Applied Mathematics and Mechanics (English Edition), 2006, 27(6):823–832c©Editorial Committee of Appl. Math. Mech., ISSN 0253-4827

DYNAMIC CHARACTERISTIC ANALYSIS OF FUZZY-STOCHASTIC TRUSS STRUCTURES BASED

ON FUZZY FACTOR METHOD ANDRANDOM FACTOR METHOD ∗

MA Juan (��), CHEN Jian-jun (���),XU Ya-lan (���), JIANG Tao (��)

(School of Electromechanical Engineering, Xidian University, Xi’an 710071, P. R. China)

(Communicated by YUE Zhu-feng)

Abstract: A new fuzzy stochastic finite element method based on the fuzzy factormethod and random factor method is given and the analysis of structural dynamic charac-teristic for fuzzy stochastic truss structures is presented. Considering the fuzzy random-ness of the structural physical parameters and geometric dimensions simultaneously, thestructural stiffness and mass matrices are constructed based on the fuzzy factor methodand random factor method; from the Rayleigh’s quotient of structural vibration, the struc-tural fuzzy random dynamic characteristic is obtained by means of the interval arithmetic;the fuzzy numeric characteristics of dynamic characteristic are then derived by using therandom variable’s moment function method and algebra synthesis method. Two examplesare used to illustrate the validity and rationality of the method given. The advantage ofthis method is that the effect of the fuzzy randomness of one of the structural parameterson the fuzzy randomness of the dynamic characteristic can be reflected expediently andobjectively.

Key words: fuzzy stochastic truss; fuzzy factor method; random factor method; fuzzyrandom dynamic characteristic

Chinese Library Classification: O327; TU3182000 Mathematics Subject Classification: 62P30; 74P10Digital Object Identifier(DOI): 10.1007/s 10483-006-0613-z

Introduction

The introduction of random variables[1] and fuzzy variables[2−5] has far effects on the de-velopment of probability theory and fuzzy mathematics respectively. Both random variablesand fuzzy variables, however, are mathematic description of some kind of uncertainties merely.When we want to construct reasonable mathematic models for vast cases with fuzzy stochas-tic factor existing objectively, the introduction of fuzzy stochastic variables[6] is necessary forthe development of fuzzy random mathematic theory, so is the fuzzy stochastic finite elementmethod.

Accordingly, many parameters in structural analysis of engineering, such as Young’s module,mass density of material, structural geometric dimensions, boundary conditions and appliedload, may be random variables, fuzzy variables or even fuzzy random variables. Stochasticfinite element deals with randomness, fuzzy finite element deals with fuzziness and fuzzy randomfinite element is used when fuzzy randomness is considered. Wang and Ou[7] did a great lot ofpioneer work in the application of theory of fuzzy random vibration to earthquake engineering,

∗ Received Jul.11, 2004; Revised Jan.3, 2006Project supported by the Natural Science Foundation of Shaanxi Province of China (No.A200214)Corresponding author CHEN Jian-jun, E-mail: jjchen@xidian.edu.cn

824 MA Juan, CHEN Jian-jun, XU Ya-lan, et al.

in addition, they presented research task of fuzzy random vibration of aseismatic structure andconstructed the theory of fuzzy-random vibration and fuzzy-random process. Some researchwas done in order to expound the category of fuzzy-random variables and corresponding fuzzymean value and fuzzy mean variance, thus establishing the base for the further research of fuzzy-random problem in Refs.[8–10]. Lu[9] solved the static problem of a kind of fuzzy stochasticstructures by means of perturbation method. But the structural dynamic analysis with fuzzystochastic parameters has not still been studied.

A new fuzzy stochastic finite element method is given in this paper to study dynamic char-acteristic of truss structures when considering the fuzzy randomness of the structural physicalparameters and geometric dimensions simultaneously (where their values are fuzzy while theirprobabilities are clear, that is, belonging to the first type of fuzzy random variables[9]), inwhich a fuzzy random variable can be described as its fuzzy main value multiplied by its ran-dom factor[11−12] and fuzzy factor[13]. From the Rayleigh’s quotient of structural vibration,upper bound and lower bound of structural fuzzy random dynamic characteristic are derivedby means of interval arithmetic[14], and its fuzzy numeric characteristics are then derived byrandom variable’s moment function method and algebra synthesis method. Through engineer-ing examples, effects of the fuzzy randomness of structural parameters on the fuzzy randomnessof dynamic characteristic are illustrated and some important conclusions are obtained.

