COMPUTATIONAL MECHANICS
ISCM2007, July 30-August 1, 2007, Beijing, China
©2007 Tsinghua University Press & Springer
A Quadrilateral Membrane Hybrid Stress Element with Drilling
Degrees of Freedom
A. P. Wang 1*, Z. S. Tian 2
1
2
Academy of Opto-Electronics, Chinese Academy of Sciences, Beijing, 100080 China
Department of Mechanics, Graduate University, Chinese Academy of Sciences, Beijing, 100080 China
Email: [email protected]
Abstract A kind of quadrilateral membrane assumed stress hybrid finite element with drilling degrees of
freedom (DOF) — which contains a traction-free inclined side — has been derived based on a modified
Hellinger-Reissner principle. The stress equilibrium conditions are introduced through the use of additional
displacements as Lagrange multipliers. The assumed stresses are expressed in complete polynomials in
natural coordinates and a rational procedure is to choose the displacement terms such that the resulting
strains are also of complete polynomials of the same order. The combination of the special elements with the
ordinary assumed displacements elements can be efficiently used for analyzing the stress concentrations
around some cutouts.
Numerical results have demonstrated that the special element can provide much more accurate stress
concentration factors and the distributions of circumferential stress along the rim of the notches than those
obtained by using other methods.
Key words: special hybrid stress element; drilling degrees of freedom; stress concentration
INTRODUCTION
Stress concentrations in the vicinity of different holes and notches have been the subject of numerous
analytical and experimental investigations. Closed-form solutions have been found only for a few
geometries and loadings. Scanning the theoretical calculations, experimental determinations and numerical
methods, most of them are limited to simple geometries and simple boundary conditions. Naturally, the
finite element methods were considered to be used for the problem.
But numerical results [1-3] show that, in general, the convergence rate for the finite element method is
dominated by the nature of the solution in the vicinity of steep stress gradient. The regular high-accuracy
assumed displacement elements and assumed stress elements obtained by using polynomials of high order as
interpolating functions cannot improve the rate of convergence. Unless using extremely fine meshes, it will
be difficult to obtain the reasonable stresses. Thus, for an efficient finite element analysis of solids with
holes and notches, it is natural to employ the special elements. For this purpose the assumed stress hybrid
element is most suitable.
It is widely recognized that membrane elements with drilling degrees of freedom have been extensively
studied to enhance element performance [4-11]. When combined with a plate bending element, this
membrane element with drilling freedom provides a simple yet versatile tool to analyze shell structures.
In this paper by the use of Allman’s interpolation scheme a kind of 4-node special hybrid stress membrane
element with drilling degrees of freedom is developed based on a modified Hellinger-Reissner principle to
study the stress distributions of V-shaped notches.
— 1060 —
ELEMENT STIFFNESS MATRIX
The special finite element with a traction-free inclined side (SFIS) is developed by a rational method [12].
The stress field is derived based on the constraint condition that the additional incompatible displacements
are used as Lagrange multipliers to enforce the equilibrium equations in a variational sense within each
element. But the traction-free inclined side condition is satisfied exactly.
In formulating the element stiffness matrix by a modified Hellinger-Reissner principle, the energy functional
for an individual element is given by Eq. (1), when the terms corresponding to applied loads are neglected
[13]:
1
e
ΠmR
(1)
= ∫ [ − σ T S σ + σ T ( Duq )]dV − ∫ ( DT σ )T uλ dV
Vn
Vn
2
where σ represent the stresses, S is the elastic compliance matrix, uq represent element displacements
expressed in terms of nodal displacements q, uλ are additional internal displacements, and Vn is volume of
the nth element. The strain displacement relation is given by ε = Du . The last term in the integral is the
Lagrange multiplier term to impose the homogeneous equilibrium equations D T σ = 0 .
The stresses σ are uncoupled and can be expressed in isoparametric coordinates
σ=Pβ
The displacements uq and uλ are expressed respectively as
uq = Nq
uλ = M λ
(2)
(3)
Here uq + uλ = u represent the total displacements in the elements. The integral
I = ∫ (D T σ )T uλ dV
(4)
Vn
can be expressed in the form
I = βT R λ
and ∂I = 0 yields the constraint equations
∂λ
T
R β=0
(5)
(6)
Now let the assumed stresses be rewritten as
σ = P ' β'
(7)
'
where β represent the independent stress parameters. Then
1
e
ΠmR
= − β 'T H ' β' + β' TG' q
2
where
H ' = ∫ P ' T S P ' dV
G' = ∫ P' T ( DN )dV
Vn
Vn
The element stiffness matrix k is then given by
k ' = G 'T H '-1G '
(8)
(9)
(10)
COMPATIBLE DISPLACEMENT FIELD [6,8]
The quadrilateral element with a traction-free inclined side 14 shown in Fig. 1 has eight in-plane translations
and two “drilling” rotations about a normal to the element plane as degrees of freedom (Only node 1 and
node 4 have rotational degrees of freedom).
