ON THE IMPERFECTION SENSITIVITY AND DESIGN OF BUCKLING CRITICAL
COMPOSITE CONICAL SHELLS UNDER AXIAL COMPRESSION
H.N.R. Wagnera,b, T. Ludwiga and R. Khakimovac
a
Technische Universität Braunschweig, Institute of Adaptronics and Function Integration, Langer
Kamp 6, 38106 Braunschweig, Germany
ro.wagner@tu-braunschweig.de
b
Siemens Mobility GmbH, MO MM R&D SYS ITV IXL, Ackerstr. 22, 38126 Braunschweig, Germany
c
Fraunhofer institute, Open Hybrid LabFactory e.V., Hermann-Münch-Straße 2, 38440 Wolfsburg,
Germany
Regina.Khakimova@iwu.fraunhofer.de
Keywords: Buckling, conical shells, new proposed design rules, experimental results,
imperfection
Abstract
A major loading scenario for conical shells is axial compression. The buckling load of these shells is very
sensitive to imperfections (geometry, loading conditions) which results in a critical disagreement between
theoretical and experimental results for axially loaded conical shells.
The design of these stability critical shells is based on classical buckling loads obtained by a linear
analysis which are corrected by a single knockdown factor (0.33 - NASA SP-8019) for all cone
geometries. This practice is well established among designers and hasn’t changed for the past 50 years
because the buckling behavior is till today not very well understood.
Within this paper a numerical lower-bound analysis for composite conical shells under axial compression
is performed. The laminate stacking sequence of the shell lead to a low imperfection sensitivity which is
verified by using a detailed reduced stiffness analysis. The results are validated by buckling experiments
and show that even in the absolute worst case the shell buckling knockdown factors is about 50 % higher
compared to the NASA SP-8019. The new design recommendation for composite conical shell structures
result in increased knockdown factors for the buckling load which in turn may lead to a significant weight
reduction potential.
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
t
TH
u
Wall thickness of a cylinder, truncated cone
Threshold
Axial displacement
1 Introduction
Conical shells are important structural elements in aerospace industry (e.g. helicopter tailboom,
airplane fuselage, launch-vehicle adapters). One of the major load cases for conical shells is axial
compression which may lead to buckling. The main geometry parameters of conical shells are
shown in Fig. 1.
Fig. 1: Geometry of a conical shell.
Multiple numerical and experimental studies for isotropic cones under axial load are for example
given by Blachut et al. [1], [2], [3] and Ifayefunmi et al. [4]. [5], [6].
However, studies for composite cones under axial compression are very rare. An experimental
study on buckling of composite conical shells under axial compression is given by Long et al.
[7]. The corresponding test specimen have a semi-vertex angle of 9° (see Fig. 2 - left). Shadmehri
[8] published recently a PhD thesis which covers numerical and experimental studies on the
buckling of a composite cone under pure bending. The composite cone is representative for a
helicopter tailboom and has a small semi vertex angle (<5 °).
Khakimova et al. published experimental results for undisturbed [9] and disturbed [10]
composite conical shells under axial compression which have a semi-vertex angle of 35 °, see
Fig. 2. The disturbed specimen were additionally loaded with a lateral perturbation load [11] and
the results show that the different between minimum disturbed and undisturbed test is about 17
%, see Fig. 3.
The experimental buckling loads are represented by a knockdown factor (KDF) which is the ratio
of experimental buckling load Nexp to the theoretical buckling load Nper, see equation (1). The
KDFs are plotted versus the radius of curvature to thickness ratio (Rc/t) which can be calculated
using equation (2).
𝑁𝑒𝑥𝑝
𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝐵𝑢𝑐𝑘𝑙𝑖𝑛𝑔 𝐿𝑜𝑎𝑑
(1)
𝐾𝐷𝐹 =
=
𝑒𝑥𝑝
𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐵𝑢𝑐𝑘𝑙𝑖𝑛𝑔 𝐿𝑜𝑎𝑑
Rc =
r
cos(𝛽)
𝑁𝑝𝑒𝑟
(2)
Fig. 2: Composite cone with ß = 9° from Tong [7] (left) composite cone with ß = 35° from Khakimova [9] (middle) composite
cone with ß = 2° from Shadmehri [8]
Weingarten et al. [1968]
Khakimova [2016] - disturbed test
Arbocz [1968]
Khakimova [2016] - undisturbed test
1
Knockdown Factor
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
200
400
600
800
1000 1200 1400 1600 1800 2000
Radius of Curvature-to-Thickness ratio, Rc/t
Fig. 3: Experimental KDF for conical shells under axial compression.
