A pre-tensioned concrete member of section 200 mm x 250 mm contains t...
The stress in the concrete can be determined using the principle of static equilibrium and the concept of modular ratio.
Given data:
- Section dimensions: 200 mm x 250 mm
- Tendon area: 500 mm2
- Pre-stress in tendons: 1000 N/mm2
- Modular ratio: 10
To find the stress in the concrete, we need to determine the force in the tendons and the force in the concrete.
1. Determining the force in the tendons:
The force in the tendons can be calculated using the formula:
Force = Stress x Area
Given that the stress in the tendons is 1000 N/mm2 and the area of the tendons is 500 mm2, we can calculate the force in the tendons as:
Force = 1000 N/mm2 x 500 mm2 = 500000 N
2. Determining the force in the concrete:
The force in the concrete can be calculated using the principle of static equilibrium. Since the tendons are located at the center of gravity of the section, the forces in the concrete above and below the tendons are equal and opposite.
Let's assume the force in the concrete is Fc. Therefore, the total force in the concrete is 2Fc.
3. Applying the principle of static equilibrium:
According to the principle of static equilibrium, the sum of all forces acting on a body in equilibrium is zero.
In this case, the force in the tendons (500000 N) and the total force in the concrete (2Fc) should balance each other out. Therefore, we can write the equation as:
500000 N + 2Fc = 0
Solving for Fc:
2Fc = -500000 N
Fc = -250000 N
Since stress is defined as force divided by area, the stress in the concrete can be calculated as:
Stress = Fc / Area
Given that the area of the concrete section is 200 mm x 250 mm = 50000 mm2, we can calculate the stress in the concrete as:
Stress = -250000 N / 50000 mm2 = -5 N/mm2
However, stress is always considered as a positive value, so taking the absolute value of the stress, we get:
Stress = | -5 N/mm2 | = 5 N/mm2
4. Taking into account the modular ratio:
The modular ratio relates the stress in the concrete to the stress in the steel. In this case, the modular ratio is given as 10.
The stress in the concrete can be related to the stress in the tendons using the modular ratio:
Stress in concrete = Modular ratio x Stress in tendons
Substituting the values:
Stress in concrete = 10 x 1000 N/mm2 = 10000 N/mm2
However, the stress in the concrete is given as 5 N/mm2, which is a positive value. So, we need to multiply the calculated stress by -1 to get a positive value:
Stress in concrete = -1 x 5 N/mm2 = -5 N/mm2
Finally, taking the absolute value of the stress, we get:
Stress in concrete = | -5 N/mm2 | = 5 N/mm2
Therefore, the stress in the concrete is 5 N/mm2, which is not the correct answer.
A pre-tensioned concrete member of section 200 mm x 250 mm contains t...
B = 200, D = 250, A
st = 500 mm
2Ps = 1000 N/mm2, m = 10, Pc = ?
Pre-tension acts on centre of gravity of the section
P = Ast.Ps = 1000 x 500 N
= 500 kN
Now, strain in concrete = strain in steel
Now, strain in concrete = 10.10/EC
Stress in steel =
= 101.01 N/mm2 (loss)
Final stress = 1000 - 101.01 = 899 N/mm2
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