In the Allman-type element [6] the tangential component ut and normal component un are selected as
s
s
u t = u t1 (1 − ) + u t2
(11)
l12
l12
s s
s
s
un = un1 (1 − ) + un2 + (1 − )( − ω1 )
l12 2
l12
l12
— 1061 —
Figure 1: Geometry of the special element with a traction-free inclined side
Figure 2: Displacement components of edge 12 and local coordinate
where l12 is the length of edge 12, s is the running distance from one end, ut1 and ut2 are the translational
components, un1, un2 are the normal components, ω1 is nodal rotation (Fig. 2).
By the use of the isoparametric coordinates, the element displacement field is
4
1
1
uq = ∑ N i (ξ ,η ) ui + [ (1 − η 2 )(1 − ξ )(y1 − y 4 ) − (1 − ξ 2 )(1 − η )(y 2 − y1 )] ω1
16
16
i =1
1
1
+ [ (1 − ξ 2 )(1 + η )(y 4 − y 3 ) − (1 − η 2 )(1 − ξ )(y1 − y 4 )] ω 4
16
16
4
1
1
vq = ∑ N i (ξ ,η ) vi + [ (1 − η 2 )(1 − ξ )(x 4 − x 1 ) − (1 − ξ 2 )(1 − η )(x1 − x 2 )] ω1
16
16
i =1
1
1
+[ (1 − ξ 2 )(1 + η )(x 3 − x 4 ) − (1 − η 2 )(1 − ξ )(x 4 − x1 )] ω 4
16
16
(12)
where
Ni (ξ ,η ) =
1
(1 + ξ iξ ) (1 + ηiη ) (i = 1, 2, 3, 4)
4
(13)
In this assumed hybrid method, as long as uq are compatible displacements, the resulting elements will pass
the patch test even if uλ is not compatible [12].
STRESS ASSUMPTIONS
The assumed stress field with 7-β is given by Eq. (14). The stress field satisfies the equations in a variational
sense but satisfies the traction-free conditions along the inclined side 14 exactly. The element is referred as
element SFIS7β.
A 2 a32ξ 2 2Aa 2 a3ξ 2
σ x = ( a 22 +
−
) β1 + a32 (1 − ξ 2 ) β 2 + 2a 2 a3 (1 − ξ 2 ) β 3
2
B
B
2 2
(Aa3 − Ba2 ) ξ
(Aa3 − Ba2 ) 2η 2
β
β 5 + a32 (ξ + ξ 2 ) β 6 + 2a 2 a3 (ξ + ξ 2 ) β 7
+
+
4
B2
B2
— 1062 —
A 2 b32ξ 2 2Ab2 b3 ξ 2
−
) β1 + b32 (1 − ξ 2 ) β 2 + 2b2 b3 (1 − ξ 2 ) β 3
B
B2
(Bb2 − Ab3 ) 2 ξ 2
(Ab3 − Bb2 ) 2η 2
β
β 5 + b32 (ξ + ξ 2 ) β 6 + 2b2 b3 (ξ + ξ 2 ) β 7
+
+
4
2
2
B
B
2
2
2
A a3 b3 ξ
AMξ
τ xy = ( a2 b2 +
−
) β1 + a3 b3 (1 − ξ 2 ) β 2 + M(1 − ξ 2 ) β 3
2
B
B
2
2
Nξ
Nη
+ 2 β 4 + 2 β 5 + a3 b3 (ξ + ξ 2 ) β 6 + M(ξ + ξ 2 ) β 7
B
B
where
1
1
1
a2 = ( − x1 + x2 + x3 − x4 ) , a3 = ( − x1 − x2 + x3 + x4 ) , b2 = ( − y1 + y2 + y3 − y4 ) ,
4
4
4
1
b3 = ( − y1 − y2 + y3 + y4 ) , A = a 2 l + b2 m , B = a3 l + b3 m , M = a3 b2 + a 2 b3 ,
4
r
r
N = (−Aa3 + Ba2 )(−Ab3 + Bb2 ) , l = cos(n ,x ) = − sin α , m = cos(n ,y ) = cosα .
