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MANE 4240 & CIVL 4240 Introduction to Finite Elements

MANE 4240 & CIVL 4240 Introduction to Finite Elements. Prof. Suvranu De. Four-noded rectangular element. Reading assignment: Logan 10.2 + Lecture notes. Summary: Computation of shape functions for 4-noded quad Special case: rectangular element Properties of shape functions

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MANE 4240 & CIVL 4240 Introduction to Finite Elements

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  1. MANE 4240 & CIVL 4240Introduction to Finite Elements Prof. Suvranu De Four-noded rectangular element

  2. Reading assignment: Logan 10.2 + Lecture notes • Summary: • Computation of shape functions for 4-noded quad • Special case: rectangular element • Properties of shape functions • Computation of strain-displacement matrix • Example problem • Hint at how to generate shape functions of higher order (Lagrange) elements

  3. y x Finite element formulation for 2D: Step 1:Divide the body into finite elementsconnected to each other through special points (“nodes”) py v3 3 px 4 3 u3 v4 2 v2 Element ‘e’ v 1 4 u u4 ST u2 v1 2 y x Su u1 1 v x u

  4. Summary: For each element Displacement approximation in terms of shape functions Strain approximation in terms of strain-displacement matrix Stress approximation Element stiffness matrix Element nodal load vector

  5. v1 v3 1 u1 (x3,y3) (x1,y1) v u3 v2 u 3 y (x,y) u2 2 (x2,y2) x Constant Strain Triangle (CST) : Simplest 2D finite element • 3 nodes per element • 2 dofs per node (each node can move in x- and y- directions) • Hence 6 dofs per element

  6. v1 v3 1 u1 (x3,y3) (x1,y1) v u3 v2 u 3 y (x,y) u2 2 (x2,y2) x Formula for the shape functions are where

  7. Approximation of displacements Approximation of the strains

  8. Element stiffness matrix t Since B is constant A t=thickness of the element A=surface area of the element Element nodal load vector

  9. Class exercise For the CST shown below, compute the vector of nodal loads due to surface traction 1 y fS3y fS2y fS2x fS3x 2 3 x (0,0) (1,0) py=-1

  10. 1 y fS3y fS2y fS2x fS3x 2 3 x py=-1 Class exercise The only nonzero nodal loads are (can you derive this simpler?)

  11. Now compute

  12. 4 3 (x3,y3) (x4,y4) v 2b u (x,y) y 1 2 2a (x2,y2) (x1,y1) x 4-noded rectangular element with edges parallel to the coordinate axes: • 4 nodes per element • 2 dofs per node (each node can move in x- and y- directions) • 8 dofs per element

  13. y l1(y) 3 4 2b 1 N1 1 2 2a x l1(x) 1 Generation of N1: At node 1 has the property Similarly has the property Hence choose the shape function at node 1 as

  14. Using similar arguments, choose

  15. Properties of the shape functions: 1. The shape functions N1, N2 , N3 and N4 are bilinear functions of x and y 2. Kronecker delta property 3. Completeness

  16. 3. Along lines parallel to the x- or y-axes, the shape functions are linear. But along any other line they are nonlinear. 4. An element shape function related to a specific nodal point is zero along element boundaries not containing the nodal point. 5. The displacement field is continuous across elements 6. The strains and stresses are not constant within an element nor are they continuous across element boundaries.

  17. The strain-displacement relationship Notice that the strains (and hence the stresses) are NOT constant within an element

  18. v3 v4 3 4 u4 u3 v y v2 u v1 (x,y) u2 2 u1 1 x v3 v4 u3 u4 u1 u2 v2 v1 Computation of the terms in the stiffness matrix of 2D elements (recap) The B-matrix (strain-displacement) corresponding to this element is We will denote the columns of the B-matrix as

  19. v4 u4 v3 u2 v2 u3 u1 v1 u1 v1 u2 v2 u3 v3 u4 v4 The stiffness matrix corresponding to this element is which has the following form The individual entries of the stiffness matrix may be computed as follows

  20.  Notice that these formulae are quite general (apply to all kinds of finite elements, CST, quadrilateral, etc) since we have not used any specific shape functions for their derivation.

  21. Example 1000 lb 300 psi y 3 4 Thickness (t) = 0.5 in E= 30×106 psi n=0.25 2 in 1 2 x 3 in • Compute the unknown nodal displacements. • Compute the stresses in the two elements. This is exactly the same problem that we solved in last class, except now we have to use a single 4-noded element

  22. Realize that this is a plane stress problem and therefore we need to use Write down the shape functions

  23. We have 4 nodes with 2 dofs per node=8dofs. However, 5 of these are fixed. The nonzero displacements are v3 u2 u3 Hence we need to solve u2 u3 v3 Need to compute only the relevant terms in the stiffness matrix

  24. Compute only the relevant columns of the B matrix

  25. Similarly compute the other terms

  26. How do we compute f3y 3 4

  27. How about a 9-noded rectangle? Corner nodes y 5 2 1 a a b 8 9 6 x Midside nodes b 7 4 3 Center node Question: Can you generate the shape functions of a 16-noded rectangle? Note: These elements, whose shape functions are generated by multiplying the shape functions of 1D elements, are said to belong to the “Lagrange” family

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