ON THE DEVELOPMENT OF DESIGN CRITERIA FOR BUCKLING OF AXIALLY LOADED
CONICAL SHELLS
H.N.R. Wagnera, C. Hühnea , R. Khakimovab and T. Ludwigc
a
Institute for Composite Structures and Adaptive Systems, German Aerospace Center (DLR),
Lilienthalplatz 7, 38108 Braunschweig, Germany
ronald.wagner@dlr.de
b
Independant Researcher, Wendenring 26, 38114 Braunschweig, Germany
c
Civil Engineering
Keywords: Buckling, conical shells, new proposed design rules, experimental results
Abstract
The buckling resistance of a isotropic conical shell under axial compression is controlled by the
geometry, material properties and the fabrication class. A simple algorithm to determine the buckling
resistance of such shells with clamped boundary conditions, in terms of the description used in EN
1993-1-6, Eurocode 3: Design of Steel Structures, Part 1.6: Strength and Stability of Shell Structures, is
presented in this article. The assessment is based on the elastic critical load NRcr, followed by
evaluation of the imperfection sensitivity for the assumed fabrication quality class.
The new procedure gives 38-117 % higher buckling resistances when compared with current design
recommendations which results in less conservative design.
The safety of the new proposals is validated within this article with many previous experimental results
which have been presented in the literature. Based on these studies it is concluded that the new
proposals are safe and more economic than current design rules for steel conical shells under axial
compression.
Abbreviations and glossary
Exp.
Experiment
F
Axial load
H
Height of a truncated cone
KDF
Knockdown factor
L
Free slant length of a truncated cone
N
Buckling load
r
Small radius of a truncated cone
R
Large radius of a truncated cone
Ra
Average radius of curvature
SBPA
t
TH
Single Boundary Perturbation Approach
Wall thickness of a cylinder, truncated cone
Threshold
u
Axial displacement

Elastic imperfection factor
1 Introduction
1.1 Buckling of conical shells under axial compression
Truncated conical shell structures are often used as adapters between cylindrical shells of
different diameters and can also be found in offshore application (legs of offshore drilling rigs)
in civil engineering applications, see Fig. 1.
Fig. 1: Conical shell civil engineering applications: storage tanks (left & middle) offshore drilling rig (right) after [1], [2], [3]
The geometry of a truncated conical shells is defined by means of the small radius r, the large
radius R, the slant length L, the height H, the shell thickness t and the semi-vertex angle , see
Fig. 2.
Small Radius
Semi-vertex angle
Slant-length L

Height h
Wall thickness t
Large Radius
Fig. 2: Geometry of a conical shell
Conical shells in civil engineering applications carry heavy axial loads and are prone to
buckling. The buckling loads of conical shells under axial compression can be extremely
sensitive to imperfections resulting in a significant reduction of the load carrying capacity.
Imperfections are commonly classified as traditional and non-traditional imperfections. Initial
geometric imperfections [4], [5], [6] are defined as shape deviations from the ideal structure
and represent traditional imperfections. There are several detailed studies regarding the
influence of initial geometric imperfection on the buckling load of unstiffened conical shells
can be found in [7], [8], [9], [10], [11]. Similar well performed studies for stiffened conical
shells can be found in [12], [13]. The imperfection sensitivity of conical shells for combined
load cases such as external pressure and axial compression was thoroughly studied in [14],
[15], [16], [17]. Interesting studies regarding combined load cases and plastic buckling of
cones are given in [18], [19]. A well-presented review of the imperfection sensitivity of conical
shells is given by Ifayefunmi and Blachut in [20].
Depending on the shape and amplitude of the geometric imperfections; a single dimple appears
within the shell during loading. This single dimple initiates the buckling process and occurs in
thin-walled shells like cylinders [21], [22] cones [23] and spheres [24], [25].
The role of geometric dimple imperfections has been thoroughly studied for cylinders under
axial compression; see for example recent well performed studies in [26], [27], [28], [29], [30].
However, for conical shells there are only some few studies concerning the influence of dimple
imperfections for example in [31], [32] and [33].
Non-traditional imperfections are for example the non-uniformity of loading around the
circumference [34], [35], [36], [37], [38], [26], [39], [40], [41] the influence of boundary
conditions [42], [43] the effect of the prebuckling deformations due to edge constraints [44]
and plastic buckling [45], [46]. The buckling load reduces significantly if non-traditional
imperfections occur; therefore they have to be considered in the design process.
1.2 Design of conical shells under axial compression
The design of conical shells in aerospace engineering relies on the application of empirical
knockdown factors. There are empirical design guidelines like the NASA SP-8019 which
proposes only one single KDF for all cone geometries, NASA = 0.33. The NASA SP-8007 [47],
[48] can also be used if an equivalent cylinder approach is applied and the semi-vertex angle
< 10° [49], [50].
The equivalent cylinder can be approximated by using the same wall thickness as the cone and
a length equal to the slant length L of the cone. The average radius of curvature re, equation (4),
was also proposed for design purposes [49], [51], [52].
Ra =
(r1 + r2 )
2 ∙ cos(𝜙)
(1)
Current design advice of conical shells in civil engineering is based on nonlinear finite element
calculations which are difficult to interpret in the context of a modern shell buckling
description, such as that of the Eurocode on shell strength and stability.
A comprehensive computational analysis has been performed to provide a more secure basis
for these design rules and the outcome has been the development of new design proposals for
the elastic buckling of conical shells under axial compression.
