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PVP2017-65724

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Proceedings of the ASME 2017 Pressure Vessels and Piping Conference
PVP2017
July 16-20, 2017, Waikoloa, Hawaii, USA
PVP2017-65724
NONLINEAR BUCKLING ANALYSIS OF CYLINDRICAL SHELL WITH NORMAL
NOZZLE SUBJECTED TO AXIAL LOADS
Qianyu Shi
Harbin Boiler Co., Ltd.
Harbin, China, 150046
Email: [email protected]
Zhijian Wang
Harbin Boiler Co., Ltd.
Harbin, China, 150046
Email: [email protected]
Hui Tang
Harbin Boiler Co., Ltd.
Harbin, China, 150046
Email: [email protected]
structures [2]. Whereas most pressure vessel codes provide
methods of stability design by rules, and only for basic
geometrical structures(e.g. cylindrical shell, conical shell,
spherical shell) and simple load conditions (e.g. axial uniform
compression, uniform external pressure and bending moment
and their combinations) [3].
ABSTRACT
Design of Large-scale and light-weight pressure vessels is
an inexorable trend of industrial development. These large thinwalled vessels are prone to buckling failure when subjected to
compression loads and other destabilizing loads. Thus, buckling
analysis is a primary and even the most important part of design
for these pressure vessels. Local buckling failure will probably
occur when cylindrical shells with nozzle subjected to axial
loads. In this paper, a FE model of cylindrical shell with a
normal nozzle is established in ANSYS Workbench. The
bifurcation buckling analysis is performed by using an elasticplastic stress analysis with the effect of nonlinear geometry, and
a collapse analysis is performed with an initial imperfection.
The axial buckling loads are obtained by these two types of
method. Some issues about nonlinear buckling analysis are
discussed through this study case.
Since 2007, ASMEⅧ-2 provides three types of method to
evaluate buckling failure using numerical solution (e.g. FEA)
as follows:
Type 1 is a bifurcation buckling analysis that is performed
using an elastic stress analysis without geometric nonlinearities
in the solution to determine the pre-stress in the component.
Type 2 is a bifurcation buckling analysis that is performed
using an elastic-plastic stress analysis with the effects of
nonlinear geometry in the solution to determine the pre-stress in
the component.
ABBREVIATIONS
ASMEⅧ-2 ASME B&PV Code Section VIII Division 2 [1]
Fx
Nozzle force in radical direction
Fy
Nozzle force in circumferential direction
Fz
Nozzle force in axial direction
node_max Maximum deformation of the lowest mode
e
Allowable maximum tolerance of shell
Type 3 is a collapse analysis in which imperfections are
explicitly considered in the analysis model geometry. It should
be noted that a collapse analysis can be performed using elastic
or plastic material behavior. It depends on the stress state in the
component.
The former two types are bifurcation buckling analysis for
perfect structures, which can be performed through eigenvalue
buckling analysis in some general FEA software (e.g. ANSYS,
ABAQUS) [4], another way for Type 2 is a nonlinear structural
analysis for all process, in which a perturbation can be used.
For example, deformation perturbation method can be used for
solving a global buckling, load perturbation method is more
suitable for solving a local buckling [5]. Collapse analysis is a
limit point buckling analysis for imperfect structures. It is same
as the elastic-plastic stress analysis method for protecting
against plastic collapse failure.
INTRODUCTION
Cylindrical shells are commonly used in engineering
structures such as aircraft, missiles, silos, pipelines, tanks, and
some submarine structures. During their service life, these
components are often subjected to axial loads. In addition, these
structures often have geometric discontinuities, such as
stiffeners and nozzles, which can lead to substantial stress
concentrations and subsequently influence the stability of the
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The effects of cylindrical shell with a cutout subjected to
axial loads were studied by some previous work [2, 6-7].
Reference [8] addresses the issue of nozzle loads, which were
applied to a circular opening in a shell, reducing the buckling
capacity. However, the nozzle itself was not modeled. The
influence of nozzle loads to shell with nozzle modeled
subjected to external pressure was investigated in reference [9].
In this paper , a FE model of cylindrical shell with a
normal nozzle is established, type 2 and type 3 are implemented
to evaluate the axial buckling loads in the case. Different
directions of nozzle force are used as the perturbation loads in
type 2. The lowest mode shape of eigenvalue buckling analysis
is used as the initial imperfection in type 3.
