Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 70

∂
∂

N
x

i

0 0

∂
∂

N
y

i

0 0

∂
∂

N
z

i

0 0

B i =

∂
∂

N
y

∂
∂
∂
∂

N
x
N
z

i i

0

∂
∂

N
y

i i

0

∂
∂

N
z

∂
∂

N
x

i i

0

Equation 3.28 expresses the relationship between strains and displace-
ments, for both an individual element and the entire domain. When the
entire domain is considered, the node number n will be the total nodes
meshed in the domain. Thus, Equation 3.27 becomes Equation 3.29 when
substituting εε with Equation 3.28,

∫

T

B DB a dv = f (3.29)

or the global equilibrium Equation 3.3 where

= ∫

T

K B DB dv (3.30)

K is the so-called global stiffness matrix. When a domain of an individual
element is considered, the results of Equation 3.30 will be the stiffness
matrix of an element.
The global equilibrium equation 3.29 or 3.3 can also be derived from the
principle of virtual works. Given any equilibrium state of a system, small
fictitious displacements—the virtual displacements—are assumed. The
virtual displacement will cause internal virtual strains. The virtual work
principle states that the virtual work done by actual external forces during
the virtual displacements is equal to the internal strain energy increased at
actual internal stresses due to the virtual strains:

∫

T T

δε σ dv = δ a f (3.31)

where δε denotes internal virtual strains corresponding to external virtual
displacements δ a . Applying Equation 3.28 into 3.31, the equilibrium equa-
tion can be obtained as