We study the transverse oscillatory modes of nanomechanical silicon nitride strings under high tensile stress as a function of geometry and mode index m≤9. Reproducing all observed resonance frequencies with classical elastic theory we extract the relevant elastic constants. Based on the oscillatory local strain we successfully predict the observed mode-dependent damping with a single frequency-independent fit parameter. Our model clarifies the role of tensile stress on damping and hints at the underlying microscopic mechanisms.