1 Description of Fuzzy Random Variables, Fuzzy Factor Method and Ran-dom Factor Method

1.1 Fuzzy random variables and their fuzzy probability characteristicsThere are several definitions of fuzzy random variables, and the definition presented by

Kwakernaak[9] is as follows: �Suppose that (Ω, A, P ) is a probability estimate space, reflection of fuzzy collect X∼ : Ω →

F0(R) = {A∼|A∼ is a limitary closed fuzzy number} is then named after a (K) fuzzy-randomvariable if the following conditions hold: �

(i) for ∀λ ∈ (0, 1], Xλ(ω) and Xλ(ω) are random variables of (Ω, A, P );(ii) for ∀λ ∈ (0, 1], Xλ(ω), Xλ(ω) ∈ Xλ(ω),

where Xλ(ω) = inf Xλ(ω) = inf{x ∈ R|X∼ (ω)(x) ≥ λ}, Xλ(ω) = sup Xλ(ω) = sup{x ∈R|X∼ (ω)(x) ≥ λ}, Xλ(ω) is the λ-level cut of X∼ (ω), X∼ (ω)(x) is the membership function of

X∼ (ω).

From definition above, if X∼ is a (K) fuzzy-random variable, for ∀λ ∈ (0, 1], Xλ(ω) =

[Xλ(ω), Xλ(ω)] is not only a closed interval number but also a random interval.Corresponding to numeric characteristics of conventional random variables, fuzzy random

variables can be described by its numeric characteristics. Its numeric characteristics are stilllimitary closed fuzzy numbers.1.2 Random factor method

The main idea of random factor method[11] is that a random variable is denoted as a deter-minate quantity (its mean value) multiplied by its random factor. Randomness of the variableis represented by random factor. The mean value of the random factor is 1, and its variancecoefficient is that of the random variable.1.3 Fuzzy factor method

The main idea of fuzzy factor method[13] is in the following. Let M = (Mm, α, β)LR bean L-R type fuzzy number[15], where Mm is its main value, α and β are its left and rightspreads, respectively. Taking the grade of membership λ as small as we can to do λ-levelcut, the approximate upper bound MR and lower bound ML of M can be obtained, and theirdeviations from Mm represent the fuzzy imprecision degree of M too. Then M can be described

Dynamic Characteristic of Fuzzy-Stochastic Truss 825

as a bound fuzzy number M = (Mm, ML, MR). Introduce a fuzzy number without dimensionγ = M/Mm as the fuzzy factor of M . Main value of γ is 1, whose value range representing thefuzzy imprecision degree of M too is [γL = ML/Mm, γR = MR/Mm]. Then M = γ · Mm.1.4 Representation of fuzzy random variables by using random factor and fuzzy

factorIf a structural parameter Y is a fuzzy random variable whose value is a fuzzy number while

its probability is clear, it can then be described as Y = y∗ · y, where y∗ is the random factorrepresenting the randomness of Y , y is an L-R type fuzzy number reflecting the fuzziness of Y .Further, the L-R type fuzzy number y is described as its main value ym multiplied by its fuzzyfactor y. Then, Y can be denoted as������������������

Y = y∗ · y = y · y∗ · ym. (1)

2 Construction of Mass Matrix and Stiffness Matrix of Fuzzy StochasticTruss Structures

Structural parameters are fuzzy stochastic variables, so they lead to the fuzzy randomnessof mass matrix, stiffness matrix and even the structural dynamic characteristic in the end. �

At first, fuzzy random finite element analysis of fuzzy random truss structures is presented.Suppose that there are ne bars in the structure and all bars are made of the same kind ofmaterial. Consider the fuzzy randomness of structural physical parameters and geometricdimensions simultaneously. For the eth bar, its Young’s module, mass density, cross-sectionalarea and length are Ee, ρe, Ae, and le, respectively. Ee of the every bar is equal, so is the massdensity ρe. Let Ee = E · E∗ · Em and ρe = ρ · ρ∗ · ρm, where Em and ρm are main values of Ee

and ρe, respectively; E and ρ are fuzzy factors of Ee and ρe respectively, whose main values areall 1.0 and value ranges are [EL/Em, ER/Em] and [ρL/ρm, ρR/ρm] respectively. EL, ER, ρL andρR are lower bound and upper bound of Ee and ρe, respectively, which are obtained by meansof the fuzzy factor method. E∗ and ρ∗ are random factors of Ee and ρe, respectively, and theirmean values are 1.0. ��

