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Fuzzy variational principle and its applications
European Journal of Mechanics A/Solids 21 (2002) 999–1018
Fuzzy variational principle and its applications L.F. Yang a , Q.S. Li b,∗ , A.Y.T. Leung b , Y.L. Zhao a , G.Q. Li c a Department of Civil Engineering, Guangxi University, Nanning, 530004 Guangxi, China b Department of Building and Construction, City University of Hong Kong, 83 Tat Chee avenue, Hong Kong c Department of Civil Engineering, Wuhan University of Technology, Wuhan, 430070 Hubei, China
Received 2 April 2001; revised and accepted 21 August 2002
Abstract Linear and non-linear peaky fuzzy numbers and their arithmetic operations are constructed for the analysis of engineering structures with fuzzy characteristic quantities. Fuzziness of the corresponding quantities is consistently incorporated into the functional of the total potential energy. A set of deterministic recursive equations is obtained as the alternative expressions of the fuzzy variational principle by means of the second-order perturbation technique. The fuzzy Ritz method and the fuzzy finite element method are presented as the applications of the fuzzy variational principle. Accordingly, the roundabout procedures frequently used in the formulations of the fuzzy finite element method are avoided. A benchmark problem of a bending beam with fuzzy Young’s modulus under fuzzy external loading is solved by the developed fuzzy numerical methods. Numerical examples show that results determined by these two fuzzy methods are both little conservative, and are in good agreement with those obtained by the analytical method. Moreover, the fuzzy Ritz method or the fuzzy finite element method can provide more valuable information than the conventional deterministic methods. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Fuzzy; Variational principle; Ritz method; Finite element method
1. Introduction Numerical methods are more frequently used than analytical methods in structural design and analysis (Zienkiewicz, 1978). Among a variety of numerical methods, the finite element method is by far the most effectively and widely used method. One of its important theoretic bases is the variational principle (Qian, 1985; Lanczos, 1964). Research works on deterministic variational principles are extensive and mature (Zienkiewicz, 1978; Qian, 1985; Lanczos, 1964), while in the field of uncertain variational principles, researchers focused their attentions on stochastic variational principles (Matthies and Bucher, 1999; Liu et al., 1988; Hien and Kleiber, 1990; de Lima and Ebecken, 2000; Yang et al., 1999; Elishakoff et al., 1996), though the relevant literatures have been fairly fewer so far. However, research work on fuzzy variational principles with application in structural analysis, to the best of our knowledge, has not been publicly reported (Yang, 1998). On the other hand, real engineering structures are usually complicated systems, which possess various uncertain quantities. Sometimes the uncertainty plays significant roles in the analysis and design of engineering structures. There are several categories of non-deterministic theories and methods: the theory of probability and random process; the fuzzy-set-based methods; the gray model theory; and the theory of convex modeling of uncertainty, etc. The theories of probability and random process are long-standing powerful tools for dealing with uncertain systems, but the usage of these theories is fairly cumbersome for situations where the uncertain quantities are of non-statistical distributions in nature or show very sophisticated random distributions (Ayyub et al., 1997). The fuzzy-set-based methods are regarded to be more appropriate when the uncertainties * Corresponding author.
E-mail address: [email protected] (Q.S. Li). 0997-7538/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 0 9 9 7 - 7 5 3 8 ( 0 2 ) 0 1 2 5 4 - 8
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of systems are due to vaguely defined system characteristics, imprecise or insufficient data (Cao and Sawyer, 1995). Then the subjective judgment and opinion of experts are crucial (Zhu, 1999) for treatment of these systems. These two theories become quite elaborate, and are the most frequently used methods in the analysis of structures with uncertain quantities. The gray model theory is proposed by Deng (1987) and has been extensively investigated (Wang, 1988). Elishakoff (1995) proposed the convex modeling of uncertainty. He had adequately demonstrated the effect of human error, as well as the influence of imprecise data, on the traditional theory of reliability. This paper incorporates the fuzziness of physical and geometrical quantities into the potential energy functional. Based on the second-order perturbation technique and the arithmetic operations constructed in this paper, the fuzzy variational principle (FVP) is proposed. The fuzzy Ritz method (FRM) and the fuzzy finite element method (FFEM) are presented as the applications of the FVP.
2. Definition and arithmetic operations of peaky fuzzy numbers Fuzzy quantities, which possess fuzziness, can be treated by means of the fuzzy number theory (Dubois and Prade, 1980). From the viewpoint of philosophy, objective things are deterministic, while the fuzziness belongs to the field of subjectivity. There may exist critical bifurcation points for any object during the process of change. When the quantitative change of an object accumulates and surpasses the critical point, the qualitative change occurs, then the objective thing A will transfer into B. This critical point, or say limit, is a criterion of demarcating different objective things. For example, a temperature of 100 ◦ C is a criterion state, which means that the liquid water may transfer into vapor. For real-life problem of engineering structures, the critical points are hardly or not necessary to be worked out exactly. Therefore this critical point is usually estimated approximately as an interval, in which the transformation from one state to another shows a progressive procedure rather than an abrupt one. In the fuzzy analysis of structures, this critical point is termed the real value of the fuzzy quantity, and the estimated interval represents the possible range of the fuzzy quantity about its real value. In the present paper the peaky fuzzy numbers are adopted. Fuzzy numbers are specific fuzzy sets pioneered by Zadeh (1965). To qualify as a peaky fuzzy number, a bounded closed fuzzy number A˜ must possess the following properties: (1) The left shape function, LA (x), of the fuzzy number A˜ are monotonic increasing and continuous from the right, while the right shape function, RA (x), are monotonic decreasing and continuous, and 0 LA (x), RA (x) < 1, as shown in Fig. 1. ˜ denoted by Ker A, ˜ there is only one element A0 . (2) Within the kernel set of the fuzzy number A, Then A˜ can be expressed as A˜ = A0 , LA (x), RA (x) .
(1)
˜ for x = A0 , attains its maximum value 1, as illustrated in Fig. 1. A0 is The membership function of the peaky fuzzy number, A, ˜ deterministic and called real value of the peaky fuzzy number A. At the point of its real value, A˜ can be decomposed as: A˜ = A0 + β˜A ,
(2)
˜ and: where β˜A is the perturbation quantity of A, β˜A = 0, LA x + A0 , RA x + A0 .
Fig. 1. A peaky fuzzy number A˜ and the governing parameters.
(3)
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
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Fig. 2. A linear peaky fuzzy number A˜ and its governing parameters.
In engineering practices, the left and right half shape functions, LA (x), RA (x), can be defined as linear or non-linear functions, sometimes the exponents may be preferable in some applications. A˜ is called a linear peaky fuzzy number when the linear shape functions are adopted, such that: 0 x − A + 1, A0 − a < x A0 , L (4a) LA (x) = aL 0, x A0 − aL , 0 A − x + 1, A0 x < A0 + a , R RA (x) = (4b) aR 0 0, x A + aR , ˜ They are termed left and right amplitudes of fluctuation, respectively, as where aL , aR represent the degree of fuzziness of A. 0 illustrated in Fig. 2. The real value, A , is usually estimated by experts according to their experience of engineering practices. ˜ denoted by Al and Ah , respectively, Similarly, the possible low value and the possible high value of the peaky fuzzy number A, are also estimated by experts. Then, aL , aR are given by aL = A0 − Al ,
(5a)
aR = Ah − A0 ,
(5b)
where
Al =
2Al − A0
if A0 < 2Al ,
0
if A0 2Al ,
Ah = 2Ah − A0 .
(6a) (6b)
It is evident that the establishment of fuzzy numbers combines the objectivity of precise mathematics with the treasurable experience of experts.
can be For binary fuzzy arithmetic operations ∗ ∈ {+, −, ×, ÷}, formulae of fuzzy operations on fuzzy numbers, A˜ and B, derived as follows, according to the fuzzy representation principle.
are considered as linear peaky fuzzy numbers, and A˜ is shown in Eqs. (1) and (4), while At first, A˜ and B 0
= B , LB (x), RB (x) , (7) B where LB (x) and RB (x) are similar in form to LA (x) and RA (x) in Eqs. (4), respectively, on condition that A and a in Eqs. (4) are replaced by B and b, respectively.
λ respectively, are crisp sets, which are given by:
denoted by A˜ λ and B For real number λ ∈ (0, 1], the λ-cut sets of A˜ and B, 0 A˜ λ = A + aL (λ − 1), A0 − aR (λ − 1) , (8a) B
λ = B 0 + bL (λ − 1), B 0 − bR (λ − 1) (8b) such that:
= A˜ + B,
one can obtain the λ-cut set of fuzzy number C
via Eqs. (8): For C 0
λ = A + B 0 + (aL + bL )(λ − 1), A0 + B 0 − (aR + bR )(λ − 1) . C
(9)
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obtained by moving A˜ along x direction. Fig. 3. Membership function of C
can be attained: Therefore, according to the fuzzy representation principle, C 0
= C , LC (x), RC (x) , C where
0 x − C + 1, C 0 − c x C 0 , L LC (x) = cL 0, x C 0 − cL , 0 C − x + 1, C 0 x C 0 + c , R RC (x) = cR 0 0, x C + cR ,
(10)
(11a)
(11b)
where C 0 = A0 + B 0 , cL = aL + bL , cR = aR + bR . Eqs. (10) and (11) show that the summation of linear peaky fuzzy numbers result in a linear peaky fuzzy number.
degenerates into a crisp number B, i.e., its left and right fuzzy amplitudes vanish: When the fuzzy number B
then
bL = bR = 0
(12)
A˜ + B = A0 + B, LA (x − B), RA (x − B) .
(13)
From this formula, one knows that the solution of a peaky fuzzy number A˜ plus a crisp number B is still a linear peaky fuzzy ˜ That is to say, the membership function of fuzzy number with real value C 0 = A0 + B, and has the same fuzzy amplitudes as A.
can be obtained by moving the membership function of A˜ a distance of B along x direction, as shown in Fig. 3. number C
= A˜ − B,
the λ-cut set of fuzzy number C
shows: (2) For C 0 0 0
λ = A − B + (aL + bR )(λ − 1), A − B 0 − (aR + bL )(λ − 1) . (14) C
can be obtained as follows: Therefore, C
= C 0 , LC (x), RC (x) , C where
(15)
0 x − C + 1, C 0 − c x C 0 , L LC (x) = cL 0, x C 0 − cL , 0 C − x + 1, C 0 x C 0 + c , R RC (x) = cR 0, x C 0 + cR , C 0 = A0 − B 0 ,
cL = aL + bR ,
cR = aR + bL .
(16a)
(16b) (16c)
from peaky fuzzy number A˜ will also result in a linear Eqs. (15) and (16) show that the subtraction of peaky fuzzy number B peaky fuzzy number.
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
= A˜ × B,
the sets of λ-cut of C
can be evaluated by: (3) For C L L
λ = C , C C for λ ∈ [0, 1], λ λ where
L Cλ = min A0 + aL (λ − 1) B 0 + bL (λ − 1) , A0 + aL (λ − 1) B 0 − bR (λ − 1) , 0 A − aR (λ − 1) B 0 + bL (λ − 1) , A0 − aR (λ − 1) B 0 − bR (λ − 1) , CλR = max A0 + aL (λ − 1) B 0 + bL (λ − 1) , A0 + aL (λ − 1) B 0 − bR (λ − 1) , 0 A − aR (λ − 1) B 0 + bL (λ − 1) , A0 − aR (λ − 1) B 0 − bR (λ − 1) .
1003
(17)
(18)
Because the multiplication of fuzzy numbers is more complicated than their addition or subtraction, only some special cases, which will be encountered in this paper, are discussed hereafter.
are positive linear peaky fuzzy numbers, then one has (a) If both A˜ and B R 0 (19a) Cλ = A − aR (λ − 1) B 0 − bR (λ − 1) , (19b) CλL = A0 + aL (λ − 1) B 0 + bL (λ − 1) . According to the fuzzy representation principle, the left and right half shape functions, LC (x) and RC (x), can be obtained:
= C 0 , LC (x), RC (x) , (20) C where
0 0 L − (bL A + aL B ) + 1, C 0 − c x C 0 , L LC (x) = 2aL bL 0, x C 0 − cL , 0 0 bR A + aR B − R + 1, C 0 x C 0 + c , R RC (x) = 2aR bR 0, x C 0 + cR , C 0 = A0 B 0 ,
where
cL = bL A0 + aL B 0 − aL bL ,
cR = bR A0 + aR B 0 + aR bR ,
L = aL B 0 + bL A0 2 − 4aL bL C 0 − x ,
= a B 0 + b A0 2 − 4a b C 0 − x . R R R R R
(20a)
(20b) (20c)
(21a) (21b)
is called non-linear peaky fuzzy number. It should be noted that, LC (x) and RC (x) are non-linear functions of x, therefore, C
is a crisp number, denoted by B, then we have Especially, when B x x 0 × B, L A , R for B > 0, A A B B
= A˜ × B = (22) C x x A0 × B, RA , LA for B < 0. B B So the solution of A˜ × B is also a linear peaky fuzzy number with real value A0 B, left fuzzy amplitude aL B, and right fuzzy amplitude aR B, respectively, for B > 0. Otherwise, the left fuzzy amplitude and the right fuzzy amplitude should be aR B and aL B, respectively, for B < 0. For B = 0, one has: A˜ × B = 0.
are both zero, as shown in Fig. 4, one has (b) When the real values of the peaky fuzzy numbers A˜ and B A˜ = 0, LA (x), RA (x) ,
= 0, LB (x), RB (x) , B where
(23)
(24a) (24b)
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Fig. 4. The peaky fuzzy number A˜ with zero real value.
