Preliminary Final Report - A.DeLugan - EWP

Understanding Buckling Behavior of a Tie Rod Using
Nonlinear Finite Element Analysis
By
Anthony DeLugan
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
In Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Dr. Ernesto Gutierrez-Miravete, Project Advisor
Rensselaer Polytechnic Institute
Hartford, CT
May 2012
CONTENTS
LIST OF FIGURES .......................................................................................................... iii
LIST OF TABLES ............................................................................................................ iv
ACRONYM LIST ............................................................................................................. v
LIST OF SYMBOLS…………………………………………………………………….vi
ACKNOWLEDGMENT .................................................................................................. vi
ABSTRACT ................................................................................................................... viii
1. Introduction.................................................................................................................. 1
2. Methodology ................................................................................................................ 2
2.1
Finite Element Model Setup .............................................................................. 3
2.2
Linear Analysis .................................................................................................. 6
2.3
Nonlinear Analysis ............................................................................................. 7
3. Results and Discussion .............................................................................................. 11
3.1
Linear Analysis ................................................................................................ 11
3.2
Nonlinear Analysis ........................................................................................... 13
4. Conclusions................................................................................................................ 21
5. References................................................................................................................. .22
6.. Appendices ............................................................................................................... .22
ii
LIST OF FIGURES
Figure 1: Tie Rod Test Setup……………………………………………………………3
Figure 2: Tie Rod Finite Element Model………………………………………………..3
Figure 3: Tie Rod Dimensions…………………………………………………………..4
Figure 4: Stress / Strain Data for 15-5PH Stainless from MMPDS-04…………………5
Figure 5: Stress / Strain Data for 2024-T3 Al from MMPDS-01…….…………………5
Figure 6: Axial Load Application on Upper Rod End…………………………………..6
Figure 7: Modified Newton Method…………………………………………………….9
Figure 8: Linear Buckling First Mode Shape and Eigenvalue………………………..12
Figure 9: Post-buckled Shape of Test Tie Rod………………………..………………13
Figure 10: Excerpt of F06 NASTRAN Output file for Modified Newton Analysis…15
Figure 11: Location of Instigative Buckling Preload…………………………………..16
Figure 12: Buckling Shape of Tie Rod at Maximum Load - Arc Length Method…...17
Figure 13: Load / Deflection Data for Node I.D. 66068………………………………17
Figure 14: First Mode Buckling Shape – Subcase 2 – NL Buckling………………….18
iii
LIST OF TABLES
Table 1: Tie Rod Material Properties for FEM Setup…………………………………..4
Table 2: F06 File Excerpt for First NL Static Run……………………………………...9
Table 3: Results Summary - Critical Buckling Load Comparison…………………...20
iv
ACRONYM LIST
AA: Aluminum Alloy
AMS: Aerospace Material Specification
CRES: Corrosion Resistant Steel
FEA: Finite Element Analysis
FEM: Finite Element Model
IN: Inch or Inches
LBS: Pounds
MMPDS: Metallic Materials Properties Development and Standardization
NL: Nonlinear
PSI: Pounds per Square Inch
v
LIST OF SYMBOLS
[ K 0 ] = stiffness of initial configuration
λCR = critical load factor multiplier or eigenvalue
[ Kσ ] = differential stiffness due to applied loads and constraints
{ϕ} = buckled mode shape or eigenvector
E = Young’s modulus (psi) = 10.5E+06 psi
I = area moment of inertia (in^4) =
π ⋅ (.Do 4 − Di 4 )
4
Do = outer rod diameter (in) = 0.653 in
Di = inner rod diameter (in) = 0.569 in
Le = Effective length (in) = 8.829 in
PCR = Euler critical buckling load (lbs) =
π2 ⋅E⋅I
Le
2
∆P = incremental load step (lbs) between applied load of subcase 1 ( Pn −1 ) and applied
load of subcase 2 ( Pn )
vi
ACKNOWLEDGMENT
To my wife Jennifer and my daughter Elizabeth.
