A box girder bridge has the span layout and cross-section shown in Fig. 6.2-7. The top flange has trough stiffeners such that Asl=b0t¼ 0:5. Determine the effective width of
top flange acting with each web at mid-span and over the supports for the main span at both SLS and ULS.
10 000
4000 4000
L2 = 80 m
L1 = 60 m L3 = 60 m
Fig. 6.2-7. Bridge deck for Worked Example 6.2-1 Considering mid-span first:
SLS
From 3-1-5/Fig. 3.1, Le¼ 0:7L2¼ 0:7 80 000 ¼ 56 000 mm
From 3-1-5/Table 3.1, the cantilever portion has effective width as follows: 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þAsl b0t s ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 0:5¼ 1:225 k¼0b0 Le ¼1:225 4000 56 000 ¼ 0:0875 ¼ 1¼ 1 1þ 6:4k2 ¼ 1 1þ 6:4 0:08752¼ 0:953
From expression 3-1-5/(3.1): beff¼ b0¼ 0:953 4000 ¼ 3813 mm
From 3-1-5/Table 3.1, the internal portion has effective width as follows: 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þAsl b0t s ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 0:5¼ 1:225 k¼0b0 Le ¼1:225 5000 56 000 ¼ 0:1094 ¼ 1¼ 1 1þ 6:4k2 ¼ 1 1þ 6:4 0:10942¼ 0:929
From expression 3-1-5/(3.1): beff¼ b0¼ 0:929 5000 ¼ 4645 mm
Hence the total width attached to each web at SLS¼ 3813 þ 4645 ¼ 8458 mm ULS
For the cantilever, from expression 3-1-5/(3.5): beff¼ k b0¼ 0:9530:0875 4000 ¼
3983 mm
For the inner part, from expression 3-1-5/(3.5): beff¼ k b0¼ 0:9290:1094 5000 ¼
4959 mm
6.2.2.3.2. Dispersion of concentrated loads (additional sub-section)
The effective flange width according to expression 3-1-5/(3.1) does not apply to the calculation of stress dispersal from concentrated axial forces. Shear lag still affects the rate of dispersal of local concentrated loads, but this rate is not connected to the bending moment profile. Consequently, where concentrated axial loads are applied to a section, such as in a cable-stayed bridge, separate calculation must be made of the effective area over which this force acts at each cross-section throughout the span.
3-1-5/clause 3.2.3(1) covers the dispersal of stress from concentrated loads in its expression (3.2). It is mainly intended for determining the distribution of stress in webs subjected to concentrated patch loads applied locally through a flange (e.g. local wheel loads or reactions during a bridge launch), but could be used to determine the dispersal of stress from longitudinal axial forces, such as from prestressing. For patch loading, the
3-1-5/clause 3.2.3(1)
Considering an internal support: SLS
From 3-1-5/Fig. 3.1, Le¼ 0:25ðL1þ L2Þ ¼ 0:25ð60 000 þ 80 000Þ ¼ 35 000 mm
From 3-1-5/Table 3.1, the cantilever portion has effective width as follows: 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þAsl b0t s ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 0:5¼ 1:225 k¼0b0 Le ¼1:225 4000 35 000 ¼ 0:140 ¼ 2 ¼ 1 1þ 6:0 k 1 2500k þ 1:6k2 ¼ 1 1þ 6:0 0:140 1 2500 0:140 þ 1:6 0:1402 ¼ 0:539
From expression 3-1-5/(3.1): beff¼ b0 ¼ 0:539 4000 ¼ 2157 mm
From 3-1-5/Table 3.1, the internal portion has effective width as follows: 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þAsl b0t s ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 0:5¼ 1:225 k¼0b0 Le ¼ 1:225 5000 35 000 ¼ 0:1750 ¼ 2 ¼ 1 1þ 6:0 k 1 2500k þ 1:6k2 ¼ 1 1þ 6:0 0:175 1 2500 0:175 þ 1:6 0:1752 ¼ 0:480
From expression 3-1-5/(3.1): beff¼ b0 ¼ 0:480 5000 ¼ 2398 mm
Hence the total width attached to each web at SLS¼ 2157 þ 2398 ¼ 4555 mm ULS
For the cantilever, from expression 3-1-5/(3.5): beff¼ k b0¼ 0:5390:140 4000 ¼
3668 mm
For the inner part, from expression 3-1-5/(3.5): beff¼ k b0¼ 0:4800:1750 5000 ¼
4397 mm
spread of load through a flange from expression (3.2) is at 1H : 1V, which is less rapid than assumed in previous UK practice. The calculated spread width, beff, is not the full extent of
spread, but is an equivalent width such that the mean stress calculated with this width equates to the peak elastic stress in the ‘real’ distribution. Since expression 3-1-5/(3.2) represents an elastic distribution of stress, it may be used for fatigue calculations as well as for ULS ones.
