Introduction
The critical areas of stress of mating screw threads are
 The effective cross section area, or tensile area, of the external thread.
 The shear area of the external thread which depends upon minor diameter of the tapped hole
 The shear area of the internal thread which depends on the major diameter of the external thread

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
The following rules of thumb are suggested for arriving at reasonable lengths of thread for steel screws used with screwed holes in weaker materials.
For steel a length of thread engagement of at least 1 x Nominal dia's of the thread
For Cast Iron or brass or bronze the thread engagement should be at least 1,5 x Nominal dia's of the thread
For Aluminium , zinc or plastices the thread engagement should be at least 2 x Nominal dia's of the thread
However for a quality safe connection, when the tapped material has a significantly lower ultimate tensile strength than the screw material,
 to ensure the screw will fail in tension before the female, it is preferable to use suitably rated nuts or engineered thread inserts.
For some notes on thread Inserts ref. Thread Inserts
Important Note:
Various studies on thread loading have established that
the shear stress is not evenly distributed across the threads. The first thread withstanding the load
is the highest stressed and the next one is much less stressed and so on... . If the thread materials were
very hard and did not yield the first thread could be withstanding nearly all
the load. However because of material yielding there is some distribution
of the load. A study (see link 2 below) has established that for a typical grade 8 nut the percentage
of the load taken by consecutive threads are about 34%, 23%, 16%,11%,9%, 7% .... This effect can be alleviated
by using very accurate threads and by using ductile materials for the components. It has been established that,for carbon steel,
there is no increase in thread shear strength by having a thread engagement length in excess of the screw diameter.
It is normal practice to use a tapped hole depth of about 1,5 x nominal diameter  this allows at least 1 diameter of good thread engagement.
A very simple rule that can be applied for that vast majority of applications is that a thread length of 80% of the
screw diameter (standard nut height) is sufficient for ensuring that the screw will fail in tension before the female thread (nut) fails in
thread stripping (assuming the screw and nut are similar materials). Equations below indicate how to make adjustments if
the tapped metal (nut) strength is lower than the screw/bolt.

Stress Area formulae
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}
When the female and male threads are
the same material.
A_{ss} = 0.5. π. d_{p}. L_{e} = 0.5 π (D  0.64952.p ). L_{e}
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} (min) = 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
More Detailed Notes
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 FEDSTDH28/2B, 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
Screw Shear Area Calculations
K _{n}max = Maximum minor diameter of internal thread.
E _{s}min = Minimum pitch dia of external thread.
E _{n}max = Maximum pitch dia of internal thread.
D _{s}min = Minimum major dia of external thread.
n = 1/p = threads per unit (mm)
Minimum Length Of Thread (Assuming male and female threads are materials of similar strength).
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
Stress area ISO 898
Note: Short derivation of nominal stress area formula from info in BS EN ISO 898..
Some calculated Stress Areas for ISO Metric Threads..medium fit (6H / 6g)
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)
All dimensions in mm
Size 

M3 
M4 
M5 
M6 
M8 
M10 
M12 
M14 
M16 
M20 
M22 
M24 
M30 
M36 
Basic Dia 
D (mm) 
3.00 
4.00 
5.00 
6.00 
8.00 
10.00 
12.00 
14.00 
16.00 
20.00 
22.00 
24.00 
30.00 
36.00 
Pitch 
p 
0.50 
0.70 
0.80 
1.00 
1.25 
1.50 
1.75 
2.00 
2.00 
2.50 
2.50 
3.00 
3.50 
4.00 
1/p 
n 
2.0000 
1.4286 
1.2500 
1.0000 
0.8000 
0.6667 
0.5714 
0.5000 
0.5000 
0.4000 
0.4000 
0.3333 
0.2857 
0.2500 
Stress Dia 
D _{s} 
2.5309 
3.3433 
4.2494 
5.0618 
6.8273 
8.5927 
10.3582 
12.1236 
14.1236 
17.6545 
19.6545 
21.1854 
26.7163 
32.2472 
Tensile Stress Area 
A _{t} 
5.0308 
8.7787 
14.1825 
20.1234 
36.6085 
57.9896 
84.2665 
115.4394 
156.6684 
244.7944 
303.3993 
352.5039 
560.5872 
816.7226 
Pitch circle dia. 
d _{p} 
2.6752 
3.5453 
4.4804 
5.3505 
7.1881 
9.0257 
10.8633 
12.7010 
14.7010 
18.3762 
20.3762 
22.0514 
27.7267 
33.4019 
Approximate Method 
Shear Area/unit Length 
A_{ss/mm} 
4.2023 
5.5690 
7.0378 
8.4045 
11.2910 
14.1776 
17.0641 
19.9506 
23.0922 
28.8653 
32.0069 
34.6383 
43.5530 
52.4676 
Shear Area

A_{ssm}
=2. A_{t} 
10.0616 
17.5574 
28.3650 
40.2468 
73.217 
115.9792 
168.533 
230.8788 
313.33568 
489.5888 
606.7986 
705.078 
1121.1744 
1633.4452 
Length of Thread (A_{ss}=2*At) 
L_{e} = A_{ss} /A _{ss/mm} 
2.3944 
3.1527 
4.0304 
4.7887 
6.4845 
8.1805 
9.8765 
11.5725 
13.5689 
16.9612 
18.9584 
20.3534 
25.7428 
31.1324 
More Accurate Method 
Max.Minor Dia (nut) 
K_{n}max 
2.5990 
3.4220 
4.3340 
5.1530 
6.9120 
8.6760 
10.4410 
12.2100 
14.2100 
17.7440 
19.7440 
21.2520 
26.7710 
32.2700 
Min Pitch Dia (Screw) 
E _{s}min 
2.5800 
3.4330 
4.3610 
5.2120 
7.0420 
8.8620 
10.6790 
12.5030 
14.5030 
18.1640 
20.1640 
21.8030 
27.4620 
33.1180 
Max Pitch dia (Nut) 
E sub>nmax 
2.7750 
3.6630 
4.6050 
5.5000 
7.3480 
9.2060 
11.0630 
12.9130 
14.9130 
18.6000 
20.6000 
22.3160 
28.0070 
33.7020 
Min Major dia (Screw) 
D _{s}min 
2.8740 
3.8380 
4.8260 
5.7940 
7.7600 
9.7320 
11.7010 
13.6820 
15.6820 
19.6230 
21.6230 
23.5770 
29.5220 
35.4650 
Shear Area/unit length (Screw) 
A _{s} /mm 
3.9034 
5.4728 
7.0731 
8.6458 
12.1612 
15.5796 
18.9762 
22.4239 
26.0969 
33.2791 
37.0302 
40.4623 
51.6384 
63.0982 
Shear Area /mm length (Nut) 
A _{n}/mm 
5.5466 
7.7691 
9.9988 
12.1909 
16.8285 
21.4769 
26.1173 
31.0335 
35.5699 
45.3881 
50.0141 
55.0098 
69.5512 
84.0601 
Length of Thread (A_{s}= 2*At) 
L_{e} 
2.5777 
3.2081 
4.0103 
4.6551 
6.0206 
7.4443 
8.8813 
10.2961 
12.0067 
14.7116 
16.3866 
17.4238 
21.7120 
25.8873 
