Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression
Buckling of steel thin walled semi-spherical shells with square cutout due to axial compressive loads has been studied by numerical simulations, and results were compared with those from the experiments.
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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irk-123456789-1731272020-11-23T01:26:20Z Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression Torabi, H. Shariati, M. Научно-технический раздел Buckling of steel thin walled semi-spherical shells with square cutout due to axial compressive loads has been studied by numerical simulations, and results were compared with those from the experiments. Выполнены численный расчет потери устойчивости стальных тонкостенных полусферических оболочек с квадратным вырезом, подвергнутых осевому сжатию, и сравнительный анализ полученных расчетных данных с экспериментальными. Виконано числовий розрахунок втрати стійкості стальних тонкостінних півсферичних оболонок із квадратним вирізом, що знаходяться під дією осьового стиску, і порівняльний аналіз отриманих розрахункових даних з експериментальними. 2014 Article Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression / H. Torabi, M. Shariati // Проблемы прочности. — 2014. — № 4. — С. 109-122. — Бібліогр.: 16 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173127 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Torabi, H. Shariati, M. Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression Проблемы прочности |
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Buckling of steel thin walled semi-spherical shells with square cutout due to axial compressive loads has been studied by numerical simulations, and results were compared with those from the experiments. |
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Article |
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Torabi, H. Shariati, M. |
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Torabi, H. Shariati, M. |
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Torabi, H. |
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Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression |
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Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression |
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Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression |
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Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression |
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Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression |
| title_sort |
buckling analysis of steel semi-spherical shells with square cutout under axial compression |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under Axial Compression / H. Torabi, M. Shariati // Проблемы прочности. — 2014. — № 4. — С. 109-122. — Бібліогр.: 16 назв. — англ. |
| series |
Проблемы прочности |
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AT torabih bucklinganalysisofsteelsemisphericalshellswithsquarecutoutunderaxialcompression AT shariatim bucklinganalysisofsteelsemisphericalshellswithsquarecutoutunderaxialcompression |
| first_indexed |
2025-07-15T09:39:49Z |
| last_indexed |
2025-07-15T09:39:49Z |
| _version_ |
1837705347470458880 |
| fulltext |
UDC 539.4
Buckling Analysis of Steel Semi-Spherical Shells with Square Cutout under
Axial Compression
H. Torabi
a,1
and M. Shariati
b
a Young Researchers Club, Mashhad Branch, Islamic Azad University, Mashhad, Iran
b Mechanical Department, Shahrood University of Technology, Shahrood, Iran
1 Torabi.mech87@gmail.com
ÓÄÊ 539.4
Ðàñ÷åò ïîòåðè óñòîé÷èâîñòè ñòàëüíûõ ïîëóñôåðè÷åñêèõ îáîëî÷åê ñ
êâàäðàòíûì âûðåçîì, ïîäâåðãíóòûõ îñåâîìó ñæàòèþ
Õ. Òîðàáè
à,1
, Ì. Øàðèàòè
á
à Êëóá ìîëîäûõ èññëåäîâàòåëåé, Ìåøõåäñêèé ôèëèàë Èñëàìñêîãî óíèâåðñèòåòà Àçàä, Ìåøõåä,
Èðàí
á Ôàêóëüòåò ìåõàíèêè Øàõðóäñêîãî òåõíîëîãè÷åñêîãî óíèâåðñèòåòà, Øàõðóä, Èðàí
Âûïîëíåíû ÷èñëåííûé ðàñ÷åò ïîòåðè óñòîé÷èâîñòè ñòàëüíûõ òîíêîñòåííûõ ïîëóñôåðè-
÷åñêèõ îáîëî÷åê ñ êâàäðàòíûì âûðåçîì, ïîäâåðãíóòûõ îñåâîìó ñæàòèþ, è ñðàâíèòåëüíûé
àíàëèç ïîëó÷åííûõ ðàñ÷åòíûõ äàííûõ ñ ýêñïåðèìåíòàëüíûìè. Ïðè ýòîì èñïîëüçîâàëèñü òðè
âàðèàíòà ïðèëîæåíèÿ ê îáðàçöàì âåðòèêàëüíûõ ñæèìàþùèõ íàãðóçîê: ÷åðåç æåñòêóþ
ïëîñêóþ ïëàñòèíó; ÷åðåç æåñòêèå áàëêè ñ òîðöàìè öèëèíäðè÷åñêîé è ñôåðè÷åñêîé ôîðìû.
