Non-probabilistic uncertain static responses of imprecisely defined structures with fuzzy parameters

Abstract

In this paper fuzzy finite element method has been used for non-probabilistic static analysis of imprecisely defined structures. Uncertainties are assumed to be present in the external load, material and geometric properties of the structures, which are modelled here through the triangular convex normalized fuzzy sets. In general fuzzy finite element method for static analysis of structures converts the problem into a Fully Fuzzy System of Linear Equations (FFSLE). As such a new method has been proposed based on a linear programming problem approach to solve FFSLE. In this approach first the sign of the solution vector is determined and then accordingly interval based fuzzy arithmetic is used with linear programming to have the final solution. Practical problems viz. six bar truss and rectangular sheet have been taken into consideration for the present analysis. Also the obtained results are compared with the existing results in special cases to illustrate the efficiency and reliability of the proposed method.

Keywords

Fuzzy number triangular fuzzy number fully fuzzy linear system of equations fuzzy finite element method six bar truss rectangular sheet

1 Introduction

From the last few decades finite element method has become a powerful tool for solving the complex systems of various scientific and engineering problems. In this method complicated structures/domains are discretized into small finite elements, giving the element wise behaviour. Assembling all the elements and applying the respective boundary conditions, it gives the output. The system parameters involved in the finite element method such as mass, geometry, material properties, external loads and boundary conditions are considered as crisp or assumed to be defined exactly. But, rather than the particular value we may have only the vague, imprecise and incomplete information about the variables and parameters being a result of errors in measurement, observations, experiment, applying different operating conditions or it may be maintenance induced errors, etc. which are uncertain in nature. Basically these uncertainties can be modelled through probabilistic approach, interval analysis and fuzzy theory.

In probabilistic approach, the variables of uncertain nature are assumed as random variables with joint probability density functions. If the structural parameters and the external load are modelled as random variables with known probability density functions, the response of the structure can be predicted using the theory of probability and stochastic processes by Elishakoff [20]. Also the probabilistic concept is already well established for the extension of the deterministic finite element method towards uncertain assessment. This has led to a number of probabilistic and stochastic finite element procedures (Holder and Mohadevan 2000) [4]. Unfortunately, probabilistic methods are not able to deliver reliable results at the required precision without sufficient experimental data. It may be due to the probability density functions involved in it. As such in the recent decades, interval analysis and fuzzy theory are becoming powerful tools for real life applications. In these approaches, uncertain variables and parameters are represented by the interval and fuzzy numbers, vectors or matrices.

Various aspects of interval analysis along with applications are explained by Moore [32]. If only incomplete information is available, it is possible to establish the minimum and the maximum favourable response of the structures using interval analysis or convex models [13, 22]. Moreover structural analysis with interval parameters using interval based approach has been studied by various authors [39 –41]. Neumaier and Pownuk [36] studied the solution procedure for linear systems with large uncertainties with the applications to truss structures with interval parameters. Sub-interval perturbed finite element method and anti-slide stability analysis method was studied by Guo-jian and Jing-bo [24]. Based on this, formula for computing the bounds of stability factor is given, which provides a basis for estimating and evaluating reasonably anti-slide stability of structures. Muhanna et al. [33] presented an interval approach for the treatment of uncertain parameter for linear static structural mechanics problems. They introduced uncertain parameters in the form of unknown but bounded quantities (intervals). Interval analysis is applied to the Finite Element Method (FEM) to analyze the system response due to uncertain stiffness and loading. A new method called the interval factor method for the finite element analysis of truss structures with interval parameters was proposed by Gao [23]. Chen and Yang [16] proposed a new interval finite element method to solve the uncertain problems of beam structures. Here the beam characteristics are assumed as interval parameters. Also Shu-xiang and Zhen-zhou [45] studied the static linear interval finite element method and proposed a solution procedure for solving the corresponding interval system of linear equations to find the interval static response. They applied the method for the uncertainty analysis of six bar truss structure. Xia and Yu [51] proposed modified interval and subinterval perturbation methods for the static response analysis of structures with interval parameters recently.

As regards, fuzzy set theoretical concept was first developed by Zadeh [53] which is successfully used by different authors in the uncertain analysis of structures. Also various book related to fuzzy mathematics have been written by various authors [30 , 55]. As discussed above, if the structural parameters and the external loads are described in imprecise terms, then the fuzzy theory can be applied. As such Valliappan and Pham [49] used fuzzy logic for the numerical modeling of engineering problems. An optimization algorithm is developed for fuzzy properties by Munck et al. [34] based on response surface for the calculation of fuzzy envelope and fuzzy response functions. Fuzzy structural analysis using α-level optimization is excellently studied by Moller et al. [31]. An important book is written by Hanss [28] in which applications of fuzzy arithmetic into engineering problems are described. Chekri et al. [17] investigated the fuzzy behaviour of mechanical systems with uncertain boundary conditions. Nonlinear membership function for fuzzy optimization of mechanical and structural systems is discussed in Dhingraet al. [19]. When the Finite Element Method (FEM) is described with fuzzy theory it is then known as Fuzzy Finite Element Method (FFEM).

Recently various generalized models of uncertainty have been applied to finite element method to solve the structural problems with fuzzy parameters. Although FEM for structural problems [38, 54] is well known and there exits large number of papers and books related to this. As such few papers that are related to fuzzy FEM are discussed here. Fuzzy finite element approach is applied to describe structural systems with imprecisely defined parameters in an excellent way by Rao and Sawyer [42]. Verhaeghe et al. [50] discussed the fuzzy finite element analysis technique for the static analysis of structures which is based on interval computation. Both fuzzy static and dynamic analysis of structures is explained by Akpan et al. [2] using the fuzzy finite element approach. Vertex method and VAST software is used in it for the fuzzy finite element analysis. Hanss and Willner [27] used fuzzy arithmetical approach for the solution of finite element problems with fuzzy parameters. Petrubation finite element method is used by Huang and Li [29] for the static analysis of structures with fuzzy environment. Very recently Balu and Rao [6] investigated both static and dynamic responses of structures with fuzzy parameters. They have used an interesting approach viz. High Dimensional Model Representation (HDMR) along with FEM for the analysis.

