Page 160 - Bridge and Highway Structure Rehabilitation and Repair

CHAPTER 4 AN ANALYTICAL APPROACH TO FRACTURE AND FAILURE 135

4.1.4 Relevant Theoretical Concepts
A bridge engineer needs to be familiar with the mathematical approach developed in the past
two centuries and how it should be applied to practical problems. A review of some analytical
methods is presented, and some newer topics are addressed. These include topics such as:

1. Nonlinear methods of analysis.
2. Arching action in slabs.
3. Shear defl ections.
4. Shear behavior of high strength concrete beams.
5. Application of theories of yielding and fracture.

6. Use of finite element methods.

7. Simplified formulae for moving loads.
4.2 STRESS ANALYSIS
4.2.1 A Review of the Basic Theory of Elasticity

1. Stress management may be defined as measuring the minutest change in length as a ratio of
the original length, multiplying strain by modulus of elasticity or rupture of the material and
making sure that the resulting stress does not exceed the bond between atomic particles.
2. Elongation gives rise to strain, which results in stress. Excessive elongation or shortening
observed at a given cross section is likely to cause separation of the atomic level bond between
particles. Failures are a result of stresses and strains exceeding the allowable resistance of
materials.
Strains can be axial strain or bending strain. Axial strain results from a linear change in

length while bending strain is due to deflection, rotation, and curvature at the section being
considered. Hence, all failures emanate from a deformation, which is a physical change in
length or rotation.

3. As stated, a change in length gives rise to a finite strain, which in turn causes a certain type
of stress. Hence, axial force gives rise to axial stress, shear force to shear stress, bending
moment to bending stress, and torsion to equivalent shear stress.
When bending stress and shear stress act in the same plane (e.g., in a vertical plane or deep
beam) they can be combined together as a principal stress acting in the resultant principal plane.
In a column subjected to bending moment, axial stress can be added to bending stress since both
act in the same direction.
At failure, the critical state of stress can be due to elongation, which is causing maximum
principal stress, principal strain, shear stress, and shear strain energy or total strain energy. Deduc-
tive reasoning concludes that external work done will have exceeded the internal strain energy.
1. Review the fundamentals of analysis: The objective is to evaluate correct defl ection and
stress under load and make the design safe enough for the life of the structure.
2. Analyze the superstructure (deck slab, parapets, and beams) and the substructure (bearings,
abutment, and piers) separately.
3. Analyze the superstructure and the substructure as a combined structure (in particular,
integral abutment bridges or arches).
4. Use reduced allowable strength in design due to fatigue stresses: Specifications review is

usually based on AASHTO LRFD and state codes.
4.2.2 External Effects Leading to Member Sizing

1. A review of the fundamentals of analysis is presented here. Knowledge of applied mechanics
can be utilized by applying laws of equilibrium, compatibility, strain displacement relations
and boundary conditions. Whether it is the finite element method or partial differential