Îïðåäåëÿëîñü âëèÿíèå îòíîøåíèé øèðèíû êâàäðàòíîãî âûðåçà ê åãî âåðòèêàëüíîìó ïîëîæå-
íèþ â îáîëî÷êå (a H) è òîëùèíû îáîëî÷êè ê åå äèàìåòðó (t D) íà ñðåäíåå çíà÷åíèå êðèòè-
÷åñêîé íàãðóçêè, ïðè êîòîðîé ïðîèñõîäèò ïîòåðÿ óñòîé÷èâîñòè ïîëóñôåðè÷åñêîé îáîëî÷êè.
Êîíå÷íîýëåìåíòíûå ìîäåëè ðåàëèçîâàíû ñ ïîìîùüþ ïðîãðàììíîãî ïàêåòà ABAQUS äëÿ íåëè-
íåéíîãî ðàñ÷åòà ïîòåðè óñòîé÷èâîñòè, à ñîîòâåòñòâóþùèå ýêñïåðèìåíòàëüíûå ðåçóëüòàòû
ïîëó÷åíû ñ èñïîëüçîâàíèåì ñåðâîãèäðàâëè÷åñêîé èñïûòàòåëüíîé óñòàíîâêè INSTRON 8802.
Ñðàâíèòåëüíûé àíàëèç ðåçóëüòàòîâ, ïîëó÷åííûõ äâóìÿ ðàñ÷åòíûìè ìåòîäàìè, ïîêàçàë
òåñíóþ êîððåëÿöèþ ìåæäó ýêñïåðèìåíòàëüíûìè è ÷èñëåííûìè íåëèíåéíûìè ðàñ÷åòíûìè
äàííûìè.
Êëþ÷åâûå ñëîâà: ïîòåðÿ óñòîé÷èâîñòè, ïîëóñôåðè÷åñêàÿ îáîëî÷êà, êâàäðàòíûé âûðåç,
ìåòîä êîíå÷íûõ ýëåìåíòîâ, ýêñïåðèìåíò.
Introduction. Thin walled semi-spherical shells usually are extensively used in many
types of structures due to their energy absorbing capacity. They are subjected to various
combinations of loading. The most critical load which challenges the stability of thin shells
is axial compression. The buckling behavior of these shells gives rise to their critical design
application, such as nose cone of aircraft, lunch vehicles and ballistic missiles due to high
energy absorbing capacity. The major deformation of rigid plastic semi-spherical shells
which were compressed between two rigid plates was first studied by Updike [1] which led
to proposal of an analytical model. The computation was restricted to the compression up to
about one-tenth of the shell radius. Deformation patterns on semi-spherical shells of R t
ratios between 36 to 420 were studied experimentally and analytically by Kitching et al.
[2]. De Oliveira and Wierzbicki [3] did similar study on crushing analysis of rotationally
symmetric plastic shells. There was also a quasi-static study on semi-spherical shells of
© H. TORABI, M. SHARIATI, 2014
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4 109
R t ratios between 36 to 420 by Kinkead et al. [4] in which the results were compared with
previous studies. Gupta et al. [5] performed experiments on metallic semi-spherical shells
of R t ratios ranging between 15 to 240 and the three levels of deformation, namely: local
flattening, inward dimpling, and multiple lobes were studied. A two-dimensional numerical
analysis for the semi-spherical shells under axial impact was presented by Gupta and
Venkatesh [6]. In this study, a very good correlation was observed between numerical
simulation and experimental results in buckling behavior related to first mode jumping
from local flattening to inward dimpling.
Shariati and Mahdizadeh Rokhi [7] studied the effect of position of elliptical cutouts
with identical dimensions on the buckling and postbuckling behavior of cylindrical shells
with different diameters and lengths and developed several parametric relationships based
on the 3 numerical and experimental results using the Lagrangian polynomial method.