1 + x , aL LA (x) = 0, 1 − x , aR RA (x) = 0, 1 + x , bL LB (x) = 0, 1 − x , bR RB (x) = 0,
−aL < x 0,
(25a)
x −aL , 0 x < aR ,
(25b)
x aR , −bL < x 0,
(25c)
x −bL , 0 x < bR ,
(25d)
x bR .
are neither positive fuzzy numbers, nor negative ones. The perturbation quantities, as mentioned in Eqs. (2) and (3), So A˜ and B belong to this kind of fuzzy numbers. The operation of multiplication on these two fuzzy numbers is given by:
= A˜ × B
= 0, LC (x), RC (x) , (26) C
are defined by Eqs. (24) and (25). According to Eqs. (18), the λ-cut set of C
is given by: where A˜ and B L R Cλ , Cλ = −cL (λ − 1)2 , cR (λ − 1)2
(27)
in which cL = max{aL bR , aR bL },
(28a)
cR = max{aL bL , aR bR }.
(28b)
can be derived as follows: According to the fuzzy representation principle, the membership function of C −x 1− , −cL < x 0, LC (x) = cL 0, x −cL , x 1 − , 0 x < cR , RC (x) = cR 0, x cR .
(29a)
(29b)
leads to (4) The arithmetic operation of addition on linear peaky fuzzy number A˜ and the non-linear peaky fuzzy number C
= A˜ + C.
D
(30)
is The λ-cut set of D L R Dλ , Dλ = −cL (λ − 1)2 + aL (λ − 1), cR (λ − 1)2 − aR (λ − 1) .
(31)
and its membership functions can be derived as follows: Thus the non-linear peaky fuzzy number D,
= 0, LD (x), RD (x) , D
(32)
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
LD (x) =
RD (x) =
1+
aL −
2 − 4c x aL L
2cL
0, 1+
aR −
2 + 4c x aR R
2cR
0,
,
−dL < x 0,
1005
(33a)
x −dL , , 0 x < dR ,
(33b)
x dR ,
where dL = aL + bL , dR = aR + bR .
i.e.: (5) When A˜ in Eq. (30) is also a non-linear peaky fuzzy number, and its membership function is similar to that of C, (34) A˜ = 0, LA (x), RA (x) , 1 − − x , −a < x 0, L LA (x) = aL 0, x −aL , x 1 − , 0 x < aR , RA (x) = aR 0, x aR .
(35a)
(35b)
where C
is still the same as that in Eqs. (29), leads to: The sum of A˜ and C,
= A˜ + C.
D
is: Then the λ-cut set of D L R Dλ , Dλ = −(aL + cL )(λ − 1)2 , (aR + cR )(λ − 1)2 .
(36)
(37)
and its membership functions, LD (x) and RD (x), According to the fuzzy representation principle, the peaky fuzzy number D are given by:
= 0, LD (x), RD (x) , (38) D 1 − − x , −d < x 0, L LD (x) = dL 0, x −dL , x 1 − , 0 x < dR , RD (x) = dR 0, x dR ,
(39a)
(39b)
respectively, and where dL and dR are the left and right fuzzy amplitudes of the fuzzy number D, dL = aL + cL ,
(40a)
dR = aR + cR .
(40b)
from A˜ The solution of subtraction of C
= A˜ − C
= 0, LD (x), RD (x) D
(41)
i.e., the membership functions LD (x) and RD (x) in Eq. (41) are the same is similar in form to that of the addition of A˜ and C, in form as those given by Eqs. (39a) and (39b) respectively, on condition that the left and right fuzzy amplitudes are evaluated by: dL = aL + cR ,
(42a)
dR = aR + cL .
(42b)
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L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
3. Fuzzy variational principle For the sake of simplicity, indicial notation is employed with indices repeated twice implying summation unless otherwise noted.
Poisson’s ratio µ, When an engineering structure possesses fuzzy characteristic parameters, such as Young’s modulus E, ˜
i , boundary traction T i , the structure are and cross sectional dimensions, etc., under fuzzy static loads, including body force F destined to yield fuzzy responses. That is to say, the stress components σ˜ ij , strain components ε˜ ij , displacement components u˜ i ,
, of the linear structures are all fuzzy quantities. and the overall potential energy Π Several investigators considered that the virtue displacements, δ u˜ i , are independent of the material properties and the loading conditions of the structure, thus independent of any fuzziness. It is noteworthy that this is only true in the case of deterministic boundary conditions, since the virtue displacements are required to satisfy the boundary conditions of the concerned structures. In actuality, the imperfection and uncertainty exist in structural boundary conditions, and hence affect the static and dynamic behaviour of the structures (Li and Li, 2001; Yang et al., 2001). Under this condition, the virtue displacements are certainly not independent of uncertainty. (Ohira, 1961), pointed out that the different boundary conditions might cause large variations in the critical buckling loads for isotropic thin-walled shells. Arbocz (2000) investigated the effect of imperfect boundary conditions on the collapse behaviour of anisotropic shells. Furthermore, the idealization of end situations of different structures, such as clamped ends, or simply supported ends, sometimes is not so reasonable, if it cannot be regarded as unacceptable. For example, the supports with finite rigidity are often idealized to be infinitely rigid. Li et al. (1993) indicated the existence of fuzziness of structural boundary conditions. Cherki et al. (2000) investigated the fuzzy behaviour of mechanical systems with uncertain boundary conditions. Since the virtue displacements are dependent on boundary conditions, they should be treated as fuzzy quantities as this paper does. The fuzzy functional of the overall potential energy of the structure is written as ˜ F
i u˜ i ) dV − T i u˜ i dS,
= (A− (43) Π V
Sσ
where displacements and strains, denoted by u˜ i , ε˜ i , respectively, are fuzzy quantities, too. A˜ denotes strain energy, and can be expressed as 1 A˜ = a˜ ij kl ε˜ ij ε˜ kl , 2 σ˜ ij = a˜ ij kl ε˜ kl .
(44a) (44b)
consisting of all the fuzzy parameters of the system including material property coefficients, geometry A fuzzy field {X}, parameters, and external forces, etc., is expressed as
2 , . . . , X
r ]T ,
= [X
1 , X {X}
(45)
can be decomposed as follows where r is the number of fuzzy parameters. The fuzzy field {X} 0
= X + {β}, ˜ {X}
(46)
where T 0 0 0 X = X1 , X2 , . . . , Xr0 ,
(47a)
˜ = [β˜1 , β˜2 , . . . , β˜r ]T , {β}
(47b)
where β˜m (m = 1, 2, . . . , r) are fuzzy perturbation quantities with zero real values. Displacement quantity u˜ i is a dependent
via
so one can expand u˜ i about the real value {X}0 of the fuzzy field {X} variable, which is a function of the fuzzy field {X}, Taylor series, and retain up to second-order terms only 1 u˜ i = u0i + β˜m ui,m + β˜m β˜n ui,mn , 2
m, n = 1, 2, . . . , r,
(48)
where u0i represents ui evaluated at the real value of the fuzzy field, {X0 }. ui,m denotes the partial derivative of ui with respect to the field variable Xm , evaluated at {X0 }. Similarly, ui,mn denotes the second partial derivative of ui with respect to Xm and ˜ Π
i , T i and A,
etc. When the second-order Xn , evaluated at {X0 }. Similar expansions are made for other fuzzy quantities u˜ i , F perturbation technique is employed, the substitution of the expanded expressions of the fuzzy quantities into Eq. (43) will yield:
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
• Zeroth-order potential energy: 0 Π0 = A − u0i Fi0 dV − u0i Ti0 dS. V
1007
(49a)
Sσ
• First-order potential energy: 0 Π,m = A,m − u0i Fi,m − ui,m Fi0 dV − ui Ti,m + ui,m Ti0 dS. V
(49b)
Sσ
• Second-order potential energy: A,mn − u0i Fi,mn − ui,m Fi,n − ui,n Fi,m − ui,mn Fi0 dV Π,mn = V
−
0 ui Ti,mn + ui,m Ti,n + ui,n Ti,m + ui,mn Ti0 dS.
(49c)
Sσ
˜ a˜ ij kl , ε˜ ij into Eq. (44a), and employing the perturbation Substituting the second-order expansion of the fuzzy quantities A, technique, one has 1 0 0 0 ε ε , A0 = aij 2 kl ij kl 1 0 ε0 + a0 ε 0 A,m = aij kl,m εij kl ij kl ij,m εkl , 2 1 0 ε0 + a 0 0 0 0 0 A,mn = aij kl,mn εij ij kl,m εij,n εkl + aij kl,n εij,m εkl + aij kl εij,m εkl,n + aij kl εij,mn εkl . kl 2
(50a) (50b) (50c)
Similarly, the following equations can be derived from Eq. (44b): 0 ε0 , σij0 = aij kl kl
(51a)
0 + a0 ε σij,m = aij kl,m εkl ij kl kl,m , 0 +a 0 σij,mn = aij kl,mn εkl ij kl,m εkl,n + aij kl,n εkl,m + aij kl εkl,mn .
(51b) (51c)
Comparing Eqs. (51) with the partial derivatives of A0 , A,m , A,mn in Eqs. (50), the following expressions will be attained. ∂A0 0 ∂εij
=
∂A,m 0 ∂εij
=
∂A,mn 0 ∂εij
∂A,m ∂A,mn = = σij0 , ∂εij,m ∂εij,mn ∂A,mn = σij,m , ∂εij,n
= σij,mn ,
(52a) (52b) (52c)
where m and n are always the free indices, and denote partial differential instead of summation even if they repeat twice in the same term. Eqs. (52a)–(52c) can be regarded as the perturbation expressions of the fuzzy constitutive equation, which is shown as ∂ A˜ = σ˜ ij . ∂ ε˜ ij
(53)
Similarly, one can transfer the fuzzy relationship equations of strain-displacement 1 ε˜ ij = (u˜ i,j + u˜ j,i ) 2 into a set of deterministic ones, by means of the perturbation technique
(54)
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L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
1 0 ui,j + u0j,i , 2 1 1 εij,m = (ui,j m + uj,im ) = (ui,j + uj,i ),m , 2 2 1 1 εij,mn = (ui,j mn + uj,imn ) = (ui,j + uj,i ),mn . 2 2 0 = εij
According to Eq. (49a), the first variation of Π 0 reads 0 ∂A 0 − F 0 δu0 dV − δε Ti0 δu0i dS. δΠ 0 = ij i i 0 ∂εij V
(55a) (55b) (55c)
(56)
Sσ
Substituting Eq. (55a) into Eq. (56), and then integrating the first term at right-hand by parts, the following expression can be attained by means of Gauss–Ostrogradski theorem 0 0 ∂A ∂A 0 δu0 dV + 0 δu0 dS. + F n − T (57) δΠ 0 = − i i i i 0 0 j ∂εij ∂εij ,j V
Sσ
The stationary condition δΠ 0 = 0 leads to
∂A0
0 ∂εij ,j
∂A0 0 ∂εij
+ Fi0 = 0 in V ,
nj − Ti0 = 0 on Sσ .
(58a)
(58b)
Substituting Eq. (52a) into Eqs. (58) yields 0 + F 0 = 0 in V , σij,j i
(59a)
σij0 nj = Ti0
(59b)
on Sσ .
These two equations are the same as the deterministic equilibrium equation and the boundary condition of surface traction. Similarly, the first-order variation of Eq. (49b) leads to ∂A,m 0 ∂A,m 0 − F 0 δu δε + δε − F δu Ti,m δu0i + Ti0 δui,m dS dV − δΠ,m = ij,m i,m i,m ij i i 0 ∂ε ∂εij ij,m V Sσ ∂A ∂A,m ,m 0+ 0 δu = − + F + F δu dV i,m i,m i i ∂εij,m ,j ∂ε0 ,j V
+ Sσ
ij
0 + ∂A,m n − T 0 δu n − T δu i,m j i,m dS. i i 0 j ∂εij,m ∂εij
∂A,m
(60)
Since δu0i , and δui,m in Eq. (60) are arbitrary and independent, the stationary condition δΠ,m = 0 leads to
σij,j m + Fi,m = 0 in V ,
(61a)
on Sσ , σij,m nj = Ti,m 0 0 σij,j + F i = 0 in V ,
(61b)
σij0 nj = Ti0
on Sσ .
(61c) (61d)
Clearly, Eqs. (61c), (61d) are equivalent to Eqs. (59a), (59b). That is to say, the stationary conditions of Π,m include the stationary conditions of Π 0 . The second-order variation of Eq. (49c) leads to ∂A,mn 0 ∂A,mn ∂A,mn 0 − F δu 0 δu δε + δε + δε − F δu − F δΠ,mn = dV ij,n ij,mn i,mn i,m i,n i,mn ij i i ∂εij,n ∂εij,mn ∂ε0 V
ij
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
−
1009
Ti,mn δu0i + Ti,m δui,n + Ti0 δui,mn dS
Sσ
∂A,mn ∂A,mn 0+ 0 δu + F + F + + F δu δu i,mn i,m i,n i,mn dV i i 0 ∂εij,n ,j ∂εij,mn ,j ∂εij ,j V ∂A,mn ∂A,mn ∂A,mn 0 0 (62) nj − Ti,mn δui + nj − Ti,m δui,n + nj − Ti δui,mn dS. + ∂εij,n ∂εij,mn ∂ε0
= −
Sσ
∂A,mn
ij
Substituting Eqs. (52) into Eq. (62), the stationary conditions δΠ,mn = 0 leads to σij,j mn + Fi,mn = 0 in V ,
σij,mn nj = Ti,mn
on Sσ ,
(63a) (63b)
σij,j m + Fi,m = 0 in V ,
(63c)
σij,m nj = Ti,m
(63c)
on Sσ ,
0 σij,j + Fi0 = 0
in V ,
(63e)
σij0 nj = Ti0
on Sσ .
