JL-88-September-October Flexural Strength of Prestressed Concrete Members

7/24/2019 JL-88-September-October Flexural Strength of Prestressed Concrete Members

1/28

Flex u ral St ren g th o f

Pres t ressed onc re te

em ers

Brian C. Skogman

Structural Engineer

Black Veatch

Kansas C ity, Missouri

Maher K. Tadros

Professor of C ivil Engineering

University of Nebraska

Omaha, Nebraska

Ronald Grasmick

Structural Engineer

Dana Larson Roubal and

Associates

Omaha, Nebraska

T

he flexural strength theory of

prestressed concrete members is

well established. The assumptions of

equivalent rectangular stress block and

plane sections remaining plane after

loading a re common ly accepted. How-

ever, the flexural strength analysis of

prestressed concrete sections is more

complicated than for sections reinforced

with mild bars because high strength

prestressing steel does not exhibit a

yield stress plateau, and thus cannot be

modeled as an elasto-plastic material.

In 1979, Mattock' presented a pro-

cedure for calculating the flexural

strength of prestressed concrete sections

on an HP-67/97 programmable cal-

culator. His procedure consisted of the

theoretically exact strain compatibil-

ity method and a power formula for

modeling the stress-strain curve of

prestressing steel. This power formula

was originally reported in Ref. 2 and is

capable of m odeling actual s tress-strain

curves for all types of steel to w ithin 1

percent.

Prior to Mattock's paper, the strain

compatibility method commonly re-

quired designers to use a graph ical so-

lution for the steel stress at a given

strain. There are computer programs for

strain compatibility analysis (see for

example Refs. 3 and 4 ). However, these

programs were developed on main

96


2/28

frame computers for research purposes,

and are not intended as design aids.

In this paper, the iterative strain com-

patibility method is coded into a user

friendly program in BASIC. The pro-

gram assumes a neutral axis depth, cal-

culates the correspond ing steel strains,

and obtains the steel stresses by use of

the power formula. 2 Force equilibrium

(T = C) is check ed, and if the d ifference

is significant, the neu tral axis depth is

adjusted and the procedure repeated

until T and C are equal. Users are al-

lowed to input steel stress-strain dia-

grams with either minimum A STM spe-

cified properties or actual ex

perimentally obtained properties.

Nonco mposite and composite sections

can b e ana lyzed, and a library of com-

mon precast con crete section sh apes is

included.

A recent survey by the a utho rs is re-

ported herein. It indicates that the ac -

tual steel stress, at a given strain, could

be as high as 12 percent over that of min-

imum ASTM values. Also, future

developments might produce steel

types with more favorable properties

than those currently covered by AS TM

standard s. With sufficient documen ta-

tion, precast concrete produ cers could

use the proposed computer an alysis to

take advan tage of these improved prop-

erties.

A secon d ob jective of this paper is to

present an approximate noniterative

procedure for calculating the

prestressed steel stress,

f

at ultimate

flexure, without a computer. The pro-

posed procedure requires a hand held

calculator with the power function y'.

Currently, such scientific calculators are

inexpensive, which makes the proposed

procedure a logical upgrade of the ap-

proximate procedure represented by

Eq. (18-3) in the ACI 318-83 Code.'

The proposed approximate procedure

is essentially a one-cycle strain-com-

patibility solution. The main approxi-

mation involves initially setting the ten-

sile steel stresses equal to the re spective

ynops s

Flexural strength theory is reviewed

and a computer program for flexural

analysis by the iterative strain com-

patibility method

is

presented. It is

available from the PCI for IBM PC/XT

and AT microcomputers and compat-

ibles.

Secondly, a new noniterative ap-

proximate method for hand calculation

of the stress f P 5

in prestressed ten-

dons a t ultimate flexure is presented.

It

is

applicable to composite and

noncomposite sections of an y shape

with any number

of

steel layers, and

any type of ASTM steel at any level of

effective prestress.

Parametric and comparative

studies indicate the proposed method

is

more accurate and more powerful

than other approximate methods.

Numerical examples are provided and

proposed ACI 318-83 Code and

Commentary revisions are given.

yield points of the steel types used in

the cross section, and setting the com-

pressive steel stress equal to zero. Ap-

proximate steel strains are then com-

puted from conditions of equilibrium

and compatibility. The final steel

stresses are obtained by substituting the

strains into the power formula. How-

ever, the main advan tage of this proce-

dure over cu rrent approximate methods

is its applicab ility to all section sha pes,

all effective prestress levels, and any

combination of steel types in a given

cross section.

The proposed approximate procedure

is compared with the precise strain

compatibility method an d two other ap-

proximate procedures: the ACI Code

method, which was developed for the

Code committee by Mattock, 6

and the

method recently proposed by Harajli

PCI JOURNAL/September-October 1988

97


3/28

A

dps

dn s

Aps

A

ns

Ens.

(a) Cross Section

CCU

0 85f c

C Asfs

c

a =arc

C

E

O.BSf cba

E

s, dec

e s

Zero

Strain

Apsfns}T

ps, dec

Ansf s

ns, dec

(b) Strains

(c) Forces

Fig. 1. Flexural strength relationships.

and Naaman. Plots of behavior of these

four methods und er various combina-

tions of concrete strength and rein-

forcement parameters are discussed.

Qua litative com parison with a recently

introduced approximate method by

Loov is also given. Results indicate that

the proposed procedure is more accu -

rate than the other approximate

methods, and it makes better use of the

actual material properties.

Num erical examples are provided to

illustrate the proposed procedure and to

compare it with the other approximate

method s. A proposal for revision of the

ACI Code and Commentary

8

is given in

Appendix B .

PROBLEM STATEMENT

AND BASIC THEORY

Referring to F ig. 1, the problem ma y

be stated as follows. Given are the

cross-sectional dimensions; the pre-

stressed, nonprestressed, and compres-

sion steel areas, A

p s , A

3 ,

and A.;, re-

spectively; the depths to these areas,

d

p s

d

8

and

d ,

respectively; the con-

crete strength f, ' and u ltimate strain

E

and the stress-strain relationsh ip(s) cf

the steel. The nominal flexural strength,

M , is required.

A procedure for obtaining the stress in

prestressed and nonprestressed tendons

at ultimate flexure can be developed as

follows. Referring to Fig. 1(c), force

equ ilibrium (T = C) may be sa tisfied by;

A

9J53

A

./1,., A Bf; = 0.85f, 1) /3, c 1)

where f3

f

and fs are the prestressed,

nonprestressed, and compression steel

stresses at ultimate flexure, respec-

tively; b is the width of the compression

face;

f3,

is a coefficient defining the

depth of the equivalent rectangular

stress block, a, in Section 10.2.7 of AC I

318-83;

and c is the distance from the

extreme compression fiber to the neutral

axis.

If the compression zone is nonrectan-

gular or if it consists of different con-

crete strengths, Eq. (1) may be rewritten

as follows:

A

,J..

+

A

nafns A

;f;

F

c

(la)

where F , is the total compressive force

in the concrete.

The equivalent rectangular stress

distribution has b een shown to be valid

for nonrectangular sections,

9 '

so the

98


4/28

area of concrete in compression may be

determined by a consideration of the

section geometry and setting the stress

in each type of concrete equal to its re-

spective 0.85 f,' value.

Assuming that plane cross sections

before loading remain plane after load-

ing, and that perfect bond exists be-

tween steel and concrete, an equ ation

can be written for the strain in steel, Fig.

1(b):

di

(2 )

i=

cu ^

1

-

t . dec

c

where i represents a steel layer des-

ignation. A steel layer is defined as a

group of bars or tendons with the same

stress-strain properties (type), the sam e

effective prestress, and that can be as-

sumed to have a combined area with a

single centroid.

