Interval based Methods for Load and 
Stiffness Uncertainty in Computer Aided Analysis 
 

Primary Researchers:
Rafi Muhanna
Department of Civil Engineering 
 University of Maryland at College Park, MD 20742, USA 
E-mail: muhannar@eng.umd.edu, Tel. (301) 405-7965 

Robert L. Mullen
Department of Civil Engineering
Case Western Reserve University, Cleveland, OH 44106, USA
E-mail: rlm@po.cwru.edu, Tel. (216) 368-2423
 
 

Powerpoint presentation (Numerical Methods and Applications 1998)

National Science Foundation Award Information
 
 

Abstract: This work extends the authors previous results on interval based uncertainty treatment in engineering problems. Uncertainty in the context of Finite Element Methods is represented in a form of fuzzy numbers. The representation of membership in the fuzzy set is limited to an interval form. Load, geometric and material uncertainty accounted for. Exact results were obtained in the case of load uncertainty and very sharp solution is achieved for the case of geometric and material uncertainty. Number of examples is introduced to illustrate this new approach.

Introduction: One of the major difficulties a designer must face is that both the external demands of the systems and its manufacturing variations are not known exactly. In order to overcome this uncertainty, the designer must provide excessive capabilities and over design the system. As analysis tools continue to be developed, the predictive skills of designers are becoming finer. The demands of the market place required that more efficient designs be developed. In order to analyze designs subject to uncertainties, the uncertainties in the performance of the system must be included in the analysis.
In this paper, we will address initial progress in representing uncertainties in finite element calculations by the use of fuzzy numbers. In our initial implementation, a fuzzy number is represented by the possible range of values the number can have. This is similar to the use of error bars in graphical presentation of data. The number is known to lie between values but the exact value is unknown. In our work, we will limit our representation of membership in a fuzzy set to an interval. An interval is a close set in R , which included the possible range of a number. In this paper, an interval will be represented by the ordered pair [a, b] = {x: a £ x £ b} where a is the lower limit of the interval and b is the upper limit of the interval.
In the present work, interval finite element calculations are implemented by first developing element stiffness matrices in parametric form (with length, modulus, and applied traction as parameters) using a standard non-interval formulation. Uncertainty is introduced by assigning an interval value for parameters. The element stiffness and load matrices are assembled and the final system of interval equations is solved using an implementation of Hanson's Algorithm [1]. This procedure has been used to solve several matrix structural and continuum problems with intervals for material properties, geometry, and loading.

Formulation: In the present work, Fuzzy finite element calculations, in level of presumption form, are implemented by first developing element stiffness matrices in parametric form (with length, modulus, and applied traction as parameters) using a standard non-fuzzy formulation. Uncertainty is introduced by assigning an interval value for parameters. The element stiffness and load matrices are assembled and the final system of interval equations is solved using an implementation of Hanson's Algorithm (Hanson 1965). This system of interval equations can be written as:

[kl , ku] [ul , uu]= [pl , pu]             (1)

This procedure has been used to solve several matrix structural and continuum problems with uncertainty for material properties, geometry, and loading (Muhanna and Mullen 1995; Mullen and Muhanna 1996; Muhanna and Mullen 1998).
In conventional finite element formulations, the nodal load is given by

                               (2)

Where pc is the vector of concentrated load and pb is the nodal load contribution from an element and has the form:

         (3)

Which is the assamblage of

                     (4)

Where pi is the generalized nodal load for node i, b(x) is the applied traction and Niis the shape function for node i.
In this paper, to achieve sharp results, two different approaches are applied. The first is used in the load uncertainty case and the second is for material and geometric uncertainty.
In the uncertain load case, the function b(x) is allowed to be fuzzy. The evaluation of fuzzy integral in equation (1) will be done in a Riemann sense. In order to correctly obtain inclusive interval values for pi , attention must be paid to the sign of the terms Ni and b(x). If the shape function is negative, the upper and lower limits of the interval value of b(x) must be interchanged. For linear displacement truss elements and cubic Euler-Bernoulli beam elements, the shape functions maintain the same sign over the length of an element. However, in the case of a distributed moment on a beam, the appropriate shape functions change sign over the length of the element. The order of multiplication of fuzzy number has a strong influence on the sharpness of the results. In order to maintain sharp results for the displacements, stress, and reaction calculations, the use of a fuzzy value should be delayed as much as possible. For the case of load uncertainty the sharpness of the interval solution (exact in the case of independent element loads) can be achieved by multiplying all non-interval values first, and the last multiplication involves the interval quantities. The load vector will be presented in the following form:

P = MF                                             (5)

Where F(m ´ 1) is the interval element load vector and m is the number of elements. Matrix M is such matrixes that ensure the proper conversion of interval element load into interval nodal one. The system of interval equations for the interval nodal displacements can be written as:

u = (k-1 M)F                                     (6)

However, in the case of material and geometric uncertainty, the problem is more complicated. Obtaining a guaranteed enclosure of the solution is achievable, but the central issue is how much the solution is sharp? Sharpness of the solution is a relative matter; we try to compare with the exact solution that can be obtained by solving for all possible combinations. The non-sharpness can be attributed to dependency, order of operations and the failure of distributive law in interval arithmetic.
It is observed that when uncertainty is included in the stiffness matrix of a system (left-hand side of the equation system), the element-based local properties can not be maintained and inhered to the final system after the assemblage. For example, if the modulus of elasticity Eis given an uncertain value in interval form, the solution of the interval system k u = p will be different from the solution of the same system after factoring out the interval constant E in any of the following forms:

E(k' ) u = p                                         (7)

k' u = (1/E) p                                     (8)

