In this work, we present an in-depth analysis of nonlocal Stress-Driven Model (SDM) in its integral formulation [1], focusing on the bending problem of Bernoulli-Euler nanobeam. In contrast to conventional approaches, which typically transform the integro-differential problem into an equivalent differential formulation using the Helmholtz kernel [2], this study develops and implements a rigorous numerical procedure for the direct solution of the integral equation via a large class of nonlocal kernels. In particular, six distinct convolution kernels are examined: Helmholtz, Gaussian, Lorentzian (or Cauchy), Triangular, Bessel and Hyperbolic Cosine, highlighting their mathematical properties (symmetry, regularity, support, decay) and their influence on the solutions of the model. The proposed approach was validated numerically by comparison with known analytical solutions obtained for the Helmholtz kernel [4]. The results show that the method provides accurate and stable solutions with negligible errors (<1%) with respect to the theoretical reference values. The comparative analysis of the different kernels further highlights the stiffening behaviour of the nonlocal model for increasing values of the nonlocal parameter, λc, with amplified or attenuated effects depending on the shape of the kernel. In the absence of renormalisation, significant deviations from the local solution are observed (especially for kernels other than the Helmholtz kernel), while the proposed method ensures proper convergence to the local behaviour in the limit λc → 0. Overall, this work contributes to the development of a general and robust formulation of the SDM model in integral form, expanding the range of kernels that can be handled numerically and providing effective tools to deal with limit effects, with particular relevance to applications in nanostructure mechanics and advanced material.