Subject: Computer Aided Design Prepared by:

Dhruvit lakhani(140050119032)

Guided by:

Ankur patel

1D Analysis(FEA)

Concept of Shape function and Natural coordinatesStrain-Displacement matrixDerivation of stiffness matrix for structural problemsProperties of stiffness matrix1-D structural problems with elimination and penalty approaches1-D thermal and fluid problems

Natural Coordinate and Shape Functions

Natural CoordinateConsider 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.

We define a natural or intrinsic coordinate system,ξ

Note: The ξ-coordinate will be used to define shape functions, required to establish interpolation function for the displacement field within the element. Shape Functions 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.

Concept of Shape function and Natural coordinates

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,

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 and

Isoparametric 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.

⇒

The two terms of the above relation are obtained as follows:

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.

Properties of 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.

The matrix analysis of one dimensional element presented here helps in understanding the general finite element method.

Figure shows one dimensional spar or rod element having two nodes.

Let l= length of element, mm

A= cross sectional area of element, mm2

E= modulus of elasticity of the material, N/mm2

f1, f2 = force acting on node 1 and 2 respectively, N

u1, u2 = displacement of node 1 and 2 respectively, mm

It is important to note that, the dimension of the element stiffness matrix [k] is (NxN), where ‘N’ is the total degree of freedom of the element.