Corner stress singularities in a high-order plate theory
C.S. Huang
*Department of Civil Engineering, National Chiao, Tung University, 1001 Ta-Hsueh Road, Hsinchu 30050, Taiwan Received 25 July 2003; accepted 18 April 2004
Available online 11 June 2004
Abstract
In the context of Lo’s high-order plate theory, the present work applies the eigenfunction expansion approach to investigating the Williams-type stress singularities at the vertex of a wedge. The characteristic equations for determining the orders of singularities in stress resultants are separately developed for plates under extension and bending. The characteristic equations of plates under extension differ from those in generalized plane stress cases when the clamped boundary condition is imposed along one of the radial edges around the vertex. For plates under bending, the presented characteristic equations are identical to those of first-order shear deformation plate theory (FSDPT) if the clamping is not involved in boundary conditions along the radial edges of the vertex. The orders of singularities in stress resultants, which vary with the vertex angle, are plotted for various types of boundary conditions. The results are also compre-hensively compared with those obtained according to other plate theories such as classical plate theory, FSDPT and Reddy’s refined plate theory.
2004 Elsevier Ltd. All rights reserved.
Keywords: Corner stress singularities; High-order plate theory; Isotropic plate; Eigenfunction expansion; Extension; Bending
1. Introduction
Plates are widely used components in engineering applications in civil engineering, mechanical engineer-ing, and aerospace engineering. The plate problem is a three-dimensional problem, but several plate theories have been proposed to simplify the three-dimensional problem into a two-dimensional one. Various plate theories were thoroughly reviewed in [1]. Re-entrant corners, at which stress singularities exist, are often encountered in analyses of plate problems. Although stress singularities are not of the real world, the exact nature of the singularities must be considered in a numerical solution to obtain an accurate and effective solution [2,3]. For example, Leissa et al. [4,5] analyzed free vibrations of circular sectorial plates with re-entrant corners or V-notches using the Ritz method, by
intro-ducing the so-called corner functions into the admissible functions to describe accurately the singular behaviors of thin plates. In the finite element approach, singular elements [6,7] are conventionally used to solve the problems with stress singularities by describing the exact order of stress singularities.
Many papers have addressed stress singularities at sharp corners in planar and three-dimensional elasticity theories (i.e., [8–12]), but only a few have considered corner stress singularities in various plate theories. According to the classical thin plate theory (CPT), Williams and his co-workers [13–15] first used the ei-genfunction expansion approach to comprehensively investigate the corner stress singularities induced by homogeneous boundary conditions around a corner for isotropic plates and orthotropic plates. Rao [16] also applied the eigenfunction expansion technique to iden-tify stress singularities at the interface corners of bi-material thin plates. Huang et al. [17] studied corner stress singularities for a sector plate with simply sup-ported radial edges by finding the closed-form solution
*
Tel.: +886-3-5712121x54962; fax: +886-3-5716257. E-mail address:cshuang@mail.nctu.edu.tw(C.S. Huang).
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for the vibration of such a plate. Sinclair [18] elucidated the occurrence of logarithmic stress singularities in angular thin plates. Using the classical lamination the-ory, Ojikutu et al. [19] applied a finite difference scheme to determine the orders of stress singularities in a com-posite wedge.
It is well known that shear deformation and rotary inertial effects have to be considered for a moderately thick plate. Within the context of first-order shear deformation plate theory (FSDPT), Burton and Sinclair [20] first examined the Williams-type corner stress sin-gularities by introducing a stress potential. Huang et al. [21] elucidated corner stress singularities in the closed-form solution for the vibration of a sector plate with simply supported radial edges. The results of Huang et al. [21] indicate the incompleteness of the solution of Burton and Sinclair [20], whose solution did not include singularities of shear forces. Recently, Huang [22] rein-vestigated corner stress singularities by adopting Xie and Chaudhuri’s approach [12] to solve the equilibrium equations in terms of displacement components, and obtained the orders of moment and shear force singu-larities identical to those obtained by Huang et al. [21] for the case of a corner with simply supported boundary conditions.
Although first-order shear deformation plate theory has been frequently applied to analyze thick plate problems, FSDPT has some limitations: the transverse shear strains are assumed to be constant through the thickness of the plate and a shear correction coefficient is needed. Some high-order plate theories (HPT) have been offered [1]. However, up to now, only one paper has considered the corner stress singularities in HPT. Huang [23] investigated the corner stress singularities by applying Reddy’s refined plate theory [24], neglecting in-plane displacement of the mid-in-plane; and revealed that different plate theories (CPT, FSDPT and Reddy’s plate theory) yields very different orders of stress singularities at a corner, under a fixed set of boundary conditions. Notably, Reddy’s refined plate theory is equivalent to the high-order plate theories proposed by Schmidt [25] and Krishna Murty [26].
There are other high-order plate theories that are not equivalent to Reddy’s. Bert [27] critically evaluated of various plate theories for nonlinear bending stress dis-tribution and concluded that the plate theory of Lo et al. [28] offered a very accurate prediction. Furthermore, this high-order plate theory has also been used to study various types of plate problems (i.e., [29–31]). This the-ory differs from CPT, FSDPT and Reddy’s plate thethe-ory not only in the assumed displacement fields, but also in the relations between strains and stresses. Unlike the generalized plane stress assumption of zero normal stress in the thickness direction used in CPT, FSDPT and Reddy’s plate theory, Lo’s theory uses the three-dimensional Hooke’s law.
The aim of this work is to derive the characteristic equations to determine the orders of Williams-type stress singularities under various boundary conditions around a corner, by applying Lo’s high-order plate theory. The characteristic equations are derived by adopting the approach of Hartranft and Sih [9] for three-dimensional elasticity problems. This paper investigates not only the singular behaviors of moments and shear forces but also the singular behaviors of in-plane forces. Notably, the singular behaviors of in-in-plane forces in FSDPT and Reddy’s plate theory are the same as those for plane stress theory, so they were not dis-cussed in [22,23]. The variations of the orders of singu-larities in stress resultants with the vertex angle of a wedge are graphically represented for different boundary conditions around the vertex. The current results are also comprehensively compared with those for CPT, FSDPT and Reddy’s refined plate theory. The obtained singular behaviors may be used when sharp corners are involved to solve free vibration, static deflection, stress intensity and buckling problems for thick plates if Lo’s high-order plate theory is applied.
