le beam. N_{1} P =
neutral plane, N A = neutral axis of section R S.]

If the bar is symmetrical and homogeneous the neutral plane is
located half-way between the upper and lower surfaces, so long
as the deflection does not exceed the elastic limit of the
material. Owing to the fact that the tensile strength of wood is
from two to nearly four times the compressive strength, it
follows that at rupture the neutral plane is much nearer the
convex than the concave side of the bar or beam, since the sum
of all the compressive stresses on the concave portion must
always equal the sum of the tensile stresses on the convex
portion. The neutral plane begins to change from its central
position as soon as the elastic limit has been passed. Its
location at any time is very uncertain.

The external forces acting to bend the bar also tend to rupture
it at right angles to the neutral plane by causing one
transverse section to slip past another. This stress at any
point is equal to the resultant perpendicular to the axis of the
forces acting at this point, and is termed the ~transverse
shear~ (or in the case of beams, ~vertical shear~).

In addition to this there is a shearing stress, tending to move
the fibres past one another in an axial direction, which is
called ~longitudinal shear~ (or in the case of beams,
~horizontal shear~). This stress must be taken into
consideration in the design of timber structures. It is maximum
at the neutral plane and decreases to zero at the outer elements
of the section. The shorter the span of a beam in proportion to
its height, the greater is the liability of failure in
horizontal shear before the ultimate strength of the beam is
reached.

_Beams_

There are three common forms of beams, as follows:

(1) ~Simple beam~--a bar resting upon two supports, one near
each end. (See Fig. 16, No. 1.)

(2) ~Cantilever beam~--a bar resting upon one support or
fulcrum, or that portion of any beam projecting out of a wall or
beyond a support. (See Fig. 16, No. 2.)

(3) ~Continuous beam~--a bar resting upon more than two
supports. (See Fig. 16, No. 3.)

[Illustration: FIG. 16.--Three common forms of beams. 1. Simple.
2. Cantilever. 3. Continuous.]

_Stiffness of Beams_

The two main requirements of a beam are stiffness and strength.
The formulae for the _modulus of elasticity (E)_ or measure of
stiffness of a rectangular prismatic simple beam loaded at the
centre and resting freely on suppor