Compression on oblique plane

Consider that a timber member sustains a compressive force with an action line that makes an oblique angle with the grain.

Let
P=allowable compressive stress parallel to grain
Q= allowable compressive stress normal to grain
N =allowable compressive stress inclined to grain
?=angle between direction of stress N and direction of grain

By Hankinson’s equation,

N=PQ/(Psin²?+Qcos²?)

From the figure it is found that member M₁ must be notched at the joint to avoid removing an excessive area from member M₂. If the member is cut in such a manner that AC and BC make an angle of ? /2 with vertical and horizontal planes, respectively, the allowable bearing pressures at these faces are identical for the two members.

Let

A =sectional area of member M₁

f₁ =pressure at AC

f₂ =pressure at BC

It may be readily shown that

AC=bsin(? /2 )/sin?

BC=bcos(? /2 )/sin?

f₁=F sin ?/A tan ( ? /2)

f₂=F sin ? tan ( ? /2)/A

This type of joint is often used in wood trusses.