Buckling of an imperfect spherical shell subjected to external pressure
M.S. Ismaila,1, J. Mahmudb, A. Jailanic
a
Jabatan Kejuruteraan Mekanikal, Politeknik Sultan Salahuddin Abdul Aziz Shah,
Shah Alam, 40150, Selangor, Malaysia
b
School of Mechanical Engineering, College of Engineering, Universiti Teknologi MARA, 40450, Shah
Alam, Selangor, Malaysia
c
Jabatan Kejuruteraan Mekanikal, Politeknik Port Dickson, Km 14 Jalan Pantai 71050 Si Rusa, Port
Dickson, Negeri Sembilan.
Abstract
This paper presents numerical results focusing on the buckling behaviour of externally pressurised imperfect
spherical shells. The numerical model followed the recommendations of the European Standard EN1993-1-6.
A good agreement between the model FE and the experimental results with an average difference of less than
7% was found. The perfect steel spherical shell is then subjected to several imperfection approaches, such as
(a) eigenmode imperfection and (b) single load indentation (SLI) imperfection. The eigenmode shape
imperfection proves to be the worst imperfection for the externally pressurised spherical shells, yet the SLI is
an attractive technique to simulate the realistic imperfection (i.e., dent/dimple). Based on the recommendation
of the European Standard EN1993-1-6 for imperfection tolerance, the SLI imperfection was used for the
realistic-case imperfection approach for different spherical shell configurations. A lower-bound closed-form
empirical formula for a spherical shell under external pressure was proposed for the spherical shell with the
single load indentation (SLI). The results show that for a shell shape parameter λ ≥ 10, the EBC provides a
much higher knockdown factor compared to the guidelines of NASA SP -8032, Wagner, and lower-bound
closed-form empirical formula (the differences are in the range of 48% - 56%).
Keywords: Spherical shell; Eigenmode imperfection; Single Load Indentation (SLI) imperfection; Closedform lower-bound empirical formula; Buckling; Design codes; Knockdown factor
1
Corresponding author: M.S. Ismail; E-mail: mohdshahromismail@gmail.com; mohdshahrom@psa.edu.my;
Phone: +6012-9676112
1
Introduction
Spherical shells are commonly used in many engineering applications. Typically, they are used as partitions
in pressure vessels/tanks, as domes to close the ends of cylindrical pressure vessels, or as hatches to cover the
access ports of variously shaped pressure vessels in subsea applications, vacuum vessels (in the chemical
industry), aerospace, and civilian applications [1–3]. In these engineering applications, the shell will most
likely be subjected to various loading conditions, with external pressure being the most common. The final
strength and stability of externally pressurised spherical shells depends strongly on their shape, material
properties, pre-buckling deformations and geometric imperfections [2,4,5]. For submersibles application, the
shell usually has to be designed to meet the ultimate strength requirements with some safety margin to achieve
the desired diving depth [6]. Some success stories in submersible design and development include Jiaolong,
operated by China NDSC [7], Shinkai, operated by JAMSTEC, Nautile, operated by IFREMER, and Consul
AS37, operated by the Russian Navy. In practice, the spherical shell is designed following contemporary
design standards such as European Convention for Constructional Steelwork [8], Det Norske Veritas [9],
British Standard [10], and American Bureau of Shipping (ABS) [11]. These design standards were derived
from the results of previous tests, particularly those conducted by the British and American naval research
institutions. The design curve in Section 3.6 of PD 5500 is derived from the lower bound of the experimental
results. Conversely, the ECCS design code also used the previous theoretical and experimental results as the
basis for the design code. Recently, a series of experiments have been conducted for (i) spherical shells [12–
16] and (ii) hemispherical shells [6,17–19] subjected to external pressure. The tests are considered to be of
high quality since they give a repeatable buckling load for identical models. Moreover, the spherical shell can
be classified as a complete, deep, and shallow shell/structure [20].
The presence of an initial structural imperfection is inevitable and poses a challenge to an engineer as it can
affect the integrity and stability of the structure. To this day, the formation of initial imperfections puzzles
engineers/designers as they have been defined in terms of its position, dent- amplitude, worst shape, etc. It is
well known that most fabricated or machined spherical shells are susceptible to the presence of an initial
imperfection. Practically, the initial geometric imperfection is most likely due to accidental damage [37]. In
this regard, the ability to understand the situation can give the engineer an advantage in his safety measures
and failure prevention. On average, structural imperfection can reduce the buckling load, Pimp, on the perfect
structure, Pperf, by 20%-50%. In general, the load-carrying capacity of the shell structure is referred to as the
knockdown factor (KDF) (i.e., Pimp/Pperf). Measured geometric imperfections (MGI) is one of the experimental
techniques to evaluate structural imperfection for spherical shells [13,15,17] and cylindrical composite shells
[21]. The MGI technique is considered to be time consuming and very expensive because the fabricated shell