Also, a special thanks to University Swaging for permission to use their tie rod buckling
experimental results.
vii
ABSTRACT
This report presents an analytical study that predicts the critical buckling load of a tie rod
that is highly loaded in axial compression. This analysis is tailored specifically to an
aerospace rod end that consists of two outer corrosion-resistant (CRES) steel rod ends
threaded into a center, hollow aluminum rod body. However, the analysis techniques
outlined in this paper can be used for any component that cannot rely on the assumptions
and boundary conditions posed by Euler’s linear buckling theory. This report provides
more evidence to existing literature that more complex structural components exhibit
nonlinear buckling behavior.
To obtain the most accurate prediction of the critical buckling load, both a linear
analysis and nonlinear analysis is performed. The linear analysis provided the mode
shape to predict the location of max displacement and instability during buckling.
Following the linear analysis, three different nonlinear analysis techniques were carried
out in this study, in this order: 1) Modified Newton’s method; 2) Arc Length method; 3)
Nonlinear eigenvalue buckling analysis. Each method was compared to the experimental
test results to verify accuracy of the calculations. For each method, a finite element mesh
sensitivity study was also conducted to verify agreement of the finite element analysis
methods and its relationship with the finite element model.
Results indicate that all three nonlinear analysis techniques are suitable to accurately
predict the critical buckling load of a structural tie rod. Amongst the three nonlinear
analysis methods, the estimated critical buckling load ranged from 12.5%-13.6% less
than the experimental buckling load. Correlation to these results with results in the
referenced literature verifies that the Arc Length method, which predicted 12.5% error
margin, is the most accurate yet least conservative of the nonlinear analysis techniques.
The Arc Length method follows the post-buckling nonlinear load/displacement curve
and predicts a slightly higher buckling load than the Modified Newton Method (13.6%
margin), which shows induced buckling at the first sign of instability or bifurcation. The
variation in safety margins amongst the nonlinear techniques is small however, so it is
recommended that the designer follows the complete methodology provided in this
report in order to obtain a high degree of confidence in the calculations.
viii
1. Introduction
For the past century a great deal of research has been invested to help predict the critical
buckling loads of cylindrical columns. Research (both theory and experimental) has indicated
that geometrical imperfections and modified boundary conditions greatly impact the critical
buckling load magnitudes and scatter of cylindrical columns. A tie rod contains such geometrical
imperfections and modified boundary conditions from a perfect cylindrical shell, since a tie rod
typically consists of two outer rod ends threaded into a cylindrical rod body, with varying end
conditions. It is very important to accurately predict the buckling loads of structural tie rods,
especially ones that are compression critical in aerospace applications.
There are several applications in the aerospace industry where a tie rod is utilized to help
secure and support equipment on an aircraft, such as on the fuselage of an airplane. These are
purely structural members, so a robust knowledge of the design loads is required to ensure the
part will satisfy its function on the aircraft. In certain cases, these tie rods need to be designed to
buckle at a specific load to avoid puncturing or damaging nearby components. Based on the
design criteria of minimizing compression margin safety coupled with the degree of difficulty to
predict buckling behavior, accurately calculating the critical buckling load is of high importance.
This report is an analytical study that gives a designer a systematic approach to accurately
predict the buckling load of a structural tie rod. To accomplish this level of accuracy, a non
linear finite element analysis is conducted. The goal of the report is to establish an acceptable
method of predicting the buckling load of a structural tie rod due to axial compression.
Two references provide good precedence for the methodology outlined in this report.
Reference [1] contains a nonlinear buckling analysis of a tie rod that utilizes arc length method.
This tie rod consists of a single material, unlike the subject tie rod of this report. One of the
focuses of this report is to see if similar outcomes are achieved using the nonlinear analysis of a
tie rod of with different materials.
