Plate analysis using classical or Reissner- Mindlin theories

Plate analysis using classical or Reissner- Mindlin theories L. Palermo Jr. Faculty of Civil Engineering, State Universiv at Campinas, Brazil Abstract Plates can be solved with classical or Reissner-Mindlin plate theory in the same computer code with an appropriate treatment of the direct boundary element formulation. The classical theory could be understood as the irrotational component of the field related to the rotations of the plate [l] and the shear deformation effect would correspond to the solenoidal component. This paper presents the fundamental solution and the direct boundary integral equations written in terms of boundary coordinates, normal (n) and tangential (S), for Reissner-Mindlin theory like it has been made for the classical analysis in order to permit the desired connection. Thus, the correction introduced by transverse shear deformation on the bending can be identified with reference to the classical approach and it is the key to establish the connection among the studied theories. An examplc will be solved to show the connections among plates theories and to present a boundary element formulation for Mindlin's theory with the irrotational tield using the Danson's fundamental solution [5]. 1 Introduction The classical theory is written with reference to the out-of-plane displacement and its derivatives or, as it was shown in [l], the rotations of the plate can be represented by an irrotational field. The integral equation contains four boundary parameters and it is necessary to satisfy two boundary conditions to obtain a single solution. It must be pointed out that, beyond the two boundary unknowns, a concentrated reaction is placed at each corner as an additional parameter in the boundary value problem when a polygonal plate is considered. The inaccuracy of

the classical theory turns out to be of practical interest in the edge zone of a plate and around holes that have a diameter not larger than the thickness of the plate. In order to overcome the above-mentioned features due to one-displacement dependence of the classical theory. Mindlin [3] and Reissner [2] developed similar theories considering the shear deformation effect. Unless those theories have been conceived to treat thin or thick plates, they are frequently referred in numerical analysis as thick plates theories. The curvatures have not been directly related with out-of-plane displacement derivative in the constitutive equations and three boundary conditions should be satisfied in the boundary value problem rather than the two of classical theory. Bezine [6], Tottenham [S], Stern171 and Danson [5] made important contributions in the plate analyses with the classical theory using the Boundary Element Method (BEM). Weeen [4] introduced a BEM formulation applied to Reissner's theory; Barcellos [l l] and Vilmann [l01 used similar formulations to treat Mindlin's theory. The boundary integral equation (BIE) to perform those analyses has used six boundary parameters and no additional unknown had to be introduced when polygonal plates were considered. Katsikadelis [l51 presented a BEM formulation for the Reissner's theory. which was expressed in terms of two potentials, one biharmonic and one Bessel potential. The plates analyzed with that strategy needed three integral equations plus three finite difference equations. Mukherjee [9] and Paiva [l21 used the classical theory with an additional degree of freedom for the tangential boundary rotation into a boundary integral formulation and showed that many problems associated with the treatment with two-boundary parameters of this theory could be overcome. 2 Plate Equations The expressions will be presented with the Latin indices taking values {l, 2. 3) and Greek indices in the range { 1, 2). A plate of uniform thickness is referred to midline coordinates X, and thickness coordinate x3. The transverse shear (Q,), the bending and twisting moments (Map), all per unit of length, have similar definition in both theories with reference to the stress. In classical theory, the constitutive relations written in terms of deflections are: Map = -D((I - v ) ~ +v-., ~ W S ~ a(j ) (1) D is the flexural rigidity and v is the Poisson's ratio. Reissner conceived his theory from an assumed stress distribution and introduced generalized displacement expressions with a weighted average across the thickness [l 31 to get the deflection (W) of the plate in the middle surface and quantities (P,) representing components equivalent but not identical to the components of changes of slope. The constitutive equations written in terms of displacements (P,, W) are given by:

h is a constant related to shear effect and it is equal to &/h, Mindlin's theory [3] was conceived from an assumed strain distribution. The "equivalent changes of slope" P, of Reissner's theory were replaced by v, which were obtained without average expressions due to assumed strain hypothesis. The integrals to get shears, bending and twisting moments were performed but the coefficients of the integrals containing y,, were replaced by constants whose magnitude were obtained from wave propagation analyses. The constitutive relations for Mindlin's theory are: Mindlin showed that h is related to the Poisson ratio and the vibration mode including the transverse shear deformation and the rotatory inertia effects. In the present study, h will be assumed as n/h independently of the Poisson ratio. Further, it is necessary to note in equation (6) with reference to equation (4) that the distributed load effect was not include. It can be shown from Reissner's studies that this feature is related to the 033 stress effect. However. since the average expressions for the displacements have been introduced by Reissner [l 31 the 033 stress was assumed to equal to k0.5q at x3 equal to +h/2. respectively. Thus, Reissner and Mindlin theories have assumed same values for o, at the faces or at x3 equal to +h/2, but Reissner's theory concerned about 033 distribution on the thickness. Mindlin showed [3] that in his theory only the linearlv weighted average effect of 033 was neglected rather than 033 itself with reference to Reissner's theory. When these points are considered it is easy to understand the absence of the distributed load effect in the Mindlin's relation (6). 3 Integral Equations The Weeen's formulation [4] for Reissner's theory can be written as:

