Note on Mathematical Development of Plate Theories

Transcription

1 Advanced Studies in Theoretical Phsics Vol. 9, 015, no. 1, HIKARI Ltd,.m-hikari.com Note on athematical Development of Plate Theories Patiphan Chantaraichit 1, Parichat Kongtong Yos Sompornjaroensuk 1,* and Jakarin Vibooljak 3 1 Department of Civil Engineering ahanakorn Universit of Technolog Bangkok, 10530, Thailand * Correspondence author Department of Automotive Engineering Thai-Nichi Institute of Technolog Bangkok, 1050, Thailand 3 Vibool Choke Ltd., Part Bangkok, 10110, Thailand Copright 014 P. Chantaraichit et al. This is an open access article distributed under the Creative Commons Attribution License, hich permits unrestricted use, distribution, and reproduction in an medium, provided the original ork is properl cited. Abstract In this article, the histor of mathematical development for plate analsis is revieed ith emphasis on providing the plate s governing differential equations in rectangular coordinates. Attention is mainl devoted to theories based on the Kirchhoff, Reissner, indlin, and Levinson theories for isotropic plates ith uniform thickness. Therefore, it is epected that this revie paper should be useful for scientists and researchers in assisting them to understand and classif relevant eisting plate s theories quickl. athematics Subject Classification: 35Q74, 74K0 Keords: Partial differential equations, Thin plate, Thick plate, Kirchhoff s plate, Reissner s plate, indlin s plate, Levinson s plate.

2 48 Patiphan Chantaraichit et al. Notation ab, dimensions of a rectangular plate (see Figure 1) h thickness of a plate q intensit of uniforml distributed load deflection function, z, rectangular co-ordinates 3 D fleural rigidit of a plate, Eh /1(1 ) E modulus of elasticit, bending moments per unit length of sections of a plate perpendicular to the - and -aes respectivel tisting moment per unit length of section of a plate perpendicular to the -ais Q, Q shearing forces parallel to the z-ais per unit length of a plate perpendicular to the - and -aes respectivel V, V reactions parallel to the z-ais per unit length of the boundar of a plate perpendicular to the - and -aes respectivel Poisson s ratio Laplace s operator in, co-ordinates 1 Introduction A ver brief surve of the historical background for mathematical development of plate is given in the present section. Hoever, the interested reader should consult the tetbook of Szilard [4]. The first impetus to a mathematical statement of plate problems as probabl done b Euler in 1776 ho studied a free vibration analsis of plate problem based on the membrane theor of ver thin plates. In 180 Chladni first eperimentall presented the various modes of free vibrations using the distributed poder to form regular patterns after induction of vibration. In 1811 the French mathematician named Sophie Germain submitted a paper to the French Academ of Science. She developed and obtained a differential equation for vibration of plates that neglected the arping term of the middle surface in the strain energ epression using the calculus of variations. After corrected Germain s results b adding the missing term, the general differential equation of the free vibration of plates as first presented b Lagrange in This mentioned equation is ell-knon and called the Germain-Lagrange plate equation. In 183, the first satisfactor theor of bending of plates as derived b Navier, ho considered the plate thickness in the general plate equation as a function of fleural rigidit D using the modern theor of elasticit. Additionall, he also introduced an eact method to transform the differential equation into algebraic

3 Note on mathematical development of plate theories 49 equations ith the use of Fourier trigonometric series. Nevertheless, this method ields mathematicall eact solutions onl for the case of rectangular plate simpl supported on four edges. In 1850, Kirchhoff published an important thesis on the theor of thin plates that stated to independent basic assumptions. These assumptions are no idel accepted in the plate-bending theor and are knon as the Kirchhoff s hpotheses. Furthermore, Kirchhoff is considered to be the founder of the etended plate theor hich takes into account the combined bending and stretching. Since the Kirchhoff s plate theor neglects the deformation caused b transverse shear, it ill lead to considerable errors if applied to moderatel thick or thick plates. Therefore, based on the Kirchhoff s theor, it underestimates deflections and overestimates frequencies and buckling loads for the shear deformable plates []. Reissner in 1945 and indlin in 1951 developed a rigorous plate theor hich considers the deformations caused b the transverse shear forces to eliminate the deficienc of the classical plate theor. A significant contribution and etensive studied in the area of plate bending analsis together ith its application have been collected and also summarized in a fundamental monograph of Timoshenko and Woinosk-Krieger [5] that represented a profound analsis of various plate bending problems. A reference book b Leissa [1] presents a set of available results for the frequencies and mode shapes of free vibrations of plates that could be provided for the design and for a research in the field of plate vibrations. Problem description The coordinate sstem to be used is shon in Figure 1. The plate has dimensions of length a and idth b in the directions of and, respectivel; and thickness h in z direction. The middle plane of the plate before bending occurs is taken as the plane and during bending; points that ere in the plane undergo small displacements normal to the plane and form the middle surface of the plate. These displacements of the middle surface are then called the deflections () of a plate. a b d d h z Figure 1 Coordinate sstem definition

4 50 Patiphan Chantaraichit et al. Since the plate is in the deformed state under the applied eternal load, the resulting action of the stresses is to cause bending and tisting of the plate. Thus, the appropriate stress resultants for the plate theor are bending and tisting moments per unit length. The can be epressed in terms of the deflection as follos: D, (1) D, () D(1 ). (3) Consider an infinitesimal element dd of the plate that presented in Figure 1, the middle plane of the element cut out of the plate is then illustrated in Figure shoing all the moments and shearing forces acting on the sides of the element and the load distributed over the upper surface of the plate. iddle Plane Q d d h d Q Q Q q (, ) d Q Q d d d d Figure Positive sign conventions for stress resultants