In Eq. (2),

i,dee

is the strain in steel

layer i at concrete decompression.

The decompression

strain,

E.dec

is a

function of the initial prestress and the

time-depend ent properties of the con-

crete and steel. In lieu of a more accu -

rate calculation, the change in steel

strain due to change in concrete stress

from effective va lue to zero (i.e., due to

concrete decompression) may be ig-

nored. Thus,

i , d e c

may be computed as

follows. If the effective prestress

f

is

known:

}_8e

3

Ei

)

. dec

Ei

or if the effective prestress is unknown:

f

i

25,000

(4 )

Ei dec

E

i

where

E

= modulus of elasticity of steel

layer i , psi

= initial stress in the tendon before

losses, psi

Note that

m

is equal to zero for non-

prestressed tendons. The constant

25,000 psi (172.4 MPa ) approximates the

prestress losses due to creep and shrink-

age plus allowance for elastic reboun d

due to decompression of the cross sec-

tion.

If the value of c from Eq. (1) is sub-

stituted into Eq. (2), then Eq. (2) be-

comes:

0.85

f

b /3, di

i .dec

icu

{ 1)+

psf

p

s

+

A

nal ns

^ sf a

(5 )

With the strain

E

i

given, the stress may

be determined from an assumed stress-

strain relationship, such as the one pre-

sented in the following se ction.

STEEL STRESS

-

STRAIN

RELATIONSHIP

In 19 79 , Mattock' used a power equa-

tion

to closely represent the

stress-strain cu rve of reinforcing steel

(high strength tendons or mild bars).

The general form of the power equa tion

is:

fi

=

iE

L

Q

+

l

+ E

{R i

Rj

f 6

where

EE

(7 )

Kfpv

and

f i

= stress in steel corresponding to a

strain

Ei

= specified tensile strength of pre-

stressing steel

and E,

K ,

Q,

and

R

are constants for any

given stress -strain curve. In lieu of ac-

tual stress-strain curves, values of

E

K ,

Q,

and

R

for the steel type of s teel layer

i may be taken from Table 1, which is

based on minimum ASTM standard

properties.

The values of

E

K ,

Q, and

R in Table

1 were determined by noting that the

yield point (,,,,, f,,,) and the ultimate

strength point

(E

P u

fn )

must satisfy Eq.

(6), where E

P

, , ,

f

,,

nd

f

are the


99


5/28

Table 1. Tendon steel stress-strain constants for Eq. 6).

p u ksi) f p y

/ f

p u E psi)

K

Q R

0.90

28,000,000

1.04

0.0151

8.449

270

strand

0.85

28,000,000

1 04

0.0270

6.598

0.90

28,000,000

1.04

0.0137

6.430

250

strand

0.85

28,000,000

1.04

0.0246

5.305

0.90

29,000,000

1.03 0.0150

6 351

250

wire

0.85

29,000,000

1 03

0.0253

5.256

0.90

29,000,000

1.03

0.0139

5.463

235

wire

0.85

29,000,000

1.03 0.0235 4.612

0.85

29,000,000

1.01

0.0161

4 991

150

bar

0.80

29,000,000

1.01

0.0217 4.224

Note: I ksi = 1000 psi = 6,895 MPa.

Qisbasedonep0=0.05.

minimum AS TM standard values for the

steel type used. A value of e p u

= 0.05 was

used for all prestressing steel types,

rather than the ASTM specified

minimum ultimate strain of 0.035 or

0.04. This is a con servative assum ption

based on experimental results; its adop-

tion results in lower stress values at in-

termediate strains.

Other assumptions were necessary to

solve for the constan ts

E, K,

Q,

and R.

These assumptions were made on the

basis of experience gained from the

shape of experimental stress-strain

curves reported in Refs. 1 and 12, and in

a separate section of this paper.

STRAIN COMPATIBILITY

APPROACH AND

COMPUTER PROGRAM

The strain com patibility method usu-

ally requires an iterative numerical so-

lution because of the interrelation of the

unknown parameters. A step-by-step

application

3 . 1 3

of this method is de-

scribed as follows:

Step 1: Assume a com pression block

depth, a, and com pute the n eutral axis

depth, c.

Step 2: Sub stitute c into E q. (2) to ob-

tain the strain for each steel layer in the

section.

Step 3: Estimate the stress in each

steel layer by use of a graphical or

analytical stress-strain relationship.

Step 4: Check satisfaction of the

equilibrium formula, Eq. (1a).

Step 5 : If Eq. (1a) is not sa tisfied, re-

peat Steps 1 through 4 with a new value

of a.

Step 6: When compatibility, Eq. (2),

and equilibrium, E q. ( la), are ach ieved

simultaneously, determine the flexural

strength,

M.

The aforemen tioned steps were used

to develop a user-friendly flexural

strength analysis program. The pro-

gram can analyze noncomposite and


6/28

B3

B3

4

Z

T3

T5

T6

SAMPLE PRECAST SECTION SHAPES

TOPPING SHAPES

Fig. 2. Sample precast section shapes and topping shapes available with the strain

compatibility computer program.

composite mem bers. Users can choose

from twelve common precast section

shapes and combine the selected sec-

tion with either of the two available top-

ping shapes (rectangular or tee) to form a

composite member. Fou r of the precast

section shapes and the two topping

shapes are shown in F ig. 2 as examples.

Ob viously, analysis is equ ally valid for

cast-in-place members constructed in

one or two stages.

Fully prestressed and partially pre-

stressed members with bonded rein-

forcement can be analyzed, and any

num ber of steel types or steel layers can

be specified. Properties for any steel

type can be taken from twelve types of

steel, built into the prog ram, that m eet

ASTM minimum standards. Ten of these

types are given in Table 1, and the oth er

two are Grades 60 (413.7 MP a) and 40

(275.8 MPa) mild bars. Alternatively,

properties for any steel type can

be as-

signed on the basis of adequately docu-

mented manufacturer supplied records.

Steel stresses are computed by E q. (6)

and force equilibrium is achieved by

selecting progressively smaller incre-

ments of a. Any system of un its may be

used. All input data can be edited as

PCI JOUR NAL /September-October 1988

01


7/28

many times as needed. This allows use of

the program for either analysis or design.

The program package is available

from the PC I for a nominal charge. The

package includes a 5.25 in. (133 mm)

diskette, and a manual containing in-

str

t

ictions, section shapes , and examples

with input/output printout.

Actual Versus Assumed Steel

Stress

-Strain Curves

In researching their paper, the authors

solicited stress-strain curves from ten-

don suppliers and m anufacturers. Se-

venty curves were received and their

breakdown is as follows: 19 curves of

Grade 270 ksi (1862 MPa) stress-re-

lieved strand, 23 curves of Grade 270 ksi

low-relaxation strand, 13 curves of

Grade 25 0 ksi (1724 MP a) low-relaxation

strand, and 15 miscellaneous curves

consisting of stress-relieved or low-re-

laxation wire of varying strengths and

0.7 in. (17.8 mm ) diameter ASTM A779

Table 2. Manufacturer legend for

stress-strain curves in Figs. 3, 4 and 5.

CURVE

MANUFACTURER

/SUPPLIER

A ARMCO INC.

BC, BU

FLORIDA WIRE AND CABLE CO.

C

PRESTRESS SUPPLY INC.

D

SHINKO WIRE AMERICA INC.

E

SIDERIUS INC.

F

SPRINGFIELD INDUSTRIES CORP.

G

SUMIDEN WIRE PRODUCTS CORP.