To overcome this difficulty in treatment of geometric and material uncertainty, a new approach is developed. An equivalent system, which includes the geometric and material uncertainty in a form of an equivalent load uncertainty, is developed. This approach is based on element-by-element technique that maintains the element-based local properties inherent to the final solution.
First the deterministic system is solved for forces in the form:

Fe,c= ke,c L kc-1 Fc                            (9)

On the element level the equilibrium equation will take the form:

ke ue = Fe,c                                          (10)

If material uncertainty is introduced in the form E = d Ec . Equation 9 could be written as:

Fe = (1/d) ke,c L kc-1 Fc                     (11)

After assemblage the following equivalent system, with interval load vector, is obtained:

k u = F                                                 (12)

In the above equations the subscripts e and c denote the element and center (deterministic) values respectively. This approach is applied for material and geometric uncertainty in trusses and plane stress problems, and for material uncertainty in beams.

Example Calculation: As examples of the method, two problems are presented: the first a two-dimensional elastic continuum. The problem consists of a 100.16 mm ´ 100.16 mm plate with a 25.4 mm radius hole in the center subject to uniaxial tension on the horizontal surfaces. The Young’s modulus and Poisson’s ratio are 200 GPa and 0.25, respectively. Due to symmetry, only ¼ of the problem is analyzed. The mesh of 1600 linear displacement triangle used to solve this problem is shown in Figure 1. The applied traction has possible values [0, 6.9] kPa. The possible values of nodal displacements at selected locations are given in Table 1.
 
 

Table 1.  Square plate with circular opening subjected to uniaxial tensile load
Corner
U ´ 10 -7 mm
V ´ 10 -7 mm
A
[-42.2, 4.96]
[0.0, 0.0]
B
[-44.02, 3.82]
[0.0, 0.0]
C
[-5.90, 16.77]
[-9.97, 16.39]
D
[0.0, 0.0]
[-2.38, 63.7]
E
[0.0, 0.0]
[-2.51, 60.05]

 

Contour plots for the bounds of the stress components are given in Figure 2 and 3 for the case of possible traction values of [0., 6.9] kPa. The plot for the maximum value of syy shows the expected stress concentration at the horizontal centerline of the hole. The corresponding plot of the minimum value of syy exhibits a compression component of a bending-like distribution along the horizontal centerline, which one could expect if the load was applied only in the center region.
 

 The zero stress region is the result of no load as a possible scenario and all other possible loading resulting in tension. It should be noted that the stress contours are not necessarily the result of any one possible loading, but are the pointwise value of the minimum and maximum stress components for all possible loading conditions.
The second problem is an equilateral triangular steel truss, given in Figure 3, with EA = [0.975, 1.025] ´  109 N, loaded with 1 kN concentrated load at the middle hinge. Table 2 shows the solutions according four different formulations. a) Interval formulation, where the element stiffness matrices are formulated as interval from the start beginning, b) E (k') u = F, c) k' u = (1/E) F, and d) equivalent load uncertainty. The results show that the interval and equivalent load solutions are the only ones, which represent an enclosure of the exact solution.  The equivalent load solution introduces the sharpest results.
 
 
Table 2. Triangular truss displacement due to different formulations.
Node
2
3
 
U ´10 -6 (m)
U ´10 -6 (m)
V ´10 -6 (m)
Combinations (exact)
[1.127, 1.184]
[0.448, 0.708]
[-3.077, -2.927]
Interval
[0.997, 1.325]
[0.318, 0.838]
[-3.150, -2.850]
Formulation b, E(k') u =p
[1.032, 1.281]
[0.483, 0.673]
[-3.123, -2.877]
Formulation c, k'u = (1/E) p
[1.127, 1.184]
[0.563, 0.592]
[-3.077, -2.927]
Equivalent load uncertainty
[1.039, 1.27]
[0.405, 0.750]
[-3.1, -2.9]

 

Conclusions: A new treatment of uncertainties in finite element solutions to mechanics problems based on fuzzy set theory has been developed. Uncertainties or fuzzy numbers are expressed as an interval of possible values. Interval representation is especially useful when components are specified to be within a given tolerance. The solution methods presented provides limit values for an exponentially growing combination of scalar solution in an effective manner.

Acknowledgment: The authors acknowledge support from the National Science Foundation's Grant DMI 97-14124.

References

[1] Hansen, E., Interval arithmetic in matrix computation, J.S. I. A. M., series B, Numerical Analysis,
[2] Kolmogorov, A.N., "Sur l'interplation et l'extraploation des suites stationnaires", Comptes Rendus de l'Academie des Sciences, Vol. 208, pp. 2043, 1939.
[3] Zadeh, L. A., "Fuzzy Sets", Information and Control, Vol. 8, 1965, pp. 338-353.
[4] Zadeh, L.A., "Fuzzy Sets as a Basis for a Theory of Possibility", Fuzzy Sets and Systems, Vol. 1, pp 3-28, 1978.
[5] Muhanna, R. L. and Mullen, R. L.(1995). "Development of Interval Based Methods for Fuzziness in Continuum Mechanics, " Proc., ISUMA-NAFIPS’95, September 17-20, 145-150.
[6] Muhanna, R. L., Mullen, R. L.(1998). "Formulation of Fuzzy Finite Element Methods for Mechanics Problems, "to appear in Special issue of Microcomputers in Civil Engineering on Fuzzy Modeling in Civil Engineering.
[7] Mullen, R. L. and Muhanna, R. L.(1996). "Structural Analysis with Fuzzy-Based Load Uncertainty, " Proc, 7th ASCE EMD/STD Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, WP I, MA, August 7-9, 310-313.
 
 

CWRU Department of Civil Engineering Communication