The new design proposals were derived from a full set of finite elements analysis (LBA, GNA
and GNIA) according to the requirements of [53]. However, they have not been validation with
test results, so this article sets out appropriate comparisons between known documents test on
the buckling of clamped conical shells and the new proposals.
2 Imperfection sensitivity of conical shells under
axial compression
2.1 Test specimen and numerical model
At the German aerospace center in Braunschweig a composite conical shell (see Fig. 4 – right)
was manufactured and tested [23].
Fig. 3: The real test specimen K8
The geometry and material parameters of the composite shell are summarized in Table 1. A
detailed report regarding manufacturing, experimental tests and test evaluation of the conical
shells can be found in [54], [55].
MESH – S4R – 5 mm
Fig. 4: Numerical model of the composite cone K8 (left) numerical model of the equivalent cylinder (right)
The conical shell was modeled by using linear shell elements (S4R in ABAQUS [56]) as
shown in Fig. 4 (left) and the equivalent cylinder is shown in Fig. 4 (right). The element length
for the meshing process is estimated with the large cone radius R and the wall thickness t
according to 0.5√𝑅 ∙ 𝑡 after [106].
The mechanical boundary conditions on both cones edges are defined as clamped by using
rigid-body interactions (Tie) which are coupled with a reference point. The displacement in
axial direction is free at the top cylinder edge for load application.
Within the testing program of the composite cone a thin equalizing layer of epoxy concrete, i.e.
epoxy reinforced with a mixture of sand and quartz power, was applied between the end plates
of the test specimens and the adjacent part of the testing rig in order to enable a uniform load
introduction into the test specimen. However, in the first test of K8, the equalizing layer was
faulty and lead to an unexpected low buckling load as shown in Fig. 5 (left). Unfortunately, no
strain and ARAMIS measurements were performed for this first buckling test and the buckling
test was also not repeated out of fear to damage the test specimen.
Table 1: Material and geometry parameters for cones K8 [23].
Material parameter
Material parameter
Ply Layup
Elasticity modulus 𝐸11 – [MPa]
Elasticity modulus 𝐸22 – [MPa]
Poisson’s ratio ν12
Shear modulus 𝐺12 – [MPa]
Geometry parameter
ß = 35°
Large radius R – [mm]
Small radius r – [mm]
Average radius of curvature Ra – [mm]
Slant Length L – [mm]
Height H – [mm]
Ply thickness tp – [mm]
Nominal Thickness t – [mm]
Average measured thickness ta – [mm]
Ra/t
L/ re
Konus K8
[40,0,-40,-40,0,40]
152400
8800
0.31
4900
400
190
360.12
366.23
300.0
0.125
0.75
0.73
480
1.01
A comparison with the results of a geometrically nonlinear analysis (GNA) for the perfect shell
show that the first buckling load in the test results to Nexp = 22.83 kN and the corresponding
KDF results to ~ 0.48.
The equalizing layers were modified and the test specimen was tested again which resulted in a
much better agreement between the axial stiffness of simulation and test as shown in Fig. 5
(right). The buckling load of the test resulted in this case to Nexp = 35.10 kN (KDF = 0.73)
which is an improvement of the buckling load by 53.7 % in comparison to the first test.
FEA, GNA for perfect shell - K8
Exp. - K8 - good equalizing layer
60
50
50
Axial Force F [N]
Axial Force F [N]
Exp. - K8 - faulty equalizing layer
60
40
30
20
10
0
FEA, GNA for perfect shell - K8
40
30
20
10
0
0
0.05
0.1
0.15
0.2
0.25
Axial Displacement u [mm]
0.3
0.35
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Axial Displacement u [mm]
Fig. 5: Load-displacement curves of cone K8: Test from [11] and GNA results.
The remaining difference between test buckling load and the buckling load of the perfect shell
according to the GNA (NGNA = 47.78 kN) are mainly caused by the geometric imperfections
and the deviation from the nominal wall thickness of the shell which are visualized in Fig. 6.
[mm]
[mm]
0.9
0.7
0.6
-0.7
Fig. 6: Thickness deviation - Ultrasonic scan results for K8 (left) Geometric deviation - ATOS
measurement results of K8 (right).
2.1.1
Reduced Stiffness method (RSM)
The reduced stiffness method (RSM) was developed by Croll et al. [54], [55] and its main
purpose is to determine a lower-bound for the buckling load of thin-walled shells. The main
hypothesis of the RSM is that stabilizing membrane energy components may be lost in the
shell due to the presence of imperfections. The RSM is a simplified energy approach in which
stabilizing membrane energy (UM) components are eliminated from the shell and only the
bending energy (Ub) remains.
Sosa et al. [56] developed steps to implement the RSM in Abaqus. For a conical shell, the first
step of the RSM requires the calculation of the 1st linear buckling eigenmode by means of a
linear buckling analysis (LBA). The corresponding nodal coordinates are then extracted for the
next step (in Visualization Module  Report Field Output  Variable (Position: Unique
Nodal  U1, U2 and U3). Then, in the second step, the stiffness of the shell is represented by
means of the ABD shell stiffness matrix (in Property Module  Section Manager  Shell
General Shell Stiffness) which allows controlling the membrane and bending components of
the shell stiffness. Also, in this second step, the coordinates of the first eigenmode extracted in
the first step are applied as a prescribed displacement field to the shell. A simple method to
apply the coordinates as a displacement field can be achieved by adding the nodal coordinates
into the input file (.inp) as shown in Table 2. These two steps of this procedure are illustrated
in Figure 4.