must be measured using optical measurement systems. An alternative way to evaluate the imperfection
sensitivity of shell structures is to apply different methods using finite element analysis (FEA), namely: (a)
eigenmode imperfection, (b) single load indentation (SLI) imperfection approaches, and (c) the reduced
Page 2 of 29
stiffness method (RSM). For years, eigenmode imperfection has been used extensively to mimic initial
structural imperfection for various forms of shell structures at an earlier design stage using FEA [22–28].
Although in practice, most imperfections found in structures do not have this buckling mode [29,30]. In some
cases, the eigenmode imperfection does not appear to be the worst imperfection. Therefore, the need for more
realistic forms of imperfections is considered essential. Hühne et al. [31] introduced the single perturbation
load analysis (SPLA) imperfection approach. In this method, the lateral concentrated force applied to the outer
surface of the shell is used to simulate the dent/ dimple shape of the structural imperfection. In [32,33], it was
reported that in addition to the SPLA, the single load indentation (SLI) imperfection method can also be used
to estimate the knockdown factor of the cone-cylinder shell assembly. This technique is identical to the SPLA
except that a displacement load is used instead of the lateral force to create the dent imperfection. Multiple
lateral loads along the shell circumference were also been used to demonstrate the worst type of imperfection
level [34,35]. This technique is referred to as the worst multiple perturbation load approach (WMPLA).
Błachut [22] investigated the externally pressurized dome knockdown factor that accounts for dimple
imperfection by using multiple approaches, namely (i) Legendre polynomials, (ii) increased-radius patch, and
(iii) a localized inward dimple. It has been reported that the location of the dimple has a strong influence on
the sensitivity of the structures [28,29]. Another well-known method, the reduced stiffness method (RSM), is
performed by reducing or eliminating the components of the membrane stiffness of the shell, leaving only the
bending stiffness. In a recent study, the membrane stiffness is reduced locally rather than globally, which is
called the locally reduced stiffness method (LRSM). Wagner et al. [20,36] investigate the imperfection
sensitivity and design of spherical domes under external pressure using LRSM.
Steel spherical domes are considered to be typically thin-walled structures and are highly susceptible to
buckling under external pressure, especially when empty or during operation. Previous studies showed
discrepancies in the buckling resistance of steel spherical shells between the design method proposed in the
European Standard EN1993-1-6 and the analytical formulas recommended in the British Standard PD 5500.
The aim of this work is twofold. First, to numerically investigate the imperfection sensitivity of a spherical
shell subjected to external pressure using different imperfection approaches - (a) eigenmode imperfection and
(b) single load indentation (SLI) imperfection - in order to determine the appropriate imperfection case in this
sense. It is worth mentioning that the numerical model followed the recommendations of the European
Standard EN1993-1-6. Based on the chosen imperfection approach, further numerical investigations were
performed on different spherical shell configurations to derive a lower-bound closed-form empirical formula
with lower limit value, which is compared with the recommended KDF according to the guidelines of (i)
NASA SP -8032 [44], (ii) Energy Barrier Criterion (EBC) [43] and (ii) Wagner [20]. This study is unique in
that there have been fewer or no studies on this topic. The information, expertise, and conclusions from this
study are intended to provide insight into the field of design or failure examination/testing of pressure
vessels/tanks, submersible structures, vacuum vessels (in the chemical industry), aerospace, and civilian
applications.
Page 3 of 29
2
2.1
Numerical modelling
Preliminary study
The numerical model of the spherical shells was benchmarked against the experimental data recorded by
[17,19,37]. Sixteen (16) laboratory scale spherical shells subjected to external pressure (see Fig. 1 (a)) are
described in the following terms: Spherical radius, R, spherical thickness, t, base radius, e, spherical height,
H, and semi-vertex angle, φ, as outlined in Fig. 1 (b). Alternatively, Table 1 outlined the geometry and
material properties of benchmark models, which are based on the experimental data [17,19,37]. The numerical
analysis assumed a fully constrained/clamped boundary condition with an elastic-plastic material model. The
numerical model uses a polar coordinate system around the shell edge. In the benchmarking study, the
nonlinear static Riks analysis is used. The setting of nonlinear static Riks analysis was Nlgeom = on, maximum
number of increments = 100, initial arc length increment = 0.01, minimum arc length increment = 1E-15 and
maximum arc length increment = 1E-36. According to the recommendation of European Standard EN19931-6 [38], the numerical model is assumed to be a perfect shell by using the geometric and material nonlinear
analysis (GMNA) approach. The numerical model in this study uses the S4R shell element, which provides a
solution to shell problems as described by classical Kirchhoff shell theory and bending stress with reference
