"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamBending Fully Plastic Momentfor Non-symmetric Bending2 Bending CESE 656 Fully Plastic Moment, M P? Just like for symmetric bending? A = A C T? C = C C T? M = C ·d = C ·dC T ? Unlike symmetric bending? N.A. is not directed along M (? plane of loads)? Finding a and locating the N.A. for a given f becomes aniterative procedure3 Bending CESE 656 Procedure for Finding M P? Estimate the location of the N.A.? It must divide the section into equal areas? Determine C (or C )C T ? Determine location of C and C C T? C is located at the centroid of the compression areaC ? C is located at the centroid of the tension areaT ? Determine f? The plane of the loads passes through C and CC T ? Iterate until f matches the plane of the loads (? M)4 Bending CESE 656 Example – MP for Non-symmetric Bending? A rectangular beam is subjected to a moment in the y-z plane.? The beam is made from a material with a yield stress of s = 30 ksiY a. Determine M if the neutral axis lies along the diagonal from C to BP b. Determine the plane of the loads (f) and the direction of the moment, M, forthis M Pß M5 Bending CESE 656 a) M P1 2 ? ? = ? = ? = 12 ? ?? ? 2 2 ? ? = ? = 30 ?? ? 9 ? ? = 360 ? ? ?? ? ? Location of C and C C T6 ? ?? ? = 3 ? ? - = 1 ? ?3 4 ? ? 2 ? ? = 2 ? ? - = ? ?3 3 y2 2 2 ß MC ? ? = 2 1 ? ? + ? ? = 2.404 ? ?C3NAdC Tz? ? = 360 ? ? ? 2.404 ? ?? ? ? = 865 ? ? · ? ? ? = 72.1 · ? ? ?? 6 Bending CESE 656??a) f and ß ? Location of C and C C T6 ? ?? ? = 3 ? ? - = 1 ? ?3 4 ? ? 2 ? ? = 2 ? ? - = ? ?3 3 ? 1 ? ? - 1 - 1 ? ? = tan = tan2 / ? ? ?3 ? ? = 56.3°y? ? + ? = 90°ß MC fC? ? = 90° - ? = 90° - 56.3°d NAC T? ? = 33.7°z7 Bending CESE 656 "