The number of pounds which concentrated at the centre will deflect a rectangular prismatic simple beam one inch may be found from the preceding formulae by substituting D = 1” and solving for P’.  The formulae then becomes: 

4 E b h^{3}
Necessary weight (P’) = -------------
l^{3}

In this case the values for E are read from tables prepared from data obtained by experimentation on the given material.

Strength of Beams

The measure of the breaking strength of a beam is expressed in terms of unit stress by a modulus of rupture, which is a purely hypothetical expression for points beyond the elastic limit.  The formulae used in computing this modulus is as follows: 

1.5 P l
R = ---------
b h{^2}

  b, h, l = breadth, height, and span, respectively, as in
    preceding formulae. 
  R = modulus of rupture, pounds per square inch. 
  P = maximum load, pounds.

In calculating the fibre stress at the elastic limit the same formulae is used except that the load at elastic limit (P_{1}) is substituted for the maximum load (P).

From this formulae it is evident that for rectangular prismatic beams of the same material, mode of support, and loading, the load which a given beam can support varies as follows: 

(1) It is directly proportional to the breadth for beams of the same length and depth, as is the case with stiffness.

(2) It is directly proportional to the square of the height for beams of the same length and breadth, instead of as the cube of this dimension as in stiffness.

(3) It is inversely proportional to the span for beams of the same breadth and depth and not to the cube of this dimension as in stiffness.

The fact that the strength varies as the square of the height and the stiffness as the cube explains the relationship of bending to thickness.  Were the law the same for strength and stiffness a thin piece of material such as a sheet of paper could not be bent any further without breaking than a thick piece, say an inch board. |---------------------------------
----------------------------------------------------| | TABLE IX | |----------------------
---------------------------------------------------------------| | RESULTS OF STATIC BENDING TESTS ON SMALL CLEAR BEAMS OF 49 WOODS IN GREEN CONDITION | | (Forest Service Cir. 213) | |---------------------------------------
----------------------------------------------| | | Fibre | | | Work in Bending | | COMMON NAME | stress at | Modulus | Modulus |-------------------------------| | OF SPECIES | elastic | of | of | To | To | | | | limit | rupture | elasticity | elastic