L/Ra <= 0.5
TH - Cone
L/Ra <= 0.5
LRSM - Cone
L/Ra > 0.5
NASA SP-8019 - Compression
L/Ra > 0.5
NASA SP-8019 - Bending
1
1
0.9
0.9
Knockdown Factor
Knockdown Factor
There is a deviation between expected theoretical buckling loads and measured experimental
buckling loads. However, the deviation is not as severe as in the case of isotropic cones. The
minimum KDF for the buckling load of the composite conical shells equals to 0.62 which is 87
% higher compared to the minimum KDF of the isotropic cones [12] under axial compression as
shown in Fig. 3. These results indicate that composite conical shells may be less imperfection
sensitive than isotropic conical shells.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0
0
0
1000
2000
3000
4000
Batdorf Parameter Z
5000
6000
0
250
500
750
1000
1250
1500
1750
2000
Batdorf Parameter Z
Fig. 4: Comparison of experimental KDFs and design lower-bounds for conical shells: compression (left) pure bending (right)
For the design of conical shells the NASA SP-8019 [13] proposes a single KDF of 0.33 for all
cone geometries for the axial compression case and a KDF of 0.41 for the pure bending case.
Wagner et al. developed improved lower-bound curves for conical shells under axial
compression [14] and pure bending [15] which depend on the Batdorf parameter Z as shown in
Fig. 4. The corresponding lower-bound curves deliver higher KDF than the NASA SP-8019 if
the shells have L/Ra > 0.5.
Z=
L2 ∙ √(1 − 𝑣 2 )
𝑅𝑎 ∙ 𝑡
(3)
(R + r)
Ra =
2 ∙ cos(𝛽)
The difference between analytical and experimental results is (like in the case of cones under
axial compression) caused by the presence of imperfections [16], [17]. First studies which link
the influence of geometric imperfections (shape deviations from the ideal shell geometry) to the
buckling load reduction were given by Koiter [18]. Over the years different concepts to consider
the influence of imperfection on the buckling load have been developed [19], [20].
One of these concepts is Koiter’s asymptotic analysis which can be used to consider the influence
of geometric imperfections for the design of slender structures [21], [22]. The most commonly
used approach is based on the application of KDFs like the NASA SP guideline and Eurocode
guidelines [23], [24]. Then there are concepts which are based on measured geometric
imperfections as proposed by for example Hilburger et al. [25], [26] or equivalent geometric
imperfections as proposed by Khakimova et al. [27] and Castro et al. [28]. Advanced
probabilistic methods which treat geometric imperfections as a random variable in order to
design thin-walled shells were proposed by Arbocz et al. [29], [30] Elishakoff [31], [32], [33]
and Kriegesman et al. [34], [35], [36].
Then there are lower-bound concepts [37], [38] which should in theory deliver a buckling load
which is equal or smaller than every buckling load caused by multiple or large amplitude
imperfections. One of the first lower-bound concepts was proposed by Croll et al. [39]. The
reduced stiffness method by Croll is based on the assumption that imperfections lead to a
degradation of the membrane stiffness or membrane energy of a shell. By reducing the complete
membrane stiffness of a shell, a lower-bound with respect to imperfections can be determined
[40].
Improved lower-bound methods were proposed by Hühne et al. [41], Hao et al. [42], Meurer et
al. [43], Tian et al. [44], Wang et al. [45], [46], Wagner et al. [47], [48], [20], [49], [50]. However,
most of the lower-bound methods were only be applied to cylindrical shells [51], [52] or conical
shells [53], [14] under axial compression.