Also, Shariati and Mahdizadeh Rokhi [8] performed a similar numerical study using
ABAQUS software to investigate the response of steel cylindrical shells with different
lengths and diameters, including elliptical cutout subjected to bending moment. They
presented some relations for finding of buckling moment of these structures.
Gupta [9] performed another study in which the semi-spherical shells of R t ratios
between 26 and 45 were analyzed experimentally and computationally. In experiments, all
the spherical shells were found to collapse in an axsymmetric mode.
Shariati and Mahdizadeh Rokhi [10] investigated numerical simulation and analysis of
steel cylindrical shells with various diameters and lengths having an elliptical cutout,
subjected to axial compression. In this work, they examined the influence of the cutout size,
cutout angle and the shell aspect ratios (L D and D t) on the pre-buckling, buckling, and
post-buckling responses of the cylindrical shells. In addition, Shariati and Mahdizadeh
Rokhi [11] did another work in which simulation and analysis of steel cylindrical shells of
various lengths, including quasi-elliptical cutout, subjected to axial compression load were
systematically carried out using the finite element method. The investigation examined the
influence of the cutout location and the shell aspect ratio (L D) on the buckling, and the
post-buckling responses of the cylindrical shells.
Shariati and Allahbakhsh [12] studied the buckling and postbuckling of steel thin-
walled semi-spherical shells under different loadings, both experimentally and numerically.
Various vertical compression loadings were applied to specimens using the following
methods: a rigid flat plate and some rigid bars with circular, square and spherical cross
sections, a rigid tube, a plate with a hole, and an indented tube.
In this work, efforts are made to determine the effect of the position and size of square
cutouts on the buckling behavior of semi-spherical shells. Various vertical compression
loadings are applied to specimens and the mean load are obtained for each other. For this
purpose, finite element (FE) models of semi-spherical shells with a square cutout having
different shell aspect ratios (t D) are generated. These FE models are analyzed using
ABAQUS linear and nonlinear analysis. In addition, several buckling tests were performed
using an INSTRON 8802 servo-hydraulic machine and the results were compared with the
results of the finite element method. A very good correlation between experiments and
numerical simulations was observed. Finally, based on the experimental and numerical
results, formulas are presented for the computation of the buckling load in such structures.
1. Numerical Analysis Using the Finite Element Method. The numerical simulations
were carried out using the finite element software ABAQUS 6.4-PR11.
1.1. Geometry and Mechanical Properties of the Shells.
In this study, thin-walled semi-spherical shells with four different thickness (t � 0.7,
0.8, 1.0, and 1.2 mm) were analyzed. A square geometry was selected for cutouts that were
created in the specimens. Figure 1 illustrates the geometry of the specimens. According to
this figure, parameters (D, d , t , and h) show the upper diameter, lower diameter, thickness
and height of the semi-spherical shells, respectively. In Fig. 1, parameter a shows the side
H. Torabi and M. Shariati
110 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4
length of the square cutout, parameter H is the distance between the center of the cutout
and the lower edge of the shell, parameter L is the arc length, and P and A are the
perimeter and area of the cutout, respectively. The values of geometric parameters for the
semi-spherical shells were d � 102 mm, D� 25 mm, and h� 52 mm. The specifications
of square cutouts are presented in Table 1.
Specimens were nominated as follows: H24-t0.8-a10. The number following t shows
the thickness value of the specimen. Furthermore, the numbers following H and a show
the distance between the center of the cutout and the lower edge of the shell and the side
value of the square cutout, respectively.
The semi-spherical shells used for this study were made of mild steel alloy. The
mechanical properties of this steel alloy were determined according to ASTM E8 standard
[13], using the INSTRON 8802 servo-hydraulic machine. The stress–strain and stress–
plastic strain curves can be found in [7]. The value of elasticity module was computed as
E � 150 GPa and the value of yield stress was obtained as � � 404 MPa. Furthermore, the
value of Poisson’s ratio was assumed to be � � 0.33.
