CHAPTER 3 STRAINS BEYOND THE ELASTIC LIMIT Summary For rectangular-sectioned beams strained up to and beyond the elastic limit,i.e.for plastic bending,the bending moments(B.M.)which the beam can withstand at each particular stage are: BD2 maximum elastic moment ME= 60 partially plastic moment MPP= Bay(3D2] 12 BD2 fully plastic moment MrP=-40y where oy is the stress at the elastic limit,or yield stress. fully plastic moment Shape factorλ= maximum elastic moment For I-section beams: BDS bd312 ME=0y 12 12]D BD2 bd27 MFP =0 4 4 The position of the neutral axis(N.A.)for fully plastic unsymmetrical sections is given by: area of section above or below N.A.=x total area of cross-section Deflections of partially plastic beams are calculated on the basis of the elastic areas only. In plastic limit or ultimate collapse load procedures the normal elastic safety factor is replaced by a load factor as follows: collapse load load factor allowable working load For solid shafts,radius R,strained up to and beyond the elastic limit in shear,i.e.for plastic torsion,the torques which can be transmitted at each stage are R3 maximum elastic torque TE=- 23 partially plastic torque TPP (yielding to radius R) 6 61CHAPTER 3 STRAINS BEYOND THE ELASTIC LIMIT Summary For rectangular-sectioned beams strained up to and beyond the elastic limit, i.e. for plastic bending, the bending moments (B.M.) which the beam can withstand at each particular stage are: maximum elastic moment BD* ME = -C 6v Mpp BUY 2 2 1 -[3D -d ] 12 partially plastic moment fully plastic moment where oy is the stress at the elastic limit, or yield stress. fully plastic moment maximum elastic moment Shape factor h = For I-section beams: BD3 bd3 2 ME = oy [y - 121 The position of the neutral axis (N.A.) for fully plastic unsymmetrical sections is given by: area of section above or below N.A. = x total area of cross-section Deflections of partially plastic beams are calculated on the basis of the elastic areas only. In plastic limit or ultimate collapse load procedures the normal elastic safety factor is replaced by a load factor as follows: collapse load allowable working load load factor = For solid shafts, radius R, strained up to and beyond the elastic limit in shear, i.e. for plastic torsion, the torques which can be transmitted at each stage are lrR3 2 maximum elastic torque TE = -ty n'5 Tpp = 2[4R3 - R:] (yielding to radius RI) 6 partially plastic torque 61