1.
1.
2.
Mesh – STRI3 – 10 mm
Create Numerical Model:
Axial Compression & Clamping
Determine 1. Linear Buckling
Eigenmode with LBA
Apply 1. Linear Buckling Eigenmode as
predefined Displacement Field and
determine strain energy (ALLSE)
Fig. 7: Sequence of steps for implementation of the RSM in Abaqus.
Also, as part of Step 2, the displacement field needs to be scaled by a factor (the amplitude of
the factor is unimportant as the strain energy is proportional to this factor). In this study a
factor equal to the shell thickness t = 0.75 mm was used. In a subsequent linear static analysis
(LSA) (Static, General, Nlgeom = OFF) the strain energy of the shell (History Output variable
– ALLSE) based on the prescribed displacement field is determined and used to compute the
KDF.
In the RSM, the buckling knockdown factor is defined by the ratio between the strain energy
obtained from the reduced stiffness calculations (Step 2) and the reference strain energy (𝑈𝑏 +
1
𝑈𝑀 ). The reduced membrane energy (𝑈𝑏 +
𝑈𝑀 ) is obtained by reducing all the components
𝛼
𝑅𝑆𝑀
of the A (membrane stiffness) of the ABD matrix with the scaling factor RSM > 1. For the
cylinder Z36, when RSM = 1000, the KDF is calculated according to equation (2).
1
𝑈 ) 1572 𝑁𝑚𝑚
𝛼𝑅𝑆𝑀 𝑀
=
= 0.481
(𝑈𝑏 + 𝑈𝑀 )
3265 𝑁𝑚𝑚
(𝑈𝑏 +
KDFRSM =
(1)
The variation of the knockdown factor as the values of the scaling factor RSM increases for the
composite cone K8 is shown in Fig. 8. From the results, it can be seen that the KDF decreases
as  increases and that the KDF approaches a plateau or lower-bound for  > 100.
Table 2: Example of input file for importing nodal coordinates as a prescribed displacement
field.
**MATERIAL DEFINITION-ABD MATRIX
*Shell General Section, Elset = Cone
A/alpha, B
B, D
*Amplitude, name=FMODAL
** The thickness of the conical shell (0.75) is used as scaling factor
0.,
0.,
1.,
0.75
** STEP: Step-1
**
*Step, name=Step-1, nlgeom = NO
*Static
**
*Boundary, type = displacement, amplitude=FMODAL
**Component U1
‘node id’,1,1,’U1 for node id’
(…)
**Component U2
‘node id’,2,2,’U2 for node id’
(…)
**Component U3
‘node id’,3,3,’U3 for node id’
(…)
**
** OUTPUT REQUESTS
*Output, history, variable=ALLSE
Knockdown Factor
1
0.8
0.6
0.4
0.2
0
1
10
100
1000
10000
Membrane Stiffness Reduction Factor 
Fig. 8: Results of RSM iteration for the cone K8
2.1.2
On nonlinear buckling of axially loaded cones
The plateau or lower-bound buckling behavior of conical shells under axial compression was
also observed in experimental testing campaigns and in comprehensive nonlinear numerical
studies.
For example Khakimova [51] realized the single perturbation load approach (SPLA) in
experiments by applying lateral perturbation loads to the composite cone K8 which was
subsequently loaded by axial compression. Perturbation loads PL = 1…5 N lead to a negligible
reduction of the buckling load as shown in Fig. 9. However, if PL is further increased the
buckling load is reduced by about 20 % until a plateau for the buckling load can be determined.
Note, that the test setup of the test specimen K8 (black line in Fig. 9) had a misalignment of the
loading plates (loading imperfection) which lead to an additional average 21 % reduction of the
buckling load in comparison to the perfect shell (red line in Fig. 9).
global buckling load - FEA, SPLA
global buckling load - Test (average)
local buckling load - FEA, SPLA
55
Buckling Load N [kN]
50
45
40
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Perturbation Load PL [N]
Fig. 9: Buckling load N vs. Perturbation Load PL: test [51] and simulation results
The transition from the reduction range to the plateau range is accompanied by a change in the
structural behavior of the cone, namely local snap-through buckling as shown in Fig. 10 for PL
= 8 N. The local snap-through leads to a reduction of the reaction force and a degradation of
the axial cone stiffness. In addition, a characteristic diamond dimple forms and the cone can be
further loaded until global collapse occurs. The local snap-through effect within buckling
experiments is described as clearly audible and visible. Consequently, the local snap-through is
highly dynamic and may already cause total collapse of the shell. The results of a nonlinear
dynamic analysis for K8 with PL = 8 N and very small time increments (i = 1E-4) are shown in
Fig. 10 (right). The shell collapses due to the snap-through and the corresponding global
buckling load is about 3 % smaller (38.11 kN / 39.14 kN) than the collapse load of the same
shell in a nonlinear static analysis. In comparison to the nonlinear static analysis, the shell in
the nonlinear dynamic analysis exhibits an additional large diamond dimple next to the initial
dimple. Consequently, the load carrying surface of the shell is reduced by factor two in
comparison to the shell in the static analysis; hence the cone collapses already by the snapthrough. Note, that the snap-through effect occurs more gradually as the perturbation load is
increased and for PL = 10 N local and global buckling coincide as shown in Fig. 9.
