Buckling_Carbon_nanotubes

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Summer Research Internship
Report
Study of Bimodal Behavior of
Buckling Load for Carbon Nanotubes
By Anshul Goyal
Pre-Final Year B.tech Civil Engineering
Indian Institute of Technology, Guwahati
Under the guidance of
Dr. Debraj Ghosh
Indian Institute of Science, Bangalore

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Acknowledgements
My sincere thanks to Dr. Debraj Ghosh who guided me in every step of whatever I could do
during these 2 months of Summer Internship. Despite the fact that I knew very little about the
field of Nano mechanics which is a very recent and modern field in Engineering, he never let me
down and encouraged and discussed every single doubt & query of mine. He always stressed on
the fundamentals which really helped me to understand the problem and its scope. Besides this
he was always very humble and kind to me. I am thankful to him that he gave me this
opportunity and it was indeed a learning experience.
I would also like to acknowledge Anoop Krishnan (Phd student under Dr. Debraj Ghosh),
Srikara P (another Phd student under Dr. Ghosh) for his help to make me understand about the
carbon nanotubes and the broad scope of the field of Nano mechanics. His talks on philosophy
and life were very motivating. My thanks to Vinay Damodaran (presently got admission to
University of Michigan, Ann Arbor) for his help regarding Python scripting used with Abaqus. I
would also express my thanks to Abhinav(Phd student under Prof Manohar) who was always
ready to help with any programming stuff & Shubhayan Dey (Presently got admission to
University of Southern California) for his motivation regarding scope of research.
Last but not the least my very sincere thanks and regards to my parents and my colleagues
(Summer Interns) who were just ‘fabulous’. I liked interacting with Abhishek(NIT Durgapur),
Somashish (IIT Kharagpur), Subham Rath (Jadhavpur University), Sabyasachi Ghosh(
presently got admission in Aalto University Finland) , Vedhus (BMS College Bangalore)
particularly his GRE tips to me, my roomie (in my PG at Bangalore) and my batch mate at IITG
Pratik raj with whom I often used to share my day to day experiences and my new Canadian
friends Lisa, Nicole and Harrison from University of Western Ontario, Canada.
It was a very pleasant stay here and was a nice learning experience.

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Contents
List of Symbols
List of figures
Scope
• About carbon nanotubes
Overview of Buckling & Stability
• Concepts of stability & Buckling
• Types of Stability
Matlab Simulations
• Study of three Spring, Rigid bar Models
• Load v/s Displacement plots (Perfect and Imperfect Structures)
• Energy Branches (Perfect & Imperfect Structures)
Monte Carlo Simulations
• Probability Density Function by varying Imperfections
• Several probability density functions by varying certain parameters
Abaqus Simulations
• Old Simulations using constant Elastic Modulus
• New Simulations having varying elastic modulus from MD Simulations
• Comparison Plots
• Imperfection Modeling
Calibrating the Asymmetric model with thin Shell model
• Introduction of Geometric Non Linearity
References

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List of Symbols
ψ: Angle of Rotation
E : Elastic Modulus
h: Thickness of the Shell
µ: Poisson’s Ratio
List of Figures
Fig1.1: Diagram showing chiral vector and chiral angle
Fig 2.1 Symmetric Stable Model
Fig2.2 Load V/s Displacement for Case 1(Perfect Structure)
Fig2.3: Load V/s Displacement for Case 1(ImPerfect Structure)
Fig2.4 Symmetric Unstable Model
Fig2.5: Load V/s Displacement for Case 2(Perfect Structure)
Fig2.7: Load V/s Displacement for Case 3(Perfect Structure) for alpha = 45 degrees
Fig2.9 Load V/s Displacement for Case 3(ImPerfect Structure) for alpha = 45 degrees
Fig2.10: Load V/s Displacement for Case 3(ImPerfect Structure) for alpha = 54 degrees
Fig2.11: Stable and Unstable branches for each of the Cases
Fig 2.12: Perfect Symmetric Stable Potential Energy as a function of ψ & Load
Fig 2.13: Perfect Symmetric Unstable Potential Energy as a function of ψ & Load
Fig 2.14: Perfect Asymmetric Potential Energy as a function of ψ & Load

