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Al-Khwarizmi 

  Engineering  

Journal 
Al-Khwarizmi Engineering Journal,Vol. 13, No. 4, P.P. 12- 21 (2017) 

 

Best Level of Parameters for a Critical Buckling Load for Circular Thin- 

Walled Structure Subjected to Bending 
 

Hussein M. H.  Al-Khafaji 
Department of Mechanical Engineering/ University of Technology/ Baghdad 

Email: hussalkhafaji@gmail.com 

 
(Received 22 February 2017; accepted 2 July 2017) 

https://doi.org/10.22153/kej.2017.07.003 

 

Abstract 
 
Circular thin walled structures have wide range of applications. This type of structure is generally 

exposed to different types of loads, but one of the most important types is a buckling. In this work, the 

phenomena of buckling was studied by using finite element analysis. The circular thin walled structure in 

this study is constructed from; cylindrical thin shell strengthen by longitudinal stringers, subjected to pure 
bending in one plane. In addition, Taguchi method was used to identify the optimum combination set of 

parameters for enhancement of the critical buckling load value, as well as to investigate the most effective 

parameter. The parameters that have been analyzed were; cylinder shell thickness, shape of stiffeners section 

and the number of stiffeners. Furthermore, to verify the contribution of parameters on buckling response, the 
analysis of variance technique (ANOVA) method was implemented, which gave the contribution weight as 

percentages. The analysis of results by these two methods showed that the more effective parameter on the 

critical buckling load was the thickness of cylinder’s shell and the lowest effective was the number of 
stiffeners The values of parameters that gave the best critical buckling load combination were: 1) the ratio of 

cylinder’s diameter to thickness of its shell was 133, 2) the ratio of the depth to thickness of stiffeners 

was1.6, and 3) the number of stiffeners was 12.  
 

Keywords: buckling, FEM, Taguchi method, thin walled structure and ANOVA. 

 

 

1. Introduction 
 
Circular hollow sections thin-walled structure 

have a wide use in many applications such as; 

aircraft lightweight aerospace, towers of wind 

turbines, chimneys, marine structure, and etc...[1]. 
One of the important phenomena for failure and 

collapse of this type of structure is the local 

buckling. this phenomenon caused by presenting 
of compression and shear forces resulting from 

applying of one or combined of pure axial 

compression, torsion, and bending loads.  Thus, it 

is desirable to know what are the circumstances 
that lead to buckle this type of thin-walled 

structures, what is the best design to reduce or 

avoid occurring of buckling and how to improve 

the stability characteristics. Many type of 
strengthening were used to reinforce the thin wall 

structure with circular cross section, but the most 

common are the longitudinal (stringers) and  
circumferential (rings). The following works [2], 

[3], [4], and [5]; were investigated theoretically 

and experimentally these two types of stiffening, 
the analysis included the shape, size and the 

enhancement of critical buckling stresses with 

different kinds of load. [6], [7], and [8] 

investigated the effect of opening and cutout 
presenting in the circular thin walled structures on 

the buckling behavior, and how to stiffen these 

opening. 
As well as, unstiffened circular thin walled 

structures were considered in the researches. 



 Hussein M. H.  Al-Khafaji                          Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 12- 21 (2017) 
 

13 

 

Reddy [9] tests unstiffened cylindrical tubes under 
pure bending experimentally to estimate the 

relation between the radius to thickness (r/t) and 

strain, also to observe the modes of deformation. 
It is observed that a small change in circular cross 

section to the oval shape, does not causes the 

ripples initiation which is the first cause of 

collapse. The Author used in test radius to 
thickness ratio (r/t) range (15-40) with nominal 

diameter 25mm and 600mm long with two 

materials aluminum alloy and stainless steel. Jiao 
and others [1] and [10] studied the bending 

capacity and buckling behavior of stiffened and 

unstiffened circular-section structures with 

different diameter to thickness ratio (17-300), 
subjected to more than one type of loading and 

supporting. 

The finite elements method represents one of 
various theoretical methods applied to study the 

circular thin wall structure, the researcher Khonke 

and Schnobrich [11] used finite element method 
to analyze the stresses on a circular cylindrical 

shell, which was reinforced by using eccentrically 

ribs strengthen longitudinally and peripherally. 

