Quadratic form of strain energy

[boeing_737]

Hi,

I have some confusion about strain energy. When using Lagrange's equations to derive the EOM of vibrating structures, the strain energy is written as :

U = $q^{T}$K$q$ ; ($q$ is the vector of generalized coordinates, and K is the stiffness matrix). Writing it in this form makes it easy to obtain $\partial U/ \partial q$ used in Lagrange's equations.

My question is, is the quadratic form of strain energy only valid for linear elastic deformation? Can we write such a quadratic form when we have geometric nonlinearities in the problem (beam undergoing large deformations for example)?

Thanks
yogesh

 

[Navin A S]

Yes, the quadratic form of strain energy is valid for nonlinear problems as well. In this case, the stiffness matrix is replaced by a nonlinear stiffness matrix that depends on the current state of deformation. This nonlinear stiffness matrix can be calculated using finite element methods or other numerical techniques. In fact, the strain energy of the nonlinear system is the sum of the linear and nonlinear strain energies.