1. The Finite Element Method A Practical Course
CHAPTER 5:
FEM FOR 2D SOLIDS

2. CONTENTS INTRODUCTION LINEAR TRIANGULAR ELEMENTS
Field variable interpolation
Shape functions construction
Using area coordinates
Strain matrix
Element matrices
LINEAR RECTANGULAR ELEMENTS
Gauss integration
Evaluation of me

3. CONTENTS LINEAR QUADRILATERAL ELEMENTS HIGHER ORDER ELEMENTS
Coordinate mapping
Strain matrix
Element matrices
Remarks
HIGHER ORDER ELEMENTS
COMMENTS (GAUSS INTEGRATION)
CASE STUDY

4. INTRODUCTION
2D solid elements are applicable for the analysis of plane strain and plane stress problems.
A 2D solid element can have a triangular, rectangular or quadrilateral shape with straight or curved edges.
2D solid element can deform only in the plane of the 2D solid.
At any point, there are two components in x and y directions for the displacement as well as forces.

5. INTRODUCTION
For plane strain problems, the thickness of the element is unit, but for plane stress problems, the actual thickness must be used.
In this course, it is assumed that the element has a uniform thickness h.
Formulating 2-D elements with given variation of thickness is also straightforward, as the procedure is the same as that for a uniform element.

6. 2D solids – plane stress and plane strain

7. LINEAR TRIANGULAR ELEMENTS
Less accurate than quadrilateral elements
Used by most mesh generators for complex geometry
Linear triangular element

8. Field variable interpolation
where
(Shape functions)

9. Shape functions construction
Assume,
i= 1, 2, 3 or

10. Shape functions construction
Delta function property:
Therefore,
Solving,

11. Shape functions construction
Area of triangle Moment matrix
Substitute a1, b1 and c1 back into N1 = a1 + b1x + c1y:

12. Shape functions construction
Similarly,

13. Shape functions construction
where
i= 1, 2, 3
J, k determined from cyclic permutation
i = 1, 2 i k j
j = 2, 3
k = 3, 1

14. Using area coordinates
Alternative method of constructing shape functions
 2-3-P:
Similarly,  3-1-P A2
 1-2-P A3

15. Using area coordinates
Partitions of unity:
Delta function property: e.g. L1 = 0 at if P at nodes 2 or 3
Therefore,

16. (constant strain element)
Strain matrix
where 
(constant strain element)

17. Element matrices
Constant matrix 

18. Element matrices For elements with uniform density and thickness,
Eisenberg and Malvern (1973):

19. Element matrices
Uniform distributed load:

20. LINEAR RECTANGULAR ELEMENTS
Non-constant strain matrix
More accurate representation of stress and strain
Regular shape makes formulation easy

21. Shape functions construction
Consider a rectangular element

22. Shape functions construction
where

23. Shape functions construction
Delta function property
Partition of unity

24. Strain matrix
Note: No longer a constant matrix!

25. Element matrices 
dxdy = ab dxdh
Therefore,

26. Element matrices
For uniformly distributed load,

27. Gauss integration
For evaluation of integrals in ke and me (in practice)
In 1 direction:
m gauss points gives exact solution of polynomial integrand of n = 2m - 1
In 2 directions:

28. Gauss integration m Gauss Point xj Gauss Weight wj Accuracy order n 12
-1/3, 1/3
1, 1 3
-0.6, 0, 0.6
5/9, 8/9, 5/9 5 4
, , ,
, , , 7
, , 0, ,
, , , , 9 6
, ,
, , ,
, , , , , 11

29. Evaluation of me

30. Evaluation of me
E.g.
Note: In practice, gauss integration is often used

31. LINEAR QUADRILATERAL ELEMENTS
Rectangular elements have limited application
Quadrilateral elements with unparallel edges are more useful
Irregular shape requires coordinate mapping before using Gauss integration

32. Coordinate mapping Physical coordinates Natural coordinates
(Interpolation of displacements)
(Interpolation of coordinates)

