A Comparison of the Vibration Response of a Stiffened Plate Using
Beam, Shell and Solid Finite Element Modeling Techniques
by
Kirsten Benamati
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
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2014
(For Graduation May, 2015)
i
CONTENTS
A Comparison of the Vibration Response of a Stiffened Plate Using Beam, Shell and
Solid Finite Element Modeling Techniques ................................................................. i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
TABLE OF SYMBOLS ................................................................................................... vi
LIST OF KEYWORDS ................................................................................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction/Background ............................................................................................. 1
2. Theory/Methodology ................................................................................................... 3
2.1
Element Theory .................................................................................................. 3
2.2
Model Creation................................................................................................... 4
2.3
2.2.1
Model Assumptions ............................................................................... 4
2.2.2
Baseline Model....................................................................................... 8
2.2.3
Case One Model ..................................................................................... 8
2.2.4
Case Two Model .................................................................................. 10
2.2.5
Case Three Model ................................................................................ 11
Analysis Methodology ..................................................................................... 13
2.3.1
Model Excitation and Response ........................................................... 13
2.3.2
Wavenumber Comparison.................................................................... 13
3. Results/Discussion ..................................................................................................... 15
3.1
Model Validation ............................................................................................. 15
3.2
Mode Identification .......................................................................................... 16
3.3
Wavenumber Comparison................................................................................ 18
4. Conclusions................................................................................................................ 24
5. References.................................................................................................................. 25
ii
6. Appendix A – Mesh Refinement Calculation ............................................................ 27
7. Appendix B – Model Work Files............................................................................... 29
7.1
HyperMesh Input Files ..................................................................................... 29
7.2
Abaqus Input Files ........................................................................................... 40
8. Appendix C – Matlab Scripts .................................................................................... 75
8.1
Result Processing Script ................................................................................... 75
8.2
Plotting Script .................................................................................................. 81
iii
LIST OF TABLES
Table 1 Geometry Assumptions ....................................................................................... 5
Table 2 Material Properties for Steel ................................................................................ 5
Table 3 Frequency Comparison of First Four Modes ...................................................... 16
Table 4 Frequencies for Each Peak (Hz) ......................................................................... 17
iv
LIST OF FIGURES
Figure 1 Boundary Conditions on Baseline Model ........................................................... 5
Figure 2 Boundary Conditions on Solid Element Model ................................................. 6
Figure 3 Baseline Model .................................................................................................... 8
Figure 4 Shell Element Plate with Beam Stiffeners .......................................................... 9
Figure 5 Shell Element Plate with Beam Stiffeners (Beam Element Profile Displayed) .. 9
Figure 6 Beam Element Offset ........................................................................................ 10
Figure 7 Shell Element Plate with Shell Element Stiffeners ........................................... 11
Figure 8 Brick Element Plate with Brick Element Stiffeners .......................................... 12
Figure 9 Mesh Refinement of Solid Model ..................................................................... 12
Figure 10 Drive Location and Response Nodes ............................................................. 13
Figure 11 Baseline First Mode Shape Profile .................................................................. 15
Figure 12 Baseline Modes: .............................................................................................. 16
Figure 13 Resonant Peaks in Response at Drive Location .............................................. 17
Figure 14 Peak 1 Wavenumber Comparison .................................................................. 18
Figure 15 Peak 2 Wavenumber Comparison .................................................................. 19
Figure 16 Peak 3 Wavenumber Comparison .................................................................. 20
Figure 17 Peak 4 Wavenumber Comparison .................................................................. 20
Figure 18 Peak 5 Wavenumber Comparison .................................................................. 21
Figure 19 Peak 6 Wavenumber Comparison .................................................................. 22
Figure 20 Peak 7 Wavenumber Comparison .................................................................. 23
v
TABLE OF SYMBOLS
a, b
length, width of plate (in.)
t
thickness (in.)
E
elastic modulus (psi)
ν
Poisson’s ratio
ρ
mass density (lbf*s2/in.4)
D
bending stiffness (lbf*in.)
f
frequency (Hz)
kf
flexural wavenumber (rad/in.)
λ
wavelength (in.)
cp
compressional wave velocity (in./s)
kp
compressional wavenumber (rad/in.)
ks
spatial wavenumber (1/in.)
F(ks)
wavenumber response
x
spatial location, (in.)
f(x)
spatial response (displacement) (in.)
ω
angular frequency (rad)
m, n
wave order
vi
LIST OF KEYWORDS
Element Type
Finite Element
Spatial Fourier Transform
Vibration
Wavenumber
vii
ACKNOWLEDGMENT
I would like to thank my project advisor, Professor Ernesto Gutierriez-Miravete, for his
support and understanding during the process of completing this project. His advice and
patience were invaluable to this project.
Also, I would like to thank Rui Botelho, Principal Engineer at General Dynamic Electric
Boat, for the technical expertise and knowledge he contributed during the formulation
and progress of this project. Without his input, this project and my own understanding
of the work would not have been complete.
I would also like to thank my husband, Christopher Benamati, who read and reread my
drafts for me, took over the dish duty while I wrote, and helped me talk through the
theory.
Finally, I would like to mention William Benamati who made life more complicated but
also more amazing. The appearance of his cute little face in the last weeks of this
project was a great blessing.
viii
ABSTRACT
This project investigates the steady state vibration response of a stiffened plate
modeled with several different element types in the Abaqus finite element analysis
software. Several modeling cases are analyzed from 1 to 1000 Hz and the results
compared for potential differences. A spatial Fourier transform on the surface
displacement along several points on the stiffened plate was performed to transform the
response into wavenumber space.
Wavenumber analysis is used to compare the
vibrational wavelengths present in the response, and thus highlight any differences in the
shape of the response due to the chosen element type. Differences in the resonant
response frequencies were also noted between the different modeling methods.
The study observed significant differences in wavenumber content between the
modeled cases for excitation frequencies above 800 Hz. Below 300 Hz, the modeling
methods provided similar results, with some minor shifts in resonant frequencies.
Between 300 and 800 Hz, correlation between modeling methods was difficult to
determine.
ix
1. Introduction/Background
Finite element analysis (FEA) is frequently used to evaluate the vibration response of
stiffened plates in a variety of applications, such as marine structures, bridges, or
buildings. A finite element model can be analyzed throughout the frequency range of
interest to determine the vibration response due to the various dynamic load conditions.
As the frequency range of interest becomes higher, the problem becomes more complex
due to the increasing number of vibrational modes. To ensure an accurate solution at
higher frequencies, mesh resolutions must increase to maintain sufficient elements per
wavelength to resolve the modes of vibration. In addition, more sophisticated element
types may become necessary to capture different types of vibratory motion. Care must
be taken to evaluate that the element type selected is appropriate to ensure an accurate
solution. More complex element types, such as solid (3D) elements and second order
elements, come with increased computational cost that does not always justify their use.
Wang, Qatu, and Yarahmadian computed the natural frequencies of simply supported
cylinders in [1], and determined that although 3D elements can provide an accurate
solution, shell elements are more practical for industrial applications of thin and thickwalled structures.
Studies comparing element types have been performed for a variety of applications.
Brown compared 3D element types at varying mesh densities for applications to turbine
engine blade frequency analysis in [2] and recommended the C3D20R (a second-order
reduced integration element) from Abaqus [3] as the most economical and accurate.
Benzley, Perry, Merkley, and Clark, [4], used eigen values and dynamic modal analysis
to compare hexagonal and tetrahedral element types for applications to elastic and
elasto-plastic bending analysis.
The objective of this project was to evaluate the
wavelengths present in the vibration response of a stiffened plate from 1 to 1000 Hz
using several finite element types of varying complexity for applications requiring
forced frequency vibrational analysis. First order element types, ranging from simple
beam and shell elements to incompatible mode solid elements, will be compared,
including the C3D8I Abaqus element type that compared well in Brown’s study [2]. The
results will be used to determine if commonly available element types predict consistent
response throughout the frequency range, or if the response diverges. This knowledge
1
can be used to approximate the frequency ranges where the solution becomes affected by
the chosen element formulation.
This study will use a Fourier analysis to characterize the response shape of a
structure driven by a steady state harmonic force by converting the displacements from
the spatial domain to the wavenumber domain.
The wavenumbers present in the
response are related to the wavelengths of vibration present in the structure. Fourier
analysis has been used before by Thompson and Pinsky, [5], to evaluate the accuracy of
p-order finite elements.
2
2. Theory/Methodology
This study investigated the vibration response of a stiffened plate using beam, shell, and
solid finite element modeling techniques in Abaqus, [3]. Abaqus was chosen due to its
widespread commercial use and robust finite element solver. Finite element results from
each model were compared using wavenumber analysis. A spatial Fourier transform of
the displacement along several points on the plate was used to evaluate the wavenumber
results in Matlab, [4]. The following sections provide the details and assumptions that
went into the model creation and analysis.
2.1 Element Theory
In general, a stiffened plate can be modeled with 2D shell elements for the plating
and 1D beam elements for the stiffeners, or entirely with shell elements using plate
theory [7]. At low frequencies when the wavelengths are long, the bending behavior is
dominant, and plate theory is appropriate to capture the effects adequately of the plate.
Modeling the stiffeners as simple beam elements accurately captures the bending
behavior since beam elements are based on exact bending solutions [7].
As the
frequency increases, plate modes of vibration can be excited within stiffeners, with the
first fundamental mode occurring at the frequency when a half wavelength forms along
the depth of the stiffener. This effect cannot be captured using beam elements based on
beam bending equations only, so stiffeners must be modeled as shell elements. At even
higher frequencies, full wavelengths can form within small structural features, such as
compression or shear waves through the thickness of a plate, or a bending wave along
the depth of a beam flange. 3D Solid elements based on three-dimensional elasticity
theory are needed to model the vibrations in this case. Although solid elements can
capture the through thickness effects at higher frequencies, solid element models will
have many more degrees of freedom and be more computationally intensive to solve
than plate and beam models.
This study observes the divergence of the predicted
response due to different element types using wavenumber analysis.
element types used for each model case are described in Section 2.2.
3
The specific
2.2 Model Creation
Abaqus was used to create a total of five models with the initial mesh being created in
HyperMesh, [8]. The first model was a simple unstiffened plate as a baseline case to
compare results to expected analytical responses. This served to validate the boundary
conditions used and the plate modes.
The other models included stiffeners on the plate. Each case was modeled using
different element types. The cases are listed below:
1. The plate was modeled with shell elements and the stiffeners were modeled with
beam elements.
2. The plate was modeled with shell elements and the stiffeners were modeled with
shell elements.
3. The plate was modeled with solid elements and the stiffeners were modeled with
solid elements. This case was broken into two subcases as follows:
a) The solid elements were first-order reduced integration solid elements.
b) The solid elements were first-order full integration solid elements with
incompatible modes.
Details about the plate geometry, materials, and common assumptions are provided
in Section 2.2.1. Descriptions of each model can be found in Sections 2.2.2 to 2.2.5.
2.2.1
Model Assumptions
The following sections outline the modeling assumptions and methods used to create the
finite element models.
2.2.1.1 Geometry and Material Assumptions
A longitudinally-stiffened steel plate was assumed as the geometry for the analysis. The
plate and stiffener dimensions are listed in Table 1. Five rectangular stiffeners running
across the length of the plate were evenly spaced 20in. apart from each other. The two
end stiffeners were 10in. from the edge of the plate. These dimensions were chosen for
the geometry as being large enough to see plate modes at low frequencies.
4
Table 1 Geometry Assumptions
Plate Dimensions
Dimension
Stiffener Dimensions
Units
Value
Length
in.
100
Width
in.
100
Thickness, t
in.
1
Dimension
Units
Value
Height
in.
5
Thickness
in.
0.5
The plate and the stiffeners were assumed to be HY-80 steel, commonly used for large
structures. The properties are listed in Table 2 [9].
Table 2 Material Properties for Steel
Property
Units
Value
Elastic Modulus, E
psi
3.00e7
Poisson’s Ratio, ν
Mass Density, ρ
0.3
lbf*s2/in4
0.000736
2.2.1.2 Boundary Conditions
The stiffened plate was assumed to be simply supported on all sides. To represent this
condition, the nodal displacements in the y-direction were fixed (forced to equal zero)
for each edge as well as the rotations for the perpendicular direction to the edge [10].
Figure 1 uses the baseline model to illustrate how the boundary conditions were applied.
Figure 1 Boundary Conditions on Baseline Model
5
The baseline model and the first two stiffened plate models used shell elements to
represent the plate. Applying the boundary conditions to the edges is equivalent to
applying them to the edges at the mid-plane location of the plate, since the 2D shell
elements are located at the mid-plane of the plate’s thickness. To fix the edges of the
solid element models, the boundary conditions were applied to the nodes located at the
mid-plane of the three-dimensional plate thickness, as seen in Figure 2. This ensures
that the boundary conditions are consistent between the models.
Figure 2 Boundary Conditions on Solid Element Model
2.2.1.3 Mesh Refinement
The finite element models needed to have a sufficient level of mesh refinement to
capture the dynamic response of the plate throughout the frequency range. If there are
not enough elements per wavelength, the waves of vibration will appear blocky, since
the model cannot accurately represent the shape of the wave. This limitation is observed
at higher frequencies, because the small wavelengths present at high frequencies cannot
be resolved.
Setting a maximum element size that ensures a minimum of twelve
elements per wavelength throughout the frequency range is recommended by the Abaqus
user guide, [11]
.To ensure sufficient mesh refinement, the element size was based on the bending
wavelength in the plate at 1000 Hz. This was calculated using the equations for an
6
infinite unstiffened plate [12]. This was a conservative calculation since the stiffening
added to the plate will increase the wavelengths, thus permitting larger elements than the
model actually uses.
Element length for bending in the plate was calculated for both shell and solid
elements using Equations 1, 2, and 3, [10].
𝐸𝑡 3
𝐷 = 12(1−𝜈2 )
𝑘𝑓 = (
𝜆=
𝜌𝑡(2𝜋𝑓)2
𝐷
[1]
1⁄4
)
2𝜋
[2]
[3]
𝑘𝑓
Equation 1 calculates the bending stiffness, D, based on the material properties of
the plate and the plate thickness.
Equation 2 uses Equation 1 and the maximum
frequency in the analysis range, f, to calculate the flexural wavenumber, kf. Equation 3
finds the flexural wavelength, λ. The full calculation is provided in Appendix A. The
element length was calculated to be 1.633in at a frequency of 1000 Hz. This was
rounded to 1.5in when creating the finite element models.
The compressional wavelength was also calculated for the solid elements. This
calculation used Equations 4, 5, and 6, [12].
1⁄2
𝐸
𝑐𝑝 = ((1−𝜈2 )𝜌)
𝑘𝑝 =
𝜆=
2𝜋𝑓
[4]
[5]
𝑐𝑝
2𝜋
[6]
𝑘𝑝
Equation 4 gives the compressional wave velocity, cp, based on the material
properties of the plate. Equation 5 uses the compressional wave velocity to calculate the
compressional wavenumber.
Finally, Equation 6 calculates the compressional
wavelength. The full calculation is provided in Appendix A. The element length was
calculated to be 17.642in at a frequency of 1000 Hz. Since the element length required
for bending was much smaller than the required element length for compression, all
models were meshed to the bending wavelength.