2. Governing equations
As in Lo et al. [28] for Cartesian coordinates, the displacement field for a sector plate (or a wedge) with cylindrical coordinates as shown in Fig. 1 is given as uðr; h; zÞ ¼ uðr; hÞ þ zwrðr; hÞ þ z 2n rðr; hÞ þ z3/rðr; hÞ; ð1aÞ vðr; h; zÞ ¼ vðr; hÞ þ zwhðr; hÞ þ z 2 nhðr; hÞ þ z 3 /hðr; hÞ; ð1bÞ wðr; h; zÞ ¼ wðr; hÞ þ zwzðr; hÞ þ z 2n zðr; hÞ; ð1cÞ
Fig. 1. Coordinate system and positive displacement compo-nents for a wedge.
where u; v, and w are the displacement components in the r, h, and z directions, respectively. This displacement field includes both in-plane and out-of-plane modes of deformation. There are 11 r and h dependent displace-ment functions, namely, u; v; w; wr;wh;wz;nr;nh;nz;/r,
and /h.
The principle of stationary potential energy is used to derive equilibrium equations through a variational ap-proach. Without assuming generalized plane stress (i.e.,rzz¼ 0), the three-dimensional Hooke’s law for
isotropic material is used. Eleven equilibrium equations for the 11 displacement functions can be found as fol-lows, without external loading:
Nr;rþ Nrh;h=rþ ðNr NhÞ=r ¼ 0; ð2Þ Nrh;rþ Nh;h=rþ 2Nrh=r¼ 0; ð3Þ Qr;rþ Qh;h=rþ Qr=r¼ 0; ð4Þ Mr;rþ Mrh;h=rþ ðMr MhÞ=r Qr¼ 0; ð5Þ Mrh;rþ Mh;h=rþ 2Mrh=r Qh¼ 0; ð6Þ Rr;rþ Rh;h=rþ Rr=r Nz¼ 0; ð7Þ Pr;rþ Prh;h=rþ ðPr PhÞ=r 2Rr¼ 0; ð8Þ Prh;rþ Ph;h=rþ 2Prh=r 2Rh¼ 0; ð9Þ Sr;rþ Sh;h=rþ Sr=r 2Mz¼ 0; ð10Þ Mr;rþ Mrh;h=rþ ðMr MhÞ=r 3Sr¼ 0; ð11Þ Mrh;rþ Mh;h=rþ 2Mrh=r 3Sh¼ 0; ð12Þ
where the subscript, ‘‘a,’’ denotes a partial differential with respect to the independent variable a. In addition, 11 boundary conditions for the r- and h-constant edges can also be derived. Along a radial edge, one member of each of the following 11 products must be prescribed: uNrh, vNh, wQh, wrMrh, whMh, wzRh, nrPrh, nhPh, nzSh,
/rMrh, and /hMh. For the r-constant edge, one member
of each of the following 11 products must also be pre-scribed: uNr, vNrh, wQr, wrMr, whMrh, wzRr, nrPr, nhPrh,
nzSr, /rMr, and /hMrh. Notably, the governing
equa-tions can also be obtained by multiplying the well-known three-dimensional equilibrium equations in continuum mechanics with 1, z, z2, or z3, and integrating
them with respect to z. To the author’s knowledge, these equilibrium equations (Eqs. (2)–(12)) have not been shown before.
The resultant forces in Eqs. (2)–(12) are defined as follows: Nr Nh Nz Nrh Qr Qh Mr Mh Mz Mrh Rr Rh ¼ Z h=2 h=2 1 z rrr rhh rzz rrh rrz rhz ð Þ dz; ð13aÞ Pr Ph Prh Mr Mh Mrh ¼ Z h=2 h=2 z2 z3 rrr rhh rrh ð Þ dz; ð13bÞ Sr Sh ð Þ ¼ Z h=2 h=2 z2 r rz rhz ð Þ dz; ð13cÞ
where rijare stress components. From Eqs. (13a–c) and
using the three-dimensional Hooke’s law for isotropic material and strain–displacement relations for infinites-imal deformation, we can obtain the following relations between the resultant forces and the displacement functions: Nr¼ h½ðk þ 2GÞu;rþ kðu þ v;hÞ=r þ kwz þh 3 12½ðk þ 2GÞnr;rþ kðnrþ nh;hÞ=r; ð14aÞ Nh¼ h½ku;rþ ðk þ 2GÞðu þ v;hÞ=r þ kwz þh 3 12½knr;rþ ðk þ 2GÞðnrþ nh;hÞ=r; ð14bÞ Nz¼ h½ku;rþ kðu þ v;hÞ=r þ ðk þ 2GÞwz þh 3 12½knr;rþ kðnrþ nh;hÞ=r; ð14cÞ Nrh¼ Gh½v;r ðv u;hÞ=r þ Gh3 12 ½nh;r ðnh nr;hÞ=r; ð14dÞ Qr¼ Ghðwrþ w;rÞ þ Gh3 12 ð3/rþ nz;rÞ; ð14eÞ Qh¼ Ghðwhþ w;h=rÞ þ Gh3 12 ð3/hþ nz;h=rÞ; ð14fÞ Mr¼ h3 12½ðk þ 2GÞwr;rþ kðwrþ wh;hÞ=r þ 2knz þh 5 80½ðk þ 2GÞ/r;rþ kð/rþ /h;hÞ=r; ð14gÞ Mh¼ h3 12½kwr;rþ ðk þ 2GÞðwrþ wh;hÞ=r þ 2knz þh 5 80½k/r;rþ ðk þ 2GÞð/rþ /h;hÞ=r; ð14hÞ
Mz¼ h3 12½kwr;rþ kðwrþ wh;hÞ=r þ 2ðk þ 2GÞnz þh 5 80½k/r;rþ kð/rþ /h;hÞ=r; ð14iÞ Mrh¼ Gh3 12 ½ðwr;h whÞ=r þ wh;r þGh 5 80 ½ð/r;h /hÞ=r þ /h;r; ð14jÞ Rr¼ Gh3 12 ð2nrþ wz;rÞ; ð14kÞ Rh¼ Gh3 12 ð2nhþ wz;h=rÞ; ð14lÞ Pr¼ h3 12½ðk þ 2GÞu;rþ kðu þ v;hÞ=r þ kwz þh 5 80½ðk þ 2GÞnr;rþ kðnrþ nh;hÞ=r; ð14mÞ Ph¼ h3 12½ku;rþ ðk þ 2GÞðu þ v;hÞ=r þ kwz þh 5 80½knr;rþ ðk þ 2GÞðnrþ nh;hÞ=r; ð14nÞ Prh¼ Gh3 12 ½v;r ðv u;hÞ=r þ Gh5 80 ½nh;r ðnh nr;hÞ=r; ð14oÞ Mr¼ h5 80½ðk þ 2GÞwr;rþ kðwrþ wh;hÞ=r þ 2knz þ h 7 448½ðk þ 2GÞ/r;rþ kð/rþ /h;hÞ=r; ð14pÞ Mh¼ h5 80½kwr;rþ ðk þ 2GÞðwrþ wh;hÞ=r þ 2knz þ h 7 448½k/r;rþ ðk þ 2GÞð/rþ /h;hÞ=r; ð14qÞ Mrh¼ Gh5 80 ½ðwr;h whÞ=r þ wh;r þGh 7 448½ð/r;h /hÞ=r þ /h;r; ð14rÞ Sr¼ Gh3 12 ðwrþ w;rÞ þ Gh5 80 ð3/rþ nz;rÞ; ð14sÞ Sh¼ Gh3 12 ðwhþ w;h=rÞ þ Gh5 80 ð3/hþ nz;h=rÞ; ð14tÞ where G is the shear modulus, k is one of the Lame’s constants, and h is the thickness of the plate.
