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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 steel area 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 due to full loads plus effective prestress. For bridges in particu- lar, the limit can not exceed 6 , a nd a n umber o f 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 FORUM PROBLEMS 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 Properties in Analysis for Required Prestressing Fig. 1. Bridge cross section.
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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.