Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 61

y ( v )

σ y

σ z

τ yx

τ yz

τ zx

τ xz τ xy

τ zy

σ x

σ x

τ zy

τ zx

τ xz

τ yz

τ xy

τ yx

σ z

x ( u )

σ y

z ( w )

Figure 3.1 Stresses and denotations on an infinitesimal cube of any point in an elastic
body.

Equation 3.5 is the geometric equation that defines the relationship
between displacements and strains. When the displacements of an elastic
body are known, its strains can be derived from the geometry equation.
However, the displacements cannot solely be defined by known strains.
Any global rigid displacement can produce the same strains according to
Equation 3.5.
The displacements described by Equation 3.4 are generic for a point on
an elastic body. When a particular type of element is discussed, compo-
nents of displacements can be simplified or modified. For example, a two-
dimensional (2D) stress or strain element will not have the w component.
A spatial beam element will have rotational displacements along three
Cartesian axes, and Equation 3.4 will become:

T

a = u v w θ θ θ (3.6)

x y z

For isotropic elastic materials, according to Hooke's Law, the relationship
between stresses and strains is defined as

σ

µ σ µ σ

σ

µ σ µ σ

σ

µ σ µ σ

x y z y z x z x y

(3.7)

ε = − − ; ε = − − ; ε = − −

x y z

E E E E E E E E E

and