MECHANICS  THEORY

Beam without Adhesion Between Layers 

Although
bending stress is generally the primary stress in beams, shear stress
can also be critical in short beams. Shear stress occurs in all beams with bending
moments and it tries to slide one horizontal beam section across another. For
example, a cantilever beam constructed with nonattached layers, as shown at
the left, will slide. If the beam is one unit, then the internal shear stress
will try to slide the attached layers. This effect can cause beams to split in
the horizontal direction. 



Shear Stress on a Horizontal Plane 

Another way to grasp how shear stress acts on a beam is to take
a small horizontal section and sum the forces. For example, the horizontal
section shown in the diagram at the left must be in static equilibrium. At the
left end, there is no force since it is a free surface. The right side has a
force, P, equal to the total of the normal bending stress at that surface. For
the section to be equilibrium, there must be a force acting to the left to counter
the bending stress load. This is the shear stress acting in the horizontal direction.
It will vary at different locations. 





Horizontal Shear Stress

Small Slice of Beam 

The first step in determining the shear stress at any location is to look at
a section in a small slice from the beam. Summing the forces due to the normal
bending stresses in the horizontal direction gives
ΣF_{x} = 0
 P + (P + dP) + τ b
dx = 0
dP/dx = τ b (1)
where b is the beam depth at the location of the shear stress being calculated. Note, P is assumed to be in tension (positive) for derivation. 



Equivalent Systems 

P can be found by integrating the normal stress over a section A (shown in diagram),
giving
But the
bending stress is σ_{b} = My/I. Substituting
and simplifying gives
Since M and I do not change, they can be moved outside the integral.






The integral now is just the first moment of the area that is commonly used
to find the centroid
of an area, and is called "Q". Substituting P into the equation
(1) and using Q, gives
Recall,
the derivative, dM/dx
is equal to the vertical shear load V. This gives,
The final horizontal shear stress equation is






Calculating Q

Determining Q by Parts at Section aa 

One of the confusing aspects of determining the shear stress τ,
is calculating Q, the first moment of the area about the neutral axis. It is
rare when the full integral,
is needed. Generally, Q can be determined using
parts which is can be written as




Examples of Q at line a
for various shapes
(Q units are distance cubed) 

In determining Q, there are a number of steps that should be considered:
 Locate the neutral axis (NA) for the full cross section.
 Select the location where the stress will be calculated (user determines
this location, generally at the midsection) and determine b (the cross section
width at that location).
 Split the area above or below the stress location into common geometric
shapes (rectangles, circles, triangles, etc.).
 Sum the product of each area with their respective centroid location from
the NA (important: from the NA, not from the stress location).
Examples of Q for various shapes and stress locations are given at the left.
Note, Q can be negative, but the shear stress equation assume all positive values. 





Vertical Shear Stress

Shear Stress on Element
Horizontal and Vertical Shear Stress
at the Same Location in a Beam 

Shear stress acts on two different parallel surfaces of any element as shown
in the diagram at the left. One side cannot be under a different shear stress
magnitude than the other. If a small element is taken from a structure under
shear, parallel sides will have shear stress loading in the opposite direction, causing
it to shear as shown in the diagram at the left. Notice, the other two sides
try to resist the sliding motion, and the stress element stays in equilibrium.
Similarly, a small element taken from a beam under a shear loading will have
equal shear stresses in the vertical and horizontal directions as shown in the
diagram at the left. The magnitude of the shear stress will depend on the location
of the stress element.
There are three possible shear stresses on a three dimensional cube. This section
has only examined one dimension since shear loading in beams is generally only
in one direction. But just like uniaxial loading, shear loading can be in three
directions. 





Rectangular Beams

Shear Stress Distribution in
Rectangular Beam 

Rectangular beams are so common, it is helpful to plot the shear stress from
top to bottom. The resulting equation is
It is a parabolic shape with the maximum at the center. The center shear stress
is