For the model meshed with solid elements, four elements through the thickness of
the plate and the stiffeners were modeled. This was to reduce hourglass effects caused
7
by solid elements [13]. It also helped to ensure that cross-sectional deformation effects
were captured.
2.2.2
Baseline Model
The baseline model was a simple plate, unstiffened, that was meshed to the same
refinement as the rest of the models and simply supported on all edges. The standard
shell element type in Abaqus, S4R, was used. Figure 3 shows the meshed baseline
model with the boundary conditions as displayed along the edges in Abaqus.
Figure 3 Baseline Model
An eigen analysis was performed on the baseline model and compared to an
analytical solution for an unstiffened simply-supported plate for validation. The results
are discussed in Section 3.1.
2.2.3
Case One Model
The case 1 model was a stiffened plate modeled with the plate modeled using S4R shell
elements and the stiffeners modeled using standard Abaqus B31 beam elements. Figure
4 shows the meshed model with the boundary conditions. The red lines running along
the plate are the beam elements representing the stiffeners.
8
Figure 4 Shell Element Plate with Beam Stiffeners
Figure 5 shows the model with the beam elements with their cross-sections
visualized so the beam placement can be seen.
Figure 5 Shell Element Plate with Beam Stiffeners (Beam Element Profile Displayed)
Beam elements are connected in the middle of the beam profile. To ensure the
beam profile was correct, the beam elements were offset by half the thickness of the
plate plus half the thickness of the beam. Figure 6 demonstrates this offset.
9
Figure 6 Beam Element Offset
In Figure 6 (a), the offset from the midpoint of the beam profile to the plate is
shown. Figure 6 (b) shows how this offset causes the beam profile to match up with the
plate edge when the plate thickness is visualized.
2.2.4
Case Two Model
The case 2 model was the stiffened plate with both the plate and the stiffeners modeled
using the standard Abaqus S4R shell elements. Figure 7 shows this model with the
boundary conditions displayed along the edges in Abaqus.
10
Figure 7 Shell Element Plate with Shell Element Stiffeners
2.2.5
Case Three Model
The case 3 model was the stiffened plate with the plate and the stiffeners modeled using
solid elements. Figure 8 shows this model with the boundary conditions displayed along
the edges in Abaqus. The application of the boundary conditions to the solid model is
described in Section 2.2.1.2.
Sub-case a) was modeled with C3D8R elements. These are first-order reduced
integration elements.
The reduced integration mitigates shear locking in the solid
elements [11]. Shear locking is the tendency of fully-integrated solid elements to be too
stiff in bending.
However, the reduced integration solid elements suffer from
hourglassing, which causes them to be too soft [13]. Abaqus inserts a small “hourglass
stiffness” into the C3D8R element formulation to counteract this.
Abaqus also
recommends that solid models consisting of C3D8R elements be modeled with four
elements through the thickness. Sub-case b) was modeled with C3D8I elements which
are first-order full integration elements with incompatible modes. The incompatible
modes formulation mitigates the shear locking that is common to fully integrated
elements while maintaining high solution accuracy.
Abaqus states that the C3D8I
elements produce results that very closely match analytical solutions for high quality
meshes (meshes with minimal distortion), [11]. Both solid element cases share the same
11
mesh, which is of high mesh quality and contains four elements through the thickness.
This maintains compatibility with both the C3D8R and C3D8I element types effects and
insure an accurate solution.
Figure 8 Brick Element Plate with Brick Element Stiffeners
The mesh refinement across the plate is similar to the previous models except
around the connections between the stiffeners and the plate.
Figure 9 shows the
refinement in one of these regions. Figure 9 also shows the four layers of elements
modeled through the thickness of the plate and the stiffeners.
Figure 9 Mesh Refinement of Solid Model
12
2.3 Analysis Methodology
2.3.1
Model Excitation and Response
A forced frequency response analysis was performed on each model using Abaqus. A
sinusoidal excitation of a one pound force was applied to the center of the plate through
a frequency range of 1 to 1000 Hz at 1 Hz increments. Figure 10 shows the baseline
model being driven at a node in the center. The vibrational displacement was saved
along the line of nodes also shown in Figure 10. Using the same method as was used for
the boundary conditions, the drive and response nodes were selected along the mid-plane
for the solid element models.
Figure 10 Drive Location and Response Nodes
2.3.2
Wavenumber Comparison
A wavenumber analysis breaks down the response of the plate into the wavelength
components and allowing characterization of the contributions of the waves. A spatial
Fourier transform (or wavenumber transform) is used to convert the response from the
spatial domain to the wavenumber domain using Equation 7 as described in Fahy, [14].
∞
𝐹(𝑘𝑠 ) = ∫−∞ 𝑓(𝑥) exp(−𝑗𝑘𝑥) 𝑑𝑥
[7]
Here f(x) is the spatial response and F(ks) is the wavenumber response. The spatial
Fourier transform can be found using a Fast Fourier Transform (FFT).
13
There were 67 response nodes across the plate.
For each model, the vertical
responses across these nodes were interpolated to 100 response points across the plate
for spacing of 1 inch. Since the plate length was 100 inches, the sample rate was 100
response points/100 inches or 1 1/in. The sample rate divided by 2 is the maximum
wavenumber supported. The wavenumber increment was determined to be the sample
rate divided by the number of response points.
The interpolated responses were
transformed to the wavenumber domain using the built-in Matlab FFT function. The full
Matlab code used for processing and plotting the results is provided in Appendix C.
This process was derived from the MathWorks Fast Fourier Transform documentation
and examples, [15].
14
3. Results/Discussion
The models were run using Abaqus and analyzed using Matlab.
The results are
compared in the following sections. Appendix B contains sample Abaqus input files for
the analysis inputs. Appendix C contains the Matlab scripts.
3.1 Model Validation
An eigen analysis was performed on the baseline model to compare the mode shapes and
frequencies of the plate against analytical values. Appendix B includes a sample Abaqus
input file for the eigen analysis.
The side profile of the first mode shape of the plate is shown in Figure 11. This sort
of profile is expected for a plate, simply-supported on all sides [10].
Figure 11 Baseline First Mode Shape Profile
The first four mode frequencies were calculated analytically as well as by the model.
Equation 8, [10], was used to find the angular frequency, ω, for the plate.
𝐷
𝑚𝜋 2
𝜔 = √𝜌 ∗ [(
𝑎
𝑛𝜋 2
) +(𝑏) ]
[8]
Here D is the bending stiffness calculated previously using Equation 1, ρ is the density
of the plate material, a and b are the length and width of the plate, and m and n are the
order of the waves traveling along each dimension. Dividing the angular frequency, ω,
by 2π gives the frequency, f, in cycles per second (Hz). Table 3 provides a comparison
15
of the first four mode frequencies calculated by the model versus the frequencies
calculated analytically. The full calculation of the analytical solution can be found in
Appendix A.
Table 3 Frequency Comparison of First Four Modes
Mode
Abaqus Solution (Hz)
Analytical Solution (Hz)
Wave Order
1
19.192
19.199
m = 1, n = 1
2
47.982
47.998
m = 2, n = 1
3
47.984
47.998
m = 1, n = 2
4
76.728
76.797
m = 2, n = 2
Figure 12 shows the first four modes of the plate.
Figure 12 Baseline Modes:
(a) First Mode at 19.192 Hz, (b) Second Mode at 47.982 Hz, (c) Third Mode at 47.987 Hz, (d) Fourth
Mode at 76.728 Hz
3.2 Mode Identification
The excitation of the plate results in a high response at the natural frequencies of the
plate. Any differences in response due to the differing element formulations will be
16
most apparent at these peak amplitudes. Figure 13 plots the response at the location of
the driving force for each frequency.
Figure 13 Resonant Peaks in Response at Drive Location
Seven significant resonant peaks were selected from this figure for comparison.
These are natural frequencies of the plate. It was observed that the element types shifted
the frequencies of the resonances with significant shifting occurring in peaks above 400
Hz. Table 4 shows the frequencies for each case.
Table 4 Frequencies for Each Peak (Hz)
Peak
Case 1
Case 2
Case 3a
Case 3b
1
37
37
38
37
2
96
96
96
96
3
227
227
223
228
4
255
255
259
262
5
279
279
288
280
6
900
900
884
881
7
966
966
945
952
All the cases share the same frequencies for the first two peaks. Some frequency
shifting between cases occurs for the next three peaks. After 400 Hz, it becomes more
difficult to determine which the frequencies are part of the same resonance. The last two
17
peaks occur in Cases 3a and 3b but it is more unclear where they occur in Cases 1 and 2.
A higher frequency resolution might make these resonances more clear and would
ensure that no resonances were missed.
3.3 Wavenumber Comparison
The spatial Fourier transform was performed on the results at the frequencies from Table
4. This gave the breakdown of the waves in each frequency. The following figures
show the spatial wavenumber comparisons at each frequency for the different models.
Note that the x-axis reports the spatial wavenumber (ks = 1/λ) as opposed to angular
wavenumber. The solid model for Case 3b was animated at the peak frequencies to
provide an example of the shape of the plate vibration.
This model used the
incompatible mode formulation brick elements, which according to the Abaqus user
guide, [11], provides a solution close to the theoretical solution. Note that the line of
nodes used for the Fourier analysis is highlighted on the model.
The models are compared for the first peak response in Figure 14. The first peak
occurs at 37 Hz for all of the models. Additionally, all of the models except case 3a
contain the same wavenumber content, and thus the same plate vibration shape, at peak
1. It is apparent from the matching results of Case 1 and Case 2 that using beam or shell
elements as stiffeners on a plate modeled with shell elements produces consistent plate
response.
Figure 14 Peak 1 Wavenumber Comparison
18
The wavenumber plot for the second peak amplitude is shown in Figure 15. As with
the first resonant response at 37 Hz, all models predict the same resonant frequency at 96
Hz. Once again Cases 1 and 2 are the same, and all cases contain the same wavenumber
content and thus shared a similar plate response shape. However, Cases 3a and 3b
predict different amplitudes, with case 3a predicting lower response than the plate and
beam models and case 3b predicting higher response. It was expected that the full-3D
solid element models would begin to diverge from the plate and beam models as the
excitation frequency of the plate increases. However, it is interesting to note that the two
different solid element formulations have a large amplitude discrepancy with each other,
and that they bound the shell and beam solutions.
Figure 15 Peak 2 Wavenumber Comparison
Figure 16 shows the wavenumber comparison for the third resonant peak. This is
the first comparison where the resonant frequencies between the modeled cases do not
match. For all cases, the wavenumber content is very similar, suggesting similar plate
motion. Case 1 and Case 2 still line up exactly, and predict the response at 227 Hz.
Case 3a has good agreement with the Case 1 and Case 2 model predictions. However,
the resonance is predicted at 223 Hz. Case 3B predicts the resonance at 228 Hz, and also
has a higher amplitude for the wavenumber content.
19
Figure 16 Peak 3 Wavenumber Comparison
Figure 17 shows the wavenumber comparison of the fourth resonance peak. The
wavenumber content is similar for all cases, but the amplitudes are different. Cases 1
and 2 continue to predict matching wavenumber content and resonance frequencies.
Cases 3a and 3b show more divergence. Cases 1 and 2 predict the lowest resonant
frequency of 255 Hz. Case 3a predicts 259 Hz, while Case 3b predicts 262 Hz.
Figure 17 Peak 4 Wavenumber Comparison
The fifth resonant peak wavenumber comparison is shown in Figure 18. Case 1 and
Case 2 match in both the wavenumber content and the predicted resonant frequency.
Cases3a and 3b are very similar in the wavenumber content, but Case 3a predicts the
20
resonant frequency at 288 Hz whereas Case 3b predicts it at 280 Hz. Cases 1 and 2
predict 279 Hz which is close to the Case 3b prediction. All cases share the same
wavenumber content trend.
Figure 18 Peak 5 Wavenumber Comparison
Figure 19 shows the wavenumber comparison for the sixth resonant peak. The
wavenumber content differs between all the cases except Case 1 and Case 2, suggesting
differences in the plate motion. Case 1 and Case 2 predict the same wavenumber
content at low amplitudes for the same resonant frequency of 900 Hz. However, the
resonant frequency for Cases 1 and 2 is approximate since it was difficult to determine
from the drive response in Figure 13. Although there was an increased amplitude at 900
Hz, the amplitude was small compared to that seen in the solid element models.
Checking the wavenumber content at surrounding frequencies did not reveal any other
possible resonances. A higher frequency resolution, such as 0.25 Hz spacing, may have
revealed a stronger resonance in the Case 1 and Case 2 models. However, it is likely
that the beam and plate elements are not capable of capturing the higher-order mode
shapes present at the frequency. The amplitude for the wavenumber content of Case 3a
is also low, but with different wavenumber content than Case 1 and Case 2. For this
case, the resonant frequency was predicted at 884 Hz. Case 3b has the highest amplitude
and predicts the resonant frequency at 881 Hz.
21
Figure 19 Peak 6 Wavenumber Comparison
The wavenumber comparison for the seventh peak is shown in Figure 20. The
wavenumber content and the resonant frequencies differ for all the cases except between
Case 1 and Case 2. Case 1 and Case 2 predict a resonant frequency of 966 Hz and share
the smallest amplitude in wavenumber content.
This is probably due to the low
amplitude of the predicted resonance at 966 Hz. This is similar to what was observed in
Figure 19. The resonant response is predicted at 945 Hz and 952 Hz for the Case 3a and
Case 3b models, respectively.
The wavenumber content for Case 3a has a large
contribution from the ks = 0.05 wavenumber, while the contribution is distributed across
more wavenumbers for Case 3b.
This means that multiple waves of different
wavelengths are being predicted by the incompatible mode solid element mesh of Case
3b, while the reduced integration solid element mesh of Case 3a predicts one dominant
wavelength.
22
Figure 20 Peak 7 Wavenumber Comparison
The last two peaks show more of the wavenumber content being dominated by the
higher wavenumbers. This corresponds to smaller wavelengths in the plate which is
expected for the higher frequencies.
23
4. Conclusions
The results of the wavenumber analysis show that there is little difference in
wavenumber content for all modeled cases for excitation frequencies below 300 Hz. By
800 Hz, the wavenumber content varies greatly between the modeled cases. Different
wavenumber content means that the actual shape of vibration is different between
models. Since the models compared have identical geometry and mesh resolution, the
changes in wavenumber content are due to the element formulations.
The Case 1 model (shell element plate with beam element stiffening) and the Case 2
model (shell element plate with shell element stiffening) provided the same response
throughout the entire frequency range. This suggests that refining the stiffeners by
switching from beams to plates did not affect the modal response of the plate. However,
at frequencies above 800 Hz, there appeared to be little to no resonant character present
in the Case 1 and Case 2 models, whereas the Case 3a (solid element, reduced
integration) and Case 3b (solid element, incompatible mode formulation) continued to
predict resonant character.
It should be noted that the small amplitude differences seen at low frequencies
between model wavenumber responses that otherwise followed the same trend is due to
the frequency resolution of the analysis. An attempt was made to plot the wavenumber
response of each model at its resonant frequency.
However, the 1 Hz frequency
resolution was not sufficient to fully align all resonant frequencies. For a future study, it
is recommended that an Eigen study be performed throughout the frequency range to
establish the exact natural frequencies of vibration for each model. Performing the
Fourier analysis at the exact resonant frequency of each model should result in better
amplitude correlation.
