1

STRUCTURAL AND GEOTECHNICAL ENGINEERING DEPARTMENT

ROCK MECHANICS 2ROCK MECHANICS 2

Giovanni Barla

Politecnico di Torino

LECTURE 11 - OUTLINEThe Finite Element Method2D and 3D Problems

(a) Triangular elements with quadraticand cubic interpolation functions(b) Triangular three noded isoparametricelement

x

y

i

k

j

P(x,y)

Free Point

Linear Function:u]k= a+bx+cyQuadratic Function:u]k= a+bx+cy+dx2+exy+fy2Cubic Function:u]k= a+bx+cy+dx2+..+iy3

Linear Function:u]k= a+bx+cyQuadratic Function:u]k= a+bx+cy+dx2+exy+fy2Cubic Function:u]k= a+bx+cy+dx2+..+iy3

y

x

x

y

x

y

Over each element the displacement components are assumed to have a simple polynomial form, where the order of the polynomial depends on the number of nodes in the element

Over each element the displacement components are assumed to have a simple polynomial form, where the order of the polynomial depends on the number of nodes in the element

TRIANGULAR ELEMENTS WITH HIGH ORDER INTERPOLATIONTRIANGULAR ELEMENTS WITH HIGH ORDER INTERPOLATIONRECTANGULAR ELEMENT - LSTRECTANGULAR ELEMENT - LST

x

y

Rectangular 4 noded elementRectangular 4 noded element

i

k

j

P(x,y) Free Point u(x,y)=1+2x + 3y + 4xy

v(x,y)=5+6x + 7y + 8xy

l

p NOTE: element with sides in the samedirection as the coordinate axes

u(x,y)

v(x,y)=

1 x y xy 0 0 0 0

0 0 0 0 1 x y xy

12345678

(11.1)

a

b

2

(a) SHAPE FUNCTION H[x,y] =[][A]-1

(11.2)H[x,y] = 1ab

(a-x)(b-y) x(b-y) xy y(a-x) 0 0 0 0

0 0 0 0 (a-x)(b-y) x(b-y) xy y(a-x)

(b) STRAIN MATRIX [B]

B[x,y] =ab1 (11.3)

(y-b) (b-y) y -y 0 0 0 0

0 0 0 0 (x-a) -x x (a-x)

(x-a) -x x (a-x) (y-b) (b-y) y -y

RECTANGULAR ELEMENT - LSTRECTANGULAR ELEMENT - LST

Isoparametric Elements - Area CoordinatesIsoparametric Elements - Area Coordinates

ISOPARAMETRIC TRIANGULAR ELEMENTISOPARAMETRIC TRIANGULAR ELEMENT

ISOPARAMETRIC ELEMENTS ( [H] can be defined directly) new formulation with respect to the generalizedcoordinates method the relationship between the nodal displacementsand the element displacements can be writtendirectly the relationship which describes the local coordinatesis used to define the shape function

ISOPARAMETRIC ELEMENTS ( [H] can be defined directly) new formulation with respect to the generalizedcoordinates method the relationship between the nodal displacementsand the element displacements can be writtendirectly the relationship which describes the local coordinatesis used to define the shape function

Derivation of shape and interpolation function (for the three noded element)

1

2

3P(x,y)

x

y

j

i

e1

e2

r

r = xi + yj = x3i + y3j + e1 + e2

x,y global coordinates, local coordinates

Introduce:

Let:

L1 = /l31 L2 = /l32

l31

l32

r = x3i + y3j + l31L1e1 + l32L2e2

where:

l31e1= (x1-x3)i+(y1-y3)j

l32e2= (x2-x3)i+(y2-y3)j

We can write for r:

r =[L1x1+L2x2+(1-L1-L2)x3]i+[L1y1+L2y2+(1-L1-L2)y3]j

continue

(11.4)

(11.5)

The link between coordinates x,y and Li (i=1,2,3) is therefore given by:

x=L1x1+L2x2+L3x3y=L1y1+L2y2+L3y3

where: L3=1-L1-L2

NOTE: 0

3

The area coordinates L1,L2,L3 can be taken as shape functions H1,H2,H3 so that we can write:

u

v=

L1 L2 L3 0 0 0

0 0 0 L1 L2 L3

u1u2u3v1v2v3

(11.11)

Therefore we have for []:

x= u L1L1 x+ uu L2 L3L2 L3x x

+

= 12A (b1u1 + b2 u2+ b3u3) and finally:

continue

(11.12)

ISOPARAMETRIC TRIANGULAR ELEMENTISOPARAMETRIC TRIANGULAR ELEMENT

x

y

xy

=

+

ux

yv

uy

vx

[]

[] =

2A=1

[] = [B]e [u]e[] = [C]e [] = [C]e [B]e [u]e

u1

v1

u2

v2v3

u3

b1 b2 b3 0 0 0

0 0 0 c1 c2 c3

c1 c2 c3 b1 b2 b3

plane strain plane stress

with constant parameters

continue

(11.13)

(11.14)

ISOPARAMETRIC TRIANGULAR ELEMENTISOPARAMETRIC TRIANGULAR ELEMENT

Ve [B]eT [C]e [B]e dV

= [B]eT[C]e [B]e A t

[k]e=

Finally, we have for the element stiffness matrix:

[k]e=

kii kij kik

kji kjj kjk

kki kkj kkk

(11.15)

(11.16)

kii kij

kji kjj 22

66

ISOPARAMETRIC TRIANGULAR ELEMENTISOPARAMETRIC TRIANGULAR ELEMENT

1 2

3

6

4

5

x

x

x= hi(r,s) xi

y= hi(r,s) yi

u= hi(r,s) ui

v= hi(r,s) vii=1

i=1

i=1

i=1

6 6

6

h1=1-r-sh2= rh3= s

h4=4r(1-r-s)

h5=4rsh6=4s(1-r-s)

SIX NODED ISOPARAMETRIC TRIANGULAR ELEMENTSIX NODED ISOPARAMETRIC TRIANGULAR ELEMENT

side nodes: corner nodes:

4

1 2

34

5

68

7

1 2

34

5

6

7

8r

s

xG

yG

x= hi(r,s) xi

y= hi(r,s) yi

8

8

i=1

i=1

(-1,-1)

(1,1)

u= hi(r,s) ui8

i=1

v= hi(r,s) vii=18

side nodes:h5=1/2 (1-r2)(1-s)h6=1/2 (1+r)(1-s2)h7=1/2 (1-r2)(1+s)h8=1/2 (1-r)(1-s2)

corner nodes:h1=1/4 (1-r)(1-s)-1/2h5-1/2h8h2=1/4 (1+r)(1-s)-1/2h5-1/2h6h3=1/4 (1+r)(1+s)-1/2h6-1/2h7h4=1/4 (1-r)(1+s)-1/2h7-1/2h8

EIGHT NODED ISOPARAMETRIC QUADRILATERAL ELEMENTEIGHT NODED ISOPARAMETRIC QUADRILATERAL ELEMENT