The peak maximum principal stress of 3,700 psi occurs at hole A.

The peak maximum principal stress of 3,700 psi occurs at hole A.


The first sketch shows the bracket before loading by force P. The middle sketch shows how the part deflects in the absence of the mounting surface and the reaction at the bolt, F = P. The final sketch shows how the mounting surface restrains deflections by pushing against the part and introduces a secondary reaction from the prying action of magnitude C. Bolt reactions are now amplified and F' = P + C.

The first sketch shows the bracket before loading by force P. The middle sketch shows how the part deflects in the absence of the mounting surface and the reaction at the bolt, F = P. The final sketch shows how the mounting surface restrains deflections by pushing against the part and introduces a secondary reaction from the prying action of magnitude C. Bolt reactions are now amplified and F' = P + C.


In this last of three parts dealing with bolted joints, David Dearth, a consulting stress analyst and president of Applied Analysis & Technology, Huntington Beach, Calif. (AppliedAT@aol.com), shows how flexible brackets deform when loaded. An example also shows how deformation changes the load path and a table compares bolt reactions to those from a rigid model.

In previous columns, simple rigid beams modeled the bracket and tube. "But physical systems in the real world are far from perfectly rigid," says Dearth. "Deflections alter load paths and consequently effect support points. Structures also push against adjacent structures and mounting surfaces. There can also be secondary reactions at attachment points that tend to 'pry bolts up' much like pulling nails with a claw hammer," he says. Accounting for the physical deflections (strain energy) of a component makes it difficult to estimate bolt reactions. Solutions to The pipe problem (also presented in the FE Update of Feb 3, 2005) use a nonlinear flexible model. It produces significant differences between the solution from the free-body equations because the physical geometry absorbs a portion of the loading in the form of strain energy. " Absorbing this energy produces deflections that change the magnitude of the reactions from using the rigid-body approach. A flexible-model solution should include nonlinear gap elements to account for deflections that would tend to put the support surface into compression against the mounting surface," says Dearth.

The flexible FEA version shows a mathematical idealization of the assembly. "The accompanying table shows that all the forces from compressive gap elements make up a relatively large portion of the total reactions, 665.8 lb in this case," he adds. But some gap elements carry small compression loads while those near the attachments carry more of the total bearing (bracket-to-mounting face) forces. The stiffness of adjacent mating parts influences the magnitude of these reaction forces because pushing the support bracket against its mounting surface introduces secondary loads into the attachment which must resist the prying action. A second illustration, How brackets react, shows the secondary prying action on a simple bracket and how it tends to amplify bolt reactions.

"Reactions at bolts vary greatly depending upon the assembly's stiffness," says Dearth. "For instance, as the support bracket and tube become more 'rigid', reactions at bolts begin to approach the values estimated from simple static analysis." Conversely, as the bracket becomes more flexible, Bolt A begins to carry more of the total external load because it takes the shortest load path (the path of least resistance) to balance the applied forces.

Dearth suggests exploring this fact by modifying the model and looking at the reactions when, for example, side gussets are 0.125-in. thick instead of 0.25 in. Or increase the flexibility of the mathematical idealization of the real geometry by increasing the number of elements. It's also interesting to increase the flexibility of the geometry by reducing the thickness of the bracket parts.

For the problem in A more flexible FEA model, loading at the bolts is not difficult to estimate using conventional equations for static equilibrium. "For these, we assumed uniform and symmetric reactions to allow for statically determinate solutions," says Dearth. "In many real-life applications, geometry and loading are not always symmetric about the attachment pattern and actual physical systems take the shortest load path to react to applied loading. This is one reason Bolt A reacts disproportionately to the total applied load. Nonlinear contact-gap elements are needed to account for the compressive reaction forces of the support bracket against the support surface. The exception is when applied loading is all in the plane of the part and there is no prying action at attachment points.

