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Beam Design: A beam is a building member that spans between
points of support and carries loads in bending. This is a uniformly loaded,
simply supported, beam
A floor or ceiling joist, a
girder, or a rafter are all examples of beams
In normal orientation, as a beam like a floor
joist bends the fibers on the top of the beam are being pushed together in
compression. The fibers on the bottom are being stretched in tension.
The magnitude
of the internal compressive and tensile forces diminish through the depth of
the beam, the center of the beam depth experiences neither compression or
tension. This neutral axis is the plane down the center of the beam in the
accompanying drawing where internal tension and compression forces pass through
a zone of zero stress.
 The point of greatest
bending moment, or rotational force, in our uniformly loaded beam, is in the
center of span. In the drawing the imaginary section cut is through that point
of greatest bending moment. Using that visualized slice through the beam the
magnitude of the internal stress in the beam is calculated. Knowing how the
stress is distributed internally within the beam, the extreme bottom fiber in
the center of span of this beam is the area of focus. The calculated stress is
compared to the allowable stress for the species and grade of the beam. If the
stress is less than the allowable limits for the material the beam checks in
bending. This process is known as "Allowable Stress Design".
Let's back up a minute and discuss
what a moment is. A moment is the tendency of a force to cause rotation. To
help visualize this we'll take a look at the simplest form of beam and the
first diagram that we know of that tried to decipher the phenomenon of flexure,
Galileo's cantilever.
 From the diagram we can
see a beam fixed at one end, extending out some distance, with a load on the
free end. If we assume the load is 100 lbs and the beam is 1' long the maximum
bending moment is 100 foot pounds. If the beam is 10 feet long and the same 100
pound load is hanging from its' end the maximum bending moment is 100lbs x 10
feet, or 1,000 ft-lbs. The point of maximum moment is about the neutral axis at
A-B.
The beam formulas and
diagrams for this condition are below;
 The moment diagram is
at the bottom and shows the magnitude of the bending stress along the length of
the beam. Notice the second formula for determining the maximum bending moment,
it is what we just worked above, concentrated load x span
length.
One more picture might
help in this description, the beam type torque wrench
The main shaft of the wrench is a round steel beam. By way of a
socket and bolt the end is fixed and a force is applied to the
free end. The pointer is not being flexed and remains straight.
We know that as long as a beam is working within its' elastic
limits that Hooke's law applies; deformations are directly
proportional to stresses. Robert Hooke was a 17th century
scientist studying clock springs when he discovered this
principle. To restate it another way, if a force produces a
certain deformation, doubling the force produces twice the
deformation. This holds true as long as we are working within the
elastic limits of the material. Notice that the increments on the wrench's scale are uniform.
The elastic limit is the point at which,when the force is removed, the material returns to its
original shape. Obviously we don't want a building to change
shape, we all know what a sagging roof looks like. Staying safely
within the elastic limits of the material prevents that. I've done one more thing in that
picture, I used my wrench that reads in inch-lbs. It looks like I'm pulling
with about 500 inch-lbs of force. To convert to foot pounds, a number that's easier for me to visualize,
divide 500 in-lbs by 12 inches/foot to give 42 ft-lbs of torque. The rotational moment on the bolt head
is 42 ft-lbs and the bending moment in the wrench's round steel beam is 42 ft-lbs.
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