A multilayer annular clamp including a plurality of sealing rings radially spaced from one another, at least two of the sealing rings including L-shaped axial-plane cross-sections nested in one another, an innermost one of the sealing rings being operative to contact an outer contour of the pipe, two clamp members disposed around a portion of the sealing rings, and a fastener, including a shank, operative to fasten the clamp members towards each other in a direction along the shank so as to apply a radially-inward clamping force on the innermost sealing ring that contacts the pipe so that the multilayer annular clamp clamps the pipe.