FINITE ELEMENT ANALYSIS
Elastic-plastic buckling analysis are carried out in ANSYS
Workbench, with a material curve derived using ASME Ⅷ -2
Annex 3D. The stress-strain curve of material is shown in
Fig.1. The elastic-plastic buckling analysis of shells subjected
to axial loads has been verified reliable in ANSYS [2].
Fig.2 The geometric model
GEOMETRY AND MESHING
A cylindrical shell is modeled with a normal nozzle at the
half height of shell in Fig.2. The parameters of the geometry are
shown in table 1. Solid 186 element with 20-node is used to
mesh the structure in Fig.3. This element is suitable for
analyzing nonlinear behaviors and stress concentrations.
Fig.3 The meshing model
BOUNDARY CONDITION
With reference to the coordinate system shown in Fig.2,
the boundary condition of the FE model in ANSYS Workbench
is shown in Fig.3. The uniform axial force is applied to the top
end of the shell, and the top end is restricted in translational
circumferential and radial directions using a local cylindrical
coordinate system, whereas the bottom end is fixed by setting
the translational DOF in x-, y- and z- directions to be zero. A
nozzle force is applied to the end of the nozzle. The different
directions of nozzle force are applied respectively in three
directions: Fx, Fy, Fz.
Fig.1 The stress-strain curve of material
STABILIZATION METHOD
Nonlinear stabilization method and arc-length method are
generally used for nonlinear buckling analysis. Nonlinear
stabilization method cannot detect the negative-slope portion of
a load-deformation curve, which is not suitable for simulating
the post-buckling behavior of structures, but it can be used for
global and local buckling. Arc-length method can detect the
negative-slope portion of a load-deformation curve, which is
suitable for simulating the post-buckling behavior of structures,
but it cannot solve problems with local buckling [4]. In this
Table 1 Parameters of the geometry (mm)
Shell diameter
2000
Nozzle diameter
500
Shell thickness
10
Nozzle thickness
20
Shell length
8000
Nozzle length
1500
2
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paper, the local buckling may occur for a shell with a nozzle,
and the post-buckling is not considered, so the nonlinear
stabilization method is used. When this method is used, the
buckling load can be obtained in load-deformation curve as
shown in Fig.5. The buckling point is indicated by the inability
to keep structure stable for a small increase in load.
Fig.6 Bifurcation axial buckling loads of shell under
different directions of nozzle loads
COLLAPSE ANALYSIS
Collapse analysis requires that initial imperfections are
considered in the model. In generally, it is difficult to measure
the imperfections of structures in the design phase. Reference
[11] has showed that the lowest buckling mode can be used to
determine the initial imperfections, and a good correlation was
observed between the buckling analysis and the experiments.
One of the popular methods is consistent mode imperfection
method. In this case, firstly, an eigenvalue buckling analysis is
carried out. The imperfection shape of structure is the lowest
buckling mode shape obtained by using command UPGEOM in
ANSYS Workbench. The magnitude of imperfection is defined
by the lowest buckling mode deformation multiplied a factor, ∆
(∆ = e / node_max) [12]. e is the allowable maximum shell
tolerance derived from requirement of ASMEⅧ-2, (e = 1% ×
Shell Diameter = 1%×2000 = 20 mm) .
Fig.4 Displacement and force boundary
condition of the FE model
The axial buckling load of collapse analysis is 7874KN in
this case. The lowest eigenvalue buckling mode shape of shell
is shown in Fig.10 and the collapse shape is shown in Fig.11.
For this analysis model, the second eigenvalue buckling mode
shape is used, as the first mode shape is affected by the nozzle
deformation.
Fig.5 The force – deformation cur
CONCLUSIONS
The nonlinear buckling analysis shows that the nozzle
loads and initial imperfections have an impact on the buckling
capacity of shells with a nozzle.
BIFURCATION BUCKLING ANALYSIS
Bifurcation buckling analysis is carried out using nonlinear
structural analysis, in which initial imperfections are not
considered. The nozzle force is regarded as the lateral
perturbation load according to reference [10].
The nozzle load Fz has the most influence on the buckling
capacity of the shells with a nozzle subjected to axial loads,
whereas the nozzle load Fx has the least influence. In actual
engineering, the nozzle loads can be controlled in a safety limit
through nonlinear buckling analysis.