The design and analysis of many engineering problems require the solution of linear system of equations. For example, the finite element formulation of equilibrium and steady state problems lead to a algebraic system of linear equations. Accordingly FFEM converts the structural problems for the static analysis to a Fuzzy System of Linear Equations (FSLE) [1 , 52] or to a Fully Fuzzy System of Linear Equations (FFSLE) [3 , 44]. There is a difference between fuzzy linear system and a fully fuzzy linear system. The coefficient matrix is treated as crisp in the fuzzy linear system, but in the fully fuzzy linear system all the parameters and variables are considered to be fuzzy numbers. Various solution methods have been proposed for the solution of FFSLE by Skalna et al. [47] and applied in structural mechanics problems. Shu-xiang et al. [46] solved the governing equation of fuzzy finite element method by means of fuzzy arithmetic approach for the static analysis of fifteen bar truss structures to find the fuzzy static responses. Behera et al. [12] has developed a method to find finite element solution of a stepped rectangular bar in presence of fuzziness in material properties. Recently [7, 9] have also proposed solution methods for fuzzy system of linear equations and applied those for the uncertainty analysis of structures using the fuzzy finite element method. Behera and Chakraverty [7] have also studied the fuzzy static responses of structures using fuzzy finite element method considering all the involved parameters asuncertain.

It is an important issue to develop mathematical models and numerical techniques that would appropriately treat the fully fuzzy linear systems because subtraction and division of fuzzy numbers are not the inverse operations of addition and multiplication respectively. So, this is an important area of research in the recent years. Accordingly this paper proposes a new method based on linear programming problem for the algebraic solution of FFSLE and applied for uncertain static analysis of structures using FFEM. In the following sections first limitations of some existing method has been discussed. Then followed by this, a new method has been proposed to solve FFSLE. Next, numerical examples viz. six bar truss and rectangular sheet with fuzzy nodal force, material and geometric properties are discussed using the proposed method to find fuzzy static responses. Lastly conclusions are drawn.

2 Limitations of the existing methods

Here we have pointed out some short comings of the existing methods for solving fuzzy and fully fuzzy system of linear equations as follows:

There exist different solution procedures [1 , 52] (Abbasbandy and Jafarian 2006; Allahviranloo 2004, 2005) for fuzzy system of linear equations where the coefficient matrix is considered as crisp real matrix. These methods are not applicable when the system is fully fuzzy.

Various methodologies [3 , 44] (Das and Chakraverty 2012; Dehgan et al., 2006) have been proposed to solve FFSLE where all the elements of fuzzy matrices are considered as non-negative. The existing methods are not suitable to solve when one may consider non-positive matrix elements as define in Example 1.

Recently [5, 37] proposed solution technique for FFSLE. They have considered, there is no restriction for the coefficient matrix. But the authors have found the non-negative solution of fuzzy system of equations. These methods are not applicable when the unknown solution vector consists only non-positive elements or both non-negative and non-positive elements.

Accordingly a new method has been proposed in the following section based on linear programming problem approach to overcome the above limitations.

3 Fully fuzzy system of linear equationsand the proposed method

The n × n fully fuzzy system of linear equations may be written as $\begin{matrix} {\tilde{a}}_{11} {\tilde{x}}_{1} + {\tilde{a}}_{12} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{1 n} {\tilde{x}}_{n} = {\tilde{b}}_{1} \\ {\tilde{a}}_{21} {\tilde{x}}_{1} + {\tilde{a}}_{22} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{2 n} {\tilde{x}}_{n} = {\tilde{b}}_{2} \\ ⋮ \\ {\tilde{a}}_{n 1} {\tilde{x}}_{1} + {\tilde{a}}_{n 2} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{nn} {\tilde{x}}_{n} = {\tilde{b}}_{n} . \end{matrix}$ (1)

In matrix notation the above system may be written as $[\tilde{A}] {\tilde{X}} = {\tilde{b}},$ where the coefficient matrix $[\tilde{A}] = ({\tilde{a}}_{kj}), 1 \leq k \leq n, j \leq n$ is a fuzzy n × n matrix, ${\tilde{b}} = {{\tilde{b}}_{k}}, 1 \leq k$ is a column vector of fuzzy numbers and ${\tilde{X}} = {{\tilde{x}}_{j}}$ is the vector of fuzzy unknowns. Here, we have considered $0 \notin {\tilde{x}}_{j}$ . This means zero is not an inner point of the elements of unknown solution vector.

Equation (1) may be written as $\sum_{j = 1}^{n} {\tilde{a}}_{kj} {\tilde{x}}_{j} = {\tilde{b}}_{k}, for k = 1, 2, \dots, n .$ (2)

Let us now define the solution set for the system (2) as follows i.e. $Ω = {x_{j} | \sum_{j = 1}^{n} a_{kj} x_{j} = b_{k} where a_{kj} \in {\tilde{a}}_{kj} and b_{k} \in {\tilde{b}}_{k}} .$ (3)

In parametric form [8], we may write the fuzzy coefficient matrix, real fuzzy unknown and the right hand real fuzzy number vector respectively as ${\tilde{a}}_{kj} = {\tilde{a}}_{kj} (α) = [{\underline{a}}_{kj} (α), {\bar{a}}_{kj} (α)],$ ${\tilde{x}}_{j} =$ ${\tilde{x}}_{j} (α) = [{\underline{x}}_{j} (α), {\bar{x}}_{j} (α)]$ and ${\tilde{b}}_{k} = {\tilde{b}}_{k} (α) = [{\underline{b}}_{k} (α),$ ${\bar{b}}_{k} (α)]$ .Substituting the above expressions in Equation(2), one may have $\sum_{j = 1}^{n} {\tilde{a}}_{kj} (α) {\tilde{x}}_{j} (α) = {\tilde{b}}_{k} (α), for k = 1, 2, \dots, n$ (4)

or $\sum_{j = 1}^{n} [{\underline{a}}_{kj} (α), {\bar{a}}_{kj} (α)] [{\underline{x}}_{j} (α), {\bar{x}}_{j} (α)] = [{\underline{b}}_{k} (α), {\bar{b}}_{k} (α)] .$ (5)

From Equation (5) one may predict the sign of the elements of solution vector by the following theorem.

