Machining fixtures which utilize screw-or strap clamps are widely used in manufacturing. Typical for them is that the cutting forces are balanced by the friction forces which act on the contact surfaces (interfaces) between clamping elements (screw- or strap clamps) and workpiece. This paper analyses load capacity and compliance of these interfaces. In order to increase their load capacity and reduce compliance, a method is proposed which is based on indenting sharp cone-shaped clamping elements into workpiece material using appropriate surfaces which are not machined, and are not expected to satisfy any particular aesthetic demands (most often castings and forgings). The results of numerical simulations and experimental investigation reveal substantial advantages of the proposed clamping method, offering possibility for industrial application and further investigation.