Buckling analysis of composite long cylinders using probabilistic finite element method/Kompoziciniu ilgu cilindru klupimo analize tikimybiniu baigtiniu elementu metodu.
Cai, Baoping ; Liu, Yonghong ; Li, Huazhou 等
1. Introduction
Fibre-reinforced composite materials are increasingly used in
marine application over the past few decades due to their light weight
and high resistance to salt water corrosion [1]. There materials are now
being applied for composite pressure hulls, autonomous underwater
vehicle, and even offshore oil equipment [2-7], which usually subject to
high pressure of seawater. For these structures, external hydrostatic
pressure-induced buckling and crushing tends to dominate structural
performance.
In recent years, composite structures, especially of cylinders,
subjected to external hydrostatic pressure have been widely studied by
using various methods, such as the mathematical formulation method and
finite element method. Hur et al. [8, 9] studied the buckling and
post-buckling behaviours of composite cylinders by performing
hydrostatic external pressure tests. Three finite element analysis
programs including ACOS, MSC.NASTRAN and MSC.MARC were used for the
failure analysis. It was identified that the results of finite element
analysis and the hydrostatic test indicated good matches. Frulloni et
al. [10] studied the failure behaviour of lattice composite hollow
structures subjected to external hydrostatic pressure. In order to
understand the failure mechanism, finite element analysis was used to
evaluate the buckling pressure of the tested tubes by using ANSYS
software. Tafreshi [11, 12] proposed a computational modelling of
delamination buckling and postbuckling of laminated composite
cylindrical shells subjected to external pressure or combined axial
compression and external pressure. The finite element analysis which was
carried out using ABAQUS verified the computational results. Graham [13,
14] developed the analytical model which were developed in conjunction
with a series of model tests and used in the design of the large scale
composite hull model. The results were validated by the finite element
analysis with ABAQUS software.
From these literatures, it can be seen that the finite element
analysis method has been widely used to study the composite pressure
vessel. Various commercial software, for example ANSYS [10], ABAQUS
[11-14, 15], ADINA [16] and SAMCEF [17], are used to perform the finite
element analysis. However, these analyses are mainly deterministic. The
reliability of composite pressure vessel does not be evaluated by using
finite element analysis method. To address the growing need for
stochastic and probabilistic finite element analysis, ANSYS Inc.
released the ANSYS Probabilistic Design System (PDS) [18]. It can be
used for an uncertainty analysis or a reliability analysis. The PDS
includes both of the Monte Carlo Simulation (MCS) method and Response
Surface Method (RSM). The method has been used to study the
probabilistic problems for various structures. Nakamura et al. [19]
demonstrated the probabilistic thermal analysis of an atmospheric
re-entry vehicle structure and investigated the probabilistic
temperature response by using MCS. Zulkifli et al. [20] evaluated the
reliability or fatigue life of the solder joints in the ball grid array
package by using RSM. Nemeth et al. [21] studied the effect of specimen
dimension of single crystal SiC on the strength response by using PDS.
Liu et al. [22] studied the strength reliability of composite laminated
high pressure hydrogen storage vessel by using MCS and RSM.
This work aims to study the effects of uncertainties of material
properties and physical dimensions on the critical buckling pressure of
composite long cylinders subjected to external hydrostatic pressure by
using probabilistic finite element analysis method. The longitudinal
modulus, transversal modulus, shear modulus, Poisson's ratio of
composite material, unsupported length, inside radius, thickness and
winding angles of inner and outer layers of composite long cylinders are
taken as random input parameters, and the critical buckling pressure is
taken as random output response. A total of four carbon-epoxy composite
long cylinders were fabricated and tested in a hyperbaric testing
chamber to validate the finite element analysis results.
2. Finite element analysis
2.1. Deterministic analysis
The structural model used to represent the composite long cylinder
under study is schematically shown in Fig. 1. The cylindrical wall is
composed of 4 inner plies with winding angle of [[alpha].sub.i] and 13
outer plies with winding angle of [[alpha].sub.0] with respect to the
axis of the cylinder. Each of the plies has equal thickness. The
stacking sequence can be denoted by
[[[alpha].sub.i4]/[[alpha].sub.o13]].sub.T], as shown in Fig. 1, a. The
geometry of the long cylinder is characterized by its overall length
[L.sub.0], unsupported length L, inside radius R and thickness T, as
shown in Fig. 1, b. The dimensions of cylinder are given in Table 1. It
is noted that there are four overall lengths and unsupported lengths,
which shows that four long cylinders with different lengths are
researched in this work. The critical external pressure of composite
long cylinder is denoted by P.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
The longitudinal modulus, transversal modulus, shear modulus,
Poisson's ratio are the four most important mechanical properties
of composite material, which are also given in Table 1. For the sake of
simplicity, the through-thickness modulus is assumed to be same as
transversal modulus [E.sub.T]; the longitudinal and transversal
through-thickness shear moduli are assumed to be same as longitudinal
in-plane shear modulus [G.sub.LT]; and the minor Poisson's ratio
and transversal Poisson's ratio are assumed to be same as major
Poisson's ratio [V.sub.LT].
