1𝑥𝑦,𝑣_1𝑥𝑦] in the case of anisotropic plates. In the absence of prescribed bending stress all along the closed boundary of the plate, in-plane 𝜏_𝑥𝑦 distribution corresponds to pure torsion where as it is different in the presence of bending load. These 𝜏_𝑥𝑦 distributions are discussed later in Section 6.2.

(j) In FSDT, (15) correspond to neglecting shear energy from transverse shear deformations. Vertical deflection 𝑤₀ has to be obtained from (14) using the solutions for [𝑢₁,𝑣₁] from (15). Reactive transverse stresses are expressed in terms of [𝑢₁,𝑣₁]. They are dependent on material constants different from prescribed nonzero transverse shear stresses along the edges of the plate.

(k) In the development of Poisson’s theory, (15) are coupled with reactive transverse stresses.

6. Poisson’s Theory

In the preliminary solution, steps (a)–(g) in Section 5 are unaltered. That is, transverse stresses and strains are zero and 𝑤=𝑤₀(𝑥,𝑦). In the absence of higher order in-plane displacement terms, reactive transverse shear stresses are parabolic and gradients of a function 𝜓(𝑥,𝑦), that is, in-plane distributions of transverse shear stresses [𝜏_𝑥𝑧,𝜏_𝑦𝑧]=𝛼[𝜓_2,𝑥,𝜓_2,𝑦]. Equation governing 𝜓(=𝜓₂) from thickness-wise integration of (2) is 𝛼²Δ𝜓+𝜎_𝑧3=0. (16) In the previous equation, 𝜎_𝑧3 is coefficient of 𝑓₃(𝑧). Satisfaction of the load condition 𝜎_𝑧=±𝑞/2 along 𝑧=±1 faces gives 𝜎_𝑧3=(3/2)𝑞 so that 𝛼²3Δ𝜓+2𝑞=0. (17) Edge condition on 𝜓 is either 𝜓=0 or its outward normal gradient is equal to the prescribed shear stress along each segment of the edge. Note that the transverse stresses thus obtained are independent of material constants. In the isotropic plate, 𝜓(𝑥,𝑦)=𝐸^′𝑒₁ in which 𝐸^′=𝐸/(1−𝜈²) and 𝑒₁=(𝜀_𝑥1+𝜀_𝑦1).

Integrated equilibrium equations (1) in terms of in-plane strains are (sum 𝑗=1,2,3): 𝑄_1𝑗𝜀_𝑗,𝑥+𝑄_3𝑗𝜀_𝑗,𝑦−𝜏_𝑥𝑧2𝑄=0,_2𝑗𝜀_𝑗,𝑦+𝑄_3𝑗𝜀_𝑗,𝑥−𝜏_𝑦𝑧2=0. (18) By substituting strains in terms of [𝑢₁,𝑣₁], (18) are two equations governing (𝑢₁,𝑣₁) and have to be solved with conditions along 𝑥- (and 𝑦-) constant edge 𝑢=0or𝜎_𝑥=𝑧𝑇_𝑥𝑦⟺𝑥,𝑦,(𝑢,𝑣),𝑣=0or𝜏_𝑥𝑦=𝑧𝑇_𝑥𝑦𝑦⟺𝑥,𝑦,(𝑢,𝑣). (19a)(19b)