Substituting Eqs. (14a)–(14t) into Eqs. (2)–(12) with careful arrangement yields the equilibrium equations in terms of displacement functions:
u;rrþ u;r r u r2þ G ð2G þ kÞ u;hh r2 þ Gþ k 2Gþ k v;rh r 3Gþ k 2Gþ k v;h r2 þ k 2Gþ kwz;rþ h2 12 nr;rr þnr;r r nr r2þ G 2Gþ k nr;hh r2 þ Gþ k 2Gþ k nh;rh r 3Gþ k 2Gþ k nh;h r2 ¼ 0; ð15Þ v;rrþ v;r r v r2þ 2Gþ k G v;hh r2 þ kþ G G u;rh r þ 3Gþ k G u;h r2 þk G wz;h r þ h2 12 nh;rr þnh;r r nh r2þ 2Gþ k G nh;hh r2 þkþ G G nr;rh r þ 3Gþ k G nr;h r2 ¼ 0; ð16Þ w;rrþ w;r r þ w;hh r2 þ wr;rþ wr r þ wh;h r þ h2 12 nz;rr þnz;r r þnz;hh r2 þ 3 /r;r þ/r r þ /h;h r ¼ 0; ð17Þ 12 h2 G 2Gþ kðw;rþ wrÞ þ wr;rrþ wr;r r wr r2þ G 2Gþ k wr;hh r2 þ Gþ k 2Gþ k wh;rh r 3Gþ k 2Gþ k wh;h r2 þ G þ 2k 2Gþ k nz;r 3G 2Gþ k/rþ 3h2 20 /r;rr þ/r;r r /r r2þ G 2Gþ k /r;hh r2 þ Gþ k 2Gþ k /h;rh r 3Gþ k 2Gþ k /h;h r2 ¼ 0; ð18Þ 12 h2 w;h r þ whþ wh;rrþwh;r r wh r2 þ 2Gþ k G wh;hh r2 þGþ k G wr;rh r þ 3Gþ k G wr;h r2 þ G þ 2k G nz;h r 3/h þ3h 2 20 /h;rr þ/h;r r /h r2 þ 2Gþ k G /h;hh r2 þ Gþ k G /r;rh r þ3Gþ k G /r;h r2 ¼ 0; ð19Þ k G u;r þu rþ v;h r þ2Gþ k G wz h2 12 wz;rr þwz;r r þ wz;hh r2 þ2G k G nr;r þnr r þ nh;h r ¼ 0; ð20Þ
u;rrþ u;r r u r2þ G 2Gþ k u;hh r2 þ Gþ k 2Gþ k v;rh r 3Gþ k 2Gþ k v;h r2 2G k 2Gþ kwz;r 4G 2Gþ knrþ 3h2 20 nr;rr þnr;r r nr r2 þ G 2Gþ k nr;hh r2 þ Gþ k 2Gþ k nh;rh r 3Gþ k 2Gþ k nh;h r2 ¼ 0; ð21Þ v;rrþ v;r r v r2þ 2Gþ k G v;hh r2 þ Gþ k G u;rh r þ 3Gþ k G u;h r2 þ2G þ k G wz;h r 4nhþ 3h2 20 nh;rr þnh;r r nh r2 þ2Gþ k G nh;hh r2 þ Gþ k G nr;rh r þ 3Gþ k G nr;h r2 ¼ 0; ð22Þ w;rrþ w;r r þ w;hh r2 þ G 2k G wr;r þwr r þ wh;h r 4ð2G þ kÞ G nzþ 3h2 20 nz;rr þnz;r r þ nz;hh r2 þ3G 2k G /r;r þ/r r þ /h;h r ¼ 0; ð23Þ 20G h2ð2G þ kÞðw;rþ wrÞ þ wr;rrþ wr;r r wr r2þ G 2Gþ k wr;hh r2 þ Gþ k 2Gþ k wh;rh r 3Gþ k 2Gþ k wh;h r2 3G 2k 2Gþ k nz;r 9G 2Gþ k/rþ 5h2 28 /r;rr þ/r;r r /r r2þ G 2Gþ k /r;hh r2 þ Gþ k 2Gþ k /h;rh r 3Gþ k 2Gþ k /h;h r2 ¼ 0; ð24Þ 20 h2 w;h r þ wh þ wh;rrþ wh;r r wh r2 þ 2Gþ k G wh;hh r2 þGþ k G wr;rh r þ 3Gþ k G wr;h r2 þ 3G þ 2k G nz;h r 9/h þ5h 2 28 /h;rr þ/h;r r /h r2 þ 2Gþ k G /h;hh r2 þ Gþ k G /r;rh r þ3Gþ k G /r;h r2 ¼ 0; ð25Þ
The above equations were developed with the aid of the symbolic logic software program MATHMATICA.
Note that Eqs. (15)–(25) partially decouple such that the equations can be combined into two groups. One group comprises Eqs. (15), (16), (20), (21), and (22) for the displacement functions u; v; wz;nr, and nh, which are
related to the extension of the middle plane of the plate. The second group consists of the other equations, which describe the behavior of a plate under bending. Conse-quently, in the following, we investigate the stress sin-gularities for these two groups separately.
