Slenderness of Prestressed Concrete Columns Journal/1983/March... · ACI Code;4 the primary moment is magnified to account for the effect of changes in geometry. 4. The Reduction

Slenderness of PrestressedConcrete Columns

Noel D. NathanProfessor of Civil EngineeringUniversity of British ColumbiaVancouver, British ColumbiaCanada

't has been found that the usualmethods of handling slenderness in

reinforced concrete columns do notapply directly to prestressed concretemembers because of differences in themoment-curvature characteristics. ThePCI Committee on Prestressed Con-crete Columns has been studying thequestion for some time, and the presentpaper is a review of the slendernessproblem from that viewpoint.

First, the behavior of reinforced andprestressed concrete columns is re-viewed, and the application of variouscode approaches is discussed. Then,attempts to develop methods specif-ically for prestressed concrete columnsare described.

Throughout these discussions, the

NOTE: This report forms part of the work of thePCI Committee on Prestressed Concrete Col-umns, of which the author is a member. It hasbeen reviewed and endorsed by the committeefor publication and discussion.

50

P

AiPpi

Cl

A2

B,

D22E z

C2

'Le]MOMENT

Fig. 1. Behavior of slender columns. Material failure and instability failure.

P

0JJ

X

PZ

questions of load factors and capacityreduction factors will be omitted. It willbe assumed throughout that loads andmoments are design values P„ and M,and it will be taken as understood thatthe necessary safety factors would beapplied to loads and capacities.

BEHAVIOR OFSLENDER COLUMNS

When the end moment of a column,loaded at equal end eccentricities, fol-lows Paths 0A1 or 0A2 of Fig. 1, themid-height moment follows paths suchas OBC, with the maximum loads P1 andP2 , corresponding to Points C l or B2,when the end moments are at Points Dland D2 , respectively.

In Case 1, the capacity of the columnis controlled by material failure of thecross section at the point of maximummoment: the mid-height moment atPoint Cl has reached the load-momentinteraction curve of the cross section. InCase 2, the capacity of the column isgoverned by instability: at Point B2 thecolumn bows sharply and the load ca-pacity is reduced.

If the load were monotonically in-creasing, the mid-height section wouldbe rapidly displaced to Point E2 andmaterial failure would occur, as a sec-ondary cause of collapse. If the verticaldisplacement were controlled or ifload-shedding were possible, the loadwould reduce as the mid-height sectionfollowed Path BQ C2 , with material fail-ure finally occurring at Point C2.

PCI JOURNAL/March-April 1983 51

2400

2000

1600a_Y

Q 12000J

J

Q 800

. e .

`rdo 20X206% Steel

o^Material Failure

O—•—•— Instability within

----^ ----\90% of material\ `,0 ---- Failure

` — Instability

N FO

\ \e0 `,9c,

1

400

00 200 400 600 800 1000

END MOMENT (KIPS-FT)

Fig. 2. Load-moment interaction curves for a heavily reinforced concrete column. The.ACI curves are based on El = E0 1 9 15 + Eel.

DIFFERENCES IN RESPONSEOF REINFORCED AND

PRESTRESSED MEMBERSFig. 2 shows the load-moment inter-

action curves for a reinforced concretecolumn with 6 percent longitudinalsteel. The short-column curve is shown,together with curves for various slen-derness (Llr) ratios, drawn throughpoints such as D1 and D2 of Fig. 1.These points were calculated by meansof a computer program which has beenfully described elsewhere.1.2

Fig. 2 also shows whether the capac-ity of the long columns is limited bymaterial failure or instability, and it willbe seen that the latter does, in fact,

govern only when the slenderness ratioapproaches 100. Fig. 3 shows similarresults for a column which has only 1.5percent of longitudinal steel. Instabilitygoverns the behavior of somewhat lessslender columns in this case, particu-larly at higher axial loads.

Figs. 4 and 5 show the same resultsfor a prestressed concrete column and adouble-tee wall section, respectively.The steel ratios for these sections are,of course, much lower than for rein-forced concrete members, and in prac-tice, the prestressed column also tendsto be more slender. Also, load-bearingwall panels often carry very low axialloads as compared to columns.

It will be seen in Figs. 4 and 5 that

52

I60C

140C

1200

(na 1000

. • is20 5 X20° Column

1.5% SteelMaterial Fail,

—• —. Instabilitywithin 90%of material

--- Failure— Insto Dility

0o 800JJ

x 6004

400

200

00

-----------------1\ N \"6

-:

\\

100 200 300 400

END MOMENT (KIPS-FT)

Fig. 3. Load-moment interaction curves for a lightly reinforced concrete column. TheACI curves are based on El = EEl9 /2.5.

these prestressed sections (the wallpanel in particular) differ from rein-forced concrete columns in that the bal-anced load is relatively much higher,and that they are governed by instabil-ity throughout most of the slendernessrange, even at low axial loads. (In com-paring the figures, however, note thatFigs. 4 and 5 extend to higher values ofslenderness.)

