Reliability-simulation of composite tubular structure by finites elements method–Mechanical loading
A.MAIZIA(1,a), A.HOCINE(1,b), H.DEHMOUS(2,c) 1 : Laboratoire Contrôles, Essais, Mesures et Simulations Mécaniques- Université Hassiba Benbouali de Chlef.BP151, Chlef 02000, Algérie.
2 : Département de Génie Civil, Université Mouloud Mammeri, Campus Hasnaoua, Tizi-Ouzou CP 15 000, Algérie.
D. CHAPELLE(3,d) ,
3 : Département de Mécanique appliquée, FEMTO ST, Université de Franche comté, Besançon, France.
Email :(a)maiziamaster2012@gmail.com,(b)hocineaek.dz@
gmail.com, (c)hocine_dehmous@ummto.dz, (d) david.chapelle@univ fcomte.fr.
Abstract— The diversified use of filamentary composites in harsh marine environments, recorded in recent years has prompted researchers to focus their work on the reliability prediction. Owing to differences between the properties of the materials used for the composite, the manufacturing processes, the load combinations and types of marine environment, the prediction of the reliability of composite materials has become a primary task. Through failure criteria, TSAI-WU and the maximum stress, the reliability of multilayer tubular structures under mechanical loading is the subject of this first part of our research project, where Monte Carlo method estimated the failure probability. A sensitivity analysis was performed in order to identify the influence of the deferent parameters as : materials properties, geometry, manufacturing and loading, on the reliability of the composite cylindrical structure studied. To achieve a high accuracy of the results, we have carried out 104 simulations. The results showed great influence of radius ratio (Ply thickness); than internal pressure loading and finally winding angle.
Keywords— Multilayer Tubular Structures; Marine structure;
Design; Reliability; Monte Carlo.
I. INTRODUCTION
The use of composite materials has become a current practice in gas transport and storage structures in harsh marine environments due to their lightweight, relative low cost and mainly their high strength over metallic materials. Because of the serious environmental loading exciting the marine structures, not only traditional static and dynamic analyses are required, but also the structure reliability design should be considered.
Nevertheless, the literature shows that, there are many design variables for the structural design of composite structures. These variables are induced first by the variations of material properties, which is made of two types of materials: matrix and fiber reinforcement. Due to their naturally anisotropic material behavior, composite material have different mechanical properties. Generally, the composite materials are characterized by a high variability of the mechanical and geometrical properties due to the heterogeneous composition of the components, the load
condition and finally the effects of the uncertainties in the manufacturing process according to the winding time, fiber tension, the winding pattern, the delivery system and the curing cycle, which can improve up to 30%, the performance of composite structure by optimizing this process factors.
Most researchers conducted deterministic studies to analyze the composites cylindrical structures behavior under various loads [1-4]. However, some of them used statistical models to describe the failure process, but neglect the randomness in mechanical properties, winding angle, loading and geometrical parameters. For the evaluation of the composite structure reliability, the probability concepts and methods must be used, in order to get an optimal design of cylindrical composite structures.
In this context, Khelif el al. [5] studied the probabilistic characterization of the lifetime of high-density polyethylene pipes, in order to assess their reliability levels under operating conditions. In this level, Zhou [6] considered the randomness of the design parameters and the fuzzy failure areas in the reliability assessment of pressure piping containing circumferential defects. Compared with the conventional reliability estimation method, which neglects the existence of fuzzy failure areas, the method proposed in this paper provides a more complete assessment of pressure piping containing defects. Amirat et al [7] characterized experimentally the residual stress distribution in large diameter pipes in order to be coupled with probabilistic corrosion model. Bouhafs et al.
present an analytical probabilistic model for the stress analysis of multilayered –wound composite pipes [8].
