PLATE GIRDERS - I 1.0 INTRODUCTION

Transcription

1 15 PLATE GIRDERS - I 1.0 INTRODUCTION A fabricated late girder shown diagrammatically in Fig. 1 is emloyed for suorting heavy loads over long sans. The bending moments and shear forces roduced in such girders are well beyond the bending and shear resistance of rolled steel girders available. In such situations the designer has the choice of one of the following solutions: Use two or more regularly available sections, side-by-side. This is an exensive solution and may still not satisfy deflection limitation. Use a cover-lated beam; i.e. weld a late of adeuate thickness to beef u each flange. This would enhance the bending resistance, in circumstances where the web has adeuate shear resistance and the rolled steel section is only marginally inadeuate. Use a fabricated late girder, wherein the designer has the freedom (within limits) to choose the size of web and flanges, or Use a steel truss or a steel-conete-comosite truss. This chater is concerned with late girders only. Plate girders are large I shaed sections built u from lates, as shown in Fig.. Transverse stiffener a Fig. 1 Tyical late girder with intermediate and end stiffeners Coyright reserved Fig. Cross-section of fabricated late girders ersion II 15-1

2 Nearly all late girders built today are welded, although they may use bolted field slices. In the West, late girders are invariably fabricated in fabrication shos, using numerically controlled welding machines [See Fig. (a)]. If the late girders are to be fabricated in the field, the tye sketched in Fig. (b) is used to minimise overhead welding. The rimary function of the flange lates is to resist bending moments by develoing axial comressive and tensile stresses. The web late resists the shear. For a given alied bending moment, the axial force deeases, as the deth of the girder (d) ineases. From this oint of view it is economical to kee the flanges as far aart as ossible. This would ensure that the flanges would have to resist smaller axial forces. Thus a smaller area of oss section would suffice than would be the case if a smaller deth were chosen. However, this would also mean that the web would be dee. To reduce the self-weight (and the corresonding self-weight bending moment), the web thickness (t) would have to be limited to slender roortions, (the web roortions are normally exressed in terms of the web slenderness ratio, d/t). Slender webs (with large d/t values) would buckle at relatively low values of alied shear loading. (It is also imortant to note that webs having a san/deth ratio smaller than 1 would result in a dee beam, wherein the structural behaviour can no longer be desibed by conventional simle beam theories). Efficient and economical design usually results in slender members. Hence advantage must be taken of the ost buckling caacity of the web i.e. the ability of the girder to withstand transverse loads considerably in excess of the load at which the web buckles under shear. A girder of high strength to weight ratio can be designed by incororating the ost buckling strength of the web in the design method emloyed. This would be articularly advantageous where the reduction of self-weight is of rime imortance. Examles of such situations arise in long san bridges, shi girders, transfer girders in building etc..0 SHEAR RESISTANCE OF TRANSERSELY STIFFENED PLATE GIRDERS Webs of late girders are usually stiffened transversely as shown in Figure 1. This hels to inease the ultimate shear resistance of the webs, as will be seen later. The stiffener sacing (a) influences both buckling and ost buckled behaviour of the web under shear. In order to allow for this, the arameter (a/d) which accounts for the geometry of the web anel is imortant. Obviously, a long san girder will have various web anels and each anel will have different combinations of bending moments and shear forces. In a long late girder, anels close to the suort will be subjected to redominant shear and those close to the centre, to redominant bending moments. In what follows, the effect of shear will be considered first, followed by the effect of coexisting bending moment and shear forces. ersion II 15 -

