• A unified variational framework is proposed for no-tension and no-compression solids. • The linearized octagonal pyramid yield surface is developed for the no-tension/no-compression materials. • A complementarity finite element formulation is proposed to deal with masonry structures and wrinkled membranes. • The algorithm achieves a total solution without Newton iteration. • The proposed model is capable of capturing the evolution of wrinkling in membranes. No-tension and no-compression constitutive models have important applications in solid mechanics, such as modelling of masonry, wrinkled membrane, unilateral contact interface, etc. Although lots of studies on no-tension and no-compression solids have been found, the variational principle constructing the cornerstone of elasticity is not studied thoroughly. The paper presents two concise variational formulations, a principle of minimum potential energy and a principle of minimum complementary energy, which are available both for no-tension and no-compression solids. Linearization of the conic yield surfaces leads to a series of linear complementary constitutive equations that are embedded into the proposed variational framework. Differing from other variational formulations, an approximate total solution rather than the Newton iteration is achieved in finite element analysis. It makes the algorithm stable. The applications include a no-tension panel benchmark test, two masonry structures and a wrinkled membrane. Compared with our previous study on bi-modulus materials, the newly developed variational formulation is capable of capturing the evolution of wrinkles in membranes, and can be used for the analysis and design of wrinkle-free structures. [ABSTRACT FROM AUTHOR]