3. Stress singularities for plates under extension
Here, a plate under extension means that in-plane loading is applied to the plate and may cause the extension of the middle plane of the plate. Some of Eqs. (15), (16), (20), (21), and (22) can be further simplified through some mathematic manipulations. From Eqs. (15) and (21), we can obtain
2G 2Gþ kwz;rþ 4G 2Gþ knr h2 15 nr;rr þnr;r r nr r2 þ G 2Gþ k nr;hh r2 þ Gþ k 2Gþ k nh;rh r 3Gþ k 2Gþ k nh;h r2 ¼ 0: ð26Þ From Eqs. (16) and (22), the following equation can be obtained 2 rwz;hþ 4nh h2 15 nh;rr þnh;r r nh r2þ 2Gþ k G nh;hh r2 þkþ G G nr;rh r þ 3Gþ k G nr;h r2 ¼ 0: ð27Þ
Consequently, the stress singularities for plates under extension will be investigated by solving Eqs. (15), (16), (20), (26), and (27) and employing some combinations of boundary conditions along radial edges.
3.1. Construction of the series solution
An eigenfunction expansion approach similar to that used in [23] is applied to solve Eqs. (15), (16), (20), (26), and (27). Let u¼X 1 m¼0 X1 n¼0;2 rkmþnUðmÞ n ðh; kmÞ; v¼X 1 m¼0 X1 n¼0;2 rkmþnVðmÞ n ðh; kmÞ; nr¼X 1 m¼0 X1 n¼0;2 rkmþnXðmÞ rn ðh; kmÞ; nh¼ X1 m¼0 X1 n¼0;2 rkmþnXðmÞ hnðh; kmÞ; and wz¼X 1 m¼0 X1 n¼0;2 rkmþnþ1WðmÞ zn ðh; kmÞ; ð28Þ
where the characteristic values km are assumed to be
constants and can be complex numbers. Notably, odd n in Eq. (28) will not give any additional solutions; therefore, they are excluded.
The regularity conditions at r¼ 0 require u, v, and w to be finite; therefore, the real part of km has to exceed
zero. Consequently, the relations between stress resul-tants and displacement functions given in Eqs. (14a)– (14d) and (14k)–(14o) reveal that the solution given by Eq. (28) yields the singularities for Nr, Nh, Nz, Nrh, Pr, Ph,
and Prh at r¼ 0 when the real part of km is below one.
Nevertheless, no singularities occur for the stress resul-tants Rr and Rh.
Substituting Eq. (28) into Eqs. (15), (16), (20), (26), and (27) and requiring that the coefficients of r with different orders equal zero yield the following recurrent equations for UðmÞ n , VnðmÞ, X ðmÞ rn , X ðmÞ hn, and W ðmÞ zn: h2 15 ðkm þ n þ 3Þðkmþ n þ 1ÞXðmÞrðnþ2Þþ G 2Gþ kX ðmÞ rðnþ2Þ;hh þ Gþ k 2Gþ kðkmþ n þ 2ÞX ðmÞ hðnþ2Þ;h 3Gþ k 2Gþ kX ðmÞ hðnþ2Þ;h ¼ 2G 2Gþ kðkmþ n þ 1ÞW ðmÞ zn þ 4G 2Gþ kX ðmÞ rn ; ð29Þ h2 15 ðkm þ n þ 3Þðkmþ n þ 1ÞXðmÞhðnþ2Þþ 2Gþ k G X ðmÞ hðnþ2Þ;hh þGþ k G ðkmþ n þ 2ÞX ðmÞ rðnþ2Þ;hþ 3Gþ k G X ðmÞ rðnþ2Þ;h ¼ 2Wn ðmÞzn;hþ 4XðmÞhno; ð30Þ ðkmþ n þ 3Þðkmþ n þ 1ÞU ðmÞ nþ2þ G 2Gþ kU ðmÞ nþ2;hh þðG þ kÞðkmþ n þ 2Þ ð3G þ kÞ 2Gþ k V ðmÞ nþ2;h þh 2 12 ðkm þ n þ 3Þðkmþ n þ 1ÞXðmÞrðnþ2Þ þ G 2Gþ kX ðmÞ rðnþ2Þ;hh þðG þ kÞðkmþ n þ 2Þ ð3G þ kÞ 2Gþ k X ðmÞ hðnþ2Þ;h ¼ k 2Gþ kðkmþ n þ 1ÞW ðmÞ zn ; ð31Þ ðkmþ n þ 3Þðkmþ n þ 1ÞV ðmÞ nþ2þ 2Gþ k G V ðmÞ nþ2;hh þðG þ kÞðkmþ n þ 2Þ þ ð3G þ kÞ G U ðmÞ nþ2;h þh 2 12 ðkm þ n þ 3Þðkmþ n þ 1ÞXðmÞhðnþ2Þ þ2Gþ k G X ðmÞ hðnþ2Þ;hh þðG þ kÞðkmþ n þ 2Þ þ ð3G þ kÞ G X ðmÞ rðnþ2Þ;h ¼ k GW ðmÞ zn;h; ð32Þ k G½ðkmþ n þ 3ÞU ðmÞ nþ2þ V ðmÞ nþ2;h h2 12 ðkm þ n þ 3Þ2 WðmÞzðnþ2Þþ WðmÞzðnþ2Þ;hhþ2G k G ½ðkmþ n þ 3ÞX ðmÞ rðnþ2Þ þ XðmÞhðnþ2Þ;h ¼ 2Gþ k G W ðmÞ zn ; ð33Þ and G 2Gþ kX ðmÞ r0;hhþ ðkmþ 1Þðkm 1ÞX ðmÞ r0 þðG þ kÞkm ð3G þ kÞ 2Gþ k X ðmÞ h0;h¼ 0; ð34Þ 2Gþ k G X ðmÞ h0;hhþ ðkmþ 1Þðkm 1ÞXðmÞh0 þðG þ kÞkmþ ð3G þ kÞ G X ðmÞ r0;h¼ 0; ð35Þ G 2Gþ kU ðmÞ 0;hhþ ðkmþ 1Þðkm 1ÞU ðmÞ 0 þðG þ kÞkm ð3G þ kÞ 2Gþ k V ðmÞ 0;h þh 2 12 G 2Gþ kX ðmÞ r0;hh þ ðkmþ 1Þðkm 1ÞXðmÞr0 þðG þ kÞkm ð3G þ kÞ 2Gþ k X ðmÞ h0;h ¼ 0; ð36Þ 2Gþ k G V ðmÞ 0;hhþ ðkmþ 1Þðkm 1ÞV0ðmÞ þðG þ kÞkmþ ð3G þ kÞ G U ðmÞ 0;h þh 2 12 2Gþ k G X ðmÞ h0;hh þ ðkmþ 1Þðkm 1ÞX ðmÞ h0 þðG þ kÞkmþ ð3G þ kÞ G X ðmÞ r0;h ¼ 0; ð37Þ k G ðkm n þ 1ÞU0ðmÞþ V ðmÞ 0;h o h 2 12 W ðmÞ z0;hh þ ðkmþ 1Þ 2 WðmÞz0 þ2G k G ½X ðmÞ h0;hþ ðkmþ 1ÞXðmÞr0 ¼ 0: ð38Þ Through lengthy and tedious mathematical operations, one can find the general solutions for Eqs. (34)–(38) given as follows:
U0ðmÞ¼ A1cosðkmþ 1Þh þ A2sinðkmþ 1Þh
þ A3cosðkm 1Þh þ A4sinðkm 1Þh; ð39Þ
V0ðmÞ¼ A2cosðkmþ 1Þh A1sinðkmþ 1Þh
þ k1½A4cosðkm 1Þh A3sinðkm 1Þh; ð40Þ
XðmÞr0 ¼ C1cosðkmþ 1Þh þ C2sinðkmþ 1Þh
XðmÞh0 ¼ C2cosðkmþ 1Þh C1sinðkmþ 1Þh
þ k1½C4cosðkm 1Þh C3sinðkm 1Þh; ð42Þ
WðmÞz0 ¼ E1cosðkmþ 1Þh þ E2sinðkmþ 1Þh