(63f)
It should be pointed out that, Eqs. (63e), (63f) are equivalent to Eqs. (61c), (61d) and Eqs. (59a), (59b), while Eqs. (63c), (63d) are equivalent to Eqs. (61a), (61b). Therefore, it can be concluded that the stationary conditions of the second-order perturbation expression Π,mn include the stationary conditions of the zeroth- and first-order perturbation expressions, Π 0 and Π,m . Accordingly, when engineering structures with fuzzy parameters are analysed by the fuzzy variational method, only the highest order perturbation expression Π,mn is necessary and efficient to derive all the required governing equations.
4. Fuzzy Ritz method In structural analysis, it is usually difficult or impossible to seek an exact analytical solution of the governing equations. One often tries to establish the equivalent functional so as to obtain readily the approximate solutions by means of the variational techniques. A simply supported elastic beam with uniform cross sections and length l, as shown in Fig. 5, is considered here
and as an example to illustrate the application of the fuzzy variational principle. For this beam, the Young’s modulus, E,
is E 0 , its possible low and high values are Et and Es , external load, q, ˜ are considered as fuzzy parameters. The real value of E respectively. While the real value of q˜ is q 0 , its possible low and high values are qt and qs , respectively. According to Eqs. (5)
denoted by eL and eR , respectively, and those of q, ˜ denoted by qL and qR , and (6), the left and right fuzzy amplitudes of E,
q˜ are respectively, can all be worked out readily. Then the expressions, or say the membership functions, of fuzzy numbers E,
and q˜ can be decomposed at their real values, respectively, as follows determined according to Eqs. (1) and (4). E
1 = E
= E 0 + β˜1 , X (64)
2 = q˜ = q 0 + β˜2 , X where, β˜1 , β˜2 are zero-real fuzzy parameters (Yang, 1998), whose real values are zero:
Fig. 5. An elastic beam with fuzzy characteristic quantities.
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β˜1 = 0, Le (x), Re (x) , β˜2 = 0, Lq (x), Rq (x) , where
1 + x , eL Le (x) = 0, 1 − x , eR Re (x) = 0, 1 + x , qL Lq (x) = 0, 1 − x , qR R q (x) = 0,
(65a) (65b)
−eL < x 0, x −eL , 0 < x eR , x eR , −qL < x 0, x −qL , 0 < x qR , x qR .
For convenience, one adopts the following identities eL = eR = γe ,
(66a)
qL = qR = γq .
(66b)
The functional of the total potential energy for the beam with fuzzy parameters and external load (fuzzy beam) is
= Π
l
1 2 Dχ˜ − q˜ w
dx, 2
(67)
0
where
= EI,
D χ˜ = −
(68a)
d2 w dx 2
.
(68b)
is obtained according to the perturbation techniques The second-order expansions of the fuzzy quantities Π l Π,ij = 0
2 1 D,ij χ 0 + D,i χ,j χ 0 + D,j χ,i χ 0 + D 0 χ,i χ,j + D 0 χ,ij χ 0 2
− q,ij w0 + q,i w,j + q,j w,i + q 0 w,ij dx,
i, j = 1, 2.
(69)
It is assumed the admissible displacement function of the beam as w
=
r m=1
mπx A˜ m sin . l
(70)
As being the fuzzy quantities, A˜ m and w
can be expanded via Taylor series w
= w0 +
2 i=1
A˜ m = A0m +
2 1 ˜ ˜ β˜i w,i + βi βj w,ij , 2
2 i=1
(71a)
i,j =1
2 1 ˜ ˜ β˜i Am,i + βi βj Am,ij . 2 i,j =1
Substituting Eqs. (71) into Eq. (70), one has
(71b)
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
r mπx 0= w A0m sin , l m=1 r mπx Am,i sin , w,i = l m=1 r mπx = Am,ij sin , w ,ij l
1011
(72a)
m=1
r 2 2 0= 0 m π sin mπx , χ A m l l2 m=1 r m2 π 2 mπx Am,i 2 sin , χ,i = l l m=1 r m2 π 2 mπx = A sin . χ ,ij m,ij 2 l l
(72b)
m=1
Substituting Eqs. (72) into Eq. (69), one obtains l r m2 n2 π 4 mπx nπx Π,ij = sin sin D,ij A0m A0n + 2D,i A0m An,j + 2D,j A0m An,i + 2D 0 Am,i An,j 4 l l 2l m,n=1
0
+ 2D 0 A0m An,ij −
r k=1
kπx 0 0 sin q,ij Ak + q,i Ak,j + q,j Ak,i + q Ak,ij dx, l
i, j = 1, 2.
(73)
The stationary condition of the variational principle leads to ∂Π,ij m4 π 4 0 0 lq 0 =0⇒ D Am + (−1)m − 1 = 0, 3 ∂An,ij mπ 2l
(74a)
4 4 lq ,j ∂Π,ij m4 π 4 0 0m π A = 0 ⇒ D,j A + D (−1)m − 1 = 0, m,j + m ∂An,i mπ 2l 3 2l 3
(74b)
m4 π 4 lq,ij 0 +D A 0A = 0 ⇒ D A + D A + D + (−1)m − 1 = 0. ,ij ,i m,j ,j m,i m,ij m 0 3 mπ 2l ∂An
∂Π,ij
(74c)
These are the deterministic recursive equations of the fuzzy Ritz method applied in the elastic beam as shown in Fig. 5, which can be solved for coefficients A0m , Am,j , Am,ij as follows: (a) when m are even numbers, one has A0m = Am,j = Am,ij = 0 for i, j = 1, 2;
(75)
(b) when m are odd numbers, one has 4l 4 q 0 , m5 π 5 D 0 4l 4 q,j 4l 4 q 0 D ,j D ,j 4l 4 Am,j = 5 5 0 − 5 5 0 2 = 5 5 0 q,j − 0 q 0 , m π D m π (D ) m π D D 4 2D,i D,j 0 4l q Am,ij = 5 5 0 2 D 0 q,ij − D,ij q 0 − q,j D,i − q,i D,j + for i, j = 1, 2. m π (D ) D0
A0m =
and q, For this beam, the fuzzy field is composed of two fuzzy quantities, E ˜ then according to Eq. (68a), one has D 0 = E 0 I, D,1 = I, D,2 = D,11 = D,12 = D,22 = 0.
(76a) (76b) (76c)
(77)
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1 = E,
such that q˜ is independent of X q,1 = 0, q,2 = 1,
(78)
q,11 = q,12 = q,22 = 0.
Substitution of Eqs. (77) and (78) into Eqs. (76) leads to the solutions of A0m , Am,j , and Am,ij , which are shown as follows (a) When m are even numbers, one has A0m = 0,
Am,i = 0,
Am,ij = 0 for i, j = 1, 2;
(79a)
(b) When m are odd numbers, one has A0m =
4l 4 q 0 , m5 π 5 D 0
Am,11 =
4l 4 I q 0 Am,1 = − 5 5 0 2 , m π (D ) Am,2 =
4l 4 , m5 π 5 D 0
8l 4 q 0 m5 π 5 D 0 (E 0 )2
,
Am,22 = 0,
(79b)
4l 4 I Am,21 = Am,12 = − 5 5 0 2 . m π (D )
Substituting coefficients A0m , Am,i , Am,ij into Eq. (72a), w0 , w,i , w,ij are given by r
w0 =
m=1
4l 4 q 0 mπx sin , l m5 π 5 D 0
(80a)
r 4l 4 I q 0 1 mπx 1 sin = − 0 w0 , w,1 = − 5 0 2 l π (D ) m5 E
(80b)
r 1 4l 4 1 mπx = 0 w0 , sin l π 5 D0 m5 q
(80c)
m=1
w,2 =
m=1 4 8l q 0
r 1 mπx 2 sin = 0 2 w0 , l π 5 D 0 (E 0 )2 m5 (E )
(80d)
r 1 4l 4 mπx 1 sin = − 0 0 w0 , w,12 = w,21 = − 5 0 0 l π D E m5 q E
(80e)
w,22 = 0.
(80f)
w,11 =
m=1
m=1
Substitution of w0 , w,i , w,ij given by Eqs. (80) into Eq. (71a), the fuzzy displacement of the beam w
is obtained 0 w
= w , Lw (x), Rw (x) , where Lw (x) =
1+
a−
a 2 − 4c(x − w0 ) , 2c
(81)
w0 − (a + c) < x w0 ,
0, x w0 − (a + c), a − a 2 + 4c(x − w0 ) 1+ , w0 x < w0 + (a + c), R w (x) = 2c 0, x w0 + (a + c) in which w,11 γE2 − 2w,12 γE γq . 2 For threshold λ ∈ [0, 1], the λ-cut of the displacement w
is 2 2 w
λ = w0 − cλ + (a + 2c)λ − a − c, w0 + cλ − (a + 2c)λ + a + c . a = −w,1 γE + w,2 γq ,
c=
(82)
(83)
Therefore the possible range of the displacement of the beam can be attained corresponding to the assumption level λ of confidence. It surely provides more useful messages for engineers to analyze and design structures.
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5. The fuzzy finite element From now on, matrix notation is adopted instead of tensor notation for the sake of clarity. The characteristic equations for the fuzzy finite element method will be derived according to the fuzzy variational principle as follows.
is shown as Eq. (45). The fuzzy functional Considering a linear engineering structure under static load, the fuzzy field {X} of total potential energy of the linear elastic structure is
ε} dV − {u}
} dV ,
= 1 {˜ε}T [D]{˜ ˜ T {T } dS − {u} ˜ T {F (84) Π 2 V
Sσ
V
{T }, {F
}, and Π
denote the structural displacement, stress, elastic matrix, surface traction vector, body where {u}, ˜ {˜ε}, [D],
force vector, and functional of total potential energy, respectively. They are all functions of the fuzzy field {X}. When analysing the structure based on the finite element method, the interested displacement field should be discretized into elements. And then the displacement field {u} ˜ of a typical element is interpolated by polynomials {u} ˜ =
t
Ni (x)˜ai = [N]{a}, ˜
(85)
i=1
where
[N] = N1 (x), N2 (x), . . . , Nt (x) , {a} ˜ = [˜a1 , a˜ 2 , . . . , a˜ t ]T
in which t denotes the number of element nodes. a˜ i denotes the value of displacement vector {u} ˜ at node i. [N] represents the matrix of shape function constituted by polynomials. Since {a} ˜ can be expanded via the second-order perturbation techniques as follows r r 1 β˜i {a}i + β˜i β˜j {a}ij . {a} ˜ = a0 + 2 i=1
One has 0 0 u = [N] a , {u}i = [N]{a}i , {u}ij = [N]{a}ij ,
(86)
i,j =1
i, j = 1, 2, . . . , r.
(87)
The strain vector {˜ε} and the displacement vector {u} ˜ satisfy the following relationship {˜ε} = L{u} ˜ = L[N]{a} ˜ = [B]{a}, ˜
(88)
where L is a differential operator; [B] denotes the strain matrix. Substituting the second-order perturbation expansions of {˜ε} and {a} ˜ into Eq. (88), one obtains 0 0 ε = [N] a , i, j = 1, 2, . . . , r. (89) {ε}i = [N]{a}i , {ε}ij = [N]{a}ij ,
{T }, {F
}, and Π
into Substituting the second-order perturbation based expansions of the fuzzy quantities {u}, ˜ {˜ε}, {D}, Eq. (84), Π,ij is attained via the perturbation techniques as follows: 0 T T 1 0 T T T 0 Π,ij = [B] [D]ij [B] a 0 + 2{a}T [B] [D]i [B]{a}j a i [B] D [B]{a}j + 2 a 2 V
T T + 2 a 0 [B]T [D]j [B]{a}i + 2 a 0 [B]T D 0 [B]{a}ij dV 0 T T T T T T 0 dV [N]T {F }ij + {a}T a − i [N] {F }j + {a}j [N] {F }i + {a}ij [N] F V
− Sσ
0 T T T T T T 0 dS. [N]T {T }ij + {a}T a i [N] {T }j + {a}j [N] {T }i + {a}ij [N] T
(90)
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According to the stationary conditions of Π,ij , ∂Πij /∂{a 0 }, ∂Πij /∂{a}i , and ∂Πij /∂{a}ij should vanish. i.e., ∂Πij = [B]T D 0 [B] dV a 0 − [N]T F 0 dV − [N]T T 0 dS = 0 ∂{a}ij V
V
one has 0 0 0 K a − P = {0}, where
where
(92)
K 0 = V [B]T D 0 [B] dV , 0 P = V [N]T F 0 dV + Sσ [N]T T 0 dS, T ∂Π,ij = [B] [D]i [B] a 0 + [B]T D 0 [B]{a}i dV − [N]T {F }i dV − [N]T {T }i dS ∂{a}j V V Sσ = [K]i a 0 + K 0 {a}i − {P }i = {0}, i = 1, 2, . . . , r,
(93)
[K]i = V [B]T [D]i [B] dV , {P }i = V [N]T {F }i dV + Sσ [N]T {T }i dS, ∂Πij [B]T [D]ij [B] a 0 + [D]i [B]{a}j + [D]j [B]{a}i + D 0 [B]{a}ij dV = 0 ∂{a } V − [N]T {F }ij dV − [N]T {T }ij dS = [K]ij a 0 + [K]i {a}j + [K]j {a}i V Sσ + K 0 {a}ij − {P }ij = {0},
where
(91)
Sσ
(94)
i, j = 1, 2, . . . , r,
[K]ij = V [B]T [D]ij [B] dV , {P }ij = V [N]T {F }ij dV + Sσ [N]T {T }ij dS.
