A concrete beam is prestressed by a cable with an initial stress of 1000 N/mm2 in the wires. The grade of concrete in the beam is M-50. The beam is located in an area having a relative humidity of 50 per cent. The beam is exposed to the environment on three sides having a depth of 400 m and a width of 300 mm. The beam was cured for seven daysbefore it was prestressed. Using the Indian Standard Code method, estimate the loss of stress in steel due to shrinkage of concrete at the age of (a) 28 days and (b) 70 years. Assume modulus of elasticity of steel as 210 kN/m
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The correct answer and explanation is:
iturn0image0turn0image2turn0image6turn0image8In prestressed concrete structures, shrinkage of concrete leads to a reduction in the prestressing force over time, known as the loss of prestress. This phenomenon occurs as the concrete contracts, imposing additional strain on the prestressing steel. The extent of this loss is influenced by factors such as the concrete mix, environmental conditions, and the duration since prestressing.
Shrinkage Strain Calculation:
According to IS 1343:2012, the total shrinkage strain (ε_sh) comprises two components: autogenous shrinkage strain (ε_ca) and drying shrinkage strain (ε_cd). The standard provides approximate values for these strains based on the grade of concrete and environmental conditions.
- Autogenous Shrinkage Strain (ε_ca): For M-50 grade concrete, the autogenous shrinkage strain is approximately 75 × 10^-6. citeturn0search2
- Drying Shrinkage Strain (ε_cd): The drying shrinkage strain depends on factors such as the relative humidity and the notional size of the member. The notional size (h_n) is calculated as twice the cross-sectional area divided by the perimeter exposed to drying. For a beam with a width (b) of 300 mm and a depth (d) of 400 mm, exposed on three sides, the notional size is: h_n = 2 × (b × d) / (b + 2d) = 2 × (300 mm × 400 mm) / (300 mm + 2 × 400 mm) = 343 mm. For a relative humidity of 50% and h_n of 343 mm, the drying shrinkage strain (ε_cd) can be estimated using the standard’s guidance, resulting in approximately 240 × 10^-6. citeturn0search2
Total Shrinkage Strain (ε_sh):
ε_sh = ε_ca + ε_cd = (75 + 240) × 10^-6 = 315 × 10^-6.
Loss of Prestress Due to Shrinkage:
The loss of prestress (Δσ_sh) due to shrinkage is calculated by multiplying the total shrinkage strain by the modulus of elasticity of the prestressing steel (E_s):
Δσ_sh = ε_sh × E_s = 315 × 10^-6 × 210,000 N/mm² ≈ 66.15 N/mm².
Considerations Over Time:
Shrinkage in concrete is a time-dependent process, with most shrinkage occurring within the first few months after casting. However, some shrinkage continues over years, albeit at a diminishing rate. For practical purposes and as per standard guidelines, the majority of shrinkage is considered to have occurred by 28 days. Therefore, the loss of prestress due to shrinkage at 28 days is approximately 66.15 N/mm². Beyond this period, additional shrinkage-induced losses are minimal and often neglected in standard design calculations.
Conclusion:
Based on the provided data and standard calculations, the estimated loss of stress in the prestressing steel due to shrinkage of concrete is approximately 66.15 N/mm² at 28 days. This value remains largely unchanged at 70 years, as the majority of shrinkage occurs within the initial months after casting.
It’s important to note that these calculations are based on standard assumptions and approximations. Actual shrinkage and corresponding prestress losses can vary based on specific material properties, environmental conditions, and construction practices.
