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A 0.025
B 120
C 10
Ep 2.00E+05
eu 0.047000001
fpu 1860
Bottom cover 60 mm
Pi 450 KN
Ap 600 mm2
fp 750 MPa
εp 0.00375 750
at bottom fiber
Fc 8.5 MPa
F'c 40 Mpa
εc 0.000225177
ε'c 0.002 8.50002
at location of PS wire
Fc 7.3 MPa
F'c 40 Mpa
εc 0.000191687
ε'c 0.002 7.300024
compressive strain
εc 0.000225177
β = 0.67316547
α = 0.160975195
c= 403.4205085 mm
C= 524590.0016 N
φ -5.58169E-07
f'c 40 MPa
ε'c 0.002
depth of section 600 mm
location of PS wires 540 mm
B 300 mm
εp 0.00375
A 0.025
B 120
C 10
Ep 2.00E+05 MPa
eu 0.047
fpu 1860 MPa
Ap 600 mm2
εt strain ratio β = α = εb c εcp ε'p F'p
-0.00023 403.4205
0.00019 0.095 0.672117 0.136869 8.54E-05 413.9335 5.79E-05 0.003808 761.5686
0.00025 0.125 0.673913 0.177755 0.000209 326.6664 0.000163 0.003913 782.6382
0.0005 0.25 0.681818 0.336111 0.000996 200.527 0.000846 0.004597 919.082
0.00075 0.375 0.690476 0.475216 0.001966 165.6959 0.001694 0.005444 1087.38
0.001 0.5 0.7 0.595238 0.002975 150.9485 0.002577 0.006328 1257.905
0.0015 0.75 0.722222 0.778846 0.005071 136.9571 0.004414 0.008164 1540.767
0.002 1 0.75 0.888889 0.007635 124.5514 0.006671 0.010421 1660.685
0.0025 1.25 0.785714 0.92803 0.010462 115.7187 0.009166 0.012916 1687.565
0.003 1.5 0.833333 0.9 0.012884 113.3193 0.011296 0.015046 1699.789
0.0035 1.75 0.9 0.810185 0.014443 117.0382 0.012649 0.016399 1706.807
0
100
200
300
400
500
600
-0.000005 0 0.000005 0.00001 0.000015 0.00002
Mo
men
t
Curvature rad/mm
M vs φ
`
M KN-m
P'p KN C equilibrium φ Ap only
-5.58E-07 0
456941.2 456941.2 6.17E-08 4.59E-07 183.185
469582.9 469582.9 -1.29E-06 7.653E-07 201.8867
551449.2 551449.2 -2.772E-06 2.493E-06 260.0847
652427.8 652427.8 -1.888E-06 4.526E-06 314.9892
754742.7 754742.7 -8.871E-08 6.625E-06 367.6865
924460.3 924460.3 -6.403E-09 1.095E-05 453.4878
996411.1 996411.1 -1.681E-05 1.606E-05 491.5229
1012539 1012539 6.9116E-07 2.16E-05 500.7401
1019874 1019874 4.6741E-07 2.647E-05 502.577
1024084 1024084 4.163E-07 2.99E-05 499.0698
0.00002 0.000025 0.00003 0.000035
Curvature rad/mm
φ
f'c 40 MPa
ε'c 0.002
depth of section 600 mm
location of PS wires 540 mm
B 300 mm
εp 0.00375
A 0.025
B 120
C 10
Ep 2.00E+05 MPa
eu 0.047
fpu 1860 MPa
Ap 600 mm2
ε top strain ratio β = α = ε bottom c εcp strain in PS due to εt
ε'p (net strain in
PS)
-0.00023 403.4205
0.00019 0.095 0.672117 0.136869 8.41E-05 415.8636 5.67E-05 0.00380684
0.00025 0.125 0.673913 0.177755 0.000204 330.6548 0.000158 0.003908405
0.0005 0.25 0.681818 0.336111 0.000932 209.4929 0.000789 0.004538951
0.00075 0.375 0.690476 0.475216 0.001796 176.7542 0.001541 0.005291441
0.001 0.5 0.7 0.595238 0.002675 163.2695 0.002307 0.00605754
0.0015 0.75 0.722222 0.778846 0.004608 147.3577 0.003997 0.007746951
0.002 1 0.75 0.888889 0.006846 135.6526 0.005962 0.009711638
0.0025 1.25 0.785714 0.92803 0.009345 126.6325 0.008161 0.011910896
0.003 1.5 0.833333 0.9 0.011518 123.9812 0.010066 0.01381662
0.0035 1.75 0.9 0.810185 0.012909 127.9756 0.011268 0.015018562
0
100
200
300
400
500
600
-0.000005 0 0.000005 0.00001 0.000015
Mo
men
t
Curvature rad/mm
M vs φ
`
M KN-m
F'p P'p KN εs (in steel
Fs (stress
in steel) Ps KN C equilibrium φ Ap , Ast
-5.582E-07 0
761.3386 456.8032 5.67E-05 11.34311 2.268621 459.0718 2.7068E-05 4.569E-07 183.7414
781.6418 468.9851 0.000158 31.65615 6.33123 475.3163 1.0677E-05 7.561E-07 203.7128
907.5871 544.5523 0.000789 157.7654 31.55307 576.1053 -3.157E-06 2.387E-06 269.9526
