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.

Thermal 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.

Heated-bolt preloading
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.
Nut-position preloading
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.

Preloading mechanism compliance
In some cases, the nut-rotation angle may be too small for this technique to be practical. It is not unusual to only require about 10° of nut rotation for preloading. For a given preload requirement, choosing an extra-fine thread over a coarse thread and a larger compliance for the preloading components will increase the nut-rotation angle.
When specifying torque for preloading, the corresponding nut-rotation angle is usually not calculated. As a result, there is no clue if the specified torque is difficult to work with because the corresponding wrench rotation angle is small.
The nut-position method is not exactly an alternative method for preloading bolted joints. Rather, it is an alternate way of controlling the magnitude of preloading. Whether specifying torque or nut rotation angle, in the end, a torque will be applied to tighten the nut. Two aspects govern the torque process: force to pull the wrench and distance to pull it. Torque specifications reflect the former; nut rotation reflects the latter.
The nut rotation angle is related to torque T by the equation:
T = CDFi
where C = the torque coefficient (a function of bolt size and coefficient of friction) and D = the nominal bolt diameter. Calculate the nut angle in degrees with:
θ = 360(N/ki)(T/CD).
Therefore, nut angle and torque are proportional, and force on the wrench and how far to rotate the nut are two indicators of the same thing.
Giving the preloading loop more compliance is an effective way to increase the preloading nut angle. Longer spacer sleeves and disc springs are some options. Perhaps more fundamentally, it is best to avoid grossly oversized bolts.

Nomenclature
(Note: Subscripts b and c refer to the bolt and clamped members, respectively.)
eb, ec = Thermal growths
kb, kc = Stiffness
Lb, Lc = Free lengths at room temperature
Δb, Δc = Initial preload deformation at room temperature
ΔTb, ΔTc = Temperature changes
αb, αc = Thermal expansion coefficients
δ = Shift in assembly position due to temperature change
δb, δc = Deformations from free length at temperature

© 2010 Penton Media, Inc.