They prove that the finite element method can be 
an effective method to simulate the thin walled 

structure. E. Kim and S. Kim [12] conducted 

analysis for buckling to cylindrical shells with 
diameter to thickness ratio of more than 800, to 

evaluate the buckling strength, finite element 

analysis was used. In addition, the researchers in 
[13],[14],[15], [16] and [17], have used finite 

element to investigate stability behavior, 

optimization and design of  unstiffened and 

stiffened  circular thin walled structures.   
Khamlichi et al. [18] and [19] showed that 

there is a further decreasing in the critical stress of 

compressed circular cylindrical shells with 
increasing one extra defect, a finite element 

method was used to model and simulate the 

problem. The model consists of circular 

cylindrical shell with two triangular defects, a 
parametric study (Taguchi method) applied to 

regard the initial geometric influence of these two 

defects.  
In this work, a finite element analysis using 

software ANSYS was applied to model the effect 

of geometry and cross section on the linear 
buckling behavior of circular thin wall structure 

(CTWS). CTWS is strengthen with longitudinal 

stringers. Furthermore, the Taguchi method was 

applied to explore the effectiveness of   
geometrical parameters and their combine sets on 

the strength of buckling. Moreover, the analysis of 

variance technique (ANOVA) was employed to 

determine the weight and contribution of each 
parameters on the critical load of buckling. 

 

 

2. Theory 
2.1. Buckling and Finite Element 

 
The Buckling is a state of structure in which   

the applied load are large enough to disturb or 

destroy the stability of structure. The linear 
buckling analysis predicts the theoretical buckling 

strength of a structure. It is modeled as an 

eigenvalue equation [15], [20] and [21]: 
                        …(1)  

Where K is stiffness matrix,   is stress 

stiffness matrix, λ is the eigenvalue (load factor) 
and ψ is the eigenvector of displacements. The 
solutions for above formula are the eigenvalues 

and corresponding eigenvectors. The smallest 

value from the eigenvalues (roots) represents the 
critical load factor of buckling. If the applied load 

is F then the critical load will be (λ*F).  If the load 
is only bending, and cross section of structure is 
symmetric, there will be   positive λ (load factors) 
and other with negative sign but equal in absolute 

value, therefore, it should be take the critical load 

factor according to the direction of applied load.  

 
2.2 Taguchi  

 

Genichi Taguchi founded the Taguchi method 

in 1949. It is a methodology for getting the 

relationship between the input process and output 
of products. It depends on using orthogonal array 

(OA) with a small number of experiments to 

acquire the most favorable setting of parameters 
for process The Taguchi method is an useful tool  

for better design, gathering concept of quality loss 

function  and design of experiment theory. It 
provides a systematically approach for optimizing 

the design performance, cost, and quality. This 

method employs a statistical measurement named 

as the ratio of the mean (called signal) to the 
standard deviation (called noise) (S/N), which 

represents a logarithmic function of wanted 

output. There are three standard types of S/N 
ratio; higher the better (HB), lower the better (LB) 

and Nominal the best (NB) [22] and [23]. In this 

study, the higher critical load of buckling is 

required, therefore, the higher the better formula 
is used: 

                                 ...(2) 

Where:  

y: is the read data. 

n: is the number of observations.  



 Hussein M. H.  Al-Khafaji                          Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 12- 21 (2017) 
 

14 

 

2.3. Analysis of Variance (ANOVA) 
 

Analysis of variance (ANOVA) is a powerful 

statistical technique, which specify the important 
parameters and demonstrate the percentage 

contribution of each parameter.  

In this research, the ratio (S/N) was used for 
making the decision. 

The technique of  ANOVA is based on  the 

total sum of squared deviations (  which is 

equal  [24] and [22]: 

                …(3) 
The percentage contribution P could be 

calculate as: 

                                  …(4) 

d.o.f of any parameter(factor) = k-1  

Total d.o.f = n-1  

  
                                  …(5) 

                      …(6) 

                       …(7) 

Where: 
N the number of observation (trails) in the 

orthogonal array. 

 ni     mean S/N ratio for the ith observation.  

 nm     mean of all parameters. 
SSd     sum of the squared deviations. 

K      amount of levels for each parameter. 

P      contribution’s percentage.  
V      parameter’s (factor) variance. 

VE      error’s variance. 

d.o.f degree of freedom. 

F    F- test, which is indicator quality 
characteristic of process. 

