# Design of cross-girders and slabs in ladder deck bridges

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2 Plate buckling action of the deck slab For the concrete slab to buckle under longitudinal compression, it must bend out of plane both longitudinally and transversely. If there were no cross-girders, the behaviour would be like that of a plate see Figure 2. Schlaich et al 1 recommend that, to prevent any significant second order effects from occurring in the slab due to such buckling, the transverse span to depth ratio should not exceed 30; the authors of this paper consider, based on precedent, this to be a reasonably pragmatic limit for typical concrete grades up to C40/50 but this limiting ratio might be a little too large for higher strength concrete grades. Figure 2. Slab buckling without restraint from cross-girders (plate-like behaviour) When the slab transverse span/ thickness ratio is within the above limit, it is sufficiently stable without any additional restraint from the cross girders. Consequently, the design of the cross-girders does not need to include any restraint forces. However, in practice, when the slab thickness is typically 250mm, the above limit is equivalent to a limiting spacing of the main girders of 7.5m (slightly less if the concrete is stronger than C40/50). Most ladder deck construction places the main girders at greater spacing than this (up to about 18m) and then the slab will need to be stiffened transversely by cross-girders. Figure 3. Slab buckling between cross-girders Where a wide slab is stiffened by the cross girders, the mode of buckling tends toward that of simple strut buckling the greater the aspect ratio, the closer the buckling is to strut-like behaviour see Figure 3. To consider whether second order effects in the slab are significant and, if they are, to evaluate the design effects, it is conservative to ignore plate action (i.e. buckling of a rectangular plate supported on four sides) and to treat the slab as a column, with a half wavelength of buckling equal to the span between cross girders. Limit on cross-girder spacing to avoid considering second order effects in the slab In PD , clause 6.1 provides a simple rule for externally prestressed members that second order effects need not be checked when the spacing between deviators does not exceed 10 times the section depth. This simple rule is considered appropriate for the slab of a ladder deck in the sagging region. The rule is less conservative than the simple rule for isolated compression members in clause of EN but even so, in practice, with a 250mm slab thickness the consequent limit of 2.5m on cross girder spacing is much lower than the preferred spacing of between 3m and 4m. In practice, therefore, second order effects in the slab will need to be considered. Second order effects in deck slab Where second order effects in the compressive region of the deck slab need to be accounted for, this can by achieved using EN , treating the slab as a simple compression member. Three methods are given for determining second order effects in reinforced concrete compression members, in clauses 5.8.6, and The clause method requires non-linear analysis and is thus not readily applicable; the clause method is not suitable for members with transverse load, such as the deck slabs; thus the clause method is used. The clause method uses a nominal stiffness of the compression member that takes account of the level of stress in the member, the concrete strength and the member slenderness. In doing so, it is important to use the actual design compression force in the slab (to avoid over-conservatism) and the eccentricity of that force from the centre of the slab (as this causes slab moment and curvature which will also be amplified by the compression). Generally, the slab force is unlikely to be at the level of its design compression resistance, as the design situation will be with combined longitudinal and transverse loading on the slab (and thus there will be significant local 94

3 130 Design of cross-girders and slabs in ladder deck bridges bending moments to be considered). Additionally, Eurocode 4 requires shear lag to be considered at ULS and this will normally mean that the design compression resistance is not that of the gross slab. Even if the full width of the slab can be mobilised, its large width makes it unlikely that it will be fully utilised in compression. The clause method derives a magnification factor that is applied to the first order moments. The factor is expressed in clause as: b M Ed M 0Ed 1+ ( NB /N ) - 1 Ed Figure 4. Eccentricity of slab force Where M 0Ed is the first order moment, which should be taken as M slab +N Ed (e+w 0 ) noted above, the value of N Ed should be based on the actual slab force calculated allowing for shear lag and the crushing resistance A c f cd should be based on the whole gross width of slab ignoring shear lag. (Note that f cd should be calculated in accordance with EN rather than EN ). The effective creep ratio j ef is determined in accordance with clause 5.8.4(2) of EN A value of zero implies all the slab moment is due to non-quasipermanent loading. Values of the magnification factor for a range of n values are given in Figure 5, for a 250mm thick slab spanning 3.0 and 4.0m. It will be noted that the curves for 4m spacing have been curtailed at n 0.6; beyond this point the curve changes gradient according to clause because the parameter k 2 reaches a limiting value of 0.2. Clearly there would be no step change in reality and calculations beyond this value of n could either be conservatively performed to clause or nonlinear analysis used. The situation where the main beam spacing is not large relative to the cross girder spacing and the behaviour is partially plate-like, rather than solely column-like, could be accommodated in the process of determining N B by substituting the buckling load for a plate, N cr,p of side B equal to the main girder spacing and length L between cross girders. k N cr,p s p 2 Et 3 k s p 2 EI 12(1-v 2 )B (1-v 2 )B 2 N Ed N B M slab e w 0 EI is the design value of the axial force in the slab, coexisting with M Ed is the buckling load, given by N B p 2 EI/L 2 is the moment in the slab due to transverse loading is the eccentricity of the axial force relative to the centroid of the slab is the geometric imperfection in the slab ( slab span/400, based on the value of θ o 1/200 in EN , clause 105 and the UK NA) is the nominal stiffness, given by clause Second order effects may be ignored if they are less than 10% of the first order effects, according to clause 5.8.2(6). The calculation of stiffness EI according to clause depends on a number of parameters, including the area of reinforcement, the effective creep ratio j ef and the relative axial force n. The relative axial force is a significant parameter: its value is defined as N Ed /A c f cd. As in which k s (B/L + L/B) 2 and v 0 for cracked concrete. In this case, EI will not be constant in the two directions, even if the reinforcement provision is the same, because the transverse direction typically has zero compressive force. A conservative approach would be to take the smaller value of the cracked stiffness derived in accordance with clause in either the longitudinal or transverse direction. This is, however, likely to be overly conservative since, for normal slab support aspect ratios, the behaviour is dominated by the stiffness in the longitudinal direction, so the slightly unconservative approach of basing EI solely on the longitudinal direction will be acceptable. In addition, the moment amplification method itself becomes more conservative the more plate-like the behaviour becomes as, for plate-like behaviour, the elastic critical buckling force is not an upper bound on strength, whereas it is for column-like behaviour. The benefit of using the plate critical force will be small but will increase as the ratio B/L reduces. Utilising this approach is likely only to be of benefit in the Structures 95

