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A post-tensioned concrete member of span 15 m and cross-section of 450 mm x 450 mm is prestressed with three steel tendons, each of cross-sectional area 200 mm2. The tendons are tensioned one after another to a stress of 1500 MPa. All the tendons are straight and located at 125 mm from the bottom of the member. Assume the prestress to be the same in all tendons and the modular ratio to be 6. The average loss of prestress, due to elastic deformation of concrete, considering all three tendons is
- a)28.32 MPa
- b)7.08 MPa
- c)14.16 MPa
- d)42.48 MPa
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A post-tensioned concrete member of span 15 m and cross-section of 450...
To determine the average loss of prestress, we need to consider the elastic deformation of the concrete due to the prestressing force. The average loss of prestress can be calculated using the following formula:
Average loss of prestress = (P * e) / A
Where:
P is the prestressing force in each tendon
e is the strain in the concrete due to the prestressing force
A is the cross-sectional area of the concrete member
Let's calculate the values step by step.
1. Calculation of Prestressing Force:
The prestressing force in each tendon can be calculated using the formula:
P = σ * A
Where:
σ is the stress in the tendon
A is the cross-sectional area of the tendon
Given:
Stress in the tendon, σ = 1500 MPa
Cross-sectional area of the tendon, A = 200 mm2
Prestressing force, P = 1500 MPa * 200 mm2 = 300000 N
2. Calculation of Strain in Concrete:
The strain in the concrete due to the prestressing force can be calculated using the formula:
εc = εs / n
Where:
εs is the strain in the steel tendon
n is the modular ratio
Given:
Modular ratio, n = 6
Since the tendons are tensioned one after another, the strain in each tendon will be the same. Therefore, we can calculate the strain in the tendon using the formula:
εs = σ / E
Where:
σ is the stress in the tendon
E is the modulus of elasticity of steel
Given:
Stress in the tendon, σ = 1500 MPa
Modulus of elasticity of steel, E = 200 GPa = 200000 MPa
Strain in the tendon, εs = 1500 MPa / 200000 MPa = 0.0075
Strain in concrete, εc = 0.0075 / 6 = 0.00125
3. Calculation of Average Loss of Prestress:
The average loss of prestress can now be calculated using the formula mentioned earlier:
Average loss of prestress = (P * e) / A
Given:
Prestressing force, P = 300000 N
Strain in concrete, e = 0.00125
Cross-sectional area of the concrete member, A = 450 mm * 450 mm = 202500 mm2
Average loss of prestress = (300000 N * 0.00125) / 202500 mm2 = 1.875 N/mm2
Converting the units to MPa:
Average loss of prestress = 1.875 N/mm2 * 1 MPa / 1000 N/mm2 = 0.001875 MPa
Therefore, the average loss of prestress, considering all three tendons, is 0.001875 MPa, which is approximately equal to 0.0019 MPa.
Average loss of prestress = (P * e) / A
Where:
P is the prestressing force in each tendon
e is the strain in the concrete due to the prestressing force
A is the cross-sectional area of the concrete member
Let's calculate the values step by step.
1. Calculation of Prestressing Force:
The prestressing force in each tendon can be calculated using the formula:
P = σ * A
Where:
σ is the stress in the tendon
A is the cross-sectional area of the tendon
Given:
Stress in the tendon, σ = 1500 MPa
Cross-sectional area of the tendon, A = 200 mm2
Prestressing force, P = 1500 MPa * 200 mm2 = 300000 N
2. Calculation of Strain in Concrete:
The strain in the concrete due to the prestressing force can be calculated using the formula:
εc = εs / n
Where:
εs is the strain in the steel tendon
n is the modular ratio
Given:
Modular ratio, n = 6
Since the tendons are tensioned one after another, the strain in each tendon will be the same. Therefore, we can calculate the strain in the tendon using the formula:
εs = σ / E
Where:
σ is the stress in the tendon
E is the modulus of elasticity of steel
Given:
Stress in the tendon, σ = 1500 MPa
Modulus of elasticity of steel, E = 200 GPa = 200000 MPa
Strain in the tendon, εs = 1500 MPa / 200000 MPa = 0.0075
Strain in concrete, εc = 0.0075 / 6 = 0.00125
3. Calculation of Average Loss of Prestress:
The average loss of prestress can now be calculated using the formula mentioned earlier:
Average loss of prestress = (P * e) / A
Given:
Prestressing force, P = 300000 N
Strain in concrete, e = 0.00125
Cross-sectional area of the concrete member, A = 450 mm * 450 mm = 202500 mm2
Average loss of prestress = (300000 N * 0.00125) / 202500 mm2 = 1.875 N/mm2
Converting the units to MPa:
Average loss of prestress = 1.875 N/mm2 * 1 MPa / 1000 N/mm2 = 0.001875 MPa
Therefore, the average loss of prestress, considering all three tendons, is 0.001875 MPa, which is approximately equal to 0.0019 MPa.
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Community Answer
A post-tensioned concrete member of span 15 m and cross-section of 450...
Given: Area of steel of each bar = 200 mm2
Initial prestress = 1500 N/mm2
∴ P-force in each tendon, Pu = poAs
As per given condition,
Loss of stress in 3rd wire = 0
Loss in wire-2, is due to P3
Loss in wire-1, is due to P2 and P3 both (Neglecting effect of self weight, because self weight will result in gain of stress, not loss)
Now, Loss of stress due to elastic shortening in wire-2,
Loss of stress in wire-2 = m(fc)avg = 6 x 2.36 = 14.16 N/mm2
Loss of stress due to elastic shortening in wire-1, due to P2 + P3 = 2P = 600 kN
Loss in stress in wire-1 = m(fc)avg = 6 x 4.72 = 28.32 N/mm2
Average loss of stress in all three wires
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A post-tensioned concrete member of span 15 m and cross-section of 450 mm x 450 mm is prestressed with three steel tendons, each of cross-sectional area 200 mm2. The tendons are tensioned one after another to a stress of 1500 MPa. All the tendons are straight and located at 125 mm from the bottom of the member. Assume the prestress to be the same in all tendons and the modular ratio to be 6. The average loss of prestress, due to elastic deformation of concrete, considering all three tendons isa)28.32 MPab)7.08 MPac)14.16 MPad)42.48 MPaCorrect answer is option 'C'. Can you explain this answer?
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