FINITE ELEMENT STUDY OF ENERGY ABSORPTION IN CORRUGATED
BEAMS
A Thesis by
Avinash P Deshpande
B.E., Bangalore University, 2001
Submitted to the Department of Mechanical Engineering
and the faculty of Graduate School of
Wichita State University
in partial fulfillment of
the requirements for the Degree of
Master of Science
December 2005
© Copyright 2005 by Avinash P Deshpande
All Rights Reserved
FINITE ELEMENT STUDY OF ENERGY ABSORPTION IN CORRUGATED
BEAMS
The following faculty members have examined the final copy of this thesis for form and
content, and recommend that it be accepted in partial fulfillment of the requirements for the
degree of Master of Science, with a major in Mechanical Engineering.
Hamid M Lankarani, Committee Chair
Ramazan Asmatulu, Committee Member
Suresh Raju Keshavanarayan, Committee Member
iii
DEDICATION
To my Parents
iv
ACKNOWLEDGEMENTS
I would like to express my deepest gratitude to my advisor, Dr. Hamid M Lankarani, for
the guidance and support he has been given me throughout the entire period of my
graduate work and research.
My special thanks to Dr. Suresh Raju Keshavanarayana for his assistance, patience, and
encouragement, which helped me translate my visions into reality.
I thank Dr. Ramazan Asmatulu for reviewing my manuscript and helping me to improve
this thesis.
Special thanks to my parents who have been a source of encouragement and inspiration
throughout my life, and who are the very reason for my existence.
v
ABSTRACT
This work is focused on the finite element study of the energy absorbing behavior of
aluminum (corrugated webs). The aim is to study the mechanism of energy absorption in
specific metallic structures by understanding these mechanisms. The sine wave web (or
corrugated web) structural component constructed of aluminum has been chosen as the
specific structural element for this study. The sine wave exhibits similar energyabsorption trend as tubes, but it represents a more realistic and efficient configuration
directly usable in design when compared to tube elements. The principal objectives are,
(1) to perform an Eigen value analysis of the sine wave web, a parametric study of the
corrugated beam is done by varying the wavelength, amplitude and young’s modulus of
the corrugated web and calculate the buckling load, and (2) comparison of the buckling
load from the finite element analysis with that of the Eigen value analysis and the
energy absorption predicted from the finite element study. The energy absorption
behavior of sine wave webs loaded in axial compression is to be
investigated in theoretical territories. Study is conducted to analyze the effect of
geometric parameter i.e. the included angle on the energy absorption of the corrugated
web. A finite element model for the energy absorption in the sine wave web is
developed by assigning aluminum properties to the sine wave web model. The included
angle of the web is varied and the energy absorption is observed for the different
included angles. The crash-impact behavior of the sine wave web is studied by
calculating the energy absorption analytically from the finite element study. Some
important conclusions are pointed out as a result of this investigation.
vi
TABLE OF CONTENTS
Chapter
1.
INTRODUCTION................................................................................................... 1
1.1
1.2
1.3
1.4
1.5
1.6
2.
Page
Background .....................................................................................................1
Crashworthiness ..............................................................................................3
Brief History on Aircraft Crash Safety ...........................................................6
Crashworthy Helicopter Design Principles ................................................... 6
Material Energy Absorption .......................................................................... 8
Study Outline ................................................................................................10
LITERATURE REVIEW ......................................................................................14
2.1
2.2
2.3
2.4
Energy Absorbed per unit mass ....................................................................14
Energy Absorbed per unit volume ................................................................15
Energy Absorbed per unit length ..................................................................15
Compressive Failure Mechanism..................................................................16
2.4.1 Euler Buckling ..................................................................................16
2.4.2 Progressive Folding ..........................................................................16
2.4.3 Brittle Fracture ..................................................................................17
2.5 Practical aspects of metallic structures crashworthiness.............................. 19
3.
EIGEN VALUE BUCKLING ANALYSIS...........................................................20
3.1
3.2
3.3
3.4
4.
Global Buckling ............................................................................................20
Local Buckling..............................................................................................20
Buckling Mode..............................................................................................21
Imposition of Eigen value buckling modes as imperfection.........................22
PARAMETRIC STUDY ........................................................................................24
4.1 Eigen Value Buckling Analyses ...................................................................24
4.1.1 Geometric Modeling .........................................................................24
4.1.2 Material Modeling ............................................................................26
4.2 Finite Element Modeling (Eigen value analysis)..........................................26
4.3 Theory ...........................................................................................................28
4.4 Buckling Modes and Eigen Values...............................................................29
4.5 Parametric Results ....................................................................................................32
4.5.1 Effect of Young’s modulus on Eigen value ..................................................32
4.5.2 Effect of wavelength on Eigen value ............................................................35
4.5.3 Effect of amplitude on Eigen value ..............................................................37
4.6 Conclusion ................................................................................................................38
4.7 Buckling Load...........................................................................................................40
vii
TABLE OF CONTENTS (cont.)
Chapter
5.
FINITE ELEMENT MODELING OF SINE WAVE WEB ..................................42
5.1
5.2
5.3
5.4
5.5
6.
Page
Finite Element Modeling ..............................................................................42
FE Analytical Tools ......................................................................................42
Sine Wave Model..........................................................................................43
Material Properties........................................................................................46
Mesh Independence Study ............................................................................47
RESULTS AND DISCUSSION................................................................................52
6.1 Energy Absorption based on Finite Element Results ...................................52
6.2 Design Curve ................................................................................................55
7.
CONCLUSION.........................................................................................................57
7.1 Recommendations for Future Research ............................................................59
REFERENCES..................................................................................................................60
APPENDIX A. Geometrical Properties for One Wave Web Cross Section……………. 64
APPENDIX B. Deformation of 60° Corrugated Web Specimen……………………….. 66
APPENDIX C. Energy absorption values for Axially Compressed Specimens…………67
viii
LIST OF TABLES
Table
Page
1
Effect of Young’s Modulus on the Eigen values.................................................... 33
2
Effect of wavelength on the Eigen values .............................................................. 35
3
Effect of Amplitude on the Eigen value ................................................................. 37
4
Result of Eigen Analyses........................................................................................ 41
5
Force of Sine Wave models ................................................................................... 49
6
Finite Element Results. .......................................................................................... 53
ix
LIST OF FIGURES
Figure
Page
1.
Fuselage Concept ...................................................................................................12
2.
Subfloor Structure ..................................................................................................13
3.
Different Crushing Modes .....................................................................................17
4.
Load Displacement Characteristics of Progressive crushing.................................17
5.
Coordinate System of Corrugated Plate.................................................................21
6.
Local buckling Mode .............................................................................................21
7.
Local buckling Mode (3D)......................................................................................22
8.
Dimensions of Sine Wave Web ..............................................................................24
9.
Sine Wave modeled in CATIA ...............................................................................25
10.
Meshed Sine wave web..........................................................................................26
11.
Sine Wave Buckling Mode 1 .................................................................................29
12.
Sine Wave Buckling Mode 2 .................................................................................30
13.
Sine Wave Buckling Mode 3 .................................................................................30
14.
Sine Wave Buckling Mode 4 .................................................................................31
15.
Sine Wave Buckling Mode 5 .................................................................................31
16.
Plot of young’s modulus vs. Eigen value...............................................................33
17.
Plot of young’s modulus vs. Eigen value with increase amplitude .......................36
18.
Plot of wavelength vs. Eigen value........................................................................36
19.
Plot of amplitude vs. Eigen value ..........................................................................39
20.
Plot of wavelength vs. Eigen value for different amplitudes.................................39
21.
Plot of amplitude vs. Eigen value for different wavelengths .................................39
x
LIST OF FIGURES (cont.)
Figure
Page
22.
Sine wave web surface...........................................................................................40
23.
Sine wave web in FEM ..........................................................................................44
24.
Ram ........................................................................................................................45
25.
FE model with Normal Mesh.................................................................................46
26.
FE model with Coarse Mesh..................................................................................47
27.
Effect of Mesh size on a 90 o included angle ..........................................................48
28.
Deformation of 90 o Corrugated Web at t = 0ms ....................................................50
29.
Deformation of 90 o Corrugated Web at t = 0.05ms ..............................................50
30.
Deformation of 90 o Corrugated Web at t = 1ms ...................................................51
31.
