16. FE-formulation of three- and
two-dimensional elasticity
Finite Element Method
Differential Equation
Weak Formulation Approximating
Functions
Weighted Residuals
FEM - Formulation
Repetition –
Weak form of heat flow in two and three dimensions
• Start with balance equation (not diff. eq.)
• 1. multiply with arbitrary weight function v=v(x,y)
• 2. integrate over region
• 3. Integrate first term by parts (Green-Gauss theorem)
Repetition - Weak form of heat flow in two and three dimensions
• Insert the rewritten first term
• Use the natural boundary condition:
• Insert Fourier’s law:
Repetition - FE formulation -two dimensional heat flow
• Approximate the temperature
• Temperature gradient
• which gives
• Insert approximation in weak form
Na ) (x
T N [N1 N2...Nn]
;
y T x T T
Tn
T T
2 1
a
y N x N
y N x N
y N x N
n n
2 2
1 1
B
Ba
T where B N
Repetition - FE formulation -two dimensional heat flow
• Choose the weight function according to Galerkin
• since v is a scalar
• Insert in weak form
• As cT is arbitrary constants
Nc v
T
v cTN v cTBT BT ( N )T
Repetition - FE formulation
-two dimensional heat flow
Fundamental Equations Elasticity
Stresses
s Equilibrium Body force
b
Displacement u
Strains
e Kinematics
Constitutive law Differential eq.
sDe ~TD~ub 0
Equilibrium equation
• Equilibrium equation
• where
• Carrying out the multiplication gives 3 equations
Boundary conditions and Constitutive relation
• Boundary conditions
– u=g on Sg – on Sh
• The second boundary condition consist of
• Constitutive relation
h n
S
t T
A preliminary result
• Consider the arbitrary vector
• According to the kinematic relation we have
and
Weak form of equilibrium equation in three dimensions
• Start with equilibrium equation in the x-direction
• Multiply with the arbitrary function vx and integrate
• Integrating first three terms using Green-Gauss gives
• Since
Weak form of equilibrium equation in three dimensions
• Weak form of all three equations of equilibrium gives
• Adding the three equations gives
Weak form of equilibrium equation in three dimensions
• Noting that
• The weak form may be written
=
=
FE formulation of three-dimensional elasticity
• Approximation
• Galerkin’s method
• Inserting in weak form gives
• and since c is arbitrary, the Finite Element form is
FE formulation of three-dimensional elasticity
• Adding the constitutive relation
• The strains are given by
• Using the approximation we get
• The constitutive relation with approximation is then
• Inserting into the FE-form
FE formulation of three-dimensional elasticity
• Boundary conditions are
• The boundary may then be split into Sh and Sg
• That we can write
Natural, (static) bc
Essential, (kinematic) bc
FE formulation of three-dimensional elasticity
• Adding the force vectors into one gives
• and we can write
• Properties of K
• Essential boundary condition must be applied!
FE formulation of three-dimensional elasticity
• Element formulation
• where
• and
• also axisymmetric cases may be analysed
• Weak form of three-dimensional elasticity
• But use
• Integrating over thickness gives
Weak form of equilibrium equation in
two dimensions
FE formulation of two dimensional elasticity
• Inserting approximation and using Galerkin method
• The constitutive relation in 2-dim is
• where D is for plane strain or plane stress
• FE formulation of two-dimensional elasticity
FE formulation of two-dimensional elasticity
• Boundary conditions are
• The boundary may then be split into Lh and Lg
• That we can write
Essential Natural
CALFEM – solid elements
Triangle 4 triangles
Melosh Turner -
Clough
CALFEM – solid elements
Isoparametric elements – treated in chapter 20 ( Don’t use yet!)
