Authored by: Moo-Zung Lee, West Hills, Calif. Edited by Kenneth J. Korane, ken.korane@penton.com Key points: Thermal effects can increase or decrease bolt load and clamping capability. Measuring the angle of nut rotation can sometimes be used to gage preload. Resources: How external loads affect bolted joints, http://machinedesign.com/article/how-external-loads-affect-bolted-joints... Modeling the effects of bolt preload, http://machinedesign.com/article/modeling-the-effects-of-bolt-preload-0907 Structural properties of bolted joints, http://machinedesign.com/article/structural-properties-of-bolted-joints-...

Thermal effects and component compliance can alter bolt preload.

Understanding the structural behavior of bolted joints is essential for sound design. For instance, forces that overcome preload can cause bolted assemblies to leak or fail. In a previous article, we learned about the effects of external loads (“How external loads affect bolted joints,” Machine Design, April 22, 2010, p. 58-60). Now let’s consider the behavior of bolted joints under various internal loads.

A preloaded bolted joint, consisting of a bolt and clamped members, is essentially a constrained bar-pair, as shown in the accompanying graphic. And if temperature changes, internal forces in the bolt and clamped members will find a new equilibrium.

Consider the situation as if the components were preloaded from free lengths at the higher temperature. Thermal expansion of the bolt length is:
eb = αbΔTbLb.
Thermal growth in the clamped members is:
ec = αcΔTcLc.
Next, length changes due to preloading at elevated temperatures are:
δb = (Δb + δ) – eb,
–δc = (–Δc + δ) – ec.
Initial preload at room temperature is:
Fi = kbΔb = –kcΔc.
The bolt force at elevated temperature is:
Fb = kbδb = kb(Δb + δ) – kbαbΔTbLb
and clamped-member force at temperature is:
–Fc = kcδc = kc(Δc – δ) + kcαcΔTcLc.
Because no external force is applied to the joint, forces in the bolt and members must be in equilibrium — that is, Fb + Fc = 0. Therefore,
(kb + kc)δ = kbαbΔTbLb + kcαcΔTcLc, or
δ = (1/(kb + kc))(kbαbΔTbLb
+ kcαcΔTcLc).
The new bolt force after the temperature change is:
Fb = (kbΔb) + (kb/(kb + kc))
(kbαbΔTbLb + kcαcΔTcLc) – (kbαbΔTbLb)
or
Fb = Fi + (αcΔTcLc – αbΔTbLb)(kbkc/(kb + kc)).
The stiffness factor at the far right is the combined internal stiffness ki defined as:
kbkc/(kb + kc) = 1/((1/kb) + (1/kc)) == ki
and 1/ki = 1/kb + 1/kc.
The expression 1/ki is the compliance of the preloading loop. Therefore, the combined internal compliance of a preloaded bolted joint is the sum of the compliances of the bolt and clamped members.
This lets us write the bolt-force equation in a more-compact form:
Fb = Fi + (ec – eb)ki.
Bolt load after a temperature change is the preload at room temperature and the product of the net thermal expansions and combined internal stiffness. This is similar to the expansion force of a bar blocked from both ends:
ΔF = ke = kα(ΔT)L.
Thermal effects may increase or decrease bolt load and clamping capability. Use this equation to weigh design options: either reducing the differential thermal expansion, giving the joint more compliance, or otherwise assessing the bolt stress at different temperatures. For example, the condition for invariant bolt load due to temperature is:
αbΔTbLb = αcΔTcLc.

Carefully designed disk springs with nearly flat force-deflection characteristics (ki ≈ 0) at operating load could make bolt loads less sensitive to temperature changes.

This method involves heating a bolt and, while elongated by thermal expansion, inserting it into the hole of the cold mating members and hand tightening. The bolt contracts as it cools and clamps the members. The task is to determine the bolt temperature required to develop specific clamping forces. The governing equations are:
δt = δbp + δcp = (1/kb + 1/kc)Fi
= (1/ki)Fi
and
t = ta + δt/Lbαb.
Here, t = heated bolt temperature, ta = ambient bolt temperature, δt = total deformation of bolt, δbp = deformation of bolt from preload, and δcp = deformation of clamped members from preload.
If the clamped members are rigid, the bolt must shrink by δbp = Fi/kb to generate preload tension Fi. Because the clamped members are not rigid, they deform by δcp = Fi/kc from compression. The bolt must shrink that much more to match member deformation and maintain the preload Fi. Therefore, the bolt must be heated to contract by δbp + δcp.
Under certain circumstances, measuring the nut-rotation angle may be an alternative to gaging torque for controlling preload. The total nut-rotation angle required to impose preload Fi in a bolt is:
θ = 360(δt/P) = 360N((1/kb + 1/kc)Fi
= 360NFi/ki
where P = the lead of the thread (for single helical screws, lead = pitch), and N = number of threads per unit length of the bolt for single threads. Nut-rotation preloading is more directly related to the bolt tension bypassing uncertainties in friction altogether. In reality, the starting point of the nut-rotation angle, usually the hand-tightened position, needs a tolerance of a few degrees. And that could be about 20% of the calculated preloading angle.