Plot of Force vs. Displacement for the 90 o specimen from FE study...................53
32.
Design Curve ........................................................................................................55
33.
Geometry of the Cross Section of One Wave Web ...............................................6
34.
Deformation of 60 o Corrugated Web at t = 0ms ....................................................66
35.
Deformation of 60 o Corrugated Web at t = 1ms ....................................................66
xi
CHAPTER 1
INTRODUCTION
1.1 Background
Crashworthy performance has emerged as a very important issue and one of fundamental
importance in the development of new generation of aircraft. Rapid progress in
understanding the structural crashworthiness of land, air and sea vehicles has occurred
over the last four decades since 1959, when Minnorsky [1] published his semi-empirical
method for predicting the damage of ships involved in major collisions. Recently,
crashworthy design has become a standard feature of many other vehicle design
processes, due in part to the realization of potential savings attainable with relatively
minor additions to the basic structure. The need for crashworthiness in the design of
aircraft and automobiles is well documented in the literature. Standard equipments like
MIL-STD-1290[2] provide design criteria for crashworthy performance. But the concept
of designing aircraft structures to sustain “crash” loads is still in its infancy [3]. In the
past, much of the research effort has been directed towards automobiles [4], other ground
transport vehicles, and helicopter crashworthiness. The Federal Aviation Agency in the
U.S performed a series of full scale crash tests on fuselage sections and complete
transport aircrafts. The effort includes analytical and experimental work and the
development of structural concepts. Tests o f t h i s nature are extremely expensive,
especially as the size of the aircraft increases, and provide limited data for a single impact
condition and one aircraft configuration.
1
During the process of design and evaluation of new structural concepts, element tests
are often conducted prior to the fabrication of the entire structure, and small scale
structures are also tested to asses the detail crashworthy performance and material
behavior.
The crash response of metallic or structures can be accurately predicted using
computational methods. The computational tools help in better understanding of the crash
design as different iterations of crushing modes can be studied. Current engineering
workstations have computational speed to analyze crush codes and stress patterns in
detail. LS-DYNA [5], does not need the solving of simultaneous equations as it uses
explicit (small) time step intervals. This eliminates the need to break down large stiffness
matrices. The time interval for an explicit solver is usually in the order of a microsecond.
But even with the small time step size large impact crashworthy models like automobile
structures would need many CPU hours to solve.
Therefore to use similar finite
element codes and solver for aerospace structures extensive analytical validation
would be required.
The objective of thesis is to study the energy absorption behavior of sine wave
webs. The principal objective is to create a suitable theoretical model for understanding
and explaining the energy absorption by doing a parametric study of the sine wave web
specimen.
The end result of this project would be to show the use explicit finite element crushing
codes to study the buckling response of airframe structures. results by Finite Element
Analysis code as LS-DYNA [5], a general purpose finite element code for structural
analysis that is widely used in the automobile and aerospace industries.
2
The MSC.PATRAN pre-processing software was used to with the LS-POST
for post processing the results. A crash simulation was executed and predictions of the
structural deformations and energy absorption of the beam at the sub floor level was
computed. The end result of this project would be to show the use explicit finite element
crushing codes to study the buckling response of airframe structures.
The energy absorption behavior of sine wave webs loaded in axial compression
is to be investigated in theoretical territories. The importance of trigger mechanism
and its effect on energy absorption is also to be observed. A theoretical model
[7] for the energy absorption mechanism in sine wave webs will be studied.
1.2 Crashworthiness
The term “crashworthiness” is generally understood to mean the ability of land, sea or air
vehicles to survive a collision without unacceptable distortion or deceleration of the part
of the structure containing the payload. The problem is therefore essentially in the low to
moderate velocity regime, and the total amount of energy to be absorbed is a major
parameter. Secondary parameters include detailed features of the structure which control
retardation and failure. The term “structural impact” is much broader; it includes many
cases of either low or high velocity impact between objects which may be vehicles,
projectiles, missiles, ground installations, etc.
3
In crashworthiness engineering, the objective in design is to minimize injuries in
the vehicle impact. To an automotive engineer, this involves design of a variety of
features ranging from bumpers (so that negligible car damage occurs at low speeds) to
restraints (so that passengers will survive high speed impact). For the aircraft designer,
this involves a similar systems approach with the landing gear, fuselage structure and
seats working together to absorb the aircraft kinetic energy and slow the occupants to rest
without injurious loading. The amount of protection that is afforded by the aircraft will
depend greatly upon the amount of thought that went crash survivability during the
original design. Crashworthiness analysis is a study of specially designed structures that
dissipate maximum energy within limited space available, while minimizing the peak
loads transmitted to the occupants [7]. The design of crashworthy structure requires
energy to be absorbed by the structures outside the occupied zone in the event of a
collision, thus protecting the people inside. The design of such structures requires that
these structures must be able to absorb large amounts of energy by controlled collapse or
crushing. The approach is to consider and design for the positive interaction of all
elements that will aid in the occupant’s survival. It starts with the design of the aircraft
structure, giving it the capability to absorb energy through controlled deformation.
Since helicopters have a relatively high number of relatively high numbers of takeoff
and landings per flight hour, they are more susceptible to accidents. Therefore,
crashworthiness studies of helicopter structures become more important and crashworthy
design of rotorcraft has become an active research field .The development of structural
concepts to limit the load transmitted to the payload has been studied as apart of crash
dynamics research conducted at the NASA Langley Research Center to determine crash
4
loads and identify structural failure mechanisms during airplane crashes. The objective of
this research was to attenuate the load transmitted by a structure, either by modifying the
structural assembly, changing geometry of its elements, or adding specific load limiting
devices to help dissipate the kinetic energy. Recent efforts in this area have concentrated
on the development of crashworthy subfloor systems. These subfloor systems provide a
high strength structural floor platform to retain the seats and a crushable zone to absorb
energy and limit vertical load by crushing. Energy-absorbing crushable material in the
lower fuselage is being designed to attenuate crash forces, absorb energy, and distribute
the ground impact forces into the fuselage in a manner that will not overstress the floor or
sidewalls [6].
The design crashworthy fuselage [8], seats and restraint systems for transport and
general aviation aircraft presents a complex engineering problem, the solution of which
can be greatly aided by sufficiently rigorous analytical techniques. The crash
environment can vary widely from one accident to another, so a great number of
conditions must be evaluated to establish those critical to occupant survival. A
crashworthy fuselage should include the capacity to absorb energy through controlled
deformation, thus reducing accompanying load [10]. Crashworthiness has traditionally
received little attention in the design of aircraft subfloor compared with the issues such as
comfort, appearance, and weight. Thus lack of attention on crashworthiness
considerations was due in part to the lack of understanding and definition of the response
of an aircraft subjected to a crash event and the lack of definitive or consistent crash
dynamics design standards [12]. However, detailed studies of accident investigation
5
reports established the fact that aircraft passenger survivability could be greatly improved
if crashworthiness was considered initially in aircraft design stage.
1.3 A Brief History on Aircraft Crash Safety.
Crash safety of aircraft, specifically the protection of the occupants of an aircraft
during a crash, has steadily become more important from the inception of the Aeronautics
Bulletins in the late 1920’s. Starting with the requirement of safety belts for occupants
and the requirement that the seats be firmly secured in place, to more recent static crash
load factors, regulations have been implemented to provide reasonable level of occupant
protection in minor crash landings. More recently the aviation industry’s efforts to
improve crashworthiness resulted in the formation of the General Aviation Safety Panel
(GASP). The GASP committee investigated the distribution of aviation related facilities
with the purpose of finding areas to improve crash safety [12].
The results of these studies, found that 68% of all aviation related fatalities occurred with
general aviation aircraft, with commuter aircraft and rotorcraft accounting for 32% of all
fatalities. The GASP committee also considered the frequency of major and fatal injuries
to specific body regions. The committee based a number of their recommendations to the
FAA on the frequency of injuries associated with the fatalities foe helicopters and U.S
Army fixed wing aircraft. The recommendation from GASP was incorporated into FAR
23.562 and 23.562 in form of occupant injuries.
1.4 Crashworthy Helicopter Design Principles
Requirements for airframe crash resistance in aircraft crashes the kinetic energy in the
horizontal direction is mainly absorbed by friction between the sliding structure and the
6
ground, and if possible by a certain amount of soil deformation. A typical fuselage of a
rotorcraft is shown in Figure 1.
