We consider the multiplicity of the solutions for a class of a system of critical growth beam equations with periodic condition on t and Dirichlet boundary condition $${u_{tt}+u_{xxxx}=av+frac{2{alpha}}{{alpha}+{eta}}u_{+}^{{alpha}-1}v_{+}^{eta}+s{phi}_{00};;in;(-frac{pi}{2},;frac{pi}{2}){ imes}R,\u_{tt}+v_{xxxx}=bu+frac{2{alpha}}{{alpha}+{eta}}u_{+}^{alpha}v_{+}^{{eta}-1}+t{phi}_{00};;in;(-frac{pi}{2},;frac{pi}{2}){ imes}R,$$ where ${alpha}$, ${eta}$ > 1 are real constants, $u_+=max{u,0}$, ${phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${lambda}_00=1$ of the eigenvalue problem $u_{tt}+u_{xxxx}={lambda}_{mn}u$. We show that the system has a positive solution under suitable conditions on the matrix $A=(array{0&a\b&0})$, s > 0, t > 0, and next show that the system has another solution for the same conditions on A by the linking arguments.