The analysis of normal and shear stresses in a cantilever beam bent by a transverse force is presented. The stress function is introduced and the governing Poisson-type partial differential equation and the accompanying boundary conditions are derived for simply and multiply connected cross sections of a prismatic beam. The exact solution to the boundary value problem is presented for circular, semi-circular, hollow-circular, elliptical, and rectangular cross sections. Approximate, but sufficiently accurate, formulas for shear stresses in thin-walled open and thin-walled closed cross sections, including multicell cross sections, are derived and applied to different profiles of interest in structural engineering. The determination of the shear center of thin-walled profiles, which is the point through which the transverse load must pass in order to have bending without torsion, is discussed in detail. The sectorial coordinate is introduced and conveniently used in this analysis. The formulas are derived with respect to the principal and non-principal centroidal axes of the cross section.

In addition to rotation, non-circular cross sections of twisted rods undergo longitudinal displacement, which causes warping of the cross section. This warping is independent of the longitudinal z coordinate and is a harmonic function of the (x,y) coordinates within the cross section. The Prandtl stress function is introduced, in terms of which the shear stresses are given as its gradients. This automatically satisfies equilibrium equations, while the compatibility conditions require that the stress function is the solution to Poisson’s equation. From the boundary condition of a traction-free lateral surface, it follows that the stress function is constant along the boundary of the cross section. The applied torque is related to the angle of twist by the integral condition of moment equilibrium. This theory is applied to determine the stress and displacement components in twisted rods of elliptical, triangular, rectangular, semi-circular, grooved-circular, thin-walled open, thin-walled closed, and multicell cross sections. The expressions for the torsional stiffness are derived in each case. The maximum shear stress and the warping displacement are also evaluated and discussed.