A systems approach should be applied whenever possible which comprises the landing
gear, the sub-floor and the high mass retention structure, seat/restraint systems with tuned
EA characteristics, and other cabin furniture (bins, galleys). However, light fixed-wing
general aviation aircraft, small passenger airplanes and helicopters, especially with
retracted landing gear, have little crushable airframe structure. Such designs typically
consist of a framework of longitudinal beams and lateral bulkheads covered by the outer
skin and cabin floor. The total structural height is often only about 200mm. The design of
intersections of beams and bulkheads (cruciforms), the beam webs and floor sections
(boxes) and the bottom parts of the frame structures contribute essentially to the overall
crash response of an airframe sub-floor assemblage, Figure 1.
Under vertical crash loads cruciforms are 'hard point' stiff columns which create high
deceleration peak loads at the cabin floor level and cause dangerous inputs to the
seat/occupant system. Frame and shell structures above the cabin floor are crucial
elements for high mass retention (transmissions, engines, rotor hubs etc.) and for
providing a livable volume in a crash sequence. Plastically deformable frames or side
shell structures of the airframe offer the possibility off load limiting concepts for large
overhead masses.
The basic requirements for crash resistant sub-floors can be summarized as follows:
•
Uniform distribution of ground reaction and seat loads.
•
Limitation of the deceleration forces by structural deformation with controlled
load concept.
7
•
The crushing characteristics of the sub-floor components should have moderate
initial stiffness and then a constant or slightly increasing force level
•
Maintain cabin floor structural integrity.
•
To minimize cost and weight penalties a dual function structural for normal
operation and EA for crash cases.
•
Water impact: outer sub-floor skins between beams and via membrane stresses
without failure to EA elements.
1.5 Energy Absorption.
Metallic structures have a considerable potential for absorbing kinetic energy
during crash. The energy absorption capability offers a unique combination of reduced
structural weight and improved vehicle safety. Crash resistance covers the energy
absorbing capability of crushing structural parts as well as the demand to provide a
protective shell around the occupants (structural integrity).
In aeronautics, especially helicopter and light fixed wing airframe structures crash
worthiness is important due to the existing and increasing application of primary
metallic structures in these aircraft. In recent years the use of Metallic structures
materials in aircraft and rotorcraft design has increased dramatically. In the last
decade several studies have demonstrated the ability of metallic structures to absorb
energy under crashworthy conditions. Some of the energy absorbing metallic structures
materials structural concepts that has been studied experimentally. The specific
energy absorption, post-crushing integrity and energy release of the candidate
materials must be known to match specific design requirements.
8
The specific energy absorption for metallic structures is primarily a function of only their
plastic behavior. To a limited extent, specimen geometry and material property effects
have been reported, with the axially loaded tube as the primary structural configuration.
In spite of the above mentioned considerable work in the intervening years, there
are no reliable standards ways to predict the energy absorption behavior. Perhaps due to
high degree of difficulty in carrying out research in this area, very little effort have been
devoted to gain an in-depth understanding of the crash-impact behavior of metallic
structures
In this context extensive tests are typically used to guarantee the crashworthiness
of metallic structures and to understand the mechanism of energy absorption and failure.
To reduce the cost of a crashworthy structure development program, the use of predictive
design tools such as Finite Element Analysis for simulating the response of metallic
structures under impact and crash loads is found to be vital [18]. Modeling of energy
absorbing structures requires that the predictive design tools be able to characterize the
complex failure events and progressive damage modes. However,
due
to
the
substantial cost of conducting repeated crash tests on large structures, tests are
conducted on sub-structures and representative specimens [19].
Similarly due to the high cost and high computational time involved in modeling
large structures, finite element analysis is carried out for simulating the crash response of
semi-scale structures and specimens. Hybrid finite element models are used to conduct
crash analysis of full scale structures and, to compensate the high computational time.
9
The data obtained from such analysis are used for simulating crash response of substructures. However, the rapid growth in computational memory and, enhanced iterative
solver techniques with no global stiffness matrix inversion has made such complex
simulations feasible [18].
1.6 Study Outline
There are several problem areas in the field of crash-impact dynamics that must
be studied in order to improve the crashworthy design of rotorcraft. To date, there have
been a number of contributions to the understanding and analysis of the energy
absorption characteristics of rotorcraft structures, but most of these tend toward one of
the two extremes. At one extreme is the effort to understand the basic material behavior
and the energy absorption mechanism in detail based on the behavior of simple
metallic structural components such as tubular structures. This can be a very effective
method to determine the precise material information at relatively little cost, and such
tests can be done in the laboratory under well controlled conditions. At the other extreme
one can try to get a physical understanding based on observations of full scale crash
simulations. It is highly realistic and can be used directly as the design basis, but of
course it is a very expensive method and can provide information for only a few limited
impact conditions and design.
In this study the objective is to explore the open fields between the extremes. The
aim is to study the mechanism of energy absorption of metallic aluminum structure and,
by understanding these mechanisms, to predict large-scale rotorcraft structural
crashworthy behavior in order to improve the crashworthy design of rotorcraft. The sine
wave web (or corrugated web) structural component constructed aluminum has
10
been chosen as the specific structural element for this study. It is known from the
previous research on the energy absorption behavior of tubes that the tubular concept is
one of the most efficient energy absorption concepts evaluated, and it is also offers
attractive simplicity from a manufacturing view point. The sine wave exhibits similar
energy- absorption trend as tubes, but it represents a more realistic and efficient
configuration directly usable in design when compared to tube elements.
The large number of variables involved in describing metallic materials makes
it very difficult to completely characterize their energy absorption behavior. An Eigen
value Buckling Analysis is performed to estimate the elastic buckling load of the
corrugated beam and to calculate the first buckling mode by Eigen value extraction using
the finite element code LS-DYNA where an implicit analysis was performed [15].
In elastic structures deformation d e p e n d s o n the applied load. As the applied
load remains constant and the structure continues to warp, this is when the structure starts to
buckle. The point at which the structure starts to buckle gives the eigen
values for the structure. Only the lowest eigen value will need to be calculated to
analyze the shape of the buckled structure.
A parametric study of the corrugated beam was done by varying the following
parameters of the sine wave beam
a) Wavelength
b) Amplitude
c) Young’s Modulus
11
The effect of these parameters on the Eigen value analysis was studied and the buckling
load was calculated. Based on the results of the parametric study i.e. the buckling load an
appropriate model with material properties was selected for the finite element analysis.
The explicit finite element code LS-DYNA was used to simulate the failure behavior of
the sine wave shaped metallic structure. The energy absorption values are predicted from
the finite element study.
Figure 1: Fuselage Concept [2]
12
Figure 2: Subfloor Structure [3]
13
CHAPTER 2
LITERATURE REVIEW
In recent years, there has been a growing realization of the importance of crashworthiness
in virtually every transportation sector. It has been increasingly accepted that traditional
structural design philosophies along the lines of the stronger, the better are far from
optimal when it comes to the protection of passengers in the event of accident. Rather, it
is preferable to design a vehicle to collapse in a controlled manner, thereby ensuring the
safe dissipation of kinetic energy and limiting the seriousness of injuries incurred by the
occupants.
In the last decade several studies have demonstrated the ability of metallic
materials to absorb energy under crashworthy conditions. Some of the energy absorbing
metallic materials structural concepts that have been studied experimentally, especially
for this application, includes sine wave web beams [14]. Moreover due to the complex
failure behavior, energy absorption and crush/crash behavior of metallic structures, there
has been a lack of experience when compared with the metallic structures [6]. In this
context analyses are typically performed on metallic structures such as Aluminum to
study the crashworthiness and to understand the mechanism of energy absorption and
failure.
2.1 Energy Absorbed Per Unit Mass.
The energy absorbed per unit mass, or specific energy absorption, E, is defined as the
energy absorbed by crushing E, per unit mass of deformed structure. Using the notation
of Fig, this can be written as [4]:
14
δ
∫0 Fdx
E
ES =
=
ρδAmat ρδAmat
(1)
For the ease of analysis, Eq.1 is often estimated using an average collapse load, F or an
average collapse stress σ . This approximate E s , given in Eq 2 is sometimes known as
specific sustained crushing stress [4].