Vertical deflection 𝑤₀ is given by (13). As mentioned earlier, cross-derivatives of [𝑢₁,𝑣₁] do not exist in (18), (19a), and (19b) due to (14). In the isotropic case, (18) are uncoupled and are simply given by 𝛼²Δ𝑢₁=𝛼𝑒_1,𝑥⟺𝑥,𝑦,𝑢,𝑦. (20) Edge conditions (19a) and (19b) are also uncoupled and they are given by 𝑢₁=0or𝐸^′(1−𝜐)𝛼𝑢_1,𝑥=𝑇_𝑥𝑦⟺,(),𝑣𝑥,𝑦𝑢,𝑣₁=0or2𝐺𝛼𝑣_1,𝑥=𝑇_𝑥𝑦𝑦⟺𝑥,𝑦,(𝑢,𝑣). (21) The condition 𝜓₀=0 in solving (17) is different from the usual condition 𝑤₀=0 in the Kirchhoff’s theory and FSDT. The function 𝜓₀ is related to the normal strain 𝜀_𝑧. In the isotropic case, 𝜓₀ is proportional to 𝑒₁(𝑥,𝑦)=(𝜀_𝑥1+𝜀_𝑦1). Gradients of 𝜓₀ are proportional to gradients of transverse strains in FSDT. If the plate is free of applied transverse stresses, that is, the plate is subjected to bending and twisting moments only, 𝑒₁ is proportional to Δ𝑤₀. The function 𝜓₀(𝑥,𝑦), thereby, Δ𝑤₀ is identically zero from (2) (in Kirchhoff’s theory, the tangential gradient of Δ𝑤₀ is proportional to the corresponding gradient of applied 𝜏_𝑥𝑦 along the edge of the plate implied from Kelvin and Tait’s physical interpretation of the contracted transverse shear condition). This Laplace equation is not adequate to satisfy two in-plane edge conditions. One needs its conjugate harmonic functions to express in-plane displacements in the form 𝑢₁,𝑣₁=−𝛼𝑧𝑤_0,𝑥+𝜑_0,𝑦,𝑤_0,𝑦−𝜑_0,𝑥. (22) The function 𝜑₀ was introduced earlier by Reissner [8] as a stress function in satisfying (2). Note that the two variables 𝑢₁ and 𝑣₁are expressed in terms of gradients of two functions 𝑤₀ and 𝜑₀.

After finding 𝑤₀ and 𝜑₀ from solving in-plane equilibrium equations (1), correction 𝑤_0𝑐 to 𝑤₀ from zero face shear conditions is given by 𝑤_0𝑐=𝜙_0,𝑦𝑑𝑥+𝜙_0,𝑥𝑑𝑦. (23) One should note here that 𝑤(𝑥,𝑦)=𝑤₀+𝑤_0𝑐 does not satisfy prescribed edge condition on 𝑤.

In FSDT, [𝑢₁,𝑣₁] are obtained from solving (1). There is no provision to find 𝑤₀ and it has to be obtained from zero face shear conditions in the form 𝛼𝑤₀=−𝑢₁𝑑𝑥+𝑣₁𝑑𝑦. (24) If the applied transverse stresses are nonhomogeneous along a segment of the edge, 𝑤₀ is coupled with [𝑢₁,𝑣₁] in (1) and (2) and edge conditions (3a), (3b), and (3c).

With reference to the present analysis, it is relevant to note the following observation: in the preliminary solution, 𝜎_𝑧, 𝛼[𝜏_𝑥𝑧,𝜏_𝑦𝑧], 𝛼²[𝜎_𝑥,𝜎_𝑦,𝜏_𝑥𝑦] are of 𝑂(1). As such, estimation of in-plane stresses, thereby, in-plane displacements is not dependent on 𝑤₀. The support condition 𝑤₀=0 along the edge of the plate does not play any role in determining the in-plane displacements.

6.1. Comparison with Analysis of Extension Problems

It is interesting to compare the previous analysis with that of extension problem. In extension problem, applied shear stresses along the faces of the plate are gradients of a potential function and they are equivalent to linearly varying body forces, independent of elastic constants, in the in-plane equilibrium equations. Rotation 𝜔_𝑧≠0 (but Δ𝜔_𝑧=0) and equations governing displacements [𝑢₀,𝑣₀] are coupled. Also, vertical deflection is given by thickness-wise integration of 𝜀_𝑧0 from constitutive relation and higher order corrections are dependent on this vertical deflection. In the present analysis of bending problems, in-plane distributions of transverse shear stresses are gradients of a function related to applied edge transverse shear and face normal load through equilibrium equation (2). They are also independent of elastic constants and derivative 𝑓_2,𝑧 of parabolic distribution of each of them is equivalent to body force in linearly varying in-plane equilibrium equations. Vertical deflection 𝑤₀ is purely from satisfaction of zero shear stress conditions along faces of the plate. Higher order corrections are dependent on normal strains and independent of 𝑤₀. It can be seen that steps in the analysis of bending problem are complementary to those in the classical theory of extension problem.

