In last week’s blog, we discussed the science behind bolted connections. Specifically their response when subjected to a combined external load and preload. In this blog we’ll work through the application of bolted joint science in a finite element analysis. We will:
- Provide an introduction to FEA modeling of pre-loaded bolted joints (e.g. meshing, connections, boundary conditions and applied loading)
- Discuss some common troubleshooting techniques.
- Establish methods to validate results of the FEA model.
- Highlight differences between bolted joint science and real-world application.
Sample Problem
Consider a simple bolted connection of a W section to a WT section. The W-section is a W8x40 and the WT-section is a WT4x20. The members are bolted together using four 5/8″ diameter structural bolts with 3/4″ diameter bolt holes. The bolts are torqued to produce an initial preload of 80,000 psi.
Bolted Connection FEA Model
The connection is meshed using solid elements. The model includes the structural members, bolts, nuts and washers. The mesh is refined at the bolted regions to capture the irregular geometry of the bolts and nuts.
Bolts, nuts and washers of the connection are modeled in full. This allows the finite element analysis to capture the local response such as bolt bearing and bending. Threads of nuts and bolts are excluded from the FEA analysis. This is done for two reasons:
- Refining the mesh to capture the threads would create a computationally exhaustive model.
- The structural capacities bolts and nuts are already well documented. Bolt loads can be compared to published nominal bolt capacities.
Preload Application
Bolt preload is typically specified in stress, often as a percentage of yield, proof or allowable tensile stress. Preload is applied in the FEA model by applying an initial tensile stress to the bolt cross section. Application of this preload stress must be verified during troubleshooting of the model.
Load Application
The connection is subjected to an external tensile load. A vertical, ramped load is applied to the hole in the WT-section. The applied loading ramps to 300 kips at 0.01 seconds and is held constant to 0.015 seconds.
Analysis Type
We’re interested in the the effect an increasing external load has on the bolted joint. Specifically the changes in total bolt load and compressive preload between the steel sections. The quickest way to obtain a single solution, modeling this behavior, is to ramp the external load as a function of time. Evaluating this time-dependent loading requires a transient analysis, and explict FEA software such as LS-DYNA. Explicit solvers are better equipped to accommodate large changes in model stiffness, such as the change in stiffness following separation of the bolted joint.
Note: An implicit solver could used to evaluate the same loading, however the resulting output would be a compilation of several steady-state solutions. To use an implicit solver the engineer must ensure that:
- If a single linear elastic model is used: The engineer must ensure that joint separation has not occurred (indicated by the loss of preload). Similar to hand calculations, if joint separation occurs there will be no member stiffness. The entire load will be passed into the bolts (Joint Stiffness Constant =1)
- If a transient implicit solver is used: The engineer must ensure that the time steps are sufficiently small to capture the response as the joint begins to separate.
Connections, Connections, Connections….
The FEA model contains all connections/interfaces to capture the preload behavior. The FEA model contains two different connection types.
- Bonded Connections: The threaded connection between bolts and nuts are modeled as a bonded contact. These connections are shown below:
Bonded Connections Between Bolt & Nut - Frictional Contacts: Interfaces between all other components are modeled as frictional contacts. Friction stabilizes the model. Contacts are modeled individually so individual connection loads can be recovered. Frictional contact surfaces are shown below:
Bolted Joint Frictional Surfaces
Boundary Condition
The bottom outside edges of the W8x40 are restrained from vertical deflection. This support scheme mimics a simply-supported beam with beam ends free to rotate.
Damping
This analysis is performed using LS-DYNA an explicit FEA code. In time dependent analysis, short duration shock loads can amplify structural response. An overdamping of 400% is applied to quell any local response. Damping is velocity dependent. These FEA results are not affected as the structure reaches a steady state response to the applied load.
Trouble Shooting the FEA Model
Verifying Contacts
We need to review the stress profile and deformation to determine if the model is behaving as expected. For example, the images below highlight the effect of a missing contact. The missing contact between a washer and nut fails to load the bolt. The missing contact results in only three of four bolts carrying the external load. This condition is evident by the lack of even stress between the four bolts. The corrected condition is also shown, with matching stresses at all four bolts.