Finally, one may comment that rein-forced concrete columns are generallybuilt into continuous frames, so that theprimary moments are affected by theloss of stiffness that accompanies ap-proaching instability. Precast columns,on the other hand, are generally stat-ically determinate during the applica-

tion of dead load and often for liveloads as well.

APPROACHES TOTHE DESIGN OF

SLENDER COLUMNSThis section considers alternative ap-

proaches to stability analysis.

Stability Analysis of Entire Frame

This is the preferred method; mostdesign codes give semi-empirical for-mulas for component design, but, whenthe applicable range of parameters isexceeded, require that a rational

PCI JOURNAL/March-April 1983 53

27001

2400

2100

cn 1800

1500

0-J

xa 900

600

300

00

24"X24" C°lumn

""—"'—"—"•"-------`--'\,\ ,^< < 400psi prestress

t I

i

^_ t

`^ f

-- Notarial Failure--- f t/ ^^ —. — Instability within

I` 90%of material Failure. ._^ . i

/. ----- — Instability

200 400 600 800END MOMENT (KIPS-FT)

Fig. 4. Load-moment interaction curves for a prestressed concrete column. The ACIcurves are based on El = E^l9/2.5.

analysis be made, presumably of theentire frame. The difficulty, of course,is that it requires a number of cycles ofanalysis and design to arrive at a satis-factory solution. MacGregor3 has sum-marized some of the aspects andmethods of making such analyses.

Stability of Components

A more practical approach to designis to estimate the influence of frame ac-tion and account for it by adopting aneffective length for the column. That isto say, the range of the points of inflec-tion is first estimated and then anequivalent pin-ended column isextracted for analysis by one of the fol-lowing procedures:

1. Rational Analysis of IndividualElements—It is now possible, bycomputer, to make numerical analysesof particular columns with as much ac-curacy as may be desired; the limit isset only by the computational effort thatis expended.

2. The Additional MomentMethod — This method has beenadopted by many European codes, in-cluding the British CP 1106 and theGerman DIN 1045. Semi-empiricalformulas give the additional momentsover and above the primary moment,which arise from changes in the geom-etry. The analysis is generally limitedto failure conditions.

3. The Moment Magnifier Method -This is the procedure adopted by the

54

a 800Y

a 600J

J

Q 400

200

8 DT 12h ory PCI Design Handbook

i 0`Ir

\i ce 7/n

\ I\

/

\^p

0

,,,

t I% / Materiol foilu re/i

/I / / /

----S AC / I r-- \/r

2

.. —• Instabilitywithin 90°/pof materialI /

-^^/ I /,

failureIInstability

0•0

20 40 60 80 100 120 140 160

END MOMENT (KIPS-FT)

Fig. 5. Load-moment interaction curves for a standard double-tee section. The ACIcurves are based on El = EE19/2.5.

ACI Code;4 the primary moment ismagnified to account for the effect ofchanges in geometry.

4. The Reduction Factor Method -In this procedure, which is permitted asan alternative by the ACI Code, boththe load and the moment are reduced.In effect, curves such as those of Fig. 2for slender columns are derived bycontracting the short column curve, re-ducing it according to an empirical for-mula depending on the slendernessratio.

5. Direct Methods — Application ofdirect methods such as energy, orweighted residual methods to deter-mine the critical load of columns gov-erned by instability, are alternatives tothe above methods.

COMMENTS ON THEAPPLICABILITY OF

METHODS TOPRESTRESSED

CONCRETE COLUMNSWhile a nonlinear stability analysis of

the entire frame may be desirable asthe final step in the design of an im-portant structure, it does not obviate theneed for an initial design procedure.Further, as noted above, prestressedprecast concrete columns are oftenparts of statically determinate systemsin which the primary moments are notdependent on stiffness. The need forsuch a complex procedure does notexist in these cases.

PCI JOURNAL/March-April 1983 55

I( Varies

000"

f,' 5KS1 . fp„= 250KS1Approx. 225psi Prestress

N N N N See PCI Design HonbookM ")' 2nd Ed.p.2-59

I—U-D:Warna-Y

a_c

-6-

0

ca

ra

0'-0

2 3 4 5 6

4 Mn( KIPS-FT PER FT )

Fig. 6. Curves for a typical hollow-core section such as might be provided by amanufacturer.

Rational analysis of individual ele-ments, however, does seem to be ahighly appropriate procedure for theprecast industry, when standardizedelements are under consideration. Itwould be relatively simple and ex-tremely useful for precasters to supplyfamilies of interaction curves, such asthose of Figs. 2 through 5 for. standardcross sections at different slendernessratios, derived by means of computerprograms such as that referred to above.An example for a typical hollow-coreslab is shown in Fig. 6.

Nevertheless, the need for a simpledesign procedure remains. The addi-tional moment and moment magnifiermethods (which are essentially thesame) both present difficulties whenthe capacity is governed by instability,

as it generally is with prestressed col-umns (Figs. 4 and 5). In the paperforming the basis of the British designprocedure, Cranston writes: "Whilstsuch a method will not deal realisticallywith instability and deflection failure, itshould be possible to ensure that themethod is conservative for such cases."This point will be discussed below.