Finite element analysis package ANSYS is used to predict the
buckling behaviour of composite long cylinders subjected to external
hydrostatic pressure. Fibre orientation, thickness distribution,
stacking sequence and number of layers are the parameters used to
describe the laminated composite structure by using linear layered
structural shell element SHELL99. The element allows up to 250 different
material layers with different orientations and orthotropic material
properties in each layer. It has six degrees of freedom at each node:
translations in the nodal x, y, and z directions and rotations about the
nodal x, y, and z-axes. Fig. 2, a shows the stacking sequence and
material orientation angles used in the long cylinder finite element
analysis. It can be found that the orientation angles and winding angles
are complementary.
The boundary conditions for the finite element analysis are fixed
along the both cylinder roots as shown in Fig. 2, b. An external
pressure of P = 1 is loaded on the external surfaces of composite
layers. Static analysis and buckling analysis are performed in sequence
in order to obtain the critical buckling pressure.
2.2. Probabilistic analysis
The probabilistic finite element analysis of composite long
cylinder is performed by mean of ANSYS/PDS. The PDS is based on the
ANSYS parametric design language, which allows users to parametrically
build a finite element model, solve it, obtain results and extract
characteristic results parameters such as the critical buckling pressure
for example. The PDS includes MCS and RSM. The MCS does not make any
simplification or assumptions in the deterministic of probabilistic
model, and the required number of simulations is not a function of the
number of input variables, whereas this method requires plenty of
computational time. The RSM replaces the true input-output relationship
of MCS by an approximation function, and the evaluation of the response
surface is much faster than a finite element solution. However, this
method is unusable when true input-output relationship is not continuous
[18].
In this work, both of MCS and RSM are used to execute the
probabilistic analysis of composite long cylinder. The mechanical
properties of composite material including longitudinal modulus,
transversal modulus, shear modulus and Poisson's ratio, and the
dimensions of cylinder including unsupported length, inside radius,
thickness and winding angles of inner and outer layers are taken as
random input parameters, and the critical buckling pressure is taken as
random output response. The statistical characteristics for them
including mean, coefficient of variation (COV) and distribution type are
given in Table 1. The uncertainties of mechanical properties and
physical dimensions are influenced significantly by the manufacturing of
composite materials and cylinders. In general, the statistical
characteristics are achieved based on the extensive data collection and
data analysis. However, in the absence of sufficient and good quality
data, professional expertise has to be employed. In this work, the
variables values estimated based on tests and engineering judgment are
used. For the winding angles, uniform distribution is assumed, and for
all the other random variables, normal distribution is assumed.
For the MCS, the Latin Hypercube Sampling is selected due to that
this technique avoids repeating samples that have been evaluated, and
also forces the tails of a distribution to participate in the sampling
process. In this work, 2000 Latin Hypercube loops are run which are
sufficient to obtain converged outputs. For the RSM, the central
composite design is used to locate the sampling points in the design
space. A total of 147 designs of experiment plus 10000 Monte Carlo
simulations are run in exploiting the response surface result. Since
each analysis iteration takes about 10 s using 2.5 GHz quad-core Intel
processor, the entire simulation requires about 5.5 h and 25 min for MCS
and RSM, respectively.
3. Experiment
According to the structural configuration and physical dimensions
shown in Fig. 1 and Table 1, four composite long cylinders with
different lengths were manufactured. All of the cylinders were made of
carbon-epoxy prepreg tape. Each ply thickness of the composite material
is 0.08 mm. The stacking sequence of [[[90.sub.4]/[0.sub.13]].sub.T] was
adopted. (Of course, some other winding angles were also selectable
[23].) In order to seal the cylinders from hydrostatic pressure, besides
the O-rings, the adhesive Aradite AW106 with Hardener HV953U were used
between the composite cylinders and the steel flanges. The measured
dimensions and errors of the manufactured composite long cylinders are
given in Table 2. The winding angles are difficult to measure, which are
not given in this paper.
The manufactured composite long cylinders are shown in Fig. 3, a.