A further development of the RSM was recently proposed, the localized reduced stiffness
(LRSM) from [54] which was applied to cylindrical shells under pure bending and axial
compression, spherical shells under external pressure [55], [56] and may be also suitable to study
the imperfection sensitivity of composite conical shells under axial compression. The purpose of
this article is to study the lower-bound buckling load of composite conical shells under axial
compression. The corresponding numerical model is described in section 2. Afterwards a lowerbound study with the LRSM is given in section 3. The lower-bound results of section 3 are
afterwards validated with experimental results from a buckling test. The results of this article are
summarized in the last section.
2 Imperfection sensitivity of composite 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. 6 – right)
was manufactured and tested [9].
Fig. 5: 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 [57], [11].
MESH – S4R – 5 mm
Fig. 6: Numerical model of the composite cone K8
The conical shell was modeled by using linear shell elements (S4R in ABAQUS [58]) as shown
in Fig. 6 (left) and the equivalent cylinder is shown in Fig. 6 (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 [59].
The mechanical boundary conditions on both cones edges are defined as clamped by using rigidbody 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. 7 (left). Unfortunately, no
strain and ARAMIS measurements were performed for this first buckling test and the buckling
test was also not repeated due to fear of damaging the test specimen.
Table 1: Material and geometry parameters for cones K8 [9].
Shell
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/ Ra
Cone 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. Note, that this KDF is even higher than the design KDF proposed by the
NASA SP-8019 (0.33).
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. 7
(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. 7: Load-displacement curves of cone K8: Tests from [9] 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. 8.
[mm]
[mm]
0.9
0.7
0.6
-0.7
Fig. 8: Thickness deviation - Ultrasonic scan results for K8 (left) Geometric deviation - ATOS
measurement results of K8 (right).
2.2 Reduced Stiffness method (RSM)
The reduced stiffness method (RSM) was developed by Croll et al. [40], [39] 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. [60] 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.
Reduced Stiffness Analysis
ABD Stiffness Matrix




Geometrical parameter
material data
Load
Mechanical boundary conditions
Create numerical model:





Shell geometry
Material definition by means of ABD stiffness matrix for whole shell
Axial compressive load
Clamping as mechanical boundary conditions
Mesh – ABAQUS – STRI3 shell elements
Mesh – STRI3 – 10 mm
Perform Linear Bifuraction Analysis (LBA):



Determine first linear buckling load
Determine first linear buckling eigenmode
Extract nodal coordinates from buckling eigenmode
Perform Linear Static Analysis (LSA):



No axial load and same mechanical boundary conditions as in LBA
Apply nodal coordinates of first linear buckling mode as prediscribed displacement field (load for this step)
Determine strain energy – ABAQUS – History Output variable  ALLSE
Perform RSM Iteration using multiple Linear Static Analysis (LSA):




Reduction of A-matrix – divide all 9 components of A-matrix by membrane stiffness reduction factor 
Reduction of B-matrix – Set all 9 components of B-matrix to zero (to prevent singular siffness matrix)
Determine strain energy – ABAQUS – History Output variable  ALLSE with respect to 
Calculate Knockdown Factor  Divide strain energy () by strain energy (reference) from process 3
Knockdown Factor  min. ?
Yes
Knockdown Factor
1
No, increase  by factor 10
0,8
0,6
0,4
0,2
0
1
10
100
1000
10000
Membrane Stiffness Reduction Factor 
Result
Fig. 9: 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 of
𝛼
𝑅𝑆𝑀
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 =
(4)
The variation of the knockdown factor as the values of the scaling factor RSM increases for the
composite cone K8 is shown in Fig. 10. 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
Membrane Stiffness Reduction Factor 
Fig. 10: Results of RSM iteration for the cone K8
10000
2.3 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 [11] 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. 11. 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. 11) 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. 11).
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. 11: Buckling load N vs. Perturbation Load PL: test [11] 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. 12 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. 12 (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 snap-through. 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. 11.
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
0.25
0
0.3
0.05
0.1
0.15
0.2
0.25
0.3
Axial Displacement u [mm]
Axial Displacement u [mm]
Fig. 12: 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. 13. 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. 13: 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. 13 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 lower-bound
buckling load. Furthermore, the snap-through is a highly dynamic event and may lead already to
global collapse of the shell.