6.2. Associated Torsion Problems

First-order shear deformation theory and higher order shear deformation theories do not provide proper corrections to initial solution (from Kirchhoff’s theory) of primary flexure problems. In these theories, corrections are due to approximate solutions of associated torsion problems. In fact, one has to have a relook at the use of shear deformation theories based on plate (instead of 3D) element equilibrium equations other than Kirchhoff’s theory in the analysis of flexure problems. In the earlier work [5], we have mentioned that this coupling of bending and torsion problems is nullified in the limit of satisfying all equations in the 3D problems. However, these higher order theories in the case of primary flexure problem defined from Kirchhoff’s theory are with reference to finding the exact solution of associated torsion problem only. In fact, one can obtain exact solution of the torsion problem (instead of using higher order polynomials in 𝑧 by expanding 𝑧 and 𝑓₃(𝑧) in sine series and 𝑓₂(𝑧) in cosine series. In a pure torsion problem, normal stresses and strains are zero implying that [𝑢,𝑣,𝑤]=𝑢𝑦,𝑧,𝑣(𝑥,𝑧),𝑤𝑥,𝑦. (25) Rotation 𝜔_𝑧≠0 and warping displacements (𝑢,𝑣) are independent of vertical displacement 𝑤. In FSDT, 𝜔_𝑧≠0 and corrections to Kirchhoff’s displacements are due to the solution of approximate torsion problem. In the isotropic rectangular plate, warping function 𝑢(𝑦,𝑧) is harmonic in 𝑦-𝑧-plane. It has to satisfy 𝐺𝛼𝑢_,𝑦=𝑇_𝑢(𝑦,𝑧) with prescribed 𝑇_𝑢 along an 𝑥= constant edge. By expressing 𝑢 in product form 𝑢=𝑓_𝑢(𝑦)𝑔_𝑢(𝑧), we have 𝛼²𝑓_{𝑢,𝑦𝑦}𝑔_𝑢+𝑓_𝑢𝑔_{𝑢,𝑧𝑧}=0. (26) By taking 𝑔_𝑢𝑛=sin𝜆_𝑛𝑧 where 𝜆_𝑛=(2𝑛+1)𝜋/2 satisfying zero face shear conditions, we get 𝑓_𝑢𝑛=𝐴_𝑢𝑛𝜆cosh^𝑛𝛼𝑦+𝐵_𝑢𝑛𝜆sinh^𝑛𝛼𝑦. (27) If prescribed 𝑇_𝑢 is 𝑓_𝑢𝑛 with specified constants (𝐴_𝑢𝑛,𝐵_𝑢𝑛), clearly there is no provision to satisfy zero in-plane shear condition along 𝑦= constant edges. It has to be nullified with corresponding solution from bending problem. That is why a torsion problem is associated with bending problem but not vice versa. In a corresponding bending problem, all stress components are zero except 𝜏_𝑥𝑦=−𝑓_𝑢𝑛(𝑦)𝑔_𝑢𝑛(𝑧). This solution is used only in satisfying edge condition along 𝑦= constant edges in the presence of specified 𝑇_𝑢𝑛 along 𝑥= constant edges. It shows that 𝜏_𝑥𝑦 distribution in flexure problem is nullified in the limit in shear deformation theories due to torsion.

It is interesting to note that the previous mentioned deficiency, due to coupling with 𝑤₀ in the plate element equilibrium equations in FSDT and higher order shear deformation theories, does not exist if applied 𝜏_𝑥𝑦 is zero all along the closed boundary of the plate. It is complementary to the fact that boundary condition paradox in Kirchhoff’s theory does not exist if tangential displacement and 𝑤₀ are zero all along the boundary of the plate.

6.3. Illustrative Example: Simply Supported Isotropic Square Plate

Consider a simply supported isotropic square plate subjected to vertical load 𝑞=𝑞₀sin𝜋𝑥𝑎sin𝜋𝑦𝑎. (28) Exact values for neutral plane and face deflections obtained earlier [7] with 𝜈=0.3, 𝛼=(1/6) are 𝐸2𝑞₀𝑤𝑎2,𝑎2𝐸,0=4.487,2𝑞₀𝑤𝑎2,𝑎2,1=4.166. (29)(30) In this example, Poisson-Kirchhoff’s boundary conditions paradox does not exist. Hence, Poisson’s theory gives the same solution obtained in the author’s previous mentioned work. Concerning face value in (30), Kirchhoff’s theory gives a value of 2.27 that is same for all face parallel planes. Poisson’s theory gives an additional value of 0.267 attributable to 𝜀_𝑧1 and it is same for neutral plane deflection since 𝑤(𝑥,𝑦,𝑧) can be expressed as 𝑤𝑥,𝑦,𝑧=𝑤_0𝐹𝑥,𝑦+𝑓₂(𝑧)𝑤₂=𝑤_0𝑁−1𝑥,𝑦2𝑧²𝜀_𝑧1. (31) In the previous expression, 𝑤_0𝑁 is deflection of the neutral plane. Higher order correction to 𝑤₀ uncoupled from torsion is 1.262 so that total correction to the value from Kirchhoff’s theory is about 1.53 [7]. Correction due to coupling with torsion is 1.45 [5] whereas it is 1.423 from FSDT and other sixth order theories [9]. With reference to numerical values reported by Lewiński [9], the previous correction is 1.412 in Reddy’s 8th-order theory and less than 1.23 in Reissner’s 12th-order and other higher order theories. It clearly shows that these shear deformation theories do not lead to solutions of bending problems (note that the value (2.27+1.423) from FSDT corresponds to neutral plane deflection and it is in error by about 17.7% from the exact value).

