EAS 453 Pre-stressed Concrete Design Pre-stress Losses Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Nature of Losses of Pre-stress The initial pre-stress in concrete undergoes a gradual reduction with time from the stage of transfer due to various causes - loss of pre-stress Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

The different types of losses encountered in the pre-tensioning and post-tensioning systems are compiled in Table below: No Pre-tensioning Post-tensioning 1. Elastic deformation of concrete. No loss due to elastic deformation if all the wires are simultaneously tensioned. If the wires are successively tensioned, there will be loss of pre-stress due to elastic deformation of concrete. 2. Relaxation of stress in steel. 3. Shrinkage of concrete. 4. Creep of concrete. 5. Friction. 6. Anchorage slip. Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Loss Due to Elastic Deformation of Concrete The loss of pre-stress due to elastic deformation of concrete depends on the modular ratio and the average stress in concrete at the level of steel. IF fc = pre-stress in concrete at the level of steel Es = modulus of elasticity of steel Ec = modulus of elasticity of concrete   = modular ratio Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Strain in concrete at the level of steel = Stress in steel corresponding to this strain = Therefore, Loss of stress in steel = If the initial stress in steel is known, the percentage loss of stress in steel due to the elastic deformation of concrete can be computed. Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Example 1 A pre-tensioned concrete beam, 100mm wide and 300mm deep, is pre-stressed by straight wires carrying an initial force of 150kN at an eccentricity of 50mm. The modulus of elasticity of steel and concrete are 210 and 35kN/mm2 respectively. Estimate the percentage loss of stress in steel due to elastic deformation of concrete if the area of steel wires is 188mm2. Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Solution: P = 150kN; A = (100*300) = 3x104 mm2; I = 225x106 mm4 Initial stress in steel = (150x103)/188 = 800 N/mm2 Stress in concrete Loss of stress due to elastic deformation of concrete = Percentage loss of stress in steel = (40x100) / 800 = 5% Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Loss Due to Shrinkage of Concrete The shrinkage of concrete in pre-stressed member results in a shortening of tensioned wires and hence contributes to the loss of stress. The shrinkage of concrete is influenced by the type of cement and aggregates and the method of curing used. Use of high strength concrete with low water cement ratios results in a reduction in shrinkage and consequent loss of pre- stress. The primary cause of drying shrinkage is the progressive loss of water from concrete. Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

The rate of shrinkage is higher at the surface of the members. The differential shrinkage between the interior and surface of large members may result in strain gradients leading to surface cracking – proper curing is essential. In the case of pre-tensioned members, generally moist curing is resorted to in order to prevent shrinkage until the time of transfer. The total residual shrinkage strain will be larger in pre-tensioned members after transfer of pre-stress in compression. For post-tensioned members, the portion of shrinkage will have already taken place by the time of transfer of stress. Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

According to cl. 4.8.4, BS 8110: Part 1:1985, Losses due to shrinkage = residual shrinkage strain x Es Recommended residual shrinkage strain in Malaysia with the temperature and high relative humidity is 200 x 10-6 Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Loss Due to Creep of Concrete The sustained pre-stress in the concrete of a pre-stressed member results in creep of concrete which effectively reduces the stress in high tensile steel. 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. According to cl. 4.8.5, BS 8110: Part 1: 1985 Pre-stress losses due to creep = creep concrete strain at tendon level x Ec Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Ec = modulus of elasticity of concrete during transfer Creep strain = (creep coefficient / Ec ) x average stress in concrete during transfer at tendon level Ec = modulus of elasticity of concrete during transfer Recommended creep coefficient in Malaysia are 1.5 for transfer time after 7 days. Ultimate Creep Strain Method If, εcc = ultimate creep strain for a sustained unit stress fc = compressive stress in concrete at the level of steel Es = modulus of elasticity of steel Then the loss of stress in steel due to creep of concrete = εcc fc Es Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Creep Coefficient Method If; φ = creep coefficient εc = creep strain εe = elastic strain αe = modular ratio fc = stress in concrete Ec = modulus of elasticity of concrete Es = modulus of elasticity of steel Creep Coefficient = (Creep Strain)/ (Elastic Strain) Thus; φ = (εc ) /(εe ), Hence, loss of stress in steel = Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Loss Due to Relaxation of Stress in Steel The loss of stress due to relaxation of steel as a percentage of the initial stress in steel. Percentage of loss is the relaxation factor multiple with relaxation value in 1000 hrs.-- from supplier Table 4.6, BS8110:Part 1: 1985: steel relaxation factor Force Type Wires and strand with relaxation class. Bar 1 2 Pre-tensioning 1.5 1.2 2.0 Post-tensioning Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Relaxation value in 1000 hrs. Strand of tendon Initial force (% characteristic strength) Maximum relaxation after 1000 hrs (Class 1) (%) (Class 2) (%) Cold-drawn steel and seven-wire strand 60 70 80 4.5 8.0 12.0 1.0 2.5 Cold-drawn steel in factory 8.5 10.0 - Alloy Steel 1.5 3.5 6.0 For initial force that less than 60%, cl. 4.8.2.2, BS 8110:Part 1: 1985 recommended that the initial value is linearly different as stated in 60% to zero for 30% initial force. The initial value for pre-tensioning must be taken as the immediate force after the tendon being stressed. The initial value for post-tensioning, the initial force is when the transfer happen. Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Example 2 A rectangular concrete beam, 300 mm deep and 200 mm wide is pre-stressed by means of fifteen 5 mm diameter wires located 65 mm from the bottom of the beam and three 5 mm wires, located 25 mm from the top of the beam. If the wire initially tensioned to a stress of 840 N/mm2, calculate the percentage loss of stress in steel immediately after transfer, allowing for the loss of stress due to elastic deformation of concrete only. Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Solutions Es = 210kN/mm2 Ec = 31.5kN/mm2 Position of the centroid of the wires from the soffit of the beam, Eccentricity, e = (150-100)mm = 50mm Area of concrete, A = (200*300) = 6 x 104 mm2 Second moment of area, I = (200*3003)/12 = 45 x 107 mm Prestressing force P = (840) (18x19.7) = 3x105 N = 300kN Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Stress in concrete: At the level of top wires = At the level of bottom wires= Modular ratio = (210)/(31.5)= 6.68 Loss of stress in wires at top = (6.68*0.83) = 5.55 N/mm2 Loss of stress in wires at bottom = (6.68*7.85) = 52.5 N/mm2 Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Percentage loss of stress For wires at top = For wires at bottom = Dr. NORAZURA MUHAMAD BUNNORI (PhD), USM

