Machine Design

Modeling the Effects of Bolt Preload

Preloading makes for robust joints. Here’s a simple way to get the analysis right.

Authored by:
Moo-Zung Lee
West Hills, Calif.

Edited by Kenneth J. Korane
[email protected]

Key points:
• Preloading stiffens bolted joints and increases an assembly’s resonant frequency.
• Using structural properties in a finite-element model simplifies analysis.

How External Loads Affect Bolted Joints,
Structural Properties of Bolted Joints,

Torquing down nuts and bolts, as any handyman or mechanic knows, makes for secure joints. The question is how much preload is just right?

External tension applied to a nonpreloaded joint transmits entirely to the bolt. The stiffness of a bolted joint without a preload equals the bolt stiffness, ke = kb. For a preloaded bolted joint, joint stiffness is the sum of the bolt and clamped-member stiffnesses, ke = kb + kc. Therefore, preloading stiffens bolted joints, which also increases the resonant frequency of an assembly.

However, once a joint is preloaded, increasing the magnitude of preload cannot make a joint stiffer. But larger preloads do handle higher external loads, and thus tend to better keep joints from separating.

If an external load consists of static and fluctuating components, the static load contributes to the mean load and the fluctuating load produces alternating loads on the bolt.

Fe = Fem + Fea
Fbm = Fi + ΦFem, and
Fba = ΦFea

where Φ = bolt load-sharing ratio.

For instance, if external force is a sine function with a unit amplitude and Φ = 0.25, then the bolt alternating load is 25% of the external load, as shown in the graph.

Preloading reduces both the mean and alternating loads on the bolt through the effect of the bolt load-sharing ratio. (The bolt load-sharing ratio, Φ, is the fraction of external load on the bolt, with 0 < Φ < 1. A typical Φ value for a bolted joint is about 0.2 to 0.5. For hard, thin, and wide clamped members, Φ approaches 0. For compliant joints, such as those with a soft gasket, Φ approaches 1.)

Thus, preloading reduces the combined bolt stress and increases the safety margin in service. Because the external load is the only source of bolt alternating load, the influence of Φ on alternating load is more significant than on the mean load. In addition, reducing alternating-load amplitude significantly increases the material’s fatigue life. Therefore, preloading is instrumental in reducing the alternating load on the bolt and increasing its fatigue life.

Consequently, a smaller Φ is better from a bolt stress and fatigue-life viewpoint. But a smaller Φ means a smaller separation load for the same amount of preload. Increasing preload, on the other hand, may result in higher assembly stress as well as the combined preload and external load stresses.

One way to better predict behavior is through a finite-element model of the preloaded bolted joint. A typical FE model of a mechanical assembly uses rigid boundary conditions for modeling bolt connections at the mounting. The reactions at boundary constraint nodes are extracted from load run outputs for detailed stress analysis of the bolts. Rigid boundary conditions, especially those with multiple degrees-of-freedom constraints, sometimes result in large unrealistic reaction load outputs. One way to mitigate this situation is to model the connections with elastic elements. The challenges in elastic modeling of a preloaded bolted joint are how to select the stiffness and how to treat the preload.

A conventional FE model of a bolted joint could be constructed to include the bolt and clamped members using many elements, and possibly simulate preload with initial strain on the elements. Joint separation might be handled by gap elements. This would require a significant number of elements to model a single joint. Simulating preload by initial strains could be complex and gap elements must be treated gingerly to have convergent solutions.

An alternative is to use the structural property of a preloaded bolted joint in the FE model via a single spring element. Begin the analysis with a simple spring element with spring rate, ke, or the sum of the bolt and clamped-member stiffnesses, kb + kc. After running the model with load, extract the joint reaction Fe at the joint node. The combined bolt load, Fb, is the sum of the bolt-preload tension, Fi, and the bolt’s share of the node reaction load, ΦFe,

Fb = Fi + ΦFe.

If the external load extracted from this model does not exceed the separation load, the simple spring element is acceptable and the analysis result is final. Otherwise, it is prudent to review the design and magnitude of preload torque to resolve joint-separation issues.

Use a nonlinear spring element only if behavior beyond joint separation is of interest. Typically, force-displacement data such as line segments OAB in the accompanying diagram are used for nonlinear spring-element input. To expedite convergence, a smoothed curve, as shown, is preferable. The nodal force output is the external force on the joint. Calculate bolt force using Fb = Fi + ΦFe for the preloaded phase and Fb = Fe for the separation phase.

Fb = Total bolt load
Fba = Fluctuating bolt load
Fb m = Static bolt load
Fe = Total external load
Fea = Fluctuating external load
Fem = Static external load
Fi = Bolt-preload tension
kb = Bolt stiffness
kc = Clamped-member stiffness
ke = Bolted-joint stiffness
Φ = Bolt load-sharing ratio

© 2010 Penton Media, Inc.

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