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Consider an elastic bar with length “L” to which it is affixed a coordinate system x (which is uniaxial in direction) with its origin arbitrarily has placed at the left end of the bar. This indicates the element coordinate system (or) the reference frame. Designating the axial displacement at any position which is along the length of the bar as u(x), and here we define node 1 and node 2 at each end as shown and it presents the nodal displacements.
u1=u (x=0)
and
u2= u (x = L)
Hence, we have the constant field variables with u(x), and which is to be expressed (roughly) in terms of two nodal variables u1 and u2. In order to accomplish this discontinuity, here we can conclude the presence of interpolation functions such as N1(x) and N2(x) (which is also known as shape functions or blending functions) such that, it is indicate as
u(x) = N1(x)*u1+N2(x)*u2
Where,
In order to determine the interpolation functions (approximate functions), here we should require that the boundary values of u(x) (which are the nodal the displacements) to be similarly satisfied by the discretization function such that it is:
u1=u(x=0) and u2= u(x = L)
Result to the following boundary conditions:
N1(0) = 1 and N2(0) = 0
N1(L) = 0 and N2(L) = 1
While we have two conditions that should be satisfied by each of the two onedimensional (1D) functions, and the simplest forms for the interpolation functions are the polynomials forms, that are:
N1(x) = a0 + a1x
N2(x) = b0 + b1x
The coefficients of the polynomials are to be determined through the satisfaction of the boundary conditions. Thus application of the conditions yields to a0 = 1, b0 = 0, and therefore a1 = –(1/L), & b1= x/L.
Thus the functions are
N1(x) = 1 − x/L
N2(x) = x/L
Therefore, the resultant statement of the blending function is as follows:
u(x) = (1 − x/L)u1 + (x/L)u2
In the matrix form, it can be written as
u(x) = [ N ] { u }
Hence, this is the expression for truss element blending function and its explanations.
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