to Koiter-Sanders shell theory. The S4R shell element is described as a four-node shell element with six
degrees of freedom (S4R in the ABAQUS element library). The numerical model also followed the J2 flow
plasticity theory because the differences in the buckling load estimated by the J2 flow theory and the J2
deformation theory are insignificant for the case of the externally pressurised cone-cylinder transition [39]
and the axially compressed cylindrical shell and plate [40,41].
Page 4 of 29
Figure 1: (a) Load and boundary condition of externally pressurised spherical shell and its (b) schematic
diagram
Table 1: Geometry and material properties of benchmark models in referring to the experimental data
R
t
E
σy
ID
R/t
v
Remark
Refs.
[mm]
[mm]
[GPa] [MPa]
D1
1816.5
1
1816.5
D2
1605.7
1.02
1574.2
D3
878.6
1.03
853.01
207
303.5 0.28 Shallow shell
[37]
D4
1166.7
1.76
662.9
D5
759.3
1.76
431.42
D6
563.4
1.76
320.11
UNHS1
60
0.77
77.922 150.8 313.63 0.29
[19]
Page 5 of 29
UNHS2
UNHS3
UNHS4
UNHS5
1#
2#
3#
4#
5#
60
60
60
60
58.84
58.77
58.99
58.77
58.75
0.76
0.76
0.76
0.77
0.432
0.422
0.423
0.406
0.415
78.947
78.947
78.947
77.922
136.2
139.27
139.46
144.75
141.57
Hemispherical
shell
193
205
Hemispherical
shell
0.28
[17]
To support the numerical model, it is common to calculate the buckling capacity of the pressurised spherical
shell using the industrial design code. The guideline proposed by the British Standard PD 5500 design code
[10] was chosen in this study to calculate and verify the buckling pressure of the tested spherical shells. This
approach is crucial for verifying the reliability of the existing design rules and their usefulness for preliminary
estimation of buckling load. In general, the available design rules use the conventional working stress to
determine each failure mode. Apart from the working stress and safety factors, the design rule also takes into
account several uncertainties, namely eccentric boundary conditions and loads, material hardening and
structural imperfections of the tested spherical shell structures.
The critical buckling formula, Pcr of a complete spherical shell under external pressure according to PD 5500
design code [10] is expressed by equation (1).
𝑃𝑐𝑟 =
2𝐸
𝑡
( )2
√3(1 − 𝜈 2 ) 𝑅
(1)
In the case of plastic buckling, the yield stress is important, and an approximate value for a plastic buckling
pressure, Pyield, is given in equation (2).
𝑡
𝑃𝑌𝑖𝑒𝑙𝑑 = 2𝜎𝑌𝑖𝑒𝑙𝑑 ( )
𝑅
(2)
The design code PD 5500 estimated the collapse pressure using equation (3)
(
1
𝑃𝑃𝐷 5500
)2 = (
1 2
1 2
) +(
)
0.3𝑃𝑐𝑟
𝑃𝑦𝑖𝑒𝑙𝑑
(3)
The benchmark analysis with experimental results for sixteen (16) models is shown in Fig. 2 (a) - (b). The
results summarise the comparison of buckling load calculated with the design code PD 5500, numerical results
(i.e., nonlinear analysis GMNA), and test results. The nonlinear Riks analysis provides good agreement with
the experimental result compared to PD 5500 design code. The average percentage difference between the
experimental results and the numerical analysis is less than 7%, for overall cases. Specifically, for the case of
Page 6 of 29
shallow shell, a more closed estimate of the buckling load was found by the D3 model with less than 1%
difference. Obviously, the subsequent numerical model overestimates the buckling load by 10% compared to
the experimental result of the D2 model. For the case of a hemispherical shell, it is again found that the UNHS1
- UNHS5 models slightly underestimated the buckling load by 8% on average. Meanwhile, models 1# - 5#
significantly overestimated the buckling load by 11% on average. This result clearly indicates that the
numerical model is appropriate for the analysis. Nevertheless, an insignificant buckling load is recorded for
the case of a shell of identical size, as shown by the UNHS1-UNHS5 and 1# - 5# models, separately. This is
a clear indication of the presence of structural defects. In this case, the structural imperfection is believed to
be in the form of dents, uneven thickness, strain hardening and many more. The severity of the imperfection
is indicated by the deviating ratio of the experimental buckling load to the calculated numerical analysis,
PExp/PColl, which is less than or greater than 1.0. It is obvious that the buckling load calculated by PD 5500
slightly underestimates the experimental results for the corresponding cases of models D1 - D6 (with an
average of 46%), UNHS1 - UNHS5 (with an average of 33%), and 1# - 5# (with an average of 10%), as
reported in [19,37]. The corresponding result shows that the buckling load of the 5# model is closer to the
experimental data to some extent, with differences of 6% calculated. A larger difference was found for the D1
model, where the buckling load deviates by 50%. From the given results, it can be said that a reasonable
agreement of the buckling load between the experiment and the design code was found for the case of a
hemispherical shell [17].
Figs. 3 - 4 show a typical plot of the external pressure against the vertical deflection of the dome apex using
specimens D6 (shallow shell) and UNHS1 (hemispherical shell) as examples. Both figures show the (i) initial
yield pressure, PYield, (ii) collapse pressure, PColl, and (iii) post-collapse pressure, PPost-Collapse. In the case of
sample D6 (i.e., shallow spherical shell), it can be seen that the spherical shell reaches its first yield pressure,
PYield, at 0.949 MPa, followed by the collapse pressure, PColl, at 1.202 MPa, and the post-collapse pressure,