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Fig3.1: Pdf for Buckling Stress when imperfections varied negatively between [-0.1, 0]
Fig3.2: Pdf for Buckling Stress when imperfections varied negatively between [0, 0.1]
Fig3.3 pdf showing variation in modal behavior by varying alpha
Fig3.4: pdf showing variation in modal behavior by varying Stiffness
Fig3.5: Variation of Imperfections, Alpha, Stiffness varying one at a time and others constant
Fig4.1: Finite Element Model of Carbon Nanotube
Fig4.2: FEM Model of the failure mode of Shell
Fig4.3: Relative Frequency Plots from the Abaqus Simulation a) Between 10-15 Angstrom b)10-
20 Angstrom c) the whole diameter range
Fig 4.4: Buckling Stress v/s Diameter from Abaqus Simulations using constant Elastic Modulus
Fig4.5: Comparison of Abaqus Simulation with Thin Shell theory
Fig 4.6: Comparison of Abaqus Simulation with Thin Shell theory
Fig 4.7: New Simulation Results a) Frequency Plots and b) Pdfs
Fig 4.8 FEM Model of an Imperfect Shell
Fig 5.1: Thin Shell, Abaqus and Calibration Results
Fig5.2: Calibration Model comparison with slight non-linearity dramatically increasing the
potential energy at the onset of the buckling.

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1 Scope of the Project
Carbon nanotubes are nanostructured carbon materials having large aspect ratios, extremely high
Young’s modulus and mechanical strength, as well as superior electrical and thermal
conductivities. These were discovered in 1991. Incorporation of a small amount of carbon
nanotube into metals and ceramics leads to the formation of high performance and functional
Nano composites with enhanced mechanical and physical properties [1]. Carbon nanotubes are
formed by rolling Graphene sheets of hexagonal carbon rings into hollow cylinders. The length
of nanotubes is in the range of several hundred micrometers to millimeters. The Graphene sheets
can be rolled into different structures, that is, zig-zag, armchair and chiral. The nanotube
structure can be defined by the chiral vector. The structure can be expressed by n, m and chiral
angle, θ. When n = m and θ =30 degrees, an armchair structure is produced. Zig-zag nanotubes
can be formed when m or n =0 and θ = 0 degrees while chiral nanotubes are formed for any
other values of n and m, having θ between 0 and 30 degrees [2].
Fig1.1 Diagram showing chiral vector and chiral angle [3]
Carbon nanotube is an amazing material with various astonishing material and electrical
properties. Berber et al. demonstrated that CNTs have an unusually high thermal conductivity on
the basis of MD simulation. The electrical properties of CNTs vary from metallic to semi-
conductor depending on the chirality and the diameter of the CNTs. There has been no book
published that deals exclusively with the fundamental issues and properties of carbon nanotube-
reinforced metals and ceramics. Hence this is a new and modern area of research.
CNTs are the best material known to man because of its extraordinary mechanical, Electrical and
Thermal properties. CNTs has a modulus of elasticity of 1000Gpa which is 5 times more than the
value of the steel and strength can be as high as 63 Gpa which is 50 times the strength of the
steel. These can also recover from large strains and can withstand high bending deformation
without any residual strain after load removal. It also improves the weakness of the matrix.

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Addition of CNTs to polymer increase the strength as well makes the polymer a conductor.
Properties of the CNTs can be studied using Molecular Dynamics or molecular structural
mechanics. But the main drawback is that the MD Simulations are computationally expensive
and fails to model very large number of atoms. This is an extremely new field with ample scope
of research. The PhDs under whom I worked are working on diverse fields in Nano mechanics.
My scope of work was to study the reason behind the bimodal behavior of Buckling Stress
observed from the Molecular Dynamics simulation and to understand the mechanics behind the
bimodality. Hence it started by studying simple spring and rigid bar systems and the nature of the
Load- Displacement plots. Later Monte Carlo Simulation was done to see the effect of
imperfections and to investigate whether some kinds of imperfections were responsible for this
or not. More over Finite Element model in Abaqus was made and mechanical as well as
geometric details of nanotubes were inputted to simulate a finite element model of shell using the
Non Linear Analysis. Interpretation of the results from Abaqus was done and processing of the
results was done using Matlab codes.
Still the study is not over and we are continuing our efforts to obtain the best explanation of the
Buckling behavior of the nanotubes from the mechanical perspective.

