Crack width for temperature and moisture effects, early age concrete. BS 8007 Appendix A
Calculates crack widths in reinforced concrete.
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Determination of Crack width for 1300mm dia Pile subjected to Axial Load & Moment. Material parameters Grade of concrete f ck
Characteristic strength of concrete Grade of steel
=
M 45
=
45 N/mm2
=
Fe 415
Yield stress of steel
f y
=
415 N/mm2
Max. Permissible stresses in concrete in compresssion (Bending)
σ cb c
=
14.5 N/mm2
=
165 N/mm2
Max. Permissible stresses for steel in flexural tension Modular ratio
m
=
6.44
1.5 x m
=
9.66
=
100200 kg
=
100000 kg.cm
cl no.8.3.4 of IS: 4651-part 4
Member forces Axial load on pile
=
100.2 T
=
Moment on the pile
=
1 T.m
=
P My
Moment on the pile
=
150.2 T.m
=
Mz
=
15020000 kg.cm
Resultant Moment on Pile
=
150.20 T.m
=
MRes
=
15020333 kg.cm
=
e
=
149.90 cm
>
16.3 cm cm
D
=
130 cm
R Nb
=
65 cm
=
60 Nos
φ rebar
=
25 mm
Area Area of the the longi ongitu tudi dina nall rei reinfor nforcm cmen entt
Ast
=
294. 294.52 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
Check for eccentricity Eccentricity
=
M/P
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 cm
deff
=
120.5 cm
Cos α
=
-0.259
Cos β
=
-0.303
Sin α
=
0.966
Sin β
=
0.953
α β
=
1.8325 ra radians
=
1.8790 ra radians
Sin 4α
=
0.866
Sin 2α
=
-0.500
Sin 2β
=
-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 =
N d
Nd − 0. 5 × D 0 . 5× D
Nd − 0 . 5 × D r
R θ
R N
α
d θ
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 f s1 532 f s1
Total tension in steel below neutral axis, Ts
Ts
=
1058 f s1
Thickness of thin shell of reinforcement Assuming Neutral axis depth Coefficient Depth of Neutral Neutral axis (N x deff )
0.845 cm 0.4000
Expressions for evaluating f s1 from ∑P
Expressions for evaluating f s2 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 f s2 25207 f s2
Moment of tension in steel about the centre line of section
MTs
=
41746 f s2
Net axial stresses on the pile section
Caxial
=
Net bending stresses on the pile section
Cben
=
1337 f s1 150655 f s2
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 f s1 f s1 = 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 f s2 f s2 = 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 k g.cm
Moment of compression in steel about the centre line of section
MCs
=
2513108 kg kg.cm
Moment of tension in steel about the centre line of section
MTs
=
4162050 kg 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. si n β c g = r × cg = 28.12 cm β radians Evaluation of extreme fibre stress in concrete The mean value of f s1 & f s2 has been adopted as the final extreme fibre stress in concrete
Maximum compressive stress,
σ cb c
0.5 x (74.95 + 99.7)
f s1
=
75 kg/cm2
f s2
=
100 kg/cm2
=
87 kg/cm2
The distance of the centroid of tension steel from neutral axis has been evaluated as under
h1 = R c g − 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
d n
<
1682 kg/cm3
=
Safe 2
7 kips/inch
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.
d c × A 0.076× γ × σ st × d 3
cw = γ= A =
=
0.103 mm
1000
Dis tan ce of extreme extreme tension tension fibre from Neutral Neutral Axis h 2
=
1.82
Dis tan ce of centroid centroid of tension tension steel from Neutral Neutral Axis Axis h1
Effective tension area of concrete surrounding the flexural tension reinforcement and having the same centroid, as that reinforcement, divided by the t otal number of bars in the pile
π × D− 2× d − φ × 2 × d φ A = N b
= =
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