σ y = (b22 +
(14)
(15)
α (Fig. 1) is the angle between edge 14 and x axis.
The 7 independent β parameters are the minimum numbers required for the suppression of kinematic
deformation modes for the 4-node special membrane element.
In the numerical examples, the stress distributions of solid with V-shaped notches are analyzed by
combining the element SFIS7β with the special elements SFCB7β and the ordinary assumed displacement
elements. The element SFCB7β with a traction-free circular boundary (Fig. 3) is developed based on the
Hellinger-Reissner principle [4] which stress field satisfies the homogeneous equilibrium equations, the
traction-free conditions along the circular boundary as well as the compatibility equations exactly.
Figure 3: Geometry of special element with a traction-free circular boundary
NUMERICAL EXAMPLES
A rectangular plate of 60˚ symmetric V-shaped notches of radius equal to R is acted by uniform tension or
bending along two opposite edges (Fig. 4), and H/W equals 0.5.
Figure 4: Thin rectangular plate with 60˚ symmetric V-shaped notches under tension or bending
— 1063 —
The stress concentration factors (SCF) are calculated by the equations:
SCF =
σ θmax
σ0
σ0 =
T
(Tension)
Wt
σ0 =
6M
W2 t
(Bending)
(16)
where σ θmax is the largest circumferential stress σ θ ; t is the thickness of the plate, W is the width of net section.
The factors are obtained by the following three arrangements:
z Combining the present both special assumed stress hybrid elements with the ordinary isoparametric
displacement elements;
z Using 4 nodes conventional displacement elements with drilling DOF at all nodes everywhere [6];
z Using 4 nodes conventional assumed displacement isoparametric elements without drilling DOF
everywhere.
The problem is analyzed by the use of two different meshes (Figs. 5, 6) for one-quarter of the plate and each
loading case. The computed stress concentration factors (SCFs) are given by Table 1 and Table 2
respectively. The total DOF used for each arrangement are also given in the Tables. The references are
obtained by Noda and Sera [14] in body force method. The distributions of σ θ / σ x along the rims of the
notch (R/W = 0.4) under tension or bending are shown in Figs. 7 and 8.
Table 1: Computed SCFs for a thin plate with 60˚ symmetric V-shaped notches under tension
R/W
Type of element
1
2
3
0.2
DOF
present special elements
SFIS7β and SFCB7β
23
the conventional assumed
displacement element with
drilling DOF at all nodes
30
4 nodes conventional assumed
displacement isoparametric
element without drilling DOF
20
0.3
0.4
Coarse mesh
SCF
Error%
SCF
Error%
SCF
Error%
2.080
1.898
−8.9
1.888
−2.6
1.694
−17.3
1.797
−13.0
1.460
−21.3
−25.1
1.692
−3.5
1.312
−25.2
1.292
−26.3
Fine mesh
1
2
3
4
present special elements
SFIS7β and SFCB7β
57
the conventional assumed
displacement element with
drilling DOF at all nodes
78
4 nodes conventional assumed
displacement isoparametric
element without drilling DOF
52
Neuber equation [15]
SCF
2.314
Error%
SCF
Error%
SCF
Error%
SCF
Error%
Reference solutions [14]
1.3
2.163
−5.3
2.160
−5.4
2.104
−7.9
2.284
R/W = 0.2
1.868
−4.1
1.843
−5.4
1.796
−7.8
1.811
−7.0
1.948
R/W = 0.4
Figure 5: Coarse meshes for one-quarter of thin plate
— 1064 —
1.747
−0.4
1.603
−8.6
1.613
−8.0
1.645
−6.2
1.754
R/W = 0.2
R/W = 0.4
Figure 6: Fine meshes for one-quarter of thin plate
Table 2: Computed SCFs for a thin plate with 60˚ symmetric V-shaped notches under bending
R/W
Type of element
1
2
3
0.2
DOF
present special elements
SFIS7β and SFCB7β
23
the conventional assumed
displacement element with
drilling DOF at all nodes
30
4 nodes conventional assumed
displacement isoparametric
element without drilling DOF
20
0.3
0.4
Coarse mesh
SCF
Error%
SCF
Error%
SCF
Error%
1.590
−8.9
1.618
−7.3
1.632
−6.4
1.438
−6.3
1.257
−18.1
1.282
−16.5
1.353
−4.4
1.179
−16.7
1.191
−15.9
Fine mesh
1
2
3
4
present special elements
SFIS7β and SFCB7β
57
the conventional assumed
displacement element with
drilling DOF at all nodes
78
4 nodes conventional assumed
displacement isoparametric
element without drilling DOF
52
Neuber equation [15]
SCF
Error%
SCF
Error%
SCF
Error%
SCF
Error%
Reference solutions [14]
1.706
−2.2
1.625
−6.9
1.596
−8.5
1.690
−3.2
1.745
1.441
−6.1
1.380
−10.1
1.328
−13.5
1.502
−2.2
1.535
1.391
−1.8
1.360
−4.0
1.301
−8.1
1.397
−1.3
1.416
Figure 7: The distributions of stresses σ x / σ 0 — under tension (R/W = 0.4)
It is seen that in comparison with the assumed displacement elements with or without drilling DOF, the
present special elements provide the stress concentration factors and stress distributions more close to the
references for both tension or bending even in very coarse meshes.
