Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 62

τ τ τ

xy yz zx

γ = γ = γ =

; ; (3.8)

xy yz zx

G G G

where:
E is the Young's modulus
G is the shear modulus
µ is the Poisson ratio

For isotropic materials, shear modulus can be derived from Young's modulus:

E

G =

(3.9)

( )

2 1 µ +

Equations 3.7 and 3.8 are elastic equations. By solving stresses in these
equations, another form of elastic equations can be written as Equation 3.10
or in matrix form as Equation 3.11.

( )

E 1 − µ

µ µ

σ = ε +

µ ε

+

µ ε

,

x x y z

( ) ( )

1 + µ 1 2 − u 1 − 1 −

( )

E 1 − µ

µ µ

σ =

µ ε

+ ε +

µ ε

,

y x y z

( ) ( )

1 + µ 1 − 2 u 1 − 1 −

(3.10)

( )

E 1 − µ

µ µ

σ =

µ ε

+

µ ε

+ ε ,

x x y z

( ) ( )

1 + µ 1 2 − u 1 − 1 −

E E E

, ,

τ = γ τ = γ τ = γ

xy

( )

xy yz

( )

yz zx

( )

zx

2 1 + µ 2 1 + µ 2 1 + µ

T

σ σ σ τ τ τ

(3.11)

x y z xy yz zx

T

= D ε ε ε γ γ γ or σ = D ε

x y z xy yz zx

where D is the so-called elastic matrix as shown in Equation 3.12.

λ + G λ λ

2

λ λ + G λ

2 0

λ λ λ + G

2

D =

(3.12)

G

0 0

G

0 0 0

G

0 0