Previous misunderstandings about the behavior of stresses in the vicinity of the core of wound rolls are resolved through the development of an explicit closed-form analytical solution for the radial and circumferential stresses which are generated during winding; the solution holds in the general case of variable winding tension. Asymptotic series are derived and then employed to compute profiles of stresses within the wound material for several cases of practical interest. A parametric analysis of the influence of core elasticity on structural integrity of the roll underscores the indispensable support provided by the core at the inner boundary of the roll. Results show that the circumferential stress in the vicinity of the core strongly depends on core stiffness. A relation that is derived between interlayer pressure and circumferential stress at the core boundary may be useful as a guide in core design and in preventing roll collapse.