Theorem 1.If $\sum_{j = 1}^{n} {\tilde{a}}_{kj} {\tilde{x}}_{j} = {\tilde{b}}_{k},$ for 1 ≤ k ≤ nwhere $0 \notin {\tilde{x}}_{j}$ , then sign of the elements of the fuzzy solution vector can be predicted by solving $\sum_{j = 1}^{n} {\tilde{a}}_{kj}^{c} x_{j} = {\tilde{b}}_{k}^{c}$ where ${\tilde{a}}_{kj}^{c} = \frac{{\underline{a}}_{kj} (α) + {\bar{a}}_{kj} (α)}{2}$ , ${\tilde{b}}_{k}^{c} = \frac{{\underline{b}}_{k} (α) + {\bar{b}}_{k} (α)}{2}$ .

Proof: The solution x_j can be obtained by solving the crisp system $\sum_{j = 1}^{n} {\tilde{a}}_{kj}^{c} x_{j} = {\tilde{b}}_{k}^{c}$ . From the definition of the solution set of Equation (2), one may easily conclude that x_j (1 ≤ j ≤ n) are the inner points of ${\tilde{x}}_{j}$ . Also we know that $0 \notin {\tilde{x}}_{j}$ . Hence one may predict the sign of the elements of fuzzy solution vector accordingly. □

It may be noted that the fuzzy solution vector may contain non-negative, non-positive or both non-negative and non-positive elements. As such following theorems may be applied to handle such situations.

Note: A fuzzy number $\tilde{U}$ is said to be non-negative (non-positive), denoted by $\tilde{U} \geq 0$ ( $\tilde{U} \leq 0$ ) if its membership function $μ_{\tilde{U}} (x)$ satisfies $μ_{\tilde{U}} (x) = 0 \forall x < 0 (x > 0) .$

Theorem 2.If the elements ofx_jfor 1 ≤ j ≤ n, are non-negative (non-positive) then the elements of fuzzy solution vector ${\tilde{x}}_{j}$ are non-negative (non-positive).

Proof: The proof of the theorem is straight forward. □

Theorem 3.Ifx_jcontains both non-negative and non-positive elements, i.e.x_jfor {j ∈ N|1 ≤ j ≤ k} are non-negative and for {j ∈ N|k + 1 ≤ j ≤ n} are non-positive for alli, where 1 ≤ i ≤ nandNis the natural number, then the fuzzy solution vector ${\tilde{x}}_{j}$ for {j ∈ N|1 ≤ j ≤ k} are non-negative and for {j ∈ N|k + 1 ≤ j ≤ n} are non-positive for alli.

Proof: The proof of the theorem is again straight forward. □

In general, obtained sign of the elements are as that of the following cases:

Case A: All ${\tilde{x}}_{j}$ are non-negative,

Case B: All ${\tilde{x}}_{j}$ are non-positive,

Case C: Few ${\tilde{x}}_{j}$ are non-negative and few are non-positive.

Next we have discussed below the solution procedure for all the above cases.

Case A: In this case we have considered all ${\tilde{x}}_{j}$ are non-negative. So, Equation (5) may be written as

$\begin{matrix} \sum_{{\tilde{a}}_{kj} \geq 0} [{\underline{a}}_{kj} (α), {\bar{a}}_{kj} (α)] [{\underline{x}}_{j} (α), {\bar{x}}_{j} (α)] \\ + \sum_{{\tilde{a}}_{kj} \leq 0} [{\underline{a}}_{kj} (α), {\bar{a}}_{kj} (α)] [{\underline{x}}_{j} (α), {\bar{x}}_{j} (α)] \\ + \sum_{0 \in {\tilde{a}}_{kj}} [{\underline{a}}_{kj} (α), {\bar{a}}_{kj} (α)] [{\underline{x}}_{j} (α), {\bar{x}}_{j} (α)] \\ = [{\underline{b}}_{k} (α), {\bar{b}}_{k} (α)] . \end{matrix}$ (6)

Applying the general rule of fuzzy multiplication we get $\begin{matrix} \sum_{{\tilde{a}}_{kj} \geq 0} [{\underline{a}}_{kj} (α) {\underline{x}}_{j} (α), {\bar{a}}_{kj} (α) {\bar{x}}_{j} (α)] \\ + \sum_{{\tilde{a}}_{kj} \leq 0} [{\underline{a}}_{kj} (α) {\bar{x}}_{j} (α), {\bar{a}}_{kj} (α) {\underline{x}}_{j} (α)] \\ + \sum_{0 \in {\tilde{a}}_{kj}} [{\underline{a}}_{kj} (α) {\bar{x}}_{j} (α), {\bar{a}}_{kj} (α) {\bar{x}}_{j} (α)] \\ = [{\underline{b}}_{k} (α), {\bar{b}}_{k} (α)] . \end{matrix}$ (7)

This can be written equivalently as

$\begin{matrix} \sum_{{\tilde{a}}_{kj} (α) \geq 0} {\underline{a}}_{kj} (α) {\underline{x}}_{j} (α) + \sum_{{\tilde{a}}_{kj} (α) \leq 0} {\underline{a}}_{kj} (α) {\bar{x}}_{j} (α) + \sum_{0 \in {\tilde{a}}_{kj} (α)} {\underline{a}}_{kj} (α) {\bar{x}}_{j} (α) = {\underline{b}}_{k} (α) \\ \sum_{{\tilde{a}}_{kj} (α) \geq 0} {\bar{a}}_{kj} (α) {\bar{x}}_{j} (α) + \sum_{{\tilde{a}}_{kj} (α) \leq 0} {\bar{a}}_{kj} (α) {\underline{x}}_{j} (α) + \sum_{0 \in {\tilde{a}}_{kj} (α)} {\bar{a}}_{kj} (α) {\bar{x}}_{j} (α) = {\bar{b}}_{k} (α) \end{matrix}}$ (8)

