1. MECH303 Advanced Stresses Analysis Lecture 5 FEM of 1-D Problems: Applications

2. Torsional Shaft Review Assumption: Circular cross section Shear stress: Deformation: Shear strain:

3. Finite Element Equation for Torsional Shaft

4. Bending Beam Review Normal strain: Pure bending problems: Normal stress: Normal stress with bending moment: Moment-curvature relationship: Flexure formula: x y M M

5. Bending Beam Review Deflection: Sign convention: Relationship between shear force, bending moment and transverse load: q(x) x y + - M M M + - V V V

6. Governing Equation and Boundary Condition Governing Equation Boundary Conditions ----- Essential BCs – if v or is specified at the boundary. Natural BCs – if or is specified at the boundary. { 0<x<L

7. Weak Formulation for Beam Element Governing Equation Weighted-Integral Formulation for one element Weak Form from Integration-by-Parts ----- (1 st time)

8. Weak Formulation Weak Form from Integration-by-Parts ----- (2 nd time) V(x 2 ) x = x 1 M(x 2 ) q(x) y x x = x 2 V(x 1 ) M(x 1 ) L = x 2 -x 1

9. Weak Formulation Weak Form Q3Q3 x = x 1 Q4Q4 q(x) y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1

10. Ritz Method for Approximation Let w(x)=  i (x), i = 1, 2, 3, 4 Q3Q3 x = x 1 Q4Q4 q(x) y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 where

11. Ritz Method for Approximation Q3Q3 x = x 1 y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 Q4Q4

12. Ritz Method for Approximation Q3Q3 x = x 1 y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 Q4Q4

13. Selection of Shape Function The best situation is ----- Interpolation Properties

14. Derivation of Shape Function for Beam Element – Local Coordinates How to select  i ??? and where Let Find coefficients to satisfy the interpolation properties.

15. Derivation of Shape Function for Beam Element How to select  i ??? e.g. Let Similarly

16. Derivation of Shape Function for Beam Element In the global coordinates:

17. Element Equations of 4 th Order 1-D Model u3u3 x = x 1 u4u4 q(x) y(v) x x = x 2 u1u1 u2u2 L = x 2 -x 1 x=x 2 x=x 1 1 1 1 1 3 3 2 2 4 4

18. Element Equations of 4 th Order 1-D Model u3u3 x = x 1 u4u4 q(x) y(v) x x = x 2 u1u1 u2u2 L = x 2 -x 1

19. Finite Element Analysis of 1-D Problems - Applications F L L L Example 1. Governing equation: Weak form for one element where

20. Finite Element Analysis of 1-D Problems Example 1. Approximation function: 3 3 2 2 1 1 4 4 x=x 1 x=x 2

21. Finite Element Analysis of 1-D Problems Example 1. Finite element model: P 1, v 1 P 2, v 2 P 3, v 3 P 4, v 4 M 1,  1 M 2,  2 M 3,  3 M 4,  4 I II III Discretization:

22. Matrix Assembly of Multiple Beam Elements Element I I

23. Matrix Assembly of Multiple Beam Elements Element I I

24. Solution Procedures Apply known boundary conditions

25. Solution Procedures

26. Shear Resultant & Bending Moment Diagram

27. Plane Flame Frame: combination of bar and beam E, A, I, L Q 1, v 1 Q 3, v 2 Q 2,  1 P 1, u 1 Q 4,  2 P 2, u 2

28. Finite Element Model of an Arbitrarily Oriented Frame  x y  x y

29. local global

30. Plane Frame Analysis - Example Rigid Joint Hinge Joint Beam II Beam I Beam Bar F FF F

31. Plane Frame Analysis P 1, u 1 P 2, u 2 Q 2,  1 Q 4,  2 Q 1, v 1 Q 3, v 2

32. Plane Frame Analysis P 1, u 2 Q 3, v 3 Q 2,  2 Q 4,  3 Q 1, v 2 P 2, u 3

33. Plane Frame Analysis