1. ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 10: Solution of Continuous Systems – Fundamental Concepts Mixed Formulations Intrinsic Coordinate Systems

2. Last TimeWeighted Residual Formulations eg. For Axial element Consider a general representation of a governing equation on a region V L is a differential operator

3. Last TimeWeighted Residual Formulations Exact Approximate Objective: Define so that weighted average of Error vanishes NOT THE ERROR ITSELF !!

4. Last TimeWeighted Residual Formulations Set Error relative to a weighting function f Objective: Define so that weighted average of Error vanishes

5. Weighted Residual Formulations f = 1 f ERROR

6. Weighted Residual Formulations f = 1 f ERROR

7. Last TimeWeighted Residual Formulations f ERROR

8. Last TimeWeighted Residual Formulations Assumption for approximate solution (Recall shape functions) Assumption for weighting function GALERKIN FORMULATION

9. Last TimeWeighted Residual Formulations fi are arbitrary and  0

10. Last TimeGalerkin Formulation Algebraic System of n Equations and n unknowns

11. Last TimeGalerkin’s Method in Elasticity Governing equations Interpolated Displ Field Interpolated Weighting Function

12. Last TimeGalerkin’s Method in Elasticity Integrate by part…

13. Last TimeGalerkin’s Method in Elasticity Virtual Work Virtual Total Potential Energy Compare to Total Potential Energy

14. Last TimeGalerkin’s Formulation • More general method • Operated directly on Governing Equation • Variational Form can be applied to other governing equations • Preffered to Rayleigh-Ritz method especially when function to be minimized is not available.

15. Displacement Based FE approximations Combine subsidiary equations to obtain G.E. G.E. in terms of displacements Stresses, Strains etc enter as natural B.C. Mixed Formulation Apply Galerkin directly on subsidiary relations Nodal dof contain displacements AND other field quantities Mixed Formulation

16. Mixed Formulation Axial Equilibrium… Stress-Displacement…

17. Mixed Formulation

18. Mixed Formulation Axial Equilibrium… Stress-Displacement… Galerkin Residual Equations

19. Mixed Formulation Axial Equilibrium…

20. Mixed Formulation A

21. Mixed Formulation Stress-Displacement… B

22. Mixed Formulation kus ksu kss A B

23. Mixed Formulation

24. Application Example

25. INTRINSIC COORDINATE SYSTEMS

26. Intrinsic Coordinate System x 1 3 2 x1 x x1=-1 x2=1 x3 x2 x Global C.S. Local C.S.

27. Intrinsic Coordinate System x 1 3 2 x x1=-1 x2=1 x1 x3 x2 x Linear Relationship Between GCS and LCS

28. Shape Functions wrt LCS x 1 3 2 x x1=-1 x2=1 u(-1)=a0 -a1 +a2 =u1 … u(1)=a0 +a1 +a2 =u2 u(0)=a0 =u3 u(x)=a0+a1 x +a2 x2

29. Shape Functions wrt Intrinsic Coordinate System x N1(x) N2(x) N3(x)

30. s, e wrt x

31. Element Strain-Displacement Matrix Cast in Matrix Form ee= B ue se= E B ue

32. Linear Stress Axial Element - In Summary 1 3 2 x3=0 x2=1 x1=-1 e = B u s = E B u

33. Linear Stress Axial Element - ke Stiffness Matrix

34. Linear Stress Axial Element - ke 1 3 2 x3=0 x2=1 x1=-1 Stiffness Matrix 3 1 2 1 2 3

35. Linear Stress Axial Element – fe,Te 1 3 2 x3=0 x2=1 x1=-1 Body Force 1 2 3 Uniformly Distributed Force 1 2 3