Let us now denote the system (8) as $\begin{matrix} \sum_{j = 1}^{n} w_{kj} = g_{k,} \\ \sum_{j = 1}^{n} q_{kj} = h_{k,} \end{matrix}} for 1 \leq k \leq n,$ (9)

where $\begin{matrix} \sum_{k = 1}^{n} w_{kj} = \sum_{{\tilde{a}}_{kj} (α) \geq 0} {\underline{a}}_{kj} (α) {\underline{x}}_{j} (α) \\ + \sum_{{\tilde{a}}_{kj} (α) \leq 0} {\underline{a}}_{kj} (α) {\bar{x}}_{j} (α) + \sum_{0 \in {\tilde{a}}_{kj} (α)} {\underline{a}}_{kj} (α) {\bar{x}}_{j} (α), \end{matrix}$ $\begin{matrix} \sum_{k = 1}^{n} q_{kj} = \sum_{{\tilde{a}}_{kj} (α) \geq 0} {\bar{a}}_{kj} (α) {\bar{x}}_{j} (α) \\ + \sum_{{\tilde{a}}_{kj} (α) \leq 0} {\bar{a}}_{kj} (α) {\underline{x}}_{j} (α) + \sum_{0 \in {\tilde{a}}_{kj} (α)} {\bar{a}}_{kj} (α) {\bar{x}}_{j} (α), \\ g_{k} = {\underline{b}}_{k} (α) and h_{k} = {\bar{b}}_{k} (α) . \end{matrix}$

Equation (9) is now converted to the following Linear Programming Problem (LPP) where we have introduced the artificial variables r_s for s = 1, 2, ⋯ , n, n + 1, ⋯ 2n, $\begin{matrix} Minimize : r_{1} + r_{2} + \dots + r_{2 n} \\ Subject to : \\ \sum_{j = 1}^{n} w_{1 j} + r_{1} = g_{1,} \\ ⋮ \end{matrix}$

$\begin{matrix} \sum_{j = 1}^{n} w_{nj} + r_{n} = g_{n,} \\ \sum_{j = 1}^{n} q_{1 j} + r_{n + 1} = h_{1,} \end{matrix}$

$\begin{matrix} ⋮ \\ \sum_{j = 1}^{n} q_{nj} + r_{2 n} = h_{n}, \end{matrix}$ (10) with the non-negative restrictions ${\underline{x}}_{j} (α), {\bar{x}}_{j} (α)$ and r_s for s = 1, 2, ⋯ , n, n + 1, ⋯ 2n ≥ 0.

Then the LPP (10) is solved and artificial variables are eliminated to have the optimum solution.

Case B: Next in this case let us consider all ${\tilde{x}}_{j}$ are non-positive. Equation (5) can similarly be written as

By changing all non-positive variables to non-negative we have

$\begin{matrix} \sum_{{\tilde{c}}_{kj} \leq 0} [{\underline{c}}_{kj} (α), {\bar{c}}_{kj} (α)] [{\underline{y}}_{j} (α), {\bar{y}}_{j} (α)] \\ + \sum_{{\tilde{c}}_{kj} \geq 0} [{\underline{c}}_{kj} (α), {\bar{c}}_{kj} (α)] [{\underline{y}}_{j} (α), {\bar{y}}_{j} (α)] \\ + \sum_{0 \in {\tilde{c}}_{kj}} [{\underline{c}}_{kj} (α), {\bar{c}}_{kj} (α)] [{\underline{y}}_{j} (α), {\bar{y}}_{j} (α)] \\ = [{\underline{b}}_{k} (α), {\bar{b}}_{k} (α)] \end{matrix}$ (12) where $[{\underline{y}}_{j} (α), {\bar{y}}_{j} (α)] = - [{\underline{x}}_{j} (α), {\bar{x}}_{j} (α)]$ and $[{\underline{c}}_{kj} (α), {\bar{c}}_{kj} (α)] = - [{\underline{a}}_{kj} (α), {\bar{a}}_{kj} (α)]$ .

Applying the general rule of fuzzy multiplication we get

$\begin{matrix} \sum_{{\tilde{c}}_{kj} \leq 0} [{\underline{c}}_{kj} (α) {\bar{y}}_{j} (α), {\bar{c}}_{kj} (α) {\underline{y}}_{j} (α)] \\ + \sum_{{\tilde{c}}_{kj} \geq 0} [{\underline{c}}_{kj} (α) {\underline{y}}_{j} (α), {\bar{c}}_{kj} (α) {\bar{y}}_{j} (α)] \\ + \sum_{0 \in {\tilde{c}}_{kj}} [{\underline{c}}_{kj} (α) {\bar{y}}_{j} (α), {\bar{c}}_{kj} (α) {\bar{y}}_{j} (α)] \\ = [{\underline{b}}_{k} (α), {\bar{b}}_{k} (α)] \end{matrix}$ (13)

Equation (14) can equivalently be written as

$\begin{matrix} \sum_{{\tilde{c}}_{kj} \leq 0} {\underline{c}}_{kj} (α) {\bar{y}}_{j} (α) + \sum_{{\tilde{c}}_{kj} \geq 0} {\underline{c}}_{kj} (α) {\underline{y}}_{j} (α) + \sum_{0 \in {\tilde{c}}_{kj}} {\underline{c}}_{kj} (α) {\bar{y}}_{j} (α) = {\underline{b}}_{k} (α) \\ \sum_{{\tilde{c}}_{kj} \leq 0} {\bar{c}}_{kj} (α) {\underline{y}}_{j} (α) + \sum_{{\tilde{c}}_{kj} \geq 0} {\bar{c}}_{kj} (α) {\bar{y}}_{j} (α) + \sum_{0 \in {\tilde{c}}_{kj}} {\bar{c}}_{kj} (α) {\bar{y}}_{j} (α) = {\bar{b}}_{k} (α) \end{matrix}}$ (14)

Next we may represent the above system as $\begin{matrix} \sum_{j = 1}^{n} w_{kj}^{*} = g_{k,} \\ \sum_{j = 1}^{n} q_{kj}^{*} = h_{k,} \end{matrix}} for 1 \leq k \leq n,$ (15) where $\begin{matrix} \sum_{k = 1}^{n} w_{kj}^{*} = \sum_{{\tilde{c}}_{kj} \leq 0} {\underline{c}}_{kj} (α) {\bar{y}}_{j} (α) + \sum_{{\tilde{c}}_{kj} \geq 0} {\underline{c}}_{kj} (α) {\underline{y}}_{j} (α) \\ + \sum_{0 \in {\tilde{c}}_{kj}} {\underline{c}}_{kj} (α) {\bar{y}}_{j} (α), \end{matrix}$ $\begin{matrix} \sum_{k = 1}^{n} q_{kj}^{*} = \sum_{{\tilde{c}}_{kj} \leq 0} {\bar{c}}_{kj} (α) {\underline{y}}_{j} (α) \\ + \sum_{{\tilde{c}}_{kj} \geq 0} {\bar{c}}_{kj} (α) {\bar{y}}_{j} (α) + \sum_{0 \in {\tilde{c}}_{kj}} {\bar{c}}_{kj} (α) {\bar{y}}_{j} (α), g_{k} \\ = {\underline{b}}_{k} (α) and h_{k} = {\bar{b}}_{k} (α) . \end{matrix}$

