Dr. D. Y. Patil Pratishthan’s DR. D. Y. PATIL INSTITUTE OF ENGINEERING,MANAGEMENT & RESEARCH

1. Dr. D. Y. Patil Pratishthan’sDR. D. Y. PATIL INSTITUTE OF ENGINEERING,MANAGEMENT & RESEARCH Ms. Vaishnavi Battul Assistant Professor

2. Prestress Loss

3. Introduction • In prestressed concrete applications, most important variable is the prestress. • Prestress does not remain constant (reduces) with time. • Even during prestressing of tendons, and transfer of prestress, there is a drop of prestress from the initially applied stress. • Reduction of prestress is nothing but the loss in prestress.

4. Early attempts to produce prestressed concrete was not successful due to loss of prestress transferred to concrete after few years. Prestress loss is nothing but the reduction of initial applied prestress to an effective value. In other words, loss in prestress is the difference between initial prestress and the effective prestress that remains in a member. Loss of prestress is a great concern since it affects the strength of member and also significantly affects the member’s serviceability including Stresses in Concrete, Cracking, Camber and Deflection. Prestress Loss

5. Loss of prestress is classified into two types: • Short-Term or Immediate Losses • immediate losses occur during prestressing of tendons, and transfer of prestress to concrete member. • Long-Term or Time Dependent Losses • Time dependent losses occur during service life of structure.

6. Immediate Losses include • Elastic Shortening of Concrete • Slip at anchorages immediately after prestressing and • Friction between tendon and tendon duct, and wobble Effect • Time Dependent Losses include • Creep and Shrinkage of concrete and • Relaxation of prestressing steel

7. Prestress Losses Immediate Time Dependent Elastic Shortening Friction Anchorage Slip Creep Shrinkage Relaxation

8. Losses in Various Prestressing Systems

9. Immediate Losses • Elastic Shortening of Concrete • In pre-tensioned concrete, when the prestress is transferred to concrete, the member shortens and the prestressing steel also shortens in it. Hence there is a loss of prestress. • In case of post-tensioning, if all the cables are tensioned simultaneously there is no loss since the applied stress is recorded after the elastic shortening has completely occurred. • If the cables are tensioned sequentially, there is loss in a tendon during subsequent stretching of other tendons.

10. Loss of prestress mainly depends on modular ratio and average stress in concrete at the level of steel. • Loss due to elastic shortening is quantified by drop in prestress (Δfp) in a tendon due to change in strain in tendon (Δεp). • The change in strain in tendon is equal to the strain in concrete (εc) at the level of tendon due to prestressing force. • This assumption is due to strain compatibility between concrete and steel. • Strain in concrete at the level of tendon is calculated from the stress in concrete (fc) at the same level due to prestressingforce.

11. Strain compatibility • Loss due to elastic shortening is quantified by the drop in prestress (∆fp) in a tendon due to change in strain in tendon (∆εp). • Change in strain in tendon is equal to strain in concrete (εc) at the level of tendon due to prestressing force, which is called strain compatibility between concrete and steel. • Strain in concrete at the level of tendon is calculated from the stress in concrete (fc) at the same level due to the prestressing force. • A linear elastic relationship is used to calculate the strain from the stress.

12. Elastic Shortening • Pre-tensioned Members: When the tendons are cut and the prestressing force is transferred to the member, concrete undergoes immediate shortening due to prestress. • Tendon also shortens by same amount, which leads to the loss of prestress.

13. Elastic Shortening • Post-tensioned Members: If there is only one tendon, there is no loss because the applied prestress is recorded after the elastic shortening of the member. • For more than one tendon, if the tendons are stretched sequentially, there is loss in a tendon during subsequent stretching of the other tendons.

14. Prestressing bed Elastic Shortening Pre-tensioned Members: operation of pre-tensioning through various stages by animation. Pre-tensioning of a member

15. Duct Anchorage jack Elastic Shortening Post-tensioned Members: complete operation of post-tensioning through various stages by animation Casting bed Post-tensioning of a member

16. Linear elastic relationship is used to calculate the strain from the stress. • Quantification of the losses is explained below. Δfp=EpΔεp =Epεc =Ep(fc/Ec) Δfp= mfc • For simplicity, the loss in all the tendons can be calculated based on the stress in concrete at the level of CGS. • This simplification cannot be used when tendons are stretched sequentially in a post-tensioned member.

17. Anchorage Slip • In most Post-tensioning systems when the tendon force is transferred from the jack to the anchoring ends, the friction wedges slip over a small distance. • Anchorage block also moves before it settles on concrete. • Loss of prestress is due to the consequent reduction in the length of the tendon. • Certain quantity of prestress is released due to this slip of wire through the anchorages. • Amount of slip depends on type of wedge and stress in the wire.

