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Institute of Structural Engineering Page 1
Method of Finite Elements I
Chapter 7
2D Elements
Book Chapters[O] V1/Ch4 & Ch6
Institute of Structural Engineering Page 2
Method of Finite Elements I
Today’s Lecture
• Continuum Elements Plane Stress
Plane Strain
Next Lecture• Structural Elements Plate Elements
Shell Elements
Extending (discretization) to higher Dimensions
Institute of Structural Engineering Page 3
Method of Finite Elements I
In the previous lectures:
The Galerkin method was presented The isoparametric concept was introduced The bar and beam elements were presented with extensions to 3D Some numerical integration methods were presented
In today’s lecture:
– Concepts from previous lectures are combined to formulate elements for the above cases
– Linear 3D elasticity equations are used as a starting point– Plane stress/Plane strain elements are presented as special cases
Today’s Lecture (in more detail)
Institute of Structural Engineering Page 4
Method of Finite Elements I
3D elasticity problem
tn
0Γ
tΓ
uΓ
Ω
Problem variables:
Displacements:
x
y
z
uuu
=
u
x xy xz
xy y yz
xz yz z
ε ε εε ε εε ε ε
=
ε
Strain tensor:
Institute of Structural Engineering Page 5
Method of Finite Elements I
3D elasticity problem
00
t
u
at
t at
at
⋅ = Γ
⋅ = Γ
= Γ
σ n
σ n
u u
tn
0Γ
tΓ
uΓ
Boundary conditions:
ΩV
xF dV
zF dV
yF dV
Traction: Distributed force per unit surface areat −
Institute of Structural Engineering Page 6
Method of Finite Elements I
3D elasticity problem
x
y
z
FFF
=
F
tn
0Γ
tΓ
uΓ
Body Forces:distributed force per unit volume (e.g., weight, inertia, etc)
Ω
NOTE: If the body is accelerating, then the inertia force, is added onto the body force vector
x x
y y
z z
F uF uF u
ρρ ρ
ρ
= − = −
F F u
ρuV
xF dV
zF dV
yF dV
Institute of Structural Engineering Page 7
Method of Finite Elements I
3D elasticity equations Internal stresses
zσ
zyτzxτ
xzτ
xyτ
xσ
yzτyσ
xyτ
z
y
x
σx, σy and σz are normal stresses. τxy, τyz and τxz are the shear stresses.
Notationτxy is the stress on the face perpendicular to the x-axis and points in the positive y direction
Stress Tensor:x xy xz
xy y yz
xz yz z
σ τ ττ σ ττ τ σ
=
σ
*total of 9 stress components of which only 6 are independent since:
xy yx
xz zx
yz zy
τ τ
τ ττ τ
=
==
Institute of Structural Engineering Page 8
Method of Finite Elements I
3D elasticity problem In FE analysis Voigt notation is typically used for strains and stresses:Here we use a variant, that is different to standard Voigt notation, which ordersthe z components of shear strains/stresses last.
