Beam Element, its Shape Function and Stiffness Matrix


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.

  1. Under the pure bending condition of beam (without any torsion (or) axial loads), which is having two degrees of freedom at any point.
  2. A Beam element as an arbitrary angle.
  3. Under a combination of loads.

beam element

Elementary Beam Theory Assumptions

The elementary beam theory has some assumptions, that which are applicable here,

  1. Any beam is loaded, that only in the direction of y-axis.
  2. As comparing to the beam dimensional characteristics, beam deflections are small.
  3. In finite element analysis (FEA), the selection of the beam material is linearly elastic, it should be isotropic in nature, and homogeneous.
  4. Beam is vibrant and the area of cross section has an symmetrical axis in the plane where bending occurs.

Let us consider a beam element with stiffness matrix which is a 4×4 matrix.

stiffness four

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 and Stiffness Matrix for Beam Element

  • Beam Element Shape Function:

Shape function for the beam element can be written as,

ν) = f(ν1, ν2, θ1θ2, x)

ν) = (1 – 3ξ² + 2ξ³)ν1 + L(ξ – 2ξ² + ξ³)θ1 + (3ξ² – 2ξ³)ν2 + Lξ²(ξ – 1)θ2

with ξ = x/L

  • Beam Element Stiffness Matrix:

Stiffness matrix for the beam element can be written as,

Beam element k

Hence, this is the shape function and stiffness matrix for a beam element.

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Hello, I am an aircraft structural analyst with industrial experience and a master degree on aerospace structures. Currently working for an aerospace company as a stress
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