The bifurcation buckling loads under different directions of
nozzle loads are shown in Fig.6. The buckling mode shapes
under 40KN nozzle loads in three directions are shown in
Fig.7~Fig.9. The nozzle load Fz has the most influence on the
buckling capacity of the shells with a nozzle subjected to axial
loads, whereas the nozzle load Fx has the least influence.
Local buckling will occur around the nozzle when shells
with a nozzle subjected to axial loads, the local reinforcement
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can be used to improve the buckling capacity (e.g. the pad
reinforced nozzles and the local stiffeners).
REFERENCES
[1] 2015 ASME Boiler & Pressure Vessel CodeⅧ Division 2,
American Society of Mechanical Engineering, New York.
[2] Han, H. P., Chen, J. Q., Taheri, F., et al, 2006, “Numerical
and Experimental Investigations of the Response of
Aluminum Cylinders with a Cutout Subject to Axial
Compression”. Thin-Walled Structures, Vol. 44, pp. 254270.
[3] Shen, J., Tang, Y. F., Liu, Y. H., 2015, “Buckling Analysis
of Pressure Vessel Based on Finite Element Method,” 14th
International Conference on Pressure Vessel Technology,
Shanghai, China, Vol. 130, pp. 355-363.
[4] ANSYS,Inc., 2015, ANSYS Mechanical User’s Guide for
Release15.0, ANSYS,Inc..
[5] Zhou, Y. Z., 2008, “Nonlinear Stability Analysis for
Single Layer Lattice Shells with Complex Surface,”
Master Thesis, Zhejiang University, China.
[6] Zhang, Z. Z., Zhao, G. W., Huang, H., et al, 2012,
“Buckling Analysis and Experiment of Open Cylindrical
Thin Shells under Axial Load,” Journal of Beijing
University of Aeronautics and Astronautics, China, Vol. 38,
No. 4, pp. 557-562.
[7] Yan, G., 2013, “The Design and Experimental Study of
Composite Cylindrical Shell under Axial Compressive
Load,” Ph.D. Thesis, Jilin University, China.
[8] Maharaj, Ashveer. 2003, “A Comparative Study on the
Effects of Internal Vs External Pressure for a Pressure
Vessel Subjected to Piping Loads at the Shell-to-nozzle
Junction.” Ph.D. Thesis, University of Natal Durban.
[9] Clarke, E., Frith, R., 2015, “The Effect of Nozzles and
Nozzle Loadings on Shell Buckling,” ASME 2015
Pressure Vessels and Piping Conference. American
Society of Mechanical Engineering, Boston, USA.
[10] Khakimova, R., Zimmermann, R., Wilckens, D., et al,
2016, “Buckling of Axially Compressed CFRP Truncated
Cones with Additional Lateral Load : Experimental and
Numerical Investigation,” Composite Structures, Vol. 157,
pp. 436-447.
[11] See, T., Mcconnel, R. E., 1986, “Large Displacement
Elastic Buckling of Space Structures,” Journal of
Structural Engineering, Vol. 112, pp. 1052-1069.
[12] Wei, X. Y., Chen, B. B., Zheng, H. Q., et al, 2015,
“Discussion of Regulation of Initial Geometric Deviations
on Cylindrical Shells under External Pressure in Pressure
Vessel Design Standard,” Pressure Vessel Technology, Vol.
32, No. 4, pp. 20-28.
One issue is worthy of being taken into consideration
when the eigenvalue buckling analysis of shells with a nozzle is
carried out for a collapse analysis. That is, a long nozzle can
affect the lower mode shape of the shells, the most portion of
large deformation would occur on the nozzle in the first or
second mode shape because of the unrestricted end of the
nozzle. These lower buckling modes cannot be used as the
imperfections for shells. A third or fourth mode may be useful
in this situation.
FUTURE WORK
 The effect that eccentric loads have on the buckling
capacity of a cylindrical shell with normal nozzle will be
investigated.

The nonlinear-based eigenvalue buckling analysis will be
used to calculate the bifurcation buckling Load.

The presence of multiple nozzles and their relative
positions could have an impact on the buckling capacity.

Different shell and nozzle diameters can be considered.
4
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FIGURE
Fig.7 The buckling mode shape of shell subjected
to axial load with nozzle load Fx
Fig.10 The lowest eigenvalue buckling mode
shape of shell (second mode)
Fig.8 The buckling mode shape of shell subjected
to axial load with nozzle load Fy
Fig.11 The collapse shape of shell subjected to axial load
Fig.9 The buckling mode shape of shell subjected
to axial load with nozzle load Fz
5
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