Buckling of thin-walled conical shells under uniform external pressure

Thin-Walled Structures 46 (2008) 516–529 

Buckling of thin-walled conical shells under uniform external pressure 

Abstract 

B.S. Golzan a , H. Showkati b, 

a Department of Civil Engineering, Urmia University, Urmia, Iran 

b Engineering Faculty, University of Urmia, Urmia, Iran 

Received 7 May 2007; received in revised form 18 October 2007; accepted 18 October 2007 

Available online 20 February 2008 

Shells are for the most part the deep-seated structures in manufacturing submarines, missiles, tanks and their roofs, and fluid 

reservoirs; therefore it is a matter of concern to bring about some basic regulations associated with the existing codes. Above all, 

truncated conical shells (frusta) and shallow conical caps (SCC) subjected to external uniform pressure when discharging liquids or wind 

loads are discussed closely in this paper concerning and thrashing out their empirical nonlinear responses along with envisaging 

numerical methods in contrast. The buckling aptitude of shells is contingent upon two leading geometric ratios of ‘‘slant-length to 

radius’’ (L/R) and ‘‘radius to thickness’’ (R/t). In this paper, developing six frusta and four shallow cap specimens and their relevant FE 

models, use is made of laboratory modus operandi to enumerate buckling elastic and plastic responses and asymmetric imperfection 

sensitivity, whose adequacy has been reckoned through comparisons with arithmetical and numerical data correspondingly. These 

obtained upshots were aimed at validating and generalizing the data for unstiffened truncated cones and SCC in full scale. 

r 2007 Elsevier Ltd. All rights reserved. 

Keywords: Truncated conical shells (frusta); Shallow conical caps (SCC); Buckling; Sole-fish buckling; Postbuckling; Nonlinear response; External 

uniform pressure 

1. Introduction 

Performing test on manufactured specimens is the most 

steadfast method in engineering researches. Buckling of a 

general conical shell depends on scores of variables, for 

instance, the geometric properties of the shell (the cone 

semi-vertex angle, the base radius, the slant length of the 

shell and the thickness), the material properties (isotropic, 

composite, laminated, etc.), and the type of the applied 

load (axial compression, hydrostatic or uniform pressure, 

torsion and combined load). The various parameters 

change the buckling behavior of the shell, making it 

difficult to achieve a general depiction. Due to the 

relatively high slenderness of the specimens, the failure is 

in all cases significantly influenced by plasticity effects. The 

elastic buckling behavior of unstiffened cones under 

compression has been the subject of early analytical studies 

based on linear theory, in which axisymmetric elastic 

buckling was investigated. 

Corresponding author. Tel.: +98 914 141 1065; fax: +98 441 277 7022. 

E-mail address: h.showkati@mail.urmia.ac.ir (H. Showkati). 

0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. 

doi:10.1016/j.tws.2007.10.011 

ARTICLE IN PRESS 

www.elsevier.com/locate/tws 

These structures encompass light weight with high 

strength in different industrial applications. The significance 

is largely due to their widespread use in tanks and 

silos [1], offshore structures, aeronautical and aerospace 

technology, ship and submarine hulls [2], pipelines and 

industrial chemical plants [3,4]. 

The critical aptitude of frusta and SCC shells under 

uniform external pressure is contingent upon geometric 

slenderness ratio of their slant length to radius (L/R) and 

radius to thickness (R/t). 

There is not enough literature devoted to the analysis 

of geometrically imperfect conical shells. Koiter’s general 

postbuckling theory provides a basis for analysis of 

geometric imperfection sensitivity. All of these imperfection 

analyses were done on the shells of constant thickness. 

Ansourian [5] presented simplified design method about 

imperfections and boundary constraint effects on shells 

subjected to wind loading. Holst et al. [6] investigated the 

method of considering the strains resulted from fabrication 

misfit of perfect and imperfect shells to attain equivalent 

residual stresses. Shen and Chen [7] studied buckling and 

postbuckling behavior of perfect and imperfect shells with

finite length which were subjected to combined axial and 

external pressure. They showed that this behavior is 

dependant on geometry, loading and initial imperfections. 

Also Yamaki [8] has studied the nonlinear behavior of 

externally pressurized cylindrical shells and effects of 

geometrical imperfections. Further, other authors have 

studied stability of the shells that are outlined in the 

perspective to come. 