PPost-Collapse, at 0.504 MPa (see Fig. 3).
Fig. 4 shows alternatively the load carrying capacity of a hemispherical shell represented by the UNHS1
model. The shell was found to reach its yield point at PYield = 3.892 MPa before collapsing at PColl = 8.19 MPa
and post-failure at PPost-Collapse = 4.5777 MPa. It was found that a significant pressure drop occurs in both
models, followed by a stable equilibrium path afterward.
The numerically tested D6 specimen (i.e., shallow spherical-cap) suffers asymmetric failure (see inserted (a)(c) in Fig. 3). It was observed that the shell experiences a pronounced yield stress in the equatorial region
before propagating to the apex region. At the peak of the buckling load, there is a strong stress concentration
in the apex region before the shell fails completely in post-collapse mode. The UNHS1 model (i.e., the
hemispherical shell) initially appears to maintain a minimal concentration of yield stress around the equatorial
edge (see Fig. 4 insert (a)). At peak of the buckling load, the stress spreads out and concentrates locally below
the apex near the equator before entering the post-collapse phase (see Fig. 4 insert (b) - (c)) and before bulging
out in an axisymmetric manner.
Page 7 of 29
Figure 2: Plot of experimental results with numerical model and PD 5500 design code [10] for (a)
shallow shell and (b) hemispherical shell
Page 8 of 29
Figure 3: Plot of load versus deflection of externally pressurized spherical shell for specimen D6 with
analysis type GMNA
Figure 4: Plot of load versus deflection of externally pressurized spherical shell for specimen UNHS1 with
analysis type GMNA
Page 9 of 29
2.2
Perfect spherical shell
In general, according to the European Standard EN1993-1-6 [38], it is common to first perform the linear
buckling analysis (LBA), Pcr, to gain insight into the analysed shell. In this analysis, the nominal parameters
used for the LBA model are spherical radius, R = 500, uniform wall thickness, t = 1 mm, spherical height, H
= 6 mm, and base radius, r = 83 mm. A stainless steel of spherical shell is considered in the analysis with an
elastic modulus = 207 GPa, a yield stress, σyield = 303.5 MPa, and a Poisson's ratio, υ = 0.28. The numerical
model is based on an elastic-plastic material model following the experiment performed by [37]. An example
of elastic-plastic material model taken from [37] is shown in Fig. 5. The linear buckling load was calculated
using the subspace solver. The subsequent pressure load of 1 MPa is applied to the spherical shell and
multiplied by the obtained eigenvalue to produce a buckling load, Pcr of the LBA.
Figure 5: An example of elastic-plastic material model taken from material properties given in [37]
In practice, it is important to study the behaviour of externally pressurised steel spherical shells as a function
of dimensionless ratios: spherical radius- to- wall thickness, (R/t)- ratio. Throughout the analysis, the wall
thickness, t, is varied accordingly and presented with a range of dimensionless ratios from 100 < R/t < 1000.
To mimic the application of a spherical shell as part of an industrial component, a fully constrained boundary
condition was considered along with three other boundary conditions, as indicated in Table 2, to determine
the most appropriate condition for the FE analysis. It is worth noting that the numerical model uses a polar
coordinate system at the edge of the shell. The terms used in the table are tangential direction, v, radial
direction, w, normal direction, u and rotational direction, θ. The rotational direction, θ is described as a
Page 10 of 29
condition that rotates about the normal direction, u of the spherical shell (see Fig. 1). The variation of boundary
conditions tested in this study is consistent with the analysis of the spherical shell in terms of (i) buckling load,
(ii) number of buckling waves (n), and, (iii) structural failure mode.
Table 3 demonstrates a convergence analysis with different numbers of finite element meshes to determine
an appropriate fine-mesh model. It is known that the finer the mesh, the more time the model takes to complete
the analysis. From the convergence analysis, it is found that 6897 elements are sufficient for the analysis. This
is reflected in the fact that the percentage deviation from the convergence analysis was reduced to almost
0.04% - 0.26%, which is due to the density of the FE model. On the other hand, when the number of elements
exceeded 10,000, insignificant buckling load was found from the analysis. The result also shows a reliable
mesh numbers with a reasonable time to completion.
This is seen as the percentage of difference from the convergence analysis was reduced to nearly 0.04% 0.26% consequently to the density of the FE model. On the other hand, with the element numbers exceeding
to 10,000, an insignificant buckling was found from the analysis. The result also shows a reliable buckling
load with a reasonable time to complete.
Fig. 6 shows the variation of the buckling load Pcr of the spherical shell pressurised from outside against the
range of dimensionless ratios of 100 < R/t < 1000 with the help of 4 different boundary conditions. It can be
clearly seen that the Type 4 boundary condition at R/t = 100 provides a conservative estimate of the critical
buckling load compared to the other boundary conditions. In contrast, boundary condition type 1 produces a
much higher buckling load for shell size R/t > 200. A significant difference in critical load decreases
