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
66 Mechanics of Materials 2 §3.2 possible to arrange for all normal loading situations to be associated with elastic stresses in the beam,the additional strength in the partially plastic condition being used as the safety margin to take account of unexpected load increases. Figure 3.4 shows the way in which moments build up with increasing depth or penetration of yielding and associated radius of curvature as the beam bends. Typical shope factor ⑦7 1.5 肠5 写18 1.0 Stress distributions at various stoges R/RE Fig.3.4.Variation of moment of resistance of beams of various cross-section with depth of plastic penetration and associated radius of curvature. Here the moment M carried by the beam at any particular stage and its associated radius of curvature R are considered as ratios of the values at the maximum elastic or initial yield condition.It will be noticed that at large curvature ratios,i.e.high plastic penetrations, the values of M/ME approach the shape factor of the sections indicated,e.g.1.5 for the rectangular section. Shape factors of other symmetrical sections such as the I-section beam are found as follows (Fig.3.5). Stress distributions 王消 A (a)Eiastic (b)Fully plostic Fig.3.5.Plastic bending of symmetrical (I-section)beam. First determine the value of the maximum elastic moment ME by applying the simple bending theory M o 1 y
66 Mechanics of Materials 2 93.2 possible to arrange for all normal loading situations to be associated with elastic stresses in the beam, the additional strength in the partially plastic condition being used as the safety margin to take account of unexpected load increases. Figure 3.4 shows the way in which moments build up with increasing depth or penetration of yielding and associated radius of curvature as the beam bends. Typical shape factor 17 @ 1.7 1.5 '5 w 1.18 2, z 1.0 Stress distrbutions vorious stages Fig. 3.4. Variation of moment of resistance of beams of various cross-section with depth of plastic penetration and associated radius of curvature. Here the moment M carried by the beam at any particular stage and its associated radius of curvature R are considered as ratios of the values at the maximum elastic or initial yield condition. It will be noticed that at large curvature ratios, i.e. high plastic penetrations, the values of MIME approach the shape factor of the sections indicated, e.g. 1.5 for the rectangular section. Shape factors of other symmetrical sections such as the I-section beam are found as follows (Fig. 3.5). Stress distributions I1 (01 Elasi~c (b) Fully plostlc Fig. 3.5. Plastic bending of symmetrical (I-section) beam. First determine the value of the maximum elastic moment ME by applying the simple bending theory
$3.3 Strains Beyond the Elastic Limit 67 with y the maximum distance from the N.A.(the axis of symmetry passing through the centroid)to an outside fibre and o =oy,the yield stress. Then,in the fully plastic condition,the stress will be uniform across the section at oy and the section can be divided into any convenient number of rectangles of area A and centroid distance h from the neutral axis. Then Mrn=∑(a,Ah (3.5) The shape factor MFp/ME can then be determined. 3.3.Application to I-section beams When the B.M.applied to an I-section beam is just sufficient to initiate yielding in the extreme fibres,the stress distribution is as shown in Fig.3.5(a)and the value of the moment is obtained from the simple bending theory by subtraction of values for convenient rectangles. ol ie. ME- y [BD3 bd312 [1212]D If the moment is then increased to produce full plasticity across the section,i.e.a plastic hinge,the stress distribution is as shown in Fig.3.5(b)and the value of the moment is obtained by applying eqn.(3.3)to the same convenient rectangles considered above. BD2 bd27 MFP=Oy 4-4 The value of the shape factor can then be obtained as the ratio of the above equations MFP/ME.A typical value of shape factor for commercial rolled steel joists is 1.18,thus indi- cating only an 18%increase in"strength"capacity using plastic design procedures compared with the 50%of the simple rectangular section. 3.4.Partially plastic bending of unsymmetrical sections Consider the T-section beam shown in Fig.3.6.Whilst stresses remain within the elastic limit the position of the N.A.can be obtained in the usual way by taking moments of area Stress distributions 门Plostic NA (o)Elastic (b)Plastic Fig.3.6.Plastic bending of unsymmetrical (T-section)beam