In view of the previous observations, it is relevant to make the following remarks: physical validity of 3D equations is constrained by limitations of small deformation theory. For 𝛼=1, this constraint is dependent on material constants, geometry of the domain, and applied loads. Range of validity of 2D plate theories arises for small values of 𝛼. Kirchhoff’s theory gives lower bound for this 𝛼. For this value of 𝛼 and for slightly higher values of 𝛼 than this lower bound, Kirchhoff’s theory is used, in spite of its known deficiencies, due to its simplicity to obtain design information. Similarly, FSDT and other sixth-order shear deformation theories are used for some additional range of values of 𝛼 due to the corrections over Kirchhoff’s theory though these corrections are due to coupling with approximate torsion problems. They serve only in giving guidance values for applicable design parameters for small range of values of 𝛼 beyond its lower bound. They do not provide initial set of equations in formulating proper sequence of sets of 2D problems converging to the 3D problem. As such, comparison of solutions from Kirchhoff theory and FSDT with the corresponding solutions in the present analysis does not serve much purpose.

Since 𝑤_0𝑁 has to be higher than 𝑤_0𝐹, one has to obtain the correction to 𝑤_0𝑁 before finding correction to internal distribution of 𝑤(𝑥,𝑦,𝑧) along with correction to 𝑤_0𝐹. Initial correction to 𝑤_0𝑁 is obtained earlier [7] in the form 𝑤=𝑤₀−𝑓₂(0)𝜀_𝑧1+𝑓₄(0)𝜀_𝑧3. (32) It is the same for face deflection (note that 𝑤₂ in the previously mentioned author’s work is required only to obtain 𝜀_𝑧3). As such, 𝑤_0𝑁 is further corrected from solution of a supplementary problem. This total correction over face deflection is about 0.658 giving a value of 4.458 which is very close to the exact value 4.487 in (29). However, we note that the error in the estimated value (=3.8) of 𝑤_0𝐹 is relatively high compared to the accuracy achieved in the neutral plane deflection. It is possible to improve estimation of 𝑤_0𝐹 by including 𝜀_𝑧1 in (𝑢₃,𝑣₃) such that (𝜏_𝑥𝑧2,𝜏_𝑦𝑧2) are independent of 𝜀_𝑧1. In the present example, correction to face deflection changes to 𝐸2𝑞₀𝑤_facemax=((6/5)((1+𝜐)/(1−𝜐)))𝛽²((2/5)((4−𝜐)/(1−𝜐)))𝛽².−1 (33) In the previous equation, 𝛽²=2𝑤_0𝐹𝛼²𝜋². Correction 𝜀_𝑧3(=1.262) to face value changes to 1.431 giving 3.97 (=2.27+0.267+1.431) for face deflection which is under 4.7% from exact value.

7. Anisotropic Plate: Second-Order Corrections

We note that 𝜀_𝑧1 does not participate in the determination of (𝑢₁,𝑣₁) and it is obtained from the constitutive relation (6) in the interior of the plate. It is zero along an edge of the plate if 𝑤=0 is specified condition. With zero transverse shear strains, specification of 𝜀_𝑧1=0 instead of 𝑤₀=0 is more appropriate along a supported edge since 𝑤₀=0 implies zero tangential displacement along a straight edge which is the root cause of Poisson-Kirchhoff’s boundary conditions paradox. As such, edge support condition on w does not play any role in obtaining the in-plane displacements and reactive transverse stresses.

