“Silver Bridge”, Point Pleasant, West Virginia

On December 15,1967 at approximately 5 p.m., the U.S. Highway 35 bridge connecting Point Pleasant, West Virginia and Kanauga, Ohio suddenly collapsed into the Ohio River. At the time of failure, thirty- seven vehicles were crossing the bridge span, and thirty-one of those automobiles fell with the bridge. Forty- six individuals perished with the buckling of the bridge and nine were seriously injured. Along with the numerous fatalities and injuries, a major transportation route connecting West Virginia and Ohio was destroyed, disrupting the lives of many and striking fear across the nation After extensive studies of the broken structure members, the cause of failure was determined. The answer was the unique eye-bar design made from the newly innovated heat treated-carbon steel. The remaining steel frame buckled and fell due to the newly concentrated stresses. The cause of failure was attributed to a cleavage fracture in the lower limb of eye-bar 330 at joint C13N of the north eye-bar suspension chain in the Ohio side span." The fracture was caused from a minute crack formed during the casting of the steel eye-bar. Over the years, stress corrosion and corrosion fatigue allowed the crack to grow, causing the failure of the entire structure. At the time of construction, the steel used was not known for subduing to corrosion fatigue and stress corrosion. Inspection prior to construction would not have been able to notice the miniature crack. Over the life span of the bridge, the only way to detect the fracture would have been to disassemble the eye-bar. The technology used for inspection at the time was not capable of detecting such cracks.

Now we can calculate the shear forces and moments within beams we can now move on to calculating stresses within the beam and deflections of beams. If a load is applied to the end of a cantilevered beam it will deflect. A curve can be drawn that represents the displacement of any point on the beam. The displacement is labeled v and is in the y-direction.

Bending is often separated into pure bending and non-uniform bending. Pure bending means that the bending is due to a uniform moment. A simple beam loaded by two couples is an example of pure bending. V=0 for this beam.

4-Point Bending Between inner contacts no shear force only constant moment. In this region pure bending occurs. The maximum strain is c/  (thickness/radius of curvature. This geometry is very often used to determine the mechanical properties of materials.

To define any point on a deformed beam we need to specify the deflection and the curvature of the bent beam: (a) beam with load, and (b) deflection curve The standard notation is shown. O’ is the centre of curvature

Sign convention for curvature Smiley Face is Positive Frowny Face is Negative

Deformations of a beam in pure bending: (a) side view of beam, (b) cross section of beam, and (c) deformed beam

Before deformation the length of a segment is  s. After deformation the new length is  s’. The strain is then given by: Using the relationships for arc-length we have

PROB Determine the maximum normal strain produced in a steel wire of diameter d=1/16 in. When it is bent around a cylindrical drum of radius R=24 in.

A copper wire having diameter d=3mm is bent into a circle and held with the ends just touching. If the maximum permissible strain in the copper is , what is the shortest length L of the wire that can be used.