The present paper based on developed work of Hocine and al [1] which studies, first the analytical and numerical mechanical response of multilayered cylindrical composite structures under mechanical loading and in the second step, reliability analysis of the sensitivity of the cylindrical composite material structures uncertainties, geometric and loading by analytical method. The sizing tool developed by Hocine and al [9] was coupled with reliability model based on Monte Carlo method. The developed analytical work has identified the main parameters, which influence the fracture of the composite structure. The reliability analysis shows the
very high increase of the probability of failure when the winding angle, thickness of layers or radius ratio Rap; and internal pressure are considered.
To identify the significant parameters on the reliability of the multilayer tubular structure, the present part I focuses on the estimation of failure probability of marine composite structure under mechanical loading by finites elements method. In order to attain this objective, two criteria are used Tsai-Wu and maximum stress. The estimation of the probability of failure is possible to identify the most influential parameters.
II. APPROACH AND FORMULATION RELIABILITY ENGINEER The theory of reliability is based on a probabilistic approach to structural safety. It aims to assess the probability of failure of the structure; knowing the structure of a limit state criterion and the variability of the parameters involved in this test. The probability of failure is defined as the probability that this criterion is exceeded. The structure is finally considered safe if the probability of failure is less than a reference value called the acceptable failure probability of [9].
To achieve our aim, we propose a reliability-mechanical study represented by the Probabilistic Design System (PDS) and an FE model implanted under the commercial code ANSYS "APDL." The following diagram (Fig.1) shows the approach of our reliability-mechanical model.
Fig. 1. Reliability-mechanical analysis steps.
The present work takes place in two main steps. Initially, the FE model is realized. Finally probabilistic modeling with Monte Carlo method is implemented to calculate the probability of failure p of the studied structure. This f probability is defined by the integral (1) as a function of a random variable X vector, which represents the uncertainty in model [5]:
0 ) (
) (
X G
x
f f X dX
p
(1)
(1)
With:
) (X
fx : Distribution function associated.
G : Performance function.
0 ) (Xi
G : Boundary domain surface between the failure domain
(
G(Xi)0)
and the security field(
G(Xi)0)
as shown in Fig.2.Fig. 2. Security and failure areas for two random variables
Xi X1, X2
[7].0 ) (Xi
G : Safe domain
0 ) (Xi
G : Failure domain
The safety area G
Xi corresponds to the difference between the limit transverse rupture Y0 and hoop stress .
Its expression is formulated by the following equation
:
X Y0G i
(2) The Monte Carlo simulation method allows to carry out a random selection of N samples X vectors to estimate the function limit state
G.
Then, we calculate the probability of failure pf,
which is defined as the ratio between the number of random samples where G(Xi)0 and the selection total number.totale 0 G
f N
p N (3)
To estimate the failure probability pf
,
it’s assumed that the tubular structure remains reliable until complete rupture of the inner layer. It is evaluated for the inner and outer layers by two failures criterions TSAI-WU and maximum stress.A. Tsai-Wu Criterion
The reliability index Tsai-Wu
INTW
is defined by equation (4) :1 F
F
INTW
i
i
ij
i
j
(4) The Fi,
Fij parameters with (i and j=1, 2, 3) are given by :13 12
3 2
23 2 1
XY 66
33 22
11
F XtXcYtYc
1 2 F 1 Yc F
1 Yt F 1
YtYc 2 F 1 Xc
1 Xt F 1 S F 1
YcYt F F 1 XcXt F 1
;
;
;
;
;
(5)
We note that:
Xc
,
Xt:
Longitudinal Limits fracture in compression and traction.Yc
,
Yt:
Transverse limits fracture in compression and traction.SXY: Shear limit.
B. Maximum Stress Criterion
For this criterion, the structure remains reliable, provided that the following inequalities are verified.
1
XY Zt Yc Yt Xc Xt
MAX INMS
Z Z r r
/ / / / / /
(6)
With :
INMS: Reliability index of maximum stress criterion.