Therefore, the stiffness equations for the fuzzy finite element method are of a set of deterministic recursive equations as follows K 0 a0 = P 0 , (95) K 0 {a}i = {P }i − [K]i a 0 , i = 1, 2, . . . , r, 0 0 K {a}ij = {P }ij − [K]ij a − [K]i {a}j − [K]j {a}i , i, j = 1, 2, . . . , r. Solving Eq. (95) recursively for coefficients {a 0 }, {a}i , and {a}ij , and substituting them into Eq. (86), the fuzzy nodal displacements of the element are attained. Then the stress vector {σ˜ } is obtained via the perturbation techniques as follows r r 1 {σ˜ } = S 0 a 0 + β˜i [S]i a 0 + S 0 {a}i + β˜i β˜j [S]ij a 0 + [S]i {a}j + [S]j {a}i + S 0 {a}ij , 2
(96)
i,j =1
i=1
where 0 0 S = D [B],
[S]i = [D]i [B],
[S]ij = [D]ij [B].
6. Numerical examples
Example 1. The first example considers the elastic beam shown in Fig. 5 with length l. The real values of Young’s modulus E and the external load q˜ are taken as E 0 and q 0 , respectively, according to the opinions of experts. The possible low and high
and q˜ are also determined by the opinions of experts, which are given by values of E
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
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El = 0.8E 0 ,
(97a)
Eh = 1.2E 0 , ql = 0.85E 0 , qh = 1.15q 0 .
(97b) (98a) (98b)
Here the fuzzy Ritz method is employed to evaluate the fuzzy displacements of the beam.
and q˜ can be evaluated, and given by According to Eqs. (5), (6) and (66), the fuzzy amplitudes of E γe = 0.4E 0 ,
(99a)
γq = 0.3q 0 .
(99b)
Substitution of Eqs. (97)–(99) into Eqs. (80) yields the solutions of w0 , w,i , w,ij . Substituting these quantities into Eqs. (82), the coefficients a and c are given by a = 0.7w0 ,
(100a)
c = 0.28w0 .
(100b)
Substituting Eqs. (100) into Eq. (81), the fuzzy displacement of the beam is determined w
= w0 , Lw (x), Rw (x) ,
(101)
where 1 4l 4 q 0 mπx sin , 5 0 l π D m5 m=1,3,5,..., 5√ 9 161 − 112x, ¯ 0.02 < x¯ 1, − Lw (x) = 4 28 0, x¯ 0.02, √ 5 9 112x¯ − 63, 1 x¯ < 1.98, − Rw (x) = 4 28 0, x¯ 1.98
w0 =
in which x¯ = x/w0 . When applying the fuzzy numerical methods to the analysis of real engineering structures, the most important information required is the λ-cut sets of the fuzzy responses of the structures. According to Eq. (83), the λ-cut of the fuzzy displacement w
shows w
λ = w0 − cλ2 + (a + 2c)λ − a − c, w0 + cλ2 − (a + 2c)λ + a + c (102) = −0.28λ2 + 1.26λ + 0.02, 0.28λ2 − 1.26λ + 1.98 w0 . Evidently, the threshold λ = 1 will lead to a result of w
= w0 . It is a deterministic result corresponding to the conventional deterministic solution when all kinds of uncertainties of the structure are neglected. If the threshold λ is less than 1, the λ-cut set of the fuzzy displacement w
is shown as an interval. For example, λ = 0.8 and λ = 0.5 will lead to 0 w)λ=0.5 = 0.58w0 , 1.42w0 , (103) (
w)λ=0.8 = 0.85w , 1.15w0 and (
w)λ=0.5 , as illustrated in Fig. 6. respectively. (
w)λ=0.8 ⊆ (
On the other hand, the deterministic analytical solution of the beam neglecting all the fuzzy properties is used herein as a benchmark to demonstrate the effective and accuracy of the fuzzy numerical methods. The analytical solution for the mid-span deflection of the beam is given by 5 ql 4 . (104) 384 EI Therefore, when the Young’s modulus E and the external load q are both regarded as fuzzy quantities, the fuzzy displacement is obtained as follows w=
w
=
˜ 4 5 ql .
384 EI
(105)
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Fig. 6. w
and its λ-cut sets at different levels of confidence. Table 1 Deflection at the centre of the beam at various levels of confidence Threshold FRM FFEM AM Multiplier
λ=1
λ = 0.9
λ = 0.8
λ = 0.7
λ = 0.6
λ = 0.5
1 1 1
[0.927,1.073] [0.925,1.073] [0.933,1.073]
[0.849,1.151] [0.846,1.150] [0.870,1.152]
[0.765,1.235] [0.760,1.235] [0.813,1.239]
[0.675,1.325] [0.669,1.326] [0.759,1.333]
[0.580,1.420] [0.468,1.424] [0.708,1.438]
0 4 w 0 = 0.01302q l /E 0 I 0
When the thresholds are given as λ = 0.8, and λ = 0.9, respectively, Eq. (105) yields w)λ=0.5 = 0.96w0 , 1.04w0 . (
w)λ=0.8 = 0.87w0 , 1.15w0 and (
(106)
Evidently, results shown in Eqs. (103) obtained by the fuzzy Ritz method are in good agreement with those obtained by the analytical method given by Eqs. (106). Detailed comparison is illustrated in Fig. 6 and Table 1. It is noteworthy that when λ < 0.5, the analytical solution will be meaningless, since the possible range of variations of fuzzy parameters proposed by the experts are enlarged during the construction of membership functions. For example, the possible ranges of the Young’s modulus E and the external load q in this example are (107a) [El , Eh ] = 0.8E 0 , 1.2E 0 , 0 0 (107b) [ql , qh ] = 0.85E , 1.15q
and q˜ given they are enlarged when γe and γq given by Eqs. (99) are evaluated. Therefore, they are equal to the 0.5-cuts of E by Eqs. (64), (65), respectively, as follows
λ=0.5 = 0.8E 0 , 1.2E 0 , (108a) (E) 0 0 (108b) (q) ˜ λ=0.5 = 0.85E , 1.15q . Example 2. The second numerical example considers the same beam discussed in Example 1. The fuzzy finite element method is employed here to evaluate the displacements of the beam with fuzzy parameters. The mid-span deflections of the beam at different levels of confidence are compared with those resulted from the fuzzy Ritz method and the analytical method, as shown in Table 1. The abbreviation of AM denotes the analytical method. The displacements along half span of the beam calculated by the FRM, the FFEM and the AM at different levels of confidence are illustrated in Fig. 7. These results show that the results obtained by the FFEM and the FRM are little conservative, and are in good agreement with those determined by the AM, when the level of confidence λ > 0.7. Moreover, the results by the fuzzy numerical methods and the analytical method at the level λ = 1 are equal to the conventional deterministic solution. These results show that the fuzzy analysis and calculation provide investigators or designers with estimation intervals of the structural responses at different levels of confidence λ. When λ = 1, the results lead to the conventional deterministic solutions. When λ is assigned a value less than 1 (and greater than 0.7), the structural displacements range over a specific interval, which often exceed the deterministic solutions. For example, if the assumption of confidence λ = 0.8 is considered, the maximum values in the interval of displacements are 15% more than the deterministic solutions. Therefore, the fuzzy numerical methods are able to provide more and valuable information in structural design and analysis.
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
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Fig. 7a. Deflection of the beam at 1.0-level of confidence by the three methods.
Fig. 7b. Fuzzy deflection w
of the beam at 0.8-level of confidence.
In the FFEM, the fuzzy field with r uncertain variables is not discretized, though the structure with fuzzy properties may be discretized into an assemblage with n degree of freedom. So the governing equations are consisted of r × r + r + 1 systems of equations, in which, there is one system with the zeroth-order, r systems with the first-order, r × r systems with the secondorder. For this beam, r = 2, so there are only 7 hierarchical systems of equations. When the random distribution function or the probabilistic characteristic parameters of the uncertain variables are known, the uncertain variables may be treated as random parameters, which are modelled to be a random field with r random variables. The perturbation based stochastic finite element method (SFEM) may be employed in this case. However, the random field has to be discretized a priori into s subdomains, which yields a random vector of r × s elements. So the governing equations of SFEM are consisted of r × s + 2 systems of equations, in which, there is one zeroth-order expanded system of equation, r × s first-order systems, one second-order system. For the beam considered in the examples, r = 2, there are 10 hierarchical systems of equations if the random field is discretized into 4 elements, i.e., s = 4. Because each system of equation comprises n equations for either FFEM or SFEM, s is greater than r for real life structures in many cases; the FFEM may reduce the computation effort considerably, as compared with SFEM.
7. Conclusion The FVP provides a solid theoretical foundation for the fuzzy numerical approaches, such as the FRM and the FFEM. The analytical solutions of a bending beam with fuzzy characteristic quantities are presented to serve as benchmark solutions. Hence, investigators or designers are able to check the effectiveness and accuracy of the FFEM and the FRM. The numerical examples reveal that the results obtained by the FFEM and the FRM are little conservative, and agree well with the analytical solutions. The level of confidence λ should be assigned with a value greater than 0.7, since the possible range of variation proposed by experts for fuzzy parameters may be enlarged during the construction of membership functions. The fuzzy analysis and calculation can provide engineers with estimation intervals of structural responses at different levels of confidence λ. Therefore, the fuzzy numerical methods are able to provide more and valuable information in structural design and analysis. The FFEM may reduce the computational efforts considerably, as compared with the perturbation based SFEM. Example 2 shows that the perturbation based FFEM may yield satisfactory results when the possible low or high values of the fuzzy parameters fluctuate within the range of about 25% smaller or greater than their real value, respectively.
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Acknowledgements The work described in this paper was supported by grants from City University of Hong Kong (Project No: 7001176), Guangxi Natural Science Foundation (Contract No: 0135001), and the Foundation of Important Science and Technology Research Project of the State Education Ministry, China.
References Arbocz, J., 2000. The effect of imperfect boundary conditions on the collapse behaviour of anisotropic shells. Internat. J. Solids Structures 37, 6891–6915. Ayyub, B.M., Guran, A., Haldar, A., 1997. Uncertainty Modelling in Vibration, Control and Fuzzy Analysis of Structural Systems. World Scientific, Singapore. Cao, S.S., Sawyer, J.P., 1995. Fuzzy finite element approach for the analysis of imprecisely defined systems. AIAA J. 33, 2364–2370. Cherki, A., Plessis, G., Lallemand, B., Tison, T., Level, P., 2000. Fuzzy behaviour of mechanical systems with uncertain boundary conditions. Comput. Methods Appl. Mech. Engrg. 189, 863–873. Deng, J.L., 1987. Basic Methods for Grey System. The Press of Huazhong University of Science and Technology, Wuhan. de Lima, B.S.L.P., Ebecken, N.F.F., 2000. A comparison of models for uncertainty analysis by the finite element method. Finite Elem. Anal. Des. 34, 211–232. Dubois, D., Prade, H., 1980. Fuzzy Sets and System: Theory and Applications. Academic Press, New York. Elishakoff, I., 1995. Essay on uncertainties in elastic and viscoelastic structures: from A.M. Freudenthal’s criticisms to modern convex modelling. Comput. and Structures 56, 871–895. Elishakoff, I., Ren, Y.J., Shinozuka, M., 1996. Variational principles developed for and applied to analysis of stochastic beams. J. Engrg. Mech. 122, 559–565. Hien, T.D., Kleiber, M., 1990. Finite element analysis based on stochastic Hamilton variational principle. Comput. and Structures 37, 893–902. Lanczos, C., 1964. The Variational Principle in Mechanics. University Toronto Press. Li, G.Q., Cao, H., Li, Q.S., Huo, D., 1993. Theories and Applications of the Reliability in Structural Dynamics. Earthquake Press, Beijing. Li, G.Q., Li, Q.S., 2001. Theory and Its Application of Time-Dependent Reliability of Engineering Structures. Science Press, Beijing. Liu, W.K., Besterfield, G.H., Belyschko, T., 1988. Variational approach to probabilistic finite elements. J. Engrg. Mech. 114, 2115–2133. Matthies, H.G., Bucher, C., 1999. Finite elements for stochastic media problems. Comput. Methods Appl. Mech. Engrg. 168, 3–17. Ohira, H., 1961. Local buckling theory of axially compressed cylindrical shells. In: Proc. of the 11th Japan National Congress of Applied Mechanics, pp. 37–40. Qian, W.C., 1985. The Generalized Variational Principles. Intellectual Press, Beijing. Wang, Q.Y., 1988. Basic element of grey mathematics and its relations with fuzzy and real number. J. Grey System (1), 69–77. Yang, L.F., 1998. Fuzzy stochastic finite element methods and their applications. Ph.D. thesis, Wuhan University of Technology. Yang, L.F., Gao, D.X., Li, G.Q., 1999. Stochastic functional of potential energy developed for stochastic structures under stochastic loads. Chinese J. Appl. Mech. 16 (4), 35–39. Yang, L.F., Leung, A.Y.T., Li, Q.S., 2001. The stochastic finite segment in analysis of shear lag effect on box girder. Engrg. Structures 23, 1461–1468. Zadeh, L.A., 1965. Fuzzy sets. Inform. Control 8, 338–353. Zienkiewicz, O.C., 1978. Numerical Methods in Offshore Engineering. Wiley. Zhu, B., 1999. On the application of reliability theory to design of concrete dams. China Civil Engrg. J. 32 (4), 10–15.
