Approximate and refined analysis methods - Computational Analysis and Design of Bridge Structures - page 38

(a) (b)
Figure 2.20 Idealized model using area elements: (a) actual structure; (b) structure ideal-
ized by 3D plane shell elements.

When applying plane shell elements in bridge analyses, it should be noted
that each node has only five, rather than six, degrees of freedom. The rota-
tional displacements along axis perpendicular to plate plane output from
a finite element analysis are faked by a technique avoiding ill-conditioned
stiffness matrix. The sixth rotational displacements are meaningful only at
nodes connecting kinked plates, and these displacements are caused only
because of geometric transformation from bending rotations in other planes.
Volume elements —A volume element is sometimes called solid element
with three, four, eight, or more nodal points. Figure 2.21 shows an eight-
node volume element as an example. In bridge superstructures, the model
usually can be built up from line or area elements or combinations of these
two types of elements. Volume elements are used rarely except for the sub-
structures with massive concrete piers or abutments. Even for the substruc-
tures, the line (beam) elements are used more frequently than the solid
elements because of easier usage and interpretation. If massive concrete is
used, it may be modeled by rigid link elements to simulate the rigid body
motion between two points.
For a typical 3D model of a slab-beam bridge, the slab is modeled as plates
(area elements) with thicknesses equal to the slab thicknesses. If the beam is
widely spaced, more nodes should be assigned between beams to simulate the
higher shear-lag effects between beams and slab deck. A good representation

σ z

Δ z

Typical at all nodes

1

τ zx

Δ y σ y

1

τ yz

Δ x

τ xy

σ x

1

(a) (b)
Figure 2.21 Finite element volume element. (a) Node displacements and (b) element
stresses.