Performing test on manufactured specimens is the most 

steadfast method in engineering researches. In this paper, 

six frusta and four shallow conical cap (SCC) specimens 

have been manufactured and tested under the effect of 

uniform external pressure. The material consisted of mild 

steel with yield stress of 277 MPa [9]. Boundary conditions 

are all simply supported in which only a radial constraint is 

provided at the edges. A loading of uniform external 

pressure is produced by gauged vacuum pump using 

suction process. The stages of prebuckling, initial buckling, 

overall buckling and collapse have been observed and 

evaluated and nonlinear response of these conical shells has 

been studied. 

2. Experimental syllabus 

2.1. Model size 

In deciding on the model size for testing, a number of 

issues were considered. Firstly, the models should not be 

too large, to avoid any undesirable inconveniences 

associated with laboratory testing. Secondly, the models 

should not be too small, so as to cause difficulties in their 

fabrication. Thirdly, the radius-to-thickness ratios (R/t) of 

the models should be analogous to those used in realistic 

structures, since the effect of interaction between yielding 

and buckling needs to be appropriately captured in the 

tests. Typical real values for the R/t ratio are within the 

range of 300–1000. As thin steel sheets of 0.5 mm and 

above can be easily obtained, welded or soldered effortlessly 

to produce high quality models with special welding 

machine or soldering apparatus, it was decided that the 

models ought to be of 600 mm in diameter. Consequently, 

Table 1 

Dimensions and aspect ratios of the specimens 

Specimen code Thickness t 

(mm) 


B.S. Golzan, H. Showkati / Thin-Walled Structures 46 (2008) 516–529 517 

Top radius 

(mm) 

Bottom radius 

(mm) 

proper (R/t) ratios can be achieved with different steel 

sheet thicknesses. 

2.2. Test specimens 

In this paper, six different frusta specimens were used, 

namely SC1, SC2, SC3, SC4, SC5, SC6 along with four 

SCC specimens represented by SCC1, SCC2, SCC3, 

and SCC4. . The properties of all models are outlined in 

Table 1. The thickness of specimens is totally constant. The 

frusta specimens have the same lower base diameter of 

600 mm whereas the top base of the first three measures 

200 mm in diameter and in the second three it is 100 mm. 

The SCC specimens have the same lower base diameter of 

600 mm except for SCC3 with a base diameter equal to 

500 mm. For the detailed geometry and slenderness ratios 

of specimens refer to Table 1. Edge conditions are all 

simply supported, in which only radial restraint was 

provided. 

Three tensile coupon tests were performed identically to 

obtain the properties of material. The yield and failure 

stresses of this mild steel are 277 and 373 Mpa, correspondingly. 

The Young modulus acquired, equals 210 GPa. 

Each specimen was assembled by cord-oriented welding 

over the rolled sheet fragment edges, as is shown in Fig. 1. 

A loading of uniform external pressure is produced by 

gauged vacuum pump using suction process. 

2.3. Fabrication modus operandi 

An important issue in shell buckling experiments is the 

fabrication of good quality specimens, including the choice 

of material and fabrication method. Many fabrication 

techniques have been developed [10,11], among which are 

electroforming (making duplicates by electroplating metal 

onto a mold of an object, then removing the mold in which 

the intricate surface details are precisely reproduced by this 

process), thermal forming of plastics (PVC, polyethylene, 

Lexan, or other materials) and cold working of metal 

(spinning, explosive forming, or hydroforming). Most of 

these are specialized laboratory techniques for fabricating 

nearly perfect model shells. Where tests are intended to 

Height h 

(mm) 

Semi-vertex 

angle (a) 

R/t R/r L/R L ¼ slant 

length 

SC1 0.6 100 300 223.6 41.81 500 3 1 

SC2 0.6 100 300 403.2 26.36 500 3 1.5 

SC3 0.6 100 300 565.7 19.47 500 3 2 

SC4 0.6 50 300 165.8 56.44 500 6 1 

SC5 0.6 50 300 374.2 33.75 500 6 1.5 

SC6 0.6 50 300 545.4 24.62 500 6 2 

SCC1 0.5 – 300 60 78.69 600 – 1.02 

SCC2 0.8 – 300 60 78.69 375 – 1.02 

SCC3 0.5 – 250 62.5 75.96 500 – 1.03 

SCC4 0.8 – 300 75 75.96 375 – 1.03

518 

Fig. 1. Sector cutting process and slant length weld lines. 

duplicate full-scale steel shell construction as closely as 

possible, the method of rolling thin steel sheets followed by 

seam welding has been commonly used (e.g. [12–16]). 

Another method in seam fusing is soldering the seams that 

resulted in a good upshot both in manufacturing and testing 

processes and inspired a good prognostication of welds’ 

performances. This method was adopted in the present work. 

