A new beam theory using first-order warping functions

Abstract

Due to a certain type of loading and geometrical boundary conditions, each beam will respond differently depending on its geometrical form of the cross section and its material definition. As an example, consider an isotropic rectangular beam under pure bending. Plane sections perpendicular to the longitudinal axis of the beam will remain plane and perpendicular to the deformed axis after deformation. However, due to the Poisson effect, particles in the planes will move relative to each other resulting in a form of anticlastic deformation. In other words, even in pure bending of an isotropic beam, each cross section will deform in the plane. If the material of the beam above is replaced by a generally anisotropic material, then the cross sections will not only deform in the plane, but also out of plane. Hence, in general, both in-plane deformation and out-of-plane warping will exist and depend on the geometrical form and material definition of the cross sections and also on the loadings. For the purpose of explanation, an analogy is made. The geometrical forms of the bodies of each individual are unique. Hence, different sizes of clothes are needed. Finding the sizes of clothes for individuals is like determining the warping functions in beams. A new beam theory using first-order warping functions is introduced. Numerical examples will be presented for an isotropic beam with rectangular cross section. The theory can be extended for composite beams.