significantly for shell size R/t > 200.
For the case of R/t = 100, the lowest and highest critical buckling loads were found to be 21.015 MPa and
54.245 MPa for boundary condition type 4 and type 3, respectively. The range of differences in critical
buckling load is between 25.15% - 61.25% for boundary condition type 1 - type 4 with shell size of 100 < R/t
< 1000. For the case of spherical shell size R/t = 1000, the shell leads to axisymmetric failure with boundary
condition type 1 and 2 as shown in Fig. 6 (see insert (a) - (b)). Conversely, for the same size, the shells failed
asymmetrically with the corresponding number of modes, namely n = 2, under boundary conditions type 3
and 4, as shown in Fig. 6 (see insert (c) - (d)).
Fig. 7 shows the evolution of the elastic and elastic-plastic material models as a function of the deflection of
a spherical shell subjected to external pressure for a perfect nominal model. It is worth mentioning that the
analysis was performed according to the Geometric Nonlinear Analysis (GNA) recommended by the European
Standard EN1993-1-6 [38]. From the analysis, it appears that the buckling behaviour of the spherical shell is
governed by plastic buckling, since the elastic analysis predicts a much larger buckling load than the elasticplastic analysis. The elastic material model also shows an unstable response after buckling at displacements
greater than 8 mm.
Page 11 of 29
Table 2: Four type of boundary condition (BS) applied at the equatorial (bottom) plane of the spherical shell,
e.g., type 1 means fully clamped support. Note c = variable is set to zero, f ≡ variable is set to free
Type of BC
1
2
3
4
U1
c
c
c
c
U2
c
c
c
c
U3
c
c
f
f
UR1
c
f
c
f
UR2
c
c
c
c
UR3
c
c
c
c
Table 3: Convergence study of externally pressurized spherical shell for nominal model
Element number
1768
4873
6897
10962
Pcr [MPa]
0.70027
0.69846
0.69817
0.69789
Figure 6: Buckling load, PLBA of LBA mode against range of dimensionless ratios of 100 <R/t < 1000 and
different boundary conditions
Page 12 of 29
Figure 7: Plot of elastic and elastic-plastic material model load response versus deflection of externally
pressurised spherical shell for perfect nominal model
3
Imperfection sensitivity of spherical shell
In this section, a numerical work was carried out to evaluate the influence of the initial geometrical
imperfection of a spherical shell under external pressure. The imperfection techniques used were also tested
to investigate the realistic case of imperfection for the spherical shell structure. Following the European
Standard EN1993-1-6 [38], the Geometric and Material Nonlinear Imperfection Analysis (GMNIA) approach
was used for the analysis. The influence of the initial geometric imperfection on the buckling strength of the
spherical shell is evaluated using two types of imperfections, namely (i) the single load indentation (SLI)
imperfection and (ii) eigenmode imperfection. These imperfections were separately superimposed on the
perfect model, with the range of imperfection amplitude versus wall thickness, wo/t, varied between 0.0 and
2.5. The imperfect spherical shell sensitivity was performed for the case of (i) a hemispherical shell and (ii)
shallow shell, which is constitutively demonstrated by the UNHS1 model, the D6 model, and the #1 model.
3.1
Single load indentation (SLI) imperfection
For the SLI case, as shown in Fig. 8 (a), a displacement-controlled perturbation load was applied laterally to
the mid-section (i.e., below the crown area) of the spherical shell until the required indentation depth is
reached. Thereafter, the magnitude of perturbation load is held constant to eliminate springback while external
pressure is applied at an incremental rate of nonlinear analysis via the Riks method until the collapse load is
reached. The bottom of the spherical shell is assumed to be in a fully constrained state during the application
Page 13 of 29
of the lateral load and external pressure. The reason for applying the lateral load in the mid-section of the
spherical shell is explained by the fact that the imperfection in this area usually occurs in the form of
depressions or dents [6,20].
3.2
Eigenmode shape imperfection
For the case of eigenmode imperfection, the modes were varied from n = 1, n = 2, and n = 3, as shown in Fig.
8 (b) - (d) using the UNHS1 model as an example. The use of selected eigenmodes is necessary to determine
the lowest knockdown factor of the shell. Then, a nonlinear static analysis using the modified static Riks
method was performed to determine the collapse pressure of the spherical shell. Again, similar to the SLI case,
the bottom of the spherical shell is assumed to be in a fully constrained state during the application of the
external pressure. Failure is observed in the regions of the apex and mid-section, as indicated by the n = 1
eigenmode shape (see Fig. 7 (b)). Fig. 7 (c) - (d) confirms that failure occurred in the mid-section of the
spherical shell below the apex.
Figure 8: Illustration of imperfection shape exemplified by UNHS1 model for (a) SLI – mid-section,
buckling mode shapes (b) n = 1, (c) n = 2, and, (d) n = 3
Page 14 of 29
3.3
Comments on imperfection sensitivity of spherical shells
The obtained results show that the spherical shell subjected to the external pressure with the SLI imperfection
in the mid-section of the shell does not have the worst imperfection in all the tested models (see Figs. 9 - 11).
