In a previous column, David Dearth, a consulting analyst and president of **Applied ****Analysis ****& ****Technology**, Huntington Beach, Calif. ( *AppliedAT@aol.com), *discussed four ways to handle fasteners and preloads with FEA in assemblies. This article continues by solving a bolt problem by hand and using an FEA program.

Dearth suggests this problem as an example of using FEA to determine reactions at bolts. "I recommend working through a few sample or warm-up problems with textbook solutions before tackling the real one. This problem simulates a 300-lb load on a bracket and tube. The task is to estimate reactions at the four mounting locations so bolts can be sized for them. This problem contains features of real-life engineering challenges and can be solved with pencil and paper using conventional static analysis — summing the forces of a free body diagram. A second task estimates reactions at the bolt locations using FEA. Then compare results," he says.

**Hand calculations **rely on equations found in most engineering textbooks with sections on finding reactions in assemblies. "To check your work, the detailed static calculations can be downloaded from *machinedesign.com *or by requesting a copy from me by e-mail," says Dearth.

**Formulating a rigid model**, the second task, generates results using an FEA model and compares them to the hand calculations. *The **rigid **model *shows the geometry of the loaded tube. "The stick-figure model uses rigid elements," says Dearth. "Solve for the reactions to four decimal places using simple equations that sum static forces and a spreadsheet to minimize roundoff in the arithmetic," he adds. The calculation-table summarizes and comparesresults from manual calculations and FEA outputs for the rigid body. The FEA model was processed using MSC/Nastran.

Bolt | X total | Y total | Z total | ||

A | |||||

B | |||||

C | |||||

D | |||||

Sum | |||||

Estimates of reactions at the bolt locations using conventional equations and FEA mathematical idealization agree well with each other. Summation of external forces from the applied loading should be F _{x}=300 cos (30°) = -259.808 lb and F_{y}=300 sin (30°) = 150.00 lb. Resultant shear = (X^{2} + Y^{2}) ^{1/2}. All loads are in lb. |

**CALCULATIONS AND NOTES FOR THE FEA MODELS **

Readers can refer to a document titled *Bolt **Reactions **HandCalcs.pdf *. There are four additional files, among them, detailed hand calculations with summary spreadsheet arithmetic (Part1_RunNotes_RigidModel_BoltReactions.pdf and Part 2_RunNotes_FlexModel_BoltReactions.pdf). The files are also available from Dearth at *AppliedAT@aol.com. *Other files include run notes and keystroke summaries for models. The 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. However, the files will also work in any version of Nastran. To obtain a free copy of this demo software, log on to: (mscsoftware.com/offers/master/contact.cfm) or telephone MSC Software at (866) 672-1549.