In the framework of our study, the choice of the hoop stress for this criterion is dictated by the axial and hoop stress ratio induced by loading, or:
2
zz
(7) On this basis, the reliability of your multilayer structure is evaluated with respect to hoop stress size. Hence the reliability index is expressed as follows :
1 Yc
INCM /
(8)
The probabilistic part was carried out under the commercial code ANSYS 15.0. This model allows us to couple the finite element method (EF), and probabilistic Monte Carlo method Direct. The first method was used to calculate the stress distribution in the cylindrical coordinate system for each layer and the second helped us to make random variables selection in order to evaluate the two limit state functions (G1 and G2) at the same time . These two functions are expressed by (9).
INCM 1 X 2 G
INTW 1 X 1 G
i i
) (
)
(
(9)
III. FINITE ELEMENT MODEL
All finite element simulations have been carried under ANSYS software. In order to define the variation of the stresses through a thickness, a bidimensionnel multilayered finite element Shell 99 with 8 nodes (Fig.3) was selected for the structure meshing process. A uniform mesh characterizes this simulation with 3648 nodes and 1200 elements.
Fig. 3. Multilayered Shell 99 element .
The current study takes two stacking configurations of 8 composite layers according to Table I with a 0.27 mm thickness for each carbon /epoxy layer. The choice of 55 ° angle is based on historical experimental and numerical research, which confirmed that this kind of stacking sequence [± 55] allowed maximizing the static rigidity of composite cylinders under internal or external pressure [11]. Additionally the aim of the presence of the circumferential windings 90 is to reduce expansion of the structure in this direction.The material properties of the material are shown in Table II.
TABLE I. STACKING SEQUENCES. Reference Sequence
Seq1
50
4Seq2
50
3
90 2TABLE II. CARBON/EPOXY PROPERTIES
Ex [GPa]
Ey [GPa]
Gxy [GPa]
yx Xt [MPa]
151 11 4 0.3 1500
Xc [MPa]
Yc [MPa]
Yt [MPa]
SXY [MPa]
1500 250 50 70
IV. DETERMINISTIC RESULTS A. Results Validation
In order to validate the numerical model developed during this work and extend it to reliability analysis to estimate the failure probability, it is essential to validate by an analytical approach developed by HOCINE et al. Table III presents a comparison of the results of the process carried out under ANSYS 15 and those of the two sequences for analytical stacking in terms of stress and radial displacement in the inner and outer cylindrical walls of the multilayer structure.
TABLE III. NUMERICAL RESULTS VALIDATION
Table III shows a good agreement between the numerical results of our model (EF) that was implanted under the commercial code ANSYS 15.0 with the analytical model in MATLAB [1], where the gap varies between 0 and 6.2%.
V. PROBABILISTIC RESULTS
The objective of this part is to find the model parameters that can cause failure of a multilayered pipe through a sensitivity analysis. In addition, the objective is to estimate the failure probability Pf, to assess the reliability of the study structure.
A. Probability model parameters
To develop a reliability analysis, it is necessary to take account of the uncertainties of the different parameters that can govern the multilayer structures behavior. These uncertainties are represented statically by a kind of distribution: Normal, Gaussian, log normal, Uniform ... etc.
The literature contains may be used for indication of the type of distribution. According to [12] can be selected laws normal, Log normal or Gaussian. In order to consolidate our reliability
analysis, the estimation of failure probability according to incremental pressure loading is performed for two types of distribution : Gaussian and Log normal.
TABLE IV. EFFECT TYPE OF DISTRIBUTION ON FAILURE PROBABILITY FOR A TYPE OF SEQ1
50
4.Pression
Interne (MPa) 10 12 14 16 18 20
Pf1TWln 0.50 0.76 0.90 0.95 0.97 0.98 Pf1TWg 0.49 0,76 0.90 0.96 0.98 99 Pf1CMln 0,009 0,067 0,2207 0,43 0,65 0,81
Pf1CMg 0,012 0,072 0,22 0,44 0,65 0,81 Pf8TWln 0,16 0,42 0,66 0,82 0,90 0,94 Pf8TWg 0,16 0,41 0,65 0,81 0,90 0,94 Pf8CMln 0,0027 0,03 0,12 0,29 0,49 0,68 Pf8CMg 0,005 0,03 0,12 0,29 0,49 0,67
Table IV shows that the results of failure probability at the extreme layers of pressure vessel, obtained through the criterion TSAI-WU, for the two Gaussian distributions and normal Log at the inner wall are almost identical. This situation is similar compared to the results obtained through the criterion maximum stress. So, for the rest of the present analysis is performed by a Gaussian kind of distribution.
