Slabs are horizontal elements that form the floor, foundation, and roof elements of different structures. They are distinctly categorized from other members base on their typical features of having their thickness disproportionately smaller than the size of their other two dimensions. Slabs can be categorized based on their structural configuration as solid, ribbed, and flat slabs. This article is particularly on the design of solid slabs.

General Concept of Slab Design

The design of slab is essentially the same as that of beams except for few modalities. Slabs are generally designed as beams of fixed width (say; 1m, 1ft etc.). The secondary span is taken to have a fixed width, while the reinforcement is designed to span along the principal/main span.  The reinforcement along the main span is spread at specific distance across the fixed width.

For better understanding of the above paragraph, consider fig (1) below of a slab spanning 3m x 7m. Let’s take the 3m length as the principal span and the 7m length as the secondary span. To design the slab, we shall take a 1m width out of the 7m span so that we have a slab of 3m x 1m (see fig(1b). We design for this 3m by 1m and then replicate this 7times so that we have our 3m x 7m slab fully designed (see fig 1(c)).

 

Effective Span of Slab

The effective span of a slab is important when estimating the internal forces and checking for deflection of the slab. The effective span of different configuration of slab is given below according to both BS 8110-1-1997, and EN 1992-1-1-2004.

Simply Supported Slab

The effective span of a simply supported slab should normally be taken as the clear distance between the faces of supports, plus one-third of their widths (Eurocode 2) or plus the effective depth (BS 8110). However, where a bearing pad is provided between the slab and the support, the effective span should be taken as the distance between the centers of the bearing pads (BS 8110 and Eurocode 2).

Continuous Slab

The effective span of a continuous slab should be taken as the distance between center of supports (for both BS 8110 and EC2).

Cantilever Slab

The effective span of a cantilever slab should be taken as the length of the cantilever from the center of the support when it is the end of a continuous slab (BS 8110 and Eurocode 2). Where the slab is an isolated cantilever, according to Eurocode 2, the effective length is the length of the cantilever from the face of the support (plus half its effective depth according to BS 8110).

 

Steps in designing a Slab

Having understood the concept of designing slab per strip width, so we shall go through the steps of designing and providing reinforcement for our main span. The following are the steps in providing reinforcement along the span

• Initial Sizing of the slab:

The preliminary thickness of the slab is determined. This can be determined using deflection criteria of the slab. Table 3.9 of BS 8110-1-1997 can be used to estimate a value of span/depth ratio to pick an effective depth for the slab. It is important to also note here that the shorter length of the slab should be taken as the controlling span to determine the effective depth. A value of 20mm may be added to the effective depth to provide for nominal cover or table 4.4N of EN 1992-1-1-2004 should be consulted for cover base on durability requirement.

• Load Estimation and Analysis of the slab for internal stresses

 The dead and live load that would be acting on the slab should be estimated. The slab should be analyzed to get the internal stresses such as bending moment, shear force etc. that are acting on it due to external load and self-weight. This are discussed in better details under “Analysis of Slab for internal forces” below.

• Flexural Design

 The slab should be designed against bending moment. The steps in designing against bending moment are enumerated below

1. Calculate the effective depth (d):

The effective depth should be calculated after a specific size of reinforcement has been assumed for design and a nominal cover chosen in accordance with the requirement for durability. The effective depth is calculated using the equation: d = h – c – dia/2

2.  Check whether the section is singly or doubly reinforced:

We check whether K (M/bd²fck) is less than K’ (0.168). Slabs are always design as singly reinforced except in special cases.