Buckling Analysis of Steel Semi-Spherical Shells ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4 111
T a b l e 1
Specification for Square Cutouts
Model
specification
Area (A),
mm2
Perimeter (P),
mm
Arc length (L),
mm
R
P
H3 � R
a
H4 �
H8-a10
H16-a10
H24-a10
H40-a10
H16-a20
H24-a20
H32-a20
H40-a20
H16-a30
H24-a30
100.42
101.03
105.43
140.01
408.72
429.46
472.24
638.03
936.50
990.20
40.15
40.31
41.21
48.00
81.13
83.20
88.01
106.60
123.05
127.41
10.06
10.14
10.59
13.97
20.42
21.43
23.79
32.80
30.99
33.06
5.019
2.519
1.717
1.200
5.071
3.467
2.750
2.667
7.691
5.309
1.2500
0.6250
0.4167
0.2500
1.2500
0.8330
0.6250
0.5000
1.8750
1.2500
Fig. 1. Geometry of specimens and cutouts.
1.2. Boundary Conditions. For applying boundary conditions on the bottom edges of
the semi-spherical shells, a rigid plate was attached to the bottom edges of the semi-
spherical shells. To analyze their buckling numerically, the specimens were subjected to
axial load similar to what was done in the experimental tests. In this process, a displacement
was applied to the center of the upper plate, or bar.
Additionally, all degrees of freedom in the lower plate and all degrees of freedom in
the upper plate, or bar, except in the direction of longitudinal axis, were constrained.
1.3. Element Formulation of the Specimens. In this study, the nonlinear element
S8R5, which is an eight-node element with six degrees of freedom per node and is suitable
for analysis of thin shells, was used. The nonlinear element was used for the analysis of the
shells, and the results were compared with each other. For rigid plate or bar the element
R3D4 was used. A friction coefficient of 0.1 has been taken. The effect of friction
coefficient ranged from 0.08 to 0.12 and affected results by less 1% [6].
1.4. Numerical Process. To analyze the buckling of semi-spherical shells, two
analysis methods, linear eigenvalue analysis and geometric nonlinear, were employed using
the “Buckle” and “Static Riks” solvers respectively. For more information about these FE
methods you can refer to Shariati and Allahbakhsh [12], Lee et al. [14] and ABAQUS user
manual [15].
2. Results of Numerical Analysis. In this Section, the numerical results of the
buckling analysis of semi-spherical shells with square cutouts of different sizes and
locations, using the finite element method, are presented. Four different shell thicknesses of
1.2, 1.0, 0.8, and 0.7 mm were analyzed.
2.1. Loading by a Rigid Plate. In this paper, for comparison, the energy absorption
capacity of specimens is a criteria that defines the mean collapse load. Mean collapse load
is calculated by dividing the area of under the load–displacement curve by the displacement
of the upper rigid plate. During loading by a rigid plate, it is seen that the collapse is
initiated by the formation of an axisymmetric ring at the smaller end. With further
compression, the mechanism of collapse changes. At this stage, its propagation is due to the
formation of stationary plastic hinges and internal lobes that is in contact with the top plate.
The number of internal lobes depends on the cutout size and location. It is clearly
noticeable as the slope of the load–deformation curves changes appreciably as shown in
Fig. 2.
112 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4
H. Torabi and M. Shariati
Fig. 2. Load–deformation curves of specimen H24-a10 and wall thickness for a semi-spherical shell
with thickness of 0.7 mm in loading by a rigid plate. (Here and in Fig. 3: A = zone representing
formation of an axisymmetric ring at the smaller end and B = zone representing formation of an
axisymmetric inward dimpling.)
2.2. Loading by Different Bars. In this section, the effect of loading conditions is
considered. Therefore, some semi-spherical shells with a diameter equal to the small
diameter of the semi-spherical shells (d � 25 mm) are loaded by a circular bar. Semi-
spherical shells are also loaded by a semi-spherical end.
Figure 3 shows the load–deformation curves that were obtained for the specimen
H32-a20 with various thicknesses in loading by a circular cross section bar. During loading
with a circular cross section, only the first mode is observed from the formation of an
axisymmetric ring towards inward dimpling. As it can be observed increasing the thickness
of shell will lead to higher values of mean collapse load. The primary part of the curve in
loading by circular cross section bar is linear shown.