In view of the complexity of theproblem and the number of relevant pa-rameters, the reduction factor methodcannot be expected to work well for thegeneral case. Approximate solutions forthe critical load of nonlinear columns,on the other hand, does attack the rootof the problem for prestressed columns,but it may not be possible to reducesuch a solution to a workable designoffice procedure.

1.1

v0C-0Cz

3sv

a'O

ODW

v

P

(2 CONDITIONS ATFAILURE ASSUMEDTO BE HERE:TOTAL MOMENT ANDCURYATURE KNOWN F-

(7̂ CP 110 w O5 CAN BE6 RIMARV MOMENT P-6MOMENT 2 CALCULATEDIS KNOWN I CALCULATE p

0 ^1 P IS KNOWN

7OACI MAGNIFIED MOMENT CURVATURE

IS CALCULATED _ O MOMENT-CURVATURE 1^RELATIONSHIP ® DISTRIBUTION OF

B®IS TOTAL MOMENT SAFEIS ASSUMED CURVATURE ALONG P

(WITHIN SHORT COLUMN CURVE I? COLUMN FOLLOWS

MOMENT

Fig. 7. Assumptions of additional moment and moment magnifier methods.

9 9 9

1

0 o f

p^^ o Q

\\ L Q

°y c ul

Pto O,

y 9^Fo

ay

' O1

o Iau I

f

Gy

NiNr

A4c

t^GN

c^R^F

C,'w

1500

U,

1000

Q0JJa_xQ

500

10110 100 200 300 400 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

MOMENT (KIPS-FT) CURVATURE

Fig. 8. CP 110 assumptions with regard to curvature at material failure for the column of Fig. 3.

ADDITIONAL MOMENTAND MOMENT

MAGNIFIER METHODS

In this section, representative exam-ples of the two methods generally usedin existing codes of practice will be dis-cussed in detail.

The basic assumptions of thesemethods, illustrated in Fig. 7, are thatthe column will deflect laterally, therewill be an additional P-A moment, andeventually, at the point of maximummoment, material failure will ensue (asopposed to instability failure of the en-tire column). Then, in a pinned-pinnedcolumn, for example, the conditions atmid-height are known at failure: for thegiven load, the moment and curvaturecan be deduced from the short columninteraction curve. Making some as-sumption about the moment curvaturerelation, the distribution of curvaturealong the column length can be approx-imated, and the deflection and the P-Amoment can then be computed. Thecapacity available for the primary mo-ment follows.

The British and American applica-tions of this approach will be used asrepresentative examples.

The British Method of CP 1106

cated by the Comite Europeen duBeton, is added to give:

ou = h (0.00575 - 5000h) (2a)

ou = —1 — (1 - 0.0035 (2b)175h \ /

where L is the effective length

This value of ultimate curvature maythen be assumed throughout the loadrange, or a linear decrease to zero maybe introduced from the balanced load tothe maximum pure axial load. Fig. 8shows that this variation of ultimatecurvature is not unreasonable above thebalanced load.

The moment curvature relationshipas moment is increased at constant loadis then taken to be linear up to the ul-timate value of Eq. (2) for any axialload. Fig. 9 shows that this assumption,too, may be considered acceptableabove the balanced load:

From this it follows that the curvaturedistribution along the column is similarto the moment distribution, and the de-flection at mid-height should lie be-tween L2/(84u ) for uniform distributionof moment and curvature and L2/(124)for triangular distribution of momentand curvature.

Using an average value of L2/(104),we obtain:

The curvature at the balanced pointof the short column interaction curve is A = h /L21 r1 - 0.0035 L2) K (3)calculated as: 1750 ` h) ( h

e, + €, (1) where K is the optional reduction duec6" h to the linear decrease in maximum cur-

vature above the balanced load point:where

e, = maximum strain of concrete (tak-en as 0.00375, which includes anallowance intended for creep)

e^ = yield strain of steel (taken as0.002)

h = overall depth of section (shouldbe effective depth)

An additional small term dependentupon the slenderness ratio, as advo-

K= Po - P (4)Po -Pb

The section must then be safe withrespect to the short column interactioncurve under the load P and a momentequal to the primary moment plus P0.

In Figs. 10 through 13, the dashedline shows the true value of the ulti-

PCI JOURNAUMarch-April 1983 59

500

400

F-U.

300a_

F-zg 2000

100

aQorro'

Qe^03Po,Qo

0.6P0 0.2Po

01 ao

0.1 P

0,8Po

O9PO

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006CURVATURE (INS-')

Fig. 9. Moment-curvature relationships for the column of Fig. 3 asgenerated by the computer program.

mate curvature of the short column, asobtained by rational analysis. Assumingthat the deflection is, in fact, equal toL2/(104 ), the value of the curvature 0uwhich would give the correct PA effectis deduced and plotted on the figures;that is, if all the assumptions of CP 110are retained, but the curvatures plottedas solid lines in Figs. 10 through 13were used in place of Eq. (2), the cor-rect Pd moment would be obtained.

