"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamBending Bending of Symmetric SectionsReview of Elastic BendingInelastic Bending2 Bending CESE 656 Elastic Bending of Symmetric Sections? Assumptions / Observations? Pure bending? Cross-section has at least one axis of symmetry? Loads act in a plane of symmetry? Beam bends into a circular arc? Cross-sections remain plane3 Bending CESE 656 Elastic Bending of Symmetric Sections? Normal strains, e, vary linearly from zero atthe neutral axis to maximum and minimumvalues at the outer edges of the cross-section? Under linear-elastic behavior (s = Ee) normalstresses, s, also vary linearly? ? ? = - ? ( ??? ) = - ? (- )? ??? ? ? ? ? ? ? ?? ? ?? ? ?? 2 ? ? = ? ?? =? ? ? ? ? ? ? ? ? =and? = - (Bending Formula)? ? ? ? ? 4 Bending CESE 656 Maximum Elastic Moment? For elastic behavior? NA passes through the centroid?? ? ? ? = =??? ? ? ? ? S = section modulus = ? ? Set s = s and solve for Mmax Y b ? ? = ? ?? ? s Yh/2 2 ? ? h ? ? ? = for a rectangle? h/2 65 Bending CESE 656 Inelastic Bending? Strains still vary linearly? Stresses vary according to the s-erelationship? Static Equilibrium? ? = - ? ? ? ?? ? ? = ? ? ? = 0 or ST = SC? 6 Bending CESE 656 Elastic-Perfectly Plastic Behavior? M = - ? ??? = ? ? ? ? ? ? ? For a rectangle1 ? ? = ? = ? ? ?1 1 ? ? 2 h ? ? = ? = ? - ? ?2 2 ? ? 2 2 2 1 h 1 h ? ? = ? ? + ? ? + ? ? + - ? + ? ? + - ??? 1 ? 1 ? 2 ? ? 2 ? ? 3 3 2 2 2 2 2 1 4 ? ? 2 ? ? = ? h ? 1 - (rectangle)? 2 4 3 h 7 Bending CESE 656??Fully Plastic Moment? ? M = -? ?? ? = ? ? ? ? ? ? For a rectangleh ? ? = ? = ? ?? 2 1 h 1 h ? ? = ? · + ? ·? 2 2 2 2 1 3 2 ? ? = ? h ? = ? (for rectangle only)? ? ? 4 2 8 Bending CESE 656 Location of Plastic Centroid? ? = ? × ? ? ?? ? ? ? ? ? ? ? and? ? ? = ? × ? ? ?? ? ? ? ? ? ? and? ? ? = ?? Therefore NA divides the beam intotwo equal areas and is not necessarilyat the centroid? NA shifts when transitioning fromelastic to plastic behavior9 Bending CESE 656??? ?? ?? ??Residual Stresses? When loaded beyond M and unloaded, residual stressesY will result since the unloading is elastic3 ? When M =M (rectangle) is removed a maximumP Y 2 3 reversal of s =s is appliedr Y 2 ? The material will exhibit elastic behavior over a reversalstress range of 2s as discussed previouslyY 10 Bending CESE 656 "