Suppose that fuzzy random imprecision degree of the every bar length le is equal. Letle = l · l∗ · lem, where lem is the main value of le, l∗ is the random factor of le and its meanvalue is 1.0. Because the main value of le(e = 1, 2, · · · , ne) is not equivalent though their fuzzyimprecision degree equals, value ranges of their fuzzy factor are not equivalent. Since the valuerange of fuzzy factor of the shortest bar includes those of fuzzy factors of other bars, fuzzyfactor of the shortest bar is taken as fuzzy factor of all bar’s length so that all bar’s fuzzinessis considered fully and completely. Let l be fuzzy factor of the shortest bar, whose main valueis 1.0 and value range is [lL/lm, lR/lm], where lm, lR, and lL are the main value, upper boundand lower bound of the shortest bar, respectively. In the same way, suppose that fuzzy randomimprecision degree of cross-sectional area Ae is equal, and let Ae = A · A∗ · Aem, where Aem

is the main value of Ae, A∗ is its random factor, A is fuzzy factor of the bar with smallestcross-sectional area whose main value is 1.0 and value range is [AL/Am, AR/Am]. Am, AR, andAL are the main value, upper bound and lower bound of the smallest cross-sectional area. ��

Using finite element method, stiffness matrix of the eth structural element in local coordi-nates is

K(e)

=

(EeAe

le

)·[

1 −1−1 1

]

=

(E

lA

)·(

E∗A∗

l∗

)EmAem

lem

[1 −1−1 1

]

=EA

l· E∗A∗

l∗· K (e)#

, (2)

826 MA Juan, CHEN Jian-jun, XU Ya-lan, et al.

where K(e)#

is the determinate part of K(e)

, that is, the stiffness matrix of the eth elementwhen E = Em, A = Aem and l = lem.

Further, stiffness matrix of the eth element in the global coordinates K(e)

∧ and structuralglobal stiffness matrix K are

K(e)

∧ =

(EA

l

) (E∗A∗

l∗

)T (e)TK (e)#T (e)

=

(E

lA

)·(

E∗ · A∗

l∗

)K

(e)#

∧ , (3)

K =ne∑

e=1

Ke

∧ =

(E

lA

)·(

E∗ · A∗

l∗

) ne∑e=1

K(e)#

∧

=

(E

lA

)·(

E∗ · A∗

l∗

)K#, (4)

where T (e) is the coordinate transforming matrix of the eth bar element, T (e)T is its transpose,K

(e)#

∧ and K# are constant matrixes, which are stiffness matrix of the eth element in theglobal coordinates and structural global stiffness matrix respectively when E = Em, A = Aem

and l = lem in Eqs.(3) and (4).In the same way, using fuzzy factor method and random factor method, mass matrix of the

eth element in the global coordinates M(e)

is

M(e)

=12(Ae leρe) · I

=12(Alρ) · (A∗l∗ρ∗)Aemlemρm

= (Alρ) · (A∗l∗ρ∗) · M (e)#, (5)

where M(e)#

is the determinate part of M(e)

when ρ = ρm, A = Aem and l = lem.Integrating mass matrixes of ne elements, the structural global mass matrix M is

M =ne∑

e=1

{12(ρeAe le) · I

}

= (Alρ) · (ρ∗A∗l∗) ·M #, (6)

where M # is constant matrix, that is, the structural global mass matrix when structuralparameters ρ = ρm, A = Aem and l = lem.

3 Analysis of Dynamic Characteristic of Fuzzy Truss Structures

Using the Rayleigh’s quotient of structural vibration, the structural natural frequency ωi is

ω2i =

φTi ·K · φi

φTi ·M · φi

, (7)

where φi is the ith mode shape corresponding to ωi, and it is fuzzy random variable too; φTi is

its transpose; K and M are the structural global stiffness matrix and mass matrix, respectively.