 

 

3. Modeling  
3.1 Structure Modeling 

 
The model of the structure consists of 

cylindrical thin wall (CTWS) cantilever beam 

structure. Two plates were mounted at terminal 
ends of the beam; to prevent ovalization. One of 

the plate to support the beam (fixed end) and the 

other at the free end of beam to support the 
applied load. The dimensions of plates at the fixed 

end edge and at free end are (560 x 560 x 20) mm 

and (560 x 560 x 10) mm, respectively, as shown 

in  
Fig. 1. The length of beam (bay between the 

two plates at the ends) was designed according to 

the ratio between plates-spacing ( l )  and  radius 

(r)  which is more than [(l / r) > (½)]  to reduce 
the strength gain of beam [2] and [25]. 

Accordingly, (l) and (r) were set as 2000 mm and 

200 mm, respectively. The diameter to thickness 
ratio of CTWS (D/t) was taken in different values 

by changing thickness and keeping diameter 

constant at (D = 400 mm). The thickness t varies 

in three values (1, 2, and 3) mm. The structure is 
strengthened with three different shape and three 

different numbers of longitudinal stringers.  The 

shape of strangers have been chosen as a 
rectangular cross section with three different 

dimensions having the same area to keep the same 

mass, as shown in  

Fig. 2.  These stringers were connected 
longitudinally along one side (connected rigidly) 

to the shell of the CTWS, and they are not 

supported at their transversal two ends (the two 
ends free). 

 

 
 

Fig. 1. The cylindrical thin wall structure 
 

 
 

Fig. 2. the detail of stiffener. 

 
 

 

 
 



 Hussein M. H.  Al-Khafaji                          Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 12- 21 (2017) 
 

15 

 

The chosen parameters for analysis are three, 
namely: diameter to thickness of the cylindrical 

shell (D/t), the Number of stringer (Ns), and the 

ratio of depth to thickness of stringer (h/ts). Each 
one of these parameters have three levels as 

shown in table 1. 

  
Table 1, 

Parameters and levels 

symbols parameter 
levels 
Level 

1 

Level 

2 

Level 

3 

A Ns 4 8 12 

B h/ts 25 6.25 1.6 

C D/t 400 200 133 

 

Where the values of Fig 2:                            

Ns  No. of stiffeners  
h  (mm)  depth of stringer ( 50, 25,12.5).  

ts  (mm)   thickness of stringer (2, 4 , 8).  

D (mm)   diameter of cylinder (400 ). 
t   (mm)   thickness of cylinder ( 1, 2 ,3 ). 
 

 

3.2  Finite Element Modeling 
 

The finite element software that used to 
perform analysis is ANSYS 16. Shell281- element 

was used to create mesh. This element has eight 

nodes. Each node has six degrees of freedom: 
rotations about the x, y, and z-axes, and 

translations in the x, y, and z-axes, and is suitable 

for thin and moderately thick structures [26]. To 
generate the mesh, the chosen elements size were; 

10 mm in the axial, and circumferential directions 

for CTWS. The element size for the  end-plates 

were 28 mm. Depending on the dimension of 
stiffener; the element size for depth ranges from 3 

to 5 mm, while in the axial direction is 10 mm. 

The fixed end of structure is supported by eight 
point on the end plate to simulate the connection 

by eight bolts. The load is applied on the free end 

in the middle of one edge of plate, as shown in the 
Fig 3.  

 
 

Fig. 3. Finite elements model 

 
 

The steel structure material was implemented, 

which is considered as a linear isotropic material. 

Table 2 lists the properties of used material. 
 

Table 2, 

Steel structure material properties 

 

 

4. Result and Dissection 
4.1 Finite Element Results 

 

A linear buckling analysis is performed to 
calculate the critical load buckling. According to 

three parameters and three levels its need an 

orthogonal array L27. Thus, twenty-seven value 

of critical load were found from different 
combinations of levels and parameters.   

Table 3 shows the levels combinations and the 

corresponding results of loads buckling. Each 
value of critical buckling load in the fifth column 

of Table 3 is obtained from the calculating of  

average value for analyzing the finite elements 
model to three times for each parameters set and 

corresponding levels (raw in the table).  