Recall the curvature which is used inCP 110 is the value at the balanced loadpoint, with the optional reductionabove the balanced load. (This variationis also shown on each of the figures, asa dash-dot line.) Thus, the diagrams re-flect the applicability of the CP 110procedure: if the solid lines (which

would give the correct result) aretightly grouped around the dash-dotline actually used by CP 110, the pro-cedure will work well; the more widelyspaced the solid lines are, the more er-ratic the procedure will be.

It will be seen that, judging by thesefigures, the accuracy of the method maybe expected to deteriorate as we movefrom heavily reinforced concrete sec-tions to more lightly reinforced sec-tions, to prestressed columns, and tothe double-tee section. This might havebeen expected since, as noted above,the method is really predicated uponmaterial failure, and Figs. 2 to 5 haveshown that instability failure becomesmore dominant as we progress throughthe same sequence of cross sections.

60

<'q\ \ __CURVATURE AT ULTIMATE MOMENT°• MAX. PERMISSIBLE AXIAL LOAD

G t^ 0dIQ^o\ REINFORCED

' CONCRETE COLUMNOF FIG.2

' WITH 6% STEELja^` mI

,^

JITHIS VALUE OF CURVATUREWOULD GIVE CORRECT RESULTIF USED IN CP 110 PROCEDURE

CEB MODIFICATION MOVESCP 110 CURVE L/r = 60TO HERE FOR{L/r:100 -\

BALANCED LOAD

111.1.1

U,

Y 1500

040JJ

1000X4

500

0 0.0001 0.0002 0.0003 0.0004 0.0005

CURVATURE (INS-')

Fig. 10. The curvature used in the CP 110 procedure compared with the ones back-figured by CP 110 formula from the correct result, for the column with 6 percent steel.

It may also be noted that, when theassumed curvature lies to the right ofthat which would give the correct re-sult, the column is being assumed to bemore flexible than it really is; that is,this will give a conservative estimate ofslenderness effects. It will be seen thatCP 110 is occasionally somewhat un-conservative but generally satisfactoryabove the balanced load. Below thebalanced load it is very unconservative.

The effect of the CEB slendernessmodification (1 – 0.0035 Llr) is shownon Fig. 10 by small marks on the bal-anced load ordinate. The variation withslenderness is reasonable, but byshifting the assumed curvature to theleft it makes the method still less con-servative.

In general, the application of CP 110procedures to prestressed sections ishampered by the following points!

1. It is more difficult to compute thecurvature at any point on the load-mo-ment interaction curve, since there isno defined yield point for the steel, andsince the strains in the steel and con-crete are not directly compatible.(However, these difficulties could beovercome.)

2. The balanced load point is rela-tively higher, so that the range in whichthe CP 110 procedures are unsatisfac-tory is greatly extended.

3. Many prestressed concrete col-umns and particularly wall panels carrylight axial loads, and so are well withinthe range referred to in Item 2 above.

PCI JOURNAL/March-April 1983 61

(___CURVATURE AT ULTIMATE MOMENT

MAX.PERMISSIBLE AXIAL LOAD

z.1 °

1 ^I REINFORCEDCONCRETE COLUMN

2 i N• OF FIG.3WITH I^g%STEELmQ

W

NO Q

m•

C

O^

\;

\

THIS VALUE OF CURVATUREu \ WOULD GIVE CORRECT RESULT

IF USED IN CP 110 PROCEDURE

BALANCED LOAD

"p

N.o

,yam

1400

1200

1000

a 8000JJa 600x

400

0 0.0002 0.0004 0.0006CURVATURE (INS -1)

Fig. 11. The curvature used in the CP 110 procedure compared withthe curvatures back-figured by CP 110 formula from the correct result,for the column with 1 1/2 percent steel.

4. These members are generally gov-erned by instability failure, so that thebasic assumption of the procedure isviolated. This is illustrated by thespread in values of curvature 1/ou thatwould be required to give the right an-swer in Figs. 12 and 13.

The Method of ACI 3184The deflection of an elastic pin-

ended beam is increased in the pres-ence of an axial load, and it can beshown that the final deflection is givento a very close approximation, by:

^o0 = 1 – PlPcrte

(5)

wheret10 = primary deflection, calculated

without regard to the axial loadP = axial loadPeru = critical value of axial load =

ir2 EI/L2

The maximum moment is then givenby:

M=Mo +PO

M=Mo+Poo1

1 – P/Perit

whereM,, = primary moment, and

0, Ao = maximum values of total andprimary deflection

62

MAX. PERMISSIBLE LOAD

1 y

1 G I

I

Q

RECTANGULARPRESTRESSED CONCRETECOLUMN OF FIG.4

1`i

BALANCED LOAD

THIS VALUE OF CURVATURE\^ WOULD GIVE THE CORRECT

/ RESULT IF USED IN1 \ CP 110 PROCEDURE

nq

2500

^O\

i

2000

Na_Y

o 150040JJ4a 1000

500

00 0.0002 0.0004 0.0006

CURVATURE (INS-')

Fig. 12. The curvature used in the CP 110 procedure compared withthe curvatures back-figured by CP 100 formula from the correct result,for a rectangular prestressed concrete column.

