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 61
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: 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
62 Mechanics of Materials 2 2πR3 fully plastic torque Tre=3-ty where ty is the shear stress at the elastic limit,or shear yield stress.Angles of twist of partially plastic shafts are calculated on the basis of the elastic core only. For hollow shafts,inside radius R1,outside radius R yielded to radius R2, Tpe= 4RR2-R号-3R1 R2 2t (R-R] TFP =3 For eccentric loading of rectangular sections the fully plastic moment is given by BD2 P2N2 MFP=4 0y-ABay where P is the axial load,N the load factor and B the width of the cross-section. The maximum allowable moment is then given by BD2 P2N M= 4N0,-4Ba) For a solid rotating disc,radius R,the collapse speed p is given by 30y 2= PR2 where p is the density of the disc material. For rotating hollow discs the collapse speed is found from wp 303 [R2-Ri LR-R」 Introduction When the design of components is based upon the elastic theory,e.g.the simple bending or torsion theory,the dimensions of the components are arranged so that the maximum stresses which are likely to occur under service loading conditions do not exceed the allowable working stress for the material in either tension or compression.The allowable working stress is taken to be the yield stress of the material divided by a convenient safety factor (usually based on design codes or past experience)to account for unexpected increase in the level of service loads.If the maximum stress in the component is likely to exceed the allowable working stress,the component is considered unsafe,yet it is evident that complete failure of the component is unlikely to occur even if the yield stress is reached at the outer fibres provided that some portion of the component remains elastic and capable of carrying load,i.e.the strength of a component will normally be much greater than that assumed on the basis of initial yielding at any position.To take advantage of the inherent additional
62 Mechanics of Materials 2 fully plastic torque where ty is the shear stress at the elastic limit, or shear yield stress. Angles of twist of partially plastic shafts are calculated on the basis of the elastic core only. For hollow shafts, inside radius RI , outside radius R yielded to radius R2, =tv 3 4 Tpp = -[4R R2 - R2 - 3R:] 6R2 For eccentric loading of rectangular sections the fully plastic moment is given by BD2 P2N2 4 4Bav MFp = -ay - - where P is the axial load, N the load factor and B the width of the cross-section. The maximum allowable moment is then given by BD~ P2N 4N 4Bo, Mz-0 -- For a solid rotating disc, radius R, the collapse speed wp is given by where p is the density of the disc material. For rotating hollow discs the collapse speed is found from Introduction When the design of components is based upon the elastic theory, e.g. the simple bending or torsion theory, the dimensions of the components are arranged so that the maximum stresses which are likely to occur under service loading conditions do not exceed the allowable working stress for the material in either tension or compression. The allowable working stress is taken to be the yield stress of the material divided by a convenient safety factor (usually based on design codes or past experience) to account for unexpected increase in the level of service loads. If the maximum stress in the component is likely to exceed the allowable working stress, the component is considered unsafe, yet it is evident that complete failure of the component is unlikely to occur even if the yield stress is reached at the outer fibres provided that some portion of the component remains elastic and capable of carrying load, i.e. the strength of a component will normally be much greater than that assumed on the basis of initial yielding at any position. To take advantage of the inherent additional
Strains Beyond the Elastic Limit 63 strength,therefore,a different design procedure is used which is often referred to as plastic limit design.The revised design procedures are based upon a number of basic assumptions about the material behaviour. Figure 3.1 shows a typical stress-strain curve for annealed low carbon steel indicating the presence of both upper and lower yield points and strain-hardening characteristics. Stress a Stroin Upper yield point hordening Tension Strain e Compression Strain hardening Fig.3.1.Stress-strain curve for annealed low-carbon steel indicating upper and lower yield points and strain- hardening characteristics. Stress o Stroin Fig.3.2.Assumed stress-curve for plastic theory -no strain-hardening.equal yield points,o=x=oy. Figure 3.2 shows the assumed material behaviour which: (a)ignores the presence of upper and lower yields and suggests only a single yield point; (b)takes the yield stress in tension and compression to be equal;
Strains Beyond the Elastic Limit 63 strength, therefore, a different design procedure is used which is often referred to as plastic limit design. The revised design procedures are based upon a number of basic assumptions about the material behaviour. Figure 3.1 shows a typical stress-strain curve for annealed low carbon steel indicating the presence of both upper and lower yield points and strain-hardening characteristics. Stress u u. "I Tension Compression Strain hardening Stroin ~ Strain L =vc Fig. 3.1, Stress-strain curve for annealed low-carbon steel indicating upper and lower yield points and strainhardening characteristics. L - Strain c 4 Fig. 3.2. Assumed stress-curve for plastic theory - no strain-hardening, equal yield points, u,, = c,,~ = c". Figure 3.2 shows the assumed material behaviour which: (a) ignores the presence of upper and lower yields and suggests only a single yield point; (b) takes the yield stress in tension and compression to be equal;
64 Mechanics of Materials 2 $3.1 (c)assumes that yielding takes place at constant strain thereby ignoring any strain-hardening characteristics.Thus,once the material has yielded,stress is assumed to remain constant throughout any further deformation. It is further assumed,despite assumption (c),that transverse sections of beams in bending remain plane throughout the loading process,i.e.strain is proportional to distance from the neutral axis. It is now possible on the basis of the above assumptions to determine the moment which must be applied to produce: (a)the maximum or limiting elastic conditions in the beam material with yielding just initiated at the outer fibres; (b)yielding to a specified depth; (c)yielding across the complete section. The latter situation is then termed a fully plastic state,or "plastic hinge".Depending on the support and loading conditions,one or more plastic hinges may be required before complete collapse of the beam or structure occurs,the load required to produce this situation then being termed the collapse load.This will be considered in detail in $3.6. 3.1.Plastic bending of rectangular-sectioned beams Figure 3.3(a)shows a rectangular beam loaded until the yield stress has just been reached in the outer fibres.The beam is still completely elastic and the bending theory applies,i.e. σl M= y BD3 2 maximum elastic moment =o x 12* D BD2 ME= 6 (3.1) Beam cross-section Stress distribution 一B✉ 9 D Yielded_ area (a）Maximum elastic (b)Partially plastic (c)Fully piastic Fig.3.3.Plastic bending of rectangular-section beam
