The critical areas of stress of mating screw threads are

The allowable stresses and screw end force and the method of applying the force in the calculation of the tensile stress are not considered on this page but are addressed on this site by tables and more importantly referenced links

If a screw threaded fastener is to fail it is preferable that the screw fails rather than the internal or external thread strips.  The length of the screw engagement should therefore be sufficient to carry the full load necessary to break the screw without the threads stripping.

The size of a screwed fastener is first established by calculating the tensile load to be withstood by the screw and selecting a suitable screw to withstand the tensile load with the appropriate factor of safety or preload.   If the joint is fixed using a nut and bolt then assuming the nut is selected from the same grade as the bolt there is little need to size the nut.   The fastener manufacture sizes the length of the nut to ensure the screw will fail before the nut. If the screw fastens into a tapped hole then a check of the depth of thread engagement is required.

Generally for female and male threads of the same material with, the female thread is stronger than the male thread in shear for the same length of engagement

D = Basic Diameter.
p = Screw Thread Pitch
L_e = Length of Thread Engagement
A _t = The screw thread tensile stress area
d _p = Pitch circle diameter of thread
A _ss =The thread shear area

The following formula for the Tensile Stress Area of the (male) screw

This is based on ISO 898 Part 1. see calculation below..

d _p = Pitch circle diameter of thread

d_p = (D - 0.64952.p )

The thread shear area = A_ss

A_ss = 0.5. π. d_p. Le = 0.5 π (D - 0.64952.p ). Le

To ensure that the screw fails before the thread strips it is necessary the the shear area is at least 2 times the tensile area. i.e

L_e = 2 . A _t / [0.5 .π .(D - 0.64952.p )]

This assumes that the male and female thread materials have the same strength. If the Female Material strength is lower i.e J as calculated below is greater than 1 then the length of engagement must be increased to prevent the female thread stripping

If the value of J is greater than than 1 then the length of engagement must be increased to at least  

The above formulae are sufficient to enable the tensile strength to be calculated and to allow the depth of thread to be confirmed for a tapped hole

Following are equations to provide more accurate evaluation of the shear strength of threads. These are equations derived from FED-STD-H28/2B, 1991 and Machinerys Handbook eighteenth Edition.  They strictly apply to UN thread series but if the relevent metric screw thread dimensions are used they will give reasonable results.   In practice when the values are calculated the value for the screw shear strength is similar to the very convenient formula provided above. These equations are only of theoretical value

K _nmax = Maximum minor diameter of internal thread.
E _smin = Minimum pitch dia of external thread.
E _nmax = Maximum pitch dia of internal thread.
D _smin = Minimum major dia of external thread.
n = 1/p = threads per unit (mm)

Length Of Thread

Shear Area For Screw

Shear Area For Female Thread

If material in which the female thread is tapped is significantly weaker that the screw material then J must be evaluated.

If the value of J is greater than than 1 then the length of engagement must be increased to at least  

Note: Short derivation of nominal stress area formula from info in BS EN ISO 898..

The purpose of this table is to show the results of the above formula.  It is clear from this table that there is no major benefit in using the detailed formula above.  The approximate formula for the screw thread shear stress area (A _ss) is generally sufficiently accurate and there is no need to use the more detailed formula for A_s. For sizes below M6 the formulas yield very similar values. For sizes M6 and above the value for A_ss provides a slightly more conservative result (20% margin at M36)

I have obtained the thread dimensions on tables in Machinery's Handbook 27th ed.   If you intend to use this information please check it against a reliable source (ref disclaimer above)