## Concrete Design By Dass Euro Code

lightly stressed when p equals 0.5 per cent, p is given by l()0Aymi/l>d where i4s,rCq i* the area of tension reinforcement required in the section. Interpolation between the values of p indicated is permissible. In the case of slabs it is reasonable to assume lha: they are lightly stressed.

Since the value of allowable span-effective depth ratio is affected by both reinforcement ratio and concrete strength it may be more convenient to use the char in figure 6.3 which is for a simply supported span with no compression steel together with a modification factor K (as shown in tabic 6.10) according to member type. Thi-approach is based on the same basic equations and offers greater flexibility than relianc. placed on tabulated values.

Figure 6.3

Graph of basic span-effective depth ratios for different classes of concrete

Figure 6.3

Graph of basic span-effective depth ratios for different classes of concrete

The basic ratios are modified in particular cases as follows:

(a) For spans longer than 7 m (except flat slabs) and where it is necessary to limit deflections to ensure that finishes, such as partitions, are not damaged, the basic values should be multiplied by 7/span.

(b) For flat slabs with spans in excess of 8.5 in, similarly multiply the basic ratios by 8.5/span.

(c) For characteristic steel strengths other than 500 N/mra2, multiply the basic ratios by 500//yk.

(d) Where more tension reinforcement is provided (/lvprov) than that calculated G4s,req) at the ultimate limit state, multiply the basic ratios by /K.prov/<4s.rcq (upper limit = 1.5).

These basic ratios assume a steel working stress of/, 310 N/mnr where /yk = 500 N/mnr f EXAMPLE 6.1 Span-effective depth ratio

A rectangular continuous beam of class C25/30 concrete spans 10 m. If the breadth is 300 mm. check the acceptability of an effective depth of 600 mm when high yield reinforcement, /yk = 500 N/mnr, is used. At the ultimate limit state it is determined that 1250mnr of tension steel is needed and 3 No. 25 mm diameter reinforcing bars (ASlprov 1470mm2) are actually provided in an interior span.

From table 6.10, for an interior span K = 1.5

Basic span -effective depth ratio (ligure 6.3) = 16

Therefore for an interior span, basic span-effective depth ratio = 1.5 x 16 24 To avoid damage to finishes for span greater than 7 m: 7

Modification for steel area provided: 1470

Modified ratio 16.8x^^=19.8

10 x I0J

Span-effeciivc depth ratio provided = ———— = 16.7

which is less than the allowable upper limit, thus deflection requirements are likely to be satisfied.

### 6.3 Calculation of deflection

The general requirement is that neither the efficiency nor the appearance of a structur. t> harmed by the deflections that will occur during its life. Deflections must thus e considered at various stages. The limitations necessary to satisfy the requirements ■> i: vary considerably according to the nature of the structure and its loadings, bu-. !ir reinforced concrete the following may be considered as reasonable guides:

1. the final deflection of a beam, slab or cantilever should not exceed span/250

2. that part of the deflection which takes place after the application of finishes or fix . of partitions should not exceed span/500 to avoid damage to fixtures and fitting-

The code suggests that deflections should be calculated under the action of the qua* permanent load combination, assuming this loading to be of long-term duration. Hencc the total loading to be taken in the calculation will be the permanent load plus . proportion of the variable load which will typically be 30 per cent of the variable loau for office-type construction. This is a reasonable assumption as deflection will be affected by long-term effects such as concrete creep, w hile not all of the variable load is likely to be long-term and hence will not contribute to the creep effects.

Lateral deflection must not be ignored, especially on tall slender structures, and limitations in these cases must be judged by the engineer. It is important to realise that there arc many factors which may have significant effects on deflections, and are difficult to allow for. Thus any calculated values must be regarded as an estimate onlv The most important of these factors are:

1. support restraint must be estimated on the basis of simplified assumptions, which will have varying degrees of accuracy;

2. the precise loading cannot be predicted and errors in permanent loading may have a significant effect;

3. a cracked member will behave differently from one that is uncracked - this may be j problem in lightly reinforced members where the working load may be near to the cracking limit;

4. the effects of floor screeds, finishes and partitions are very difficult to assess -frequently these are neglected despite their 'stiffening' effect.

It may be possible to allow for these factors by averaging maximum and minimum estimated effects and, provided that this is done, there are a number of calculation methods available which will give reasonable results. The method adopted by EC2 i-. based on the calculation of curvature of sections subjected to the appropriate moments with allowance for creep and shrinkage effects where necessary. Deflections are then calculated from these curvatures. A rigorous approach to deflection is to calculate the curvature at intervals along the span and then use numerical integration techniques to estimate the critical deflections, taking into account the fact that some sections along the span will be cracked under load and others, in regions of lesser moment, will be uncracked. Such an approach is rarely justified and the approach adopted below, based on EC2, assumes that it is acceptably accurate to calculate the curvature of the beam or slab based on both the cracked and uncracked sections and then to use an 'average' value in estimating the final deflection using standard deflection formulae or simple numerical integration based on elastic theory.

The procedure for estimating deflections involves the following stages which are illustrated in example 6.2.

### 6.3.1 Calculation of curvature

Curvature under the action of the quasi-permanent load should be calculated based on both the cracked and uncracked sections. An estimate of an 'average' value of curvature can then be obtained using the formula:

where average curvature values of curvature calculated for the uncracked case and cracked case respectively coefficient given by 1 - /?(<rsr/<7s)2 allowing for tension stiffening load duration factor (I for a single short-term load; 0.5 for sustained loads or cyclic loading)

stress in the tension steel for the cracked concrete section stress in (he tension steel calculated on the basis of a cracked section under llie loading thai will just cause cracking at the section being considered.

Appropriate values of concrete tensile strength to be used in the calculation of rrsr can be obtained from table 6.11. In calculating (,. the ratio can more conveniently be replaced by (Ma/M) where Ma is the moment that will just cause cracking of the section and M is the design moment for the calculation of curvature and deflection.

In order to calculate the 'average* curvature, separate calculations have to be carried out for both the cracked and uncracked cases.

Uncracked section

The assumed elastic strain and stress distribution for an uncracked section is shown in figure 6.4.

For a given moment. M, and from elastic bending theory, the curvature of the section,

CAL-is g'ven by

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