Approximate and refined analysis methods - Computational Analysis and Design of Bridge Structures - page 30

2.4 REFINED ANALYSIS METHODS
2.4.1 Grillage analogy method
Grillage (or grid) analysis has been used by the bridge engineers for quite
some time and is in the same category as the FEM (see Section 2.4.5).
Grillage method can be regarded as a special case of FEM (Jategaonkar
et al. 1985) with deck slab structure idealized as a 2D model consisting
of beam elements. In this type of model, the deck is cut, theoretically, into
pieces in both directions with each piece considered as a beam element
(Hambly 1991). Choice of these imaginary cuts is based on experience and
should be made with caution.
The general approach for the grillage analysis is to model the longitudi-
nal girders as beam elements, straight or curved. If the intermediate dia-
phragms are present within the spans, the transverse beam elements are
placed at the same locations as the diaphragms. If intermediate diaphragms
are far apart or not present, deck slab is modeled as transverse beam ele-
ments with deck's moment of inertia based on a certain effective width. In
this grillage model, each node has three degrees of freedom, one vertical
translational and two planar rotational degrees of freedom.
2D grillage analysis is simulated by 2D grillage of beams with different
section properties. The basic principle is the same as defined in Section 2.4.5.
The difference from a generic type of finite element is that only vertical flexure
and torsion of a beam are considered in a grillage element, as how most decks
behave. Therefore, each node in a grillage model has a vertical translational
displacement and two rotational displacements along axes in deck plane,
and a grillage element has only vertical bending moment, vertical shear, and
torque defined. When the grillage model is used, a suitable grillage mesh
should be defined to get meaningful results. Figure 2.13 shows examples for
four different types of bridges. Figure 2.13a shows stiffness to be about equal
along the longitudinal and transverse directions and the beam elements coin-
cide with the real longitudinal and transverse beams. Figure 2.13b shows
longitudinal beams that are more predominant and coincident with the beam
elements. The placing of the transverse beam elements is recommended that
a proper aspect ratio be maintained between transverse and longitudinal ele-
ments, at diaphragm locations if diaphragms are present and at equal spacing
to simulate the plate transverse distribution if no diaphragms are present.
Figure 2.13c is a bridge with closely spaced beams. For practical purposes,
each longitudinal beam element can represent more than one beam. The rule
of thumb is to place longitudinal beam elements no farther apart than about
one-tenth of the span (Hambly 1991). Figure 2.13d has wider beams with
two longitudinal members per beam. Usually for this type of structure, the
longitudinal members are much stiffer than the transverse members, which
may be representing just the thin slab on top.