Reinforced Concrete Design of Simply Supported Solid Slabs

strength of materials used in this article

weight of concrete = 24kN/m³

Characteristic strength of concrete fcu = 25N/mm²

Characteristic strength of steel 𝑓y = 460N/mm²

General procedure for design of solid slabs to BS Code.

1.Determine a suitable slab depth;  you can easily estimate this by using the formula stated below;

Effective depth = span/( basic ratio × modification factor)

Span/effective depth ratios

For a simply supported design, the basic ratio is = 20 ( table 3.9, page 35 of BS 8110 part 1 : 1997). Use an initial modification factor (m.f) of 1.4.

Also the architect/ engineer may specify the slab thickness for you. 

However, your slab design must satisfy deflection requirements otherwise you will have to redesign. 

Ways to make your design slab satisfy deflection requirements if failed initially is to either increase slab thickness or amount of reinforcement or both. 

2. Calculate the main and secondary reinforcement areas; you can do this by using the formulas stated in the BS code. ( Details later)

3. Check for excessive deflection ; as stated earlier in number 1, your design must satisfy deflection requirements. 

4. Detailing requirements;  the economical arrangement of steel reinforcement. You can refer to clause 3.12.10.3, BS 8110. An explanatory diagram is shown below for a simply  supported case.

Detailing requirements for simply supported (a) & continuous ( b) slabs

It means that 40% of the reinforcement placed around the center of the slab ( which is critical) only, should extend to the edges of the slab ( which is less critical). 

Explanation of other important points in solid slab design to BS 8110

Effective span of slab:

Referring to the diagram above, the effective span of slab refers to ‘A’ the distance between the centers of bearings,  or the clear distance between supports ‘D’, plus effective depth of slab ‘d’.

Calculating steel areas.

In calculating the slab self weight, the overall depth of the slab referred to as ‘h’, should be used. h is the effective depth of slab plus allowance for cover to reinforcement plus half the assumed main steel diameter bar.

The self weight of the slab together with the dead and live load is used to determine the design moment ‘M’.

The value of M must be checked against the value of Mú which is the ultimate moment of resistance

Mú = 0.156fcubd²

Where b=1000mm and fcu = strength of concrete.

If Mú ≥ M then the slab doesn’t need compression reinforcement which is usually the case.

Main reinforcement steel areas

The area of reinforcement Aₛ can be determined using Aₛ= M÷(0.87𝑓yz) where,

M = design moment= WL²/8 (for simply supported slab)

W = design load of slab in kN/m²

L = span of slab in meters.

𝑓y= strength of steel

 z = d[0.5 + √ (0.25 – K/0.9)] and 

K= M / fcubd²

Secondary reinforcement steel areas

Secondary reinforcement also refers to distribution steel. As per BS 8110, it is calculated as follows;

Aₛ min = 0.24% Ac  when 𝑓y = 250N/mm²

Aₛ min = 0.13% Ac  when 𝑓y = 460N/mm²

Check that your slab design meets deflection requirements

Design service stress, fs = (2 𝑓y As req.) ÷ ( (3 As prov), (table 3.10 page 36 of BS 8110 part 1 1997.)

Where As req & As prov is the area of steel calculated and the area of steel provided, respectively.

Modification factor;

 m.f  =  0.55 + (477 – fs) ÷[120 ( 0.9 + M/bd²)] ≤ 2

Now this m.f is your actual m.f since it is a function of  the area of steel calculated and area of steel provided.

Use this new m.f to replace the m.f you initially assumed to be 1.4 in the equation;

Effective depth; d = span/( basic ratio × modification factor).

you can also determine m.f from the table below

If the assumed  value of d is greater than actual d, then deflection requirements are satisfied.

Crack widths

The BS code specifies that crack width should not exceed 0.3mm. except by calculation, the following rules should ensure crack width requirements are satisfied;

fy = 250N/mm² and depth of slab ≤ 250mm 

or

fy  = 460N/mm² and depth of slab ≤ 200mm

or

the percentage of reinforcement ( 100As/bd) ≤ 0.3%

Maximum bar spacing of reinforcement

BS code specifies that the clear distance between tension bars should be less than 3d or 750mm whichever is the lesser.

Example design of a simply supported solid slab

From the diagram of part of a floor plan shown above, I will be designing the panel labeled A. 