Reference [2] contains analysis of a completely different configuration with a much smaller
number of elements, however the general approach and methodology used to perform non-linear
analysis in Reference [2] is the same as what is executed in this paper.
2. Methodology
Finite element linear modal analysis is first conducted in order to obtain the first mode
eigenvalue, which when multiplied by the applied compressive load provides the estimate linear
buckling load. Knowing the linear buckling load is important as it provides a good starting point
load to use in the ensuing nonlinear analysis [1]. Linear and nonlinear analysis is conducted
using Unigraphics NX7.5 Advanced Simulation module with NASTRAN solver. Linear buckling
analysis uses NASTRAN solver SOL105, and nonlinear analysis uses NASTRAN solver
SOL106.
Nonlinear finite element analysis is needed to obtain the most accurate prediction of the
critical buckling load.
Linear analysis does not account for geometric imperfections and
permanent material deformation. Material buckling is largely a material strain / deformation
phenomenon. Its responses cannot be controlled and measured through a traditional linear finite
element analysis. Linear analysis assumes small pre-buckling deflections and linear stress strain
relationships; these assumptions do not represent true material behavior.
The nonlinear analysis described in this study consists of three different approaches
which are each evaluated and compared. The first nonlinear analysis method is load ramp step up
using Newton’s method. The second method is using the arc-length convergence method. The
third method is employing a nonlinear eigenvalue buckling analysis. Analysis iterations
incorporating changes of the FE mesh will verify convergence response and mesh sensitivity.
The critical buckling load is calculated for each case and compared to the average experimental
buckling load of the tie rod in question. Experimental buckling data is provided courtesy of
Primus International, University Swaging division, Woodinville, Washington. The tie rod in
question is shown in figure 1, in the compression test setup.
2
Figure 1: Tie Rod Test Setup (image courtesy of University Swaging) [3]
The tie rod consists of two outer CRES steel rod ends made of 15-5PH material. The outer rod
ends each contain a spherical bearing. These rod ends are threaded into a 2024 T3 hollow
aluminum rod body.
2.1 Finite Element Model Setup
The tie rod FEM consists of 5,531 tetrahedral 10-node 3D elements. Since the rod end and
rod body materials differ, they have separate material properties in the FEM. The rod ends are
connected to the rod body using glue-coincident mesh mating conditions, so mating nodes are
lined up as if it is a continuous mesh. Figure 2 shows a picture of the tie rod FEM.
Figure 2: Tie Rod Finite Element Model
3
The tie rod dimensions are shown in Figure 3. The thickness of the rod body is 0.084” nominal.
Figure 3: Tie Rod Dimensions [3]
For the FEM, material properties for the rod ends and center rod body are listed in Table 1
below.
Tie Rod Detail
Rod Ends
Rod Body
Material
15-5PH per
AMS5659
AA2024-T3 per
AMS4088
Compression
Elastic
Modulus (psi)
Modulus (psi)
29.2E+06
28.5E+06
0.272
10.7E+06
10.5E+06
0.33
Poisson’s Ratio
Table 1: Tie Rod Material Properties for FEM Setup ([4], [5])
The nonlinear analysis requires stress-strain information for the tie rod materials. The stress
strain information for the 15-5PH rod end material used in the nonlinear FEM is taken from the
data in Figure 5, which is from MMPDS-04 [4]. Data is taken from the H1025 condition which is
4
what the part was heat treated to. The stress-strain info for the 2024-T3 rod body used in the
FEM is shown in Figure 6. This data is from MMPDS-01 [5]. The L-compression curve data
applies in this case.
Figure 4: Stress / Strain Data for 15-5PH Stainless from MMPDS-04 [4]
Figure 5: Stress / Strain Data for 2024-T3 Al from MMPDS-01 [5]
5
To best simulate the constraints of the test setup, the top and bottom rod ends are pinned. Load
application occurs along the bottom half on the inner rod end diameter on the upper rod end, as
Figure 6 illustrates. To simplify the FEM, the bearing inserts are removed, and the machined
keyways are removed from each rod end.
