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

f=6M / bh2

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,


For uniform loading,


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.

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