Calculating Load Balancing Cable Eccentricity for Prestressed Beams

The question asks us to determine the eccentricity required for a load balancing cable in a prestressed concrete beam. Load balancing is a concept where the upward force components from the draped cable are designed to exactly counteract the downward external loads on the beam, effectively creating a state where the beam experiences minimal bending due to those specific loads.

We are given the following information:

• Beam size: 350 mm × 750 mm
• Span (L): 9 m
• Live load (wl): 10 KN/m
• Prestressing force (P): 1700 KN
• Location: Centre of the beam

To achieve load balancing, the bending moment caused by the prestressing force eccentricity must counteract the bending moment caused by the external loads at the point of interest (the centre of the beam in this case). The external loads consist of the live load and the beam's own self-weight (dead load).

Step 1: Calculate the Self-Weight (Dead Load) of the Beam

The self-weight depends on the cross-sectional area of the beam and the density of concrete. Assuming a standard density for reinforced concrete is approximately 25 kN/m³.

Cross-sectional area (A) = Width × Depth

\( A = 0.350 \, \text{m} \times 0.750 \, \text{m} = 0.2625 \, \text{m}^2 \)

Self-weight per unit length (wd) = Area × Density

\( w_d = 0.2625 \, \text{m}^2 \times 25 \, \text{kN/m}^3 = 6.5625 \, \text{kN/m} \)

Step 2: Calculate the Total Uniformly Distributed Load (UDL)

The total UDL acting on the beam is the sum of the live load and the self-weight.

Total load (w) = Live load (wl) + Self-weight (wd)

\( w = 10 \, \text{kN/m} + 6.5625 \, \text{kN/m} = 16.5625 \, \text{kN/m} \)

Step 3: Calculate the Maximum Bending Moment at the Centre

For a simply supported beam subjected to a uniformly distributed load, the maximum bending moment occurs at the centre of the span.

Maximum bending moment (M) = \( \frac{wL^2}{8} \)

\( M = \frac{16.5625 \, \text{kN/m} \times (9 \, \text{m})^2}{8} \)

\( M = \frac{16.5625 \times 81}{8} \)

\( M = \frac{1341.5625}{8} \)

\( M \approx 167.695 \, \text{kN-m} \)

Step 4: Determine the Required Eccentricity for Load Balancing

At the centre of the beam, the bending moment caused by the eccentric prestressing force (Mp) is given by \( M_p = P \times e \), where P is the prestressing force and e is the eccentricity.

For load balancing at the centre, the moment due to prestressing must equal the bending moment due to the external loads:

\( P \times e = M \)

Now, we can solve for the required eccentricity (e):

\( e = \frac{M}{P} \)

\( e = \frac{167.695 \, \text{kN-m}}{1700 \, \text{kN}} \)

\( e \approx 0.09864 \, \text{m} \)

Step 5: Convert Eccentricity to Millimetres

The options are given in millimetres, so we convert the calculated eccentricity from meters to millimetres.

\( e = 0.09864 \, \text{m} \times 1000 \, \text{mm/m} \)

\( e \approx 98.64 \, \text{mm} \)

The calculated eccentricity for load balancing at the centre of the beam is approximately 98.64 mm.

Comparing this value with the given options, 98.6 mm is the closest value.