This paper proposes a numerical solution method for upper bound shakedown analysis of perfectly elasto-plastic thin plates by employing the C\(^{1}\) natural element method. Based on the Koiter’s theorem and von Mises yield criterion, the nonlinear mathematical programming formulation for upper bound shakedown analysis of thin plates is established. In this formulation, the trail function of residual displacement increment is approximated by using the C\(^{1}\) shape functions, the plastic incompressibility condition is satisfied by introducing a constant matrix in the objective function, and the time integration is resolved by using the König’s technique. Meanwhile, the objective function is linearized by distinguishing the non-plastic integral points from the plastic integral points and revising the objective function and associated equality constraints at each iteration. Finally, the upper bound shakedown load multipliers of thin plates are obtained by direct iterative and monotone convergence processes. Several benchmark examples verify the good precision and fast convergence of this proposed method.

本文提出了一种数值求解方法，该方法采用C \（^ {1} \）自然元素方法对完全弹塑性薄板的上限振动进行分析。基于柯伊特定理和冯·米塞斯屈服准则，建立了薄板上限振动分析的非线性数学编程公式。在此公式中，剩余位移增量的尾迹函数通过使用C \（^ {1} \）来近似形状函数，通过在目标函数中引入常数矩阵来满足塑性不可压缩条件，并使用柯尼希（König）技术解决时间积分问题。同时，通过将非塑性积分点与塑性积分点区分开，并在每次迭代时修改目标函数和相关的等式约束，使目标函数线性化。最后，通过直接的迭代和单调收敛过程获得了薄板的上限减振载荷乘数。几个基准示例验证了该方法的良好精度和快速收敛性。