Structural modelling for analysis 1 Structural modelling and basic assumptions

The basic requirement of 3-2/clause 5.1.1(1) for analysis is that it should realistically model the true behaviour. The Note to 3-2/clause 5.1.1(4) acknowledges that reference may be necessary to other parts of EN 1993 to achieve this. Where stiffness in analysis might be affected by shear lag or plate buckling effects, reference needs to be made to 3-1-5/clause 2.2. This gives rules for when and how to take these effects into account. For steel-only bridges, it will generally only be necessary to consider these effects for box girders with an orthotropic deck or other steel beams with a common steel top flange. For steel and concrete composite members, slightly different rules for shear lag apply for concrete flanges. These are given in EN 1994-2.

The Note to 3-2/clause 5.1.1(4) also refers to EN 1993-1-11 for the design of cable- supported structures. Specific guidance on modelling joints, ground–structure interaction and cable-supported structures is given in sections 5.1.2 to 5.1.4 respectively below.

Shear lag

In wide flanges, in-plane shear flexibility leads to a non-uniform distribution of bending stress across the flange width. This effect is known as shear lag and is illustrated in Fig. 5.1-1 for a simply supported box girder with knife edge load applied at midspan. The elastic distribution of shear stress across the box top flange leads to a transverse strip of flange deforming as shown. The free ends of the box top flange therefore adopt a similar deflected shape arising from this shear deformation together with axial shortening from the compressive bending stresses. The distorted box top flange is shorter along the webs

3-2/clause 5.1.1(1) 3-2/clause 5.1.1(4)

than along its centre so the axial compressive stress must therefore be greater at the webs than in the middle of the flange. The stress in the flange adjacent to the web is consequently found to be greater than expected from analysis with gross cross-sections, while the stress in the flange remote from the web is lower than expected. Similar results are produced with continuous beams with the maximum in-plane shear lag displacements occurring at points of contraflexure. This shear lag also leads to a loss of stiffness of a section in bending, which can be important in determining realistic distributions of moments in analysis.

The determination of the actual distribution of stress is a complex problem which depends on the loading configuration, the stiffening to the flanges and any plasticity occurring. The stress distribution at the serviceability limit state can be modelled using elastic finite- element analysis with shell elements. At the ultimate limit state, plasticity usually occurs and non-linear finite-element analysis is required to produce an accurate representation of the stress distribution.

The Eurocodes account for both the loss of stiffness and localized increase in flange stresses by the use of an effective width of flange which is less than the actual available flange width. The effective flange width concept is artificial but, when used with engineering bending theory, leads to uniform stresses across the whole reduced flange width that are equivalent to the peak values adjacent to the webs in the true situation. It follows from the above that if finite-element modelling of flanges is performed with sufficient detail for the flange elements, shear lag will be taken into account and the additional use of an effective flange in accordance with this clause would be unnecessary.

For global analysis, 3-1-5/clause 2.2(3) allows the effective width of flange acting on each side of a web to be taken as the lower of the full available width and L/8 where L is the span

3-1-5/clause 2.2(3)

View on top flange

Shear deformation of strip Net deformation of free end Axial stress distribution

Elastic shear stress distribution across strip

or twice the length of a cantilever. This width may be taken as constant throughout the entire span. Alternatively, the values for serviceability limit state (SLS) cross-section design from 3-1-5/clause 3 could be used. These are discussed later in section 6.2.2.3, together with worked examples.

Plate buckling

Slender plates (Class 4 according to 3-1-1/clause 5.5) also exibit a loss of stiffness under load. The stiffness of perfectly flat plates suddenly reduces when the elastic critical buckling load is reached. In ‘real’ plates that have imperfections, there is an immediate reduction in stiffness from that expected from the gross plate area because of the growth of geometric imperfec- tions under load. This stiffness continues to reduce with increasing load. This arises because non-uniform stress develops across the width of the plate as shown in Fig. 5.1-2. The non-uniform stress arises because the development of the buckle along the centre of the plate leads to a greater developed length of the plate along its centreline than along its edges. Thus the shortening due to membrane stress, and hence the membrane stress itself, is less along the centreline of the plate.