In the first stage of iterative procedure, second-order reactive transverse stresses have to be obtained by considering higher order in-plane displacement terms 𝑢₃ and 𝑣₃. However, one has to consider limitation in the previous Poisson’s theory like in Kirchhoff’s theory; namely, reactive 𝜎_𝑧 is zero at locations of zeros of 𝑞. It is identically zero in higher order approximations since 𝑓_2𝑛+1(𝑧) functions are not zero along 𝑧=±1 faces. Moreover, it is necessary to account for its dependence on material constants. To overcome these limitations, it is necessary to keep 𝜎_𝑧5 as a free variable by modifying 𝑓₅ in the form 𝑓^∗₅(𝑧)=𝑓₅(𝑧)−𝛽₃𝑓₃(𝑧). (34) In the previous equation, 𝛽₃=[𝑓₅(1)/𝑓₃(1)] so that 𝑓^∗₅(±1)=0. Denoting coefficient of 𝑓₃ in 𝜎_𝑧 by 𝜎^∗_𝑧3, it becomes 𝜎^∗_𝑧3=𝜎_𝑧3−𝛽₃𝜎_𝑧5. (35) Displacements (𝑢₃,𝑣₃) are modified such that they are corrections to face parallel plane distributions of the preliminary solution and are free to obtain reactive stresses 𝜏_𝑥𝑧4, 𝜏_𝑦𝑧4, 𝜎_𝑧5 and normal strain 𝜀_𝑧3.

We have from constitutive relations 𝛾_𝑥𝑧2=𝑆₄₄𝜏_𝑥𝑧2+𝑆₄₅𝜏_𝑦𝑧2⟺𝑥,𝑦,(4,5). (36) Modified displacements and the corresponding derived quantities denoted with ∗ are 𝑢^∗₃=𝑢₃−𝛼𝑤_2,𝑥+𝛾_𝑥𝑧2⟺𝑥,𝑦,(𝑢,𝑣). (37) Strain-displacement relations give 𝜀^∗_𝑥3=𝜀_𝑥3−𝛼²𝑤_2,𝑥𝑥+𝛼𝛾_{𝑥𝑧2,𝑥}⟺,𝛾𝑥,𝑦_∗𝑥𝑦3=𝛾_𝑥𝑦3−2𝛼²𝑤_2,𝑥𝑦+𝛼𝛾_{𝑥𝑧2,𝑦}+𝛾_{𝑦𝑧2,𝑥},𝛾^∗_𝑥𝑧3=𝑢₃+𝛾_𝑥𝑧2⟺𝑥,𝑦,(𝑢,𝑣). (38) In-plane and transverse shear stresses from constitutive relations are 𝜎^∗_𝑖(3)=𝑄_𝑖𝑗𝜀^∗_𝑗(3),𝜏𝑖,𝑗=1,2,3_∗𝑥𝑧2=𝑄₄₄𝑢₃+𝑄₄₅𝑣₃+𝜏_𝑥𝑧2⟺𝑥,𝑦,(4,5). (39)(40) Suffix (3) in (39) is to indicate that the stresses and strains correspond to 𝑢₃ and 𝑣₃.

We get from (2), (15), (35), (37), and (40) with reference to coefficient of 𝑓₃(𝑧): 𝛼𝑄₄₄𝑢₃+𝑄₄₅𝑣_3,𝑥+𝑄₅₄𝑢₃+𝑄₅₅𝑣_3,𝑦=𝛽₃𝜎_𝑧5. (41) Note that 𝑤₂is not present in the previous equation.