Table 2 present the results from numerical simulations with the rigid plate (RP) and
bar with a circular cross section (CC).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4 113
Buckling Analysis of Steel Semi-Spherical Shells ...
Fig. 3. Load–deformation curves of specimen H32-a20 and wall thickness for a semi-spherical shell
with thickness of 0.7 mm in loading by a circular cross section bar.
T a b l e 2
Summary of Numerical Analysis for Semi-Spherical Shells Including a Square Cutout
in Loading with Rigid Plate and Circular Cross Section Bar
Model
specification
Deformation
height
(mm)
Mean collapse load (kN)
t � 0 7. mm t � 0 8. mm t �10. mm t �12. mm
RP CC RP CC RP CC RP CC
Perfect
H8-a10
H16-a10
H24-a10
H40-a10
Perfect
H16-a20
H24-a20
H32-a20
H40-a20
Perfect
H16-a30
H24-a30
28
28
28
28
28
23
23
23
23
23
21
21
21
14.85
13.28
13.32
13.42
13.72
13.06
11.58
12.04
10.60
10.33
12.37
9.22
10.67
3.82
3.29
3.50
3.44
3.60
3.41
3.12
3.11
2.83
2.34
3.24
3.11
2.94
17.84
16.21
16.39
17.32
16.67
14.89
14.40
14.49
12.80
12.41
15.69
12.86
13.67
4.67
4.73
4.37
4.28
4.36
4.17
3.88
3.87
3.55
2.90
3.97
3.68
3.63
23.95
22.90
22.88
23.51
22.79
20.18
20.78
19.66
17.90
17.32
21.22
17.81
18.24
6.47
6.65
6.20
6.12
6.09
5.75
5.53
5.51
5.11
4.14
5.44
5.22
5.10
30.79
30.24
30.11
30.07
29.51
26.05
26.81
25.36
23.30
23.44
27.36
23.22
23.24
8.97
8.60
8.28
8.16
8.10
7.98
7.38
7.36
6.88
5.85
7.68
7.25
6.84
Figure 4 shows loading by a rigid bar with a semi-spherical cross section. As for
loading by a circular bar, it is seen that the primary part of the curve is linear. Also, the
formation of an axisymmetric ring is not observed and a mode jump is observed, namely,
from inward dimpling to formation of stationary plastic hinges.
Table 3 presents the results from numerical simulations for the bar with a semi-
spherical cross section.
2.3. Loading by a Rigid Tube. In this section, the loading is carried out by a rigid tube
with thickness of 5 mm. Figure 5 shows the load–deformation curve and wall thickness for
the H40-t0.7-a10 which has been loaded by tube with two � values. It is clear that with
decreasing �, the mean collapse load decreases.
2.4. The Effects of Cutout Size, a H and t D Ratios.
2.4.1. Analysis of the Effect of Change in Cutout Height on the Mean Collapse Load.
To study the effect of a change in cutout height on the buckling load of semi-spherical
shells, cutouts with constant sides (10, 20, and 30 mm) were created in different positions
of shells. Then, with changing the height of the cutouts from 8 to 40 mm, the change in
mean collapse load was studied. The results of the analysis are shown in Table 2.
2.4.2. Analysis of the Effect of Change in Cutout Side on the Mean Collapse Load. In
this section, the effect of changing the cutout side on the mean collapse load of
semi-spherical shells is studied. For this reason, cutouts with fixed height (8, 16, 24, 32,
114 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4
H. Torabi and M. Shariati
T a b l e 3
Summary of Numerical Analysis for Semi-Spherical Shells Including a Square Cutout
in Loading with a Spherical Cross Section Bar
Model specification Deformation (mm) Mean collapse load (kN)
H16-t0.7-a10
H24-t0.7-a10
H40-t0.7-a10
H16-t0.7-a20
H24-t0.7-a20
H32-t0.7-a20
H40-t0.7-a20
H16-t0.7-a30
H24-t0.7-a30
25
25
25
20
20
20
20
35
35
2.99
2.92
3.71
2.56
2.57
2.99
2.53
4.82
3.14
Fig. 4. Load–deformation curves of specimen H32-t0.7-a20 and wall thickness for a semi-spherical
shell in loading by a spherical cross section bar (SPC).
and 40 mm) were created in different positions of shells. Then, with changing the side of
the cutouts from 10 to 30 mm, the change in mean collapse load was studied. The results of
this analysis are also presented in Table 2. The results show when the cutout height is
constant, an increase in cutout side decreases the mean collapse load. It is evident from
Table 2 that an increase in the cutout side when cutout height is constant causes a
considerable reduction in the mean collapse load.
