An Engineering Guide to the Design of Post-Tensioned Concrete Slabs

Section 1.0: Principles of Post-Tensioned Slab Systems

Prestressed concrete represents a significant advancement in structural engineering. The core principle is the intentional creation of permanent internal stresses in a structure to counteract the stresses that will be induced by external loads. Concrete is strong in compression but weak in tension. Prestressing actively uses concrete's compressive strength while mitigating its weakness.

In post-tensioning, high-strength steel tendons are housed in ducts within the concrete formwork. After the concrete gains sufficient strength, these tendons are tensioned with hydraulic jacks and mechanically anchored, transferring a compressive force into the concrete. This "pre-compression" must be overcome by tensile stresses from applied loads before any net tension can develop, leading to superior performance with minimal cracking.

1.2 System Components and Materials (Indian Context)

A post-tensioning system is an assembly of specialized components. Standard materials available in the Indian market include:

• Tendons: High-strength, seven-wire steel strands, typically conforming to IS 14268 with an ultimate tensile strength ($f_{pu}$) of 1860 MPa. A common size is 12.7 mm diameter.
• Ducts (Sheathing): For unbonded systems, common for building slabs, the tendon is coated in grease and encased in a seamless plastic sheath. For bonded systems, corrugated metal or HDPE ducts are used as per IS 1343.
• Anchorages: A critical assembly of an anchor head, wedges, and a bearing plate that transfers the tendon force to the concrete.
• Concrete: As per IS 1343, a minimum grade of M30 is required for post-tensioned work. Commonly, M35 or M40 grades are used for better durability and performance. A minimum strength at transfer ($f'_{ci}$), often 21-25 MPa, is specified.

Section 2.0: The Load Balancing Method

The design of post-tensioned slabs is governed by the "load balancing method." This approach replaces the effect of a draped tendon with an equivalent upward external load. This upward force is used to "balance" a portion of the downward-acting gravity loads.

2.1 Conceptual Framework

The design begins by selecting a target percentage of the gravity load to be balanced. For two-way floor slabs, it is customary to balance between 60% and 80% of the slab's self-weight. The portion of the load not balanced is the "unbalanced load," and the slab is analyzed for this net load. The analysis must distinguish between primary moments (from the PT force at an eccentricity, Pe) and secondary (hyperstatic) moments which arise in continuous, indeterminate structures.

Section 3.0: Systematic Design Approach (as per IS Codes)

The design is an iterative process that synthesizes analysis, code requirements, and practical considerations.

1. Preliminary Sizing: Establish an initial slab thickness using a span-to-depth ratio (e.g., L/h = 40 to 45).
2. Load Determination: Calculate dead, superimposed dead, and live loads as per IS 875 (Parts 1 & 2).
3. Tendon Force Selection (Iterative): Use the load balancing method to find the required PT force. Check the resulting average precompression ($P/A$) against allowable limits (typically 0.8 MPa to 2.5 MPa). If outside this range, adjust the slab thickness and repeat.
4. Prestress Loss Calculation: Account for immediate and long-term losses as per IS 1343, Clause 19.5.
5. Final Analysis: Use software to find design moments and shears for all load combinations specified in IS 456.

Section 4.0: Design Verifications per IS 1343 & IS 456

The design must be verified against code requirements at two limit states: Serviceability and Ultimate.

4.1 Serviceability Limit State (SLS) Checks

• Flexural Stress Analysis: Stresses in the concrete are checked at transfer and service against allowable limits in IS 1343, Table 10.
• Crack Control: IS 1343 defines three types of prestressed structures based on allowable tensile stress, which dictates crack control requirements.
• Deflection Analysis: Deflections are checked against limits in IS 456, Clause 23.2.

4.2 Ultimate Limit State (ULS) Checks

• Flexural Strength ($\phi M_n \geq M_u$): The factored moment resistance must exceed the factored applied moment.
• Shear Strength ($\phi V_n \geq V_u$): One-way and two-way (punching) shear must be checked as per IS 456, Clause 31.6. Punching shear is often the controlling failure mode.

