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.


Comparing a Static Calculation and Rigid-Body FEA Results
 
BOLT AXIAL
BOLT SHEAR
BOLT SHEAR
RESULTANT SHEAR
Bolt
Solutions
X total
Y total
Z total
Net shear
A
By hand
-367.617
67.500
-60.000
90.312
 
Rigid FEA model
-367.617
67.500
-60.000
90.312
 
% difference
0.0
0.0
0.0
0.0
B
By hand
151.999
7.500
-60.000
60.467
 
Rigid FEA model
151.9999
7.500
-60.000
60.467
 
% difference
0.0
0.0
0.0
0.0
C
By hand
-281.902
67.500
60.000
90.312
 
Rigid FEA model
-281.903
67.500
60.000
90.312
 
% difference
0.0
0.0
0.0
0.0
D
By hand
237.713
7.500
60.000
60.467
 
Rigid FEA model
237.713
7.500
60.000
60.467
 
% difference
0.0
0.0
0.0
0.0
Sum
Net forces
-259.808
150.000
0.000
N/A
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 Fx=300 cos (30°) = -259.808 lb and Fy=300 sin (30°) = 150.00 lb. Resultant shear = (X2 + Y2) 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.