Structural analysis
5.3 Idealization of the structure
5.3.1 Structural models for overall analysis
2-1-1/clause 5.3.1(1)Plists typical elements comprising a structure and states that rules are given in EC2 to cover the design of these various elements. While detailing rules are provided by element type in section 9 of EN 1992-2, rules for resistances are generally presented by resistance type rather than by element type in section 6. For example, 2-2/clause 6.1 covers the design of sections in general to combinations of bending and axial force. These rules apply equally to beams, slabs and columns.
2-1-1/clause 5.3.1(3), (4), (5) and (7) provide definitions of beams, deep beams, slabs and columns. The definitions given are self-explanatory and are often useful in defining the detail-ing and analysis requirements for the particular element. For example, the distinction made between a beam and deep beam is useful in determining the appropriate verification method and detailing rules. A beam can be checked for bending, shear and torsion using 2-2/clause 6.1 to 6.3, while deep beams are more appropriately treated using the strut-and-tie rules of 2-2/clause 6.5. No distinction is, however, made between beams with axial force and columns in EC2 in cross-section resistance design. A distinction, however, remains necessary when selecting the most appropriate detailing rules from section 9. Sometimes it may be appropriate to treat parts of a beam, such as a flange in a box girder, as a wall or column for detailing purposes, as discussed in section 9.5 of this guide.
5.3.2. Geometric data
5.3.2.1. Effective width of flanges (all limit states)
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. The stress in the flange adjacent to the web is consequently found to be greater than expected from section analysis with gross cross-sections, while the stress in the flange remote from the web is lower than expected. This shear lag also leads to an apparent loss of stiffness of a section in bending.
The determination of the actual distribution of stress is a complex problem which can, in theory, be determined by finite element analysis (with appropriate choice of elements) if realistic behaviour of reinforcement and concrete can be modelled. For un-cracked concrete, the behaviour is relatively simple but becomes considerably more complex with cracking of the concrete and yielding of the longitudinal reinforcement, which both help to redistribute the stress across the cross-section. The ability of the transverse reinforcement to distribute the forces is also relevant.
2-1-1/clause 5.3.2.1(1)Paccounts 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 2-1-1/clause
5.3.1(1)P
2-1-1/clause 5.3.1(3), (4), (5) and (7)
2-1-1/clause 5.3.2.1(1)P
Buckled shape
a
L
a Kink imperfection
Sinusoidal imperfection
θl
Fig. 5.2-5. Imperfection for in-plane buckling with ‘rigid’ foundations
peak values adjacent to the webs in the ‘true’ situation. It follows from the above that if finite element modelling of flanges is performed using appropriate analysis elements, shear lag will be taken into account automatically (the accuracy depending on the material properties specified in analysis as discussed above) so an effective flange need not be used.
The rules for effective width may be used for flanges in members other than just ‘T’ beams as suggested by 2-1-1/clause 5.3.2.1(1)P; box girders provide an obvious addition. The physical flange width is unlikely to be reduced for many typical bridges, such as precast beam and slab decks where the beams are placed close together. The effect of shear lag is greatest in locations of high shear where the force in the flanges is changing rapidly. Conse-quently, effective widths at pier sections will be smaller than those for the span regions.
2-1-1/clause 5.3.2.1(1)P notes that, in addition to the considerations discussed above, effective width is a function of type of loading and span (which affect the distribution of shear along the beam). These are characterized by the distance between points of zero bending moment. 2-1-1/clause 5.3.2.1(2) and (3), together with 2-1-1/Figs 5.2 and 5.3 (not reproduced), allow effective widths to be calculated as a function of the actual flange width and the distance, l0, between points of zero bending moment in the main beam adjacent to the location considered. This length actually depends on the load case being considered and the approximations given are intended to save the designer from having to determine actual values of l0 for each load case. The total effective width acting with a web, beff, is given as follows:
beff¼X
beff;iþ bw b 2-1-1/(5.7)
beff;i¼ 0:2biþ 0:1l0 bi and 0:2l0 2-1-1/(5.7a)
where b is the total flange width available for the particular web and biis the width available to one side of the web, measured from its face.