Nevertheless, it can be seen that the imperfect spherical shell decays slightly at wo/t = 1 in the case of model
UNHS1, model D6 and model #1. It can be seen that in the range of wo/t = 1.0 - 2.5, the decrease in knockdown
factors is about 10% - 20% for all models. Remarkably, at wo/t = 2.5, the shell can withstand only about 21%
- 22% for the hemispherical shell case and 8% for the case of shallow shell.
In Figs. 9 - 11, the reduction of knockdown factor is plotted against the ratio of imperfection amplitude and
wall thickness (wo/t). It can be seen from the figures that the eigenmode shape imperfection is more
conservative compared to the SLI for all the externally pressurised spherical shells tested, namely the UNHS1,
D6 and #1 models. It can be noted that the eigenmode shapes n = 1 - 3 produce the largest imperfection
sensitivity for the UNHS1, D6 and #1 models, respectively. From the given results, it is clear that the mode
shape, n = 1 does not produce the lowest knockdown factor for the UNHS1, D6 and #1 models. The results
also show that the first mode shape cannot be considered as the worst case of geometric imperfection. Here,
the knockdown factor of each shell is defined at the beginning of the plotted threshold (i.e., w o/t ≥ 0.5). The
knockdown factors estimated using the eigenmode approach for mode shape n = 1 - 3 were in the range of
0.125 - 0.219 for model UNHS1, 0.295 - 0.488 for model D6, and 0.113 - 0.187 for model #1. It must be
emphasised here that the eigenmode imperfection approach is able to estimate the lower-bound knockdown
factor, but depends strongly on the mode shape and the imperfection amplitude.
For the case of a hemispherical shell (i.e., model UNHS1 and model #1), wo/t = 0.8 shows that the imperfect
spherical shell with mode shapes n = 3 can only support about 40% of the load-carrying capacity of the perfect
shell. Beyond this imperfection amplitude (i.e., wo/t = 0.8), the knockdown factor appears to be marginal. On
the other hand, at wo/t = 1.0, the hemispherical shell can only support about 35% of the load-carrying capacity
of the perfect shell. For the case of shallow spherical shell, represented by the D6 model, an identical trend
was also observed. The corresponding D6 model also shows that for wo/t = 0.8, the bearing capacity decreases
by almost 59% compared to the perfect shell, and for wo/t = 1.0, it slightly increases everywhere by 60%.
Although the trend seems to be similar, it is worth noting that a sharp decrease in structural strength is observed
for the shallow shell (i.e., the D6 model) compared to the hemispherical shell. This peculiarity is due to the
shallow depth of the spherical shell, which is less able to support a larger load and therefore fails faster than
the other shells. On the other hand, the obtained results are in agreement with the references [23,34], in which
it was found that the first mode shape cannot be considered as the worst geometric imperfection. Meanwhile,
the SLI has tried to produce a more realistic imperfection in the form of a dent failure in the region of the
spherical shell mid-section, as observed in the eigenmode failure at n = 2-3.
Page 15 of 29
Figure 9: Spherical shell knockdown factor against imperfection amplitude to wall thickness ratio, wo/t for
the case of eigenmode shape imperfection and SLI with UNHS1 model
Figure 10: Spherical shell knockdown factor against imperfection amplitude to wall thickness ratio, wo/t for
the case of eigenmode shape imperfection and SLI with D6 model
Page 16 of 29
Figure 11: Spherical shell knockdown factor against imperfection amplitude to wall thickness ratio, wo/t for
the case of eigenmode shape imperfection and SLI with #1 model
4
Imperfection sensitivity of spherical shell – parametric study
From the imperfection sensitivity analysis performed in the previous section, it can be concluded that the
eigenmode shape imperfection is the worst imperfection for the spherical shells tested, but this is not
necessarily true for the case of a realistic imperfection. In practice, most imperfections in structures do not
exhibit buckling modeshape (i.e., deformation). This is confirmed by the fact that some researchers have
argued that the eigenmode imperfection method is not suitable for the worst case of geometric imperfection
[23,34]. Finally, eigenmode imperfection is often used only tentatively to estimate the lowest buckling load
[29]. In contrast, SLI is an attractive technique for simulating defects (e.g., dent) that resemble real defects
found in various industrial structures. This argument supports the work done by [42,43] for the postbuckling
behaviour of spherical shells under pressure.
Finally, in this section, the imperfection sensitivity of the externally pressurised spherical shell is further
investigated by applying the SLI imperfection technique. On the other hand, referring to the European
Standard EN1993-1-6 [38], the nonlinear analysis based on the geometric and material nonlinear imperfection
analysis (GMNIA) was performed.
Page 17 of 29
4.1
Suggested imperfection based on European Standard EN1993-1-6
In accordance with the report of [20], the spherical shell in this study is categorised according to the following
structure: (i) hemispherical shell - H/R = 1, (ii) spherical shell - H/R = 0.75, (iii) deep spherical shell - H/R =
0.5, and (iv) shallow spherical shell - H/R = 0.07. The imperfect spherical shells subjected to external pressure
are tested with the following setup:

