midas有限元理论-3 平面应力单元&平面应变单元

For triangular element
fi
(ξ
,η
)
=

1 2 1 4
(1+η ) (1+ ξiξ
)
(1
−
η
)
for i = 3 for i = 1, 2
For quadrilateral element
fi
(ξ
,η
)
=
1 4
(1+
ξiξ
)
(1
+
ηiη)
for i = 1, 2,3, 4
σ = Dε = DBq
where, D is the elasticity matrix defining mechanical properties of the material. Also, the matrix D is the
only difference that distinguishes plane stress elements from plain strain elements in finite element
∂v ∂x

=

∂x 0
∂ ∂y

∂
0

∂x
∂ u
∂y

v

=

0
∂
∂
∂x
∂y

0

∂ f1
∂y


0
∂
∂x
0 f1
f2 0
0 ... fn
f2
0
u1

v1

0

u2

f
n


v2 #


un
vn
∴ε = du = dfq = Bq
where
B = strain-displacement matrix q = vector of nodal displacements
Then, the stress-strain relation become,
the triangular and quadrilateral elements are shown in Table bellow.
Table Gauss-Legendre Integration for 2D Plane Elements
Integration order
2×2 1
η
3
34 1
Now, the strains at any point within element are expressed in terms of nodal displacements as,
∂u ∂



ε ε
x y
γ xy

=

∂x ∂v ∂y ∂u + ∂y
ui , vi = displacements at i − th node along global X ,Y − axes, respectively
fi (ξ ,η )
= interpolation function related to i − th node and defined in the natural coordinates

∑ u

∑ v
= =
n
i =1 n
i =1
fi (ξ ,η ) ui + α1P1 + α3P2 fi (ξ ,η ) vi + α2P1 + α4P2
where P1, P2 are the extra shape functions related to the additional incompatible modes expressed as,
nodal force vector due to initial strain ε 0
range of the integration (e.g. edge length for edge load, no integration for point load, volume for temperature load, etc.).
Location of integration points
η
4 3
ξ
2 2
1 1
3 4
ξ
2 2
η
η
3×3
3
8 79
4 7
4
3 89 56
4 56
ξ
1
23
ξ
1
23
1
2
1
2
Here it should be noted that the compatible 4-node quadrilateral element based on standard isoparametric formulation does not produce accurate results in many cases for both displacements and stresses. In order to improve the performance of this element are added so called the incompatible displacements modes. Now we consider the procedure, which involves the incompatible modes. Taking into account the incompatible modes, the displacement field is defined as,
Plane Stress & Plane Strain Elements
The Plane Stress and Plane Strain Elements incorporated in the Finite Element Library of the MIDAS Family Programs are the 3-node triangular and 4-node quadrilateral elements shown in Fig. 1. Finite Element Formulation of both element types is based on the isoparametric procedure (the element geometry and displacements are interpolated in the same way).
m, n = number of integration points in direction of ξ and η , respectively.
The appropriate order of numerical integration and corresponding locations of integration points used in

∑
x
=


∑
y
=
n
i =1 n
i =1
fi (ξ ,η ) xi fi (ξ ,η ) yi
∑ u =


∑
v
=
n
i =1 n
i =1
fi (ξ ,η )ui fi (ξ ,η ) vi
⇒ u = fq
where
xi , yi = global x-y coordinates at i − th node
procedure, based on the Gauss-Legendre quadrature. Thus the numerical integration formulas used for 2D
plane elements are:
( ) ( ) ( ) n m
∑ ∑ K = t
WjWkBT ξ j ,ηk DB ξ j ,ηk J ξ j ,ηk
1−ν ν
0
D
=
(1
+ν
E
) (1 −
2ν
)

ν 0
1−ν
0

0 (1− 2ν ) 2
in which
E= ν=
Young modulus Poisson’s ratio
Accordingly, the stiffness matrix and force vectors for a typical isoparametric 2D plane element are
Up to this moment we have reviewed the standard isoparametric formulation procedure, which is identical for both types of the 2D plane elements (3-node triangular and 4-node quadrilateral). It should be noted that the 3-node element presented here is in fact a degenerated form of the 4-node quadrilateral element. Formulation of the triangular element by collapsing the quadrilateral element allows evaluating the above integrals for both element types by use of the same standard 2D numerical integratio