5 Note on mathematical development of plate theories 51 Because the plate must be in the equilibrium states, three conditions for the equilibrium ma be ritten b summing the forces in the z-direction and summing the moments about the and aes, setting them equal to zero as follos: Q Q q 0, (4) Q 0, (5) Q 0. (6) Using (1) to (3), the shearing forces can be ritten in terms of the deflection, hich are 3 3 Q D 3, (7) 3 3 Q D 3. (8) The above to shearing forces mean the shearing forces acting in an element of the plate. For the shearing forces distributed along the free edge or the support of the plate, the are called the supplemented or Kirchhoff shearing forces or vertical edge reactions that can be determined in the folloings belo, 3 3 V Q D ( ) 3, (9) 3 3 V Q D ( ) 3. (10) Hoever, a more details of discussion for to equations above have been given in Timoshenko and Woinosk-Krieger [5]. In order to determine the governing differential equation of plate, it can be derived from (4) ith using (1) to (3) that results in

6 5 Patiphan Chantaraichit et al. here q, D (11). (1) The obtained (11) is represented for the governing differential equation of Kirchhoff s plate theor. 3 Shear deformable plate theories Although the classical Kirchhoff s plate theor ields sufficientl accurate results for thin plates, its accurac decreases ith groing thickness of the plate. Noadas man plate theories eist that account for the effects of transverse shear forces. Therefore, at least three ell-knon plate theories are presented and listed in the folloing details. Reissner s Theor Folloing the theor of plate introduced b Reissner, there are to assumptions to be used. In the first, the displacement field is varied linearl through the plate thickness. The second is involved ith the middle surface here plane sections remain plane but lines originall perpendicular to the middle surface do not remain perpendicular to it after bending. Utilizing the Castigliano s theorem of least ork, three simultaneous partial differential equations can be obtained as h D q q, 10 1 (13) h h 1 q Q D Q, (14) h h 1 q Q D Q (15) indlin s Theor This theor is the displacement-based theor folloing kinematic assumptions for the in-plane displacements in hich thickness stretch is not alloed. It is interesting to note that the mentioned plate theor fails to satisf the zero-stress condition on the top and bottom surfaces of the plate. Therefore, it is necessar to introduce a correction factor that depends on the Poisson s ratios.

7 Note on mathematical development of plate theories 53 Utilization of the minimum potential energ theorem, a set of three differential equations can eplicitl be derived as follos: Gh( ) q 0, (16) D (1 ) (1 ) Gh 0 D Gh (1 ) (1 ) 0, (17), (18) here and represent the rotations at the middle plane and defines as. (19) It can be noted that to foregoing theories proposed b Reissner and indlin are knon to be the first-order shear deformable plate theories for hich the in-plane displacements u and v var linearl. Both theories provide for further development. Levinson s Theor In order to avoid the use of shear correction factor, a higher-order shear theor is based on the assumed in-plane displacement fields to higher-order variations. Levinson introduced the kinematic assumptions satisfing the requirements for shear-free conditions on the top and bottom surfaces of the plates and also a parabolic distribution of shear stresses across the plate thickness. These assumptions allo the cross sections at the middle plane to arp [3]. B proposing higher-order of in-plane displacement functions to cubic function of z, Levinson derived an improved approimation to the theor of shear deformable plate as follos: ( ) 0 3 Gh q, (0) D 1 (1 ) (1 ) ( ) Gh 0 5, (1) 3 D 1 (1 ) (1 ) ( ) Gh 0 5. () 3 It can be clearl seen that three unknons presented in (0) to () are the

8 54 Patiphan Chantaraichit et al. deflection, and the rotations at the middle plane (16) to (18) for the indlin s plate theor. and similar to those of 4 Discussion In the previous sections, the thin plate and three different shear deformable plates are revieed. The Kirchhoff s plate theor gives a reasonable result in analzing the plates hich has the ratio of thickness to governing span length (h/l) beteen 1/50 and 1/10. For the moderatel thick plates (1/10 < h/l < 1/5), the Reissner s plate and iddlin s plate theories can be used properl to obtain the accurate results. If the ratio h/l of the plate is larger than 1/5 (h/l > 1/5), the plate becomes thick plate and the use of Levinson s theor appears to be more accurate. Additionall, it is noted that different higher-order of displacement fields can result in obtaining different refined plate theories. 5 Conclusion The main goal of the present paper is to revie some important plate s theories. A classical plate theor as ell-knon to be the Kirchhoff s plate, to different firstorder shear deformable plate theories, namel, the Reissner s plate and indlin s plate, and the third-order shear deformable plate of Levinson s theor are then considered. Their governing partial differential equations are also provided. Acknoledgements. The authors ould like to epress their sincere appreciation to Asst. Prof. Dr.Sutja Boonachut for all his careful reading the manuscript of this ork and especiall, openhanded invaluable help since the authors need. References [1] A.W. Leissa, Vibration of Plates, Reprinted ed., Acoustical Societ of America, Washington, D.C., [] K.. Lie, Y. Xiang and S. Kitipornchai, Research on thick plate vibration: A literature surve, Journal of Sound and Vibration, 180(1995), [3] V. Panc, Theories of Elastic Plates, Noordhoff International Publishing, Leden, [4] R. Szilard, Theories and Applications of Plate Analsis: Classical, Numerical and Engineering ethods, John Wile & Sons, Inc., Ne Jerse, 004.

9 Note on mathematical development of plate theories 55 [5] S.P. Timoshenko and S. Woinosk-Krieger, Theor of Plates and Shells. nd ed., cgra-hill, Singapore, Received: December, 014; Published: Januar 19, 015