Figure 6: Axial Load Application on Upper Rod End
2.2 Linear Analysis
The linear analysis will prove whether or not the tie rod exhibits linear behavior during the
buckling phenomenon. The linear analysis contains the following key assumptions:
•
There is a linear relationship between stress and strain
•
There are small deflections prior to buckling
•
The reference equilibrium position is the initial geometry of the part
Linear buckling analysis theory is represented by the following eigenvalue equation [6]:
([ K 0 ] + λ CR ⋅ [ K σ ] ⋅ {ϕ} = 0
6
[1]
The first mode eigenvalue represents the most accurate critical load factor, such that when
multiplied by the applied load, produces the critical buckling load. The eigenvector associated
with the first mode eigenvalue produces the first mode shape that can be seen in a postprocessing buckling plot. This information is valuable as it should simulate the buckled shape of
the tie rod in question.
Since linear buckling theory in SOL105 employs the same set of conditions as those
assumed in Euler’s columnar buckling equation, a calculation is made using Euler’s buckling
equation to predict the theoretical linear critical buckling load. This critical load will be
compared to the SOL105 critical load to verify accuracy of the simulated SOL105 linear critical
buckling load. Euler’s buckling equation for a cylindrical column is shown below [6]:
PCR =
π 2 ⋅E⋅I
Le
2
[2]
For a typical rod, the effective length is the pin center-to-pin center length of the rod multiplied
by a constant, which is dependent on the boundary conditions. In this case, a pinned-pinned
boundary condition is assumed. This has a constant equal to one.
2.3 Nonlinear Analysis
Nonlinear analysis is required when a structure under applied loading no longer has a linear
elastic stress / strain relationship. It is anticipated that the tie rod in question exhibits geometric
nonlinear effects due to large displacements and rotation, and material nonlinear effects due to
plastic strain. Verification that the tie rod buckling has material and geometric nonlinear
characteristics is when the deformed shape of the buckled tie rod after experimental test is
visibly different that the initial configuration.
Solution convergence becomes more difficult in nonlinear analysis as the structure
experiences plastic strain and geometry changes, which forces an update to the stiffness matrix.
For this reason, solution convergence must be reached by enforcing applied loads in incremental
steps.
7
As mentioned in section 2.1, additional material stress/strain information is required to
run nonlinear static analysis. This information is needed to compute the actual finite elemental
displacements and ensuing stiffness matrix updates while these matrices track the nonlinear load
/ deflection curve.
A static nonlinear analysis solution is run to predict a more accurate tie rod buckling load.
There are three specific nonlinear analysis techniques used in the SOL106 solution, each one is
described in more detail below:
1. Modified Newton Method
The first SOL106 analysis technique used to obtain the critical buckling load uses an incremental
load ramp up via the Modified Newton method. With this method, a total applied load is divided
into a specific number of increments, and gradually ramped up. For each iteration, unbalanced
loads and reaction forces are evaluated, and a linear solution is performed using unbalance loads.
If the solution does not converge within a load increment, the unbalance loads are re-evaluated,
the stiffness matrix is updated and a solution is reached [7]. A pictorial description of the
modified Newton method is shown in Figure 7.
Figure 7: Modified Newton Method [8]
8
When the program reads that convergence is not possible, divergence processing is initiated
through the bisection method. The bisection method reduces the load step increment in half until
convergence is reached. Bisection method gets initiated when the structure experiences a large
nonlinearity or when a load step is deemed to be too large. It is estimated that the applied load at
which the bisection method is activated is close to the critical buckling load [2].