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Overview of Stability & Buckling
Every structure is in equilibrium whether it is static or dynamic. If a structure or a body is not in
equilibrium it becomes a mechanism. Hence a mechanism cannot resist any loads and is of no
use to the Civil Engineers. The resulting loss in ability of the structure to resist loading due to
change in geometry of a structure or a structural component is known as Instability. There are
basically three kinds of equilibrium states for a body:
• Stable Equilibrium: Structure is in stable equilibrium when small perturbations do not
cause large movements like a mechanism. Structure vibrates about it equilibrium
position.
• Unstable Equilibrium: Structure is in unstable equilibrium when small perturbations
produce large movements – and the structure never returns to its original equilibrium
position.
• Neutral Equilibrium: In this case small perturbations can cause large deflections but the
structure can be brought back to its original position with no work.
Buckling is a phenomenon which usually occurs for structures under compressive loads. As
the compressive load is increased the structure shifts from the stable equilibrium state to
neutral and unstable states. The equilibrium state becomes unstable either due to large
deformation of the structure or due to inelasticity of the structure material.
Any structure subjected to compressive forces can undergo buckling which is nothing but the
bifurcation from state of stable equilibrium to the state of unstable equilibrium. The other one
is the material inelasticity. [4]
While in bifurcation buckling as the load reaches the critical value the deformation changes
its nature of equilibrium from stable to unstable as well as the equilibrium load-deformation
path bifurcates. The bifurcation may be Stable Symmetric Bifurcation , Unstable Symmetric
Bifurcation or an Asymmetric Bifurcation depending on the symmetry of the post buckling
paths after bifurcation. The symmetry or asymmetry is about the load axis.
However there is an Instability failure in which no bifurcation occurs in the load deflection
path. The stiffness of the structure decreases due to large deformations or material
inelasticity. The structural stiffness reduces to zero which implies that the load carrying
capacity has reached. This becomes negative indicating unstable equilibrium. The example
are Beam-Column failure, Snap-through Buckling and Shell buckling which is very sensitive
to the initial imperfections[5] which will be studied in details later.
In my study three spring-column models were taken to understand the concept using these
simple models each of which will be described using the plots of Load v/s Displacement.

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There are basically three approaches to study the stability namely
• Bifurcation Approach: This involves writing the equilibrium equation and solving
these equations to determine the onset of buckling. This is useful for determining the
critical load for the perfect system subjected to loads. The method is assumes very
small deformations and is not valid for the imperfect structures. It cannot provide any
information regarding the Post Buckling behavior.
• Energy Approach: This involves writing the expression for potential energy for the
entire system, analyzing this total potential energy expression to establish the
equilibrium path and examining the stability of the equilibrium states. This is
generally considered best when establishing the equilibrium equation and examining
the stability. The deformations can be large or small and the the system can even have
imperfections. It also provides information regarding the Post Buckling behavior if
we assume large deformations.
• Dynamic Approach: This involves writing the equations for the dynamic equilibrium
of the system. The equations are then solved to determine the natural frequency
(omega) of the system. Instability corresponds to reduction of omega to zero.

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2 Matlab Simulations
Case I Symmetric Stable
Perfect Structure: The model (Fig) consist of a rigid bar and a rotational spring with spring
constant K1. The length of the model is L and a load P is applied axially. Bifurcation analysis is
used to solve the problem in which a small amount of deformation is given to the model and
using the equation of static equilibrium the buckling load can be obtained as K1/L.
Fig 2.1 Symmetric Stable Model
Results from the analysis: The results of analysis can be plotted in the form of Load v/s
Displacement plots. The similar results can be obtained by writing the expression for potential
energy and differentiating it with respect to delta to obtain the equilibrium load and displacement
profile. The bifurcation path is stable equilibrium and hence is known as Symmetric Stable
Bifurcation path.
U = ½*K*ψ2
−P*L*(1-cos(ψ))
Fig2.2 Load V/s Displacement for Case 1(Perfect Structure)

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Imperfect structure: For the imperfect structure energy equations can be written and solved for
the imperfect geometry with very small initial imperfections. This assumes the structure is
initially titled to a small angle ψ0 prior to the perturbations. The load v/s displacement curve is
shown as
The strange thing about the Imperfect structure of case 1 is that the load carrying capacity keeps
on increasing and hence it becomes difficult to define buckling for such a case. However the
stable and the stable branches can be found out using the second derivative of potential energy
and checking it for the maximum and minimum. The behavior is sometimes attributed similar to
flat plate which shows post buckling behavior of the similar nature.
Equilibrium paths whether stable or unstable will be showed in a separate figure for all the three
cases.
Case II Symmetric Unstable
Perfect Structure: The model (Fig) consists of a rigid bar with a translation spring attached to the
end. The spring has spring constant as K2 and the length as L and P as the axial load applied.
Similar to the first model bifurcation analysis is performed from the equations of static
equilibrium the critical load can be obtained as K2*L.
Fig2.4 Symmetric Unstable Model
P
K L