The first thing to check is if the slab is a one or two way slab. By observing the dimensions relating to the slab panel, the longer side is

3401 + 587 + 1225 = 5213mm

The shorter side of the panel is

1150 + 1225 = 2375mm

Therefore longer side ÷ shorter side =

5213÷2375=  2.19 ≥ 2;

Panel A is a one way slab.

This means that the main reinforcement bars will span along the y- axis ( shorter side) and the distribution bar will span along the x – axis ( longer side).

direction of main reinforcement

Note that in a one way slab, the span of the slab is the shorter side. The span of the slab panel A being designed is 2375mm.

Further observation of the slab panel will show that its thickness  (overall depth of slab) has been given as 150mm;

h = 150mm

Your duty as the designer is to calculate the required steel reinforcement and check if the design satisfies deflection requirements. If the slab thickness wasn’t given then you are free to assume a thickness and then check that it works.

Calculating the Design load.

The formula stated by BS code is

1.4Gk + 1.6Qk

where Gk = dead load

Qk = live load

dead load is the self weight of the slab plus finishes

Live load refers to variable or movable loads such as people, furniture etc, the slab carries. Live loads for different categories of buildings are stated in BS 6399 part 1: 1997.

Gk = weight of concrete x slab thickness

     = 24kN/m³ x 0.15m ( slab thickness of 150mm)

= 3.6kN/m²,  plus finishes of say 1.2 kN/m² which gives a total gk of

4.8kN/m² ;

Gk = 4.8kN/m²

For private dwellings, Qk = 1.5kN/m² (see table 1 of BS 6399 part 1: 1997)

Design load = 1.4gk + 1.6qk

=( 1.4x 4.8 + 1.6 x 1.5) kN/m²

= 9.12kN/m² per m width ( slabs are designed per m width.)

Design moment M = WL²/8

where W = design load and L = span of slab

W=[ 9.12kN/m² x ( 2.375m)² ] / 8

= 6.43kNm

ultimate moment of resistance

Mu= 0.156fcubd²

Now d = effective depth of slab which can be estimated as overall depth of slab minus concrete cover to reinforcement minus half of main steel diameter

I.e. d = 150mm – 25mm – 6mm = 119mm.

( 25mm is concrete cover to reinforcement and 6mm is half the diameter of 12mm steel rod)

b = 1000mm ( slabs are designed per m width)

fcu  = 25N/mm² ( strength of concrete)

Mu = 0.156 x 25N/mm² x 1000mm x (119mm)² = 55.2279 x 10⁶ Nmm 

= 55.23kNm

Since M < Mu no compression reinforcement is required.

K= M / fcubd²

= 6.43 x 10⁶/ ( 25 x 1000 x 119²)

0.0182

z = d[0.5 + √ (0.25 – K/0.9)]

= 119[ 0.5 + ✓ ( 0.25 – 0.0182/0.9)]

= 116.54mm ≤ 0.95d

Limiting to 0.95d

0.95d = 113.05; use z= 113.05

Hence, 

Aₛ= M÷(0.87fyz)

= 6.43 x 10⁶ ÷ ( 0.87 x 460 x 113.05) 

Aₛ required. = 142.12 mm²/m width of slab

From fig. 1 shown above, use Y10 bars at 200mm spacings as main steel reinforcement

I.e.

Y10- 200

Aₛ provided = 393 mm²/m

Distribution reinforcement

Minimum Steel reinforcement specified by code  = 0.13%bh

= 0.13% x 1000 x 150

Aₛ minimum required.= 195 mm²/m

With reference to fig 1, use Y10 – 250 bars. Aₛ = 314mm²/m

Note that the calculated required reinforcement is less than the minimum reinforcement specified by code, in this case the minimum reinforcement specified by code supersedes and should be used to select provided steel. Also using Y10- 200 as main and distribution steel is OK. It’s your design, you are in charge, just make sure you follow the code requirements and design laying emphasis on economy.

Checks

Deflection check

Design service stress, 

fs = 2 fy As req./ (3 As prov)

 = 2×460×142.12÷(3×393)

=110.9N/mm²

fs = 110.9N/mm²

m.f  =  0.55 + (477 – fs) ÷[120 ( 0.9 + M/bd²)] ≤ 2

= 0.55 + (477- 110.9)÷[120(0.9 + (6.43 x 10⁶/100 x119²)]

=2.80 ≤ 2

Since m.f is limited to 2,

Hence, m.f = 2

dmin = span / basic ratio x m.f

          = 2375mm / (20 x 2)

          = 59.375mm 

59.375mm ≤ 119mm hence deflection is satisfied.

