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 supports at either end is:[10]

[Footnote 10:  Only this form of beam is considered since it is the simplest.  For cantilever and continuous beams, and beams rigidly fixed at one or both ends, as well as for different methods of loading, different forms of cross section, etc., other formulae are required.  See any book on mechanics.]

P’ l^{3}
E = -------------
4 D b h^{3}

b = breadth or width of beam, inches.
h = height or depth of beam, inches.
l = span (length between points of supports) of beam, inches. 
D = deflection produced by load P’, inches. 
P’ = load at or below elastic limit, pounds.

From this formulae it is evident that for rectangular beams of the same material, mode of support, and loading, the deflection is affected as follows: 

(1) It is inversely proportional to the width for beams of the same length and depth.  If the width is tripled the deflection is one-third as great.

(2) It is inversely proportional to the cube of the depth for beams of the same length and breadth.  If the depth is tripled the deflection is one twenty-seventh as great.

(3) It is directly proportional to the cube of the span for beams of the same breadth and depth.  Tripling the span gives twenty-seven times the deflection.