To compute the deformation and the change in force in the tendon and concrete for a concentrically loaded prestressed member, we need to follow these steps:

Given Data:

• Area of prestressing steel (Aps): $0.5 \, \text{in}^2$
• Area of concrete section (Ac): $100 \, \text{in}^2$
• Initial length of the member (L0): $720 \, \text{in}$
• Modulus of elasticity of concrete (Ec): $4400 \, \text{kips/in}^2$
• Modulus of elasticity of steel (Es): $25000 \, \text{kips/in}^2$
• Initial stress in the tendon (fsi): $180 \, \text{kips/in}^2$

Step 1: Calculate the Initial Prestressing Force in the Tendon

The initial force in the tendon before transfer can be computed using: $P_{si} = f_{si} \times A_{ps}$ $P_{si} = 180 \, \text{kips/in}^2 \times 0.5 \, \text{in}^2 = 90 \, \text{kips}$

Step 2: Compute the Deformation in Concrete

The deformation (shortening) in the concrete can be calculated using: $\Delta L_c = \frac{P_{si} \times L_0}{A_c \times E_c}$ $\Delta L_c = \frac{90 \, \text{kips} \times 720 \, \text{in}}{100 \, \text{in}^2 \times 4400 \, \text{kips/in}^2}$ $\Delta L_c = \frac{64800 \, \text{kips} \cdot \text{in}}{440000 \, \text{kips}} = 0.1473 \, \text{in}$

Step 3: Compute the Deformation in the Tendon

Since the tendon and the concrete are bonded, the deformation in the tendon will be the same as the deformation in the concrete: $\Delta L_s = \Delta L_c = 0.1473 \, \text{in}$

Step 4: Calculate the Change in Force in the Tendon

The change in force in the tendon due to this deformation can be calculated using Hooke's Law: $\Delta P_s = \frac{\Delta L_s \times E_s \times A_{ps}}{L_0}$ $\Delta P_s = \frac{0.1473 \, \text{in} \times 25000 \, \text{kips/in}^2 \times 0.5 \, \text{in}^2}{720 \, \text{in}}$ $\Delta P_s = \frac{1841.25 \, \text{kips} \cdot \text{in}}{720 \, \text{in}} = 2.56 \, \text{kips}$

Step 5: Calculate the Final Force in the Tendon After Transfer

The final force in the tendon after transfer is: $P_{sf} = P_{si} - \Delta P_s$ $P_{sf} = 90 \, \text{kips} - 2.56 \, \text{kips} = 87.44 \, \text{kips}$

Summary of Results:

• Deformation in concrete and tendon: $\Delta L_c = \Delta L_s = 0.1473 \, \text{in}$
• Change in force in tendon: $\Delta P_s = 2.56 \, \text{kips}$
• Final force in tendon after transfer: $P_{sf} = 87.44 \, \text{kips}$

Questions:

1. What is the significance of the change in force in the tendon after the transfer?
2. How would the deformation change if the modulus of elasticity of the concrete was higher?
3. How does the area of the prestressing steel affect the final force in the tendon?
4. What would be the impact on the results if elastic shortening was considered?
5. Can the same method be applied if the load was eccentric?

Tip:

Ensure that when performing these calculations, the units are consistently maintained throughout to avoid errors in the final results.