the imperfections of the spherical shell (i.e. shape, dent and amplitude) are calculated according to the
European standard EN1993-1-6 [38], based on equation (4)
(4)
𝐿𝑔𝑥 = 4√𝑅𝑡

the spherical shell is assumed to be completely clamped at the equatorial edge

the dent is created by applying the lateral displacement load in the middle of the spherical curvature
(i.e. below the apex)

the response of the tested shells to the buckling load is recorded by considering the respective
dimensionless ratio of Pimp/Py to Pcr/Py according to PD 5500 design code [10]. The dimensionless
ratio, which can be calculated based on equations (1) and (2)
The input parameters used for the parametric study are listed in Table 4. Parameter variation was set in 3
dimensionless-ratio groups, namely, (i) R/t, (ii) H/R, and (iii) σYield. As mentioned below, 4 types of spherical
shells were tested, with each type of shell consisting of 12 cases. Therefore, 48 numerical models were
successively run via FE nonlinear analysis in the range of input parameters.
Table 4: The input parameters used for the parametric study
Variable
Description
Variation range
R/t
Spherical radius to thickness ratio
250<R/t<1000
H/R
Dome height to spherical radius ratio
0.075<H/R<1
σYield
Yield stress
200 MPa<σYield<400 MPa
Fig. 12 (a) - (b) shows a plot of the dimensionless- ratio of Pimp/Py (y-axis) versus Pcr/Py (x-axis) for the case
of (i) hemispherical shells, (ii) spherical shells, (iii) deep spherical shells, and (iv) shallow spherical shells.
The corresponding results highlight several findings that can be outlined below:

the interactive behaviour of the dimensionless- ratio of Pimp/Py versus Pcr/Py follows the format of the
PD 5500 design code. Fig. 12 (a) shows that the hemispherical shell produces a larger value of Pimp/Py
= 0.891, followed by the spherical shell calculated with a value of Pimp/Py = 0.385. Fig. 12 (b) again
shows that the deep spherical shell produces a larger value of Pimp/Py = 0.910, followed by the shallow
spherical shell with a value of Pimp/Py = 0.797

it is obvious that the hemispherical shell overestimates the dimensionless-ratio of Pimp/Py by almost
42% compared to the spherical shell with Pcr/Py = 2.321 (see Fig. 12 (a)). Even for the similar case
Page 18 of 29
with Pcr/Py = 0.290, the calculated differences are about 70 %. On the other hand, Fig. 12 (b) shows a
nominal difference in Pimp/Py of about 12 % for the case of a deep spherical shell and a shallow
spherical shell at Pcr/Py = 2.321. At Pcr/Py = 0.290, the shallow spherical shell estimates Pimp/Py of about
17 % compared to the deep spherical shell

Fig. 11 (a) - (b) shows a polynomial curve fitting method for the scatter plot of each tested case after
regression analysis. It can be seen that each of the derived equations appears to be reliable with an R2
value of 0.9186 - 0.9694
Page 19 of 29
Figure 12: A plot of dimensionless-ratio of Pimp/Py against Pcr/Py for the case of (i) Hemispherical shell, (ii)
Spherical shell, (iii) deep spherical shell, and, (iv) Shallow spherical shell
Page 20 of 29
5
Lower-bound imperfection
As mentioned earlier, the literature shows that knowledge of the imperfection sensitivity of spherical shells
subjected to external pressure is still limited in the open literature in terms of design formulations, curvature
guidelines, etc. Based on the available experimental results and actual FEA results, a closed-form lower-bound
empirical formula for the knockdown factor (KDF) of the externally loaded spherical shell is derived. In the
design guideline, the knockdown factor (KDF) is derived by a ratio of P imp/Pcrit. To obtain the buckling
pressure of the imperfect spherical shell, the KDF was multiplied by Pcrit. The derived empirical formula may
be limited in terms of its scope. Nevertheless, it provides a first estimate of the buckling load of an imperfect
spherical shell.
5.1
NASA SP-8032
The empirical formula was developed based on the format NASA SP-8032 [44], as it is very general and
flexible when dealing with different shell buckling problems. The use of the NASA SP-8032 guideline is still
relevant in the aerospace or subsea industries at the preliminary design stage, as all aerospace or subsea
regulatory agencies have adopted this procedure as a safe and conservative approach equivalent to
implementing the lower bound of the buckling knockdown factor curve. The NASA SP-8032 guideline is
calculated under the following condition:

according to NASA SP-8032 [44] the shell parameter, λ are calculate by the following equation (5)
𝑅
𝜙
4
𝜆 = √12(1 − 𝜈 2 )√ 2𝑠𝑖𝑛 ( )
𝑡
2

