A simply supported rectangular concrete beam of span 8m has to be pres...
Given data:
- Span of the beam = 8m
- Prestressing force = 1600kN
- External uniformly distributed load = 30kN/m
- Tendon profile: parabolic with zero eccentricity at supports
To determine the maximum dip of the tendon at the mid-span to balance the external load, we need to consider the equilibrium condition of the beam.
1. Calculation of the total external load:
The total external load acting on the beam can be calculated by multiplying the intensity of the load (30kN/m) by the span of the beam (8m):
Total external load = 30kN/m x 8m = 240kN
2. Calculation of the centroid of the external load:
The centroid of the external load is located at the mid-span of the beam. Since the load is uniformly distributed, the centroid can be calculated by dividing the total load by 2:
Centroid of external load = 240kN / 2 = 120kN
3. Calculation of the eccentricity at mid-span:
Since the tendon has a parabolic profile with zero eccentricity at the supports, the eccentricity at mid-span will be half of the maximum eccentricity. To calculate the maximum eccentricity, we can use the formula for the parabolic profile:
Maximum eccentricity = (4/5) x (span/2) = (4/5) x (8/2) = 3.2m
Eccentricity at mid-span = 3.2m / 2 = 1.6m
4. Calculation of the maximum dip of the tendon:
To balance the external load, the prestressing force should generate a moment that is equal in magnitude and opposite in direction to the moment caused by the external load. The moment caused by the external load can be calculated by multiplying the centroid of the load by the span of the beam:
Moment caused by external load = 120kN x 8m = 960kNm
To create an equal and opposite moment, the tendon should have a dip at mid-span. The relationship between the dip of the tendon and the moment generated by the prestressing force can be expressed as:
Dip of the tendon = Moment caused by external load / Prestressing force
Dip of the tendon = 960kNm / 1600kN = 0.6m
However, since the tendon has a parabolic profile, the actual dip at mid-span will be half of the calculated value:
Actual dip of the tendon at mid-span = 0.6m / 2 = 0.3m
Therefore, the maximum dip of the tendon at the mid-span to balance the external load is 0.3m or 0.15m (rounded to two decimal places).