moment orientation or cross-sectional shape. Note that do not change from one point to another on the cross section. Large bending deformation For large deformations of the body, the stress in the cross-section is calculated using an extended version of this formula. First the following assumptions must be made: Assumption of flat sections – before and after deformation the considered section of body remains flat (i.e., is not swirled). Shear and normal stresses in this section that are perpendicular to the normal vector of cross section have no influence on normal stresses that are parallel to this section. Large bending considerations should be implemented when the bending radius is smaller than ten section heights h: With those assumptions the stress in large bending is calculated as: where is the normal force is the section area is the bending moment is the local bending radius (the radius of bending at the current section) is the area moment of inertia along the x-axis, at the place (see Steiner's theorem) is the position along y-axis on the section area in which the stress is calculated. When bending radius approaches infinity and , the original formula is back: . Timoshenko bending theory In 1921, Timoshenko improved upon the Euler–Bernoulli theory of beams by adding the effect of shear into the beam equation. The kinematic assumptions of the Timoshenko theory are: normals to the axis of the beam remain straight after deformation there is no change in beam thickness after deformation However, normals to the axis are not required to remain perpendicular to the axis after deformation. The equation for the quasistatic bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions is where is the area moment of inertia of the cross-section, is the cross-sectional area, is the shear modulus, is a shear correction factor, and is an applied transverse load. For materials with Poisson's ratios () close to 0.3, the shear correction factor for a rectangular cross-section is approximately The rotation () of the normal is described by the equation The bending moment () and the shear force () are given by Beams on elastic foundations According to Euler–Bernoulli, Timoshenko or other bending theories, the beams on elastic foundations can be explained. In some applications such as rail tracks, foundation of buildings and machines, ships on water, roots of plants etc., the beam subjected to loads is supported