þ k2½A3cosðkm 1Þh þ A4sinðkm 1Þh
þ k3½C3cosðkm 1Þh þ C4sinðkm 1Þh; ð43Þ where k1¼3þk3þkm4t mþ4t, k2¼ 24t h2ð3þk mþ4tÞ, k3¼ 2ð13tÞ 3þkmþ4t, and t
is the Poisson’s ratio. In Eqs. (39)–(43), the character-istic value km and the coefficients E1; E2; Ai, and Ci
(i¼ 1; 2; 3; 4) are determined by satisfying the boundary conditions along radial edges.
UðmÞ n , VnðmÞ, X ðmÞ rn , X ðmÞ hn, and W ðmÞ
zn for n P 1 are
deter-mined by solving Eqs. (29)–(33). However, these solu-tions are not related to the singularities of the stress resultants, so they will not be considered further here.
Notably, one can also start to construct the series solution by assuming the following form of the general solutions for Eqs. (15), (16), (20), (26), and (27):
u¼X 1 m¼0 X1 n¼0;2 rkmþnþl1UðmÞ n ðh; kmÞ; v¼X 1 m¼0 X1 n¼0;2 rkmþnþl2VðmÞ n ðh; kmÞ; nr¼X 1 m¼0 X1 n¼0;2 rkmþnþl3XðmÞ rn ðh; kmÞ; nh¼ X1 m¼0 X1 n¼0;2 rkmþnþl4XðmÞ hnðh; kmÞ; wz¼X 1 m¼0 X1 n¼0;2 rkmþnþl5WðmÞ zn ðh; kmÞ; ð44Þ
where li ði ¼ 1; 2; . . . ; 5Þ are either one or zero.
Follow-ing the above procedure of constructFollow-ing the solutions, one discovers that the solution form given by Eq. (28) is the only one resulting in the Williams-type singularities of stress resultants.
3.2. Characteristic equations and singularities for stress resultants
As mentioned earlier, the characteristic values, km,
are determined by satisfying the boundary conditions
along the radial edges of a corner. Two types of boundary conditions along a radial edge are considered here:
free: Nh; Nrh; Rh; Ph; and Prh¼ 0; ð45aÞ
clamped: u; v;nr; nh; and wz¼ 0: ð45bÞ
These two cases can be combined to give three distinct problems concerning a wedge.
Substituting Eqs. (39)–(43) and Eq. (28), with n¼ 0, into Eqs. (45a) or (45b) with h¼ 0 and a, where a is the vertex angle of the wedge, yields the vanishing deter-minant of a 10· 10 coefficient matrix. Then, the char-acteristic equations for km can be established. Notably,
when the same boundary conditions are imposed along the two radial edges, the symmetry of the problem can be taken advantage of, by considering a=2 6 h 6 a=2, and separately considering the symmetric and anti-symmetric parts of the solution given in Eqs. (39)–(43). For example, considering the symmetric behavior of a wedge with two fixed radial edges, Eqs. (39)–(43), with A2¼ A4¼ C2¼ C4¼ E2¼ 0, are substituted into Eq.
(45b), the determinant of the resulting 5· 5 coefficient matrix is expanded, and
sin kma km 3 4tsin a cosðkmþ 1Þa=2 ¼ 0 ð46Þ
is obtained. However, cosðkmþ 1Þa=2 ¼ 0 yields
A1¼ A3¼ C1¼ C3¼ 0 and does not give any stress
singularities at the vertex of the wedge. Accordingly, it is discarded. Table 1 lists the characteristic equations for three possible combinations of boundary conditions along the two radial edges. These characteristic equa-tions are not related to the thickness of the plate. Pois-son’s ratio is the only material property involved in these characteristic equations.
For comparison, Table 1 also summarizes the char-acteristic equations given by Williams [8] for generalized plane stress in thin plates. The characteristic equations in this work differ from Williams’ when the clamped boundary condition is applied, because Lo’s plate theory does not use the generalized plane stress condition. Note that if t in the characteristic equations presented herein is replaced by t=1þ t, then the resulting characteristic equations are identical to Williams’.
Table 1
Characteristic equations for plates under extension
Case no. Boundary conditions Present Thin plate theory [8]
1 Free–free sin kma¼ kmsin a sin kma¼ kmsin a
2 Clamped–clamped sin kma¼ 3þ4tkm sin a sin kma¼ ð3þt1þtÞkmsin a
3 Clamped–free sin2k ma¼4ð1þtÞ 2 3þ4t þ k2 m 3þ4tsin 2a sin2k ma¼ð3tÞð1þtÞ4 ð1þt3tÞk2msin 2a
Fig. 2 shows the smallest positive real part of km
varies with the vertex angleðaÞ for three combinations of boundary conditions after the roots of the characteristic equations given in Table 1 are numerically determined. The results were computed for t¼ 0:3. In the legend of Fig. 2, ‘‘C’’ and ‘‘F’’ denote clamped and free boundary conditions, respectively.
