Design Chartsfor Proportioning

Rectangular PrestressedConcrete Columns

John R. SalmonsProfessor of Civil EngineeringUniversity of Missouri-ColumbiaColumbia, Missouri

Donald G. McLaughlinStructural Engineer

Burns and McDonnel Engineering Co.Columbia, Missouri

(Formerly Graduate StudentUniversity of Missouri-Columbia)

W ith the increased use of precastprestressed concrete conies the

challenge to the engineer to more effi-ciently design such products, If theproduct is a prestressed column the de-sign engineer has few options. Somereferences include load versus momentinteraction curves for these columns,but most curves are for a specific sizecolumn and a specific prestressing steelpercentage. If the designer cannot de-sign within these limitations, the alter-natives are to calculate and draw spe-cific interaction diagrams for the partic-ular column under consideration or usethe computer to evaluate the columncapacity. In either case, the process canbe time consuming and costly.

The design aids presented here de-

scribe many different steel percentagesand prestressing levels in a few simplecurves. These curves are applicable tosymmetrically prestressed short col-umns (no length effects arc included)for both preliminary and final design.

Nondimensiona] load versus momentinteraction curves and associated equa-tions for rectangular columns with 270ksi (1861.7 N/mmx ) prestressing steel inall four faces have been developed.These curves are based on particularconcrete strengths, effective stress inprestressing steel after losses, and thedistance between centers of steel in thetension and compressive faces. Withineach chart there are curves which rep-resent different percentages of pre-stressing steel that can be present in

120

the column. With an additional equa-tion these charts can be used to obtain amodified effective stress in prestressingsteel after losses. The generalized in-teraction curves are presented in theAppendix as well as design examplesillustrating how the interaction curvesand the stress modification equationsshould be used.

Governing Equations

The derivation of equations gener-ated for a general prestressed concretecolumn parallels the development ofthose for load-moment interaction rela-tionships for any specific column. Forthe general case, the equations dependupon the assuming of a neutral axis lo-cation and satisfying the compatibilityof strains. The concrete compressionforce and steel forces are calculatedfrom the strain distribution. Theseforces are nondimensionalized and,when used in the appropriate equationsof equilibrium, produce a single pointon the generalized load-moment curve.`The entire interaction curve is obtainedby varying the neutral axis locationwithin an appropriate range.

The concrete compression force iscalculated from a concrete stress-strainrelationship which accounts for a por-tion of the nonlinear region of thecurve. This solution is more accurateover the entire range of the interactioncurve than one based on the rectangularstress block approximation. The generalequations are designed so that any con-crete stress-strain curve can easily besubstituted for the one used for the so-hition presented.

The steel forces are calculated interms of a steel percentage rather thanfor a particular number of strands. Foranalysis, the prestressing strands havebeen replaced by a thin rectangulartube of prestressing steel with anequivalent area of steel. The total strainin the tube consists of the steel strain

SynopsisA series of nondimensional ulti-

mate load- moment interactioncurves for short, rectangular con-crete columns with axial prestress-ing are presented. Several designexamples are included to show theapplication of the design charts.

General nondimensional equa-tions are given which are based onnonlinear approximations for boththe concrete and prestressing steelstress-strain curves, linear strainsacross the cross section, straincompatibility and equilibrium. Thesteel location is generalized by re-placement of individual strands witha rectangular prestressed steeltube.

The series of curves presentedare for a steel strength of 270 ksi(1861.7 Nlmm2), concrete strengthsof 5, 6, 7, and 8 ksi (34.48, 41.37,48.26, and 55.16 N/mm 2 ), andgeometric parameters correspond-ing to the more commonly used col-umns. In addition, full and partialstressing of the prestressing steelwere considered.

due to prestressing, precompression ofthe concrete and bending of the mem-ber. The force in the steel tube is cal-culated from these strains using an ac-tual stress-strain relationship for pre-stressing strand.

The stress stage in the steel isanalyzed using three specific cases:Case 1 is appropriate when all the steelstrains are in the linear range of thestress-strain curve, Case 3 is used whenall steel strains are in the nonlinearrange, Case 2 is a combination of Cases1 and 3 and deals with steel strains inboth the linear and nonlinear portionsof the stress-strain curve. Other pre-

PCI JOURNAL/January-February 1982 121

N

a)mU0U

G,= 1.7f^/E, EC_I

Concrete Strain

E l ( f zForO c E^ . E p : f= 0.85f^ ^2 `

1 – I `^

`Ep 1Ep

For €< < E, -- Ecu : tc = 0.85f'

Fig. 1. Assumed approximation for the concrete stress-strain relationship.

stressing steel stress-strain relation-ships could he used in the analysis pre-sented here; however, the translonna-tion would be more involved than thatnecessary for the replacement of theconcrete stress-strain relationship.

NONDIMENSIONAL PARAMETERSThe governing equations are used to

generate nondimensional load versusmoment interaction curves for pre-stressed concrete columns. The evalu-ation of the equations is accomplishedby incrementing the depth of the com-pressed concrete area (k xt) which ismeasured from the extreme fiber on thecompression face. The solution of eachincrement of k„t produces nondimen-sional loads (K) and correspondingnondimensional moments (R) whichrepresent a point on an interactioncurve. Each of the two nondimensionalterms can be considered in two partswhich correspond to the contribution ofthe concrete and steel respectively.

These nondimensional terms K and Rare defined as:

P

K fcb = K, + Kt8

R = f bts

R` + R,

whereP„ = ultimate axial load acting on

columnM„ = ultimate moment acting on

column= concrete strength

b = width of cross sectiont = depth of cross sectionK, = portion of K due to compres-

sive force in concreteRc = portion of R die to compres-

sive force in concreteK8 = portion ofK due to steel forcesR s = portion ofR due to steel forces

CONCRETE COMPRESSIONPARAMETERS

The parameters KC and R1 are depen-dent upon the compressive force of theconcrete (C c) and the moment (M e) ofthis force about the bending axis of the

122

b

1 t/2

t/2

Cck„t

N.A. YcB.A.

Fig. 2 Concrete stress distribution

column cross section. These quantitiesare derived based on the following as-sumptions and approximations.

1. The concrete area displaced bythe prestressing steel in the com-pression zone has been neglected,

2. The tensile strength of the con-crete has been neglected.

3. The approximation for the stress-strain relationship for concrete isshown in Fig. 1.2

A rectangular column cross sectionwithout the prestressing steel is shownin Fig. 2 where the concrete stress dis-tribution and the depth of the neutralaxis (k„t) are also illustrated.

The following equations can bewritten from Fig. 2.

Cr. = faaebkat (1)

M e = ycC C (2)where

fa re = average stressye = distance between centroid of

stress distribution and bendingaxis

In order to obtain K, both sides ofthe equation for C e are divided by f htand to obtain R, both sides of the equa-tion for M, are divided by_f^ht 2. The re-sulting expressions are:

K c G.=

(3?}fbt

M` (4)1'ebt2

STEEL FORCE PARAMETERSThe dimensionless expressions (KB

and R 8) for the prestressing steel arebased on the following assumptions andapproximations:

1. The bending strain in the pre-stressing steel and in the concreteare compatible.

2. Complete bond exists between theprestressing steel and the con-crete.

3. The prestressing steel is assumedto be a thin rectangular tube asshown in Fig. 3. One-fourth of thetotal steel area is in each side ofthe rectangular tube.

The total strain in the steel at any lo-cation is evaluated in terms of threecomponents; the strain in the steel dueto prestressing (En8 ), the strain in theconcrete due to prestressing (€), thestrain due to bending (EB ). The strainsE„ and E,. e are defined as follows:'

pp f" 1 a,ESQ = and Epg =

E, E,wherefHe = effective stress in prestressing

steel after losses

PCI JOURNALJJanuary-February 1982 123

b

b

Steel Tube

B.At t.9t

' i Prestressing Strandsb'

Fig. 3. Steel distributed as a thin tube.

gt

iLI b' j

E ps = modulus of elasticity of pre-stressing steel

E e = modulus of elasticity of con-crete

p, = Ap t l(bt) = total steel percent-age

A, = total area of prestressing steelThe strain due to bending is that

which exists in the column concrete asa result of a failure strain at the com-pression face of the column. The di-mensions and variables required to cal-culate the steel strains due to bendingare shown in Fig. 4.

From the linear strain distribution, itis possible to calculate the bendingstrain in the top steel (E R4 ) and the bot-tom steel (€B1):

_ d, – k„ t _ 1 +g-2k

E81 — k rEcu —

2kEcu

y LL

_k„t-d4_1-g-2kgEE4 k t Ecv — 2k Ecu

u ,,

The total strain in the top steel (ep4)

and bottom steel (e px ), respectively, is:

= l.e + pg Le + 1 + g – 2kv EEp1

ru (5)Eve E 2 ku

__ f,, Pnfse 1 – g – 2k„E } E } 2kw Eau (6)

va r

Since the stress-strain curve for pre-stressing steel becomes nonlinear athigher strains, three possibilities ofstress-strain relationships occur in thecross section. All, part, or none of thesteel strains can be in the linear rangeof the stress-strain curve. Each of thesepossibilities is examined in the follow-ing three cases.

Case 1 — Steel Stresses Within theLinear Range

The first case considers a rectangularsection in which all of the prestressingsteel stresses are within the linear por-tion of the stress-strain curve.

fn1 ^ ENL Ea4 C FNL

where ENL is the strain at which thestress-strain curve becomes nonlinear(see Fig. 5).

For this case the nondimensionalforce and moment due to steel (K„ Rs)can be calculated by the following ex-pressions:

Ks --E 8 + €84) (7)

s /R= 6 fc E Da (€81 — EY4) (8)

124

1.

B.A.

e

___ (1-g)t/2

Ccu

!1Iigt /

t N.A.

gt/2

CBI

b

Fig. 4. Bending strain diagram.

Case 2 -- Steel Stresses in BothLinear and Nonlinear Ranges

The second case considers a rectan-gular section in which the prestressingsteel strains are in both the linear andnonlinear portions of the stress-straincurve:

EPI > E NL EP4 C ENL

It now becomes essential to adopt astress-strain relationship for the pre-stressing steel. The stress-strain curveshown in Fig. 5 has been used in thesederivations.* The following equationsdescribe this curve.

With Ep < FNL = 0.008:f, = EpEps (9)

With €1,> ENL = 0.008:

0.075Ip =268- C0.9$fP,

E P - 0.0065 (10)

where f„ is the stress in the prestressingsteel.

For this case, the nondimensionalforce and moment due to the pre-stressing steel (K,, R,) can be calcu-lated by the following expressions:

xt _ Pp Ep s lD,+

( ENL — Ep4 )+ Ep4

4j; EP. (En, — £p4

2 ( C NL — EPI ) f7 1(1.1)

(Ep1 — Ep4 ) Epa J

R, = 9PP Eye L f" — Ep4 + 2i Epq x

o.fl, Epp

( ENL — Epi)( ENL — Ep4) +

( Eyl — E1^4 ^S

(4ENL — Ep 4 — 3Ep 1) (EN — E ) 2 — 2'fP7 XL p4

3 Epl — E EP4 E,

12EP7 — E, 4 ) ( E NL — CPT ) ] (12)( Ep l — E4 14

wherefp , is calculated from Eq. (10).f 7 is the average stress between

strains of ENL, and EP, (correspondsto ?,, , see Fig. 5)

fp, is the strain at the centroid of thearea under the stress-strain curvebetween ENL and Epi (A7).

Case 3 — Steel Stresses Within theNonlinear Range

The third case considers a rectangu-

PCI JOURNAUJanuary-February 1982 125

N

NN

mu)

).0065) ksi.

6P4 6NL eel

Steel Strain

Fig. 5. Stress-strain curve for 270 ksi (1861.7 Nlmrre ) prestressing steel.

lar section in which all the prestressingsteel stresses are within the nonlinearportion of the stress-strain curve.

E vl > E NL E 94 > EYL

The nondimensional farce and mo-ment due to steel (K„ R 1 ) can be cal-culated for this case with the followingexpressions:

K

pp (3 Al + f94 + 2177) (13)4 f'

R• gpP IJ PI S48f

+ 2€ — Eg1 Jl\ !PT

The quantity fA7 is the average stressbetween strains of E94 and E p , ; Ep7 is thestrain at the centroid of the area underthe stress-strain curve between €94 andEpp,

The equations presented here corre-spond directly to those used to develop

load versus moment interaction curvesfor a specific column. Points on an in-teraction curve are calculated by as-suming a neutral axis location, satisfy-ing the compatibility of strains, calcu-Iating the concrete compression forceand steel forces, and solving the equa-tions of equilibrium. For general use,the concrete compression force and thesteel forces were nondimensionalized.With this one deviation a general de-sign aid was developed for rectangularprestressed concrete columns.

For all points above the balancepoints on the interaction curves, theprestressing steel strain was found to bein the linear portion of the stress-strainrelationship. Therefore, a change in theeffective prestressing level causes avertical translation of the interactioncurve in the compression control re-

Modification for Partial(1 4) Prestressing

126

gion. The amount the curve translates(AK) for a change in the effective pre-stressing level (Af, ) can be found bymodifying the basic nondimensionalequation for the nondimensional loadK:

K =K, +K&

Since all prestressing steel strains arein the linear portion of the stress-straincurve, equations for modification toK(AK) were developed in terms of thechange in prestressing (Afte ),

AK = P. O .f e I + (15)

r Pp

This equation can be solved for Af1e :

4 }ae =.fc AK

(16)

Pv f ET

L Er.

With Eqs. (15) and (16) and the inter-action curves, a prestressed concretecolumn can be designed with any levelof effective prestressing provided allthe strands are stressed to the samelevel. Complete derivations of allequations are available in a previouspublications

Effects of DerivationApproximations

Concrete Stress-StrainRelationship

A nonlinear approximation for theconcrete stress-strain relationship isused in order to produce more accurateresults regardless of the location of theneutral axis. This capability is impor-tant since the development of an inter-action diagram requires the depth tothe neutral axis to vary from zero to in-finity. The details of the concretestress-strain approximation are shownin Fig. 1.

The concrete stress-strain approxi-mation is especially useful as the depthto the neutral axis becomes larger than

the depth of the section. This allows asmooth reliable curve between the bal-ance point and the load axis. However,at higher steel percentages and lower gvalues, the concrete stress-strain rela-tionship used in this derivation alsoshows a slight inconsistency. This iscaused by the inability of the stress-strain approximation to give accurateresults for the concrete force and its lo-cation as the nonlinear portion of thecurve moves oil' the cross section. Athigher steel percentages and lower gvalues, the concrete compression forceand its location is a major determinantof the load and moment capacity of thecolumn. The result of this inconsistencyis a small discontinuity in the curvesnear the load axis.

Prestressed Steel TubeApproximation

The replacement of the prestressingstrand with a prestressed steel tube isthe primary approximation in the deri-vation of the equations used to generatethe load vs. moment interaction curves.The accuracy of the solutions obtainedwith this approximation was checkedby calculating points on the interactioncurves for specific columns. These spe-citie columns consisted of various steelpercentages, numbers of strand, tub val-ues, and values of g. The values ob-tained were then nondimensionalizedand checked with the general interactioncurves.

For the case in which all the pre-stressing steel strains are in the linearportion of the stress-strain curve, the er-rors were in the range of 2 percent.However, certain factors determine themagnitude of the error. For a givensteel percentage, the column with thelargest number of strands had the leasterror. Increasing the g value or the tlbvalue increased the error produced bythe prestressed steel tube, Increasingthe steel percentage increased themagnitude of the error.

For the case in which all the pre-

PCI JOURNAIJJanuary-February 1982 127

stressing steel strains are in the nonlin-ear portion of the stress-strain approxi-mation, the errors were on the order of4 percent. The increase in error is dueto the inability of the prestressed steeltube to accurately evaluate the discreteforces through the nonlinear portion ofthe stress-strain relationship. The max-imum error occurs at large steel per-centages and the higher values ofg and

tlb. The errors decrease as the value of'g, tlb, and the steel percentage de-creases.

The case in which the prestressingsteel strains are in both the linear andnonlinear portion of the stress-straincurve resulted in errors on the order of3 percent. The magnitude of the errorsfollow the same pattern mentioned inthe two previous cases.

CONCLUDING REMARKSThe purpose of this study was to pro-

vide the engineer with a versatile de-sign aid for rectangular prestressedconcrete columns without accountingfor length effects. This was accom-plished by the development of a gen-eral formulation for a prestressed col-umn analysis and of nondimensionalload versus moment interaction curves.

The input information necessary inorder to use these design aids is only

the concrete and steel strengths and theultimate load and moment acting on thecolumn. From these parameters variouscolumn sizes, steel percentages andeffective prestressing levels can beevaluated until a combination is foundwhich satisfies the design require-ments. This method offers efficiencyand versatility in either preliminary orfinal design of rectangular prestressedconcrete columns.

REFERENCES1. Everard, N, J., and Cohen, E., Ultimate

Strength Design of Reinforced ConcreteColumns, Interim Report of ACI Com-mittee 340, ACI Publications SP-7.

2. Notes on ACI 318-71 Building Code Re-quirements With Design Applications,Portland Cement Association, 1972,Chapter 10, p. 26.

3. Nilson. A. H., Design of Prestressed Con-crete, John Wiley and Sons, 1978, Chap-

ter 1I, pp. 389-399.PCI Design Handbook -Precast Pre-stressed Concrete, 2nd Edition, Pre-stressed Concrete Institute, Chicago, Il-linois, 1978, Chapter 8, p. 17.McLaughlin, D. G., "Prestressed Con-crete Column Design," MS Thesis, De-partment of Civil Engineering, Univer-sity of Missouri-Columbia, December1980.

128

APPENDIX A - DEVELOPMENT OFDESIGN CHARTS

The following charts were formulatedusing the numbered equations in theprevious presentation. Two variables f,and g are considered in various combi-nations for rectangular columns withequal prestressing steel in each face.The steel percentage pp , is based onthe gross cross section of the column.

The following data were used in thedevelopment of the design charts:

= OA03 in./in. (0.003 cm/cm)E, = 57.5 v' .) (psi) (ksi)fp„ = 270 ksi (1861.7 Nlmm2 )fde = 154.9 ksi (1068.0 N/mm2)E= 27,500 ksi (189.61 kN/mm2).The charts were developed without

accounting for column length effects.Also, the workmanship factor, 0, has notbeen applied to the curves. Therefore,the first step in design would be to ad-just the loads for slenderness effects byusing a moment magnifier approachand applying the workmanship factor,

. The appropriate design charts can

then be used as illustrated in the designexamples.

The design curves for prestressedconcrete columns are similar to thosefor regular reinforced columns in shape.The prestressed concrete columncurves exhibit a less defined balancedpoint but do have a concrete compres-sion control region and a tension con-trol region for each steel percentage(see Fig. B2, Appendix B).

An interesting feature of the chartspresented here is that the curves forprestressing steel percentages crosseach other. This occurs because in thecompression control region, the higherthe prestressing steel percentage, thelower the column capacity. Hence, inthe compression control region it is de-sirable to use less prestressing steel.However, in the tension control regionthe higher the percentage of prestress-ing steel, the higher the moment car-rying capacity of the column.

FULL-SIZED DESIGN CHARTSBecause of the PCI JOURNAL size and the necessityto save expensive PCI JOURNAL space, the sixteendesign charts in this article were reduced 50 percent.Readers interested in securing (at cost of reproductionand handling at time of request) photostats of theoriginal charts should contact PCI Headquarters.

PCI JOURNAL/January-February 1982 129

0-90

0.8O

0. 7C

0.6C

0.5C

o204c

03C

0 2(

0,! C

0.90

cc

0.70

0.60

0.50U

0.4C

0.3C

0.2C

0.10

0.00 0.02 004 006 0.06 0.10 0.12 ^i4 0.00

K e/t =Fjo/f'bt2

CHART NO.

f^ = 5.O k safpu =270.0ksi .

p fse =154.9 ksi _o000 j g = 0 . 6 t

e

OQ?

0no

/2

CHART NO. 2

f;0 = 5.0ksi • I

^Pvz2700 ksi .fSe =154.9 k 9tmoo/g = 0.7 t

ePu

o°°so

p0)

rr

NN

//

0.02 0.04 0.06 0.06 0.10 0.12 D.ia

Ke/t=Pue/f^bt2

W

0.90

0.80

0.70

0.60

U 050

0.40

0.30

020

0. 0

CHART NO. 3

f^ = 5.0 k s i

• fpU -2700 ksifSe =154.9 ksi9 = 0.8

^ I p

P,

P;Q

^oa3'9t

t

O ^

///7 __

CHART NO_ 4

= 5.0ks1 ^•

fpm =270.0 ksifSe =154.9 ksi

g = 0.9

I

p^

p p00

Qc

gt

t

oop S0 ^

00/

7/__

_ 7/ _012 0.140.00 0.02 0.04 0.06 0.08 0.10

Ke/t=pie/f'bt2

012 014 0.02 0.04 0.06 0.08 0.10

Ke/t=%e^f^bt2

o_scN

0.8c

o.7(

0.6i

U 0.5{

QJ

Y 0.4C

0.3c

0.2C

0-IC

0.90

0.80

0.7C

0.6C

0.50

7aIf

0.4C

0.3C

0 2C

OHC

0.00 0.02 0,04 0.06 008 0.10 012 0.14 0.00

Ke/t=P"e/f^bt2

r^-aPT yin

f^ = 6.Oksi ^•fp =270.0 ksi LfSe =154.9 ksi

9 = 0.6

._

fe

P^

p o0

° loa'o°

ao °s

ogo°

rr

CHART NJO. 6

f^ = 6.0 ks ^fpu =270.0 ksi bfse=154.9 ksi9 = 0.7

.,^-ePv

I

te

R^

^,op°^ r

9t

oo°o ^°O>S°o s

00r

0.02 0.04 0.06 0.06 0.10 0.12 0.14

Ke/t=PUe^f^bt2

CHART NO. 7 CHART NO. 8

Ww

_0.00 0.02 0.04 0.06 0.06 0.10

Ke/t=Pue^ffbt2

bfPv =270.0 ksi

se =154.9 k ig = 0.8

_

u

PU

e

o ^^^qo^9

7 _LL

0.90

080

0.7(

0.61

0.S(

II0.41

0.3'

02

0.1

= 6.0ksi •

fPu =270.0 ksi b

fse=154.9 ks.l c^#

9 = 0.9#

e

I "

Pu

°•^a

0040)pad a9

012 0.14 0.00 0.02 0.04 0,06 0.06 OJU

Ke/t=Pue/f"bt2

0.12 0i4

0.91

0.81

07'

0.61

0.51

0.4(

0.3(

0 2(

O,IC

0 9(

Os(

0.71

0.5(

Na?

0.4c

03C

02(

0.€(

r'I-1LQT Kin 0

f^ = 7.0 ksibfp U =270.0 ksi

fse=154.9 ksi 9t

= 0.sOs

ePu

Q o

f'= 7.0ksi

fpu =270.0 ksi

I54.9 ksi 9t_____fse^ t 5'9 = 0.7 t

^

^ao°

0o a

40.00 0.02 0.04 0.06 O.OB 010 0.12 0.14 0.0p 002 0.04 006 0.08 0.10 012 0.14

Ke/t=Pue,/f^b t2 Ke/t=Pue/f^'bt2

0.5C

0.80

070

0.60

-U 0.50

0.40

030

0.20

0.10

W

CHART NO_ II

fC = 7.0 ks ifpu =270.0 ksi .fSe=154.9 kni tg = 0.8 #

e

Li

Pu

^Ea°PI

Qo ̂ ^

7/_

C^1l-1\_imil[o7f

f'= 7.0ksifP^ =270.0 ksfSe=154.9 ksi 9. °°

P ^^ 1

9 = 0.9 tea

_ _ Ii0.12 0J40.00 002 004 0.06 0.08 0.10

Ke/t=PUe^ffbt2

0.12 0.14 0.00 0.02 0.04 0.06 0.08 0.40

Ke/t = PLie/ fbt2

CHART NO. 13 CHART NO. 140.0

Cn

0.5C

0 7(

0.6(

0.5C

0.4(

0.3(

0.21

01(

090

080

0 7C

0.60

050

dJ

040

0.30

0.20

01C

0.00 0.02 004 0.06 0.06 0.10 0.12 0.14 0'00

Ke/t=pie/rbt2

fC = 8.0 k s i ^

l =270.0 ksifse=154.9 ksitg = 0.6

• •

.

° 4OO t

f^ = 8.Oksifpu =270.0 ksi

fse=154,9 k9 = 0.7

--* -, -®per

gt

te

pu

' •aOf

Q

0.02 0.04 0.06 0.08 0.10 0.12 0.14

Ke/t=Pue^f"bt2

o 0.9n0c

03 0.7

mQ

0.6

mN r

U0.51

d

0.41

0.3(

0.21

0.1c

0.0c

W

0.9{

0.81

0.71

0.61

0.5(

NII

0.41

0.3C

02

0.Ic

o.oc0.14

CHART M( I

fc = 8. ksi

fSe =154.9 ksi 9t9p- 0.8 t

eo °Q,

//.

V-tie U.U4 U.Ub U.Ut$ U.IU 0.12

Ke/t=Pe^ffbt2

1-APT MO I^

f^ = 8.0 ks;fpufSel 54.9 k si gt

0.9 ' f

pu

%,a

Qa a^qa, °•

// /

U.uc c1.U4 u.Ub U.Ua U.w 0.12 0.i4

Ke/t=Fbe^ffbt2

APPENDIX B — DESIGN EXAMPLES

4-318 Strands

i N (D

N2'I 1 2" , 2"I

16"

Fig. Bi. Trial design 1.

DESIGN EXAMPLE 1A square prestressed concrete col-

umn in the concrete compression con-trol region is to be designed using thefollowing data:

Pa = 550 kips; f1 = 270 ksiM. = 875 kips; f e = 154.9 ksi

f 6ksi; 0 =0.7The minimum cover to center of pre-

stressing steel is 2 in.

Trial Design 1By initially assuming that b = f = 16

in. and taking the design parametersinto account the following parameterscan be calculated.

K- Pu = 550 =0.512

O f/ ht 0.7(6)16(16)

R = K e = M1

t doff bt2

_ 875 = 0.0510.7(6)16(16)2

gt = (16 in.) - 2(2 in.) = 12 in.Therefore, g = 12/16 = 0.75.The (A factor is applied to the param-

eters K and R since this workmanshipfactor has not been applied to the inter-action curves.

Enter Chart No. 6 with K = 0.512,K(elt) = 0,051 and find that a steel per-centage of 0.001 is adequate (see Fig.B2). This steel percentage is also ade-quate when checked with Chart No. 7with ag of 0.8. For a steel percentage of0.001, the area of prestressing required(A,,) is given by the following equa-tion.

A,, = pp bt = 0.001(16)16 = 0.256 in.'This requires four %-in. diameter

strands (A9 = 0.34 in. ). The results oftrial design 1 are shown in Fig. B1.

Analysis of Trial DesignThe interaction curves can be used to

find the load and moment capacity of acolumn. For trial design 1 the followingparameters are calculated:

K=0.5120.01- _- =

R = 0.051

_ 0.34g = 0.75 Pp = t lfi(lfi) = 0.0013

A line is drawn from the origin,through the point K = 0.512, R = 0.051to a steel percentage of 0.0013 (see Fig.B2). The coordinates corresponding tothis end point are as follows:

138

fc = 6.Oksib

fC =270.0 ksi ^,^^aPU

fse=1`^4-9 ksi gt'°o

'

09 = 0.7 t

eP

qo s

0.6 3 °o>qo ^'

00^'

_ 0.512

^^ I

10I

10IO O

OII II

/ Tv'7__

0.91

:gel

0.71

K=

a 6,

K0.51

041

:.3C

0 2(

0.1C

0.00

00

00COU,U,

aE0

o VO0

II Balance Point

O U)

6

O

0

r'I-IGQT NIfl r,

^.vc u,U^+ vun U.Ua 0.10 0.12 0.14

Ke/t=PUe/f^bt2

Fig. B2. Design chart for Example 1.

K = 0.63 and K(e/t) = 0.063The actual design load capacity of the

column is given by the following egi . ia-tions:

Pu = (^ K fG f7t=0.7(0.63)6(16)j6= 677 kips > 550 kips

M= O(K t )S, b(t)^

= 0.7(0.063)6(16)(16)2= 1083 in. -kips > 875 in. -kips

The 16 x 16-in. column with four3/s-in. diameter strands is adequate.

DESIGN EXAMPLE 2

Design a 14 x 14-in. prestressed con-crete column with four ½-in, diameterstrands with the following data:

P„ = 425 kips = 270 ksiM„ = 700 in.-kips f P =f, = 5ksi = 0.7

First calculate K, K(elt), g and pn.

K = P°

425 = 0.620.7(5)14(14)

PCI JOURNAUJanuary-February 1982 139

0.90

040

0.7C

K

AKi V 6C

05Cu

`S 34C

0.3C

0.2C

0. IC

CHART NO. 2

fc = 5.O ksi Ib

^^ P

r =270.0 ksi^

HLgttPz4 fSe=154.9 ksi

0003' g = 0.7

R^= 0.62 ° oys0

JK

b

II I

0.10 0.12 0!40.00 0.0Z 0.04 0.06 006

Ke/t=%e^ffbt2

Fig. B3. Design chart for Example 2.

K e _ M,t Offb2

7000.7(5)(14)(14)2

= 0.073

gt = (14 in.) - 2(2 in.) = 10 in.

10g = 14 = 0.71

_ 4(0.153)

Ap 14(14)0.0031

The column will be checked for thefully prestressed case. Enter Chart No.2 with K = 0.62 and K(elt) = 0.073 (seeFig. B3). This point lies outside the pp= 0.0031 interaction curve, hence thecolumn will not work when fully pre-stressed &, = 154.9 ksi).

To calculate the required level ofprestressing, locate the intersection ofelt = 0.12 line and the pp = 0.0031 in-teraction curve. This point is the lowerlimit of AK as shown in Fig. B3.

AK=0.62-0.58=0.04The differential stress Af, e can be

calculated from Eq. (16):

140

8-1/2" Strands

4-Lr+ ^NI}

Fig. B4. Trial Design 1.

f ee =

5(0.04)

0.0031 I 1 + 0.0031(27500 ) j

L ( 57.5 V1 5000 .

= 63.2 ksi

The required prestressing level is:

fe= 154.9-63.2=91.7ksi

The 14 x 14-in, column with four1/z-in, diameter strand will work with f8e

91,7 ksi.

DESIGN EXAMPLE 3

Design a rectangular column usingthe following data:

Assume strong axis bending.

P„ = 140 kips M 2250 in.-kips

= 5 ksi f = 270 ksi

fire = 154.9ksi bit — 0.8 ¢=0.7

Trial Design 1Initiall y assume that b = 16 in. and t

= 20 in. and calculate the following pa-rameters:

K = P.f^ bt

— 140 = 0.125

0.7(5)(16)20

e M . 2250

^t = K t Of,' bt 20.7(5)(15)20)2

= 0.100gt = (20 in.) — 2(2 in.)g = 16120 = 0.8Enter Chart No. 3 with K — 0.125 and

R = 0.100; the required prestressingsteel percentage is 0.0035. The re-quired area of prestressing (A p1 ) is cal-culated as follows:

A, = pbt = 0.0035 (16)20 = 1.12 in.RThis relates to eight ½-in, diameter

strands with a total prestressing steel

METRIC (SI) UNITEQUIVALENTS

1 in. = 25.4 mm1 in. 2 — 645.6 mm21 ft = 0.3048m1 f2= 0.0929 m21 psf = 47.99 NIm21 psf = 47.88 Pa1 psi = 0.006895 MPa1 psi — 0.006895 N/mm21 pcf — 16.02 kg/m21 kip = 4.448 kN1 kip/ft = 14.594 kN/m1 in.-kip= 113.0N•m1 ft-kip = 1.356 kN•m

PC! JOURNALUJanuary-February 1982 141

090

0.80

0.70

0.6C

05C

a

Y 04C

o.9C

0.2C

010

0.00

CHART NO. 3

fG = 5.0 ks i ' I

fpu =270.0 ksifse-154.9 k si°'a gt

Q/0 g = 0.8 tea

l e >^O9

Dp

// -iK = 0.130

K = 0.125 cvor o

o toY L ii

002 004 006 006 cc 0.10 Cr 0.12 0.14

Ke/t=Pie/f'bt2

Fig. B5. Design chart for Example 3.

area of 1.224 in.' The results of trial de-sign 1 are shown in Fig. B4.

Analysis of Trial Design 1The interaction curves can be used to

find the load and moment capacity ofthe column by using the following pa-rameters:

K = 0.125 R=0.100 g=0.8An , __ 1.224

= 0.0038P° = bt 16(20)

A line is drawn from the origin onChart No. 3 through the point K =

0.125, R = 0.100 to a steel percentage of0.0038 (see Fig. B5).

The coordinates corresponding to thisend point are as follows:

K = 0.13 and K(elt) = 0.012The actual design load capacity of the

column is given by the following equa-tions:

Pa = 0.7 (0.13) (5) (16)20= 145.6 kips > 140 kips

0.7 (0.102) (5) (16) (20)2= 2285 in.-kips > 2250 in.-kips

The 16 x 20-in, column with eightb-in. diameter strands is adequate.

142

APPENDIX C — NOTATION

A, = total area of prestressing steelb = column widthB.A. = bending axisC, = compressive concrete farced (.,cr ip, 7 = distance between ex-

treme compression fiberand subscript location

E = modulus of elasticity of con-crete

E, = modulus of elasticity of pre-stressing steel

er = eccentricity of ultimate load^ltiN^Ascrept7 = prestressing steel bend-

ing strain at subscript lo-cation

E^^ = strain in concrete due to pre-stressing

fek = ultimate concrete strainE.KL = strain at which prestressing

steel stress-strain curve be-comes nonlinear

Eo = concrete strain at which con-crete stress-strain curves be-come linear

En[,ur,Krrip[1 = prestressing steel strainat subscript location

E (,(Nunsnrapr7 = strain at centroid of non-Iinear portion at pre-stressing steel stress-strain curve

fe = concrete stress

fz = compressive strength of con-crete

fp{.ue*eripI) = prestressing steel stressat subscript Iocation

f. = effective stress in prestressingsteel after losses

gt = distance between top and bot-tom steel layer

K = nondimensional loadK, = portion of nondimensional Ioad

due to concreteK 8 = portion of nondimensional load

due to prestressing steelk„, = depth of concrete compression

blockM = moment due to concrete com-

pression forceM„ = ultimate moment acting on col-

uninN.A. = neutral axisPa = ultimate axial load acting on

columnR = nondiniensional momentR^ = portion of nondimensional

moment due to concreteR, = portion of nondimensional

moment due to steelpa = prestressing steel percentaget = column depthJc = distance between centroid of

concrete stress distribution andbending axis

NOTE: Discussion of this paper is invited. Please submityour discussion to PCI Headquarters by September 1, 1982.

PC E JOURNAL,January-February 1962 143