Flexural Strength and Ductility Analysis of Spirally Reinforced Prestressed Concrete Piles

Pam Hoat Joen Design Engineer

Beca Carter Hollings and Ferner Consulting Engineers

Auckland, New Zealand

Robert Park Professor and Head of Civil Engineering University of Canterbury Christchurch, New Zealand

A method to analyze the flexural strength and ductility of prestressed concrete piles contain­ing spiral reinforcement is presented. The meth­od is based on moment-curvature analysis. Account is taken of the increase in flexural strength and ductility of the pile due to con­crete confinement from the spirals. The meth­od is applicable to all sections of the pile. In the end regions of the pile, allowance is made for the possibility that the stresses in the pre­tensioned strands may not have fully devel­oped. Design tables and charts for the en­hanced flexural strength, available ultimate curvature and curvature ductility factor are derived for prestressed concrete piles with 400 mm (15. 7 in.) octagonal cross sections and with various spiral contents and axial load levels.

64

D uring severe earthquakes, piles are subjected to large cyclic lateral loads as well as vertical loads from the structure. These earthquake-induced

lateral loads and resulting soil deformations may cause large curvatures in the critical regions of piles.'-4 The regions of the pile where high curvatures occur need to be designed to possess adequate ductility, where ductility may be defined as the ability to undergo large amplitude cyclic deformations in the post-elastic range without a sig­nificant reduction in strength.

The maximum induced curvatures in the pile will depend on the properties of the soil layers, the moment­curvature relations of the pile and the characteristics of the earthquake motions. Soil-pile interaction analyses have been conducted. For example, Banerjee et al.' determined that the curvature demand is especially high at the inter­face between stiff and soft layers of soils. They predicted curvatures at such interfaces of up to 0.0087 radians/m (0.22 x 10-3 radians/in.) for extreme soil and seismic con­ditions. It is evident that, at pile-pile cap connections, the curvature demand could be significantly higher than that value.

It is important for designers to be able to calculate the required amount of transverse confining reinforcement necessary in prestressed concrete piles in order to achieve adequate curvature ductility. Also, it is known that the presence of transverse confining reinforcement enhances the flexural strength of prestressed concrete piles. This ef­fect is not considered in the current code approach for the calculation of flexural strength. Hence, an approach which includes the effect of confinement on the flexural strength of piles would lead to savings in the quantity of lon­gitudinal steel required.

An approach for the seismic design of reinforced con­crete columns for adequate flexural strength and ductility has already been deriveds-7 based on moment-curvature analyses. It should be noted that the moment-curvature relationship for a structural concrete section gives infor­mation on the flexural stiffness and the moment of resis­tance of the section at all curvature levels. Therefore, it enables determination of the flexural strength and avail­able ductility of the section.

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Previous paperss.9 have described tests and theoretical analyses conducted in New Zealand to obtain information on the flexural strength and ductility of prestressed con­crete piles. The piles were subjected to compressive load­ing and cyclic lateral loading which simulated severe earthquake loading. The piles had 400 mm (15.7 in.) oc­tagonal cross sections and contained longitudinal preten­sioned strands, longitudinal nonprestressed reinforcement in some cases, and spiral reinforcement.

This paper presents a theoretical method for determin­ing the flexural strength and ductility of prestressed con­crete piles containing spiral reinforcement. The method takes into account the effect of possible inadequate anchorage of the pretensioned strands in the development regions at the ends of piles. The method is based on monotonic moment-curvature analysis. Design tables and charts are derived for a range of cases of prestressed con­crete piles with 400 mm ( 15.7 in.) octagonal cross sections and with various axial load levels.

The objective is to present a method which enables designers to determine directly the amount of confining spiral reinforcement required to achieve a particular cur­vature ductility level and the associated flexural strength. The results summarized in this paper are reported in more detail elsewhere.w

STRESS-STRAIN MODELS FOR THE MATERIALS

In order to conduct moment-curvature analyses, it is necessary to establish models for the stress-strain be­havior of the materials.

