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Compatibility conditions are investigated for planar network structuresconsisting of nodes and connecting bars; these conditions restrict theelongations of bars and are analogous to the compatibility conditions ofdeformation in continuum mechanics. The requirement that the deformationsremain planar imposes compatibility. Compatibility for structures withprescribed lengths and its linearization is considered. For triangulatedstructures, compatibility is expressed as a polynomial equation in the lengthsof edges of the star domain surrounding each interior node. The continuumlimits of the conditions coincide with those in the continuum problems. Thecompatibility equations may be summed along a closed curve to give conditionsanalogous to Gauss-Bonnet integral formula. There are rigid trusses withoutcompatibility conditions in contrast to continuous materials. The compatibilityequations around a hole involve the edges in the neighborhood surrounding thehole. The number of compatibility conditions is the number of bars that may beremoved from a structure while keeping it rigid; this number measures thestructural resilience. An asymptotic density of compatibility conditions isanalyzed.

Compatibility Conditions for Discrete Planar Structure