2-1-1/Expression (5.7) and 2-1-1/Expression (5.7a) differ from similar ones in EN 1994-2, as a minimum of 20% of the actual flange width may always be taken to act each side of the web where the span is short. The reference in 2-1-1/clause 5.3.2.1(3) to ‘T’ or ‘L’ beams is only intended to describe webs with flanges to either one or both sides of a web. It is not intended to limit use to main beams of these shapes.
The limitations on span length ratios for use of 2-1-1/Fig. 5.2, given in the Note to 2-1-1/
clause 5.3.2.1(2), are made so that the bending moment distribution within a span conforms with the assumption that spans have hog moments at supports and sag moments in the span.
The simple rules do not cater for other cases, such as entire spans that are permanently hogging. If spans or moment distributions do not comply with the above, then the distance between points of zero bending moment, l0, should be calculated for the actual moment distribution. This is iterative because analysis will have to be done first with cross-section properties based on the full flange width to determine the likely distribution of moment.
The same effective width for shear lag applies to both SLS and ULS. This is unlike previous design to BS 5400, where it was permissible to neglect shear lag at ULS on the basis that the effects of concrete cracking and reinforcement yielding discussed above allow stresses to redistribute across a flange. EC2, however, bases effective widths at ULS on widths approximating more closely to the elastic values, thus avoiding the complexity of providing rules to calculate effective widths which allow for these redistribution effects.
This differs from the approach in EN 1993-1-5 for steel flanges, where consideration of plasticity is allowed at ULS and greater effective widths can be achieved. The practical significance of using the same effective width at ULS and SLS will not usually be great for concrete bridges and often the full width will be available.
For global analysis, 2-1-1/clause 5.3.2.1(4) permits section properties to be based on the mid-span value throughout that entire span. Quite often this will lead to the full flange width being used. It can, however, be advantageous to use the actual distribution of effective widths near to supports and at mid-span in continuous beams to reduce the stiffness at supports and hence also the support hogging moment. A more accurate prediction of stiffness, and hence effective width distribution throughout the span, may also be necessary where prediction of
2-1-1/clause 5.3.2.1(2) and (3)
2-1-1/clause 5.3.2.1(4)
deflections is important to the construction process, such as in balanced cantilever construc-tion. For section design, the actual effective width at the location being checked must always be used.
Where it is necessary to determine a more realistic distribution of longitudinal stress across the width of the flange, as may be required in a check of combined local and global effects in a flange, the formula in EN 1993-1-5 clause 3.2.2 could be used to estimate stresses. This is explicitly permitted for slabs in steel–concrete composite construction in EN 1994-2. A typical location where this might be necessary would be at a transverse diaphragm between main beams at a support where the deck slab is in tension under global bending and also subjected to a local hogging moment from wheel loads. The use of the formula in EN 1993-1-5 can be beneficial here as often the greatest local effects in a slab occur in the middle of the slab between webs where the global longitudinal stresses are lowest.
The effective flange widths in 2-1-1/clause 5.3.2.1 do not apply to the introduction of axial loads, such as those from prestressing or anchorages in cable-stayed bridges. The phenom-enon of shear lag still applies to local concentrated axial loads, but stresses spread out across the section at a rate un-connected to the bending moment profile. Where concentrated axial loads are applied to a section, separate assessment must be made of the area over which this force acts at each cross-section through the span – see 2-1-1/clause 8.10.3.
Worked example 5.3-1: Effective flange width for a box girder
A box girder bridge has the span layout and cross-section shown in Fig. 5.3-1. Determine the effective width of top flange acting with an outer web at mid-span and supports for the main span.
Considering mid-span first:
From 2-1-1/Fig. 5.2, l0¼ 0:7l2¼ 0:7 40 000 ¼ 28 000 mm.
From 2-1-1/(5.7a), the cantilever portion has effective width given by:
beff;i¼ 0:2biþ 0:1l0¼ 0:2 4000 þ 0:1 28 000 ¼ 3600 mm < 0:2l0¼ 5600 mm Similarly, the internal flange associated with the web has effective width:
beff;i¼ 0:2biþ 0:1l0¼ 0:2 5700
2 þ 0:1 28 000 ¼ 3370 mm < 0:2l0¼ 5600 mm but this is greater than the available width of 5700=2¼ 2850 mm so the effective width is taken as 2850 mm.
