CST ELEMENT Constant Strain Triangular Element Decompose twodimensional

CST ELEMENT • Constant Strain Triangular Element – Decompose two-dimensional domain by a set of triangles. – Each triangular element is composed by three corner nodes. – Each element shares its edge and two corner nodes with an adjacent element – Counter-clockwise or clockwise node numbering – Each node has two DOFs: u and v – displacements interpolation using the shape functions and nodal displacements. – Displacement is linear because three nodal data are available. – Stress & strain are constant. y v 3 3 u 3 v 1 1 u 1 v 2 2 u 2 x 1

DISPLACEMENT INTERPOLATION • Assumed form for displacements – Components u(x, y) and v(x, y) are separately interpolated. – u(x, y) is interpolated in terms of u 1, u 2, and u 3, and v(x, y) in terms of v 1, v 2, and v 3. – interpolation function must be a three term polynomial in x and y. – Since we must have rigid body displacements and constant strain terms in the interpolation function, the displacement interpolation must be of the form – The goal is to calculate αi and βi, i = 1, 2, 3, in terms of nodal displacements. 2

CST ELEMENT cont. • Equations for coefficients – x-displacement: Evaluate displacement at each node 3 v 3 u 3 v 1 – In matrix notation 1 u 1 v 2 2 u 2 – When is the coefficient matrix singular? 3

SOLUTION • Explicit solution: – where – Area: 4

INTERPOLATION FUNCTION • Insert to the interpolation equation N 1(x, y) N 2(x, y) N 3(x, y) 5

SIMILARLY FOR V • Displacement Interpolation – A similar procedure can be applied for y-displacement v(x, y). Shape Function – N 1, N 2, and N 3 are linear functions of x- and y-coordinates. – Interpolated displacement changes linearly along the each coordinate direction. – Tent functions 6

Quiz-like questions • Given the two finite-element model in shear below with the only non-zero displacements given below Calculate the shape function of Node 1 in the top element. • From an exam • Which node in which element has the shape function y? • Solutions in the notes page 7

MATRIX EQUATION • Displacement Interpolation – [N]: 2× 6 matrix, {q}: 6× 1 vector. – For a given point (x, y) within element, calculate [N] and multiply it with {q} to evaluate displacement at the point (x, y). 8

STRAINS • Strain Interpolation – differentiating the displacement in x- and y-directions. – differentiating shape function [N] because {q} is constant. Strains are constant inside! 9

B-MATRIX FOR CST ELEMENT • Strain calculation – [B] matrix is a constant matrix and depends only on the coordinates of the three nodes of the triangular element. – the strains will be constant over a given element 10

PROPERTIES OF CST ELEMENT • Since displacement is linear in x and y, the triangular element deforms into another triangle when forces are applied. • a straight line drawn within an element before deformation becomes another straight line after deformation. • Consider a local coordinate x such that x = 0 at Node 1 and x = a at Node 2. 3 • Displacement on the edge 1 -2: 1 • Since the variation of displacement is linear, the displacements should depend only on u 1 and u 2, and not on u 3. a x 2 11

PROPERTIES OF CST ELEMENT cont. 3 • Property of CST Element 1 1 1 a x Element 2 • Inter-element Displacement Compatibility 2 3 2 – Displacements at any point in an element can be computed from nodal displacements of that particular element and the shape functions. – Consider a point on a common edge of two adjacent elements, which can be considered as belonging to either of the elements. – Because displacements on the edge depend only on edge nodes, both elements will produce the same displacement for a point on a common edge. 12

EXAMPLE - Interpolation • nodal displacements {u 1, v 1, u 2, v 2, u 3, v 3, u 4, v 4} = {− 0. 1, 0, 0. 1, 0} y • Element 1: Nodes 1 -2 -4 (1, 1) 3 (0, 1) 4 2 1 Calculate strains directly 1 (0, 0) 2 (1, 0) x 13

EXAMPLE – Interpolation cont. y • Element 2: Nodes 2 -3 -4 (1, 1) 3 (0, 1) 4 2 1 1 (0, 0) 2 (1, 0) x Strains are discontinuous along the element boundary 14

Quiz-like questions • To determine εxx at any point in a CST element which nodal displacements are needed? • For a tent function in a CST element N 2=1 at node 2. At which edge of element will N 2 be zero? • Will the strain at a point on common edge of two elements be same when calculated from both the elements ? • For Element 1 on slide 14 we are give that u 1=0, u 2=2 mm. What is εxx ? • Find location of the point on edge of CST element with displacement u=0. 3 mm. Nodal displacement u 1 =0. 2 mm , u 2=0. 5 mm. Edge length is 3 cm. 15