Saint-Venant solutions for functionally graded piezoelectric beams with transverse polarization

The symplectic method is employed to investigate the functionally graded piezoelectric beam which is polarized in the thickness direction. The material constants vary along the length in an identical exponential form. The method uses both the displacements and their conjugate stress as variables so that the problem is reformulated in a state space. Using the method of separation of variables, the problem is further reduced to analyzing eigenvalues and their eigensolutions. The Saint-Venant eigensolutions corresponding to the eigenvalues of zero and -¿ which may be classified into 0-group and -¿-group are solved. They satisfy the adjoint symplectic orthogonality relations. The 0-group eigensolutions are found identical with those for the homogenous materials, representing rigid-body translations and rotations and uniform electric potential. But the -¿-group eigensolutions are different. However, they degenerate to the ones for the homogeneous materials with the related rigid motions restrained.