Master Thesis

Behavior of polygonal semi-closed
thin-walled cross-section
A study based on finite strip analysis
Hamse Abdi
Jimmy Adamo
Civil Engineering, masters level (120 credits)
2017
Luleå University of Technology
Department of Civil, Environmental and Natural Resources Engineering

I
MASTER’S THESIS
Behavior of polygonal semi-closed thin-walled
cross-section
A study based on finite strip analysis
Hamse Abdi
Jimmy Adamo
2017
Luleå University of Technology
Department of Civil, Environmental and Natural resources engineering,
Division of Structural and Construction Engineering – Steel Structures
SE- 971 87 LULEÅ
www.ltu.se/sbn

I
Forward
The work in this master’s thesis has carried out in affiliation with the Department of Civil,
Environmental and Natural resources engineering, Division of Structural and Construction
Engineering – Steel Structures. Hopefully this work will give the reeder some understanding
and new knowledge about the bahevior of polygonal thin-walled semi-closed crossection.
The work has been perfomered at LTU in collibration with senior lecture Efthymois Koltsakis,
PhD student Panagiotis Manoleas and Ove Lagerqvist as eximiner. We would like to express our
deep gratitude to our supervisor Panagiotis Manoleas for his help and for his guidaness,
encourgment and valubule disscussions. We would also wish to thank Efthymois Koltsakis for
help and support.
Finally we would like to thank our family and friends for all the support.
Luleå, January 2017
Hamse Abdi
Jimmy Adamo

Abstract
The acceptance and the use of cold-formed steel sections has significantly increased in recent
years due to advantages such as consistency and accuracy of profile, ease of fabrication, high
strength and stiffness to the lightness in the weight. For thin-walled columns, made by folding a
plane plate into a section, it is possible that when they are subjected to compression loads they
may buckle either locally, if the member is very short, or globally if the member is very long. In
addition to local and global buckling, a thin-walled member of an open cross section may also
show buckling involving a “distortion” of the cross section. Compared to local and global buck-
ling, distortional buckling is not very familiar and has been discovered only in thin-walled mem-
bers of open cross sections such as cold-formed steel section columns.
The objective of this study is to investigate the behavior of polygonal semi-closed cross-section
with pure compression. The study comprise to only elastic buckling and the methodology is
consisted by using CUFSM analysis. In order to execute CUFSM of polygonal profiles, the
scripts have created which match the Matlab script files (m-files) downloaded from CUFSM 4
open source.
The distortional buckling mode is governing as a buckling failure, which occur and dominate in
the cases where spring values are 100 kN or higher. However, the contrary result reveals by a
decreasing of the spring values. The behavior of the cross-section is dependent on how the
interaction of different buckling modes prevails at the corresponding critical half-wavelength.
Considering the predomination of distortional buckling mode indicates that the most of polyg-
onal cross-section do not behave as rigid, i.e. as whole cross-section. A reducing of distortional
mode and increasing of local mode as well as global mode gives indication that the behavior of
the cross-section has changed and turned significantly into more rigid and thus is expected to
behave more as whole cross-section. The more spring values decrease, the higher global mode
arises and dominates for the lower slenderness range. The critical half-wavelength for each profile
illustrates the needed density between bolts on the longitudinal part of the member. In the in-
terest of eliminating distortional buckling failure, due the fact that distortional buckling is un-
predictable, the bolt-density should be lower than the corresponding half-wavelength for the
profile where the distortional mode is predominating.
Keywords: Buckling mode, Matlab script, CUFSM analysis, Half-wavelength, Spring value,
Hexagonal, Nonagonal, Dodecagonal

III
Sammanfattning
Acceptansen och användningen av kallformade stålprofiler har ökats kraftigt under de senaste
åren på grund av de fördelar de besitter, såsom konsekvent och noggrannhet av profilen, enkel
tillverkning, hög hållfasthet och styvhet i förhållande till lätt vikt. För tunnväggiga pelare, till-
verkade genom att böja en plan platta till en sektor, kan det ske buckling då de utsätts för tryck-
belastningar. Pelaren kan bucklas antingen lokalt, om den är mycket kort, eller globalt om den
är mycket lång. Förutom lokal och global buckling för ett tunnväggigt element av ett öppet
tvärsnitt kan dessutom uppvisa en knäckning som involverar en "distorsion" av tvärsnittet. I
jämförelse med lokal och global buckling, är distorsionsknäckning (buckling) inte välbekant och
har endast upptäckts i tunnväggiga element av öppna tvärsektioner såsom kallformade stål ele-
ment.
Syftet med denna studie är att undersöka beteendet hos polygonala halvslutna tvärsnitt som utsätts
för rent tryck. Studien omfattar endast elastisk buckling och metodiken utfördes genom CUFSM
analys. För att genomföra CUFSM av polygonala profiler, har skript skapats så att dem samspelar
med Matlab skriptfiler (m-filer) hämtade från CUFSM 4 tillgängliga källan.
Distorsionsknäckning är ledande som ett knäckningsbrott, som förekommer och dominerar i de
fall där fjärdens värde är 100 kN eller högre, men däremot visar resultatet motsatsen vid mins-
kande av fjärdens värde. Beteendet hos tvärsnittet är beroende av hur interaktionen mellan olika
bucklingsmoder som råder vid tillhörande kritiska halv våglängd. Med avseende på dominansen
av distorsionsknäckning, detta tyder på att de flesta av polygonala tvärsnitt inte beter sig som
stela, dvs. som sammansatta tvärsnitt. Reducerande av distorsionsknäckning samt ökande av
bägge lokal och global buckling, ger indikation på att beteendet hos tvärsnittet har förändrats
och övergått till något styvare tvärsnitt. Tvärsnittet förväntas därmed bete sig mer som samman-
satt tvärsnitt. Ju mer fjärdensvärde minskar, uppstår högre global knäckning och därmed bli mer
dominerande för lägre slankhet intervall. Den kritiska halva våglängden för varje profil, illustrerar
det nödvändiga avståndet mellan bultarna på den längsgående delen av elementet. I syfte med att
eliminera distorsionsknäcknings brott, på grund av att distorsions knäckning är oförutsägbar, bör
bult avståndet vara något lägre än motsvarande halv våglängd för profilen där distorsionsknäck-
ning dominerar.
Keywords: Buckling mode, Matlab script, CUFSM analysis, Half-wavelength, Spring value,
Hexagonal, Nonagonal, Dodecagonal

Notations
Notations
Latin upper case letters
A Cross-sectional area of the member [𝑚𝑚2
]
𝐴 𝑒𝑓𝑓 Effective cross-sectional area of the member [𝑚𝑚2
]
E Elastic module of steel [MPa]
I Moment of inertia [𝑚𝑚4
]
K Euler’s coefficient [-]
L Length of the member [𝑚𝑚]
𝐿 𝑏 Euler’s length of the member [𝑚𝑚]
𝑁𝑏𝑅𝑑 The designed buckling load [𝑁]
𝑁𝑋
̅̅̅̅ Euler’s critical buckling load, positive in compression [𝑁]
Latin small letters
a The length of the plate [𝑚𝑚]
b The width of the plate [𝑚𝑚]
𝑓𝑦 Yield strength [𝑀𝑃𝑎]
𝑚 The number of half sine waves in the buckling mode [-]
𝑛 The number of half sine waves in the buckling mode [-]
𝑝 𝐶𝑟 The Euler’s critical buckling load [𝑁]
r Radius of the profile [𝑚𝑚]
t Thickness of the profile [𝑚𝑚]
Greek small letters
𝛾 𝑚1 Partial factor [-]
𝜆̅ Relative slenderness [-]
𝜐 Poisson ratio [-]
𝜒 Reduction Factor [-]

V
Contents
FORWARD................................................................................................................. I
ABSTRACT ................................................................................................................II
SAMMANFATTNING ..............................................................................................III
NOTATIONS ............................................................................................................IV
1 INTRODUCTION ............................................................................................1
1.1 Background.......................................................................................................................1
1.2 Research questions ............................................................................................................2
1.3 Goal and objectives............................................................................................................2
1.4 Scientific approach.............................................................................................................2
1.5 Limitations ........................................................................................................................2
2 LITERATURE VIEW.........................................................................................3
2.1 Thin-walled structures .......................................................................................................3
How cold-formed steel are manufactured .................................................................. 3
The primary features of hot rolled and cold-formed sections ...................................... 4
2.2 Wind tower.......................................................................................................................5
2.3 Critical buckling load.........................................................................................................6
2.4 The stability behavior of bars, plates and shells under axial compression...............................8
2.5 Elastic Buckling ...............................................................................................................11
Local buckling ........................................................................................................ 12
Distortional buckling............................................................................................... 12
Global buckling ...................................................................................................... 12
2.6 Numerical methods .........................................................................................................13
Finite strip method.................................................................................................. 13
CUFSM ................................................................................................................. 14
MATLAB............................................................................................................... 14
3 METHOD (PARAMETRIC STUDY) .............................................................15
4 RESULTS..........................................................................................................24
5 ANALYSIS ........................................................................................................25
5.1 Spring value - 100 kN (approximated value).....................................................................25
5.2 Increasing of the spring value ...........................................................................................27
Increasing of the spring value by 100 % ................................................................... 27
Increasing of the spring value by 200 %. .................................................................. 29
Increasing of spring value by 300 % ......................................................................... 32
5.3 Decreasing of the spring value..........................................................................................34
Decreasing spring value by 50 %.............................................................................. 34
Decreasing of spring value by 66,6 %....................................................................... 37

Decreasing of spring value by 75 %.......................................................................... 41
5.4 Summary of profiles of interest.........................................................................................45
6 CONCLUSION ................................................................................................48
7 FUTURE WORK.............................................................................................50
8 REFERENCES..................................................................................................51
APPENDIX A.............................................................................................................53
8.1 Approximated spring value - 100 kN ...............................................................................53
Hexagonal .............................................................................................................. 53
Nonagonal.............................................................................................................. 55
Dodecagonal........................................................................................................... 57
8.2 Increasing ........................................................................................................................60
Increasing of spring value by 100 % ......................................................................... 60
Increasing the spring value by 200 % ....................................................................... 66
Increasing the spring value by 300 % ....................................................................... 72
8.3 Decreasing.......................................................................................................................78
Decreasing the spring value by 50 %........................................................................ 78
Decreasing the spring value by 66,6 % ..................................................................... 85
Decreasing the spring value by 75 %........................................................................ 92
APPENDIX B.............................................................................................................99

1
1 Introduction
This chapter provides an overview of the development of cold-formed steel including the be-
havior of local, distortional and global buckling. Moreover, the research questions, limitation
and methodology are also presented in this section.
1.1 Background
The acceptance and the use of cold-formed steel sections has significantly increased in recent
years due to advantages that such as consistency and accuracy of profile, ease of fabrication, high
strength and stiffness to the lightness in the weight. Another advantage is flexibility, by producing
variety of the sectional profile without waste reduction.
For thin-walled columns, made by folding a plane plate into a section, it is possible that when
they are subjected to compression loads they may buckle either locally, if the member is very
short, or globally if the member is very long. In addition to local and global buckling, a thin-
walled member of an open cross section may also show buckling involving a “distortion” of the
cross section. Compared to local and global buckling, distortional buckling has not been very
familiar and has been discovered only in thin-walled members of open cross sections such as
cold-formed steel section columns. The three different buckling modes which has mentioned
previously, are defined by the shape that a member buckles into and occur at different half
wavelengths. The local buckling arises in the short half wavelength, the distortional buckling at
the intermediate half wavelengths and the flexural/ flexural torsional buckling at the long half
wavelengths
The expression “distortional buckling” has been first made up by Hancock to distinguish it from
local and lateral torsional buckling. Later on, Hancock published the first numerical study on the
distortional buckling of columns with edge stiffened cross-sections. He showed that the corre-
sponding buckling curves display two local minima, related with local buckling for short columns
and distortional buckling for intermediate columns. Furthermore, he showed that for some cases
the distortional buckling may be critical [15].
The background of the investigation is lack of formal calculation for the polygonal cross-sections
and better understanding of the distortional buckling phenomena which is particularly unpre-
dictable. Furthermore, the column investigated in this thesis is made of thin-walled polygonal
cross-section and is a part of the legs in a truss towers. The legs are made of either three, six or
nine. This kind of columns are more cost effective when it comes to production and transpor-
tations.

