An analytical singular element with arbitrary high-order precision is constructed using the analytical symplectic eigenfunctions of an annular sector thin plate with both straight sides free. These values can be used to describe the local stress singularities near an arbitrary V-notch or a crack tip. Numerical examples of Kirchhoff’s plate bending problem with V-shaped notches are given by applying the Local-Global method. This method combines the present analytical singular element and the conventional finite element method. The numerical results show that the present method is an effective numerical technique for analysis of Kirchhoff plate bending problems with boundary stress singularities.

References

1.
Williams
,
M. L.
, 1961, “
The Bending Stress Distribution at the Base of a Stationary Crack
,”
ASME J. Appl. Mech.
,
28
, pp.
78
82
.
2.
Sih
,
G. C.
,
Paris
,
P. C.
, and
Erdogan
,
F.
, 1962, “
Crack Tip Stress-Intensity Factors for Plane Extension and Plate Bending Problems
,”
ASME J. Appl. Mech.
,
29
, pp.
306
310
.
3.
Wilson
,
W. K.
, and
Thompson
,
D. G.
, 1971, “
On the Finite Element Method for Calculating Stress Intensity Factors for Cracked Plates in Bending
,”
Eng. Fract. Mech.
,
3
(
2
), pp.
97
102
.
4.
Su
,
R. K. L.
, and
Sun
,
H. Y.
, 2002, “
Numerical Solution of Cracked Thin Plates Subjected to Bending, Twisting and Shear Loads
,”
Int. J. Fracture
,
117
, pp.
323
335
.
5.
Wang
,
Y. H.
,
Tham
,
L. G.
,
Lee
,
P. K. K.
, and
Tsui
,
Y.
, 2003, “
A Boundary Collocation Method for Cracked Plates
,”
Comput. Struct.
,
81
, pp.
2621
2630
.
6.
Zehnder
,
A. T.
, and
Viz
,
M. J.
, 2005, “
Fracture Mechanics of Thin Plates and Shells Under Combined Membrane, Bending, and Twisting Loads
,”
Appl. Mech. Rev.
,
58
(
1
), pp.
37
48
.
7.
Palani
,
G. S.
,
Iyer
,
N. R.
, and
Dattaguru
,
B.
, 2006, “
A Generalised Technique for Fracture Analysis of Cracked Plates Under Combined Tensile, Bending and Shear Loads
,”
Comput. Struct.
,
84
, pp.
2050
2064
.
8.
Leung
,
A. Y. T.
,
Xu
,
X. S.
,
Zhou
,
Z. H.
, and
Wu
,
Y.F.
, 2009, “
Analytic Stress Intensity Factors for Finite Elastic Disk Using Symplectic Expansion
,”
Eng. Fract. Mech.
,
76
, pp.
1866
1882
.
9.
Barsoum
,
R. S.
, 1976, “
A Degenerate Solid Element for Linear Fracture Analysis of Plate Bending and General Shells
,”
Int. J. Numer. Meth. Eng.
,
10
, pp.
551
564
.
10.
Ahmad
,
J.
, and
Loo
,
F. T. C.
, 1979, “
Solution of Plate Bending Problems in Fracture Mechanics Using a Specialized Finite Element Technique
,”
Eng. Fract. Mech.
,
11
, pp.
661
673
.
11.
Chen
,
W. H.
, and
Chen
,
P. Y.
, 1984, “
A Hybrid-Displacement Finite Element Model for the Bending Analysis of Thin Cracked Plates
,”
Int. J. Fract.
,
24
, pp.
83
106
.
12.
Leung
,
A. Y. T.
, and
Su
,
R. K. L.
, 1994, “
Mode I Crack Problems by Fractal Two Level Finite Element Methods
,”
Eng. Fract. Mech.
,
48
(
6
), pp.
847
856
.
13.
Leung
,
A. Y. T.
, and
Su
,
R. K. L.
, 1995, “
Mixed-Mode Two-Dimensional Crack Problem by Fractal Two Level Finite Element Method
,”
Eng. Fract. Mech.
,
51
(
6
), pp.
889
895
.
14.
Leung
,
A. Y. T.
, and
Su
,
R. K. L.
, 1996, “
Fractal Two-Level Finite Element Method for Cracked Kirchhoff’s Plates Using DKT Elements
,”
Eng. Fract. Mech.
,
54
(
5
), pp.
703
711
.
15.
Leung
,
A. Y. T.
, and
Su
,
R. K. L.
, 1996, “
Fractal Two-Level Finite Element Analysis of Cracked Reissner’s Plate
,”
Thin-Walled Struct.
,
24
(
4
), pp.
315
334
.
16.
Jiang
,
C. P.
, and
Cheung
,
Y. K.
, 1995, “
A Special Bending Crack Tip Finite Element
,”
Int. J. Fract.
,
71
, pp.
55
69
.
17.
Liu
,
C. T.
, and
Jiang
,
C. P.
, 2001,
Fracture Mechanics for Plates and Shells
, 2nd ed.,
National Defence Industry Press
,
Beijing, China
.
18.
Hung
,
N. D.
, and
Ngoc
,
T. T.
, 2004, “
Analysis of Cracked Plates and Shells Using “Metis” Finite Element Model
,”
Finite Elem. Anal. Design
,
40
, pp.
855
878
.
19.
Munaswamy
,
K.