60
50
Axial Force N [kN]
Axial Force N [kN]
60
40
30
20
10
50
40
30
20
10
0
0
0
0.05
0.1
0.15
0.2
Axial Displacement u [mm]
0.25
0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
Axial Displacement u [mm]
Fig. 10: Load-shortening curve for K08 with PL = 8 N: nonlinear static analysis (left) nonlinear
dynamic analysis (right)
The post-buckling mode in the plateau range is characterized by a specific diamond shaped
dimple. The plateau behavior remains even if the perturbation load is further increased and is
based on a specific distribution of the membrane stresses in the cone.
In the case of P = 1 N, there is a sligthly disturbed membrane stress state if the shell collapses
as shown in Fig. 11. As the perturbation load increases the membrane stress state becomes
more and more disturbed.
F
48 kN
F
42 kN
F
F
F
38.6 kN
39 kN
38 kN
u
PL = 1 N
u
PL = 5 N
-70 MPa
u
PL = 8 N
35 MPa
u
PL = 8 N
u
PL = 15 N
0 MPa
Fig. 11: Membrane stresses of the cone K8 (right before buckling) for different PL
For PL = 8 N, right before snap-through buckling occurs (39 kN), there are still residual
membrane stresses during axial compression. However, after the snap-through has occurred
(38 kN), the shell surface below and above the diamond dimple has nearly no membrane
stresses. Consequently, if the perturbation or imperfection measure in this area is further
increased, there is no further reduction of the buckling load, because the membrane stresses in
this area are already approximately zero as shown in Fig. 11 for PL = 15 N.
In summary, imperfections may lead to local snap-through buckling in a cone under axial
compression. The snap-through reduces the membrane stresses above and below its origin to
approximately zero, hence a further increase of imperfections in this region doesn’t reduce the
buckling load additionally. This structural behavior is associated with the plateau or lowerbound buckling load. Furthermore, the snap-through is a highly dynamic event and may lead
already to global collapse of the shell.
2.1.3
Localized Reduced Stiffness Method (LRSM)
In this section, a variant of the reduced stiffness method (RSM) is introduced. This variant is
defined as localized reduced stiffness method (LRSM) and based on geometrically nonlinear
analyses (GNA) which as opposed to the RSM does not require the use of the first buckling
eigenmode. Similarly to the RSM presented in Section 2.2, the membrane stiffness components
are eliminated from the shell, and only the bending stiffness remains. However, unlike the
RSM, the membrane stiffness in the LRSM is reduced in a localized fashion rather than
globally. The purpose of this localized reduction is to induce the specific membrane stress state
which is associated with the plateau for the buckling load as described in Section 2.3.
Main Shell Surface
Rs
Rs
Reduced Membrane Stiffness Surface
Fig. 12: Schematic of the conical and equivalent cylindrical shell model used in the LRSM
A schematic representation of the region considered for reducing the membrane stiffness in a
cylindrical shell is shown in Figure 10. The cylindrical shell has two sections, the main shell
surface, and a reduced membrane stiffness surface. On one side, The main shell stiffness is
modeled in Abaqus by using the general shell stiffness matrix (ABD – matrix). On the other
side, the reduced membrane stiffness surface, the components of the membrane stiffness matrix
(A-matrix) are reduced significantly (by a factor of 1000). Also, the area of the reduced
membrane stiffness surface in incrementally increased by increasing the radius Rs so its
influence on the buckling load can be studied.
Simulation results corresponding to the implantation of the LRSM are shown in Fig. 11. In Fig.
11 (left), the buckling load versus Rs/R ratio diagram has 2 zones. In the first zone, for Rs/R ≤
0.05, a linear reduction of the buckling load occurs. In the second zone, a local snap-through
occurs beneath the reduced membrane stiffness surface for Rs/R > 0.05 as shown in Fig. 13
(left). The global buckling load (red x in Fig. 13-left) reduces gradually as the Rs radius is
increased. However, the local buckling load (black circles in Fig. 13-left) approaches a plateau
(buckling load is nearly constant ~ 50.9 kN) for RS/R in the range of 0.2 to 0.3. Both, local and
global buckling load continue to reduce after the RS/R ratio exceeds 0.3. The LRSM leads to
an about 8% reduction of the axial stiffness in comparison to the perfect shell as shown in Fig.
13 (right).
local buckling load - LRSM
perfect shell - Z36
100
90
90
80
80
Axial Force F [kN]
Buckling Load N [kN]
global buckling load - LRSM
100
70
60
50
40
30
20
10
LRSM - Rs/R = 0.25
70
60
50
40
30
20
10
0
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
LRSM radius-to-shell radius ratio, Rs/R
0.4
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Axial Displacement u [mm]
Fig. 13: First local and final global buckling loads of the LRSM iteration (left) and
corresponding load-displacement curve (right).
local buckling load - LRSM
perfect shell - Z36
100
90
90
80
80
Axial Force F [kN]
Buckling Load N [kN]
global buckling load - LRSM
100
70
60
50
40
30
20
10
LRSM - Rs/R = 0.25
70
60
50
40
30
20
10
0
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
LRSM radius-to-shell radius ratio, Rs/R
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Axial Displacement u [mm]
Fig. 14: First local and final global buckling loads of the LRSM iteration (left) and
corresponding load-displacement curve (right).