1.2 Research questions
Answering the research questions below will provide a better understanding of the behavior of
semi-closed polygonal cross sections.
(a) Does the cross section behave as a whole (is the assumption of considering individual
sectors of it as rigid sound?)
(b) Under what bolting density is the rigid assembly assumption (a) correct?
1.3 Goal and objectives
The goal and objective of this master’s thesis are to investigate the behavior of the polygonal
profiles by studying the interaction between different buckling modes for polygonal sections
and also eliminating the distortional buckling of individual sectors (density of bolting).
1.4 Scientific approach
Initially, a study has been performed based on the literature that followed with the CUFSM.
Furthermore MATLAB scripts have been used to create an input of cross-sections to CUFSM.
Research has been done on elastic buckling and the behavior of polygonal cross-section for wind
towers.
(a) Use Matlab software by taking consideration CUFSM 4 Matlab-to assess the buckling of
each sector individually for every type of cross-sectional profile
(b) Start working at the Cl-3 - Cl-4 limit of a similar cylindrical section (D/t=90ε2
) and push
slenderness upwards.
(c) How dense should the bolting be to do that by calculating the half-wavelength at the
corresponding critical buckling load?
1.5 Limitations
In this study, pure compression will be considered on three different types of cross-sections:
hexagonal, nonagonal and dodecagonal. The residual stress will be neglected at the corners which
occur generally during the bending process. By assuming it, the corners will be considered having
the same structural steel grade as rest of the cross-section. All cross-sections are made of S355
steel.

3
2 Literature View
2.1 Thin-walled structures
Thin-walled structures comprise an important and growing proportion of engineering construc-
tion such as bridges, industrial buildings and warehouses. Many factors contribute to this growth,
such as including cost and weight economy and need of new materials [17]. Thin-walled struc-
tural elements are extremely efficient because of the reduced thickness-to-width ratio [10]. The
manufacturing process plays a governing role for some characteristics that have an influence on
the buckling of the profiles, which leads first to a modification of the strain curve of the steel
[20].
How cold-formed steel are manufactured
Cold-formed steel members are made of steel plate, strip or sheet material formed at room temper-
ature. There are two different production processes that cold-formed steel members undergo to
achieve the desired shape, which are cold rolling and press-braking , see figure 1 (a) and (b). Press-
braking is a relatively simple manufacturing procedure and it is commonly used for production of
simple cross sectional shapes and of sections that are relatively wide such as roof sheets and decking
units). However, cold rolling is used for production of wall panels, roofs, and floors and it is also
used in the fabrication of window and door frames, gutters and pipes .The sheets are put through
different roller dies which form and bend them into the final needed shape. A simple section needs
about sex roller dies, but for more complicated ones, as many as fifteen might be required. There are
several limitations for the size range and shape, e.g. the cut lengths shall be maximum 12 meter and
the thickness between 0.2 to 7.6 mm [22].
(a)

(b)
Figure 1: Cold rolling for cold formed members (a) press braking for roll formed shapes (b) [22]
The primary features of hot rolled and cold-formed sections
The use of cold formed steel members in construction is new compered to hot rolled. The
technique of cold-form profiles for constructional use has been developed in USA during the
time of World War II. During 1960s, the technique transferred to Europe and Sweden where it
has been further developed in 1970s and later became the foundation for Eurocode 3, part 1-3
[8].
The main difference between the behavior of cold-formed and hot rolled structural sections is
that the cold-formed members are mostly thin-walled and have open cross-section. They tend
to buckle locally under compression before reaching yield strength. Furthermore, cold-formed
members, are susceptible to distortional buckling due to nature of open sections.
The use of thin-walled members and cold-forming manufacturing effects can results in design
problems such as buckling strength, low torsional stiffness, web crippling, low ductility and more.
These problems are not normally faced when tick hot-rolled members are used [20]. Different
buckling curves are used in European design codes for design of hot-rolled profiles because of
the effect of residual stresses which have significant influence of the buckling strength [20].
The structural behavior of cold-formed members can be influenced by the reduction of the
thickness in the bent corner and the residual stresses that depend on the bending radius. This can
lead to an increasing of the strength which have the negative effect that the stiffness decreases
[8]. However, this may be neglected if the radius is less than five times the thickness according
to EN 1993 part 1-3 [4]. .

5
2.2 Wind tower
Another use of thin-walled structures is on towers such as telecommunication masts and wind
turbines. Wind energy has been harvested for more than 3000 years and today the high demand
for wind energy is leading to the development of more powerful wind energy converters [11].
Renewable energy is vital for the fight against climate change and thus the demand for wind
power is increasing. During 2015, 12.800 MW worth of wind power were connected to the
EU grid. The wind power portion was bigger than any of the other forms of power generated
in the same year according to The European Wind Energy Association [6].
Most of the wind towers are built from steel due to its manufacturing and transportation ad-
vantages [1]. There are two common types of wind towers of steel: tubular towers and lattice
(truss) towers. For both types, the height of the tower is an important factor in order to reach a
higher height where there is more stable wind, especially onshore [7]. Furthermore, higher tow-
ers are required for larger rotors (larger radius) that gives grater wind gathering and thus yield
more energy. Tubular towers have limitations for higher hub heights and larger turbines with
larger head masses. Transportation issues limit the base diameter of tubular towers, a fact that
does not apply on lattice towers [12]. .
In this investigation, we are interested in truss towers, which are more cost efficient than tubular
towers due to use of less material. Another advantage of truss towers is that they can be trans-
ported in smaller parts which can simply be installed in the construction site [7]. The trusses are
made of different numbers of legs (columns), most commonly three, six (see figure 2 below) or
nine. Depending on the number of columns different polygons are eligible for the shape of the
cross-section. Hexagonal, nonagonal and dodecagonal cross-sections will be studied herein.
Figure 2: Lattice tower with six legs [11].

2.3 Critical buckling load
The structural member has a major function to carry the loads that is designed for [16]. For
members which has higher compressive stress than the ultimate compressive stress, the member
will collapse. In some cases the buckling, which is defined as instability issue, arise prior to
yield/ultimate stress. For compressed structural members, buckling occurs unexpectedly as a
sideway deformation. However, the determination, if a member is stable or unstable, dependents
on the material, geometry, boundary conditions and the imperfection of the structure.
The initial theory of buckling of columns was worked out by Leonhard Euler in 1757 [2]. That
theory is based on ideal column with elastic material, e.g. the axial compression load is applied
centrally on the center of gravity for the cross section of the member, which leads to the result
that the imperfections of the cross section are neglected. That assumption is rather poor and
unrealistic considering the impossibility of manufacturing perfect cross section without any va-
riety of imperfections.
According to [16], the stability of a structural member is mainly characterized by its critical load
or critical stress, when the member is subjected to compression load. The figure shows the rela-
tionship between the critical load and the deformation under the axial load and also illustrates
the bifurcation point at which two equilibrium paths intersect. At bifurcation points, the deflec-
tion when subjected to compressive load, changes from one direction to a different one. The
load at which bifurcation occurs is the critical buckling load [13].
The deflection path that occurs before it reaches the bifurcation is defined as stable Path and that
after bifurcation is called as unstable or post buckling path, see figure 3 below.
Figure 3: illustration of bifurcation points at which different paths intersect during buckling process
Since the Euler’s buckling method will not lead to precisely accurate results of the critical buck-
ling load. The calculation of the buckling resistance should be obtained by other special methods.
By taking into account the post buckling behavior and effect of the imperfections.

7
Further There are four classes to identify the extent to which the resistance and rotation capacity,
see figure 4. The classification depends on the material yield strength, fy, and the width to thick-
ness ratio. Eurocode 1993-1-1 [3] defines these four classes in Clause 5.5.2 as following:
 Class 1 - cross-sections are those which can form a plastic hinge with the rotation capacity
required from plastic analysis without reduction of the resistance.
 Class 2 - cross-sections are those which can develop their plastic moment resistance, but
have limited rotation capacity because of local buckling.
 Class 3 - cross-sections are those in which the stress in the extreme compression fiber of
the steel member assuming an elastic distribution of stresses can reach the yield strength,
but local buckling is liable to prevent development of the plastic moment resistance.
 Class 4 - cross-sections are those in which local buckling will occur before the attainment
of yield stress in one or more parts of the cross-section.
Figure 4: the relation between the critical moment and the rotation of different classes of a cross-section
The design buckling resistance of members
The non-dimensional slenderness is given by:

λ̅ = √
A∗fy
pcr
𝒇𝒐𝒓 𝒄𝒍𝒂𝒔𝒔 𝟏, 𝟐 𝒂𝒏𝒅 𝟑 𝒄𝒓𝒐𝒔𝒔𝒆𝒄𝒕𝒊𝒐𝒏
λ̅ = √
𝐴 𝑒𝑓𝑓∗fy
pcr
𝒇𝒐𝒓 𝒄𝒍𝒂𝒔𝒔 𝟒 𝒄𝒓𝒐𝒔𝒔𝒆𝒄𝒕𝒊𝒐𝒏
The Euler’s buckling load can be taken as:
pcr =
π2
EI2
Lb
2 where 𝐿 𝑏 = 𝐾 ∗ 𝐿
α = 0,49 - Buckling curve c (cold formed)
The reduction factor for the relevant buckling mode is be given by:
χ =
1
ϕ + √ϕ2 − λ̅2
Where
Φ = 0,5[1 + α(λ̅ − 0,2) + λ̅2
]
The design buckling resistance of a compression member should be taken as
𝐍 𝐛𝐑 𝐝
=
𝛘∗𝐀∗𝐟 𝐲
𝛄 𝐌𝟏
𝒇𝒐𝒓 𝒄𝒍𝒂𝒔𝒔 𝟏, 𝟐 𝒂𝒏𝒅 𝟑 𝒄𝒓𝒐𝒔𝒔𝒆𝒄𝒕𝒊𝒐𝒏
𝐍 𝐛𝐑 𝐝
=
𝛘∗𝐴 𝑒𝑓𝑓∗𝐟 𝐲
𝛄 𝐌𝟏
𝒇𝒐𝒓 𝒄𝒍𝒂𝒔𝒔 𝟒 𝒄𝒓𝒐𝒔𝒔𝒆𝒄𝒕𝒊𝒐𝒏
2.4 The stability behavior of bars, plates and shells under axial compression
Those three structural elements have different buckling phenomena with respect to the post
buckling and the effect of the initial imperfections. Initial imperfections tend to cause consider-
ably large deformation than what would occur for an ideal element. The important attributes of
the buckling phenomena are deflection of the mode shape, sudden buckling occurrence and
change in path on the load deformation curve.