Section 5.0: Understanding Key Design Parameters

The interaction between Slab Thickness, Tendon Drape, and Prestressing Force is the core of post-tensioned slab design. This section explains these critical parameters.

Slab Thickness (h)

• What it is: The vertical depth of the concrete slab. This is the first parameter estimated in the design process, typically using Calculator 1.
• Why it's important: The thickness is fundamental to the design. It dictates the slab's self-weight, its strength and stiffness against bending, its capacity to resist punching shear, and critically, the amount of space available to drape the tendons.
• How it's chosen: An initial estimate is made using a span-to-depth ratio (e.g., L/h = 42). However, this is just a starting point. The final thickness is determined iteratively. If later checks for stress (Calculator 4) or shear (Calculator 3) fail, the most common solution is to increase the slab thickness.

Tendon Drape (a)

• What it is: Post-tensioning tendons are not flat; they are profiled to run near the top of the slab over columns and swoop down to run near the bottom of the slab at mid-span. The **drape** is the vertical distance between the tendon's high point and its low point.
• Why it's important: Drape acts as the lever arm for the prestressing force. A larger drape allows the tendon to create the required upward balancing force more efficiently. The load balancing equation, $P = (w_{bal} \times L^2) / (8a)$, shows that the required force `P` is inversely proportional to the drape `a`. A larger drape means less force is needed.
• What limits it: Drape is physically limited by the slab thickness and the required concrete cover (the minimum concrete needed to protect the tendon). The drape cannot be larger than the slab thickness minus the required top and bottom cover.

Prestressing Force (P)

• What it is: This is the total compressive force applied by the tensioned steel tendons within a design strip. This force squeezes the concrete, putting it into "pre-compression."
• Why it's important: This is the active force that makes the system work. It counteracts the slab's weight, controls deflections, and keeps the concrete in compression, preventing cracks under normal service loads.
• How it's calculated: The force is not arbitrary. As shown in Calculator 2, it is calculated based on the load to be balanced, the span length, and the available tendon drape. Once the total required force `P` is known, it is divided by the force per tendon (e.g., 112.5 kN) to find the required number of tendons.

The Design Loop

These three parameters are locked in an iterative cycle. You start with a guess for **Slab Thickness**, which defines the maximum **Tendon Drape**. These two values then determine the required **Prestressing Force**. You then check if the resulting precompression (`P/A`) is within acceptable limits. If it's too high, you must go back and **increase the slab thickness**, which starts the cycle over again until a balanced and efficient design is achieved.

Section 6.0: Comprehensive Design Example (IS Codes)

This section applies the principles to a representative interior bay using IS code provisions.

6.1 Problem Statement

• Bay Dimensions: 9.14 m (E-W) x 7.62 m (N-S)
• Materials: Concrete M35 ($f_{ck}=35$ MPa), PT Tendons = 12.7 mm dia. (112.5 kN effective force/strand)
• Loads: SDL = 1.0 kN/m², LL = 2.4 kN/m²
• Design Targets: Balance 75% of DL, P/A between 0.8 and 2.5 MPa.

6.2 Design Steps

A preliminary thickness of 200 mm is tried, but the precompression is too high (2.73 MPa). The design is iterated, and a final thickness of 230 mm is selected.

For the 230 mm slab, the required force is 3935 kN, requiring 35 tendons. The average precompression is P/A = 2.25 MPa. This is within the acceptable range.

From analysis, the moments at the interior support are: M_dl = -292 kN-m, M_ll = -128 kN-m. The balancing moment M_bal is +329 kN-m. These values can be entered into the Flexural Stress Check calculator to verify serviceability.

Section 7.0: References

This guide is compiled from industry-standard sources, primarily IS 1343:2012, IS 456:2000, and related engineering literature.