For design purposes, there are two possibilities: One is to use the average local buckling load
in the plateau range (Rs/R = 0.2 to 0.3 - Fig. 13-left) or, the second one, to use the minimum
local buckling load at Rs/R = 0.11 (Fig. 13 - left) which is about 7 % smaller than the plateau
buckling load (47.7 kN / 50.9 kN). A comparison with experimental results is required to
define a conservative design load for the LRSM.
The results show that a plateau for the buckling load can be identified in a geometrically
perfect shell by eliminating the membrane stiffness in a small portion of the shell surface. The
corresponding buckling load represents a conservative lower-bound with respect to the
experimental results. Among the advantages of the proposed LRSM is that it delivers similar
KDFs for the buckling load as the SBPA [57] and about 42% higher KDFs than the RSM.
Also, the LRSM does not require nonlinear contact definitions nor additional convergence
studies like the SBPA and is, therefore, relatively easier to implement in finite element codes.
The corresponding KDFs are also 85 % above the NASA SP-8007 recommendations. Table 3
summarizes the KDFs obtained with different methods for the composite cylinder Z36.
2.1.4
On the absolute minimum buckling load limit
The results of the previous sections indicate that the “worst” test result of the cone K8 can be
approximated if the post-buckling load of the perfect shell or the RSM is used. However, more
cases need to be investigated to further substantiate the thesis, that the absolute minimum
buckling load can be approximated with the RSM or the post-buckling load of the perfect shell.
In this section the results of a parametric study are presented which are based on the postbuckling load of the perfect isotropic shell and the RSM. The corresponding lower-bounds
were obtained in [x] and are compared with experimental results for isotropic conical and
cylindrical shells. The experimental KDFs are shown versus the Batdorf parameter Z which
means that the shell length as well as the slenderness is considered.
The RSM KDFs for isotropic shells are most of the time not conservative and are also
approximately 2 times higher than the post-buckling KDFs. The post-buckling lower-bound
agrees well with the minimum test KDFs for cones and especially cylinders.
From these results it is concluded that the RSM is not suitable to estimate absolute minimum
buckling load. However, the post-buckling load of the perfect shell is a reasonable measure to
approximate the absolute minimum buckling load.
Knockdown Factor
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
TH - Postbuckling
TH - RSM
Flügge [1932]
Lundquist [1933]
Donnel [1934]
Ballerstedt[1936]
Bruhn [1945]
Harris [1957]
Weingarten [1965]
Arbocz et al. [1968]
Esslinger [1970]
1
0.9
Knockdown Factor
TH - Postbuckling
TH - RSM
Lackman (1960)
Schnell (1962)
Seide (1965)
Arbocz (1968)
Foster (1987)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0
1000
2000
3000
4000
5000
6000
7000
0
1000
Batdorf Parameter Z
2000
3000
4000
5000
6000
Batdorf Parameter Z
Fig. 15: Comparison KDFs for buckling experiments with “absolute” lower-bound according
to RSM and post-buckling load for: cones (left) and cylinders (right)
2.1.5
Summary
Table 3: Buckling load and KDFs for the composite conical shell K8.
Shell
GNA
K8
Buckling Load [kN]
47.78
KDF
1.00
Post-buckling load of perfect shell
Experiment (good equalizing layer)
Experiment (faulty equalizing layer)
GNIA (geometric imperfections)
GNIA (geometric and thickness imperfections)
26.08
35.10
22.82
37.72
33.46
0.54
0.73
0.48
0.79
0.70
RSM
22.93
0.48
LRSM
NASA SP-8019 [60]
15.76
0.33
3
In this section the SBPA and SPDA are applied to large-scale composite sandwich cones which
are representative primary structures of NASA Space Launch System (SLS), see Fig. 16. The
first shell is the Universal Stage Adapter (USA) and the second shell is the Payload Attach
Fitting (PAF). The geometry data of representative sandwich cones are given in Table 4 and
are based on an imperfection sensitivity study by Sleight et al. [39].
The sandwich composite shells consist of an aluminum honeycomb-core with two
unidirectional IM7/8552 facesheets. The corresponding material properties are given in Table 5
and Table 6.
Fig. 16: NASAs Space Launch System with structural elements retrieved from [39]
Table 4: Geometry data for the shells SLS PAF and SLS USA
SLS PAF
Geometry parameter
Large Radius R - [mm]
4203.7
Small Radius r - [mm]
787.4
Slant length L - [mm]
4831.4
Ply Thickness tp - [mm]
0.13716
Core Thickness tc - [mm]
12.7…38.1
Ra/teff
~52…147
L/Ra
~1.37
Table 5: CFK plies material data from CTA8.1 [40]
Material parameter - CFK
elasticity modulus 𝐸11 - [MPa]
elasticity modulus 𝐸22 - [MPa]
elasticity modulus 𝐺12 - [MPa]
Poisson’s ratio ν12
Table 6: Sandwich core material data from CTA8.1 [40]
SLS USA
4203.7
2910.8
4996.2
0.13716
12.7…38.1
~54…154
~1.36
149915
9369
5309
0.36
Material parameter - CORE
elasticity modulus 𝐸11 - [MPa]
elasticity modulus 𝐸22 - [MPa]
elasticity modulus 𝐸33 - [MPa]
elasticity modulus 𝐺12 - [MPa]
elasticity modulus 𝐺13 - [MPa]
elasticity modulus 𝐺23 - [MPa]
Poisson’s ratio ν12
Poisson’s ratio ν12
Poisson’s ratio ν12
0.344
0.262
413.69
0.12
203.4
82.74
0.45
0.0001
0.0001
The core material was modeled in ABAQUS using the material editor [Elastic  Engineering
Constants]. The sandwich composite cones were represented by linear shell elements (see Fig.