9
Bars
The Euler’s buckling critical load equation for bars [9].
𝑝 𝐶𝑟 = 𝑚2
∗ 𝜋2
∗
𝐸𝐼
𝐿2
, 𝑤ℎ𝑒𝑟𝑒 𝑚 𝑖𝑠 𝑚𝑜𝑑𝑒 𝑠ℎ𝑎𝑝𝑒 (ℎ𝑎𝑙𝑣𝑒 𝑠𝑖𝑛𝑒 𝑤𝑖𝑣𝑒)
As it is shown in figure 5, the bar is initially loaded and deformed in axial compression into
straight stable configuration until the critical buckling load is reached. Then the deformation
path changes at the bifurcation point where the loaded bar does not deform into unstable con-
figuration, but instead deforms into the stable buckling configuration due to lateral bending
under buckling. A relatively perfect bar under axial load increases slightly beyond the Euler’s
buckling because lateral stress cannot be developed enough to contribute to the lateral defor-
mation.
Figure 5: load deformation behavior for an axially loaded bar
Plates
The buckling load of plates depend on the following plate geometry, material properties and the
buckling mode. The critical buckling load for plates is [9].
𝑁̅𝑥 = 𝑁̅𝑥(𝑎, 𝑏, 𝑡, 𝐸, 𝑣, 𝑚, 𝑛)
Where
𝑁̅𝑥 is positive in compression
a is length of the plate
b is width of the plate
n and m are the numbers of half sine waves in the buckling mode
For plate as shown in the figure, the load increases above the Euler’s buckling and the initial
imperfection can affect the post-buckling behavior. There are actually numerous buckling pat-
terns for a plate and each critical load has its own buckling mode. The buckling modes have

waviness in both x and y directions of the plate which transfer according to the buckling mode
of the plate. However, the nodal line, which is a straight line on the plate surface, does not
transfer in any directions during the buckling pressure, see figure 6.
Figure 6: load deformation behavior for an axially loaded plate
The degrees of freedom for a plate is translation in three direction (u, v, w) and one rotation (θ).
That gives four degrees of freedom at each node, see figure 7.
Figure 7: degrees of freedom of a plate

11
Shells
The buckling load of shells is expressed by the following shell geometry, material properties and
the buckling mode [9],
𝑁̅𝑥 = 𝑁̅𝑥(𝐿, 𝑟, 𝑡, 𝐸, 𝑣, 𝑚, 𝑛)
Axially loaded shells have lower post-buckling behavior than the Euler’s buckling, which de-
pends on the initial imperfections. At a certain load below Euler’s buckling, the cylindrical shell
buckles rapidly into a deformed shape where the nodal circles do not move perpendicular the
shell surface [9], see figure 8. However, the flat shell i.e. curved panel has mainly same post
buckling behavior as an ideal plate. Initial imperfections are unpreventable during the manufac-
turing, since the impossibility of unmaking a certain degree of irregular waviness of circular
cross section along the cylinder`s length.
Figure 8: load deformation behavior for an axially loaded shell
2.5 Elastic Buckling
The stability of a structural element, is the ability to withstand a specified load without under-
going a sudden change in its configuration or its equilibrium state. However, the instability of
the structural element occurs when any small disturbance of the system results in a sudden change
in deformation mode or displacement value after which the system does not return to its original
equilibrium state [9], see figure 9.

Figure 9: An axially loaded bar with or without disturbances s
Buckling is an instability failure, a sideway failure, that occurs for slender members subjected to
axial compressive load and can occur at loads that are far smaller than the material failure. Buck-
ling on a non-ideal member leads to eccentricity, which in turn leads to additional moments to
the member. For thin walled profiles in compression, three types of buckling modes can mainly
occur: Local buckling, distortional buckling and global buckling [8]
An elastic material is a material that have a linear relationship between stress and strain. Defor-
mation caused by loading recovers when unloading i.e. deformation without yielding. Most
structural steels are linear, at least for stresses less than the yield stress while many other structural
are regarded as being linear over most of the range of working load [18].
Steel members may be subjected to local, global and distortional buckling. Local and distortional
buckling affect the shape and the resistance of member cross sections [20].
Local buckling
Local buckling occurs at the level of individual plate elements. The half-wavelengths are less
than the largest dimension of the cross section member. The local buckling contains only rotation
at the fold lines of the member [14]. Furthermore, local buckling of a thin plate element of a
structural member involves deflections of the plate out of its original plane and it is usually
concentrated near a particular cross-section [18]. Moreover, local buckling is prevalent in cold-
formed steel sections. However, these phenomena does not generally results in failure of the
section as does Euler buckling [20].
Distortional buckling
The distortional buckling’s phenomena arises for thin-walled open members where the half-
wavelength is several times higher than the largest dimension of the cross-section. Distortional
buckling involves rotation but also, unlike local buckling, translation at the fold line of the mem-
ber [14]. Distortional buckling is buckling which takes place as a consequence of distortional of
the cross section and it is characterized by relative movement of the fold-lines of the cold-formed
sections [20].
Global buckling
Global buckling modes are seen as flexural, torsional or flexural – torsional buckling. All three
modes occur as the minimum mode at long half-wavelengths. These involve translation and/or
rotation of the whole cross-section [14]. Flexural buckling of a member may involve transverse
displacement of the cross-section and is resisted by the flexural rigidity [18].

13
2.6 Numerical methods
The thin-walled elements are susceptible to the reducing of the thickness to width ratio of the
structural cross section, which plays a key factor in the design and the behavior. The increasing
slenderness leads to insufficiency of both Vlasov beam theory and ordinary Euler-Bernoulli beam
theory with St.venat torsion for thin walled structures [19].
There are many proper numerical tools which can be employed to account for different essential
phenomena which occur to the thin-walled structures. The Finite Element Method (FEM),
Finite Strip Method (FSM), and Fourier series method can be used in different engineering
simulations programs regarding buckling analysis for thin walled members program [5].The FEM
allows for generating imperfection configurations so therefore it can solve problems ranging from
the elastic linear analysis to more general nonlinear analysis. However, FSM which is a speciali-
zation of the FEM, is a very useful tool for the cross sections which have complex geometry but
are simple along the length. Furthermore, the classical Fourier series solutions are useful because
they have the potential to yield closed form solutions, which can be seen as advantage for design
approximation [5].
Finite strip method
The finite strip method is variation of the finite element method in both methodology and
theoretical basis. It was originally developed by Y.K Chenung and the details behind the analysis
of method can be found in his book [5]. The use of the finite trip method for cold-formed steel
members has been greatly extended by G. Hancock [5].It has been shown that finite strip method
is a sufficient tool to determine the elastic buckling stress and corresponding nodes in terms of
the nodal degrees of freedom for regular geometric section and simple boundary conditions.
Employing the shape function, the displacement and the strain will be defined in terms of the
nodal degrees of freedom. Since strain-stress relationship and the displacement are known, the
stiffness coefficient of the nodal degree of freedom can be obtained. The difference between the
finite element method and the finite strip method is how meshes are discretized .The finite strip
method employs as a single element to the model in the longitudinal direction instead of a series
full elements [5] as it is shown in figure 10. The advantages and the accuracy of the finite strip
solution depends on a judicious choice of the shape function for the longitudinal displacement
field [5].

CUFSM
A CUFSM (Cornell University Finite Strip Method) is software for finite strip analysis and is
developed by Professor B.W. Schafer and its solution provides an approach for exploring elastic
buckling behavior and thus stability solutions for a given buckling mode of arbitrarily shaped and
thin-walled [5]. It has been originally written to support research on the behavior and design of
cold formed steel members with a variety of different types of longitudinal stiffeners [5].. Using
the finite strip method of CUFSM provides diverse benefits that allow the software-users to
obtain a better understanding of the elastic buckling behavior and accurately determine the elastic
buckling stress (e.g. distortional buckling and local buckling), contrasting the traditional hand-
method design of plate structures which often ignore compatibility at plate junctures and do not
provide the appropriate calculation of important buckling modes. Moreover, the CUFSM de-
termines inputs such as the elastic buckling moment and the elastic buckling load for the direct
strength method [5].
MATLAB
MATLAB stands for matrix laboratory and is a high-performance language for numerical com-
puting. It is a software that is used for solving problems by computation, programming and
visualization. MATLAB can be used for simple calculation and for more complex problems by
writing scripts and programs.
For this investigation, MATLAB has been used to create database which is employed as input to
CUFSM. The input consisted of nodes and elements which built cross-sections for analyzing in
CUFSM. In order to get the cross-section of interest, a script has been made for regular polygonal
geometries and then programed to suit a semi-closed hexagonal, nonagonal and dodecagonal
respectively.
Figure 10: The meshes of the finite element and finite strip methods [5].

15
3 Method (Parametric study)
CUFSM is a software program created in Matlab and employs the semi-analytical FSM to pro-
vide solutions for the cross-section stability (elastic buckling) for thin-walled members. The pro-
gram and its Matlab files (m-files) is freely available to download.
CUFSM 4 is programed with a C and Z cross-section where you can choose the dimensions.
Another alternative is to give an input, nodes and elements, of an interest cross-section manually.
This is efficient for simple cross sections and when studying few number of cross sections. In this
study a more complex cross-sections (polygons) are investigated. Furthermore, for each cross-
section several types are studied due to the multiple combination given by the number of corner,
diameter and slenderness ranges that are chosen for these task, see table 1. Therefore, the pre-
programmed CUFSM, as it is, is not an alternative. Hence an automatized and more efficient
script that are suitable for this study has been constructed in Matlab. Four Matlab-scripts has been
created where the first two creates x and y coordinates for the profiles and calculates the points
of all the profiles within a range of values and returns them as a cell array. The other two executes
CUFSM analysis and finds minimum values for CUFSM results respectively. The analysis gives
the signature curves in regards to the load factor and the half wavelength.
Initially the cross-section has been divided in three sections where each section has meant to be
analyzed separately. Thereafter, the three sections were put together with constraints (connects
two elements) which symbolized the bolts connecting the sections. However, it shows that the
constrained Finite Strip Method (cFSM), which gives the participation of the buckling modes,
cannot generate and classify the results. In the interest of running cFSM and obtaining the per-
centage of each buckling modes, the sectors has been studied separately. The bolts, which have
a partial rigid behavior, are simulated by adding springs that are connected to the lips and the
“ground” (origin in the coordinated system).
Slenderness of the member.
There are different types of failure that occur for steel structure elements, such as instability
failure and material / plastic failure. The material failure is considered for the range of (0 to 0.2)
for non-dimensional slenderness λ̅ , whereas the reduction factor X is equal to the value of 1.
However, the instability failure is considered for range of (0.2 to 3) for non-dimensional slen-
derness λ̅, see figure 11. The higher cross-sectional area the element takes the lower slenderness
λ̅ the element tends to have, which leads to a higher value for the reduction factor χ and thus
a higher buckling resistance. However, the higher Euler’s critical length the element takes the
higher slenderness λ̅ element tends to have, which leads to a lower reduction factor and thus
more less buckling resistance.

Figure 11: reference European design buckling curve
The concerned range of non- dimensional slenderness regarding this study is between {0, 65 to
1, 25}, considering that these following range is commonly used for thin walled cross sectional
profiles. The critical buckling length is varied between polygonal profiles based on the chosen
geometry of the polygonal cross section. Also by assuming a certain value of the slenderness, the
critical length of which value of slenderness is employed, be obtained by using these following
equations which are derived for the equations above.
Creating polygonal profiles of interest
Below follows description of the Matlab scripts what has created and used for this study.
Pcoords.m
Initially, the study has performed by using Matlab and creating a script named
pcoords.m that examines a geometric problem and has a function to calculate x and y coordinates
of one sector of a semi-closed polygonal cross-section. All cross-sections are consisting of three
identical sector, 120 degrees each.
To start with, the characteristics of the radius (R), thickness of the profile (t), thickness of the
gusset plate (tg) and the length of the lip (l_lip) has calculated and rounded up to obtain the full
number.
Polygon sector:

17
The sectors differ due to the number of corners they have. The geometry described below is for
a nonagonal, the others two types of sectors are obtained similarly but with different number of
the angles theta and phi. Phi are corners in respect to the x-axis which is the inclination of the
vector for the points of the corner. The thetas are corners to one edge of the polygon and thus
the relative angle between the vectors. The x and y coordinates for the corners of the polygonal,
see figure 12 and 13, are calculated by multiple the radius R with cos (phi) and sin (phi) respec-
tively. The origin of the coordinate system is in the middle of the circle with the diameter d.
Figure 12: x and y coordinates of a nonagonal sector
Figure 13: Points on the bends

Bends:
The corners have a shape of an arc and is calculated from the geometry. The bending radius is
obtained by the thickness of the plate multiplied by a certain coefficient which is chosen to be 6
in this study.
To find the center of the bending arc (xc and yc), the distance lc between bending center and the
corners should be obtained, see figure 14. Furthermore, the x and y coordinates of the points
along the arc, are generated of a loop function.
Figure 14: bends