(5)
simultaneously the spherical shell knockdown factor (KDF) is calculated by following the equation
(6)
𝐾𝐷𝐹𝑁𝐴𝑆𝐴 = 0.14 +
3.2
𝜆2
(6)
In addition to the results from FE, additional data from available experimental results from [45–50] were
included and compared with other available knockdown factor formulas, namely: (i) Energy Barrier Criterion
(EBC) from [43] and (ii) Wagner [20], as indicated in Table 5. Finally, 343 KDF samples of tested spherical
shells subjected to external pressure were recorded and evaluated accordingly.
Table 5: Knockdown factor formulas with its corresponding description
Formulas
Ref.
0.693
𝐾𝐷𝐹𝐸𝐵𝐶 =
2
EBC, λ > 5 [43]
5
√(1 − 𝜈)𝜆5
Page 21 of 29
𝐾𝐷𝐹𝑊𝑎𝑔𝑛𝑒𝑟 = 5.172𝜆−1.464 + 0.1296
Wagner, λ > 5.5 [20]
Fig. 13 shows a comparison of the knockdown factor estimated according to NASA SP-8032, EBC [43],
Wagner [20] with experimental results for spherical shells subjected to external pressure. A curve of the lowerbound of imperfection is derived from the scatter of the KDF data and fitted using the mathematical expression
in equation (7). The value of the closed-form empirical formula for the lower-bound fitted to the curve is
determined based on trials and regression analysis. Finally, a closed-form lower-bound empirical formula of
a spherical shell subjected to external pressure is derived using the following equation (7):
𝐾𝐷𝐹𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑤𝑜𝑟𝑘 = 0.07 +
3.2
𝜆2
(7)
The comparison of the knockdown factor estimated according to NASA SP-8032 [44], EBC [43], Wagner
[20] and the experimental results for spherical shells subjected to external pressure, as shown in Fig. 13, will
now be discussed in detail:

from the figure, it can be seen that Wagner's [20] approach predicts a much higher knockdown factor
at shell shape parameter λ ≤ 10

nevertheless, for a shell shape parameter λ ≥ 10, the EBC yields a much higher knockdown factor
compared to NASA SP-8032, Wagner and the proposed equation (7) (where the differences are in the
range of 48% - 56%)

interestingly, the present nonlinear parametric FEA results, denoted by the symbol (▲), have been
found to be inconsistently insensitive to the imperfection by producing a much larger knockdown
factor for a shell shape parameter λ ≥ 10 compared to the other experimental results within the range
of the shell shape parameter

an important point to emphasise here is that the proposed equation (7) accounts for much less than the
other equations
Figs. 14 - 15 show a comparison of the knockdown factor calculated using (i) NASA SP-8032 and (ii) the
proposed equation (7) with a total of 343 samples of experimental results. The black solid line in the figure
represents the estimated knockdown factor as a function of shell shape parameter, λ and is expressed as a
linear function. From the given results, it can be summarised that:

the knockdown factor predicted by NASA SP-8032 does not agree with several tested experimental
results, as shown in Fig. 14. This confirms that NASA SP-8032 cannot serve as a lower bound for the
knockdown factor.
Page 22 of 29

similarly, the proposed equation (7) anticipates the problem that it is more conservative compared to
NASA SP-8032 as shown in Fig. 15.

in conclusion, although the proposed equation (7) is more conservative than the other guidelines, it
can still serve as an initial estimate of the knockdown factor for imperfect spherical shells under
external pressure.
Figure 13: Comparison of knockdown factor estimated by NASA SP-8032, EBC [43], Wagner [20] and
experimental results for spherical shells subjected to the external pressure
Page 23 of 29
Figure 14: Knockdown factor of the experimental results versus NASA SP-8032
Figure 15: Knockdown factor of the experiment results versus proposed equation
6
Conclusion
The results of the numerical study of the imperfection sensitivity of a spherical shell pressurised from the
outside are presented in this paper. The following conclusions can be drawn from the preceding analysis:
Page 24 of 29
1. The numerical calculation results of the imperfection sensitivity of a spherical shell structure
pressurised from the outside are presented in this paper. From the results, it is found that the buckling
strength of externally loaded spherical shells is reduced in the presence of a structural imperfection.
2. The imperfection approach also plays a major role in determining the buckling strength of the structure.
Through the different types of imperfection approaches considered in this study, it was shown that the
load-bearing capacity of a spherical shell superimposed with an eigenmode imperfection is greatly
reduced. This was followed by single load indentation (SLI) imperfection. The result also highlights
the catastrophic nature of the dent/dimple amplitude generated by the SLI (Single Load Indentation)
imperfection, especially in the case of an externally pressurised spherical shell typically found in
industry.
3. Finally, a lower-bound closed-form empirical formula with lower limit value for a spherical shell under
external pressure was proposed for the spherical shell with the appropriate imperfection approach (i.e.,
the single load indentation (SLI)). The results show that for a shell shape parameter λ ≥ 10, the EBC
provides a much higher knockdown factor compared to the guidelines of NASA SP -8032, Wagner,
and lower-bound closed-form empirical formula (the differences are in the range of 48% - 56%).
In general, the results of the study are highly useful and contribute to the body of knowledge on the effects of
structural imperfections in the preliminary stages of design, fabrication, and analysis of the externally
pressurised spherical shell structure.
Author statement
The research data can be shared at the request of the editor.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could
have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to acknowledge the financial support from Politeknik Sultan Salahuddin Abdul Aziz
Shah and the Ministry of Higher Education Malaysia. The first author would like to thank H.N.R. Wagner for
the spherical shell experiment data.
Page 25 of 29
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