so that

–0.234<I– P`MA° <+ 0.178

Noting that, for practical columns, P1

Pcra will also be small, we can ignore

M=M° 1 – P /Pcru (1 – Pcrtt 0°/M°) (6)

the term:

1 — PlPcrtt

Peru D°P/Pcrtt (1 – M

Assuming, as in the case of CP 110, °that the distribution of primary moment in comparison with unity. Thus, it is ar-lies between a uniform and a triangular gued that, for the elastic case, the mag-distribution, and again confining our- nified moment is given with sufficientselves to the elastic case, we have. accuracy by:

M° L2 < D < M° L2 M = M° 1 (7)12E1 8E1 1–P/Pcrtt

M = M° ^1 +-:-PA1 1 – P/P",.,1)

(1_P/Pci.u+PoIMo)M=M1–PIPGrit

PCI JOURNAUMarch-April 1983 63

1200

1000

800a_

a0 600JJ

Xa 400

200

CURVATURE AT ULTIMATE MOMENT

BASIC ASSUMTION OF CP 110

MAX. PERMISSIBLE LOAD

'II 8 DT 12OF FIG.5

T BALANCED

THIS VALUE OF CURVATURE

/

WOULD GIVE CORRECT RESULT

IF USED IN CP 110 PROCEDURE

^2s

0•0 0.0005 0.001 0.0015 0.002 0.0025 0.003

CURVATURE (INS-')

Fig. 13. The curvature used in the CP 110 procedure comparedwith the curvatures back-figured by CP 110 formula from thecorrect result, for a double-tee section.

We note, once again, that this expres-sion is applied to the elastic case, with:

AEIPcTu = L2 (8)

The term EI, of course, is the curva-ture constant; for the elastic case, cur-vature 0 = MIEI. The ACI procedureadapts this expression for the inelasticcase of reinforced concrete columns byincluding an empirical expression forthe rigidity EI. Once again, the columnmust be safe with respect to the shortcolumn interaction curve under theload P and the magnified moment M.

Fig. 14 shows the load path OA at the

end of a pinned column loaded underequal end eccentricities. OBC is theload path followed by the mid-heightsection of the column. As the load in-creases, the magnification increases assuggested by the formula developedabove; but the moment-curvature rela-tion eventually becomes nonlinear, andthis leads to a slightly more rapid in-crease in the moment magnification.

The ACI formula, however, assumesa linear relationship between momentand curvature; it is desirable that thepredicted load should end at the correctpoint C, so the empirical value of EI ispresumably chosen so as to exaggeratethe moment magnification slightly at

64

A C (MATERIAL FAILURE)

A. `p INSTABILITY4? 1 q FAILURE

ro ^^ 0 E F J

s

4.04() 4+0 ^^ \OHS\GHQ pC\\

P M

o paeoO^ ^o 6

MOMENT

Fig. 14. Actual and fictitious load-paths for columns subject tomaterial failure or instability.

0J

A

lower loads, to follow a path such asODC. The British procedure, of course,does not attempt to predict the loadpath, but merely to calculate additionalmoment AC when failure occurs.

In the case of a column which fails inthe instability mode, the end follows aload path such as OE, while the mid-height section follows the path OFG. Ifthe two methods really worked exactly,the additional moment of CP 110 or themagnification factor of ACI 318 wouldboth increase without bound quite sud-denly when the load corresponding to

Point F was reached. However, CP 110is predicated upon material failure, andACI 318 upon linear moment-curvaturerelationships, so they must be adjustedin some empirical manner to accountfor instability failure. In the ACI proce-dure, the value of El must be adjustedto predict a load path such as OHJ, sothat it appears that material failure oc-curs at the load corresponding to PointsF and J.

As pointed out above, this is a muchmore serious problem for prestressedcolumns than it is for reinforced con-

PCI JOURNAUMarch-April 1983 65

crete members. In the British proce-dure, the deflection when the mid-height section is really at Point F mustbe exaggerated to give aP-0 moment ofEJ instead of EF. This happens auto-matically, to some degree, since thecurvature is assumed to be the ultimatevalue, corresponding to Point J, insteadof that corresponding to Point F; butFigs. 12 and 13 show that the self-cor-rection of the method is far from per-fect.

It is seen, then, that if these methodsare to be used when failure is generallygoverned by instability rather than bymaterial failure, the rigidity (EI in theACI procedure; curvature at ultimatemoment, 4, in the British procedure)must be artificially modified to give theappearance of material failure when in-stability actually occurs. It was decidedto attempt a modification of the ACIprocedure, as detailed in the next sec-tion.

MODIFICATION OF THEACI MOMENT-MAGNIFIER

PROCEDURE FORPRESTRESSED COLUMNSIn this section, current efforts at em-

pirical modification of code procedures,to fit them to prestressed concrete col-umns, are described. The ACI methodwas selected for attention. In order todevelop a data base for comparisons,the previously mentioned computerprogram was used. (Note that the ex-perimental verification of the programhas been discussed by Alcock. 2 ) Forty-eight sections as shown in Fig. 15 wereanalyzed, each at ten load levels and sixeffective lengths, that is the equivalentof 2880 laboratory experiments.