U, is p, u3 is W, t, is the product Mapa, t3 is the product Q,& and the differentials ds(y) and dr(y) denotes boundary and domain differentials, respectively. Uij represents the rotation Cj=1,2) or the deflection Cj=3) due to a unit point couple (i=1,2) or a unit point force (i=3). 2 I * -~{-[A(z)-B(~)r,~r,p]-;8~~~:z-~)-~r,~r,p] - ZnD I-" With z equal to hr It is necessary to remember that the integrand of the domain integral in equation (8) contains the additional factor from Reissner's theory related to the distributed load (linearly weighted average effect of c ~ that ~ is ~ turned ) null in Mindlin's theory. The following domain integral replaces the domain integral for Reissner's theory in equation (8) to analyze plates using Mindin's hypotheses: r 1 The boundary integral equation for Mindlin's theory written with reference to boundary coordinates ns (normal-tangential) is given by: With Cij equals to Kronecker delta for a smooth boundary, uj is related to the fundamental solution used and it corresponds to the deflection for the solution due to a unit point load (i=3) or the rotation for the solution due to a unit point couple (i=l or 2). 8, and 8, are v, and v,, respectively. The integral equation for classical analysis can be written as: With Cap equals to Kronecker delta for a smooth boundary, up is related to the fundamental solution used and it corresponds to the deflection for the solution due to a unit point load (a=l) or the rotation for the solution due to a unit point couple (a=2). The parameters introduced are the equivalent shear (V,) and a corner reaction (R,) at each corner when polygonal plates are considered.

The expression (13) presents the corner reaction (R,) at corner i as the difference between the twisting moments at the corner neighborhood on the forward side (M,:) and the backward side (M,:). The classical analysis can be performed with the fundamental solution due to Danson or Bezine-Stern for a unit point load. A general expression for these solutions is given by: Bezine-Stern's solution is obtained from equation (14) when k is turned null or h is equal to e (the base to which Napierian logarithms are calculated). Danson's solution is obtained when k is equal to 0.5 or h can be turned equal to eo.'. The fundamental solution for a unit couple in the classical analysis is obtained from the derivative of the fundamental solution for a unit point load with respect to the field argument, as it was shown by Timoshenko [l 41. When equation (10) and (l l) are compared, a connection among classical and Mindlin theories can be noted. The boundary integral equation (11) could be obtained fiom equation (10) when classical analysis was considered or 0, and 0, would be equal to -W,, and -W,,, respectively. The other end of the connection can be established when the field decomposition [l] is taken into account in the fundamental solution. Thus, a vector containing the rotations of the plate can be written as [l]: (vl;v2)= (m.,.mj+ (H.~.-q) (15 ) 41 and H represent functions whose components are independent of X, [l]. The obtained functions W, 4 and H for a unit couple F, are given by [l]: F: can be understood as the result from domain integration on a circle containing the Dirac-delta force in the direction a (it is equal to 1) and it was introduced to distinguish each couple effect. [l]. The obtained functions W and 4 for a unit point load are given by [l]: r- --

The fimction $ is related to the irrotational field and it can be shown that the solution of H is null due to the radial symmetry of the unit point load problem on an infinite plate. The solutions (16) to (19) can be turned equal to Weeen9s solution but a rigid body motion must be included in (1 8) to get it. On the other hand, the fundamental solution can be written with reference to boundary coordinates ns in order to use (10) and to identify the corrections introduced by the shear deformation effect: a) Solution for a unit point load The bending and twisting moments have similar expressions to those used for the classical theory and their expressions written with reference to coordinates ns are given by: * (I-")&& 471 2 471 & The transverse shear defined in eqn (7) can be written with reference to coordinates ns with the following expression (Q,=Q,.n,): The rotations v, and v, (or 8, and OS, respectively) are given by: The main difference with reference to classical theory due to Reissner- Mindlin hypothesis appears in the deflection, expression (18). The first term in that expression corresponds to the deflection obtained from classical hypotheses and the second term corresponds to the shear deformation effect. b) Solution for a unit couple The deflection due to a unit couple in the a direction is similar to that obtained under classical hypotheses and it is given by equation (16). Further, the relation (15) for rotations can be putted into a particular expression because the deflection (W) is -$ in this solution and it can be written in terms of ns as: (v,.%) = (- W,,,.-W.,)+ (H.,.-H,,) (20) The notation presented in expression (20) shows the explicit connection among classical and the Reissner-Mindlin theories. When the expression (20) is introduced in the relations for the transverse shear (7), they can be written as: The distributed shears Qi can be written in terms of boundary coordinates ns: The vector given by expression (2 l) could be used in the relation (20):

