This paper deals with both divergence and second order work criteria and the kinematical structural stability called ki.s.s. In this context, kinematical structural stability means that the criterion remains valid even if the system is subjected to additional kinematic constraints. First some developments about the effect of additional kinematics constraints are presented on divergence instability. Secondly, divergence and second order work criteria are addressed. Using a variational formulation, previous results from a usual algebraic formulation using Schur's complement formula are highlighted and finally translated through the ki.s.s. concept: unconditional ki.s.s. for the second order work and the divergence of conservative systems and conditional ki.s.s. for the divergence of nonconservative elastic systems.