2.4. Process of model fabrication 

To build the conical shell, it is first made by cutting and 

rolling a plate into the desired shape and soldering the 

meridional seams. Sheet cutting in the present work is done 

using a manually controlled shears and stonecutter cutting 

installation. A ‘‘beam compass’’, Fig. 1, consisting of a 

precisely machined aluminum strip with two end units was 

employed for quality sector cutting. One end unit is 

equipped with a small bearing to center the required circle 

at a small hole pre-drilled on the sheet and the other end 

unit is used to position the tip of the liner. Circles of 

different sizes can be obtained by using aluminum strips of 

different lengths. Accordingly, sectors can be obtained with 

a cutting accuracy within 70.1 mm. 


B.S. Golzan, H. Showkati / Thin-Walled Structures 46 (2008) 516–529 

It is quite intricate to obtain clear-cut conical shapes 

using the manual rolling process since a number of things 

have to be carefully controlled during the process. Firstly, a 

small angle (the dip angle) is required between the axis of 

the top roll and those of the lower rolls. Secondly, the 

rolling speed should be smaller at the small end than that at 

the large end of the cone. Consequently, special heed is 

required during the cone rolling process and the same 

modus operandi needs to be recurring a few times until the 

desired shape is achieved. Many radial lines were drawn on 

the panel. Such lines have to be kept parallel to the axes of 

the rolls when the lines pass through the rolls. To weld 

together the shell components is another arduous task. The 

meagerness of the models insinuates that special concern is 

required in assembling these models to guarantee that the 

weld or solder is sturdy enough so that structural failure 

precedes joint malfunction, and that the level and form of 

geometric imperfections bear some resemblance to those in 

real structures. 

3. Empirical set-up 

3.1. SCC testing system 

Fig. 2 shows an overall view of the experimental set-up 

for SCC specimens. Concerning the application of this 

machine, we coined the name ‘‘machine of detection and 

investigation of yield-lines and failure in bending behavior 

of steel plates and shells’’ that was invented by the authors 

in the preceding year to come up with some empirical 

features in accordance with the plates and conical shells 

with different shapes and aspect ratios. The full guide and 

explanation to this machine has been presented in another 

paper and is not the issue of concern in this study. 

The base brink of the model junction is placed in the 

groove of a rigid circular rim, which in turn sits on vertical 

Fig. 2. View of test rig for the cap specimens.

supports. To ensure that the brink is properly tenable in the 

groove it is covered by a grooved rubber and both the 

groove of rubber and the rim is filled with silicone sealant. 

Then connecting the vacuum pump to the rig the process of 

air suction is conducted under the specimen. This trend is 

performed in such a way that the loading is exerted 

incrementally and in all stages every thing is under the very 

control so that the specimen is not destroyed abruptly to 

not let us study the process hesitantly and exhaustively. 

3.2. Frusta testing system 

The test rig of the frusta specimens is composed of two 

parts (Fig. 3), which was invented by the authors. The first 

part is designed to hold the test specimen at the desired place, 

which is composed of two rigid circular grooved plates. 

These grooves are entrenched in both sides of specimen. 

Four threaded long bars are provided to adjust the plates for 

the specimen height. The second part of the rig consisted of a 

small platform to be used for installation of vacuum pump. 

This pump is employed to generate uniform external pressure 

over the shell surface. Careful measurements of the test 

results were done by six circumferentially and meridionaly 

mounted strain gauges, a manometer and four transducers. 

All collected data were processed using a data logger and a 

software named UCAM-20PC. 

3.2.1. Setting up the frusta 

The upper and lower brinks of the frusta were covered 

by grooved rubber and then silicon glue was used over all 

openings to prevent any possible air seepage during the 

suction process. The frusta were placed on the lower 

grooved rigid plate. For specimens SC1 and SC4, as the 

slant of their inclined surfaces exceeded far more than 

vertical position, they could luxate from the grooves; so a 

special ring truncated on the edges equal to the slant of the 

surfaces was employed to not let the frusta edges luxate 

outwardly in case of higher loading and disarticulation. 

For the top edges, as there is liability to luxate inwardly, 

another specific round plate chamfered inwardly at the 

edges is located to prop up this susceptible location 

(Fig. 3). Using four threaded bars, the upper plate was 

Supporting ring 

Fig. 3. View of the frusta test rig and chamfered ring to support the lateral 

luxating of edges. 



placed exactly over the top edge of the frusta in which the 

simply supported boundary conditions were geared up at 

both trimmings. The modification nuts could foil any axial 

load to be applied to the specimens. On the top plate, three 

holes were drilled for the purpose of air suction, 

manometer installation and air release valve assembly to 

control the rate of loading and unloading on shell 

specimens. The produced pressure was measured by the 

above-mentioned monometer. Fig. 3 shows a total view of 

test provision. 

