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Q1: For bridge girder design, our computer computations
using transformed section analysis reduce the number of
prestressing strands required by as many as 10 strands. Can
we save this much strand just by converting from gross sec-
tion to transformed section analysis?
A1: The use of transformed section analysis is more accu-
rate than gross section analysis. As long as we have steel
that is assumed to be bonded to the concrete, any loading
applied, including prestress transfer to the concrete member,
acts on a composite section of steel and concrete. This inter-
action is taken into account by converting the steel area to
an equivalent precast concrete area, by multiplying the steelarea by (n – 1), where n is the modular ratio E ps / E c, where
E ps and E c are the modulus of elasticity of prestressing steel
and concrete, respectively, at the time of application of the
load being considered. The (–1) value in the (n – 1) term
recognizes that steel is replaced with concrete, and the
“hole” left from the removal of steel is filled with concrete.
Thus the (n – 1) factor should be used in conjunction with
gross concrete section properties. If net concrete section
properties are to be used, the gross area has to be reduced by
the area of steel, and then the net area has to be supple-
mented by a transformed steel to concrete area equal to n
times the steel area.
In older days of prestressed concrete, the effect of the
more precise transformed section was not considered be-
cause prestressing forces were smaller and electronic com-
putation tools, such as spreadsheet programs, were not
available. Thus, it was conservative to ignore the differences
between gross concrete section, net concrete section, and
transformed concrete section.
Since the analysis being considered is a linear elastic
analysis, it only affects service limit states. The most impor-
tant stress limit is the concrete tension at final condition dueto full loads plus effective prestress. For bridges in particu-
lar, the limit cannot exceed 6 , and a number of states
limit it even further. Transformed section analysis reduces
the bottom fiber tensile stress due to gravity loads and thus
reduces the demand for prestressing in order to meet the
code stress limit.
Prestress loss estimates by AASHTO formulas were
based on the assumption that gross section properties are
used in the concrete stress analysis. Unless these formulas
′ f c
2 PCI JOURNAL
OPEN FORUMPROBLEMS AND SOLUTIONS
The comments and opinions expressed herein are those of the contributing
author and do not necessarily reflect official PCI policy. Some of the provided
answers may have alternate solutions. Reader comments are invited.
Significance of Transformed Section Propertiesin Analysis for Required Prestressing
Fig. 1. Bridge crosssection.
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are modified, transformed section analysis may be incorrect
and misleading. If the proper loss components are accounted
for, the difference in results between the approximate gross
section analysis and the more accurate transformed section
analysis is not expected to be large.
Q2: Please explain what effects should be taken into ac-
count in transformed section analysis compared to gross
section of analysis.
A2: Elastic shortening loss at prestress transfer and elastic
elongation gain at the time of application of gravity loads
must be isolated and subtracted from the total prestress loss
to obtain the long-term loss due to creep and shrinkage of
concrete and relaxation of steel. When external loads, in-
cluding initial prestress just before transfer to concrete, are
introduced to a transformed section, the elastic losses or
gains are automatically accounted for. The long-term loss
should be applied to the net concrete section that exists dur-
ing the time of its development. Long-term loss between
transfer and deck placement should be applied to the net
precast section. Long-term loss between time of deck place-ment and final time should be applied to the net composite
girder/deck section. However, the latter long-term loss com-
ponent is a small fraction of the total long-term loss and is
generally not separated out in most loss prediction methods.
It is therefore acceptable and slightly more conservative to
apply the total long-term loss to the net precast section or
even the gross precast section.
Q3: Please give an example illustrating your point.
A3: Consider Example 9.4 of the PCI Bridge Design
Manual (BDM), a single-span AASHTO-PCI bulb-tee
girder bridge (BT-72), with bridge cross section as shown in
Fig. 1. Cross-sectional dimensions of the BT-72 girder are
November-December 2002 3
provided in Fig. 2. Dimensions of the composite section are
given in Fig. 3. The properties of various cross sections and
midspan moments for the example are given in Tables 1 and
2, respectively. The following data summarize the material
properties of the prestressing strand, precast bridge girder,
and cast-in-place deck:
Fig. 3. Dimensions
of the compositesection.
Fig. 2. AASHTO-PCI BT-72 dimensions.