"Results between pencil and paper calculations versus flexible body techniques in FEA that estimate reactions at bolts can differ greatly, sometimes by an order of magnitude," says Dearth. "That's because FEA models account for the bolts' secondary loading due to physical deformations of the geometry. These secondary reactions are the result of strain energy being reacted by the structure's physical geometry and bolt locations constrained in all directions.

"Think of a simply supported beam. Hand estimates for this condition produce no reaction in the direction of the beam length when force is exerted normal to the beam. But an FEA model with both ends pinned restrains the deflection and generates reactions in the longitudinal direction."

The differences between FEA models and hand estimates stem from assuming that reactions at the bolts are statically determinate for hand analysis, thus eliminating reactions due to deflections. "Constraining reactions in FEA models in all three directions typically introduces additional reaction forces due to deflections from external loads," he adds. To complicate matters further, the bolt axes may not align with the global X, Y, and Z axis. In these applications, FEA software can calculate loads at the bolts in terms of the local bolt-coordinate system.

In actual applications, conventional equations found in engineering literature can be used to estimate a magnitude for bolt preload using the stiffness (spring rate) of the bolt and assembled components. Design tables from manufacturers typically size bolts based on axial load (preload) and resultant shear. "When FEA users incorporate boundary conditions at the bolt that allows bending moments and torsional reactions, these moments and reactions can cause a mismatch for selecting the correct bolt," says Dearth.

Comparing Results From Hand Solutions And A Flexible Fea Model
 
Bolt axial
Bolt shear
Bolt shear
Resultant shear
Bolt
Solutions
Net total, X
Net total, Y
Net total, Z
Net shear
 
Pencil and paper
367.617
67.500
60.000
90.312
A
Flexible FEA model
498.610
167.876
74.872
183.815
 
% difference
32.917
146.598
23.707
101.804
 
Pencil and paper
151.999
7.500
60.000
60.467
B
Flexible FEA Model
36.630
32.398
11.159
34.266
 
% difference
123.182%
519.949%
79.954%
44.243%
 
Pencil and paper
281.902
67.500
60.000
90.312
C
Flexible FEA Model
337.364
35.038
31.246
46.946
 
% difference
22.945
46.639
49.077
47.701
 
Pencil and paper
237.713
7.500
60.000
60.467
D
Flexible FEA model
53.046
20.516
54.785
58.500
 
% difference
78.976
379.682
7.171
1.572
Bearing
Sum comp. gap
665.842
0.000
0.000
N/A
Sum
Net sum forces
259.808
150.000
0.000
N/A
All loads are in pounds. The sum of external forces from the applied loading should equal: Fx = 300 cos (30°) = 259.808 lb and Fy = 300 sin (30°) = 150.00 lb.

Calculations and notes for the FEA models

Readers will find several PDFs such as the one titled Bolt Reactions HandCalcs.pdf at MachineDesign.com/MD/bolts. The files contain detailed hand calculations with arithmetic for the models in this series. The files are also available from Dearth at AppliedAT@aol.com. FEA models, RigidMdl_BoltReactions_v2004.mod and FlexMdl_BoltReactions_v2004.mod, are small enough to process using limited-node or demo versions of MSC/Nastran v2004. To obtain a free copy of the limited-node software, log onto: mscsoftware.com/offers/master/contact.cfm or telephone MSC Software at (866) 672-1549. However, the files will also work in any version of Nastran.


FOR FURTHER READING John H. Bickford, An Introduction to the Design and Behavior of Bolted Joints, Chapter 15, Introduction to Joint Failure, Marcel Dekker Inc., 1981.

J.E. Shigley & C.R. Mischke, Standard Handbook of Machine Design, Chapter 23, Bolted and Riveted Joints, McGraw-Hill Book Co., 1986.

Erik Oberg, Franklin D. Jones, and Holbrook L. Horton, Machinery's Handbook 26th Edition, "Bolts, Screws, Nuts and Washers Formulas for Stress Areas and Lengths of Engagement of Screw Threads," Industrial Press, 2000.