Abstract: Linear form functions are commonly used in a long time for a toroidal volume element swept by a triangle revolved about the symmetrical axis for general axisymmetrical stress problems. It is difficult to obtain the rigidity matrix by exact integration, and instead, the method of approximate integration is used. As the locations of element close to the symmetrical axis, the accuracy of this approximation deteriorates very rapidly. The exact integration have been suggested by various authors for the calculation of rigidity matrix. However, it is shown in this paper that these exact integrations can only be used for those axisymmetric bodies with central hole. For solid axisymmetric body, it can be proved that the calculation fails due to the divergent property of rigidity matrix integration. In this paper a new form function is suggested. In this new form function, the radial displacement u vanishes as radial coordinates r approach to zero. The calculated rigidity matrix is convergent everywhere, including these triangular toroidal element closed to the symmetrical axis. This kind of element is useful for the calculation of axisymmetric elastic solid body problems.