th
8 ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability
PMC2000-187
RELIABILITY-BASED OPTIMIZATION OF THIN COMPOSITE PIPE N. Kogiso, and S. Katou, Osaka Prefecture University, Sakai, Osaka, JAPAN 599-8531
[email protected],
[email protected] Y. Murotsu Osaka Prefectural College of Technology, Neyagawa, Osaka, JAPAN 572-8572
[email protected] Abstract This study is concerned with the reliability-based optimization of a thin laminated composite pipe subject to bending, tension and torsion loadings. The total volume of the pipe is minimized in terms of each ply thickness subject to mode reliability constraints. The reliability is evaluated by modeling the pipe as a series system consisting of each ply failure and an elastic deformation. The reliability of a ply failure is evaluated based on the Tsai-Wu criterion. The reliability of the elastic deformation is evaluated under the assumption that the pipe will fail when a tip displacement is beyond an allowable limit. The mode reliabilities are obtained by the first order reliability method (FORM), where the material properties and the applied loads has probabilistic variations. Through numerical calculations, the effect of variations of the random variables is discussed. Also, the importance of considering the structural reliability in designing the laminated composite structure is demonstrated.
Introduction The laminated composite material is widely used in structural applications because of its high specific strength and stiffness. It is also known as a tailored material whose properties can be designed by selecting the ply orientation angles or the ply thicknesses as design variables. Hence, a lot of studies are conducted on the optimum design of a composite laminated structure (Güradal et al., 1999). However, such an optimum design is strongly anisotropic and sensitive to the change in the loading conditions. Therefore, it is necessary to consider the effect of such variations by applying the structural reliability theory (Thoft-Christensen and Murotsu, 1986). For the composite plate under the in-plane stress, the reliability-based design subject to the first-ply failure criterion is clarified to approach a quasi-isotropic laminate construction (Shao et al., 1993; Murotsu et al., 1994). For the case of considering the buckling failure, the reliability reaches maximum when the mode reliabilities are well balanced. It is a good contrast to the deterministic optimum design which has a duplicated buckling mode (Kogiso et al., 1998). However, the study on the reliability of the actual composite laminated structure is still in progress (Liu and Mahadevan, 1998). In this study, the reliability-based optimization of the composite laminated pipe subject to bending, tension and torsion loadings is addressed. The ply failure is evaluated by the Tsai-Wu criterion (Tsai, 1988). The mode reliabilities are evaluated by the FORM, where the applied loads and the material properties have probabilistic variations. Additionally, the reliability of the elastic deformation is evaluated under the assumption that the pipe
Kogiso, Katou and Murotsu
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will fail when a tip displacement is beyond an allowable limit. The reliability is evaluated by modeling the pipe as a series system consisting of each ply failure and the elastic tip deformation. Then, the total volume is minimized in terms of the ply thickness under the mode reliability constraints. Through numerical calculations, it is demonstrated that the reliability-based optimum design is different from the deterministic optimum design. Also, the effect of the variation of the material properties and the applied loads on the reliability is clarified. Deterministic Strength Analysis The cantilevered laminated composite pipe with the length L, the radial thickness h, and the radius at the root and the tip, Rroot, and Rtip, respectively, is shown in Figure 1. The axial force F, the transversal flexural force P, and the torsion torque T is applied at the tip. Under the plane stress condition, the stresses at the root and the tip are obtained as follows (Miki et al., 1993):
σ 1root = − σ 1tip =
F A
PLRroot Iz
cos α +
F A
, σ 2root = 0 ,
σ 2tip = 0 ,
,
σ 6root = σ 6tip =
TRroot Ip
TRtip Ip
P
+
πRroot
sin α , (1)
+
P
πRtip
sin α ,
where A, I z, and I p are the cross sectional area, the moment of inertia, and the polar moment of inertia, respectively. α denotes a cylindrical coordinate on the cross section as illustrated in Figure 1. The ply failure is evaluated by the Tsai-Wu criterion (Tsai, 1988) as follows:
1 − [ Fxxσ x + 2 Fxu σ xσ y + Fyyσ y + Fssσ s 2
2
2
> 0 safety + Fxσ x + Fy σ y ]= 0 limit state < 0 failure