In addition to isolating the exact natural frequencies, future work should include
evaluation of different plate geometries and stiffening arrangements, such as an isogrid
stiffening arrangement, curved plates, and thick plate structures (plate thickness less than
1/10 the length). Higher order elements, such as the 20-node brick element, should also
be evaluated. Finally, a two-dimensional wavenumber analysis technique could be used
to characterize the entire surface of the plate.
24
5. References
[1] Wang, Wenchao, Mohamad S. Qatu, and Shantia Yarahmadian. “Accuracy of shell
and solid elements in vibration analyses of thin–and thick–walled isotropic cylinders”.
International Journal of Vehicle Noise and Vibration 8.3 (2012): 221-236.
[2] Brown, Jeff. "Characterization of MSC/NASTRAN & MSC/ABAQUS elements for
turbine engine blade frequency analysis." Proc. MSC Aerospace Users' Conference.
1997.
[3] Dassault Systèmes Simulia. 2012. Abaqus Software. Abaqus/CAE 6.12-1
[4] Benzley, Steven E., et al. "A comparison of all hexagonal and all tetrahedral finite
element meshes for elastic and elasto-plastic analysis." Proceedings, 4th International
Meshing Roundtable. Vol. 17. Albuquerque, NM: Sandia National Laboratories, 1995.
[5] Thompson, Lony L., and Peter M. Pinsky. "Complex wavenumber Fourier analysis
of the p-version finite element method." Computational Mechanics 13.4 (1994): 255275.
[6] Mathworks Software, R2012a, MATLAB Student Version 7.14.0.739.
[7] Cook, R., Malkus, D. Plesha, M., and Witt, J. 2002. Concepts and Applications of
Finite Element Analysis, 4th Ed. John Wiley & Sons, Inc.
[8] Altair Engineering Inc., Altair HyperWorks, Version 12.0, HyperMesh v12.0
[9] MatWeb. HY-80 Steel Material Properties. [Online] [Cited: October 15, 2014.]
[10] Leissa, Arthur. 1993. Vibration of Plates. Acoustical Society of America
[11] Dassault Systèmes Simulia, 2013. Abaqus 6.13. Abaqus/CAE User’s Guide
25
[12] Junger, Miguel C., Feit, David. 1993. Sound, Structures, and Their Interaction.
Acoustical Society of America.
[13] Sun, Eric Qiuli. “Shear Locking and Hourglassing in MSC Nastran, ABAQUS, and
ANSYS”. Msc software users meeting. 2006.
[14] Fahy, Frank.1985. Sound and Structural Vibration: Radiation, Transmission and
Response. Academic Press Inc. (London) Ltd.
[15] MathWorks. R2014b Documentation of Fast Fourier Transform (FFT). [Online]
[Cited December 11, 2014]
26
6. Appendix A – Mesh Refinement Calculation
Note: All these calculations were performed in Excel Spreadsheet
Table 6-1 Geometry Assumptions
Plate Dimensions
Dimension
Units Value
Length, a
in
100
Width, b
in
100
Thickness, t
in
1
Stiffener Dimensions
Height, h
in
5
Thickness, ts
in
0.5
Table 6-2 Material Properties
Property
Units
Value
Elastic Modulus, E
psi
3.00E+07
Poisson’s Ratio, ν
Mass Density, ρ
0.3
Lbf*s2/in4 (0.284lbm/in3)/(32.174ft/s2)*(12in/ft) = 0.000736
Table 6-3 Wavelength Calculations for Infinite, Unstiffened Plate (Reference 10)
Description
Equation
1. Bending Stiffness
D = E*t^3/(12*(1-ν^2))
(Equ. Around 7.59)
3.00E+07*(1^3)/(12*(1-(0.3^2))) =
2. Plate Flexural
Wavenumber
2747253
kf = [ρ*t*(2*π*f)^2/D]^1/4
(0.000736*1*((2*PI()*1000)^2)/
(Equ. 7.62)
2747253)^(1/4) =
3. Flexural
Value
λ = 2*π/kf
27
0.320644127
Wavelength
2*PI()/0.320644127 =
(Equ. 2.21)
19.59551036
19.59551036/12 =
Element Length for Bending (Shell, Brick)
4. Compressional
Wave Velocity
1.633
cp = [E/((1-ν^2)*ρ)]^1/2
(3.00E+07/((1-(0.3^2))*
211701
(Equ. 2.53)
0.000736))^(1/2) =
5. Compressional
Wavenumber
(Equ. Around 2.21)
6. Compressional
Wavelength
(Equ. 2.21)
c = 2*π*f/k -> kp = 2*pi*f/cp
2*PI()*1000/211701 =
0.029679457
λ = 2*π/kp
2*PI()/0.029679457 =
211.7014891
211.7014891/12 =
Element Length for Compression (Brick)
17.642
Table 6-4 Eigen Modes of SSSS Baseline Plate
m, n
ω = SQRT(D/ρ)*
Value
[(m*π/a)^2 + (n*π/b)^2]
m=1
SQRT(2747253/0.000736)*
n=1
(((1*PI()/100)^2)+((1*PI()/100)^2))
m=2
SQRT(2747253/0.000736)*
n=1
(((2*PI()/100)^2)+((1*PI()/100)^2))
m=1
SQRT(2747253/0.000736)*
n=2
(((1*PI()/100)^2)+((2*PI()/100)^2))
m=2
SQRT(2747253/0.000736)*
n=2
(((2*PI()/100)^2)+((2*PI()/100)^2))
28
f = ω/2*π
Value
120.6321 120.6321/(2*PI()) 19.1992
301.5803 301.5803/(2*PI()) 47.9980
301.5803 301.5803/(2*PI()) 47.9980
482.5286 482.5286/(2*PI()) 76.7968
7. Appendix B – Model Work Files
7.1 HyperMesh Input Files
Baseline Model
Example Input File from HyperMesh:
**
** ABAQUS Input Deck Generated by HyperMesh Version : 12.0.112-RENUMBER-HWDesktop
** Generated using HyperMesh-Abaqus Template Version : 12.0.112-RENUMBER-HWDesktop
**
** Template: ABAQUS/STANDARD 3D
**
*NODE
1, 0.0
, 0.0
, 100.0
2, 1.4285714285714, 0.0
, 100.0
3, 2.8571428571429, 0.0
, 100.0
4, 4.2857142857143, 0.0
, 100.0
5, 5.7142857142857, 0.0
, 100.0
6, 7.1428571428571, 0.0
, 100.0
7, 8.5714285714286, 0.0
, 100.0
8, 10.0
, 0.0
, 100.0
9, 10.0
, 0.0
, 98.507462686567
10, 10.0
, 0.0
, 97.014925373134
…
4547, 95.714285714286, 0.0
, 97.014925373134
4548, 97.142857142857, 0.0
, 98.507462686567
4549, 98.571428571429, 0.0
, 98.507462686567
4550, 95.714285714286, 0.0
, 98.507462686567
4551, 97.142857142857, 0.0
, 94.029850746269
4552, 97.142857142857, 0.0
, 95.522388059702
4553, 98.571428571429, 0.0
, 94.029850746269
4554, 98.571428571429, 0.0
, 95.522388059702
4555, 95.714285714286, 0.0
, 94.029850746269
4556, 95.714285714286, 0.0
, 95.522388059702
**HWCOLOR COMP
1
57
*ELEMENT,TYPE=S4R,ELSET=plate
208,
37,
334,
299,
36
207,
326,
333,
331,
325
29
206,
301,
300,
333,
326
205,
332,
38,
39,
204,
331,
332,
328,
327
203,
325,
331,
327,
324
202,
149,
150,
329,
330
201,
41,
149,
330,
40
200,
330,
329,
327,
328
199,
40,
330,
328,
39
4413,
4551,
4552,
4554,
4553
4414,
4553,
4554,
4084,
4085
4415,
4552,
4546,
4545,
4554
4416,
4554,
4545,
4083,
4084
4417,
4511,
4536,
4555,
4512
4418,
4512,
4555,
4551,
4513
4419,
4536,
4535,
4556,
4555
4420,
4555,
4556,
4552,
4551
4421,
4535,
4534,
4547,
4556
4422,
4556,
4547,
4546,
4552
328
…
**HM_comp_by_property "shell"
11
*SHELL SECTION, ELSET=plate, SHELL THICKNESS = , MATERIAL=Steel
,
*MATERIAL, NAME=Steel
*DENSITY
7.3590E-04,0.0
*DAMPING, ALPHA = 0.01
*ELASTIC, MODULI = LONG TERM, TYPE = ISOTROPIC
30000000.0,0.3
,0.0
*****
30
Case 1 Model (Shell Element Plate and Beam Element Stiffeners)
Example Input File from HyperMesh:
**
** ABAQUS Input Deck Generated by HyperMesh Version : 12.0.112-RENUMBER-HWDesktop
** Generated using HyperMesh-Abaqus Template Version : 12.0.112-RENUMBER-HWDesktop
**
** Template: ABAQUS/STANDARD 3D
**
*NODE
1, 0.0
, 0.0
, 100.0
2, 1.4285714285714, 0.0
, 100.0
3, 2.8571428571429, 0.0
, 100.0
4, 4.2857142857143, 0.0
, 100.0
5, 5.7142857142857, 0.0
, 100.0
6, 7.1428571428571, 0.0
, 100.0
7, 8.5714285714286, 0.0
, 100.0
8, 10.0
, 0.0
, 100.0
9, 10.0
, 0.0
, 98.507462686567
10, 10.0
, 0.0
, 97.014925373134
…
4547, 95.714285714286, 0.0
, 97.014925373134
4548, 97.142857142857, 0.0
, 98.507462686567
4549, 98.571428571429, 0.0
, 98.507462686567
4550, 95.714285714286, 0.0
, 98.507462686567
4551, 97.142857142857, 0.0
, 94.029850746269
4552, 97.142857142857, 0.0
, 95.522388059702
4553, 98.571428571429, 0.0
, 94.029850746269
4554, 98.571428571429, 0.0
, 95.522388059702
4555, 95.714285714286, 0.0
, 94.029850746269
4556, 95.714285714286, 0.0
, 95.522388059702
**HWCOLOR COMP
2
52
*ELEMENT,TYPE=B31,ELSET=beam_stiffener
4487,
11,
10
4486,
12,
11
4485,
13,
12
4484,
14,
13
4483,
15,
14
31
4482,
16,
15
4481,
17,
16
4480,
18,
17
4479,
19,
18
4478,
20,
19
4748,
3254,
3255
4749,
3255,
3256
4750,
3256,
3257
4751,
3257,
3258
4752,
3258,
3259
4753,
3259,
3260
4754,
3260,
3261
4755,
3261,
3262
4756,
3262,
3263
4757,
3263,
3264
…
**HWCOLOR COMP
1
61
*ELEMENT,TYPE=S4R,ELSET=plate
208,
37,
334,
299,
36
207,
326,
333,
331,
325
206,
301,
300,
333,
326
205,
332,
38,
39,
204,
331,
332,
328,
327
203,
325,
331,
327,
324
202,
149,
150,
329,
330
201,
41,
149,
330,
40
200,
330,
329,
327,
328
199,
40,
330,
328,
39
4413,
4551,
4552,
4554,
4553
4414,
4553,
4554,
4084,
4085
4415,
4552,
4546,
4545,
4554
4416,
4554,
4545,
4083,
4084
4417,
4511,
4536,
4555,
4512
4418,
4512,
4555,
4551,
4513
4419,
4536,
4535,
4556,
4555
4420,
4555,
4556,
4552,
4551
328
…
32
4421,
4535,
4534,
4547,
4556
4422,
4556,
4547,
4546,
4552
**HM_comp_by_property "shell"
11
*SHELL SECTION, ELSET=plate, SHELL THICKNESS = , MATERIAL=Steel
,
**HM_comp_by_property "beam"
11
*BEAM SECTION, ELSET=beam_stiffener, MATERIAL= Steel, SECTION=BOX
0.0
,0.0
,0.0
,0.0
,0.0
,0.0
*MATERIAL, NAME=Steel
*DENSITY
7.3590E-04,0.0
*DAMPING, ALPHA = 0.01
*ELASTIC, MODULI = LONG TERM, TYPE = ISOTROPIC
30000000.0,0.3
,0.0
*****
33
Case 2 Model (Shell Element Plate and Stiffeners)
Example Input File from HyperMesh:
**
** ABAQUS Input Deck Generated by HyperMesh Version : 12.0.112-RENUMBER-HWDesktop
** Generated using HyperMesh-Abaqus Template Version : 12.0.112-RENUMBER-HWDesktop
**
** Template: ABAQUS/STANDARD 3D
**
*NODE
1, 0.0
, 0.0
, 100.0
2, 1.4285714285714, 0.0
, 100.0
3, 2.8571428571429, 0.0
, 100.0
4, 4.2857142857143, 0.0
, 100.0
5, 5.7142857142857, 0.0
, 100.0
6, 7.1428571428571, 0.0
, 100.0
7, 8.5714285714286, 0.0
, 100.0
8, 10.0
, 0.0
, 100.0
9, 10.0
, 0.0
, 98.507462686567
10, 10.0
, 0.0
, 97.014925373134
…
6247, 10.0
, -1.375
, 95.522388059702
6248, 10.0
, -2.75
, 94.029850746269
6249, 10.0
, -4.125
, 94.029850746269
6250, 10.0
, -1.375
, 94.029850746269
6251, 10.0
, -2.75
, 98.507462686567
6252, 10.0
, -2.75
, 97.014925373134
6253, 10.0
, -4.125
, 98.507462686567
6254, 10.0
, -4.125
, 97.014925373134
6255, 10.0
, -1.375
, 98.507462686567
6256, 10.0
, -1.375
, 97.014925373134
**HWCOLOR COMP
1
57
*ELEMENT,TYPE=S4R,ELSET=plate
208,
37,
334,
299,
36
207,
326,
333,
331,
325
206,
301,
300,
333,
326
205,
332,
38,
39,
204,
331,
332,
328,
328
327
34
203,
325,
331,
327,
324
202,
149,
150,
329,
330
201,
41,
149,
330,
40
200,
330,
329,
327,
328
199,
40,
330,
328,
39
4413,
4551,
4552,
4554,
4553
4414,
4553,
4554,
4084,
4085
4415,
4552,
4546,
4545,
4554
4416,
4554,
4545,
4083,
4084
4417,
4511,
4536,
4555,
4512
4418,
4512,
4555,
4551,
4513
4419,
4536,
4535,
4556,
4555
4420,
4555,
4556,
4552,
4551
4421,
4535,
4534,
4547,
4556
4422,
4556,
4547,
4546,
4552
…
**HWCOLOR COMP
3
5
*ELEMENT,TYPE=S4R,ELSET=plate_stiffener
6097,
6256,
6247,
6096,
10,
6095,
6255,
6094,
9,
6093,
6058,
6092,
8,
6091,
6254,
6245,
6052,
6053
6090,
6252,
6246,
6245,
6254
6089,
6253,
6254,
6053,
6054
6088,
6251,
6252,
6254,
6253
4767,
4708,
4718,
4631,
4632
4766,
4709,
4717,
4718,
4708
4765,
4716,
4625,
4626,
4714
4764,
3263,
3264,
4625,
4716
4763,
4713,
4716,
4714,
4712
4762,
3262,
3263,
4716,
4713
4761,
4715,
4627,
4628,
4629
4760,
4714,
4626,
4627,
4715
11,
6247,
6256,
10,
6252,
6251
6255
6251,
6255,
6252
6256
6256,
6255,
9,
6246,
6057
6058
…
35
4759,
4711,
4715,
4629,
4630
4758,
4712,
4714,
4715,
4711
**HM_comp_by_property "shell"
11
*SHELL SECTION, ELSET=plate, SHELL THICKNESS = , MATERIAL=Steel
,
*MATERIAL, NAME=Steel
*DENSITY
7.3590E-04,0.0
*DAMPING, ALPHA = 0.01
*ELASTIC, MODULI = LONG TERM, TYPE = ISOTROPIC
30000000.0,0.3
,0.0
*****
36
Case 3 Model (Solid Element Plate and Stiffeners)
Note: Element types are not assigned in HyperMesh so subcases a and b are from
the same input file.
