Understanding Prestressed Concrete Beam Stresses

In prestressed concrete beams, the primary goal is to introduce compressive stresses into the concrete to counteract the tensile stresses induced by external loads (dead load and live load). This improves the beam's load-carrying capacity and durability.

The resultant stress at any point in the beam is the algebraic sum of stresses caused by:

• The prestressing force (P).
• The bending moment resulting from the eccentricity (e) of the prestressing force.
• The bending moment due to the dead load (M_g).
• The bending moment due to the live load (M_q).

Calculating Stress at the Top Fiber at Mid-Span

We need to determine the stress at the top fiber specifically at the mid-span of the simply-supported beam.

Stress due to Prestressing Force (P)

The prestressing force P acts axially on the concrete section. This results in a uniform compressive stress across the entire cross-section:

Direct Stress = \( \frac{P}{A} \)

Where:

• P is the prestressing force.
• A is the area of the cross-section.

Stress due to Eccentric Prestressing (e)

When the prestressing force P is applied with an eccentricity e, it creates a moment \( M_p = P \times e \). This eccentric moment causes bending stress in the concrete section. At the mid-span of a simply-supported beam, if the eccentricity is such that it causes sagging (e.g., tendon draped downwards), the moment \( P \times e \) induces tensile stress at the top fiber and compressive stress at the bottom fiber.

The magnitude of the bending stress at the extreme fiber is given by:

Bending Stress due to Eccentricity = \( \frac{M_p}{Z} = \frac{P \times e}{Z} \)

Where:

• e is the eccentricity of the prestressing force.
• Z is the section modulus of the cross-section.

Since this stress is tensile at the top fiber, it typically reduces the compressive stress or can even cause tension if it's large enough. Representing it as a negative contribution to the overall compressive stress, we consider it as \( -\frac{P \times e}{Z} \) when calculating the resultant compressive stress, or as a positive tensile stress term to be algebraically summed.

Therefore, the combined effect of prestressing (direct and eccentric) at the top fiber is often represented as:

Prestress Effect = \( \left( \frac{P}{A} - \frac{P \times e}{Z} \right) \)

Here, \( \frac{P}{A} \) is the compressive stress, and \( \frac{P \times e}{Z} \) is the magnitude of tensile stress at the top fiber.

Stress due to Dead Load Moment (M_g)

The bending moment due to the dead load (M_g) acting on a simply-supported beam typically causes compression at the top fiber and tension at the bottom fiber.

Bending Stress due to Dead Load = \( \frac{M_g}{Z} \)

This stress is compressive at the top fiber.

Stress due to Live Load Moment (M_q)

Similarly, the bending moment due to the live load (M_q) also causes compression at the top fiber of a simply-supported beam.

Bending Stress due to Live Load = \( \frac{M_q}{Z} \)

This stress is also compressive at the top fiber.

Resultant Stress Calculation

The resultant stress at the top fiber at mid-span is the algebraic sum of all these stresses. Assuming compressive stresses are represented as positive and tensile stresses as negative when summing up for the top fiber:

Resultant Stress = (Stress from Direct P) + (Stress from Eccentric P) + (Stress from M_g) + (Stress from M_q)

Resultant Stress = \( \left( \frac{P}{A} \right)_{\text{compressive}} + \left( -\frac{P \times e}{Z} \right)_{\text{tensile}} + \left( \frac{M_g}{Z} \right)_{\text{compressive}} + \left( \frac{M_q}{Z} \right)_{\text{compressive}} \)

If we consider the magnitudes and algebraic signs convention as implied by the options, where \( \frac{P}{A} \) is compressive, \( \frac{Pe}{Z} \) represents the magnitude of tensile stress at the top fiber due to eccentricity, and \( \frac{M_g}{Z} \) and \( \frac{M_q}{Z} \) are compressive stresses at the top fiber:

Resultant Stress = \( \left( \frac{P}{A} - \frac{P \times e}{Z} \right) + \left( \frac{M_g}{Z} \right) + \left( \frac{M_q}{Z} \right) \)

This formula represents the compressive stress from the prestressing force, reduced by the tensile stress from the eccentric prestressing component, and increased by the compressive stresses from the dead and live load moments, all at the top fiber.