ECIV 720 A Advanced Structural Mechanics and Analysis Lecture 15: Quadrilateral Isoparametric Elements (cont’d) Force Vectors Modeling Issues Higher Order Elements

Integration of Stiffness Matrix BT(8x3) D (3x3) ke (8x8) B (3x8)

Integration of Stiffness Matrix Each term kij in ke is expressed as Linear Shape Functions is each Direction Gaussian Quadrature is accurate if We use 2 Points in each direction

Integration of Stiffness Matrix h

Choices in Numerical Integration Numerical Integration cannot produce exact results Accuracy of Integration is increased by using more integration points. Accuracy of computed FE solution DOES NOT necessarily increase by using more integration points.

FULL Integration A quadrature rule of sufficient accuracy to exactly integrate all stiffness coefficients kij e.g. 2-point Gauss rule exact for polynomials up to 2nd order

Reduced Integration, Underintegration Use of an integration rule of less than full order Advantages Reduced Computation Times May improve accuracy of FE results Stabilization Disadvantages Spurious Modes (No resistance to nodal loads that tend to activate the mode)

Spurious Modes Consider the 4-node plane stress element 1 t=1 E=1 v=0.3 1 8 degrees of freedom 8 modes Solve Eigenproblem

Spurious Modes Rigid Body Mode Rigid Body Mode

Spurious Modes Rigid Body Mode

Spurious Modes Flexural Mode Flexural Mode

Spurious Modes Shear Mode

Uniform Extension Mode Spurious Modes Stretching Mode Uniform Extension Mode (breathing)

Element Body Forces Total Potential Galerkin

Body Forces Integral of the form

Body Forces In both approaches Linear Shape Functions Use same quadrature as stiffness maitrx

Element Traction Total Potential Galerkin

Element Traction Similarly to triangles, traction is applied along sides of element 3 u v Tx Ty h 4 4 x 2 1

Traction For constant traction along side 2-3 Traction components along 2-3

More Accurate at Integration points Stresses h x Stresses are calculated at any x,h

Modeling Issues: Nodal Forces In view of… A node should be placed at the location of nodal forces Or virtual potential energy

Modeling Issues: Element Shape Square : Optimum Shape Not always possible to use Rectangles: Rule of Thumb Ratio of sides <2 Larger ratios may be used with caution Angular Distortion Internal Angle < 180o

Modeling Issues: Degenerate Quadrilaterals Coincident Corner Nodes 1 2 3 4 2 1 4 x x Integration Bias 3 Less accurate

Modeling Issues: Degenerate Quadrilaterals Three nodes collinear 1 2 3 4 Integration Bias 1 2 3 4 x x Less accurate

Modeling Issues: Degenerate Quadrilaterals 2 nodes Use only as necessary to improve representation of geometry Do not use in place of triangular elements

All interior angles < 180 A NoNo Situation y 3 (7,9) Parent x h (6,4) 4 1 2 J singular (3,2) (9,2) x All interior angles < 180

Another NoNo Situation x, y not uniquely defined x

FEM at a glance It should be clear by now that the cornerstone in FEM procedures is the interpolation of the displacement field from discrete values Where m is the number of nodes that define the interpolation and the finite element and N is a set of Shape Functions

FEM at a glance m=2 x x1=-1 x2=1 x1=-1 1 x 3 x2=1 2 m=3

FEM at a glance 1 2 3 q6 q5 q4 q3 q2 q1 v u x h m=3 x h 4 1 2 3 m=4

FEM at a glance In order to derive the shape functions it was assumed that the displacement field is a polynomial of any degree, for all cases considered 1-D 2-D Coefficients ai represent generalized coordinates

FEM at a glance For the assumed displacement field to be admissible we must enforce as many boundary conditions as the number of polynomial coefficients x1=-1 1 x 3 x2=1 2 e.g.