Fig. 2 depicts that no singularities exist for Nr, Nh, Nz,
Nrh, Pr, Ph, and Prh when the vertex angle is less than
approximately 57. However, singularities always arise for a > 180, regardless of which of the three combina-tions of boundary condicombina-tions around the vertex is con-sidered. Among the three combinations of boundary conditions considered here, boundary condition C_F provides the strongest singularity. As the vertex angle approaches 2p (i.e., a crack), the singular order of stress resultants at the tip of the crack approaches r3=4for the
boundary condition C_F, while the boundary conditions C_C and F_F yield an order of r1=2. Comparing the
present results with Williams’ reveals that the orders of stress singularities associated with Lo’s plate theory are very close to those according to generalized plane stress theory, except those that involve the boundary condition C_F with approximately a < 142.
4. Stress singularities for plates in bending
Recall that Eqs. (17)–(19) and (23)–(25) are for plates under bending. Through mathematical manipulation of these equations, we can obtain the following equa-tions, which are simpler than Eqs. (17)–(19) and (23)– (25): 2k G wr;r þwr r þ wh;h r þ4ð2G þ kÞ G nz h 2 15 nz;rr þnz;r r þ nz;hh r2 þh 2 20 4G þ 6k G /r;r þ/r r þ /h;h r ¼ 0; ð47Þ 8G h2ð2G þ kÞðw;rþ wrÞ þ 2G 2Gþ knz;rþ 6G 2Gþ k/r h 2 35 /r;rr þ/r;r r /r r2 þ G 2Gþ k /r;hh r2 þ Gþ k 2Gþ k /h;rh r 3Gþ k 2Gþ k /h;h r2 ¼ 0; ð48Þ 8 h2 w;h r þ wh þ2nz;h r þ 6/h h2 35 /h;rr þ/h;r r /h r2 þ2Gþ k G /h;hh r2 þ Gþ k G /r;rh r þ 3Gþ k G /r;h r2 ¼ 0; ð49Þ 6 7h2 w;h r þ wh þ 19 70 þ 2k 35G nz;h r þ 57 70/h þ 1 35 /h;rr þ/h;r r /h r2 þ 2Gþ k G /h;hh r2 þ Gþ k G /r;rh r þ3Gþ k G /r;h r2 ¼ 0; ð50Þ 6G 7h2ð2G þ kÞðw;rþ wrÞ þ 19G 70 þ2k 35 1 2Gþ knz;r þ 57G 70ð2G þ kÞ/rþ 1 35 /r;rr þ/r;r r /r r2 þ G 2Gþ k /r;hh r2 þ Gþ k 2Gþ k /h;rh r 3Gþ k 2Gþ k /h;h r2 ¼ 0; ð51Þ 1 15 w;rr þw;r r þ w;hh r2 þ 1 15 þ k 6G wr;r þwr r þ wh;h r þ2Gþ k 3G nzþ h2k 40G /r;r þ/r r þ /h;h r ¼ 0: ð52Þ
For example, Eq. (47) was obtained by subtracting Eq. (23) from Eq. (17), while Eq. (48) was obtained by subtracting Eq. (24) from Eq. (18).
4.1. Construction of the series solution
To establish the series solution for Eqs. (47)–(52) using the eigenfunction expansion approach, the fol-lowing solution form is used:
Fig. 2. Variation of minimum ReðkmÞ with vertex angle (extension).
w¼X 1 m¼0 X1 n¼0;2 rkmþnþ1WðmÞ n ðh; kmÞ; nz¼ X1 m¼0 X1 n¼0;2 rkmþnþ1XðmÞ zn ðh; kmÞ; wr¼X 1 m¼0 X1 n¼0;2 rkmþnWðmÞ rnðh; kmÞ; wh¼X 1 m¼0 X1 n¼0;2 rkmþnWðmÞ hnðh; kmÞ; /r¼ X1 m¼0 X1 n¼0;2 rkmþnUðmÞ rn ðh; kmÞ; /h¼ X1 m¼0 X1 n¼0;2 rkmþnUðmÞ hnðh; kmÞ; ð53Þ
where the characteristic values kmare again assumed to
be constants and can be complex numbers. It is also noted that odd n in the above equation will not produce any additional solution, so they are not included.
The real part of km must exceed zero to satisfy the
regularity conditions at the apex of the wedge. Notably,
kmwith a real part smaller than one leads to singularities
of Mr, Mh, Mz, Mrh, Mr, Mhand Mrhat the vertex but no
singularities for Qr, Qh, Sr and Sh, which can be
dis-covered from the relationships between these stress resultants and the displacement functions given in Eqs. (14e)–(14j) and (14p)–(14t).
