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In this paper, we study optimal experimental design problems with a broad class of smooth convex optimality criteria, including the classical A-, D-, and pth mean criterion. In particular, we propose an interior point (IP) method for them and establish its global convergence. Further, by exploiting the structure of the Hessian matrix of the optimality criteria, we derive an explicit formula for computing its rank. Using this result, we then demonstrate that the Newton direction arising in the IP method can be computed efficiently via the Sherman-Morrison-Woodbury formula when the size of the moment matrix is small relative to the size of the design space. Finally, we compare our IP method with the widely used multiplicative algorithm introduced by [S. D. Silvey, D. M. Titterington, and B. Torsney, Commun. Statist. Theory Methods, 7 (1978), pp. 1379-1389] and the standard IP solver SDPT3 [K. C. Toh, M. J. Todd, and R. H. Tütüncü, Optim. Methods Softw., 11/12 (1999), pp. 545-581], [R. H. Tütüncü, K. C. Toh, and M. J. Todd, Math. Program. Ser. B, 95 (2003), pp. 189-217]. The computational results show that our IP method generally outperforms these two methods in both speed and solution quality. [ABSTRACT FROM AUTHOR]