3.  Calculate the lever arm: Use Z = $
Z\,\,=\,\,d\left( 0.5+\sqrt{\text{0.25}-\,\,\frac{K}{1.134}} \right)
$

4. Calculate the area of steel: use  $A_{st\,\,=\,\,\frac{M_{Ed}}{0.87f_{ck}Z}}
$

         . Shear design

Since slabs are often subjected to light uniformly distributed load, the shear capacity of the concrete section without shear reinforcement (V_Rdc) is often sufficient to resist the maximum design shear force (V_{Ed )}

         . Deflection Check

The deflection check is carried by checking the actual span-effective depth ratio against the limit span-effective depth ratio. The formula for the limit span-effective depth ratio is given in the code as:

l/d = K[11 + 1.5√f_ck ρ0/ρ + 3.2√f_ck (ρ^o/ρ – 1)3/2]       if ρ ≤ ρ^o

l/d = K[11 + 1.5√fck ρ^o/ρ + 3.2√fck √ρ^o/ρ ]                   if ρ > ρ^o

 

Analysis of Slab for internal forces

After estimating a slab thickness, the next point of call is to analyse the slab for internal forces as stipulated in the aforementioned “steps in designing slabs”. So how do we analyze our slabs?

It is actually easy to analyze slabs using commercial software which are equipped with high-power analysis techniques such as Finite Element Method (FEM), strip analysis etc. With software, the user only needs to define the slab geometry, assign loads, define the slabs boundary conditions, then at the press of a button – boom! – the internal forces in the slab are produced so that we can proceed with our design (steps 3 to Step 5 in “steps in designing slab”).

We can also analyse slabs using various hand calculations. However, due to complexities of internal forces in slabs, especially two-way slabs, some form of design aids is necessary to make the calculation less daunting. Eurocodes are, however, generally less prescriptive; they only states binding principles but provides less design guides and aids to serve as direct assistance to the designer in performing calculations. Eurocodes leave out the prescriptive guides as well as design aids for reference textbooks and manuals. Manual for the design of concrete structures to Eurocode 2 by IstructE provides tabular coefficients similar to that present in BS 8110-1-1997 for shear force and bending moments. Alternatively, some designers just all together reach out for coefficients directly from BS 8110-1-1997 unmodified to estimate the bending moments and shear forces. This article shall adopt this later approach and subsequently reference coefficients tables from BS 8110-1-1997 when appropriate for use.

Categorization of Slabs

To perform hand analysis of bending moment and shear force in slabs using coefficients method, the slab shall be categorized based on its behavior in resisting bending moment. Solid Slab can be categorized as one-way spanning or two-way spanning.

One-way spanning

A one-way spanning slab resist bending moment in one span. This behavior is induced in a slab when the ratio of its longer length to that of the shorter length is greater than 2.

Mathematically: ly/lx ≥ 2.

Additionally, according to clause 5.3.1(5) of Eurocode 1992-1-1-2004, a slab can also be considered as one-way if it contains two free and sensibly parallel edges. Conversely, even if the ratio of the longer length to the shorter length is less than two, a slab shall be considered as one-way spanning if it has two parallel edges unsupported.

Analysis of one-way spanning slab

For ease of understanding for design purpose, one-way spanning slab shall be categorized a) simply-supported b) continuous one-way spanning slab

1. Simply-supported one-way slab

Analysis and design of a one-way simply supported slab is exactly the same as that of a beam except that the slab has a fixed breadth of 1m as explained earlier. The main span along which the main reinforcement is provided is the shorter span, then the other orthogonal span is only provided with minimum area of reinforcement as per BS 8110-1-1997 and EN 1992-1-1-2004. However, for EN 1992-1-1-2004, the minimum area of reinforcement should not be less than 20% of the main reinforcement.

The internal stresses are calculated as follows:

Maximum moment of the slab (Mmax) = wl²/8

Maximum shear force at support (V) = vl/2

After obtaining the maximum internal stresses due to loadings on the slab, the slab is designed as a normal beam of 1m width.

        2. Continuous one-way spanning slab   

Ordinarily continuous slabs should be made to resist the most unfavorable load arrangement on its spans. The most unfavorable effects of trying different pattern of load arrangement on a slab are adopted for design. However, trying different arrangement of the loads can be onerous and time consuming, clause 3.5.2.3 of BS 8110-1-1997 and UK national annex to Eurocode 2 allow that the effects of a single load case of maximum design load acting on all spans with 20% load redistribution can be designed for as long as the following three conditions are met:

2. In a one-way spanning slab, the area of each bay exceeds 30m2.
3. The ratio of the characteristic imposed load to the characteristic dead load does not exceed 1.25.
4. The characteristic imposed load does not exceed 5kN/m2 excluding partitions

Provided that the above conditions are met, for easy analysis the coefficients in table 3.12 of BS 8110-1-1997 can be used to evaluate the bending moments and shear force in a continuous one-way slab. These coefficients already allow for the 20% redistribution, so no further redistribution is required. The table is reproduced below.