18. The magnitude of slip can be known from the tests or from the patents of the anchorage system. • Loss of stress is caused by a definite total amount of shortening. • Percentage loss is higher for shorter members. • Due to setting of anchorage block, as the tendon shortens, there develops a reverse friction. • Effect of anchorage slip is present up to a certain length, called the setting length lset.

19. Anchorage loss can be accounted for at the site by over-extending the tendon during prestressing operation by the amount of draw-in before anchoring. • Loss of prestress due to slip can be calculated:

20. Frictional Loss • In Post-tensioned members, tendons are housed in ducts or sheaths. • If the profile of cable is linear, the loss will be due to straightening or stretching of the cables called Wobble Effect. • If the profile is curved, there will be loss in stress due to friction between tendon and the duct or between the tendons themselves.

21. Friction Post-tensioned Members • Friction is generated due to curvature of tendon, and vertical component of the prestressing force. A typical continuous post-tensioned member (Courtesy: VSL International Ltd.) 5

22. Friction Variation of prestressing force after stretching Post-tensioned Members P0 Px

23. The magnitude of prestressing force, Px at any distance, x from the tensioning end follows an exponential function of the type,

24. Time Dependent Losses • Creep of Concrete • Time-dependent increase of deformation under sustained load. • Due to creep, the prestress in tendons decreases with time. • Factors affecting creep and shrinkage of concrete • Age • Applied Stress level • Density of concrete • Cement Content in concrete • Water-Cement Ratio • Relative Humidity and • Temperature

25. For stress in concrete less than one-third of the characteristic strength, the ultimate creep strain (εcr,ult) is found to be proportional to the elastic strain (εel). • The ratio of the ultimate creep strain to the elastic strain is defined as the ultimate creep coefficient or simply creep coefficient, θ. εcr,ult= θεel • IS: 1343 considers only the age of loading of the prestressed concrete structure in calculating the ultimate creep strain.

26. The loss in prestress (Δfp ) due to creep is given as follows. Δfp = Epεcr, ult=Ep θ εel Sinceεcr,ult = θ εel Epis the modulus of the prestressing steel • Curing the concrete adequately and delaying the application of load provide long-term benefits with regards to durability, loss of prestress and deflection. • In special situations detailed calculations may be necessary to monitor creep strain with time. • Specialized literature or standard codes can provide guidelines for such calculations.

27. Following are applicable for calculating the loss of prestress due to creep. • Creep is due to sustained (permanent) loads only. Temporary loads are not considered in calculation of creep. • Since the prestress may vary along the length of the member, an average value of the prestress is considered. • Prestress changes due to creep, which is related to the instantaneous prestress. • To consider this interaction, the calculation of creep can be iterated over small time steps.

28. Shrinkage of Concrete • Time-dependent strain measured in an unloaded and unrestrained specimen at constant temperature. • Loss of prestress (Δfp) due to shrinkage is as follows. • Δfp = Epεsh • where Epis the modulus of prestressing steel. • The factors responsible for creep of concrete will have influence on shrinkage of concrete also except the loading conditions.

29. The approximate value of shrinkage strain for design shall be assumed as follows (IS 1383): For pre-tensioning = 0.0003 For post-tensioning = Where t = age of concrete at transfer in days.

30. Relaxation • Relaxation is the reduction in stress with time at constant strain. • decrease in the stress is due to the fact that some of the initial elastic strain is transformed in to inelastic strain under constant strain. • stress decreases according to the remaining elastic strain.

31. Factors effecting Relaxation : • Time • Initial stress • Temperature and • Type of steel. • Relaxation loss can be calculated according to the IS 1343-1980 code.

32. Losses in Prestress Notation Geometric Properties • Commonly used Notations in prestressed member are • Ac = Area of concrete section = Net c/s area of concrete excluding the area of prestressing steel. • Ap = Area of prestressing steel = Total c/s area of tendons. • A = Area of prestressed member = Gross c/s area of prestressed member = Ac + Ap

33. At = Transformed area of prestressed member = Area of member when steel area is replaced by an equivalent area of concrete = Ac + mAp= A + (m – 1)Ap Here, m = the modular ratio = Ep/Ec Ec = short-term elastic modulus of concrete Ep = elastic modulus of steel.

34. Areas for prestressed members CGC, CGS and eccentricity of typical prestressed members

35. CGC = Centroid of concrete = Centroid of gravity of section, may lie outside concrete • CGS = Centroid of prestressing steel = Centroid of the tendons. • CGS may lie outside the tendons or the concrete • I = MoI of prestressed member = Second moment of area of gross section about CGC. • It = Moment of inertia of transformed section = Second moment of area of the transformed section about the centroid of the transformed section. • e = Eccentricity of CGS with respect to CGC = Vertical distance between CGC and CGS. If CGS lies below CGC, e will be considered positive and vice versa

36. Load Variables • Pi = Initial prestressing force = force applied to tendons by jack. • P0=Prestressing force after immediate losses = Reduced value of prestressing force after elastic shortening, anchorage slip and loss due to friction. • Pe = Effective prestressing force after time-dependent losses = Final prestressing force after the occurrence of creep, shrinkage and relaxation.