Verifying Preload
Preliminary FEA results show each bolt has an initial preload of 25 kips (refer to graph below). The nominal area of a 5/8″ bolt is 0.307 square inches. Considering a prestress of 80,000 psi, the nominal bolt preload is calculated to be 24.5 kips. The bolts are sufficiently preloaded. Convergence can be increased by reducing the tolerance in the dynamic relaxation phase of the FEA run. Prestress in the bolts is evident in the figure below. Tensile stresses are positive while compressive stresses are negative.
Results of Finite Element Analysis
The bolted connection is shown below. Load path from the hole in the WT-section to supports is evident by the highly stressed regions. Load travels from the point of application to the bolted connection. Then from the bolted connection through the beam W-section beam web to the support points.
As shown in the figure below, increasing external load results in separation of the joint. The partial separation of the joint is caused by the flexibility of the flanges of the section. This response is commonly referred to as prying. Prying results in an extra bolt load and residual contact at the compressive joint face.
Comparison with Bolted Joint Science
The science of bolted joints subjected to external loads is well documented. Structural response can be divided into two steps:
- Pre-separation: Prior to joint separation the compressive preload diminishes significantly. The bolts undergo a small increase in tensile load for a larger decrease in preload. The ratio of preload loss is to bolt load increase is known as the Joint Stiffness Constant.
- Post-separation: After separation, extra loading results in a direct increase of bolt load.
Finite Element Analysis vs. Science
Results of the finite element analysis are extracted and shown below. The graph tracks the load in the following:
- Bolt Load: Tracked via bonded contact between bolt and nut
- Compressive Preload: Tracked via frictional contact between the W and WT section
- Applied External Load: Tabular FEA input data
- Cumulative Bolt Load: Summation of individual bolt loads
- Difference between compressive preload and cumulative bolt load: Cumulative bolt load minus compressive preload
Validation:
The FEA model needs to be validated before results can be considered correct. The model can be validated by ensuring the joint remains in static equilibrium. That is, the summation of all forces across the bolted joint equals zero. Rewriting, the difference between the total bolt load and compressive preload should equal the applied external load.
The figure, above, shows the difference between bolt load and compressive preload. The applied external load is overlaid on the graph. As shown in the figure above the applied external load equals the difference between bolt load and compressive preload. The joint satisfies static equilibrium and the results are valid.
Agreements with Bolted Joint Science
The finite element model behaves as predicted by bolt joint science. Two behaviors are noteworthy:
- Initial Loading: The bolted joint carries a significant external load of 40,000 lbs with negligible increase in bolt load. This load is accompanied by a decrease in the compressive preload. This behavior agrees with the analytical predictions for bolted joints.
- Loading with Separation: Following separation, the increase in external loading is directly proportional to the increase in bolt load. This behavior also agrees with the analytical predictions for bolted joints.
Deviations from Bolted Joint Science?
The results of the finite element model generally agree with bolted joint science. One apparent exception is the behavior of the compressive load between the W and WT section. In the FEA model, the compressive preload decreases but is never lost. Joint science suggests that the preload be completely lost as the joint separates. So what’s going on?
This difference can be explained in one word, Prying. Bolted joint science assumes that connected parts are substantially stiff. Stiff members create even separation across the entire joint face. In our FEA example the flanges of the members are much more flexible and bend. This bending (prying) amplifies bolt loads. A residual compressive load develops the joint to satisfy static equilibrium. These compressive prying stresses are evident at the edges of the WT section near bolts, as shown in the figure above.
Limitations of this FEA Model:
This finite element model is developed for informational purposes only. The FEA model uses a linear elastic material model with no failure criteria included.
Conclusion
FEA requires thoughtful selection of boundary conditions and connections to represent joint behavior. The results of the finite element model correlate well with bolted joint science. Finite element analysis has several additional benefits including the ability to probe secondary joint responses such as prying.
One thought on “Modeling Bolted Connections Under External Load with Finite Element Analysis”
very interesting and informative article. Thanks a lot for sharing.