Buckling Of Isogrid Plates
by
Jeffrey Lavin
An Engineering 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:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
June 2010
i
© Copyright 2010
by
Jeffrey Lavin
All Rights Reserved
ii
CONTENTS
List of Figures ................................................................................................................... iv
List of Tables ..................................................................................................................... v
List of Symbols ................................................................................................................. vi
Abstract ............................................................................................................................ vii
1. Introduction/Background ............................................................................................. 1
1.1
Problem Description ........................................................................................... 1
1.2
Methodology ....................................................................................................... 2
1.3
Expected Outcome .............................................................................................. 3
2. Buckling Theory And Analytical Solutions ................................................................ 4
2.1
Isogrid Simplification ......................................................................................... 5
2.2
Plate Buckling Analytical Solutions ................................................................... 8
3. Numerical Results and Discussion ............................................................................ 13
3.1
Model Creation and Explanation ...................................................................... 13
3.2
Model Parameters and Boundary Conditions ................................................... 14
3.3
Plate Critical Load Comparison........................................................................ 19
3.4
Isogrid Critical Load Comparison .................................................................... 20
3.5
Mode Shape Visualization ................................................................................ 21
3.6
Mode Shape Comparison .................................................................................. 23
3.7
Rib Geometry Variation ................................................................................... 26
4. Conclusions................................................................................................................ 29
References........................................................................................................................ 32
Appendixes ...................................................................................................................... 33
iii
List of Figures
Figure 1: Free body diagram............................................................................................. 2
Figure 2: Off design condition free body diagram ........................................................... 2
Figure 3: Fixed-free beam ................................................................................................ 4
Figure 4: Column buckling example ................................................................................ 5
Figure 5: Typical isogrid pattern ...................................................................................... 6
Figure 6: Transformed isogrid structure ........................................................................... 7
Figure 7: Isogrid geometry ............................................................................................... 9
Figure 8: FBD load case 1 and 2 ...................................................................................... 9
Figure 9: FBD load case 3 .............................................................................................. 11
Figure 10: Loading applied to edge a and b ................................................................... 12
Figure 11: Cross section geometry comparison.............................................................. 14
Figure 12: Isogrid edge conditions ................................................................................. 15
Figure 13: Isogrid point constraints ................................................................................ 15
Figure 14: Initial isogrid finite element model ............................................................... 16
Figure 15: Final isogrid geometry .................................................................................. 17
Figure 16: Final isogrid boundary conditions ................................................................. 18
Figure 17: Final isogrid load case 1 ................................................................................ 18
Figure 18: Element count vs. Isogrid 1st Critical Buckling Load ................................... 18
Figure 19: Plate model showing first buckling mode shape and load ............................ 22
Figure 20: Plate model showing second buckling mode shape and load (m=2, n=1) .... 22
Figure 21: Second mode shape at section A-A (mid span) m=2 .................................... 22
Figure 22: Second mode shape at section B-B (max displacement) n=1 ....................... 22
Figure 23: Isogrid and plate model mode shape 1-5 comparison loaded on edge b ....... 24
Figure 24: Isogrid and plate model mode shape 1-5 comparison loaded on edge a ....... 25
Figure 25: Isogrid and plate model mode shape 1-5 comparison loaded on both edges 26
Figure 26: Example of rib buckling mode ...................................................................... 27
Figure 27: 1st Critical buckling load as a function of  .................................................. 28
iv
List of Tables
Table 1: Transformed section properties using parallel axis theorem .............................. 7
Table 2: Material properties for 1” x 1” plate................................................................. 10
Table 3: 1” x 1” Critical buckling loads .......................................................................... 10
Table 4: Buckling load/mode change with edge loading................................................ 11
Table 5: Critical values for loading on both edges 1.000” x 1.1547” plate .................... 12
Table 6: Material property comparison .......................................................................... 15
Table 7: Model Element Count Comparison .................................................................. 17
Table 8: 4.000” x 4.618” Critical load (lb) on edge b .................................................... 19
Table 9: 4.000” x 4.618” Critical load (lb) on edge a and b ........................................... 19
Table 10: Isogrid model critical loads (lb) comparison with load applied to edge b ..... 20
Table 11: Isogrid model critical loads (lb) comparison load with applied to edge a...... 20
Table 12: Isogrid model critical loads (lb) comparison, load applied to both edges ...... 21
Table 13: Rib study parameter variation ........................................................................ 27
v
List of Symbols
Symbol
Description
Units
E0
Elastic Modulus
psi
E*
Equivalent Elastic Modulus
psi
t
Skin thickness
in
t*
Equivalent plate thickness
in
D
Bending Stiffness
lb/in
K
Tensile Stiffness
lb/in
P, F
Critical Buckling Load
lb
a
Horizontal Edge Length
in
b
Vertical Edge Length
in
l
Column Length
in

Dimensionless Rib Height
-

Dimensionless Cap Parameter
-

Dimensionless Cap Height
-

Dimensionless Rib Parameter
-
h
Isogrid height
in

Poisson Ratio
-
A
Area
in2
I
Cross Sectional Inertia
in4

Centroid Distance
in
vi
Abstract
The focus of this report will be the buckling of an isogrid structures.