x xy xz
xy y yz
xz yz z
σ τ ττ σ ττ τ σ
=
σ
x xy xz
xy y yz
xz yz z
ε ε εε ε εε ε ε
=
ε 222
xx x
yy y
zz z
xy xy
xz xz
yz yz
ε εε εε εε γε γε γ
= =
ε
x
y
z
xy
xz
yz
σσστττ
=
σ
engineering strain
Institute of Structural Engineering Page 9
Method of Finite Elements I
3D elasticity equations
Equilibrium equations:
0
0
0
xyx xzx
xy y yzy
yzxz zz
Fx y z
Fx y z
Fx y z
τσ τ
τ σ τ
ττ σ
∂∂ ∂+ + + =
∂ ∂ ∂∂ ∂ ∂
+ + + =∂ ∂ ∂
∂∂ ∂+ + + =
∂ ∂ ∂
0∇⋅ + =σ F
In vector formComponent wise
where 𝑭𝑭 is the applied body force:
x
y
z
FFF
=
F
Institute of Structural Engineering Page 10
Method of Finite Elements I
3D elasticity equations Strain definition:
( )12
s T= ∇ = ∇ +∇ε u u u
In vector form
Component wise
1 12 2
1 12 2
1 12 2
yx x x z
y y yx z
yxz z z
uu u u ux y x z x
u u uu ux y y z y
uuu u ux z y z z
∂ ∂ ∂ ∂ ∂ + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + ∂ ∂ ∂ ∂ ∂
∂ ∂∂ ∂ ∂ + + ∂ ∂ ∂ ∂ ∂
ε
Using Voigt notation
x
y
z
yx
x z
y z
uxuyuz
uuy x
u uz x
u uz y
∂ ∂
∂ ∂
∂ ∂ = ∂ ∂
+ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂
ε
Institute of Structural Engineering Page 11
Method of Finite Elements I
3D elasticity equations Constitutive equation, using Voigt notation can be conveniently written as:
=σ Eε
1 0 0 01 0 0 0
1 0 0 01 20 0 0 0 0
2(1 )(1 2 )1 20 0 0 0 0
21 20 0 0 0 0
2
x x
y y
z z
xy xy
xz xz
yz yz
E
ν ν νν ν νσ εν ν νσ ε
νσ ετ γν ν
ντ γτ γ
ν
− −
− − = + − − −
Assumption: Linear elastic material (Hooke’s Law) & isotropic
E
Institute of Structural Engineering Page 12
Method of Finite Elements I
3D elasticity equations The weak form of the problem can be obtained using the Galerkin method as:
( ) ( )T T Td d dΩ Ω Γ
Ω = ⋅ Ω+ ⋅ Γ∫ ∫ ∫ε w Eε u w F w t
where is the weight functionx
y
z
www
=
w
Institute of Structural Engineering Page 13
Method of Finite Elements I
Plane stress/strain
In several cases of practical interest:
• The third dimension of the problem is: either very small
or very large but includes no variation in the problem parameters
• The problem equations can be simplified resulting in a 2D problem
Institute of Structural Engineering Page 14
Method of Finite Elements I
Definition of strains in the 2D domain
[O] §4 Fig. 4.3
Institute of Structural Engineering Page 15
Method of Finite Elements I
Definition of stresses in 2D
[O] §4 Fig. 4.4: Definition of stresses σx; σy; σxy and principal stresses σI ; σII in 2D solids
Institute of Structural Engineering Page 16
Method of Finite Elements I30-Apr-10
First let’s consider a structure where:• The length in one dimension is
much smaller than the other two
• Loads are applied only within a plane
• The in-plane stresses, strains and displacements are constant in the third dimension
• Normal and shear stresses in the third dimension are negligible
Plane stress assumptions
Such a structure is said to be in a state of plane stress
Institute of Structural Engineering Page 17
Method of Finite Elements I
Plane stress equationsAssuming that stresses along the third dimension are zero:
0),(),(),(
=τ=τ=σ
τ=τ
σ=σσ=σ
yzxzz
xyxy
yy
xx
yxyxyx
0
0
=+∂σ∂
+∂τ∂
=+∂τ∂
+∂σ∂
yyxy
xxyx
Fyx
Fyx
, 00
xz
zx
yy xz
xy yzyx
uuxzu
yuu
y x
εεε εγ ε
∂ ∂=∂ ∂ ∂ = = = ∂ = ∂ ∂ + ∂ ∂
ε
( )
2
1 01 0
110 0
2
1
x x
y y
xy xy
z x y
Eσ ν εσ ν ε
νσ ν γ
νε ε εν
= − −
= − +−
Constitutive equationStrain definition
Equilibrium equations:Assumptions:
How is this derived?