19
Start-end extensions:
The same approach and methodology as in bends, is applied for calculating of the extensions
center coordinates and hence x and y coordinates along the extensional arc, which has smaller
bending radius than the bends of the rest of the profile. After that, x and y coordinates can be
collected and plotted for the profile.
The obtained geometry for the sectors belonging to hexagonal, nonagonal and dodecagonal are
shown in the figures below 15 to 17.
Figure 15: One sector of hexagonal cross-section
Figure 16: One sector of nonagonal cross-sections

Figure 17: One sector of dodecagonal cross-section
Polygoner.m
This script calls previous script and makes a series of polygonal cross sections with different
numbers of corners, diameter thickness and slenderness. Furthermore, following input arguments
is giving:
nrange (number of corners), drange (cross-sectional diameter), Slendrange (slenderness of the
cross-section, i.e. a ratio between thickness and diameter of the profile). The class limits is chosen
between 3 & 4, that gives slenderness 70 to 150 according to the formula
D/t=90ε2
which is based on EC3 part 1-1 [3]. These three parameters are range of values, see
table 1. The others are single values which can be locked as certain values.
The rest of the input parameters are as following: rcoef (Bending arc radius to thickness ratio), fy
(yield strength), nbend (number of points along the bending arcs), l-ratio ( ratio between diam-
eter and extension length) and t-ratio (ratio of thickness of the gusset plate and thickness of the
plate of the sector), lambda is the slenderness of the flexural buckling for the column
The following values has chosen for the input arguments
Table 1: Inputs values for the script.
Parameter value
number of corners 6, 9 and 12
diameter 300,500,700 and 900 mm
slenderness 70 to 150 with an increase of
10

21
Bending radius to thickness ratio 6
yield strength 355 MPa
number of points along the bending arcs 4
Ratio between the diameter and the lip extension length 0,14
Ratio between thickness of the gusset plate and the plate
of the sector
1,2
The return of the output is given by the [profiles and meta data]. Furthermore, the profile ini-
tializes a cell array which hosts data for x and y coordinates of the polygonal profiles and thus are
stored in a database called profile. The metadata initializes a cell array to host meta data which
has one more dimension then the profiles data. Similar dimensions for the two different databases
[profiles and meta data] are number of corners, diameter and slenderness. However, in meta data
an additional dimension lambda, is added which takes into consideration the flexural buckling
of the column. The characteristic variables are obtained by a function loop of the values within
the given ranges of both [profiles and meta data].
The cross-section properties area A and moment of inertia I in the strong and weak axis are
obtained by calling a function in CUFSM called cutwp_prop2 which has nodes and elements as
an input and returns cross-section properties. This function returns to the meta data more prop-
erties, such as the center of gravity, the shear center and the rotation moment of inertia. How-
ever, only the needed properties A and I is collected and stored in the meta database. Nodes and
elements for input to this function are obtained for the whole profile, including all three sectors,
which gives area and moment of inertia for the whole cross section. These parameters are needed
in the interest of calculating the slenderness of the member.
Scripts that execute CUFSM of polygonal profiles of interest
Polygoner CFSM.m
In order to execute CUFSM of polygonal profiles, one have to create scripts that match the
Matlab script files (m-files) downloaded from CUFSM 4 open source. Polygoner CFSM is a
script created to analyze the profiles and return curves, shapes and classification (clas). This script
corresponds to the input which is required of CUFSM in order to run the analysis.
The 4D cell array 'meta' is converted to 3D cell array. By applying it, the lambda of different
lengths will be neglected. The CUFSM analyses for many sub-lengths for which the eigenvalues
are calculated. The highest slenderness will be selected in favor of avoiding to rerun the analysis
for different physical lengths and also will return the same signature curves. Concerning hosting
the results, the curves, shapes and classification should be initialized to cell array with the same
amount of cells.

For node data you have to define a column of node numbers and for each node give nodal
coordinates (x and y coordinates). Degree of freedom, dof, must also be defined for x-direction,
y-direction, z-direction and rotation. The values 1, for free, and 0, for fixed, can be chosen,
however, 1 is generally used. Stress at the node must also be defined where 1.0 is used to ignore
or create stress distribution of interest. Regarding this study 100 MPa has been chosen.
For element data you have to enter element number and node i and node j which gives how
the elements are connected. Furthermore, the thickness for each element and material number
must be given. The material number refers back to the generally property input.
Material properties allow to define material number, young’s modulus E, Poisson’s ratio v and
shear modulus G. Moreover, the material properties have to be constructed to the prop array.
The selection of the boundary conditions for these studies would be the signature curve (tradi-
tional solution) which is a special case of the boundary conditions. Whereas the longitudinal
shape function terms are orthogonal and is separable and thus the problem may be approached
as a series of q separate solutions. The longitudinal term is employed as m =1 and the boundary
conditions is set as simply supported (S-S). Usually the lengths in the signature curve are distin-
guished as a sweep of half-wavelengths. Thus, the solution of signature curve is in terms of load
factor versus half-wavelengths. A large number of the half-wavelength, which is known as half
sine wave, are analyzed in order to understand the different possible buckling shapes. Moreover,
the number of eigenvalues are specified as 10 eigen solutions in order to ensure that the solution
includes accurately all the three buckling modes (local, distortional and global modes). Concern-
ing interfacing to the CUFSM data to obtain participation of different buckling modes, the initial
GBT parameters for the unconstrained analysis has been defined.
For the investigated model there are no constraints acting on the cross sectional profile, which
leads to the constraints are set to zero. The analysis has approached differently by replacing the
constraints with springs and hence run the script in order to obtain the participation of different
buckling mode which have not generated by applying constraints.
There are three different springs that are applied at the end of the extensions parts of the cross-
section with respect to the three degrees of freedom. The parallel project, which run by other
students, has among other purposes to run different models in Abaqus to obtain the translations
springs (k1) and the significant value of the translational springs (k2) thus the data would be
finalized for CUFSM. The rotational springs is assumed to be neglected, considering the value
of the moment of the rotational springs are significantly small. However, the values of the springs
have not received from the parallel project before the appointed time for the presentation of our
master’s thesis. By confronting these issue, the limitation of running with a certain value of
springs has to be done in order to proceed with analyses. The value of the spring has a good
approximation which is 100 kN. In addition to that, the translation springs breaks into two
components with an angel of pi/6.
Springs in the required input for CUFSM are as following. Each spring is the node number
where the spring will act (node#). The global degree of freedom in which the spring will act
(DOF,x=1,z=2,y=3, q=4). The stiffness of the spring (kspring) and lastly the ‘kflag’ is to indicate
if the entered value is the total stiffness (0) or a foundation (1). Here the rotation (q=4) is not
considered and ‘kflag’ is chosen to be 0.

23
To run and obtain the curves and shapes for all the profiles of interest a function loop has created
which in turn calls the strip.m, a script among CUFSM 4 m-files, with all the parameter above
as an input. The classification analysis for all the profiles are run by a separate loop that calls a m-
file called mode_class.m with several input data including mode which in turn calls shapes. All
the input are found as m-files in in CUFSM files.
Min finder.m
Min finder is a script that extracts minimum values of the load factor and their corresponding
half-wavelength. Furthermore, it collects the participation values for the half-wavelength where
the minimum values occur.

4 Results
All results that are obtained by running CUFSM analysis in the Matlab software, i. e. minimum
load factor and the corresponding half-wavelengths, participation percentage of the local (L)
buckling mode, distortional (D) buckling mode, global (G) buckling mode, are all presented as
tables in Appendix A. These CUFSM analysis are run by a start value of the spring and cases
where the spring value are increased and decreased respectively. Furthermore, a graphic presen-
tation of the results of the half-wavelength for each diameter and slenderness are presented for
hexagonal, nonagonal and dodecagonal cross-section can be found in the analysis chapter.

25
5 Analysis
This chapter presents the results obtained from CUFSM analysis. Half-wavelengths for each
diameter and slenderness are presented for hexagonal, nonagonal and dodecagonal cross-section.
The rest of the results, the percentage of the buckling modes, i.e. local (L), distortional (D),
global (G) and the corresponding minimum load factor are presented as tables in Appendix A.
5.1 Spring value - 100 kN (approximated value)
These first results are calculated with a spring value of 100 kN. This value is approximated and
is chosen as a start value.
Hexagonal
For all profiles that belong to hexagonal, are shown that the distortional buckling mode is pre-
dominate. The corresponding half-wavelengths are between 125 to 450 mm. This values can be
considered as short half- wavelengths which exhibit that local buckling mode has more partici-
pation percentage than the global buckling mode. The higher range cross-sectional slenderness
gives a lower minimum load factor and the lower slenderness range gives the highest half-wave-
length due to impact of a higher moment of inertia as profiles thickness increases. Moreover,
increasing of the diameter results in to a higher half-wavelength due to higher moment of inertia
of the cross-section. The range from the lowest slenderness (70) to the highest slenderness (150)
gives almost four times decreasing of the minimum load factor, see figure (15) and Appendix
Figure 15: Hexagonal cross-section with spring value of 100 kN

Nonagonal
The distortional buckling mode is predominating in all nonagonal profiles by applying spring
values of 100 kN, however with the exception of profile (2,4,1) which demonstrates more gov-
erning global buckling behavior with a half-wavelength up to 3,8 meter. Another deviating
profile is the profile (2,3,3) which has 100% local buckling mode with a considerably short half-
wavelength. The lowest slenderness value for profile with diameter 500, 700 and 900 mm are
deviating remarkably from the rest of slenderness values, in regards to participation of the buck-
ling mode and the corresponding half-wavelength. The profiles which have the lowest slender-
ness are the most critical profiles, due to the lowest corresponding minimum load factor. Fur-
thermore, for the same diameter, the half-wavelength is almost in the same range excluding the
lowest value of slenderness as it mentioned previously. The range from the lowest slenderness
(70) to the highest slenderness (150) gives about three to four times decreasing of the minimum
load factor.
Figure 16: Nonagonal cross-section with spring value of 100 kN
Dodecagonal
The same behavior is observed for the dodecagonal profiles as for hexagonal and nonagonal i.e.
the distortional buckling mode is predominating in all profiles. Furthermore, the lower slender-
ness range has higher corresponding half-wavelength then the higher range (see figure 17). The
most critical profile is profile (3,3,9), due to the lowest corresponding minimum load factor (See
Appendix A). Moreover, the range from the slenderness (70) to the highest slenderness (150)
gives about two to two and half times decreasing of the minimum load factor.

27
Figure 17: Dodecagonal cross-section with spring value of 100 kN
5.2 Increasing of the spring value
The CUFSM analysis is as well run for a higher spring value due to that the chosen start value
is an approximated one and there is a possibility that this approximated value is lower than the
actual spring value.
Increasing of the spring value by 100 %
Hexagonal
The minimum load factor is more or less identical for hexagonal profiles with a spring value of
100 kN and 200 kN. The corresponding half-wavelength in this two cases are the same for
higher slenderness range and slightly lower for lower slenderness range. The distortional mode
participation is reduced by few percentage points and the local mode participation are increased
by few percentage points. That depends on increasing of the spring values which results in that
profiles get likely more resistance for distortional buckling and be more susceptible to local buck-
ling (see figure 18 and appendix A)

Nonagonal
An increasing by 100% of the spring value for nonagonal profiles leads to a lower participation
percentage for global buckling mode in the profiles. However, the participation percentage for
distortional buckling is considerably higher than the case with 100 kN. For instance, profile
(2.4.1) get distortional failure whereas it fails in global buckling for the assumed spring value.
Decreasing and increasing for the percentage of distortional and local participation mode respec-
tively, occur for short half-wavelengths. Moreover, the profile (2,3,3) fails with 100 % local
buckling as previously (see figure 19 and Appendix A).