From integration of equilibrium equations, we have with sum on 𝑗=1,2,3𝜏_𝑥𝑧4=𝛼𝑄_1𝑗𝜀^∗_𝑗,𝑥+𝑄_3𝑗𝜀^∗_𝑗,𝑦(3),𝜏_𝑦𝑧4=𝛼𝑄_2𝑗𝜀^∗_𝑗,𝑦+𝑄_3𝑗𝜀^∗_𝑗,𝑥(3). (42) Note that𝑤₂is present in the previous expression. 𝛼𝜏_{𝑥𝑧4,𝑥}+𝜏_{𝑦𝑧4,𝑦}+𝜎_𝑧5=0Coefficientof𝑓₅𝜀_𝑧3,=𝑆_6𝑗𝜎_𝑗+𝑆₆₆𝜎_𝑧(3). (43)(44) One gets one equation governing in-plane displacements (𝑢₃,𝑣₃) from (41), (43), noting that 𝑓_5,𝑧𝑧+𝑓₃=0, in the form 𝛽₃𝜏_{𝑥𝑧4,𝑥}+𝜏_{𝑦𝑧4,𝑦}=𝛼²𝑄₄₄𝑢₃+𝑄₄₅𝑣_3,𝑥𝑥+𝑄₅₄𝑢₃+𝑄₅₅𝑣_3,𝑦𝑦. (45) The second equation governing these variables is from the condition 𝜔_𝑧=0. Here, it is more convenient to express (𝑢₃,𝑣₃) as 𝑢₃,𝑣₃=−𝛼𝜓_3,𝑥−𝜙_3,𝑦,𝜓_3,𝑦+𝜙_3,𝑥 (46) so that Δ𝜑₃=0 and (45) becomes a fourth-order equation governing 𝜓₃. This sixth-order system is to be solved for 𝜓₃ and 𝜑₃ with associated edge conditions 𝑢^∗₃=0or𝜎^∗₃𝑣=0,_∗3=0or𝜏^∗_𝑥𝑦3𝜓=0,₃=0or𝜏^∗_𝑥𝑧3=0, (47) along an 𝑥=constant edge and analogous conditions along a 𝑦= constant edge.

In (45) expressed in terms of 𝜓₃ and 𝜑₃, we replace 𝑤₂ by 𝜓₃ for obtaining neutral plane deflection since contributions of 𝑤₂ and 𝜓₃ are one and the same in finding [𝑢₃,𝑣₃].

7.1. Supplementary Problem

Because of the use of integration constant to satisfy face shear conditions, vertical deflection of neutral plane is equal to that of the face plane deflection. This is physically incorrect since face plane is bounded by elastic material on one side whereas neutral plane is bounded on both sides. This lack in interior solution is rectified by adding the solution of a supplementary problem based on leading cosine term that is enough in view of 𝑓₂(𝑧) distribution of reactive shear stresses. This is based on the expansion of 𝜎_𝑧=𝑓₃(𝑧)(3/2)𝑞 in sine series. Here, load condition is satisfied by the leading sine term and coefficients of all other sine terms are zero.

Corrective displacements in the supplementary problem are assumed in the form 𝑤=𝑤_2𝑠𝜋2𝜋cos2𝑧,𝑢_𝑠=𝑢_3𝑠𝜋sin2𝑧⟺(),𝜎𝑢,𝑣_3𝑠𝑖=𝑄_𝑖𝑗𝜀_3𝑠𝑗𝑖,𝑗=1,2,3. (48) We have from integration of equilibrium equations𝜏_{2𝑠𝑠𝑥𝑧}2=−𝜋𝛼𝑄_1𝑗𝜀_{3𝑠𝑗,𝑥}+𝑄_3𝑗𝜀_{3𝑠𝑗,𝑦},𝜏_{2𝑠𝑦𝑧}2=−𝜋𝛼𝑄_2𝑗𝜀_{3𝑠𝑗,𝑦}+𝑄_3𝑗𝜀_{3𝑠𝑗,𝑥}. (49a)(49b)

In-plane distributions 𝑢_3𝑠 and 𝑣_3𝑠 are added as corrections to the known 𝑢^∗₃ and 𝑣^∗₃ so that (𝑢,𝑣) in the supplementary problem are 𝑢=𝑢^∗₃+𝑢_3𝑠𝜋sin2𝑧⟺(𝑢,𝑣). (50) We note that 𝜎_𝑧 from integration of equilibrium equations with 𝑠 variables is the same as 𝜎_𝑧 from static equilibrium equations with ∗ variables. Hence, 2𝜋²𝛼²𝑄_1𝑗𝜀_{3𝑠𝑗,𝑥𝑥}+2𝑄_3𝑗𝜀_{3𝑠𝑗,𝑥𝑦}+𝑄_2𝑗𝜀_{3𝑠𝑗,𝑦𝑦}+𝛽₃𝜎_𝑧5=0. (51) It is convenient to use here also 𝑢_3𝑠,𝑣_3𝑠=−𝛼𝜓_3𝑠,𝑥−𝜙_3𝑠,𝑦,𝜓_3𝑠,𝑦+𝜑_3𝑠,𝑥. (52) Correction to neutral plane deflection due to solution of supplementary problem is from 𝜀_𝑧3𝑠=𝑆_6𝑗𝜎_3𝑠𝑗+𝑆₆₆𝜎_𝑧3𝑠sum𝑗=1,2,3. (53)