Click here to study a worked example on the design of a continuous one-way slab using table 3.12 of BS 8110-1-1997

Two-way spanning slab

A two-way spanning slab resist moment in the two spans. This type of slabs has the ratio of its longer length to that of the shorter length to be less than 2.

Mathematically: ly/lx ≤ 2

Analysis of two-way spanning slab

Since two-way spanning resist moment in both orthogonal directions, then the reinforcement spanning along both the longer and shorter direction are main reinforcement. By this, the shorter length is taken as main span while the longer span is taken as 1m width, then the reinforcement is designed for the shorter span. This is repeated such that the longer span is taken as the main span with 1m width along the shorter span.  Although but spans are considered main spans however the shorter span is more critical, hence reinforcement along the shorter span is placed further from the neutral axis for larger effective depth to be achieved.

For ease of understanding for design purpose, two-way spanning slab shall be categorized as a) simply-supported b) continuous/restrained two-way spanning slab.

1. Simply supported two-way Spanning Slab

A simply supported two-way spanning slab is barely supported by beams or walls without restraint at the corners to prevent lifting or adequate provision against torsion. The coefficients given in table 3.13 of BS 8110-1-1997 can be used to obtain the maximum moments in both spans as follows:

Msx = αsx n lx²

Msy = αsy n lx²

Where n = maximum ultimate load (1.4gk + 1.6qk for BS 8110, 1.3gk + 1.6qk for Eurocode)

lx = length of shorter side

ly = length of longer side

αsx, and αsy are bending moment coefficients to obtained from table 3.13 of BS 8110-1-1997.

 

             2. Continous/Restrained Two-way spanning slab

These are two-way spanning slabs that are restrained from curling up and adequate reinforcement for torsion is provided. The maximum design moments per unit width can be obtained using:

Msx = 𝛽sx n lx²

Msy = 𝛽sy n lx²

𝛽sx and 𝛽sy are coefficients given in table 3.14 of BS 8110-1-1997

Where n = maximum ultimate load (1.4gk + 1.6qk for BS 8110, 1.3gk + 1.6qk for Eurocode)

lx = length of shorter side

Ideally, restrained/continuous slab should be divided into edge strip and middle strip. Msx and Msy are provided at the middle strip while the edge strip is provided with nominal reinforcement. However, in practice the middle strip reinforcement is provided across both strips (middle and end) for convenience.

The above equations can be applied to continuous slab when the following conditions are met:

1. a) The characteristic dead and imposed loads on adjacent panels are approximately the same as on the panel being considered.
2. b) The span of adjacent panels in the direction perpendicular to the line of the common support is approximately the same as the span of the panel considered in that direction.

Other conditions in 3.5.3.5 and clause 3.5.3.6 from BS 8110-1-1997 state conditions that has to be met in different circumstances for the application of the equation states to a restrained slab – continuous or discontinuous.

       Shear force

The shear force can be evaluated using the value in table 3.15 of BS 8110-1-1997

Reinforcement detailing requirements for Slabs.

• Minimum area of reinforcement (Asmin) = 0.26 x fctm/fyk x b x d ≥0013bd
• Maximum Area of reinforcement is 0.04Ac
• Maximum Spacing of main bars is 3h or 400mm, whichever is lesser.
• Minimum spacing should not be less than maximum bar size diameter, or 20mm, or maximum aggregate size plus 5mm, whichever is greater.
• Transverse reinforcement must not be less than 20% of the main reinforcement in one-way slab.

 

References

EN 1992-1-1:2004 (Eurocode 2): Design of concrete structures – Part 1-1: General rules and rules for buildings

BS 8110-1-1997: Structural use of concrete — Part 1: Code of practice for design and construction

IStructE: Manual for the design of concrete building structures to Eurocode 2