Chapter 3 Prestressing with Post-Tensioning
5.7 Prestressing Losses and Tendon Sizing for Final Design (Pjack)
The required prestressing force at the center of the middle span is 8,172 kips. Assuming three tendons in each of the five webs of the example problem cross section, the final post-tensioning force in each of the 15 tendons is 545 kips. The number of prestressing strands required in these tendons is a function of the amount of instantaneous and long-term prestressing force loss. Losses will be computed in accordance with the information presented in chapter 4.
5.7.1 Losses from Friction, Wobble, and Anchor Set
Using the CG Profile, the forces along the tendon after two-end stressing are computed. The assumed values for friction, wobble coefficient, and anchor set are 0.25, 0.0002/ft, and 0.375”, respectfully. The geometry required for determining the losses is shown in table 5.2. The friction diagram, expressed in terms of stress is shown in figure 5.27.
Point b (ft) h (ft)
Angular Deviation θ=2h/b (rad)
Angular Force Coefficient
Cumulative Force Coefficient
Tendon Stress
(ksi)
1 0 0.0000 0.00000 0.0000 1.0000 202.50
2 50 2.5000 0.10000 0.9656 0.9656 195.54
3 55 2.9464 0.10714 0.9629 0.9298 188.28
4 15 0.8036 0.10714 0.9707 0.9025 182.76
5 20 0.9375 0.09375 0.9729 0.8781 177.81
6 60 2.8125 0.09375 0.9652 0.8475 171.62
7 60 2.8125 0.09375 0.9652 0.8180 165.65
8 20 0.9375 0.09375 0.9729 0.7959 161.16
9 15 0.8036 0.10714 0.9707 0.7725 156.43
10 55 2.9464 0.10714 0.9629 0.7439 150.63 11 50 2.5000 0.10000 0.9656 0.7183 145.45
Table 5.2 –Data for Friction Diagram for the CG Profile Tendon
( )
( )
28, 637 10, 451
4, 752 (0.3596L Span 1) 0.6288 2.732 2.5 0.391
F + − kips
= =
+ −
( )
( )
36, 037 10, 451
8,172 (0.5L Span 2) 0.6288 2.732 2.5 1.087
F + − kips
= =
+ −
48, 426 ( 14, 415)
7, 227 (At Piers 2 and 3) 1.25 0.6288(3.768) 1.087
F = − − − = kips
− − −
Chapter 5 – Preliminary Design 104 of 369 Figure 5.27 – Friction Diagram for the CG Profile Tendon
5.7.2 Losses from Elastic Shortening
Equation 4.24, repeated here, presented the AASHTO LRFD equation for losses from elastic shortening.
(Eqn. 5.27)
Difficulty arises here, because the value of fcgp, the concrete stress at the center of gravity of the prestressing, is computed after jacking and before any long-term losses. This requires an area of steel to be multiplied by the tendon stresses in figure 5.29, to produce a prestressing force by which concrete stresses can be computed.
This difficulty is overcome by making an initial assumption of all losses equal to 20 ksi.
Subtracting this from the tendon stress at the center of the middle span (171.62 ksi) produces an estimate of final stress in the tendon at this location of 151.62 ksi (56 percent of fpu). The estimate of final force per strand in the tendon, using 0.6” diameter strands with an area of 0.217 in2 is 32.9 kips. Dividing the 550 kips per tendon by the force per strand results in 16.7 strands required in each tendon. Rounding up to 17 strands per tendon, forces after stressing can now be computed as the stresses in figure 5.29 times the tendon area of 0.217(17) = 3.689 in2.
Chapter 5 – Preliminary Design 105 of 369 All of the post-tensioning tendons will be grouted within their ducts after installation and stressing. During construction, however, tendons are unbonded when elastic shortening occurs, assuming all tendons are stressed in one phase of construction.
AASHTO LRFD Article 5.9.5.2.3b states that:
For post-tensioned structures with unbonded tendons, the fcgp value may be calculated as the stress at the center of gravity of the prestressing steel averaged along the length of the member.
