Inteoduction to the finite element method - Triangular Finite Elements.pptx

A general introduction of triangular Finite elements used in the finite element method

1. TRIANGULAR ELEMENTS

2. Triangular Solid Elements:

Now that we have the expressions for the stiffness matrix and nodal forces vector for
general solid elements, we can begin to look at some common elements:

tkel= furicisia — fo= if (ends + [ NI" obax

> Depends on the element type and geometric order!

To begin, we will examine triangular elements:
X X

> 3 Nodes > 6 Nodes

X xX
Linear Triangular Element Quadratic Triangular Element aa

3. Linear Triangular Elements (Constant Strain Triangle):

Let’s start with the linear triangular element (constant strain triangle):

v3

X
> > 3 Nodes us ~ Each node has 2 DOF
1 N v V2
1 1 2 X Lek uz

A linear approximation function is assumed between nodes (hence linear):
2 [| pe R +aXı + #4 = E = (ane +uN, + Ee
220 = [ol = [bg + b,X, + bèXz = X20 = [vl = Lon, + v2N2 + va,

> Want to express the approximation in terms ofnodal
displacements (u; and v;) and shape functions (N;)

4. Shape Functions:

Using the same process as 1D elements, we can determine the shape functions:

M X Ne X N %

x Xi

N=1-X-Xz Nz =X, Nz = X2

m 0 M 0 M 0 = 177% 0 X 0 X 0
[0 M 0 M 0 N = 0 1-X,-X, 0 X 0 X
From the shape functions, we can then determine the B Matrix for the element:
AN, ON, aN,
OX, OX, DA
> z : Ad al

E 0 0 0

ON, ON, ON;
lee 0001]
1.10.2018 1550.

0X2 OX, OX? OX, 0X2 OX > Constant! = (lu. gpl

5. Stiffness Matrix:

With the B matrix known, we can now calculate the stiffness matrix:
(ke) = fisrricitsiax
e

> Depends on if plane stress or strain is assumed!

Assuming a constant into the page thickness (t) and plane strain:

-1 0 -1
-1 -1 1-v v 0
1% E 4 -1 0 1 0 0 0
a Wed y 0
hel =e f ral ES 0 -1 0 0 0 1|dX,dX,
0 0 1|[4+v04-2) 1-2v
0 40 = E
OMe On EL 0. 7 190215100,
Opt O
[BJ” [c] [8]

— Results in a 6x6 matrix! ro

6. Nodal Forces:

In addition to the stiffness matrix, we can also determine the nodal forces due to body

forces (pb) and traction vectors (t,):

1. Body Forces:

feow = [IM pb dx

1-X,-X, 0
0 1-X,-X,

rl fj A
fono = pt ff ji o x Fal dX,dX,
x 0

0 da

> Integrate over Volume

2. Traction Vectors:

far = | INV tn dx

nef

AA, 0
0 ty
a 0 ty
o x, lejos
x 0

0 sE

> Integrate over Surface

a

7. Quadratic Triangular Elements:

Now let's look at the quadratic triangular element:

X2 us
> > 6 Nodes ve vs —>Each node has 2 DOF
16 s => us
v
Y
1 14 2 X u us

A quadratic approximation function is assumed between nodes (hence quadratic):
Serie (‘| _ [ao + aıXı + a2X2 + 03X,X, + a4X? + asX?
20% Lol [hy 4B X, Ebo Xa + DXi Xo + bg Xe Hg XS

> ik Y = Er + U,N2 + UgN3 + UN, + USNs + UgNg
*20 = lo] = | v,N, + v2Nz + v3N3 + vaNa + U5Ns + V6No

8. Shape Functions:

Using the same process as 1D elements, we can determine the shape functions:
N, = (1—X, — X2)(1 — 2X, — 2X.)

x Np = X,(2X; — 1)

N3 = X2(2X2 — 1)

Na = 4X, (1 — X1 — X2)

Ns = 4X,X2

Ne = 4X2(1 — X1 — X2)

SN ON Nz PONS TN E 310) 4
IM=|0 Ni 0 N 0 N 0 N 0 Ns 0 | (Running Out of Colors!)

From the shape functions, we can then determine the B Matrix for the element:

M

Going to get messy!

ON, ON, ON; ON, ON; ONG
ee A Fra QU pra AM tra 0
ito aM, y Mm 5 M aN, 4 Ns (OM
= DA OX OX, DA Ox, 0X,

9. Stiffness Matrix and Nodal Forces:

With the new shape function matrix [N] and B matrix [B] know, we can calculate the
stiffness matrix and nodal forces vector:

Stiffness Matrix:
[ke] = [verifier ex > Results in a 12x12 matrix!

> Depends on if plane stress or strain is assumed!
Nodal Forces:

fe= | [Nt,ds + i [N]’ pbdx — Results in a 12 component vector!
de e

As we see, the complexity of the calculations is proportional to the degrees of freedom of
the element!

> This was 2D elements! Imagine 3D! 3

10. 3D Triangular Elements:

The 3D version of the triangular element is referred to as the tetrahedron element:

Linear Tetrahedron Element: Quadratic Tetrahedron Element:
> 4 Nodes > 10 Nodes
> Each node has 3 DOF > Each node has 3 DOF

Remember that the process is the same! Shape Functions > B Matrix > Stiffness Matrix a