Page 164 - Bridge and Highway Structure Rehabilitation and Repair

CHAPTER 4 AN ANALYTICAL APPROACH TO FRACTURE AND FAILURE 139

direction. With the use of diaphragms in the transverse direction, a load distribution factor
is applied. The longitudinal beam is assumed stiffer than the transverse diaphragm. This
approach is commonly used in the U.S. The primary mode of deformation is by bending
and torsion.

The simplified beam theory uses AASHTO LRFD design code for two or more design
lanes loaded (AASHTO Table 4.6.2.2.2.b-1). A set of tables for distribution coeffi cients for
computing bending moments are given in AASHTO 2007. The distribution formula Eq.
(4.6a), takes into account spacing, span, and beam stiffness such as:
DF 3 0.075 4 (S/9.5) (S/L) (Kg /12 L t ) (4.6a)
0.2
0.6
3 0.1
s
where: S 3 beam spacing, L 3 span, t 3 thickness of slab.
s
In older AASHTO codes, the distribution of loads in the transverse direction was based
on a simpler equation for the distribution factor. The model used for original analysis was
a typical longitudinal interior girder, which was generally simply supported between abut-
ments or not fully continuous (with two lines of bearings over the piers), namely
DF 3 Girder spacing/5.5 (4.6b)
Spacing generally varied between 6 and 12 ft and the empriicaldistribution factor varia-
tion differed widely for each spacing. A similar approach is used for DF of timber bridges
(AASHTO Table 4.6.2.2.2.a-1).
• In the grid or grillage theory in which the transverse spanning deck strips (and diaphragms)
allow full two-way bending action. Transverse diaphragms are idealized as truss members.
The bridge deck is idealized as a grid in plan (line girders in longitudinal directions and
line diaphragms in transverse directions). The grid or grillage theory is commonly used
in European countries.
2. Comparison between the simplified beam theory and grillage theory.

With a longer span and small spacing, longitudinal girder bending will be controlled. For
small spans and wider spacing, transverse distribution of moments and forces will be higher
(approaching a grillage type bending). A fi nite element analysis for the bridge deck using
the grillage model will give a more accurate estimate of girder deflection, moment, shear

distribution, and diaphragm forces.
• In both the girder and grillage theories, deck slabs are analyzed as a thin plate. However,
for lower L/d ratios, such as a 9-inch thick deck slab with a continuous span of 6 feet, for
example, it is likely to behave as a thick plate.
• Due to composite action between girders and the deck slab, arching action may result,
thereby increasing the compressive forces in the top flange of girders.

3. Compact rolled steel joists (compactness defined by Iyc/Iyy) have proven to perform better
transverse stress distribution than the fabricated plate girders. Use of non-compact plate
girder sections will only allow partial moment and shear distributions compared to girders
with compact sections. Due to the relative thickness of the web and the gentle transition

between the rolling of flange and web profile, there are fewer instances of rolled steel joist

failures than those of plate girders resulting from local web or fl ange buckling.
4. Basic assumptions for analysis:
• Diaphragms can be modeled as co-existing beam or truss in the transverse direction. Truss ac-

tion is by X-frame, and K-frame configurations of the diaphragm need to be considered.
• The concrete deck may be assumed isotropic, i.e., having equal strength in all directions.
Certain types of grid decks may be orthotropic. Poisson’s ratio for the slab and beam
system for one-way action is neglected in the longitudinal beam theory.
• Vertical shear deformation in the deck slab is considered negligible.
• Wheel loads may be approximated as an equivalent patch load acting under the wheel
width. A 45 degree distribution in both directions may be assumed, located away from the