Resultant forces at by sanmelody


									“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
The standard notation is shown.
O’ is the centre of curvature
Smiley Face
is Positive

        Sign convention for curvature

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:
          s  s
  lim
    s 0   s
Using the relationships for arc-length we have

  lim
               y      
      0                        
   m ax 

      y
    m ax
              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.

PROB. 5.4-1
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 0.0024,
what is the shortest length L of the wire that can be

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