The LPP as mentioned below obtained from the above system may be solved to have the corresponding fuzzy solution vector. $\begin{matrix} Minimize : r_{1} + r_{2} + \dots + r_{2 n}, Subject to : \\ \sum_{j = 1}^{n} w_{1 j}^{*} + r_{1} = g_{1,} \end{matrix}$

$\begin{matrix} ⋮ \\ \sum_{j = 1}^{n} w_{nj}^{*} + r_{n} = g_{n,} \\ \sum_{j = 1}^{n} q_{1 j}^{*} + r_{n + 1} = h_{1,} \\ ⋮ \\ \sum_{j = 1}^{n} q_{nj}^{*} + r_{2 n} = h_{n}, \end{matrix}$ (16)

with the non-negative restrictions ${\underline{y}}_{j} (α), {\bar{y}}_{j} (α)$ and r_s for s = 1, 2, ⋯ , n, n + 1, ⋯ 2n ≥ 0.

Case C: Finally the solution vector ${\tilde{x}}_{j}$ is assumed to contain both non-negative and non-positive fuzzy numbers. Hence we consider ${\tilde{x}}_{j}$ for {j ∈ N|1 ≤ j ≤ i} as non-negative and for {j ∈ N|i + 1 ≤ j ≤ n} as non-positive for all k, where 1 ≤ k ≤ n and N is the natural number. As such from Equation (4) we have

$\begin{matrix} \sum_{j = 1}^{i} {\tilde{a}}_{kj} (α) {\tilde{x}}_{j} (α) + \sum_{j = i + 1}^{n} {\tilde{a}}_{kj} (α) {\tilde{x}}_{j} (α) \\ = {\tilde{b}}_{k} (α), for k = 1, 2, \dots, n . \end{matrix}$ (17)

The above equation is now expressed as

$\begin{matrix} \sum_{j = 1}^{i} [{\underline{a}}_{kj} (α), {\bar{a}}_{kj} (α)] [{\underline{x}}_{j} (α), {\bar{x}}_{j} (α)] \\ + \sum_{j = i + 1}^{n} [{\underline{a}}_{kj} (α), {\bar{a}}_{kj} (α)] [{\underline{x}}_{j} (α), {\bar{x}}_{j} (α)] \\ = [{\underline{b}}_{k} (α), {\bar{b}}_{k} (α)] . \end{matrix}$ (18)

Equation (18) is now converted to the following crisp system in the similar fashion as discussed in previous two cases

$\begin{matrix} \underset{for 1 \leq j \leq i}{\underset{︸}{\sum_{{\tilde{a}}_{kj} (α) \geq 0} {\underline{a}}_{kj} (α) {\underline{x}}_{j} (α) + \sum_{{\tilde{a}}_{kj} (α) \leq 0} {\underline{a}}_{kj} (α) {\bar{x}}_{j} (α) + \sum_{0 \in {\tilde{a}}_{kj} (α)} {\underline{a}}_{kj} (α) {\bar{x}}_{j} (α)}} \\ + \underset{for i + 1 \leq j \leq n}{\underset{︸}{\sum_{{\tilde{c}}_{kj} \leq 0} {\underline{c}}_{kj} (α) {\bar{y}}_{j} (α) + \sum_{{\tilde{c}}_{kj} \geq 0} {\underline{c}}_{kj} (α) {\underline{y}}_{j} (α) + \sum_{0 \in {\tilde{c}}_{kj}} {\underline{c}}_{kj} (α) {\bar{y}}_{j} (α)}} = {\underline{b}}_{k} (α), \\ \underset{for 1 \leq j \leq i}{\underset{︸}{\sum_{{\tilde{a}}_{kj} (α) \geq 0} {\bar{a}}_{kj} (α) {\bar{x}}_{j} (α) + \sum_{{\tilde{a}}_{kj} (α) \leq 0} {\bar{a}}_{kj} (α) {\underline{x}}_{j} (α) + \sum_{0 \in {\tilde{a}}_{kj} (α)} {\bar{a}}_{kj} (α) {\bar{x}}_{j} (α)}} \\ + \underset{for i + 1 \leq j \leq n}{\underset{︸}{\sum_{{\tilde{c}}_{kj} \leq 0} {\bar{c}}_{kj} (α) {\underline{y}}_{j} (α) + \sum_{{\tilde{c}}_{kj} \geq 0} {\bar{c}}_{kj} (α) {\bar{y}}_{j} (α) + \sum_{0 \in {\tilde{c}}_{kj}} {\bar{c}}_{kj} (α) {\bar{y}}_{j} (α)}} = {\bar{b}}_{k} (α) . \end{matrix}}$ (19)

Which may again be written as $\begin{matrix} \sum_{j = 1}^{i} w_{kj} + \sum_{j = i + 1}^{n} w_{kj}^{*} = g_{k,} \\ \sum_{j = 1}^{i} q_{kj} + \sum_{j = i + 1}^{n} q_{kj}^{*} = h_{k,} \end{matrix}} for 1 \leq k \leq n .$ (20)

Corresponding LPP for the above system (20) can be expressed as $\begin{matrix} Minimize : r_{1} + r_{2} + \dots + r_{2 n} \\ Subject to : \\ \sum_{j = 1}^{i} w_{1 j} + \sum_{j = i + 1}^{n} w_{1 j}^{*} + r_{1} = g_{1,} \\ ⋮ \\ \sum_{j = 1}^{i} w_{nj} + \sum_{j = i + 1}^{n} w_{nj}^{*} + r_{n} = g_{n,} \\ \sum_{j = 1}^{i} q_{1 j} + \sum_{j = i + 1}^{n} q_{1 j}^{*} + r_{n + 1} = h_{1,} \\ ⋮ \\ \sum_{j = 1}^{i} q_{nj} + \sum_{j = i + 1}^{n} q_{n 1 j}^{*} + r_{2 n} = h_{n,} \end{matrix}$ (21)

with the non-negative restrictions that is ${\underline{x}}_{j} (α),$ ${\bar{x}}_{j} (α), {\underline{y}}_{j} (α), {\bar{y}}_{j} (α)$ and r_s for s = 1, 2, ⋯ , n, n + 1, ⋯ 2n ≥ 0.