93.3 Strains Beyond the Elastic Limit 67 with y the maximum distance from the N.A. (the axis of symmetry passing through the centroid) to an outside fibre and (T = oY, the yield stress. Then, in the fully plastic condition, the stress will be uniform across the section at oY and the section can be divided into any convenient number of rectangles of area A and centroid distance h from the neutral axis. Then MFP = x(cyA)h (3.5) The shape factor MFp/ME can then be determined. 33. Application to I-section beams When the B.M. applied to an I-section beam is just sufficient to initiate yielding in the extreme fibres, the stress distribution is as shown in Fig. 3.5(a) and the value of the moment is obtained from the simple bending theory by subtraction of values for convenient rectangles. (TI Y i.e. ME = - BD3 bd3 2 =uy [T - -4 5 MFP ‘Cy [T - -4 If the moment is then increased to produce full plasticity across the section, i.e. a plastic hinge, the stress distribution is as shown in Fig. 33b) and the value of the moment is obtained by applying eqn. (3.3) to the same convenient rectangles considered above. BD2 bd2 The value of the shape factor can then be obtained as the ratio of the above equations MFP/ME. A typical value of shape factor for commercial rolled steel joists is 1.18, thus indicating only an 18% increase in “strength” capacity using plastic design procedures compared with the 50% of the simple rectangular section. 3.4. Partially plastic bending of unsymmetrical sections Consider the T-section beam shown in Fig. 3.6. Whilst stresses remain within the elastic limit the position of the N.A. can be obtained in the usual way by taking moments of area Fig. 3.6. Plastic bending of unsymmetrical (T-section) beam
68 Mechanics of Materials 2 §3.4 about some convenient axis as described in Chapter 4.7A typical position of the elastic N.A.is shown in the figure.Application of the simple blending theory about the N.A.will then yield the value of ME as described in the previous paragraph. Whatever the state of the section,be it elastic,partially plastic or fully plastic,equilibrium of forces must always be maintained,i.e.at any section the tensile forces on one side of the N.A.must equal the compressive forces on the other side. ∑stress×area above N.A.=∑stress×area below N.A. In the fully plastic condition,therefore,when the stress is equal throughout the section, the above equation reduces to ∑areas above N.A.=∑areas below N.A. (3.6) and in the special case shown in Fig.3.5 the N.A.will have moved to a position coincident with the lower edge of the flange.Whilst this position is peculiar to the particular geometry chosen for this section it is true to say that for all unsymmetrical sections the N.A.will move from its normal position when the section is completely elastic as plastic penetration proceeds.In the ultimate stage when a plastic hinge has been formed the N.A.will be positioned such that eqn.(3.6)applies,or,often more conveniently, area above or below N.A.total area (3.7) In the partially plastic state,as shown in Fig.3.7,the N.A.position is again determined by applying equilibrium conditions to the forces above and below the N.A.The section is divided into convenient parts,each subjected to a force average stress x area,as indicated,then F1+F2=F3+F4 (3.8) Yielded area Fig.3.7.Partially plastic bending of unsymmetrical section beam. and this is an equation in terms of a single unknown yp,which can then be determined,as can the independent values of F1,F2,F3 and F4. The sum of the moments of these forces about the N.A.then yields the value of the partially plastic moment MPP.Example 3.2 describes the procedure in detail. EJ.Hearn,Mechanics of Materials 1,Butterworth-Heinemann,1997
