Friction Losses in
Prestressed Steel by
Equivalent Load Method
Engin Keyder
Associate Professor of
Civil Engineering
Middle East Technical University
Ankara, Turkey
A new approach to friction
loss calculations in prestressed steel is presented.
Tendon profile is converted
into equivalent external loads
as in load balancing. Friction
forces are then obtained by
multiplying the normal components of equivalent loads
by the coefficient of friction.
The method is applicable for
any measurable curvature,
intended or unintended.
Losses can be calculated for
intermediate points along the
length of the tendon as well
as the total length. The approach gives very close results with the classical friction
loss method.
L
asses in the prestressing
force of tendons have been
investigated in detail in the
past.L 2 In one experimental study, 3
Campbell and Batchelor measured
friction losses on a model two-span
continuous bridge girder. Their
measurements indicated that wobble and friction coefficients varied
mainly with the degree of prestressing levels. For final level of
prestress values, wobble coefficient, K, and friction coefficient, J.L,
were measured to be K = 0.0006
andJ.L = 0.25.
Along with stress level in the
tendons, the wobbling effect has
been shown to be influenced by the
length of the tendon, coefficient of
friction between the contact materials, and workmanship in the
alignment of the duct. The friction
coefficient, in tum, was found to be
dependent on the type of steel used
and the kind and condition of surface. Values for K and J.L vary
greatly . Typical range of values for
normal conditions for K and J.L are
given in the PCI Recommendations4 and ACI Code Commentary.5
The loss in the prestressing force
due to friction is determined from
where
P = prestressing force at jack-
ing end
Px
= prestressing force after
friction losses at a distance
l from jack
K = wobble coefficient
l = length
coefficient of friction
J.L
=
(J
= angle change in radians
~
_j
1+------x/2
t - - - - -- - x - - - - ---t
Fig. 1. Curvature calculation by classic approach.
74
PCI JOURNAL
For small friction losses:
P =Px (1
+ Kl + J1l))
(2)
The() term in Eqs. (1) and (2) is
obtained by considering the tangents to the tendon curve or by
determining the radius of curvature.
Determination of 8 by
Classical Approach
With reference to Fig. 1.:
()
m
x/2
2m
x
tan-=--
2
m=2y
Fig. 2. Friction force in a harped tendon due to curvature.
tan .!_ = ()
2
2
()
4y
=
2
X
() = ~
radians
X
()l
()2
=
~fiih~;~~;-~__-.P-·P
y
xl
=L
~
~
()=
y
xl
#77
L
+L
radians
x2
a)
P
Determination ofK and11- Values
In friction loss calculation for K
and 11- values, usually average recommended values are used. If
measurements are possible at the
job site, K and 11- values may be
more accurately established.
EQUIVALENT LOAD
APPROACH
The equivalent load approach
presents a new look at the friction
loss problem. The tendon profile is
converted into an equivalent load
that causes the tendon to bear
against the surrounding material.
The equivalent load thus causes
equal and opposite normal or contact forces. Friction force, then, is
simply the normal force times the
coefficient of friction.
In the case of a bent tendon, the
normal force is N (Fig. 2) and the
March-Aoril 1990
'\
I
I
~
w
--P-dP
~n
b)
Fig. 3. Friction forces in a curved tendon due to curvature.
friction force will be 11- N.
For a curved tendon, the equivalent load will be a distributed load
(uniform for a parabolic tendon).
The normal force will be n (Fig. 3)
and friction force, 11- n. Over a total
length (L), the total friction loss
will be 11- nL (Fig. 3). For an intermediate point, at distance /, the
curvature friction loss will be 11- nl.
The problem of friction loss thus
reduces to finding the equivalent
load as in load balancing. The fol-
lowing two examples illustrate the
proposed method and compare the
results with the classic approach.
EXAMPLE 1
Determine the friction losses due
to curvature only for the prestressed concrete beam shown in
Fig. 4. The prestress jacking force
is 300 kips (1334 kN). "A" is the
jacking end and "D" is the anchorage end. 11- = 0.25.
75
Equivalent Load Method
Normal force at B:
8
N = - - x300
9xl2
= 22.2 kips (98 .7 kN)
0 ~
0-y
clBJ
j •. .•.