, and
Pullela
,
R.
, 2008, “
Computation of Stress Intensity Factors for Through Cracks in Plates Using p-Version Finite Element Method
,”
Commun. Numer. Meth. Eng.
,
24
, pp.
1753
1780
.
20.
Williams
,
M. L.
, 1951, “
Surface Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates Under Bending
,”
Proceedings of the First US National Congress of Applied Mechanics
, pp.
325
329
.
21.
Williams
,
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
.
22.
Hasebe
,
N.
, and
Iida
,
J.
, 1983, “
Intensity of Corner and Stress Concentration Factor
,”
J. Eng. Mech.
,
109
(
1
), pp.
346
356
.
23.
Hasebe
,
N.
, 1986a, “
Stress Analysis of a Blunted Notch in a Clamped Edge
,”
J. Eng. Mech.
,
112
(
1
), pp.
142
153
.
24.
Hasebe
,
N.
,
Sugimoto
,
T.
, and
Nakamura
,
T.
, 1986b, “
Stress Concentration in Clamped Edge of Thin Plate
,”
J. Eng. Mech.
,
112
(
7
), pp.
642
653
.
25.
Hasebe
,
N.
, and
Iida
,
J.
, 1990, “
Notch Mechanics for Plane and Thin Plate Bending Problems
,”
Eng. Fract. Mech.
,
37
(
1
), pp.
87
99
.
26.
Pengfei
,
H.
,
Ishikawa
,
H.
, and
Kohno
,
Y.
, 1995, “
Analysis of the Orders of Stress Singularity at the Corner Point of a Diamond-Shape Rigid Inclusion or Hole in an Infinite Plate Under Anti-Plane Bending by Conformal Mapping
,”
Int. J. Eng. Sci.
,
33
(
11
), pp.
1535
1546
.
27.
Maucher
,
R.
, and
Hartmann
,
F.
, 1999, “
Corner Singularities of Kirchhoff Plates and the Boundary Element Method
,”
Comput. Methods Appl. Mech. Eng.
,
173
, pp.
273
285
.
28.
Treifi
,
M.
,
Oyadiji
,
S. O.
, and
Tsang
,
D. K. L.
, 2009, “
Computations of the Stress Intensity Factors of Double-Edge and Centre V-Notched Plates Under Tension and Anti-Plane Shear by the Fractal-Like Finite Element Method
,”
Eng. Fract. Mech.
,
76
(
13
), pp.
2091
2108
.
29.
Savruk
,
M. P.
, and
Kazberuk
,
A.
, 2010, “
Two-Dimensional Fracture Mechanics Problems for Solids With Sharp and Rounded V-Notches
,”
Int. J. Fract.
,
161
, pp.
79
95
.
30.
Zhong
,
W. X.
, 1994, “
Plane Elasticity in Sectorial Domain and Hamiltonian System
,”
Appl. Math. Mech.
,
15
(
12
), pp.
1113
1123
.
31.
Yao
,
W. A.
,
Zhong
,
W. X.
, and
Su
,
B.
, 1999, “
New Solution System for Circular Sector Plate Bending and its Application
,”
Acta Mech. Solida Sinica
,
12
(
4
), pp.
307
315
.
32.
Yao
,
W. A.
,
Zhong
,
W. X.
, and
Lim
,
C. W.
, 2009,
Symplectic Elasticity
, 1st ed.,
World Scientific
,
Singapore
.
33.
Lim
,
C. W.
,
Yao
,
W. A.
, and
Cui
,
S.
, 2008, “
Benchmark Symplectic Solutions for Bending of Comer-Supported Rectangular Thin Plates
,”
The IES Journal Part A: Civil & Structural Engineering
,
1
(
2
), pp.
106
115
.
34.
Lim
,
C. W.
, 2010a, “
Symplectic Elasticity Approach for Free Vibration of Rectangular Plates
,”
Advances in Vibration Engineering
,
9
, pp.
159
163
.
35.
Lim
,
C. W.
,
,
C. F.
,
Xiang
,
Y.
, and
Yao
,
W.
, 2009, “
On New Symplectic Elasticity Approach For Exact Free Vibration Solutions of Rectangular Kirchhoff Plates
,”
Int. J. Eng. Sci.
,
47
(
1
), pp.
131
140
.
36.
Lim
,
C. W.
, 2006, “
Symplectic Elasticity Exact Analytical Approach for Piezoelectric Composite Thick Beams
,”
Second Symposium on Piezoelectricity, Acoustic Waves, and Device Applications
, Hangzhou, PRC, Dec., pp.
14
17
.
37.
Zhang
,
H. W.
, and
Zhong
,
W. X.
, 2003, “
Hamiltonian Principle Based Stress Singularity Analysis Near Crack Corners of Multi-Material Junctions
,”
Int. J. Solids Struct.
,
40
(
2
), pp.
493
510
.
38.
Wang
,
C. Q.
, and
Yao
,
W. A.
, 2003, “
Application of the Hamilton System to Dugdale Model in Fracture Mechanics
,”
Chinese Journal of Applied Mechanics
,
20
(
3
), pp.
151
154
.
39.
Lim
,
C. W.
, and
Xu
,
X. S.
, 2010b, “
Symplectic Elasticity: Theory and Applications
,”
Appl. Mech. Rev.
,
63
(
050802
), pp.
1
10
.
You do not currently have access to this content.