2. Arc Length Method
The Arc Length method is chosen as it is deemed appropriate for snap-through or post-buckling
problems. The Modified Newton method may produce an incorrect convergence at a particular
load step with snap-through problems (structure buckles completely to another stable
configuration), forcing erroneous stiffness updates so it may not accurately predict buckling by
itself [8]. The Arc Length method takes the Modified Newton method and requires it to converge
along an arc, which prevents unrealistic divergence.
The Arc Length method helps eliminate bifurcation points, which are points along a load /
deflection curve where deformation changes slope differently than the pre-buckled curve.
However, bifurcation points can be avoided if the structure is either pre-deformed to match the
first mode buckling shape or if a preload is applied which will induce the structure to begin to
deform in the first mode buckling shape [1]. It is important to note that this ‘instigative’ load
must be removed before evidence of buckling occurs because that will produce underestimated
critical buckling loads [2].
The Arc Length method gets initiated when a negative factor diagonal in the stiffness matrix is
encountered by the FEA solver. This is at the same time that the solver automatically bisects the
load step in half. The Arc Length method controls the load by reducing the load steps as the
displacements increase [1]. To obtain the critical buckling load using the Arc Length method, the
node with the most pronounced displacement needs to be traced. The critical buckling load is the
load that corresponds to the maximum observed deflection on the load / deflection curve [1], [2].
3. Nonlinear Eigenvalue Buckling
9
The third and final nonlinear analysis technique in SOL106 is an eigenvalue buckling analysis.
The governing equation for the nonlinear eigenvalue analysis starts out as the same equation for
the linear buckling analysis [2]:
([ K 0 ] + λCR ⋅ [ Kσ ]) ⋅ {ϕ} = 0
[3]
For the nonlinear buckling analysis to converge properly a pre-buckle instigative load must be
applied to the structure to induce the buckling shape, or as in the case of the analysis in ref [1],
the geometry has an initial modification that would influence its movement towards the buckling
shape. This report utilizes the former method. This preload forces the following update to the
eigenvalue equation [2]:
σ
([ K 0 ] + K PRELOAD + λ CR ⋅ [ K BUCKLE ]) ⋅ {ϕ} = 0
[4]
There is a differential stiffness for both the variable buckling load and the preload.
Eigenvalue extrapolation requires using two incremental solutions because stiffness matrices
need evaluation at two consecutive solution points near the instability point of the structure [2].
Once the solver provides the eigenvalue, the equation to calculate the critical buckling factor is
shown below [2]:
{PCR } = {Pn } + λ ⋅ {∆P}
With {∆P} = {Pn } − {Pn−1}
10
[5]
[6]
3. Results and Discussion
3.1 Linear Analysis
It is known from the experimental data that the average applied load which caused the test tie rod
to buckle is 7061 lbs. This value is used as a starting applied load in the linear analysis phase.
SOL105 is run with the above estimated buckling load and the first mode eigenvalue extraction
provides an eigenvalue of 4.026. The first mode buckling shape along with eigenvalue
calculation is provided in Figure 8 below.
Figure 8: Linear Buckling First Mode Shape and Eigenvalue
The critical buckling load (the estimated load that induces buckling) as calculated using the
methodology of the SOL105 linear buckling analysis is equal to the first mode eigenvalue
multiplied by the applied load. In this case, the critical buckling load is equal to 4.026 x 7061 lbs
or 28,428 lbs. The SOL105 linear buckling analysis predicts a buckling load that is about 4 times
11
the magnitude of the experimental buckling load. This discrepancy indicates that the tie rod
buckling phenomenon does not behave linearly and must be nonlinear in nature due to geometric
and material nonlinear characteristics (see section 2.3). A similar study carried out in ref [1]
discovered a linear buckling load equal to about 3.6 times the experimental buckling load,
proving that the linear analysis in this report is consistent with the assertion that tie rod buckling
is largely non-linear behavior.