The
simplification reduces the isogrid structure to a single sheet with an equivalent bending
and tensile stiffness by creating and equivalent modulus of elasticity (E*) and thickness
(t*). This simplification is often used in structural analysis models to reduce engineering
lead-time. However, the results from these models must still produce accurate results. A
simply supported isogrid and equivalent stiffness plate model will both be subjected to
varying load orientations. The orientation of the load relative to the isogrid pattern will
vary to ensure the simplification is applicable to parts subject to varying load conditions.
The accuracy of the critical buckling loads and predicted mode shape for both the isogrid
and the plate models will be compared to an analytical solution with similar boundary
conditions and loads. Finally, a recommendation for when the simplification process is
no longer valid due to a change in failure mode will also be discussed.
vii
1. Introduction/Background
Isogrid structures are used in a variety of aerospace components. The design of
the isogrid structure allows the part to maintain isotropic properties even though material
has been removed for weight reduction. Due to their wide use in applications they are
subjected to a variety of boundary constraints and loading conditions all of which can
vary during operation. Initial designs are completed with the best available predicted
loads that must be validated during engine test. Even with advances in technology the
predicted temperature and pressures are not exact and these changes vary the
component-to-component load direction and orientation. In order to maintain structural
integrity the isogrid must continue to function properly if load orientation is changed
during operation. Thus the simplification process must also capture the change in
critical buckling load and mode shape if load orientation is changed.
1.1 Problem Description
The problem is formulated from an existing isogrid part experiencing
deformation attributed to a change in boundary conditions and loading. To reduce
design lead-time the isogrid part was simplified to a single sheet. This simplification
was developed for NASA and is known as the E*t* method. The simplification process
of the isogrid is noted as a potential cause of failure and is being reviewed to ensure this
is not the cause.
The part is designed with specific load directions and boundary conditions. The
change in load direction is also a potential cause of failure. A simplified free body
diagram of the component with the correct loading condition is seen in figure 1. Figure
2 shows a possible change in the boundary conditions that could change the input loads
to the part.
These changing load directions are also a potential cause of the part
deformation.
For this reason three different load orientations are used in the report to
ensure the load orientation relative to the isogrid is not a reason for failure.
1
y
x
a
b
Figure 1: Free body diagram
y
x
a
b
Figure 2: Off design condition free body diagram
1.2 Methodology
A closed form solution is solved for each loading condition for comparison to the
numerical analysis. The E*t* method is used to simplify the isogrid to a plate. This
simplification will be used as an input to the analytical solution as well as used in a
numerical analysis of the plate. These two solutions will be compared to ensure proper
boundary condition modeling technique as well as mesh density. The plate model will
also be used for comparison to the modeled isogrid geometry. Each numerical model is
created using the Comsol finite element code.
In Comsol both a static solution and a linear buckling solution is completed. The
static solution is used to verify the input load and that the initial boundary conditions do
not impart any additional constraint or load.
Upon verification of the boundary
conditions and input loads the linear buckling analysis is completed. This analysis is
completed with a unit load so that the calculated eigen value is the critical buckling load.
These loads are then compared to the analytical solution for verification of the E*t*
2
methodology. Analytical buckling mode shapes will also be compared to numerical
mode shapes of the plate and isogrd to ensure the numerical model captures the correct
buckled shape.
1.3 Expected Outcome
It is expected that the change in plate size and load orientation will cause a change
in the critical buckling loads and mode shapes. It is expected that the E*t* method can be
used to accurately predict the buckling mode shapes and critical loads of isogrid plates to
a certain extent. It is also expected that there will be additional modes not captured by
the E*t* method due to the ribs and smaller panels created by the ribs.
3
2. Buckling Theory And Analytical Solutions
The problem will begin with the basics of buckling and steadily progress in
complexity. All of the problems considered below have a closed form solution. In order
to verify the numerical results and modeling approach each model result will be
compared to the closed form solution. This allows verification of each analysis step to
ensure the outcome is accurate.
The first problem focused on creating a fixed-free beam with a single load applied
in the vertical direction. A diagram of this can be seen in figure 3. The critical load for
this problem is shown in equation 1.