It was desired to modify the rigidity(and hence the curvature) of the sec-tion, as discussed previously, to achievethe fictitious load path OJ of Fig. 14. Inthe ACI Code for reinforced concrete

columns at low levels of reinforce-ment ratio, the rigidity is set equal to:

E,1, /2.5 (9)

where E, I, = EI for the gross concretesection.

Since the steel ratio is generally verylow for prestressed concrete columns, itwas hoped that the rigidity could be setequal to:

E,I,/x (10)

where A would be a scalar function of alimited number of parameters.

The affect on A of the following fac-tors was investigated:

(a) Amount, distribution, and level ofprestress.

(b) Shape of cross section.(c) Load level P/P0 , where P,, is the

pure axial load capacity.(d) Slenderness ratio L/r.

Effect of PrestressWithin a practical range of prestress,

up to about 600 psi (4.14 MPa), in-creasing the amount of prestressing ordistributing it more widely across thesection appears to increase the shortcolumn bending capacity more than itincreases the moment at instability.This makes the proposed proceduresomewhat less conservative, but withinthe stated range of stress the variation isnot large.

The variation is greater at low axialloads, since the prestress, and particu-larly its distribution about the centroid,affects bending capacity more than axialload. However, higher levels of pre-stress have the effect of reducing thebalanced load, and finally of eliminat-ing the tension failure region of thecolumn interaction curve, drasticallychanging the behavior of the cross sec-tion.

The same effect is brought about ifexcess high strength prestressing steelis used at a low level of stress in the

66

SQUARE SECTIONS12'x 12' OR 24'x24'

PRESTRESSED TO 200 PSI 400 PSI OR 600 PSI

PRESTRESS CONCENTRATED AT CENTROID

OR DISPERSED VERTICALLY

(12 SECTIONS)

8`—

TEE SECTIONSFLANGE 40'x4' OR 80'x2' o/A DEPTH 12' OR 24'

PRESTRESSED TO 200 PSI 400 PSI OR 600 PSI

PRESTRESS CONCENTRATED AT CENTROID

OR DISPERSED VERTICALLY

(24 SECTIONS)

INVERTED TEE SECTIONSFLANGE 40'x4' OR 80'x2' o/A DEPTH 12' OR 24'

PRESTRESSED TO- 200 PSI 400 PSI OR 600 PSI

PRESTRESS CONCENTRATED AT CENTROID

(12 SECTIONS)

Fig. 15. Sections used for developing computed data.

steel, to induce low initial prestress inthe concrete. These phenomena can beobserved in analyzing some of the testsdone by Aroni.8 Nevertheless, it wasfelt that a value of X which applied onlyto columns with initial prestress up to600 psi (4.14 MPa), achieved by use ofprestressing steel utilized at high stress,would be of practical value.

Effect of Cross SectionThe shape of the cross section has a

marked effect on column behavior. Awide compression flange again in-creases the bending capacity more thanit affects the initial curvature, or the in-stability moment. Thus again, the ap-parent magnification factor should belarger.

PCI JOURNAL/March-April 1983 67

Effect of Load Ratio andSlenderness

The principal variables affecting therequired values of A are, however, theload ratio P/P0 and the slenderness ratioL/r. In the interests of developing apractical procedure, therefore, it wasdecided to base A upon P/P0 , Llr, and onthe presence or absence of a compres-sion flange, and to account for other in-fluences by adopting a sufficiently con-servative form for X.

The value of A to give the "correct"computed long column moment wasdeduced for each of the 2880 casesmentioned above, and tabulated againstthe ratios P/P. and L/r. A trial functionfor A was then adopted; the differencebetween the moment capacity obtainedfrom the computer program and thevalue based on the trial A was deter-mined and expressed as a ratio of thecomputer value.

The trial function was varied in anendeavor to get small differences on theconservative side. This was done forsections with and without a compres-sion flange; no precise definition of acompression flange could be arrived at,but essentially the question seems to bewhether or not the compression stressblock will be rectangular at lower loads.

The expressions finally arrived atwere as follows:

X=778-1.5 (11)

where

1100 forP/PO _- 0.02

P3 ^ p.02 2 for0.02<P/P0<0.42

7forP/P0%0.42

o = L7

for sections with no compres-sion flange

B = 31 – 0.09 for sections with a com-Llr pression flange

The magnification factor is thengiven by the familiar ACI formula:

8 = Cm (12)1 – P/Peru

where

7r2[EI/(1 + Pd)]Peru = L2

andEl = ECI0/A

In Figs. 16 to 18 the curves representlong column capacities in a diagram-matic sense. (The curves for real col-umns do terminate, for example, at theappropriate P/P,, ratios shown.) Usingthese curves as a base line, the dis-tributions of errors, arising from use ofEqs. (11) and (12), are plotted for the 48columns of Fig. 15 for each load andslenderness ratio, i.e., the distributionof errors is shown relative to the longcolumn moment in each case, but, toput the situation in perspective, thelong column moments are shown in aschematic relationship to each other. Itwill be noted that:

1. There is considerable scatter, ex-tending to the unconservative side, atvery low loads (P/P, = 0.02). This ap-parently reflects the influence of sec-tion shape, amount, and distribution ofprestressing steel when the load is al-most pure bending. The easiest way toavoid this problem would be to basedesign on a minimum axial load, in ad-dition to prestress, of 0.05 P0.

