A class of axisymmetric and asymmetric problems related to an annular inclusion embedded in bonded contact with a transversely isotropic elastic layer, elastically supported at both faces is considered. The displacements on the contact area are induced in one of four ways: (i) translation and (ii) rotation in a direction normal to the plane face of the annulus; (iii) rotation and (iv) translation in a direction parallel to the face. The results give immediate solutions of some bonded contact problems of the annulus and layer under arbitrary load. The crack problem in an elastically supported layer, damaged by an annular crack subjected to pressure, shear and torsion across its faces is also analysed and the essential feature of the singular stress field near the crack is preserved, and the stress intensity factors of the Mode-I, -II, -III cases are easily evaluated.