Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 59

∂
∂
∂
∂

Π

a

1

∂
∂

Π Π

(3.1)

= =

0

a a

2



where a , a 1 , a 2 , … denote displacements and Π is the total potential energy—
the sum of the total internal strain energy and total potential energy of
external loads is

Π = U W + (3.2)

In Equation 3.2, U and W are the total strain energy and the total potential
energy of external loads, respectively. For a given domain, they can be sub-
divided into many regular or well-formed elements. The total strain energy
U is the sum of strain energies of individual elements. Given any admissible
displacements at the nodes of an element, if an appropriate displacement
pattern can be assumed based on these nodal displacements, displacements
at any point within the element can be expressed as a function of nodal
displacements. Strain can then be derived as a function of nodal displace-
ments. Considering the relationship between stress and strain, stress can be
expressed as a function of nodal displacements as well. The total potential
energy due to external loads is a simple function of nodal displacements.
Therefore, the total potential energy Π is a function of nodal displacements.
Applying variations over nodal displacements as in Equation 3.1 or the
well-known Rayleigh-Ritz process piecewise over all elements, a relation-
ship between unknown nodal displacements and known external loads can
be established as

(3.3)

Ka = f

where:
K is the so-called global stiffness matrix
f is external nodal loads
a is nodal displacements

The procedures of applying FEM for structural analysis are standardized
and can be summarized as follows:

1. Subdivide the continuum or structure into small elements. This step
is also called system discretization . Element types and mesh density
have to be determined in this step.

2. Determine an appropriate displacement pattern of an element. This
is critical to the solution as it derives how displacements at any point