Geometric Uncertainty in Truss Systems: An Interval Approach

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Geometric Uncertainty in Truss Systems An
Interval Approach

• Rafi L. Muhanna and Ayse Erdolen
• Georgia Institute of Technology

Robert L. Mullen Case Western Reserve
University
NSF Workshop on Modeling Errors and Uncertainty
in Engineering Computations February 22-24,
2006,Georgia Institute of Technology, Savannah,
USA
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Outline

• Introduction
• Interval Finite Elements
• Geometric Uncertainty
• Examples
• Conclusions

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Center for Reliable Engineering Computing (REC)
We handle computations with care
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Introduction- Uncertainty

• Uncertainty is unavoidable in engineering system
• structural mechanics entails uncertainties in
  material, geometry and load parameters

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Introduction- Engineering Systems

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

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Introduction- Uncertainty

• Geometric uncertainty due to fabrication and/or
  thermal changes in engineering systems
• tolerances (geometrical uncertainty)
• uncertainty in the components' length

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Introduction- Truss Systems
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Introduction- Uncertainty
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Introduction- Uncertainty
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Introduction- Uncertainty
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Introduction- Uncertainty

• 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

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Introduction- Why Interval?

• 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)

• Provides guaranteed enclosures

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Introduction- Finite Element Method

• Finite Element Method (FEM) is a numerical method
  that provides approximate solutions to partial
  differential equations

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Interval arithmetic

• Interval number
• Interval vector and interval matrix, e.g.,
• Notations

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Interval Finite Elements

• Local and global coordinate systems for a truss
  bar element
• ccos? and ssin?

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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

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Interval Finite Elements

• Nodal forces induced by a given bar due
  fabrication error or temperature change
• P0i Interval vector of nodal forces obtained as
  a result of missfitting problem

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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

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Interval Finite Elements
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Interval Finite Elements
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Interval Finite Elements

• 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

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Interval Finite Elements

• Internal force in each bar
• Si interval force of the ith bar of truss
• Ki ith element stiffness matrix
• Li Boolean matrix with 1 and 0 entries

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Numerical example-Truss structure

• A1, A2, A3, A4, A5, A6 0.01m2 cross-sectional
  area
• E 200 GPa modulus of elasticity of all
  elements
• ?L-0.001, 0.001 same fabrication error for
  all members

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Truss structure-results

• Table 1 One Bay Truss (6 Elements)-nodal
  displacement-Present solution

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Truss structure-results
Table 2. One Bay Truss (6 Elements), nodal
displacement for all possible ?L
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Truss structure-results

• Table 3. One Bay Truss (6 Elements) nodal
  displacement without accounting for dependency

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Conclusions

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

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• Interval finite elements for infinite beauty.