EVALUATION OF TIME DEPENDENT PRESTRESS LOSSES IN PRETENSIONED CONCRETE MEMBER WITH TOP AND BOTTOM TENDONS

K e y w o r d s : Creep; Prestress losses; Pretensioned concrete member; Relaxation; Shrinkage; Stress in prestressing reinforcement; Tendons. 3/2019 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T 105 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T The Si les ian Univers i ty of Technology No. 3/2019 d o i : 1 0 . 2 1 3 0 7 / A C E E 2 0 1 9 0 4 0 V . P a r k h a t s , R . K r z y w o ń bottom tendons to analyse an error between the results according to the deduced equations and those obtained on the basis of the equation (5.46) from Eurocode 2 [4] for the common substitute cross-section of the prestressing reinforcement: where: Δσp,c+s+r(t) (marked as Δσp(t) in the rest of the paper) is the variation of stress in the prestressing steel at time t; Ap is the area of the cross-section of the prestressing steel; ε cs(t, t0) is the shrinkage strain at time t (the loading is applied at time t0); Ep is the modulus of elasticity for the prestressing steel; Δσpr(t) is the variation of stress in the tendons at time t due to relaxation of the prestressing steel (determined for the initial stress in the tendons due to initial prestress and quasi-permanent actions); Ecm is the modulus of elasticity for the concrete; φ(t, t0) is the creep coefficient at time t (the loading is applied at time t0); σc(t0) is the stress in the concrete at the time when the loading is applied t0; Acs is the area of the transformed cross-section; Ics is the second moment of the area of the transformed cross-section; zcp is the distance between the centre of gravity of the transformed cross-section and the prestressing steel. 2. EVALUATION OF THE TIME DEPENDENT PRESTRESS LOSSES SEPARATELY FOR THE TOP AND BOTTOM TENDONS OF A PRETENSIONED CONCRETE MEMBER The variation of strain of the concrete at time t is given by [5] where: Δσc(t) is the variation of stress in the concrete at time t; χ is the aging coefficient [6, 7]. The variation of strain of the prestressing steel at time t is where: χr is the reduced relaxation coefficient [8]. It is assumed that on the level of the centroid of the tendons the variation of strain of the concrete is equal to that of the prestressing steel, Δε c(t) =Δεp(t), so the variation of stress in the prestressing steel at time t is given by The equation (4) can be transformed to: By the omitting the effect of relaxation of the prestressing steel, it might be assumed that a ratio of the variation of stress in the concrete to that in the prestressing steel is equal to a ratio of the stress in the concrete (on the level of the centroid of the tendons) Δσc,p to that in the prestressing steel Δσp,p. By using the equation (6), the equation (5) for the top tendons can be expressed as: For the bottom tendons: 106 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T 3/2019 ( ) ( ) ( ) ( ) ( )