Page 184 - Bridge and Highway Structure Rehabilitation and Repair

CHAPTER 4 AN ANALYTICAL APPROACH TO FRACTURE AND FAILURE 159

as a channel. An I-shape conforms better to conventional steel beam theory, in which the two

flanges resist bending moment and the web resists shear force.
Design of an I-shaped girder consists of the design of its individual components, i.e., com-

pression flange, tension flange, and web. Length, depth, width, and thickness and the relative

proportions of each component of an I-girder determine local stress distribution and buckling.
Sizing of girders plays an important role in both the elastic and plastic behavior of girders.
In steel beam theory, bending moment can be replaced by two axial forces, one compres-
sion and one tension, each acting at the other side of the neutral axis. Axial compressive stress
in fl ange will cause local buckling if the fl ange area is small and also if the fl ange is long and
slender (having high L/r ratio).

For a compact steel section, flange thickness needs to exceed a certain minimum, so that the

width to thickness ratio is less than 16. Typical values of flange thickness are greater than 1 inch.

Stress requirements must also be satisfied. At supports when bottom flanges are in compression

and there is no lateral bracing available, the thickness of the bottom flange needs to be controlled

by the slenderness ratio of the flange width to thickness. The compression part of web will also

help prevent buckling by making a T-section with the fl ange.
In addition, torsion may occur at girder ends when girder ends are restrained at bearing sup-
ports. Torsional moment occurs due to eccentricity of load application when the result of vertical
loads is not located at the center line of the web, especially at fascia girders.
4.9.11 Lateral Buckling of Girders
Failure mode of a steel girder is either by yielding or by buckling. If adequate lateral sup-
port is provided, failure will be by yielding. If the member does not satisfy slenderness (kL/r)
requirement, failure will occur by out-of-plane buckling due to compression.

Failure can also occur by torsion due to lateral deflection. This mode of failure is known as
“lateral buckling” or “lateral torsional buckling.” Twisting is resisted by a combination of St.
Venant (pure) torsion or warping torsion. In closed sections such as boxes and tubes, two webs
are available; torsional stiffness is very high in resisting warping torsion, and lateral buckling
is not important.
All I-shaped girders with thin webs are subjected to additional stress from lateral buckling
and need to be designed for combined stresses. Warping torsion is higher for cantilever beams

which have high deflection at the free end.
Formulae to calculate lateral bucking stress are provided by AASHTO LRFD specifi ca-
tions.

4.10 NONLINEAR ANALYSIS IN STEEL AND CONCRETE

4.10.1 Relationships between Load and Deflection, Stress, and Strain

Relationships between load and deflection, stress, and strain are initially linear. After a certain
magnitude of applied load, the behavior of the beam or column becomes nonlinear.
For example, dead load or self weight analysis is usually linear. When response of the mem-
ber under a higher load, as observed in the laboratory, shows second or third degree nonlinearity
of load deflection or stress-strain relationships, the analysis is no longer straight forward. Also,

significant changes in geometry of the structure can result in overstress, cracking or yielding

of material.
Geometric nonlinearity: Examples are load deflection characteristics of suspension cables,

long span trusses, and slender columns (P-Delta effect). Tied arches and trusses are not designed
to undergo large deflections. In other structures long spans and heavy loads may lead to nonlinear

load-defl ection behavior of members. Hence, analysis should correspond to a series of unique
loads under which the deformed structure is in equilibrium and has geometric compatibility.
Material non-linearity: When analysis is based on plastic deformation of certain materials,
material nonlinearity is considered. Nonlinear analysis is required when studying post-elastic