Flat slab structures are defined as slabs (solid or coffered) supported on point supports. Unlike two-way spanning slabs on line supports, flat slabs can fail by yield lines in either of the two orthogonal directions (Fig. 3.24). For this reason, flat slabs must be capable of carrying the total load on the panel in each direction.
Methods of analyses
Many recognized methods are available. These include: (1) the equivalent frame method
(2) simplified coefficients (3) yield line analysis (4) grillage analysis.
In this guide, only the first two of these methods will be considered. For other approaches
specialist literature should be consulted (e.g. CIRIA Report 1105).
Equivalent frame method
Division into frames
The structure may be divided in two orthogonal directions into frames consisting of columns and strips of slabs acting as ‘beams’. The width of the slab to be used for assessing the stiffness depends on the aspect ratio of the panels and whether the loading is vertical or horizontal.
Vertical loading. When the aspect ratio is less than 2 the width may be taken as the distance between the centre lines of the adjacent panels. For aspect ratios greater than 2, the width may be taken as the distance between the centre lines of the adjacent panels when considering bending in the direction of longer length spans of the panel and twice this value for bending in the perpendicular direction. (See Fig. 3.25.)
The width of beams for frame analysis is as follows:
Table 3.6. Shear force coefficients for uniformly loaded rectangular panels supported on four sides with provision for torsion at corners
Type of panel and location
βvxfor values ofγy/γx
βvy
1.0 1.1 1.2 1.3 1.4 1.5 1.75 2.0
Four edges continuous
Continuous edge 0.33 0.36 0.39 0.41 0.43 0.45 0.48 0.50 0.33
One short edge discontinuous
Continuous edge 0.36 0.39 0.42 0.44 0.45 0.47 0.50 0.52 0.36
Discontinuous edge – – – – – – – – 0.24
One long edge discontinuous
Continuous edge 0.36 0.40 0.44 0.47 0.49 0.51 0.55 0.59 0.36
Discontinuous edge 0.24 0.27 0.29 0.31 0.32 0.34 0.36 0.38 –
Two adjacent edges discontinuous
Continuous edge 0.40 0.44 0.47 0.50 0.52 0.54 0.57 0.60 0.40
Discontinuous edge 0.26 0.29 0.31 0.33 0.34 0.35 0.38 0.40 0.26
Two short edges discontinuous
Continuous edge 0.26 0.30 0.33 0.36 0.38 0.40 0.44 0.47 –
Discontinuous edge – – – – – – – – 0.40
Two long edges discontinuous
Continuous edge – – – – – – – – 0.40
Discontinuous edge 0.26 0.30 0.33 0.36 0.38 0.40 0.44 0.47 –
Three edges discontinuous (one long edge continuous)
Continuous edge 0.45 0.48 0.51 0.53 0.55 0.57 0.60 0.63 –
Discontinuous edge 0.30 0.32 0.34 0.35 0.36 0.37 0.39 0.41 0.29
Three edges discontinuous (one short edge continuous)
Continuous edge – – – – – – – – 0.45
Discontinuous edge 0.29 0.33 0.36 0.38 0.40 0.42 0.45 0.48 0.30
Four edges discontinuous
When lx< ly< 2lx: Wx= (lx1+ lx2)/2 Wy= (ly1+ ly2)/2 When ly> 2lx: Wx= (lx1+ lx2)/2 Wy= 2Wx
Horizontal loading. Horizontal loading in the frame will be considered only in unbraced structures. In these cases, the question of restraint to the columns, and hence the effective length of columns, is a matter of judgement. If the stiffness of the slab framing into the column is overestimated, the effective length of the column will reduce correspondingly. As the stiffness at slab-column junctions is a grey area, codes of practice adopt a cautious approach. For the slab, half the stiffness applicable to vertical loading is used.
Fig. 3.24. Possible failure modes of flat slabs
Stiffness properties are generally based on the gross cross-section (ignoring the reinforcement). Additional stiffening effects of drops or solid concrete around columns in coffered slabs may be included, but this will complicate hand calculations. It should be noted that drops (and solid areas) should only be taken into account when the smaller dimension of the drop (and solid areas) is at least 33% of the smaller dimension of the surrounding panels.
Analysis
The equivalent frames may be analysed using any of the standard linear elastic methods such as moment distribution (see Section 3.7.1). Braced structures may be partitioned into subframes consisting of the slab at one level continuous with columns above and below. The far ends of the columns are normally taken as fixed unless this assumption is obviously wrong (e.g. columns with small pad footings not designed to take moments).
The load combinations given in Section 3.2 may be used.
The bending moments obtained from the analysis should be distributed laterally in the ‘width’ of the slab in accordance with the allocation of moments between strips (see p. 47).
Simplified coefficients
In braced buildings with at least three approximately equal bays and both slabs subject predominantly to uniformly distributed loads, the bending moments and shear forces may be obtained using the coefficients given in Section 3.8.1 (Table 3.3).