 

 
 

 

 
 

 

 

Young’s 

modulus 
(MP) 

Poisson’s 

ratio 
 

tensile 

yield 
strength 

(MP) 

tensile 

ultimate 
strength 

(MP) 

2x105 0.3 250 460 



 Hussein M. H.  Al-Khafaji                          Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 12- 21 (2017) 
 

16 

 

Table 3, 

Paramters and  resulting critical buckling load 

No. 

of  

tests 

No. of 

Stiffeners  

( Ns ) 

shape of 

stiffeners 

 ( h/ts ) 

D/t 

Critical 

Buckling 

Load (MN) 

1 4 25 400 34.52 

2 8 25 400 37.78 

3 12 25 400 40.28 

4 4 6.25 400 42.72 

5 8 6.25 400 44.78 

6 12 6.25 400 49.82 

7 4 1.6 400 45.28 

8 8 1.6 400 48.55 

9 12 1.6 400 50.37 

10 4 25 200 76.63 

11 8 25 200 77.86 

12 12 25 200 80.22 

13 4 6.25 200 137.58 

14 8 6.25 200 140.79 

15 12 6.25 200 159.83 

16 4 1.6 200 147.53 

17 8 1.6 200 150.01 

18 12 1.6 200 161.51 

19 4 25 133 113.51 

20 8 25 133 115.18 

21 12 25 133 118.06 

22 4 6.25 133 319.47 

23 8 6.25 133 327.52 

24 12 6.25 133 334.82 

25 4 1.6 133 332.11 

26 8 1.6 133 339.27 

27 12 1.6 133 343.14 

 
 

The Fig. 4 and Fig. 5 shows the variation of 

load with variation in two parameters and the 

third one is constant. These figures show that the 
most effective parameter is the thickness of 

cylinder especially in the Fig. 5, while the less 

effective parameter is the Number of stiffeners; 
this is attributed to the load, which is applied in 

one plane (vertical plane) as shown in Fig 3.  

 
 

 
 

 
 

 
 

Fig. 4. The variation of critical load with 

parameters at constant D/t 

 
 

 
 



 Hussein M. H.  Al-Khafaji                          Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 12- 21 (2017) 
 

17 

 

 
 

 
 

Fig. 5. The variation of critical load with 

parameters at constant h/ts ratio 

 

 
Fig.7 shows samples of local buckling 

deformation in cylinder shell and stiffeners. The 

buckling occurs in the side of compression stress 

due to bending. it could be seen that mode of the 
buckling where the cylinder’s shell of structure is 

wrinkled, and the stiffener buckled in sine wave 

manner. 
 

 
 

Fig. 6. Loacal buckling of four stiffneres  

 

 
 



 Hussein M. H.  Al-Khafaji                          Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 12- 21 (2017) 
 

18 

 

 
 

Fig.7.  (8 and 12) stiffeners with different h/ts. 

 

 

4.2 .Taguchi and ANOVA Results 
 

The orthogonal array is shown in Table 4 , 

while the results of mean based on S/N were 
presented in Table 5 and Fig. 8. 

 
Table 4, 

Orthogonal array and S/N 

No.

of  

tests 

-A- 

 

sN 

-B- 

 

h/ts 

-C- 

 

D/t 

Critical 

Buckling 

Load 

MN 

S/N 

1 4 25 400 34.52 30.761 

2 8 25 400 37.78 31.545 

3 12 25 400 40.28 32.102 

4 4 6.25 400 42.72 32.613 

5 8 6.25 400 44.78 33.022 

6 12 6.25 400 49.82 33.948 

7 4 1.6 400 45.28 33.118 

8 8 1.6 400 48.55 33.724 

9 12 1.6 400 50.37 34.043 

10 4 25 200 76.63 37.688 

11 8 25 200 77.86 37.826 

12 12 25 200 80.22 38.086 

13 4 6.25 200 137.58 42.771 

14 8 6.25 200 140.79 42.971 

15 12 6.25 200 159.83 44.073 

16 4 1.6 200 147.53 43.378 

17 8 1.6 200 150.01 43.522 

18 12 1.6 200 161.51 44.164 

19 4 25 133 113.51 41.101 

20 8 25 133 115.18 41.228 

21 12 25 133 118.06 41.442 

22 4 6.25 133 319.47 50.089 

23 8 6.25 133 327.52 50.305 

24 12 6.25 133 334.82 50.496 

25 4 1.6 133 332.11 50.426 

26 8 1.6 133 339.27 50.611 

27 12 1.6 133 343.14 50.709 

 

 

From the Table 5 and Figure 8 it is seen that 
the optimum set of levels is (A3 B3 C3) which 

gives the best critical buckling load. This means 

that the Number of stiffeners is 12 and the (h/ts 
=1.6) and (D/t =133). Delta column represents the 

difference between the highest and lowest level 

b- 12 stiffeners 
 

a- 8 stiffeners 



 Hussein M. H.  Al-Khafaji                          Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 12- 21 (2017) 
 

19 

 

value. It indicates that the ratio D/t (cylinder 
thickness) has more effect than the other 

parameters. 