17) in Abaqus [36] (S4R) and the honeycomb-core was modeled as an additional ply (middle
between facesheets) in the composite layup. The mechanical boundary conditions were defined
as ideal clamped.
Fig. 17: Numerical model of SLS USA (left) and SLS PAF (right)
The numerical KDFs of the SBPA and the SPDA were determined for equivalent cylinders
with different re/teff ratios, a fixed L/re = 2.0 and different laminate stacking sequences
corresponding to either quasi-isotropic or axially stiff material behavior. The KDFs were fitted
with power law in order to determine different lower-bound curves.
1
Pot.(SBPA (numerical))
Pot.(SPDA (numerical))
0.6
y = 1.7113x-0.168
R² = 0.9881
0.4
SBPA (numerical)
SPDA (numerical)
0.2
SPDA (numerical)
y = 1.248x-0.066
R² = 0.9957
0.8
Knockdown Factor
0.8
Knockdown Factor
1
SBPA (numerical)
y = 1.1734x-0.066
R² = 0.9957
Pot.(SBPA (numerical))
Pot.(SPDA (numerical))
0.6
y = 1.999x-0.168
R² = 0.9881
0.4
0.2
0
0
0
200
400
600
800
re/teff
1000
1200
1400
1600
0
500
1000
1500
2000
2500
3000
3500
4000
Batdorf Parameter Z
Fig. 18: Lower-bound curves (L/re = 2) for quasi-isotropic and axially stiff sandwich composite shells with numerical KDFs
from [x]
The threshold KDFs deliver similar lower-bounds for the sandwich composite
re −0.06
KDFSPDA (Quasi−Isotropic) = 1.0589 ∙ ( )
t eff
re −0.18
KDFSBPA (Quasi−Isotropic) = 1.5814 ∙ ( )
t eff
(2)
KDFSPDA (Axially Stiff)
re −0.067
= 1.1472 ∙ ( )
t eff
re −0.168
KDFSBPA (Axially Stiff) = 1.5963 ∙ ( )
t eff
SPDA - Quasi-Isotropic
1
SPDA - Axially Stiff
1
SBPA - Quasi-Isotropic
SLS PAF - 1x Imperfection
SLS PAF - 2x Imperfection
0.6
SLS PAF - 5x Imperfection
SLS USA - 1x Imperfection
0.4
SLS USA - 2x Imperfection
SLS USA - 5x Imperfection
0.2
NASA SP-8019
0.8
Knockdown Factor
Knockdown Factor
SBPA - Axially Stiff
NASA SP-8019
0.8
SLS PAF - 1x Imperfection
SLS PAF - 2x Imperfection
0.6
SLS PAF - 5x Imperfection
SLS USA - 1x Imperfection
0.4
SLS USA - 2x Imperfection
SLS USA - 5x Imperfection
0.2
0
0
0
50
100
150
200
0
50
re/teff
100
150
200
re/teff
Fig. 19: Lower-bound curves (L/re = 2) for quasi-isotropic and axially stiff sandwich composite shells with numerical KDFs
from [x]
KDFSPDA (Quasi−Isotropic) = 1.1475 ∙ Z −0.06
KDFSBPA (Quasi−Isotropic) = 2.0124 ∙ Z −0.18
(3)
KDFSPDA (Axially Stiff) = 1.2549 ∙ Z −0.067
KDFSBPA (Axially Stiff) = 1.999 ∙ Z −0.168
4 Eurocode EN 1993-1-6
In this section the previously described design approaches are applied within the framework
of the Reference Resistance Design (RRD), recently developed by Rotter as a method to
design thin-walled conical shells under axial load. The RRD is based on the capacity curve
which relates a shell’s dimensionless characteristic resistances to its dimensionless
slenderness.
4.1 Current Buckling strength verification
The new proposals are set out in terms of the shell buckling Eurocode requirements, which are
described in [54], [55], [56], [57], [58], [59]. The design buckling stresses for conical shells
which are required for the buckling strength verification according to EN 1993-1-6 can be
determined by applying an equivalent cylinder approach (ECA). The ECA is an approximate
method for the analysis and design of conical shells. This method allows the application of
design methods for cylindrical shells to conical shells and is based on the following geometry
assumptions:
1. equivalent cylinder length le = slant length L
2. equivalent cylinder radius re = average radius of curvature Ra
In addition only conical shells with a uniform wall thickness t and a semi-vertex angle  < 65°
are covered by the following rules. In the first step the shell segment length parameter  is
determined in order to define the shell length type.
ω=
𝐿
√𝑅 ∙ 𝑡
=
𝑙𝑒
√𝑟𝑒 ∙ 𝑡
ω < 1.7 for short shells
1.7 < ω < 0.5 ∙
𝑟𝑒
𝑡
(2)
for medium length shells
The elastic critical buckling stress should be determined by using equation (3) which depends
on the parameter Cx .
σx,Rcr = 0.605 ∙ 𝐸 ∙
Cx = 1.36 −
1.83
𝜔
+
2.07
𝜔2
𝑡
∙𝐶
𝑟𝑒 𝑥
for short shells
(3)
Cx = 1 for medium length shells
In the next step the relative slenderness  (ratio of yield stress 𝑓𝑦,𝑘 to buckling stress σx,Rcr ) is
determined with equation (4).