Next, a critical load using Euler’s column buckling equation is checked against the SOL105
load. Euler’s buckling equation assumes a constant material and cross sectional area throughout
the length of the stressed section. This is not true for the tie rod, so assumptions are made for the
outer and inner diameters. The outer diameter and inner diameters are averaged based on twothirds the smaller diameter and one-third of the larger diameter from the rod body dimensions of
figure 3. The effective length is considered to be the rod end centerpoint to rod end centerpoint
length. The rod body AA2024 T3 material is chosen. The tie rod parameters for Euler’s equation
are shown in the List of Symbols section.
The Euler critical load calculated at 80,240 lbs is about 182% higher than the SOL105 simulated
load. Because the representative tie rod geometry does not meet the Euler equation assumptions,
this discrepancy is not surprising. For this reason, this comparison is not significant as the
SOL105 critical buckling load more closely matches the SOL105 critical buckling load from ref
[1].
The experimental test buckling shape has a similar profile to the shape shown in figure 6.
Although the bending is not as pronounced, the location of max displacement is about the same,
as Figure 9 indicates.
Figure 9: Post-buckled shape of test tie rod (courtesy of University Swaging) [3]
12
3.2 Nonlinear Analysis
Nonlinear analysis results are presented individually for each technique described in the
Methodology section. Afterward, results from the three techniques are compared to each other.
Modified Newton Method
Several nonlinear iterations using the Modified Newton method were needed to ensure the
applied load and number of load increments used resulted in the best estimate of the point of
instability. Two subcases for load ramp up were chosen. The first subcase loaded the tie rod up
to 6000 lbs in 5 increments, and also enforced an instigative preload of 500 lbs to be incremented
in 5 steps also. The preload allows the tie rod to get into the first mode buckling shape and
prevents any bifurcation phenomena. The second subcase loaded the tie rod from 6000 lbs to
7000 lbs in five increments. The instigative load is released in the second subcase.
Previous iterations found that the tie rod was not buckling at a load equal or greater than the
average experimental buckling load of 7061 lbs. This is why the second subcase falls into a load
range less than 7061 lbs. Results of the SOL106 run with Modified Newton method shows the
bisection method activated in the first increment of the second subcase. An excerpt of the F06
NASTRAN output file is shown as Figure 10 below. The bisection method stops at a point that is
considered to be the critical buckling point. In this case that is load factor 0.1 of the second
subcase. The critical buckling load based on the Modified Newton method can be calculated as
follows:
Critical Buckling Load, Modified Newton Method = 6000 + 0.1 ⋅1000 = 6100 lbs
To substantiate the validity of the results, a mesh sensitivity study for performed. A much
finer mesh with approximately three times the number of elements was applied to the FEM and
the analysis was repeated. Results indicated that bisection method converged at the same exact
load step in the second subcase.
13
Figure 10: Excerpt of F06 NASTRAN Output file for Modified Newton Analysis
Critical Buckling Load, Modified Newton Method = 6100 lbs
The critical buckling load using the Modified Newton method underestimates the average
experimental buckling load by about 13.6%.
The complete NASTRAN input and output files for the Modified Newton nonlinear analysis are
contained in Appendix A for reference. This is the case for all linear and nonlinear analysis
conducted.
Figure 11 below shows the location of the instigative buckling preload, designated by the red
arrow. The load is oriented in the direction of max displacement based on the linear analysis first
buckling mode.
14
Figure 11: Location of Instigative Buckling Preload
1. Arc Length Method
Nonlinear analysis using the Arc Length method takes over where the Modified Newton
procedure left off. The idea is to ramp up the load just before the buckling point, and then let the
arc length method govern the load control post-buckling, until a solution converges. The node
with the max deflection is followed and the load / deflection data is plotted as the option for
NASTRAN to plot specific data at each load step is turned on.