3. Experimental Verification. Experimental tests were conducted on a large number
of specimens, in order to confirm some of the cases analyzed in the numerical simulations.
As shown in Fig. 6, for these tests a servo-hydraulic INSTRON 8802 machine was used.
The specimens were constrained by steel sleeve fixtures inserted at both ends, which
models the fixed-fixed boundary condition used in the finite element simulations (Fig. 6).
Various specimens were tested for each loading and almost identical results were obtained
compared to those obtained from the numerical simulations. The experimental results are
compared to numerical findings in Tables 4 and 5 in loading by the rigid plate and circular
cross section bar, respectively. The comparison shows that there is a little difference
between the two sets of data.
The load–deformation curves and deformed shape of specimens in the buckling and
postbuckling states in numerical and experimental tests in loading by the rigid plate and
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Fig. 5. Load–deformation curves for the specimen H40-t0.7-a10 at loading by rigid plate with a hole
at � �15 and 30�.
Fig. 6. A servo-hydraulic INSTRON 8802 machine in loading on semi-spherical shell including a
square cutout by rigid plate.
circular cross section bar are compared in Figs. 7–15 and Fig. 16, respectively. It can be
seen that the slope of linear part of the curves is higher in numerical analysis than in
experimental results. This is maybe due to the presence of internal defects in the material
which reduce the stiffness of the specimens in the experimental method, while the materials
are assumed to be ideal in the numerical analyses.
Comparison of deformations resulted by numerical and experimental methods for the
specimens shown in Fig. 16 in the buckling state, shows that almost identical results were
obtained.
4. Empirical-Numerical Equations. Based on the numerical and experimental
dimensionless mean collapse loads of shells, formulas are presented here using the
Lagrangian polynomial for the computation of the mean collapse load of semi-spherical
shells with square cutouts subject to axial compression. To get these formulas, surfaces
were fitted to the dimensionless mean collapse load values using the Lagrangian polynomial
method [16]. Value of K cutout is introduced as a mean collapse load reduction factor for
semi-spherical shells with cutout (dimensionless mean collapse load), � � t D and � � a H.
H. Torabi and M. Shariati
T a b l e 4
Comparison of the Experimental and Numerical Results for Semi-Spherical Shells
Including a Square Cutout in Loading with a Rigid Plate
Model
specification
Deformation
height (mm)
Mean collapse load (kN) | |
%
F Fnum
exp
numF
100
Experimental Numerical
H16-a10
H24-a10
H40-a10
H16-a20
H24-a20
H32-a20
H40-a20
H16-a30
H24-a30
20
20
20
20
20
20
20
20
20
10.66
10.65
9.99
10.44
10.80
10.04
8.30
9.74
8.41
10.72
10.75
10.85
10.63
11.17
9.90
8.37
9.07
9.20
0.56
0.93
7.93
1.79
3.31
1.41
0.84
7.39
8.59
T a b l e 5
Comparison of the Experimental and Numerical Results for Semi-Spherical Shells
Including a Square Cutout in Loading with a Circular Cross Section Bar
Model
specification
Deformation
height (mm)
Mean collapse load (kN) | |
%
F Fnum
exp
numF
100
Experimental Numerical
H8-a10
H16-a10
H24-a10
H40-a10
H16-a20
H24-a20
H32-a20
H40-a20
H16-a30
H24-a30
25
25
25
25
25
25
25
25
25
25
3.33
3.29
3.16
2.98
3.10
3.20
2.76
2.36
3.04
2.89
3.18
3.34
3.30
3.28
3.26
3.28
2.89
2.46
3.22
3.18
4.72
1.50
4.24
9.15
4.91
2.44
4.50
4.07
5.59
9.11
116 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4
Value of K cutout is defined according to Eq. (1):
K
F
F
cutout
cutout
perfect
� , (1)
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Buckling Analysis of Steel Semi-Spherical Shells ...