Example Input File from HyperMesh:
**
** ABAQUS Input Deck Generated by HyperMesh Version : 12.0.112-RENUMBER-HWDesktop
** Generated using HyperMesh-Abaqus Template Version : 12.0.112-RENUMBER-HWDesktop
**
** Template: ABAQUS/STANDARD 3D
**
*NODE
1, 0.0
, 0.0
, 100.0
2, 1.4285714285714, 0.0
, 100.0
3, 2.8571428571429, 0.0
, 100.0
4, 4.2857142857143, 0.0
, 100.0
5, 5.7142857142857, 0.0
, 100.0
6, 7.1428571428571, 0.0
, 100.0
7, 8.5714285714286, 0.0
, 100.0
76, 8.5714285714286, 0.0
, 0.0
77, 7.1428571428571, 0.0
, 0.0
78, 5.7142857142857, 0.0
, 0.0
…
43169, 90.125
, 0.5
, 79.10447761194
43170, 90.125
, 0.5
, 97.014925373134
43171, 90.125
, 0.5
, 95.522388059702
43172, 90.125
, 0.5
, 94.029850746269
43173, 90.125
, 0.5
, 100.0
43174, 90.125
, 0.5
, 98.507462686567
43175, 90.125
, 0.5
, 89.55223880597
43176, 90.125
, 0.5
, 88.059701492537
43177, 90.125
, 0.5
, 92.537313432836
43178, 90.125
, 0.5
, 91.044776119403
**HWCOLOR COMP
4
56
*ELEMENT,TYPE=C3D8R,ELSET=solid_plate
14941,
10812,
10803,
10793,
10808,
15368,
15359,
15349,
10787,
10781,
10803,
10812,
15343,
15337,
15359,
15364
14940,
37
15368
14939,
10811,
10812,
10808,
10805,
15367,
15368,
15364,
10791,
10787,
10812,
10811,
15347,
15343,
15368,
10773,
10811,
10805,
10768,
15329,
15367,
15361,
10755,
10791,
10811,
10773,
15311,
15347,
15367,
10809,
10796,
10798,
10810,
15365,
15352,
15354,
10808,
10793,
10796,
10809,
15364,
15349,
15352,
10806,
10809,
10810,
10807,
15362,
15365,
15366,
10805,
10808,
10809,
10806,
15361,
15364,
15365,
29558,
29567,
29570,
29559,
31258,
31267,
31270,
29553,
29564,
29567,
29558,
31253,
31264,
31267,
29571,
29572,
29573,
29574,
31271,
31272,
31273,
20438,
6073,
6076,
29562,
29571,
29574,
20435,
6076,
5985,
29577,
29578,
29579,
29580,
31277,
31278,
31279,
29574,
29573,
29578,
29577,
31274,
31273,
31278,
29568,
29577,
29580,
29569,
31268,
31277,
31280,
29563,
29574,
29577,
29568,
31263,
31274,
31277,
15361
14938,
15367
14937,
15324
14936,
15329
14935,
15366
14934,
15365
14933,
15363
14932,
15362
…
50534,
31259
50535,
31258
50536,
31274
50545,
20435,
31285,
24802,
24811,
31281
50538,
29563,
31262,
31271,
31274,
31263
50544,
20439,
31281,
24811,
24812,
31284
50540,
31280
50541,
31277
50542,
31269
50543,
38
31268
*MATERIAL, NAME=Steel
*DENSITY
7.3590E-04,0.0
*DAMPING, ALPHA = 0.01
*ELASTIC, MODULI = LONG TERM, TYPE = ISOTROPIC
30000000.0,0.3
,0.0
*****
39
7.2 Abaqus Input Files
Baseline Model (Eigen Analysis)
Example Abaqus Input File:
*Heading
Eigen analysis of baseline plate
** Job name: Eigens Model name: PlateBaseline_Eigen
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=PART-1
*Node
1,
0.,
0.,
100.
2, 1.42857146,
0.,
100.
3, 2.85714293,
0.,
100.
4, 4.28571415,
0.,
100.
5, 5.71428585,
0.,
100.
6, 7.14285707,
0.,
100.
7, 8.5714283,
0.,
100.
8,
10.,
0.,
100.
9,
10.,
0., 98.5074615
10,
10.,
0., 97.0149231
…
4547, 95.7142868,
0., 97.0149231
4548, 97.1428604,
0., 98.5074615
4549, 98.5714264,
0., 98.5074615
4550, 95.7142868,
0., 98.5074615
4551, 97.1428604,
0., 94.0298538
4552, 97.1428604,
0., 95.5223846
4553, 98.5714264,
0., 94.0298538
4554, 98.5714264,
0., 95.5223846
4555, 95.7142868,
0., 94.0298538
4556, 95.7142868,
0., 95.5223846
*Element, type=S4R
1, 174, 176, 178, 173
40
2, 173, 178, 168, 169
3, 11, 179, 177, 10
4, 179, 178, 176, 177
5, 12, 167, 179, 11
6, 167, 168, 178, 179
7, 5, 6, 180, 175
8, 175, 180, 176, 174
9, 9, 181, 7, 8
10, 181, 180, 6, 7
…
4413, 4551, 4552, 4554, 4553
4414, 4553, 4554, 4084, 4085
4415, 4552, 4546, 4545, 4554
4416, 4554, 4545, 4083, 4084
4417, 4511, 4536, 4555, 4512
4418, 4512, 4555, 4551, 4513
4419, 4536, 4535, 4556, 4555
4420, 4555, 4556, 4552, 4551
4421, 4535, 4534, 4547, 4556
4422, 4556, 4547, 4546, 4552
*Elset, elset=PLATE, generate
1, 4422,
1
** Section: Section-1-PLATE
*Shell Section, elset=PLATE, material=STEEL
1., 5
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=PART-1-1, part=PART-1
*End Instance
**
*Nset, nset=Set-1, instance=PART-1-1
1, 2, 3, 4, 5, 6, 7, 8, 545, 625, 626, 627, 628, 629, 630, 631
41
632, 633, 634, 635, 636, 1429, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518
1519, 1520, 2313, 2393, 2394, 2395, 2396, 2397, 2398, 2399, 2400, 2401, 2402, 2403, 2404, 3197
3277, 3278, 3279, 3280, 3281, 3282, 3283, 3284, 3285, 3286, 3287, 3288, 4081, 4155, 4156, 4157
4158, 4159, 4160
*Nset, nset=Set-2, instance=PART-1-1
75, 76, 77, 78, 79, 80, 81, 82, 612, 613, 614, 615, 616, 617, 618, 619
620, 621, 622, 623, 624, 1496, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504, 1505, 1506
1507, 1508, 2380, 2381, 2382, 2383, 2384, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 3264
3265, 3266, 3267, 3268, 3269, 3270, 3271, 3272, 3273, 3274, 3275, 3276, 4148, 4149, 4150, 4151
4152, 4153, 4154
*Nset, nset="Plate Center", instance=PART-1-1
1462,
*Nset, nset=ResponseZ, instance=PART-1-1, generate
1429, 1496,
1
*Nset, nset=ResponseX, instance=PART-1-1
41, 116, 149, 150, 151, 152, 153, 154, 578, 1063, 1084, 1089, 1090, 1095, 1096, 1123
1128, 1129, 1132, 1136, 1139, 1462, 1947, 1968, 1973, 1974, 1979, 1980, 2007, 2012, 2013, 2016
2020, 2023, 2346, 2831, 2852, 2857, 2858, 2863, 2864, 2891, 2896, 2897, 2900, 2904, 2907, 3230
3715, 3736, 3741, 3742, 3747, 3748, 3775, 3780, 3781, 3784, 3788, 3791, 4114, 4379, 4384, 4385
4392, 4393, 4394
*Nset, nset=BCset3, instance=PART-1-1
1, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144
145, 146, 147, 148
*Nset, nset=BCset4, instance=PART-1-1, generate
4081, 4148,
1
*End Assembly
**
** MATERIALS
**
*Material, name=STEEL
*Damping, alpha=0.01
*Density
0.0007359,
*Elastic
42
3e+07, 0.3
**
** BOUNDARY CONDITIONS
**
** Name: SimplySupported Type: Displacement/Rotation
*Boundary
Set-1, 2, 2
Set-1, 6, 6
** Name: SimplySupported2 Type: Displacement/Rotation
*Boundary
Set-2, 2, 2
Set-2, 6, 6
** Name: SimplySupported3 Type: Displacement/Rotation
*Boundary
BCset3, 2, 2
BCset3, 4, 4
** Name: SimplySupported4 Type: Displacement/Rotation
*Boundary
BCset4, 2, 2
BCset4, 4, 4
** ---------------------------------------------------------------**
** STEP: EigenStep
**
*Step, name=EigenStep, perturbation
Finding plate modes
*Frequency, eigensolver=Lanczos, acoustic coupling=on, normalization=displacement
10, , , 1., ,
**
** OUTPUT REQUESTS
**
*Restart, write, frequency=0
**
** FIELD OUTPUT: F-Output-1
**
*Output, field
*Node Output
43
U, UR, UT
*End Step
44
Baseline Model (Forced Analysis)
Example Abaqus Input File:
*Heading
** Job name: BaselineForced Model name: PlateBaseline
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=PART-1
*Node
1,
0.,
0.,
100.
2, 1.42857146,
0.,
100.
3, 2.85714293,
0.,
100.
4, 4.28571415,
0.,
100.
5, 5.71428585,
0.,
100.
6, 7.14285707,
0.,
100.
7, 8.5714283,
0.,
100.
8,
10.,
0.,
100.
9,
10.,
0., 98.5074615
10,
10.,
0., 97.0149231
…
4547, 95.7142868,
0., 97.0149231
4548, 97.1428604,
0., 98.5074615
4549, 98.5714264,
0., 98.5074615
4550, 95.7142868,
0., 98.5074615
4551, 97.1428604,
0., 94.0298538
4552, 97.1428604,
0., 95.5223846
4553, 98.5714264,
0., 94.0298538
4554, 98.5714264,
0., 95.5223846
4555, 95.7142868,
0., 94.0298538
4556, 95.7142868,
0., 95.5223846
*Element, type=S4R
1, 174, 176, 178, 173
2, 173, 178, 168, 169
3, 11, 179, 177, 10
4, 179, 178, 176, 177
45
5, 12, 167, 179, 11
6, 167, 168, 178, 179
7, 5, 6, 180, 175
8, 175, 180, 176, 174
9, 9, 181, 7, 8
10, 181, 180, 6, 7
…
4413, 4551, 4552, 4554, 4553
4414, 4553, 4554, 4084, 4085
4415, 4552, 4546, 4545, 4554
4416, 4554, 4545, 4083, 4084
4417, 4511, 4536, 4555, 4512
4418, 4512, 4555, 4551, 4513
4419, 4536, 4535, 4556, 4555
4420, 4555, 4556, 4552, 4551
4421, 4535, 4534, 4547, 4556
4422, 4556, 4547, 4546, 4552
*Elset, elset=PLATE, generate
1, 4422,
1
** Section: Section-1-PLATE
*Shell Section, elset=PLATE, material=STEEL
1., 5
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=PART-1-1, part=PART-1
*End Instance
**
*Nset, nset=Set-1, instance=PART-1-1
1, 2, 3, 4, 5, 6, 7, 8, 545, 625, 626, 627, 628, 629, 630, 631
632, 633, 634, 635, 636, 1429, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518
1519, 1520, 2313, 2393, 2394, 2395, 2396, 2397, 2398, 2399, 2400, 2401, 2402, 2403, 2404, 3197
3277, 3278, 3279, 3280, 3281, 3282, 3283, 3284, 3285, 3286, 3287, 3288, 4081, 4155, 4156, 4157
46
4158, 4159, 4160
*Nset, nset=Set-2, instance=PART-1-1
75, 76, 77, 78, 79, 80, 81, 82, 612, 613, 614, 615, 616, 617, 618, 619
620, 621, 622, 623, 624, 1496, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504, 1505, 1506
1507, 1508, 2380, 2381, 2382, 2383, 2384, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 3264
3265, 3266, 3267, 3268, 3269, 3270, 3271, 3272, 3273, 3274, 3275, 3276, 4148, 4149, 4150, 4151
4152, 4153, 4154
*Nset, nset="Plate Center", instance=PART-1-1
1462,
*Nset, nset=ResponseZ, instance=PART-1-1, generate
1429, 1496,
1
*Nset, nset=ResponseX, instance=PART-1-1
41, 116, 149, 150, 151, 152, 153, 154, 578, 1063, 1084, 1089, 1090, 1095, 1096, 1123
1128, 1129, 1132, 1136, 1139, 1462, 1947, 1968, 1973, 1974, 1979, 1980, 2007, 2012, 2013, 2016
2020, 2023, 2346, 2831, 2852, 2857, 2858, 2863, 2864, 2891, 2896, 2897, 2900, 2904, 2907, 3230
3715, 3736, 3741, 3742, 3747, 3748, 3775, 3780, 3781, 3784, 3788, 3791, 4114, 4379, 4384, 4385
4392, 4393, 4394
*Nset, nset=BCset3, instance=PART-1-1
1, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144
145, 146, 147, 148
*Nset, nset=BCset4, instance=PART-1-1, generate
4081, 4148,
1
*End Assembly
**
** MATERIALS
**
*Material, name=STEEL
*Damping, alpha=0.01
*Density
0.0007359,
*Elastic
3e+07, 0.3
**
** BOUNDARY CONDITIONS
47
**
** Name: SimplySupported Type: Displacement/Rotation
*Boundary
Set-1, 2, 2
Set-1, 6, 6
** Name: SimplySupported2 Type: Displacement/Rotation
*Boundary
Set-2, 2, 2
Set-2, 6, 6
** Name: SimplySupported3 Type: Displacement/Rotation
*Boundary
BCset3, 2, 2
BCset3, 4, 4
** Name: SimplySupported4 Type: Displacement/Rotation
*Boundary
BCset4, 2, 2
BCset4, 4, 4
** ---------------------------------------------------------------**
** STEP: LoadStep
**
*Step, name=LoadStep, perturbation
*Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO
1., 1000., 1000, 1.
**
** LOADS
**
** Name: CenterLoad Type: Concentrated force
*Cload, real
"Plate Center", 2, 1.