33. Coordinate mapping
where ,

34. Coordinate mapping
Substitute x = 1 into or
Eliminating ,

35. Strain matrix or
where
(Jacobian matrix)
Since ,

36. Strain matrix Therefore,
(Relationship between differentials of shape functions w.r.t. physical coordinates and differentials w.r.t. natural coordinates)
Therefore,
Replace differentials of Ni w.r.t. x and y with differentials of Ni w.r.t.  and 

37. Element matrices
Murnaghan (1951) :
dA=det |J | dxdh

38. Remarks
Shape functions used for interpolating the coordinates are the same as the shape functions used for the interpolation of the displacement field. Therefore, the element is called isoparametric element.
Note that the shape functions for coordinate interpolation and displacement interpolation do not have to be the same.
Using the different shape functions for coordinate interpolation and displacement interpolation, respectively, will lead to the development of so-called subparametric or superparametric elements.

39. HIGHER ORDER ELEMENTS Higher order triangular elements
nd = (p+1)(p+2)/2
Node i,
Argyris, 1968 :

40. HIGHER ORDER ELEMENTS Higher order triangular elements (Cont’d)
Quadratic element
Cubic element

41. HIGHER ORDER ELEMENTS Higher order rectangular elements Lagrange type:
(Zienkiewicz et al., 2000)

42. HIGHER ORDER ELEMENTS Higher order rectangular elements(Cont’d)
(9 node quadratic element)

43. HIGHER ORDER ELEMENTS Higher order rectangular elements(Cont’d)
Serendipity type:
(eight node quadratic element)

44. HIGHER ORDER ELEMENTS Higher order rectangular elements(Cont’d)
(twelve node cubic element)

45. ELEMENT WITH CURVED EDGES

46. COMMENTS (GAUSS INTEGRATION)
When the Gauss integration scheme is used, one has to decide how many Gauss points should be used.
Theoretically, for a one-dimensional integral, using m points can give the exact solution for the integral of a polynomial integrand of up to an order of (2m-1).
As a general rule of thumb, more points should be used for higher order of elements.

47. COMMENTS (GAUSS INTEGRATION)
Using smaller number of Gauss points tends to counteract the over-stiff behaviour associated with the displacement- based method.
Displacement in an element is assumed using shape functions. This implies that the deformation of the element is somehow prescribed in a fashion of the shape function. This prescription gives a constraint to the element. The so- constrained element behaves stiffer than it should be. It is often observed that higher order elements are usually softer than lower order ones. This is because using higher order elements gives less constraint to the elements.

48. COMMENTS (GAUSS INTEGRATION)
Two gauss points for linear elements, and two or three points for quadratic elements in each direction should be sufficient for many cases.
Most of the explicit FEM codes based on explicit formulation tend to use one-point integration to achieve the best performance in saving CPU time.

49. CASE STUDY
Side drive micro-motor

50. Elastic Properties of Polysilicon
CASE STUDY
10N/m
Elastic Properties of Polysilicon
Young’s Modulus, E
169GPa
Poisson’s ratio, 
0.262
Density, 
2300kgm-3
10N/m
10N/m

51. CASE STUDY
Analysis no. 1: Von Mises stress distribution using 24 bilinear quadrilateral elements (41 nodes)

52. CASE STUDY
Analysis no. 2: Von Mises stress distribution using 96 bilinear quadrilateral elements (129 nodes)

53. CASE STUDY
Analysis no. 3: Von Mises stress distribution using 144 bilinear quadrilateral elements (185 nodes)

54. CASE STUDY
Analysis no. 4: Von Mises stress distribution using 24 eight-nodal, quadratic elements (105 nodes)

55. CASE STUDY
Analysis no. 5: Von Mises stress distribution using 192 three-nodal, triangular elements (129 nodes)

56. CASE STUDY Analysis no. Number / type of elements
Total number of nodes in model
Maximum Von Mises Stress (GPa) 1
24 bilinear, quadrilateral 41
0.0139 2
96 bilinear, quadrilateral
129
0.0180 3
144 bilinear, quadrilateral
185
0.0197 4
24 quadratic, quadrilateral
105
0.0191 5
192 linear, triangular
0.0167