The lowest-order interpolation model consistent with two integration constants is linear, in the form 9.2 We seek to identify Φm1 in terms of the nodal values of u... 122 Finite Element
Trang 1Element Fields in Linear Problems
This chapter presents interpolation models in physical coordinates for the most part, for the sake of simplicity and brevity However, in finite-element codes, the physical coordinates are replaced by natural coordinates using relations similar to interpola-tion models Natural coordinates allow use of Gaussian quadrature for integrainterpola-tion and, to some extent, reduce the sensitivity of the elements to geometric details in the physical mesh Several examples of the use of natural coordinates are given
9.1 INTERPOLATION MODELS 9.1.1 O NE -D IMENSIONAL M EMBERS 9.1.1.1 Rods
The governing equation for the displacements in rods (also bars, tendons, and shafts) is
(9.1)
in which u(x, t) denotes the radial displacement, E, Α and ρ are constants, x is the spatial coordinate, and t denotes time Since the displacement is governed by a second-order differential equation, in the spatial domain, it requires two (time-dependent) constants of integration Applied to an element, the two constants can
be supplied implicitly using two nodal displacements as functions of time We now approximate u(x, t) using its values at x e and x e+1, as shown in Figure 9.1
The lowest-order interpolation model consistent with two integration constants
is linear, in the form
(9.2)
We seek to identify Φm1 in terms of the nodal values of u Letting u e=u(x e) and
u e+1=u(x e+1), furnishes
(9.3)
9
EA u
u t
∂
∂
∂
2 2
2 2
u t
m T
e e
m T
( )
+
ϕϕ 1 ΦΦ γγ1 1 γγ 1 ϕϕ
1
,
u t e( )=(1 x e)ΦΦ γγm1 m1( ),t u e+1( )t =(1 x e+1)ΦΦ γγm1 m1( ).t
Trang 2122 Finite Element Analysis: Thermomechanics of Solids
However, from the meaning of γγγγm1(t), we conclude that
(9.4)
9.1.1.2 Beams
The equation for a beam, following Euler-Bernoulli theory, is:
(9.5)
in which w(x, t) denotes the transverse displacement of the beam’s neutral axis, and
I is a constant In the spatial domain, there are four constants of integration In an element, they can be supplied implicitly by the values of w and w′=∂w/∂x at each
of the two element nodes Referring to Figure 9.2, we introduce the interpolation model for w(x, t):
(9.6)
FIGURE 9.1 Rod element.
FIGURE 9.2 Beam element.
ue
ue+1
x
wet
we
wet +1
we+1
x
Φm
e
x
1
1
1
1
1
1 1
1
=
=
−
−
+
−
+
+
,
EI w
u t
∂
∂
4 4
2
2 0
w w w w
e e e e
− ′
+ +
2 3
1
1 1
1
0749_Frame_C09 Page 122 Wednesday, February 19, 2003 5:09 PM
Trang 3Element Fields in Linear Problems 123
Enforcing this model at x e and at x e+1 furnishes
(9.7)
9.1.1.3 Beam Columns
Beam columns are of interest, among other reasons, in predicting buckling according
to the Euler criterion The z–displacement w of the neutral axis is assumed to depend
only on x and the x–displacement Also, u is modeled as
(9.8)
in which u0(x) represents the stretching of the neutral axis It is necessary to know
u0(x), w(x) and at x e and x e+1 The interpolation model is now
(9.9)
and
9.1.1.4 Temperature Model: One Dimension
The temperature variable to be determined is T − T0, in which T0 is a reference
temperature assumed to be independent of x The governing equation for a
one-dimensional conductor is
(9.10)
Φb1
2
2
1 3
2
1
1
1
−
u x z u x z w x
x
( , )= ( )− ∂ ( ),
∂
0
∂
∂
w x x
( )
u x z t( , , )=(1 x) m1 m1−z(1 x x2 x3) b b,
1 1
Φ
e e
b
e e e e
u t
u t
w w w w
l= l ,
− ′
+
( ) ( ) ,
1
Φm1 1 Φb1
2
2
1 3
2
1
1
1
1
−
+
−
l
e
kA
x Ac e t
∂
∂
∂
2 2
Trang 4124 Finite Element Analysis: Thermomechanics of Solids
This equation is formally the same as for a rod equation (see Equation 9.1),
furnishing the interpolation model for the element as
(9.11)
9.1.2 I NTERPOLATION M ODELS : T WO D IMENSIONS
9.1.2.1 Membrane Plate
Now suppose that the displacements u(x, y, t) and v(x, y, t) are modeled on the
triangular plate element in Figure 9.3, using the values u e(t), v e(t), u e+1(t), v e+1(t), u e+2(t),
and v e+2(t) This element arises in plane stress and plane strain, and is called a
membrane plate element A linear model suffices for each quantity because it provides
three coefficients to match three nodal values The interpolation model now is
(9.12)
9.1.2.2 Plate with Bending Stresses
In a plate element experiencing bending only, the in-plane displacements, u and v,
are expressed by
(9.13)
FIGURE 9.3 Triangular plate element.