3 .1 Shear resistance of a web.1.1 Pre-buckling behaviour (Stage 1) When a web late is subjected to shear, we can visualise the structural behaviour by considering the effect of comlementary shear stresses generating diagonal tension and diagonal comression. Fig. 3 Unbuckled shear anel Consider an element E in euilibrium inside a suare web late subject to a shear stress. The reuirements of euilibrium result in the generation of comlementary shear stresses as shown in Fig. 3. This results in the element being subjected to rincial comression along the direction AC and tension along the direction of BD. As the alied loading is inementally enhanced, with corresonding ineases in, very soon, the late will buckle along the direction of comressive diagonal AC. The late will lose its caacity to any further inease in comressive stress; the corresonding shear stress in the late is the itical shear stress. The value of can be determined from classical stability theory if the boundary conditions of the late are known. As the true boundary conditions of the late girder web are difficult to establish due to restraint offered by the flanges and stiffeners we may conservatively assume them to be simly suorted. The itical shear stress in such a case is given by = k s 1 1 π E ( ν ) t d (1) ersion II 15-3

4 where, k s is the shear buckling coefficient given by d a k s = where 1, i. e. for wide anels a d d a k s = where 1, i. e. for webs with closely a d saced transverse stiffeners alues of for various values of web asect ratios are tabulated in Table 1. d/t a/d Table 1: (Pa) alues Buckling does not govern Buckling does not govern E = 00,000 Pa ν = 0.3 π = When the value of (d/t) is sufficiently low (d/t < 85) ineases above the value of yield shear stress and the web will yield under shear before buckling..1. Post buckled behaviour (Stage ) The comression diagonal (AC) is unable to resist any more loading beyond the one corresonding to the elastic itical stress. Once the web has lost its caacity to sustain inease in comressive stresses, a new load carrying mechanism is develoed. Alications of any further ineases in the shear load are suorted by a tensile membrane field, anchored to the boundaries, viz. the to and bottom flanges and the adjacent stiffener members on either side of the web. The angle of inclination of the membrane stress (θ) is unknown at this stage (See Fig. 4). Thus the total state of stress in this web late may be obtained by suerimosing the ost-buckled membrane tensile stresses ( t ) uon those set u when the alied shear stress reached the itical value. The state of stress in the web in the ost-buckled stage is shown in Fig. 5. ersion II 15-4

5 Fig. 4 Post buckled behaviour Fig. 5 State of stress in the web in the ost buckled stage Resolving these stresses in the direction along and erendicular to the inclination θ we get, θ ( θ θ =.sin θ + t + 90) = sin θ =.cosθ () Since the flanges are of finite rigidity, the ull exerted by the tensile membrane stresses in the web will cause the flanges to bend inwards..1.3 Collase behaviour (Stage 3) When the load is further ineased, the tensile membrane stress ( t ) develoed in the web continues to exert an ineasing ull on the flanges. Eventually the resultant stress ( θ ) (obtained by combining the buckling stress in Euation (1) and the membrane stress t ) reaches the yield value for the web. This value (of the membrane stress at yield) may be denoted by, and may be determined by on-ises yield iterion. θ + + = ( θ + 90) θ. ( θ 90) + 3θ (3) where, = tensile yield stress of the web By substituting the values of θ, (θ+90) and θ from Euation (), the above euation may be resented in a non-dimensional form as follows: ersion II 15-5

6 yt = sin 4 θ 3. sin θ (4) where, is obtained from Euation (1) tensile yield = i. e. shear yield = 3 3 (5) Tensile membrane stress at yield = yt Fig. 6 Collase of the anel Once the web has yielded, final collase of the girder will occur when four lastic hinges are formed in the flanges as shown in Fig. 6. The lastic moment caacity of the flange late is f. By using the virtual work method, Rockey and his team at Cardiff have shown that the failure load can be comuted from S =. d. t + yt. t sin θ θ + f ( d cot a + c) 4 (6) where c = distance between the hinges given by c c = sinθ yt f. t (7) This euation can be non-dimensionalised by using the shear load reuired to roduce yielding in the entire web ( =. d.t.) ersion II 15-6