29
Dodecagonal
The noticeable difference for an increasing of the spring value is for profile (3,2,1) which shows
100 % participation of local buckling with a half-wavelength of 549 mm (see figure 20). Fur-
thermore, decreasing and increasing of the percentage for the distortional and local participation
mode respectively, occur at short half-wavelengths (Appendix A).
Figur 20: Dodecagonal cross-section with spring value of 200 kN
Increasing of the spring value by 200 %.
Hexagonal
The minimum load factor is more or less identical for hexagonal profiles with a spring value of
100 kN and 300 kN. The corresponding half-wavelength are the same for higher slenderness
range and slightly lower for lower slenderness range. The distortional participation percentage is
reduced with few percentage points. This behavior is assumed to be influenced by increasing of
the spring values which results in that the profiles get likely more resistance regarding distortional
buckling and be more susceptible to local buckling (see figure 21 and Appendix B).

Nonagonal
percentage for global buckling mode in the profiles whereas the participation percentage for
distortional buckling is considerably higher for the approximated spring value. For instance, pro-
file (2.4.1) reaches distortional failure whereas it fails in global buckling for the approximated
spring value. Furthermore, similar decreasing and increasing for the percentage for the distor-
tional and local participation mode respectively, occur for short half-wavelengths. Additionally,
the profile (2,3,3) leads to failure mode with 100 % local buckling as previously case (see figure
19 and Appendix A).

31
Figure 22: Nonagonal cross-section with spring value of 300 kNs
Dodecagonal
The distinct difference for an increasing of the spring value is for profile (3,2,1) which shows
100 % participation of local buckling with a half-wavelength of 549 mm (see figure 19 and
Appendix A). Moreover, decreasing and increasing of the percentage for the distortional and
local participation mode respectively, occur at short half-wavelengths. The minimum load factor
increases slightly or stays the same whereas the corresponding half-wavelength decreases for same
profiles and are identical for others (Appendix A).

Increasing of spring value by 300 %
Hexagonal
The minimum load factor is slightly higher for hexagonal profiles with a spring value of 400 kN
in comparison to the approximated value 100kN. The distortional mode participation is reduced
by few percentage points and the local mode participation are increased by few percentage points.
That is dependent on increasing of the spring values which results in that profiles get likely more
resistance for the distortional buckling and be more susceptible to the local buckling. The cor-
responding half-wavelength in this two cases are the same for higher slenderness range and
slightly lower for lower slenderness range. Furthermore, the same buckling mode behavior oc-
curs for the two cases ( see figure 24 and Appendix A )
Nonagonal
distortional buckling is significantly higher for the approximated value of springs. For instance,
profile (2.4.1) reaches majorly distortional failure whereas it leads to global mode failure for the
approximated spring value. Similar decreasing and increasing of the percentage for distortional
and local participation mode respectively, occur for short-wavelengths.

33
Dodecagonal
The noticeable difference for an increasing of the spring value is for profile (3, 2, 1) which shows
100 % participation of local buckling with a half-wavelength of 549 mm (see figure 26). Fur-
thermore, decreasing and increasing of the percentage for distortional and local participation
mode respectively, occur at short half-wavelengths. The minimum load factor increases slightly
whereas the corresponding half-wavelength decreases for same profiles and stays the same for
others (see Appendix A).

5.3 Decreasing of the spring value
Likewise, an increasing of the spring value is necessary, a decreasing of the spring value is of in-
terest to understand how such case effect The CUFSM analysis.
Decreasing spring value by 50 %
Hexagonal
The minimum load factor is to some extent reduced for hexagonal profiles with a spring value
of 50 kN in comparison to 100 kN. The corresponding half-wavelength in this case is slightly
higher or stays the same in regards to the first case (100 kN). The distortional mode participation
is increased by few percentage points and the local mode participation are decreased by few
percentage points which results in that profiles get mainly more susceptible for distortional buck-
ling and more resistance to local buckling (see figure 27 and Appendix A).

35
Nonagonal
A decreasing by -50% of the spring value for nonagonal profiles leads to a lower participation
global buckling is likely higher for the approximated value of springs. The minimum load factor
decreased at some extent whereas the corresponding half-wavelength increases marginally for
slenderness 70 & 80 and stays the same for the rest. The distortional buckling mode participation
is predominate for all profiles except for profiles (2,1,1), (2,3,1), (2,3,2), (2,4,1) and (2,4,2). These
profiles are dominated by global buckling mode with around 70 %. Profile (2,3,3) shows pure
local mode behavior with 100 percentage.

The figure below shows a reduced z-axis.
Figure 2: Nonagonal cross-section with spring value of 50 kN (reduced z-axis)

37
Dodecagonal
The minimum load factor decreased at some extent whereas the corresponding half-wavelength
increases or stays the same (see Appendix A). The distortional buckling mode participation is
predominate for all profiles except for profiles (3,2,1), (3,3,2), (3,3,1), (3,4,1),(3,4,2) and (3,4,3)
which dominated by global buckling mode(see figure 30). Furthermore, increasing and decreas-
ing of the percentage of the distortional and local participation mode respectively, occur at short
half-wavelengths.
Decreasing of spring value by 66,6 %
Hexagonal
of 66,6 kN in comparison to hexagonal profiles with 100 kN. The corresponding half-wave-
length in this case is slightly higher or stays the same in regards to the first case (100 kN). The
distortional mode participation is increased by few percentage points and the local mode partic-
ipation are decreased by few percentage points which results in that profiles get likely more
susceptible for distortional buckling and more resistance to local buckling. The distortional buck-
ling mode participation is predominating for all profiles except for profile (1,4,1), see figure 31.
This profile behavior is influenced and dominated by global buckling mode which leads to a
significantly high wave length (see Appendix A)

Figure 31: Hexagonal cross-section with spring value of 33,3 kN
Nonagonal
A decreasing by 66,6% of the spring value for nonagonal profiles leads to a lower participation
percentage for distortional buckling mode in the profiles whereas the participation percentage
for global buckling is considerably higher for the approximated value of springs. The minimum
load factor decreased at some extent whereas the corresponding half-wavelength increases mar-
ginally for slenderness 70 & 80 and stays the same for the rest of the profiles (see Appendix A).
The distortional buckling mode participation is predominating for all profiles, except for profiles
(2,1,1), (2,1,2), (2,2,1), (2,2,2), (2,3,1), to (2,3,3) and (2,4,1) to (2,4,3) (see figure 32 and 33).
These profiles are dominated by global buckling mode with a range between 50 and 70 %.
Furthermore, profile (2,4,4) shows pure local mode behavior with 100 percentages.

39
Figure 32: Nonagonal cross-section with spring value of 33,3 kN
Figure 33: Nonagonal cross-section with spring value of 33,3 kN (reduced z-axis)

Dodecagonal
predominate for all profiles except for profiles (3,2,1), (3,2,2),(3,3,1), to (3,3,3) and (3,4,1) to
(3,4,4) which are dominated by global buckling mode(See figure 32). Furthermore, similar in-
creasing and decreasing of percentage of the distortional and local participation mode respec-
tively, occur at the short half-wavelengths.
Figure 32: Dodecagonal cross-section with spring value of 33.3 kN
Figure 34: Dodecagonal cross-section with spring value of 33,3 kN (reduced z-axis)

41
Decreasing of spring value by 75 %
Hexagonal
of 25 kN in comparison to hexagonal profiles with 100 kN. The corresponding half-wavelength
in this case is slightly higher or stay the same in regards to the first case with100 kN, see Appendix
A. The distortional mode participation is increased by few percentage points and the local mode
participation are decreased by few percentage points which results in that profiles get likely more
susceptible for distortional buckling and more resistance to local buckling. The distortional buck-
ling mode participation is predominating for all profiles except for profiles (1,2,1), (1,3,1) and
(1,4,1) which are majorly influenced by global buckling mode and thus leads to significantly high
corresponding half- wavelengths (see figure 35 and 36).

Figure 36. Hexagonal cross-section with spring value of 25 kN (reduced z-axis)
Nonagonal
A decreasing by 75% of the spring value for nonagonal profiles leads to a lower participation
percentage for distortional buckling mode in the profiles. However, the participation percentage
for global buckling is significantly higher for the approximated value of springs. The minimum
load factor decreases at some extent whereas the corresponding half-wavelength increases mar-
ginally for slenderness 70 & 80 and stays the same for the rest of profiles (see Appendix A). The
distortional buckling mode participation is predominate for all profiles except for profiles (2,1,1),
(2,1,2), (2,2,1), (2,2,2),(2,3,1) to (2,3,4) and(2,4,1) to (2,4,4) (see figure 37 and 38). These pro-
files are dominated by global buckling mode with a range between 71 and 93 %. No sign of pure
local mode for any profile due to influence of non-rigid behavior as the spring values decrease.

43
Figure 38: Nonagnal cross-section with spring value of 25 kN (reduced z-axis)
Dodecagonal

predominate for all profiles except for profiles (3,1,1), (3,1,2), (3,2,1) to (3,2,4), (3,3,1) to
(3,3,5)and (3,4,1) to (3,4,5) which are dominated by global buckling mode (see figure 39). Fur-
thermore, increasing and decreasing in the hexagonal and nonagonal profiles for the percentage
of the distortional and local participation mode respectively, occur at short half-wavelengths.
Figure 40: Dodecagonal cross-section with spring value of 25 kN (reduced z-axis)

45
5.4 Summary of profiles of interest
The following profiles are of interest due to these profiles show a different behavior than distortional buckling
predominating behavior (see the table 2)
Table 2: summery of profiles that fails in local and global mode
Profile Diameter
(mm)
Slenderness Minimum
load factor
Half-
Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
100 kN
(2,3,3 ) 700 90 22,82 220,20 0,00 0,00 100,00 0,00
(2,4,1 ) 900 70 27,83 3798,00 70,41 28,63 0,58 0,38
Increasing by 100 %
(2,3,3 ) 700 90 23,27 220,20 0,00 0,00 100,00 0,00
(3,2,1 ) 500 70 35,43 548,50 0,00 0,00 100,00 0,00
Increasing by 200 %
(2,3,3 ) 700 90 23,42 220,20 0,00 0,00 100 0,00
(3,2,1 ) 500 70 37,42 548,50 0,00 0,00 100,00 0,00
Increasing by 300 %
(3,2,1 ) 500 70 38,61 548,50 0,00 0,00 100,00 0,00

Profile Diameter
(mm)
Slenderness Mini-
mum
load fac-
tor
Half-Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
Decreasing by 50%
(2,2,1 ) 500 70 26,27 2153,90 71,74 27,39 0,51 0,37
(2,3,1 ) 700 70 22,50 3108,20 74,09 25,03 0,52 0,36
(2,3,2 ) 700 80 23,41 3106,40 71,53 26,51 0,61 0,35
(2,3,3 ) 700 90 21,93 220,20 0,00 0,00 100,00 0,00
(2,4,1 ) 900 70 19,95 4215,70 76,08 23,41 0,44 0,34
(2,4,2 ) 900 80 20,68 4214,10 73,65 25,47 0,55 0,32
(3,2,1 ) 500 70 26,12 668,00 51,59 46,87 0,28 1,26
(3,1,1 ) 300 70 28,52 358,90 34,47 63,13 0,36 2,05
(3,1,2) 300 80 28,52 358,90 34,47 63,13 0,36 2,05
(3,4,1 ) 900 70 20,14 4266,90 77,57 20,78 0,67 0,97
(3,4,2 ) 900 80 20,74 1351,30 56,68 41,97 0,40 0,95
(3,4,3 ) 900 90 19,97 1350,50 49,55 49,22 0,13 1,09
Decreasing by 66,6%
(1,4,1 ) 900 70 15,80 4526,20 78,04 21,28 0,51 0,17
(2,1,1 ) 300 70 31,90 87,80 6,40 60,56 24,86 8,18
(2,1,2) 300 80 31,90 87,80 6,40 60,56 24,86 8,18
(2,2,1 ) 500 70 21,58 2153,90 75,69 23,51 0,51 0,39
(2,2,2 ) 500 80 21,88 2374,80 71,19 27,93 0,58 0,30
(2,3,1 ) 700 70 18,32 3441,20 77,91 21,36 0,42 0,31
(2,3,2 ) 700 80 19,25 3439,30 74,47 24,68 0,55 0,30
(2,3,3 ) 700 90 19,72 3438,60 72,92 26,20 0,58 0,29
(2,4,1 ) 900 70 16,11 4679,30 80,22 19,07 0,42 0,29
(2,4,2 ) 900 80 16,85 4677,50 77,25 22,00 0,47 0,28
(2,4,3 ) 900 90 17,57 4675,40 74,36 24,82 0,55 0,27
(2,4,4 ) 900 100 17,57 310,30 0,00 0,00 100,00 0,00
Decreasing by 75%