4 (a) (b) Figure 5. Magnification factor, to allow for second order effects (250 mm slab, C40/50), (a) H20 bars at 150 mm centres, top and bottom, (b) H16 bars at 150 mm centres, top and bottom 96

5 130 Design of cross-girders and slabs in ladder deck bridges situation where a small overstress was produced using the column buckling approach. An example of the calculation of the magnifying factor, for both simple strut buckling and for plate panel buckling is given below. Assume a slab thickness 250mm, cross girder spacing 3500mm and main girder spacing 12000mm. The slab concrete is C40/50 and the reinforcement is H16 bars at 150mm centres (take distance between top and bottom bars 160mm, symmetrically placed about h the slab centroid). Take j ef 0 and 0.20 and f cd a cc f ck / g m 22.7 MPa with a cc 0.85 and g m 1.5. Properties per metre width of slab I c mm 4 I s mm 4 Material properties E s MPa E cd E cm /g ce / MPa The effective stiffness is given by clause of EN as: EI K c E cd I c + K s E s I s K s 1 K 1 K 2 f /20 ck 40/ h l 3500/ K c k k x (1+j ef ) (1+0) EI x x x x x 17.2 x x Nmm 2 The design value of axial load per metre width is: N Ed ha c f cd 0.2 x x 22.7 x kn The critical force per metre width is: N B p 2 EI/L 2 p 2 x 6.50 x / x kn Conservatively assume that the distribution of the first order moments is parabolic, this gives a b value of 1.0. The moment magnification factor is: b 1 m (N B /N Ed )-1 (5237/1133)-1 Alternatively, N B can be replaced by N cr,p k s (B/L+L/B) 2 (3500/ /3500) v 0 for cracked concrete. k x p 2 x 6.50 x N cr,p s p 2 EI x (1-v 2 )B kn And then 1 m (6166/1133)-1 This reduces the moment magnification factor. Destabilizing effects on transverse cross girders When the presence of cross-girders is needed to prevent buckling of the slab in compression, they must be designed for stiffness and strength so that they do in fact restrict the buckling length of the slab to the distance between cross-girders. This is achieved by designing them as transverse stiffeners to EN clause 9.2. The stiffness of the composite transverse girder may conservatively be based on its fully cracked section properties but it will usually be adequate to use uncracked properties for the slab, as the permanent cross girder moment will usually be sagging. This sagging moment will be increased by the destabilising effect of the slab if the cross girder being checked has a net downward deflection compared to the adjacent girders. If an adjacent girder is loaded, the destabilising effect would tend to push the cross girder up and relieve the sagging moment (and does not therefore need to be considered in verifying the strength of the cross girder). The Designers Guide to EN gives a method of calculation to combine the destabilising effect of the deck slab and the transverse moments acting on the transverse members in the strength check. Stiffness requirement Based on the requirement in EN , the stiffness requirement may be expressed as: d +w B Ed < 300 where w Ed d d w 0 where I st w 0 s m is the deflection of the cross girder under transverse load is the extra deflection arising due to the compression in the slab and, in the absence of transverse axial force in the cross girder, is given by: EI st s m B 4 4 p w 0 w 0 + w Ed -1-1 is the second moment of area of the composite section is the initial imperfection which, according to EN , Annex D1.6(5), for one cross girder level relative to the adjacent girders, is 2L/400 L/200 is the destabilizing effect of the load in the slab, which can conservatively be taken, for the case of cross girders, as: 2N Ed 2s c bt slab 2s c t s slab m BL BL L Structures 97

6 In which s c is the mean longitudinal stress in the slab. This value can either be determined according to the actual design effects or to the value if resistance is fully mobilised. Strength requirement The destabilising load on the cross girder depends on longitudinal imperfection and its peak value (at the centre of the cross girder) is (w 0 d)s m. Assuming that this is a sinusoidal imperfection, this slab applies an additional moment on the cross girder of: Acknowledgement The authors thank Mabey Bridge for the supply of the photograph in Figure 1. References 1. J.Schlaich and H Scheef (1982), Concrete box girder bridges, Structural engineering Documents 1e, IABSE, Switzerland. 2. PD (2008), Recommendations for the design of structures to BS EN :2005, BSI, England 3. C.R.Hendy and C.J.Murphy (2007), Designers Guide to EN Eurocode 3: Part 2: Design of Steel Structures Part 2, Steel Bridges, Thomas Telford, UK. (w 0 d) s m B2 p 2 This additional moment must be included in the ULS verification of the cross girder. 98

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