AbstractRelations between displacement models using reduced integration and hybrid mixed models based on the Hellinger–Reissner and the Herrmann principles have been investigated. Five different isoparametric elements are considered for the case of plane stress, plane strain and plate bending elasticity problems.It is proved that the following relations hold for both models, provided the number of terms of a polynomial expressing the stress or strain component in the formulation of the mixed models is equal to the number of integration points of the reduced integration. If the stiffness matrix of the mixed model is evaluated by the identical integration rule with the reduced integration, the stiffness matrices of both models are identical. Thus, both models are equivalent, if the stiffness matrix of the mixed model is integrated exactly by the identical integration rule with the reduced integration.The displacement models using uniform reduced integration and selective reduced integration are equivalent to the mixed models based on the Hellinger–Reissner and the Herrmann principles except for the 6‐node element, respect