Substituting Eq. (53) into Eqs. (47)–(52), the van-ishing of the coefficients corresponding to the smallest order in r for each equation results in
2k Gðkmþ 1ÞW ðmÞ r0 þ 2k GWh0;h h2 15ðX ðmÞ z0;hhþ ðkmþ 1Þ2XðmÞz0 Þ þh 2 20 4G þ 6k G ðkm n þ 1ÞUðmÞr0 þ UðmÞh0;ho¼ 0; ð54Þ G 2Gþ kU ðmÞ r0;hhþ ðkmþ 1Þðkm 1ÞU ðmÞ r0 þðG þ kÞkm ð3G þ kÞ 2Gþ k U ðmÞ h0;h¼ 0; ð55Þ 2Gþ k G U ðmÞ h0;hhþ ðkmþ 1Þðkm 1ÞUðmÞh0 þðG þ kÞkmþ ð3G þ kÞ G U ðmÞ r0;h¼ 0; ð56Þ G 2Gþ kW ðmÞ r0;hhþ ðkmþ 1Þðkm 1ÞWðmÞr0 þðG þ kÞkm ð3G þ kÞ 2Gþ k W ðmÞ h0;h¼ 0; ð57Þ 2Gþ k G W ðmÞ h0;hhþ ðkmþ 1Þðkm 1ÞWðmÞh0 þðG þ kÞkmþ ð3G þ kÞ G W ðmÞ r0;h¼ 0; ð58Þ 1 15 W ðmÞ 0;hh n þ ðkmþ 1Þ 2 W0ðmÞ o þ 1 15 þ k 6G WðmÞh0;h n þ ðkmþ 1ÞWðmÞr0 o þ h 2k 40G U ðmÞ h0;h n þ ðkmþ 1ÞUðmÞr0 o ¼ 0: ð59Þ The general homogeneous solutions for the above dif-ferential equations are
W0ðmÞ¼ eA1cosðkmþ 1Þh þ eA2sinðkmþ 1Þh
þ k6½eB3cosðkm 1Þh þ eB4sinðkm 1Þh
þ k7½eE3cosðkm 1Þh þ eE4sinðkm 1Þh; ð60Þ
WðmÞr0 ¼ eB1cosðkmþ 1Þh þ eB2sinðkmþ 1Þh
þ eB3cosðkm 1Þh þ eB4sinðkm 1Þh; ð61Þ
WðmÞh0 ¼ eB2cosðkmþ 1Þh eB1sinðkmþ 1Þh
þ k1½eB4cosðkm 1Þh eB3sinðkm 1Þh; ð62Þ
UðmÞr0 ¼ eE1cosðkmþ 1Þh þ eE2sinðkmþ 1Þh
þ eE3cosðkm 1Þh þ eE4sinðkm 1Þh; ð63Þ
UðmÞh0 ¼ eE2cosðkmþ 1Þh eE1sinðkmþ 1Þh
þ k1½eE4cosðkm 1Þh eE3sinðkm 1Þh; ð64Þ
XðmÞz0 ¼ eD1cosðkmþ 1Þh þ eD2sinðkmþ 1Þh
þ k4½eB3cosðkm 1Þh þ eB4sinðkm 1Þh
þ k5½eE3cosðkm 1Þh þ eE4sinðkm 1Þh; ð65Þ
where k4¼ h2ð3þ60tk mþ4tÞ, k5¼ 3ð15tÞ 3þkmþ4t, k6¼ 1þ3t 3þkmþ4t, and k7¼ 3th 2
4ð3þkmþ4tÞ. The characteristic value km and the
coefficients eA1, eA2, eD1, eD2, eBi, and eEi ði ¼ 1; 2; 3; 4Þ are
determined from the boundary conditions along the two radial edges of the wedge.
4.2. Characteristic equations
Four types of homogeneous boundary conditions along a radial edge are considered here to elucidate the stress singularities at the vertex of the wedge:
clamped: w¼ wr¼ wh¼ /r¼ /h¼ nz¼ 0; ð66aÞ
free: Qh¼ Mrh¼ Mh¼ Mrh¼ Mh¼ Sh¼ 0; ð66bÞ
type I simply supported: w¼ wr¼ /r¼ nz
¼ Mh¼ Mh¼ 0; ð66cÞ
type II simply supported: w¼ Mrh¼ Mh
For simplicity, in the following, the clamped and free boundary conditions are denoted by C and F, respec-tively, while type I and type II simply supported boundary conditions are represented by S(I) and S(II), respectively.
Substituting Eqs. (60)–(65) and Eq. (53) with n¼ 0 into the prescribed boundary conditions along two ra-dial edges yields twelve linear homogeneous algebraic equations in eA1, eA2, eD1, eD2, eBi, and eEi ði ¼ 1; 2; 3; 4Þ.
The vanishing determinant of the 12· 12 coefficient matrix from the twelve equations yields the character-istic equations for km.
Table 2 lists the characteristic equations for 10 com-binations of the boundary conditions. These character-istic equations are again not related to the thickness of the plate. Again, Poisson’s ratio is the only material property involved in these characteristic equations. Some boundary conditions yield the same characteristic equa-tions. The boundary conditions F_F, S(II)_S(II), and S(II)_F lead to the same characteristic equation, while the boundary conditions S(I)_F and S(I)_S(II) also share the same characteristic equation. The C_F boundary condition gives the same characteristic equation as does
Table 2
Characteristic equations for plates under bending
Case no. Boundary conditions Present Reddy’s refined plate theory
[23]
1 Simply supported (I)–simply supported (I) cos kma¼ cos a cos ka¼ cos aa;b
2 Clamped–free sin2k ma¼4ð1þtÞ 2 3þ4t þ k2 m 3þ4tsin 2a sin2k ma¼4 k2 mð1þtÞ2sin2a ð3tÞð1þtÞ a sin2k ma¼4 k2 mð1tÞ2sin2a ð3þtÞð1tÞ b
3 Simply supported (I)–free sin 2kma¼ kmsin 2a sin 2kma¼ kmsin 2aa
sin 2kma¼ kmð1tÞ
3t sin 2a b
4 Simply supported (I)–clamped sin 2kma¼
km 3þ4tsin 2a sin 2kma¼ kmð1þtÞ 3þt sin 2a a sin 2kma¼ kmsin 2ab
5 Free–free sin kma¼ kmsin a sin kma¼ kmsin aa
sin kma¼ kmð1tÞ 3t sin a b 6 Clamped–clamped sin kma¼ km 3þ4tsin a sin kma¼ kmð1þtÞ 3þt sin a a sin kma¼ kmsin ab 7 Simply supported (II)–simply supported (II) sin kma¼ kmsin a sin kma¼ kmsin aa cos kma¼ cos a
8 Clamped–simply supported (II) sin2k
ma¼4ð1þtÞ 2 3þ4t þ k2 m 3þ4tsin 2a sin2k ma¼4 k2 mð1þtÞ2sin2a ð3tÞð1þtÞ a sin 2kma¼ kmsin 2a 9 Simply supported (I)–simply supported (II) sin 2kma¼ kmsin 2a sin 2kma¼ kmsin 2aa
cos 2kma¼ cos 2a
10 Simply supported (II)–free sin kma¼ kmsin a sin kma¼ kmsin aa
sin 2kma¼ kmð1þtÞ
3þt sin 2a Note: ‘‘a’’ means that the equation can be recovered in FSDPT.
‘‘b’’ means that the equation can be regained in CPT.
Fig. 3. Variation of minimum ReðkmÞ with vertex angle (bending).
the C_S(II) boundary condition. Moreover, the charac-teristic equations for the bending and extension cases are the same for the boundary conditions C_F and C_C.
For comparison, Table 2 also summarizes the char-acteristic equations corresponding to various boundary conditions based on CPT [13], FSDPT [20,22] and Reddy’s plate theory [23]. Notably, the characteristic equations of Reddy’s plate theory include those of CPT and FSDPT, except for those related to the S(II) boundary condition, which does not apply in CPT. Comparing the characteristic equations in Table 2 indicates that different plate theories usually yield dif-ferent characteristic equations. Interestingly, the
char-acteristic equations obtained herein are exactly the same as those for FSDPT when the clamped boundary con-dition is not imposed along the radial edges. The boundary condition S(I)_S(I) results in the same char-acteristic equation for different plate theories.