47
(1,2,1 ) 500 70 18,20 2300,40 75,95 23,33 0,52 0,20
(1,3,1 ) 700 70 15,37 3329,8 79,92 19,35 0,53 0,20
(1,4,1 ) 900 70 13,44 4526,20 82,62 16,62 0,52 0,18
(2,1,1 ) 300 70 24,70 1311,70 71,57 27,50 0,57 0,36
(2,1,2) 300 80 24,70 1311,70 71,57 27,50 0,57 0,36
(2,2,1 ) 500 70 18,60 2376,60 78,23 20,99 0,45 0,33
(2,2,2 ) 500 80 19,92 2374,80 73,61 25,50 0,57 0,32
(2,3,1 ) 700 70 15,68 3441,20 81,67 17,59 0,41 0,32
(2,3,2 ) 700 80 16,65 3439,30 77,73 21,41 0,54 0,31
(2,3,3 ) 700 90 17,15 3438,60 75,84 23,28 0,57 0,31
(2,3,4 ) 700 100 17,62 3805,90 93,23 25,95 0,57 0,24
(2,4,1 ) 900 70 13,70 4679,30 84,13 15,16 0,41 0,30
(2,4,2 ) 900 80 14,45 4677,50 80,93 18,31 0,47 0,29
(2,4,3 ) 900 90 15,20 4675,40 77,58 21,60 0,54 0,28
(2,4,4 ) 900 100 15,58 4674,40 75,89 23,20 0,63 0,28
(3,1,1 ) 300 70 23,97 394,10 49,46 48,77 0,42 1,35
(3,1,2) 300 80 23,97 394,10 49,46 48,77 0,42 1,35
(3,2,1 ) 500 70 18,76 2405,30 79,58 18,80 0,64 0,98
(3,2,2 ) 500 80 19,83 736,50 55,60 42,95 0,46 0,99
(3,2,3 ) 500 90 19,16 812,30 50,41 48,51 0,14 0,94
(3,2,4 ) 500 100 19,16 812,30 50,41 48,51 0,14 0,94
(3,3,1 ) 700 70 15,79 3483,10 82,81 15,61 0,60 0,97
(3,3,2 ) 700 80 16,80 3480,20 79,28 19,07 0,69 0,45
(3,3,3 ) 700 90 17,35 3478,90 77,57 20,73 0,75 0,94
(3,3,4 ) 700 100 17,13 1133,30 56,29 42,46 0,42 0,83
(3,3,5 ) 700 110 16,71 1132,90 51,13 47,78 0,16 0,92
(3,4,1 ) 900 70 13,79 4736,90 85,10 13,37 0,62 0,91
(3,4,2 ) 900 80 14,56 4733,90 82,26 16,21 0,62 0,90
(3,4,3 ) 900 90 15,35 4730,80 79,21 19,18 0,73 0,89
(3,4,4 ) 900 100 15,71 1663,80 63,39 35,58 0,46 0,57
(3,4,5 ) 900 110 15,19 1663,00 54,16 45,07 0,08 0,70

6 Conclusion
The objective of this thesis has investigated the behavior of polygonal cross-section with pure
compression. The study has comprised to only elastic buckling and the methodology is consisted
by using Finite Strip Method (FSM) and the general guidance of Eurocode. The Finite element
models has implemented with help of the CUFSM software. There is no available theoretical
formula to understand and calculate the resistance for the polygonal cross section in regards to
the distortional buckling resistance and the interaction between the different bucking modes.
This means that only the numerical result will be reliable for the thesis. Several important agree-
ments have demonstrated from these study by running the analysis and plotting the result con-
sidering the variation of the different cross-sectional parameters for every single polygonal profile.
The analyzed result has as aim to examine the interaction of the different buckling mode and
hence how the interaction introduces a certain unstable behavior for the polygonal member
during the elastic buckling failure. Additionally, in which bolt density the member performs yet
as rigid i.e. whole. The main conclusions that has drawn, is discussed as following below.
By analyzing and comparing results, the distortional buckling mode is governing as a buckling
failure, which occur and dominate for the approximated value (100 kN) and a similar behavior
occur, by increasing the spring values by 100, 200 and 300 %. However, the global buckling is
more predominate as the spring values decrease by 50, 66,6 and75 %. The behavior of the cross-
section is dependent on how the interaction of different buckling modes prevails at the corre-
sponding critical half-wavelength. Considering the predomination of distortional buckling
mode, indicates the most of polygonal cross-section do not behave as rigid, i.e. whole. The more
spring values increase, the lower distortional participation mode arises and also the higher local
participation mode arises at short half-wavelengths. A reducing of distortional mode and increas-
ing of local mode gives indication that the behavior of the cross-section has changed and turned
likely into more rigid and thus it is expected to behave more as whole cross-section. The contrary
result reveals by a decreasing the spring values. The behavior of the cross-section shows distinctly
by employing a different value of the springs that implies in plot figures and Appendix A.
The lower slenderness range gives higher half-wavelength. These type of slenderness tends to
generate higher global buckling. This is aligned with understanding of the theory in regards to
thin walled buckling behavior, due to the fact that the global buckling occur commonly for
higher half-wavelength. The global buckling mode has a greater impact as the spring values
increase and this reveals the influence of Euler’s buckling length. Due to the boundary condition
which in turn become more clamped as spring values increase. The contrary result indicates by
decreasing the spring values. Furthermore, a larger diameter yields larger half-wavelength due to
the fact that a higher diameter gives more slender member which leads to lower local mode but
higher distortional and global mode.

49
The critical half-wavelength for each profile illustrates the needed density between bolts on the
longitudinal part of the member. In the interest of eliminating distortional buckling failure, due
the fact that distortional buckling is unpredictable, the bolt-density should be lower than the
corresponding half-wavelength for the profile where the distortional mode is predominating.
The profiles that do not get to failure for distortional buckling mode are of interest, due to
understanding and familiarity about local and global buckling. These two buckling modes have
been studied in decades and are well-known how to approach a solution for them.

7 Future work
Using and comparing different types of structural steel e.g. in high strength. In favor of deter-
mining how the member, by applying different structural steel, behaves variously accounting to
the effect of different buckling modes.
Running nonlinear buckling analysis by using elastic plastic buckling finite element method
and the theory of nonlinear buckling, in order to observe the influence of the initial geometric
imperfection in the buckling behavior of the member.
Analyzing the whole polygonal section as a rigid body and classifying the participations of
buckling modes by applying constraints which visualize the bolts, in contrary to our analysis
where only one section has studied in each profile by means of springs. It cannot clearly be
managed in CUFSM. However, it can possibly be achieved by using other FEM programs.
Preforming laboratory test and comparing the laboratory values with achieved numerical values
from the strip method in order to validate our obtained results.
A further study should be made to create a new method to calculate theoretically the distor-
tional buckling resistances for the polygonal cross- sectional profiles to interfere the unpredicta-
ble behavior of the distortional buckling and validate the achieved numerical results.

51
8 References
[1] A. T. Tran, “Resistance of Circular and Polygonal Steel Towers for Wind Turbines”,
Luleå: Luleå University of Technology, 2014.
[2] S. R. Calinger, “Leonhard Euler: Mathematical Genius in the Enlightenment”,
Princeton University Press, 2015.
[3] CEN, Eurocode 3: design of steel structures, part 1-1: general rules., Brussels:
European committee for standardization, 2005.
[4] CEN, Eurocode 3: design of steel structures, part 1-3: general rules., Brussels:
European committee for standardization, 2002.
[5] Excerpted from Chapter 2 of B.W. Schafer, “Cold-Formed Steel Behavior and Design:
Analytical and Numerical Modeling of Elements and Members with Longitudinal
Stiffeners”, Ph.D. Thesis, 1998.
[6] EWEA, The European Wind Energy Association, “Wind in power: 2015 European
Statistics”, 2016.
[7] O. Garzon, “Resistance of polygonal cross-sections – application on steel towers for wind tur-
bines”, Luleå University of Tecknology, 2013.
[8] J. Strömberg & T. Höglund, “Kallformade profiler, Modul 7 2rd
”, Stockholm:
Stålbyggnadsinstitutet, Luleå Tekniska Universitet, Kungliga tekniska
Högskolan, 2006.
[9] M. R. Jones, “Buckling of Bars, Plates and Shells”, Bull Ridge Publishing, Blacksburg,
Virginia, 2006.
[10] J. Jonsson & M. Andreassen, “Distortional eigenmodes and homogeneous solutions
forsemi -discretized thin-walled beams”, Thin- Walled Structures,2011.
[11] S. Jovasevic, et al., “Global fatigue life modelling of steel half-pipes bolted
Connections”, Procedia Engineering, 2016.
[12] K. Hüsemann, “Ruukki Wind Towers: High truss towers for wind turbine generators”,
RUUKKI, Helsinki, 2010.
[13] B. L. Rekha & G. Kalurkar, “Study of Buckling Behaviour of Beam and Column
Subjected To Axial Loading for Various Rolled I Sections”, International Journal of

Innovative Research in Science, Engineering and Technology, 2014.
[14] B. W. Schafer & S. Ádány, “Buckling analysis of cold-formed steel members using
CUFSM:Conventional and constrained finite strip methods” Eighteenth International
Specialty Conference on Cold-Formed Steel Structures, Orlando, FL, 2006.
[15] B. W. Schafer& G. Hancock, “Distortional buckling of cold-formed steel columns”,
American Iron and Steel Institute, 2000, Revision 2006.
[16] J. M. Gere & S. P. Timoshenko, “Theory of Elastic Stability 2rd
”, New York:McGraw-
Hill Book Company, 1963.
[17] Thin-walled structures 2016-10-22. [Online]. Available:www.elsevier.com/locate/tws
[18] N. S. Trahair, “Flexural torsional buckling of structures”, The University of Sydney
Australia, 1996.
[19] V. Z. Vlasov, “Thin-walled elastic beams 2rd
”, Jerusalem, Israel: Israel Program for
Scientific Translations, 1961.
[20] V. Ungureanu, D. Dubina & M. Kotełko, “Design of cold formed steel structures”,
Eurocode 3 Design of cold formed steel structures, Part 1-3 – Design of Cold-formed
Steel structures 2012.
[21] B. W. Schafer & C. Yu, “Distortional buckling of cold-formed steel members in
bending”, American Iron and Steel Institute, 2005.
[22] W. W. Yu, “Cold-Formed Steel Design 3rd
”. John Wiley and Sons, New-York. Review,
2000.