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Results from the Analysis: The results obtained can be displayed in the form of the Load v/s
Displacement plot. The critical load can also be obtained using the energy approach as done for
the first case. The bifurcation path is unstable and hence it is known as Symmetric Unstable
Bifurcation.
Fig2.5: Load V/s Displacement for Case 2(Perfect Structure)
Imperfect Structure: For the imperfect structure energy equations can be written and solved for
the imperfect geometry with very small initial imperfections. This assumes the structure is
initially titled to a small angle ψ0 prior to the perturbations. The load v/s displacement curve is
shown as

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The buckling load is reduced slightly from its theoretical value without imperfections which was
2.5. Hence it also validates the experimental fact with a practical specimen which is bound to
have imperfection the buckling load reduces. The behavior is typical to cylinders.
Case III Asymmetric Structure
The Asymmetric structure consists of a spring with stiffness K3, with the length of the rigid bar
as L and the spring is inclined at any angle alpha with horizontal. For simplicity of the analysis
alpha is chosen as 45 degrees. However varying alpha has other important conclusions which
are understood from the Load Displacement behavior of the model.
Perfect Structure: This is a special case where there is no symmetry and hence the nature of the
bifurcation is also not symmetric. Not only the shape is different but also the equilibrium state
for both the bifurcation branches is different. The branch on the right is unstable while the
one on the left is stable.

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The strange thing about the plot is that only the left one is affected by changing the alpha.
It sharply turns to give a smooth peak unlike for alpha 45 which does not show any peak.
The nature is also complimentary i.e. for complementary angles like 30 and 60 the nature
of the plot is same. The model is viewed as a close representative for the shell structure which is
highly sensitive to the Imperfections and the buckling load drastically decreases with slight
introduction of imperfections .
Imperfect Structure: Slight introduction of imperfection make things interesting and the model
now buckles comparatively at a lower buckling load. In the present set of arbitrary values it is
2.5 for the perfect structure and 2.04 for the imperfect one.
Fig2.9 Load V/s Displacement for Case 3(ImPerfect Structure) for alpha = 45 degrees

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Fig2.10: Load V/s Displacement for Case 3(ImPerfect Structure) for alpha = 54 degrees
In this two distinct buckling peaks are observed here. The second peak on the left become
visible when the angle alpha is any angle other than 45 degrees. The left side peak behavior
is still not fully understood since the right peak seems very compatible with the shell which
states that the Buckling behavior should decrease on the introducing imperfections. Hence
the model becomes very interesting.
Combined behavior of all the 6 models can be explained with the help of the figure showing the
nature of equilibrium at every point on the load displacement curve. The red ones show unstable
equilibrium while the blue ones show stable equilibrium.

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Fig2.11: Stable (Blue) and Unstable (Red) branches for each of the Cases
The 3D plots for the potential energy delta and load were also generated and have been put
in final results folder.
Fig2.12: Perfect Symmetric Stable Potential Energy as a function of ψ & Load

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Fig 2.13: Perfect Symmetric Unstable Potential Energy as a function of ψ & Load
Fig 2.14: Perfect Asymmetric Potential Energy as a function of ψ & Load

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3 Monte Carlo Simulation
This was particularly done for case 3 (asymmetric model) to investigate the effect of
imperfections which were varying randomly on the buckling stress obtained by the taking the
maximum of the load displacement value from the stable equilibrium paths. Random
imperfections were generated between -0.1 to 0.1 which resulted in random values of
imperfections. A function file was made to calculate the critical Buckling Load corresponding to
each imperfection and a probability density function (pdf) was plotted which showed a clear
bimodal behavior when the imperfections varied between -0.1 to 0.1 keeping the other
parameters such as alpha and stiffness as constant.
However a unimodal behavior was obtained when the imperfections varied positively but it
is showing an interesting comparison for negative imperfections where a slight bimodality
is observed.
Fig3.1: Pdf for Buckling Stress when imperfections varied negatively between [-0.1, 0]

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Fig3.2: Pdf for Buckling Stress when imperfections varied negatively between [0, 0.1]
A subplot can be attached showing the effect of varying the stiffness and alpha one at a time
while keeping the imperfections randomly varying between [-0.1 0.1].
Fig3.3 pdf showing variation in modal behavior by varying alpha

