Roymech engineering encyclopedia

Screw thread Calculations

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Screw Stress Areas


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
Le = 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

dp = (D - 0.64952.p )

The thread shear area = Ass

When the female and male threads are the same material.

Ass = 0.5. π. dp. 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

Le (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 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

Screw Shear Area Calculations

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)

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 As. For sizes below M6 the formulas yield very similar values. For sizes M6 and above the value for Ass 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 Ass/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 Assm
=2. At
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
Le =
Ass /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) Knmax 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 smin 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 smin 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
(As= 2*At)
Le 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

Relevant Links
  1. Bolt Science..A site dedicated to the Science & Technology of bolted joints
  2. F.E.D.S Screw Thread design .A very useful and informative document- You must read this if you are doing detaile design.
  3. Tribology-ABC Screw Joints metric Lots of very useful notes and calculators
  4. Heli-coil Catalogue Wire thread Insert - the simple method of providing high thread strength with thread locking.
  5. Bolted Joints..An informative paper published on the Web
  6. Metric Bolt Strength..Bolt Sizes Strengths- American bias
  7. Croberts Com..Very useful section on the consequence of Bolt Failure
  8. Fastener Design Manual...NASA GRC RP-1228 (9.6 Mbyte pdf file). Design info on bolt + rivet joints
  9. Bolt Council Publications...->Guide to Design Criteria for Bolted and Riveted Joints.(6.7 Mbyte pdf file).  Excellent ....
  10. The yielding of fasteners during tightening...An article with surprising conclusions
  11. Keeping It All Together ...Practical notes on bolted joints from an enthuiast
  12. International Thread Standards .. A comprehensive set of thread tables including BSP