3.3. Measurement of imperfections and deformations 

Buckling of shells is generally known to be sensitive to 

geometric imperfections; so precise surveys of initial 

geometric imperfections are an essential step in any high 

quality shell buckling experiments. In addition, it is also 

desirable to have precise measurements of deformed shapes 

of the shell during its loading so that the buckling/collapse 

mode can be accurately determined and compared with 

theoretical predictions. Many shell imperfection measurement 

techniques have been developed [18]. LVDTs or other 

contacting probes were usually used in most of the earlier 

measurement systems [12,13]. For very thin shells with a 

relatively low transverse stiffness, the small probe force 

may induce distortions of the shell surface, so non-contact 

probes are favored. 

A simpler way has been applied in the present measurement 

system for appraising both initial imperfections and 

displacements. Seeing that the complete measurements of a 

conical surface require a three-dimensional survey of the 

radial, circumferential and meridional coordinates, manual 

scanning was implemented as the measurement technique. 

At first a number of meridians were drawn on the 

expanded surfaces of the cones at specified degrees, and 

then they were assembled, conducting their meridional 

joints. After fabricating, circumferential segments were 

segregated on the surface and then the cone was installed in 

its place. Subsequently, at the contiguous of each meridian 

a ruler was mounted and another ruler was employed to 

measure the horizontally projected distance between nodes 

of drawn meshes and the edge of the specimens identified 

by the ruler rim. In each node of obtained mesh, three 

coordinates of r, y and z are measured carefully in all 

specimens. Therefore, a real geometry of shell is obtained 

and then is used in finite element modeling of the structure 

for further comparative analyses. 

Despite the relatively stocky geometry of the specimens, 

initial geometric imperfections were recorded on all specimens, 

with the method and mesh outlined above. In order to 

render these measurements functional for comparative 

studies and numerical modeling, the unrefined imperfections 

were subjected to some data processing techniques that 

enable the identification of dominant modes and facilitate 

comparisons of imperfections with observed buckling and 

collapse modes. Fig. 4 shows typical imperfection layouts for 

some of the models, (inward/outward) in two different views.

520 

It is worth bearing in mind that the measured imperfections 

on the models are too large and this is attributed to the small 

scale of the models but should be taken into account in 

correlating the experimental results with analytical and 

design dealings. The first geometric imperfections in the 

main lead to a considerable difference between theoretical 

and empirical consequences. 

A number of strain gauges were installed on the specimens, 

with some of them being used to measure circumferential 

strains and the others to measure meridional strains all 

over the specimens and near the weld transitions. The exact 

locations of circumferential (SH) and meridional (SV) strain 

gauges and transducers are provided in Table 3. 

It is worth saying that the difference in external pressure is 

less sensitive than that of axial loading. This is because of 

postbuckling capacity, which is available in lateral pressure. 

4. Exhaustive executions of the tests 

The steps of test implementation were as follows: 

Installation and calibration of instrumental apparatus. 

Calculation of approximated critical buckling load by 

Jawad [19] equation to determine load steps all through 

the tests. 



288 

270 

252 

306 

234 

324 

216 

342 

198 

300 

250 

200 

150 

100 

50 

0 

0 

180 

SCC4 

Fig. 4. Initial imperfection layout for SC3, SC6 and SCC4. 

18 

162 

Gauging initial geometric imperfections. 

Applying an initial external load up to approximately 

20% of calculated buckling load, and unloading for the 

purpose of system conditioning. 

Applying the external gradually increasing load to reach 

to initial buckling and ongoing until the incidence of 

overall buckling and failure mode of shell is attained; 

Measuring all required records throughout the test 

progress. 

5. Failure behavior and potency 

Most of the circumferential strains were also similar 

and approximately proportional to the load in the initial 

stage of loading (Figs. 5(a) and (b)), indicating linear 

and dominantly axisymmetric behavior. However, a small 

number of circumferential strain gauges had different 

readings right from the beginning, which is attributable 

to the attendance of relatively large local imperfections 

close by. As the load increased, the strain readings at 

different locations gradually diverged from each other. 

This divergence is a reflection of the growth of nonsymmetric 

deformations. At a certain increment of the 

load, non-periodical deformations could be observed on 

36 

144 

54 

126 

72 

90 

108

the shells by naked eyes. These deformations continued to 

grow with further loading, leading to obvious buckles of 

non-similar wavelengths. The development of these buckling 

lobes was associated with a reduction in the load 

carrying capacity as it led to some deformations at the 

radial base edge that caused seepage of the air. Explicitly, 

the specimens indicated a stable postbuckling path. 

