Beam Design

Introduction
          This module looks at the failure mechanisms for simply supported beams with a uniformly distributed load and a rectangular cross-section.  This is a simple case but clearly shows the relationship between the three limiting conditions discussed here.  Although the examples shown use wood beams, the beam design theory is not specific to wood engineering.  Rather, the concepts presented here are grounded in basic mechanics principals that can be applied to any material.
 
          The bending stress that a beam is required to resist is a function of the load (or moment, Mx)  and the cross-sectional geometry (section modulus Sx).  The resulting stress is compared to the adjusted capacity of the beam, F'b.  The equation shown below is for a simply supported beam that is fully braced against lateral buckling of the compression edge. It is uniformly loaded  with bending about the strong (x-x) axis.

Therefore the maximum load (w) based on bending strength of the beam is:

          For a given (i.e., 2x6) beam the above equation can be graphed on a load (w) vs. span (L) table.   Because the span length is squared, the resulting plot would be quadratic.
 

          The shear equation is also a function of the load and the cross-sectional geometry.  In this case however, the plot on the load-span chart would be linear.
 
Therefore the maximum load (w) based on shear strength of the beam is:
          Similarly, the equation for deflection can also be plotted on the load-span plot.   The modulus of elasticity (E) is a material property of the beam and the moment of inertia (I) is function of the cross-section.  Because of the exponent of the length term the plot is cubic.
If the maximum allowable deflection is then:

Failure Envelope

          When the three equations listed above are plotted simultaneously on the load-span chart, they form a failure envelope specific for that beam.  The following chart shows the failure envelope for a Select Structural 2x6.  The failure envelope is the "lower bound"  formed by the three curves generated from the bending, shear, and deflection equations above.

          The failure envelope defines the loads that a given span can support.  For example; if you were to design a beam that you knew would span 12 feet.  Then by looking at the chart you would know that it's capacity would be about 850 lbs/ft.