Again the LPP (21) may be solved to have the required solution vector.

4 Numerical examples and discussions

The above method has now been implemented in the following example problems to demonstrate the efficiency of the proposed method. Example 1 includes a purely mathematical problem whereas Examples 2 and 3 describe two application problems. Examples 2 and 3 illustrate the applicability of the proposed method to obtain the bounds of the fuzzy static response of structure using fuzzy finite element method.

Example 1. Let us consider a 2 × 2 fully fuzzy system of linear equations $\begin{matrix} (- 2, 3, 4) {\tilde{x}}_{1} + (- 2, 2, 3) {\tilde{x}}_{2} = (- 13, 2, 14) \\ (1, 2, 2) {\tilde{x}}_{1} + (4, 4, 5) {\tilde{x}}_{2} = (- 14, - 4, 0) . \end{matrix}$ (22)

Suppose ${\tilde{x}}_{1} = [{\underline{x}}_{1} (α), {\bar{x}}_{1} (α)]$ and ${\tilde{x}}_{2} = [{\underline{x}}_{2} (α),$ ${\bar{x}}_{2} (α)]$ , hence Equation (22) in parametric form may represented as

$\begin{matrix} [5 α - 2, - α + 4] {\tilde{x}}_{1} \\ + [4 α - 2, - α + 3] {\tilde{x}}_{2} = [15 α - 13, - 12 α + 14] \\ [α + 1, 2] {\tilde{x}}_{1} + [4, - α + 5] {\tilde{x}}_{2} = [10 α - 14, - 4 α] . \end{matrix}$ (23)

Using the proposed method we have ${\tilde{x}}_{1} = (1, 2, 2)$ and ${\tilde{x}}_{2} = (- 3, - 2, - 1)$ .

Example 2 (Six bar truss). In this example a 6-bar truss structure as shown in Fig. 1, has been considered. The structure consists of 6 elements. Material, geometric properties and the applied load parameter viz. $\tilde{P}$ are assumed as uncertain. The values of the input variables with different cases have been shown in Table 1 for the present analysis.

From Table 1 one may clearly observe that in Case 1 all the input parameters are considered as crisp value. Applied load parameter is only assumed as uncertain in Case 2. Cross sectional area for elements 5 and 6 along with load parameter are considered only as fuzzy in Case 3. Lastly in Case 4 all the input variables are assumed as uncertain.

Usual finite element method for static analysis of structures with crisp parameters converts the problem into an algebraic system of linear equations. Hence the corresponding equilibrium equation for the present problem, for Case 1 of Table 1 is represented as Kδ = F, where K, F and δ are the reduced stiffness matrix, load and displacements vector respectively.

Here, $K = [\begin{matrix} \frac{E_{1} A_{1}}{L_{1}} + 0.36 \frac{E_{5} A_{5}}{L_{5}} & - 0.48 \frac{E_{5} A_{5}}{L_{5}} & - \frac{E_{1} A_{1}}{L_{1}} & 0 \\ - 0.48 \frac{E_{5} A_{5}}{L_{5}} & \frac{E_{3} A_{3}}{L_{3}} + 0.64 \frac{E_{5} A_{5}}{L_{5}} & 0 & 0 \\ - \frac{E_{1} A_{1}}{L_{1}} & 0 & \frac{E_{1} A_{1}}{L_{1}} + 0.36 \frac{E_{6} A_{6}}{L_{6}} & 0.48 \frac{E_{6} A_{6}}{L_{6}} \\ 0 & 0 & 0.48 \frac{E_{6} A_{6}}{L_{6}} & \frac{E_{4} A_{4}}{L_{4}} + 0.64 \frac{E_{6} A_{6}}{L_{6}} \end{matrix}]$ , $F = [\begin{matrix} P \\ 2 P \\ 2.5 P \\ - 1.5 P \end{matrix}]$ and $δ = [\begin{matrix} x_{2} \\ y_{2} \\ x_{3} \\ y_{3} \end{matrix}]$ to be determined. For this case we have assumed, ${\tilde{A}}_{i} = A_{i}$ , ${\tilde{E}}_{i} = E_{i}$ and $\tilde{P} = P$ for all i = 1 to 6. Substituting the corresponding values in the above expression as defined in Case 1 one may have $\begin{matrix} K = [\begin{matrix} 429380 & - 105840 & - 350000 & 0 \\ - 105840 & 403620 & 0 & 0 \\ - 350000 & 0 & 429380 & 105840 \\ 0 & 0 & 105840 & 403620 \end{matrix}] \end{matrix}$ $\begin{matrix} (N / mm), F = [\begin{matrix} 20500 \\ 41000 \\ 51250 \\ - 30750 \end{matrix}] (N) . \end{matrix}$

As such, horizontal and vertical displacements at nodes 2 and 3 along with the comparison of Shu-xiang and Zhen-zhou [45] and Qiu and Elishakoff [39] are shown in Table 2. From the results it can be seen that present results are found in good agreement.

Next for Cases 2, 3 and 4, fuzzy finite element method is used with the proposed methodology to compute the uncertain static displacements. Corresponding results are depicted in Figs. 2 to 5.

For special case α = 0 obtained results for Case 3 are also compared with Shu-xiang and Zhen-zhou [45] and Qiu and Elishakoff [39] and are depicted in Table 3 and again a good agreement may be seen.

One may notice from the results of six bar truss for Case 1 as depicted in Table 2 that horizontal and vertical displacements at node three are maximum and minimum respectively. From Table 3 for interval parameter (special case of Case 3 for α = 0), uncertainty in the horizontal displacement at node two is found to be maximum and minimum at node three. Similarly it can also be observed that uncertainty in the vertical displacement at node three is maximum and minimum at node two. From Figs. 2 to 5 one can observe that larger width has been obtained when fuzziness appears for all the parameters (Case 4). It can clearly be seen from Figs. 2, 3 and 5 that minimum width has been obtained for the displacements when fuzziness appears only in external load (Case 2) and spread in the fuzzy displacements are gradually increases when we have introduced fuzziness in the cross sectional area along with other parameters viz. for Cases 3 and 4 respectively. But in Fig. 4 minimum spread been obtained for Case 3. As discussed above obtained results are compared in Table 3 with the results of existing methods [39, 45] in special cases to show the effectiveness of the proposed method. Moreover we found that the proposed solution method estimates narrow and exact bounds for the structural responses.