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Fig3.4: pdf showing variation in modal behavior by varying Stiffness
However more results can be generated by keeping the imperfections constant and varying the
stiffness and alpha. The bimodal behavior still persists if the imperfections varies randomly from
[-0.1 0.1].
Fig3.5: Variation of Imperfections, Alpha, Stiffness varying one at a time and others constant

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4 Abaqus Simulations
Abaqus is a software application used for both the modeling and analysis of mechanical
components and assemblies (pre-processing) and visualizing the finite element analysis result.
Abaqus/Standard is a general-purpose Finite-Element analyzer that employs implicit integration scheme
(traditional). Abaqus/Explicit is a special-purpose Finite-Element analyzer that employs explicit
integration scheme to solve highly nonlinear systems with many complex contacts under transient loads.
It uses Python as the scripting language.
The purpose of doing the Abaqus Simulations was to capture the trend of Buckling Load with
diameter of the nanotube. Though Molecular dynamics simulations were performed for the
carbon Nanotube which showed a strange bimodal behavior which could not be deciphered and
hence we were interested to study the mechanics by comparing nanotubes to a thin shell and
using Finite Element Method which also computationally cheaper than the MD Simulations. It
was suspected that the bimodal behavior could be explained by studying the mechanics and
hence there was the need to perform Abaqus Simulations.
The first towards that was choosing the diameter, length and the thickness of the nanotube. The
diameter of the nanotube was calculated using m-n formula as 0.0783*(m2
+ n2
+ mn) 1/2
. The
values of m and n varied from 4 to 25. The L/D ratio was kept as 5 and the thickness of the
nanotube was chosen as 0.34 angstrom. A python script was written to generate the set of
nanotubes. In all a total of 368 nanotubes were simulated two times once with the modulus of
elasticity as 1012
Pa and the other time using varying elasticity taken from MD Simulations. Non-
Linear Analysis was performed using the Static Riks [6] method. The finite element model for
the carbon nanotube is shown below:
Fig4.1: Finite Element Model of Carbon Nanotube

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The shell consists of 15900 nodes and S4R element is a four node quadrilateral. Nonlinear
analysis is taking into account large deformation such as displacements and rotations. The output
from the .odb file is written in the form of a data file using Python scripting and the further
processing of the data using the Matlab code. One typical mode of failure of Shell from
simulation is shown as:
Fig4.2: FEM Model of the failure mode of Shell
The results of the Abaqus can be shown in the form Load v/s Displacement plots and the relative
frequency plots for various diameter ranges can also be plotted.
Fig4.3: Relative Frequency Plots from the Abaqus Simulation a) Between 10-15 Angstrom b)
10-20 Angstrom c) the whole diameter range

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The behavior 10-15 Angstrom shows a clear bimodal behavior while for the whole diameter
range the entire distribution is unimodal. One reason which could be behind this bimodality
can be the nonlinear Static Riks analysis used in Abaqus. However it does not end here
only since the exact mechanics behind the process is still until study.
Fig 4.4: Buckling Stress v/s Diameter from Abaqus Simulations using constant Elastic Modulus
The Abaqus Simulation results were also compared with results of the thin shell theory which
showed that the Abaqus Simulation satisfies most of the points of the Thin Shell Theory except
for few points which shows unusually very high buckling loads. The buckling formula for the
Thin Shell theory [7] is:
E*h/[a*(3*(1-µ2
)0.5
)]
Fig4.5: Comparison of Abaqus Simulation with Thin Shell theory

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The similar set of simulation was performed for the 368 Carbon nanotubes but the elastic
modulus for each se was taken from the MD Simulation. This was done to know the effect of
Elastic Modulus on the buckling stress of the Carbon Nanotubes which are highly susceptible to
these small changes. The results of the new simulations were compared with that of the old ones
using constant elastic modulus and it was found that new simulations showed a higher Buckling
Stress. However possibility of constant scaling was checked it was found that there is no constant
scaling in Buckling Stress like doubling the Elastic modulus does not doubles the buckling Stress
of CNTs. Hence possibility of some other kind of mechanics governing their failure is there. The
new simulation results were also compared with the thin shell formula in which the elastic
modulus was used from MD simulations and some close results were observed as majority of
points still followed the behavior of Thin Shell. Still there were good numbers of data points
showing the deviation particularly on the higher side.
Fig 4.6: Comparison of Abaqus Simulation with Thin Shell theory
However the Frequency plots and the Pdfs still show the same trend of bimodality between the
same diameter range and the results can be presented below as:

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Fig 4.7: New Simulation Results a) Frequency Plots and b) Pdfs
Some interesting results were also showed up while observing the mode shapes of the
buckled Shell which showed a more global trend of buckling for higher diameters shells
and a more local trend of failure for the lower diameters ones.