**
** OUTPUT REQUESTS
**
**
** FIELD OUTPUT: ResponseX
**
*Output, field
48
*Node Output, nset=ResponseX
U,
**
** FIELD OUTPUT: ResponseZ
**
*Node Output, nset=ResponseZ
U,
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
49
Case 1 Model (Shell Element Plate and Beam Element Stiffeners)
Example Abaqus Input File:
*Heading
** Job name: PlateBeam_Forced Model name: Plate_wBeamStiffeners
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=PART-1
*Node
1,
0.,
0.,
100.
2, 1.42857146,
0.,
100.
3, 2.85714293,
0.,
100.
4, 4.28571415,
0.,
100.
5, 5.71428585,
0.,
100.
6, 7.14285707,
0.,
100.
7, 8.5714283,
0.,
100.
8,
10.,
0.,
100.
9,
10.,
0., 98.5074615
10,
10.,
0., 97.0149231
…
4547, 95.7142868,
0., 97.0149231
4548, 97.1428604,
0., 98.5074615
4549, 98.5714264,
0., 98.5074615
4550, 95.7142868,
0., 98.5074615
4551, 97.1428604,
0., 94.0298538
4552, 97.1428604,
0., 95.5223846
4553, 98.5714264,
0., 94.0298538
4554, 98.5714264,
0., 95.5223846
4555, 95.7142868,
0., 94.0298538
4556, 95.7142868,
0., 95.5223846
*Element, type=S4R
1, 174, 176, 178, 173
2, 173, 178, 168, 169
3, 11, 179, 177, 10
4, 179, 178, 176, 177
50
5, 12, 167, 179, 11
6, 167, 168, 178, 179
7, 5, 6, 180, 175
8, 175, 180, 176, 174
9, 9, 181, 7, 8
10, 181, 180, 6, 7
…
4413, 4551, 4552, 4554, 4553
4414, 4553, 4554, 4084, 4085
4415, 4552, 4546, 4545, 4554
4416, 4554, 4545, 4083, 4084
4417, 4511, 4536, 4555, 4512
4418, 4512, 4555, 4551, 4513
4419, 4536, 4535, 4556, 4555
4420, 4555, 4556, 4552, 4551
4421, 4535, 4534, 4547, 4556
4422, 4556, 4547, 4546, 4552
*Element, type=B31
4423, 75, 74
4424, 74, 73
4425, 73, 72
4426, 72, 71
4427, 71, 70
4428, 70, 69
4429, 69, 68
4430, 68, 67
4431, 67, 66
4432, 66, 65
…
4748, 3254, 3255
4749, 3255, 3256
4750, 3256, 3257
4751, 3257, 3258
4752, 3258, 3259
4753, 3259, 3260
4754, 3260, 3261
4755, 3261, 3262
51
4756, 3262, 3263
4757, 3263, 3264
*Elset, elset=BEAM_STIFFENER, generate
4423, 4757,
1
*Elset, elset=PLATE, generate
1, 4422,
1
*Elset, elset=Set-3, generate
4423, 4757,
1
*Elset, elset=Set-4, generate
4423, 4757,
1
*Elset, elset=Set-5, generate
4423, 4489,
1
** Section: Section-1-PLATE
*Shell Section, elset=PLATE, material=STEEL
1., 5
** Region: (Section-2:Set-3), (Beam Orientation:Set-5)
*Elset, elset=_I2, internal, generate
4423, 4489,
1
** Section: Section-2 Profile: Profile-2
*Beam Section, elset=_I2, material=STEEL, temperature=GRADIENTS, section=I
-0.5, 5., 0.5, 0.5, 0.5, 0.5, 0.5
-1.,0.,0.
** Region: (Section-2:Set-3), (Beam Orientation:Set-4)
*Elset, elset=_I3, internal, generate
4490, 4757,
1
** Section: Section-2 Profile: Profile-2
*Beam Section, elset=_I3, material=STEEL, temperature=GRADIENTS, section=I
-0.5, 5., 0.5, 0.5, 0.5, 0.5, 0.5
1.,0.,0.
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=PART-1-1, part=PART-1
52
*End Instance
**
*Nset, nset=Set-1, instance=PART-1-1
1, 2, 3, 4, 5, 6, 7, 8, 545, 625, 626, 627, 628, 629, 630, 631
632, 633, 634, 635, 636, 1429, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518
1519, 1520, 2313, 2393, 2394, 2395, 2396, 2397, 2398, 2399, 2400, 2401, 2402, 2403, 2404, 3197
3277, 3278, 3279, 3280, 3281, 3282, 3283, 3284, 3285, 3286, 3287, 3288, 4081, 4155, 4156, 4157
4158, 4159, 4160
*Nset, nset=Set-2, instance=PART-1-1
75, 76, 77, 78, 79, 80, 81, 82, 612, 613, 614, 615, 616, 617, 618, 619
620, 621, 622, 623, 624, 1496, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504, 1505, 1506
1507, 1508, 2380, 2381, 2382, 2383, 2384, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 3264
3265, 3266, 3267, 3268, 3269, 3270, 3271, 3272, 3273, 3274, 3275, 3276, 4148, 4149, 4150, 4151
4152, 4153, 4154
*Nset, nset=CenterNode, instance=PART-1-1
1462,
*Nset, nset=ResponseZ, instance=PART-1-1, generate
1429, 1496,
1
*Nset, nset=ResponseX, instance=PART-1-1
41, 116, 149, 150, 151, 152, 153, 154, 578, 1063, 1084, 1089, 1090, 1095, 1096, 1123
1128, 1129, 1132, 1136, 1139, 1462, 1947, 1968, 1973, 1974, 1979, 1980, 2007, 2012, 2013, 2016
2020, 2023, 2346, 2831, 2852, 2857, 2858, 2863, 2864, 2891, 2896, 2897, 2900, 2904, 2907, 3230
3715, 3736, 3741, 3742, 3747, 3748, 3775, 3780, 3781, 3784, 3788, 3791, 4114, 4379, 4384, 4385
4392, 4393, 4394
*Nset, nset=BCset3, instance=PART-1-1
1, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144
145, 146, 147, 148
*Nset, nset=BCset4, instance=PART-1-1, generate
4081, 4148,
1
*End Assembly
**
** MATERIALS
**
*Material, name=STEEL
53
*Damping, alpha=0.01
*Density
0.0007359,
*Elastic
3e+07, 0.3
**
** BOUNDARY CONDITIONS
**
** Name: BC-2 Type: Displacement/Rotation
*Boundary
Set-2, 2, 2
Set-2, 6, 6
** Name: SimplySupported1 Type: Displacement/Rotation
*Boundary
Set-1, 2, 2
Set-1, 6, 6
** Name: SimplySupported3 Type: Displacement/Rotation
*Boundary
BCset3, 2, 2
BCset3, 4, 4
** Name: SimplySupported4 Type: Displacement/Rotation
*Boundary
BCset4, 2, 2
BCset4, 4, 4
** ---------------------------------------------------------------**
** STEP: LoadStep
**
*Step, name=LoadStep, perturbation
*Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO
1., 1000., 1000, 1.
**
** LOADS
**
** Name: CenterLoad Type: Concentrated force
*Cload, real
CenterNode, 2, 1.
54
**
** OUTPUT REQUESTS
**
**
** FIELD OUTPUT: ResponseX
**
*Output, field
*Node Output, nset=ResponseX
U,
**
** FIELD OUTPUT: ResponseZ
**
*Node Output, nset=ResponseZ
U,
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
55
Case 2 Model (Shell Element Plate and Stiffeners)
Example Abaqus Input File:
*Heading
** Job name: PlatePlate_Forced Model name: Plate_wPlateStiffeners
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=PART-1
*Node
1,
0.,
0.,
100.
2, 1.42857146,
0.,
100.
3, 2.85714293,
0.,
100.
4, 4.28571415,
0.,
100.
5, 5.71428585,
0.,
100.
6, 7.14285707,
0.,
100.
7, 8.5714283,
0.,
100.
8,
10.,
0.,
100.
9,
10.,
0., 98.5074615
10,
10.,
0., 97.0149231
…
6247,
10.,
-1.375, 95.5223846
6248,
10.,
-2.75, 94.0298538
6249,
10.,
-4.125, 94.0298538
6250,
10.,
-1.375, 94.0298538
6251,
10.,
-2.75, 98.5074615
6252,
10.,
-2.75, 97.0149231
6253,
10.,
-4.125, 98.5074615
6254,
10.,
-4.125, 97.0149231
6255,
10.,
-1.375, 98.5074615
6256,
10.,
-1.375, 97.0149231
*Element, type=S4R
1, 174, 176, 178, 173
2, 173, 178, 168, 169
3, 11, 179, 177, 10
4, 179, 178, 176, 177
56
5, 12, 167, 179, 11
6, 167, 168, 178, 179
7, 5, 6, 180, 175
8, 175, 180, 176, 174
9, 9, 181, 7,
8
10, 181, 180, 6, 7
…
6088, 6251, 6252, 6254, 6253
6089, 6253, 6254, 6053, 6054
6090, 6252, 6246, 6245, 6254
6091, 6254, 6245, 6052, 6053
6092, 8, 9, 6255, 6058
6093, 6058, 6255, 6251, 6057
6094, 9, 10, 6256, 6255
6095, 6255, 6256, 6252, 6251
6096, 10, 11, 6247, 6256
6097, 6256, 6247, 6246, 6252
*Elset, elset=PLATE, generate
1, 4422,
1
*Elset, elset=Set-2, generate
4758, 6097,
1
** Section: Section-1-PLATE
*Shell Section, elset=PLATE, material=STEEL
1., 5
** Section: Section-2
*Shell Section, elset=Set-2, material=STEEL
0.5, 5
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=PART-1-1, part=PART-1
*End Instance
**
57
*Elset, elset=PLATE_STIFFENER, instance=PART-1-1, generate
4758, 6097,
1
*Nset, nset=Set-2, instance=PART-1-1
1, 2, 3, 4, 5, 6, 7, 8, 545, 625, 626, 627, 628, 629, 630, 631
632, 633, 634, 635, 636, 1429, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518
1519, 1520, 2313, 2393, 2394, 2395, 2396, 2397, 2398, 2399, 2400, 2401, 2402, 2403, 2404, 3197
3277, 3278, 3279, 3280, 3281, 3282, 3283, 3284, 3285, 3286, 3287, 3288, 4081, 4155, 4156, 4157
4158, 4159, 4160
*Nset, nset=Set-3, instance=PART-1-1
75, 76, 77, 78, 79, 80, 81, 82, 612, 613, 614, 615, 616, 617, 618, 619
620, 621, 622, 623, 624, 1496, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504, 1505, 1506
1507, 1508, 2380, 2381, 2382, 2383, 2384, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 3264
3265, 3266, 3267, 3268, 3269, 3270, 3271, 3272, 3273, 3274, 3275, 3276, 4148, 4149, 4150, 4151
4152, 4153, 4154
*Nset, nset=CenterPlate, instance=PART-1-1
1462,
*Nset, nset=ResponseZ, instance=PART-1-1, generate
1429, 1496,
1
*Nset, nset=ResponseX, instance=PART-1-1
41, 116, 149, 150, 151, 152, 153, 154, 578, 1063, 1084, 1089, 1090, 1095, 1096, 1123
1128, 1129, 1132, 1136, 1139, 1462, 1947, 1968, 1973, 1974, 1979, 1980, 2007, 2012, 2013, 2016
2020, 2023, 2346, 2831, 2852, 2857, 2858, 2863, 2864, 2891, 2896, 2897, 2900, 2904, 2907, 3230
3715, 3736, 3741, 3742, 3747, 3748, 3775, 3780, 3781, 3784, 3788, 3791, 4114, 4379, 4384, 4385
4392, 4393, 4394
*Nset, nset=BCset3, instance=PART-1-1
1, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144
145, 146, 147, 148
*Nset, nset=BCset4, instance=PART-1-1, generate
4081, 4148,
1
*End Assembly
**
** MATERIALS
**
*Material, name=STEEL
58
*Damping, alpha=0.01
*Density
0.0007359,
*Elastic
3e+07, 0.3
**
** BOUNDARY CONDITIONS
**
** Name: BC-1 Type: Displacement/Rotation
*Boundary
Set-2, 2, 2
Set-2, 6, 6
** Name: SimplySupported Type: Displacement/Rotation
*Boundary
BCset3, 2, 2
BCset3, 4, 4
** Name: SimplySupported2 Type: Displacement/Rotation
*Boundary
Set-3, 2, 2
Set-3, 6, 6
** Name: SimplySupported4 Type: Displacement/Rotation
*Boundary
BCset4, 2, 2
BCset4, 4, 4
** ---------------------------------------------------------------**
** STEP: LoadStep
**
*Step, name=LoadStep, perturbation
*Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO
1., 1000., 1000, 1.
**
** LOADS
**
** Name: CenterLoad Type: Concentrated force
*Cload, real
CenterPlate, 2, 1.
59
**
** OUTPUT REQUESTS
**
**
** FIELD OUTPUT: ResponseX
**
*Output, field
*Node Output, nset=ResponseX
U,
**
** FIELD OUTPUT: ResponseZ
**
*Node Output, nset=ResponseZ
U,
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
60
Case 3a Model (Solid Element Plate and Stiffeners, Type C3D8R)
Example Abaqus Input File:
*Heading
** Job name: Solid_Forced Model name: SolidPlate_wSolidStiffeners
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=PART-1
*Node
1,
0.,
0.,
100.
2, 1.42857146,
0.,
100.
3, 2.85714293,
0.,
100.
4, 4.28571415,
0.,
100.
5, 5.71428585,
0.,
100.
6, 7.14285707,
0.,
100.
7, 8.5714283,
0.,
100.
76, 8.5714283,
0.,
0.
77, 7.14285707,
0.,
0.
78, 5.71428585,
0.,
0.
…
43169,
90.125,
0.5, 79.1044769
43170,
90.125,
0.5, 97.0149231
43171,
90.125,
0.5, 95.5223846
43172,
90.125,
0.5, 94.0298538
43173,
90.125,
0.5,
43174,
90.125,
0.5, 98.5074615
43175,
90.125,
0.5, 89.5522385
43176,
90.125,
0.5,
43177,
90.125,
0.5, 92.5373154
43178,
90.125,
0.5, 91.0447769
100.