Curve BL represents a lower bound of 10 curves and curve

BU represents an upper bound of the sam e 10 curves.

prestressing strand.

Six curves for Grade 270 ksi stress-re-

l ieved strand, six curves for Grade 270

ksi low-relaxation strand, and two

curves for Grade 25 0 ksi low-relaxation

strand were con sidered representative

of the data received. These curves are

reproduced in F igs. 3, 4, and 5, respec-

tively, and a manufacturer legend is

given in Table 2. Differences in the

290

G

280

270

_ PCI HANDBOOK EQ.

260

,-

C

W

u

250

, ^

n

240

/CEO. 6) FITTED TO ASTM

SPECIFICATIONS WITH K=1.04

230

2200

.01

. 02

.03

.04

. 05

.06

. 07

. 08

. 09

STRAIN in./in.)

Fig. 3. Manufacturer stress-strain curves for ASTM A416, 270 ksi, 7-wire,

stress-relieved strand.

102


8/28

300l

BU

290

B L'`

28

^

E

0)

s_ 270

pCI

HANDBOOK EQ.

w 260

TED TO ASTM

250

IONS

WITH

K=1.04

240

2300

.01

. 02

. 03

. 04

. 05

. 06

. 07

. 08

. 09

STRAIN in./in.)


low-relaxation strand.

280

C

270

E

260

250

r,(6F

cn PC

ANDBOOK EQ.

40

F-

TO

ASTM

230

WITH

K=1.04

22

210

.01

. 02

. 03

. 04

. 05

. 06

. 07

. 08

. 09

STRAIN (in. /in. )


low-relaxation strand.

PCI JOURNAUSeptember-October 1988

103


9/28

shape of the curves beyond the yield

strain,

E

py

=0.01, are attributable to an

absence of data for Curves A, C, D, E,

and G for strains greater than 0.015 and

less than the u ltimate strain,

E vu

,

and for

Curves BL and BU for strains greater

than 0.035 and less than

Epu

The figures also show plots of the PCI

Design Han dbook

1 5

equations and Eq.

(6) set to ASTM minimum specifica-

tions. F or convenience, the PC I Design

Handbook equations are reproduced

here.

If

E

ps

< 0 .008 then f,,, = 28,000

E

ksi)

If

E >

0.008:

For 250 ksi (1724 MP a) strand:

0.058

3 = 248

< 0.9 8 f^ (ksi)

E

ps

0.006

(9 )

For 270 ksi (1862 MP a) strand:

fps

=

268

0.075

0 and E

0.01 is shown in Ta ble 3, Part (a). The

results of similar analyses for the PCI

Design Handbook equations and Eq. (6)

set to ASTM minimum specifications

are show n in Table 3, Parts (b) and (c),

respectively.

Table 3, Part (a) reveals that very

small errors are obtained when E q. (6) is

f itted to a given ma nufacturer's curve.

This is in close agreement with Mat-

tock's' and Naaman's

4

findings. The P CI

Design Handbook equations and the

minimum ASTM Standard values can

underestimate the steel stress by as

mu ch as 10.82 an d 12.31 percent, re-

spectively.

Prestressed concrete producers tend

to buy their tendons from a limited

number of manufacturers. Therefore,

they are in a position to take advan tage

of higher tendon capacities with ade-

quate documentation of the actual

stress-strain curves and use of the

aforementioned computer program.

Proposed Approximate Method

The proposed approximate method is

essentially one cycle of the iterative

strain compatibility approach. In order

to get accurate results at the end of one

cycle, initial param eters mu st be care-

fully selected. It is difficult to assume an

accurate initial value for the neutral axis

depth, c, due to its wide variation.

Rather, th e steel stresses are initially as-

sumed to be at the yield point for the

tensile reinforcement, and at zero for the

compressive reinforcement. These ini-

tial assumptions are based on numerous

trials and param etric studies discussed

in a separate section.

The proposed approximate method

can b e performed by using the following

steps:

tep

1: Set f , = f5

, f

13 = s or and

f8 = 0 in Eq. (1a) and com pute the total

compressive force in the concrete,

F.

=Fc

(la)

Step 2: Set the quantity

F, equal to

0.85f A,, where A, is the area in com-

pression for a type of concrete, and solve

for the compression block d epth, a. For

composite sections, there are as man y

0.85f A, terms as the nu mber of types

of concrete in compression.

Step 3: Compute the depth of the

neutral axis c

= al,.

For composite

104


10/28

Table 3. Maximum percent deviation between manufacturer stress-strain

curves and a reference curve.

TYPE OF STRAND

REFERENCE

270

K S I b

270 KSI

250 KSId

CURVE

STRESS-RELIEVED LOW-RELAXATION

LOW-RELAXATION

>0

?0.01

>0

?0.01

>0

>_0.01

(a) EQ.

6)

MANUFACTURER

-0.79e

-0.79

-1.36 -1.36 -1.65 -0.77

CURVE

b) PCI

HANDBOOK

-6.34

-6.34 -10.82 -10.82 -7.63

-3.81

EQUATIONS

(c) EQ.

6)

SET TO ASTM

MINIMUM

-12.21 -12.21 -12.31 -12.12

-11.96

-11.96

STANDARDS

K = 1 . 0 4

Note: 1 ksi = 6.895 MPa.

a

All strand is ASTM A4 16; b 6 curves, see Fig. 3; c

6

curves, see Fig.

4 . d 2

curves, see Fig. 5; e

a

negative value

indicates the stress by the reference curve is less than the actual stress.

sections, assume an average I3, as fol- whichever is applicable. For nonpre-

lows:

stressed steel

f

,

= 0 .

Step 5: Compute the stress in each

/3, ave. _0 85 (f^A

C

3

)

II)

steel layer a i by use of Table 1 and

F

qs. (6) and (7):

where k is the concrete type number.

Step 4: Compute the strain in each

steel layer i by Eq. (2). In general,

mild tension reinforcement, if any,

yields for practical applications. Thus,

Step 4 may be omitted for this type of

steel.

E

c 0

E

{,dec

(2 )

Note for mild reinforcement, it is

where

easier to use the relationship

f; =

E{E

f,,

than to apply Eqs. (6) and (7).

Step 6 : With the steel stresses at ulti-

,dec =Ee

(3 )

mate flexure known, apply the standard

2

equilibrium relationships to get the

or

flexural capacity,

M.

To illustrate the above procedure, two

f

25,000

num erical examples are worked out on

4 )

i,dec

E,,

the next few pages.

ft= ESE

Q +

*, ,] --fr,..

6)

(I

+E

Q

i

)

and

EtE

(7 )

_

Kf.


105


11/28

NUMERICAL EXAMPLES

Two numerical examples are now

shown to illustrate the calculation of the

nominal moment capacity using the

proposed method and to compare the re-

sults with existing ana lytical methods.

In the first examp le (a precast inverted

T-beam with cast-in-place topping), the

proposed mom ent capacity is compared

with the value obtained u sing the strain

compatibility method. In the second

example (a precast inverted T-beam

without topping), the proposed momen t

capacity is compared with the results

obtained using the ACI 318-83 Code

method, the Harajli-Naaman method,

and the strain compatibility method.

EXAMPLE 1

The nom inal moment capacity of the

T-beam shown in F ig. 6 is calculated by

the proposed approximate method an d

the strain compatibility method.

Given: f

(precast) = 5 ksi (34.5 M Pa),

f

(topping) = 4 ksi (27.6 MPa). Rein-

forcemen t is 20 -

/2in. (12.7 mm) d iam-

eter 270 ksi (1862 M Pa ) low-relaxation

prestressed strands, A

s = 3.06 in.