The first subcase loads the structure up to 6000 lbs with a 200 lb preload using the
Modified Newton method. The preload is reduced to ensure that it does not induce buckling. The
second subcase employs the Arc Length method to catch the post-buckling behavior. The arclength method is initiated in the second subcase. Node I.D. 66865 exhibits the largest deflections
in the instability zone, and its deflections are traced as the arc length method gradually reduces
the load. Figure 12 shows the tie rod shape at maximum deflection of node I.D 66865
15
Figure 12: Buckling Shape of Tie Rod at Maximum Load – Arc Length Method
Load / deflection data is captured at each load step using the INTOUT = YES parameter in
NASTRAN. This data is collected and displayed in figure 13 below.
16
Figure 13: Load / Deflection data for Node I.D. 66068
Based on the maximum load at deflection, the critical buckling load calculated using the Arc
Length method equals 6,134 lbs. To substantiate the validity of the results, a mesh sensitivity
study for performed using the same finer mesh model that was used in the Modified Newton
analysis. Load / displacement for the mesh-sensitive model was plotted on figure 13 with the
baseline FEM. Results indicated that the finer mesh produced a slightly higher estimated critical
buckling load of 6,177 lbs.
Critical Buckling Load – Arc Length Method = 6,177 lbs
The critical buckling load using the Arc Length method underestimates the average
experimental buckling load by about 12.5%.
17
Nonlinear Buckling Method
To run the nonlinear eigenvalue buckling analysis, the NASTRAN input file is modified in
accordance with reference [2], see NASTRAN input file in Appendix A for reference. The most
accurate eigenvalue extraction will occur if the applied loads in the subcases fall right under the
expected critical buckling load. Since the critical buckling load has already been estimated to be
slightly above 6000 lbs, two subcase loads of 5000 lbs and 6000 lbs respectively with 5
increments each are used.
Eigenvalue extraction from both the NASTRAN F06 output file and post processing
results are provided. The first mode eigenvalues are the critical values for each subcase and are
extracted for this purpose of obtaining the critical buckling load per equation [5]. Below are the
first mode eigenvalue calculations for both subcases.
Subcase 1 1st Mode Eigenvalue = 0.812
Subcase 2 1st Mode Eigenvalue = 0.562
The buckling mode shapes that correspond to these eigenvalues should be checked to ensure that
the deformation due to buckling is realistic. The first buckling mode shape from the more
important eigenvalue in subcase 2 is shown in figure 13. This buckling mode shape is verified as
acceptable for the nonlinear buckling problem, so the eigenvalue is valid.
18
Figure 14: First Mode Buckling Shape – Subcase 2 – NL Buckling
The critical buckling load based on equation (5) can now be calculated.
Subcase 1: PCR = 5000 + .811 ⋅ 200 = 5162 lbs
Subcase 2: PCR = 6000 + .562 ⋅ 200 = 6112 lbs
The nonlinear buckling analysis was repeated with the finer mesh FEM. The extracted
eigenvalues for the first and second subcases are .856 and .598 resp.
Subcase 1: PCR = 5000 + .856 ⋅ 200 = 5171 lbs
Subcase 2: PCR = 6000 + .598 ⋅ 200 = 6120 lbs
Despite slightly higher eigenvalue calculations, the mesh sensitive FEM produced similar critical
buckling loads using equation [5].
Critical Buckling Load - Nonlinear Eigenvalue Buckling Analysis = 6120 lbs
19
The critical buckling load using the nonlinear eigenvalue buckling analysis underestimates the
average experimental buckling load by about 13.3%.
The critical buckling loads calculated via nonlinear analysis are all very closely matched to
each other. Table 3 below summarizes the comparison.
Analysis
Critical
Error
Method
Buckling Load
Margin
Modified
6100 lbs
13.6%
Arc Length
6177 lbs
12.5%
NL Buckling
6120 lbs
13.3%
Newton
Table 3: Results Summary – Critical Buckling Load Comparison
The buckling load calculated from the Arc Length method is the closest to the experimental
average buckling load, although only by a few tenths of a percentage point. This relation is
consistent with reference [2] which asserts the Arc Length method to be the least conservative,
yet potentially the most accurate result. In reference [1] the critical load calculated via FEA (arc
length method) overstated the experimental buckling load by about 9%. In this report, the critical
load calculated via FEA understated the experimental buckling load by about 13%.