Institute of Structural Engineering Page 18
Method of Finite Elements I30-Apr-10
Examples of plane stress problems
Circular Plate UnderEdge Loadings
Thin Plate WithCentral Hole
Institute of Structural Engineering Page 19
Method of Finite Elements I30-Apr-10
Examples of plane stress problems
[O] §4 Fig. 4.1
Institute of Structural Engineering Page 20
Method of Finite Elements I30-Apr-10
Next we consider a structure where:• The length in one dimension is
much larger than the other two
• Loads are applied only within a plane
• Loads are constant in the third dimension
• Displacements and strains along the third dimension are negligible
Such a structure is said to be in a state of plane strain
Plane strain assumptions
Institute of Structural Engineering Page 21
Method of Finite Elements I
Plane strain equationsNext we assume that strains along the third dimension are zero:
0
0
=+∂σ∂
+∂τ∂
=+∂τ∂
+∂σ∂
yyxy
xxyx
Fyx
Fyx
0, 0
0
x
x zy
y xz
xy yzyx
uxuy
uuy x
ε εε γγ γ
∂ ∂ = ∂ = = = ∂ = ∂ ∂ + ∂ ∂
ε
( )
1 01 0
(1 )(1 2 )1 20 0
2
x x
y y
xy xy
z x y
Eσ ν ν εσ ν ν ε
ν νσ ν γ
σ ν σ σ
− = − + − −
= +
Constitutive equationStrain definition
Equilibrium equations:Assumptions:
( , )( , )
0
x x
y y
z
u u x yu u x yu
==
=
Institute of Structural Engineering Page 22
Method of Finite Elements I30-Apr-10
Examples of plane strain
x
y
z
x
y
z
P
Long CylindersUnder Uniform Loading
Semi-Infinite Regions Under Uniform Loadings
Institute of Structural Engineering Page 23
Method of Finite Elements I30-Apr-10
Examples of plane strain[O
] §4
Fig
. 4.2
Institute of Structural Engineering Page 24
Method of Finite Elements I
Weak form of the problemThe weak form of the problem is similar to the one used for the 3D problem:
( ) ( )T T Td d dΩ Ω Γ
Ω = ⋅ Ω+ ⋅ Γ∫ ∫ ∫ε w Eε u w F w t
where:
stresses, strains, displacements and constitutive matrices correspond to the plain stress/strain case
integration is carried out over an area instead of a volume
Institute of Structural Engineering Page 25
Method of Finite Elements I
Discretization Several options are available for discretizing the
plane stress/strain problem
One of the most common choices is isoparametricLagrange elements
In the following, we will briefly review some basic properties of isoparametric elements
Some specific, and widely used, elements will be presented for plane stress/strain analysis
Institute of Structural Engineering Page 26
Method of Finite Elements I30-Apr-10
1 1( , ) ; ( , )
n n
i i i ii i
x h r s x y h r s y= =
= =∑ ∑
Using the isoparametric concept, geometry can be discretized as:
Isoparametric formulation
and displacements as:
1 1( , ) ; u ( , )
n n
x i xi y i yii i
u h r s u h r s u= =
= =∑ ∑
where are nodal values of the spatial coordinates and displacement components
, , ,i i xi yix y u u
Institute of Structural Engineering Page 27
Method of Finite Elements I
Isoparametric quadrilaterals
1 11 ( , )x y
2 22 ( , )x y
3 33 ( , )x y
4 44 ( , )x y
1 ( 1, 1)− − 2 (1, 1)−
4 ( 1,1)− 3 (1,1)
r
s
x
y
For a 4 noded linear isoparametric quadrilateral (q4), coordinates r and s are defined based on the following transformation:
Institute of Structural Engineering Page 28
Method of Finite Elements I30-Apr-10
4 4
1 1( , ) ; ( , ) i i i i
i ix h r s x y h r s y
= =
= =∑ ∑
Then the displacement and geometry approximations specialize to:
Isoparametric quadrilaterals
4 4
1 1( , ) ; u ( , ) x i xi y i yi
i iu h r s u h r s u
= =
= =∑ ∑
with
1 2
3 4
1 1(1 )(1 ); (1 )(1 );4 41 1(1 )(1 ); (1 )(1 );4 4
h r s h r s
h r s h r s
= − − = + −
= + + = − +
Institute of Structural Engineering Page 29
Method of Finite Elements I30-Apr-10
