Page 166 - Bridge and Highway Structure Rehabilitation and Repair

CHAPTER 4 AN ANALYTICAL APPROACH TO FRACTURE AND FAILURE 141

Table 4.1 Types of mathematical models.
Coordinate Dimensions
of Mathematical Model Numerical Model Type of Structural Theory
1 Stiffness method/fl exibility method Beam theory
Slope defl ection/moment distribution
Strain energy method
2 Grillage analysis Beam, two-dimensional thin
Finite difference/Finite elements plate theory
Harmonic analysis
Finite strip analysis
Classical plate solutions
3 Grillage analysis Three-dimensional theory of
Finite difference/fi nite elements elasticity for beams, thick and
Harmonic analysis simplifi ed thin plate theory
Finite strip
Classical plate solutions

• Pony trusses acting as through bridges and unbraced at the top. A through truss bridge
has a low degree of redundancy and greater risk of collapse.

4.5.3 Analysis of Box Beams
Three-dimensional analysis is required for transverse frame action and for modeling of
boundary conditions. Both St. Venant’s torsion stress and torsional warping stress need to be
considered. The following well-known stiffness matrix method is used by most bridge analysis
software. Using matrix notations, {F} 3 [k] { }.
4.5.4 Three Dimensional/Triaxial Behavior of Flat Plates and Overhead Sign Structures

A slab (or plate) offers the greatest benefits to the user, whether it is a bridge deck sup-
porting moving vehicles, a roof slab offering protection from the elements of nature, or a fl oor
slab in a residential building. Hence, stress distribution in slabs is of paramount importance in
structural engineering.
A thin plate element is subjected to beam type bending in two directions and the accom-
panying twisting moments and shear. A plate may be idealized as a series of beams placed in
two directions at right angles which interact with each other according to Poisson’s ratio, i.e., a
fraction of the load is also shared by beam bending at right angles to primary bending.
S. P. Timoshenko has expressed bending of a plate or slab using partial differential equa-
tions as follows:
Equation 4.4a for line element of a beam in the x direction can be generalized for a (plate)
surface element by allowing secondary curvature in the y direction as:
2
2
2
6M 3 D (; w/; x ) 4 ' D (; w/; y ) (4.6a)
2
x
where flexural rigidity D is similar to EI value of the beam. Moment of inertia of plate

3
includes Poisson’s ratio ', and while beam I 3 b h /12, Plate I 3 h /12 (1 6 ' ) for unit width
3
2
where ' is Poisson’s ratio. A plate will be subjected to maximum number of moments and forces
and is truly a 3-dimensional problem. As given by Timoshenko, theory of plates and shells
Chapter 1, Equation 3.
For unit width, D 3 E h /12 (1 6 ' )
3
2