Fuzzy variational principle and its applications L.F. Yang a , Q.S. Li b,∗ , A.Y.T. Leung b , Y.L. Zhao a , G.Q. Li c a Department of Civil Engineering, Guangxi University, Nanning, 530004 Guangxi, China b Department of Building and Construction, City University of Hong Kong, 83 Tat Chee avenue, Hong Kong c Department of Civil Engineering, Wuhan University of Technology, Wuhan, 430070 Hubei, China
Received 2 April 2001; revised and accepted 21 August 2002
Abstract Linear and non-linear peaky fuzzy numbers and their arithmetic operations are constructed for the analysis of engineering structures with fuzzy characteristic quantities. Fuzziness of the corresponding quantities is consistently incorporated into the functional of the total potential energy. A set of deterministic recursive equations is obtained as the alternative expressions of the fuzzy variational principle by means of the second-order perturbation technique. The fuzzy Ritz method and the fuzzy finite element method are presented as the applications of the fuzzy variational principle. Accordingly, the roundabout procedures frequently used in the formulations of the fuzzy finite element method are avoided. A benchmark problem of a bending beam with fuzzy Young’s modulus under fuzzy external loading is solved by the developed fuzzy numerical methods. Numerical examples show that results determined by these two fuzzy methods are both little conservative, and are in good agreement with those obtained by the analytical method. Moreover, the fuzzy Ritz method or the fuzzy finite element method can provide more valuable information than the conventional deterministic methods. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Fuzzy; Variational principle; Ritz method; Finite element method
1. Introduction Numerical methods are more frequently used than analytical methods in structural design and analysis (Zienkiewicz, 1978). Among a variety of numerical methods, the finite element method is by far the most effectively and widely used method. One of its important theoretic bases is the variational principle (Qian, 1985; Lanczos, 1964). Research works on deterministic variational principles are extensive and mature (Zienkiewicz, 1978; Qian, 1985; Lanczos, 1964), while in the field of uncertain variational principles, researchers focused their attentions on stochastic variational principles (Matthies and Bucher, 1999; Liu et al., 1988; Hien and Kleiber, 1990; de Lima and Ebecken, 2000; Yang et al., 1999; Elishakoff et al., 1996), though the relevant literatures have been fairly fewer so far. However, research work on fuzzy variational principles with application in structural analysis, to the best of our knowledge, has not been publicly reported (Yang, 1998). On the other hand, real engineering structures are usually complicated systems, which possess various uncertain quantities. Sometimes the uncertainty plays significant roles in the analysis and design of engineering structures. There are several categories of non-deterministic theories and methods: the theory of probability and random process; the fuzzy-set-based methods; the gray model theory; and the theory of convex modeling of uncertainty, etc. The theories of probability and random process are long-standing powerful tools for dealing with uncertain systems, but the usage of these theories is fairly cumbersome for situations where the uncertain quantities are of non-statistical distributions in nature or show very sophisticated random distributions (Ayyub et al., 1997). The fuzzy-set-based methods are regarded to be more appropriate when the uncertainties * Corresponding author.
E-mail address: [email protected] (Q.S. Li). 0997-7538/02/$ – see front matter 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 0 9 9 7 - 7 5 3 8 ( 0 2 ) 0 1 2 5 4 - 8
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of systems are due to vaguely defined system characteristics, imprecise or insufficient data (Cao and Sawyer, 1995). Then the subjective judgment and opinion of experts are crucial (Zhu, 1999) for treatment of these systems. These two theories become quite elaborate, and are the most frequently used methods in the analysis of structures with uncertain quantities. The gray model theory is proposed by Deng (1987) and has been extensively investigated (Wang, 1988). Elishakoff (1995) proposed the convex modeling of uncertainty. He had adequately demonstrated the effect of human error, as well as the influence of imprecise data, on the traditional theory of reliability. This paper incorporates the fuzziness of physical and geometrical quantities into the potential energy functional. Based on the second-order perturbation technique and the arithmetic operations constructed in this paper, the fuzzy variational principle (FVP) is proposed. The fuzzy Ritz method (FRM) and the fuzzy finite element method (FFEM) are presented as the applications of the FVP.
2. Definition and arithmetic operations of peaky fuzzy numbers Fuzzy quantities, which possess fuzziness, can be treated by means of the fuzzy number theory (Dubois and Prade, 1980). From the viewpoint of philosophy, objective things are deterministic, while the fuzziness belongs to the field of subjectivity. There may exist critical bifurcation points for any object during the process of change. When the quantitative change of an object accumulates and surpasses the critical point, the qualitative change occurs, then the objective thing A will transfer into B. This critical point, or say limit, is a criterion of demarcating different objective things. For example, a temperature of 100 ◦ C is a criterion state, which means that the liquid water may transfer into vapor. For real-life problem of engineering structures, the critical points are hardly or not necessary to be worked out exactly. Therefore this critical point is usually estimated approximately as an interval, in which the transformation from one state to another shows a progressive procedure rather than an abrupt one. In the fuzzy analysis of structures, this critical point is termed the real value of the fuzzy quantity, and the estimated interval represents the possible range of the fuzzy quantity about its real value. In the present paper the peaky fuzzy numbers are adopted. Fuzzy numbers are specific fuzzy sets pioneered by Zadeh (1965). To qualify as a peaky fuzzy number, a bounded closed fuzzy number A˜ must possess the following properties: (1) The left shape function, LA (x), of the fuzzy number A˜ are monotonic increasing and continuous from the right, while the right shape function, RA (x), are monotonic decreasing and continuous, and 0 LA (x), RA (x) < 1, as shown in Fig. 1. ˜ denoted by Ker A, ˜ there is only one element A0 . (2) Within the kernel set of the fuzzy number A, Then A˜ can be expressed as A˜ = A0 , LA (x), RA (x) .
(1)
˜ for x = A0 , attains its maximum value 1, as illustrated in Fig. 1. A0 is The membership function of the peaky fuzzy number, A, ˜ deterministic and called real value of the peaky fuzzy number A. At the point of its real value, A˜ can be decomposed as: A˜ = A0 + β˜A ,
(2)
˜ and: where β˜A is the perturbation quantity of A, β˜A = 0, LA x + A0 , RA x + A0 .
Fig. 1. A peaky fuzzy number A˜ and the governing parameters.
(3)
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
1001
Fig. 2. A linear peaky fuzzy number A˜ and its governing parameters.
In engineering practices, the left and right half shape functions, LA (x), RA (x), can be defined as linear or non-linear functions, sometimes the exponents may be preferable in some applications. A˜ is called a linear peaky fuzzy number when the linear shape functions are adopted, such that: 0 x − A + 1, A0 − a < x A0 , L (4a) LA (x) = aL 0, x A0 − aL , 0 A − x + 1, A0 x < A0 + a , R RA (x) = (4b) aR 0 0, x A + aR , ˜ They are termed left and right amplitudes of fluctuation, respectively, as where aL , aR represent the degree of fuzziness of A. 0 illustrated in Fig. 2. The real value, A , is usually estimated by experts according to their experience of engineering practices. ˜ denoted by Al and Ah , respectively, Similarly, the possible low value and the possible high value of the peaky fuzzy number A, are also estimated by experts. Then, aL , aR are given by aL = A0 − Al ,
(5a)
aR = Ah − A0 ,
(5b)
where
Al =
2Al − A0
if A0 < 2Al ,
0
if A0 2Al ,
Ah = 2Ah − A0 .
(6a) (6b)
It is evident that the establishment of fuzzy numbers combines the objectivity of precise mathematics with the treasurable experience of experts.
can be For binary fuzzy arithmetic operations ∗ ∈ {+, −, ×, ÷}, formulae of fuzzy operations on fuzzy numbers, A˜ and B, derived as follows, according to the fuzzy representation principle.
are considered as linear peaky fuzzy numbers, and A˜ is shown in Eqs. (1) and (4), while At first, A˜ and B 0
= B , LB (x), RB (x) , (7) B where LB (x) and RB (x) are similar in form to LA (x) and RA (x) in Eqs. (4), respectively, on condition that A and a in Eqs. (4) are replaced by B and b, respectively.
λ respectively, are crisp sets, which are given by:
denoted by A˜ λ and B For real number λ ∈ (0, 1], the λ-cut sets of A˜ and B, 0 A˜ λ = A + aL (λ − 1), A0 − aR (λ − 1) , (8a) B
λ = B 0 + bL (λ − 1), B 0 − bR (λ − 1) (8b) such that:
= A˜ + B,
one can obtain the λ-cut set of fuzzy number C
via Eqs. (8): For C 0
λ = A + B 0 + (aL + bL )(λ − 1), A0 + B 0 − (aR + bR )(λ − 1) . C
(9)
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obtained by moving A˜ along x direction. Fig. 3. Membership function of C
can be attained: Therefore, according to the fuzzy representation principle, C 0
= C , LC (x), RC (x) , C where
0 x − C + 1, C 0 − c x C 0 , L LC (x) = cL 0, x C 0 − cL , 0 C − x + 1, C 0 x C 0 + c , R RC (x) = cR 0 0, x C + cR ,
(10)
(11a)
(11b)
where C 0 = A0 + B 0 , cL = aL + bL , cR = aR + bR . Eqs. (10) and (11) show that the summation of linear peaky fuzzy numbers result in a linear peaky fuzzy number.
degenerates into a crisp number B, i.e., its left and right fuzzy amplitudes vanish: When the fuzzy number B
then
bL = bR = 0
(12)
A˜ + B = A0 + B, LA (x − B), RA (x − B) .
(13)
From this formula, one knows that the solution of a peaky fuzzy number A˜ plus a crisp number B is still a linear peaky fuzzy ˜ That is to say, the membership function of fuzzy number with real value C 0 = A0 + B, and has the same fuzzy amplitudes as A.
can be obtained by moving the membership function of A˜ a distance of B along x direction, as shown in Fig. 3. number C
= A˜ − B,
the λ-cut set of fuzzy number C
shows: (2) For C 0 0 0
λ = A − B + (aL + bR )(λ − 1), A − B 0 − (aR + bL )(λ − 1) . (14) C
can be obtained as follows: Therefore, C
= C 0 , LC (x), RC (x) , C where
(15)
0 x − C + 1, C 0 − c x C 0 , L LC (x) = cL 0, x C 0 − cL , 0 C − x + 1, C 0 x C 0 + c , R RC (x) = cR 0, x C 0 + cR , C 0 = A0 − B 0 ,
cL = aL + bR ,
cR = aR + bL .
(16a)
(16b) (16c)
from peaky fuzzy number A˜ will also result in a linear Eqs. (15) and (16) show that the subtraction of peaky fuzzy number B peaky fuzzy number.
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
= A˜ × B,
the sets of λ-cut of C
can be evaluated by: (3) For C L L
λ = C , C C for λ ∈ [0, 1], λ λ where
L Cλ = min A0 + aL (λ − 1) B 0 + bL (λ − 1) , A0 + aL (λ − 1) B 0 − bR (λ − 1) , 0 A − aR (λ − 1) B 0 + bL (λ − 1) , A0 − aR (λ − 1) B 0 − bR (λ − 1) , CλR = max A0 + aL (λ − 1) B 0 + bL (λ − 1) , A0 + aL (λ − 1) B 0 − bR (λ − 1) , 0 A − aR (λ − 1) B 0 + bL (λ − 1) , A0 − aR (λ − 1) B 0 − bR (λ − 1) .
1003
(17)
(18)
Because the multiplication of fuzzy numbers is more complicated than their addition or subtraction, only some special cases, which will be encountered in this paper, are discussed hereafter.
are positive linear peaky fuzzy numbers, then one has (a) If both A˜ and B R 0 (19a) Cλ = A − aR (λ − 1) B 0 − bR (λ − 1) , (19b) CλL = A0 + aL (λ − 1) B 0 + bL (λ − 1) . According to the fuzzy representation principle, the left and right half shape functions, LC (x) and RC (x), can be obtained:
= C 0 , LC (x), RC (x) , (20) C where
0 0 L − (bL A + aL B ) + 1, C 0 − c x C 0 , L LC (x) = 2aL bL 0, x C 0 − cL , 0 0 bR A + aR B − R + 1, C 0 x C 0 + c , R RC (x) = 2aR bR 0, x C 0 + cR , C 0 = A0 B 0 ,
where
cL = bL A0 + aL B 0 − aL bL ,
cR = bR A0 + aR B 0 + aR bR ,
L = aL B 0 + bL A0 2 − 4aL bL C 0 − x ,
= a B 0 + b A0 2 − 4a b C 0 − x . R R R R R
(20a)
(20b) (20c)
(21a) (21b)
is called non-linear peaky fuzzy number. It should be noted that, LC (x) and RC (x) are non-linear functions of x, therefore, C
is a crisp number, denoted by B, then we have Especially, when B x x 0 × B, L A , R for B > 0, A A B B
= A˜ × B = (22) C x x A0 × B, RA , LA for B < 0. B B So the solution of A˜ × B is also a linear peaky fuzzy number with real value A0 B, left fuzzy amplitude aL B, and right fuzzy amplitude aR B, respectively, for B > 0. Otherwise, the left fuzzy amplitude and the right fuzzy amplitude should be aR B and aL B, respectively, for B < 0. For B = 0, one has: A˜ × B = 0.
are both zero, as shown in Fig. 4, one has (b) When the real values of the peaky fuzzy numbers A˜ and B A˜ = 0, LA (x), RA (x) ,
= 0, LB (x), RB (x) , B where
(23)
(24a) (24b)
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Fig. 4. The peaky fuzzy number A˜ with zero real value.