7.2. Summary of Results from the First Stage of Iterative Procedure

The previous analysis gives displacements [𝑢𝑣]=𝑓₃[𝑢₃,𝑣₃], consistent with [𝜏_𝑥𝑧,𝜏_𝑦𝑧]=𝛼𝑓₂[𝜓_,𝑥,𝜓_,𝑦], and corrective transverse stresses from the first stage of iteration 𝑤=𝑤₀−𝑓₂𝜀_𝑧1+𝑓₄𝜀_𝑧3−𝜀_𝑧𝑠3𝜋2𝜋cos2𝑧,𝑢=𝑓₁𝑢₁+𝑓₃𝑢^∗₃+𝑢^∗₃+𝑢_3𝑠𝜋sin2𝑧⟺(),𝜏𝑢,𝑣_𝑥𝑧=𝑓₂𝜏_2𝑥𝑧+𝜏_{2𝑠𝑥𝑧}𝜋2𝜋cos2𝑧⟺,𝜎𝑥,𝑦_𝑧=𝑓₃(𝑧)𝜎^∗_𝑧3(From(35)). (54) It is to be noted that the dependence of transverse stresses on material constants is through the solution of supplementary problem. One may add 𝑓₄ terms in shear components and 𝑓₅ term in normal stress component. These components are also dependent on material constants, but they need correction from the solution of a supplementary problem. Successive application of the previous iterative procedure leads to the solution of the 3D problem in the limit.

8. Concluding Remarks

Poisson’s theory developed in the present study for the analysis of bending of anisotropic plates within small deformation theory forms the basis for generation of proper sequence of 2D problems. Analysis for obtaining displacements, thereby, bending stresses along faces of the plate is different from solution of a supplementary problem in the interior of the plate. A sequence of higher order shear deformation theories lead to solution of associated torsion problem only. In the preliminary solution, reactive transverse stresses are independent of material constants. In view of layer-wise theory of symmetric laminated plates proposed by the present author [10], Poisson’s theory is useful since analysis of face plies is independent of lamination which is in confirmation of (8)-(9) in a recent NASA technical publication by Tessler et al. [11].

One significant observation is that sequence of 2D problems converging to 3D problems in the analysis of extension, bending, and torsion problems are mutually exclusive to one other.

Highlights

(i)Poisson-Kirchhoff boundary conditions paradox is resolved. (ii)Transverse stresses are independent of material constants in primary solution.(iii)Edge support condition on vertical deflection has no role in the analysis.(iv)Finding neutral plane deflection requires solution of a supplementary problem.(v)Twisting stress distribution in pure torsion nullifies its distribution in bending.

Appendix

 𝑎 = side length of a square plate 2ℎ = plate thickness 𝑓𝑛(𝑧) = through-thickness distribution functions, 𝑛=0,1,2… 𝐿 = characteristic length of the plate in 𝑋-𝑌-plane 𝑞(𝑥,𝑦) = applied face load density 𝑄_𝑖𝑗 = stiffness coefficients 𝑆_𝑖𝑗 = elastic compliances [𝑇_𝑥,𝑇_𝑥𝑦,𝑇_𝑥𝑧] = prescribed stress distributions along an 𝑥= constant edge [𝑈,𝑉,𝑊] = displacements in 𝑋,𝑌,𝑍-directions, respectively [𝑢,𝑣,𝑤] = [𝑈,𝑉,𝑊]/ℎ 𝑂−𝑋𝑌𝑍 = Cartesian coordinate system [𝑥,𝑦,𝑧] = nondimensional coordinates [𝑋/𝐿,𝑌/𝐿,𝑍/ℎ] 𝛼=ℎ/𝐿 𝛽_2𝑛+1=[𝑓_2𝑛+3(1)/𝛼²𝑓_2𝑛+1(1)] Δ = Laplace operator (𝜕²/𝜕𝑥²+𝜕²/𝜕𝑦²) [𝜀_𝑥,𝜀_𝑦,𝛾_𝑥𝑦] = in-plane strains [𝛾_𝑥𝑧,𝛾_𝑦𝑧,𝜀_𝑧] = transverse strains  𝜔_𝑧=𝛼(𝑣_,𝑥−𝑢_,𝑦), rotation about 𝑧-axis [𝜎_𝑥,𝜎_𝑦,𝜏_𝑥𝑦] = bending stresses  [𝜏_𝑥𝑧,𝜏_𝑦𝑧,𝜎_𝑧] = transverse stresses.