P
l
Figure 3: Fixed-free beam
Pcr 
 2 EI
4l 2
[1]
This problem can be turned into the most common or fundamental case, which
consists of a beam pinned at both ends. An example of this can be seen in figure 4. This
problem has the same solution as the fixed-free case assuming the critical length is now
l 2 and a symmetrical boundary condition at the center of the column. The solution to
this fundamental case is seen in equation 2 in which the critical value occurs at n = 1.
This fundamental case is the most often assumed condition in analysis [2] and will be the
basis for the simply supported plate boundary condition.
4
l/2
l/2
Figure 4: Column buckling example
n 2 2 EI
P
l2
[2]
The symmetry condition allows the finite element model to be simplified. A freefree beam will have a rigid body motion due to the lack of constraint in one direction.
Thus this simplified symmetric model is better suited for numerical analysis when
applicable.
The problem under consideration is not a 2D structure but rather a complex 3D part
that contains an isogrid pattern. As previously stated the isogrid is used to reduce the
weight of the structure while still maintaining the isotropic material properties of a single
sheet of material.
2.1 Isogrid Simplification
A typical isogrid structure is used to reduce weight while maintaining structural
efficiency. However it is difficult to model and often requires multiple iterations to
obtain the correct stiffness required in the design. Thus it saves design iteration time if
the structure can be turned into an equivalent single sheet with representative stiffness.
A typical isogrid structure can be seen below in figure 5.
The plate will not have the same geometric dimensions as the isogrid but it will
have the same stiffness in both the tensile and bending directions. This simplification
allows multiple design iterations to be completed by changing only the stiffness of the
part and not the model geometry. The simplification also allows for a reduction in
computational time.
5
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Figure 5: Typical isogrid pattern
Assuming the isogrid plate is in a state of uniaxial stress it can be shown that the
structure is equivalent to a single sheet in plane stress [1]. The process to simplify the
sheet uses several non-dimensional parameters (  , ,  , , h ) for a unit width of isogrid.
The procedure also uses the parallel axis theorem to reduce the isogrid to a single sheet.
The non-dimensional parameters are defined in equation 3. The single sheet will have
an equivalent tensile stiffness (K) and bending stiffness (D), where is the material
Poisson’s ratio and E0 is the material elastic modulus. The different stiffness equations
are shown in equation 4 and equation 5.

3
d
c
bd
wc
,  ,
, 
,h  a
t
t
th
th
2
D
1
E0 I
1 2
[4]
K
1
E0 A
1 2
[5]
[3]
The procedure reduces any isogrid geometry down to a unit width of isogrid, which
is created from the variables in equation 3. The transformed isogrid is shown in figure 6.
6
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Figure 6: Transformed isogrid structure
Ai
i
Aii
Aii2
I0
1
t
t
(1   )
2
t2
(1   )
2
t3
(1   )2
4
t 2
t
12
2
t
0
0
0
t
( t)2
12
3
t
t
  (   )
2
t3
 (   )2
4
t
(t)2
12
Total
t(1     )
t3
(1   )2   (   )2 
4
t3
1   2   2 
12
Part

t2
 (   )
2
t2
(1   )  (   )
2
Table 1: Transformed section properties using parallel axis theorem
To complete the transformation the individual areas (Ai), centroids (i) and moment
of inertia (I0) are calculated with the equations in table 1. This table contains all the
required information to calculate the appropriate stiffness properties for the equivalent
thickness plate. The final stiffness for the plate is calculated from the total section
properties. These are seen in equations 6, 7 and 8.
A   Ai
t 
[6]
 A
i i
A
I   Ai i2   I 0i  A i2
7
[7]
[8]
The equations for both I and A are now in terms of variable t only. Thus with two
equations for stiffness (K, D) and two unknowns (E, t) equation 4 and 5 are solved
simultaneously to determine an equivalent thickness and stiffness.
The solution of the two equations creates a sheet with an equivalent thickness (t*)
and an equivalent elastic modulus (E*) that must be used together to create the required
tensile and bending stiffness. By producing the correct stiffness with t* and E* these
variables can be used together to predict loads but not stress. Thus the combination is
used for the calculation of critical buckling loads. Equations for t* and E* are shown in
equation 9 and equation 11. Equation 10 can be used to calculate the internal parameter
, which can be used to calculate t* from only non-dimensional parameters. The
parameters E* and t* will be used when comparing an equivalent single sheet to the
isogrid structure.
t* 
12I

t
A
1  
[9]
2
2
2
2
2
  (1     )  3(1   )  3 (   )  1       3(1   )   (   )


E *  E0
A
t*
2
[10]
[11]
With equation 9 and 11 it can be shown that equation 4 and equation 5 can be put in
terms of both E* and t* where D and K are now represent the correct stiffness for the
isogrid. These can be seen in equation 12 and equation 13.
