x,u

y,v

z,w

Fx Fx

Fy

Fy

PELAT MEMBRAN

M A T R IK H O O K E [H ] & [C ]

+

+

+

)2(100000

0)2(10000

00)2(1000

0001--

000-1-

000--1

E

1

υ

υ

υ

υυ

υυ

υυ

−

2

2-100000

02

2-10000

002

21000

000-1

000-1

000-1

υ

υ

υυυυ

υυυ

υυυ

)2-)(1(E

E

υυ+

[C] =

[H] =

ρ = massa volumik; E= modulus Young; υ = koefisien Poisson

pres. 7

P L A N E S T R A IN

HIPOTESA : u = u (x,y) ; v = v(x,y) ; w = 0εz = ε xz = ε yz = 0

matrik Hooke menjadi :

−

−

+

2

2-100

01

01

)2-)(1(1

E

υυυ

υυ

υυ

−

−+

200

01

01

E

1υυ

υυυ

[ C ] = [ H ]-1 =

[H] =

y

z x Penampang dianalisa sbg plane

strain

PRES. 8

PLANE STRAIN

y,v

x,u

z

Kondisi Plane Strain

Retainning Wall

{ }

+

+

+=

=

=

yz

xz

xy

z

y

x

yz

xz

xy

zz

yy

xx

w,v,

,wu,

v,u,

w,

v,

u,

2

2

2

yz

xz

xy

zz

yy

xx

γ

γ

γ

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

xy

yy

xx

2ε

ε

ε

xy

yy

xx

γ

ε

ε

+ xy

y

x

v,u,

u,

u,

{ε } = = =

DEFORMASI LINIERDEFORMASI LINIERDARI TEORI ELASTIS 3 DDARI TEORI ELASTIS 3 D

PROBLEM 2D :PROBLEM 2D :

PRES. 5

H U B U N G A N

T E G A N G A N - R E G A N G A N

{σσ } = {H} { εε}

{H} = Matrik Hooke

atau :

{ εε } = [ H ]-1{σσ } = [C] { σσ }

εx = )( zyE1 νσνσσ −−x

ε y = )( zyE1 νσσνσ −+− x

εz = )( zyE1 σνσνσ +−x

xyG1

xy σγ = ; xzG1

xz σγ = ; yzG1

yz σγ =

G = ) 2(1

E

υ+

MATERIAL ISOTROP :

PRES. 6

PLANE STRESS

Struktur Membrane

x

y

z

xy

σ

σ

σ

σ

σ

σ

σσ

xy

xy

xy

x

x

y

y

σ

z

y

σ

σ

y

y

z= 0x

dx

dy

dA

Komponen tegangan suatu pelat tipis dibebani sejajar pelat

h

L

i

b

a

ba

elemen ke i

nodal elemen

P

P

P

P

σ z = σ xz = σ yz = 0

P L A N E S T R E S S

HIPOTESA :

+

−

)2(100

01

0-1

E1

υ

υ

υ

2

-100

01

01

υυ

υ

)-(1E

2υ

[C] =

[H] =

PRES. 9

X

Y

P

P

HUBUNGAN TEGANGAN REGANGAN

{ σ } = [Hσ] {ε}

[ ]HE

συ

υυ

υ=

− −

1

1 01 0

0 0 12

2

{ }σσστ

=

xyxy

{ }εεε

γ=

xyxy

AXISYMMETRIC

r

r,u

z,v

θθ,w

MATRIKS HOOKEUNTUK MATERIAL ISOTROP-HOMOGEN & LINIER

ELASTIS

Plane Stress

[ ]HE

=− −

1

1 01 0

0 01

2

2υ

υ

υυ

Plane Strain

[ ]

( )( )( )

( )

( )( )

( )

HE

=−

+ −

−

−−

−

11 1 2

11

0

11 0

0 01 2

2 1

υυ υ

υυ

υυ

υ

υ

Axisymmetric

[ ]( )

( )( )

( ) ( )

( )

( )

HE

Simetric=

−+ −

− −

−

−−

11 1 2

11 1

0

1 1 01 0

1 22 1

υυ υ

υυ

υυ

υυ

υυ

E = Modulus Young ; υ = Poisson’s ratio

X

Y

σ x

σ yτyx

τxy

σ

τ

τxy

τxy0

σ y

σ σx y+

2

σ σx y−

2

2θ

σ x

σ 1

σ 2A

B

OC

R

D

F

E

( )Rx y

xy=−

+

σ στ2

22

( )

σσ σ

τσ σ

1 2

1 2

2

2

,

max

=+

±

=−

x y

xy

R

LINGKARAN MOHR

E L E M E N 2 DIM E N SI

TRIANGULAR

QUADRILATERAL

3 nodal 6 nodal9 nodal

4 nodal 8 nodal12 nodal

PRES. 10

η

λ=1-ξ-η

ξ

3

1

2

N1 = λ =1-ξ-η

N2 = ξN3 = η

1

ξ

2 3

46

5η

N1=λ(2λ-1)

N2=4λ ξ

N3=ξ(2ξ-1)

N4=4ξ η

N5=η (2η -1)

N6=4λ η

Triangular 6 nodal

η

ξ

9

8

7

6

5 10

1 2 3 4

λ = 1-ξ -η N1= ½ λ (3λ -1)(3λ -2)

N2 =½ 9λ ξ (3λ -1)

N3=½ 9λ ξ (3ξ -1)

N4=½ ξ (3ξ -1)(3ξ -2)

N5=½ 9ξ η (3ξ -1)

PRES. 11

Triangular 3 nodal

Triangular 10 nodal

F U N G S I B E N T U KE L E M E N S E I T I G A

N6=½ 9ξ η (3η -1)

N7=½ η (3η -1)(3η -2)

N8=½ 9 λ η (3η -1)

N9=½ 9λ η (3λ -1)

N10=½ 54λ ξ η