ES ≈
σ
F
=
ρAmat ρ
(2)
Specific energy absorption is an especially useful measure for comparing the energy
absorption capabilities of different materials for structures in which weight is an
important consideration.
2.2 Energy Absorbed Per Unit Volume.
The energy absorbed per unit volume will be of interest in situations in which the
space available for energy absorption deformation zone or device is in some way
restricted. It may also be appropriate when mechanisms other than deformation of the
parent material contribute significantly to a structure’s overall energy absorption
capability.
2.3 Energy Absorbed Per Unit Length.
The energy absorbed per unit length, E L is defined as the energy absorbed per unit of
deformation distance. This can be expressed as [4];
EL =
E
δ
(3)
The energy absorbed per unit length provides a convenient and easily measured way of
quantifying the crashworthiness of structures where collapse is restricted to a well
15
defined crumple zone. A relatively straightforward crashworthiness specification such as
this allows for the ready verification of structures through the use of appropriate test
procedures or finite element simulation.
It can therefore be seen that the choice of the most suitable normalized energy absorption
parameter for a given circumstance will depend upon the material and geometry of the
crushed structure, as well as particular application under consideration.
2.4 Compressive Failure Mechanism.
Upon the application of an axial compressive force, there are, according to Farley and
Hull [9], three general ways in which failure can occur. These are:
1) Euler Buckling
2) Shell Buckling, leading to hinge formation and progressive folding.
3) Brittle Fracture.
2.4.1 Euler Buckling
Long thin tubes are likely to fail by Euler buckling before the onset of any other failure
mechanism. The buckling will occur at a local non-uniformity in material or geometry.
Euler buckling gives rise to very low levels of energy absorption as failure is restricted to
a much localized area of the tube material. It is easily avoided by as suitable choice of
tube length, diameter and wall thickness. This is shown in fig 3
2.4.2 Progressive Folding
Progressive folding involves the successive formation of local buckles such that the tube
folds axially in a manner similar to that of a concertina Fig 3. The corresponding load
displacement curve, also shown in the Fig 4, exhibits three main regimes of collapse.
There is an initial linear-elastic response which terminates in a peak load before
16
dropping. This is then followed by a series of oscillations about a mean crush level in
which each oscillation corresponds to the formation of one complete fold in the tube.
Finally, the load rises rapidly as the tube becomes fully crushed.
Figure 3: Different crushing modes [8]
2.4.3 Brittle Fracture
Provided unstable catastrophic failure of the specimen can be avoided by using a collapse
trigger mechanism, the overall failure mode will be one of progressive crushing with very
high level of energy absorption.
Figure 4: Load Displacement Characteristics of Progressive crushing [3]
17
Fig 4 shows a typical load-displacement characteristic for a brittle FRP tube which fails
in this manner. It can be seen that, in contrast to the regular oscillations exhibited by
progressive folding tubes, the load fluctuates in a much more random fashion with
serrations of small amplitude. According to Hull [13], these serrations arise due to the
stick slip nature of the crushing mechanism in which the stresses required to initiate the
crack growth are higher than those required for propagation.
By definition, the energy absorbed during crushing is given by the area under the load
displacement curve. Therefore, for given upper limits on force and displacement, the
energy absorption will be maximized by a rectangular profile. The implications of small
amplitude serrations shown in fig 4 are that of the tubes failing by progressive crushing
approach this idea very closely. Their behavior is certainly superior to that of progressive
folding tubes as shown in fig 3 which exhibit relatively large oscillations in load about a
mean crush level.
Both splaying and fragmentation exhibit analogies with tube inversion, a mode of failure
associated with axial collapse of metal tubes.
Of the two brittle fracture failure modes, there is some evidence to suggest that the
fragmentation mode of failure generally results in higher energy absorptions than the
splaying modes Hamada et al, [7. The superior energy absorption of tubes that exhibited
the fragmentation failure modes was related to the higher failure strain of the matrix used.
However the issue is complicated by the large differences in the areas of the crush zone
generated by the two modes of failure. It can be seen from Fig 6 that the lamina bending
failure mode tends to generate a large crush area and so has higher potential for absorbing
energy by frictional effects at the platen specimen interface.
18
It should be noted that high energy absorption values will only be attained when a tube
crushes in a stable progressive manner. However, as Farley and Jones [9] pointed out, not
all tubes will crush this way. Whether a tube will crush progressively or not will depend
upon the structure of the tube. Should stable crushing in some way inhibited, then
catastrophic unstable failure will occur with much lower energy absorption.
2.5 Practical Aspects of Metallic Structures Crashworthiness.
The above discussion has concentrated largely on the energy absorption properties of
tubular elements as such geometries have been the primary focus of research in this field.
However, while tubes can be considered structurally representative up to a point, it is
clear that broader understanding of the behavior of real structural components is required
Several authors like Hanagud et al [8] have reported energy absorption studies on more
practically applicable geometries for use in automotive and aerospace structures. The
review has demonstrated that metallic structures due to compressive failures can be
designed to exhibit higher normalized energy absorption capabilities.
19
CHAPTER 3
EIGEN VALUE BUCKLING ANALYSIS
The Analysis of the buckling of metallic structures can offer useful information on the
properties and failure characteristics, as necessitated by the variety of their applications.
3.1 Global Buckling
Global buckling of a sine wave web is a three dimensional problem. Buckling of the sine
wave beam is caused by bending, and it first occurs about the principal axis associated
with the least moment of inertia; this axis approaches the z axis as the wavelength
increases. As shown in the Fig 8 the coordinate system is defined with the compression
load in the x direction. It is known that the buckling load for a homogeneous isotropic
column with both ends simply supported (Euler’s column formula) is [7];
π 2 EI
P cr = 2
L
(4)
Where L is the length and EI is the flexural stiffness for the homogeneous isotropic beam.
3.2 Local Buckling.
The local buckling of a sine wave beam is assumed to be based on progressive
development of edge delaminations in the local crush zone and subsequent buckling
collapse of each laminate when a critical load is exceeded. The behavior leads to the local
collapse or crushing of the loaded edge which can lead to the efficient crushing of the
structure.
20
Figure 5: Coordinate System of Corrugated Plate [7]
3.3 Buckling Mode.
When the applied load in the structure remains constant and it starts to warp, this mode is
defined as the buckling mode. Usually the structure bends outward usually taking the shape
of a Zee. The buckling mode usually represents two modes of deformation, the primary and
secondary mode. The secondary buckling mode is usually defined as the point at which the
structure starts to buckle under the applied load. Both the primary and the secondary given
buckling modes have the same laminar energy at bending. A typical buckling mode is shown
below in fig 9.
Figure 6: Local buckling [8]
21
The fig 10 below shows a 3D representation of the local buckling mode shape. Local
buckling occurs at smaller sized time interval as shown here.
Figure 7: Local buckling (3D) [8]
3.4 Imposition of Eigenvalue Buckling Modes as Imperfection (LS-Dyna Online
Manual Version 970)
In elastic structures deformation d e p e n d s o n the applied load. As the applied
load remains constant and the structure continues to warp, this is when the structure starts to
buckle. The point at which the structure starts to buckle gives the eigen
values for the structure. Only the lowest eigen value will need to be calculated to
analyze the shape of the buckled structure.
Eigenvalue buckling analysis is often used to predict the critical buckling load and failure
mode of a structure. In the general Eigen value buckling problem the critical load is given
when the stiffness matrix becomes singular i.e. Kν =0, where K is the tangent stiffness
matrix when the loads are applied and ν are the nontrivial displacement solutions.
22
In an eigenvalue buckling problem, we look for the loads for which the model stiffness
matrix becomes singular, so that the problem;
K MN ν M =0
(6)
has a nontrivial solution. In the equation K MN is the tangent stiffness matrix when the
loads are applied, and ν M is a nontrivial displacement solution.
In LS-DYNA, an incremental loading pattern, Q N , is defined in the beginning of the Eigen
value buckling analysis. The magnitude of this loading is not important; it will be scaled by
the load multipliers, found in the eigen value problem. The buckling mode shapes are often
the most useful outcome of the eigenvalue analysis, since they predict the likely failure mode
of the structure.
23
CHAPTER 4
PARAMETRIC STUDY
In this chapter are the Eigen values buckling analyses of a sine wave web in Ls-Dyna as
well as the parametric study of the sine wave beams to determine the buckling load.
