This paper thoroughly examines the singularity of stress resultants of the form $r−ξFθ$ for $0<ξ⩽1$ as $r→0$ (Williams-type singularity) at the vertex of an isotropic thick plate; the singularity is caused by homogeneous boundary conditions around the vertex. An eigenfunction expansion is applied to derive the first known asymptotic solution for displacement components, from the equilibrium equations of Reddy’s third-order shear deformation plate theory. The characteristic equations for determining the singularities of stress resultants are presented for ten sets of boundary conditions. These characteristic equations are independent of the thickness of the plate, Young’s modulus, and shear modulus, but some do depend on Poisson’s ratio. The singularity orders of stress resultants for various boundary conditions are expressed in graphic form as a function of the vertex angle. The characteristic equations obtained herein are compared with those from classic plate theory and first-order shear deformation plate theory. Comparison results indicate that different plate theories yield different singular behavior for stress resultants. Only the vertex with simply supported radial edges (S(I)_S(I) boundary condition) exhibits the same singular behavior according to all these three plate theories.

1.
Rice, J. R., and Tracy, D. M., 1973, “Computational Fracture Mechanics,” Numerical and Computer Methods in Structural Mechanics, S. J. Fenves et al., eds., Academic Press, San Diego, CA, pp. 585–623.
2.
Aminpour
,
M. A.
, and
Holsapple
,
K. A.
,
1991
, “
Finite Element Solutions for Propagating Interface Cracks With Singularity Elements
,”
Eng. Fract. Mech.
,
39
(
3
), pp.
451
468
.
3.
Leissa
,
A. W.
,
McGee
,
O. G.
, and
Huang
,
C. S.
,
1993
, “
Vibrations of Sectorial Plates Having Corner Stress Singularities
,”
ASME J. Appl. Mech.
,
60
, pp.
134
140
.
4.
McGee
,
O. G.
,
Leissa
,
A. W.
, and
Huang
,
C. S.
,
1992
, “
Vibrations of Cantilevered Skewed Plates With Corner Stress Singularities
,”
Int. J. Numer. Methods Eng.
,
35
(
2
), pp.
409
424
.
5.
William
,
M. L.
,
1952
, “
Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension
,”
ASME J. Appl. Mech.
,
19
, pp.
526
528
.
6.
Hein
,
V. L.
, and
Erdogan
,
F.
,
1971
, “
Stress Singularities in a Two-Material Wedge
,”
Int. J. Fract. Mech.
,
7
(
3
), pp.
317
330
.
7.
Dempsey
,
J. P.
, and
Sinclair
,
G. B.
,
1979
, “
On the Stress Singularities in the Plate Elasticity of the Composite Wedge
,”
J. Elast.
,
9
(
4
), pp.
373
391
.
8.
Ting
,
T. C. T.
, and
Chou
,
S. C.
,
1981
, “
Edge Singularities in Anisotropic Composites
,”
Int. J. Solids Struct.
,
17
(
11
), pp.
1057
1068
.
9.
Hartranft
,
R. J.
, and
Sih
,
G. C.
,
1969
, “
The Use of Eigenfunction Expansions in the General Solution of Three-Dimensional Crack Problems
,”
J. Math. Mech.
,
19
(
2
), pp.
123
138
.
10.
Xie
,
M.
, and
Chaudhuri
,
R. A.
,
1998
, “
Three-Dimensional Stress Singularity at a Bimaterial Interface Crack Front
,”
Compos. Struct.
,
40
(
2
), pp.
137
147
.
11.
William, M. L., 1952, “Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates Under Bending,” Proceedings of 1st U.S. National Congress of Applied Mechanics, ASME, New York, pp. 325–329.
12.
Williams, M. L., and Owens, R. H., 1954, “Stress Singularities in Angular Corners of Plates Having Linear Flexural Rigidities for Various Boundary Conditions,” Proceedings of 2nd U.S. National Congress of Applied Mechanics, ASME, New York, pp. 407–411.
13.
William, M. L., and Chapkis, R. L., 1958, “Stress Singularities for a Sharp-Notched Polarly Orthotropic Plate,” Proceedings of 3rd U.S. National Congress of Applied Mechanics, ASME, New York, pp. 281–286.
14.
Rao
,
A. K.
,
1971
, “
Stress Concentrations and Singularities at Interfaces Corners
,”
Z. Angew. Math. Mech.
,
51
, pp.
395
406
.
15.
Ojikutu
,
I. O.
,
Low
,
R. O.
, and
Scott
,
R. A.
,
1984
, “
Stress Singularities in Laminated Composite Wedge
,”
Int. J. Solids Struct.
,
20
(
8
), pp.
777
790
.
16.
Huang
,
C. S.
,
Leissa
,
A. W.
, and
McGee
,
O. G.
,
1993
, “
Exact Analytical Solutions for the Vibrations of Sectorial Plates With Simply-Supported Radial Edges
,”
ASME J. Appl. Mech.
,
60
, pp.
478
483
.
17.
Sinclair
,
G. B.
,
2000
, “
Logarithmic Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates Under Bending
,”
ASME J. Appl. Mech.
,
67
, pp.
219
223
.
18.
Burton
,
W. S.
, and
Sinclair
,
G. B.
,
1986
, “
On the Singularities in Reissner’s Theory for the Bending of Elastic Plates
,”
ASME J. Appl. Mech.
,
53
, pp.
220
222
.
19.
Huang
,
C. S.
,
McGee
,
O. G.
, and
Leissa
,
A. W.
,
1994
, “
Exact Analytical Solutions for the Vibrations of Mindlin Sectorial Plates With Simply Supported Radial Edges
,”
Int. J. Solids Struct.
,
31
(
11
), pp.
1609
1631
.
20.
Huang, C. S., 2001, “Stress Singularities at Angular Corners in First-Order Shear Deformation Plate Theory,” Int. J. Mech. Sci., submitted for publication.
21.
Reddy, J. N., 1999, Theory and Analysis of Elastic Plates, Taylor and Francis, London.
22.
Schmidt
,
R.
,
1977
, “
A Refined Nonlinear Theory for Plates With Transverse Shear Deformation
,”
J. Indust. Math. Soc.
,
27
(
1
), pp.
23
38
.
23.
Krishna Murty
,
A. V.
,
1977
, “
Higher Order Theory for Vibration of Thick Plates
,”
AIAA J.
,
15
(
2
), pp.
1823
1824
.
24.
Reddy, J. N., 1984, Energy and Variational Methods in Applied Mechanics, John Wiley and Sons, New York.
25.
Leissa
,
A. W.
,
McGee
,
O. G.
, and
Huang
,
C. S.
,
1993
, “
Vibrations of Circular Plates Having V-Notches or Sharp Radial Cracks
,”
J. Sound Vib.
,
161
(
2
), pp.
227
239
.
26.
Huang, C. S., 1991, “Singularities in Plate Vibration Problems,” Ph.D. dissertation, The Ohio State University.