The engineering beam bending theory summarized in Chapter 1 assumed that the beam cross-sectional area has a plane of symmetry and that bending moments were acting along a single axis. In this chapter we want to remove those restrictions and to examine the multiaxis bending of beams with nonsymmetrical cross-sections. This will lead to a generalization of the flexure formula for the normal stress in the beam.In Chapter 1 we also obtained an expression for the shear stresses induced in symmetrical beams. It is difficult to obtain similar analytical shear-stress forms for beams with general unsymmetrical cross-sections. However, we will show that when the cross-section is thin one can obtain explicit expressions for the shear stresses. Analysis of the bending of thin beams will demonstrate that the shear force in the beam must pass through a specific point, called the shear center, if the beam is to bend without twisting. A new cross-sectional area property, called the principal sectorial area function, will be shown to play a key role in locating the shear center.

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.