Dynamic Characteristic of Fuzzy-Stochastic Truss 827

Substituting Eq.(4) and Eq.(6) into Eq.(7), then

ω2i =

(E · A) · (E∗A∗)(ρ · A · l2)(ρ∗A∗l∗2)

· φTi ·K# · φi

φTi ·M # · φi

=

(E

ρ · l2

)·(

E∗

ρ∗ · l∗2)· K#

i

M#i

=

(E

ρ · l2

)·(

E∗

ρ∗ · l∗2

)· (ω#

i )2, (8)

where K#i = φT

i · K# · φi is the ith main stiffness when K = K#; M#i = φT

i · M# · φi is theith main mass when M = M#. Since fuzzy-randomness of φi, K

#i and M#

i are fuzzy randomvariables and their fuzzy randomness is correlative completely, their quotient is deterministic,which is fuzzy main value of ω2

i .Because the same fuzzy factor A and random factor A∗ can be counteracted in Eq.(8), it is

expressed asω2

i = ω · ω∗ · ω#2i , (9)

where ω is the fuzzy factor of ω2i , and its value range will determine the fuzzy imprecision

degree of ω2i ; ω∗ = E∗/(ρ∗l∗2) is random factor of ω2

i , and its mean value is still 1.0 accordingto the random variable’s moment function method.

When fuzzy factor l, E, ρ and A are taken as their main values 1.0 simultaneously, the fuzzymain value of ω is obtained, obviously, it is still 1.0. But the value range of ω needs to bedetermined again. Due to l ∈ [lL, lR], E ∈ [EL, ER], ρ ∈ [ρL, ρR] and lL = lL/lm, lR = lR/lm,EL = EL/Em, ER = ER/Em, ρL = ρL/ρm, ρR = ρR/ρm, according to the multiplication anddivision of interval arithmetic[14], then ωL = EL/(ρR l2R), ωR = ER/(ρL l2L), where ωL and ωR

are the lower and supper bounds of ω, respectively. Further, from the extraction of intervalarithmetic[14], it holds that

ωi =√

ω · ω∗ · ω#i

=ω∧ · (ω∗ · ω#2i )1/2, (10)

where ω∧ is fuzzy factor of the ith natural frequency ωi, and its value range is [√

ωL,√

ωR]; ω#i

is main value of ωi; when random factor ω∗ is taken as its mean value 1.0, range value of ωi is[√

ωL · ω#i ,

√ωR · ω#

i ], that is, fuzziness of ω∧ is returned to ωi.In addition, ω2

i possesses fuzzy-randomness and closed fuzzy value interval of ω2i is a random

interval at the same time. According to random variable’s moment function method and algebrasynthesis method[11], mean value μωi and mean variance σωi of fuzzy-random variable ωi areclosed interval fuzzy numbers and they are expressed as

μωi =ω∧ · (ω#i ) · ( μE

μρμZ)

12 {[1 + ν2

Z + ν2ρ + ν2

Zν2ρ − CEρ · νE · (ν2

Z + ν2ρ + ν2

Zν2ρ)1/2]2

− 12[ν2

E + ν2Z + ν2

ρ + ν2Zν2

ρ − 2CEρ · νE · (ν2Z + ν2

ρ + ν2Zν2

ρ)1/2]}1/4, (11)

σωi =ω∧ · (ω#i ) · ( μE

μρμZ)

12 {[1 + ν2

Z + ν2ρ + ν2

Zν2ρ − CEρ · νE · (ν2

Z + ν2ρ + ν2

Zν2ρ)1/2]

− {[1 + ν2Z + ν2

ρ + ν2Zν2

ρ − CEρ · νE · (ν2Z + ν2

ρ + ν2Zν2

ρ)1/2]2

− 12[ν2

E + ν2Z + ν2

ρ + ν2Zν2

ρ − 2CEρ · νE · (ν2Z + ν2

ρ + ν2Zν2

ρ)1/2]}1/2}1/2, (12)

νZ =νl2 =

√4ν2

l + 2ν4l

1 + ν2l

. (13)

828 MA Juan, CHEN Jian-jun, XU Ya-lan, et al.

When ω∧ is taken as its main value 1.0, the upper bound√

ωR and lower bound√

ωL,respectively, then main values, upper bounds and lower bounds of μωi and σωi are obtainedrespectively. CEρ is correlation coefficient of variable E and ρ; ν represents variable’s variationcoefficients; mean values of all random factors are 1.0.

