Abstract: This paper proposes an effective second-order optimal design method for minimizing the mass of structure and satisfying the dynamic displacement and stress constraints. An efficient algorithm of the first and second derivatives of dynamic displacement and stress with respect to design variables is formulated based on the finite element method and the Newmark method. The time-dependent mathematical model for achieving minimum mass design with the dynamic displacement, dynamic stress and design variable constraints is formulated. The inequality time-dependent constraint problem is converted into a sequence of appropriately formed time-independent unconstrained problems using the integral interior point penalty function method. Gradient and Hessian matrix of the integral interior point penalty function are also computed using the information of the first and second derivatives of dynamic displacement and dynamic stress with respect to design variables. The efficient and effective second-order optimal algorithm is constructed to solve the optimal design problem using the gradient and Hessian matrix. The numerical results show that the optimal design method proposed in this paper can obtain the local optimum design of frame structures and is more efficient than the augmented Lagrange multiplier method.