FEA of a Transverse Beam

Set-Up

The local mass matrix (for linear-elastic material in elastic deformation) is

and the local stiffness matrix under the same restrictions is

Note that different shape functions may be more appropriate for more complex materials and loadings and will yield different local matrices. These local matrices are for Euler-Bernoulli (i.e., “slender”) beams. The shape function and thus stiffness and mass matrices will be different for Timoshenko beams. Damping is neglected.

Examples

Natural Frequencies of a Fixed-Free Beam in Transverse Vibration

Natural Frequencies of a Fixed-Pinned Beam in Transverse Vibration

Natural Frequencies of a Tapered Fixed-Free Beam in Transverse Vibration

Dynamics of a Fixed-Free Beam with an Initial Point Load at the Free End

Limitations

We restricted the analysis to systems with

• Homogeneous boundary conditions

• No forcing, only initial conditions

• Deformation in only linear-elastic region

• Negligible damping

• No axial loading