Corrugated beam can buckle in two different ways, globally or locally. Local buckling
occurs when the beam facing buckles between the corrugations and global is when the
entire beam buckles. The theory is the same in the two cases, it is only the geometry of
the structure and the boundary and loading conditions that are different.
4.1 Eigen value buckling analyses.
An Eigen value buckling analyses was done in Ls-Dyna to determine the buckling mode
shapes and thereby determine the buckling load. The specimen buckled locally.
4.1.1 Geometric Modeling
To study the buckling mode shapes, the sine wave web is modeled with the parameters as
shown in the fig 11.
Fig 8: Dimensions of the sine wave web [4]
24
Since the objective is to perform a parametric study of the sine wave web the amplitude
and the wavelength is varied. The amplitude is varied from 1 inch to 3 inch in steps of
0.25 inches and the wavelength is varied from 6 inch to 12 inch in steps of two inch. The
thickness of the sine wave web is kept constant at 0.01 inches. Since the sine wave web is
desired to buckle locally the height of the sine wave is kept constant at 5 inches. The web
configuration was chosen since it is an arrangement that could be directly used in
airframe structures as a part of built up assembly.
The sine wave web was modeled using Catia (Part Design) software. The fig 12 below
shows the sine wave web modeled in Catia with a wavelength of 6 inches; amplitude 1
inch and height as 5 inches.
Figure 9: Sine Wave modeled in Catia
25
4.1.2 Material Modeling.
The material behavior is isotropic, and approximated as linear-elastic. This
approximation is made to reduce the computation time. The Elastic Modulus for the
parametric study is varied. From the results of the parametric study the appropriate elastic
modulus, depending on the buckling load computed from the Eigen value, is taken for the
analysis. A poisons ratio of 0.3 is used for the analysis.
4.2 Finite Element Modeling (Eigen Value Analysis)
Finite element modeling provides the engineer with a powerful tool that consistently
predicts the physical behavior of a particular structural member without having to
conduct numerous laboratory tests. The first step in a FEA is to decide what kind of
elements to use and how dense the FE mesh has to be in order to obtain valid results. As
mentioned above the sine wave web was modeled using Catia as shown in the fig 12. The
model was then imported to the Finite element software Patran for buckling analysis. For
the Eigen value analysis the sine wave web was meshed using Isomesh as shown in the
fig 13 below.
Figure 10: Meshed Sine wave web
26
The material properties are assigned to the sine wave model. The young’s modulus is
varied from 2 Msi to 10 Msi for the parametric study. Since the material is isotopic, shell
elements are chosen for the analysis. The sine wave model is constrained in the following
manner. Since the sine wave web is desired to buckle locally to study the different mode
shapes the constraints for the model are described as follows.
All the nodes at the bottom of the sine wave web are fixed i.e. all the nodes are
constrained in x, y & z directions thereby there is no translational as well as rotational
movement of the bottom nodes. The nodes at the top of the sine wave model are
constrained such that there is no translation and rotational movement in the y & z
direction. All the nodes at the top are left unconstrained in the x direction. This is because
the load is applied in the x direction. The remaining nodes in the sine wave model are left
free for translation and rotational movement in the x, y & z directions. After describing
the boundary conditions the next step is to apply load to the sine wave model. The
objective of doing a parametric study for the Eigen value buckling analysis is to calculate
the buckling load. Since the buckling load is to be determined a reference load of 1lbf is
applied to the sine wave web. This load is applied to only one node at the top of the web
in the x direction. The node to which the load is to be applied can be anyone of the nodes
at the top of the sine wave web. A concept called as multi point constraints, which will be
described later, is used which will distribute the force equally to all the nodes at the top of
sine wave web. After applying the load and the boundary conditions described above the
model is submitted for analysis and a key file is generated.
27
4.3 Theory.
To perform the Eigen value buckling analysis the key file generated from the FEM
software Patran has to be solved using the solver Ls-Dyna. Since an Eigen value buckling
analysis is an implicit analysis implicit cards, such as control_implicit_general,
control_implicit_eigen have to be used in the key file to compute the number of buckling
modes and Eigen values. The implicit method is activated by selecting IMFLAG=1 on the
control_implicit_general and a non zero value for the NEIG flag in the card assigned. By
default the lowest NEIG Eigen values is found. For NEIG>0 Eigen values will be
computed at time=o and Ls-Dyna will terminate. Buckling analysis is performed at the
end of a static implicit solution, the simulation is linear. After the load is applied to the
model, the buckling Eigen value problem is solved.
The lowest n Eigen values and
Eigen vectors are computed. The geometrical stiffness terms needed for the buckling
analysis are automatically computed when the termination time is reached. A curve id is
defined in the control_implicit_general card which gives the solution method as a
function of time. The curve id indicates when to extract the Eigen values and how many
to extract. A curve value of one has been defined for implicit phase. Ad3plot database is
produced for transient solution results. Consecutively d3eigv and eigout database is
produced for each intermittent extraction. The multi point constraint, mentioned in sec
4.2, is defined using the constrained_node_set card. This card defines nodal constraint
sets for translational motion in global coordinates. In this analysis all the nodes in the
nodal set are constrained to move equivalently in the x direction. As mentioned in sec 4.2
a load of 1 lbf is applied on any one of the to nodes of the sine wave model, by defining
28
this card in the key file the nodal displacement in all the nodes will be equal to the
displacement of the node to which the reference load is applied.
4.4 Buckling modes and Eigen value
The key file is solved in Ls-Dyna The Eigen values are written to a text file “eigout”
represent multipliers to the applied loads which give the buckling loads The Eigen
vectors are written to a binary database “d3eigv” represent the buckling mode shapes.
These modes are viewed using LS-PREPOST.
In the figures below the buckling modes for a sine wave beam of 6 inch wavelength, 1
inch amplitude and a young’s modulus of 2 Msi are shown:
Figure 11: Sine wave buckling mode 1
29
Figure 12: Sine wave buckling mode 2
Figure 13: Sine wave buckling mode 3
30
Figure 14: Sine wave buckling mode 4
Figure 15: Sine wave buckling mode 5
The eigen value buckling modes of the sine wave beam from FEA predication is
indicated in the above figures. The number of buckling modes corresponds to the number
31
of eigen values extracted. The height of the sine wave beam is short; therefore local
buckling can be seen in the figures (14-18) above. The wrinkling of the sine wave beam
can also be seen from the above figures. The wrinkles on the side of the sine wave beam
are equally spaced i.e. the distance between the wrinkles are equal. This can be seen in
fig 16. The magnitude of the buckling modes i.e. the Eigen values is very important in
determining the buckling load for the sine wave beams.
A parametric study is carried out on sine wave beams taking the local buckling in to
consideration. The influence of wavelength, amplitude of the sine wave beam is
considered by applying the Eigen value buckling analysis and how the buckling load
varies with respect to these parameters.
4.5 Parametric Results.
The parametric study is conducted by doing an Eigen value buckling analyses on the sine
wave beam, fig 13, by varying the following parameters:
a) Wavelength of the sine wave web
b) Amplitude of the sine wave web
c) Young’s modulus of the material
The objective is to see how these parameters effect the eigen values, from the eigen value
analysis, which in turn effect the buckling load. From the parametric study the optimum
value of the wavelength, amplitude and the young’s modulus is to be selected depending
on the buckling load determined from the eigen values.
4.5.1 Effect of Young’s Modulus on Eigen Value
To study the effect of young’s modulus on the Eigen values, obtained from the Eigen
value buckling analysis, the young’s modulus is varied from 2Msi to 10Msi in steps of
32
1Msi. The wavelength of the sine wave beam is kept constant at 6 inch, amplitude at 1
inch, thickness at 0.01 inch and the height at 5 inch. For the same value of
the wavelength, amplitude, thickness and height the young’s modulus is varied and
the different Eigen values are recorded from the Eigen value analysis as show in table 3.
WAVELENGTH
HEIGHT
(in)
AMPLITUDE
(in)
6
6
6
6
6
6
6
6
6
5
5
5
5
5
5
5
5
5
1
1
1
1
1
1
1
1
1
YOUNGS
MODULUS
(msi)
2
3
4
5
6
7
8
9
10
THICKNESS
(in)
EIGENVALUE
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
1.93E+04
5.06E+04
6.74E+04
8.43E+04
1.01E+05
1.18E+05
1.35E+05
1.52E+05
1.69E+05
Table 1: Effect of Young’s Modulus on the Eigen values
The values of the young’s modulus are plotted against the Eigen values obtained from the
analysis. The plot of young’s modulus versus Eigen values is shown in fig 16.