This loss of stiffness must be considered in the global analysis, where significant, and can also be represented by an effective width of plate. The reduction in ultimate strength (caused both by the non-uniform axial membrane stress and the out-of-plane bending stresses due to the deflections in Fig. 5.1-2) is also accounted for by using effective widths for the plate panels, but these widths are smaller than those appropriate for stiffness in global analysis; the reduction in strength due to plate buckling is greater than the reduction in stiffness. The same effective widths as used for strength calculation can however be used for global analysis (3-1-5/clause 2.2(4)) or more accurate effective widths for global analysis can be determined from 3-1-5/Annex E. Alternatively, 3-1-5/clause 2.2(5) allows the effects of plate buckling to be ignored in global analysis where the effective areas of compression elements at the ultimate limit state are greater than
limtimes the gross area.
limis a limiting

value of the ultimate limit state (ULS) reduction factor for plate buckling discussed in section 6.2.2.5 of this guide and is a nationally determined parameter whose recommended value is 0.5. This value will ensure that plate buckling effects rarely need to be considered in global analysis.

A similar loss of stiffness occurs from bowing of any longitudinal stiffeners present and further modifications to the effective areas are used to model this effect also. The rules in 3-1-5/clause 4.3 are used to do this and these are discussed later in sections 6.2.2.5 and 6.2.2.6 of this guide where strength is also discussed.

Shear lag combined with plate buckling effects

Since the concept of effective widths for both shear lag and plate buckling can be confusing, EN 1993-1-5 distinguishes between effective widths for shear lag and for plate buckling and

3-1-5/clause 2.2(4) 3-1-5/clause 2.2(5)

for the combined effective widths using the following notation: effectivep– effective width for plate buckling

effectives– effective width for shear lag

effective – effective width for plate buckling and shear lag.

The combination of the two effects is achieved by first calculating the effectivepwidth for plate buckling and then considering only that part of the area which is in the effectiveswidth for shear lag.

5.1.2. Joint modelling

3-2/clause 5.1.2(1)refers to both EN 1993-1-1 and EN 1993-1-8. 3-1-1/clause 5.1.2(1) and 3-1- 1/clause 5.1.2(2)state that it is generally permissible to ignore detailed considerations of joint stiffness in analysis of bridges, with joints treated as either pinned or rigid as appropriate. One exception to this is where ‘semi-continuous’ joints, as defined in EN 1993-1-8, are used. These are joints which are neither rigid nor pinned but have a certain amount of flexibility when resisting load. An example of such a joint might include a connection made via bolted end plates, where flexure of the end plates gives joint flexibility but the joint still is capable of carry- ing moment. It is recommended that semi-continuous joints are not used for bridges so that fatigue can be assessed using the detail categories in EN 1993-1-9. This is the reason for the Note to 3-2/clause 5.1.2(5). Semi-continuous joints may still, in some cases, be unavoidable, such as end plate connections in some U-frame bridges. In this latter specific case, the flexibility would have to be considered in deriving the restraint provided to the compression flange by the U-frame. EN 1993-1-8 provides methods of determining the joint stiffness.

Another apparent exception to the above rule, where joint behaviour must be considered, is in the consideration of bolt slip discussed in section 5.2.1 of this guide.

5.1.3. Ground–structure interaction

3-1-1/clause 5.1.3(1)refers to ‘deformation characteristics of supports’, so the stiffness of the bearings, piers, abutments and ground have to be taken into account in analysis. This also includes consideration of stiffness in determining effective lengths for buckling or in calculating buckling resistances directly from the analysis. For further guidance on the latter, see section 5.2 of this guide.

5.1.4. Cable-supported bridges (additional sub-section)

A detailed treatment of the design of cable-supported bridges is outside the scope of this guide but a few salient points are noted here. The general guidance in sections 5.2 to 5.4 of this guide are also relevant.

5.1.4.1. Analysis

EN 1993-1-11 covers the design of cable-supported bridges. The analysis of cable-supported bridges needs to consider non-linearities arising from second-order effects under axial load, from large deflections altering the overall bridge geometry and from the sag of cables. The latter may be covered by a simple correction to the ‘E ’ value of the cables as discussed in section 3.4 of this guide. Where there are significant non-linearities, the design at the ultimate limit state needs to be performed by applying factored loads to the analysis model in the same way as discussed in section 5.2 for second-order effects.

In general, the analysis should consider the build-up of load effects throughout the construction sequence. An analysis should be performed using characteristic values of actions to determine an intended design profile. This allows the deformed shape to be monitored on site and cables adjusted to achieve this profile if necessary. An important distinction must therefore be made between bridges where the cables are to be adjusted on site to achieve the assumed design profile of the bridge and those where no adjustment is to be made as discussed below.

