Problem 5.8. Losses of prestress due to creep

Consider the pre-tensiond cross section given in the previous problem (Problem 5.7). Applying the Trost model determine the percentage of the loss in prestressing force due to creep, when the creep coefficient is φ = 2.

The cross section is symmetrically reinforced, it contains 2 × 6Φ12.5 steel bars, elastic moduli of concrete and steel are: E_c = 31 GPa and E_s = 200 GPa, tensile stiffness of concrete is f_ct = 1.2 N/mm².

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Loss of prestress [%]=

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Steps

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Step 1. Calculate section properties according to the Trost model.

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Eq.(5-60)

A_{c} = 300 \times 400 = 1.2 \times 10^{5} {mm}^{2} A_{s 1} = A_{s 2} = 6 \frac{ϕ^{2} π}{4} = 6 \frac{12 . 5^{2}}{4} π = 736 {mm}^{2} A_{eχ} = A_{c} + \frac{E s}{E_{c}} (A_{s 1} + A_{s 2}) (1 + χ φ) = 120000 + 2 \frac{200}{31} 736 (1 + 0.8 \times 2) = 1.45 \times 10^{5} mm

Step 2. Determine loss of prestress.

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Eq.(5-68)

σ_{s} = σ_{so} - ∆ σ_{s} = σ_{so} (1 - φ \frac{E_{s}}{E_{c}} \frac{A_{s}}{A_{eχ}}) \to \frac{∆ σ_{s}}{σ_{so}} = φ \frac{E_{s}}{E_{c}} \frac{A_{s}}{A_{eχ}} = 2 \frac{200}{31} \frac{2 \times 736}{1.45 \times 10^{5}} = 0.131 = 13.1 %

Results

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Area of the replacement homogeneous cross section is calculated considering creep according to the Trost model:

Eq.(5-60)

A_{c} = 300 \times 400 = 1.2 \times 10^{5} {mm}^{2} A_{s 1} = A_{s 2} = 6 \frac{ϕ^{2} π}{4} = 6 \frac{12 . 5^{2}}{4} π = 736 {mm}^{2} A_{eχ} = A_{c} + \frac{E s}{E_{c}} (A_{s 1} + A_{s 2}) (1 + χ φ) = 120000 + 2 \frac{200}{31} 736 (1 + 0.8 \times 2) = 1.45 \times 10^{5} mm

Loss of prestress is

Eq.(5-68)

σ_{s} = σ_{so} - ∆ σ_{s} = σ_{so} (1 - φ \frac{E_{s}}{E_{c}} \frac{A_{s}}{A_{eχ}}) \to \frac{∆ σ_{s}}{σ_{so}} = φ \frac{E_{s}}{E_{c}} \frac{A_{s}}{A_{eχ}} = 2 \frac{200}{31} \frac{2 \times 736}{1.45 \times 10^{5}} = 0.131 = 13.1 %