From Eqs.(11) and (12), variation coefficient νωi of ωi is

νωi =σωi

μωi

. (14)

From above, numeric characteristics of fuzzy-random variable of dynamic characteristics areL-R fuzzy number as well. When fuzzy factor ω∧ is taken as its main value, lower limit andupper limit, then main values, lower limits and upper limits of μωi and σωi can be obtained.

4 Examples

Example 1 28-bar planar fuzzy random truss structureAccording to the solution and computational formula above, analytical program of structural

dynamic characteristic is programmed. The material of the truss structure in Fig.1 is steel. Allstructural parameters are fuzzy random variables. Here their fuzzy values are symmetric normalfuzzy numbers M = (Mm, α, β)LR: Young’s module E = (7.0 × 1010 Pa, 0.02 × 1010 Pa, 0.02 ×1010 Pa)LR; mass density ρ = (7.8 × 103 kg/m3

, 0.02 × 103 kg/m3, 0.02 × 103 kg/m3)LR; the

most possible length of bar is showed in Fig.1 and their fuzzy random imprecision degreeequals to {L} = ({l}, 0.02{l}, 0.02{l})LR; bars’ cross-sectional area A = (3.0 × 10−4 m2, 0.02 ×10−4 m2, 0.02 × 10−4 m2)LR. The fuzzy random imprecision degrees of the bars’ length andcross-sectional area are equal each other respectively. Taking grade of membership λ = 1.365×10−11 to do λ-level cut, then fuzzy values of structural parameters are respectively transferredinto bound fuzzy numbers M = (Mm, ML, MR): E = (7.0 × 1010 Pa, 6.9 × 1010 Pa, 7.1 ×1010 Pa); ρ = (7.8 × 103 kg/m3

, 7.7 × 103 kg/m3, 7.9 × 103 kg/m3); A = (3.0 × 10−4 m2, 2.9 ×

10−4 m2, 3.1 × 10−4 m2); {L} = ({l}, 0.9{l}, 1.1{l}). Additionally, probability distributions ofall fuzzy random parameters are normal distributions.

Fig.1 128-bar truss structure (unit: mm)

Dynamic Characteristic of Fuzzy-Stochastic Truss 829

According to the deduction above, the fuzzy factors of all structural parameters and dynamiccharacteristic and the first two fuzzy natural frequencies are listed in Table 1.

Table 1 Value range of fuzzy factors of fuzzy random variables

ModelsStructural parameters Fuzzy factor Natural frequencies

l E ρ ω∧ ω1 ω2

Main value 1 1 1 1 12.81 51.10

Lower bound 0.99 0.986 0.987 0.977 12.52 49.92

Upper bound 1.01 1.014 1.013 1.024 13.12 52.33

In order to compare the effects of the fuzzy randomness of structural parameters on fuzzyrandomness of structural dynamic characteristic, the deterministic model and fuzzy randommodels are adopted. All structural parameters are taken as their main values simultaneously indeterministic model. In fuzzy random models I–VII, structural physical parameters are takenas fuzzy random variables, geometric dimensions are taken as fuzzy random variables and bothof them are fuzzy-random variables, respectively. The mean values and mean square variancesof the first two natural frequencies are computed. All computational results are listed in Table2 and given in the form of M = (Mm, ML, MR), where Mm, MR, and ML are the main value,upper bound and lower bound, respectively.

Table 2 Analytical results of 28-bar planar fuzzy random truss structure (ω/Hz)

Models Characteristics

Natural frequency ω1 ω2

Deterministic model μω1 = 12.81 μω2 = 51.10

Fuzzy stochastic model I μω1 = (12.82, 12.21, 13.98) μω2 = (51.13, 48.70, 53.82)

νE = νρ = 0.1, E = ρ = [0.95, 1.05] σω1 = (0.283, 0.270, 0.297) σω2 = (1.133, 1.077, 1.191)

Fuzzy stochastic model II μω1 = (12.99, 12.55, 13.88) μω2 = (51.82, 50.09, 55.36)

νA = νl = 0.1, A = l = [0.95, 1.05] σω1 = (1.250, 1.190, 1.316) σω2 = (4.986, 4.747, 5.250)

Fuzzy stochastic model III μω1 = (12.81, 12.20, 13.48) μω2 = (51.10, 48.67, 53.79)

νE = νρ = 0.01, E = ρ = [0.95, 1.05] σω1 = (0.058, 0.055, 0.061) σω2 = (0.230, 0.219, 0.242)