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Imperfection Modeling in Abaqus: Thin Shells are very susceptible to the slightest of the
imperfections. To understand this Imperfect shell was simulated and the buckling stress was
measured with corresponding to the scale of imperfections normalized between 0 to 1.
Fig 4.8 FEM Model of an Imperfect Shell
The idea to generate imperfection is based on superimposition of Eigen modes of buckling.
Various modes can be superimposed with the corresponding scale factors acting as the weights to
generate a disturbed geometry. The analysis was done in two steps. In the first step Linear
Buckling Analysis was done to generate various modes which were superimposed to create the
disturbed geometry. The scale factor governed the magnitude of the imperfections. The second
analysis was a nonlinear step using Static Riks method.
However the imperfection option is itself not supported in Abaqus/CAE so
*IMPERFECTION [8] is used in the input file to generate it. The result was that with the
slightest introduction of imperfection the Buckling Load decreased to three times the
perfect structure. Hence the simulation justified the practical results with thin shell. Moreover
superimposing various modes using appropriate factor which is calculated using another
FORTRAN script and using parametric study using python script can be used to evaluate the
lowest Buckling Load and the suitable combination generating it.
Various other simulations with constant strain rates were performed to see the effect of strain
rates on buckling. The results of these simulations have been presented in a separate folder.
However buckling load was almost observed to be constant with different strain rates. Though
CNTs are very susceptible to strain rates but with Abaqus Simulations the similar effect was not
observed.

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5 Calibrating the Asymmetric Model with
Thin Shell Model
Since the reason of Bimodality is suspected to lie in the non-linear analysis so an interest was
developed to understand the real physics behind the problem and hence a study of the
asymmetric model (closely represent the thin shell) which is a simple single degree of freedom
system was done. The model was calibrated by setting the buckling load of the model equal to
thin shell load and taking the stiffness of the model from the Molecular Dynamics Simulation.
Solving this equation we obtained a constant length for the asymmetric model. Now the
asymmetric model was equal to the thin shell model. After calibrating it a slight imperfection
was done in the model to trace the geometric non-linearity which was supposed to be a reason of
our problem. Matlab simulations were done and Buckling Stress v/s diameter plots were
generated. More over potential energy of the buckling was also observed and was plotted against
the diameters.
Fig 5.1: Thin Shell, Abaqus and Calibration Results
The calibration results overestimate the value of Buckling Stress with introduction of
geometric nonlinearity. However the sensitivity of imperfection to the fluctuation in

28
buckling load from Abaqus is significant. Similar sensitivity is shown by the potential
energy curves at the onset of Buckling.
Fig5.2: Calibration Model comparison with slight non-linearity dramatically increasing the
potential energy at the onset of the buckling.
However the plot for Elastic Modulus from MD Simulation shows the similar behavior as
that of the potential energy from the calibrated model.
However the reason to the problem is still not fully understood and our efforts are still continuing
to find the best answer of the bimodal behavior of the Carbon Nanotubes often known as one of
the weirdest material known to man with some amazing properties.

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References
1. Sie chin Tjong Carbon Nanotube Reinforced composites
2. Sie chin Tjong Carbon Nanotube Reinforced composites
3. Image copyright Elsevier
4. Lecture notes on Structural Stability & Design by Dr Amit H Verma, Assistant preofessor,
Purdue university
5. Lecture notes on Structural Stability & Design by Dr Amit H Verma, Assistant preofessor,
Purdue university
6. MEMON Bashir-Ahmed & SU Xiao-zu Arc-length technique for nonlinear finite element analysis
7. Theory of Elastic Stability Stephen P Temoshenko, James M Gere
8. Abaqus 6.11 online Documentation & Manual
Books referred in general
1. Stress, Stability & Chaos in Structural Engineering by E.L. Nachie
2. Theory of Elastic Stability by Stephen P Timoshenko & James M Gere
3. Computerized Buckling analysis of Shell by D Bushnell
4. Advanced Engineering Mathematics by Erwin Kreszig

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