The extreme fiber stress in bending for a rectangular timber beam is

f=6M / bh2

=M/S

A beam of circular cross section is assumed to have the same strength in bending as a square beam having the same cross-sectional area.

The horizontal shearing stress in a rectangular timber beam is

H=3V /2bh

For a rectangular timber beam with a notch in the lower face at the end, the horizontal shearing stress is

H=(3V /2bd1) (h / d1)

A gradual change in cross section, instead of a square notch, decreases the shearing stress nearly to that computed for the actual depth above the notch.

In the above equation:

f= maximum fiber stress, lb/in2 (MPa)

M= bending moment, lb in (Nm)

h= depth of beam, in (mm)

b= width of beam, in (mm)

S= section modulus ( bh2/6 for rectangular section), in3 (mm3)

H= horizontal shearing stress, lb/in2 (MPa)

V= total shear, lb (N)

d1= depth of beam above notch, in (mm)

l= span of beam, in (mm)

P= concentrated load, lb (N)

V1= modified total end shear, lb (N)

W= total uniformly distributed load, lb (N)

x= distance from reaction to concentrated load in (mm) For simple beams, the span should be taken as the distance from face to face of supports plus one-half the required length of bearing at each end; and for continuous beams, the span should be taken as the distance between the centers of bearing on supports.

When determining V, neglect all loads within a distance from either support equal to the depth of the beam.

In the stress grade of solid-sawn beams, allowances for checks, end splits, and shakes have been made in the assigned unit stresses.

For concentrated loads,

V1=[10P(l-x)(x/h)2]/9l[2+(x/h)2]

For uniform loading,

V1=W[(1-2h/l)]/2

The sum of the V1 values from these equations should be substituted for V in the very first equation, and the resulting H values should be checked against those given in tables of allowable unit stresses for end-grain bearing. Such values should be adjusted for duration of loading.