88.0597
*Element, type=C3D8R
6099, 326, 333, 331, 325, 6261, 6262, 6263, 6264
6100, 301, 300, 333, 326, 6265, 6266, 6262, 6261
6102, 331, 332, 328, 327, 6263, 6267, 6270, 6271
6103, 325, 331, 327, 324, 6264, 6263, 6271, 6272
61
6104, 149, 150, 329, 330, 6273, 6274, 6275, 6276
6106, 330, 329, 327, 328, 6276, 6275, 6271, 6270
6108, 323, 329, 150, 151, 6279, 6275, 6274, 6280
6109, 324, 327, 329, 323, 6272, 6271, 6275, 6279
6110, 316, 315, 320, 322, 6281, 6282, 6283, 6284
6111, 306, 316, 322, 307, 6285, 6281, 6284, 6286
…
125647, 39244, 39176, 39178, 39246, 10733, 15289, 15285, 10729
125648, 39238, 39170, 39171, 39239, 10785, 15341, 15340, 10784
125649, 39163, 39238, 39239, 39164, 4538, 10785, 10784, 4540
125650, 39239, 39171, 39169, 39237, 10784, 15340, 15342, 10786
125651, 39241, 39173, 39167, 39235, 10758, 15314, 15348, 10792
125652, 39166, 39235, 39236, 39165, 4544, 10792, 10789, 4542
125653, 39235, 39167, 39168, 39236, 10792, 15348, 15345, 10789
125654, 39236, 39168, 39170, 39238, 10789, 15345, 15341, 10785
125655, 39162, 39240, 39242, 39160, 4523, 10761, 10757, 4519
125656, 39240, 39172, 39174, 39242, 10761, 15317, 15313, 10757
*Elset, elset=Set-1
6099, 6100, 6102, 6103, 6104, 6106, 6108, 6109, 6110, 6111, 6112, 6113, 6114, 6115,
6116, 6117
6118, 6119, 6120, 6121, 6122, 6123, 6124, 6125, 6126, 6128, 6130, 6131, 6132, 6134,
6136, 6137
6138, 6139, 6140, 6141, 6142, 6143, 6144, 6145, 6146, 6147, 6148, 6149, 6150, 6151,
6152, 6153
6154, 6156, 6158, 6159, 6160, 6162, 6164, 6165, 6166, 6167, 6168, 6169, 6170, 6171,
6172, 6173
6174, 6175, 6176, 6177, 6178, 6179, 6180, 6181, 6182, 6184, 6186, 6187, 6188, 6190,
6192, 6193
…
125585, 125586, 125587, 125588, 125589, 125590, 125591, 125592, 125593, 125594, 125595, 125596,
125597, 125598, 125599, 125600
125601, 125602, 125603, 125604, 125605, 125606, 125607, 125608, 125609, 125610, 125611, 125612,
125613, 125614, 125615, 125616
125617, 125618, 125619, 125620, 125621, 125622, 125623, 125624, 125625, 125626, 125627, 125628,
125629, 125630, 125631, 125632
125633, 125634, 125635, 125636, 125637, 125638, 125639, 125640, 125641, 125642, 125643, 125644,
125645, 125646, 125647, 125648
62
125649, 125650, 125651, 125652, 125653, 125654, 125655, 125656
** Section: Section-1
*Solid Section, elset=Set-1, material=STEEL
,
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=PART-1-1, part=PART-1
*End Instance
**
*Elset, elset=SOLID_PLATE, instance=PART-1-1
6099, 6100, 6102, 6103, 6104, 6106, 6108, 6109, 6110, 6111, 6112, 6113, 6114, 6115,
6116, 6117
6118, 6119, 6120, 6121, 6122, 6123, 6124, 6125, 6126, 6128, 6130, 6131, 6132, 6134,
6136, 6137
6138, 6139, 6140, 6141, 6142, 6143, 6144, 6145, 6146, 6147, 6148, 6149, 6150, 6151,
6152, 6153
6154, 6156, 6158, 6159, 6160, 6162, 6164, 6165, 6166, 6167, 6168, 6169, 6170, 6171,
6172, 6173
6174, 6175, 6176, 6177, 6178, 6179, 6180, 6181, 6182, 6184, 6186, 6187, 6188, 6190,
6192, 6193
…
125585, 125586, 125587, 125588, 125589, 125590, 125591, 125592, 125593, 125594, 125595, 125596,
125597, 125598, 125599, 125600
125601, 125602, 125603, 125604, 125605, 125606, 125607, 125608, 125609, 125610, 125611, 125612,
125613, 125614, 125615, 125616
125617, 125618, 125619, 125620, 125621, 125622, 125623, 125624, 125625, 125626, 125627, 125628,
125629, 125630, 125631, 125632
125633, 125634, 125635, 125636, 125637, 125638, 125639, 125640, 125641, 125642, 125643, 125644,
125645, 125646, 125647, 125648
125649, 125650, 125651, 125652, 125653, 125654, 125655, 125656
*Elset, elset=SOLID_STIFFENER, instance=PART-1-1
63
45184, 45186, 45188, 45190, 45191, 45192, 45193, 45194, 45195, 45196, 45198, 45200, 45201, 45202,
45203, 45204
45206, 45208, 45209, 45210, 45211, 45212, 45214, 45216, 45217, 45218, 45219, 45220, 45222, 45224,
45225, 45226
45227, 45228, 45230, 45232, 45233, 45234, 45235, 45236, 45238, 45240, 45241, 45242, 45243, 45244,
45246, 45248
45249, 45250, 45251, 45252, 45254, 45256, 45258, 45259, 45260, 45261, 45262, 45263, 45264, 45266,
45268, 45269
45270, 45271, 45272, 45274, 45276, 45277, 45278, 45279, 45280, 45282, 45284, 45285, 45286, 45287,
45288, 45290
…
51804, 51805, 51806, 51807, 51808, 51809, 51810, 51811, 51812, 51813, 51814, 51815, 51816, 51817,
51818, 51819
51820, 51821, 51822, 51823, 51824, 51825, 51826, 51827, 51828, 51829, 51830, 51831, 51832, 51833,
51834, 51835
51836, 51837, 51838, 51839, 51840, 51841, 51842, 51843, 51844, 51845, 51846, 51847, 51848, 51849,
51850, 51851
51852, 51853, 51854, 51855, 51856, 51857, 51858, 51859, 51860, 51861, 51862, 51863, 51864, 51865,
51866, 51867
51868, 51869, 51870, 51871, 51872, 51873, 51874, 51875, 51876, 51877, 51878, 51879, 51880, 51881,
51882, 51883
*Nset, nset=Set-3, instance=PART-1-1
76, 77, 78, 79, 80, 81, 82, 613, 614, 615, 616, 617, 618, 619, 620, 621
622, 623, 624, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504, 1505, 1506, 1507, 1508,
2381
2382, 2383, 2384, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 3265, 3266, 3267, 3268,
3269
3270, 3271, 3272, 3273, 3274, 3275, 3276, 4148, 4149, 4150, 4151, 4152, 4153, 4154, 34016,
34624
34964, 35304, 35644, 35701, 36238, 36578, 36918, 37258, 37399, 37739, 38079, 38419, 38759, 39099,
39531, 39871
40211, 41014, 41354, 41694, 42359, 42699, 43039
*Nset, nset=Set-4, instance=PART-1-1
1,
2,
3,
4,
5,
6,
7, 625, 626, 627, 628, 629, 630, 631, 632, 633
634, 635, 636, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518, 1519, 1520,
2393
64
2394, 2395, 2396, 2397, 2398, 2399, 2400, 2401, 2402, 2403, 2404, 3277, 3278, 3279, 3280,
3281
3282, 3283, 3284, 3285, 3286, 3287, 3288, 4081, 4155, 4156, 4157, 4158, 4159, 4160, 34017,
34633
34973, 35313, 35653, 35702, 36086, 36426, 36766, 37106, 37400, 37740, 38080, 38420, 38760, 39100,
39544, 39884
40224, 41003, 41343, 41683, 42192, 42532, 42872
*Nset, nset=SolidCenter, instance=PART-1-1
39903,
*Nset, nset=ResponseZ, instance=PART-1-1
39783, 39785, 39791, 39792, 39793, 39796, 39805, 39806, 39811, 39813, 39817, 39818, 39819, 39822,
39827, 39830
39833, 39838, 39839, 39842, 39843, 39844, 39847, 39856, 39857, 39862, 39865, 39866, 39869, 39870,
39871, 39872
39875, 39877, 39878, 39880, 39883, 39884, 39887, 39888, 39890, 39891, 39894, 39896, 39897, 39899,
39902, 39903
39905, 39907, 39910, 39912, 39952, 39957, 39958, 39961, 39962, 39965, 39970, 39972, 39975, 39978,
39981, 39982
39985, 39986, 39989, 39993
*Nset, nset=ResponseX, instance=PART-1-1
116, 149, 150, 151, 152, 153, 154, 1063, 1084, 1089, 1090, 1095, 1096, 1123, 1128, 1129
1132, 1136, 1139, 1947, 1968, 1973, 1974, 1979, 1980, 2007, 2012, 2013, 2016, 2020, 2023,
2831
2852, 2857, 2858, 2863, 2864, 2891, 2896, 2897, 2900, 2904, 2907, 3715, 3736, 3741, 3742,
3747
3748, 3775, 3780, 3781, 3784, 3788, 3791, 4114, 4379, 4384, 4385, 4392, 4393, 4394, 34015,
34498
34838, 35178, 35518, 35743, 36146, 36486, 36826, 37166, 37441, 37781, 38121, 38461, 38801, 39139,
39563, 39903
40243, 40895, 41235, 41575, 42371, 42711, 43051
*Nset, nset=BCset3, instance=PART-1-1
1, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144
145, 146, 147, 148
*Nset, nset=BCset4, instance=PART-1-1, generate
65
4081, 4148,
1
*End Assembly
**
** MATERIALS
**
*Material, name=STEEL
*Damping, alpha=0.01
*Density
0.0007359,
*Elastic
3e+07, 0.3
**
** BOUNDARY CONDITIONS
**
** Name: SimplySupported1 Type: Displacement/Rotation
*Boundary
Set-3, 2, 2
Set-3, 6, 6
** Name: SimplySupported2 Type: Displacement/Rotation
*Boundary
Set-4, 2, 2
Set-4, 6, 6
** Name: SimplySupported3 Type: Displacement/Rotation
*Boundary
BCset3, 2, 2
BCset3, 4, 4
** Name: SimplySupported4 Type: Displacement/Rotation
*Boundary
BCset4, 2, 2
BCset4, 4, 4
** ---------------------------------------------------------------**
** STEP: LoadStep
**
*Step, name=LoadStep, perturbation
*Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO
1., 1000., 1000, 1.
66
**
** LOADS
**
** Name: CenterLoad Type: Concentrated force
*Cload, real
SolidCenter, 2, 1.
**
** OUTPUT REQUESTS
**
**
** FIELD OUTPUT: ResponseX
**
*Output, field
*Node Output, nset=ResponseX
U,
**
** FIELD OUTPUT: ResponseZ
**
*Node Output, nset=ResponseZ
U,
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
67
Case 3b Model (Solid Element Plate and Stiffeners, Type C3D8I)
Example Abaqus Input File:
*Heading
** Job name: Solid_ForcedB Model name: SolidPlate_wSolidStiffeners_Hex8i
** Generated by: Abaqus/CAE 6.12-1
*Preprint, echo=NO, model=NO, history=NO, contact=NO
**
** PARTS
**
*Part, name=PART-1
*Node
1,
0.,
0.,
100.
2, 1.42857146,
0.,
100.
3, 2.85714293,
0.,
100.
4, 4.28571415,
0.,
100.
5, 5.71428585,
0.,
100.
6, 7.14285707,
0.,
100.
7, 8.5714283,
0.,
100.
76, 8.5714283,
0.,
0.
77, 7.14285707,
0.,
0.
78, 5.71428585,
0.,
0.
…
43169,
90.125,
0.5, 79.1044769
43170,
90.125,
0.5, 97.0149231
43171,
90.125,
0.5, 95.5223846
43172,
90.125,
0.5, 94.0298538
43173,
90.125,
0.5,
43174,
90.125,
0.5, 98.5074615
43175,
90.125,
0.5, 89.5522385
43176,
90.125,
0.5,
43177,
90.125,
0.5, 92.5373154
43178,
90.125,
0.5, 91.0447769
100.