53
APPENDIX A
This Appendix presents all results obtained from the FSM analysis performed by Matlab soft-
ware by taking consideration CUFSM 4 Matlab-scripts. The tables below show the parameters
for each type of polygonal cross-section. The profiles has the combination of (number of cor-
ners, diameter, slenderness). For each profile the type, diameter and slenderness are presented.
Furthermore, minimum load factor and the corresponding half-wavelength and also the per-
centage of the buckling modes, i.e. local (L), distortional (D), global (G) and others are pre-
sented.
8.1 Approximated spring value - 100 kN
The assumed spring value is chosen to be 100 kN and this value is approximated to be roughly
expected forces from the bolt
Hexagonal
Table 1 Approximated Spring value – 100 kN for hexagonal section
Hexagonal
Profile Diameter
(mm)
Slen-
derness
Minimum
load factor
Half-
Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
(1,1,1 ) 300 70 17,13 137,00 11,33 51,94 30,48 6,25
(1,1,2) 300 80 17,13 137,00 11,33 51,94 30,48 6,25
(1,1,3 ) 300 90 12,16 124,90 8,14 52,15 34,17 5,54
(1,1,4) 300 100 12,16 124,90 8,14 52,15 34,17 5,54
(1,1,5 ) 300 110 7,95 124,90 6,04 52,92 37,53 4,5
(1,1,6 ) 300 120 7,95 124,90 6,04 52,92 37,53 4,5
(1,1,7 ) 300 130 4,55 113,79 3,71 52,87 40,20 3,22
(1,1,8 ) 300 140 4,55 113,79 3,71 52,87 40,20 3,22
(1,1,9 ) 300 150 4,55 113,79 3,71 52,87 40,20 3,22
(1,2,1 ) 500 70 19,95 241,60 13,48 56,05 24,99 5,48
(1,2,2 ) 500 80 13,76 219,00 9,35 55,28 30,19 5,18
(1,2,3 ) 500 90 11,06 219,10 8,16 54,32 32,68 4,85
(1,2,4 ) 500 100 11,06 219,10 8,16 54,32 32,68 4,85
(1,2,5 ) 500 110 8,58 198,60 6,17 54,35 35,11 4,37

(1,2,6 ) 500 120 6,39 198,70 5,08 53,72 37,41 3,78
(1,2,7 ) 500 130 6,39 198,65 5,08 53,72 37,41 3,78
(1,2,8 ) 500 140 4,49 198,66 3,94 53,56 39,39 3,11
(1,2,9 ) 500 150 4,49 198,66 3,94 53,56 39,39 3,11
(1,3,1 ) 700 70 18,67 323,40 11,80 59,78 23,65 4,78
(1,3,2 ) 700 80 14,34 323,40 10,12 58,01 27,18 4,68
(1,3,3 ) 700 90 12,35 323,50 9,27 57,07 29,10 4,57
(1,3,4 ) 700 100 10,51 292,30 7,30 56,97 31,33 4,39
(1,3,5 ) 700 110 8,79 292,30 6,58 55,95 33,31 4,16
(1,3,6 ) 700 120 7,21 292,30 5,81 55,12 35,22 3,85
(1,3,7 ) 700 130 5,77 292,30 5,00 54,55 36,99 3,47
(1,3,8 ) 700 140 5,77 292,30 5,00 54,55 36,99 3,47
(1,3,9 ) 700 150 4,46 292,34 4,15 54,30 38,54 3,01
(1,4,1 ) 900 70 17,94 414,70 11,13 62,41 22,24 4,22
(1,4,2 ) 900 80 14,56 414,70 9,91 60,81 25,05 4,23
(1,4,3 ) 900 90 11,57 414,70 8,71 58,99 28,15 4,15
(1,4,4 ) 900 100 10,20 414,70 8,09 58,09 29,76 4,05
(1,4,5 ) 900 110 7,63 373,90 5,91 56,95 33,40 3,75
(1,4,6 ) 900 120 6,48 373,90 5,35 56,13 35,00 3,53
(1,4,7 ) 900 130 5,42 373,87 4,75 55,50 36,50 3,25
(1,4,8 ) 900 140 5,42 373,87 4,75 55,50 36,50 3,25
(1,4,9 ) 900 150 4,44 373,88 4,14 55,07 37,86 2,92

55
Nonagonal
Table 2 Approximated Spring value – 100 kN for nonagonal section
Nonagonal
Profile Diameter
(mm)
Slenderness Minimum
load factor
Half-
Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
(2,1,1 ) 300 70 31,50 87,80 6,30 63,58 22,52 7,61
(2,1,2) 300 80 31,50 87,80 6,30 63,58 22,52 7,61
(2,1,3 ) 300 90 22,07 87,80 4,33 72,05 17,86 5,76
(2,1,4) 300 100 22,07 87,80 4,33 72,05 17,86 5,76
(2,1,5 ) 300 110 14,24 87,70 2,71 82,10 11,16 4,02
(2,1,6 ) 300 120 14,24 87,70 2,71 82,10 11,16 4,02
(2,1,7 ) 300 130 8,07 87,71 1,42 92,87 3,29 2.42
(2,1,8 ) 300 140 8,07 87,71 1,42 92,87 3,29 2.42
(2,1,9 ) 300 150 8,07 87,71 1,42 92,87 3,29 2.42
(2,2,1 ) 500 70 33,97 599,40 42,21 55,95 0,31 1,53
(2,2,2 ) 500 80 25,36 151,10 4,83 71,88 17,48 5,82
(2,2,3 ) 500 90 20,21 151,00 3,80 76,20 14,97 5,02
(2,2,4 ) 500 100 20,21 151,00 3,80 76,20 14,97 5,02
(2,2,5 ) 500 110 15,55 151,00 2,82 82,04 11,11 4,03
(2,2,6 ) 500 120 11,50 151,00 2,04 88,23 6,57 3,16
(2,2,7 ) 500 130 11,50 150,97 2,04 88,23 6,57 3,16
(2,2,8 ) 500 140 8,04 150,94 1,33 93,41 2,92 2,29
(2,2,9 ) 500 150 8,04 150,94 1,33 93,41 2,92 2,29
(2,3,1 ) 700 70 30,86 916,20 41,58 50,19 0,15 1,08
(2,3,2 ) 700 80 26,57 220,20 4,77 75,22 14,87 5,14
(2,3,3 ) 700 90 22,82 220,20 0,00 0,00 100,00 0,00
(2,3,4 ) 700 100 19,34 220,20 3,34 79,83 12,49 4,30
(2,3,5 ) 700 110 16,11 220,10 2,77 82,58 10,86 3,79

(2,3,6 ) 700 120 13,15 220,10 2,25 86,83 7,63 3,29
(2,3,7 ) 700 130 10,45 198,76 1,97 88,42 6,62 2,99
(2,3,8 ) 700 140 10,45 198,76 1,97 88,42 6,62 2,99
(2,3,9 ) 700 150 8,02 198,74 1,44 93,29 2,90 2,37
(2,4,1 ) 900 70 27,83 3798,00 70,41 28,63 0,58 0,38
(2,4,2 ) 900 80 26,99 279,70 4,78 77,96 12,68 4,58
(2,4,3 ) 900 90 21,41 279,60 3,79 79,66 12,28 4,27
(2,4,4 ) 900 100 18,81 279,60 3,31 81,20 11,47 4,02
(2,4,5 ) 900 110 14,00 279,50 2,40 86,00 8,26 3,33
(2,4,6 ) 900 120 11,84 279,50 2,01 89,13 5,89 2,97
(2,4,7 ) 900 130 9,84 279,41 1,64 91,91 3,89 2,56
(2,4,8 ) 900 140 9,84 279,41 1,64 91,91 3,89 2,56
(2,4,9 ) 900 150 8,01 279,37 1,28 93,82 2,76 2,14

Table 3 Approximated Spring value – 100 kN For dodecagonal section
Dodecagonal
Profile Diameter
(mm)
Slenderness Minimum
load factor
Half-
Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
(3,1,1 ) 300 70 32,24 358,90 20,57 76,23 0,67 2,53
(3,1,2) 300 80 32,24 358,90 20,57 76,23 0,67 2,53
(3,1,3 ) 300 90 28,25 393,80 14,21 82,18 1,21 20,39
(3,1,4) 300 100 28,25 393,80 14,21 82,18 1,21 20,39
(3,1,5 ) 300 110 22,76 73,30 3,09 56,33 26,94 13,63
(3,1,6 ) 300 120 22,76 73,30 3,09 56,33 26,94 13,63
(3,1,7 ) 300 130 13,03 73,26 1,48 78,57 11,00 8,95
(3,1,8 ) 300 140 13,03 73,26 1,48 78,57 11,00 8,95
(3,1,9 ) 300 150 13,03 73,26 1,48 78,57 11,00 8,95
(3,2,1 ) 500 70 31,17 605,30 35,34 62,45 0,24 1,97
(3,2,2 ) 500 80 27,76 667,40 25,36 71,82 0,83 1,99
(3,2,3 ) 500 90 25,73 736,10 20,06 76,85 1,26 1,83
(3,2,4 ) 500 100 25,73 736,10 20,06 76,85 1,26 1,83
(3,2,5 ) 500 110 23,33 812,00 15,30 81,37 1,67 1,66
(3,2,6 ) 500 120 18,54 131,10 2,36 62,45 23,21 11,98
(3,2,7 ) 500 130 18,54 113,15 2,36 62,45 23,21 11,98
(3,2,8 ) 500 140 12,99 113,11 1,54 76,68 12,58 9,19
(3,2,9 ) 500 150 12,99 113,11 1,54 76,68 12,58 9,19
(3,3,1 ) 700 70 28,49 835,80 41,16 56,76 0,26 1,82
(3,3,2 ) 700 80 26,50 924,90 33,97 63,82 0,45 1,77
(3,3,3 ) 700 90 25,37 924,60 30,26 67,23 0,62 1,88
(3,3,4 ) 700 100 24,08 1023,50 25,73 71,45 1,11 1,71
(3,3,5 ) 700 110 22,68 1023,10 22,21 74,48 1,52 1,79
(3,3,6 ) 700 120 21,00 1132,5 17,92 78,59 1,89 1,60
(3,3,7 ) 700 130 16,82 163,29 1,88 67,75 19,88 10,57
(3,3,8 ) 700 140 16,82 163,29 1,88 67,75 19,88 10,57

59
(3,3,9 ) 700 150 12,96 163,25 1,34 78,27 11,71 8,68
(3,4,1 ) 900 70 26,54 1217,90 47,06 51,48 0,11 1,36
(3,4,2 ) 900 80 25,23 1217,20 40,50 57,75 0,22 1,53
(3,4,3 ) 900 90 23,77 1216,50 34,12 63,64 0,53 1,71
(3,4,4 ) 900 100 22,90 1350,10 30,26 67,30 0,91 1,53
(3,4,5 ) 900 110 20,87 1498,00 23,00 73,92 1,65 1,42
(3,4,6 ) 900 120 19,08 205,8 1,95 65,69 21,70 10,66
(3,4,7 ) 900 130 15,90 205,79 1,62 69,59 19,15 9,64
(3,4,8 ) 900 140 15,90 205,79 1,62 69,59 19,15 9,64
(3,4,9 ) 900 150 12,96 205,75 1,34 77,91 12,19 8,56

8.2 Increasing
An increasing of the spring value is of interest due to the assumed spring value calculated for
the results above.
Increasing of spring value by 100 %
Table 4 Increasing of spring value by 100 % for Hexagonal
Hexagonal k*+100%
Profile Diameter
(mm)
Slen-
derness
Minimum
load factor
Half-
Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
(1,1,1 ) 300 70 17,59 124,80 10,07 49,27 33,45 7,21
(1,1,2) 300 80 17,59 124,80 10,07 49,27 33,45 7,21
(1,1,3 ) 300 90 12,42 124,90 7,86 49,64 36,39 6,11
(1,1,4) 300 100 12,42 124,90 7,86 49,64 36,39 6,11
(1,1,5 ) 300 110 8,08 124,90 5,69 50,44 39,06 4,81
(1,1,6 ) 300 120 8,08 124,86 7,86 49,64 36,39 6,11
(1,1,7 ) 300 130 4,59 113,79 3,52 52,12 41,01 3,35
(1,1,8 ) 300 140 4,59 113,79 3,52 52,12 41,01 3,35
(1,1,9 ) 300 150 4,59 113,79 3,52 52,12 41,01 3,35
(1,2,1 ) 500 70 20,62 219,00 12,14 51,50 29,43 6,93
(1,2,2 ) 500 80 14,18 219,00 9,44 50,96 33,51 6,08
(1,2,3 ) 500 90 11,36 198,60 7,37 51,78 35,37 5,49
(1,2,4 ) 500 100 11,36 198,60 7,37 51,78 35,37 5,49
(1,2,5 ) 500 110 8,75 198,60 6,16 51,74 37,29 4,82
(1,2,6 ) 500 120 6,49 198,65 4,95 51,94 39,03 4,08
(1,2,7 ) 500 130 6,49 198,65 4,94 51,94 39,03 4,08
(1,2,8 ) 500 140 4,54 198,66 3,77 52,45 40,51 3,27
(1,2,9 ) 500 150 4,54 198,66 3,77 52,44 40,51 3,27
(1,3,1 ) 700 70 19,35 323,40 12,49 53,26 28,06 6,19
(1,3,2 ) 700 80 14,86 292,20 9,31 53,48 31,41 5,80
(1,3,3 ) 700 90 12,74 292,30 8,46 53,03 33,01 5,50