λ=√
𝑓𝑦,𝑘
σx,Rcr
(4)
In the subsequent step the elastic imperfection factor after equation is required which
depends on the characteristic imperfection amplitude wk for different manufacturing qualities
(excellent quality Q = 40, high quality Q = 25, normal quality Q = 16).
α=
0.62
1 + 1.91 ∙ (∆wk )1.44
(5)
∆wk =
1
𝑅 1
𝑟𝑒
∙√ = ∙√
𝑄
𝑡 𝑄
𝑡
Next the shell class has to be determined, which is defined by comparing the relative
slenderness  with the squash limit  and plastic limit relative slenderness p. For cylindrical
and conical shells under axial compression the squash limit relative slenderness  is defined as
= 0.2 and the plastic limit relative slenderness p is given by the following equation (6):
λp = √
α
1 − βr
(6)
The term r in equation (6) is the plastic range factor and is defined as r = 0.6 and the shell
class equals to elastic imperfect buckling if p 0. The stability reduction factor  can be
determined with equation (7).
χ= 1−β∙[
χ=
λ − λ0 𝜂
]
λP − λ0
α
λ2
(7)
Finally the design buckling resistance σx,Rd can be determined with equation (8).
σx,Rd = χ ∙ σx,Rcr
(8)
4.2 New Buckling strength verification
α=
1
1 + 2.6 ∙ (∆wk )0.8
(4)
1
𝑅 1
𝑟𝑒
∆wk = ∙ √ = ∙ √
𝑄
𝑡 𝑄
𝑡
The elastic imperfection sensitivity factors  according to equation (4) is based on the single
boundary perturbation approach (SBPA) which induces a characteristic buckling mode in thinwalled shells (see Fig. 20 - left) and has been well proven for the design of thin-walled shells
under axial compression [62], [63], [64], [65].
The elastic imperfection factor PB for normal shells is based on the post-buckling equilibrium
load (see Fig. 20 - right) of a conical shell under axial compression.
Q = 40 (current)
Q = 25 (current)
Q = 16 (current)
Q = 40 (new)
Q = 25 (new)
Q = 16 (new)
1
Elastic Imperfection Factor 
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
2000
Equivalent Cylinder Radius-to-Thickness ratio, re/t
Fig. 20: Diamond dimple mode in a conical shell (left) – post-buckling load of a conical shell (right)
The elastic imperfection factors according to equation (3) depend on the Batdorf parameter Z
and the average radius of curvature Ra. The Batdorf parameter is effectively a function of the
square of the length and more difficult for a designer to assimilate. Therefore an easier to
assimilate representation of the elastic imperfection factors was defined by means of the Ra/t
ratio. The imperfection sensitivity of conical shells also depends on the L/Ra ratio and was
defined as L/Ra = 2.0 which covers a wide range of civil engineering applications.
A comparison of the new and current elastic imperfection factors is given in Ошибка!
Источник ссылки не найден. and the results indicate that the new design factors are
significantly above the current design recommendations.
4.3 Comparisons with test results
4.3.1
Introduction
This section presents a large number of comparisons of test results with the new proposals
defined above. The elastic buckling resistance, which dominates many of these tests, is very
sensitive to the average radius-to-thickness ratio (Ra/t) leading to very small values for thin
shells. The experiments are presented by means of the dimensionless buckling resistance which
is defined as the ratio of the test buckling load NRk to the elastic critical buckling load NRcr.
4.3.2
Lackman, Penzien (1960)
One of the first test campaigns on conical shells under axial compression was performed by
Lackman and Penzien in 1960 [66]. The conical shells (15) were manufactured by
electroplating pure nickel on steel mandrels and have semi-vertex angle = 20° & 40°. The
corresponding experimental results are plotted in Fig. 21 vs. the Ra/t ratio and range from
Dimensionless buckling resistance pRk / p Rcr
0.2…0.5. The new and current imperfection sensitivity factors for different fabrication classes
are shown in Fig. 21 also.
1
Q = 40 (current)
0.9
Q = 25 (current)
0.8
Q = 16 (current)
0.7
Q = 40 (new)
0.6
Q = 25 (new)
0.5
Q = 16 (new)
0.4
Lackman (1960)
0.3
0.2
0.1
0
0
500
1000
1500
2000
Equivalent Cylinder Radius-to-Thickness ratio, re/t
Fig. 21: The test results of Lackman & Penzien compared with the new predictions
The results show that the experimental buckling loads are all higher than the current capacity
curve for Q = 25. Most of the experimental results can be well approximated by the new
capacity curve for Q = 25 and three thin shells with Ra/t > 900 are slightly above the new
capacity curve for Q = 16. This comparison indicates that the new proposals are conservative
relative to these tests while still delivering on average 38 % higher buckling resistances.
4.3.3
Schnell, Schiffner (1962)
Schnell and Schiffner presented a test series of 14 Mylar conical shells with semi-vertex
angle= 10°, 20° and 30°. Unfortunately, no information regarding the manufacturing of the
test specimen is given in [67]. The corresponding experimental results are plotted in Fig. 22 vs.
the Ra/t ratio and range from 0.35…0.62.