68 Mechanics of Materials 2 g3.4 about some convenient axis as described in Chapter 4.t A typical position of the elastic N.A. is shown in the figure. Application of the simple blending theory about the N.A. will then yield the value of ME as described in the previous paragraph. Whatever the state of the section, be it elastic, partially plastic or fully plastic, equilibrium of forces must always be maintained, i.e. at any section the tensile forces on one side of the N.A. must equal the compressive forces on the other side. 1 stress x area above N.A. = 1 stress x area below N.A. In the fully plastic condition, therefore, when the stress is equal throughout the section, the above equation reduces to areas above N.A. = areas below N.A. (3.6) and in the special case shown in Fig. 3.5 the N.A. will have moved to a position coincident with the lower edge of the flange. Whilst this position is peculiar to the particular geometry chosen for this section it is true to say that for all unsymmetrical sections the N.A. will move from its normal position when the section is completely elastic as plastic penetration proceeds. In the ultimate stage when a plastic hinge has been formed the N.A. will be positioned such that eqn. (3.6) applies, or, often more conveniently, area above or below N.A. = total area (3.7) In the partially plastic state, as shown in Fig. 3.7, the N.A. position is again determined by applying equilibrium conditions to the forces above and below the N.A. The section is divided into convenient parts, each subjected to a force = average stress x area, as indicated, then -- Yielded ----- =* Fig. 3.7. Partially plastic bending of unsymmetrical section beam. and this is an equation in terms of a single unknown 7,. which can then be determined, as can the independent values of F 1, F2, F3 and F4. The sum of the moments of these forces about the N.A. then yields the value of the partially plastic moment Mpp. Example 3.2 describes the procedure in detail. E.J. Hearn, Mechanics of Materials 1, Butterworth-Heinemann, 1991
$3.5 Strains Beyond the Elastic Limit 69 3.5.Shape factor-unsymmetrical sections Whereas with symmetrical sections the position of the N.A.remains constant as the axis of symmetry through the centroid,in the case of unsymmetrical sections additional work is required to take account of the movement of the N.A.position.However,having deter- mined the position of the N.A.in the fully plastic condition using eqn.(3.6)or(3.7),the procedure outlined in $3.2 can then be followed to evaluate shape factors of unsymmetrical sections-see Example 3.2. 3.6.Deflections of partially plastic beams Deflections of partially plastic beams are normally calculated on the assumption that the yielded areas,having yielded,offer no resistance to bending.Deflections are calculated therefore on the basis of the elastic core only,i.e.by application of simple bending theory and/or the standard deflection equations of Chapter 5f to the elastic material only.Because the second moment of area I of the central core is proportional to the fourth power of d, and I appears in the denominator of deflection formulae,deflections increase rapidly as d approaches zero,i.e.as full plasticity is approached. If an experiment is carried out to measure the deflection of beams as loading,and hence B.M.,is increased,the deflection graph for simply supported end conditions will appear as shown in Fig.3.8.Whilst the beam is elastic the graph remains linear.The initiation of yielding in the outer fibres of the beam is indicated by a slight change in slope,and when plastic penetration approaches the centre of the section deflections increase rapidly for very small increases in load.For rectangular sections the ratio MFP/ME will be 1.5 as determined theoretically above. Theoretical collapse lood Theoretical initial yield/ _ading condition Deflection Fig.3.8.Typical load-deflection curve for plastic bending. 3.7.Length of yielded area in beams Consider a simply supported beam of rectangular section carrying a central concentrated load W.The B.M.diagram will be as shown in Fig.3.9 with a maximum value of WL/4 at EJ.Hearn,Mechanics of Materials 1.Butterworth-Heinemann,1997