(2)
U2 L=1000[mm] 2 Rroot
1
α
T
y θ
0°
Failure
x Rtip
270° h
90° F P
u* h(U)=0
β
180°
Safety O Figure 1 Beam structural model
Kogiso, Katou and Murotsu
Limit State U1
Figure 2 Illustration of β point
2
T where (σ x, σ y, σ s) is the ply stress vector along the material principal axis, which is T obtained by the coordinate transformation from the stress vector (σ1, σ2, σ6) alo ng the beam axis. F (. ) are the strength parameters which are defined as follows:
Fxx =
1 Xt Xc
, Fyy =
1 Yt Yc
, Fss =
1 S2
, Fx =
1 Xt
−
1 Xc
, Fy =
1 Yt
−
1 Yc
, Fxy =
1 * Fxy ( Fxx Fyy ) 2
(3)
X and Y represents the strength along the fiber and the transverse directions, respectively. The subscript T and C mean the tensile and the compression sides, respectively. S is the shear strength. The normalized coupling term, Fxy* is assumed to be -0.5 (Tsai, 1988). The strength ratio of i-th ply ri(X) is defined as the ratio between the the ply failure stress T T vector σ f = (σ f x, σ f y , σ fs) and the applied stress vector σa = (σx, σy, σs) und er the proportional loading assumption, which is obtained by substitute σf =ri(X) σa int o Eq.(2). The ply strength ratio is evaluated as the minimum of those at several critical points; the top and the bottom of the cross section at the root and the tip. In this study, the first ply failure criterion is adopted. Thus, the pipe will fail when the minimum of the ply strength ratios is less than unity. Reliability Analysis The mode reliability is evaluated by FORM, where the material properties and the applied loads have probabilistic variations (Thoft-Christensen and Murotsu, 1986). The ply failure at the bottom and the top in the root and the tip of the pipe are considered as the failure mode. Additionally, the elastic failure mode is introduced, which is defined under the assumption that the pipe will fail when a tip displacement ω(X) is beyond an allowable limit, ω a. The limit state functions are defined as follows: g i ( X ) = ri ( X ) − 1 = 0 , g ω ( X ) = ω a − ω ( X ) = 0
(4)
The random vector X is transformed into the standardized normal distribution vector U. Then, the minimum distance from the origin to the limit state surface gi(X)=hi(U)=0 (see Figure 2) in U-space is obtained as a reliability index β which is the solution to a following nonlinear programming problem: Minimize : β i = ui ui , T
subject to : hi (u) = 0.
(5)
Reliability-Based Optimization The total volume of the composite laminated pipe made up of 0°, ±45°, and 90° plies is minimized in terms of each ply thickness and the tip radius subject to the mode reliability
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constraints. The pipe is modeled as the series system consisting of each ply failure and the elastic deformation failure. Therefore, the design problem is formulated as follows: Minimize : π ( Rroot + Rtip )(h0 + h90 + h± 45 ) subject to : β ij ( x,u) ≥ β a ,
(6)
β ω ( x,u) ≥ β a′ , h0 , h90 , h± 45 ≥ 0,
0 ≤ Rtip ≤ Rroot
where x=(h0, h90, h±45, Rtip)T is the design vector consisting of the ply thickness and the tip radius and u denotes random vector. The first and the second constraints are concerned with the reliabilities of the ply failure and the tip deformation, respectively. The subscript i (i=1,...,4) denotes the position along the radius and j denotes the plies (j = 0°, 90°, +45°, -45°). β a and β a ' are the specified reliability limits of each failure mode. The sequential quadratic programming (SQP) method is used to search for the optimum solution (Ibaraki and Fukushima, 1991). Numerical Example The total volume of the (0°, ±45°, 90°) pipe with L=1000(mm) and Rroot=20(mm) made up of Graphite/Epoxy (T300/5208) is minimized subject to the reliability constraints of β a=β a ' = 3.0. The material properties are assumed to have normal distribution, as listed in Table 1. The applied load is also assumed to have normal distribution, as listed in Table 2. At first, consider a straight pipe whose tip radius is fixed at Rtip = Rroot. The change of the reliability-based and the deterministic optimum ply thicknesses in terms of the allowable tip displacement are compared in Figure 3. The symbol “ ∞ ” at the right end indicates that the elastic failure mode is not considered. The 90°-ply thickness is not plotted because the ply thicknesses are vanished in all of the cases. As the allowable tip displacement is smaller, the reliability-based design requires more 0°-ply and less ±45°Random variables Xt Xc Strength Yt Yc S Ex Elasticity Ey Es ν
Mean 1500 [Mpa] 1500 [Mpa] 40 [Mpa] 246 [Mpa] 68 [Mpa] 181 [Gpa] 10.3 [Gpa] 7.17 [Gpa] 0.28
COV 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Table 1 Material properties of T300/5208.