Crack width check

Since the slab is less than or equal to 200mm thick, crack width check is satisfied

Maximum spacing check

The reinforcement spacing of 200mm for main steel and 250 mm for distribution steel is less than  3d ( 3 x 119 = 357mm) hence maximum spacing check is satisfied.

Types of Wells

Different types of constructed wells

People dig wells to serve as an independent water supply system. These days before a well is dug, a survey is carried out on the site to determine the amount of water beneath the ground, its location, and yielding capacity. This will determine the type of well to be dug.

Types of wells

Generally wells can be categorized into 2 types;

Deep well and shallow well.

Deep well.

From an economical point of view,A well is considered deep when the length of the well is over 25ft or 7.62m deep. What causes the digging or boring of a deep well is as a result of the water table being too far from the top soil. Another reason for a deep well is if although, the water table is not far from the top soil, the soil beneath the water table doesn’t yield water, then the well has to be dug further down to locate the water yielding soil. Types of soil that hold and yield water are those of sand or rock formations. A type of soil that holds water but doesn’t yield it is a soil that consists mostly of shale.

 Another definition of deep well is a well that extends several meters below the water table. Meaning that its source of water is not from the water table but from other water bearing stratum beneath the water table. 

A deep well can extend up to 300m.

Shallow well

A well is considered shallow if the length is less than 25ft from the surface. This is because the water table is not too far from the surface and the soil directly under the water table yields water. 

Shallow wells are also referred to as water table wells. Meaning, the source of water for these types of well is the water table. It also implies that if the water table is further down the earth ( over 25ft), the shallow well extends about the same length or more. Based on the location of water table, shallow wells can be over 35m in extreme cases.

Types of constructed wells

Generally they are of 3 types;

Dug well.

Driven well.

Bored well.

Hand Dug well.

A dug well is done using hand tools. A pick and shovel to dig the shaft, and a bucket and rope to lift out soil. This type of well cannot be dug a few feet below the water table. If the water table is very far from the surface  ( over  10m deep),then a hand dug well may not be economically and practically viable.  Most hand dug wells for private dwellings are less than 10m deep. Hand Dug wells are always shallow wells and they are the least costly.

Driven well

A driven well consists of pipes driven into the ground. The drive point is usually twisted into the ground by hand or powered auger. This type of well is suitable for Sandy soils and cannot be dug through rock formations. The well can be driven up to 100m. It can be shallow or deep,depending on the nature of the soil and water yielding characteristics.

Bored or drilled well

This type of well is also called bore hole. It also consists of pipes much larger than that used in a driven well, bored into the ground using a heavy drilling machine. Bore holes can reach depths of up to 300m and can be drilled through rocky formations. Boreholes can either be shallow or deep, depending on depth of water table and water yielding characteristics of the soil. It is the most expensive of the types mentioned in this article.

The Economics of Reinforced Concrete Slab Design (In-situ)

slab construction

Reinforced concrete slabs can be divided into 2 types. They are In situ and precast slab. Insitu means a slab cast in place. Precast means that the slab is produced somewhere else and then transported to site for installation.

Cast in place slabs are further divided into 3 types. They are solid slab, flat slab and ribbed slab.

Solid slabs refers to slabs supported by beams or walls. They are economical up to spans of 5m

Flat slab refers to  a slab supported by columns only. The main advantage flat slabs have over solid slabs is that they are easier to construct. Also, widows can extend to the underside of a flat slab since it doesn’t require beams for support. The main disadvantage for these types of slab is that they are prone to punching shear. Flat slabs are economical for spans ranging from 5m to 9m.

Ribbed slabs are Slabs in which part of the slab in the tension zone has been omitted thereby reducing its weight considerably. They have higher formwork costs compared to other slab systems.Ribbed slabs are economical for spans ranging from 8m – 12m. 

Types of Insitu tests for density and shear strength of soil

Types of Tests to obtain the density and shear strength of soils include;

Standard penetration

Vane

Unconfined compression test

Plate bearing test

California bearing ratio test (CBR)

Standard penetration test

This test measures the relative density of the soul. lt is done by driving a 35mm diameter rod into the soil. The number of controlled blows it takes for the rod to be driven to a particular depth is measured and then the result is checked against a standard chart to determine the strength and classification of the soil.

