The purpose of this paper is to present a method for calculating the upper bound limit loads of plate bending using a conforming Hsieh-Clough-Tocher (HCT) element. These limit loads can be obtained from Koiter’s kinematic shakedown theorem for the case of one load vertex instead of using the kinematic limit theorem. When combining this theorem with the approximated displacement field, the limit analysis turns into an optimization problem and can be effectively solved by Second-Order Cone Programming (SOCP). Several benchmark plate problems such as square, rectangular, and L-shape plates are investigated to illustrate the effectiveness of the proposed solution. The results of the proposed method show good agreement with the results of previous studies. The maximum error is only 2.91% for the fully clamped rectangular plate problem.