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We set up the statistical mechanics for a nearly flat, thermally equilibratedfluid membrane, attached to an elastic network through one of its sides. Wepredict that the resulting structural (inversion) asymmetry of the membrane,notably due to the elastic network attached to one of its sides, can generate alocal spontaneous curvature $C_0$, that may in turn destabilize the otherwiseflat membrane. As $C_0$ rises above a threshold at a fixed temperature, a flattethered membrane in the thermodynamic limit becomes structurally unstable,signaling {\em crumpling} of the flat membrane. In-vitro experiments on redblood cell membranes after depletion of adenosine-tri-phosphate molecules andartificial deposition of spectrin filaments on lipid bilayers may be used toverify our results.

Tethered membranes do not remain flat for strong structural asymmetry