Ultimate failure occurred by the formation of a plastic 

collapse mechanism with nearly non-uniform plastic 

deformations over a large part of the circumference that 

resulted in failure of the supports to sustain the proper 

function. Figs. 6 and 7 show the models after the failure. 

Table 2 

Buckling pressure and mode of the specimens 

Specimens Buckling 

load 

obtained by 

experiments 

(KPa) 

Buckling 

load 

obtained by 

FEA (KPa) 

Buckling 

load 

obtained by 

Jawad 

equation 

(KPa) 


Mode 

numbers by 

experiments 

SC1 25 28 35.4 7 (skirt mode) 

SC2 20 25 32.3 7 (skirt mode) 

SC3 14 22 24.68 6 (skirt mode) 

SC4 20 27 24.1 6 (skirt mode) 

SC5 25 40 31.35 6 (skirt mode) 

SC6 21 35 27.27 6 (skirt mode) 

SCC1 5.7 6.2 3.83 Sole-fish mode 

SCC2 8 14.5 20.3 Sole-fish mode 

SCC3 7.5 13 4.02 Sole-fish mode 

SCC4 10 25 16.2 Sole-fish mode 

Table 3 

Layout of strain gauges and transducers on all specimens 

These results show that most of the buckles on the 

models were amplified from initial geometric imperfections 

(Fig. 4). The buckling load can be defined as the pinnacle 

load of a nonlinear load–displacement curve. Such a 

buckling load incorporates the effect of imperfections. 

For the in attendance models, the buckling load of the 

corresponding perfect structure is believed to be a good 

assess of its veracity, but the determination of buckling 

load is not straightforward. A rough approach may be, to 

take the load at which the strain readings started to 

diverge, the same as the buckling load. This, on the other 

hand, does not allocate a precise definition of the buckling 

load as the strain readings had some differences right from 

the beginning of loading due to the presence of initial 

imperfections. Another drawback of using strain readings 

is that due to the cost and installation considerations, 

normally, merely part of the cone circumference is installed 

with strain gauges. So the most sought-after locations of 

strain measurements for buckling load determination may 

have been disregarded. For these reasons, the use of 

displacements of the cones, which are appraised around the 

whole circumference in the current set-ups, is preferred for 

attaining the buckling loads. 

In Table 2, the buckling pressures and modes of six 

frusta and four SCC specimens in diverse test stages are 

tabulated. Before initial buckling the behavior of shell 

is quite static with no pragmatic buckle lobe. As a 

comparison three obtained buckling loads from different 

approaches have been presented in this table. It is quite 

apparent that the experimental outcomes are taking up the 

lower range of load owing to the presence of initial 

Strain gauges Transducers 

SH1 horizontal 

gauge 

SV2 vertical 

gauge 


gauge 

SV4 vertical 

gauge 


gauge 

SV6 vertical 

gauge 

SC1 3061, 140 mm 2451, 110 mm 1501, 90 mm 1221, 60mm 541, 50mm 181, 80 mm 3121, 

90 mm 

SC2 771, 30mm 961, 350 mm 1341, 120 mm 1981, 220 mm 2581, 270 mm 3421, 170 mm 181, 

150 mm 

SC3 1901, 230 mm 2471, 385 mm 3071, 110 mm 81, 210 mm 691, 60 mm 1251, 310 mm 1501, 

310 mm 

SC4 1751, 120 mm 1751, 170 mm 1501, 70mm 481, 220 mm 3481, 50 mm 2641, 70 mm 2051, 

70 mm 

SC5 91, 120 mm 3511, 120 mm 2791, 30 mm 2421, 270 mm 1891, 120 mm 991, 320 mm 3421, 

70 mm 

SC6 8.51, 70mm 261, 170 mm 941, 270 mm 1631, 370 mm 2051, 95 mm 2911, 220 mm 3511, 

120 mm 

SCC1 3421, 30 mm 3061, 280 mm 2701, 120 mm 1621, 150 mm 1081, 30mm 541, 280 mm 3421, 

120 mm 

SCC2 3261, 80 mm 3401, 30 mm 1851, 130 mm 2751, 10mm 701, 20 mm 1231, 230 mm 3151, 

80 mm 

SCC3 511, 220 mm 2831, 220 mm 2311, 30 mm 2051, 120 mm 1771, 70mm 771, 120 mm 3471, 

70 mm 

SCC4 1081, 30 mm 1081, 288 mm 91, 120 mm 2971, 170 mm 2611, 285 mm 1891, 70 mm 1441, 

30 mm 

All distances are measured from the bigger base on the slant length. 