7 S = + a yt yt f 3 sin θ cotθ sinθ. d d t. (8) Euation (6) or (8) can be solved if θ is known. As these euations are based on Energy ethod, the correct solution will be obtained by maximising s with resect to θ. By a systematic set of arametric studies, Evans has established that θ is aroximately eual to /3 of the inclination of diagonal of the web. θ 1 tan 3 d a Notice that Euations (6) and (8) are obtained by adding 3 uantities: web-buckling strength, ost-buckling membrane strength of the web late and the lastic moment caacity of the flange. In this context, it must be noted that in order for the flanges to develo hinges, the flanges must be classifiable as "lastic" sections. If flanges can not develo lastic hinges because they are comact, semi-comact or slender, this method of analysis is NOT alicable..1.4 Weak flanges When a late girder has weak flanges, f is a small uantity in comarison with the other terms. Hence (9) S = + 3 sin θ cotθ a d yt (10) In this case the tension field is NOT suorted by flanges. Fig. 7 Weak flange The field anchors entirely on transverse stiffeners as shown in Fig. 7. ersion II 15-7

8 .1.5 ery Strong flanges When ery Strong flanges are emloyed, the distance (c) of the lastic hinge away from the end anel ineases. When c = a, the hinges will form at the four corners, constituting a icture frame tye mechanism (See Fig. 8) and the tension field angle (θ) is 45. Ultimate shear in this case is given by: S 1 = d f. a d t (11) Fig. 8 Picture frame mechanism of strong flange.1.6 ery Thick webs In case the web is thick, it will yield before buckling failure will form by a icture frame mechanism, with =. S = d f 3. a d t (1).1.7 ery Slender webs ery slender webs are rarely used as they cause anxieties for the users due to high levels of buckling. In very slender webs / is extremely small and there will be significant ost-buckled tension field, the value of membrane stress yt will be very large. The general exression given in Euation (8) is valid and θ can be evaluated using euation (9). ersion II 15-8

9 3.0 WEBS SUBJECTED TO CO-EXISTENT BENDING AND SHEAR When a girder is subjected to redominant bending moments and low shear, its ultimate caacity is conditioned by the interaction between the effects of the bending moment and shear force. Fig. 9(a) Fig. 9(b) Fig. 9 Interaction between bending and shear effects ersion II 15-9

10 The interaction diagram is generally exressed in the form seen in Fig. 9, where the shear caacity is lotted in the Y-axis and the bending caacity in the X-axis. Any oint in the interaction diagram shows the co-existent values of shear and bending moment that the girder can sustain. The vertical ordinates are non-dimensionalised using (Yield shear of the web) and the horizontal ordinates by (the fully lastic moment resistance of the oss section). The ortion of the curve between oints A and C is the region in which the girder will fail by redominant shear, i.e. shear mechanism of the tye reresented in Fig. 6 will develo at collase. The vertical ordinate at A resents the shear caacity ( s ) given by Euation 8. This shear caacity will reduce gradually due to the resence of co-existent bending moment. Beyond oint C, when the alied moment is high, the failure will be triggered by the collase of flanges by one of the following: (i) by yielding of flange material or (ii) by inward buckling of the comression flange or (iii) by lateral buckling of the flange. Thus there is a distinct change in failure iterion reresented by line OC in Fig. 9(a); the left of OC reresents shear failure and the right of OC, flexural failure. Generally the flange failure mode will be triggered, when the alied bending moment is aroximately eual to the lastic moment resistance F, rovided by the flange lates only, neglecting the contribution from the web. F = bf. T. yf + T ( d ) (13) where, b f - Breadth of flange T - Thickness of flange yf- Design stress of flange d - deth of web late This value reresents the horizontal co-ordinate of the oint C, i.e. the oint F. In zone ABC, the resence of additional bending moment reuires the following three factors to be considered. The reduction in the web buckling stress due to the resence of bending stresses. The influence of bending stresses on the value of membrane stress reuired causing yield in the web. The reduction of lastic moment caacity of flanges due to the resence of axial flange stresses caused by bending moment. 3.1 odified web buckling stress The modified web buckling stress due to coincident bending stress may be comuted from the following interaction Euation: m + f f mb b = 1 (14) where, m = modified shear buckling stress in web ersion II 15-10