For an unstiffened flange, with a load applied through the flange, the spread width simplifies to: beff¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 eþ ðz=0:636Þ2 q (D6.2-1) and the design transverse stress at depth z below the loaded flange is:
z;Ed¼FEd befft
(D6.2-2) where seis the loaded width at the top of the web under the loaded flange and t is the web thickness. The angle of spread through an unstiffened web tends to a constant value of 0.785H : 1V when remote from the loaded area (which is approximately at a distance equal to twice the loaded width at flange level) as shown in Fig. 6.2-8. However, the initial stress trajectory beneath the flange is vertical, so there is no simple idealized spread angle that can be used throughout as was previous UK practice.
Care is needed when using expression 3-1-5/(3.2) for stiffened plates where the stiffener spacing is large compared to the loaded width, as the formula is derived assuming the stiffeners to be closely spaced and smeared. The Note to 3-1-5/clause 3.2.3(1) consequently limits its use to situations where sst=se 0:5, where sstis the stiffener spacing. Outside this
limit, equation (D6.2-2) above for unstiffened plates should be used.
6.2.2.4. Effective properties of cross-sections with Class 3 webs and Class 1 or 2 flanges
The method given in 3-1-1/clause 6.2.2.4(1) is often referred to as ‘the hole in the web’ method. In beams subjected to hogging bending, it often happens that the bottom flange is in Class 1 or 2, and the web is in Class 3. The initial effect of local buckling of the web would be a small reduction in the bending resistance of the section. The assumption that a defined depth of web, the ‘hole’, is not effective in bending enables the reduced section to be upgraded from Class 3 to Class 2, and removes the sudden change in the bending resistance that would otherwise occur. The method is analogous to the use of effective areas for Class 4 sections, to allow for local buckling. The Designers’ Guide to EN 1994-27 gives more detail on this method and an example of its use.
It should be noted that if a Class 3 cross-section is treated as an equivalent Class 2 cross- section for section design, it should still be treated as Class 3 when considering the actions to
3-1-1/clause 6.2.2.4(1) se beff Flange 1:1 0.785H:1V z
consider in its design. Indirect actions, such as differential settlement, which may be neglected for true Class 2 sections, should not be ignored for effective Class 2 sections. When indirect actions contain both primary and secondary components, such as differential shrinkage acting on statically indeterminate structures, the primary self- equilibrating stresses could reasonably be neglected, but not the secondary effects.
6.2.2.5. Class 4 members – general and effective section method
6.2.2.5.1. Methods of approach
Class 4 members are those that are unable to attain the full yield stress under the loading considered because of the onset of local buckling. Plate buckling is discussed generally in section 5.1.1. The method for dealing with Class 4 cross-sections is given in EN 1993-1-5. Two methods are presented and 3-2/clause 6.2.2.5(1) requires that one of these methods is followed:
(i) Use of section properties based on effectivepwidths to allow for both plate and stiffener buckling – 3-1-5/clause 4 covers this method.
(ii) Use of section properties based on the gross cross-section but with a reduced allowable stress limit (less than yield) – 3-1-5/clause 10 covers this method.