Dimensionless buckling resistance pRk / p Rcr
1
Q = 40 (current)
0.9
Q = 25 (current)
0.8
Q = 16 (current)
0.7
Q = 40 (new)
0.6
Q = 25 (new)
0.5
Q = 16 (new)
0.4
Schnell (1962)
0.3
0.2
0.1
0
0
500
1000
1500
2000
Equivalent Cylinder Radius-to-Thickness ratio, re/t
Fig. 22: The test results of Schnell & Schiffner compared with the new predictions
Most tests far exceed the old predictions for Q = 40 though there are two thin shells with Ra/t >
1300 which are slightly below the new capacity curve for Q = 40. The new design proposals
for excellent quality shells deliver on average 47 % higher buckling resistances when
compared to the old design proposal for shells with Q = 40.
4.3.4
Seide, Weingarten, Morgan (1965)
Within the testing program by Seide et al. [68] about 130 different conical specimens were
tested. The conical shells had different semi-vertex angles (10°, 20°, 30°, 45° and 60°) and
were manufactured by using Mylar sheets and lap-joints. The corresponding experimental
results are plotted in Fig. 23 vs. the Ra/t ratio and range from 0.33…0.79.
All tests in Fig. 23 lie on or above the new high (Q = 25) fabrication quality line which delivers
on average 117 % higher buckling resistances when compared to the old capacity curve for Q =
25.
Dimensionless buckling resistance pRk / p Rcr
1
Q = 40 (current)
0.9
Q = 25 (current)
0.8
Q = 16 (current)
0.7
Q = 40 (new)
0.6
Q = 25 (new)
0.5
Q = 16 (new)
0.4
Seide (1965)
0.3
0.2
0.1
0
0
500
1000
1500
2000
Equivalent Cylinder Radius-to-Thickness ratio, re/t
Fig. 23: The test results of Seide, Weingarten and Morgan compared with the new predictions
4.3.5
Arbocz (1968)
Dimensionless buckling resistance pRk / p Rcr
Arbocz presented in 1968 [5] a test series of conical shells which have semi-vertex angle =
5°,10°,15°,20° and 25°. The conical shells were manufactured by electroplating copper on wax
mandrels. For each semi-vertex angle two quality classes of shells were manufactured,
“perfect” and “imperfect” shell. The imperfect shells had additional axisymmetric imperfection
halfway between the boundary conditions and have significantly reduced buckling loads when
compared to the “perfect” shell.
The corresponding experimental results are plotted in Fig. 24 vs. the Ra/t ratio and range from
0.33…0.47 for the “imperfect” shells and 0.59…0.8 for the “perfect” shells.
1
Q = 40 (current)
0.9
Q = 25 (current)
0.8
Q = 16 (current)
0.7
Q = 40 (new)
0.6
Q = 25 (new)
0.5
Q = 16 (new)
0.4
Arbocz (1968)
0.3
0.2
0.1
0
0
500
1000
1500
2000
Equivalent Cylinder Radius-to-Thickness ratio, re/t
Fig. 24: The test results of Arbocz compared with the new predictions
The results show that all experimental buckling loads for the “perfect” shells are far above the
new capacity curves for Q = 40. The remaining experimental results for “imperfect” shells are
between the new capacity curve for Q = 25 and Q = 40.
This comparison shows that the new proposals for Q = 25 are conservative for all conical shells
even for the “imperfect” specimen and delivers on average 76% higher buckling resistances
when compared with the old capacity curve for Q = 25.
4.3.6
Foster (1987)
Dimensionless buckling resistance pRk / p Rcr
Foster [69] conducted experiments on conical shells made of epoxy in 1987. The shells had
semi-vertex angle ~ 15° and were manufactured by a spin – casting technique. The
corresponding experimental results are plotted in Fig. 25 vs. the Ra/t ratio and very high as they
range from 0.7…1.
All experimental buckling loads exceed the new capacity curves for Q = 40 by far. The new
capacity curve for Q = 40 is on average 14 % higher compared to the old capacity curve for Q
= 40.
1
Q = 40 (current)
0.9
Q = 25 (current)
0.8
Q = 16 (current)
0.7
Q = 40 (new)
0.6
Q = 25 (new)
0.5
Q = 16 (new)
0.4
Foster (1987)
0.3
0.2
0.1
0
0
500
1000
1500
2000
Equivalent Cylinder Radius-to-Thickness ratio, re/t
Fig. 25: The test results of Foster compared with the new predictions
5 Conclusion and Outlook
In the first part of this paper a short literature review regarding the buckling and imperfection
sensitivity of axial loaded conical shells is presented.
A new procedure for the buckling resistance assessment of isotropic conical shells subjected to
axial compression was presented. The procedure is relatively simple, clear and does not require
an engineer to have special skills. It was devised to be consistent with the general rules of the
shell buckling standard Eurocode 3 Part 1-6.
The main objective of this article was to validate the new design proposal with experimental
results.
The new design procedure for the elastic buckling of conical shells under axial compression
was compared with about 220 experimental results from 5 different testing campaigns
(different materials, different geometries, different manufacturing techniques). The results
show that the new proposal is always conservative when compared with experimental results
and delivers on average 14 – 117 % higher buckling resistances in comparison to the old
capacity curves.
The average radius of curvature-to-thickness ratio (Ra/t) covers a wide range of application
(~100 – 4000). In all comparisons, the test results were usually well above the capacity curve
values. In several cases some test results fell into the lower fabrication quality classes but none
lay below the lowest class prediction. The tests confirm that the new proposals as well
supported by very many experiments in the past.
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