11 = elastic itical shear stress in web (ure shear case as defined reviously) f mb = comressive bending stress in the extreme fibre at the mid anel due to the bending moment. f b = buckling stress for the late due to a ure bending moment given by f b = π E ( ν ) t d (15) 3. odified membrane stress for web yielding The bending membrane stress ( yt ) to be added to the itical shear stress ( ) was calculated in the ure shear case, using Euations (4) and (1) resectively. The modified exression for the membrane stress ytm, in the resence of alied bending moment is given by ytm 1 = A+ ( A 4( + 3 ) (16) 1 b m where, A = 3 m sin θ + b sin θ - b cos θ b = The value of bending stress, which varies over both the deth and width of the web anel. As ytm varies for various values of b, it may be necessary to comute ytm at a number of locations in order to comute the resultant of the membrane stresses. This could be timeconsuming. To simlify the design calculations an adeuately accurate aroximate rocedure is suggested a little later. 3.3 Reduction of lastic moment caacity of flanges When high axial forces are develoed in the flanges due to bending moments, their effects in reducing lastic moment caacity of flange lates must be taken into account. From lasticity theory, the reduced caacity (' f ) is given by ' f = f 1 f yt (17) where, f is the average axial stress for the ortion of the flange between hinges. 3.4 Design rocedure The simlified design rocedure suggested by Rockey et. al (1978) is validated by them by exeriments and arametric studies. This rocedure is summarised below: ersion II 15-11

12 The shear load caacity at oint C of the interaction diagram may be obtained aroximately from an emirical relationshi given below. c = + yt 4 sin θ d f F b d 1 8 (18) This euation gives the vertical ordinate of the oint C in the interaction diagram [Fig. 9(a)]. The horizontal ordinate as stated reviously is given by the value of F (See Euation 13). The interaction diagram is constructed in stages as follows [See Fig. 9(b)]: (i) (ii) (iii) Between A and B, the curve is horizontal. The horizontal ordinate B is given by maximum bending moment in the end anel given by s.b, but limited to a value of 0.5 P. Between B and C, the curve may be straight (for simlicity). The moment corresonding to C is given by F = b f. T. yf (d + T) The oint D reresents nearly the ultimate caacity of the flanges ( u ) and the shear values when high bending is resent. This is discussed in the next section. 3.5 Webs subjected to ure bending The region beyond C of the interaction diagram reresents a high bending moment, so the failure is by bending moment, rather than by shear mode. In a thin walled girder, the web subjected to comressive bending stress will buckle, thereby losing its caacity to carry further comressive stresses. The comression flange will therefore carry ractically all the comressive stresses, as the web is unable to be fully effective. Conseuently the girder is unable to develo full lastic moment of resistance ( ) of the oss section. If no lateral buckling occurs (e.g. by rovision of adeuate lateral suorts), the girder will fail by inward collase of comression flange at an alied moment ( u ) which is aroximately eual to the moment reuired to roduce first yield in the extreme fibres of comression flange. This moment is of course reduced because of the effects of web buckling. Though the concet is simle, the resulting calculations are comlex. The research of Cooer in 1971 enables the ultimate moment caacity to be determined by a simle formula: u y = A A w f d t 5.7 E yf F y (19) y = Bending moment reuired to roduce yield in the extreme fibre of flange assuming fully effective web (i.e. neglecting web buckling) ersion II 15-1

