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A pre-tensioned rectangular concrete beam 150 mm wide and 300 mm depth is prestressed with three straight tendons, each having a cross-sectional area of 50 mm2, to an initial stress of 1200 N/mm2. The tendons are located at 100 mm from the soffit of the beam. If the modular ratio is 6, the loss of prestressing force (in kN, up to one decimal place) due to the elastic deformation of concrete only is ______. 
    Correct answer is '4.8'. Can you explain this answer?
    Verified Answer
    A pre-tensioned rectangular concrete beam 150 mm wide and 300 mm depth...

    Stress = 1200 N / mm2


    Loss due to elastic deformation = m.f c = 6 × 5.33 = 31.98
    Prestress force = 31.98 × 3 × 50 = 4797 N = 4.8 KN
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    A pre-tensioned rectangular concrete beam 150 mm wide and 300 mm depth...
    Given data:
    - Width of beam (b): 150 mm
    - Depth of beam (d): 300 mm
    - Cross-sectional area of each tendon (A): 50 mm2
    - Initial stress in tendons (σp): 1200 N/mm2
    - Distance between tendons and soffit (h): 100 mm
    - Modular ratio (m): 6

    Calculating prestressing force:
    The prestressing force (P) can be calculated using the formula:
    P = A × σp

    Given that there are three tendons, the total prestressing force (Ptotal) is:
    Ptotal = 3 × P
    Ptotal = 3 × (A × σp)

    Calculating loss of prestressing force due to elastic deformation of concrete:
    The loss of prestressing force due to elastic deformation of concrete (PL) can be calculated using the formula:
    PL = (Ptotal × e) / m

    Where:
    - e is the strain in the concrete due to the initial stress in the tendons, and
    - m is the modular ratio.

    To calculate e, we need to find the stress in the concrete (σc) at the level of the tendons. This can be calculated using the formula:
    σc = σp × (d - h) / d

    Substituting the given values:
    σc = 1200 N/mm2 × (300 mm - 100 mm) / 300 mm
    σc = 1200 N/mm2 × 200 mm / 300 mm
    σc = 800 N/mm2

    The strain in the concrete can be calculated using the formula:
    εc = σc / Ec

    Where Ec is the modulus of elasticity of concrete. Assuming Ec as 30,000 N/mm2, we can calculate εc:
    εc = 800 N/mm2 / 30,000 N/mm2
    εc = 0.0267

    Substituting the values in the formula for PL:
    PL = (Ptotal × e) / m
    PL = (3 × (A × σp) × εc) / m
    PL = (3 × (50 mm2 × 1200 N/mm2) × 0.0267) / 6
    PL = 4.8 kN

    Therefore, the loss of prestressing force due to the elastic deformation of concrete is 4.8 kN.
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    A pre-tensioned rectangular concrete beam 150 mm wide and 300 mm depth is prestressed with three straight tendons, each having a cross-sectional area of 50 mm2, to an initial stress of 1200 N/mm2. The tendons are located at 100 mm from the soffit of the beam. If the modular ratio is 6, the loss of prestressing force (in kN, up to one decimal place) due to the elastic deformation of concrete only is ______.Correct answer is '4.8'. Can you explain this answer?
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