1
Ar-----------------~
Loss at B due to curvature:
41 PB = IL N = 0.25 X 22.2
= 5.55 kips (24.7 kN)
J •.
36.0'
Remaining prestressing force :
300- 5.55 = 294.45 kips (1310 kN)
~00-llP
Loss at C due to curvature:
8
N = - - X 294.45
9xl2
= 21.81 kips (97 .0 kN)
41 PC = IL N = 0.25 X 21.81
= 5.45 kips (24.2 kN)
Fig. 4. Prestressed beam with harped tendon.
Final prestressing force:
294.45- 5.45 = 289 kips (1285 kN)
Classic Approach*
Curvature changes at B:
= -89x12
()
12"
. ~J~lt2],_k-=-~== ., =-J l!BlJ
= 0.074
Loss atB:
41 p = p.() P
= 0.25 X 0.074 X 300
0
= 5.55 kips (24.7 kN)
t
!
1
36.0
Remaining prestressing force:
300- 5.55 = 294. 45 kips (1310 kN)
Loss atC:
Fig. 5. Prestressed beam with parabolically draped strand.
() = 0.074
AP =
(J
P = 0.25
0.074 X 294.45
= 5.45 kips (24.2 kN)
X
Final prestressing force:
294.45 - 5.45 = 289 kips (1285 kN)
- b---~~..?'.;.;..'---11-
EXAMPLE2
Determine the friction losses due
to curvature alone in the parabolic
tendon of the prestressed beam
shown in Fig. 5. The prestressing
force P is 300 kips (1334 kN) at A
andp. = 0.25.
e (at center) = 8 + 4
=
12 in. (305 mm)
• Note that both the classic approach and the
equivalent load method give identical results since
N = 8 P. In a curved tendon the results differ
slightly, as shown in the calculations of Example 2.
76
Fig. 6. Wobble effect converted to equivalent loads.
Equivalent Load Method
Equivalent load
8 Pe
8x300x12/l2
w = - - = - - -2 - V
=
(36)
1.85 kips per ft
(27kN/m)
AP
= w L p. = 1.85x36x0.25
= 16.65 kips
(74kN)
RemainingP:
300- 16.65 = 283.35 kips
(1260 kN)
PC! JOURNAL
Classic Approach
8
= ! [ = 8x12/22 = 0 .22
36
d P = () P = 0.25x0.22x300
= 16.67 kips (74 kN)
X
Remaining P:
300 - 16.67 = 283.33 kips (1260
kN)
WOBBLE EFFECT
Wobble or unintended curvature,
if measurable, can be converted to
an equivalent load and the approach described may be applied.
Fig. 6 illustrates the principle. In
practice , however, unintended
curvatures are usually too small to
be measured. Measurements give
average wobble values over a
length rather than local curvatures.
March-April 1990
For average conditions, recommended K values from the ACI
Code tables may be used as in the
classic approach.
In the example problems, losses
due to wobble effect (taking K =
0.0006) may be approximated as:
36 X 300 X 0.0006 = 6.48 kips
(29 kN)
for each problem.
REFERENCES
l. Cooley, E. H., "Friction in Post-
2.
3.
CONCLUDING REMARKS
1. The equivalent load method
provides a direct and simple approach to curvature friction loss
calculations.
2. The method is as accurate as
the classical approach.
3. The method may be used for
loss calculations at intermediate
points.
4.
5.
tensioned Prestressing Systems,''
and "Estimation of Friction in Prestressed Concrete," Cement and
Concrete Association, London,
1953.
Lin, T . Y., "Cable Friction in Posttensioning," Journal of Structural
Division, American Society of Civil
Engineers, November 1956.
Campbell, T. 1., and Batchelor, B.
deY., "Load Testing of a Model
2-Span Continuous Prestressed
Concrete Trapezoidal Bridge
Girder," PCI JOURNAL, V. 22,
No. 6, November-December 1977,
pp. 62-79.
PCI Committee on Prestress Losses,
"Recommendations for Estimating
Prestress Losses," PCI JOURNAL,
V. 20, No. 4, July-August 1975, pp.
43-75.
ACI Committee 318. "Building
Code Requirements for Reinforced
Concrete (ACI 318-83)," Commentary, American Concrete Institute,
Detroit, MI, 1983.
77