Fuzzy stochastic model IV μω1 = (12.81, 12.38, 13.69) μω2 = (51.10, 49.40, 54.60)

νA = νl = 0.01, A = l = [0.95, 1.05] σω1 = (0.128, 0.122, 0.135) σω2 = (0.511, 0.486, 0.538)

Fuzzy stochastic model V μω1 = (12.82, 12.77, 12.90) μω2 = (51.13, 50.94, 51.45)

νE = νρ = 0.1, E = ρ = [0.995, 1.005] σω1 = (0.284, 0.283, 0.286) σω2 = (1.133, 1.129, 1.140)

Fuzzy stochastic model VI μω1 = (14.37, 13.03, 15.91) μω2 = (57.3, 51.95, 63.45)

νA = νl = 0.1, A = l = [0.995, 1.005] σω1 = (1.250, 1.244, 1.258) σω2 = (4.986, 4.961, 5.016)

Fuzzy stochastic model VIIμω1 = (14.33, 13.01, 15.87) μω2 = (57.25, 51.78, 63.42)

νA = νl = νE = νρ = 0.1,

E = ρ = A = l = [0.95, 1.05]σω1 = (1.367, 1.239, 1.514) σω2 = (5.462, 4.954, 6.047)

830 MA Juan, CHEN Jian-jun, XU Ya-lan, et al.

From results of Table 2, it can be seen that(i) From results of model I and model II, when physical parameters and geometric dimen-

sions are fuzzy random variables respectively and their fuzzy random imprecision degrees areequal (value range of fuzzy factor of every parameter and their variance coefficients are equiv-alent respectively), the fuzzy randomness of geometric dimensions has the greater impacts onthe fuzzy randomness of structural natural frequency than that of the physical parameters does(that is, the main value of mean square variance of natural frequency in model II is greater thanthat in model I, in addition, the fuzzy value ranges of mean value and mean square varianceof natural frequency in model II are greater than that in model I). Thus, the fuzzy random-ness of geometric dimensions should not be neglected in the analysis of structural dynamiccharacteristic.

(ii) In models I, III and V, physical parameters are fuzzy random variables. When fuzzyimprecision degree decreases (V) or random dispersion degree decreases (III), compared withthe results of model I, decrease of the randomness (that is, the decrease of variance coefficients)affects more on the randomness of natural frequency (that is, the main values of mean valueand mean square variance), while the fuzziness of natural frequency is always determined bythe fuzzy imprecision of parameters (that is, the value ranges of fuzzy factors).

(iii) In models II, IV and VI, geometric dimensions are fuzzy random variables. Whenfuzzy imprecision degree of geometric dimensions decreases (VI) or their random dispersiondegree decreases (IV), compared with model II, the decrease of the randomness of geometricdimensions affects more on the randomness of natural frequency, while fuzzy dispersal degreeof natural frequency is determined by that of the geometric dimensions (that is, the valueranges of fuzzy factors). Compared with models I and II, considering fuzzy randomness of allstructural parameters simultaneously (model VII), the fuzzy randomness of structural dynamiccharacteristic is greater.

Example 2 4-bar space fuzzy random truss structureIn Fig.2, structural physical parameters and geometric dimensions are still fuzzy random

variables and their fuzzy values and probabilistic distribution are the same as those of Example1. In order to illustrate the influences of the fuzzy randomness of structural parameters on thatof structural dynamic characteristic, two models VIII and IX are taken.

From the results in model VIII and model IX (see Table 3), with the decrease of the fuzzyrandomness of physical parameters and geometric dimensions, the fuzzy randomness of thestructural natural frequency decreases greatly.

Fig.2 4-bar spatial truss (Unit: in; 1 in=25.4 mm)

Dynamic Characteristic of Fuzzy-Stochastic Truss 831

Table 3 Analytical results of 4-bar space fuzzy random truss structure (ω/Hz)

Models Characteristics

Natural frequency ω1 ω2

Deterministic model μω1 = 147.88 μω2 = 156.13

Fuzzy stochastic model VIII μω1 = (147.88, 134.13, 163.77) μω2 = (156.13, 141.61, 172.84)νl = νE = νρ = 0.1,

E = ρ = l = [0.95, 1.05]σω1 = (12.95, 11.75, 14.34) σω2 = (14.89, 13.51, 16.48)

