Beam element is defined as a slender body with uniform cross-sectional area, which has a six degrees of freedom at each node as shown in the below figure. But this beam element is not suitable for structures which is having complex geometries, voids, and stress concentration points.
If we combining the stiffness constants of a beam which is under the pure buckling condition, a truss element,and a torsion bar, then we can get the beam constant of stiffness. Representation of a beam can be made by a beam element under bending condition, a torsional bar, and a truss element. In the finite element analysis (FEA), it is a common practice to use the beam elements to constitute the any of the load conditions.
The elementary beam theory has some assumptions, that which are applicable here,
Let us consider a beam element with stiffness matrix which is a 4×4 matrix.
Above equation shows element stiffness matrix for the beam element which is a 4×4 matrix. It relates to the fact that that beam element can exhibits 4 degrees of freedom (DOF), and the displacements for the beam are dependent (it means the structure is not having any discontinuities and it is elastic in nature).
Moreover, the matrix is symmetric matrix and also it is a singular matrix and therefore it is invertible. That is why, the problem is defined as incomplete and it does not having a solution.
Shape function for the beam element can be written as,
ν( x ) = f(ν1, ν2, θ1, θ2, x)
ν( x ) = (1 – 3ξ² + 2ξ³)ν1 + L(ξ – 2ξ² + ξ³)θ1 + (3ξ² – 2ξ³)ν2 + Lξ²(ξ – 1)θ2
with ξ = x/L
Stiffness matrix for the beam element can be written as,
Hence, this is the shape function and stiffness matrix for a beam element.
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