The external hydrostatic pressure tests of them were carried out in a
hyperbaric testing chamber (Fig. 3, b) in Rongsheng Machinery
Manufacture Co., Ltd., Huabei oilfield, Hebei, China. A high pressure
pump was used to supply hydrostatic pressure. Each of the manufactured
long cylinders (No. 1, No. 2, No. 3 and No. 4) was submerged in water
for testing consecutively, starting from No. 1 and ending in No. 4. The
applied external pressure was increased by 0.01 MPa step by step, and in
each step the pressure maintained 5 seconds, till buckling and
subsequent collapse behaviours occurred.
[FIGURE 3 OMITTED]
4. Results and discussions
4.1. Comparison of analysis and experimental results
Fig. 4, a and b shows the amplificative buckling mode shape with
scale factor of 20 for the composite long cylinder No. 2 with
unsupported length of 550 mm according to the finite element method. It
is obvious that a two circumferential lobe mode is present. Similarly,
for the other three cylinders, two circumferential lobe modes are also
observed.
In general, when the buckling and subsequent collapse behaviours
occur, the applied pressure drops sharply. In this work, it was found
that it was different to distinguish buckling pressure and collapse
pressure; therefore, the two pressures are considered as uniform, that
is, the long cylinder collapsed immediately after the buckling behaviour
occurred. In addition, a louder sound was heard when the long cylinder
burst. The experimental failures of composite long cylinders are shown
in Fig. 4, c. It can be seen that longitudinal crack is the main failure
mode, and in each long cylinder there are two long fracture lines. The
experimental buckling mode shapes are similar to the predicted shapes
but are slightly different due to the collapse behaviour.
[FIGURE 4 OMITTED]
Table 3 gives the predicted and experimental critical buckling
(collapse) pressure. It can be seen that with the increasing of
unsupported length of composite long cylinders, the buckling pressure
decreases slightly, for both of the finite element analyses and
hydrostatic pressure tests. The predicted bucking pressures for the four
cylinders by using MCS and RSM are almost the same, which are a little
lower than the deterministic results. The deviations predicted by
deterministic and probabilistic finite element analyses are less than
10% in comparison with the experimental results. For the composite long
cylinder No.2, the error is even less than 1%. This shows that the
finite element analysis and experimental results show a good agreement.
4.2. Probabilistic analysis results
The cumulative distribution function of critical buckling pressure
with 95% confidence limit for composite long cylinder No.2 by using RSM
is shown in Fig. 5, a. The value of cumulative distribution function at
each point states the probability that the related parameter lays under
the point. Therefore, when the external pressure is 1.78 MPa, the
probability of buckling is 49.79%, which indicates that the buckling
behaviour of composite long cylinder highly likely occurs.
Fig. 5, b shows the sensitivity of critical buckling pressure to
random input variables. It can be seen that the thickness of composite
layers, transversal modulus, inside radius and longitudinal modulus have
significant effects on the performance of composite long cylinders. The
four variables are responsible for more than 95% of the effect on the
failure probability, with the other five variables together making up
for the remaining parts. Therefore, the four parameters should be paid
more attention when designing composite long cylinders. They are
followed by shear modulus, Poisson's ratio, winding angle of outer
layers, winding angle of inner layers and unsupported length (T >
[E.sub.T] > R > [E.sub.L] > [G.sub.LT] > [v.sub.LT] >
[[alpha].sub.o] > [[alpha].sub.i] >L). For each simulation,
although the orders of sensitivity for some random input variables are
different, they have no significant influences on the critical buckling
pressure, which can be ignored.
In order to research the effects of COV (boundary) of random input
variables on the sensitivity of critical buckling pressure, computations
are made when each of the COV (boundary) is varied between [+ or -]40%
of the values given in Table 1. The results of composite long cylinder
No. 2 by using RSM are given in Fig. 6. It is obvious that the
rank-order correlation coefficients of thickness, transversal modulus,
inside radius and longitudinal modulus are high, which indicates that
the four variables have significant effects on the output response.
It can be seen from Fig. 6, a that with the increasing COV of
longitudinal modulus, the correlation coefficient of longitudinal
modulus increases rapidly, from 0.15 to 0.35. The correlation
coefficients of thickness, transversal modulus and inside radius
decrease slightly, whereas the correlation coefficients of other five
variables are almost changeless. Similar trends are observed for
transversal modulus, inside radius and thickness, as shown in Fig. 6, b,
f and g, respectively. For the other five variables, the variation of
COV almost has no effects on the correlation coefficients of all of the
input variables, as shown in Fig. 6, c, d, e, h and i. It is noted that
the critical buckling pressure should be strongly dependent on the
winding angel. However, in this work, the uncertainty of winding angle
which arises from the geometric imperfections during filament winding
process is not so large; therefore, the little change of winding angel
causes its little effect on the correlation coefficients, as shown in
Fig. 6, h and i.
