Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 71

∫

( ) =

T

ψ a B σ dv − = f 0 (3.32)

where ψ is the sum of general internal and external forces. Equation 3.32
can be stated as that at any equilibrium point internal forces due to inter-
nal stresses should balance external loads that cause internal strains.
Furthermore, when physical equation 3.13 is substituted into Equation 3.32,
a more generic equilibrium equation can be obtained as

∫ ∫ ∫

( ) =

T T T

ψ a B DB a dv − B D ε dv + B σ dv − = f

0

(3.33)

0 0

Equation 3.33 illustrates the balance between internal and external forces
when initial strains and initial stresses exist.
Each element's stiffness matrix K e can be obtained by integration over
the entire element body. The physical meaning of any element at row i and
column j of K , , ( ) is the force caused at i ith degree of freedom because

of a unit displacement happening at j th degree of freedom, as the existence
or contribution of the element. The variables i and j are the order num-
ber of degree of freedom of an individual element. Because the total strain
energy of a continuum is the sum of strain energies of subdivided elements,
assembling all elements' stiffness matrices in an appropriate order can form
the global stiffness matrix in Equation 3.30. Obviously, if all elements con-
nected at a global node have the same local coordinate systems as the global
coordinate system, stiffness elements corresponding to this global node in
K can be obtained by summing the contributions ( K e i j

K i j

e e

( , ) ) from all con-
nected elements. This process is the assembly of global stiffness matrix,
which reveals the implementation of the approach by meshing a continuum
into finite regular-shaped elements.

3.2.5 displacement relationship processing
when assembling global stiffness matrix
As discussed in the Section 3.2.4, an element stiffness matrix will be
assembled into a global stiffness matrix. The assembly is done by matching
element nodes with their global order. For example, an element has two
nodes, i and j , and its element stiffness matrix is shown in Figure 3.7b.
When assembling, each submatrix in Figure 3.7b will be added into its
corresponding submatrix in the global matrix in Figure 3.7a. It should be
noted that the element stiffness matrix must be transformed into the global
coordinate system before adding it into the global matrix. The element
stiffness matrix is established in its local coordinate system, which is often
different from the global coordinate system. Because stiffness of a degree of
freedom is a vector in space, the transformation of the stiffness matrix can
be taken as a simple standard space transformation process.