Analysis of Rectangular and Circular Cross-section Power Hardening Elements Under Pure Bending

Abstract Engineering theory of elastic plastic bending is used in solving a grate variety of strength and durability problems. The accuracy of analysis depends on stress-strain curve idealizat ion model. The elastic linear hardening stress-strain relation is simple and frequently used. But the actual stress-strain behaviour of ductile material in plastic reg ion is non-linear and, therefore, elastic power hardening material model is more p referab le. Moreover, the properties of monotonic tension and compression as proportional limits and parameters of hardening can differ. In this regard, analytical solution for rectangular and circular cross-section elements loading by monotonic elastic plastic pure bending is presented in the paper. The simple power relation of stress and strain response is used. The equations describing deviation of the stress neutral axis from centroidal axis of an element and dimensionless bending moment are derived.


Introduction
The study of elastic-plastic bending has a wide field of application. Engineering and mathemat ical theories of plastic bending are well studied in T. X. Yu at al. book [1]. The authors include many experimental verifications as well as theoretical results. Elastic-p lastic analysis of beam-like structures of different cross-sections under bending is considered in [2][3][4][5][6][7][8][9][10]. The authors employ different models of non-linear stress-strain idealization. The most used models are linear and power hardening. M. Daunys performed theoretical analysis of rectangular cross-section linear hardening beam subjected to static and cyclic pure bending [11]. The author analyses the drift of the stress neutral a xis fro m the centroidal axis of beam, if parameters of plastic tension and compression (proportional limits, modulus of hardening) are not equal. Circular cross-section linear hardening element under pure bending is studied in [12,13]. Derived equations are fitted to analysis of cyclic pure bending.
Power hardening stress-strain curve model is more preferable in analytical researches of strength of structures than linear hardening one because of better accuracy. Therefore, using of such a kind of idealizat ion in pure bending analysis has not only theoretical but practical importance and applications as well.

Pure Bending of Rectangular Cross-Section Element
It is assumed that in case of pure bending a transverse plane section perpendicular to the centroidal axis of the beam before deformation remains plane after defo rmation and that a cross-section is perpendicular to the centroidal axis after bending. It is also assumed that norma l stresses and strains in the transverse and lateral d irections are neglected and every fiber of the beam is either under uniaxial tension or under uniaxial co mpression. Therefore stress and strain distribution in the longitudinal cross-section of an element fit the schemes presented in where b is a width of an element and y is a distance fro m the fiber under analysis to the centroidal a xis of an element. ( It is evident that In the plastically deformed fibers on the convex side of the beam, as in case of the monotonic tension, stress-strain curve with proportional limit  (12) and ( ) In the fibers deformed elastically (14) Hence, integral (4) can be written as a sum of three integrals representing three different reg ions of strains (16) Dimensionless distant from extreme fiber with By referring to integrals (15) ( The pure bending mo ment corresponding to the proportional limit stress is (21) Substituting σ (Eqs. (12)-(14)) in (21) we get:  In case of elastic plastic pure bending of circu lar cross-section beam, the stress and strain distribution in the longitudinal cross-section also fit the schemes presented in Figs. 1 and 2. But in the transversal cross-section some differences are seen (see Fig. 5).  In case of pure bending of circular cross-section element, the bending mo ment corresponding to the proportional limit is ( ) (32)

Pure Bending of Circular Cross-Section Element
The final exp ression of M is Mechanical properties of majority of ductile materials determined by tensile and compressive testing usually differ. Consequently constants of idealized stress-strain curves (proportional limits and parameters of hardening) are also different. Analysis of pure bending of rectangular and circular cross-section element can be realized with regard to this. The values of dimensionless bending mo ment can be found directly fro m presented diagrams or calculated by deduced formulas.
The developed analytical solution of monotonic elastic plastic pure bending can be used to analyse the cyclic elastic plastic pure bending as it is presented by formulas [11,12].

Conclusions
Elastic-power hardening idealizat ion of uniaxial stress strain curve is used in theoretical research of rectangular and circular cross-section element under elastic p lastic pure bending. Presented derivations allow to calcu late deviation of stress neutral axis fro m centroidal axis of an element and pure bending mo ment versus monotonic strain if parameters of plastic tension and compression (proportional limits, exponents of hardening) are not equal. Derived expressions of bending mo ment can be used in theoretical research of cyclic elastic plastic pure bending.