3-2/clause 5.1.2(1) 3-1-1/clause 5.1.2(1) 3-1-1/clause 5.1.2(2) 3-2/clause 5.1.2(5) 3-1-1/clause 5.1.3(1)

Design for in-service condition

The intention in 3-1-11/clause 5.3 is that if cables are to be adjusted to achieve the assumed design deflection profile, then the self-weight and cable preloads are combined into a single permanent action, ‘Gþ P’, whose application to the structure corresponds to the intended permanent profile of the bridge. For ultimate limit states, this single entity is then multiplied by either the favourable or unfavourable load factor Gas appropriate to determine action

effects. It is however essential that cables are adjusted on site if necessary to achieve the intended design profile if the actions are to be combined in this way. This is because the combined effects of dead load and cable preload (e.g. bending moments) are formed from the difference between two large opposing actions whose net effect will typically be designed to be as near to zero throughout the bridge as possible. A load factor applied to a near zero effect will obviously still give a near zero effect at ULS. If there is no control on deflections, by adjusting cables, the real (as opposed to calculated) difference between these two large numbers can become very large if, for example, the bridge’s self-weight is greater than the characteristic value assumed in the design.

There are however some problems with this approach in certain structures and it additionally contradicts the general format for effects of actions in EN 1990 as discussed in section 2.3 of this guide. In some situations, the required deflection control will be automatically achieved through normal site controls on profile. For example, it would not be possible to achieve an unintended differential between ‘G’ and ‘P’ in a large cable- stayed bridge with a flexible deck because the deflections during construction would become excessive and the cables would have to be adjusted. Application of separate favour- able and unfavourable partial factors to self-weight and prestress in this situation would be unrealistic as the deflections and stresses found from such an analysis could not actually occur in practice due to site profile controls.

If the deck was however very stiff compared to the cables, such as might occur in a short- span cable-stayed bridge with a stiff concrete deck, an unintended differential between ‘G’ and ‘P’ might not be noticed as the difference from predicted deflections would be less measurable. The same would apply to a bridge with external prestressing, where it is unlikely that cables would be adjusted in any case. In both the latter cases, combination of ‘Gþ P’ into one entity with a common load factor is potentially unsafe.

The authors would prefer that the presumption should always initially be for separate combination unless there is a demonstrable reason to do otherwise. A cautionary note is therefore given as follows. For some structural types, combination of P and G into a single action (Gþ P) is not appropriate because normal site monitoring of deflections and adjustment of cables will be insufficient to guarantee that there is no significant unintended imbalance between G and P. This will be the case for structures where the deflections from an unintended out of balance of P and G would be small, where the bridge deck is stiff in flexure compared to the support offered by the cables or where the profile of the structure is unaffected by the prestressing force. Such structures could include cable-stayed bridges with stiff decks, externally post-tensioned bridges and guyed towers and masts. In such cases, the actions P and G should have partial factors applied to them separately. In all cases, the method of applying partial factors should be agreed with the appropriate Oversee- ing Authority.

3-1-11/clause 2.3.5(3) does acknowledge that if cable adjustment is not intended, the effects of possible variations in prestress force should be considered. No numerical guidance is however provided so the above approach is recommended.

Design during construction

Further to the discussion above, it would also be necessary to treat self-weight and cable preloads separately with separate favourable and unfavourable load factors to determine the possible differential effects for ultimate limit states for stages of construction before cables have been adjusted or where it is not possible to detect the differential effect. This is the basis of 3-1-11/clause 5.2(3), which requires the partial factor Pfor prestressing to be

defined for this situation in the National Annex.

3-1-11/clause 5.3

3-1-11/clause 5.2(3)

Cable replacement

Cables should normally be replaceable and the design should consider both a controlled replacement and an accidental removal. The load combination for controlled replacement can be defined in the National Annex to EN 1993-1-11 via clause 2.3.6. Often, these conditions will be project-specific. The load combination for accidental removal should be considered in an accidental combination but the National Annex may again define the relevant loading.

The dynamic effect of a sudden accidental cable removal should be considered. 3-1-11/ clause 2.3.6(2)suggests this can be done by calculating the design effects for the structure with the cable in place, Ed1, and with the cable removed, Ed2, and calculating a dynamic

design effect to add to Ed1given by:

E_d¼ kEd2
Ed1 3-1-11/(2.4)

EN 1993-1-11 sets the value of k at 1.5.

This formula produces incorrect results, particularly for cables remote from the removed cable where there are no effects from the cable removal so that E_d1¼ Ed2. In this case, the

formula still predicts that the additional dynamic force to consider is 0.5E_d2. It is suggested here that a more appropriate formula is:

Ed¼ k Eð d2
Ed1Þ (D5.1-1)

This ensures the system is designed for additional effects equal to the change in static internal effects caused by cable removal, multiplied by a dynamic factor. k¼ 2:0 corresponds to zero damping and k¼ 1:8 would be a reasonable value for most structures to make allow- ance for some damping. k¼ 1:5 would probably be too optimistic with this formulation.