1 + x , aL LA (x) = 0, 1 − x , aR RA (x) = 0, 1 + x , bL LB (x) = 0, 1 − x , bR RB (x) = 0,
−aL < x 0,
(25a)
x −aL , 0 x < aR ,
(25b)
x aR , −bL < x 0,
(25c)
x −bL , 0 x < bR ,
(25d)
x bR .
are neither positive fuzzy numbers, nor negative ones. The perturbation quantities, as mentioned in Eqs. (2) and (3), So A˜ and B belong to this kind of fuzzy numbers. The operation of multiplication on these two fuzzy numbers is given by:
= A˜ × B
= 0, LC (x), RC (x) , (26) C
are defined by Eqs. (24) and (25). According to Eqs. (18), the λ-cut set of C
is given by: where A˜ and B L R Cλ , Cλ = −cL (λ − 1)2 , cR (λ − 1)2
(27)
in which cL = max{aL bR , aR bL },
(28a)
cR = max{aL bL , aR bR }.
(28b)
can be derived as follows: According to the fuzzy representation principle, the membership function of C −x 1− , −cL < x 0, LC (x) = cL 0, x −cL , x 1 − , 0 x < cR , RC (x) = cR 0, x cR .
(29a)
(29b)
leads to (4) The arithmetic operation of addition on linear peaky fuzzy number A˜ and the non-linear peaky fuzzy number C
= A˜ + C.
D
(30)
is The λ-cut set of D L R Dλ , Dλ = −cL (λ − 1)2 + aL (λ − 1), cR (λ − 1)2 − aR (λ − 1) .
(31)
and its membership functions can be derived as follows: Thus the non-linear peaky fuzzy number D,
= 0, LD (x), RD (x) , D
(32)
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
LD (x) =
RD (x) =
1+
aL −
2 − 4c x aL L
2cL
0, 1+
aR −
2 + 4c x aR R
2cR
0,
,
−dL < x 0,
1005
(33a)
x −dL , , 0 x < dR ,
(33b)
x dR ,
where dL = aL + bL , dR = aR + bR .
i.e.: (5) When A˜ in Eq. (30) is also a non-linear peaky fuzzy number, and its membership function is similar to that of C, (34) A˜ = 0, LA (x), RA (x) , 1 − − x , −a < x 0, L LA (x) = aL 0, x −aL , x 1 − , 0 x < aR , RA (x) = aR 0, x aR .
(35a)
(35b)
where C
is still the same as that in Eqs. (29), leads to: The sum of A˜ and C,
= A˜ + C.
D
is: Then the λ-cut set of D L R Dλ , Dλ = −(aL + cL )(λ − 1)2 , (aR + cR )(λ − 1)2 .
(36)
(37)
and its membership functions, LD (x) and RD (x), According to the fuzzy representation principle, the peaky fuzzy number D are given by:
= 0, LD (x), RD (x) , (38) D 1 − − x , −d < x 0, L LD (x) = dL 0, x −dL , x 1 − , 0 x < dR , RD (x) = dR 0, x dR ,
(39a)
(39b)
respectively, and where dL and dR are the left and right fuzzy amplitudes of the fuzzy number D, dL = aL + cL ,
(40a)
dR = aR + cR .
(40b)
from A˜ The solution of subtraction of C
= A˜ − C
= 0, LD (x), RD (x) D
(41)
i.e., the membership functions LD (x) and RD (x) in Eq. (41) are the same is similar in form to that of the addition of A˜ and C, in form as those given by Eqs. (39a) and (39b) respectively, on condition that the left and right fuzzy amplitudes are evaluated by: dL = aL + cR ,
(42a)
dR = aR + cL .
(42b)
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L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
3. Fuzzy variational principle For the sake of simplicity, indicial notation is employed with indices repeated twice implying summation unless otherwise noted.
Poisson’s ratio µ, When an engineering structure possesses fuzzy characteristic parameters, such as Young’s modulus E, ˜
i , boundary traction T i , the structure are and cross sectional dimensions, etc., under fuzzy static loads, including body force F destined to yield fuzzy responses. That is to say, the stress components σ˜ ij , strain components ε˜ ij , displacement components u˜ i ,
, of the linear structures are all fuzzy quantities. and the overall potential energy Π Several investigators considered that the virtue displacements, δ u˜ i , are independent of the material properties and the loading conditions of the structure, thus independent of any fuzziness. It is noteworthy that this is only true in the case of deterministic boundary conditions, since the virtue displacements are required to satisfy the boundary conditions of the concerned structures. In actuality, the imperfection and uncertainty exist in structural boundary conditions, and hence affect the static and dynamic behaviour of the structures (Li and Li, 2001; Yang et al., 2001). Under this condition, the virtue displacements are certainly not independent of uncertainty. (Ohira, 1961), pointed out that the different boundary conditions might cause large variations in the critical buckling loads for isotropic thin-walled shells. Arbocz (2000) investigated the effect of imperfect boundary conditions on the collapse behaviour of anisotropic shells. Furthermore, the idealization of end situations of different structures, such as clamped ends, or simply supported ends, sometimes is not so reasonable, if it cannot be regarded as unacceptable. For example, the supports with finite rigidity are often idealized to be infinitely rigid. Li et al. (1993) indicated the existence of fuzziness of structural boundary conditions. Cherki et al. (2000) investigated the fuzzy behaviour of mechanical systems with uncertain boundary conditions. Since the virtue displacements are dependent on boundary conditions, they should be treated as fuzzy quantities as this paper does. The fuzzy functional of the overall potential energy of the structure is written as ˜ F
i u˜ i ) dV − T i u˜ i dS,
= (A− (43) Π V
Sσ
where displacements and strains, denoted by u˜ i , ε˜ i , respectively, are fuzzy quantities, too. A˜ denotes strain energy, and can be expressed as 1 A˜ = a˜ ij kl ε˜ ij ε˜ kl , 2 σ˜ ij = a˜ ij kl ε˜ kl .
(44a) (44b)
consisting of all the fuzzy parameters of the system including material property coefficients, geometry A fuzzy field {X}, parameters, and external forces, etc., is expressed as
2 , . . . , X
r ]T ,
= [X
1 , X {X}
(45)
can be decomposed as follows where r is the number of fuzzy parameters. The fuzzy field {X} 0
= X + {β}, ˜ {X}
(46)
where T 0 0 0 X = X1 , X2 , . . . , Xr0 ,
(47a)
˜ = [β˜1 , β˜2 , . . . , β˜r ]T , {β}
(47b)
where β˜m (m = 1, 2, . . . , r) are fuzzy perturbation quantities with zero real values. Displacement quantity u˜ i is a dependent
via
so one can expand u˜ i about the real value {X}0 of the fuzzy field {X} variable, which is a function of the fuzzy field {X}, Taylor series, and retain up to second-order terms only 1 u˜ i = u0i + β˜m ui,m + β˜m β˜n ui,mn , 2
m, n = 1, 2, . . . , r,
(48)
where u0i represents ui evaluated at the real value of the fuzzy field, {X0 }. ui,m denotes the partial derivative of ui with respect to the field variable Xm , evaluated at {X0 }. Similarly, ui,mn denotes the second partial derivative of ui with respect to Xm and ˜ Π
i , T i and A,
etc. When the second-order Xn , evaluated at {X0 }. Similar expansions are made for other fuzzy quantities u˜ i , F perturbation technique is employed, the substitution of the expanded expressions of the fuzzy quantities into Eq. (43) will yield:
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
• Zeroth-order potential energy: 0 Π0 = A − u0i Fi0 dV − u0i Ti0 dS. V
1007
(49a)
Sσ
• First-order potential energy: 0 Π,m = A,m − u0i Fi,m − ui,m Fi0 dV − ui Ti,m + ui,m Ti0 dS. V
(49b)
Sσ
• Second-order potential energy: A,mn − u0i Fi,mn − ui,m Fi,n − ui,n Fi,m − ui,mn Fi0 dV Π,mn = V
−
0 ui Ti,mn + ui,m Ti,n + ui,n Ti,m + ui,mn Ti0 dS.
(49c)
Sσ
˜ a˜ ij kl , ε˜ ij into Eq. (44a), and employing the perturbation Substituting the second-order expansion of the fuzzy quantities A, technique, one has 1 0 0 0 ε ε , A0 = aij 2 kl ij kl 1 0 ε0 + a0 ε 0 A,m = aij kl,m εij kl ij kl ij,m εkl , 2 1 0 ε0 + a 0 0 0 0 0 A,mn = aij kl,mn εij ij kl,m εij,n εkl + aij kl,n εij,m εkl + aij kl εij,m εkl,n + aij kl εij,mn εkl . kl 2
(50a) (50b) (50c)
Similarly, the following equations can be derived from Eq. (44b): 0 ε0 , σij0 = aij kl kl
(51a)
0 + a0 ε σij,m = aij kl,m εkl ij kl kl,m , 0 +a 0 σij,mn = aij kl,mn εkl ij kl,m εkl,n + aij kl,n εkl,m + aij kl εkl,mn .
(51b) (51c)
Comparing Eqs. (51) with the partial derivatives of A0 , A,m , A,mn in Eqs. (50), the following expressions will be attained. ∂A0 0 ∂εij
=
∂A,m 0 ∂εij
=
∂A,mn 0 ∂εij
∂A,m ∂A,mn = = σij0 , ∂εij,m ∂εij,mn ∂A,mn = σij,m , ∂εij,n
= σij,mn ,
(52a) (52b) (52c)
where m and n are always the free indices, and denote partial differential instead of summation even if they repeat twice in the same term. Eqs. (52a)–(52c) can be regarded as the perturbation expressions of the fuzzy constitutive equation, which is shown as ∂ A˜ = σ˜ ij . ∂ ε˜ ij
(53)
Similarly, one can transfer the fuzzy relationship equations of strain-displacement 1 ε˜ ij = (u˜ i,j + u˜ j,i ) 2 into a set of deterministic ones, by means of the perturbation technique
(54)
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L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
1 0 ui,j + u0j,i , 2 1 1 εij,m = (ui,j m + uj,im ) = (ui,j + uj,i ),m , 2 2 1 1 εij,mn = (ui,j mn + uj,imn ) = (ui,j + uj,i ),mn . 2 2 0 = εij
According to Eq. (49a), the first variation of Π 0 reads 0 ∂A 0 − F 0 δu0 dV − δε Ti0 δu0i dS. δΠ 0 = ij i i 0 ∂εij V
(55a) (55b) (55c)
(56)
Sσ
Substituting Eq. (55a) into Eq. (56), and then integrating the first term at right-hand by parts, the following expression can be attained by means of Gauss–Ostrogradski theorem 0 0 ∂A ∂A 0 δu0 dV + 0 δu0 dS. + F n − T (57) δΠ 0 = − i i i i 0 0 j ∂εij ∂εij ,j V
Sσ
The stationary condition δΠ 0 = 0 leads to
∂A0
0 ∂εij ,j
∂A0 0 ∂εij
+ Fi0 = 0 in V ,
nj − Ti0 = 0 on Sσ .
(58a)
(58b)
Substituting Eq. (52a) into Eqs. (58) yields 0 + F 0 = 0 in V , σij,j i
(59a)
σij0 nj = Ti0
(59b)
on Sσ .
These two equations are the same as the deterministic equilibrium equation and the boundary condition of surface traction. Similarly, the first-order variation of Eq. (49b) leads to ∂A,m 0 ∂A,m 0 − F 0 δu δε + δε − F δu Ti,m δu0i + Ti0 δui,m dS dV − δΠ,m = ij,m i,m i,m ij i i 0 ∂ε ∂εij ij,m V Sσ ∂A ∂A,m ,m 0+ 0 δu = − + F + F δu dV i,m i,m i i ∂εij,m ,j ∂ε0 ,j V
+ Sσ
ij
0 + ∂A,m n − T 0 δu n − T δu i,m j i,m dS. i i 0 j ∂εij,m ∂εij
∂A,m
(60)
Since δu0i , and δui,m in Eq. (60) are arbitrary and independent, the stationary condition δΠ,m = 0 leads to
σij,j m + Fi,m = 0 in V ,
(61a)
on Sσ , σij,m nj = Ti,m 0 0 σij,j + F i = 0 in V ,
(61b)
σij0 nj = Ti0
on Sσ .
(61c) (61d)
Clearly, Eqs. (61c), (61d) are equivalent to Eqs. (59a), (59b). That is to say, the stationary conditions of Π,m include the stationary conditions of Π 0 . The second-order variation of Eq. (49c) leads to ∂A,mn 0 ∂A,mn ∂A,mn 0 − F δu 0 δu δε + δε + δε − F δu − F δΠ,mn = dV ij,n ij,mn i,mn i,m i,n i,mn ij i i ∂εij,n ∂εij,mn ∂ε0 V
ij
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
−
1009
Ti,mn δu0i + Ti,m δui,n + Ti0 δui,mn dS
Sσ
∂A,mn ∂A,mn 0+ 0 δu + F + F + + F δu δu i,mn i,m i,n i,mn dV i i 0 ∂εij,n ,j ∂εij,mn ,j ∂εij ,j V ∂A,mn ∂A,mn ∂A,mn 0 0 (62) nj − Ti,mn δui + nj − Ti,m δui,n + nj − Ti δui,mn dS. + ∂εij,n ∂εij,mn ∂ε0
= −
Sσ
∂A,mn
ij
Substituting Eqs. (52) into Eq. (62), the stationary conditions δΠ,mn = 0 leads to σij,j mn + Fi,mn = 0 in V ,
σij,mn nj = Ti,mn
on Sσ ,
(63a) (63b)
σij,j m + Fi,m = 0 in V ,
(63c)
σij,m nj = Ti,m
(63c)
on Sσ ,
0 σij,j + Fi0 = 0
in V ,
(63e)
σij0 nj = Ti0
on Sσ .
