Machine Design

One of the most critical factors when it comes to reliability of a threaded-fastener joint is preload—the force the tightened fastener exerts on an assembly. Preload is a function of many variables, including fastener material and finish, head style, and lubrication. So, in the end, there is no simple, totally reliable way to compute the precise preload needed for every application. However, the following equations serve as useful guides in relating preload to three of the more important factors in threaded-joint design: total bolt load, tensile stress, and tightening torque.

The tension on a fastener for a given preload and external load is:

Pt = Pi = Pe(Ks/(Ks + Kc))

This is a conservative expression that is accurate enough for most joint designs. But it is intended only as a guide because it ignores factors such as bending, heating, and impact loading.

When fasteners are torqued to increase preload, torsional stresses placed on them reduce the tensile force they can exert before yielding. The total tensile stress felt by the bolt is:

St= Pt/2A + ((Pt/2A)2 - (tr/J)2)0.5

The required tightening torque for the fastener can be estimated from the empirical expression for the fastener:

T = KDPt

Constant K is normally about 0.2 for a black screw. For a lubricated fastener or one with cadmium plaiting, K is about 0.15. Unlubricated zinc-plated screws may have a K as high as 0.35

For rigid steel parts, the conservative practice is to tighten the fasteners to 75% of yield. Lower torques should be considered for flexible joints, joints with gaskets, or assemblies subject to high temperatures.

These equations may aid engineers and designers in determining the proper required preload, which is often a major factor controlling a joint’s fatigue life. In a typical rigid assembly, an external load below that of the preload has little effect on fastener tension. Thus, the fastener generally does not fail in fatigue even if such a load is repeatedly applied. However, a repeated external load higher than the preload produced cyclic tensioning that may lead to fatigue failure.

For example, a socket screw with a rated tensile strength of 180,000 psi may have an average endurance limit of only 15,000 psi. This means the fastener can withstand a one-time applied stress of 180,000 psi, but a cyclic stress of more than 15,000 psi could induce fatigue failure within a given number of cycles.

The most common way to avoid such failures is to increase the size of the fastener. However, this usually requires changes to hole preparation, tightening methods, and assembly fixtures. Frequently, the problem can instead be solved by merely preloading the fastener above the external load.

### Nomenclature

A = Thread stress area (in.2)

D = Nominal screw diameter (in.)

J = Polar moment of inertia, pi*r4/2

K = Constant from 0.05 to 0.35

Kc = Assembly spring constant, lb./in.

Ks = Screw spring constant, lb./in.

Pt = Total bolt load, lb.

St = Total tensile stress felt by bolt, psi

T = Tightening torque, lb.-in.

t = Torsion felt by screw, lb.-in.