EN 1993-2 allows the National Annex to choose which method to use, but there are restrictions on the applicability of method (i) given in EN 1993-1-5 in some cases. It is therefore logical to permit both methods to be used. In this guide, the effective section method is discussed in detail under section 6.2.2.5 and the reduced stress method is discussed in section 6.2.2.6, although a brief comparison of the methods is first given below. (i) Effective sections to EN 1993-1-5 clause 4
The first method differs significantly from UK practice to date. This is because the use of effectivepwidths for web and flange elements allows load shedding between all the various elements such that their combined strength is optimally used.
The load shedding implicit in the effective width model of EN 1993-1-5 implies that there is sufficient post-buckling strength and ductility to permit this redistribution. Figure 6.2-9 gives definitions of panel components. Unstiffened plates and sub-panels can maintain their peak resistance for a reasonable strain increase after their maximum resistance is reached, so such load shedding is possible. The post-buckling strength stems from an unstiffened plate panel’s ability for load to concentrate along its longitudinal supported edges after elastic buckling. Stiffened panels undergoing overall buckling generally have less post-buckling strength however, and for short wide panels, buckling is largely column-like where the elastic critical buckling load is an upper bound to the resistance. The effective width method still implicitly assumes there is adequate deformation capacity to shed load to other plate elements. Details of the test results that were used by the EN 1993-1-5 Project Team in the calibration of this method are not known to the authors of this guide. It represents a significant change from previous UK practice.
3-2/clause 6.2.2.5(1) a b b Typical sub-panel Longitudinal stiffeners Direct stress
The above assumptions of post-buckling strength and ductility certainly do not apply where local torsional buckling (sometimes known as tripping) of open stiffeners occurs, as there is insufficient post-buckling strength in such an element with a free edge to maintain its load over any strain increase. The load drops off rapidly when buckling occurs, which can lead to progressive failure. It is therefore essential to prevent torsional buckling when the method of effective sections is used. A method for ensuring its prevention is given in 3-1-5/clause 9.2.1. Torsional buckling is discussed in section 6.9 of this guide.
There are further restrictions on method (i) given in 3-1-5/clause 4.1(1):
(a) The panels should nominally be rectangular and the flanges should be parallel (to within 108). However, it is possible to square off panels based on their largest dimensions to calculate a lower bound on the effective width fraction, , to overcome this limitation. (b) Stiffeners must be provided longitudinally and/or transversely, i.e. not skewed. (c) An unstiffened open hole in a panel should not have diameter exceeding 5% of the panel
width, b. This is because large holes can limit post-buckling strength and ductility of panels. Secondary bending stresses are also set up, particularly around web openings, which should be accounted for. No rules are given as to how heavily a hole would have to be stiffened (both transversely and longitudinally) to permit a relaxation of this limit or how to consider the secondary bending stresses. This is therefore a matter for judgement by individual designers.
(d) Members must be of uniform cross-section. Haunched members with haunch angle less than 108 can be treated as uniform for consistency with (a) above. If flanges are continu- ously curved in elevation, the resulting pressure imposed on the web can be dealt with using 3-1-5/clause 8, but EN 1993 provides no means of considering the interaction with other effects. It is difficult therefore to use the effective section method for beams with continuously curved flanges without some judgement – see the discussion in section 6.10.1.1 of this guide.
(e) The web should be adequate to prevent buckling of the compression flange into the plane of the web. Rules are given in 3-1-5/clause 8 which are discussed and extended in section 6.10 of this guide.
Another restriction not specifically mentioned in EN 1993-1-5 is that the effective section method cannot be used (without modification) where there is a uniform transverse direct stress accompanying the longitudinal stress. The rules and interactions for transverse loading in 3-1-5/clause 6 and 3-1-5/clause 7 may be applied for concentrated loads, but the effect of more uniform transverse stress would need to be evaluated using the method of reduced stresses in 3-1-5/clause 10.
The effective section method may be used where the flange has a greater yield strength than the web, provided that the flange yield stress is not more than a recommended limit of twice that of the web – 3-1-5/clause 4.3(6) refers. The web stresses must then not exceed the yield strength of the web and the effective widths of the web should be determined using the higher flange yield strength.