Example 3 (Rectangular sheet)

A uniform rectangular sheet as shown in Fig. 6 is considered in this example. One of its ends is fixed and in the other end, uniform force $\tilde{q}$ is applied. Applied force and elastic modulus are considered as uncertain. Input for all the variables and parameters are shown in Table 4 for different cases.

Applying usual finite element method with crisp parameters and applying the boundary conditions, the reduced equilibrium equation for the above structure may be obtained as Kδ = F, where

$K = \frac{3 Eh}{32} [\begin{matrix} 7 & - 4 & - 4 & 2 \\ - 4 & 13 & 2 & - 12 \\ - 4 & 2 & 7 & 0 \\ 2 & - 12 & 0 & 13 \end{matrix}]$ , $F = [\begin{matrix} F_{2} \\ F_{3} \end{matrix}] = [\begin{matrix} \frac{q}{2} \\ 0 \\ \frac{q}{2} \\ 0 \end{matrix}]$ and $δ = [\begin{matrix} δ_{2} \\ δ_{3} \end{matrix}] = [\begin{matrix} x_{2} \\ y_{2} \\ x_{3} \\ y_{3} \end{matrix}]$ . Here we have assumed $\tilde{E} = E$ and $\tilde{q} = q$ .

Hence solving the corresponding above system with the input values as defined in Case 1 of Example 3 obtained crisp displacements are given in Table 5 with the comparison of existing results.

Next for Cases 2 and 3, fuzzy finite element method is used with the proposed methodology and results are depicted in Tables 6 and 7 respectively.

From Table 5 one may observe that horizontal and vertical displacements of rectangular sheet for crisp parameters at nodes two and three are maximum and minimum respectively. Obtained results are compared with Huang and Li [29] and Li et al. (2003) and found to be equal. One may observe from Tables 6 and 7 that horizontal and vertical displacement at node two and horizontal displacement at node 3 suffers more uncertainty for Case 3. Also vertical displacement at node 3 for Case 2 suffers more uncertainty. Similarly horizontal and vertical displacements at node two and horizontal displacement at node 3 suffers less uncertainty for Case 2. Vertical displacement at node 3 for Case 3 suffers less uncertainty. It is worth mentioning that in special case fuzzy displacements for α = 1, are exactly same with the deterministic results as shown in Table 5.

5 Conclusions

Uncertain static analysis of imprecisely defined structures has been investigated here using fuzzy finite element method. Material, geometric properties and applied forces of the structures are assumed as uncertain. Triangular convex normalised fuzzy sets are considered to model the uncertainty. Fuzzy finite element method converts the problem to a fully fuzzy system of linear equations as such a new method has been proposed here to obtain fuzzy static responses. Fuzzy arithmetic and linear programming concepts are used in the solution procedure. There are no restrictions on the coefficient matrix of the corresponding system. The method found to be efficient when the elements of the fuzzy solution vector are both non-negative and non-positive. Investigation presented here may find well in other real applications too where the material, geometric and applied forces may not be obtained in term of crisp but a vague value in term of fuzzy is known. In this regard one mathematical example and two application problems viz. 6-bar truss and rectangular sheet have been analysed using the proposed method. Results obtained by the proposed method are compared with the results of different techniques to show the efficiency and powerfulness of the proposed method.

References

Abbasbandy

, Jafarian

and Ezzati

, Conjugate gradient method for fuzzy symmetric positive-definite system of linear equations, Applied Mathematics and Computation171 (2005), 1184–1191.

Akpan

U.O.

, Koko

T.S.

, Orisamolu

I.R.

and Gallant

B.K.

, Practical fuzzy finite element analysis of structures, Finite Elem Anal Des38 (2001), 93–111.

Allahviranloo

and Mikaeilvand

, Non-zero solutions of the fully fuzzy linear systems, Applied and Computational Mathematics10 (2011), 271–282.

Antonio

C.C.

and Hoffbauer

L.N.

, Uncertainty propagation in inverse reliability-based design of composite structures, Int J Mech Mater Des6(1) (2010), 89–102.

Babbar

, Kumar

and Bansal

, Solving fully fuzzy linear system with arbitrary triangular fuzzy numbers (m, α, β), Soft Computing17 (2013), 691–702.

Balu

A.S.

and Rao

B.N.

, High dimensional model representation based formulations for fuzzy finite element analysis of structures, Finite Elem Anal Des50 (2012), 217–230.

Behera

and Chakraverty

, Fuzzy analysis of structures with imprecisely defined properties, Comp Model Eng Sci96 (2013a), 317–337.

Behera

and Chakraverty

, Fuzzy finite element analysis of imprecisely defined structures with fuzzy nodal force, Eng Appl Artificial Intell26 (2013b), 2458–2466.

Behera

and Chakraverty

, Fuzzy finite element based solution of uncertain static problems of structural mechanics, Int J Comp Appl69 (2013c), 6–11.

10.

Behera

and Chakraverty

, Solution to fuzzy system of linear equations with crisp coefficients, Fuzzy Inf Eng5 (2013d), 205–219.

11.

Behera

and Chakraverty

, Solution of fuzzy system of linear equations with polynomial parametric form, Applications Appl Math: An Int J7 (2012), 648–657.

12.

Behera

, Datta

and Chakraverty

, Development of a finite element solution of a stepped rectangular Bar in presence of fuzziness in material properties, Proc 5th Int Conf Adv Mech Eng, SVNIT, Surat-395 007, India, ICAME- 2011, 2011, pp. 205–209.

13.

Ben-Haim

and Elishakoff

, Convex Models of Uncertainty in Applied Mechanics, Elsevier Science, Amsterdam, 1990.

14.

Chakraverty

and Behera

, Fuzzy system of linear equations with crisp coefficients, J Intelligent Fuzzy Syst25 (2013), 201–207.

15.

Chen

and Rao

S.S.

, Fuzzy finite element approach for the vibration analysis of imprecisely defined systems, Finite Elem Anal Des27 (1997), 69–83.

16.

Chen

S.H.

and Yang

X.W.

