Computer Aided Engineering Design

328 COMPUTER AIDED ENGINEERING DESIGN

whereB is the strain-displacement matrix which is a constant and depends on the position of nodal
coordinates. Thus, a triangular element is sometimes referred to as the constant strain triangular or
CST element. From linear elasticity, strains are related to stresses in three dimensions as

εσνσσxxyz

E

=^1 [] – ( + )

εσνσσyyzx

E

=^1 [] – ( + ) (11.10j)

εσνσσzzxy

E

=^1 [] – ( + )

γ

τ

γ

τ

γ

τ

xy

xy

yz

yz

xz

xz

= GGG, = , =

whereE is the elastic modulus, ν is the Poisson’s ratio and G is the shear modulus defined as

G = E

2 (1 + )ν

. Also σx,σy and σz are the normal stresses along the subscript directions and τxy is the

shear stress in the x plane along the y direction. For a plane stress case, where the stresses are non-
zero only in a plane, say, the xy plane (that is, σz = τxz = τyz = 0), we have

εσνσ σ

νν

ε

νν

ε

εσνσ σ

νν

xxy x x y

yyx y

E

E

E

E

=^1 [ – ()] =

2

1

1 –

+

1

1 +

+^1

1 –

• 
  

1

1 +

=^1 [ – ()]or =

2

1

1 –

• 
  

1

1 +

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

⎛

⎝

⎜⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

ε ννε ⎥

γ

ντ

τ

γ

ν

xy

xy

xy

xy

xy

E

E

+^1

1 –

+

1

1 +

=

2 (1 + )

=

2 (1 + )

(11.10k)

which in matrix form becomes

 = =

(1 – )

10

10

00 1 –

2

=

σ

σ

τ

ν

ν

ν

ν

ε

ε

γ

x

y

xy

x

y

xy

E

⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

⎛

⎝

⎜

⎜

⎜⎜

⎞

⎠

⎟

⎟

⎟⎟

⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

2 D (11.10l)

whereD is the elasticity matrix for the plane stress case. The strain energy stored in the element then
is

SE dV d V dV dV

V

T

V

T

V

TT T

V

=^1 T

2

=^1

2

=^1

2

=^1

∫∫ ∫ 2 ∫

⎧

⎨

⎩

⎫

⎬

⎭

  D u B DBu u B DB u

The element stiffness matrix ke is

ke B DB B DB

V

= TTdV At=

∫

(11.10m)

wheret is the out-of-plane thickness and the constant matrices B and D are given by Eqs. (11.10i) and
(11.10l).