(ii) Reduced stress limits to EN 1993-1-5 clause 10
Where the conditions above for the use of effective widths are not met, a method based on stress analysis with gross cross-section properties and subsequent plate buckling checks may be used according to 3-1-5/clause 10. This method may always be used as an alternative to the effective width approach, but it takes no account of the beneficial shedding of load from overstressed panels. The method is discussed further in section 6.2.2.6 of this guide.
For greatest structural economy, it is generally better to use 3-1-5/clause 4, although there are some exceptions as discussed in section 6.2.2.6 below.
6.2.2.5.2. Method using effective sections
Effective widths are determined on the basis of the distribution of stresses acting on the individual parts of the cross-section. 3-1-5/clause 4.3(3) and 3-1-5/clause 4.3(4) allow section properties to be developed separately for axial loads and for bending, or
3-1-5/clause 4.1(1) 3-1-5/clause 4.3(6) 3-1-5/clause 4.3(3) 3-1-5/clause 4.3(4)
alternatively they may be based on the overall stress distribution caused by combined axial load and bending. The latter option is less convenient because the section properties will vary with each load case.
The basic procedure, outlined in 3-1-5/clause 4.4(3), is to determine the effective section for the flanges first, based on stresses computed with gross-section properties but allowing for shear lag if relevant. The effective section for a web should then be calculated using section properties comprising the gross web and the effective flanges (including shear lag effects). If the cross-section has longitudinal stiffeners, then the derivation of the effective section has to consider both local buckling of the plate sub-panels and overall buckling of the stiffened plates. If the stress in a cross-section builds up in stages with the cross- section changing throughout (as in steel–concrete composite construction), 3-1-5/clause 4.4(3) allows the stresses to first be built up with effective flanges and gross web. The total stress distribution so derived in the web may then be used to determine an effective section for the web and the resulting effective cross-section can be used for all stages of con- struction to build up the final stresses. This is a convenient approximation which overcomes the problem that otherwise the effective section of the web would keep changing throughout construction.
Where there is biaxial bending, what constitutes a flange or a web is not defined. However, the precise classification matters less with the uniform approach to webs and flanges in EN 1993-1-5 than it would have done to BS 5400: Part 3.4
6.2.2.5.2.1. Effective widths for unstiffened plates and plate sub-panels
Effective widths for unstiffened plate panels, including sub-panels between stiffeners, are calculated using 3-1-5/clause 4.4. According to 3-1-5/clause 4.4(1), the effective area of the plate is given by:
Ac;eff¼ Ac 3-1-5/(4.1)
where is a reduction factor which depends on whether the plate panel considered is internal (and therefore has both longitudinal edges stiffened) or is an outstand (and therefore has only one longitudinal edge stiffened). The distribution of the effective area within the plate panel is determined from either 3-1-5/Table 4.1 or 4.2 for internal or outstand elements respectively. 3-1-5/clause 4.4(2)gives formulae for the reduction factors which are reproduced below.
For internal elements: ¼p 0:055ð3 þ Þ
2p
1:0 but ¼ 1:0 for p 0:673 3-1-5/(4.2)
and for outstands: ¼p 0:188
2p 1:0 but ¼ 1:0 for p 0:748 3-1-5/(4.3)
where is the stress ratio across the plate shown in 3-1-5/Tables 4.1 and 4.2. The format of expression 3-1-5/(4.2) for pure compression ( ¼ 1) was originally proposed by Winter.8The definition of slenderness, p, follows the usual Eurocode notation of being the square root of
a ratio of a yield resistance to an elastic critical buckling resistance and is therefore: p¼ ffiffiffiffiffiffi fy cr s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fy k 2Et2 12ð1 2Þb2 v u u u t ¼ b=t 28:4" ffiffiffiffiffik p
where "¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffi235= fy, kis a buckling coefficient, determined from 3-1-5/Tables 4.1 and 4.2,
which depends on stress distribution and panel edge support conditions, and b=t is the plate width-to-thickness ratio. The values of k assume simply supported edges (except at free
edges), but benefit could be taken in deriving higher values where significant edge rotational support stiffness could be guaranteed. They also assume infinitely long plates, which is discussed further below.