With :
Pf1 TWln: Layer 1 failure probability, using criterion of Tsai- Wu, with a log normal distribution.
Pf1CMln: Layer 1 failure probability, using maximum stress test, with a log normal distribution.
Pf1TWg: Layer 1 failure probability, using criterion of Tsai- Wu, with a Gaussian distribution.
Pf1CMg: Layer 1 of probability of failure, using maximum stress criterion, with a Gaussian distribution.
On 8 by analogy represents the last layer.
Furthermore the random variables are also defined by their coefficients of variance (CV%), which is limited to a value of 12% for all random parameters.
B. Sensitivity
A sensitivity analysis was performed in order to identify the influence of the deferent parameters such as : materials properties, geometry, manufacturing and loading, on the reliability of the composite cylindrical structure studied.
The results (Fig. 4 and 5), showed great influence of Rap radius ratio; the applied load (Print) the orientation angle;
followed by the material parameters: Elastic
E
x and Limit fractureY
c in the hoop direction).Sequence Analytical
Model [1]
Numerical
Model Gap (%)
50
4
MPa
Inside wall 161.7 151.67 6.2
Outside wall 145 137.95 4.8
MPa
r Inside wall -9.98 -9.87 1.1
Outside wall -0.015 -0.012 2
mm
Ur Inside wall 0.1718 0.168 2.2
Outside wall 0.1727 0.168 2.7
50
3
902
MPa
Inside wall 90 86.55 3.8
Outside wall 350 349 0.28
MPa
r Inside wall -9.98 -9.95 0.30
Outside wall 0 0 0
mm
Ur Inside wall 0.0838 0.0809 3.4
Outside wall 0.0833 0.0809 2.8
Fig. 4. Maximal stress sensitivity for Seq1 ;(PRINT=10 MPa, RAP=1.0818).
Fig.5. Tsai-Wu criterion sensitivity for Seq2;(PRINT=10 MPa, RAP=1.0818).
The sensitivity approach carried out in this work was able to identify the most influential parameters on the random failure probability and which can be summarized as:
1) Load pressure;
2) Thickness of layers or radius ratio Rap;
3) Orientation angle;
4) Coefficient of variation CV;
C. Failure Probability
The main objective of fiabilistes study is to estimate the failure probability (Pf). So we're looking in this part of valuing the effects of two parameters (Internal Pressure and Ply Thickness), on the variation of (Pf) .
C.1 Internal Pressure Loading Effect
The results (Figures 6 and 7) show a great influence of the load applied on the change (Pf). Moreover, the inner wall is most susceptible to any failure compared to the outer wall per the reliability index Tsai-Wu for Seq1. By cons for the second sequence (Seq2), we note that the external layer cracks before the inner layer.
It is important to note that the estimated default probability by the criterion of Tsai-Wu differs compared with that of the criterion of the maximum stress. Relative to deterministic computation, the reliability engineer approach shows that the
structure always ensures the rigidity although the pressure is around 30 MPa.
Fig. 6 Internal pressure effect on failure probability
P
fby Tsai-Wu criterion ( CV% =12, RAP=1.0818).
Fig. 7. Internal pressure effect on failure probability Pf by Maximum stress criterion ( CV% =12, RAP=1.0818).
C.2 Effect of Ply Thickness
The shapes represented by Fig. 8, indicate that the failure probability is very sensitive to the ply thickness uncertain and particularly those of the inner wall. Any error of increase on thickness ply reduces the failure probability and every mistake on thickness reduction affect the structure safety. To prevent failure of a multilayer pressure pipe, it is important to ensure optimal dimensioning in terms of radius ratio.
