
MECHANICS  THEORY



Knowing the
shear and moment at any location in a beam is only useful if that information
can be used to design a beam. The shear and moment need to
be used to determine the stresses which can be used to find if the material will
fail
This section will examine bending stress and how it can be calculated from the
bending moment. The following section will consider shear stress.






Beam Bending Strain

Small Beam Element in Bending


To understand the bending stress in an arbitrary loaded beam, consider a small
element cut from the beam as shown in the diagram at the left. The beam type
or actual loads does not effect the derivation of bending strain equation. Recall,
the
basic definition of normal strain is
ε = ΔL/L
Using the line segment, AB, the before and after length can be used to give
The line length AB is the same for all locations before bending. However, the
length A'B' becomes shorter above the neutral axis (for positive moment) and
longer below. The line AB and A'B' can be described using the
radius of curvature,
ρ, and the differential angle, dθ.
Notice that the y coordinate is assumed upward from the neutral axis, where
there is no strain. Putting these together gives,
This relationship gives the bending strain at any location as a function of
the beam curvature and the distance from the neutral axis. However, this equation
is of little use, and needs to be converted to stress. Also, radius of curvature
is difficult to determine at a given beam location.






Beam Bending Stress



The strain equation above can be converted to stress by using
Hooke's law, σ = Eε, giving,
σ = Ey/ρ (1)
There is still the issue of not knowing the radius of curvature, ρ. If one thinks
about it, the radius of curvature and the bending moment should be related.




Beam Section Cut 

This relationship can be determined by summing the moment due to
the normal stresses on an arbitrary beam cross section and equating it to the
applied internal moment. This is the same as applying the moment equilibrium
equation about the neutral axis (NA).
For a positive moment, the top stresses will be in compression (negative stress)
and the bottom stresses will be in tension (positive stress) and thus the negative
sign in the equation. This equation can be changed by using equation (1),
It is interesting to note that the integral is the
area moment of inertia, I,
or the second moment of the area. Many handbooks list the moment of inertia of
common shapes (see
Sections appendix). A review of moment of inertia is given below in the next subsection.
Using the area moment of inertia gives
E I / ρ = M 



Bending Stress Distribution 

But the radius of curvature, ρ, is still there. But
equation (1), ρ = Ey/σ ,
can be used again to eliminate ρ, giving,
E I /(Ey/σ ) = M
Simplifying and rearranging gives,
This equation gives the bending normal stress, and is also commonly called the
flexure formula. The y term is the distance from the neutral axis (up
is positive). The I term is the moment of inertia about the neutral axis.






Locating the Neutral Axis



If the cross section is symmetrical about the horizontal axis, then the neutral
axis is halfway between the top and bottom. However, for nonsymmetrical beam,
such as a "T" cross
section, the neutral axis is not halfway between the top and bottom, and needs
to be determined before the bending stress equation can be used.




Centroid for Arbitrary Shape 

The neutral axis is located at the centroid (geometric center) of the cross
section. Recall from Statics, the
centroid can be found using two methods. The first is by integration,




Centroid Based on Subshapes 

The second, and more common method, is the method of parts. The beam cross section
is split into geometric shapes that are common (rectangle, triangle, circle,
etc.). The centroid of basic shapes can be found in handbooks (see
Appendix Sections), eliminating the need for integration. The centroid is
If there is a hole, then that area is considered to be negative, and the same
equation can still be used. As an example, the diagram at the left would be,






Area Moment of Inertia

Moment of Inertia around Neutral Axis
using Integration


Similar to the centroid, the
area moment of inertia can be found by either integration
or by parts. The moment of inertia is also called the "second moment of the area"
since that describes the integration equation,
When using this with the bending stress equation, I is about
the neutral axis and not the xaxis.




Moment of Inertia around the
Neutral Axis using Parts


A more common method to find the moment of inertia is by parts.
Like finding the centroid (needs to be done first), the object is split into
smaller basic shapes. The moment of inertia about the centroid of each part can
be found in a handbook (see Appendix
Sections). Then the individual moment of inertia's are moved to the neutral
axis using the
parallel axis theorem. For a particular subshape, this gives

I_{NAi} = I_{i} + y_{i}^{2} A_{i} (parallel
axis theorem)


where I_{i} is the moment of inertia about its own shape and I_{NAi} is
the moment of inertia about the object's neutral axis.
All the moment of inertia terms can then be added together to give,
I_{NA} = Σ I_{NAi}
For the diagram at the left, the parts method gives,
I = (I_{1} + y_{1}^{2} A_{1})
+ (I_{2} + y_{2}^{2} A_{2})  (I_{3} +
y_{3}^{2} A_{3})
Notice, for a hole, the moment of inertia is subtracted for that shape.