61
(1,3,4 ) 700 100 10,81 292,30 7,59 52,66 34,61 5,14
(1,3,5 ) 700 110 9,02 292,30 6,71 52,42 36,15 4,73
(1,3,6 ) 700 120 7,38 292,29 5,80 52,35 37,58 4,27
(1,3,7 ) 700 130 5,89 292,30 4,90 52,47 38,88 3,75
(1,3,8 ) 700 140 5,89 292,30 4,90 52,47 38,88 3,75
(1,3,9 ) 700 150 4,53 264,16 3,73 53,12 39,97 3,18
(1,4,1 ) 900 70 18,59 414,70 12,10 55,29 26,93 5,68
(1,4,2 ) 900 80 15,12 414,70 10,63 54,39 29,53 5,45
(1,4,3 ) 900 90 11,98 373,80 8,05 54,44 32,41 5,10
(1,4,4 ) 900 100 10,52 373,80 7,42 53,96 33,75 4,86
(1,4,5 ) 900 110 7,84 373,90 6,10 53,29 36,32 4,28
(1,4,6 ) 900 120 6,64 373,87 5,43 53,13 37,50 3,93
(1,4,7 ) 900 130 5,53 373,87 4,75 53,12 38,59 3,55
(1,4,8 ) 900 140 5,53 373,87 4,75 53,12 38,59 3,55
(1,4,9 ) 900 150 4,52 373,88 4,06 53,25 39,55 3,14

Table 5 Increasing of spring value by 100% for Nonagonal
Nonagonal k*+100%
Profile Diameter
(mm)
Slenderness Minimum
load factor
Half-
Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
(2,1,1 ) 300 70 31,80 87,80 6,38 61,30 24,28 8,05
(2,1,2) 300 80 31,80 87,80 6,38 61,30 24,28 8,05
(2,1,3 ) 300 90 22,21 87,80 4,37 70,73 18,9 6,00
(2,1,4) 300 100 22,21 87,80 4,37 70,73 18,90 6,00
(2,1,5 ) 300 110 14,30 87,70 2,72 81,48 11,67 4,13
(2,1,6 ) 300 120 14,30 87,75 2,72 81,48 11,67 4,13
(2,1,7 ) 300 130 8,09 87,71 1,41 92,73 3,41 2,46
(2,1,8 ) 300 140 8,09 87,71 1,41 92,73 3,41 2,46
(2,1,9 ) 300 150 8,09 87,71 1,41 92,73 3,41 2,46
(2,2,1 ) 500 70 38,00 151,20 7,51 59,18 24,59 8,72
(2,2,2 ) 500 80 25,72 151,10 4,95 68,87 19,76 6,42
(2,2,3 ) 500 90 20,42 151,00 3,88 74,28 16,44 5,40
(2,2,4 ) 500 100 20,42 151,00 3,88 74,28 16,44 5,40
(2,2,5 ) 500 110 15,67 151,00 2,86 80,88 12,00 4,26
(2,2,6 ) 500 120 11,57 151,97 2,05 87,63 7,02 3,29
(2,2,7 ) 500 130 11,57 150,97 2,05 87,63 7,02 3,29
(2,2,8 ) 500 140 8,07 150,94 1,33 93,23 3,09 2,25
(2,2,9 ) 500 150 8,07 150,94 1,33 93,23 3,09 2,25
(2,3,1 ) 700 70 35,84 220,30 6,69 63,98 21,81 7,52
(2,3,2 ) 700 80 27,23 220,20 5,01 70,04 18,73 6,22
(2,3,3 ) 700 90 23,27 220,20 0,00 0,00 100,00 0,00
(2,3,4 ) 700 100 19,65 220,20 3,49 77,26 14,41 4,84
(2,3,5 ) 700 110 16,31 198,80 3,18 77,27 15,05 4,50
(2,3,6 ) 700 120 13,25 198,80 2,61 83,94 9,57 3,88
(2,3,7 ) 700 130 10,51 198,76 1,99 87,69 7,19 3,13
(2,3,8 ) 700 140 10,51 198,76 1,99 87,69 7,19 3,13

63
(2,3,9 ) 700 150 8,06 198,74 1,44 93,01 3,09 2,45
(2,4,1 ) 900 70 34,12 1086,00 37,67 59,89 0,76 1,68
(2,4,2 ) 900 80 27,89 279,70 5,12 70,42 18,37 6,09
(2,4,3 ) 900 90 21,92 279,60 3,98 75,30 15,54 5,17
(2,4,4 ) 900 100 19,18 279,60 3,46 77,97 13,88 4,70
(2,4,5 ) 900 110 14,18 279,50 2,48 84,37 9,45 3,69
(2,4,6 ) 900 120 11,96 279,46 2,06 88,08 6,63 3,22
(2,4,7 ) 900 130 9,92 279,41 1,67 91,25 4,34 2,73
(2,4,8 ) 900 140 9,92 279,41 1,67 91,25 4,34 2,73
(2,4,9 ) 900 150 8,07 251,70 1,49 92,89 3,14 2,47

Table 6 Increasing of spring value by 100% for Dodecagonal
Dodecagonal k*+100%
Profile Diameter
(mm)
Slenderness Minimum
load factor
Half-
Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
(3,1,1 ) 300 70 35,02 358,90 9,59 86,63 0,83 2,95
(3,1,2) 300 80 35,02 358,90 9,59 86,63 0,83 2,95
(3,1,3 ) 300 90 29,92 432,30 5,54 90,68 1,44 2,34
(3,1,4) 300 100 29,92 432,30 5,54 90,68 1,44 2,34
(3,1,5 ) 300 110 22,79 73,30 3,27 55,84 26,98 13,90
(3,1,6 ) 300 120 22,79 73,30 3,27 55,84 26,98 13,90
(3,1,7 ) 300 130 13,04 66,23 1,87 74,08 14,45 9,59
(3,1,8 ) 300 140 13,04 66,23 1,87 74,08 14,45 9,59
(3,1,9 ) 300 150 13,04 66,23 1,87 74,08 14,45 9,59
(3,2,1 ) 500 70 35,43 548,50 0,00 0,00 100,00 0,00
(3,2,2 ) 500 80 30,57 667,40 13,30 83,21 1,12 2,37
(3,2,3 ) 500 90 20,81 736,10 9,98 86,50 1,41 2,11
(3,2,4 ) 500 100 27,81 736,10 9,98 86,50 1,41 2,11
(3,2,5 ) 500 110 24,74 812,00 7,49 88,95 1,71 1,86
(3,2,6 ) 500 120 18,57 113,15 2,58 61,54 23,40 12,48
(3,2,7 ) 500 130 18,57 113,15 2,58 61,54 23,40 12,48
(3,2,8 ) 500 140 13,01 113,11 1,70 76,27 12,54 9,48
(3,2,9 ) 500 150 13,01 113,11 1,70 76,27 12,54 9,48
(3,3,1 ) 700 70 32,93 835,80 26,38 70,89 0,40 2,31
(3,3,2 ) 700 80 30,00 924,90 19,85 77,03 0,92 2,19
(3,3,3 ) 700 90 28,29 924,60 17,29 79,36 1,08 2,28
(3,3,4 ) 700 100 26,48 1023,50 13,99 82,56 1,44 2,02
(3,3,5 ) 700 110 24,56 1023,10 11,91 84,21 1,82 2,06
(3,3,6 ) 700 120 21,19 163,33 2,94 58,44 25,12 13,50
(3,3,7 ) 700 130 16,87 163,29 2,15 66,41 20,10 11,34
(3,3,8 ) 700 140 16,87 163,29 2,15 66,41 20,10 11,34

65
(3,3,9 ) 700 150 12,98 163,25 1,52 77,76 11,56 9,15
(3,4,1 ) 900 70 31,14 1097,00 30,92 66,65 0,33 2,10
(3,4,2 ) 900 80 29,15 1096,50 26,04 71,23 0,44 2,28
(3,4,3 ) 900 90 26,81 1216,50 20,40 76,37 1,14 2,10
(3,4,4 ) 900 100 25,57 1350,10 17,26 79,56 1,33 1,85
(3,4,5 ) 900 110 22,67 1498,00 12,60 83,84 1,91 1,65
(3,4,6 ) 900 120 19,21 205,83z 2,54 62,18 22,72 12,56
(3,4,7 ) 900 130 15,96 205,79 1,96 67,86 19,44 10,74
(3,4,8 ) 900 140 15,96 205,79 1,96 67,86 19,44 10,74
(3,4,9 ) 900 150 12,99 205,75 1,55 77,15 12,04 9,25

Increasing the spring value by 200 %
Table 7 Increasing of spring value by 200 % for Hexagonal
Hexagonal k*+200%
Profile Diameter
(mm)
Slen-
derness
Minimum
load factor
Half-
Wave
length
(mm)
G
buckling
mode
(%)
D
buckling
mode
(%)
L
buckling
mode
(%)
Others
(%)
(1,1,1 ) 300 70 17,82 124,80 9,74 47,91 34,72 7,63
(1,1,2) 300 80 17,82 124,80 9,74 47,91 34,72 7,63
(1,1,3 ) 300 90 12,55 124,90 7,55 48,73 37,36 6,37
(1,1,4) 300 100 12,55 124,90 7,55 48,73 37,36 6,37
(1,1,5 ) 300 110 8,14 113,80 5,21 50,62 39,27 4,90
(1,1,6 ) 300 120 8,14 113,80 5,21 50,62 39,27 4,90
(1,1,7 ) 300 130 4,60 113,79 3,42 51,8 41,35 3,41
(1,1,8 ) 300 140 4,60 113,79 3,42 51,8 41,35 3,41
(1,1,9 ) 300 150 4,60 113,79 3,42 51,8 41,35 3,41
(1,2,1 ) 500 70 20,98 219,00 12,00 49,13 31,28 7,59
(1,2,2 ) 500 80 14,40 219,00 9,19 49,33 34,99 6,48
(1,2,3 ) 500 90 11,48 198,60 7,24 50,46 36,52 5,78
(1,2,4 ) 500 100 11,48 198,60 7,24 50,46 36,52 5,78
(1,2,5 ) 500 110 8,83 198,60 6,00 50,79 38,19 5,02
(1,2,6 ) 500 120 6,53 198,65 4,79 51,29 39,72 4,21
(1,2,7 ) 500 130 6,53 198,65 4,79 51,29 39,71 4,21
(1,2,8 ) 500 140 4,57 198,65 3,63 52,02 41,00 3,35
(1,2,9 ) 500 150 4,57 198,65 3,63 52,02 41,00 3,35
(1,3,1 ) 700 70 19,74 323,40 12,45 50,46 30,19 6,89
(1,3,2 ) 700 80 15,09 292,20 9,36 51,14 33,19 6,31
(1,3,3 ) 700 90 12,92 292,30 8,43 51,04 34,61 5,92
(1,3,4 ) 700 100 10,95 292,30 7,51 51,02 36,00 5,47
(1,3,5 ) 700 110 9,13 292,30 6,57 51,11 37,33 4,98
(1,3,6 ) 700 120 7,45 292,29 5,65 51,34 38,57 4,45

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