1. Geometric Uncertainty in Truss Systems: An Interval Approach Rafi L. Muhanna and Ayse Erdolen Georgia Institute of Technology NSF Workshop on Modeling Errors and Uncertainty in Engineering Computations February 22-24, 2006,Georgia Institute of Technology, Savannah, USA Robert L. Mullen Case Western Reserve University

2. Outline Introduction Introduction Interval Finite Elements Interval Finite Elements Geometric Uncertainty Geometric Uncertainty Examples Examples Conclusions Conclusions

3. Center for Reliable Engineering Computing (REC) We handle computations with care

4.  Uncertainty is unavoidable in engineering system  structural mechanics entails uncertainties in material, geometry and load parameters Introduction- Uncertainty

5. Engineering systems are usually designed with a pre-described geometry in order to meet the intended function for which they are designed Introduction- Engineering Systems

6.  Geometric uncertainty due to fabrication and/or thermal changes in engineering systems  tolerances (geometrical uncertainty)  uncertainty in the components' length Introduction- Uncertainty

7. Introduction- Truss Systems

8. Introduction- Uncertainty

11.  Interval number represents a range of possible values within a closed set L  L,  Represents an uncertain quantity by giving a range of possible values L = [Lo   L, Lo +  L]  How to define bounds on the possible ranges of uncertainty?  experimental data, measurements, statistical analysis, expert knowledge Introduction- Uncertainty

12.  Simple and elegant  Conforms to practical tolerance concept  Describes the uncertainty that can not be appropriately modeled by probabilistic approach  Computational basis for other uncertainty approaches (e.g., fuzzy set, random set) Introduction- Why Interval?  Provides guaranteed enclosures

13. Finite Element Method (FEM) is a numerical method that provides approximate solutions to partial differential equations Introduction- Finite Element Method

14. Interval arithmetic  Interval number:  Interval vector and interval matrix, e.g.,  Notations

15. Interval Finite Elements Local and global coordinate systems for a truss bar element c=cos  and s=sin 

16. Interval Finite Elements  Interval axial forces for bar element E = modulus of elasticity A = cross sectional aria  L = [-  L, +  L] interval deviation from the nominal value of the bar’ s length  L = [-  T, +  T] interval of the temperature change  = coefficient of thermal expansion

17. Interval Finite Elements  Nodal forces induced by a given bar due fabrication error or temperature change P 0i = Interval vector of nodal forces obtained as a result of missfitting problem

18. Interval Finite Elements  In the absence of external loading the final interval finite element system of equations K = stiffness matrix of the system U = vector of interval displacements

19. Interval Finite Elements

21.  In the absence of external loading the final interval finite element system of equations M = a matrix that relates the system’ s degrees of freedom with elements loads

22. Interval Finite Elements  Internal force in each bar S i = interval force of the ith bar of truss K i = ith element stiffness matrix L i = Boolean matrix with 1 and 0 entries

23. Numerical example-Truss structure A1, A2, A3, A4, A5, A6 : 0.01m 2 cross-sectional area E: 200 GPa modulus of elasticity of all elements  L=[-0.001, 0.001] same fabrication error for all members  m          

24. Truss structure-results Table 1: One Bay Truss (6 Elements)-nodal displacement- Present solution NodeUxUx UyUy 1[0, 0] 2[-0.00150,0.00150][0, 0] 3[-0.003414, 0.003414][-0.00150,0.00150] 4[-0.0032071, 0.0032071][-0.00150,0.00150]

25. Truss structure-results NodeUxUx UyUy 1[0, 0] 2[-0.00150,0.00150][0, 0] 3[-0.003414, 0.003414][-0.00150,0.00150] 4[-0.0032071, 0.0032071][-0.00150,0.00150] Table 2. One Bay Truss (6 Elements), nodal displacement for all possible  L

26. Truss structure-results Table 3. One Bay Truss (6 Elements) – nodal displacement without accounting for dependency NodeUxUx UyUy 1[0,0] 2[-0.03,0.03][0,0] 3[-0.08742,0.08742][-0.03,0.03] 4[-0.08431,0.08431][-0.0331,0.0331]

27. Conclusions Geometric uncertainty for truss systems in form of tolerances is presented IFEM is used Exact enclosure on the deformed geometry is obtained

28. “ Interval finite elements for infinite beauty…. ”