$3.5 Strains Beyond the Elastic Limit 69 35. Shape factor - unsymmetrical sections Whereas with symmetrical sections the position of the N.A. remains constant as the axis of symmetry through the centroid, in the case of unsymmetrical sections additional work is required to take account of the movement of the N.A. position. However, having determined the position of the N.A. in the fully plastic condition using eqn. (3.6) or (3.7), the procedure outlined in $3.2 can then be followed to evaluate shape factors of unsymmetrical sections - see Example 3.2. 3.6. Deflections of partially plastic beams Deflections of partially plastic beams are normally calculated on the assumption that the yielded areas, having yielded, offer no resistance to bending. Deflections are calculated therefore on the basis of the elastic core only, i.e. by application of simple bending theory t and/or the standard deflection equations of Chapter 5 to the elastic material only. Because the second moment of area I of the central cors is proportional to the fourth power of d, and I appears in the denominator of deflection formulae, deflections increase rapidly as d approaches zero, i.e. as full plasticity is approached. If an experiment is carried out to measure the deflection of beams as loading, and hence B.M., is increased, the deflection graph for simply supported end conditions will appear as shown in Fig. 3.8. Whilst the beam is elastic the graph remains linear. The initiation of yielding in the outer fibres of the beam is indicated by a slight change in slope, and whtn plastic penetration approaches the centre of the section deflections increase rapidly for very small increases in load. For rectangular sections the ratio MFP/ME will be 1.5 as determined theoretically above. P J x Theoretical collapse load Theoret lcal w -2oding condition L Deflection / Fig. 3.8. Typical load-deflection curve for plastic bending 3.7. Length of yielded area in beams Consider a simply supported beam of rectangular section carrying a central concentrated load W. The B.M. diagram will be as shown in Fig. 3.9 with a maximum value of WL/4 at E.J. Hearn, Mechanics of Materials I, Butterworth-Heinemann, 1997
70 Mechanics of Materials 2 $3.7 W/2 一L/2一儿/2 w/2 B.M.Diagram WL/4 Fig.3.9. the centre.If loading is increased,yielding will commence therefore at the central section when (WL/4)=(BD2/6)oy and will gradually penetrate from the outside fibres towards the N.A.As this proceeds with further increase in loads,the B.M.at points away from the centre will also increase,and in some other positions near the centre it will also reach the value required to produce the initial yielding,namely BD2/6.Thus,when full plasticity is achieved at the central section with a load Wp,there will be some other positions on either side of the centre,distance x from the supports,where yielding has just commenced at the outer fibres;between these two positions the beam will be in some elastic-plastic state.Now at distance x from the supports: 2 2WpL B.M.=w,2=MFP=54 L x=3 The central third of the beam span will be affected therefore by plastic yielding to some depth.At any general section within this part of the beam distance x'from the supports the B.M.will be given by xBa (3D2-d2] BM.=W2=12 (1) BD WpL Now since 0,=w4 0y= BD2 Therefore substituting in (1), x' w,2= 23D2-4 pL BD2 (3D-)L 6D2 L d21 术= 2 3D2 This is the equation of a parabola with x'=L/2 when d =0 (i.e.fully plastic section)
70 Mechanics of Materials 2 03.7 w/2 F-% L/2 & L/2 w/2 Fig. 3.9. the centre. If loading is increased, yielding will commence therefore at the central section when (WL/4) = (BD2/6)o, and will gradually penetrate from the outside fibres towards the N.A. As this proceeds with further increase in loads, the B.M. at points away from the centre will also increase, and in some other positions near the centre it will also reach the value required to produce the initial yielding, namely BD20,/6. Thus, when full plasticity is achieved at the central section with a load W,, there will be some other positions on either side of the centre, distance x from the supports, where yielding has just commenced at the outer fibres; between these two positions the beam will be in some elastic-plastic state. Now at distance x from the supports: x2 2 W,L B.M. = W,- = -MFp = -- 23 34 L 3 .. x=- The central third of the beam span will be affected therefore by plastic yielding to some depth. At any general section within this part of the beam distance x’ from the supports the B .M. will be given by Now since XI Bo B.M. = W,- = ‘[3D2 - d2] 2 12 L WPL --(Tv=wp- -(T -- BD~ 4 4 ’- BD2 Therefore substituting in (l), W,,- x’ B WPL = -[3D2 - d2]- 2 12 BD2 L I (3D2 -d2) 6D2 x= 2 This is the equation of a parabola with XI = L/2 when d = 0 (i.e. fully plastic section)
