Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 60

within an element are interpolated from nodal displacements. Together
with mesh density, the displacement pattern affects the convergence of
the FEM solution. Displacement pattern is defined by different types of
elements. Therefore, once the types of elements used to discrete the sys-
tem are decided, displacement patterns are automatically determined.

3. Compute the stiffness matrix of every element and assemble the global
stiffness elements.

4. Prepare the global stiffness matrix according to known boundary con-
ditions. As any arbitrary rigid movements can satisfy Equation 3.3,
the global stiffness matrix K becomes singular. To solve Equation 3.3,
K has to be condensed to contain only unknown nodal displacements.

5. Solve Equation 3.3.

6. Compute strains and stresses of each element. Once nodal displace-
ments are solved, displacements at any point within an element can be
interpolated by assumed displacement patterns. Furthermore, strains
and stresses at any point of element can be obtained.

Theories and literatures on FEM are widely available. In this chapter, the
key procedures like a generic FEM and some other special topics regarding
its numerical application in bridge structural analyses will be discussed.

3.2.2 Geometric and elastic equations
When external loads are acting on an elastic body, displacements and
deformations * will be induced. The displacement at any point a is described
by its projection on the Cartesian axes, u v w , , , respectively, as shown in

Figure 3.1. These three displacement components are functions of coordi-
nates x y z , respectively.

, ,

a = [ v w T

]

(3.4)

u

The deformation at any point in the elastic body is described by three
direct strains and three shear strains, which are the first derivations of
displacements. †

T

ε = ε ε ε γ γ γ

x y z xy yz zx

T

u
x

v
y

w
z

u
y

v
x

v
z

w
y

w
x

u
z

∂
∂

∂
∂

∂
∂

∂
∂

+ ∂
∂

∂
∂

+ ∂
∂

∂
∂

∂
∂

= +

(3.5)

* Displacement refers to translational or rotational movement along a direction and is used
to measure the absolute geometric change at a point in structure. Deformation refers to
shape change in a direction and is used to measure the strain at a point in the elastic body.
† When geometric nonlinearity is considered, the second order derivatives will be included as
in Equation 3.35.