 
Table 5, 

Analysis of mean results based on S/N 

S
y
m

b
o
ls 

P
a
r
a
m

e
te

r
s 

L
e
v
e
l - 1

 

L
e
v
e
l - 2

 

L
e
v
e
l - 3

 

D
e
lta

 

R
a
n

k
 

A Ns 40.216 40.528 41.007 0.791 3 

B h/ts 36.864 42.254 42.633 5.769 2 

C D/t 32.764 41.609 47.379 14.62 1 

 

 

 
 

Fig. 8. The parameters- S/N curves 

 
 

Table 6 shows the results of ANOVA. These 

results indicate the importance and contribution of 

parameters as a percentage. The D/t ratio has the 
highest contribution (80.814%) and the lowest and 

approximately no effect is number of stiffeners 

(0.133%).   
 

Table 6, 

 ANOVA result 

Source Sum sq. d.o.f 
Mean 

sq. 
F P  % 

A 1.67 2 0.834 0.33 0.133 

B 188.45 2 94.225 37.15 15.02 

C 1014.21 2 507.10 199.9 80.81 

Error 50.73 20 2.536  4.042 

Total 1255 26    

 

 

5. Conclusion  
 
This paper has presented the analysis of   linear 

buckling which results from pure bending. In this 

work, the bending was applied on hollow 

cylindrical thin walled structure strengthen by 
longitudinal stiffeners. The finite elements 

methods used to carry on the analysis. It was 

shown that   the thickness of the cylinder is more 
effective parameter on the critical buckling load 

which is followed by the shape of cross section 

for stiffeners.  The results indicated that the 

number of stiffeners shows approximately no 
effect to the critical load of buckling. These 

results approved and conformed by using the 

Taguchi and ANOVA methods. Taguchi gaves the 
weight of parameters and the best-set combination 

of parameters, while ANOVA showed the 

percentage of contribution for each parameter. 

The results demonstrated that Taguchi and 
ANOVA are power full tool to investigate the 

importance of parameters during the design of 

structures. 
 

 
6. References 

 
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thin-walled circular hollow section tubes 

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140–155, 2012. 



 Hussein M. H.  Al-Khafaji                          Al-Khwarizmi Engineering Journal, Vol. 13, No. 4, P.P. 12- 21 (2017) 
 

20 

 

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 )2017( 12-21، صفحة 4، العدد13دجلة الخوارزمي الهندسية المجلم                                                          حسين محمد حسين

21 

 

 

 

  ة االلتواء الحرج لهيكل رقيق الجدران دائري معرض لالنحناءحمولل المستوى االفضل
 

  حسين محمد حسين الخفاجي
 بغداد /قسم الهندسة الميكانيكية / الجامعة التكنولوجية 

 hussalkhafaji@gmail.comالبريد االلكتروني : 
 

 

  

  

  الخالصة
  
يتعرض عادة ألنواع مختلفة من االحمال، ولكن واحدة من اهم نوع هو حمل  الهياكل الرقيقة الجدران االسطوانية لها تطبيقات واسعة. هذا النوع 

راسة مركب من: االنبعاج. في هذا العمل، تم دراسة ظاهرة االنبعاج باستخدام التحليل بالعناصر المحددة، الهيكل الرقيق الجدران االسطواني في هذه الد
بأضالع تقوية طولية معرض فقط الى انحناء في مستوي واحد. المتغيرات التي تم تحليلها هي سمك قشرة األسطوانة، شكل  أسطوانة ذات جدرن رقيقة مقواه

كذلك للتحقق مقطع اضالع التقوية وعدد اضالع التقوية. طريقة تاكوجي استخدمت لتحديد أفضل تركيب من المتغيرات لتحسين قيمة الحمل الحرج لالنبعاج و
 .مئويةيرات أكثر فاعلية. ألثبات مساهمة المتغيرات على استجابة االنبعاج تم تطبيق طريقة التحليل المتباين والتي تعطي وزن المساهمة كنسبة أي من المتغ

الية هو عدد اضالع األكثر فعالية على حمل االنبعاج الحرج كانت سمك قشرة االسطوانة وأدنى فع أن المتغيرهاتين الطريقتين  باستخدامأظهر تحليل النتائج 
)، نسبة العمق إلى سمك 133هي ( قشرتهاهي: نسبة قطر األسطوانة إلى سمك  حرج وكانت قيم المتغيرات التي تعطي تركيب ألفضل حمل انبعاج التقوية.
  ).12( هي ) ألضالع التقوية، وعدد التقوية1.6(هي الطبقة