2. At an Llr ratio of 150, the resultshave become erratic.

3. At high axial loads, the results tendto separate into two groups, which arefound to represent the flanged andnon-flanged sections. This is not signif-icant, as the minimum eccentricity re-quired by codes precludes design inthis region.

4. Between these regions the distri-bution of errors is about as good as can

68

0.7

0.6

0.5

0.4

Qo

a

0.3

0.2

0,1

0

Fig. 16. Schematic representation of load-moment curves showing distribution of,results obtained by proposed modification to ACI formula. Slenderness ratios of25 and 100.

be hoped for from an empirical proce-dure which does not account in detailfor all the parameters.

Figs. 19 and 20 show possible designaids giving the value of X in Eqs. (11)and (12). Figs. 21-24 show the results ofapplying Eqs. (11) and (12) to the

double-tee of Fig. 5 and to a rectangularload-bearing wall section.

It will be noted that A as given by Eq.(11) decreases as P/Pa increases and alsoL/r increases. This implies that the ef-fective rigidity El of a column crosssection increases as P/P0 increases or as

PCI JOURNAUMarch-April 1983 69

0.7

0.6

0.5

0.4

a0

a

0.3

0.2

0.l

0

Fig. 17. Schematic representation of load-moment curves showing distribution ofresults obtained by proposed modification to ACI formula. Slenderness ratios of50 and 125.

L/r increases, and these trends may de-serve comment.

The above can be explained by refer-ence to Fig. 25, which shows the mo-ment-curvature relationships at twodifferent loads for the square pre-

stressed concrete column of Fig. 2. Thecurve for the 500-kip load is higher andsteeper than that for the 100-kip load.(Note: 1 kip = 4.448 kN.) This kind ofrelationship has already been shown inFig. 9. But Fig. 25 also shows the range

70

0.7

0.6

0.5

0.4

a

0.3

0.2

0.1

0

Fig. 18. Schematic representation of load-moment curves showing distribution ofresults obtained by proposed modification to ACI formula. Slenderness ratios of75 and 150.

of curvature that was actually exhibitedby the two columns, with Llr equal to75 and 150, respectively, at the momentof instability, according to a rationalanalysis.

The shorter column naturally reached

higher bending moments, and thereforemoved further out along the moment-curvature relationship. The slopes ofthese curves are the effective rigiditiesEl. Following the work of Shanley, 9 thestability of the columns can be ex-

PCI JOURNAL/March -April 1983 71

■I■■I■

1000

800a_

00 600JJQ

a 400

200

8DT12Sh ory Cod PCI DESIGN HDBK

p.2-6

1

/S./

1 ^ /

_,//J /

/

COMPUTER SOLUTION

\ ----- PROPOSED MODIFIED ACI

0 20 40 60 80 100 120 140 160END MOMENT (KIPS-FT)

Fig. 21. Solution obtained by proposed modification to ACI formula compared withcomputer solution for a double-tee section. Slenderness ratios of 25, 75, and 125.

pected to be related to the tangentmodulus. Although they cannot be de-termined with great accuracy, values ofthe tangent modulus at the centerlineor mid-height of the column at the mo-ment of instability are given in Table 1.

The secant values of rigidity wouldclearly follow the same pattern as thosein Table 1, and they are seen to exhibitthe trends reflected in Eq. (11), in-creasing both with load and with slen-derness. However, it must be recalledthat Eq. (11) is not intended to give theactual values of rigidity, but rather togive artificial values leading to the fic-titious load path OJ of Fig. 14.

It is interesting to note the valuesgiven for the critical loads when thetangent rigidities of Table 1 are used

with the Euler formula. For the columnwith Llr = 75 they give 97.5 and 360kips (instead of 100 and 500 kips) andfor the column with Llr = 150, theygive 90.5 and 463 kips. (Note: 1 kip =4.448 kN.)

Again, following Shanley, one mightexpect them to give slightly low esti-

Table 1. Computer values of slope ofM/¢ curve at point reached at columnmid-height at time of instability (kip-in.).

Load L/r = 75 Llr = 150

100 kips 2.67 x 101 9.9 x 106

500 kips 9.85 x 106 50.74 x 106

Note: 1 kip-in. = 0.113 kN-m.

PCI JOURNAL/March-April 1983 73

8 DT 12Sh Orf C0^Umn PCI DEDSIGNSHDBK

--------------^

No

/ J2 /

J

200f J COMPUTER SOLUTION

/ ----PROPOSED_\` ' MODIFIED ACI

1000

U) 800a_

0 600JJ

Q 400

00 20 40 60 80 100 120 140 160

END MOMENT (KIPS-FT)

Fig. 22. Solution obtained by proposed modification to ACI formula compared withcomputer solution for a double-tee. section. Slenderness ratios of 50, 100, and 150.

mates of the actual loads at instability.The average tangent rigidity in the col-umn is somewhat higher than the valuereached at midheight.

