"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamCurved Beams Bending Formula for CurvedBeams – Circumferential Stress2 Curved Beams CESE 656 Curved Beams? A beam whose longitudinal axis has a radius of curvatureless than approximately 5 times the beam depth3 Curved Beams CESE 656 Geometry of a Curved Beam? R = Distance from center of curvature to the cross-section centroid? R = Distance from center of curvature to the neutral axisn ? r = Distance from center of curvature to differential area, dAR nRRR n4 Curved Beams CESE 656 Assumptions? Plane section remain plane(R – r)n ? Cross-section is constant and has aplane of symmetry perpendicular to(R – r)(R – r)n n the axis of the moment? Material is homogeneous andisotropic? Linear-elastic behavior? +M tends to flatten the beam(increase the radius of curvature)5 Curved Beams CESE 656 Geometry of Bending Deformation? Original length of a line segment(R – r)n ? ? = ?0 ? Deformed length of a line segment(R – r)(R – r)n n ? ? = ? + ? ? ? - ?? ? ? Strain?? ? - ? ? ? ? ? ? = = ? - 1? ?? ? ? Strain is not a linear function of r? Hooke’s Law? ? ? ? = ?? - 1? 6 Curved Beams CESE 656?? ??Location of the Neutral Axis? ? ? ? = ? ? - 1? (R – r)n ? Equilibrium(R – r)(R – r)n n ? ? ? ? ? = ? ? ? = ? ? - 1 ? ? = 0? ? ? ? ? ? ? ? - ? ? = 0? ? ? ? ? ? ? ? ? ? = =? ? ? ? ? ? ? R = location of the NA measured fromn the center of curvature7 Curved Beams CESE 656??Evaluating A , A, and R m ? ? ? ? =? ? ? ? ? A = Cross-sectional area? R = Location of Centroid w.r.t. the center of curvature8 Curved Beams CESE 656 Bending Formula of Curved Beams? ? ? ? = ? ? - 1? (R – r)n ? Equilibrium(R – r)(R – r)n n ? ? ? = ? ? ? ?? ? ? ? ? = ? - ? ? ? - 1 ? ?? ? ? ? ? 2 ? ? = ? ? ? - 2 ? ? ? + ?? ? ? ? ? ? ? ? ? ? st ? 1 integral is Am, which is ? ? nd ? 2 integral is A? ?? ? rd ? 3 integral is A·R since ? =? ? ? = ? ? - ?? 9 Curved Beams CESE 656?? ??Bending Formula of Curved Beams? ? = ? ? - ? (1)? ? ? ? ? = ? ? - 1 (2) (R – r)n ? (R – r)(R – r)n n ? ? - ? ? ? ? =?? ? - ? ? ? M = Moment on cross-section? r = Location in cross-section where s isdetermined w.r.t. the center of curvature? A = Area of cross-section? R = Location of NA w.r.t. center ofn curvature? R = Location of centroid w.r.t. center ofcurvature10 Curved Beams CESE 656??"