Structural periodic lattices made of two or more dissimilar materials can be viewed conceptually in terms of a spectrum of structure types, bounded by stretch-dominated lattices of a single material at one end and tensegrities of tensile and compressive struts at the other. The present study probes the unexplored domain within this lattice-tensegrity spectrum with a focus on dissipative bi-material lattices. To begin, a primitive structural motif that exhibits the desired behavior is identified and its compressive response is analyzed. A 2D multi-cell lattice based on the primitive motif is designed and several material variants are fabricated and tested. Analysis of test results addresses effects of finite node dimensions, constraints on strut rotation at the nodes, free edges, and friction with the loading platens as well as limits dictated by rupture of tensile struts or buckling or yielding of compressive struts. The study culminates with guidelines on design of bi-material lattices with high strength and high straining capability.