The document explains the use of linear shape functions in finite element modeling to interpolate solutions between discrete values at mesh nodes. It outlines the concepts of natural coordinates, the strain-displacement matrix, and properties of the stiffness matrix, emphasizing the importance of linear interpolation to estimate displacement fields within elements. Applications include stress-strain approximations, nodal displacement determination, and establishing relations between local and global coordinates.

1. Finite Element Modelling
Expression for shape function of linear
element

2. Introduction
The shape function is the function which interpolates the
solution between the discrete values obtained at the mesh
nodes. Therefore, appropriate functions have to be used
and, as already mentioned, low order polynomials are
typically chosen as shape functions. In this work linear
shape functions are used
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3. Contents
 Concept of shape function and natural co-ordinates.
 Strain-Displacement Matrix.
 Properties of stiffness matrix.
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4. Shape functions and coordinates.
Natural Coordinate and Shape Functions
Natural Coordinate
Consider a single element. Local node 1 is at distance
x1 from a datum, and node 2 is at x2, measured from
the same datum point.
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5. Shape functions
We define a natural or intrinsic coordinate system,ξ
The defined shape functions, required to establish interpolation
function for the displacement field within the element.
In the finite element method, continuous models are approximated
using information at a finite number of discrete locations. Dividing
the structure into discrete elements is called discretization.
Interpolation within the elements is achieved through shape
functions.
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6. The displacement field, u(x), within the element is not known. For
simplicity, it is assumed that the displacement varies linearly from
node 1 to node 2 within the element.
We establish a linear interpolation function to represent the linear
displacement field within the element. To implement this, linear
shape functions are defined, given by,
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7. The linear displacement field, u(x), within the element can now be
expressed in terms of the linear shape functions and the local nodal
displacement q1 and q2 as:
In matrix form:
Where:
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8. Isoperimetric Formulation
Coordinate x of any point on the element (measured from the same
datum point as x1 and x2) can be expressed in terms of the same
shape functions, N1 and N2 as
When the same shape functions N1 and N2 are used to establish
interpolation function for coordinate of a point within an element
and the displacement of that point, the formulation is specifically
referred to as an isoparametric formulation .
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9. Strain displacement relation
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The two terms of the above relation are
obtained as follows

10. Thus the normal strain relation can be written as
which can be written in matrix form as
where [B] is a row matrix called the strain-displacement matrix, given
by since x2 – x1 = element length = le
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11. Properties od stiffness matrix
 The dimension of the global stiffness matrix is (nxn),
where n is total number d.o.f. of the body(or structure).
 It is symmetric matrix.
 It is singular matrix, and hence [K]-1 does not exist.
 For global stiffness matrix, sum of any row or column
is equal to zero.
 It is positive definite i.e. all diagonal elements are
nonzero and positive.
 I tis banded matrix. That is, all elements outside the
band are zero.
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12. Applications
• Stress strain approximations can be done.
• Analysis od shapes can be done.
• Nodal displacements can be determined.
• Relation between local and global co-ordinates can be
established.
• Linear interpolation can be determined.
• Shape functions of any order can be determined.
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