z
Y
middle surface
Xe+1,Ye+1
Xe+2,Ye+2
( )
t t
e e
−
+
0
ϕϕT ΦΦ θθ θθ
u x y t
v x y t
t t
m m m m u v
( , , ) ( , , )
( ) ( )
=
ϕϕ ϕϕ
Φ Φ
γγ γγ
ΤΤ ΤΤ
ΤΤ ΤΤ 2
2 2 2
0 0
0 0
T T
T T
( ) ( ) ( )
( ) ( ) ( )
t
u t
u t
u t
t
v t
v t
v t
x y
e e e
e e e
=
+ +
+ +
1 2
1 2
1
1
x e+ y e+
−
u x y z t z w
x v x y z t z
w y
( , , , )= − ∂ , ( , , , ) ,
0749_Frame_C09 Page 124 Wednesday, February 19, 2003 5:09 PM
Trang 5in which z = 0 at the middle plane The out-of-plane displacement, w, is assumed
to be a function of x and y only.
An example of an interpolation model is introduced as follows to express w(x, y)
throughout the element in terms of the nodal values of :
(9.14)
It follows that
(9.15)
9.1.2.3 Plate with Stretching and Bending
Finally, for a plate experiencing both stretching and bending, the displacements are assumed to satisfy
(9.16)
w, ∂w x and w y
w x y t( , , )= ϕϕb2( , )x yΦΦ γγb2 b2( )t
T
ϕϕb2 x y 1 x y x2 xy y2 x3 1 x y2 y x2 y3
2
T
w x
w y
w x
w
T
( )
∂
∂
Φb
x y x x y y x x y x y y
x y x x y y
x x x x y y
x y x x
2
1
1
−
=
+
y y x x y x y y
x y x x y y
+
1
1 1
1 1
1
1 2
3 1
y y
x y x x y y x x y x y y
x y x x y y
− +
)
x e y e ( x e x e y e ) y e
u(x, y, z, t) v(x, y, z, t) w(x, y, z, t)
z
b
b
b
b b
x y
=
−
−
∂
∂
∂
∂
ϕϕ ϕϕ
ΤΤ
ΤΤ
ΤΤ
ϕϕ
ΦΦ γγ
2
2
2
2 2( )
u x y z t u x y z t z w
x v x y z t v x y z t z
w y
( , , , )= ( , , , )− ∂ , ( , , , ) ( , , , )
Trang 6126 Finite Element Analysis: Thermomechanics of Solids
and w is a function only of x, y, and t(not z) Here, z = 0 at the middle surface, while
u0 and v0 represent the in-plane displacements Using the nodal values of u0, v0, and
w0, a combined interpolation model is obtained as
(9.17)
9.1.2.4 Temperature Field in Two Dimensions
In the two-dimensional, triangular element illustrated in Figure 9.3, the linear inter-polation model for the temperature is
(9.18)
9.1.2.5 Axisymmetric Elements
An axisymmetric element is displayed in Figure 9.4 It is applicable to bodies that are axisymmetric and are submitted to axisymmetric loads, such as all-around
pressure The radial displacement is denoted by u, and the axial displacement is denoted by w The tangential displacement v vanishes, while radial and axial
dis-placements are independent of θ Now u and w depend on r, z, and t.
There are two distinct situations that require distinct interpolation models In
the first case, none of the nodes are on the axis of revolution (r = 0), while in the second case, one or two nodes are, in fact, on the axis In the first case, the linear
FIGURE 9.4 Axisymmetric element.
