Elastic Spring Constants for Running Shoes: A Mathematical Model

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##### Elastic Spring Constants for Running Shoes: A Mathematical Model

Background: Running shoe compliance and track surface stiffness can reduce peak vertical foot forces. It is therefore of interest to measure directly the force-deflection curve for running shoes in the heel and forefoot areas. This study compares these measurements with similar work on track and field surfaces, and derives a mathematical stress-strain model useful over the entire force range. Methods: Six different running shoes from 4 popular brands are measured to determine vertical spring stiffness. The heel and ball areas are tested with 3.8 and 5.1 cm (1.5 and 2.0 in.) diameter heels in the force range of 0 to 0.13 kN and 2.0 to 2.6 kN (0 to 30 lbf. and 450 to 600 lbf.). The results show a factor of 2 difference from one shoe to the next, holding test area, heel diameter and force range constant. Load increments are applied on a time scale of 0.1 seconds, comparable to typical foot contact times during running. Results: The measured spring constants are essentially independent of plunger area, a useful simplification. For a given shoe, the ball area can be three times less compliant than the heel. Conclusion: Heel spring constants at the high force levels fall in the range from 290 kN/m to 600 kN/m (20,000 to 42,000 lbf/ft.), and thus approximate optimal track stiffness. In terms of theory, an exponential function derives from the observation that the data fall along a straight line on semi-log co-ordinates. This mathematical model enables calculation of running shoe compressive response and spring constant at physiological force levels.