Approximate and refined analysis methods - Computational Analysis and Design of Bridge Structures - page 36

2.4.5 Finite element method
The mathematical theory and formulation of the FEM are well docu-
mented in many textbooks, and they are not in the scope of this chapter.
Regarding element types, a structure can be modeled using 1D, 2D, or 3D
elements or even the combinations of these three elements (Jategaonkar
et al. 1985).
Line elements —Line elements for modeling the bridge members include
two main types. The first is the bar type with only axial tension or compres-
sion with one degree of freedom at each node; usually it is used for modeling
a truss member, a bearing, or an individual member of the cross-frame. The
second type is the beam element, as shown in Figure 2.17, which has six
degrees of freedom. It is used usually to model the beam or column that has
axial stiffness as well as bending stiffness. For more simplified line elements,
certain degrees of freedom can be excluded for cases where only others are
of concern or predominant. For example, a 3D frame element can be ret-
rograded to a 2D beam element in cases that only two degrees of freedom,
vertical translational and rotational displacements, are considered.
A grillage model, in Section 2.4.1, is another example of this simplifica-
tion. For a grillage model, another degree of freedom, torsion, is added back
into the model. Because of the translational bending of the slab and dia-
phragm action, the main beams will be under torque. For highly skewed,
curved bridges, or with long overhang, this torsional action may be signifi-
cant. Therefore, displacements of vertical translational, vertical flexural rota-
tion, and axial torsional rotation are dominant in deck behavior, and when
only these displacements are considered, a 3D frame element is retrograded
to a grillage element. In this aspect, a grillage model and a plane frame model
are the same, except that the exclusion of displacements is different.

ω

θ z

μ

θ x

ω

υ

θ y

θ z

Beam element
Six degrees of freedom
at each node

θ x

μ

υ

θ y

Figure 2.17 Degrees of freedom of a 3D frame element.