Chapter 3 Prestressing with Post-Tensioning
4.1.1 Friction and Wobble Losses (AASHTO LRFD Article 5.9.5.2.2b)
Friction between the strands and duct during stressing is related to intended angular changes in the tendon geometry. The top sketch in figure 4.1 shows the trajectory of a tendon within a desired duct profile. As the tendon is stressed, friction where the tendon contacts the duct wall reduces the force in the tendon. The friction coefficient (à), defined to predict losses of this type, is a function of the duct material.
Predicting friction losses along the length of a tendon using the friction coefficient alone has not proven to correlate well with field results. Another coefficient of friction loss, wobble (K), is used to account for unintended friction between strand and duct as a result of unintended duct misalignments. The concept of duct wobble is shown in the bottom sketch of figure 4.1.
Figure 4.1 – Friction and Wobble
Chapter 4 – Prestressing Losses 61 of 389 The equation to predict losses due to friction and wobble is found by considering a small section of tendon following a circular path as shown in figure 4.2.
Figure 4.2 – Section of Tendon with Radial Alignment
The variables in figure 4.2 are defined as:
dα = angle change
dl = tendon length through angle dα
F = is the force at the stressing end of the section of tendon dF = loss in tendon force P resulting from friction and wobble
n = distributed radial force resisting the angle change of the tendon force.
N = resultant of distributed radial force, n μ = friction coefficient
For an infinitesimally small element of tendon, the vertical equilibrium can be expressed as:
(Eqn. 4.1) Or, (Eqn. 4.2)
For small angles, (Eqn. 4.3) And, (Eqn. 4.4)
So that Equation 4.1 can be reduced to, (Eqn. 4.5)
Chapter 4 – Prestressing Losses 62 of 389 Summing the forces horizontally,
(Eqn. 4.6)
Again, for small angles, (Eqn. 4.7)
Equation 4.6 then reduces to, (Eqn. 4.8)
Combining expressions 4.5 and 4.8, (Eqn. 4.9)
We define the wobble coefficient, K, as a function of the prestressing force, per unit length, l, along the tendon so that the total loss in prestress force becomes:
(Eqn. 4.10)
The change in prestress force as a function of the original force is, (Eqn. 4.11)
Which can be integrated,
(Eqn. 4.12) To find, (Eqn. 4.13)
which can be expressed as, (Eqn. 4.14)
Or in terms of stress, (Eqn. 4.15)
Equation 4.15 relates tendon stress at a length along the tendon to the jacking force, tendon geometry, and the coefficients of friction and wobble. Expressions similar to equation 4.15 are found in AASHTO LRFD Article 5.9.5.2.2.
Chapter 4 – Prestressing Losses 63 of 389 Design example 1 (Appendix C) presents the design of a three-span cast-in-place box girder bridge with span lengths of 120’, 160’ and 120’. The cross section of the box girder superstructure, shown in figure 4.3, has five webs, each containing three post-tensioning tendons, each comprised of 19, 0.6” diameter prestressing strands.
Figure 4.3 – Cross Section of Superstructure for Design Example 1
Figure 4.4 shows details of the tendon profiles of design example 1 in the end spans and center span. The profiles follow a series of parabolic segments as defined in the previous chapter. The tendon elevations shown take into account the location of the strands within the ducts as shown in the detail of figure 4.3. For this tendon size, a 1” offset from the center of the duct to the center of gravity of the strands is specified (AASHTO LRFD Article 5.9.1.6). When the tendon is low at mid-spans, the strands are pulled to the top of the duct and are 1” above the center of gravity of the duct. Over the piers the strands are pulled to the bottom of the duct and their cg is modeled 1” below the cg of the duct. At points between the parabolic segments, the tendons are modeled at the cg of the duct.
Chapter 4 – Prestressing Losses 64 of 389 Figure 4.4 – Tendon Profiles For Design Example 1
Figure 4.5 shows the elevation of Tendon T2 only, highlighting the deviation angles that the tendon makes over the three spans. Assuming that the tendon is stressed from the left end, End A in figure 4.5, the calculation of the force along the length of the tendon can be made using equation 4.14. The coefficient of friction, μ, is assumed to be 0.25 (1/rad), and the wobble coefficient, k, is 0.0002 (1/ft).
Figure 4.5 – Tendon T2 Profile and Angular Deviations
The calculation of the force along the length of Tendon T2 is summarized in the table shown in table 4.1. The steps taken to compute the values in this table are:
1. Define points, in this case points 1 through 11, at the beginning and end of the tendon and at transitions between parabolic segments.
Chapter 4 – Prestressing Losses 65 of 389 2. Compute the lengths and heights of the parabolic segments from the
tendon profile with appropriate adjustments for strand location within the ducts. The heights in this calculation are the absolute value of the difference in dimensions from the bottom fiber to the tendon centroid.
3. Compute the angular deviation that the tendon makes through each parabolic segment as computed using equation 3.32 from chapter 3.
4. Using equation 4.14 compute the incremental force coefficient as a fraction of the force at the beginning of the parabolic segment. (Fi/F0).
5. Beginning with unit value at End A of the tendon, successively multiply the incremental force coefficients to determine the cumulative force coefficients at the end of each parabolic segment, in this case expressed as a fraction of the jacking force.
6. Multiply each cumulative force coefficient by the jacking stress. For this example the jacking stress was 75 percent of the ultimate strength of the tendon.
7. Find the tendon force along its length by multiplying the stress at that location times the area of the tendon. In this example the tendon is comprised of 19, 0.6” diameter strands. The area of the tendon is 0.217 x 19 = 4.123 in2.
Point b (ft) h (ft)
Angular Deviation θ=2h/b (rad)
Angular Force Coefficient
Cumulative Force Coefficient
Tendon Stress
(ksi)
Tendon Force (kips)
1 0 0.0000 0.00000 0.0000 1.0000 202.50 835
2 50 1.2500 0.05000 0.9778 0.9778 197.99 816
3 55 2.4063 0.08750 0.9677 0.9461 191.59 790
4 15 0.6563 0.08750 0.9754 0.9229 186.88 771
5 20 0.9479 0.09479 0.9727 0.8977 181.78 749
6 60 2.8438 0.09479 0.9649 0.8662 175.04 723
7 60 2.8438 0.09479 0.9649 0.8358 169.25 698
8 20 0.9479 0.08750 0.9727 0.8130 164.63 679
9 15 0.6563 0.08750 0.9754 0.7930 160.58 662
10 55 2.4063 0.05000 0.9677 0.7674 155.39 641
11 50 1.2500 0.05000 0.9778 0.7503 151.93 626
Table 4.1 – Tendon Loss Calculations – Friction and Wobble
Figure 4.6 shows a graphical representation of the force along the length of Tendon T2 at full jacking force and with losses resulting from friction and wobble. The force at the left end of the tendon is equal to the jacking force of 835 kips. The force at the anchorage at End B is 626 kips.
Chapter 4 – Prestressing Losses 66 of 389 Though the radii within the parabolic segment are changing over their lengths, the tendon force is assumed to vary linearly over the length of the parabolic segment. This results in the straight- line representation shown in figure 4.6.
It is important to note that the lengths used for computing losses for wobble in this example were the horizontal lengths of the parabolic segments. This simplification is made because the horizontal projection of the tendon length is not significantly different from the actual tendon length for most cast-in-place box girder bridges. When this assumption cannot be made, the losses should be computed along the length of the tendon.
Figure 4.6– Tendon Loss Calculations – Friction and Wobble