The formula for effective prestress force is a fundamental equation in the design and analysis of prestressed concrete structures, representing the force remaining in the prestressing tendons after all losses have occurred.
The formula for effective prestress force is:
$$Np = \sigma{pe} A_p$$
Understanding the Components of the Formula
To fully grasp the effective prestress force, it's essential to understand each component of the formula:
- $N_p$ (Effective Prestressing Force): This is the actual axial force carried by the prestressing steel that remains effective in the concrete member. It is the force that counteracts external loads and improves the structural performance of the concrete element.
- $\sigma_{pe}$ (Effective Prestress Stress): This represents the stress in the prestressing steel after accounting for all types of instantaneous and time-dependent losses. It is a critical value, as it directly influences the magnitude of the effective prestressing force.
- $A_p$ (Area of Prestressing Steel): This is the total cross-sectional area of all the prestressing tendons, wires, or strands used within the concrete member.
The Significance of Effective Prestress Stress ($\sigma_{pe}$)
The effective prestress stress, $\sigma{pe}$, is not the initial stress applied during tensioning. Instead, it is the result of deducting various "losses" from the initial stress. These losses occur due to several factors, both immediately upon tensioning and over the service life of the structure. Understanding these losses is crucial for accurately determining $\sigma{pe}$.
Types of Prestress Losses
Prestress losses are categorized into instantaneous and time-dependent losses:
- Instantaneous Losses: These occur at the time of prestressing or concrete transfer.
- Elastic Shortening: The concrete shortens elastically when the prestressing force is applied, leading to a reduction in tendon stress.
- Friction Losses: In post-tensioned systems, friction between the tendon and the duct, especially around curves, reduces the force transmitted along the tendon.
- Anchor Set Losses: A small amount of slippage occurs when the tendon is seated in the anchorage device, resulting in a slight reduction of force.
- Time-Dependent Losses: These occur gradually over the life of the structure.
- Creep of Concrete: Concrete slowly deforms under sustained compressive stress, causing the concrete to shorten further and reducing the prestress in the tendons.
- Shrinkage of Concrete: As concrete dries and hardens, it undergoes volumetric changes (shrinks), which shortens the concrete member and reduces the prestress.
- Relaxation of Steel: Prestressing steel, under high sustained tension, experiences a gradual loss of stress over time, even if the strain remains constant.
The following table summarizes these common types of prestress losses:
| Type of Loss | Description | Impact on $\sigma_{pe}$ |
|---|---|---|
| Elastic Shortening | Shortening of concrete due to the immediate application of prestressing force. | Reduces effective stress (immediate) |
| Friction | Loss of force due to friction between tendons and ducts, particularly significant in long or curved tendons (post-tensioning). | Reduces effective stress (immediate) |
| Anchor Set | Loss of force due to the slight slippage of the tendon at the anchorage during the transfer of prestress. | Reduces effective stress (immediate) |
| Creep of Concrete | Long-term deformation (shortening) of concrete under sustained compressive stresses from prestressing. | Reduces effective stress (time-dependent) |
| Shrinkage of Concrete | Reduction in concrete volume over time due to drying and chemical reactions, causing the member to shorten. | Reduces effective stress (time-dependent) |
| Relaxation of Steel | Gradual decrease in stress within the prestressing steel under constant strain over extended periods. | Reduces effective stress (time-dependent) |
The Role of Eccentricity in Prestressing
While the effective prestressing force ($Np$) quantifies the axial force, its application is rarely perfectly axial. Most prestressed concrete elements utilize eccentricity to induce beneficial bending moments. For a parabolic tendon, the eccentricity, often denoted as $e{px}(x)$, refers to the varying distance between the centroid of the prestressing steel and the centroidal axis of the concrete member at any given point $x$ along the member.
This eccentricity is crucial for:
- Generating Bending Moments: An eccentric force creates an internal bending moment ($M_p = Np \cdot e{px}(x)$) that effectively counteracts the moments caused by external applied loads (like gravity). This moment helps to control tensile stresses and deflections in the concrete.
- Equivalent Loads: In design, the effect of an eccentric prestressing force can be visualized as an "equivalent load." For instance, a parabolic tendon with varying eccentricity creates an equivalent upward distributed load, a concept often used in load balancing design to simplify analysis.
Practical Implications
The accurate calculation of effective prestress force is paramount for several reasons:
- Structural Integrity: Ensures the concrete remains in compression or within allowable tensile limits under service loads, preventing cracking and enhancing durability.
- Deflection Control: The magnitude and eccentric application of $N_p$ are carefully designed to control deflections, making structures stiffer and more serviceable.
- Optimized Material Use: By leveraging the high strength of prestressing steel and the compressive strength of concrete, prestressing allows for more slender and efficient structural members.
Understanding the formula for effective prestress force and the factors influencing its components is fundamental for any engineer involved in the design and construction of prestressed concrete structures.