4.3. Singularities of stress resultants
Fig. 3 displays the minimum positive real part of the characteristic value km, which varies with the vertex
angle of the wedge, a, and the boundary conditions. The results were determined for t¼ 0:3. Recall that kmwith a
real part less than one, leads to the singularities of Mr,
Fig. 4. Comparison of minimum ReðkmÞ for different plate theories: (a) for boundary conditions S(I)_F, S(I)_C and F_F, (b) for boundary conditions C_F and C_C, (c) for boundary conditions S(II)_S(II) and C_S(II) and (d) for boundary conditions S(I)_S(II) and S(II)_F.
Mh, Mz, Mrh, Mr, Mh and Mrh at the vertex. No
singu-larities arise for approximately a < 57, while singular-ities always exist for a > 180. The boundary conditions C_F and C_S(II) produce the strongest singularities at 57 < a < 109, while the S(I)_S(I) boundary condition leads to the strongest singularities for a P 109. The boundary condition C_C yields the weakest singularities among the considered boundary conditions.
Fig. 4a–d compare the minimum positive ReðkmÞ
obtained using different plate theories. Again, the results were computed for t¼ 0:3. Recall that the characteristic equations based on Reddy’s plate theory consist of those of CPT and FSDPT except for those related to the S(II) boundary condition. Hence, the smallest positive ReðkmÞ
for Reddy’s plate theory in Fig. 4a and b is the smaller of the characteristic values for CPT and FSDPT. The boundary condition S(II) does not occur in CPT, so Fig. 4c and d do not display the results for CPT.
The results in Fig. 4a reveal several important find-ings. Under the boundary condition S(I)_F, CPT and Reddy’s theory produce stronger singularities than does FSDPT or Lo’s high-order plate theory (referred to as HPT in the legend) for 90 < a < 180 and 270 < a < 360. Lo’s theory generates the strongest singularities for 90 < a < 180 and 270 < a < 360 in the case of the S(I)_C boundary condition. However, HPT and FSDPT generate very similar minimum positive Re(km) for the
boundary condition S(I)_C. For a wedge with two free radial edges, CPT produces weaker singularities at the vertex than does FSDPT, Reddy’s theory or HPT.
Fig. 4b indicates that under the C_F boundary con-dition, HPT yields a smaller positive Re(km) than the
other theories when a is below about 142, while all the theories generate almost identical orders of stress sin-gularities at other angles. More severe sinsin-gularities occur with CPT and Reddy’s theory than with FSDPT and HPT for the C_C boundary condition. Nevertheless, all the theories share the same orders of stress singularities when the vertex angle approaches 2p.
Fig. 4c and d present the minimum positive Re(km)
for boundary conditions involving S(II). Fig. 4c displays that the minimum positive Re(km) for Reddy’s theory
differs greatly from that for FSDPT and HPT under the S(II)_S(II) boundary condition. The former produces much stronger singularities than the latter. For the C_S(II) boundary condition, Reddy’s theory and FSDPT generate identical minimum positive Re(km)
except for 180 < a < 270. For the C_S(II) boundary condition, HPT produces stronger singularities than do FSDPT and Reddy’s theory for a below roughly 142, while HPT and FSDPT produce almost identical mini-mum positive Re(km) for other angles. Fig. 4d shows
that Reddy’s theory produces more severe singularities at the vertex than do FSDPT and HPT for the boundary conditions S(I)_S(II) and S(II)_F. Notably, when the vertex angle approaches 2p, the orders of stress
singu-larities obtained using Reddy’s theory are different from those obtained using FSDPT and HPT in the cases of the S(II)_S(II), S(II)_F, and S(I)_S(II) boundary con-ditions, while all the theories share the identical orders of stress singularities for the other boundary conditions.
5. Concluding remarks
This work presented an eigenfunction expansion approach to investigating corner singularities of stress resultants in thick plates based on Lo’s high-order plate theory. The singular behaviors of stresses at a sharp corner of a plate under extension and bending were thoroughly studied. The characteristic equations for determining the orders of stress singularities were de-rived for various combinations of boundary conditions around the vertex of a wedge. The thickness of the plate does not affect the orders of stress singularities, while Poisson’s ratio is the only material property that can possibly influence the singularity behavior.
For a wedge under extension, Lo’s high-order plate theory and the theory that uses the generalized plane stress assumption yield the same characteristic equation for the F_F boundary condition along the radial edges of the wedge, but they give different characteristic equations for the cases of the C_F and C_C boundary conditions. This difference may follow from the high-order plate theory’s using the three-dimensional Hoo-ke’s law for stress-strain relations. Nevertheless, these different characteristic equations for the C_C boundary condition produce very similar orders of stress singu-larities when the Poisson’s ratio is 0.3, and generate al-most identical singular orders for the C_F boundary condition when the vertex angle exceeds roughly 142.
For a plate (wedge) under bending, Lo’s high-order plate theory gives the same characteristic equations as FDSPT when the boundary conditions along the radial edges do not involve clamping. Generally, different plate theories, such as CPT, FSDPT, Reddy’s refined plate theory, and Lo’s high-order plate theory, yield different characteristic equations. However, these plate theories yields identical characteristic equations for the S(I)_S(I) boundary condition. For a plate with a Poisson’s ratio of 0.3, Lo’s theory does not generate stress singularities when the vertex angle is below 57, while a singularity always arises when the vertex angle exceeds p. More-over, the boundary conditions C_F and C_S(II) produce the strongest singularities for 57 < a < 109, while the S(I)_S(I) boundary condition leads to the strongest singularities for a P 109. Finally, it should be specially noted that Lo’s theory does not produce the kind of shear force singularity generated by CPT and FSDPT.
The characteristic equations and the numerical re-sults shown here are very important for dealing with thick plates having sharp corners when Lo’s theory is
applied. The stress singularities at the corners have to be appropriately taken into account to obtain accurate solution when such numerical techniques as the finite element method, the finite difference approach, and the Ritz method are used to solve complex plate problems with sharp corners. Notably, Lo’s theory may also produce logarithmic stress singularities, and therefore requires further research.
Acknowledgements
This work reported herein was supported by the National Science Council, R.O.C. through research grant no. NSC91-2211-E-009-038. The supported is gratefully acknowledged. The appreciation is also ex-tended to author’s graduate student, Mr. Y. H. Tsai, for preparing the figures shown in the paper.
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