88.0597
*Element, type=C3D8I
6099, 326, 333, 331, 325, 6261, 6262, 6263, 6264
6100, 301, 300, 333, 326, 6265, 6266, 6262, 6261
6102, 331, 332, 328, 327, 6263, 6267, 6270, 6271
6103, 325, 331, 327, 324, 6264, 6263, 6271, 6272
68
6104, 149, 150, 329, 330, 6273, 6274, 6275, 6276
6106, 330, 329, 327, 328, 6276, 6275, 6271, 6270
6108, 323, 329, 150, 151, 6279, 6275, 6274, 6280
6109, 324, 327, 329, 323, 6272, 6271, 6275, 6279
6110, 316, 315, 320, 322, 6281, 6282, 6283, 6284
6111, 306, 316, 322, 307, 6285, 6281, 6284, 6286
…
125647, 39244, 39176, 39178, 39246, 10733, 15289, 15285, 10729
125648, 39238, 39170, 39171, 39239, 10785, 15341, 15340, 10784
125649, 39163, 39238, 39239, 39164, 4538, 10785, 10784, 4540
125650, 39239, 39171, 39169, 39237, 10784, 15340, 15342, 10786
125651, 39241, 39173, 39167, 39235, 10758, 15314, 15348, 10792
125652, 39166, 39235, 39236, 39165, 4544, 10792, 10789, 4542
125653, 39235, 39167, 39168, 39236, 10792, 15348, 15345, 10789
125654, 39236, 39168, 39170, 39238, 10789, 15345, 15341, 10785
125655, 39162, 39240, 39242, 39160, 4523, 10761, 10757, 4519
125656, 39240, 39172, 39174, 39242, 10761, 15317, 15313, 10757
*Elset, elset=Set-1
6099, 6100, 6102, 6103, 6104, 6106, 6108, 6109, 6110, 6111, 6112, 6113, 6114, 6115,
6116, 6117
6118, 6119, 6120, 6121, 6122, 6123, 6124, 6125, 6126, 6128, 6130, 6131, 6132, 6134,
6136, 6137
6138, 6139, 6140, 6141, 6142, 6143, 6144, 6145, 6146, 6147, 6148, 6149, 6150, 6151,
6152, 6153
6154, 6156, 6158, 6159, 6160, 6162, 6164, 6165, 6166, 6167, 6168, 6169, 6170, 6171,
6172, 6173
6174, 6175, 6176, 6177, 6178, 6179, 6180, 6181, 6182, 6184, 6186, 6187, 6188, 6190,
6192, 6193
…
125585, 125586, 125587, 125588, 125589, 125590, 125591, 125592, 125593, 125594, 125595, 125596,
125597, 125598, 125599, 125600
125601, 125602, 125603, 125604, 125605, 125606, 125607, 125608, 125609, 125610, 125611, 125612,
125613, 125614, 125615, 125616
125617, 125618, 125619, 125620, 125621, 125622, 125623, 125624, 125625, 125626, 125627, 125628,
125629, 125630, 125631, 125632
125633, 125634, 125635, 125636, 125637, 125638, 125639, 125640, 125641, 125642, 125643, 125644,
125645, 125646, 125647, 125648
69
125649, 125650, 125651, 125652, 125653, 125654, 125655, 125656
** Section: Section-1
*Solid Section, elset=Set-1, material=STEEL
,
*End Part
**
**
** ASSEMBLY
**
*Assembly, name=Assembly
**
*Instance, name=PART-1-1, part=PART-1
*End Instance
**
*Elset, elset=SOLID_PLATE, instance=PART-1-1
6099, 6100, 6102, 6103, 6104, 6106, 6108, 6109, 6110, 6111, 6112, 6113, 6114, 6115,
6116, 6117
6118, 6119, 6120, 6121, 6122, 6123, 6124, 6125, 6126, 6128, 6130, 6131, 6132, 6134,
6136, 6137
6138, 6139, 6140, 6141, 6142, 6143, 6144, 6145, 6146, 6147, 6148, 6149, 6150, 6151,
6152, 6153
6154, 6156, 6158, 6159, 6160, 6162, 6164, 6165, 6166, 6167, 6168, 6169, 6170, 6171,
6172, 6173
6174, 6175, 6176, 6177, 6178, 6179, 6180, 6181, 6182, 6184, 6186, 6187, 6188, 6190,
6192, 6193
…
125585, 125586, 125587, 125588, 125589, 125590, 125591, 125592, 125593, 125594, 125595, 125596,
125597, 125598, 125599, 125600
125601, 125602, 125603, 125604, 125605, 125606, 125607, 125608, 125609, 125610, 125611, 125612,
125613, 125614, 125615, 125616
125617, 125618, 125619, 125620, 125621, 125622, 125623, 125624, 125625, 125626, 125627, 125628,
125629, 125630, 125631, 125632
125633, 125634, 125635, 125636, 125637, 125638, 125639, 125640, 125641, 125642, 125643, 125644,
125645, 125646, 125647, 125648
125649, 125650, 125651, 125652, 125653, 125654, 125655, 125656
*Elset, elset=SOLID_STIFFENER, instance=PART-1-1
70
45184, 45186, 45188, 45190, 45191, 45192, 45193, 45194, 45195, 45196, 45198, 45200, 45201, 45202,
45203, 45204
45206, 45208, 45209, 45210, 45211, 45212, 45214, 45216, 45217, 45218, 45219, 45220, 45222, 45224,
45225, 45226
45227, 45228, 45230, 45232, 45233, 45234, 45235, 45236, 45238, 45240, 45241, 45242, 45243, 45244,
45246, 45248
45249, 45250, 45251, 45252, 45254, 45256, 45258, 45259, 45260, 45261, 45262, 45263, 45264, 45266,
45268, 45269
45270, 45271, 45272, 45274, 45276, 45277, 45278, 45279, 45280, 45282, 45284, 45285, 45286, 45287,
45288, 45290
…
51804, 51805, 51806, 51807, 51808, 51809, 51810, 51811, 51812, 51813, 51814, 51815, 51816, 51817,
51818, 51819
51820, 51821, 51822, 51823, 51824, 51825, 51826, 51827, 51828, 51829, 51830, 51831, 51832, 51833,
51834, 51835
51836, 51837, 51838, 51839, 51840, 51841, 51842, 51843, 51844, 51845, 51846, 51847, 51848, 51849,
51850, 51851
51852, 51853, 51854, 51855, 51856, 51857, 51858, 51859, 51860, 51861, 51862, 51863, 51864, 51865,
51866, 51867
51868, 51869, 51870, 51871, 51872, 51873, 51874, 51875, 51876, 51877, 51878, 51879, 51880, 51881,
51882, 51883
*Nset, nset=Set-3, instance=PART-1-1
76, 77, 78, 79, 80, 81, 82, 613, 614, 615, 616, 617, 618, 619, 620, 621
622, 623, 624, 1497, 1498, 1499, 1500, 1501, 1502, 1503, 1504, 1505, 1506, 1507, 1508,
2381
2382, 2383, 2384, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 3265, 3266, 3267, 3268,
3269
3270, 3271, 3272, 3273, 3274, 3275, 3276, 4148, 4149, 4150, 4151, 4152, 4153, 4154, 34016,
34624
34964, 35304, 35644, 35701, 36238, 36578, 36918, 37258, 37399, 37739, 38079, 38419, 38759, 39099,
39531, 39871
40211, 41014, 41354, 41694, 42359, 42699, 43039
*Nset, nset=Set-4, instance=PART-1-1
1,
2,
3,
4,
5,
6,
7, 625, 626, 627, 628, 629, 630, 631, 632, 633
634, 635, 636, 1509, 1510, 1511, 1512, 1513, 1514, 1515, 1516, 1517, 1518, 1519, 1520,
2393
71
2394, 2395, 2396, 2397, 2398, 2399, 2400, 2401, 2402, 2403, 2404, 3277, 3278, 3279, 3280,
3281
3282, 3283, 3284, 3285, 3286, 3287, 3288, 4081, 4155, 4156, 4157, 4158, 4159, 4160, 34017,
34633
34973, 35313, 35653, 35702, 36086, 36426, 36766, 37106, 37400, 37740, 38080, 38420, 38760, 39100,
39544, 39884
40224, 41003, 41343, 41683, 42192, 42532, 42872
*Nset, nset=SolidCenter, instance=PART-1-1
39903,
*Nset, nset=ResponseZ, instance=PART-1-1
39783, 39785, 39791, 39792, 39793, 39796, 39805, 39806, 39811, 39813, 39817, 39818, 39819, 39822,
39827, 39830
39833, 39838, 39839, 39842, 39843, 39844, 39847, 39856, 39857, 39862, 39865, 39866, 39869, 39870,
39871, 39872
39875, 39877, 39878, 39880, 39883, 39884, 39887, 39888, 39890, 39891, 39894, 39896, 39897, 39899,
39902, 39903
39905, 39907, 39910, 39912, 39952, 39957, 39958, 39961, 39962, 39965, 39970, 39972, 39975, 39978,
39981, 39982
39985, 39986, 39989, 39993
*Nset, nset=ResponseX, instance=PART-1-1
116, 149, 150, 151, 152, 153, 154, 1063, 1084, 1089, 1090, 1095, 1096, 1123, 1128, 1129
1132, 1136, 1139, 1947, 1968, 1973, 1974, 1979, 1980, 2007, 2012, 2013, 2016, 2020, 2023,
2831
2852, 2857, 2858, 2863, 2864, 2891, 2896, 2897, 2900, 2904, 2907, 3715, 3736, 3741, 3742,
3747
3748, 3775, 3780, 3781, 3784, 3788, 3791, 4114, 4379, 4384, 4385, 4392, 4393, 4394, 34015,
34498
34838, 35178, 35518, 35743, 36146, 36486, 36826, 37166, 37441, 37781, 38121, 38461, 38801, 39139,
39563, 39903
40243, 40895, 41235, 41575, 42371, 42711, 43051
*Nset, nset=BCset3, instance=PART-1-1
1, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112
113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128
129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144
145, 146, 147, 148
*Nset, nset=BCset4, instance=PART-1-1, generate
72
4081, 4148,
1
*End Assembly
**
** MATERIALS
**
*Material, name=STEEL
*Damping, alpha=0.01
*Density
0.0007359,
*Elastic
3e+07, 0.3
**
** BOUNDARY CONDITIONS
**
** Name: SimplySupported1 Type: Displacement/Rotation
*Boundary
Set-3, 2, 2
Set-3, 6, 6
** Name: SimplySupported2 Type: Displacement/Rotation
*Boundary
Set-4, 2, 2
Set-4, 6, 6
** Name: SimplySupported3 Type: Displacement/Rotation
*Boundary
BCset3, 2, 2
BCset3, 4, 4
** Name: SimplySupported4 Type: Displacement/Rotation
*Boundary
BCset4, 2, 2
BCset4, 4, 4
** ---------------------------------------------------------------**
** STEP: LoadStep
**
*Step, name=LoadStep, perturbation
*Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO
1., 1000., 1000, 1.
73
**
** LOADS
**
** Name: CenterLoad Type: Concentrated force
*Cload, real
SolidCenter, 2, 1.
**
** OUTPUT REQUESTS
**
**
** FIELD OUTPUT: ResponseX
**
*Output, field
*Node Output, nset=ResponseX
U,
**
** FIELD OUTPUT: ResponseZ
**
*Node Output, nset=ResponseZ
U,
**
** FIELD OUTPUT: F-Output-1
**
*Output, field, variable=PRESELECT
**
** HISTORY OUTPUT: H-Output-1
**
*Output, history, variable=PRESELECT
*End Step
74
8. Appendix C – Matlab Scripts
8.1 Result Processing Script
result_processing.m
% result_processing.m
% Description: This script reads the results from the Abaqus .rpt files and
%
processes it as .mat files for further comparison.
%
% Kirsten Benamati
% October 2014
%
%%
clear; clc; close all;
InputPath
=
'C:\Users\Kirsten\Documents\Master_Degree\Final_Masters_Project\Website\Abaqus_Work_Files\input_f
iles\';
FilePath
=
'C:\Users\Kirsten\Documents\Master_Degree\Final_Masters_Project\Website\Abaqus_Result_Files\report
_files\';
SavePath
=
'C:\Users\Kirsten\Documents\Master_Degree\Final_Masters_Project\Website\Matlab_Work_Files\';
%% Sort Response Locations on Plate with Shell Elements
% Sort the response coordinates along the plate by location
% This applies to the Baseline, Case 1, and Case 2 models
% Read the nodal coordinates from the Abaqus input file
FileName = [InputPath, 'BaselineForced.inp'];
fid = fopen(FileName);
PlateCoords = textscan(fid,'%f %f %f %f','Delimiter',',','HeaderLines',9);
fclose(fid); clear FileName fid ans
% Response nodes on plate
X_Nodes = [41, 116, 149, 150, 151, 152, 153, 154, 578, 1063, 1084, 1089,...
1090, 1095, 1096, 1123, 1128, 1129, 1132, 1136, 1139, 1462, 1947,...
1968, 1973, 1974, 1979, 1980, 2007, 2012, 2013, 2016, 2020, 2023,...
2346, 2831, 2852, 2857, 2858, 2863, 2864, 2891, 2896, 2897, 2900,...
75
2904, 2907, 3230, 3715, 3736, 3741, 3742, 3747, 3748, 3775, 3780,...
3781, 3784, 3788, 3791, 4114, 4379, 4384, 4385, 4392, 4393, 4394];
% Find the response nodes and sort by x-direction
for ii=1:length(X_Nodes)
Xidx = find(PlateCoords{1,1}==X_Nodes(ii));
XCoords(ii,:) = [PlateCoords{1,1}(Xidx) PlateCoords{1,2}(Xidx)...
PlateCoords{1,3}(Xidx) PlateCoords{1,4}(Xidx)];
clear Xidx
end; clear ii
[Xsort Xidx] = sort(XCoords(:,2)); % Sort along x-direction
XCoords_Shell = XCoords(Xidx,:);
AxialCoords = XCoords_Shell(:,2); % Axial location of all response nodes
% Find the drive node index
% Drive node: 1462
DPidx = find(XCoords_Shell==1462);
clear PlateCoords X_Nodes XCoords Xsort
%% Baseline Case
% Response: Imaginary, Real
% Read the vertical response along the x-axis
FileName = [FilePath, 'BaselinePlate_U2_Complex_1000.rpt'];
Temp = importdata(FileName, ' ', 4);
BaseFreq = Temp.data(:,1);
% Frequency List
BaseImag = Temp.data(:,2:68);
% Imaginary Response
BaseReal = Temp.data(:,69:end); % Real Response
% Sort the response by location
BaseImag_Resp = BaseImag(:,Xidx);
BaseReal_Resp = BaseReal(:,Xidx);
BaseImag_DP = BaseImag_Resp(:,DPidx); % Response at drive location
BaseReal_DP = BaseReal_Resp(:,DPidx); % Response at drive location
save([SavePath, 'Baseline_Forced_Response'],'BaseFreq','BaseImag_DP',...
'BaseReal_DP','BaseImag_Resp','BaseReal_Resp','AxialCoords');
76
clear FileName Temp BaseImag BaseReal
%% Case 1: Shell Element Plate, Beam Element Stiffeners
% Response: Imaginary, Real
% Read the vertical response along the x-axis
FileName = [FilePath, 'BeamPlate_U2_Complex_1000.rpt'];
Temp = importdata(FileName, ' ', 4);
Case1Freq = Temp.data(:,1);
Case1Imag = Temp.data(:,2:68);
% Frequency List
% Imaginary Response
Case1Real = Temp.data(:,69:end); % Real Response
% Sort the response by location
Case1Imag_Resp = Case1Imag(:,Xidx);
Case1Real_Resp = Case1Real(:,Xidx);
Case1Imag_DP = Case1Imag_Resp(:,DPidx); % Response at drive location
Case1Real_DP = Case1Real_Resp(:,DPidx); % Response at drive location
save([SavePath, 'Case1_Forced_Response'],'Case1Freq','Case1Imag_DP',...
'Case1Real_DP','Case1Imag_Resp','Case1Real_Resp','AxialCoords');
clear FileName Temp Case1Imag Case1Real
%% Case 2: Shell Element Plate, Shell Element Stiffeners
% Response: Imaginary, Real
% Read the vertical response along the x-axis
FileName = [FilePath, 'BeamPlate_U2_Complex_1000.rpt'];
Temp = importdata(FileName, ' ', 4);
Case2Freq = Temp.data(:,1);
Case2Imag = Temp.data(:,2:68);
% Frequency List
% Imaginary Response
Case2Real = Temp.data(:,69:end); % Real Response
% Sort the response by location
Case2Imag_Resp = Case2Imag(:,Xidx);
Case2Real_Resp = Case2Real(:,Xidx);
Case2Imag_DP = Case2Imag_Resp(:,DPidx); % Response at drive location
Case2Real_DP = Case2Real_Resp(:,DPidx); % Response at drive location
save([SavePath, 'Case2_Forced_Response'],'Case2Freq','Case2Imag_DP',...
77
'Case2Real_DP','Case2Imag_Resp','Case2Real_Resp','AxialCoords');
clear FileName Temp Case2Imag Case2Real Xidx DPidx
%% Sort Response Locations on Plate with Solid Elements
% Sort the response coordinates along the plate by location
% This applies to the Case 3a and Case 3b models
% Read the nodal coordinates from the Abaqus input file
FileName = [InputPath, 'Solid_ForcedA.inp'];
fid = fopen(FileName);
PlateCoords = textscan(fid,'%f %f %f %f','Delimiter',',','HeaderLines',9);
fclose(fid); clear FileName fid ans
% Response nodes on plate
X_Nodes = [116, 149, 150, 151, 152, 153, 154, 1063, 1084, 1089, 1090,...
1095, 1096, 1123, 1128, 1129, 1132, 1136, 1139, 1947, 1968, 1973,...
1974, 1979, 1980, 2007, 2012, 2013, 2016, 2020, 2023, 2831, 2852,...
2857, 2858, 2863, 2864, 2891, 2896, 2897, 2900, 2904, 2907, 3715,...
3736, 3741, 3742, 3747, 3748, 3775, 3780, 3781, 3784, 3788, 3791,...
4114, 4379, 4384, 4385, 4392, 4393, 4394, 34015, 34498, 34838,...
35178, 35518, 35743, 36146, 36486, 36826, 37166, 37441, 37781,...
38121, 38461, 38801, 39139, 39563, 39903, 40243, 40895, 41235,...
41575, 42371, 42711, 43051];
% Find the response nodes and sort by x-direction
for ii=1:length(X_Nodes)
Xidx = find(PlateCoords{1,1}==X_Nodes(ii));
XCoords(ii,:) = [PlateCoords{1,1}(Xidx) PlateCoords{1,2}(Xidx)...