a b
Fig. 7. Comparison of the experimental and numerical results for specimen H16-t0.7-a10 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
a b
Fig. 8. Comparison of the experimental and numerical results for specimen H24-t0.7-a10 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
a b
Fig. 9. Comparison of the experimental and numerical results for specimen H40-t0.7-a10 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
where Fperfect is the mean collapse load for perfect semi-spherical shells without cutouts
and Fcutout is the mean collapse load for semi-spherical shells with cutout.
The general form of K cutout is according to Eq. (2):
K A B C D E Fcutout ( , ) ... .� � � � � � ��� � � � � � �2 2
(2)
The coefficients A, B, C , … are computed using the Lagrangian polynomial method.
To use these expressions, the mean collapse load for semi-sherical shells without cutout
must be known.
118 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4
H. Torabi and M. Shariati
a b
Fig. 10. Comparison of the experimental and numerical results for specimen H16-t0.7-a20 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
a b
Fig. 11. Comparison of the experimental and numerical results for specimen H24-t0.7-a20 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
a b
Fig. 12. Comparison of the experimental and numerical results for specimen H32-t0.7-a20 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
The exact form of the resulting equations is summarized in Eqs. (3)–(6). Both
experimental and numerical results (in situations where experimental data were not
available) are used in these equations.
177807 24 972 1217934 39252 397 72115363 2. . . . . .� � � �� � 5 2� �
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Buckling Analysis of Steel Semi-Spherical Shells ...
a b
Fig. 13. Comparison of the experimental and numerical results for specimen H40-t0.7-a20 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
a b
Fig. 14. Comparison of the experimental and numerical results for specimen H16-t0.7-a30 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
a b
Fig. 15. Comparison of the experimental and numerical results for specimen H24-t0.7-a30 in loading
with a rigid plate (a) and a rigid tube with a circular cross section (b).
�
�
2581 10 5738 10 3705 10 43914315 2 3 5 2 2 5 2. . . .� � � � � � ��3 �
� 9730298 62374072. . ,�� �� (3)
7838 21805 1766218 1007 10 1664 103 5 2 5 2. . . . .� � � � � �� �
�
3
�
�4 592 10 3763 10 265185 75293215 2 2 5 2 3 2. . . .� � � � �� ��
6335756 5767 24187 2. . . ,�� � (4)
1593001715 5552 556068 1199 5644 308343 2. . . . . .� � � �� � 261 2� �
�
� 45637580 1223 10 98656650 70081782 3 5 2 2 2. . . .� � � � � � ��3 �
� 1989794 16882492. . ,�� �� (5)
120 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4
H. Torabi and M. Shariati
Fig. 16. Comparison of the experimental and numerical results for specimens: (a) H40-t0.7-a10 (CC);
(b) H32-t0.7-a20 (CC); (c) H40-t0.7-a20 (CC); (d) H40-t0.7-a10 (RP); (e) H24-t0.7-a20 (RP); (f)
H24-t0.7-a30 (RP).
a b
c d
e f
507 19126 1196966 88830365 8264 2883 2 2 3. . . . .� � � � � �� �
� � 4479194 34014 989 362212 15923912 2 2 3 2. . . .� � � � �� ��
� 2081485 2863 17282 2. . . .�� � (6)
Equations (3) and (4) yield the reduction factor for the semi-spherical shells with the
deformation height of 28 mm and various lengths (0.006863� �t D 0.011765), with a
square cutout of fixed side (a� 10) and various heights (0.25� �a H 1.25) in different
positions of the shell for loading by a rigid plate and a circular cross section bar,
respectively.
Equations (5) and (6) yield the same for a square cutout of fixed side (a� 20) and
various heights (0.5� �a H 1.25).