— 1065 —
Figure 8: The distributions of stresses σ x / σ 0 — under bending (R/W = 0.4)
CONCLUSIONS
A kind of 4-node special hybrid stress membrane element with drilling DOF at two nodes which has a
traction-free inclined side is derived for stress distributions analysis in thin plate with V-shaped notches.
Numerical results have demonstrated that the special membrane element is not only superior to the ordinary
4-node assumed displacement element with or without drilling DOF, but also possess high computational
efficiency.
Acknowledgements
This work is supported by the National Science Foundation of China (10072064)
REFERENCES
1. Tian ZS. A study of stress concentrations in solids with circular holes by 3-D special hybrid stress
2.
3.
4.
5.
6.
7.
elements. J. Strain Anal., 1990; 25: 29-35.
Tian ZS, Tian Z. Improved hybrid solid element with a traction-free cylindrical surface. Int. J. Num.
Mech. Engrg, 1990; 29: 801-809.
Tian ZS, Liu JS, Pian THH. Studies of stress concentration by using special hybrid stress elements. Int.
J. Num. Meth. Engrg, 1997; 40(18): 1399-1412.
Wang AP, Tian ZS. Special hybrid element with drilling degrees of freedom. In: Computational
Mechanics, Proc. of Sixth World Congress on Computational Mechanics (WCCM VI), CD-ROM,
2004, RS-30.
Wang AP, Tian ZS. A 3-dimensional assumed stress hybrid element with drilling degrees of freedom.
EPMESC X, Sanya, Hainan, China, CD-ROM, Aug. 21-23, 2006, R-89.
Allman DJ. A compatible triangular element including vertex rotations for plane elasticity analysis.
Comput. & Struct., 1984; 19: 1-8.
Bergen PG, Felippa CA. A triangular membrane element with rotational degrees of freedom. Comput.
Meth. Appl. Mech. Engng., 1985; 50: 25-69.
8. Cook RD. On the Allman triangular and a related quadrilateral element. Comput. & Struct., 1986; 22:
1065-1067.
9. Yunus SM, Saigal S, Cook RD. On improved hybrid finite elements with rotational degrees of freedom.
Int. J. Num. Mech. Engng., 1989; 28: 785-800.
10. Macneal RH, Harder RL. A refined four-noded membrane element with rotational degree of freedom.
Comput. & Struct., 1988; 28: 75-84.
11. Pan YS, Chen DP. A 4-node membrane element with rotational degree of freedom based on hybrid
stress model. Proc. of EPMESC’IV, 1992; II, pp. 700-706.
— 1066 —
12. Pian THH, Sumihara K. Rational approach for assumed stress finite elements. Int. J. Num. Meth.
Engng., 1984; 20: 1685-1695.
13. Pian THH, Chen DP. Alternative ways for formulation of hybrid stress element. Int. J. Num. Meth.
Engng., 1982; 18: 1679-1684.
14. Noda NA, Sera M, Takase Y. Stress concentration factors for round and flat test specimens with
notches. Int. J. Fatigue, 1995; 17(3): 163-178.
15. Neuber H. Kerbspannungslehre. Springer-Verlag, Berlin, Germany, 1958 (in German).
— 1067 —