PlateCoords{1,3}(Xidx) PlateCoords{1,4}(Xidx)];
clear Xidx
end; clear ii
[Xsort Xidx] = sort(XCoords(:,2)); % Sort along x-direction
XCoords_Solid_Full = XCoords(Xidx,:);
% Find the drive node index
% Drive node: 39903
DPidx = find(XCoords_Solid_Full==39903);
78
clear PlateCoords X_Nodes XCoords Xsort
%% Case 3a: Solid Element Plate, Solid Element Stiffeners, Type 1
% Response: Imaginary, Real
% Read the vertical response along the x-axis
FileName = [FilePath, 'SolidA_U2_Complex_1000.rpt'];
Temp = importdata(FileName, ' ', 5);
Case3aFreq = Temp.data(:,1);
% Frequency List
Case3aImag = Temp.data(:,2:88);
% Imaginary Response
Case3aReal = Temp.data(:,89:end); % Real Response
% Sort the response by location
Case3aImag_Resp = Case3aImag(:,Xidx);
Case3aReal_Resp = Case3aReal(:,Xidx);
Case3aImag_DP = Case3aImag_Resp(:,DPidx); % Response at drive location
Case3aReal_DP = Case3aReal_Resp(:,DPidx); % Response at drive location
clear FileName Temp Case3aImag Case3aReal
%% Case 3b: Solid Element Plate, Solid Element Stiffeners, Type 2
% Response: Imaginary, Real
% Read the vertical response along the x-axis
FileName = [FilePath, 'SolidB_U2_Complex_1000.rpt'];
Temp = importdata(FileName, ' ', 5);
Case3bFreq = Temp.data(:,1);
Case3bImag = Temp.data(:,2:88);
% Frequency List
% Imaginary Response
Case3bReal = Temp.data(:,89:end); % Real Response
% Sort the response by location
Case3bImag_Resp = Case3bImag(:,Xidx);
Case3bReal_Resp = Case3bReal(:,Xidx);
Case3bImag_DP = Case3bImag_Resp(:,DPidx); % Response at drive location
Case3bReal_DP = Case3bReal_Resp(:,DPidx); % Response at drive location
clear FileName Temp Case3bImag Case3bReal Xidx DPidx
%% Find Corresponding Response Locations
% To compare Case 3a and Case 3b results with the baseline, Case 1, and
79
% Case 2 results, the response nodes from the solid plate must be found for
% the corresponding response nodes from the shell plate.
% corresponding response locations from the shell plate on the solid plate.
% Find index for matching response nodes
nResp = length(XCoords_Shell(:,1));
% Number of response nodes
for ll = 1:nResp
LocInd(ll) = find(XCoords_Solid_Full(:,2)==XCoords_Shell(ll,2));
end; clear ll
XCoords_Solid = XCoords_Solid_Full(LocInd,:);
% Check that coordinates are the same for both sets
CoordComp = isequal(XCoords_Shell(:,2:4),XCoords_Solid(:,2:4));
if CoordComp == 1
disp('Response locations for the Solid and Shell plates are the same');
end
% Find response
Case3aImag_Resp = Case3aImag_Resp(:,LocInd);
Case3aReal_Resp = Case3aReal_Resp(:,LocInd);
Case3bImag_Resp = Case3bImag_Resp(:,LocInd);
Case3bReal_Resp = Case3bReal_Resp(:,LocInd);
save([SavePath, 'Case3a_Forced_Response'],'Case3aFreq','Case3aImag_DP',...
'Case3aReal_DP','Case3aImag_Resp','Case3aReal_Resp','AxialCoords');
save([SavePath, 'Case3b_Forced_Response'],'Case3bFreq','Case3bImag_DP',...
'Case3bReal_DP','Case3bImag_Resp','Case3bReal_Resp','AxialCoords');
80
8.2 Plotting Script
wavenumber_analysis_and_plotting.m
% wavenumber_analysis_and_plotting.m
% Description: This script loads processed .mat files from the Abaqus
%
analysis, performs a spatial FFT on the data, and plots the
%
results for comparison.
%
% Kirsten Benamati
% December 2014
%
%%
clear; clc; close all;
FilePath
=
'C:\Users\Kirsten\Documents\Master_Degree\Final_Masters_Project\Website\Matlab_Work_Files\';
%% Load Results
% Baseline
FileName = [FilePath, 'Baseline_Forced_Response.mat'];
load(FileName); disp(['Loaded ',FileName]);
FreqList = BaseFreq;
% Frequency list is the same for all cases
clear FileName AxialCoords BaseFreq BaseImag_DP BaseReal_DP
% Case 1
FileName = [FilePath, 'Case1_Forced_Response.mat'];
load(FileName); disp(['Loaded ',FileName]);
clear FileName AxialCoords Case1Freq
% Case 2
FileName = [FilePath, 'Case2_Forced_Response.mat'];
load(FileName); disp(['Loaded ',FileName]);
clear FileName AxialCoords Case2Freq
% Case 3a
FileName = [FilePath, 'Case3a_Forced_Response.mat'];
load(FileName); disp(['Loaded ',FileName]);
clear FileName AxialCoords Case3aFreq
81
% Case 3b
FileName = [FilePath, 'Case3b_Forced_Response.mat'];
load(FileName); disp(['Loaded ',FileName]);
clear FileName Case3bFreq
%% Colors
plotcolors = [0 0 0;
% Black
1 1 1;
% Gray
10 0 0;
% Red
0 7 0;
% Green
0 0 10;
% Blue
5 0 5]./10;
% Purple
%% Plot Drive Location Response vs. Frequency
% Create complex results of the drive location response
Case1DP = Case1Real_DP + Case1Imag_DP.*1i;
Case2DP = Case2Real_DP + Case2Imag_DP.*1i;
Case3aDP = Case3aReal_DP + Case3aImag_DP.*1i;
Case3bDP = Case3bReal_DP + Case3bImag_DP.*1i;
clear Case1Real_DP Case1Imag_DP Case2Real_DP Case2Imag_DP
clear Case3aReal_DP Case3aImag_DP Case3bReal_DP Case3bImag_DP
figure; hold; grid;
plot(FreqList,Case1DP,'-','color',plotcolors(3,:),'linewidth',1.5)
plot(FreqList,Case2DP,':','color',plotcolors(5,:),'linewidth',1.5)
plot(FreqList,Case3aDP,'-.','color',plotcolors(4,:),'linewidth',1.5)
plot(FreqList,Case3bDP,'--','color',plotcolors(6,:),'linewidth',1.5)
axis([0 1000 -1e-4 1e-4]);
xlabel('Frequency, Hz');
ylabel('Displacement, in.');
title(['Response at Drive Location']);
legend('Case 1: Shell Element Plate, Beam Element Stiffeners',...
'Case 2: Shell Element Plate, Shell Element Stiffeners',...
'Case 3a: C3D8R Solid Element Plate, Solid Element Stiffeners',...
'Case 3b: C3D8I Solid Element Plate, Solid Element Stiffeners');
set(gcf,'position',[0,0,840,450]);
saveas(gcf,[FilePath, 'Drive Point Response vs Frequency.fig']);
saveas(gcf,[FilePath, 'Drive Point Response vs Frequency.emf']);
82
%% Find Complex Results
BaseResp = BaseReal_Resp + BaseImag_Resp.*1i;
Case1Resp = Case1Real_Resp + Case1Imag_Resp.*1i;
Case2Resp = Case2Real_Resp + Case2Imag_Resp.*1i;
Case3aResp = Case3aReal_Resp + Case3aImag_Resp.*1i;
Case3bResp = Case3bReal_Resp + Case3bImag_Resp.*1i;
clear BaseReal_Resp BaseImag_Resp Case1Real_Resp Case1Imag_Resp
clear Case2Real_Resp Case2Imag_Resp Case3aReal_Resp Case3aImag_Resp
clear Case3bReal_Resp Case3bImag_Resp
%% Interpolate Values
% The spacing between the response points is a little uneven due to the
% mesh accomodating the stiffeners. A linear interpolation on the data
% will smooth out the spatial fourier transform.
RespVector = 0:1:100;
% Interpolation response spacing
for ii = 1:length(FreqList)
BaseRespI(ii,:) = interp1(AxialCoords,BaseResp(ii,:),RespVector,'linear');
Case1RespI(ii,:) = interp1(AxialCoords,Case1Resp(ii,:),RespVector,'linear');
Case2RespI(ii,:) = interp1(AxialCoords,Case2Resp(ii,:),RespVector,'linear');
Case3aRespI(ii,:) = interp1(AxialCoords,Case3aResp(ii,:),RespVector,'linear');
Case3bRespI(ii,:) = interp1(AxialCoords,Case3bResp(ii,:),RespVector,'linear');
end; clear ii
% Check Interpolation ans Sort by Plotting Figure
IIncr = 100/100; %X-axis increment after interpolation
keyFreq=[37 96 228 262 280 881 952];
for ff = 1:length(keyFreq)
figure;
plot((0:100)*IIncr,real(Case3bRespI(keyFreq(ff),:)),'-r')
legendStr1 = '100 Axial Points (Interpolated)';
hold;
plot(AxialCoords,real(Case3bResp(keyFreq(ff),:)),'-b');
legendStr2 = '67 Axial Points (As Modeled)';
legend(legendStr1,legendStr2,4);
xlabel('Position, In');
83
ylabel('Displacement, In');
title(['Response Line Vertical Displacement f=',num2str(keyFreq(ff))]);
end; clear ff IIncr
%% Wavenumber Range
Length = 100;
% Length of plate (in.)
nSample = length(RespVector);
% Number of samples (interpolated)
SampleRate = (nSample-1)/Length; % Samples per unit length (1/in.)
% Nyquist Cut-off (to remove the mirroring effect)
nqKs = SampleRate/2;
% Maximum wavenumber supported (1/in.)
nqLambda = 1/nqKs;
% Maximum wavelength supported (in.)
WaveIncr = SampleRate/nSample;
% Wavenumber increment (1/in.)
WaveNumber = (0:nSample-1)*WaveIncr; % Wavenumber range
clear SampleRate nSample WaveIncr
%% Spatial FFT Analysis
% Using the fft function, the responses are transformed to wavenumber
% space.
for ff = 1:length(FreqList)
BaseFFT(ff,:) = fft(BaseRespI(ff,:));
Case1FFT(ff,:) = fft(Case1RespI(ff,:));
Case2FFT(ff,:) = fft(Case2RespI(ff,:));
Case3aFFT(ff,:) = fft(Case3aRespI(ff,:));
Case3bFFT(ff,:) = fft(Case3bRespI(ff,:));
end; clear ff
% Find the magnitudes for plotting
BaseData = abs(BaseFFT);
Case1Data = abs(Case1FFT);
Case2Data = abs(Case2FFT);
Case3aData = abs(Case3aFFT);
Case3bData = abs(Case3bFFT);
clear BaseFFT Case1FFT Case2FFT Case3aFFT Case3bFFT
%% Plots
84
% Baseline Comparison Plot: All cases against the Baseline at 10 Hz
% The stiffened plates have a similar character to the baseline plate
% Case 1, Case 2, and Case 3b are very similar, but Case 3a is not
figure; hold; grid;
plot(WaveNumber,BaseData(10,:),'-k','linewidth',1.5)
plot(WaveNumber,Case1Data(10,:),'-','color',plotcolors(3,:),'linewidth',1.5)
plot(WaveNumber,Case2Data(10,:),':','color',plotcolors(5,:),'linewidth',1.5)
plot(WaveNumber,Case3aData(10,:),'-.','color',plotcolors(4,:),'linewidth',1.5)
plot(WaveNumber,Case3bData(10,:),'--','color',plotcolors(6,:),'linewidth',1.5)
xlim([0 nqKs]);
xlabel('Wavenumber, 1/in');
ylabel('Displacement Amplitude');
title(['Response Comparison in Wavenumber Space, ',num2str(FreqList(10)),' Hz']);
legend('Baseline: Unstiffened Plate',...
'Case 1: Shell Element Plate, Beam Element Stiffeners',...
'Case 2: Shell Element Plate, Shell Element Stiffeners',...
'Case 3a: C3D8R Solid Element Plate, Solid Element Stiffeners',...
'Case 3b: C3D8I Solid Element Plate, Solid Element Stiffeners');
set(gcf,'position',[0,0,840,450]);
saveas(gcf,[FilePath, 'Baseline Response Comparison, ',num2str(FreqList(10)),' Hz.fig']);
saveas(gcf,[FilePath, 'Baseline Response Comparison, ',num2str(FreqList(10)),' Hz.emf']);
% Comparison Plots (without Baseline)
% Around 37, 96, 228, 262, 280, 881, 952 Hz
% Case1F = [37 96 227 255 279 881 952];
Case1F = [37 96 227 255 279 885 966];
% Case2F = [37 96 227 255 279 881 952];
Case2F = [37 96 227 255 279 900 966];
Case3aF = [38 96 223 259 288 884 945];
Case3bF = [37 96 228 262 280 881 952];
for ff = 1:length(keyFreq)
figure; hold; grid;
plot(WaveNumber,Case1Data(Case1F(ff),:),'-x','color',plotcolors(3,:),'linewidth',1.5)
legendStr1 = ['Case 1: Shell Element Plate, Beam Element Stiffeners, ',num2str(Case1F(ff)),' Hz'];
plot(WaveNumber,Case2Data(Case2F(ff),:),':o','color',plotcolors(5,:),'linewidth',1.5)
legendStr2 = ['Case 2: Shell Element Plate, Shell Element Stiffeners, ',num2str(Case2F(ff)),' Hz'];
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plot(WaveNumber,Case3aData(Case3aF(ff),:),'-.s','color',plotcolors(4,:),'linewidth',1.5)
legendStr3 = ['Case 3a: C3D8R Solid Element Plate, Solid Element Stiffeners, ',num2str(Case3aF(ff)),'
Hz'];
plot(WaveNumber,Case3bData(Case3bF(ff),:),'--^','color',plotcolors(6,:),'linewidth',1.5)
legendStr4 = ['Case 3b: C3D8I Solid Element Plate, Solid Element Stiffeners, ',num2str(Case3bF(ff)),'
Hz'];
%
xlim([0 nqKs]);
xlim([0 0.2]);
xlabel('Wavenumber, 1/in');
ylabel('Displacement Amplitude');
title(['Response Comparison in Wavenumber Space, Peak ',num2str(ff)]);
legend(legendStr1,legendStr2,legendStr3,legendStr4);
set(gcf,'position',[0,0,840,450]);
saveas(gcf,[FilePath, 'Response Comparison in Wavenumber Space, Around ',num2str(keyFreq(ff)),'
Hz.fig']);
end; clear ff
% % Image Plots for Each Case
% figure;
% imagesc(FreqList,RespVector,log10(BaseData))
% set(gca,'YDir','normal');
% xlabel('Location Along Plate, in.');
% ylabel('Frequency, Hz');
% title('Log(DFT) Response Across Frequency Range');
%
% figure;
% imagesc(FreqList,RespVector,log10(Case1Data))
% set(gca,'YDir','normal');
% xlabel('Location Along Plate, in.');
% ylabel('Frequency, Hz');
% title('Log(DFT) Response Across Frequency Range');
%
% figure;
% imagesc(FreqList,RespVector,log10(Case2Data))
% set(gca,'YDir','normal');
% xlabel('Location Along Plate, in.');
% ylabel('Frequency, Hz');
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% title('Log(DFT) Response Across Frequency Range');
%
% figure;
% imagesc(FreqList,RespVector,log10(Case3aData))
% set(gca,'YDir','normal');
% xlabel('Location Along Plate, in.');
% ylabel('Frequency, Hz');
% title('Log(DFT) Response Across Frequency Range');
%
% figure;
% imagesc(FreqList,RespVector,log10(Case3bData))
% set(gca,'YDir','normal');
% xlabel('Location Along Plate, in.');
% ylabel('Frequency, Hz');
% title('Log(DFT) Response Across Frequency Range');
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