u x y z t
v x y z t
w x y z t
u x y t
v x y t
w x y t
z z
b b
b
x y
( , , , )
( , , , )
( , , , )
( , , ) ( , , ) ( , , )
=
+
−
−
∂
∂
∂
∂
0
0
0
2
2
2
ϕϕ ϕϕ
ΤΤ
ΤΤ
ΤΤ
ϕϕ
=
−
−
∂
∂
∂
∂
ΦΦ γγ
ϕϕ ϕϕ
ϕϕ
Φ Φ Φ
γγ γγ γγ
ΤΤ
ϕϕ ϕϕ
b b
m
b
m
b
b
m m b
u v w
t
z z
t t t
x y
2 2
2
2
2
2
2
2
( )
( ) ( ) ( )
0 0
T T
T−T0=ϕϕ ΦΤΤ 2Φ θθ2 2 θθ2 ΤΤ= T −T0 T+1−T0 T+2−T0
r
e+2
e e+1
θ
Trang 7interpolation model is given by
(9.19)
Now suppose that there are nodes on the axis, and note that the radial displace-ments are constrained to vanish on the axis For reasons shown later, it is necessary
to enforce the symmetry constraints a priori in the formulation of the displacement model In particular, suppose that node e is on the axis, with nodes e + 1 and e + 2
defined counterclockwise at the other vertices The linear interpolation model is now
(9.20)
A similar formulation can be used if two nodes are on the axis of symmetry so
that the u displacement in the element is modeled using only one nodal displacement,
with a coefficient vanishing at each of the nodes on the axis of revolution
9.1.3 I NTERPOLATION M ODELS : T HREE D IMENSIONS
We next consider the tetrahedron illustrated in Figure 9.5
A linear interpolation model for the temperature can be expressed as
(9.21)
u r z t
w r z t
t t
a a a a ua wa
( , , ) ( , , )
( ) ( ) .
=
ϕϕ ϕϕ
Φ Φ
γγ γγ
1 1 1 1 1 1
T T
T T
T T
0 0
0 0
ua e e e
wa
e e e
r z
u u u
w w w
1
2
2
1
1 1 1
T
=
−
+ +
+ +
u r z t
w r z t
t t
a a a a ua wa
( , , ) ( , , )
( ) ( )
=
00 ϕϕ
γγ γγ
2 2 2 2 2 2
T T
T T
r z z
1
T
−
−
T−T0=ϕϕ Φ3TΦ θθ3T 3
T
ϕϕ
Φ
3
3
1
1 1 1 1 1
T
T
T T
T =
=
−
θ
Trang 8128 Finite Element Analysis: Thermomechanics of Solids
For elasticity with displacements u, v, and w, the corresponding interpolation
model is
(9.22)
9.2 STRAIN-DISPLACEMENT RELATIONS
AND THERMAL ANALOGS
9.2.1 S TRAIN -D ISPLACEMENT R ELATIONS : O NE D IMENSION
For the rod, the strain is given by An estimate for ε implied by the interpolation model in Equation 9.3 has the form
(9.23)
FIGURE 9.5 Tetrahedral element.
e+2
e+3
y e
z
0
u x y z t
v x y z t
w x y z t
t t t
u v w
( , , , ) ( , , , ) ( , , , )
( ) ( ) ( )
=
γγ γγ γγ
3 3 3
3 3 3
e e e e
v
e e e e
w
e
e
t
u t
u t
u t
u t
t
v t
v t
v t
v t
t
w t
w t
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) (
=
=
=
+ + +
+ + +
+ 1
2 3
1 2 3
1
)) ( ) ( )
w t
e e
+ +
2 3
ε = = ∂
∂
E11 u x
ε( , )x t = ββmT1( )xΦΦ γγm1 m1( ),t
Trang 9For the beam, the corresponding relation is
(9.24)
from which the consistent approximation is obtained:
(9.25)
For the beam column, the strain is given by
(9.26)
9.2.2 S TRAIN -D ISPLACEMENT R ELATIONS : T WO D IMENSIONS
In two dimensions, the (linear) strain tensor is given by
(9.27)
We will see later the two important cases of plane stress and plane strain In the
latter case, E zz vanishes and s zz is not needed to achieve a solution In the former
case, s zz vanishes and E zz is not needed for solution
In traditional finite-element notation, we obtain [cf (Zienkiewicz and Taylor, 1989)]
(9.28)
ββm ϕϕ
m
d dx
T T
ε( , , )x z t z w,
x
= − ∂
∂
ε( , , )x z t = − ββz b1( )xΦΦ γγb1 b1( ),
T
t
ββb ϕϕ
b
d
2
T T
ε( , , )x z t = − ββz mTb1( )x ΦΦ γγmb1 mb1( ),t
γγ γγ
m mb m b
z
t t
1 1 1 1
( ) ( )
xx xy
xy yy
u
x u y x v u
y v x v y
, =
∂
∂
1 2 1
2
′ =
γγ
E E E
xx yy xy
m T m u v
2 2
2 2
ˆ
Trang 10130 Finite Element Analysis: Thermomechanics of Solids
The prime in e′ is introduced temporarily to call attention to the fact that it does
not equal VEC(EL) Hereafter, the prime will not be displayed