D
E *t *3
12(1  v2 )
E *t *
K
1 2
[12]
[13]
2.2 Plate Buckling Analytical Solutions
Similar to the initial 2D applications the initial problem 3D plate problems were
compared to known analytical solutions to ensure completeness. The first case is a
simply supported plate in uni-axial compression. The plate was 1” x 1” with a thickness
8
of .040”. The second case used was a 1.000” x 1.1547” plate with a thickness of .046”.
This thickness is derived from the isogrid geometry shown in figure 7 and the equations
from section 2.1. The isogrid is representative of the geometry found in the failing
component. The face sheet thickness is .030” and the ribs are .050” tall by .040” wide
with a triangle side length (a) of 1.154” and a triangle height (h) of 1.000”. The two
cases will be used to verify the numerical results for a plate with varying height to width
ratio as well as thickness.
.040У
.030У
.050У
1.000У
Figure 7: Isogrid geometry
The plate has all edges simply supported. This free body diagram is similar to a
section of the component in which the suspected change in boundary conditions is being
observed. The part is free in the y direction and interference is predicted in the x
direction. See figure 8 for a free body diagram of the structure.
The buckling load for this problem is calculated using equation 14. The critical
value will occur with n equal to 1, where both m and n are integers. This is similar to the
beam model of figure 4 and equation 2. The value of m corresponds to the number of
half waves parallel to the direction of loading while n determines the number of half
waves perpendicular to the direction of loading.
These equations can be further
simplified for m  n  1 in a square plate and are shown in equation 15.
y
a
x
b
Figure 8: FBD load case 1 and 2
9
2
a 2 2 D  m 2 n 2 
Px  b

m 2  a 2 b 2 
Px  b
4 2 D
a2
where D 
[14]
Et 3
12(1   2 )
[15]
For the 2 cases above the first 5 critical buckling load values are shown in table 3.
The material properties used in the calculations are shown in table 2. The modulus of
elasticity (E*) is calculated from the isogrid simplification of the previous section
however for this comparison there is no requirement to use the value of (E*). The
calculated values will be compared to the numerical results of the next section.
E*
20.83e6 psi
v
.3
Table 2: Material properties for 1” x 1” plate
1.000” x 1.1547” Plate
1.000” x 1.000” Plate
b = 1.000
Fcritical (lb)
m
n
Fcritical (lb)
m
n
4818
1
1
7529
1
1
7529
2
1
9833
2
1
13385
3
1
16406
3
1
19274
2
2
25966
4
1
21758
4
1
30115
2
2
Table 3: 1” x 1” Critical buckling loads
From the data in table 3 it is seen that the mode shapes of the plate change as a
function of the ratio of a/b. This can be seen by the change in critical load and the
values of m and n between the fourth and fifth values of the calculated examples.
The third case used was a change in orientation of the loading on the plate. This
loading is shown in figure 9. For the 1x1 plate there is no change in critical buckling
10
loads but for the rectangular plate of case 2 (1.000 x 1.1547) there is a change to the
critical loads. Again this is caused by the ratio of a/b and the change in load orientation.
The comparison of calculated loads can be seen in table 4.
y
x
a
b
Figure 9: FBD load case 3
Load on edge a
Load on edge b
a = 1.000”
Fcritical (lb) m
b = 1.1547”
n
Fcritical (lb)
m
n
6520
1
1
7529
1
1
12008
2
1
9833
2
1
22487
3
1
16406
3
1
26080
2
2
25966
4
1
34064
3
2
30115
2
2
Table 4: Buckling load/mode change with edge loading
Additionally the critical loads were calculated for loading on both of the edges of
the plate. This load case is most similar to the loading of the failing part in the field.
The additional load on the plate further reduces the load required to buckle the plate.
The applied loads are seen in figure 10.
11
y
x
a
b
Figure 10: Loading applied to edge a and b
This load case has a solution shown in equation 16. Again the changing values of m
and n change the mode shape and the critical load required to produce buckling in the
plate. As expected the combined load case has reduced the critical value below the
previous two load cases. The calculation of the critical values can be seen in table 5.
2
(mb a)2  n 2 
 2D
Px 
mb a 2  (Pyb Px a)n2 b
Calculated (lb)
3494
9872
7631
13976
14541
Kcc
1.9
5.4
4.1
7.6
7.9
m
1
1
2
2
3
[16]
n
1
2
1
2
1
Table 5: Critical values for loading on both edges 1.000” x 1.1547” plate
Each of these load cases will be compared to the simplified isogrid structure to
determine the applicability of the E*t* method.