3-1-5/clause 4.4(3) 3-1-5/clause 4.4(1) 3-1-5/clause 4.4(2)
It can be seen from Fig. 6.2-10 that the real plate strength is less than the elastic critical buckling load at low slenderness due to imperfections and occurrence of full plasticity. However, at high slenderness, the real plate strength exceeds the elastic critical value because of the post-buckling strength of the parts of the plate near to the supported edges. The elastic critical buckling values presented in 3-1-5/clause 4 assume that the plate panels are much longer than they are wide. For internal plates, the lowest mode of buckling will have one transverse half wave of buckling and an integral number of half waves in the longitudinal direction. Minimum buckling load occurs where the length of the panel is an integer multiple of the width as shown in Fig. 6.2-11. For uniform compression, this results in k¼ 4 for a=b ¼ 1, 2, 3, etc. For length-to-width ratio, 1 < a=b < 3, the buckling
load is affected slightly by non-integer values of a=b and k rises to approximately 4.5 at
a=b¼ 1:42. For non-integer values of a=b greater than 3, any fluctuation in buckling load is minimal.
For panels that are shorter than they are wide, the buckling load begins to rise (although EN 1993-1-5 does not provide a formula for kin this case) and the buckling mode becomes
more and more like strut buckling of an isolated strip of plate without transverse edge restraint. For very low values of a=b < 1, the restraint from transverse bending of the plate is small and the idealized strut buckling mode is accurate and gives a critical stress cr;c approximately the same as would be obtained for plate buckling, cr;p. As a=b
increases towards 1.0, this approximation becomes more conservative as the restraint from transverse bending of the plate increases and cr;p is greater than the column critical stress cr;c.
The reduction factor needed for column-type buckling is greater than for plate buckling at a given slenderness (because plates have some reserve of strength beyond the elastic critical buckling load whereas for struts the elastic critical load is an upper bound on strength), so
2.5 2.0 1.5 1.0 0.5 0 0 50 100 150 200 250 300 b/t
Calculated strength/yield strength
EC3-1-5 strength Elastic critical strength
Fig. 6.2-10. Comparison of elastic critical and real strength of internal plates in S355 steel
kσ a/b 20 10 4 0.3 1.0 2.0 3.0
the two situations have to be considered where a=b < 1. It is however always safe to ignore this column-type buckling behaviour for low a=b if is derived using the slenderness for long panels. 3-1-5/clause 4.4(6) allows column type buckling to be considered for plates by using 3-1-5/clause 4.5.3(2), where the column buckling load for an unstiffened plate is given as:
cr;c¼
2
Et2
12ð1 2Þa2 3-1-5/(4.8)
If benefit is taken from the increased buckling resistance associated with short panel length, it is important that the transverse stiffeners providing the reduced length must be checked for their ability to provide such support in accordance with 3-1-5/clause 9.2.1. This is discussed in section 6.6 of this guide.
The slenderness for column-type buckling is then given by 3-1-5/clause 4.5.3(4): c¼ ffiffiffiffiffiffiffiffiffi fy cr;c s 3-1-5/(4.10) The column-type reduction factor, c, is then determined from the flexural buckling curves
of 3-1-1/clause 6.3.1.2 using the imperfection parameter ¼ 0.21 in accordance with 3-1-5/ clause 4.5.3(5). Finally 3-1-5/clause 4.5.4(1) requires interpolation to be performed between the reduction for plate behaviour, , and the reduction for column behaviour, c, according to:
c¼ ð cÞ ð2 Þ þ c 3-1-5/(4.13)
with ¼ cr;p=cr;c 1ð0 1Þ where cr;pis the elastic critical buckling stress for plate
behaviour and cr;c is the elastic critical buckling stress for column buckling. c can
conservatively be taken as c by assuming cr;p¼ cr;c. Within the application rules
presented in EN 1993-1-5, this conservative approximation of taking c¼ c will be
necessitated by the absence of a formula for short plates that considers plate behaviour as in Fig. 6.2-11. Solutions can however be found, such as those by Bulson9 or from IDWR10 (Fig. 6.2-17), which give values of k for short plates. For pure compression only and