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CIVE50003 Computational Methods II

Linear eigenvalue problems for 2D frames

Fig. 1 – Frame ABCDEF with inclined members under various point loads

Q1. Frequency  analysis. Consider the  frame  structure  in  Fig.  1  above.  The mass matrix for a 2D Euler-Bernoulli general beam element is given by:

where ρ = 7850 kg/m3  = 7850×10-9  kg/mm3 is the material density (steel).

Assemble the global mass matrix [M] at the same time as the global stiffness matrix [K]  and extract the lowest four  eigenvalues  and  associated  eigenvectors  from  the system [KFF]{ϕ} = ω2 [MFF]{ϕ}. Here each {ϕ}i is a general eigenvector representing the i-th natural vibration mode and  each ωi is  its  corresponding natural angular vibration frequency (radians per second). The cyclic frequencies f = ω/2π should be 0.396, 1.041, 2.360 and 2.763 Hz. Plot each one in its own properly-annotated figure. Hint: you may find it useful to revise your notes from CM1. You will also need to apply a transformation matrix [Te] to account for each element’s orientation.

Q2. Buckling analysis. Consider the frame structure in Fig. 1 above. The geometric stiffness matrix for a 2D Euler-Bernoulli general beam element is given by:

Work out  the  axial  force N in  each  element  and  on  this  basis build the  global geometric stiffness matrix  [Kg]. With this, extract the lowest four eigenvalues and associated eigenvectors from the system  [KFF]{u} = λ[Kg,FF]{u}, where  [K]  is the ‘usual’  (material)  stiffness  matrix  you  already  have.  Here  each  {u}i is  a  general eigenvector   representing   the i-th critical   buckling   mode and   each λi is   its corresponding critical buckling load (Newtons). How many possible modes are there in your FE model? Plot each one in its own properly-annotated figure. What is the critical buckling load factor and is this structure safe? Is there any resemblance at all between the buckling and frequency eigenmodes?