12
3. Numerical Results and Discussion
Comsol was used to solve the isogrid buckling and equivalent plate buckling
numerical analysis. The results in this section are then compared to the analytical
solutions from section 2 to determine the validity of both numerical models. The
numerical results are also compared to each other to ensure the use of the equivalent
plate method will provide similar numerical answers to the isogrid model.
Each model created is required to compute a static solution prior to calculating a
buckling solution. This is required so that Comsol can calculate the pre-stress in the
model. The pre-stress is required for the calculation of the stiffness matrix needed in the
eigen buckling value solution.
3.1 Model Creation and Explanation
There were four different models created to verify the applicability of the E*t*
method. Each model was created using the gravitational IPS units. The two main
variables elastic modulus (E and E*) and thickness (t and t*) have units of psi and inches.
The first model created was a 1.000” x 1.1547” x 0.046” plate. This plate model
corresponds to a case completed in section 2.2 and was the beginning of the E*t*
verification. The plate model was also used to calibrate the Comsol modeling technique
to the analytical solution. The plate is a constant thickness (t* = 0.046”) and given
modulus of elasticity (E* = 20.83e6 psi).
To match the plate model described above a single isogrid panel was created for
comparison. The geometry was created using a single block (1.000” x 1.1547” x 0.030”)
and adding individual ribs. The ribs were created at the center of the plate and then
rotated 60 about the center in either direction. Each rib was 1.500” x 0.040” x 0.050”.
These ribs were then trimmed with separate blocks to create the proper length. The final
rib was then created at the center of the plate and formed the last piece of the isogrid.
The third model created was a 4.000” x 4.618” x 0.046” plate. The plate was again
modeled from the simplification of the isogrid, which required a constant thickness (t*)
of 0.046”. The modulus of elasticity (E* = 20.83e6 psi) used was also the same as the
original 1.000” plate and calculated using the E*t* process from section 2.
13
The final model created was an isogrid geometry that maintained the same rib
length (s) and height (h) of the initial 1.000” model but contained more isogrid cells.
The model was created using a similar technique to the first model but after rotating the
diagonal ribs each rib was arrayed in the x direction to create multiple entities from the
original. The array process allowed for a reduced number of modeling steps. Once the
rotated ribs were created the final horizontal ribs were added and united to the rotated
ribs. Creating a composite object of just the ribs allowed for a reduction in the number
of subtractions required to create diagonal ribs with the proper length. The model now
had two components. The face sheet of constant thickness and the rib structure were left
as separate entities so that a rib height variation could also be completed.
The rib height study models were each created separately by scaling the ribs of the
original isogrid model in the z direction. This allowed for the face sheet thickness to
remain unchanged for each separate analysis so that only the change in rib geometry was
evaluated.
3.2 Model Parameters and Boundary Conditions
The isogrid dimensions seen in figure 11 were used along with the procedure
described in section 2.1 to produce the appropriate thickness (t*) and modulus of
elasticity (E*). These computations were completed in excel to simplify the calculation
effort. A comparison of the geometries is shown in figure 11 and table 6 shows the
comparison of thickness and elastic modulus.
.046У
1.000У
.040У
.030У
.050У
1.000У
Figure 11: Cross section geometry comparison
14
Thickness Elastic Modulus (lb/in2)
Plate
.046”
20.83e6 (E*)
30.00e6 (E0)
Isogrid As drawn
Table 6: Material property comparison
The boundary conditions for the isogrid were identical to the plate. The isogrid
boundary conditions are shown in figure 12.
Each side of the isogrid is simply
supported to match the analytical solution constraints. This requires support in the
vertical z direction with additional point constraints at specific points in order to prevent
a rigid body motion. The point constraints are at the center of each edge and constrain
movement in the direction parallel to the edge. The point constraints are seen in figure
13.
Figure 12: Isogrid edge conditions
Figure 13: Isogrid point constraints
The geometry for both the equivalent sheet and the isogrid were meshed with
tetrahedral 3D solid elements. The mesh geometry for the isogrid is shown in figure 14.
The mesh contained 25423 elements and 126690 degrees of freedom. The analytical
15
solution of a plate with a single edge load produces a critical buckling load of 7529 lb.
The critical buckling load for the isogrid finite element model was computed at 6735 lb
and the finite element model of the plate calculated a critical buckling load of 7180 lb.
This is within 10.5% and 4.5% respectively of the analytical solution for a flat plate of
equivalent stiffness.
Figure 14: Initial isogrid finite element model
These predictions were not acceptable for verification of the equivalent stiffness
method so a larger model was completed to remove the influence that the boundary
conditions may have on the results.
The final model was a 4.000” x 4.618” rectangular isogrid. The lengths a and h
(figure 5) for the isogrid were kept the same so that the E*t* simplification did not
change between the small and large models. The isogrid simplification reduces the
system to a unit width, which remains applicable to the larger system, provided the
geometry is produced properly. All final results presented will be from the larger model.