The expression (22) shows a physical meaning for the correction introduced by Reissner-Mindlin theory. The final expression for the transverse shear Q, due to a unit couple in the a direction is given by: The first term between the square brackets in expression (23) is equal to the transverse shear due to a couple obtained from the classical theory. The second term is the correction due to the Reissner-Mindlin hypotheses. The bending and twisting moments are given by: The moments expressions (24 and 25) contain the classical relation (eqn 1) plus a correction introduced by the shear deformation effect. The deflection. rotations and efforts were written with reference to boundar) coordinates ns at field points and for a unit couple in the a direction at the source point. Appropriate dot products must be employed when one of the boundary directions at the source point is used for the couple direction. As an example, the deflection and the bending moment due to a couple in the direction would be: W!= wa.ila (26) M," Mna.lla (27) Boundary directions at the source point are noted by C and 2 in equations (26, 27). The deflection and the bending moment due to a couple in the direction were noted by W" and M," respectively. The deflection and the bending moment due to a couple in the direction Q were noted by W" and M,", respectively. Thus, theconciusions about the present development can be summarized as: The rotations and efforts obtained from the fundamental solution due to a unit point load with the Reissner-Mindlin theory were similar to those obtained from the classical theory. The shear deformation effect appeared in the deflection as an additional term to correct the classical expression. The deflection obtained from the fundamental solution due to a unit couple with the Reissner-Mindlin theory was similar to that obtained from the classical theory. However, rotations and efforts had the classical expression plus an additional term introduced by the shear deformation effect. The BEM permits that the shear deformation effect can be switched on or off in plate analyses. When the boundary equation (1 1) is used with two boundary unknowns, it would be necessary to introduce an additional unknown because Reissner-Mindlin theory works with three boundary

unknowns. In this way, it would be necessary to suppress the unknowns related to corner reactions (R,) because they corresponds to a particular feature of the classical analysis and to return from V,* to Q,* in the boundary parameters. On the other hand. when the equation (10) is used, the soft restraint condition (M,, is released on the boundary) and the fundamental solution according to the classical theory can be employed. The integral equation could be written as: i C 'J' U J + ir(q;.w - b$.w,n-h'(,s.w,s).6 = - Mn.w,b).a+ jjwi.q.& (28) The equation (28) is the alternative way to perform classical analyses, which is similar to studies developed by Mukherjee [9] and Paiva [12]. It could be understood as a weak formulation from plate theories with the shear deformation effect. Thus, the responses would be obtained by an irrotational approach of the plate behavior and the terms related to the shear correction would be disregard. 4 Numerical Example A square plate was analyzed with classical and Reissner-Mindlin theories. The boundary was divided into 40 elements (44 nodes) and double nodes were employed at the corners. All nodal parameters were placed at the ends of the element and when discontinuous boundary elements were used, the collocation points were shifted to inside the element at a distance equal to a quarter of the element length. The classical approach used the equation (1 1) with Danson's fundamental solution for a unit point load. The source points were placed on the boundary and outside the domain in the normal direction of the boundary at a distance from the nodes equals to a quarter of the element length. Reissner's theory used the equation (8) with Weeen's formulation. Mindlin's theory used the equation (10). The responses from Mindlin's theory were labeled by Weeen and Danson when h was equal to nlh and eo", respectively. The source points were placed outside in the same points used for the classical analysis. The plate was loaded by an uniform load equal to 1 ~.m-~, the Young modulus was 205.000 MPa and the side of the plate had 2 m in length. Table 1 presents the results for a simply supported plate from all sides. The results presented in Table 2 were obtained when the plate had one side clamped and was simply supported from others sides. The responses were obtained at the center of the plate with two kinds of boundary conditions related to the twisting moments: "soft" when the twisting moments were released on the boundary and "hard" when they were restrained. The features presented in Tables 1 and 2 can be summarized as. 0 The differences among results obtained from Reissner and Mindlin theory were less than 1 % for internal points and efforts distribution on the boundary. The result obtained for internal points with "hard condition" were closer to the classical approach than those obtained with the "soft condition", even for the thick plates. On the other hand, the efforts distribution on the boundary with the "soft" condition were closer to the classical approach than those

obtained with the "hard" condition. It agrees with Timoshenko's explanation [l41 and the approaches contained when one of the components of the field is employed alone [l]. The use of h equal to eo.' for the irrotational field did not present significant differences in the results from those obtained with h equal to dh. Other boundary conditions are being studied and the results will be reported soon. Table 1 - Simply Supported plate BEM / Soft 1 0.4424 1 0.4897 1 0.1946 1 0.2048 1 Table 2 - One edge clamped and simply supported from others edges ' Obtained at the center W l M11 Thickness 1 0.02 0.2 0.02 0.2 The author is grateful to FAPESP due to the support for the development of studies on plates. References [l] Palermo Jr., L., (2000). A Study About Fundamental Solution in Plates, BEM 22. Editors C.A. Brebbia, H. Power, [2] Reissner. E., (1945). The Effect of Transverse Shear Deformation on the Bending of Elastic Plates. Jozirnal of Applled ~Mechan~cs, [3] Mindlin, R.D., (195 1). Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates, Journal ofapplled Mechunlcs,

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