1057.195 634.3172 0.001541 308.2634 61.65268 695.9699 -1.186E-05 4.243E-06 333.354
1206.75 724.0499 0.002307 461.4831 92.29662 816.3474 -0.0008092 6.125E-06 394.1778
1491.108 894.6648 0.003997 500 100 994.6648 -1.018E-06 1.018E-05 484.1904
1642.035 985.2207 0.005962 500 100 1085.221 4.5772E-06 1.474E-05 530.8143
1680.057 1008.034 0.008161 500 100 1108.034 -7.524E-06 1.974E-05 543.2155
1693.052 1015.831 0.010066 500 100 1115.831 -9.859E-07 2.42E-05 544.9062
1699.644 1019.786 0.011268 500 100 1119.787 -0.0003 2.735E-05 540.1973
0.000015 0.00002 0.000025 0.00003
Curvature rad/mm
M vs φ
f'c 40 MPa
ε'c 0.002
depth of section 600 mm
location of PS wires 540 mm
B 300 mm
εp 0.00375
A 0.025
B 120
C 10
Ep 2.00E+05 MPa
eu 0.047
fpu 1860 MPa
Ap 600 mm2
εt strain ratio β = α = εb c εcp ε'p F'p
-0.00023 403.4205
0.00019 0.095 0.672117 0.136869 8.91E-05 408.4236 6.12E-05 0.003811 762.2371
0.00025 0.125 0.673913 0.177755 0.000216 321.5561 0.00017 0.00392 783.9511
0.0005 0.25 0.681818 0.336111 0.001026 196.6335 0.000873 0.004623 924.3989
0.00075 0.375 0.690476 0.475216 0.002021 162.3865 0.001744 0.005494 1097.186
0.001 0.5 0.7 0.595238 0.003058 147.8645 0.002652 0.006402 1271.818
0.0015 0.75 0.722222 0.778846 0.005243 133.4619 0.004569 0.008319 1556.49
0.002 1 0.75 0.888889 0.008001 119.9915 0.007001 0.010751 1666.549
0.0025 1.25 0.785714 0.92803 0.011049 110.7103 0.009694 0.013444 1690.866
0.003 1.5 0.833333 0.9 0.013717 107.6766 0.012045 0.015795 1703.705
0.0035 1.75 0.9 0.810185 0.015572 110.1108 0.013665 0.017415 1711.971
0
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-0.000005 0 0.000005 0.00001 0.000015 0.00002
Mo
men
t
Curvature rad/mm
M vs φ
M KN-m
P'p KN εsc Fsc Psc KN C equilibrium φ Ap , Asc
-5.582E-07 0
457.3423 0.000162 32.4175599 6.483511987 450.8588 -6.83498E-06 4.652E-07 184.6935
470.3706 0.000203 40.6703685 8.134073703 462.2369 -0.000348427 7.775E-07 203.4286
554.6393 0.000347 69.4863763 13.89727526 540.7421 -2.65392E-05 2.543E-06 262.4232
658.3119 0.000473 94.5766618 18.91533237 639.3967 -0.000179123 4.619E-06 318.5077
763.0906 0.000594 118.844593 23.76891862 739.3223 -0.000621368 6.763E-06 372.3811
933.8939 0.000826 165.130059 33.02601185 900.8679 -1.5777E-06 1.124E-05 458.9042
999.9293 0.001 199.985853 39.9971706 959.9321 -4.16041E-06 1.667E-05 494.3681
1014.52 0.001145 229.022531 45.80450612 968.7152 -1.51268E-06 2.258E-05 502.9597
1022.223 0.001328 265.665602 53.13312048 969.0896 -1.01279E-06 2.786E-05 505.3338
1027.183 0.001593 318.565989 63.71319787 963.4694 -8.47983E-07 3.179E-05 503.116
0.00002 0.000025 0.00003 0.000035
Curvature rad/mm
φ
f'c 40 MPa
ε'c 0.002
depth of section 600 mm
location of PS wires 540 mm
B 300 mm
εp 0.00375
A 0.025
B 120
C 10
Ep 2.00E+05 MPa
eu 0.047
fpu 1860 MPa
Ap 600 mm2
εt strain ratio β = α = εb c εcp ε'p F'p
-0.00023 403.4205
0.00019 0.095 0.672117 0.136869 8.77E-05 410.4562 6E-05 0.00381 761.9884
0.00025 0.125 0.673913 0.177755 0.000211 325.6746 0.000165 0.003915 782.8898
0.0005 0.25 0.681818 0.336111 0.000958 205.7156 0.000812 0.004563 912.3082
0.00075 0.375 0.690476 0.475216 0.001844 173.5006 0.001584 0.005334 1065.688
0.001 0.5 0.7 0.595238 0.002745 160.2154 0.00237 0.006121 1218.798
0.0015 0.75 0.722222 0.778846 0.004761 143.7442 0.004135 0.007885 1508.714
0.002 1 0.75 0.888889 0.007167 130.8992 0.006251 0.010001 1650.873
0.0025 1.25 0.785714 0.92803 0.009883 121.1312 0.008645 0.012395 1683.941
0.003 1.5 0.833333 0.9 0.012293 117.7036 0.010763 0.014514 1696.936