The isogrid model mesh, boundary conditions and loads can be seen in figures 1517. The final mesh consisted of 16358 elements and 96660 degrees of freedom. The
free mesh parameters were set to “coarser” to create the mesh. A study was completed
using the “extremely coarse” and “extra coarse” free mesh parameter option. Element
count versus percent error to the first critical buckling load is shown in table 7. As the
element count increased the accuracy of the solution increased. A graph of the data from
table 7 can be seen in figure 18. The slope of the graph shows little change in accuracy
for the increase in element count once ~10000 elements are used. The final element
count provided accurate results while allowing the model to solve in approximately 5
minutes.
16
A similar study was also completed for the plate model. The model was run with
the “coarser”, “coarse”, “normal” and “fine” free mesh parameters. This study showed
that the model converged to a solution and the “normal” free mesh parameter was used.
This produced a model with 12203 elements. Unlike the isogrid model, the plate model
under-predicts the first critical buckling load.
Model
Element
Count
Isogrid
6716
10076
16358
4408
7668
12203
17432
Plate
Analytical
Solution
(lb)
1882
1882
1882
1882
1882
1882
1882
First Critical
Buckling Mode
(lb)
2049
1972
1946
1870
1862
1858
1857
Percent
Error
8.86
4.77
3.39
-.65
-1.07
-1.28
-1.33
Table 7: Model Element Count Comparison
The boundary conditions were again applied to approximate a simply supported
plate and boundary loads were applied to the main face of the plate. The loads were
applied to the boundary face and applied so that the total input load was 1 lb per each
side. Additional models were created with loads applied to additional faces to simulate
each of the load cases in section 2.2. This included loading on the long edge and both
edges.
Figure 15: Final isogrid geometry
17
Figure 16: Final isogrid boundary conditions
Figure 17: Final isogrid load case 1
st
Critical Buckling
2100
2000
Numerical Solution
Anaytical Solution
1900
1
st
Critical Buckling
Value (lb)
Element Count vs. 1
1800
6000
8500
11000
13500
16000
18500
Element Count
Figure 18: Element count vs. Isogrid 1st Critical Buckling Load
18
3.3 Plate Critical Load Comparison
A comparison of the buckling mode shapes and the corresponding loads are shown
in table 8 for the 4.000” x 4.618” plate with loading on edge b. As the mode shapes
increase in complexity the accuracy of the model does not reduce. This shows the model
is capable of predicting accurate displacement with the mesh. For the loads applied to
edge b the first 4 modes should correspond to m  1  4 with the value n remaining
constant at n = 1. The fifth mode being the first mode were the value of n = 2.
Calculated
1882
2458
7529
4102
6492
Percent Error
-1.28
-1.11
-1.27
-0.94
-0.87
Comsol Plate
1858
2431
7433
4063
6435
m
1
2
2
3
4
n
1
1
2
1
1
Table 8: 4.000” x 4.618” Critical load (lb) on edge b
The change in loading direction will change the critical buckling value as well as the
mode shape ordering. This was seen in the calculations completed in section 2.2. No
additional modeling was completed for loads applied to edge a due to the accuracy of the
loads on edge b.
The model was also evaluated for loading applied to both sides (a and b) of the plate
and the solution was compared to the analytical value using equation 16. The results
compared well to the calculated value. The results can be seen in table 9.
Calculated
874
2468
5128
1908
3494
3635
5215
Percent Error
-0.75
0.13
1.38
0.02
0.71
1.18
-2.07
Comsol Plate
867
2471
5199
1908
3519
3678
5107
Table 9: 4.000” x 4.618” Critical load (lb) on edge a and b
19
3.4 Isogrid Critical Load Comparison
To ensure the E*t* isogrid simplification method produces accurate critical loads all
three load cases were run with the 4.000” x 4.618” size isogrid model. This will also
provide substantiation that load orientation into the isogrid can be neglected.
The 4.000” x 4.618” isogrid results are compared to the E*t* Comsol plate and the
analytical solution. The model for the isogrid accurately predicted the critical buckling
loads as compared to both the analytical value and the Comsol plate model. The results
for the case with loads applied to edge b (short edge) can be seen in table 10.
Calculated
1882
2458
7529
4102
6492
Percent Error
-1.28
-1.11
-1.27
-0.94
-0.87
Comsol Plate
1858
2431
7433
4063
6435
Isogrid
1946
2543
7620
4223
6568
Percent Error
3.39
3.44
1.21
2.96
1.18
m
1
2
2
3
4
n
1
1
2
1
1
Table 10: Isogrid model critical loads (lb) comparison with load applied to edge b
To ensure the model will capture the change in geometry (i.e. non-square) and load
orientation, the model was run with a load applied to edge a (long edge). The error
results are similar to the model with load applied to edge b. This showed the isogrid
model and the E*t* method accurately predicted loads with a changing length ratio. The
results for loading on edge a can be seen in table 11.