37. Strain compatibility • Loss due to elastic shortening is quantified by the drop in prestress (∆fp) in a tendon due to change in strain in tendon (∆εp). • Change in strain in tendon is equal to strain in concrete (εc) at the level of tendon due to prestressing force, which is called strain compatibility between concrete and steel. • Strain in concrete at the level of tendon is calculated from the stress in concrete (fc) at the same level due to the prestressing force. • A linear elastic relationship is used to calculate the strain from the stress.

38. The quantification of the losses is explained below • For simplicity, the loss in all the tendons can be calculated based on the stress in concrete at the level of CGS. • This simplification cannot be used when tendons are stretched sequentially in a post-tensioned member.

39. Original length of member at transfer of prestress Pi Length after elastic shortening P0 Elastic Shortening Pre-tensioned Axial Members Elastic shortening of a pre-tensioned axial member

40. Elastic Shortening • The stress in concrete due to prestressing force after immediate losses (P0/Ac) can be equated to the stress in transformed section due to the initial prestress (Pi /At). • The transformed area At of the prestressed member can be approximated to the gross area A. • The strain in concrete due to elastic shortening (εc) is the difference between the initial strain in steel (εpi) and the residual strain in steel (εp0).

41. Length of tendon before stretching εpi Pi εp0 εc P0 Elastic Shortening Pre-tensioned Axial Members Elastic shortening of a pre-tensioned axial member 25

42. The following equation relates the strain variables. εc=εpi- εp0 • The strains can be expressed in terms of the prestressing forces. • Substituting the expressions of the strains • Thus, the stress in concrete due to the prestressing force after immediate losses (P0/Ac) can be equated to the stress in the transformed section due to the initial prestress (Pi /At).

43. Problem • A prestressed concrete sleeper produced by pre-tensioning method has a rectangular cross-section of 300mm × 250 mm (b × h). It is prestressed with 9 numbers of straight 7mm diameter wires at 0.8 times the ultimate strength of 1570 N/mm2. Estimate the percentage loss of stress due to elastic shortening of concrete. Consider m = 6.

44. Solution • Approximate solution considering gross section The sectional properties are. • Area of a single wire, Aw = π/4 × 72 = 38.48 mm2 • Area of total prestressing steel, Ap= 9 × 38.48 = 346.32 mm2 • Area of concrete section, A = 300 × 250 = 75 × 103 mm2 • Moment of inertia of section, I = 300 × 2503/12 = 3.91 × 108 mm4 • Distance of centroid of steel area (CGS) from the soffit,

45. Prestressing force, Pi= 0.8 × 1570 × 346.32 N = 435 kN • Eccentricity of prestressing force, e = (250/2) – 115.5 = 9.5 mm • The stress diagrams due to Piare shown. • Since the wires are distributed above and below the CGC, the losses are calculated for the top and bottom wires separately.

46. Stress at level of top wires (y = yt = 125 – 40) • Stress at level of bottom wires (y = yb= 125 – 40),

47. Loss of prestress in top wires = mfcAp(in terms of force) = 6 × 4.9 × (4 × 38.48) = 4525.25 N • Loss of prestress in bottom wires = 6 × 6.7 × (5 × 38.48) = 7734.48 N • Total loss of prestress = 4525 + 7735 = 12259.73 N ≈ 12.3 kN • Percentage loss = (12.3 / 435) × 100% = 2.83%

48. b) Accurate solution considering transformed section. Transformed area of top steel, A1 = (6 – 1) 4 × 38.48 = 769.6 mm2 Transformed area of bottom steel, A2 = (6 – 1) 5 × 38.48 = 962.0 mm2 Total area of transformed section, AT= A + A1+ A2= 75000.0 + 769.6 + 962.0 = 76731.6 mm2 Centroid of the section (CGC) = 124.8 mm from soffit of beam

49. Moment of inertia of transformed section, IT= Ig+ A(0.2)2 + A1(210 – 124.8)2 + A2(124.8 – 40)2 = 4.02 × 108mm4 • Eccentricity of prestressing force, e = 124.8 – 115.5 = 9.3 mm • Stress at the level of bottom wires, • Stress at the level of top wires,

50. Loss of prestress in top wires = 6 × 4.81 × (4 × 38.48) = 4442 N • Loss of prestress in bottom wires = 6 × 6.52 × (5 × 38.48) = 7527 N • Total loss = 4442 + 7527 = 11969 N ≈ 12 kN • Percentage loss = (12 / 435) × 100% = 2.75 % • It can be observed that the accurate and approximate solutions are close. Hence, the simpler calculations based on A and I is acceptable.