1. 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. u1 v1 u2 v2 u3 v3 3 1 2 x y

2. 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 u1, u2, and u3, and v(x,y) in terms of v1, v2, and v3.
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.

3. CST ELEMENT cont. Equations for coefficients
x-displacement: Evaluate displacement at each node
In matrix notation
When is the coefficient matrix singular? u1 v1 u2 v2 u3 v3 3 1 2

4. SOLUTION
Explicit solution:
where
Area:

5. INTERPOLATION FUNCTION
Insert to the interpolation equation
N1(x,y)
N2(x,y)
N3(x,y)

6. SIMILARLY FOR V Displacement Interpolation
A similar procedure can be applied for y-displacement v(x, y).
N1, N2, and N3 are linear functions of x- and y-coordinates.
Interpolated displacement changes linearly along the each coordinate direction.
Tent functions
Shape Function

7. Quiz-like questions From an exam
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
The shape function is first
The shape function N=y is zero at nodes 1 and 2 and is equal to 1 at nodes 3 and 4. it cannot be a shape function for the top element, because it is equal to 1 at two nodes of this element, so it must be N3 for the bottom element.

8. 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).

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

10. 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

11. 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.
Displacement on the edge 1-2:
Since the variation of displacement is linear, the displacements should depend only on u1 and u2, and not on u3. 3 1 2 x a

12. PROPERTIES OF CST ELEMENT cont.3 1 2 x a
Element 1
Element 2
Property of CST Element
Inter-element Displacement Compatibility
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.

13. EXAMPLE - Interpolation
nodal displacements {u1, v1, u2, v2, u3, v3, u4, v4} = {−0.1, 0, 0.1, 0, −0.1, 0, 0.1, 0}
Element 1: Nodes 1-2-4 1 2 3 4
(0,0)
(1,0)
(0,1)
(1,1) x y
Calculate strains directly

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

15. 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 N2=1 at node 2. At which edge of element will N2 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 u1=0, u2=2mm. What is εxx ?
Find location of the point on edge of CST element with displacement u=0.3mm. Nodal displacement u1 =0.2mm , u2=0.5mm. Edge length is 3cm.
Only the x direction displacement(u) of the three nodes
At the edge that does not contain node 2. That is edge 1-3
Not all strain components. The strain is constant in each element, and if adjacent elements had to have strain continuity, the strain will be the same everywhere. The strain components are in general determined by all three nodes, and the two elements share only two nodes. However, the strain in the direction of the common edge will be the same, as hinted to by the next question.
The strain along the 1-2 edge is 0.2mm/1m= Because the edge is in the x-directions this is the desired strain. It is possible instead to assume any u displacement at the third node and calculate from the general equations and get the same result more laboriously.
So ξ =(1/3) * 3 = 1cm from node 1
𝑢(𝜉)= 1− 𝜉 𝑎 𝑢 1 + 𝜉 𝑎 𝑢 2
𝜉 𝑎 =1/3