2.4 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 2.1.1, 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 2.1.2.
Main Shell Surface
Rs
Rs
Reduced Membrane Stiffness Surface
Fig. 14: Schematic of the LRSM surface for thin walled shells
A schematic representation of the region considered for reducing the membrane stiffness in a
conical and cylindrical shell is shown in Fig. 14. The conical 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 (Amatrix) 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. 15. In Fig.
15 (left), the buckling load versus Rs/R ratio diagram has 2 zones. In the first zone, for Rs/R ≤
0.04, a reduction of the buckling load occurs. In the second zone, a local snap-through occurs
near the reduced membrane stiffness surface for Rs/R > 0.05 as shown in Fig. 15 (left). The
global buckling load (red + in Fig. 15-left) approaches a plateau as the Rs radius is increased.
However, the local buckling load (black diamond in Fig. 15-left) reduces and the minimum
buckling load ~ 30.9 kN. Both, local and global buckling load coincide after the Rs/R ratio
exceeds 0.07.
Local Buckling Load
50
45
45
40
40
Axial Force F [kN]
Buckling Load N [kN]
Global Buckling Load
50
35
30
25
20
15
10
35
30
25
20
15
10
5
5
0
0
0
0,02
0,04
0,06
0,08
0,1
LRSM radius-to-shell radius ratio, Rs/R
0
0,05
0,1
0,15
0,2
0,25
Axial Displacement u [mm]
Fig. 15: First local and final global buckling loads of the LRSM iteration (left) and
corresponding load-displacement curve (right).
For design purposes, the minimum local buckling load is used which is conservative with respect
to the undisturbed test result (compare with Fig. 7 - right) and also the disturbed test (compare
with Fig. 11).
The LRSM delivers KDFs for the buckling load which are about 33% higher compared to the
RSM. The corresponding KDFs are also 93 % above the NASA SP-8019 recommendations.
Table 3 summarizes the KDFs obtained with different methods for the composite cone K8.
Table 3: Buckling load and KDFs for the composite conical shell K8.
Shell
GNA
Post-buckling load of perfect shell
Experiment (good equalizing layer)
Experiment (faulty equalizing layer)
GNIA (geometric imperfections)
GNIA (geometric and thickness imperfections)
RSM
LRSM
NASA SP-8019 [13]
K8
Buckling Load [kN]
47.78
26.08
35.10
22.82
37.72
33.46
22.93
30.91
15.76
KDF
1.00
0.54
0.73
0.48
0.79
0.70
0.48
0.64
0.33
3 Conclusion and Outlook
This article presents a detailed analysis regarding the imperfection sensitivity of a composite
conical shell under axial compression. The composite cone K8 considered has a semi-vertex
angle of 35 °, Ra/t ~480 and L/Ra ~ 1.01. The experimental KDF of the cone K8 was measured
as 0.73 which is significantly higher (more than double) compared to the design KDFs of the
NASA SP-8019 (0.33).
A variant of the reduced stiffness method (RSM) is introduced for the design of composite
conical shells under axial compression, the localized reduced stiffness method (LRSM). Within
the framework of the LRSM, the membrane stiffness is reduced in a localized fashion rather than
globally. The purpose of this localized reduction is to induce a localized dimple on the surface
of the shells which may lead to snap-through buckling. It was shown that snap-through buckling
reduces the membrane stresses above and below its origin to approximately zero. Hence a further
increase of imperfections in this region does not reduce the buckling load additionally. This
structural behavior is associated with the plateau or lower-bound buckling load and corresponds
well the assumptions of the RSM in its original definition. For finite element implementation,
and in contrast with RSM, the LRSM does not require the application of the first buckling
eigenmode as a possible geometric imperfection.
The composite cone was analyzed with the RSM and the LRSM and the resulting lower-bound
KDFs are conservative with respect to the experimental result and deliver improved design KDFs
(45-92 % higher) when compared with the NASA SP-8019.
The results of this article show that there is potential to increase the KDF for the buckling of
composite conical shells under axial compression which in turn results in structural weight
reduction potential.
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