For a plate with bending stresses only,
(9.29)
from which
For a plate experiencing both membrane and bending stresses, the relations can
be combined to furnish
(9.31)
9.2.3 A XISYMMETRIC E LEMENT ON A XIS OF R EVOLUTION
For the toroidal element with a triangular cross section, it is necessary to consider two cases If there are no nodes on the axis of revolution, then
(9.32)
ββ
00
ΤΤ
ϕϕ
ϕϕ
m
m
m
m m m
x
y
2
2
2
2 2 2
1 2
1 2
=
=
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂ ∂
e ( , , , ) x y z t z ,
w x w y w
x y
2 2 2 2 2
∂
∂
∂
∂
∂
∂ ∂
e ( , , , ) x y z t z ( )x ( ),t
x y
x y
b T
b b b
ϕϕ ϕϕ ϕϕ
2 2 2 2 2 2 2
2
e ( , , , ) x y z t ββmb2( , , )x y zΦΦ γγmb2 mb2( )t
T
γγ γγ
m b mb
m b
t t
2 2 2
2 2
( ) ( )
e( , , )
( ) ( )
r z t
t t
u r u r w z u z
w r
a a a ua wa
=
∂
∂
∂
∂
∂
1 2
1 1 1 1 1
ββ Φ
Φ
γγ γγ
0
Trang 11in which the prime is no longer displayed If element e is now located on the axis
of revolution, we obtain
(9.33)
9.2.4 T HERMAL A NALOG IN T WO D IMENSIONS
The thermal analog of the strain is the temperature gradient satisfying
(9.34)
9.2.5 T HREE -D IMENSIONAL E LEMENTS
Recalling the tetrahedral element in the previous section, the strain relation can be written as
(9.35)
ββa
z r
1
1 2
1
T
=
e( , , ) b
( ) ( )
r z t
t t
a a a ua wa
2 2 1 2 1
0
Φ Φ
γγ γγ
ββa
z ze r
2
T
=
−
T ββ ΦT Φ θθ ββ
T
2 2 2 2
e=
=
=
∂
∂
∂
∂
∂
∂
∂
∂ +∂∂
∂
∂ +∂∂
∂
E E E E E E
xx yy zz xy yz zx
u x v y w z u y
v x v
z w y w
x u z
1 2 1 2 1 2
3
ββT
Φ Φ Φ
γγ γγ γγ
3 3 3
3 3 3
u v w
Trang 12132 Finite Element Analysis: Thermomechanics of Solids
9.2.6 T HERMAL A NALOG IN T HREE D IMENSIONS
Again referring to the tetrahedral element, the relation for the temperature gradient is
(9.36)
9.3 STRESS-STRAIN-TEMPERATURE RELATIONS
IN LINEAR THERMOELASTICITY
9.3.1 O VERVIEW
If S is the stress tensor under small deformation, the stress-strain relation for a
linearly elastic, isotropic solid under small strain is given in Lame’s form by
(9.37)
in which I is the identity tensor The Lame’s coefficients are denoted by λ and µ,
and are given in terms of the familiar elastic modulus E and Poisson ratio ν as
(9.38)
Letting s = VEC(S) and e = VEC(E L), the stress-strain relations are written using Kronecker product operators as
(9.39)
and D is the tangent-modulus tensor introduced in the previous chapters.
9.3.2 O NE -D IMENSIONAL M EMBERS
For a beam column, recalling the strain-displacement model,
(9.40)
The cases of a rod and a beam are recovered by setting γγγγm1 or γγγγb1 equal to zero vectors, respectively
T ββ3 3 θθ3 ββ3
S=2µE L+λtr( E L) ,I
µ
ν
=
E E
2 1( ), (1 2 )(1 ).
s=De, D=2µI9+λii T,
S11=σ( , , )x z t =Eε= −zEββmbT1( )x ΦΦ γγmb1 mb1( ).t
Trang 139.3.3 T WO -D IMENSIONAL E LEMENTS
9.3.3.1 Membrane Response
In two-dimensional elements, several cases can be distinguished We first consider elements in plane stress It is convenient to use Hooke’s Law in the form
(9.41)
Under plane stress, S xz = S yz = S zz = 0 Now, E zz ≠ 0, but we will see that it is not of present interest since it does not influence the solution process Later on, for reasons such as tolerances, we may wish to calculate it, but this is a postprocessing issue Consequently, in plane stress, the stress-strain relations reduce to
(9.42)
In traditional finite-element notation, this can be written as
(9.43)
in which
(9.44)
= +
= +
= +
1
1
1
1
1
1
E
E
E
E
E
E
ν ν ν ν ν ν
= +
1
1
1
E
E
E
ν ν ν
S S S
E E E
xx yy xy m xx yy xy
=
D m21
1
2
1
=
−
−
+
=
−
+
−
ν ν
ν ν
ν