The Structural Engineer Volume 62A No. 5 May 1984
Continuous r.c. slabs -
A. N. Beal BSc CEng MlStructE MICE
R. H. Thomason & Partners
The design of continuous reinforced concrete slabs is a routine item -
‘every bar should extend, except at end supports, beyond the point at which it is no longer needed for a distance equal to the effective depth of the member, or twelve times the size of the bar, whichever is greater.’
However simple moment analysis shows that, for slabs on free supports, even a small live load can produce hogging moments more than 0.25L from a support. The standard detailing rules do not comply with the ‘theoretical’ rules: CP110 cl. 220.127.116.11 seems to contradict CP110 cl. 18.104.22.168. In addition, Beeby pointed out that, curiously, standard design moments require more reinforcement for a one-
On the assumptions (a) that curtailment of top steel at 0.25L has been proven satisfactory in service and (b) that one-
(a) slabs outside the stated limits must still comply with the ‘theoretical’ rules -
(b) the difference between the requirements of the simplified rules and the ‘theoretical’ rules is so great that it does not inspire confidence in either of them.
There must also be a question as to whether a detailing practice that has proved satisfactory in the past will necessarily remain trouble-
In fact there are grounds for concern on this last point. In the past, internal bays of continuous slabs were commonly designed for moments of ±wL²/12 (±0.0833wL², where w is loading) . However design moments have reduced over the years: a slab designed to the latest proposed rules would have a hogging moment of only -
In the circumstances, the performance of new slabs detailed to the traditional rules cannot be taken for granted.
We need a method of analysis which determines the true safety factors of existing designs and allows a design method to be used that:
Design moments and loadings
The standard live load patterns are:
(i) any two adjacent spans loaded;
(ii) alternate spans loaded.
(CP114 cl. 312, CP110 cl. 22.214.171.124).
CP110 is unusual in requiring part of the fixed dead load to be considered as a mobile live load. Loading (ii) is critical for determining curtailment of top reinforcement.
A continuous slab can be considered as a plastic mechanism -
The total load on each of the loaded spans can then be calculated from the sagging moment capacity and the hogging moment -
MSUPPORT = -
for a typical internal span (wD is dead load).
The maximum load on the loaded spans at failure can be calculated from
wmax = (MSUPPORT + MSPAN)/(L²/8) (MSPAN calculated with reinforcement at yield).
The load on alternate spans at failure can be expressed either as a proportion of total design load (wmax/(wD + wL)) or as a failure live load compared with the design live load ((wmax -
If supports have some rotational stiffness, this can increase resistance to ‘alternate spans’ loading substantially. For an internal span, the extra support moment generated can be taken approximately as
MSUPP' = -
where K = EI/L and MSUPP' is additional support moment. If support stiffness is not known, it is suggested here that a reasonable value for one-
For flat slab design, the moment can be calculated using the column stiffness, although there is some uncertainty over the degree of moment transfer . (It is interesting to note that over the years slabs have tended to become thicker, in order to control deflection at higher steel stresses; columns have tended to become thinner, because of increases in concrete strength. Thus the column/slab stiffness ratio, critical to the performance of flat slabs under non-
The author has analysed various typical slabs as they would have been designed at different times in the past from prewar flat slabs to 1960s BRC-
When the proposals for slab design in the revised CP110 are compared, there is some cause for concern: an internal bay of slab on ‘free’ supports, designed to the new rules and with design live load equal to dead load, could only support an ‘alternate spans’ live load of 1.45 times the design value (a total load on loaded spans of only 1.22 times the design total load).
While it might be argued that this would be acceptable, it is clearly less safe than any design the author has analysed which could be regarded as typical of designs that have been proven in service. (For ‘stiff’ supports, the new rules give quite acceptable results within their limits of application, although they require more extensive top reinforcement than traditional practice.
It is suggested here that the basis of design for curtailment of reinforcement should be revised as follows:
(a) Reinforcement should extend as far as required by the elastic moment diagram with full load applied to all spans.
(b) A plastic analysis, with reinforcement stressed up to its guaranteed yield stress (according to BS4449 and BS4461, this is 0.93 times the ‘characteristic’ strength), should show that at least 1½ times the design imposed load can be carried when applied to alternate spans, with others carrying only dead load. In both cases, bars should extend at least 12 bar diameters beyond theoretical cut-
Curtailment based on ordinary elastic analysis and working loads meets these requirements, except in the case of cantilevers, where a check on overturning stability should be done.
If these rules are accepted, they could be used as the basis for revised tables of standard design moments and detailing rules for common situations. The rules for flat slab design could also be revised in a consistent fashion; it is likely here that limits on minimum column stiffness will be necessary for standard ‘empirical’ designs.
Thanks are due to A. R. Alexander, A. W. Beeby, L. Bullock, A. A. Park, and W. E. A. Skinner for their assistance, advice, and comments.
1. CP114 The structural use of reinforced concrete in buildings, London, British Standards Institution, 1969.
2. CP110 The structural use of concrete, London, British Standards Institution, 1972.
3. Beeby, A. W.: ‘Are our Code provisions for slabs safe?’, The Structural Engineer, 59A, No. 11, November 1981.
4. Draft for comment ‘The structural use of concrete’, London, British Standards Institution, 10 February 1982.
5. Reynolds, C. E.: Reinforced concrete designers’ handbook (7th edition), Table 20 p. 175, London, Cement & Concrete Association, 1971.
6. Long, A. E., Cleland, D. J., Kirk, D. W.: ‘Moment transfer and the ultimate capacity of slab column structures’, The Structural Engineer, 56A, No. 7, April 1978.
7. Bowie, P. G.: ‘Moments in flat slab’, The Structural Engineer, January 1938.