, Interval finite element method for beam structures, Finite Elem Anal Des34 (2000), 75–88.

17.

Chekri

, Plessis

, Lallemand

, Tison

and Level

, Fuzzy behavior of mechanical systems with uncertain boundary conditions, Comput Methods Appl Mech Eng189 (2000), 863–873.

18.

Dehghan

, Hashemi

and Ghatee

, Solution of the fully fuzzy linear systems using iterative techniques, Chaos, Solitons & Fractals34 (2007), 316–336.

19.

Dhingra

A.K.

, Rao

S.S.

and Kumar

, Nonlinear membership function in multi-objective fuzzy optimization of mechanical and structural systems, AIAA J, 1992, pp. 251–260.

20.

Elishakoff

, Probabilistic Methods in the Theory of Structures, Wiley, New York, 1983.

21.

Friedman

, Ming

and Kandel

, Fuzzy linear systems, Fuzzy Sets Syst96 (1998), 201–209.

22.

Ganzerli

and Pantelides

C.P.

, Optimum structural design via convex model superposition, Comput Struct74 (2000), 639–647.

23.

Gao

, Interval finite element analysis using interval factor method, Comput Mech39 (2007), 709–717.

24.

Guo-jian

and Jing-bo

, Interval finite element method and its application on anti-slide stability analysis, Appl Math Mech28(4) (2007), 521–529.

25.

Guo

and Gong

, Block Gaussian elimination methods for fuzzy matrix equations, Int J Pure App Math58 (2010), 157–168.

26.

Haldar

and Mahadevan

, Reliability Assessment Using Stochastic Finite Element Analysis, John Wiley & Sons, New York, 2000.

27.

Hanss

and Willner

, A fuzzy arithmetical approach to the solution of finite element problems with uncertain parameters, Mech Res Commun27 (2000), 257–272.

28.

Hanss

, Applied Fuzzy Arithmetic: An Introduction with Engineering Applications, Springer-Verlag, Berlin, 2005.

29.

Huang

and Li

, Perturbation finite element method of structural analysis under fuzzy environments, Engineering Applications of Artificial Intelligence18 (2005), 83–91.

30.

Kaufmann

and Gupta

M.M.

, Introduction to Fuzzy Arithmetic: Theory and Applications, Van Nostrand Reinhold Company, New York, 1985.

31.

Moller

, Graf

and Beer

, Fuzzy structural analysis using α-level optimization, Comput Mech26(6) (2000), 547–565.

32.

Moore

R.E.

, Methods and Applications of Interval Analysis, SAIM Publication, Philadelphia, 1979.

33.

Muhanna

R.L.

, Zhang

and Mullen

R.L.

, Interval finite elements as a basis for generalized models of uncertainty in engineering mechanics, Reliable Comp13 (2007), 173–194.

34.

Munck

M.D.

, Moens

, Desmet

and Vandepitte

, A response surface based optimisation algorithm for the calculation of fuzzy envelope FRFs of models with uncertain properties, Comput Struct86(10) (2008), 1080–1092.

35.

Muzzioli

and Reynaerts

, The solution of fuzzy linear systems by non-linear programming: A financial application, European Journal of Operational Research177 (2007), 1218–1231.

36.

Neumaier

and Pownuk

, Linear systems with large uncertainties, with applications to truss structures, Reliable Comp13 (2007), 149–172.

37.

Otadi

and Mosleh

, Solving fully fuzzy matrix equations, Applied Mathematical Modelling36 (2012), 6114–6121.

38.

Petyt

, Introduction to Finite Element Vibration Analysis, Cambridge University Press, New York, 2010.

39.

Qiu

and Elishakoff

, Antioptimization of structures with large uncertain-but-nonrandom parameters via interval analysis, Comput Methods Appl Mech Engrg152 (1998), 361–372.

40.

Qui

, Wang

and Chen

, Exact bounds for the static response set of structures with uncertain-but-bounded parameters, Int J Sol Struct43 (2006), 6574–6593.

41.

Rao

S.S.

and Berke

, Analysis of uncertain structural systems using interval analysis, AIAA J34(4) (1997), 727–735.

42.

Rao

S.S.

and Sawyer

J.P.

, Fuzzy finite element approach for the analysis of imprecisely defined systems, AIAA J33(12) (1995), 2364–2370.

43.

Ross

T.J.

, Fuzzy Logic with Engineering Applications, John Wiley & Sons, New York, 2004.

44.

Senthilkumar

and Rajendran

, New approach to solve symmetric fully fuzzy linear systems, Sadhana36 (2011), 933–940.

45.

Shu-xiang

and Zhen-zhou

, Interval arithmetic and static interval finite element method, Appl Math Mech22 (2001), 1390–1396.

46.

Shu-xiang

, Zhen-zhou

and Li-fu

, Fuzzy arithmetic and solving of the static governing equations of fuzzy finite element method, Appl Math Mech23 (2002), 1054–1061.

47.

Skalna

, Rama Rao

M.V.

and Pownuk

, Systems of fuzzy equations in structural mechanics, J Comp Appl Math218 (2008), 149–156.

48.

Sun

and Guo

, Solution to general fuzzy linear system and its necessary and sufficient condition, Fuzzy Inform Eng3 (2009), 317–327.

49.

Valliappan

and Pham

T.D.

, Fuzzy logic applied to numerical modeling of engineering problems, Comput Mech Adv2 (1995), 213–281.

50.

Verhaeghe

, Munck

M.D.

, Desmet

, Vandepitte

and Moens

, A fuzzy finite element analysis technique for structural static analysis based on interval fields, 4th Int Workshop Reliable Eng Comp (2010), 117–128.

51.

Xia

and Yu

, Modified interval and subinterval perturbation methods for the static response analysis of structures with interval parameters, J Struct Eng (2013). DOI: 10.1061/(ASCE)ST.1943-541X.0000936.

52.

Yin

J.F.

and Wang

, Splitting iterative methods for fuzzy system of linear equations, Comp Math Model20 (2009), 326–335.

53.

Zadeh

, Fuzzy sets, Inf Control8(3) (1965), 338–353.

54.

Zienkiewicz

O.C.

, The Finite Element Method, Tata McGraw Hill Edition, 1979.

55.

Zimmermann

H.J.

, Fuzzy Set Theory and its Application, Kluwer Academic Publishers, London, 2001.