Post tension concrete slabs are a type of prestressed concrete structure, wherein high-strength steel cables or wires are used to apply compressive force to concrete slabs before its service life. This prestressing force counteracts the tensile stress that would otherwise develop in the concrete and also reduces or eradicates deflection which often times is the governing criteria in non-prestressed slabs. Post tension slabs are typically used in the construction of bridges, parking structures, high-rise buildings, slab on expansive soil, and other structures where enhanced durability and strength are required.
Advantages of PT Slabs
Post-tensioned slabs have numerous advantages over reinforced concrete slabs. Some of these advantages are enumerated below:
- Thinner Slabs: Post-tension Slabs are always substantially thinner than Reinforced Concrete slabs, this consequently result to overall lighter structures which also consequently impact positively on cost
- Reduced storey height: Slabs are thin and also beams can be omitted altogether thereby reducing storey height. This is of specific advantage in multi-storey buildings as the overall building height is reduced.
- Reduce Shrinkage: Post-tensioning reduces shrinkage cracking-therefore no joints, or fewer joints, are needed
- It allows us to build slabs on expansive or soft soils
- Increased floor Spans: Post tensioned slabs span much greater length without support. This leads to reduction in number of required columns and in-turn few obstructions
- Low serviceability issues: Post-tension Slabs have low serviceability issues like deflection and cracking
- Highly durable: Since there is low serviceability issues, PT slabs are more durable and require very much less maintenance. Cracks are also reversible preventing corrosion of steel which is the foremost durability concern in concrete structures.
Post-tensioning with Bonded or Unbonded Tendons
Post tensioned slabs are design and constructed using either bonded or unbonded tendons. Either of the two have gained tractions in certain countries due to old practice and local experience, however some engineers make informed decision of choosing one over the other depending on merit of use. We shall discuss the two types of tendons and also their advantages and disadvantages.
Bonded Tendons
In bonded tendons, post-tensioned strands are installed in plastic ducts that have been laid in-place in the required profile and have been cast together with the concrete. At the ends a combined anchorage casting is provided which anchors all of the strands within the duct. The strands are stressed after installation and then grouted with concrete suspension to form a bond with the surrounding hardened concrete. The duct together with the strands contained within are collectively called a tendon.
Advantages of Bonded Tendons
- Local failure due to abrasion, accidental cut, explosion, earthquake, etc. has no or limited effect.
- Higher flexural strength and better crack control than unbonded tendons
Disadvantages of Bonded Tendons
- Desired maximum eccentricity cannot be achieved due to the relative difference in the centroid of the duct and tendons when installed. Flat ducts are sometimes adopted over circular ones to limit this effect.
- It is more expensive than unbonded tendons.
Unbonded Tendons
In an unbonded system mono-strands are encapsulated in a plastic sheath and the voids between the sheath and the strands are filled with a rust-inhibiting grease. The sheath and grease are applied under factory conditions, and they prevent the strands from abrasion, corrosion and of course bonding with the concrete as earlier mentioned. They also reduce friction between the strands and sheath during stressing operation.
During construction, the individual tendons are anchored at each end with anchorage castings. The tendons are cast into the concrete section and are jacked to apply the required prestress force once the concrete has achieved the required strength.
Advantages of Unbonded Tendons
- Maximum eccentricity desired can be achieved due to the absence of duct
- Installation of tendon is fast and easy.
- Grouting is absence hence fewer labour activities
- More economical
Disadvantages of Unbonded Tendons
- Local failure due to abrasion, accidental cut, explosion, earthquake, etc. renders the tendon completely useless.
- It leads to reduced ultimate strength of slab and lower crack control
Types of Post-tensioned Slabs
Like reinforced concrete slabs, post-tensioned slabs are broadly categorized as one-way and two-way slabs base on aspect ratio and behaviour under loads.
One Way Post Tensioned (PT) Slabs
One-way slabs span and majorly resist bending in one direction and are supported on beams or walls. This one-way behavior is induced in slabs when the ratio of their longer lengths to that of their shorter lengths are greater than 2 ( ie: mathematically: ly/lx ≥ 2). According to clause 5.3.1(5) of EN 1992-1-1-2, a slab can also be considered as one-way if it contains two free and sensibly parallel edges. Conversely, even when 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.
One-way slabs can either be single span or continuous over supports. If the supports are beams, they can either be standard or banded beams. Band beams are wide shallow beams that have their width to be smaller than their depths.
The prestressing cables in one-way slabs run in the span direction between beams or walls. In the transverse direction, unstressed reinforcement is used to control cracking due shrinkage and thermal stress.
Two-way Post Tensioned (PT) Slabs
Two-way slabs span and resist moment in both directions. This behavior is induced in slabs when the ratio of their longer lengths to that of the shorter lengths are less than 2 (i.e: mathematically: ly/lx ≤ 2). Two-way slabs are supported on all sides or are supported on discrete columns so that they are called flat slabs.
