Abstract: The finite integral method is a numerical solution by which Brown and Trahair analyzed some differential equations. The kernel mechanism of the finite integral method is how to calculate z = z(x) numerically when some values of z′ = z′(x) are known. Function z′ = z′(x) is the derivative of z = z(x). Essentially, the deflection calculation by curvatures φ is a mathematical process to calculate z from z″. Based on the relations between z-z″ and z′-z″ in the finite integral solutions, the deflection-curvature matrix is derived by matrix operations, and the curvature-deflection equation is derived by the relation φ = - z″. The curvature-deflection equations for some kinds of common boundary conditions are discussed. The finite integral solution for deflections of structures with complicated distribution of curvature is obtained.