Calculated
1630
8516
3002
6520
5622
Percent Error
-1.47
-1.10
-1.21
-1.47
-1.08
Comsol Plate
1606
8422
2966
6424
5561
Isogrid
1684
8839
3114
6592
5760
Percent Error
3.3
3.8
3.7
1.1
2.5
m
1
1
2
2
3
n
1
2
1
2
1
Table 11: Isogrid model critical loads (lb) comparison load with applied to edge a
The final case used to verify the E*t* method accurately calculates critical load is the
combined loading on both edge a and edge b. This load case also produces critical
20
buckling loads similar to both the plate numerical and analytical analysis. The result for
the model with loads applied to both edges is seen in table 12.
Calculated
874
2468
1908
3494
3635
Percent Error
-0.75
0.13
0.02
0.71
1.18
Comsol Plate
867
2471
1908
3519
3678
Isogrid
902.00
2556
1971.00
3532.00
3737.00
Percent Error
3.26
3.57
3.32
1.08
2.80
m
1
1
2
2
3
n
1
2
1
2
1
Table 12: Isogrid model critical loads (lb) comparison, load applied to both edges
3.5 Mode Shape Visualization
Equations 14 and 16 can be used to calculate multiple buckling loads for the part
depending on boundary conditions, the lowest load being the most important.
As
described in the sections above, the mode shape changes based on the value of m and n
in equation 14 and equation 16. Mode shapes are reviewed to verify that Comsol is
computing not only critical load but also the correct mode shapes. This is done by
comparing the calculated buckling load and mode shape based on the values of m and n
to the shapes and loads produced by Comsol. An example of the first mode shape for
the plate can be seen in figure 19. The second mode shape for the plate can be seen in
figure 20. The second mode shape is used as an example to show how the shape varies
with the value of m and n. The corresponding displacements in figure 21 and figure 22
help visualize the values of m and n.
21
Figure 19: Plate model showing first buckling mode shape and load
B
A
A
B
Figure 20: Plate model showing second buckling mode shape and load (m=2, n=1)
Figure 21: Second mode shape at section A-A (mid span) m=2
Figure 22: Second mode shape at section B-B (max displacement) n=1
22
3.6 Mode Shape Comparison
Despite accurate critical loads being calculated the mode shape of the linear
buckling solution must be verified in order to ensure there are no modeling issues
present. These issues can arise due to imperfections in the model geometry, boundary
conditions or from anti-symmetric loads. They can also arise from abrupt stiffness
changes that may be seen in the transition between the skin thickness and the ribs.
Thus the final step to verifying the E*t* method is to compare the predicted mode
shapes for the isogrid geometry with the mode shapes produced by the plate model and
the analytical solution. Each mode produced from the isogrid model is compared to the
plate model to verify the shapes are correct. The plate models can be compared to the
analytical solution with specific values of m and n to determine if they are accurate. A
comparison of the first 5 mode shapes for loading in along edge b is seen in figure 23. A
mode shape comparison for loading along edge a is seen in figure 24. The combined
loading condition mode shape comparison is seen in figure 25.
23
Figure 23: Isogrid and plate model mode shape 1-5 comparison loaded on edge b
24
Figure 24: Isogrid and plate model mode shape 1-5 comparison loaded on edge a
25
Figure 25: Isogrid and plate model mode shape 1-5 comparison loaded on both
edges
3.7 Rib Geometry Variation
Changing rib geometry relative to the plate changes the critical buckling loads
without changing the buckling mode shapes until the ribs begin to dominate the stiffness
of the structure. An example of the change in buckling mode shape for varying rib
geometry can be seen in figure 26. This change in failure mode signifies when the E*t*
method is no longer applicable to use to determine the critical buckling load or mode
shape of the part.
To determine the change in mode shape with respect to rib geometry a single load
case was completed with loads applied to both edge a and edge b. Four different rib
geometries are modeled with an increasing height. The width of the rib, height of the
isogrid triangle and the skin thickness is all held constant. The heights used for the study
26
can be seen in table 13. The table also contains the rib geometry and the calculated
value for 
As stated previously, the ribs were created in the Comsol model as one composite
object and the skin was created as a separate object, which allowed for rib scaling in the
z direction. For each new model a new value of E* and t* are calculated for use in the
analytical solution.
Each model was then compared to the analytical solution for
verification of the critical buckling load.