(1974

mm

),andf

f

= 16 2 ksi (1117MP a);4

I

a

in. (12.7 mm) diameter 270 ksi (1862

MPa) low-relaxation nonprestressed

strands,A

n 3

= 0.612 in.

2

(395 mm2).

Solution:

1. Proposed method

Step :

From Eq. (1a):

F,=

3.06(0.9)270+0.612(0.9)270

= 892.30 kips (396 9 kN)

Step 2: Compute depth of stress block a.

0.85(4)(56 )(2.5) + 0.85 (5)(16)(a - 2.5) _

892.30

a = 8.6 2 in. (218.9m m)> 2.5 in.

(63.5 mm) (ok)

Step 3: Compute average /3, from Eq

(11).

R

ave. =

0.85 4) 56) 2.5) 0.85

+

892.30

0.85 (5) (16) (8.62 - 2.5) 0.80

892.30

= 0.83

c = a//3

= 8.62/0.83 = 10.39 in.

(263.9 mm)

Step 4: Compu te strains in prestressed

and nonprestressed steel.

F rom Eqs. (3) and (2):

ep s d e c =

162/28,000 = 0.00578

and

P S = 0.003 (

358

-1

+ 0.00578

10.39

)

= 0.01312

Similarly, from Eqs . (4) and (2):

E ns dec = -

0.00089 and e n s

= 0.006 07

Step 5: Com pute stress in prestressed

steel.

From Table 1:

E = 28,000 ksi (19 3,060 M Pa)

K = 1.04

Q = 0.0151

R = 8.449

From Eqs. (7) and (6):

E

= 0.01312 28,000)

= 1.4536

p 8

1.04 (0.9) 270

f

= 0.01312 (28,000) 10.0151+

1-0.0151

(1 + 1.4536

8 4 4 9 1

8 449

= 253.23 ksi (1746 M Pa)

Similarly, e*

= 0.6725 and

f =

169 .28 ksi (1167 MPa)

Step 6 : Substituting the values of

f

3

and

fns into Eq. (1a) yields:

F,

= 878.48 kips (39 07 kN)

Correspond ing a = 8.42 in. (213.9 mm)

Taking moments abou t mid-thickness of

the flange yields:

M .

=

A

nsfns

dr. - ^

I

A

nsfns

d.

2f

0.8

5f/,p,bn, a -

h.) -.-)

= 2377 kip-ft (3223 kN-m)

106


12/28

Fig. 6 . Precast inverted T-beam with cast-in-place topping for Example 1.

2. Strain compatibility

Analysis by the aforementioned com-

puter program yields:

= 253.41 ksi (1747 MPa)

= 173.23 ksi (1194 M Pa) and

M = 2383 kip-ft (3231 kN-m)

EXAMPLE 2

The nominal moment capacity of the

precast inverted T-beam shown in F ig. 7

is calculated by the proposed meth od,

Therefore, the proposed method gives

answ ers that are very close to those of

the strain compatibility analysis. The

other approximate methods are not ca-

pable of calculating tend on stresses in

sections containing both prestressed

and nonprestressed tendons.

the ACI 318-83 Code method, Harajli

and Naa man's method, and the strain

compatibility method. A discussion of

the features of the other two approxi-

d

ps

=33 24

dns=33.5

6

16., 6

36

LAps

2

Ans

Fig. 7. P recast inverted T-beam for Example 2.

PCI JOURNAUSeptember-October 1988

107


13/28

Table 4. Summary of results for Examples 1 and 2.

METHOD

PARAMETER

EXAMPLE

2

VALUE P E R C E N T '

D I F F E R E N C E

VALUE

P E R C E N T

D I F F E R E N C E

STRAIN

COMPATIBILITY

f

k s i)

253.41 0

247.91

0

fns ksi)

173.23

0

60 0

M n

kip-f t)

2383 0

79 1 0

PROPOSED

METHOD

f

ksi)

253.23 -0.07 248.80 +0.4

f ns ksi)

169.28

-2.3 60 0

Mn kip-f t)

2377 -0.2

793 +0.2

ACI

318-83

f

ksi)

NA*

NA 254.11 +2.5

f5 ksi)

NA

NA

60 0

Mn kip-f t)

NA

NA

805 +1.8

HARAJLI

NAAMAN

f

ksi)

NA NA

256.50 +3.5

fns ksi)

NA NA

60

Mn kip-f t)

NA

NA

810

+2.4

Note: 1 ksi = 6.895 MPa; 1 kip-ft = 1.356 kN-m.

Relative to the strain compatibility analysis.

Not applicable.

mate methods is given in the next sec-

3

Harajli and Naaman s method?

tion.

Given:

5 ksi (34.5 MPa). Rein-

forcement is 6 -

/2

in. (12.7 mm) diameter

270 ksi (186 2 MP a) stress-relieved pre-

stressed strands, A p8

= 0.918 in.

2 (592.2

mm 2 ),

f

e =

150 ksi (1034 M Pa ); 2 - #7

(22.2 mm) Grade 6 0 (414 MP a) bars, Any

= 1.20 in.

2

(774.2 mm2).

Solution:

1

Proposed method

Decompression strain in prestressed

steel:

8 ps,dec =

0.00536 and strain, e

p s

= 0.0220

Stress in prestressed steel:

f

s

=

248.80 ksi (1715 MP a)

Corresponding n ominal f lexural capac-

ity:

M.

= 79 3 kip-ft (1075 kN-m)

2

ACI Code methods

f

3 =

254.11 ksi (1752 MP a) and

M

n

= 805 kip-ft (1092 kN-m)

Compute depth to center of tensile

force, assuming

f. = fp U

d,. = 33.89 in.

(860.8 mm).

Neutral axis depth, c = 5 .65 in. (143.5

mm) and f a

=

256 .50 ksi (1769 MP a).

Depth to cen ter of tensile force:

e

=

33.88 in. (860.5 mm) and

M

= 810 k ip-ft (109 8 kN-m)

4. Strain compatibility

Analysis by aforementioned computer

program yields:

f8

= 247.9 1 ksi (1709 M Pa)

fee = 60 ksi (413.7 M Pa) and

M

= 79 1 kip-ft (1072 kN-m)

A summary of the results of Examples

1 and 2 is given in Table 4. It shows that

all three approximate methods give rea-

sonable accuracy for the section consid-

ered in Example 2; however, the pro-

posed method has a slight edge. A major

advantage of the proposed m ethod is its

wide range of applicability, as dem on-

108


14/28

Table 5. Parameters used in developing Figs. 8 through 16 .

TYPE OF BEAMa

RECTANGULAR

TEE

Figure No.

8 9

10

11

12 13 14

b

15

c

16

C

ksi)

5 7

5

5

5

5

7

d

5/3

Grade of

Ans

ksi)

N/A

60

60

60

270

270

N/A

N/A 60

A

ns

ps

0

2

2 2

0.5

0.5 0

0

0.5

f p y

f u

0.85

0.85

0.85

0.9

0.85

0.85

0.85 0.85

0.9

d

n 5

d

ps

N/A

N/A

N/A

1.04

f e f pu 0.56

0.56

0.56

0.56

0.56

0.56

VARIES

0.56

0.56

f ns, e ks,)

N/A

-25

-25

-25

-25

-25

N/A

N/A

-25

Note: 1 ksi = 6.895 MPa.

a

For all beams: E

p s = E

n s

= 28 000 ksi A 5

= 0, c

c u

= 0.003 cp u

= 0.05 f

p u

= 270 ksi.

b

Typical 8 ft. x 24 in. PCI Double Tee.

c

Section dimensions correspond to beam in E xample 4.2.6 of R ef. 15.

d

precast/topping strength.

strated by Exam ple 1, and furth er dis-

cussed in the following sections.

Parametric Studies

The proposed approximate method

includes assumption of initial values for

the steel stresses. Numerous trials were

mad e, for a wide range of applications,

with initial steel stresses varying from

u to well below

f, .

It was found that

the best accuracy was ach ieved by as-

suming the tensile steel stresses equal

to the respective yield points of the steel

types used, and the compressive steel

stress -equal to zero. The following d is-

cussion of Figs. 8 through 16 further il-

lustrates this finding.

Sample plots of the resu lts of the pro-

posed method, the strain compatibility

method , Eq. (18-3) of ACI 318-83,

5

and

Eqs. (21), (22), and (24) of Harajli and

Naaman' are shown in F igs. 8 through

16. A summary of the concrete and

reinforcement parameters used in de-

veloping Figs. 8 through 16 is given in

Table 5. Loov

1 6

has recen tly proposed

an approximate method . Unfortuna tely,

the final draft of Loov's paper was not

available in time to include his meth od

in Figs. 8 through 16 . For readers' con-

venience, the methods of Refs. 5, 7, and

16 are summ arized in the following sec-

tion. In addition, their main features are

compared with those of the proposed

approximate method.

For the parameters considered in

Figs. 8 through 11, all three approximate

metho ds are applicable. The proposed

method plots within abou t 1.5 percent of

the strain compatibility curve, and it

performs better than E q. (18-3) of ACI

318-83 and Harajli and Naaman's

method. In Figs. 9 through 11, f

was

taken equal to

f

in the proposed method

becau se the m ild reinforcement yields

before the prestressed reinforcement

reaches

fPS.

F igs. 12 and 13 show the relationship

between steel stress at ultimate flexure

PCI

JOURNAL/September-October

1988

09


15/28

fp

u 270 ksi, A ns

=

0,f = 5 ksi, fpy

/f

pu

= 0.85

1

-,

ss

fps

\\

fpu

\ \


.85

------------

PROPOSED

ACI 318 83

- HARAJLI NAAMAN

.8

0

.8 5

.1

.1 5

.2

.25

.3

(Apsfpu+Ansfy Asfy)/fcbdps

Fig. 8. Stress in prestressed tendon at ultimate flexure vs. total steel index.

f pu

=270 ksi, A

ns

/A

ps

= 2,f

c

=5 ksi, f

y= 60ksi, fpy/fpu=0.85

. ss

f

P

Pu


85

------------PROPOSED

- - - ACI 318-83

- HARAJLI NAAMAN

8'

0

.0 5

.1

.1 5

.2

.25

.3

(A

psf A

ns

fy Asfy)

/f^bdps


110


16/28

fpu = 270 ksi, A

ns /A

ps =

2,fc=7 ksi, f

y

= 60ksi, fpy/fpu=0.85

1

95

fps

f

's

pu

85

V

0

.05

.1

.15

.2

.rb

A

ps

f

pu

Ans fy

A

S

f )/ fC bdpS


f

ps 270ksi, A

ns

/Aps

=2,f' =5ksi,f

y =60ksi,fpy /fps 0.9

\

.9 5

ps

pu

f

TRAIN COMPATIBILITY

85

------------ PROPOSED

ACI 318 83

HARAJLI NAAMAN

0

.05

.1

.1 5

.2

.2 5

.3

(A

ps

f

pu

+A

ns fy

As

f

y

)/f

cfps



1 1 1


17/28

f

p

270ksi, Ans

/A

ps

=0.5, f'=5ksi,f

py

/fp

s 0.85

1

--,

.ss

fps

f

's

pu

--

TRAIN COMPATIBILITY

.85

------------ PROPOSED

.8

0

.05

1

.1 5

2

(Apsfpu+Ansfpu A^sfy)/fcbdps

.25

3


fpu

=270ksi,A

ns

/A

ps =

0.5, f, = 5ksi,f

py

/fp

s 0.85

.7

'.

fns

f

5

^'^

pu

4

3


PROPOSED

2

0

.05

.1

.1 5

.2

.25

.3

(Aps

fp u

+A

ns

f

pu As fy)/ff bdns

Fig. 13. Stress in nonprestressed tendon at ultimate flexure vs. total steel index.

112


18/28

f

pu

= 270 ksi, f' = 5ksi, f

py

/f

pu

=0.85, A

ns

= 0, (A

ps

f pu /ff bd

ps ) = 0.15

9 6

.94

92

f p s .88

-

fpu

86 STRAIN COMPATIBILITY

LOWER LIMIT OF

------------ PROPO SED

ACI 318-83

ACI 318-83

84

HARAJLI NAAMAN

.82

80

1

.2

.3

.4

.5

.6

.7

fse/fpu

Fig. 14. Stress in prestressed tendon at ultimate flexure vs. effective prestress.

f

pu

=270 ksi,A

ns

=0,f =7ksi,f

py

/ f

pu

=0.85 b/ b

W

top=8.35,

1

h/ h=0 083

f 5.75

5.75

.

95

\

^ I^

3 75

3.75

f

f pu

.s


.85

------------ PROPOSED

ACI 318-83

HARAJLI NAAMAN

B

0

.025

.0 5

.075

1

.125

.1 5

A

ps

f pu /

^bdps

Fig. 15. Stress in prestressed tendon at ultimate flexure vs. prestressed steel index for a

typical 8 ft x 24 in. PCI double T-section.

PCI JOURNAL

/September-October 198 8

113


19/28

f p u

= 270 k si, A

n s

/ A PS = 0. 5, f

top = 3 k si, f

PC = 5 ksi, f

y

= 60 ksi

fI f=0.9 ds

/ d p s

=1.04,b/ b

tp

3.66,h

/ h=0.077,d/ h=0.154

p

dps / h = 0 . 885

95

b

ht

:............

^`^

d

ns

dps

b^

top

fpu

.s


.85 ------------ PROPOSED

ACI 318 83

.8

0

.0 5

.1

.1 5

.2

.25

.3

(A

p s f

p u +A n s f

y

A'

s

f y

) / fG top bdps

Fig. 16. S tress in prestressed ten don a t ultim ate flexure vs. total steel index for a

com posite T-section.

and tota l re in forcem ent inde x when pre-

s tressed tendons a re supplem ented wi th

nonprestressed tendons. In this case

neither Eq. (18-3) of ACI 318-83 nor

Haraj li an d Naam an 's m ethod i s app l i -

cable . In F ig . 12, the p roposed curve h as

a m ax im um dev ia t ion o f abou t 1 . 5 pe r -

cent . In F ig . 13 , the proposed cu rve de -

v i a tes by no m ore than abou t 2 percent

in the lower two- th i rds o f the re in force-

m ent range , which is where m ost prac ti-

cal designs would fall. It yields very

conservat i ve s tr ess va lues in the up per

third.

Fig. 14 shows the relationship be-

tween prestressed steel stress at ulti-

m ate f lexure and e f f ec tive pres t ress

when the reinforcement index is held

constant. The steel stress by the pro-

posed method is in close agreement

with the stra in com pat ib il ity me thod fo r

all values of effective prestress. The

other approximate methods for deter-

mining ff

are l im ited to cases where the

effective prestress is not less than

0.5

fem.

Figs . 15 and 16 sho w the re la t ionsh ip

between prestressed stee l s tress at u l ti -

mate flexure and total reinforcement

index fo r T -sect ions. In both f i gures the

proposed method offers better results

than the o ther approx imate m ethods . It

should be noted from Fig. 15 that the

AC I Code m e thod becomes i nc r easing l y

uncon serva tive as the dep th o f the com -

pression block, a, exceeds the flange

thickness,

hr.

Harajli and Naaman's

m etho d correct ly ad justs fo r th is T-sec-

t ion e f fect .

In Fig. 16, Harajli and Naaman's

m ethod was om itted because the ir equa-

t ions d o not ex pl ic i t ly show h ow to ca l -

cula te

f

when the depth of the com-

pression block a includes more than

114


20/28

one concrete strength. An example in

the i r paper , h owev er , ind ica tes how to

app ly the a ssum pt ions o f the ir m ethod

to com pos it e m em bers . I f the i r m ethod

were inc luded in F ig . 16 , it wou ld ind i -

cate trends similar to those shown in

Fig . 15 .

At this point, an im portan t observation

concern ing the p roposed m ethod can be

m ade . A lthough the p roposed m ethod is

s ligh t l y uncon serva t ive , in som e ca ses ,

with respect to the s t ra in com pat ib il ity

m e thod in F i gs . 8 -16 , it m us t b e no t ed

that these f igures are based on stee l with

m in im um AS TM prope r t ie s . In r ea l it y,

steel properties are signi f icantly greater

than m in imu m AS TM proper ties , as d is -

cussed ear l ier .

Comparison of Approximate

Methods

A d escr ip t ion o f four app rox im ate pro-

cedu res fo r ca lcu lat ion o f

f,,

at ult im ate

f lexu re i s g i ven in Tab le 6 . Discuss ion

of the features of these m ethod s is g iven

in Table 7 . It is shown that the m ain ad-

vantage o f the proposed p rocedu re i s its

flexibility. It is applicable to current

m ater ia l an d construct ion technology, as

wel l as poss ib le future d eve lopm ents .

The ACI Code method is reasonably

accu rate and s imp le t o use if the com -

pression block is of con stant width . Use

of steel indexes can be confusing for

non rec tangu lar sec tion shapes . An im -

provement of the current form was

sug gested by Mattock, in h is discussion

of Ref . 7, as fol lows:

fp3 = f

11

0.85 y p I

(12)

9

whe re c, , is the n eutral ax is depth ca lcu-

lated assuming f

=.

This modified form would combine

the benefits of both the ACI Code and

Hara jli and Naam an ' s me thod . The au -

thors agree with Mattock's statement

that the use o f

d

rather than d .

or

d

a s

sugge sted in Ref . 7, is more th eoret ical ly

correct . Fu rther , Eq . (12) takes into ac-

count the effect of

f/f

a n d t h u s

br ings ou t the advan tage o f us ing l ow-

relaxation steel.

Loov's method appears to have a

mathematical form that would give a

better fit than the predominantly

straight-line relationships of the ACI

C ode m ethod (see F igs . 8 -11 an d 14 -16 ),

and Hara jl i and Naam an ' s me thod ( s e e

Fig . 8-11, an d 1 4 ) . I t is l im ited in scope ,

how ever , to the sam e appl icat ions as the

other two m ethods.

ON LUSION

The flexural strength theory of

bonded prestressed and partially pre-

s tr essed concre t e m em bers is r ev iewed

and an alys is by the stra in com pat ib il ity

method is described. A computer pro-

gram for f lexu ral ana lys is by the s t ra in

compatibility method is provided in

BASIC for IBM PC/XT and AT mi-

crocompu ters and com pat ib les. Program

users can take advan tage o f h igher t en-

don c p cities with dequ te

documentation of actual stress-strain

curves . The p rog ram and i t s m anua l a re

available from the PCI for a nominal

charge .

A new approximate method for cal-

culating the stress in prestressed and

nonprestressed tendons a t u l t im ate f lex-

ure is a lso presen ted . I t is ap pl icable to

sect ions o f an y sha pe, com posite or non-

composite, with any number of steel

l aye rs , and w i th a ny type o f AS TM ten -

dons stressed to any level. Parametric

and comparative studies indicate that

t h e p r oposed m e thod is m or e accura t e

and more powerful than Eq. (18-3) of

ACI 318-83 and other available ap-

prox im ate m ethods.

Th e proposed m ethod is il lustra ted by

two num er ica l exam p les and resu l ts a re

compared with those of the iterative

strain compatibility method and with

o the r app rox im at e m e thods . P r oposed

AC I 318 -83 Code and Com m en tary r e vi-

s ions are g iven in Appen d ix B.


115


21/28

rn

Table 6. Summary of approximate methods for determining

fps.

(1) PROPOSED

Steps:

(1 )

Assume tensile steel stresses =

respective yield points and

compressive steel stress = 0,

and use force equilibrium to

compute F0.

(2)

Set F

= 10.85 1'

A

for all

C

c

c

concrete types in compression,

and compute a.

(3)

c=a/ (11.

For composite sections assume

YO.

8

5

(

f c

Ac

(3 )

k

1k

i ave. =

F

c

(4)

Compute steel strains in each

layer T.

d lI

E =E

i cu c

/

i, dec

where

E,dec=fse/ E

or E

i,

=(f t

-25, 0001/ Ei

lP

)

whichever is applicable.

(5)

Use power formula to compute

steel stresses.

I

=E,E O+

1-O

R1/Rl

hf / p1

otherwise treat as a rectangular section.

2)

c>d/

(1

-Ey

/ cc)

Otherwise ignore compression steel

in the c

formula.

St

where

E y

=

yed stran of compression stee

E=

ultimate concrete stran.

c

* To obtain c

e

, change f pu

to f

ps

and d

u

to de


22/28

Table 7. Comparison of the features of the approximate methods for determining

fFS.

METHOD

FEATURE

(1) PROPOSED

(2)

ACI

318-83

(3) HARAJLI

NAAMAN

(4) LOOVt

Slightly

lengthier

than

SIMPLICITY

Method (2) for the same

Simplest

where

appl icable

Same as (1)

Same as (1)

applications

Slightly

less

Expected to

be

slightly

ACCURACY

Very accurate

Reasonable wher

accurate than

more accurate than

applicable

method (2)

method (2)

Developed for rectangular

Rectangular and T

CROSS SECTION Any shape sections.

May be inaccurate sections.

Must be mod if ied

Same as (3)

SHAPE

for other shapes.

for other shapes.

COMPOSIT

Yes

N o

No.

Must be mod if ied for

Same as 3)

SECTIONS

more than one concrete type.

STRESSED

Any type

Mild bars only

Same as (2)

Same as (2)

TEEL

STEEL

NUMBER OF

Ali ASTM steels.

Power fomula constants

Steels with f

py /

f p u

No

distinction

between

Valid for all

TENDON STEEL

TYPES

can be easily determined

y

= 0.80, 0.85, 0.90

steel

types

p

f

py

f pu values

for

future types.

NUMBER OF

No

limit

Maximum = 3

Same as (2)

Same as (2)

STEEL LAYERS

Not

part

of

original

proposal,

COMPRESSION Automaticall

y

Conditions

for

ieldin

y g

t

but conditions were developed

Condition placed on (c / d )

STEEL YIELDING checked

are given

later to match Method (2)

to guarantee yielding.

CONDITION ON

EFFECTIVE

No cond itions

f se > 0.5 f

pu

f

Se

> _ 0 5 f

pu f Se

> _ 0 6 0 f p y

PRESTRESS

Relative to the strain compatibility method with conditions of Section 10.2 of ACI 318-83, and minimum ASTM standard steel properties.


23/28

R F R N S

1.

Mattock, A. H., Flexural Strength of

Prestressed Concrete Sections by Pro-

grammable Calculator, PCI JOURNAL,

V. 24, No. 1, January-February 1979, pp.

32-54.

2. Menegotto, M., and Pinto, P. E.,

Method of Analysis for Cyclically

Loaded R. C. Plane Frames, Including

Changes in Geometry and Non-Elastic

Behavior of Elements Under Combined

Normal Force and Bending, Interna-

tional Association for Bridge and Struc-

tural Engineering, Preliminary Report

for Symposium on Resistance and Ulti-

mate Deformability of Structures Acted

on by Well-Defined Repeated Loads,

Lisbon, Portuga l, 1973, pp. 15-22.

3.

Naaman, A. E., Ultimate Analysis of

Prestressed and Partially Prestressed

Sections by Strain Compatibility, PCI

JOURNAL, V. 22, No. 1, January-Feb-

ruary 1977, pp. 32-51.

4.

Naaman, A. E., An Approximate Non-

linear Design Procedure for Partially

Prestressed Beams,

Computers and

Structures, V.

17, No. 2, 1983, pp. 287-

293.

5.

ACI Committee 318, `Building Code

Requirements for Reinforced Concrete

(ACI 318-83), American Concrete In-

stitute, Detroit, Michigan, 1983.

6.

Mattock, A. H., Modification of ACI

Code Equation for Stress in Bonded Pre-

stressed Reinforcement at Flexural Ul-

timate,

ACI Journal,

V. 81, No. 4, July-

Au gust 1984, pp. 331-339.

7.

Harajli, M. H., and Naaman, A. E.,

Evaluation of the Ultimate Steel Stress

in Partially Prestressed Flexural Mem-

bers, PCI JOURNAL, V. 30, No. 5,

September-October 1985, pp. 54-81. See

also discussion by A. H. Mattock and

Authors, V. 31, No. 4, July-August 1986,

pp. 126-129.

8.

ACI Committee 318, Commentary on

Building Code Requirements for Rein-

forced Concrete (ACI 318-83), (ACI

318R-83), American Concrete Institute,

Detroit , Michigan, 1983 , 155 p p. See a lso

the 1986 Supplement.

9.

Ma ttock, A. H., and Kriz, L. B., Ultimate

Strength of Structural Concrete Members

with Nonrectangular Compression

Zones,

ACI Journal,

Proceedings V. 57,

No. 7, Janu ary 1961, pp. 737-766.

10.

Mattock, A. H., Kriz, L. B., and Hognes-

tad, E., Rectangular Concrete Stress

Distribution in Ultimate Strength De-

sign,

ACI Journal,

Proceedings V. 57,

No. 8, Februa ry 1961, pp. 875-928.

11.

Tadros, M. K., Expedient Service Load

Analysis of Cracked Prestressed Con-

crete Sections, PCI JOURNAL, V. 27,

No. 6, November-December 1982, pp.

86-111. See also discussion by Bach-

mann, Bennett, Branson, Brondum-Niel-

sen, Bruggeling, Moustafa, Nilson,

Prasada Rao and Na ta ra jan , Ramaswamy,

Shaikh, and Author, V. 28, No. 6, No-

vember-December 1983, pp. 137-158.

12.

Naaman, A. E., Partially Prestressed

Concrete: Review and Recommenda-

tions, PCI JOURNAL, V. 30, No. 6, No-

vember-December 1985, pp. 30-71.

13.

Notes on ACI 318 83 Building Code Re

quirements for Reinforced Concrete

with Design Applications,

Fourth Edi-

tion, Portland Cement Association,

Skokie, Illinois, 1984, pp. 25-31 to 25-34.

14.

Skogman, B. C., Flexural Analysis of

Prestressed Concrete Members, M. S.

Thesis, Department of Civil Engineer-

ing, University of Nebraska, Omaha,

Nebraska, 1988.

15.

PCI Design Handbook,

Third Edition,

Prestressed Concrete Institute, Chicago,

Illinois, 1985, p. 11-1 8.

16.

Loov, R. E., A General Equation for the

Steel Stress,

ft,,,

for Bonded Members,

to be published in the November-

December 1988 PCI JOURNA L.

17.

Proposal to ACI-ASCE Committee 423,

Prestressed Concrete, on changes in the

Code provisions for prestressed and par-

tially prestressed concrete. Submitted by

A. E. Naa ma n, on Ma rch 8, 1987.

118


24/28

APPENDIX A - NOTATION

The symbo ls lis ted be low supp l emen t

and supercede those given in Chapter

18

of AC 1318-83.

a

= depth of equivalent rectan-

gu lar stress b lock as de f ined

in Section 10.2.7 of ACI

318-83

A,

= area in com pression for a type

o f concrete . The re i s on ly one

concre te type in noncom pos-

ite constru ction.

A

n 3

,A^,

= areas of nonprestressed and

prestressed tension rein-

f o r c ement

b

= width of compression face of

m e m b e r

c

= distance from extreme com-

pression f iber to neutral axis

C

= total compressive force in

cross sect ion o f m em ber

d

= distance from extreme com-

pression fiber to centroid of

steel layer i

d n

d

p i =

is tances from ex t rem e com -

press ion f iber to centro ids o f

nonprestressed and prestressed

tension rein forcemen t

d

o p

=

overall depth of concrete

topping

d =

d i st a nc e fr om e x t rem e c om -


com press ion s tee l

E

modulus of elasticity; sub-

scr ipt i re fers to re in force -

m ent layer number .

E p =

moduli of elasticity of non-

pres t r essed a nd p res t r essed

re in fo rcement

f^ specified

compressive

s t rength o f concre t e ; s econd

subscr ip ts pc

and

top

refer to precast (first stage)

and topping (second stage)

con cretes, respect ive ly.

F,

= total compressive force in

concrete at u lt im ate f lexure

=

t r ess in t en don s tee l cor re -

sponding to a stra in

,

Sign convention: Tensile stress in

steel and com pressive stress in conc rete

are posit ive .

fnB,fP

=

stress in nonprestressed an d

pres t ressed re in fo rcemen t a t

ult imate f lexure

fn8,e,fse

=

stress in nonprestressed an d

prestressed reinforcement

after allowance for time-de-

pend ent e f f ec ts

E

=

nitial tendon stress before

losses

= specified tensile strength of

prestressing tendons

fp s

=

pecified yield strength of

prestressing tendons

f8

stress in compressive rein-

f orcem ent a t u l tim ate f lexure

f

v =

pecified yield strength of

nonprestressed mild rein-

f o r c emen t

h

overa l l th ickness o f mem ber

h f

=

hickness o f f lang e o f f lan ged

sections

i

= a subscript identifying the

steel layer number. A steel

layer i i s de f ined as a g roup

of bars or tendons with the

same stress-stra in propert ies

(type), the same effective

prestress, an d that can be as-

sumed to have a combined

area w ith a single centro id .

K, Q, R

= constants u sed in Eq. (6 )

T

= total tensile force in cross

section

3

1

= a/ c f a c t o r de f ined in S ec t ion

10.2.7 o f ACI

318 83

= [0.85 0.05 (f,

4 ksi )]

0.85

and , 0 .65

e c u

=

maximum usable compres-

sive strain at extreme con-

crete fiber, normally taken

equa l to

0.003

t =

train in steel layer i at u l-

t imate f lexure

e e e

=

train in steel layer i at

concrete decompress ion


119


25/28

ns dec

ps.dec

=

stra in in non prestressed

and pres t ressed tens ion

reinforcement at con-

cre te decom press ion

es

strain in prestressed tendon

reinforcement at ultimate

f lexure

Epu

= st ra in in h igh st rength tend on

at stress fr,.

e p y

= yield strain of prestressing

t endon

e s

= s tra in in com press ion s tee l a t

ult imate f lexure

Es

,dec

= s tra in in com press ion s tee l a t

concre te decompress ion

ACKNOWLEDGMENT

The au thors w ish to thank Rona ld G.

Dull chairperson of the PCI Commit-

tee on Prestressing Steel, the Union

Wire Rope Division of Armco Inc.,

Florida Wire and Cable Co., Prestress

S upp ly Inc . , S h inko Wi re Am er ica Inc . ,

Siderius Inc., Springfield Industries

Corp., and Sumiden Wire Products

Corp. for supplying the stress-strain

curves used in the preparation of this

paper . Th e authors a lso wish to express

their appreciation to the reviewers of

this article for their many helpful

suggestions.

COMPUTER PROGRAM

A package ( com pr is ing a p r in tout o f the com pute r p rog ram , user 's m anu a l ,

and d i ske tt e su it ab le f o r IBM P C /XT a nd AT m ic ro com pu te rs ) is a va i lab l e

f r om P C I Headqu ar te rs f o r $20 .00 .

120


26/28

APPENDIX B - PROPOSED ACI 318-83 CODE

AND COMMENTARY REVISIONS

I f the prop osed rev is ions are incorpo-

ra ted in t o the C ode

s

an d C o m m e n ta ry ,'

the r e f e rence , equat ion , and tab le nu m -

bers given herein will need to be

changed .

Proposed Code Revisions

I t is proposed tha t the fo l lowing n ota-

t ion be ch anged in S ec t ion 1 8 .0 o f the

Code: Replace A

3

with A

3

,

with d8

and

d

p

with

d

o m

Delete

yp.

It is proposed that Sections 18.7.1,

18.7 .2, and 18 .7 .3 o f the C ode be rev ised

to read as fo l lows:

18.7.1 Design moment strength of

f lexura l m em bers sha l l be comp uted by

the strength design methods of this

Code. The stress in steel at ultimate

f lexure is

f

fo r prestressed tendons an d

fs

for nonprestressed tendons.

18 .7 .2 In l ieu o f a m ore a c cu ra te de -

term inat ion o f

f

3

and f

8

based on stra in

compatibility, the following approxi-

m ate va lues of

f8 3

and

f,,

3

shal l be used.

(a )

For members with bonded pre-

stressing tend ons, f ,, , an d

fb

m ay

be closely approximated by the

m ethod g i v en i n th e C om m en t a ry

to this Cod e.

(b )

The fo rmu las in S ect ions 18 .7 .2 (c )

and 18.7 .2 (d ) shal l be used on ly if

f

is not less than 0.5f8,, .

(c )

Use Section 18.7.2 (b) of ACI

318-83.

(d )

Use Section 18.7.2 (c) of ACI

318-83.

18.7.3 Nonprestressed mild rein-

forcemen t con forming to S ect ion 3 .5 .3 , i f

used with prestress ing tend ons, m ay be

con s idere d to con t r ibute to the t en s il e

force and may be included in moment

streng th com putat ions at a st ress equa l

to the speci f ied yie ld streng th f3 .

Proposed Commentary Revisions

tion be added to Appendix C of the

Commen ta ry :

A, = area in compression for a type

of concrete . Th ere is on ly one

conc r e te t ype in noncom pos -

i te construction.

d =

is ta nc e f r om e x tr em e c om -


steel layer i

_

od ulu s o f e last ic ity o f re in-

f o rcem ent (Chapte r 18 )

F,

total compressive force in

concrete at u l tim ate f l exure

fi _

stress in stee l layer i corre-

spond ing to a strain

Et

f

n

nitial tendon stress before

losses

a subscript identifying the

steel layer number. A steel

layer i is de f ined as a group

of bars or tendons with the

sam e stress-strain propert ies

(type), the same effective

prestress, and that can be as-

sumed to have a combined

area w ith a single cen tro id .

KQR=

constants defined in Table

B-1* for the AS TM propert ies

of the steel of layer i

Ei

strain in stee l layer i at u l -

t im ate f l exure

E j d e c

= strain in steel layer i at

concre te decom pression

E Y

= yield strain of mild rein-

f o r c ement

I t is proposed tha t the f i rst parag raph

o f Sect ion 1 8 .7 .1 and the f irs t four para-

graphs of Section 18.7.2 of the Com-

m entary be rev ised to read as fo l lows:

18.7.1 Design moment strength of

prestressed flexural members may be

computed using the same strength

equations as those for conventionally

reinforced concrete members. Equa-

tions given in Sections 18.7.1.A and

It is proposed that the following nota-

* Same as Table 1 of this paper.


1 21


27/28

18.7.1.B of the Commentary are valid

except when nonprestressed tendon

reinforcement is used in place of mild

t ens ion re in fo rcem ent . In th i s case the

stress in the n onprestressed tend on re-

in forcement, f.^s

shou ld be used instead

offs.

18 .7 .2 A m ic r o com pute r p rog ram f o r

determining flexural strength by the

s t ra in com pat ib il ity m ethod , us ing th e

assumptions given in Section 10.2 is

ava i lab le f rom Refs . A an d B.* In l ieu o f

the i tera t ive com pu ter ana lys is , the fo l -

l ow ing ap p rox im at e p r ocedu r e m ay be

used for determining the stress

f

n

an y stee l layer i . A layer i i s de f ined

as a g roup o f ba rs o r tendon s w i th th e

sam e stress-strain propert ies ( type) , the

sam e ef fect ive prestress, and that can be

assum ed to have a com bined area wi th a

single centroid. The procedure given

be low i s va l id regard l ess o f the sec t i on

shape , nu m ber o f conc re te types in the

section, number of steel layers, and

leve l o f e f fect ive prestress, f3 .

A. General Case Noncomposite

or Composite Cross Sections of

General Shape with any Number of

Steel Layers

Step 1 : In i tia lly assum e the tensi le s tee l

stresses equal to the respective yield

points of the steel types used and the

com press ive s t ee l s t ress equ a l t o ze ro ,

and use force equilibrium (T = C) to

com pu te th e t o ta l c om press iv e f o r c e i n

concrete,

Step 2 : Us ing the p rov is ions o f Se c t ion

10.2 .7 , com pu te the d epth o f the s tress

block, a. For composite sections, the

f o rce

Fe

m a y h a v e m o r e t h a n o n e co m -

pon ent , 0 .85 f , A

e

, whe re

f,

a n d A

e

a re

the s t reng th an d a rea in com pression o f

each concrete part in the sect ion.

Step 3 : Com pute the neu t ra l ax is dep th

Refs. A and B correspond to this paper and Ref. 14,

respe tively

t Same as Table 1 of this paper

c = al f3,.

For composite sections as-

sume an average /3

as fol lows:

10 .85 ( f ^Ac/31 )k

a 1

ave.=

F

(B-1)

c

where k is the concrete type nu m ber .

Step 4 : Com pute the s t ra in in each s tee l

l ayer i by:

E

i

= 0.003

) +

i dec

B-2

\

c

where l dec may be approximated as

f

/E;. I f a layer con sists o f part ia l ly ten-

s ioned tendons,

E

{ d e

,

m ay be taken =

fr,

25,000

psi /E

where f initial pre-

s tress , ps i. For no nprestressed tend ons

or mild bars,

E i dec

may be taken

25,000

psi/E1.

Step 5 : Com pute the stress in each ten-

don s tee l layer i by :

.fi

= EtE

I Q

+

1Q 1 R

J

JL-88-September-October Flexural Strength of Prestressed Concrete Members

Documents