Understanding this difference between analyses is not exactly clear; there are many different
parameters and variables in the nonlinear analysis. The fact that the tie rod in this report
consisted of different materials which had glue-coincident meshes may have modified the
dynamics of the nonlinear response, compared to analysis of a single material in reference [1].
In general, the different non-linear analysis techniques to predict the critical buckling load
show very good agreement amongst each other.
20
4. Conclusions
In conclusion, the techniques carried out in this report to accurately predict the critical
buckling load of a structural tie rod are proven to be reliable based on the precision and accuracy
of the calculated margins. Tie rod buckling exhibits nonlinear behavior and accurately predicting
the critical buckling load can only be evaluated through the techniques described in this report,
which are closely related to those techniques from references [1] and [2].
Based on the results, the Arc Length method produces the most accurate buckling load. In
situations where the compression margin of safety needs to be minimized, it is recommended to
use the critical buckling load calculated from the Arc Length method. The Modified Newton
method will produce the most conservative buckling load because the analysis will show indicate
buckling at the first sign of bifurcation or instability. For added conservatism it is recommended
that the designer use the buckling load from this analysis. The nonlinear eigenvalue buckling
analysis is expected to produce a buckling load in between the buckling loads calculated by the
Arc Length and Modified Newton method. This is a convenient analysis technique that is easy to
set up, but requires some knowledge of the estimated buckling load in advance. This is why it is
recommended that the designer analyze the tie rod using all the methodologies outlined in this
report in the same exact order. This will help verify the accuracy of the calculated critical
buckling load by checking it against all three analysis techniques. In order to use the It has been
made clear that linear buckling analysis alone is not appropriate for predicting the buckling load
of a tie rod.
21
5. References
[1] Campbell, G., Ting, W., Aghssa, P., & Hoff, C. (1994). Buckling and Geometric Nonlinear
Analysis of a Tie Rod in MSC/NASTRAN Version 68. MSC 1994 World Users' Conference (pp.
1-15). Lake Buena Vista, FL: MSC Software Corporation
[2] Lee, S. H. (2001). Essential Considerations for Buckling Analysis. World Aerospace
Conference and Technology Showcase. Toulouse, France: MSC Software Corporation.
[3] Primus International – University Swaging Division (2007). Tie Rod Test Article and Test
Data. Woodinville, WA.
[4] Federal Aviation Administration (2008). MMPDS-04: Metallic Materials Properties
Development and Standardization. Washington, D.C.,: Federal Aviation Administration.
[5] Federal Aviation Administration (2001). MMPDS-01: Metallic Materials Properties
Development and Standardization. Washington, D.C.: Federal Aviation Administration.
[6] Tindal, U.C. (2010). Machine Design. New Delhi, India,: Dorling Kindersley Pvt. Ltd.
[7] Jain, Ritu. (2003). Solution Procedure for Non-Linear Finite Element Equations. Project
Report. University of California, Davis.
[8] Siemens Cast Online Library. (2010). Solutions and Solution Processes. Siemens Product
Lifecycle Management Software Inc.
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6. Appendices
Appendix A
NASTRAN Input & F06 Output Files - Linear Buckling Analysis
linear_buckling_in.txt
linear_buckling_out.f06
NASTRAN Input & F06 Output Files - Modified Newton Method
nl_Newtons_in.txt
nl_Newtons_out.f06
NASTRAN Input & F06 Output Files - Arc Length Method
arc_length_in.txt
arc_length_out.f06
NASTRAN Input & F06 Output Files - Nonlinear Buckling Analysis
nl_buckling_in.txt
nl_buckling_out.f06
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