Concrete stresses will be computed at four locations along the bridge and then a weighted average will be used for fcgp. The equation used for computing the concrete stress is:
(Eqn. 5.28)
where: MSW = bending moment from self weight only (ft-kips) At end of bridge:
9, 976
100.3 99.45
fcgp = = ksf At the point of maximum eccentricity in Span 1 and 3
10, 362 10, 362(2.5)2 14, 058(2.5)
150.2
99.45 643.7 643.7
fcgp = + − = ksf
Over Piers 2 and 3
10,113 10,113(1.25)2 30, 532(1.25) 99.45 643.7 643.7 66.9
fcgp = + − = ksf
At the center of the middle Span
Using a straight-line average between the stresses at these locations, the average stress in the tendon is 106.4 ksf or 0.74 ksi. The loss associated with elastic shortening, using a concrete strength at the time of stressing of 4 ksi, is then:
5.7.3 Losses from Concrete Shrinkage
The loss of prestress force resulting from shrinkage of the concrete superstructure is given by AASHTO LRFD Equation 5.9.5.4.3a-1 as:
9, 497 9, 497(2.5)2 17, 788(2.5)
118.6
99.45 643.7 643.7
fcgp = + − = ksf
15 1 2,8500
(.74) 2.6 2(15) 3,834
fpES − ksi
∆ = =
Chapter 5 – Preliminary Design 106 of 369 (Eqn. 5.29)
The strain εbdf, is the shrinkage strain from the time of prestressing to final time as provided by AASHTO LRFD Equation 5.4.2.3.3-1:
(Eqn. 5.30)
The factors in Equation 4.25 are defined by AASHTO LRFD Equations 5.4.2.3.2-2, 5.4.2.3.3-2, 5.4.2.3.2-4, and 5.4.2.3.2-5. Evaluating these for the example problem:
( )
1.45 0.13 1.45 0.13 12.57 .18 ( 1.0)
s V
k Use
S
= − = − = −
(2.00 0.014 ) (2.00 0.014(75)) 0.95
khs = − H = − =
'
5 5
1 4 1
f 1
ci
k = f = =
+ +
'
10, 000
0.996 61 4(5) 10, 000
td 61 4
ci
k t
f t
= − + = − + =
This results in a shrinkage strain of:
(1.0)(0.95)(1.0)(0.996)0.48 10 3 0.000454
εbdf = × − =
Evaluation of the transformed section coefficient Kdf requires computing the creep coefficient:
0.118
( , ) 1.9
b f it t k k k k ts hc f td i−
Ψ =
1.56 0.008 1.56 0.008(75) 0.96
khc = − H = − =
0.118
( , ) 1.9(1.0)(0.96)(1.0)(0.996)(28) 1.226
b f it t −
Ψ = =
The transformed section coefficient is then:
2
1
1 1 1 0.7 ( , )
df
p ps c pc
b f i
ci c c
K E A A e
Ψ t t
E A I
=
+ + +
Chapter 5 – Preliminary Design 107 of 369
[ ]
2
1 1.054
28500 55.335 99.45(2.5)
1 1 1 0.7(1.226)
3834 99.45(144) 643.7(144)
Kdf = =
+ + +
Multiplying the components of equation 5.29, the loss of prestress stress resulting from the shrinkage of concrete is:
0.000454(28500)(1.054) 13.6
fpSD ksi
∆ = =
5.7.4 Losses from Concrete Creep
The loss of stress in the prestressing steel resulting from the creep of concrete is:
28500
( , ) (.66)(1.226)(1.054) 6.3 3834
pCD p cgp b f i df
ci
f E f Ψ t t K ksi
∆ = E = =
5.7.5 Losses from Steel Relaxation
Using the AASHTO LRFD permissible value for low relaxation steel, the loss of prestressing stress is resulting from relaxation of the prestressing steel is:
pR 2.4
f ksi
∆ =
5.7.6 Total of Losses and Tendon Sizing
The total loss of prestressing force after jacking is the sum of elastic shortening, shrinkage, creep and relaxation:
2.6 13.6 6.3 2.4 24.9
fp ksi
∆ = + + + =
The resulting stress in the prestressing steel at the center of the middle span is then:
171.62 24.9 146.72
fp = − = ksi
The resulting force in a 0.6” diameter strand would be 31.9 kips. The number of strands in the 15 tendons, based on the force requirement at the center of Span 2 of 8,172 kips, would be 17.2. Therefore, use 18 strand tendons in final design. The jacking stressed assumed in the loss calculations was 75 percent of the ultimate strength of the strand. The resulting jacking force (Pjack) for each tendon would be 791 kips.