RECOMMENDATIONSIt is recommended that the design of

prestressed concrete columns by meansof Eqs. (11) and (12) be permitted,subject to the limitations listed below.When these limitations cannot be met,some rational procedure must be sub-stituted, and use of a computer programsuch as that referred to above would beconsidered acceptable.Limitations:

1. The intial prestress in the concrete

should not exceed 600 psi (4.14 MPa).2. The prestressing steel should be

initially stressed to at least 50 percentof its ultimate strength.

3. The slenderness ratio Llr shouldnot exceed 150.

ACKNOWLEDGMENTSThe research reported herein was

funded by the Natural Sciences andEngineering Research Council ofCanada. Some work was done under aPCI Fellowship, and computing facili-ties were made available, during theauthor's sabbatical, by the University ofthe Witwatersrand, Johannesburg,South Africa.

74

48"

• . 4'

700 3— 2~4> STRANDS

f,= 5KSI f• „= 270KS1

600 ---------------`^^ Initial Prestress = 368psi

^ S

a_ 500Y

a 4000JJQ 300x4

• •

n

\ IL/r = 125 1

/ ,^ / COMPUTER SOLUTION

j .—.—PROPOSED MODIFIED ACI

0 5 10 15 20 25 30 35

END MOMENT(KIPS-FT)

Fig. 23. Solution obtained by proposed modification of ACI formula compared withcomputer solution for a rectangular wall section. Slenderness ratios of 25, 75, and 125.

REFERENCES

1. Nathan, N.D., "Slenderness of Pre-stressed Concrete Beam-Columns," PCIJOURNAL, V. 17, No. 6, November-December 1972, pp. 45-57.

2. Alcock, W. J., and Nathan, N. D., "Mo-ment Magnification Tests of PrestressedConcrete Columns," PCI JOURNAL, V.22, No. 4, July-August 1977, pp. 50-61.

3. MacGregor, J. G., "Stability of Multi-storyConcrete Buildings," ASCE-IABSEConference on Tall Buildings,Bethlehem, Pennsylvania, SoA — Rep23-3, Planning and Design of Tall Build-ings, Vol. III, 1972, pp. 517-536.

4. ACI Committee 318, `Building Code Re-quirements for Reinforced Concrete (ACI318-77)," American Concrete Institute,

Detroit, Michigan, 1977.5. Cranston, W. B., "Analysis and Design of

Reinforced Concrete Columns," ReportNo. 20, Cement and Concrete Associa-tion, London, 1972, 54 pp.

6. British Standards Institution, "Code ofPractice for the Structural Use of Con-crete (CP110)," Part 1, 1972.

7. Timoshenko, S. P., and Gere, J. M.,Theory of Elastic Stability, McGraw-Hill,New York, N.Y., 1961.

8. Aroni, S., "Slender Prestressed ConcreteColumns," Report No. 67-10, Structuresand Material Research, Department ofCivil Engineering, University of Califor-nia, Berkeley, May 1967, pp. 231.

9. Shanley, F. R., "Inelastic ColumnTheory," Journal of Aeronautical Sci-ences, V. 4, No. 5, May 1947.

PCI JOURNAUMarch-April 1983 75

700

600

U) 500

Y

0 400JJQX 300Q

I••

0

48

• 4

3-,STRANDSf^= 5KSI. , fs^' 27OKSI

Initial Prestress=368 psi

Sti

1

! O

! COMPUTER SOLUTION

/ ----- PROPOSED

'^ I MOOIFIE ACI

0 5 10 15 20 25 30 35

END MOMENT (KIPS-FT)

Fig. 24. Solution obtained by proposed modification to ACI formula compared withcomputer solution for a rectangular wall section. Slenderness ratios of 50, 100 and 150.

APPENDIX - NOTATIONE = modulus of elasticity Pb = balanced loadE, = modulus of elasticity of concrete PC,.;, = critical load

El = flexural rigidity Pn = design load

h = overall depth of section P° = pure axial load capacity

I = moment of inertia of cross section r = radius of gyration of cross sectionA = maximum deflection

I, = moment of inertia of concrete = primary deflectioncross section E, = strain in concrete at crushingIa = moment of inertia of gross cross Ev = yield strain in steelsection = function ofP/P° defined in Eq. (11)

K = quantity defined in Eq. (4); used 0 = function ofLlr defined in Eq. (11)in CP 110 A = function of 0 and q, defined in

L = effective length Eq. (11), bywhichE^Ia is dividedM„ = design moment to get effective rigidityM° = primary moment ¢ = curvatureP = load Oti = ultimate curvature

76

800g00 KIPS

700 16v

600 ^^ P - 100KIPS

.1^

LL 500 o ^l^

U)a_h0

400 h v^^zw202 300

200

O0 0.0002 0.0004 0.0006

CURVATURE (INS-')

Fig. 25. Curvatures at instability in the square prestressed concrete column of Fig. 4.

NOTE: Discussion of this paper is invited. Please submityour discussion to PCI Headquarters by Nov. 1, 1983.

PCI JOURNALJMarch-April 1983 77