Stress-Strain Model for Concrete

Several investigators, for example Iyengar et al., 11 Park and Leslie,l2 Shah et a1., 13 and Mander et al.,•4 have proposed stress-strain relationships for concrete subjected to concentric compression and confmed by spiral rein­forcement.

The model used here for spirally confmed concrete is that proposed by Mander et al. 14 and is illustrated in Fig. 1.

~

V) V)

~ (/)

c: ~ fo V)

t ~

-8

I I I I I

Confined concrete

~Cover concrete / 1 (linear relation)

I f.' E:co 2£co E:spa/1 t

Compression Strain , E:c

Fig. 1. Confined and unconfined concrete stress-strain curve.14

July-August 1990

The whole range of the monotonic compressive stress­stress relation is defined by:

(1)

where 1c; = compressive strength of spirally confined con­

crete (peak stress reached in the stress-strain curve)

and the concrete strain ratio is:

(2)

where Ecc = longitudinal concrete strain corresponding to

stressfc Ecc = longitudinal concrete strain corresponding to

peak stress fc~ and the concrete modulus parameter is:

(3)

in which the modulus of elasticity of concrete is:

Ec =4700 fi: (4)

where t: is in MPa (1 MPa = 145 psi). The concrete secant modulus is:

(5)

The compressive strength of spirally confined concrete is given as:

f~ =J: { 2.254-v I+ 7.94 ~- 2~- 1.254} (6)

which assumes that the compressive strength of uncon­fined concrete is given by the cylinder strength t:. The ef­fective lateral confining pressure J,. due to the spiral in Eq. (6) is given by:

(7)

where k, is the confinement effectiveness coefficient. This coefficient can be expressed as the ratio of effectively con­fined core area to core area of concrete section:

s' 1 -0.5([

k = s e 1- P ee (8)

in which Ps = ratio of volume of spiral reinforcement to

volume of concrete core measured to centerline of spiral bars

hh = yield strength of spiral reinforcing steel s' = clear vertical spacing between spiral bars

65

ds diameter of spiral measured to centerline of spiral bars

Pee = ratio of total area of longitudinal steel to total area of concrete core measured to centerline of spiral bars

Eq. (5) was determined from a model for the ultimate strength failure surface for concrete subjected to triaxial compressive stresses. 14 Eq. (7) takes into account the reduction due to arching of concrete between the spiral bars (see Fig. 2).

Cover

Spira£.~oncref: }

bar ds

Core -" concrete ~

(a) Confinement stresses

I A-A I

Effectively confined core

(b) Confinement from spiral reinforcement

Fig. 2. Confinement of concrete core by spiral reinforcement. 14

The longitudinal concrete compressive strain whenf/c is reached is given by:

(9)

where

66

Eco = longitudinal concrete compressive strain when compressive strength f/ is reached by uncon­fined concrete

R = ratio of increase in strain at peak stress to in­crease in strength at peak stress due to confine­ment

The parameter R is an experimentally determined valuei4 which can vary from 3 to 6.

fr/f(; 0.095

----0.075

0.057

------0.038

LONGITUDINAL STRAIN

(a) For various effective lateral confining pressures

0 ~ 72 ;;; 64

t;l56 g: 48

"' -.J 40 ~ 32

3 24 ~ 16 i'5 8

<co= 0.002 fr/f(; = 0.095 f(; = 45MPa IMPa = 145psi

-.J 0 o~~~a~o,~L_o~.o~2~~ao~3~L_o~.o~4~~a~o5~~o~.06~~--LONGITUDINAL STRAIN

(b) For various values of R

Fig. 3. Examples of stress-strain curves for spirally confined concrete given by model of Mander et al. 14

Fig. 3a shows monotonic stress-strain curves for con­crete given by the above stress-strain model for various ratios of the effective lateral confining pressure to the un­confined cylinder strength, with the other variables held constant. It is evident that a substantial increase in the strength and ductility of the concrete occurs with an in­crease in confinement. Fig. 3b shows monotonic stress­strain curves for concrete for various values of R with the other variables held constant. As R decreases, the slope of the ascending and descending branches increases, but the strain at peak stress decreases.

