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.

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.