We develop the interior penalty virtual element method (IPVEM) to tackle the Kirchhoff plate obstacle problem, which incorporates in-plane stress effects and contact problem with frictional forces. The plate obstacle problem is described as a fourth-order elliptic variational inequality of the first kind, with the plate lying over an obstacle which requires us to find the solution in a closed convex subset and the plate contact problem, classified as second kind, including a non-differentiable component as a result of the frictional contact. Through modifications to the existing \(H^2\) -conforming VE, we design an \(H^1\) -nonconforming VE that preserves the same degrees of freedom as the \(H^1\) -conforming VE and guarantees local \(H^2\) -regularity on every polygon element. In comparison to other alternative methods, this method presents clear advantages, including fewer degrees of freedom and effectively handling of hanging nodes. Following the IPVE framework, a priori error estimates for the two kinds of VIs have been provided, demonstrating that both problems achieve optimal convergence order in the case of the lowest-order approximation. Numerical results are presented to substantiate the theoretical expectations regarding the convergence order of the IPVEM. In the numerical experiments, following the approach presented in Beirão da Veiga et al. (Comput Fluids 141:2–12, 2016), we propose serendipity IPVEM (SIPVEM) by removing the internal degrees of freedom from IPVEM. We also compare IPVEM with other types of VEMs, and the results show that IPVEM and SIPVEM achieve the smallest errors in solving the VI, outperforming all other VEMs.