Conclusions. Semi-spherical shells of different loading with square cutouts of various
sizes and positions were investigated experimentally and numerically. The load–deformation
curves at different stages of compression are found to match well with those obtained from
experiments. The predicted deformed shapes at different stages of compression and various
loadings are also found to be in good agreement with the actual deformed profiles. Finally,
formulas were presented for the computation of the mean collapse load of semi-spherical
shells with square cutouts based on the mean collapse load of perfect semi-spherical shells.
These relationships are applicable to a wide range of semi-spherical shells with square
cutouts. The following results were found in this study:
1. The mean collapse load is higher in loading by circular bar, in comparison with that
by a rigid bar with a spherical cross section.
2. The thickness of shell changes during compression and thickness strain is more in
stationary plastic hinges, in comparison to rolling plastic hinges.
3. Rolling plastic hinges increase with increasing of thickness.
4. Among semi-spherical shells of four different thickness values with an identical
square cutout, the best geometry, which ensures the maximal value of the mean collapse
load under compression conditions, is that with the maximal thickness.
When the cutout side length is constant and height of the cutout increases, the mean
collapse load reduces. However, the amount of reduction in the mean collapse load is
negligible. Increasing the side length of the cutout while the cutout height is constant
decreases the mean collapse load extremely.
Ð å ç þ ì å
Âèêîíàíî ÷èñëîâèé ðîçðàõóíîê âòðàòè ñò³éêîñò³ ñòàëüíèõ òîíêîñò³ííèõ ï³âñôåðè÷-
íèõ îáîëîíîê ³ç êâàäðàòíèì âèð³çîì, ùî çíàõîäÿòüñÿ ï³ä 䳺þ îñüîâîãî ñòèñêó, ³
ïîð³âíÿëüíèé àíàë³ç îòðèìàíèõ ðîçðàõóíêîâèõ äàíèõ ç åêñïåðèìåíòàëüíèìè. Ïðè
öüîìó âèêîðèñòàíî òðè âàð³àíòà ïðèêëàäàííÿ äî çðàçê³â âåðòèêàëüíèõ ñòèñêàëüíèõ
íàïðóæåíü: ÷åðåç æîðñòêó ïëîñêó ïëàñòèíó; ÷åðåç æîðñòêó áàëêó ç òîðöÿìè öèë³íä-
ðè÷íî¿ ³ ñôåðè÷íî¿ ôîðìè. Âèçíà÷àâñÿ âïëèâ â³äíîøåíü øèðèíè êâàäðàòíîãî âèð³çó
äî éîãî âåðòèêàëüíîãî ïîëîæåííÿ â îáîëîíö³ (a H) ³ òîâùèíè îáîëîíêè äî ¿¿
ä³àìåòðà (t D) íà ñåðåäíº çíà÷åííÿ êðèòè÷íîãî íàâàíòàæåííÿ, çà ÿêîãî â³äáóâàºòüñÿ
âòðàòà ñò³éêîñò³ ï³âñôåðè÷íî¿ îáîëîíêè. Ñê³í÷åííîåëåìåíòí³ ìîäåë³ ðåàë³çîâàíî çà
äîïîìîãîþ ïðîãðàìíîãî ïàêåòà ABAQUS äëÿ íåë³í³éíîãî ðîçðàõóíêó âòðàòè ñò³é-
êîñò³, à â³äïîâ³äí³ åêñïåðèìåíòàëüí³ ðåçóëüòàòè îòðèìàíî ç âèêîðèñòàííÿì ñåðâî-
ã³äðàâë³÷íî¿ âèïðîáóâàëüíî¿ óñòàíîâêè INSTRON 8802. Ïîð³âíÿëüíèé àíàë³ç ðåçóëü-
òàò³â, îòðèìàíèõ äâîìà ðîçðàõóíêîâèìè ìåòîäàìè, ïîêàçàâ ò³ñíó êîðåëÿö³þ ì³æ
åêñïåðèìåíòàëüíèìè ³ ÷èñëîâèìè íåë³í³éíèìè ðîçðàõóíêîâèìè äàíèìè.
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Buckling Analysis of Steel Semi-Spherical Shells ...
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Received 10. 07. 2013
122 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 4
H. Torabi and M. Shariati
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