The bending moments obtained from the above, should be distributed laterally in the ‘width’ of the slab in accordance with h the allocation of moments between strips (see p. 47, Table 3.7).
Lateral distribution of moments in the width of the slab
In order to control the cracking of the slabs under service conditions, the bending moments obtained from the analysis should be distributed taking into account the elastic behaviours of the slab. As can be imagined, the strips of the slab on the lines of the columns will be stiffer than those away from the columns. Thus, the strips closer to the column lines will attract higher bending moments.
Division of panels
Flat slab panels should be divided into column and middle strips, as shown in Fig. 3.26.
As drawn, Fig. 3.26 applies to slabs without drops (or solid areas around columns in coffered slabs). When drops (or solid areas around columns in coffered slabs) of plan
dimensions greater than lx/3 are used, the width of the column strip may be taken as the
width of the drop. The width of the middle strip should be adjusted accordingly.
Allocation of moments between strips
The bending moments obtained from the analysis should be distributed between the column and middle strips in the proportions shown in Table 3.7.
In some instances the analysis may show that hogging moments occur in the centre of a span (e.g. the middle span of a three-bay structure, particularly when it is shorter than the adjacent spans). The hogging moment may be assumed to be uniformly distributed across the slab, if the negative moment at midspan is less than 20% of the negative moment at supports. When the above condition is not met, the moment is concentrated more in the middle strip.
Moment transfer at edge columns
As a result of flexural and torsional cracking of the edge (and corner) columns, the effective width through which moments can be transferred between the slabs and the columns will be much narrower than in the case for internal columns. Empirically, this is allowed for in design by limiting the maximum moment the slab (without edge beams) can transfer to the columns:
Mt max= 0.17bed2f ck
where beis the effective width of the strip transferring the moment as defined in Fig. 3.27,
and d is the effective depth of the slab.
Mt maxshould not be less than 50% of the design moment obtained from an elastic analysis
or 70% of the design moment obtained from grillage or finite element analysis. If Mt maxis less
than these limits, the structure should be redesigned.
When the bending moment at the outer support obtained from the analysis exceeds Mt max,
then the moment at the outer support should be reduced to Mt max, and the span moment
should be increased accordingly.
The reinforcement required in the slab to transfer the outer support moment to the
column should be placed on a width of slab Cx+ 2r (Fig. 3.28). For slabs with a thickness less
than 300 mm, r = Cy, and for thicker slabs, r = 1.67Cy. In the latter case, it is essential to
provide torsional links along the edge of the slab. However, U bars (as distinct from L bars) with longitudinal anchor bars in the top and bottom may be assumed to provide the necessary torsional reinforcement (Fig. 3.29).
Table 3.7. Distribution of design moments in panels of flat slabs
Apportionment between column and middle strip expressed as a percentage of the total negative or positive design moment
Column strip (%) Middle strip (%)
Negative 60–80 40–20
Positive 50–70 50–30
For the case where the width of the column strip is taken as equal to that of the drop, and the middle strip is thereby increased in width, the design moments to be resisted by the middle strip should be increased in proportion to its increased width. The design moments to be resisted by the column strip may be decreased by an amount such that the total positive and the total negative design moments resisted by the column strip and middle strip together are unchanged.
Fig. 3.27. Effective slab widths for moment transfer
Fig. 3.28. Flat slabs: detailing at outer supports
Bending moments in excess of Mt max may be transferred to the column only if an edge beam (which may be a strip of slab) is suitably designed to resist the tension.
3.8.4. Beams
For the design of beams, the bending moment coefficients given in Fig. 3.20 or Table 3.3 may be used, noting the conditions to be complied with, which are discussed in Section 3.8.1.
3.8.5. Simplifications
EN 1992-1-1 permits the following simplifications regardless of the method of analysis used: (1) At a support assumed to offer no restraint to rotation (e.g. over walls), a beam or a slab
which is continuous over it may be designed for a support moment which is less than the moment theoretically calculated on the centre line of the support. The permitted
reduction in moment is then FEd, suppt/8, where FEd, suppis the design support reaction and t
is the breadth of the support. This recognizes the effect of the width of support and arbitrarily rounds off the peak in the bending moment diagram.
(2) Where a beam or a slab is cast monolithically into its supports, the critical moment may be taken as that at the face of the supports (but see also (3) below). This provision is quite reasonable, as failure cannot occur within the support.
(3) The design moment at the faces of rigid supports should not be less than 65% of the support moment calculated assuming full fixity at the faces of support. This ensures a minimum design value for the support moment, particularly in the case of wide supports. (4) Loads on members supporting one-way-spanning continuous slabs (solid and ribbed) and beams (including T beams) may be assessed on the assumption that supports offer no rotational restraint. This is reasonable; but the effect of continuity should be considered when designing the support such as columns or walls.