Figure 16: Plot of young’s modulus vs. Eigen value
33
From the plot shown in the figure above it can be inferred that for a given value of
wavelength and amplitude with the increase in young’s modulus the Eigen value
increases.
To substantiate the results more another plot of young’s modulus and Eigen value can be
seen in the fig 20, where the wavelength is kept at 6 inches but this time the amplitude is
increased to 1.5 inches. Again it can be seen from the plot that the increase in young’
modulus increases the Eigen value irrespective of the wavelength and the amplitude.
Even though in this case the wavelength was kept same as in the previous case and the
amplitude increased, a higher young’s modulus predicted a higher Eigen value from the
analysis. The plot is as shown in the fig 20 below. It can be seen that either with higher or
a lower values of wavelength and amplitude the increase in young’s modulus increases the
Eigen value.
34
4.5.2 Effect of Wavelength on Eigen Value
To study the effect of wavelength on the Eigen values, obtained from the Eigen value
buckling analysis, the wavelength is varied from 6 inch to 8inch in steps of 2 inches. The
amplitude of the sine wave beam is kept constant at 1 inch, young’s modulus at 2 Msi,
thickness at 0.01 inch and the height at 5 inch. For the same value of the young’s
modulus, amplitude, thickness and height the wavelength is varied and the different
Eigen values are recorded from the Eigen value analysis.
WAVELENGTH
(in)
HEIGHT
(in)
AMPLITUDE
(in)
6
8
10
12
5
5
5
5
1
1
1
1
YOUNGS
MODULUS
(msi)
2
2
2
2
THICKNESS
(in)
EIGENVALUE
0.01
0.01
0.01
0.01
6.77E+04
4.98E+04
3.94E+04
2.94E+04
Table 2: Effect of wavelength on the Eigen values
The values of the wavelength are plotted against the Eigen values obtained from the analysis.
The plot of wavelength versus Eigen values for amplitude of 1 inch is shown in fig 18.
35
Figure 18: Plot of wavelength vs. Eigen value
From the plot shown in the figure above it can be inferred that for a given value of
amplitude, thickness and young’ modulus the increase in wavelength decreases the Eigen
value.
36
4.5.3 Effect of Amplitude on Eigen Value
To study the effect of amplitude on the Eigen values, obtained from the Eigen value
buckling analysis, the amplitude is varied from1 inch to 3 inch in steps of 0.25 inches
To study the effect of amplitude on the Eigen values, obtained from the Eigen value
buckling analysis, the amplitude is varied from1 inch to 3 inch in steps of 0.25 inches.
The wavelength of the sine wave beam is kept constant at 8 inch, young’s modulus at 2
Msi, thickness at 0.01 inch and the height at 5 inch. For the same value of the young’s
modulus, wavelength, thickness and height the amplitude is varied and the different
Eigen values are recorded from the analysis.
YOUNGS
MODULUS
(msi)
WAVELENGTH
(in)
HEIGHT
(in)
AMPLITUDE
(in)
THICKNESS
(in)
8
5
1
8
5
1.25
2
0.01
1.62E+04
8
5
1.5
3
0.01
1.91E+04
8
5
1.75
4
0.01
2.22E+04
8
5
2
5
0.01
2.48E+04
8
5
2.25
6
0.01
2.67E+04
8
5
2.5
7
0.01
2.79E+04
8
5
2.75
8
0.01
2.86E+04
8
5
3
9
0.01
2.90E+04
EIGENVALUE
1.35E+04
Table 3: Effect of Amplitude on the Eigen value
The values of amplitude are plotted against the Eigen values obtained from the analysis.
The plot of wavelength versus Eigen values for a wavelength of 8 inch is shown in fig 22.
The values of amplitude are plotted against the Eigen values obtained from the analysis. The
plot of wavelength versus Eigen values for a wavelength of 8 inch is shown in fig 22. From
the plot shown it can be inferred that for a given value of wavelength, thickness and young’
modulus the increase in amplitude increases the Eigen value.
37
Figure 19 Plot of Amplitude vs. Eigen value
4.6 Conclusion
The plots shown in fig 20 and fig 21 summarize the results of the parametric study. It can
be seen that for a given value of amplitude and thickness of the sine wave web
an increase in the wavelength decreases the Eigen value.
Conversely for a given value of wavelength and thickness of the sine wave web an
increase in the amplitude increases the Eigen value. Also for a given value of the
wavelength and amplitude of the sine wave web an increase in young’s modulus
increases the Eigen value. Since a high value of young’s modulus resulted in a very high
Eigen value from the analysis the least value of the young’s modulus which is 2 Msi was
considered for the parametric study of wavelength and amplitude. The plots of amplitude
and wavelength against the Eigen value are shown below:
38
Figure 20: Plot of wavelength vs. Eigen value for different amplitudes
amp v eigenvalue
1.20E+05
1.00E+05
eigenvalue
8.00E+04
wavelength=6'
6.00E+04
wavelength=8'
wavelength=10'
wavelength=12'
4.00E+04
2.00E+04
0.00E+00
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
amplitude
Figure 21: Plot of amplitude vs. Eigen value for different wavelengths
39
4.7 Buckling Load
The main objective of performing an Eigen value analysis is to calculate the buckling
load. The Eigen values obtained from the analysis are used to calculate the buckling load.
As mentioned in section 4.2 a reference load of 1lbf which was applied to the sine wave
web for performing the Eigen value buckling analysis is used to calculate the buckling
load. The equation used to calculate the buckling load is as shown [14]:
N= (1/S)*Eigen value
Where;
N=buckling load
1=reference load
S=perimeter of the sine wave web.
To calculate the perimeter of the sine wave web the formula for the perimeter of a
parabolic section was considered as shown in the fig 25.
Figure 22: Sine wave web surface
The perimeter of the sine wave web is calculated using the following equation [22]:
Where:
a= amplitude of the sine wave; b=wavelength of the sine wave
40
From the results of the parametric study (section 4.6) an optimum Eigen value was
considered. The Eigen value was obtained from the analysis of a sine wave web, with a
wavelength of 6 inch, amplitude of 1 inch, young’s modulus 2Msi, thickness 0.01 inch
and a height of 5 inches. These values were selected from the parametric study done on
the wavelength, amplitude and young’s modulus of the sine wave web. The parametric
values gave an optimum result from the Eigen analysis that could be used to calculate the
buckling load, which can then be validated by FE results.
Substituting the values of the wavelength and amplitude of the sine wave web in equation
the buckling load is calculated. The result is as shown in table 4.
WAVELENGTH
(inches)
HEIGHT
(inches)
AMPLITUDE
(inches)
EIGENVALUE
S
(inch)
N
(lb/inch)
6
5
1
9.82E+03
6.42
1.53E+03
Table 4: Result of Eigen Analyses
A buckling load of 1500lb is predicted from the study for a specimen height of 5 inches.
41
CHAPTER 5
FINITE ELEMENT MODELING OF SINE WAVE WEB
5.1 Finite Element Modeling.
Plastic deformation is a mode of energy absorption for metallic structures which can be
shown by finite element modeling with relative ease. Metallic structures also use plastic
deformation to absorb energy but showing it in a finite element model is difficult. There
is on going validation to distinguish the energy absorption of metallic structures using
finite element analysis.
To evaluate crashworthiness using finite element analysis a
momentary dynamic solution at short time intervals is required.
Crashworthiness can be shown in a finite element model by explicit or implicit
techniques. The implicit and explicit techniques differ in their time step prerequisite. The
explicit solution is constant only at small time steps at large natural frequencies. The
explicit solution is stable at large time intervals and does not depend on the time step size.
Even though the explicit solution uses large time intervals it is still preferred for short
momentary solutions. The explicit method has the advantage of reducing computational
time as it does not need solving instantaneous equations.
5.2 FE Analytical Tools
The crash response for metallic structures is determined using finite element methods. To
determine the buckled modes and shapes of metallic structures they are broken down in to
smaller element size for analysis.
