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The Structural Engineer Volume 62A No. 5 May 1984

Viewpoint

Continuous r.c. slabs - reconciling theory with practice

A. N. Beal BSc CEng MlStructE MICE

R. H. Thomason & Partners

Introduction

The design of continuous reinforced concrete slabs is a routine item - moments are taken from CP114 [1] Table 15 or CP110 [2] Table 4 (according to taste), reinforcement is calculated and it is then placed according to normal detailing prac­tice: it is customary to stop off top reinforcement at 0.25L or 0.3L from a support (L = span). Yet, as Beeby has pointed out in a recent paper [3], this situation contains a paradox. Both Codes specify similar theoretical rules for curtailing reinforcement - e.g. CP110 cl. 3.11.7.1:

‘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. 3.11.7.3 seems to contradict CP110 cl. 3.11.7.1. In addition, Beeby pointed out that, curiously, standard design moments require more rein­forcement for a one-way slab supported on beams or walls than that required for an equivalent beamless ‘flat slab’.

On the assumptions (a) that curtailment of top steel at 0.25L has been proven satisfactory in service and (b) that one-way slabs should not re­quire more reinforcement than flat slabs, he pro­posed new design rules, and these have been adopted in modified form in the revised CP110 [4]. (Curtailment is at 0.3L from face of support.) These rules apply only if certain limits on live load (LL ≤ DL) and bay size are met. This leads to new problems:

(a) slabs outside the stated limits must still comply with the ‘theoretical’ rules - so a slight increase in live load could re­quire a completely disproportionate increase in reinforcement and

(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-free in new designs based on new design moments and stresses.

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) [5]. However design moments have reduced over the years: a slab designed to the latest proposed rules would have a hogging mo­ment of only -0.0667wL² and a sagging mo­ment of +0.0583wL². Furthermore, although reinforcement properties have changed little over the years, allowable stresses have risen from 124N/mm² (mild steel), 138N/mm² (high yield steel) before 1948, through 124N/mm² and 186N/mm² (CP114:1948), 124N/mm² and 207N/mm² (CP114:1957), 140N/mm² and 210/230N/mm² (CP114:1965) to approximately 145N/mm² (MS), 247/267N/mm² (HYS) and 281N/mm² (fabric) in current designs to CP110. In other words, some recent slab designs may have 23% less steel than those designed for the same materials and moments 20 years ago. A slab designed to the latest proposed moments and using steel fabric would have less than half as much bottom reinforcement as an otherwise identical slab designed for ±wL²/12 before 1957.

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:

- is consistent in safety with older designs;

- is theoretically consistent - allowing ‘theoretical’ and ‘empirical’ designs to be reconciled;

- can be applied in both ‘standard’ and unusual situations.

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. 3.2.2.1).

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.

Proposed analysis

A continuous slab can be considered as a plastic mechanism - under ‘alternate spans’ loading, it will form hinges at the curtailment points of top reinforcement (Fig 1) and these will rotate until yield is reached in the midspan steel.

The total load on each of the loaded spans can then be calculated from the sagging moment capacity and the hogging moment - the latter being limited by cracking to a value of

MSUPPORT = - (a/2)(1 - a)wDL²

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 - wD)/wL).

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' = -(WLL²/12)/(2KSUPPORT/(KSPAN + 2KSUPPORT))

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-way slabs in rc framed structures is KSUPPORT = ¼KSPAN, reducing the formula for additional moment to MSUPP = -WLL²/36.

For flat slab design, the moment can be calculated using the column stiffness, although there is some uncertainty over the degree of moment transfer [6]. (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-uniform loads [7], has been reduced considerably in recent years.)

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-designed slabs and modern CP114 and CP110 designs; the same slabs as they would have been designed to CP110 and CP114 ‘theoretical’ rules were also compared. This revealed that, although traditional detailing practice does not comply with the Code ‘theoretical’ requirements, all the designs analys­ed showed quite acceptable factors of safety on ‘alternate spans’ live load - even old flat slabs designed to modest stresses for a total moment of wL²/10! In all cases, the slabs analysed proved able to carry a load applied to alternate spans equivalent to at least 12/3 times the design live load (total load at least 11/3 times design total load), with steel at ‘characteristic’ stress.

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.

Proposals

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-off points.

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.

Acknowledgements

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.

References

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.

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