(63f)
It should be pointed out that, Eqs. (63e), (63f) are equivalent to Eqs. (61c), (61d) and Eqs. (59a), (59b), while Eqs. (63c), (63d) are equivalent to Eqs. (61a), (61b). Therefore, it can be concluded that the stationary conditions of the second-order perturbation expression Π,mn include the stationary conditions of the zeroth- and first-order perturbation expressions, Π 0 and Π,m . Accordingly, when engineering structures with fuzzy parameters are analysed by the fuzzy variational method, only the highest order perturbation expression Π,mn is necessary and efficient to derive all the required governing equations.
4. Fuzzy Ritz method In structural analysis, it is usually difficult or impossible to seek an exact analytical solution of the governing equations. One often tries to establish the equivalent functional so as to obtain readily the approximate solutions by means of the variational techniques. A simply supported elastic beam with uniform cross sections and length l, as shown in Fig. 5, is considered here
and as an example to illustrate the application of the fuzzy variational principle. For this beam, the Young’s modulus, E,
is E 0 , its possible low and high values are Et and Es , external load, q, ˜ are considered as fuzzy parameters. The real value of E respectively. While the real value of q˜ is q 0 , its possible low and high values are qt and qs , respectively. According to Eqs. (5)
denoted by eL and eR , respectively, and those of q, ˜ denoted by qL and qR , and (6), the left and right fuzzy amplitudes of E,
q˜ are respectively, can all be worked out readily. Then the expressions, or say the membership functions, of fuzzy numbers E,
and q˜ can be decomposed at their real values, respectively, as follows determined according to Eqs. (1) and (4). E
1 = E
= E 0 + β˜1 , X (64)
2 = q˜ = q 0 + β˜2 , X where, β˜1 , β˜2 are zero-real fuzzy parameters (Yang, 1998), whose real values are zero:
Fig. 5. An elastic beam with fuzzy characteristic quantities.
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L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
β˜1 = 0, Le (x), Re (x) , β˜2 = 0, Lq (x), Rq (x) , where
1 + x , eL Le (x) = 0, 1 − x , eR Re (x) = 0, 1 + x , qL Lq (x) = 0, 1 − x , qR R q (x) = 0,
(65a) (65b)
−eL < x 0, x −eL , 0 < x eR , x eR , −qL < x 0, x −qL , 0 < x qR , x qR .
For convenience, one adopts the following identities eL = eR = γe ,
(66a)
qL = qR = γq .
(66b)
The functional of the total potential energy for the beam with fuzzy parameters and external load (fuzzy beam) is
= Π
l
1 2 Dχ˜ − q˜ w
dx, 2
(67)
0
where
= EI,
D χ˜ = −
(68a)
d2 w dx 2
.
(68b)
is obtained according to the perturbation techniques The second-order expansions of the fuzzy quantities Π l Π,ij = 0
2 1 D,ij χ 0 + D,i χ,j χ 0 + D,j χ,i χ 0 + D 0 χ,i χ,j + D 0 χ,ij χ 0 2
− q,ij w0 + q,i w,j + q,j w,i + q 0 w,ij dx,
i, j = 1, 2.
(69)
It is assumed the admissible displacement function of the beam as w
=
r m=1
mπx A˜ m sin . l
(70)
As being the fuzzy quantities, A˜ m and w
can be expanded via Taylor series w
= w0 +
2 i=1
A˜ m = A0m +
2 1 ˜ ˜ β˜i w,i + βi βj w,ij , 2
2 i=1
(71a)
i,j =1
2 1 ˜ ˜ β˜i Am,i + βi βj Am,ij . 2 i,j =1
Substituting Eqs. (71) into Eq. (70), one has
(71b)
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
r mπx 0= w A0m sin , l m=1 r mπx Am,i sin , w,i = l m=1 r mπx = Am,ij sin , w ,ij l
1011
(72a)
m=1
r 2 2 0= 0 m π sin mπx , χ A m l l2 m=1 r m2 π 2 mπx Am,i 2 sin , χ,i = l l m=1 r m2 π 2 mπx = A sin . χ ,ij m,ij 2 l l
(72b)
m=1
Substituting Eqs. (72) into Eq. (69), one obtains l r m2 n2 π 4 mπx nπx Π,ij = sin sin D,ij A0m A0n + 2D,i A0m An,j + 2D,j A0m An,i + 2D 0 Am,i An,j 4 l l 2l m,n=1
0
+ 2D 0 A0m An,ij −
r k=1
kπx 0 0 sin q,ij Ak + q,i Ak,j + q,j Ak,i + q Ak,ij dx, l
i, j = 1, 2.
(73)
The stationary condition of the variational principle leads to ∂Π,ij m4 π 4 0 0 lq 0 =0⇒ D Am + (−1)m − 1 = 0, 3 ∂An,ij mπ 2l
(74a)
4 4 lq ,j ∂Π,ij m4 π 4 0 0m π A = 0 ⇒ D,j A + D (−1)m − 1 = 0, m,j + m ∂An,i mπ 2l 3 2l 3
(74b)
m4 π 4 lq,ij 0 +D A 0A = 0 ⇒ D A + D A + D + (−1)m − 1 = 0. ,ij ,i m,j ,j m,i m,ij m 0 3 mπ 2l ∂An
∂Π,ij
(74c)
These are the deterministic recursive equations of the fuzzy Ritz method applied in the elastic beam as shown in Fig. 5, which can be solved for coefficients A0m , Am,j , Am,ij as follows: (a) when m are even numbers, one has A0m = Am,j = Am,ij = 0 for i, j = 1, 2;
(75)
(b) when m are odd numbers, one has 4l 4 q 0 , m5 π 5 D 0 4l 4 q,j 4l 4 q 0 D ,j D ,j 4l 4 Am,j = 5 5 0 − 5 5 0 2 = 5 5 0 q,j − 0 q 0 , m π D m π (D ) m π D D 4 2D,i D,j 0 4l q Am,ij = 5 5 0 2 D 0 q,ij − D,ij q 0 − q,j D,i − q,i D,j + for i, j = 1, 2. m π (D ) D0
A0m =
and q, For this beam, the fuzzy field is composed of two fuzzy quantities, E ˜ then according to Eq. (68a), one has D 0 = E 0 I, D,1 = I, D,2 = D,11 = D,12 = D,22 = 0.
(76a) (76b) (76c)
(77)
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1 = E,
such that q˜ is independent of X q,1 = 0, q,2 = 1,
(78)
q,11 = q,12 = q,22 = 0.
Substitution of Eqs. (77) and (78) into Eqs. (76) leads to the solutions of A0m , Am,j , and Am,ij , which are shown as follows (a) When m are even numbers, one has A0m = 0,
Am,i = 0,
Am,ij = 0 for i, j = 1, 2;
(79a)
(b) When m are odd numbers, one has A0m =
4l 4 q 0 , m5 π 5 D 0
Am,11 =
4l 4 I q 0 Am,1 = − 5 5 0 2 , m π (D ) Am,2 =
4l 4 , m5 π 5 D 0
8l 4 q 0 m5 π 5 D 0 (E 0 )2
,
Am,22 = 0,
(79b)
4l 4 I Am,21 = Am,12 = − 5 5 0 2 . m π (D )
Substituting coefficients A0m , Am,i , Am,ij into Eq. (72a), w0 , w,i , w,ij are given by r
w0 =
m=1
4l 4 q 0 mπx sin , l m5 π 5 D 0
(80a)
r 4l 4 I q 0 1 mπx 1 sin = − 0 w0 , w,1 = − 5 0 2 l π (D ) m5 E
(80b)
r 1 4l 4 1 mπx = 0 w0 , sin l π 5 D0 m5 q
(80c)
m=1
w,2 =
m=1 4 8l q 0
r 1 mπx 2 sin = 0 2 w0 , l π 5 D 0 (E 0 )2 m5 (E )
(80d)
r 1 4l 4 mπx 1 sin = − 0 0 w0 , w,12 = w,21 = − 5 0 0 l π D E m5 q E
(80e)
w,22 = 0.
(80f)
w,11 =
m=1
m=1
Substitution of w0 , w,i , w,ij given by Eqs. (80) into Eq. (71a), the fuzzy displacement of the beam w
is obtained 0 w
= w , Lw (x), Rw (x) , where Lw (x) =
1+
a−
a 2 − 4c(x − w0 ) , 2c
(81)
w0 − (a + c) < x w0 ,
0, x w0 − (a + c), a − a 2 + 4c(x − w0 ) 1+ , w0 x < w0 + (a + c), R w (x) = 2c 0, x w0 + (a + c) in which w,11 γE2 − 2w,12 γE γq . 2 For threshold λ ∈ [0, 1], the λ-cut of the displacement w
is 2 2 w
λ = w0 − cλ + (a + 2c)λ − a − c, w0 + cλ − (a + 2c)λ + a + c . a = −w,1 γE + w,2 γq ,
c=
(82)
(83)
Therefore the possible range of the displacement of the beam can be attained corresponding to the assumption level λ of confidence. It surely provides more useful messages for engineers to analyze and design structures.
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5. The fuzzy finite element From now on, matrix notation is adopted instead of tensor notation for the sake of clarity. The characteristic equations for the fuzzy finite element method will be derived according to the fuzzy variational principle as follows.
is shown as Eq. (45). The fuzzy functional Considering a linear engineering structure under static load, the fuzzy field {X} of total potential energy of the linear elastic structure is
ε} dV − {u}
} dV ,
= 1 {˜ε}T [D]{˜ ˜ T {T } dS − {u} ˜ T {F (84) Π 2 V
Sσ
V
{T }, {F
}, and Π
denote the structural displacement, stress, elastic matrix, surface traction vector, body where {u}, ˜ {˜ε}, [D],
force vector, and functional of total potential energy, respectively. They are all functions of the fuzzy field {X}. When analysing the structure based on the finite element method, the interested displacement field should be discretized into elements. And then the displacement field {u} ˜ of a typical element is interpolated by polynomials {u} ˜ =
t
Ni (x)˜ai = [N]{a}, ˜
(85)
i=1
where
[N] = N1 (x), N2 (x), . . . , Nt (x) , {a} ˜ = [˜a1 , a˜ 2 , . . . , a˜ t ]T
in which t denotes the number of element nodes. a˜ i denotes the value of displacement vector {u} ˜ at node i. [N] represents the matrix of shape function constituted by polynomials. Since {a} ˜ can be expanded via the second-order perturbation techniques as follows r r 1 β˜i {a}i + β˜i β˜j {a}ij . {a} ˜ = a0 + 2 i=1
One has 0 0 u = [N] a , {u}i = [N]{a}i , {u}ij = [N]{a}ij ,
(86)
i,j =1
i, j = 1, 2, . . . , r.
(87)
The strain vector {˜ε} and the displacement vector {u} ˜ satisfy the following relationship {˜ε} = L{u} ˜ = L[N]{a} ˜ = [B]{a}, ˜
(88)
where L is a differential operator; [B] denotes the strain matrix. Substituting the second-order perturbation expansions of {˜ε} and {a} ˜ into Eq. (88), one obtains 0 0 ε = [N] a , i, j = 1, 2, . . . , r. (89) {ε}i = [N]{a}i , {ε}ij = [N]{a}ij ,
{T }, {F
}, and Π
into Substituting the second-order perturbation based expansions of the fuzzy quantities {u}, ˜ {˜ε}, {D}, Eq. (84), Π,ij is attained via the perturbation techniques as follows: 0 T T 1 0 T T T 0 Π,ij = [B] [D]ij [B] a 0 + 2{a}T [B] [D]i [B]{a}j a i [B] D [B]{a}j + 2 a 2 V
T T + 2 a 0 [B]T [D]j [B]{a}i + 2 a 0 [B]T D 0 [B]{a}ij dV 0 T T T T T T 0 dV [N]T {F }ij + {a}T a − i [N] {F }j + {a}j [N] {F }i + {a}ij [N] F V
− Sσ
0 T T T T T T 0 dS. [N]T {T }ij + {a}T a i [N] {T }j + {a}j [N] {T }i + {a}ij [N] T
(90)
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According to the stationary conditions of Π,ij , ∂Πij /∂{a 0 }, ∂Πij /∂{a}i , and ∂Πij /∂{a}ij should vanish. i.e., ∂Πij = [B]T D 0 [B] dV a 0 − [N]T F 0 dV − [N]T T 0 dS = 0 ∂{a}ij V
V
one has 0 0 0 K a − P = {0}, where
where
(92)
K 0 = V [B]T D 0 [B] dV , 0 P = V [N]T F 0 dV + Sσ [N]T T 0 dS, T ∂Π,ij = [B] [D]i [B] a 0 + [B]T D 0 [B]{a}i dV − [N]T {F }i dV − [N]T {T }i dS ∂{a}j V V Sσ = [K]i a 0 + K 0 {a}i − {P }i = {0}, i = 1, 2, . . . , r,
(93)
[K]i = V [B]T [D]i [B] dV , {P }i = V [N]T {F }i dV + Sσ [N]T {T }i dS, ∂Πij [B]T [D]ij [B] a 0 + [D]i [B]{a}j + [D]j [B]{a}i + D 0 [B]{a}ij dV = 0 ∂{a } V − [N]T {F }ij dV − [N]T {T }ij dS = [K]ij a 0 + [K]i {a}j + [K]j {a}i V Sσ + K 0 {a}ij − {P }ij = {0},
where
(91)
Sσ
(94)
i, j = 1, 2, . . . , r,
[K]ij = V [B]T [D]ij [B] dV , {P }ij = V [N]T {F }ij dV + Sσ [N]T {T }ij dS.