T1 T2 T3 T4 

1441, 

190 mm 

431, 

100 mm 

1501, 

160 mm 

601, 

220 mm 

2881, 

220 mm 

171, 

170 mm 

2341, 

30 mm 

2251, 

20 mm 

2571, 

30 mm 

721, 

220 mm 

841, 3481, 

90 mm 90 mm 

2311, 3021, 

130 mm 50 mm 

2701, 3451, 

360 mm 210 mm 

3481, 2641, 

70 mm 120 mm 

1981, 1261, 

170 mm 120 mm 

1201, 2491, 

270 mm 220 mm 

901, 1261, 

70 mm 280 mm 

1691, 301, 

80 mm 230 mm 

Apex 1031, 

120 mm 

3061, 1441, 

285 mm 150 mm

522 

pressure(kpa) 




t1 

t2 

t3 

t4 

-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 

t1 

t2 

t3 

t4 

total deformation(mm) 

-21 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 



8 

6 

4 

2 

0 

7 

6 

5 

4 

3 

2 

1 

0 

14 

12 

10 

10 

9 

8 

7 

6 

5 

4 



sh5 

-0.002 -0.001 0 0.001 0.002 0.003 

0 

0.004 

sv2 

sv4 

sv6 

strain 

-0.0023 -0.0018 -0.0013 -0.0008 -0.0003 

t1 

t2 

t3 

t4 

3 

2 

1 

sh1 

sh3 

sh5 

2 

1 

0 

0 

-22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 -0.002 -0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 


t1 

t2 

4 

t3 

t4 

2 

0 

-19 -17 -15 -13 -11 -9 -7 -5 -3 -1 



A 

A 

A 

A 

20 

18 

16 

14 

12 

10 

8 

6 

(i) 

(ii) 


(iii) 


(iv) 

strain 

strain 

-0.0035 -0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005 0 

Fig. 5. (a) SCC total deformation and strain values vs. pressure. (i) SSC1, (ii) SSC2, (iii) SSC3, (iv) SSC4 (b) load–deformation and load–strain graph for 

specimens SC1 and SC3 in different coordinates. 

sv2 

sv4 

sv6 

strain 

B 

B 

B 

B 

sh1 

sh3 

7 

6 

5 

4 

3 

2 

1 

12 

10 

8 

6 

4 

2 

0 

8 

7 

6 

5 

4 

3 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

0

Digital transducer 

Failure area 



Fig. 5. (Continued) 

Fig. 6. Failure modes of specimens SC5 and SC6. 

Digital strain gauge 

Wresting of apex point at the 

bifurcation

524 

geometrical imperfections, apparatus shortcomings and 

other human and instrument-related factors. On the other 

hand, for six frusta, taking into account both the buckling 

load obtained from FEA and the equation developed by 

Jawad it is noticed that for the first three specimens the 

latter amount of load exceeds the one that of the previous 

one whereas for the second three the outcome is completely 

vice versa. Bearing in mind that the all initial imperfections 

have been entered to FE models too, one can understated 

that the results of the first three specimens are closer to 

reality than the second three; that is because the equation 

emanated by Jawad formulation is based on the transformed 

geometrical shape from cylinder to cone; and 

according to Table 2, (R/r) ratio for the first three is more 

similar to a cylinder than the second three which have a 

higher tapering ratio. 



Fig. 7. General layout of failure by formation of plastic displacements in the circumference and supports. 

6. Buckling of conical shells 

The derivation of the equations for the buckling of 

conical shells is practically convoluted. The derivation 

for the buckling pressure of the cone, shown in Fig. 8, 

comprises obtaining expressions for the work carried out 

by the applied pressure, membrane forces, stretching of the 

middle surface and bending of the cone. The total work is 

then minimized to obtain a critical pressure expression. 

Seide [20] indicated that the buckling of a cone is affected 

by the function f (1 r/R) and is expressed as 

p cr ¼ ¯pf ð1 r=RÞ, (1) 

where ¯p is the pressure of equivalent cylinder as defined 

above, f the cone function as defined in Fig. 9.

p cr p 

125 

120 

115 

110 

105 

By various substitutions [19], it can be shown that Eq. (1) 

can be transferred to the form 

0:92Eðte=RÞ 25 

pcr ¼ , (2) 

Le=R 

where te is the effective thickness of cone t cos a; t the 

thickness of cone; and Le the effective length of cone L/2 

(1+r/R). Thus, conical shells subjected to external 

pressure may be analyzed as cylindrical shells with an 

effective thickness and length. 