42
To analyze the buckled behavior and modes of metallic structures, which exhibit nonlinear material and geometric properties, finite element software is used which until
recently, were not commonly available.
MSC PATRAN & LS-DYNA are examples of finite element software tools that are used in
applications such as aerospace and automotive for predicting crashworthiness. The
advantage of using LS-DYNA is in the fact that it has material models which aids in
analyzing metallic structures. LS-DYNA was selected for the current modeling and
analyzing efforts. To gain an initial understanding of the working of the buckled model
in LS-DYNA, a s i m i l a r model from “ Reference 13” is studied. The sine wave web
is then modeled using the dimensions from the parametric study and the buckling
load aft er wh i ch i t is validated. Then the aluminum sine wave web model is used to
predict the crush energy absorption for comparative study.
5.3 Sine Wave Model
The buckling load predicted from the parametric study is analyzed using LS-DYNA. The
specimens are elasto-plastic model of Aluminum. The specimen crossection geometry is
composed of tangentially joined circular arc segments as shown in fig 11 which represent
the sine wave web specimens. The sine wave web geometry is closer in form to an actual
structural element.
43
For the finite element study elasto-plastic model of Aluminum has been used. The
specimens, as mentioned in the parametric study (Chapter 4), were 5 inch high and 2
waves wide with a wavelength of 6 inch. The thickness of the specimens was 0.01 inch
with an R/t ratio < 10 for local buckling. The specimens were crushed under quasistatic
conditions at uniform rate.
The crushing energy absorption and the failure behavior of the sine wave web were
obtained. The present FE modeling was conducted for a two wave web specimen, for
different included angle of the web. The load predicted from the FE study is compared
with the load obtained from the parametric study. The average crushing predicted from
the FE study along with the maximum force (buckling load) is studied .The Finite
Element Analysis done on the sine wave web is a velocity (displacement) based analysis.
The FE model consists of a corrugated beam and a ram as shown in the fig 23 and fig 24.
Figure 23: Sine wave web in FEM
44
Figure 24: Ram
A two wave corrugated beam is analyzed. A velocity/displacement motion is defined for
the ram such that it simulates experimental conditions. The sine wave web is modeled as
a shell element. The Sine wave beam consisting elasto-plastic model of Aluminum is
analyzed. The degrees of freedom of the bottom nodes of the sine wave beam are
fixed such that there is no translation and rotation motion in the x, y and z
directions i.e. w(x,y,L) = 0, ∂w/∂x(x,y,L) = 0 The ram, which is a rigid body, is
modeled as a solid element. A velocity motion in the X-translation direction is imposed
on the ram using the boundary prescribed motion card. A surface to surface contact is
defined between the sine wave web and the ram using the contact card. The softening
factor as defined before is estimated to be 1. The sine wave model with 90 o included
angle is analyzed with an initial velocity of 2in/ms. The value of 2in/ms for
velocity is arbitrary chosen for
crushing simulation
45
5.4 Material Properties
In this study, the material used is Aluminum:
E = 1.0152 e7 lbf/in 2
σ y = 36012.86 lbf/in 2
ν 12 = 0.33
Mass Density = 2.64e-04 lb/in 3
.
Figure 25: FE model with Normal Mesh
46
Figure 26: FE model with Coarse Mesh
5.5 Mesh Independence Study
The appropriate mesh size for the analysis is determined by running repeated simulations
with normal and coarse mesh sizes . First the model with 90 o included angle is studied
with coarse mesh, fig 28, and subjected to crushing simulation. The results show
variations with peak to valley of 2000lb, at which the sine wave crushes due to
localized buckling. A second model of 90 o included angle is studied with normal mesh,
fig 27. There is an improvement due to mesh refinement (Normal mesh) in
the force curve for peaks and valleys. The result still shows a force of 2800lb, at which
the sine wave crushes, due to mesh refinement.
Force generated due to crushing increases with the reduction in element size. The
damage model in LS-DYNA uses a crash front procedure to represent the
47
crushing of metallic structures. The FE model with normal mesh is observed to have
slightly higher percentage of failed shell elements when compared with coarse mesh for a
90 o included angle. Normal mesh was chosen for analysis considering that SSCS results
for coarse mesh was essentially the same.
Fig 27: Effect of Mesh size on a 90 o included angle
The FE models used to establish the mesh for sine wave models are analyzed for crushing
simulation with an initial velocity of 2in/ms. The normal mesh was selected based on the
SSCS value comparison between the mesh sizes as described above. The displacement
rate of 2in/ms was selected. FE models are generated with different included angle (60 o
and 90 o). The force generated by the crushing of the sine wave with an included angle of
90 o is very close to the buckling load predicted from the Eigen analysis. Similarly model
48
with 60 o included angle subjected to crushing simulation. The force generated by the
crushing of the sine wave (due to velocity motion imposed on the ram) 800lb.
The force due to the crushing of the sine wave for 60 o included angle specimen is less
when compared to the specimen with 90 o included angle. The peak load values of the sine
wave webs and the average crushing force obtained from the simulation of the sine wave
webs is given in table 5.
The deformation of all models is due to the local buckling mode, more than the element
deletion that is expected from the damage model. The deformation of for 90 o included
angle are shown in the figures 28-30.
νo
in/msec
Included
Angle
Peak Load.
lb
60 o
2
800
90 o
2
1900
Table 5: Force of Sine Wave models
Also the force due to the crushing of the sine wave beam with a 60 o included angle is less
when compared to the force (buckling load) calculated from Eigen value analysis. It can
be said that for a lower included angle of the web the load decreases and also the specific
sustained crushing stress.
49
Figure 28: Deformation of 90 o Corrugated Web at t = 0ms
Figure 29: Deformation of 90 o Corrugated Web at t = 0.05ms
50
Figure 30: Deformation of 90 o Corrugated Web at t = 1ms
51
CHAPTER 6
RESULTS AND DISCUSSION
6.1 Energy Absorption based on finite element Results
FE simulation of the crushing is attempted using LS-DYNA’s piecewise linear plasticity
model. The failure of shell elements and crash front elements were expected to play a
vital role in simulating failure modes. However the buckling behavior dominated
the FE models with lesser tendency towards formation of crash front elements. The
included angle of the web is an important geometric parameter as it defines the
effective width to thickness ratio of the web. The flat plate geometry can be considered to
be one extreme with zero amplitude. It has been established that tube type specimens
exhibit high energy absorption, and because of sufficient similarity in geometry between
circular cross-sections tube specimens and sine wave (tangent half circle) beams. The
energy absorption values obtained from the FE study show that the energy absorption for
the 60 o specimens is negligible when compared to the 90 o specimens as shown in Table 5.
The failure mode is local buckling and of the same type as studied for the flat
plate specimens. This suggests the existence of stability boundary in the range of
60 o-90 o included angle (for the study performed) where the failure mode switches from
efficient crushing to an inefficient local failure. This result is important for design
purposes in order to avoid inefficient energy absorption performance. The FE models
triggered only at loading end. From the observations of the FE model results the
energy absorption values tend to increase with the increase in included angle. The
difference can also be attributed to the use of default values for parameters in the
material model.
52
and displacement rates used for the FE analysis. The failure initiator in FE models was
observed to be of the buckling type.
Table 6: Finite Element Results
Figure 31: Plot of Force v/s Displacement from FE study
53
The Force v/s Displacement graph for the 90 degree Aluminum model is shown in Fig 31 above.
It can be seen from the above graph that the failure of the Aluminum model represents a ductile
type of failure. The peak crushing load for the specimen is seen at 1900 lbf (8.45 kN) at a given
displacement of 0.05 inch. The peak load then drops to 600 lbf which is then followed by a series
of oscillations about the mean crushing load of 900 lbf
54
6.2 Design Curve
Finite element analysis is carried out on the corrugated web specimens for included
angles 75 o and 120 o. The analysis justifies the results obtained from the finite element
method for the included angles of 60 o and 90 o. The remaining parameters i.e. the
wavelength and amplitude is the same for these specimens. The peak load predicted for
the 75 o sine wave specimen is more when compared to the peak load predicted for the 60 o sine
wave specimen but less than the load predicted for the 90 o corrugated web specimen. The plot
shows that the 120 o sine wave specimen predicts a load of 5200 lbs. The peak load predicted
for the 120 o sine wave specimen is more when compared to the peak load
predicted for the 90 o sine wave specimen.
