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### Combined Thread and Body Bolt Elongation Under Preload Formula and Calculator

Bolt & Screw Torque Charts and Equations

With torque or turn on a bolt thread, we’re trying to control the tightening process through the forces applied to, or the motion of, the nut. What we’re really interested in is the bolt, however, since this is the thing which is being stretched to produce the clamping force on the joint.

We can consider the bolt to be a stiff spring. The relationship between the change in length of the bolt and the preload within it can be described by

Equation 1
ΔL = Fp ( 1 / KB )

where
ΔL = the change in length of the bolt (in., mm)
KB = the stiffness of the bolt (lb/in., N/mm)
Fp = the preload in the bolt (lb, N)

This equation says that the change in length of a bolt is equal to the preload within it times a constant. We can use Hooke’s law to determine the constant in terms of the bolt properties and dimensions. For a common hex bolt, with a body, for example,

Equation 2
ΔLc = Fp [ Lbe / ( E AB ) + Lse / ( E As ) ]

where
Lbe = effective body length (in., mm)
AB = body cross-sectional area (in.2, mm2)
E = modulus of elasticity (psi, MPa)
Lse = effective thread length (in., mm)
As = effective cross-sectional area of threads (in.2, mm2)
ΔLc = combined change in length of all sections (in., mm)

Equation 2 suggests that if we can measure the change in length of the bolt accurately, we can determine the preload with the same degree of accuracy.

Related:

Reference:

Introduction to the Design and Behavior of Bolted Joints Non-Gasketed Joints, Forth Edition
Founding Editor
L. L. Faulkner
Columbus Division, Battelle Memorial Institute,
Department of Mechanical Engineering
The Ohio State University
Columbus, Ohio