6.1. Sole-fish buckling mode 

Fig. 8. Parametric considerations. 

100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

1-r/R 

Fig. 9. Cone function. 

An important parameter in buckling analysis of axisymmetrically 

loaded shells, for instance, the models thrashed 

out herein, is the number of circumferential waves in the 

buckling mode layout. Notionally speaking, the buckling 

mode is in the form of a single harmonic mode around the 

circumference, but in a test, owing to the incidence of 

imperfections, miscellaneous modes may be implicated. A 

well-established way of construing geometric imperfection 

and deformation measurements to decipher the dominant 

harmonic modes and their relationship is to carry out 



Fourier decompositions. Such decompositions were not 

carried out for the imperfection or deformation measurements 

on these models, since the form of buckling that was 

developed all over the cones was drastically outlying from a 

well-ordered wave configuration. On such specimens there 

was an overall buckling predisposed by position of weld 

lines and efficiency of the supports and formation of the 

initial leakage point. So as to set forth a better depiction of 

this buckling, principally based on empirical remarks and 

for the first time we coined the name ‘‘sole-fish buckling 

mode’’ on this phenomenon. The spine of this animal can 

better illustrate the weld line and its impact on the other 

parts of the shell. The appearance of non-periodical waves 

can be easily spotted from this plot. 

7. Observations and milestones 

7.1. Frusta specimens 

In Figs. 10 and 11 a contrast is carried out between 

initial and ultimate geometry of specimen SC5 in which the 

maximum deformation is located roughly at the height 

of 1 

3 h. 

In all shells of this study the loading was continued to 

farther than the range of postbuckling behavior. It is 

observed that a ‘‘V’’ shape yield line is developed in the 

region close to the restrained boundaries of the frusta, 

before failure takes place. The same phenomenon was 

reported by Showkati [21]. InFig. 6 a typical behavior is 

represented for specimen SC5 and SC6. 

By escalating the external pressure, the failure mode 

was gradually approached. In most specimens, incidence 

of a very large displacement in one edge caused an 

288 

270 

252 

306 

234 

324 

216 

342 

198 

300 

280 

260 

240 

220 

200 

180 

160 

140 

120 

100 

80 

60 

40 

20 

0 

0 

180 

18 

162 

36 

144 

54 

126 

Fig. 10. Polar plot of final geometry measured on specimen SC5. 

72 

90 

108

526 

radius 

250 

240 

230 

220 

210 

200 

190 

180 

170 

160 

150 

140 

130 

120 

110 

100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

0 

Radius (mm) 

204 

201 

198 

195 

192 

189 

186 

183 

180 

177 

174 

171 

168 

unmanageable outflow on vacuum function and then the 

test was impeded. Fig. 6 shows plainly the breakdown of 

frusta SC5 and SC6. 

It is worth noting in externally pressurized frusta shells 

that the inward deformations are as well, larger than 

outward ones. Comparable geometries of this fact in 

specimen SC5 and SCC1 have been plotted in Figs. 11 

and 12. Graphs of load–deformation and load–strain paths 

of specimens SC1 and SC3 are presented in Fig. 5(b) for 

buckling and postbuckling stages. 

In this experimental study it is observed that the longer 

shells have more deformation and lesser buckling load than 

shorter ones. 



buckled form 

initial imperfection 

0 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 

Degree 

Fig. 11. Initial and ultimate radial deformations in SC5 at the height of 141.4 mm. 

Initial imperfection 

Buckled layout 

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 

degree 

Fig. 12. Initial imperfection and buckled form of SCC1. 

As another comparison Fig. 13 shows the diversity in 

behavior between experimental and FEA methods for two 

specimens SC1 and SC4. 

7.1.1. Yield line wresting 

As shown in Fig. 6 in specimens SC5 and SC6 along 

with an increase in pressure at first, initial buckling 

lobes were formed, then up hilling the load general 

deformation developed in the whole body of the specimens 

to the extent that all lobes were complete. Exceeding 

the ultimate buckling load and as we neared the failure 

load the climax of some of the ‘‘V’’ shape yield lines 

that were formed began to wrest. This trend kept

acting until the bifurcation point at the apex of two 

or three yield lines caused resurgence of buckling lobes 

in the opposite side of the frusta and eventually it 

led to complete failure of one side of the frusta which 

took place in the supporting point with luxating the lower 

edge. 