13 This value of u is the moment reuired to roduce yield in the extreme fibres of the flange. The corresonding stresses in the web will be below yield. (Point D in the interaction diagrams). The ordinate of D can be calculated aroximately from D C = w u (0) where, is the fully lastic moment caacity of the comlete oss section w = lastic moment resistance of the web late alone. = 0.5 td. The comlete interaction diagram can now be drawn. 4.0 ULTIATE BEHAIOUR OF TRANSERSE WEB STIFFENERS The shear failure mechanism desibed so far has been extensively verified by exeriments. Before ost-buckling action in webs can develo the members in the boundary (viz. the flanges and the stiffeners) must be able to suort the forces imosed on them by the web tension field. The transverse stiffeners lay an imortant role in allowing the full ultimate caacity of the girder to be achieved (a) by ineasing the web buckling stress (b) by suorting the tension field after web buckling and (c) by reventing the tendency of flanges to get ulled towards each other. The stiffeners must therefore ossess sufficient rigidity to ensure that they remain straight, while restricting buckling to the individual web anels. Fig. 10 Force imosed on transverse stiffeners by tension field 4.1 Analysis of loads imosed on the transverse stiffener Fig. 10 reresents the loads acting on tyical stiffener CD ositioned between two adjacent anels, each of which have develoed shear failure mechanism. This is erhas the most itical form of loading of an intermediate stiffener. Let us consider the loads on the intermediate stiffener ersion II 15-13

14 The resultant of the loads acting on ortion W 1 C of to flanges is F fw, inclined at an angle of θ 1 and DY of bottom flange is F f., inclined at an angle of θ. The vertical comonent of these forces will tend to ull the flanges together and the stiffener will resist this by develoing end loads ( C and D ). C D = F = + F f.1 f. sinθ1 sinθ (1) oreover, the loading imosed directly uon the stiffener by web tension field can be slit into 3 zones: the to art CG is subject to a ull to the left by the left hand anel, the bottom art HD is similarly subject to a ull to the right by the right hand anel. The art GH is ulled to the left and to the right (these two forces more or less balancing each other). Thus the central region remains virtually unloaded by the tension field action. The vertical comonent of forces on zones CG and DH are resectively obtained as 1 = = yt1 yt. t. CG.sinθ1.cosθ1. t. HD.sinθ.cosθ () Thus once the ultimate shear loading and the geometry of failure mechanism has been determined, calculations of forces on the stiffeners by emloying Euations (1) and () may be made. Unfortunately, the actual behaviour of stiffeners (as evidenced by exerimental studies by Rockey in 1981 and Puthali in 1979) is somewhat different from the simle model desibed above. Firstly, a ortion of the web late acts with the stiffeners in resisting the axial load, desite the fact the web has theoretically yielded due to tension field. The effective oss section is in the form of a T section or a uciform section, if the stiffeners are on both sides of the web late. Secondly the theory exlained reviously assumed that the axial loading is alied to the stiffener oss section at its mid thickness. The true osition of load alication is unknown and some degree of eccentricity of alication of loads is inevitable. Thus the stiffener is subjected to both axial load (P) and bending moment arising due to the eccentricity of the alied load from the centroidal axis ( x) and is given by x. There will inevitably be some imerfections (δ o ) in the stiffener giving rise to conseuent moments of Pδ o. The disturbing action on the stiffener due to the web buckling is difficult to uantify but nevertheless is resent. Horne (1979) has roosed a suitable exression to define the combination of axial load and bending moment (making allowance for the above comlexities) which can be sustained by a stiffener: ersion II 15-14