Fuzzy stochastic model IX μω1 = (147.88, 146.40, 149.39) μω2 = (156.13, 154.57, 157.69)νl = νE = νρ = 0.01,

E = ρ = l = [0.995, 1.005]σω1 = (1.431, 1.417, 1.445) σω2 = (1.515, 1.500, 1.530)

5 Conclusions

(1) Fuzzy random finite element modeling of truss structures with fuzzy random physicalparameters and geometric dimensions is presented in this paper, which is a general modeling.Conventional deterministic modeling (fuzzy factors and random factors of fuzzy random vari-ables are taken as their main values 1.0 and mean values 1.0 respectively), fuzzy modeling(random factors of fuzzy random variables are taken as their mean values 1.0) and randommodeling (fuzzy factors of fuzzy random variables are taken as their main value 1.0) are onlyspecial cases of the modeling given here.

(2) Based on concept and property of fuzzy random variables, a first type of fuzzy randomvariable can be described as its fuzzy main value multiplied by its random factor and fuzzy fac-tor. Compared with perturbation method in structural analysis with fuzzy random parameters,one of the merits of the method given here is that the effect of the fuzzy randomness of any oneof structural parameters on the fuzzy randomness of structural dynamic characteristic can beillustrated expediently and objectively, and its computational work is small and conveniently.Additionally, the method given can be extended to the structural static or dynamic responseanalysis directly. Through examples, the method and finite element modeling in this paper arecorrect and valid.

References

[1] Zhao Lei, Chen Qiu. Neumann dynamic stochastic finite element method of vibration for structureswith stochastic parameters to random excitation[J]. Computers & Structures, 2000, 77(6):651–657.

[2] Guo Shuxiang, Lu Zhenzhou. Procedure for computing the possibility and fuzzy probability offailure of structures[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(3):338–343.

[3] Wang Guangyuan. Review of the development of uncertain structural mechanics[J]. Advances inMechanics, 2002, 32(2):205–211 (in Chinese).

[4] Guo Shuxiang, Lu Zhenzhou. A fuzzy reliability approach for structures in the possibility con-text[J]. Chinese Journal of Computational Mechanics, 2002, 19(1):189–193 (in Chinese).

[5] Rao S S, Cao L. Fuzzy boundary element method for the analysis of imprecisely defined system[J].AIAA Journal, 2001, 39(9):1788–1797.

[6] Lu Zhenzhou, Yue Zhufeng, Feng Yunwen. Random-fuzzy model for the ductile/brittle transi-tion[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(10):1158-1167.

[7] Wang Guangyuan, Ou Jingping. Structural fuzzy random vibration under earthquake[J]. ActaMechanica Sinica, 1988, 20(2):173–180 (in Chinese).

832 MA Juan, CHEN Jian-jun, XU Ya-lan, et al.

[8] Lu Enlin, Zhong Youming. Mathematic description about random variable with fuzzy densityfunction(RVFDF)[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(8):957-964.

[9] Lu Enlin. Perturbational solution for fuzzy-stochastic finite element equilibrium equa-tions(FSFEEE)[J]. Applied Mathematics and Mechanics (English Edition), 1997, 18(7):679–688.

[10] Lu Enlin, Zhong Youming. Random variable with fuzzy probability[J]. Applied Mathematics andMechanics (English Edition), 2003, 24(4):491–498.

[11] Chen Jianjun, Che Jianwen, Sun Huaian, Ma Hongbo, Cui Mingtao. Probabilistic dynamic anal-ysis of truss structures[J]. Structural Engineering and Mechanics, 2002, 13(2):231–239.

[12] Gao Wei, Chen Jianjun, Ma Juan. Optimal placement of active bars in active vibration controlfor piezoelectric intelligent truss structures with random parameters[J]. Computers & Structures,2003, 81(1):53–60.

[13] Ma Juan, Chen Jianjun, Huang Ping, Gao Wei. Dynamic response analysis of fuzzy truss underfuzzy excitation[J]. Acta Mechanica Sinica, 37(3):378–384 (in Chinese).

[14] Alefeld G, Claudio D. The basic properties of interval arithmetic, its software realizations andsome applications[J]. Computers & Structures, 1998, 67(1/3):3–8.

[15] Dubois D, Prade H. Fuzzy Sets and System: Theory and Applications[M]. Academic Press, NewYork, 1980.