Therefore, the stiffness equations for the fuzzy finite element method are of a set of deterministic recursive equations as follows K 0 a0 = P 0 , (95) K 0 {a}i = {P }i − [K]i a 0 , i = 1, 2, . . . , r, 0 0 K {a}ij = {P }ij − [K]ij a − [K]i {a}j − [K]j {a}i , i, j = 1, 2, . . . , r. Solving Eq. (95) recursively for coefficients {a 0 }, {a}i , and {a}ij , and substituting them into Eq. (86), the fuzzy nodal displacements of the element are attained. Then the stress vector {σ˜ } is obtained via the perturbation techniques as follows r r 1 {σ˜ } = S 0 a 0 + β˜i [S]i a 0 + S 0 {a}i + β˜i β˜j [S]ij a 0 + [S]i {a}j + [S]j {a}i + S 0 {a}ij , 2
(96)
i,j =1
i=1
where 0 0 S = D [B],
[S]i = [D]i [B],
[S]ij = [D]ij [B].
6. Numerical examples
Example 1. The first example considers the elastic beam shown in Fig. 5 with length l. The real values of Young’s modulus E and the external load q˜ are taken as E 0 and q 0 , respectively, according to the opinions of experts. The possible low and high
and q˜ are also determined by the opinions of experts, which are given by values of E
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
1015
El = 0.8E 0 ,
(97a)
Eh = 1.2E 0 , ql = 0.85E 0 , qh = 1.15q 0 .
(97b) (98a) (98b)
Here the fuzzy Ritz method is employed to evaluate the fuzzy displacements of the beam.
and q˜ can be evaluated, and given by According to Eqs. (5), (6) and (66), the fuzzy amplitudes of E γe = 0.4E 0 ,
(99a)
γq = 0.3q 0 .
(99b)
Substitution of Eqs. (97)–(99) into Eqs. (80) yields the solutions of w0 , w,i , w,ij . Substituting these quantities into Eqs. (82), the coefficients a and c are given by a = 0.7w0 ,
(100a)
c = 0.28w0 .
(100b)
Substituting Eqs. (100) into Eq. (81), the fuzzy displacement of the beam is determined w
= w0 , Lw (x), Rw (x) ,
(101)
where 1 4l 4 q 0 mπx sin , 5 0 l π D m5 m=1,3,5,..., 5√ 9 161 − 112x, ¯ 0.02 < x¯ 1, − Lw (x) = 4 28 0, x¯ 0.02, √ 5 9 112x¯ − 63, 1 x¯ < 1.98, − Rw (x) = 4 28 0, x¯ 1.98
w0 =
in which x¯ = x/w0 . When applying the fuzzy numerical methods to the analysis of real engineering structures, the most important information required is the λ-cut sets of the fuzzy responses of the structures. According to Eq. (83), the λ-cut of the fuzzy displacement w
shows w
λ = w0 − cλ2 + (a + 2c)λ − a − c, w0 + cλ2 − (a + 2c)λ + a + c (102) = −0.28λ2 + 1.26λ + 0.02, 0.28λ2 − 1.26λ + 1.98 w0 . Evidently, the threshold λ = 1 will lead to a result of w
= w0 . It is a deterministic result corresponding to the conventional deterministic solution when all kinds of uncertainties of the structure are neglected. If the threshold λ is less than 1, the λ-cut set of the fuzzy displacement w
is shown as an interval. For example, λ = 0.8 and λ = 0.5 will lead to 0 w)λ=0.5 = 0.58w0 , 1.42w0 , (103) (
w)λ=0.8 = 0.85w , 1.15w0 and (
w)λ=0.5 , as illustrated in Fig. 6. respectively. (
w)λ=0.8 ⊆ (
On the other hand, the deterministic analytical solution of the beam neglecting all the fuzzy properties is used herein as a benchmark to demonstrate the effective and accuracy of the fuzzy numerical methods. The analytical solution for the mid-span deflection of the beam is given by 5 ql 4 . (104) 384 EI Therefore, when the Young’s modulus E and the external load q are both regarded as fuzzy quantities, the fuzzy displacement is obtained as follows w=
w
=
˜ 4 5 ql .
384 EI
(105)
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L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
Fig. 6. w
and its λ-cut sets at different levels of confidence. Table 1 Deflection at the centre of the beam at various levels of confidence Threshold FRM FFEM AM Multiplier
λ=1
λ = 0.9
λ = 0.8
λ = 0.7
λ = 0.6
λ = 0.5
1 1 1
[0.927,1.073] [0.925,1.073] [0.933,1.073]
[0.849,1.151] [0.846,1.150] [0.870,1.152]
[0.765,1.235] [0.760,1.235] [0.813,1.239]
[0.675,1.325] [0.669,1.326] [0.759,1.333]
[0.580,1.420] [0.468,1.424] [0.708,1.438]
0 4 w 0 = 0.01302q l /E 0 I 0
When the thresholds are given as λ = 0.8, and λ = 0.9, respectively, Eq. (105) yields w)λ=0.5 = 0.96w0 , 1.04w0 . (
w)λ=0.8 = 0.87w0 , 1.15w0 and (
(106)
Evidently, results shown in Eqs. (103) obtained by the fuzzy Ritz method are in good agreement with those obtained by the analytical method given by Eqs. (106). Detailed comparison is illustrated in Fig. 6 and Table 1. It is noteworthy that when λ < 0.5, the analytical solution will be meaningless, since the possible range of variations of fuzzy parameters proposed by the experts are enlarged during the construction of membership functions. For example, the possible ranges of the Young’s modulus E and the external load q in this example are (107a) [El , Eh ] = 0.8E 0 , 1.2E 0 , 0 0 (107b) [ql , qh ] = 0.85E , 1.15q
and q˜ given they are enlarged when γe and γq given by Eqs. (99) are evaluated. Therefore, they are equal to the 0.5-cuts of E by Eqs. (64), (65), respectively, as follows
λ=0.5 = 0.8E 0 , 1.2E 0 , (108a) (E) 0 0 (108b) (q) ˜ λ=0.5 = 0.85E , 1.15q . Example 2. The second numerical example considers the same beam discussed in Example 1. The fuzzy finite element method is employed here to evaluate the displacements of the beam with fuzzy parameters. The mid-span deflections of the beam at different levels of confidence are compared with those resulted from the fuzzy Ritz method and the analytical method, as shown in Table 1. The abbreviation of AM denotes the analytical method. The displacements along half span of the beam calculated by the FRM, the FFEM and the AM at different levels of confidence are illustrated in Fig. 7. These results show that the results obtained by the FFEM and the FRM are little conservative, and are in good agreement with those determined by the AM, when the level of confidence λ > 0.7. Moreover, the results by the fuzzy numerical methods and the analytical method at the level λ = 1 are equal to the conventional deterministic solution. These results show that the fuzzy analysis and calculation provide investigators or designers with estimation intervals of the structural responses at different levels of confidence λ. When λ = 1, the results lead to the conventional deterministic solutions. When λ is assigned a value less than 1 (and greater than 0.7), the structural displacements range over a specific interval, which often exceed the deterministic solutions. For example, if the assumption of confidence λ = 0.8 is considered, the maximum values in the interval of displacements are 15% more than the deterministic solutions. Therefore, the fuzzy numerical methods are able to provide more and valuable information in structural design and analysis.
L.F. Yang et al. / European Journal of Mechanics A/Solids 21 (2002) 999–1018
1017
Fig. 7a. Deflection of the beam at 1.0-level of confidence by the three methods.
Fig. 7b. Fuzzy deflection w
of the beam at 0.8-level of confidence.
In the FFEM, the fuzzy field with r uncertain variables is not discretized, though the structure with fuzzy properties may be discretized into an assemblage with n degree of freedom. So the governing equations are consisted of r × r + r + 1 systems of equations, in which, there is one system with the zeroth-order, r systems with the first-order, r × r systems with the secondorder. For this beam, r = 2, so there are only 7 hierarchical systems of equations. When the random distribution function or the probabilistic characteristic parameters of the uncertain variables are known, the uncertain variables may be treated as random parameters, which are modelled to be a random field with r random variables. The perturbation based stochastic finite element method (SFEM) may be employed in this case. However, the random field has to be discretized a priori into s subdomains, which yields a random vector of r × s elements. So the governing equations of SFEM are consisted of r × s + 2 systems of equations, in which, there is one zeroth-order expanded system of equation, r × s first-order systems, one second-order system. For the beam considered in the examples, r = 2, there are 10 hierarchical systems of equations if the random field is discretized into 4 elements, i.e., s = 4. Because each system of equation comprises n equations for either FFEM or SFEM, s is greater than r for real life structures in many cases; the FFEM may reduce the computation effort considerably, as compared with SFEM.
7. Conclusion The FVP provides a solid theoretical foundation for the fuzzy numerical approaches, such as the FRM and the FFEM. The analytical solutions of a bending beam with fuzzy characteristic quantities are presented to serve as benchmark solutions. Hence, investigators or designers are able to check the effectiveness and accuracy of the FFEM and the FRM. The numerical examples reveal that the results obtained by the FFEM and the FRM are little conservative, and agree well with the analytical solutions. The level of confidence λ should be assigned with a value greater than 0.7, since the possible range of variation proposed by experts for fuzzy parameters may be enlarged during the construction of membership functions. The fuzzy analysis and calculation can provide engineers with estimation intervals of structural responses at different levels of confidence λ. Therefore, the fuzzy numerical methods are able to provide more and valuable information in structural design and analysis. The FFEM may reduce the computational efforts considerably, as compared with the perturbation based SFEM. Example 2 shows that the perturbation based FFEM may yield satisfactory results when the possible low or high values of the fuzzy parameters fluctuate within the range of about 25% smaller or greater than their real value, respectively.
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Acknowledgements The work described in this paper was supported by grants from City University of Hong Kong (Project No: 7001176), Guangxi Natural Science Foundation (Contract No: 0135001), and the Foundation of Important Science and Technology Research Project of the State Education Ministry, China.
References Arbocz, J., 2000. The effect of imperfect boundary conditions on the collapse behaviour of anisotropic shells. Internat. J. Solids Structures 37, 6891–6915. Ayyub, B.M., Guran, A., Haldar, A., 1997. Uncertainty Modelling in Vibration, Control and Fuzzy Analysis of Structural Systems. World Scientific, Singapore. Cao, S.S., Sawyer, J.P., 1995. Fuzzy finite element approach for the analysis of imprecisely defined systems. AIAA J. 33, 2364–2370. Cherki, A., Plessis, G., Lallemand, B., Tison, T., Level, P., 2000. Fuzzy behaviour of mechanical systems with uncertain boundary conditions. Comput. Methods Appl. Mech. Engrg. 189, 863–873. Deng, J.L., 1987. Basic Methods for Grey System. The Press of Huazhong University of Science and Technology, Wuhan. de Lima, B.S.L.P., Ebecken, N.F.F., 2000. A comparison of models for uncertainty analysis by the finite element method. Finite Elem. Anal. Des. 34, 211–232. Dubois, D., Prade, H., 1980. Fuzzy Sets and System: Theory and Applications. Academic Press, New York. Elishakoff, I., 1995. Essay on uncertainties in elastic and viscoelastic structures: from A.M. Freudenthal’s criticisms to modern convex modelling. Comput. and Structures 56, 871–895. Elishakoff, I., Ren, Y.J., Shinozuka, M., 1996. Variational principles developed for and applied to analysis of stochastic beams. J. Engrg. Mech. 122, 559–565. Hien, T.D., Kleiber, M., 1990. Finite element analysis based on stochastic Hamilton variational principle. Comput. and Structures 37, 893–902. Lanczos, C., 1964. The Variational Principle in Mechanics. University Toronto Press. Li, G.Q., Cao, H., Li, Q.S., Huo, D., 1993. Theories and Applications of the Reliability in Structural Dynamics. Earthquake Press, Beijing. Li, G.Q., Li, Q.S., 2001. Theory and Its Application of Time-Dependent Reliability of Engineering Structures. Science Press, Beijing. Liu, W.K., Besterfield, G.H., Belyschko, T., 1988. Variational approach to probabilistic finite elements. J. Engrg. Mech. 114, 2115–2133. Matthies, H.G., Bucher, C., 1999. Finite elements for stochastic media problems. Comput. Methods Appl. Mech. Engrg. 168, 3–17. Ohira, H., 1961. Local buckling theory of axially compressed cylindrical shells. In: Proc. of the 11th Japan National Congress of Applied Mechanics, pp. 37–40. Qian, W.C., 1985. The Generalized Variational Principles. Intellectual Press, Beijing. Wang, Q.Y., 1988. Basic element of grey mathematics and its relations with fuzzy and real number. J. Grey System (1), 69–77. Yang, L.F., 1998. Fuzzy stochastic finite element methods and their applications. Ph.D. thesis, Wuhan University of Technology. Yang, L.F., Gao, D.X., Li, G.Q., 1999. Stochastic functional of potential energy developed for stochastic structures under stochastic loads. Chinese J. Appl. Mech. 16 (4), 35–39. Yang, L.F., Leung, A.Y.T., Li, Q.S., 2001. The stochastic finite segment in analysis of shear lag effect on box girder. Engrg. Structures 23, 1461–1468. Zadeh, L.A., 1965. Fuzzy sets. Inform. Control 8, 338–353. Zienkiewicz, O.C., 1978. Numerical Methods in Offshore Engineering. Wiley. Zhu, B., 1999. On the application of reliability theory to design of concrete dams. China Civil Engrg. J. 32 (4), 10–15.