Isoparametric quadrilaterals
1h 2h
3h 4h
Institute of Structural Engineering Page 30
Method of Finite Elements I30-Apr-10
1
1
2
1 2 3 4 2
1 2 3 4 3
3
4
4
0 0 0 00 0 0 0
xyx
h h h h yxh h h h xy
yxy
=
x
x
N
The shape functions can also be written in matrix form as:
Isoparametric quadrilaterals
1
1
2
21 2 3 4
31 2 3 4
3
4
4
0 0 0 00 0 0 0
x
y
x
x y
y x
y
x
y
uuu
u uh h h hu uh h h h
uuu
=
u N
u
=u N u =x N x
Institute of Structural Engineering Page 31
Method of Finite Elements I30-Apr-10
Derivatives with respect to the spatial coordinates can be obtained via use of the Jacobian (Lecture 4):
Shape function derivatives
1
−
∂ ∂= ⇒ =
∂ ∂=
J Γr x x rΓ J
∂ ∂∂ ∂
x yxr r r
x yys s s
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= ∂∂ ∂ ∂
∂ ∂ ∂ ∂ J
r x
∂ ∂∂ ∂
In addition, the infinitesimal surface element can be transformed as:
det( )d dx dy dr dsΩ = = J
Institute of Structural Engineering Page 32
Method of Finite Elements I30-Apr-10
Using the above transformations, strain components can be obtained, for instance:
4
,1
( , )xx i x xi
i
u h r s ux
ε=
∂= =∂ ∑ , 11 12
( , ) ( , ) ( , )( , ) i i ii x
h r s h r s h r sh r sx r s
∂ ∂ ∂= = +
∂ ∂ ∂Γ Γwith
The above can be written in matrix form as: 1
1
2
21, 2, 3, 4,
3
3
4
4
0 0 0 0
x
y
x
yx x x x x
x
y
x
y
uuuu
h h h huuuu
ε
=
Shape function derivatives
Institute of Structural Engineering Page 33
Method of Finite Elements I30-Apr-10
All strain components can be similarly computed and gathered in a matrix using Voigt notation:
1
1
21, 2, 3, 4,
21, 2, 3, 4,
31, 1, 2, 2, 3, 3, 4, 4,
3
4
4
0 0 0 00 0 0 0
x
y
xx x x x
yy y y y
xy x y x y x y x
y
x
y
uuu
h h h hu
h h h hu
h h h h h h h huuu
=
u
B
ε
=ε B u
Shape function derivatives
Institute of Structural Engineering Page 34
Method of Finite Elements I30-Apr-10
Next the strains can be substituted in the weak form to obtain the stiffness matrix as:
Stiffness matrix and load vectors
T dΩ
= Ω∫K B EB
In the above, the domain of integration is defined with respect to the xy system and the shape functions with respect to r and s. Therefore a change of variables has to be performed as follows:
1 1
1 1
det( )T Td drdsΩ − −
= Ω =∫ ∫ ∫K B EB B EB J
Institute of Structural Engineering Page 35
Method of Finite Elements I30-Apr-10
In a similar manner load vectors can be obtained, for instance due to a body force:
1 1
1 1
det( )T drds− −
= ∫ ∫Ff N F J
For surface tractions, integration has to be carried out along the sides of the quadrilateral, e.g.:
1
1
det( )T dr−
= ∫t s1f N t J
where is the Jacobean determinant of the side where the load is applied
)det(𝐉𝐉𝐬𝐬𝐬𝐬
Stiffness matrix and load vectors
Institute of Structural Engineering Page 36
Method of Finite Elements I30-Apr-10
By reviewing the derived expressions:
1 1
1 1
det( )T drds− −
= ∫ ∫K B EB J
It can be observed that: integration over a general quadrilateral was reduced to
integration over a square the expressions to be integrated are now significantly more
complicated due to the use of the chain rule analytical integration might not be possible
1 1
1 1
det( )T drds− −
= ∫ ∫Ff N F J
1
1
det( )T dr−
= ∫t s1f N t J
Stiffness matrix and load vectors
Institute of Structural Engineering Page 37
Method of Finite Elements I30-Apr-10
To overcome this difficulty, numerical integration is typically employed, thus reducing integrals to sums:
1 1
1 1
det( ) ( , ) ( , )det[ ( , )]T Ti j i j i j i j
i jdrds w w r s r s r s
− −
= =∑∑∫ ∫K B EB J B EB J
where are the weights and coordinates of the Gauss points used
, ,i i iw r s
Notice that the Jacobian is also evaluated at the different Gauss points since it is not constant in general!
