Determination of Crack width for 1300mm dia Pile subjected to Axial Load & Moment. Material parameters Grade of concrete fck Characteristic strength of concrete Grade of steel = M 45 = 45 N/mm2 = Fe 415 Yield stress of steel fy = 415 N/mm2 Max. Permissible stresses in concrete in compresssion (Bending) σ cbc = 14.5 N/mm2 = 165 N/mm2 m = 6.44 1.5 x m = 9.66 P My = 100200 kg = = 100000 kg.cm = 15020000 kg.cm = 15020333 kg.cm Max. Permissible stresses for steel in flexural tension Modular ratio cl no.8.3.4 of IS: 4651-part 4 Member forces Axial load on pile = Moment on the pile = 100.2 T = 1 T.m Moment on the pile = 150.2 T.m = Mz Resultant Moment on Pile = 150.20 T.m = MRes = e Check for eccentricity Eccentricity = M/P = 149.90 cm > 16.3 cm D = 130 cm R Nb = 65 cm = 60 Nos φ rebar = 25 mm Area of the longitudinal reinforcment Ast = 294.52 cm2 Clear cover to reinforcement c dc = 7.5 cm = 9.55 cm Assuming that the steel bars are equivalent to a thin shell of the same cross sectional area Dshell Diameter of thin shell of reinforcement = 110.9 cm Limit of eccentricity for the entire cross section to be in compression Geometrical parameters used in the evaluation of crack width of the pile Diameter of Pile = 1300 mm = Distance between centre of the section to the outer most fibre No of longitudinal rebars Diameter of longitudinal rebars Effective cover = clear cover + cg of the rebar ir = 55.5 cm deff = 120.5 cm Cos α = -0.259 Cos β = -0.303 Sin α Sin β = 0.966 = 0.953 α β = 1.8325 radians = 1.8790 radians Sin 4α Sin 2α Sin 2β = 0.866 = -0.500 = -0.578 Distance between centre of section to cg of main steel (inner radius) Effective depth of pile cross section Secondary parameters used in the evaluation of crack width of the pile Cos α = Cos β = Nd−0.5×D 0 .5×D Nd−0. 5× D r Nd R θ R N α dθ β A Determination of neutral axis Ast / (2 π r) tshell = N dn = = 48.18 cm Total compression in concrete above neutral axis, Cc Cc = Total compression in steel above neutral axis, Cs Cs = 1862 fs1 532 fs1 Total tension in steel below neutral axis, Ts Ts = 1058 fs1 Thickness of thin shell of reinforcement Assuming Neutral axis depth Coefficient Depth of Neutral axis (N x deff) 0.845 cm 0.4000 Expressions for evaluating fs1 from ∑P Expressions for evaluating fs2 from ∑M 1 of 2 (0.125 x D) Determination of Crack width for 1300mm dia Pile subjected to Axial Load & Moment. Moment of compression in concrete about the centre line of section MCc = Moment of compression in steel about the centre line of section MCs = 83703 fs2 25207 fs2 Moment of tension in steel about the centre line of section MTs = 41746 fs2 Net axial stresses on the pile section Caxial = Net bending stresses on the pile section Cben = 1337 fs1 150655 fs2 Evaluation of extreme fibre stress in concrete Evaluation of extreme fibre stress in concrete by first condition of equilibrium, i.e, ∑P = 0 Axial Load, P = (Cc + Cs - Ts) x fs1 fs1 = P/Caxial = 75 kg/cm2 Evaluation of extreme fibre stress in concrete by second condition of equilibrium, i.e, ∑M = 0 Moment, M = (MCc + MCs + MTs) x fs2 fs2 = M/Cben = 100 kg/cm2 Evaluation of internal forces & moments on cross section of pile Total compression in concrete above neutral axis Cc = 139578 kg Total compression in steel above neutral axis Cs = 39888 kg Total tension in steel below neutral axis Ts = 79269 kg Moment of compression in concrete about the centre line of section MCc = 8345164 kg.cm Moment of compression in steel about the centre line of section MCs = 2513108 kg.cm Moment of tension in steel about the centre line of section MTs = 4162050 kg.cm Evaluation of distance of centroid of tensile steel from centre of cross section of the pile The distance of centroid of tensile steel, which is in the form of an arc of a circle, from the centre of the cross section of the pile has been evaluated. sinβ cg = r× β ra dians cg = 28.12 cm Evaluation of extreme fibre stress in concrete The mean value of fs1 & fs2 has been adopted as the final extreme fibre stress in concrete σ cbc Maximum compressive stress, 0.5 x (74.95 + 99.7) fs1 = 75 kg/cm2 fs2 = 100 kg/cm2 = 87 kg/cm2 The distance of the centroid of tension steel from neutral axis has been evaluated as under h1 = R cg − d n (65 + 28.12 - 48.18) = 44.94 cm The tensile stress at the centroid of tensile steel is evaluated as under σ st = m×σ cbc × h1 = 524 kg/cm2 dn < 1682 kg/cm3 Safe = 7 kips/inch2 The distance from neutral axis to extreme fibre (h2), where crack width is calculated has been evaluated as under h2 = 2× R −d n = 81.82 cm Determination of crack width of concrete on tensile face of pile The crack width of the pile has been evaluated by the following Gerely - Lutz Equation, given in ACI 318R-95 Commentary of Building code requirements for structural concrete published by the American Concrete Institute. cw = γ= A = 0. 076×γ×σ st × 3 d c × A = 0.103 mm 1000 Dis tan ce of extreme tension fibre from Neutral Axis h 2 = 1.82 Dis tan ce of centroid of tension steel from Neutral Axis h 1 Effective tension area of concrete surrounding the flexural tension reinforcement and having the same centroid, as that reinforcement, divided by the total number of bars in the pile A= π × D− 2× d−φ × 2×d φ Nb = = 103.08 cm2 15.98 sq.inches Summary Actual Crack width Permissible Crack width = 0 inches = 0.10 mm < 0.30 mm (Refer Cl 8.3.4 of IS: 4651 (Part-4) - 1989) Conclusion As the actual crack width is less than the permitted crack width, the design is safe in Limit state of serviceability. 2 of 2 Hence safe