15 ( b x 0.5t ) ys ts s = s s P P s (3) where, s = Full lastic moment caacity of the section where there is no axial loading P s = Suash load (i.e. full axial yield load) b s = Width of stiffener t s = Thickness of stiffener ys = Design stress of stiffener t = Thickness of web For any girder, the axial load to which the stiffener is subjected can be comuted from Euations (1) and (). Then, rovided the co-existent moment is less than the allowable moment defined by Euation (3), the stiffener will be able to suort the loads to which it is subjected. The theory governing the design of stiffeners is given above; but the design codes make simlifications to ease the task of the designers and enable uick sizing of the stiffeners. 5.0 GENERAL BEHAIOUR OF LONGITUDINALLY STIFFENED GIRDERS In order to obtain greater economy and efficiency in the design of late girders, slender webs are often reinforced both longitudinally and transversely. The longitudinal stiffeners are generally located in the comression zones of the girder. The main function of the longitudinal stiffeners is to inease the buckling resistance of web. The longitudinal stiffener remains straight thereby subdividing the web and limiting the web buckling to smaller web anels. In the ast it was usually thought that the resulting inease in ultimate strengths could be significant. Recent studies have shown that this is not always the case, as the additional cost of welding the longitudinal stiffeners invariably offsets any economy resulting in their use. The main effect of longitudinal stiffeners is to inease the elastic itical buckling strength. Studies by Rockey et al have shown that a longitudinally reinforced late girder subject redominantly to shear would develo a collase mechanism, similar to the one desibed reviously, rovided the stiffeners remained rigid u to failure. Once a tension field develos it extends over the comlete deth of the girder. In other words, once one of the sub anels has buckled, the ost buckling tension field develos over the whole deth of the web anel and the influence of the stiffeners may be neglected. Thus the euations established reviously are valid, keeing in mind values are enhanced due to the smaller anel dimensions. In the design of longitudinal stiffeners, linear buckling theories are often used to establish the minimum value of stiffener rigidity reuired to ensure that the longitudinal stiffeners remain straight at first buckling. This would ensure that the buckling is limited to individual sub-anels of the web. This invariably involves the rovision of stiffeners of substantial sizes, just so that they would remain straight without themselves buckling. ersion II 15-15

16 Normally the rigidity of such stiffeners has to be ineased by 4-6 times, to satisfy this reuirement. The heavy stiffeners thus designed, though adeuate, will naturally inease the weight of steel used and therefore the cost. It seems that it is more sensible to inease the web thickness in these cases. In general, the extra cost of roviding longitudinal stiffeners is rarely justified. In western countries numerically controlled machines are used extensively for fabricating late girders. In these countries the rovision of longitudinal (and transverse) stiffeners involves manual welding and contributes to the rise in the cost of fabrication. Indeed, in Scandinavian countries, the current trend is to eliminate or minimise the use of transverse stiffeners as far as ossible. The use of longitudinal stiffeners has also been discontinued for this reason in these countries. In other words, if they can hel it, they do not use any stiffeners at all! Longitudinal stiffeners are rarely if ever used in buildings. They are sometimes used in bridges, articularly when the Elastic Design is emloyed. Fig. 11 shows a tyical late girder with longitudinal and transverse stiffeners. Fig. 11 Longitudinal and Transverse stiffeners 6.0 CONCLUSIONS This chater has considered the ultimate behaviour of late girders in some detail. Fundamental theoretical relationshi based on buckling and ost-buckling theories have been established. In some case, semi-emirical rocedures have been suggested to ease the tedium of lengthy calculations. Transverse stiffeners have been considered in some deth. The use of longitudinal stiffeners has also been desibed and the reasons for not using these extensively have been discussed. ersion II 15-16

17 7.0 REFERENCES 1. Narayanan R. Plated Structures, Stability & Strength. Chater 1 Longitudinally and transversely reinforced Plate Girders, (by EANS H.R.) Elsevier Alied Science Publishers, London, Owens D.R.G. Skaloud,. and Rockey K.C. Ultimate Load behaviour of longitudinally reinforced web lates, IABSE emories 1970 Zurich Cooer P.B The ultimate bending moment for Plate Girders, Proceedings of IABSE Colloum, London, ele and Puthali R : University of Rome Reorts CECAR / 136, Rockey K.C., Evans H.R. and Porter D.. A design method for redicting collase behaviour of Plate Girders, Proceedings of the Institution of Civil Engineers 1978, Rockey K.C. and Skaloud : Influence of flange stiffeners on load carrying caacity of webs in shear. Proceedings of the Eighth Congress IABSE, New York, ersion II 15-17