A prestressed rectangular concrete beam of size 150 × 450 mm is prestressed by wires of area 150 mm² at an eccentricity of 50 mm. The initial pre stress in the wires is 1300 N/mm². What is the loss of stress in steel due to creep of concrete? Take E_s 210 KN/mm2 , E_c 35 KN/mm², ultimate creep strain is 41×10^-6 mm/mm per N/mm2 .

Explanation-

Loss due to creep of concrete-

The loss of stress in steel due to creep of concrete can be estimated if the magnitude of ultimate creep strain or creep coefficient is known.

1. Ultimate Creep Strain Method

ε_cc = ultimate creep strain for a sustained unit stress

f_c = compressive stress in concrete at the level of steel

E_s = modulus of elasticity of steel

Then the loss of stress in steel due to creep of concrete = ε_ccf_cE_s.

2. Creep Coefficient Method

ϕ = creep coefficient

ε_c = creep strain

ε_e = elastic strain

α_e = modular ratio

f_c = stress in concrete

E_c = modulus of elasticity of concrete

E_s = modulus of elasticity of steel

Creep coefficient = \(\rm \left(\frac{creep\ strain}{elastic \ strain}\right)\)

∴ \(\rm ϕ=\left(\frac{\varepsilon_c}{\varepsilon_e}\right)\)

∴ εc = ϕ εe = ϕ (f_c/E_c)

Hence loss of stress in steel = εcE_cϕE_s = ϕ(fc/Ec)E_s = ϕf_cα_e

Calculation:

Given data:

Size of the beam, b = 150 mm, D=450 mm

Area of wire = 150 mm²

e=50 mm

Initial prestress = 1300 N/mm2

 Es =210 KN/mm2 , Ec =35 KN/mm2,

ultimate creep strain = 41×10-6 mm/mm per N/mm²

Presressing force = P₁ = σ_i × A = 1300 × 150 = 195 kN

Stress in concrete at the level of steel = \(\rm \frac{P}{A}+\frac{Pe+e}{I}\)

\(=\frac{195×10^3}{150×450}+\frac{195×10^3×50^2}{150×\frac{450^3}{12}}=3.32\) N/mm²

Loss of stress in steel = 41 × 10^-6 × 3.32 × 210 × 10³ N/mm²

Loss of stress in steel = 28.5 N/mm²