Introduction to the Finite Element Method.pptx

A general introduction to the Finite element method

1. INTRO TO THE FINITE ELEMENT METHOD

2. Solid Elements:

Until this point, we have created elements for unique scenarios such as bars/trusses or
Euler-Bernoulli beams.

deu dty
cal) > —el($)=a

Now lets consider a 2D general solid element based on continuum equilibrium:

À … Se

General Approximation:
_ [Go + aıXı + a2X2 + 03X1X2 +"
D Lbo +b,X, + b2X2 + b3X,X2 +

—

X2

wo

3. Element Degrees of Freedom:

Recall in the Finite Element Method, we want to express the displacement approximation
in terms of nodal displacements (u; and v;) and shape functions (N;):

v3
v
uz
vi A rene UN, + UgN2 ++ UnNn
u 2 2D. [9,N, + V2N2 ++ Nn
u U2

From here, we can separate the nodal displacements and shape functions:
Uy

[MO No 0 EN, =0 oa
X20 = 1 q O SN 0 Nn 72 220 = |] = Niue

; u
> Functions of X, and X, ES = Constant

4. Strain Vector:

Recall that from the displacement function, we can determine the strain components:

Le du
a DA
M 0 N 0 + Nh 0]|* dv
Xv =| MEN Den, 72 e= ta |- AUS (2D Case)
A Ej
Un cs 1 we
Un + ox,
We know that u, is constant, therefore the differentiation only Sr to [N]:
ON, ON, ON u
aX, 2 aX, Cae aX, AA
SORA o M =>
= aX, aX, x, ||:
ON, AN, ON, ON, ONn 0Np||y, => 5 Matrix
OX, OX, 0X, OX, 0X, OX,

Un (Engineer Creating

5. Stress Vector:

From the strains (£), we know we can calculate the stresses (0):

= (ea) - [ec] le £22 :|- [Cle = | o = [C1[Blu. |

2&2
In 2D, all of the stresses or strains act in the same plane!

+ Can use plane stress/strain constitutive laws

Plane Stress (033 = 0): Plane Strain (£33 = 0):
1

a lege E
Vv
ani Ale)

[c] =

> E = Young's Modulus
> v = Poisson's Ratio

6. Principle of Virtual Work:

Knowing the stresses and strains, we can utilize the principle of virtual work:

IVW, = EVW, > Continuum Case

3
| > Ejoijdx = | tn wWds+ foo «u dx
de e

€ ¡j=1

Internal Virtual Work:

[eraacıte1Ja = |; tows + [pb-urax

[ut = | tn: Wds+ foo -u*dx > Ue, ug = Constant
e de e
Factor out displacement vectors (u, and uj):

uz Juricusiaxu, = Le -u'ds + fo «uv dx

a

7. External Virtual Work:

us | (21 [c]tBlaxu, -[ tn wds+ Jo -u’dx
e de e
Replace the general displacement vectors with nodal displacement vectors:
uz fiericusiaru = [ty ust)" )as + [pb (usina
e de e

> u; is Constant -. Factor Out

YN [BI"[C](Bldx u, Y, [NJ"tnds Y [NJ"pbdx

With u; appearing in every term, we can cancel it out:

[terieneiaxu. = [ (eds + [Ivi pba

(Ikelue = fe) a

8. Stiffness Matrix and Nodal Force Vector:

Virtual Work equation after simplification:
[terrteitziaxue = | Nas + [IN pbax = Ikelue = f
e de e
— A

Stiffness Matrix Nodal Forces Vector

Stiffness Matrix:
= Both are needed to determine the nodal
B]"[C][B]dx displacements!

[k
€ = However, we have yet to define the
shape functions (N;) needed for [B] and
IN]
fe= | [NT tnds+ N [NT pbdx = These shape functions will depend on
de e element type and geometric order

(discussed later) a

Nodal Forces Vector:

9. 3D Elements:

Now that we have the stiffness matrix and nodal forces vector for a 2D case, how about

ea MH

u Ay + A,X, + a2X2 + A3X3 + 4X1 X2 +
X3D = | =
w.

bo + b1X1 + b2X2 + b3X3 + b4XX2 +

Co + C1X1 + C2X2 + CaX3 + C4X1X2 ++"
Like the 2D case, we will modify the approximation function to be in terms of the nodal
displacements (u;, vj, and w;) and the shape functions (Nj):

UN, + U2N2 ++ Un Ny
X3p = | vıNı + V2N2 +--+ U,Ny
WN, + w2N2 ++ Wan r

10. 3D Elements (Nodal Displacements):
Now we will separate the shape functions from the nodal displacements:

u
V1
Wi
M 0 0 M 0 0 + N 0 0]
xp =|0 M 0 0 N 0 + 0 m 0||[2 =>
DTO: NIFSD 0 Ns EN,
Un
Un
Wa

> Notice how this results in the same expression with the only
exception being the additional row in the [N] matrix!

ro

11. 3D Elements (Stress and Strain):

Strain Components:
du ON ON; ON,
Ox, Fra CE CT ARE ner CPU
de an, ON, ON,
tu DA Crag omar a
£22 ow Fk SENS Laa oes
_ | €33 |_| 2x LA OX; X, DA
= l2&2| "|, | “Jan, on 0 2 2% CALE
2e3| [2% 2%] fax. Ox, OX, OX, OX, 0%
2e. ou 0w| [OM 5 am OM 9 OM OM ¿OM
232 (33, OXi| |3x, OX, OX; HA Oxy ox,
av , aw} |, 2M aM 5 ON: Nz o In Mn
IX, * 3%, x, Ox, x, OX, x OX,
Stress Components:
E

ou =
ie

=101|2e,,

2813

2823,

=> [c)=

(A-2v)(1+v)

= [Blue

> Gross!

0
0
0
0

12. 3D Elements (Principle of Virtual Work):

Principle of Virtual Work:

3
IVW, = EVW, = [Ye
E

ij=1

de = | end was + [ob u dx

> Same as 2D Case!

[terteieiaxu. = f (tds + INT pas

Stiffness Matrix:
BI"[C][B]dx = Same as 2D Case!
= The difference is the [B] and [N] matrix
Nodal Forces Vector: = However like before, we still have not

defined the shape functions (Nj)
f= | [NT tnds + Jen obax required for the [B] and [N] matrices
de e

13. General Element Requirements:

As you will see, different shape functions will be proposed for different element types
and geometric orders. Although different, each element shown will follow the general
element requirements:

1. Isotropy (Does not favor one direction over another)
U = ao + a,X, +a2X2 > Change in X, direction greater than X}!
2. Ability to Model Rigid Body Motion & Constant Strains

u =1X, +4,X, — Displacement changes along the element

3. Element Compatibility > CO Continuous
u

Fl