6000
5000
Force (lb)
4000
3000
2000
1000
0
15
30
45
60
75
90
105
120
Included Angle
Figure 32: Design Curve (Included angle vs. Peak Load)
Based on the finite element results available for the 60 o, 90 o, 75 o &120 o corrugated web
specimens a design curve is plotted. The design curve is a plot of the included angle
55
versus the peak load predicted by these specimens from the finite element study. It can be
inferred from the curve that as the included angle increases the peak load also increases.
Thus it can be said that a corrugated web specimen with a lower included angle predicts a
peak load which is less than that predicted by a corrugated web specimen with a higher
included angle.
56
CHAPTER 7
CONCLUSIONS
In this study, the objective was to investigate the mechanism of energy absorption in
metallic structures so that by using the understanding of these mechanisms, designers
will be able to predict large scale rotorcraft structural crashworthy behavior, and
therefore improve the crash worthy design of the rotorcraft.
The sine wave web (or corrugated web) structural element constructed of aluminum was
chosen for examination in this study. It represents a realistic configuration directly
usable in practical structural designs, and also it is a simple metallic structural
component from which basic material behavior and the energy absorption mechanisms
can be studied in detail.
One of the important methods for achieving high energy absorption performance with
ductile material like aluminum is the stabilization and triggering of an effective failure
mode. Thus, a relatively small loss of load carrying capacity can be traded for large
gain in energy absorption capability.
A FE study of the energy absorption behavior of aluminum sine wave webs was done
using the finite element software code LS-DYNA. The finite element study demonstrated
that sine wave webs (90 o included angles) can provide nearly the energy absorption
capability of tubular specimens, while exhibiting superior characteristics such as ease of
construction and space efficiency. The included angle of the web is an important
geometric parameter since it defines an effective width to thickness ratio of the web. A
stability boundary was detected in the 60 o - 90 o interval, and an abrupt transition from
57
the global buckling to the local crushing mode. This result is important for design
purposes in order to avoid inefficient energy absorption performance.
Also the force due to the crushing of the sine wave beam with a 90 degree included angle is
slightly higher when compared to the force (buckling load) calculated from Eigen value.
analysis. It can be said that for a higher included angle of the web the load increases the
specific sustained crushing stress of the specimen.
Knowledge of metallic structures and their failure behavior under impact and
quasistatic conditions is necessary to model and simulate the crashworthiness of
metallic structures. The specific compressive failure modes observed in crushing
have a direct influence on the energy absorption of metallic structures. The Failure
modes observed in crushing consist of compressive failure mechanism.
Energy Absorption values obtained from the FE calculation could not be associated with
the failure behavior observed in the FE models since the failure mode observed in the FE
models were predominantly of compressive failure type. This indicated a deficiency of
operation in the crash front procedure in the model.
58
7.1 Recommendations for Future Research
As indicated in Chapter 1 there are several problems in the field of crash-impact behavior
of metallic structures . They must be solved in order to improve the crashworthy design
of rotorcraft that require further studies to quantify the use of such parameters..
A detailed study on the effects of different loading conditions such as low
constrained velocity would be required to make sure that the SSCS values obtained from
the finite element calculations are accurate. A study on procedures like boundary prescribed
motion instead of rigid wall method would be required to confirm whether the procedure
has any effect on the buckling behavior of the FE models.
59
REFERENCES
60
LIST OF REFERENCES
[1]
Saczalski K., Singely G. T. III, Pilkey W.D. and Huston, R.L., “Aircraft
Crashworthiness”, Univ. Press of Virginia, Charlottesville, 1975.
[2]
Thomson, R.G. and Hayduk, R.J., “Light Aircraft Crash Safety Program,”
Proceedings of the 1974 SAE Business Aircraft Meeting , Wichita,
Kansas, 1974.
[3]
Bannerman, D.C. and Kindervater, C.M., “Crash Impact Behavior of
Simulated Composite and Aluminum Helicopter Fuselage Elements,”
Proceedings of 9th European Rotorcraft Forum, 1983.
[4]
Weiyu, Z., “Crash-Impact Behavior of Graphite-Epoxy Composite Sine
Wave Webs,” PhD. Thesis, Department of Aerospace Engineering,
Georgia Institute of Technology, November 1989.
[5]
Hanagud, S., Craig, J.I., Schrage, D. and Sriram, P., “Crashworthy Design
of Rotorcraft: A Basic Research Approach,” American Helicopter Society
41st Annual Forum Proceedings, 1985.
[6]
McCarthy, M.A., Harte, G., Wiggenraad, J.F.M., Michielsen. A.L.P.J.,
Kohlgruber, D., and Kamaolakos, A., “Finite Element Modeling of Crash
Response of Aerospace Sub-floor Structures, Computational Mechanics,
Vol. 26, 2000, pp. 250-258.
[7]
Anon., LS-DYNA Keyword User’s Manual, Nonlinear Dynamic Analysis
of Structures, Livermore Software Technology Corporation, May 1999.
61
LIST OF REFERENCES (cont.)
[8]
Farley, G.L., “Crash Energy Absorbing Sub-Floor Structures” 27th
AIAA/ASME/ASCE/AHS Structures Dynamics and Materials Conference 1986.
[9]
Johnson, A.F., Kindervater, C.M., Kohlgruber, D., and Lutzenburger, M.,
“Predictive Methodologies for the Crashworthiness of Aircraft Structures”
Proceedings of the American Helicopter Society 52nd Annual Forum,
Washington DC, June 4-6, 1996.
[10]
Bannerman, D.C. and Kindervater, C.M., “Crashworthiness investigation of
Aircraft Subfloor Beam Sections,” Structural Impact and Crashworthiness,
Vol. 2, Pub. Elsevier Applied Science Publishers, 1984.
[11]
Yin, W.L., “Cylindrical Buckling of Laminated and Delaminated Plates,”
27th SDM Conference, Paper No. 86-0883, 1986.
[12]
Jones, R.M., Mechanics of Composite Materials, Mc-Graw Hill, New
York, 1975.
62
APPENDICES
63
Appendix A
Geometrical Properties for One Wave Web Cross Section
For one wave sine wave web with included angle α , the geometry of cross section is
shown in Figure 33.
The normalized cross section area and moments of inertia are listed following:
A ∗ = ∫ ds = 2R α
∫z
I ∗y =
2
ds
c
π +α
2
∫
= 2R 3
(sin
π −α
2
α 1
α
− sin α + α sin 2 )
2 2
2
= 2R 3 (
I z∗ =
∫y
2
α
+ cos θ ) 2 d θ
2
ds
c
π +α
2
= 2R 3
α
(sin θ − cos ) 2 dθ
2
π −α
∫
2
= 2R 3 (
I ∗yz =
∫ zy
α
α 3
+ α cos 2 − sin α )
2
2 2
ds
c
π +α
2
= 2R 3
∫
(sin
π −α
2
= 2R 3 (2 sin 2
α
α
+ cos θ )(sin θ − cos ) d θ
2
2
α α
− sin α )
2 2
64
Figure 33: Geometry of the Cross Section of One Wave Web [4]
65
Appendix B
Deformation of 60 o Corrugated Web Specimen
Figure 34: Deformation of 60 o Corrugated Web at t = 0ms
Figure 35: Deformation of 60 o Corrugated Web at t = 1ms
66
Appendix C
Energy absorption values for Axially Compressed Specimens
Table 8 shows examples of energy absorption values recorded for axially compressed
FRP and metal tubes.
Fiber
Lay-up
Matrix
Thickness to E s
Outside
Er
Et
Ref
(kj/kg) (MJ/m 3 ) (kJ/m)
Diameter
Ratio
Carbon-
[0/ ± 15] 4
0.033
99
20
26
Farley(1983)
[ ± 45] 4
0.021
50
6
8
Farley(1991)
[ ± 45] 4
0.066
60
22
19
Thornton(1979)
[0/ ± 15] 4
0.020
9
1
1
Farley(1983)
-
0.070
45
92
12
Thornton(1979)
Epoxy
CarbonEpoxy
AramidEpoxy
AramidEpoxy
1015 Steel
&
Magee(1977)
6061 Al
-
0.070
60
42
6
Thornton(1979)
&
Magee(1977)
67