Influence of Elastic-Plastic Bending on the Relationship Between Applied Load and Maximum Bending Stress for Straight and Curved Bars

Bending capacity in excess of the load required to cause yielding is due to a combination of work hardening and the effect of the plastic zone spreading toward the neutral axis. For materials of sufficiently high ductility, a fully developed plastic zone is achieved and the bulk of the section is stressed beyond yield. For lower ductility materials, failure may occur prior to full development of the plastic zone such that only a fraction of the cross section is at or above the yield stress. In such cases, the relationship between applied load and maximum bending stress becomes sensitive to the shape of the stress-strain curve near the yield point. This relationship is examined for straight and curved bars of rectangular and trapezoidal cross-section for tensile stress-strain curves characterized by nonlinear functions. The stress distribution as a function of applied load is determined analytically by enforcing moment equilibrium across the section. The strain distribution is determined through the classical condition of “planes remain plane” during deformation. The solutions provide analytically smooth load curves such that maximum stress can be directly plotted as a function of applied load. These plots exhibit three distinct regimes of response: 1) elastic, 2) development of plastic zone, and 3) fully developed plastic zone. Since the response is analytically smooth, the detailed relationship through the knee of the tensile curve can be examined. The results indicate that bending capacity is influenced significantly by the development of small amounts of plastic strain prior to reaching a yield point defined by the usual 0.2% plastic strain offset method. The results also show how loss of ductility with respect to tensile elongation translates into reduced bending load capacity in a non-linear relationship.