Vane test

Vane test

This test measures the shear strength of soft cohesive soils. The vane is pushed into the soil and rotated by hand. The amount of torque necessary for rotation is measured and then shear strength of the soil is calculated

Unconfined compression test

unconfined compression test equipment

This is a device used to measure the compressive strength of soil.  A soil sample of 76mm long and 38m in diameter is placed in the apparatus under applied loading. The the sample is then sheared under load & shear stress us automatically recorded

Plate bearing test

plate bearing test

This process involves digging a portion of the ground to foundation level. Placing a 600mm by 600 mm steel plate at the bottom and subjecting it to vertical loading. A displacement meter is used to check the displacement of the steel plate into the soil. This is done until the place has sinked in the soil to about 15% of the breadth of the steel plate. The safe load is taken as ⅓ of the load that caused the steel plate to be vertically displaced by 15% of its breadth. Readings are taken every 6 hours and increments to loading are done every 24 hours.

California bearing ratio test (CBR)

The test shows the load penetration of soils relative to that of a standard crushed stone sample. It is stimulated in the laboratory by driving a cylindrical plunger of known cross sectional area  into the soil at a given rate.

Types of samples in soil investigation related to civil engineering works

Soil samples

The two types of soil samples collected for further study are;

  • Disturbed sample
  • Undisturbed sample

Disturbed Soil Sample

As the name implies; the sample is disturbed by the method used in collecting it. Soil samples collected using the auger, percussion or rotary drill, shell or by hand can result in samples in which their natural structure has been substantially interfered with. Such samples are therefore generally useful for determining moisture content and visual grading.

Undisturbed Soil Sample

These are samples removed by methods which preserve as far as possible the natural structure and properties of the soil material. Methods of collecting undisturbed samples include, by hand, core drilling.

Of the two types of soil samples discussed so far, the undisturbed soil sample gives the best test results.

Methods of Soil Exploration

Machine operated soil drill

Borings are the main procedure used for soil exploration on site. There are different types of borings. They can generally be grouped into 3 types, they are:

  • Trial pits of up to 3m deep
  • Borings of up to 30m deep
  • Headings and Shafts

Trial pits

Trial pits are pits of about 1.2m by 1.2 m. They can be dug with the use of labour or excavating machines. Trial pits should be dug  at distances of 20m apart and clear of the positions of the building’s foundation. Trial pits are economical up to a depth of 3m. Trial pits are suitable for small-scale works. The main advantage of this method is that soil and rock samples can easily be exposed and examined.

Deep boring.

This is the process by which hand or mechanical/machine boring tools are used to drill holes into the soil for investigative purposes. Deep borings are suitable for medium to  large scale works such as high rise buildings.

Hand or mechanical auger borings are cheap methods of boring soils which can stand unsupported. Holes can be sunk up to 3m. For soils that cannot stand on their own such as loose soils, steel casing can be dug surrounding the bored area.

Boring rig

A steel casing called shell is used to collect soil samples for testing.

Auger sampling shell

It is done at regular intervals as hole is being dug.

For deeper borings of up to 30m, machine operated drilling must be employed. They are of many types. There is the percussion boring in which the soil formation is broken up by repeated blows after which water is added to the hole as the work proceeds. The water helps to wash out the soil debris by pressure washing or by shell auger. The samples collected by this method are disturbed samples because of the repeated blows to the soil formation.

Wash boring method uses only a strong jet of water or drilling mud to bore deep holes in the soil. This method produces undisturbed samples. However, it is suitable if the soil does not contain boulders or large gravels.

Rotary drilling is a method used for boring of soil with rock formations. The drilling bit is fitted with industrial diamond to break the rock formations. Drilling mud, water or compressed air is jetted into the drilled hole to flush out the rock debris.

Headings and Shafts.

Headings and Shafts are employed to explore steeply dipping strata. It is best suited for very large scale works such as dams, power stations etc. The method is used to create exploration tunnels. The advantage they have over pits drills is that they can be easily drained of water and also easy removal of soil debris. Shafts are bored using large power driven augers. The sides of the shafts must be supported to protect the work personnel engaged in soil inspection.

Choice of soil exploration methods.

The choice of different exploration methods will depend on the following factors:

  • Topography.
  • Nature of ground.
  • Cost.