A Note on the Pure Bending of Nonhomogeneous Prismatic Bars

Abstract

The paper examines the pure bending of a linearly elastic, isotropic, nonhomogeneous bar. The bending stress, elastic strain energy and the end cross-section rotations are determined for in-plane variation of the Young's modulus with small strains and displacements. It is shown that the governing formulae for elastic pure bending of nonhomogeneous bars have same forms as formulae for symmetrical bending (bending in the principal planes) of homogeneous bars. Two examples illustrate the application of the developed formulae. In the first, a composite beam is considered; the second deals with the determination of the maximum tensile and compressive stresses in a bent functionally graded elastic bar.

The kern of non-homogeneous cross sections

“…1) and let the planes of cross sections be parallel with the plane of axes x and y. In this case axis z connects the E-weighted centers of cross sections and we say that the axis (center-line) of the non-homogeneous bar is axis z and we have [1] …”

“…Therefore it is enough to consider only one cross section and to use only the cross-sectional coordinates x and y. The deformation of the bar -according to the beam theory of Bernoulli-Euler -can be described by the equation [1] e z ¼ e 0 þ jg:…”

“…We note that, in paper [1] J is called the E-weighted inertia tensor of non-homogeneous cross section A about point C E . Using Eq.…”

A Note on the Shear Formula

“…In a previous paper [1], an extension of the fl exure (bending) formula was derived for a non-homogeneous bar (see Fig. 2…”

“…In equation 4, the unit vector n = n x e x + n y e y gives the neutral axis, n, of bending (its determination is presented in [1]). In equation 4, the scalar product of two vectors has been denoted by a dot.…”