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Natural Frequencies and Buckling Load of Beams
1. FEM formulation of the Beam
The column is discretized using two-noded Euler beam elements of length ‘l’ with two degrees of freedom namely transverse displacement and rotation at each node as shown in Fig 1. Let I be the moment of inertia of the beam cross sectional area. To describe the displacement at intermediate nodal points Hermite polynomial shape functions are used
[1]
Figure 1: Two noded beam element
The transverse displacement w(x) can be written as
[2]
2. Element Stiffness Matrix
Elemental potential energy Ue of the beam is given by
[3]
Substituting eq. (2) in eq. (3) and applying Galerkin’s method, results in the stiffness matrix of the beam element.
Elemental stiffness matrix given by
[4]
3. Element Mass Matrix
Elemental kinetic energy Te of the beam element is given by
[5]
Where ρ is the mass density per volume of the beam and A is the cross sectional area of the beam. Substituting eq. (5) in eq. (8) and applying Galerkin’s method results in the elemental mass matrix for the beam.
[6]
4. Elemental Geometric Stiffness Matrix
The beam is subjected to an external axial periodic force p ,the elemental work done by the external periodic force p is given by
[7]
Substituting eq. (2) in eq. (7) and applying the Galerkin’s method yields the geometric stiffness matrix.
Elemental geometric stiffness matrix given as follows
[8]
5. Natural Frequencies and Mode Shapes
On solving the Eigenvalue equation
[9]
Where K is the assembled stiffness matrix of the beam and M is the assembled mass matrix of the beam, we get Eigenvalues / natural frequencies and Eigenvectors/mode shapes.
6. Buckling Load
On solving the Eigenvalue equation
[10]
Where K is the assembled stiffness matrix of the beam and KG is the assembled geometric stiffness matrix of the beam, we get Eigenvalues / Euler buckling load and Eigenvectors/buckling mode shapes of the beam.
7. Code Natural Frequencies and Buckling Load
A code is written in MATLAB to find the Natural frequencies and Buckling loads of a beam for four different boundary conditions. The values obtained with FEM are compared with theoretical formulas and are in good agreement. The present code can be used to find the natural frequencies and buckling load and plot the mode shapes for the given beam. Four different boundary conditions as shown in Fig 2 are considered in the code. User can change the type of boundary condition. User can change the number of elements and geometric, physical properties of the beam accordingly. At present the following are the values used in the code
Figure 2: Columns with different boundary conditions
Number of Elements, nel = 50
Material properties (MKS system):
Young s modulus, E=2.1*1011
Moment of inertia of cross-section, I=2003.*10-8
Mass density of the beam, mass = 61.3
Total length of the beam, tleng = 7
To validate the code the values obtained are compared with the theoretical formulae’s available. Error percentage is also shown. The following table shows the result obtained for a clamped-free beam/cantilever beam.
Table 1:Natural frequencies (rad/S) and Buckling load (N) for the beam
Figures 3 and 4 shows the first four mode shapes of the natural frequencies and buckling loads.
Figure 3: First Four Mode Shapes (frequencies in rad/s)
Figure 4: First four Buckling mode shapes (buckling load in N)
Code can be obtained from the following link.
Natural Frequencies and Buckling loads of beams :
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