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
5. Conclusions
The buckling behaviours of composite long cylinders subjected to
external hydrostatic pressure are researched by using deterministic and
probabilistic finite element analyses. The effects of uncertainties of
material properties and physical dimensions on the critical buckling
pressure are also researched. Four composite long cylinders with
different lengths are manufactured. The external hydrostatic pressure
tests performed in order to validate the finite element analysis
results.
1. The deterministic and probabilistic finite element analyses
predict the similar critical buckling pressure, which is a little higher
than the experimental results.
2. The probability of buckling predicted by using RSM is correct
according to the experimental critical buckling pressure.
3. The thickness of composite layers, transversal modulus, inside
radius and longitudinal modulus have significant effects on the
performance of composite long cylinders, whereas shear modulus,
Poisson's ratio, winding angle of outer layers, winding angle of
inner layers and unsupported length have small influences.
4. The buckling behaviours of finite element analyses by using
deterministic and probabilistic methods and hydrostatic pressure tests
indicate good matches.
Acknowledgment
The authors wish to acknowledge the financial support of the
National High-Technology Research and Development Program of China
(No.2007AA09A101), National Natural Science Foundation of China
(No.50874115) and Incubation Programme of Excellent Doctoral
Dissertation of China University of Petroleum (No.2010-02).
Received April 07, 2011
Accepted September 21, 2011
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Baoping Cai, College of Mechanical and Electronic Engineering,
China University of Petroleum, Dongying, Shandong, 257061, China,
E-mail: caibaoping987@163.com
Yonghong Liu, College of Mechanical and Electronic Engineering,
China University of Petroleum, Dongying, Shandong, 257061, China,
E-mail: liuyh@upc.edu.cn
Huazhou Li, Petroleum Systems Engineering, Faculty of Engineering
and Applied Science, University of Regina, Regina, Saskatchewan S4S0A2,
Canada, E-mail: li385@uregina.ca
Zengkai Liu, College of Mechanical and Electronic Engineering,
China University of Petroleum, Dongying, Shandong, 257061, China,
E-mail: liuzengk@163.com
Table 1
Statistical characteristic of dimensions and mechanical properties
Property Symbol Mean
Longitudinal modulus [E.sub.L], MPa 135000
Transversal modulus [E.sub.T], MPa 10000
Shear modulus [G.sub.LT], MPa 5000
Poisson's ratio [v.sub.LT] 0.3
Overall length [L.sub.0] mm 390/580/680/780
Unsupported length L, mm 360/550/650/750
Inside radius R, mm 21.5
Thickness T, mm 1.36
Winding angle of [[alpha].sub.i], 90
inner layers [omicron]
Winding angle of [[alpha].sub.0], 0
outer layers [omicron]
Property COV/Boundary Distribution
type
Longitudinal modulus 0.10 Normal
Transversal modulus 0.06 Normal
Shear modulus 0.08 Normal
Poisson's ratio 0.08 Normal
Overall length -- --
Unsupported length 0.01 Normal
Inside radius 0.01 Normal
Thickness 0.02 Normal
Winding angle of [+ or -] 3 Uniform
inner layers
Winding angle of [+ or -] 3 Uniform
outer layers
Table 2
Dimensions of the manufactured composite long cylinders
Cylinder Overall Error, Unsupported Error,
length % length %
[L.sub.0], mm L, mm
No. 1 392.01 0.52 361.24 0.34
No. 2 580.85 0.15 551.75 0.32
No. 3 678.24 -0.26 649.52 -0.07
No. 4 779.51 -0.06 752.02 0.27
Cylinder Inside Error, Thickness Error,
radius % T, mm %
R, mm
No. 1 21.48 -0.09 1.34 -1.47
No. 2 21.78 1.30 1.38 1.47
No. 3 21.85 1.63 1.35 -0.74
No. 4 21.32 -0.84 1.39 2.21
Table 3
Predicted and experimental critical buckling pressure of
composite long cylinders
Cylinder Experimental Deterministic Error,
Pcr, MPa Pcr, MPa %
No. 1 1.81 1.906 5.30
No. 2 1.78 1.787 0.39
No. 3 1.69 1.775 5.03
No. 4 1.61 1.770 9.94
Cylinder MCS Error, RSM Error,
Pcr, MPa % Pcr, MPa %
No. 1 1.9014 5.05 1.9011 5.03
No. 2 1.7850 0.28 1.7846 0.26
No. 3 1.7735 4.94 1.7734 4.93
No. 4 1.7660 9.69 1.7668 9.74