7.2. SCC specimens 

SC1 

In Fig. 12 a contrast is carried out between initial and 

ultimate geometry of test specimen SCC1 in which the 

utmost deformation is positioned at the height of 1 

4L of the 

specimen. 


-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 

SC4 

FEA 

Expev 

FEA 

Exper. 


Radial displacement (mm) 

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 

Radial displacement (mm) 

Fig. 13. Radial displacement vs. external pressure for specimens, SC1 and SC4. 

0 

0 

The load–displacement curves for the net displacements 

are plotted for quite a few points in Fig. 5(a). In some of 

the models the displacements were similar in the initial 

stage of loading, while in the other ones the displacements 

started to differ early in the loading stage of SCC1. These 

two genuses of models were thus selected to contrast the 

two types of behavior. These curves show similar 

divergence as observed from the load–strain curves, 

Fig. 5(a). The first set, Fig. 5(a) which had similar 

displacements in the initial stage of loading, experienced 

rapid increases in displacements at increased loads. It is 

hence recommended that for the determination of the 

buckling load of a model conical shell in its corresponding 

0 

70 

65 

60 

55 

50 

45 

40 

35 

30 

25 

20 

15 

10 

5 

65 

60 

55 

50 

45 

40 

35 

30 

25 

20 

15 

10 

5 

0 

Pressure (KPa) 

Pressure (KPa)

528 

perfect state, a suitable load–deflection curve which shows 

an obvious slope change be identified. This is likely to be 

for a point in a relatively more perfect region of the shell. 

The intersection point between the initial slope of this 

curve and a tangent to the postbuckling part of the curve 

can be taken as a good rough figure to the buckling load of 

a corresponding perfect model. To identify such a suitable 

point, load–displacement curves of all points at or near 

wave crests and troughs can be contrived. This method 

emerges to offer a rational approach for the determination 

of buckling loads for conical caps. 

8. Concluding remarks 

This paper has described a recently developed experimental 

facility for buckling experiments on conical shells. 

The facility consists of a loading system, a simple 

measurement strategy for rather accurate geometric 

imperfection and deformation surveys, and compulsory 

equipment for the fabrication of quality test models. 

Distinctive results of sample tests have been presented to 

illustrate the competence of this facility. Procedures for 

processing the test results to verify both the buckling load 

and the modes of buckling have also been presented. 

The deliberate data and obtained domino effects are 

reported for six frusta and four SCC specimens with simply 

supported ends subjected to uniform peripheral pressure. 

The salient concluding tips are as follows [9]: 

Fabrication and testing of small-scale models have been 

undertaken to examine the buckling behavior of 

unstiffened shallow conical shells. In all models, smallscale 

manufacturing has produced relatively high 

imperfection values. However, since full imperfection 

scans have been recorded, the test results can be used to 

validate numerical or other models. Since the predictions 

are influenced by imperfection amplitudes, a 

reasonable assumption must be made in the absence of 

test models. 

In all specimens, the initial buckling occurred when one 

or more buckling lobes were detected. The applied 

pressure increased until an overall buckling mode was 

formed. 

The yield lines in the lower part of the frusta in all 

specimens are in the form of ‘‘V’’ shape, which were 

recognized in the range of postbuckling. 

In all specimens, the inward deformations are so larger 

than the outward deformations. 

In all specimens, difference between initial buckling and 

overall buckling loads was substantial. 

Postbuckling potency exists apparently in all specimens 

under the effect of external pressure. 

In all specimens, it is experimentally corroborated that 

the longer shells are more flexible in radial direction and 

therefore, they are weaker than shorter shells. 

The buckling loads obtained from experiments are lower 

than the ones derived from FEA and Jawad equation 



while the two latter ones are different in some aspects 

related to the tapering ratio of specimens (R/r) and 

initial imperfections present in FE models but absent in 

the arithmetical come up, which is worth noticing as 

mentioned in the context. 

Clearly, for more slender shells the collapse mode will 

change and the kinematical assumptions of the mechanism 

approach would be inappropriate. 

Acknowledgments 

The work depicted herein outlined part of a scheme on 

‘‘Stability and Strength of Conical Cones’’ subsidized by 

the Ministry of Science, Technology and Research in 

the I.R. Iran and carried out in collaboration with the 

Structural Research Center at Urmia University. We would 

like to put across gratitude to the technicians in the 

Structures Laboratory of Urmia University, in particular 

Mr. Jafar Azim Zadeh and our best friend Mr. Emad 

Jahangiri for their enthusiasm and professionalism in 

conducting the probes. The authors are so appreciative to 

Prof. J.G. Teng, for his great favors in providing 

commentaries and presenting his constitutive remarks 

regarding this research. 

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