Flat slabs are majorly the type of two-way slabs used in practice. They provide uncluttered flat soffit for easy unobstructive installation of services and better illumination. When flat slabs have uniform soffit then they are called flat plate. Sometimes, owing to the fact that flat slabs transfer loads to supporting columns via a narrow area, shear stress around these columns are extremely high which necessitate increasing the thickness of the slabs around columns to prevent punching shear failure. Another alternative to thickening the slab around columns is to increase the size of the column head near the slab. When flat slabs have local thickening around columns, they are called flat slabs with drops.
Preliminary Sizing and Structural Schemes
Determining the structural scheme involves picking a type of slab and the arrangement of the elements depending on the slab type. This decision is influenced by various factors such as span, economy, loading, aesthetic, architectural form, objective of building, special requirement, etc.
Having determined the type of slab, what next is to determine the thickness of the slab. The slab thickness must be such that the requirement of strength, deflection, and vibration are met. The concrete society in the publication “Post tensioned Concrete Floors” provided graphs and tables for preliminary sizing of various types of post tensioned slabs. These graphs can be adopted as tools for determining the preliminary thickness of slab, Preliminary shear check for slab thickness at internal column, and Ultimate shear check for flat slab at face of internal column. The publication also provides a table that can be used in determining the span and thickness of all types of post-tensioned slab, the table is reproduced below
Estimation of Prestress Force
Having determine the structural scheme and slab thickness, the amount of prestress necessary to achieve the desired slab behaviour should be estimated. This is done by using the load balancing method where a percentage of the slab load is balanced off. This is done by controlling the prestress and the tendon drape in the spans. It is however desirable to achieve the desired slab behaviour by increasing the tendon drape rather than subjecting the slab to high prestress which might lead to axial shortening thereby causing distress in supporting walls and columns.
Care should also be taken when balancing external loads to avoid over balancing – this is a scenario where more than the load present on the slab are balanced which may lead to uplifting of the slab from the supports and bursting of the top fiber of the slab. Generally, to avoid over-balancing, the live load and additional/supper dead load (partition loads, finishes, etc.) are not to be balanced as they would not be present during the stressing operation
Another issue to avoid is under-balancing where the prescribed prestress is too low to prevent unsightly cracks from developing in the slab. Should this happen, it would have defeated two of the major objectives of post-tensioned construction which are: Improved serviceability and enhanced durability.
As a good practice, balanced load should be between 70 – 100% of the dead load, likewise, as recommended by the concrete society, the average prestress should be 0.7MPa to 3MPa. The prestress can however be as high as 6MPa for ribbed and waffle slabs. These recommendations most times keep the stresses in the slab within limits, reduce deflection, and also reduce or eliminate cracks.
Balancing Loads in One-way and Two-way Slabs
Balancing loading in one-way slab is straight forward. The cables are placed to run in a single direction to balance the external transverse loads using the expression below:
w = P x 8e/l²
Unstressed reinforcements are provided in the other span against cracks due to thermal and shrinkage stress.
As for two-way slabs supported by beams or walls, the external transverse loads are to be balanced in both directions. The proportion of the loads to be balanced in either direction depends on the engineer’s discretion as long as statics is obeyed. However, Gilbert et al recommends that the expression below is adopted to estimate the proportion of the balanced load to be balanced in the short direction.
$$
w_{py}\,\,=\,\,\frac{l_x^4}{\delta l_y^4\,\,+\,\,l_x^4}\,\,w_{unbal}
$$
Where δ depends on the support conditions and is given by:
- for 4 edges continuous or discontinuous
- 0 for 2 adjacent edges discontinuous
- 0 for 1 long edge discontinuous
- 5 for 1 short edge discontinuous
- 5 for 2 long +1 short edge discontinuous
- 4 for 2 short +1 long edge discontinuous
- 0 for 2 long edges discontinuous
- 0.2 for 2 short edges discontinuous
According to statics, the proportion of the load to be balanced in the longer direction equals:
wpx = wbal – wpy
Having determine the amounts of load to balance in each span, then the equation for a parabolic tendon can be used to estimate the amount of prestress in both directions
As for flat slabs, the full balanced load is to be balanced in each orthogonal span (longitudinal and transverse) and each span is to be considered separately.
Analysis of Post-tensioned Slabs
There are several techniques that can be deployed to analyse post-tensioned slabs. These includes: Moment coefficient method, moment distribution method, equivalent frame method, grillage Analysis, and finite element analysis. The propriety of any of this analysis technique depends on slab types, slab geometry, loading regime on the slab, and the preference of the designer
Analysis of One-way, and Two-way beam and slab systems
One-way post-tensioned beam and slab systems can be analysed as a continuous beam with a fixed width using moment-distribution method. Also, the moments coefficients table in BS 8110-1-1997 or similar tables in Eurocode 2 compliant materials can also be adopted provided that the attendant conditions peculiar to those tables are met.
