Natural Frequenies.docx

Natural Frequencies and Buckling Load Siva Srinivas Kolukula Structural Mechanics Laboratory IGCAR, Kalpakkam INDIA

E-mail: [email protected]

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

N1  1 

N3 

3x2 2x3 2x3 2x2 x3 N  x    2  ; 2 l l2 l3 l l3

3x2 2x3  3 l2 l

; N4  

x 2 x3  2 l l

(1)

Figure 1: Two noded beam element

The transverse displacement w(x) can be written as

w(x)  N1 N2 N3

 q1    q  N4  2  q3  q4 

(2)

2. Element Stiffness Matrix Elemental potential energy Ue of the beam is given by l

Ue 

1 2   2 w  EI 2 x2  x 



(3)

0

Substituting eq. (2) in eq. (3) and applying Galerkin’s method, results in the stiffness matrix of the beam element. l

Ke 

 B EIBdx T

0

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Elemental stiffness matrix given by

6l 12 6l   12   2 6l 4l  6l 2l2  EI Ke  3  l  12  6l 12  6l   2l2  6l 4l2   6l

(4)

3. Element Mass Matrix Elemental kinetic energy Te of the beam element is given by l

1 2w T  ρA 2 2 t



e

(5)

0

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. l

 N ρANdx

m 

T

e

0

 156 22l  4l2 ρAl  22l Me  13l 420  54  2  13l  3l

54 13l   13l  3l2  156  22l   22l 4l2 

(6)

4. Elemental Geometric Stiffness Matrix The beam is subjected to an external axial periodic force p(t) ,the elemental work done by the external periodic force p(t) is given by l

we 

1 2w p(t) 2 dx 2 x



(7)

0

Substituting eq. (2) in eq. (7) and applying the Galerkin’s method yields the geometric stiffness matrix. l

T

 dN   dN  k eg     dx  dx   dx  0



Elemental geometric stiffness matrix given as follows 3l 36 3l   36   2 3l 4l  3l  l2  1  K eg  30l  36  3l 36  3l   2  3l 4l2   3l  l

(8)

Where 𝐾𝑔𝑒 is called geometric stiffness matrix or stability matrix or initial stress matrix.

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5. Natural Frequencies and Mode Shapes On solving the Eigen value equation |𝐾 − 𝜔2 𝑀| = 0

(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 Eigen value equation |𝐾 − 𝜆𝐾𝐺 | = 0

(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. About the present Code 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

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Number of Elements nel = 50 Material properties (MKS system) Youngs modulus, E=2.1*10^11 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 are the result obtained for a clamped-free beam/cantilever beam: Table 1 Natural frequencies (rad/S) and Buckling load (N) for the beam

Mode

Theory

FEM

Error %

1

18.8177

18.7964

-0.1132

2

117.6106 117.7989

0.1568

3

329.8443 329.9134

-0.0045

Buckling load

211808

211808

-0.0010

Figure 3: First Four Mode Shapes (frequencies in rad/s)

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Figure 4: First four Buckling mode shapes (buckling load in N)

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