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4 PCI JOURNAL
Prestressing strands:
48 – 1 / 2 in. dia., 270 ksi, low-relaxation strand, A ps =7.344 sq in., y ps relative to bottom fibers = 6.92 in., E ps =
28,500 ksi, f pi = 202.5 ksi
Precast girder:
f ′ci = 5.8 ksi, E ci = 4617 ksi, ni = E ps / E ci = 6.17
f ′c = 6.5 ksi, E c = 4888 ksi, n = E ps / E c = 5.83
C.I.P. deck:
f ′c = 4.0 ksi, E cd = 3834 ksi, nd = E cd / E c = 0.78
Relative humidity = 70 percent
There are two options for calculating elastic loss or gain
due to various effects: Option A, using transformed section
analysis, and Option B, using an iterative net section analy-
sis. Using Option B with gross section properties used to ap-
proximate net section properties is the common practice at
present. The values for this example are summarized in
Table 3. The initial elastic loss of 18.90 given in Table 3
was calculated using transformed section properties. The
elastic shortening loss of 18.60 ksi reported in the BDM,
which was calculated iteratively using gross precast section
properties, is slightly different.
The prestress loss according to AASHTO LRFD Specifi-
Section A (sq in.) I (in.4) Y b (in.) α = 1 + A*e p*e p / I α b = 1 + A*e p* y b / I
Gross precast cross section 767 545,894 36.60 2.2377 2.5263
Net precast cross section 760 539,362 36.89 2.2648 2.5569
Gross composite section 1419 1,089,063 54.59 3.9530 4.3801
Net composite section 1395 1,064,857 54.59 4.0147 4.4484
Transformed section at release 805 577,779 35.20 2.1142 2.3869
Transformed section at service 802 575,765 35.29 2.1216 2.3952
Transformed composite section 1454 1,167,708 53.42 3.6874 4.0858
Table 1. Properties of various sections.
Loads M (kip-in.)
Beam weight, M g 17,258
Deck weight, M s 19,915
Superimposed dead load, M sd 6480
Live load, M l 25,666
Table 2. Applied moments at midspan for Service IIIstress analysis, for calculating bottom fiber stress.
Note: 1 in. = 25.4 mm; 1 sq in. = 645 mm2; 1 in.4 = 416,231 mm4.
Note: 1 kip-in. = 0.113 kN-m.
Concrete stress at
Loading centroid of strands (ksi) Elastic loss/gain
Initial 3.061 18.90 ksi
Deck weight –0.981 –5.72 ksi
SIDL –0.256 –1.50 ksi
LL –1.016 –5.92 ksi
Table 3. Elastic losses and gains due to external loads,(initial prestress just before transfer is considered anexternal load).
Note: 1 ksi = 0.006895 kN-mm2.
cations is made up of four components:
Elastic loss at transfer = 18.90 ksi
Shrinkage = 17 – 0.15H
= 17 – 0.15(70)
= 6.50 ksi
Creep = 12 f cgp – 7 f cdp
= 12(3.061) – 7(–0.981 – 0.256)
= 28.07 ksi
Relaxation = 6 – 0.12∆ f pES – 0.06(∆ f pSR + ∆ f pCR)
= 6 – 0.12(18.9) – 0.06(6.5 + 28.07)= 1.66 ksi
The total long-term loss according to AASHTO LRFD is
6.50 + 28.07 + 1.66 = 36.23 ksi. Slightly different values for
creep and relaxation losses are given in the Bridge Design
Manual because gross section properties are used to approx-
imate net section properties. The long-term loss in the BDM
is 6.50 + 26.60 + 1.80 = 34.90 ksi.
Since the loss formulas in AASHTO LRFD were devel-
oped on the assumption that designers use gross section
properties, the long-term loss includes an allowance for
elastic gain due to deck weight and superimposed dead load(SIDL). These two components must be excluded if trans-
formed section properties are used to calculate concrete
stresses.
An exact analysis can be undertaken in one of two ways:
1. Apply an initial prestress of 202.50 ksi plus dead and
live loads to the transformed section properties. Apply true
long-term losses of 43.45 ksi to net section properties.