0.0035 1.75 0.9 0.810185 0.013957 120.2935 0.012212 0.015962 1704.564
0
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-0.000005 0 0.000005 0.00001 0.000015 0.00002 0.000025
Mo
men
t
Curvature rad/mm
M vs φ
P'p KN εs Fs Ps KN εsc Fsc Psc KN C equilibrium
457.193 6E-05 11.99315 2.398631 0.000162 32.44521 6.489041 453.1026 4.27968E-05
469.7339 0.000165 32.90483 6.580967 0.000204 40.78835 8.15767 468.1573 -8.58576E-05
547.3849 0.000812 162.4983 32.49967 0.000354 70.83352 14.1667 565.7179 -2.81215E-06
639.4127 0.001584 316.8571 63.37142 0.000491 98.12699 19.6254 683.1587 -4.45979E-06
731.2788 0.00237 474.0925 94.81849 0.000626 125.1008 25.02017 801.0771 2.93249E-05
905.2286 0.004135 500 100 0.000874 174.7775 34.9555 970.2731 -7.38904E-07
990.5236 0.006251 500 100 0.001083 216.6528 43.33055 1047.193 -0.000288996
1010.365 0.008645 500 100 0.001262 252.3346 50.46691 1059.898 -5.32054E-06
1018.162 0.010763 500 100 0.001471 294.1469 58.82939 1059.332 -6.43451E-07
1022.739 0.012212 500 100 0.001754 350.8539 70.17078 1052.568 -0.00026618
0.000025 0.00003 0.000035
Curvature rad/mm
M KN-m
φ Ap , Asc , Ast
-5.58E-07 0
4.629E-07 185.2904
7.676E-07 205.3459
2.431E-06 272.6137
4.323E-06 337.4054
6.242E-06 399.6706
1.044E-05 490.3615
1.528E-05 534.8792
2.064E-05 546.1312
2.549E-05 548.3245
2.91E-05 545.091
PRESTRESSED CONCRETE STRUCTURES
ASSIGNMENT 6
P Sameer Kumar Sarma
Ce13p1007
The given cross section is analysed for different cases of Asc, Ast by assuming initial top strain.
For this particular top strain the depth of neutral axis is found by trail and error by equating
tensile force and compressive force. Then the resisting moment is found for that particular top
strain. Corresponding curvature is found and plotted.
Algorithm
Assume εt
Find β , α
Assume εb
Find c (depth of neutral axis) = (εt)*depth of section/(εt+εb)
Find strain at level of PS wires εcp= (location of PS wires 'd'-c)*εt/c
Net strain in PS ε'p= εcp +εpi (εpi is found from intial PS force and Ramberg Osgood
curve)
Increase in stress in PS wires 'fcp' is found from Ramberg Osgood curve by using ε'p
The tensile force in 'T' PS wires is found from fcp by multiplying with area of PS wires
The compressive force 'C' is found by c*fck*b*β*α
For equilibrium C=T else change εb and repeat above steps till C=T
When C=T , Resiting moment of section = C*(D/2-.5*c*β) +T*(d-D/2) (D is depth of section)
If Asc or Ast is present , strain in these reinforcement is found by similar traingles concept
as done for PS wires and force is found by respective stress strain curves. The force of Asc is
added to C and force in Ast is added to T and equilibrium shall be checked. Else, bottom strain
is varied solved again. When T= C , resisting moment is found by adding moment due to Asc
and Ast
Curvature φ = εt/c rad/mm
The Moment vs curvature for various cases is plotted and shown below. Following inferences
can be made by observing the values in calculations and graph
1- The curvature value increases due to presence of steel in compressive area
i.e increases ductility
2- Steel in tensile area stiffens the beam i.e it have more moment capacity and less
curvature for same maximum strain
3- Presence of steel in compressive region do not increase capacity of section
0
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400
500
600
-0.000005 0 0.000005 0.00001 0.000015 0.00002 0.000025 0.00003 0.000035
Mo
me
nt
KN
-m
Curvature rad/mm
M vs φ
Ap only
Ap , Ast
Ap , Asc
Ap , Asc , Ast