Abstract The Lobatto-Chebyshev method for the numerical solution of the Cauchy type singular integral equation of crack problems in two-dimensional elasticity, plates and shells and the determination of the values of the stress intensity factors at the crack tips is shown to converge for non-differentiable Hölder-continuous or even discontinuous loading distributions as far as the values of the stress intensity factors are concerned. Moreover, in all cases of differentiable loading distributions it is shown to converge more rapidly than believed up to now. The problems of a simple straight crack and a periodic array of cracks loaded by three non-differentiable loading distributions are used for the application of the present results. The displayed numerical results for these problems verify and further corroborate the theoretical results. The extension of the present results to the Gauss-Chebyshev method is also quite possible.