Kogiso, Katou and Murotsu
Components Tensile F Bending P Torque T
Mean COV 8000 [N] 0.2 150 [N] 0.2 150 [Nm] 0.2
Table 2 Applied load.
4
2.4
0ﾟ (reliability) ±45ﾟ (reliability)
1.2
0.6
Tip radius
2 1.5 1.6 1.2
1
Volume 0.8
h0 0.4
0
60 ∞
20 40 Allowable tip displacement [mm]
0
Fig.3 Change of reliability-based design of the straight pipe in terms of the allowable tip displacement.
Mode Root Tip Deform.
UEx 0.132 0.232 -2.001
UEy 0.950 0.894 -0.004
0.5
h45
0
0
2
Tip radius [cm]
0ﾟ (deterministic) ±45ﾟ (deterministic)
3] Thickness[mm] Volume[cm
Ply thickness [mm]
1.8
0 ∞ 50 100 Allowable tip displacement [mm]
Fig.4 Change of reliability-based design of the tapered pipe in terms of the allowable tip displacement.
UEs
U Yt
UP
UT
-1.359 -1.411 -0.004
-1.150 -1.071 0.0
0.099 0.082 2.282
2.063 2.063 0.0
Table 3 Design points of critical failure modes of –45° ply of the reliability-based optimum tapered pipe at the allowable limits; wa = 10 mm.
ply than the deterministic design. As the next example, consider a tapered pipe, where the tip radius Rtip is included in the design variables. The change of the reliability-based optimum ply thicknesses in terms of the allowable tip displacement is shown in Figure 4. The reliability-based design requires more 0° and less ±45° plies, as the elastic condition becomes severe. This is the same tendency as the straight pipe. The reason of the tendency is investigated from the design points. Coordinates of the design points in some critical failure modes of the reliability-based optimum tapered pipe under ω a =10(mm) are shown in Table 3. The first two rows indicate the design points of –45°-ply failure mode of the tension side at the root and the tip. It is seen that the variations of the shear modulus, the longitudinal tensile strength and the applied torque are important. The third row indicates that of the elastic deformation failure mode. The variations of Young’s modulus in fiber direction and the applied bending force have large effects on the reliability of the elastic failure mode.
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Conclusions In this study, the reliability-based optimum design of the composite laminated pipe subject to the bending, tension and torsion loads is obtained by evaluating the mode reliability by the FORM, where material properties and the applied loads have probabilistic variations. Through numerical calculations, the following conclusions are remarked. 1. The deterministic and the reliability-based optimum design are not so different from each other, when only the ply failure is considered. 2. However, the reliability-based design is different from the deterministic design, when the elastic deformation failure mode is considered. 3. It is clarified that variations of Young’s modulus in fiber direction and the bending load have a large effect on the elastic failure mode. On the other hand, the variations of the shear modulus, the longitudinal tensile strength and the applied torque have large effect on the ply failure mode. The future work is to develop the efficient method to design an actual composite laminated pipe structure considering the reliability. References Gürdal, Z., R.T. Haftka, and P. Hajela (1999), Design Optimization of Laminated Composite Materials, John Wiley & Sons, New York. Ibaraki, T., and M. Fukushima (1991), FORTRAN 77 Optimization Programming, Iwanami-Shoten, Tokyo (in Japanese). Kogiso, N., S. Shao, and Y. Murotsu (1997), “Reliability-Based Optimum Design of a Symmetric Laminated Plate Subject to Buckling,” Structural Optimization, 14, 184-192. Liu, X., and S. Mahadevan (1998), “System Reliability-Based Optimization of Composite Structures,” Proc. th of AIAA 7 Symposium on Multidisciplinary Analysis and Optimization, 856-860, AIAA-98-4814. Miki, M., Y. Murotsu, M. Onuki, and T. Yamaguchi (1993), “Passive Deformation Control of Structures with Structurally-Unsymmetric Laminate Configuration Pipes,” Proc. of AIAA Structures, Structural Dynamics and Materials Conference, 1790-1795, AIAA-93-1514. Murotsu, Y., M. Miki, and S. Shao (1994), “Reliability Design of Fiber Reinforced Composites,” Structural Safety, 15, 35-49. Shao, S., Y. Murotsu, and M. Miki (1993), “Optimum Fiber Orientation Angle of Multiaxially Laminated Composites Based on Reliability,” AIAA J., 31, 919-920. Thoft-Christensen, P., and Y. Murotsu (1986), Application of Structural Systems Reliability Theory, Springer-Verlag, Berlin. Tsai S.W. (1988), Composites Design, 4th edition, Think Composites, Dayton.
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