Page 239 - Bridge and Highway Structure Rehabilitation and Repair

214 SECTION 2 STRENGTHENING AND REPAIR WORK

3. Usually, the center of gravity of truck load coincides with the peak moment at the section to
give maximum positive bending moment. Similarly, maximum negative moment will occur
at a support due to different positions of axle loads.
4. Using the law of superposition, any section of beam magnitude of bending moment may
be added for each axle load to produce the combined bending moment from three or more

axles. The first axle is placed at the unit peak value and multiplied by the axle load to give
peak moment due to that load. Values of other axle loads, say spaced at 14 ft, are multiplied
by the corresponding unit values and added. For certain positions of truck axles, absolute
maximum moment may be obtained by comparison. Therefore, many positions of the truck
need to be considered.

5. For convenience, influence lines may be divided into regions to extract maximum positive
and negative moments. Maximum moments so generated will be multiplied by distribution
factors and impact factors to give design moment.

5.5.8 The Need for Live Load Distribution Factors
1. As discussed in Chapter 4, Section 4.4, the theoretical approach for slab and beam systems
should be considered. The true physical model is three-dimensional, with multiple beams
located under multiple lanes. Truck loads, with two or more rows of wheels placed 6 feet
apart transversely, will be distributed to more than one beam and will be shared by all the
beams to varying degrees. Wheel load is one-half of axle load.
2. For analysis of the superstructure, the designer has the option to use:
• A three-dimensional model including diaphragms and deck slab thickness using the fi nite
element method, Fourier Series, and Finite difference method. The time spent in model-
ing with appropriate boundary conditions is usually beyond the scope of a regular bridge
project. Complex bridges and bridges with unusual features such as very long spans and
very tall piers may warrant a complete model.
• A two-dimensional grillage model of beams in longitudinal direction and diaphragms in

transverse direction using differential equations: The finite difference method or Fourier

series approach is used to compute deflections, moments, and shear forces. Based on

previous investigations, simplified methods are available, and a modified grillage analysis
is used for saving time.
3. A significant contribution to the use of distribution factors was made in 1956 by Morice and

Little in England when a distribution coefficient method similar to that proposed by Guyon

and Massonnet in France was generalized. Harmonic analysis of orthotropic plates was

used. Cusens and Pama refined the method by considering higher order terms in harmonic
analysis.
As a most general case of two-directional bending of orthotropic slab-beam systems, both

flexural and torsional rigidity of longitudinal and transverse strips were considered. For a
bridge deck of span L and width 2b,
= 3 b/L (D / D ) 0.25 ;
y
x
3 (D 4 D )/ 2 D . D ; where
xy
x
y
yx
D 3 Longitudinal fl exural rigidity per unit width corresponding to
x
EI of longitudinal beam
D 3 Transverse fl exural rigidity per unit length corresponding to
y
EI of transverse beam
D 3 Longitudinal torsional rigidity per unit width corresponding to
xy
GJ of longitudinal beam
D 3 Transverse torsional rigidity per unit length corresponding to
yx
EI of longitudinal beam.