
MECHANICS  THEORY

Beam with Same Area but Different
Load Capacity Due to Different I 

The geometric cross section is critical in beam design. For example, a solid
rectangular cross section will support substantially less load than an efficient
Ibeam. The reason for this can be understood by examining the
bending
stress equation, σ = My/I. As I becomes
large, the stress reduces. Thus, it is advantageous to have most of the crosssectional
area away from the xaxis which increases I. A good example of this is
an Ibeam. However, too much of a good thing is bad. For example, with
an I beam, if the web section is too thin, shear stress will cause failure. 





Section Modulus



In the bending stress equation, there are two parameters, y and I, that are
based on the cross section geometry. It would be easier to design beams if these
two values were combined into a single parameter as
S = I/c
where c is the maximum y value (top or bottom edge) and S is the section modulus.
The bending stress equation then changes to




Basic Structural Steel Shapes 










The S value, the section modulus, is listed in handbooks for all major structural
shapes (see Structural Shapes appendix).
Thus, if the moment and allowable stress are known, then a structural member
can be chosen that meets or exceeds the section modulus. This eliminates working
with the moment of inertia and the distance from the neutral axis to the outside
edge. 



Structural Steel Shapes


There are a large variety of structural shapes. The major ones include Ibeams
(W and S types), channels, tubes, angles, pipes, and tees. The Ibeam is generally
the most efficient, but may not fit the space or needs of a particular structure.
Many times, structural shapes are combined, such as two angles, to give new
shapes.
One note of interest is that there are two Ibeam types, W and S shapes. The
S or Standard beam is narrower than the W or Wideflange beam and was the first
standardized Ibeam sections. With the increased quality of steel and manufacturing
techniques, the W beams have largely displaced the S beams.
All the major shapes are shown at the left. Each one can be clicked and a list
of popular shapes will be shown. 




Lumber Shapes (US Units only)

Basic Lumber Sizes
(other
sizes in appendix) 
Nominal
Dimensions
in 
Net
Dimensions
in 
2 x 4 
1.5 x 3.5 
2 x 6 
1.5 x 5.5 
2 x 8 
1.5 x 7.25 
2 x 10 
1.5 x 9.25 
4 x 4 
3.5 x 3.5 
6 x 6 
5.5 x 5.5 


For various reasons, actual lumber dimensions (net) are not the same as the
name (nominal) dimension. For example, a "2x4" is not 2 in by
4 in. It is important that the correct size be considered when designing any structure
with wood.
An additional issue with lumber is the inconsistent quality. Unlike metals that
are manufactured, wood is natural which introduces a higher variation of stiffness
and strength. Most handbooks that give material properties for wood use low values,
but the design should also consider using a higher
factor of safety. 