Two-way beam and slab system can also be analysed using moment distribution method with each orthogonal span of the slab considered separately. Two-way slabs can also be analysed using moment coefficients tables provided that the attendant conditions peculiar to those tables are met.
Using any of the analysis techniques above, the equivalent load due to prestressing, permanent loads, and live loads are analysed separately for each relevant spans and their effects of actions derived are used to design the slabs considering the interactions between these effects.
Generally, the finite element analysis and grillage analysis can also be used to analyse post-tensioned beam and slab system using advanced computer software.
Analysis of Flat Slab
Analysis of post-tensioned flat slabs are often undertaken using any of the following analysis techniques
- equivalent frame analysis
- Grillage analysis
- Finite element analysis
Equivalent frame analysis is a traditional analysis method where the flat slab system is divided into discrete frames consisting of columns and slab strips in both longitudinal and transverse spans of the slab. Each orthogonal span (longitudinal and transverse) is considered separately with the full load acting on it and the resulting frame is analyzed like that of a multi-span frame or continuous beam using elastic method like moment distribution method.
Grillage and finite element methods are advanced analytical techniques often deployed to analyse flat slabs with complex geometry and irregular loadings.
Verification of Post-tensioned Slabs for Limit States
Having sized and analysed post-tensioned slabs for relevant forces, the next and conclusive design step is to verify the slab for relevant limit states. This is discussed briefly as follows:
Servicecerbility Limit State
The serviceability limit state of stress, cracking and deflection are checked for the post-tensioned slabs. This are discussed further below:
Stress
The permissible stress ultimately controls the quantity of tendons, amount of prestress, and drapes of tendons in post-tensioned concrete design. Although the initial estimation of prestress and tendon drapes are governed by the percentage of permanent load on the slab to be balanced, after stress check and it is observed that the stress in the slab exceeds the limiting stress in the slab, then the amount of prestress that would achieve stresses below the limiting stresses shall govern. The slab is checked for limiting stress at critical sections such as mid-span and over supports at both transfer condition and service condition.
Stress Check
σb = $\left(\frac{-P}{A}+\frac{M_{unb }}{Z_b}\right) $
σt = $\left(\frac{-P}{A} – \frac{M_{unb }}{Z_t}\right) $
The unbalance moment at transfer is the difference in moment caused by prestress and moment caused by the self-weight of the slab. While the unbalance moment at service is the difference in moment caused by prestress and moment caused by the self-weight and imposed load on the slab, this is equivalent to stress in the slab under normal working condition.
Cracking
At sections where tensile stress exceeds the permissible stress, the designer has to assess whether the crack widths are within the allowable limits provided in table 7.2N in EN 1992-1-1. Where the crack width is within the permissible limits, then the design is deemed satisfactory and no further cracking check is required. However, if the crack width exceeds the permissible limit, then the section has to be resized or un-tensioned bonded reinforcement may have to be provided to limit the cracks.
Ultimate limit State
Having determine the amount of prestressing steel required for serviceability limit states, then the design for ultimate strength shall be considered. The slab shall be checked for flexural strength and shear strength.
Flexural Strength
Verification of flexural strength is generally carried out using the provided tendons for serviceability at both mid-spans and over supports of the slab. When the strength is inadequate, then the tendons are supplemented with non-prestressed reinforcement to augment the strength of the member. It is always more economical if only the span over interior supports requires additional non-prestressed reinforcement as thelength of slab associated with the high local moment at each interior support is relatively small, so that only short lengths of non-prestressed reinforcement are usually required.
To calculate the ultimate moment on the slab, the ultimate moment caused by dead and live load are added to the secondary moment caused by prestressed force. The secondary moment from prestress action is treated as an external load with a partial load factor of 1.0
Shear Strength
Shear is rarely critical for prestressed beam and slab systems due to favorable effect of prestress action. They often have the shear strength of the slab without reinforcement to be greater than the shear force. The shear strength should be checked for both cracked and uncracked section of the slab at critical sections.
Another form of shear critical but common with flat slabs is punching shear. This occurs as a result of concentrated load or reaction acting on localized area of a slab. The design of punching shear for post-tensioned concrete slab is similar to that of reinforced concrete slabs. For more details on punching shear design read ‘Punching Shear Design to Eurocode 2’ and ‘Design of Punching Shear to Eurocode 2 – Worked Example’
References
Post tensioned Concrete Floors By Concrete Society
Design of Prestressed Concrete to Eurocode 2 by Gilbert, Mickleborough, & Ranzi
