MCE 561 Computational Methods in Solid Mechanics One-Dimensional Problems

1. MCE 561 Computational Methods in Solid Mechanics One-Dimensional Problems

2. One-Dimensional Bar Element Axial Deformation of an Elastic Bar x A = Cross-sectional AreaE = Elastic Modulus f(x) = Distributed Loading Typical Bar Element W (j) (i) L (Two Degrees of Freedom) Virtual Strain Energy = Virtual Work Done by Surface and Body Forces For One-Dimensional Case

3. One-Dimensional Bar Element

4. x (local coordinate system) (2) (1) L u(x) x (2) (1) Linear Approximation Scheme y1(x) y2(x) 1 x (2) (1) yk(x) – Lagrange Interpolation Functions

5. Element EquationLinear Approximation Scheme, Constant Properties

6. x (3) (1) (2) u(x) L x (3) (1) (2) Quadratic Approximation Scheme y2(x) y3(x) y1(x) 1 x (3) (1) (2)

7. (2) (1) (2) (3) (1) (3) (2) (4) (1) Lagrange Interpolation FunctionsUsing Natural or Normalized Coordinates

8. P A1,E1,L1 A2,E2,L2 1 2 (1) (3) (2) Simple Example

9. P A1,E1,L1 A2,E2,L2 1 2 (1) (3) (2) Simple Example Continued

10. f(x) = Distributed Loading I = Section Moment of InertiaE = Elastic Modulus Typical Beam Element W (1) (2) L One-Dimensional Beam Element Deflection of an Elastic Beam x (Four Degrees of Freedom) Virtual Strain Energy = Virtual Work Done by Surface and Body Forces

11. Beam Approximation Functions To approximate deflection and slope at each node requires approximation of the form Evaluating deflection and slope at each node allows the determination of ci thus leading to

12. Beam Element Equation

13. (2) (3) (1) 1 2 FEA Beam Problem f Uniform EI a b

14. (2) (3) (1) 1 2 FEA Beam Problem Solve System for Primary Unknowns U3 ,U4 ,U5 ,U6 Nodal Forces Q1and Q2Can Then Be Determined

15. Special Features of Beam FEA Analytical Solution GivesCubic Deflection Curve Analytical Solution GivesQuartic Deflection Curve FEA Using Hermit Cubic Interpolation Will Yield Results That Match Exactly With Cubic Analytical Solutions

16. Truss Element Generalization of Bar Element With Arbitrary Orientation y k=AE/L x

17. W (1) (2) L Frame Element Generalization of Bar and Beam Element with Arbitrary Orientation Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation