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.
