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A simply supported prestressed beam of 300 × 500 mm in c/s is subjected to a superimposed load of 20 kN/m over a span of 10m. If a prestressing force of 1000 kN is applied through a straight tendon located along centroidal axis then what is the extreme top fiber stress at the mid section:
  • a)
    26
  • b)
    27
Correct answer is between '26,27'. Can you explain this answer?
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Solution:

Given data:

Cross-sectional dimensions of the beam = 300 × 500 mm

Span of the beam = 10 m

Superimposed load = 20 kN/m

Prestressing force = 1000 kN

We need to find the extreme top fiber stress at the mid-section of the beam.

Assumptions:

1. The beam is made up of homogeneous and isotropic material.

2. The beam is initially stress-free.

3. The beam is simply supported.

4. The effective prestressing force is the difference between the initial prestressing force and the force due to the losses.

Calculation:

1. Calculation of Effective Prestressing Force:

Initial Prestressing Force = 1000 kN

Losses in Prestressing:

a) Losses due to friction = 0.2% of the initial prestressing force

Loss due to friction = 0.2/100 × 1000 = 2 kN

b) Losses due to anchorage slip = 0.5% of the initial prestressing force

Loss due to anchorage slip = 0.5/100 × 1000 = 5 kN

Effective Prestressing Force = Initial Prestressing Force – Losses in Prestressing

Effective Prestressing Force = 1000 – (2 + 5) = 993 kN

2. Calculation of Bending Moment:

Superimposed Load = 20 kN/m

Span of the beam = 10 m

Maximum bending moment occurs at the mid-span of the beam.

Bending Moment (M) = wl^2/8 = 20 × 10^2/8 = 250 kNm

3. Calculation of Extreme Top Fiber Stress:

Section Modulus (Z) = (bd^2)/6 = (300 × 500^2)/6 × 10^6 = 12.5 × 10^-3 m^3

Extreme Top Fiber Stress (σ) = (M/Z) + (Pe/A)

Where,

P = Effective Prestressing Force = 993 kN

e = eccentricity of the prestressing force

A = Cross-sectional area of the beam = 300 × 500 × 10^-6 = 0.15 m^2

e = (P × y)/A

where y is the distance of the centroid of the section from the extreme top fiber.

y = (500/2) - 300/2 = 100 mm = 0.1 m

e = (993 × 0.1)/0.15 = 662 mm

σ = (250 × 10^6/12.5 × 10^-3) + (993 × 10^3/0.15) × (0.1 + 662/1000)

σ = 26.22 MPa (approx)

Therefore, the extreme top fiber stress at the mid-section of the beam is 26.22 MPa (approx).
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A simply supported prestressed beam of 300 × 500 mm in c/s is subjected to a superimposed load of 20 kN/m over a span of 10m. If a prestressing force of 1000 kN is applied through a straight tendon located along centroidal axis then what is the extreme top fiber stress at the mid section:a)26b)27Correct answer is between '26,27'. Can you explain this answer?
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