Finally, from 2-1-1/(5.7) the total width of flange acting with an outer web is:
beff¼X
beff;iþ bw¼ 3600 þ 2850 þ 300 ¼ 6750 mm This is almost the whole available width.
Considering the supports:
From 2-1-1/Fig. 5.2, l0¼ 0:15ðl1þ l2Þ ¼ 0:15 ð30 000 þ 40 000Þ ¼ 10 500 mm.
From 2-1-1/(5.7a), the cantilever portion has effective width given by:
beff;i¼ 0:2biþ 0:1l0¼ 0:2 4000 þ 0:1 10 500 ¼ 1850 mm < 0:2l0¼ 2100 mm Similarly, the internal flange associated with the web has effective width:
beff;i¼ 0:2biþ 0:1l0¼ 0:2 5700
2 þ 0:1 10 500 ¼ 1620 mm < 0:2l0¼ 2100 mm Finally, from 2-1-1/(5.7) the total width of flange acting with an outer web is:
beff;i¼X
beff;iþ bw¼ 1850 þ 1620 þ 300 ¼ 3770 mm
This represents only 53% of the available width of 7150 mm, illustrating that shear lag can be significant at supports for wide flanges and short spans.
5.3.2.2. Effective spans of beams and slabs
2-2/clause 5.3.2.2 gives requirements for the effective span, leff, of beams and slabs. Examples are given in 2-1-1/Fig. 5.4. The main cases of interest in bridge design are:
(i) Beams monolithic with supports
Where a horizontal member is built in monolithically to another vertical member, the effec-tive span is taken to a point within the vertical member which is the minimum of half the vertical member thickness, t, and half the horizontal member depth, h, from the edge of the vertical member according to Figs 5.3-2 (a) and (b). The limitation to h/2 is intended to keep the centre of reaction in a realistic position where the supporting member is very thick. Cases (a) and (b) could apply, for example, to integral bridges and to the transverse cantilever design of box girder bridges respectively. In this situation, 2-1-1/clause 5.3.2.2(3)and its Note state that the design moment should be taken as that at the face of the support but not less than 65% of the actual maximum end moment.
(ii) Beams on bearings
2-1-1/clause 5.3.2.2(2)permits rotational restraint from bearings to generally be neglected.
This is the usual assumption made for most mechanical bearings but the designer needs to exercise judgement. Clearly, for example, the torsional restraint provided by linear rocker bearings should not be ignored.
The effective span is measured between centres of bearings as shown in Fig. 5.3-2(c). The moment so obtained may, however, be rounded off over each bearing in accordance with 2-2/
clause 5.3.2.2(104). This has the effect of reducing the moment by FEd;supt/8, where t can be defined in the National Annex. The recommended value in the Note to 2-2/clause 5.3.2.2(104) is the ‘breadth of the bearing’. This is intended to be the dimension in the direction of the span of the bearing contact surface with the deck. FEd;sup is the support reaction coexisting with the moment case considered.
The definition of t above is intended to be used for a rectangular contact area. For a bearing with circular contact area, using the diameter for t is slightly unconservative and, for consistency, the moment should be reduced by FEd;supD/3, based on the centroid of half of a circular area, where D is the contact diameter. The contact dimensions should be taken as the lesser of the physical top plate size or the dimension obtained by spreading from the edges of the stiff part of the bearing through the top plate. A spread of 1 : 1 is suggested, in keeping with that given in EN 1993-1-5 clause 3.2.3, although a greater spread at 608 to the vertical was allowed by BS 5400 Part 3.7
The wording of 2-2/clause 5.3.2.2(104) differs from that in 2-1-1/clause 5.3.2.2(4) in order to clarify that the rounding can only be made if the analysis assumes point support. The wording in EC2-1-1, which starts with ‘Regardless of method of analysis . . .’, was considered undesirable as the support width could be included in a more detailed analysis model.
2-1-1/clause 5.3.2.2(3)
2-1-1/clause 5.3.2.2(2)
2-2/clause 5.3.2.2(104)
4000 5700 5700 4000
300 300 300
30 m 40 m 30 m
Fig. 5.3-1. Box girder for Worked example 5.3-1