Question:  How is bolt installation preload calculated?

Answer:  Bolt pretension, also called preload or prestress, comes from the installation torque T you apply when you install the bolt.  The inclined plane of the bolt thread helix converts torque to bolt pretension.  Bolt preload is computed as follows.

    P_i = T/(K D)        (Eq. 1)

where P_i = bolt preload (called F_i in Shigley).
T = bolt installation torque.
K = torque coefficient.
D = bolt nominal shank diameter (i.e., bolt nominal size).

Torque coefficient K is a function of thread geometry, thread coefficient of friction m_t, and collar coefficient of friction m_c.  Look up K for your specific thread interface and collar (bolt head or nut annulus) interface materials, surface condition, and lubricant (if any).  ("Torque specs for screws," Shigley, and various other sources discuss various K value estimates.)  If you cannot find or obtain K from credible references or sources for your specific interfaces, then you would need to research to try to find the coefficients of friction for your specific interfaces, then calculate K yourself using one of the following two formulas listed below (Shigley, Mechanical Engineering Design, 5 ed., McGraw-Hill, 1989, p. 346, Eq. 8-19, and MIL-HDBK-60, 1990, Sect. 100.5.1, p. 26, Eq. 100.5.1, respectively), the latter being far simpler.

    K = {[(0.5 d_p)(tan l  +  m_t sec b)/(1  –  m_t tan l sec b)]   +   [0.625 m_c D]}/D        (Eq. 2)

    K = {[0.5 p/p]   +   [0.5 m_t (D – 0.75 p sin a)/sin a]   +   [0.625 m_c D]}/D        (Eq. 3)

where D = bolt nominal shank diameter.
p = thread pitch (bolt longitudinal distance per thread).
a = thread profile angle = 60° (for M, MJ, UN, UNR, and UNJ thread profiles).
b = thread profile half angle = 60°/2 = 30°.
tan l = thread helix angle tan = p/(p d_p).
d_p = bolt pitch diameter.
m_t = thread coefficient of friction.
m_c = collar coefficient of friction.

D and p can be obtained from bolt tables such as Standard Metric and USA Bolt Shank Dimensions.

The three terms in Eq. 3 are axial load component (coefficient) of torque resistance due to (1) thread helix inclined plane normal force, (2) thread helix inclined plane tangential (thread friction) force, and (3) bolt head or nut washer face friction force, respectively.

However, whether you look up K in references or calculate it yourself, the engineer must understand that using theoretical equations and typical values for K and coefficients of friction merely gives a preload estimate.  Coefficient of friction data in published tables vary widely, are often tenuous, and are often not specific to your specific interface combinations and lubricants.  Such things as unacknowledged surface condition variations and ignored dirt in the internal thread can skew the results and produce a false indication of preload.

The engineer and technician must understand that published K values apply to perfectly clean interfaces and lubricants (if any).  If, for example, the threads of a steel, zinc-plated, K = 0.22, "dry" installation fastener were not clean, this might cause K to increase to a value of 0.32 or even higher.  One should also note that published K values are intended to be used when applying the torque to the nut.  The K values will change in relation to fastener length and assembly running torque if the torque is being read from the bolt head.

One should measure the nut or assembly "running" torque with an accurate, small-scale torque wrench.  ("Running" torque, also called prevailing torque, is defined as the torque when all threads are fully engaged, fastener is in motion, and washer face has not yet made contact.) The only torque that generates bolt preload is the torque you apply above running torque.

A few more things to be aware of are as follows.  Bolt proof strength S_p is the maximum tensile stress the bolt material can withstand without encountering permanent deformation.  Published bolt yield strengths are determined at room temperature.  Heat will lower the yield strength (and proof strength) of a fastener.  Especially in critical situations, you should never reuse a fastener unless you are certain the fastener has never been yielded.

1.1   Bolt Preload Measurement

If a more accurate answer for bolt preload is needed than discussed above, the specific combination and lubricant would have to be measured instead of calculated.  Measurement methods are generally involved, time-consuming, and expensive, and are beyond the scope of this article.  But perhaps one of the simplest and least expensive methods, to test specific combinations and lubricants, is to measure the installed fastener with a micrometer, if possible, and compute torque coefficient K as follows, per Shigley, op. cit., p. 345, para. 2.

    K = T L/(E A delta D)        (Eq. 4)

where T = bolt installation torque, L = bolt grip length, E = bolt modulus of elasticity, A = bolt cross-sectional area, D = bolt nominal shank diameter, and delta = measured bolt elongation in units of length.