Complex astigmatic elliptical beams possess very high orbital angular momentum. This kind of beams has important applications in many fields, such as atomic physics and biology. An experimental method of optical torsion balance to measure the orbital angular momentum of the beams is presented in this paper. In our experiment, the absorption slice rotates under the irradiation of the beam. The angular momentum of beams is obtained by measuring the rotation angle of the absorption slice. The angular momentum of complex astigmatic elliptical beam with different parameters is measured in this paper. The dependences of orbital angular momentum on the parameters of the elliptical beams are investigated. The experimental results are consistent with the theoretical predictions.