2. Apply all the forces mentioned in Item 1 to net section
properties. In addition, apply the elastic loss at transfer,
18.90 ksi, and the elastic gain due to deck weight (–5.72
ksi), SIDL (–1.50 ksi), and live load (–5.92 ksi) separately
as external loads on net section properties.
The results of these two procedures are compared in
Table 4 with the results of the current practice of calculation
using gross section properties.
Q4: What do you recommend as the best approach for
calculation of bottom fiber stress at service, assuming that I
would like to use the AASHTO LRFD Refined Method for
loss prediction?
A4: The best solution is the most exact one within the
limitations of the AASHTO LRFD loss formulas. First, cal-
culate initial stress due to prestress force just before release
and member weight. Use transformed section properties
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November-December 2002 5
based on the concrete modulus of elasticity at time of pre-
stress transfer. Disregard the LRFD formula for elastic loss
at transfer. Then calculate the stress due to deck weight ap-
plied to the precast transformed section at time of deck
placement (i.e., using E c of the girder) and the stress due to
superimposed dead load (SIDL) applied to the transformed
composite section. The stresses due to these three loading
cases should be calculated at the bottom fiber and again at
the centroid of the strand. The stresses at the centroid of the
strand should then be used to calculate the LRFD long-termloss. Before the long-term loss is used in concrete stress cal-
culation, elastic gain due to deck and SIDL must be ex-
cluded (increasing the quantity you assign to long-term
loss). The long-term loss should be applied to the net precast
concrete section (which is very close to the gross section).
Since the LRFD long-term loss does not explicitly give a
breakdown of the loss before and after composite action oc-
curs, it is reasonable to apply the entire long-term loss to the
precast-only section. Finally, calculate the concrete bottom
fiber stress due to live load using the transformed composite
section.
The approximate solution currently used in common prac-
tice is to use the gross precast and gross composite sections.
For that solution to be equivalent to the exact solution, one
should account for the elastic loss at transfer, which is gen-
erally done now. In addition, the designer should include an
allowance in the long-term loss formula for the elastic gain
due to deck weight and superimposed dead and live loads.
These elastic gain values may be estimated using gross sec-
tion properties, in place of the more accurate transformed
section properties, by multiplying the concrete stress at steel
centroid by the modular ratio. Each steel stress gain is thenmultiplied by the steel area, and the resulting force is as-
sumed as an additional prestress force applied to the gross
concrete section. The results of both the preferred exact so-
lution and the acceptable approximate solution are summa-
rized in Table 5.
[Contributed by Shane A. Hennessey, Project Engineer,
Tadros Associates, LLC, Omaha, Nebraska, and
Maher K. Tadros, Cheryl Prewett Professor of Civil
Engineering, University of Nebraska-Lincoln,
Omaha, Nebraska.]
Table 4. Bottom fiber stress by various methods.
Net section (exact) Approximate gross section (current practice)
Transformed Elastic Elastic
Cause section (exact) (ksi) Loading (ksi) loss/gain (ksi) Net (ksi) Loading (ksi) loss/gain (ksi) Net (ksi)
Initial prestress 4.410 5.006 –0.596 4.410 4.898 –0.450 4.448
Self-weight –1.051 –1.180 +0.129 –1.051 –1.157
Deck weight –1.221 –1.362 0.141 –1.221 –1.335 – –1.335
SIDL –0.294 –0.329 0.035 –0.294 –0.323 – –0.323
Long-term loss –1.074 –1.074 – –1.074 –0.844 – –0.844
LL –1.166 –1.303 0.137 –1.166 –1.278 – –1.278
Net –0.397 –0.397 –0.488
Loading Transformed section (exact) (ksi) Gross section (approximate) (ksi)
Initial prestress plus self-weight 4.410 – 1.051 = 3.359 4.898 – 0.583 – 1.157 + 0.126 = 3.284
Long-term loss –1.074 0.138 + 0.036 – 1.051 + 0.143 = –0.734*
Deck weight –1.221 –1.335
Superimposed dead loads –0.294 –0.323
Live load –1.166 –1.278
Net –0.397 –0.386
Table 5. Proposed options for analysis.
* Includes long-term losses and elastic gains due to deck weight, superimposed DL, and LL.
Note: 1 ksi = 0.006895 kN-mm2
.
Note: 1 ksi = 0.006895 kN-mm2.