FEM at a glance This yields a system of as many equations as the number of generalized displacements that can be solved for ai

FEM at a glance Substituting ai in the assumed displacement field and rearranging terms…

FEM at a glance 1 3 2 u(x)=a0+a1 x +a2 x 2 u(-1)=a0 -a1 +a2 =u1 u(0)=a0 =u3 …

Let’s go through the exercise 1 x2 2 Assume an incomplete form of quadratic variation

Incomplete form of quadratic variation x1 1 x2 2 We must satisfy

Incomplete form of quadratic variation And thus,

Incomplete form of quadratic variation And substituting in

Incomplete form of quadratic variation Which can be cast in matrix form as

Isoparametric Formulation The shape functions derived for the interpolation of the displacement field are used to interpolate geometry x1 1 x2 2

Intrinsic Coordinate Systems Intrinsic coordinate systems are introduced to eliminate dependency of Shape functions from geometry 1 (-1,-1) 2 (1,-1) 4 (-1,1) 3 (1,1) x h The price? Jacobian of transformation Great Advantage for the money!

Field Variables in Discrete Form Geometry Displacement e = B un Strain Tensor s = DB un Stress Tensor

FEM at a glance Element Strain Energy Work Potential of Body Force Work Potential of Surface Traction etc

Higher Order Elements Quadrilateral Elements Recall the 4-node Complete Polynomial 4 Boundary Conditions for admissible displacements 4 generalized displacements ai

Assume Complete Quadratic Polynomial Higher Order Elements Quadrilateral Elements Assume Complete Quadratic Polynomial 9 generalized displacements ai 9 BC for admissible displacements

BT18x3 D3x3 B3x18 ke 18x18 9-node quadrilateral 9-nodes x 2dof/node = 18 dof BT18x3 D3x3 B3x18 ke 18x18

9-node element Shape Functions Following the standard procedure the shape functions are derived as 1 2 3 4 Corner Nodes x h 5 6 7 8 Mid-Side Nodes 9 Middle Node

9-node element – Shape Functions Can also be derived from the 3-node axial element x1=-1 1 x x2=1 3 2

Construction of Lagrange Shape Functions x (1,h) (1,1) 1 (-1,-1)

N1,2,3,4 Graphical Representation

N5,6,7,8 Graphical Representation

N9 Graphical Representation

Polynomials & the Pascal Triangle Degree 1 x y 1 2 x2 xy y2 3 x3 x2y xy2 y3 4 x4 x3y x2y2 xy3 y4 5 x5 x4y x3y2 x2y3 xy4 y5 …….

Polynomials & the Pascal Triangle To construct a complete polynomial Q1 4-node Quad 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5 Q2 9-node Quad etc

Incomplete Polynomials 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5 3-node triangular

Incomplete Polynomials 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5

8 coefficients to determine for admissible displ. 8-node quadrilateral Assume interpolation 1 2 3 4 x h 5 6 7 8 8 coefficients to determine for admissible displ.

BT16x3 D3x3 B3x16 ke 16x16 8-node quadrilateral 8-nodes x 2dof/node = 16 dof BT16x3 D3x3 B3x16 ke 16x16

8-node element Shape Functions Following the standard procedure the shape functions are derived as 1 2 3 4 Corner Nodes h 5 6 7 8 Mid-Side Nodes x

N1,2,3,4 Graphical Representation

N5,6,7,8 Graphical Representation

Incomplete Polynomials 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5

6 coefficients to determine for admissible displ. 6-node Triangular Assume interpolation 1 2 3 4 5 6 6 coefficients to determine for admissible displ.

BT12x3 D3x3 B3x12 ke 12x12 6-node triangular 6-nodes x 2dof/node = 12 dof 1 2 3 4 5 6 BT12x3 D3x3 B3x12 ke 12x12

6-node element Shape Functions Following the standard procedure the shape functions are derived as Corner Nodes 1 2 3 Mid-Side Nodes 4 5 6 Li:Area coordinates

Other Higher Order Elements 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x5 x4y x3y2 x2y3 xy4 y5 12-node quad x h 1 2 3 4

Other Higher Order Elements 1 x y x2 xy y2 x3 x2y xy2 y3 x4 x3y x2y2 xy3 y4 ……. x3y2 16-node quad x h 1 2 3 4 x5 x4y x3y2 x2y3 xy4 y5