Additionally the mode shapes for each
geometry change are evaluated to ensure the failure mode did not change from plate
buckling to rib buckling. The variable chosen to determine when the simplification
could no longer be used due to the change in failure mode is . This variable takes into
account the rib cross-section as well as the length of the rib and skin thickness. A plot of
 vs. critical buckling load is seen in figure 27.
Rib Width
.040"
.040"
.040"
.040"
Rib Height
.050"
.100"
.250"
.500"

0.07
0.13
0.33
0.67
Table 13: Rib study parameter variation
Figure 26: Example of rib buckling mode
27
1st Critical Buckling
Load (lb)
 vs. 1 st Critical Buckling
250000
200000
150000
Plate mode
Isogrid Mode
100000
50000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8

Figure 27: 1st Critical buckling load as a function of 
28
4. Conclusions
Comsol accurately predicts both the critical buckling loads and mode shapes for
simply supported flat plates.
Several models of varying size were compared to
analytical solutions and the final results were within 2%. There is good correlation
between the two solution techniques. The 4” x 4.618” plate numerical model underpredicts the buckling load relative to the analytical solution by approximately 1.3%,
while the isogrid model over-predicts the critical buckling load relative to the analytical
solution by approximately 3.5%. The plate model is converged and it is assumed that
with additional computational resources the isogrid model error could be reduced. This
is shown in the comparison between element count and percent error for the isogrid
model. This comparison and correlation provides the basis for using Comsol to predict
the validity of the E*t* method.
The model was completed as a full symmetry model for all cases to ensure the
proper boundary conditions were established. Initially the model was completed with a
symmetry boundary condition on two sides to simplify the modeling constrain. Upon
comparison to the analytical solutions and the full symmetry model the symmetric model
did not produce answers that matched either the analytical solution or the full symmetry
numerical solution. It was assumed that the symmetry modeling constraints were either
incorrectly applied or calculated incorrect loads and mode shapes.
No further
investigation was completed into why the symmetric model did not compare favorably
to the analytical or full symmetry solutions.
An improvement to the Comsol modeling package would include the ability to use
2D plate or shell elements and calculate buckling load. Using tetrahedral elements
required a larger number of elements and degrees of freedom when compared to a
similar model completed in Ansys with shell elements. This added to the computational
time required for this analysis. Without the ability to complete a buckling run with plate
or shell elements most sheet metal structures can be better analyzed with another finite
element code.
The isogrid design manual report [1] contained an error that was found during the
creation of the E*t* excel sheet in the appendix.
This error was confirmed with
unpublished work completed at Pratt and Whitney. The error was in the calculation of
29
the distance to the centroid of the transformed isogrid cap. The problems involved in
this report did not contain a capped isogrid, however the correction was included in the
report for completeness. Without a capped isogrid model the correction has not been
verified numerically.
The load orientation into the isogrid model does not change the result from the E*t*
method. The change in load orientation completed in section 3.5 and 3.6 shows that the
simplification process does not effect the calculation of critical load or mode shape
between the isogrid model and plate model. This allows the simplification process to be
used despite load orientation into the isogrid.
For the specific isogrid geometry modeled in figure 11 the E*t* method can be used
to accurately predict both critical buckling loads and mode shapes. The prediction for
both the critical loads are within 4% of the analytical values and within 6% of the
numerically calculated critical loads. Comsol correctly produced the first five mode
shapes when compared to the analytical solution. Only when buckling of the ribs
occurred were incorrect mode shapes produced. The incorrect mode shapes were used to
determine the applicability of the E*t* method to the geometry.
With the geometry presented in this report the structure is dominated by skin
buckling and not rib buckling. Rib buckling can occur with larger rib geometry as
shown in the rib buckling section. With larger and stiffer ribs the isogrid simplification
process does not predict the correct buckling load or mode shapes and the E* t* method
should not be used.
Based on the information in this report the E*t* method should not be used for
values of  > 0.23. For values of  < 0.23 rib buckling must be verified prior to design
finalization however the E*t* method should provide sufficient data. If the value of  >
0.23 the model will no longer predict plate buckling modes but will instead predict rib
buckling modes. These modes are also of concern but are not applicable to the E*t*
method of analysis.
Additionally care should be taken when using eigenvalue buckling during
component design. The predicted critical load is incorrectly predicts values that are
higher than the actual buckling value. For this reason eigen buckling should not be used
for component design without adding additional safety margin during the design phase.
30
31
References
[1] McDonnell Douglas Astronautics Company. 1973. Isogrid Design Handbook. CA.
McDonnell Douglas Astronautics Company.
[2] Timoshenko and Greer. 1961. Theory of Elastic Stability. NY. McGraw Hill Inc.
[3] Brush and Almroth. 1975. Buckling of Bars, Plates, and Shells. NY McGraw-Hill
Inc.
32
Appendixes
33