Loss of Stress Due to Friction In post-tensioning systems there will be movement of the greater part of the tendon relative to the surrounding duct during the tensioning operation. If the tendon is in the contact with either the duct or any spacer provided, friction will cause a reduction in the pre-stressing force as the distance from the jack increases. Whether the desired duct profile is straight or curved or a combination of both, there will be a slight variations in the actual line of the duct, which may cause additional points of contact between the tendon and the sides of the duct ---- FRICTION

There are two types of friction losses: Loss of stress due to wobble effect, which depends upon the local deviations in the alignment of the cable. The wobble or wave effect is the result of accidental or unavoidable misalignment, since ducts or sheaths cannot be perfectly located to follow a predetermined profile throughout the length of the beam. Po Magnitude of the pre-stressing force, Px x

Px = magnitude of pre-stressing force Po = pre-stressing force at the jacking end e = Napier logarithm (2.718) K = profile coefficient depending on the type of duct or sheath employed, the nature of it’s inside surface, the method of forming it and the degree of vibration employed in placing the concrete (cl. 4.9.3.3 BS 8110:Part 1: 1985) K value per meter length Contact types K/m Normal condition 33 x 10-4 Greased strands running in plastics sleeves 25 X 10-4 Strong rigid sheaths or duct formers, closely supported so that they did not displaced during the concreting operation 17 x 10-4

b) Friction due to curvature of tendons The loss of tension due to friction is depend on the angle turned through and the coefficient of friction μ between the tendon and its support. The pre-stressing force, Px, at any distance x along the curve from the tangen point may be calculated from the following equation: θ rps x straight curve Po Pre-stress in x distance from curvature tangent, Px

Px = pre-stressing force Po = pre-stressing force at the jacking end μ = coefficient of friction rps = radius of curvature e = Napier logarithm (2.718) θ = x/rps = tendon curvature angle (radian) The value of μ depends upon the type and the surface conditions of the tendon and the duct (cl 4.9.4.3 BS 8110) Contact type μ Lightly rusted strand running on unlined concrete duct 0.55 Lightly rusted strand running on lightly rusted steel duct 0.30 Lightly rusted strand running on galvanized duct 0.25 Bright strand running on galvanized duct 0.20 Greased strand running on plastic sleeve 0.12

Combination of the frictions. Kx +μθ e -(kx+μθ) 0.01 0.990 0.02 0.980 0.03 0.970 0.04 0.961 0.05 0.951 - 0.19 0.827 0.20 0.819

Draw-in during anchorage Cl 4.8.6 BS 8110:Part 1:1985 In post tensioning systems allowance should be made for any movement of the tendon at the anchorage when the pre-stressing force is transferred from the tensioning equipment to the anchorage. The loss due to this movement is particularly important in short members and the allowance made in design should be checked on site. The loss in anchorage system usually in a range of 3mm. For a longer member, loss of pre-stress can be encounter with giving an extra force to the tendon---- 5-10%.