Fig.8 Effect of ply thickness for Seq1 on failure probability ( CV% =12, PRINT=14 Mpa ).
VI. CONCLUSION
This paper focuses, by way of the finite element method, on the estimation of failure probability for a multilayer composite tubular structure under internal pressure for two stacking sequence Seq1=
50
4and Seq2=
50
3
90 2.
A sensitivity analysis was performed in order to identify the influence of the deferent parameters such as: materials properties, geometry, manufacturing and loading, on the reliability of the composite cylindrical structure studied. The results showed great influence of Rap radius ratio (ply thickness); the internal pressure loading and winding angle.
The results obtained clearly show that the rigidity of the structure is very sensitive to random variables loading, thickness and a little less to the winding angle. Any uncertainty can lead to failure of the layered composite structure.
According to the results of the probability of failure Pf obtained, we can conclude that:
• The probabilistic study allows us to consider the uncertainties of the studied structure.
• Compared to the results of [8] the deterministic failure does not necessarily include a probabilistic failure.
•
We can see that the use of hoop windings at 90° to the outside of the tubular structure, improves the reliability of multilayer tubes.In our research project, the present composite structure is subjected to marine environmental conditions during the service life. Moisture and temperature have an adverse effect on the performance of composites. Stiffness and strength are reduced with the increase in moisture concentration and temperature. The previous part II of this research project is devoted to develop reliability-numerical model, which can take into account a different loading and marine environmental conditions. This work aims to provide a reliability-sizing tool dedicated to optimal design of tubular structures of gaseous conveying in marine environments.
References
[1] A.Hocine,D.Chapelle, L.Boubakar,A. Benamar, Bezazi A. Experimental and analytical investigation of the cylindrical part of a metallic vessel reinforced by filament winding while submitted to internal pressure. International Journal of Pressure Vessels and Piping 86 (2009) 649–655.
[2] A. Ghouaoula, A. Hocine, D. Chapelle, D. K. Achira, M.L.
Boubakar. Analytical damage prediction in the composite part of a type III hydrogen storage vessel. Mechanics of Composite Materials, Vol. 48, No. 1, March, 2012.
[3] J.M. Lifshitz, H. Dayan : Filament–wound pressure vessel with thick metal liner, Composite Structures, Vol 32, pp. 313-323, 1995.
[4] A. Hocine, F.K Achira, A. Ghouaouala. Coupled structural/thermal analysis of cylindrical part of multilayered composite vessel. Advanced Materials and Research Vol. 445 (2012) pp 589-594.
[5] R. Khelif, A. Chateauneuf, K. Chaoui. Reliability-based assessment of polyethylene pipe creep lifetime. International Journal of Pressure Vessels and Piping 84 (2007) 697–707.
[6] J . Zhou. Reliability assessment method for pressure piping containing circumferential defects based on fuzzy probability. International Journal of Pressure Vessels and Piping 2005;82:669-78
[7] A . Amirat, M.A. Chateauneuf, K. Chaoui. Reliability assessment of underground pipelines under the combined effect of active corrosion and residual stress. International Journal of Pressure Vessels and Piping 2006;83:107-17.
[8] M. Bouhafsa, Z. Sereir, A. Chateauneuf. Probabilistic analysis of the mechanical response of thick composite pipes under internal pressure.
International journal of pressure vessel and piping 95 (2002) 7-15.
[9] A. Hocine, M. Chérifi, A. Maizia, D. Chapelle. A contribution of the reliability based approach for cylindrical composite of sensitivity analytical design. AMPT 2015 Madrid, Spain 14- 17th December 2015.
[10] M.W.K. Rosenow. Winding angle effects in glass fiber-reinforced polyester filament wound pipes, Composites, Vol. 15, pp. 144-52, 1984.
[11] S.W. Tsai : Composite design, Think composite, 4th edition, Dayton, 1988.
[12] D.M. Frangopol,Se. Recek, “ Reliability of fiber-reinforced composite laminate plates”, Probabilistic Engineering Mechanics 18 (2003) 119–
137.