Stiffness matrix and load vectors
Institute of Structural Engineering Page 38
Method of Finite Elements I30-Apr-10
Gauss point weights and coordinates are typically precomputed can be found in tables. For instance for a linear quadrilateral:
Numerical integration
1 1
2 2
11, ,3
11,3
w r
w r
= = −
= =
In 1D:
The above can be combined to obtain the 2D coordinates and weights
12
34
1 2
4 3
r
s
x
y
Institute of Structural Engineering Page 39
Method of Finite Elements I30-Apr-10
The procedure presented above can be performed for different kinds of elements just by changing the shape functions and Gauss points used, for instance:
8 noded quadratic quadrilateral (q8) Shape functions
Other isoparametric elements
1
2
34
1 2
4 3
r
s
x
y
5
6
7
8
5
6
7
8
( )
( )
( )
( )
( )( )
( )( )
( )( )
( )( )
1
2
3
4
2
5
2
6
2
7
2
8
1 (1 )(1 ) 141 (1 )(1 ) 141 (1 )(1 ) 141 (1 )(1 ) 141 1
21 1
21 1
21 1
2
h r s r s
h r s r s
h r s r s
h r s r s
r sh
r sh
r sh
r sh
= − − − − −
= + − − + −
= + + − + +
= − + − − +
− −=
+ −=
− +=
− −=
Institute of Structural Engineering Page 40
Method of Finite Elements I30-Apr-10
The procedure presented above can be performed for different kinds of elements just by changing the shape functions and Gauss points used, for instance:
8 noded quadratic quadrilateral (q8) Gauss points
Other isoparametric elements
12 3
4
1 2
4
3r
s
x
y5 6
7 8
5 6
7 89
9
1 1
2 2
3 3
5 / 9, 0.68 / 9, 0
5 / 9, 0.6,
w rw r
w r
= = −= =
= =
In 1D:
The above can be combined to obtain the 2D coordinates and weights
Institute of Structural Engineering Page 41
Method of Finite Elements I30-Apr-10
Two-dimensional Gauss quadrature for rectangular elements
P. W
rigg
ers,
Com
puta
tiona
l Con
tact
Mec
hani
cs (2
006)
Institute of Structural Engineering Page 42
Method of Finite Elements I30-Apr-10
The procedure presented above can be performed for different kinds of elements just by changing the shape functions and Gauss points used, for instance:
Linear or constant strain triangle (CST) Shape functions
Other isoparametric elements
1
2
3
1 2
3
r
s
x
y
1
2
3 1
h rh sh r s
=== − −
Institute of Structural Engineering Page 43
Method of Finite Elements I30-Apr-10
The procedure presented above can be performed for different kinds of elements just by changing the shape functions and Gauss points used, for instance:
Linear or constant strain triangle (CST) Gauss points
Other isoparametric elements
1
2
3
1 2
3
r
s
x
y
1 1 11, 1/ 3, 1/ 3w r s= = =
Institute of Structural Engineering Page 44
Method of Finite Elements I30-Apr-10
Two-dimensional Gauss quadrature for triangular elements
P. W
rigg
ers,
Com
puta
tiona
l Con
tact
Mec
hani
cs (2
006)