64 Mechanics of Materials 2 53.1 (c) assumes that yielding takes place at constant strain thereby ignoring any strain-hardening characteristics. Thus, once the material has yielded, stress is assumed to remain constant throughout any further deformation. It is further assumed, despite assumption (c), that transverse sections of beams in bending remain plane throughout the loading process, i.e. strain is proportional to distance from the neutral axis. It is now possible on the basis of the above assumptions to determine the moment which must be applied to produce: (a) the maximum or limiting elastic conditions in the beam material with yielding just (b) yielding to a specified depth; (c) yielding across the complete section. initiated at the outer fibres; The latter situation is then termed a fully plastic state, or “plastic hinge”. Depending on the support and loading conditions, one or more plastic hinges may be required before complete collapse of the beam or structure occurs, the load required to produce this situation then being termed the collapse load. This will be considered in detail in 53.6. 3.1. Plastic bending of rectangular-sectioned beams Figure 3.3(a) shows a rectangular beam loaded until the yield stress has just been reached in the outer fibres. The beam is still completely elastic and the bending theory applies, i.e. .. DI M=- Y BD3 2 maximum elastic moment = cry x - x - 12 D BD~ ME = - 6 uy Beam Stress Cross-section dis+,,bvtion (3.1) (a) Maximum elastic (b) Partially plastic (c) Fully plastic Fig. 3.3. Plastic bending of rectangular-section beam
$3.2 Strains Beyond the Elastic Limit 65 If loading is then increased,it is assumed that instead of the stress at the outside increasing still further,more and more of the section reaches the yield stress oy.Consider the stage shown in Fig.3.3(b). Partially plastic moment, MPp moment of elastic portion total moment of plastic portion M=g,+2{o,x[B-引(？-)+ Bd2 stress area moment arm Bd2 B Mpp=+(D-d)(D+d) 2d+3D2-4P1=D2-的 212 (3.2) When loading has been continued until the stress distribution is as in Fig.3.3(c)(assumed), the beam with collapse.The moment required to produce this fully plastic state can be obtained from eqn.(3.2),since d is then zero, i.e. fully plastic moment,MP=12 Ba.x 3DBD 49 (3.3) This is the moment therefore which produces a plastic hinge in a rectangular-section beam. 3.2.Shape factor -symmetrical sections The shape factor is defined as the ratio of the moments required to produce fully plastic and maximum elastic states: MFP shape factor=ME (3.4) It is a factor which gives a measure of the increase in strength or load-carrying capacity which is available beyond the normal elastic design limits for various shapes of section,e.g. for the rectangular section above, B /BD2 shape factor = 4r/60,=1.5 Thus rectangular-sectioned beams can carry 50%additional moment to that which is required to produce initial yielding at the edge of the beam section before a fully plastic hinge is formed.(It will be shown later that even greater strength is available beyond this stage depending on the support conditions used.)It must always be remembered,however, that should the stresses exceed the yield at any time during service there will be some associated permanent set or deflection when load is removed,and consideration should be given to whether or not this is acceptable.Bearing in mind,however,that normal design office practice involves the use of a safety factor to take account of abnormalities of loading,it should be evident that even at this stage considerable advantages are obtained by application of this factor to the fully plastic condition rather than the limiting elastic case.It is then
$3.2 Strains Beyond the Elastic Limit 65 If loading is then increased, it is assumed that instead of the stress at the outside increasing still further, more and more of the section reaches the yield stress o,,,. Consider the stage shown in Fig. 3.3(b). Partially plastic moment, Mpp = moment of elastic portion + total moment of plastic portion stress area moment arm 1 B Mpp = cy [T + x(D - d)(D +d) = BO, -[2d2 + 3(D2 - d2)] = B -[3D2 or - d2] 12 12 (3.2) When loading has been continued until the stress distribution is as in Fig. 3.3(c) (assumed), the beam with collapse. The moment required to produce this fully plastic state can be obtained from eqn. (3.2), since d is then zero, i.e. BU,, BD~ fully plastic moment, MFP = - x 3D2 = - 12 4 uy (3.3) This is the moment therefore which produces a plastic hinge in a rectangular-section beam. 3.2. Shape factor - symmetrical sections The shape factor is defined as the ratio of the moments required to produce fully plastic (3.4) and maximum elastic states: MFP shape factor A = ~ ME It is a factor which gives a measure of the increase in strength or load-carrying capacity which is available beyond the normal elastic design limits for various shapes of section, e.g. for the rectangular section above, BD~ 4 shape factor = -cy/? cy = 1.5 Thus rectangular-sectioned beams can carry 50% additional moment to that which is required to produce initial yielding at the edge of the beam section before a fully plastic hinge is formed. (It will be shown later that even greater strength is available beyond this stage depending on the support conditions used.) It must always be remembered, however, that should the stresses exceed the yield at any time during service there will be some associated permanent set or deflection when load is removed, and consideration should be given to whether or not this is acceptable. Bearing in mind, however, that normal design office practice involves the use of a safety factor to take account of abnormalities of loading, it should be evident that even at this stage considerable advantages are obtained by application of this factor to the fully plastic condition rather than the limiting elastic case. It is then