To define the stress-strain behavior of unconfined con­crete and the cover concrete,Je~ and Ece are replaced by f/ and Eeo. respectively, and also the part of the falling branch in the region when I Ee I > I 2Eeo I is assumed to be a straight line which reaches zero stress at the spalling strain Espall (see Fig. 1).

In tension, a linear stress-strain relation is assumed provided that the concrete tensile strength// has not been exceeded. The longitudinal concrete stress whenfc </(is given by:

(10)

Note that once /(has been exceeded,fc = 0. Relationships for the cyclic stress-strain behavior have

also been derived by Mander et al.I4 It was assumed that the monotonic stress-strain curve forms an envelope for the cyclic stress-strain branches.

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Stress-Strain Model for Nonprestressed Reinforcing Steel

Several investigators have proposed stress-strain relationships for nonprestressed reinforcing steeLs The model used here is that proposed by Mander et al.Is The monotonic stress-strain curve is assumed to consist of three branches as shown in Fig. 4, for which the relations between the steel stress..fs and steel strain Es are:

For 0 ~ Es ~ Ey :

where

and

/s=Es Es

Es = modulus of elasticity h = yield strength

For Ey < Es ~ Esh: ..fs = h

where Esh = strain at commencement of strain hardening

where Esu = strain at ultimate (peak) strength !su = ultimate strength

and

p = Esh ( Esu - Esh l !s.-/y)

Esh = strain hardening modulus at Esh

For Es > Esu: ..fs = 0

(11)

(12)

(13)

(14)

(15)

(16)

The values of the parameters defining the above three regions may be different for tension and compression loading.

Ey Esh

Steel Strain , E5

Esu

E­pu

It is assumed that the transverse reinforcement present will prevent premature buckling of the longitudinal non­prestressed reinforcing steel.

When the load is reversed, the stress-strain curve will form a softened curved branch without a well defined yield point due to the Bauschinger effect. Relationships for the cyclic stress-strain behavior have also been derived by Mander et al. 1s It was assumed that the monotonic curve forms an envelope curve for the cyclic stress-strain branches, but with a readjusted origin to account for cyclic loading history.

Stress-Strain Model for Prestressing Steel

Stress-strain models for prestressing steel have been proposed by Blakeley et al.,16 Thompson et al.,17 Menegot­to et aJ.Is and others. The model used here is that proposed by Blakeley, Thompson et al. The monotonic stress-strain curve in tension and compression are each assumed to consist of three branches as shown in Fig. 5, comprising two straight lines and a transition curve. The relations be­tween the stressfp and strain Ep are:

Tension branch: For 0 ~ Ep ~ Epa :

where Epa = strain at end of elastic range in tension Ep = modulus of elasticity

w For Epa < Ep ~ Epb: /p = - + Y

Ep

where

(17)

(18)

Epb = strain at end of curved transition range in tension

and

W = Epa Epb (/pa - /pb)

Epb- Epa

y = /pb Epb- /pa Epa

epb- epa

(19)

(20)

Epu Ep

Fig. 4. Stress-strain curve for nonprestressed reinforcing steel. Fig. 5. Stress-strain curve for prestressing steel.

July-August 1990 67

where ha = stress at strain of Epa

hb = stress at strain of Epb

ForEpb < Ep ~ Epu :/p =/pb + U (Ep- Epb)

U = hu - hb Epu- Epb

For Ep > Epu: h=O

Compression branch:

(21)

(22)

(23)

Prestressing strands are more likely than reinforcing bars to buckle in compression, due to their higher stress levels and smaller diameters. The assumptions made to model the stress-strain curve in compression are:

1. The value of the ultimate compressive strain Ep;; is equal to its tensile counterpart, Epu·

2. The value of the buckling stress, Sb , is equal to one­half of the ultimate tensile stress,hu· That is:

(24)

This assumption is close to that of Blakely et al.,16 who assumed that the buckling of the tendon in compression starts at a strain of 0.005, which results in a buckling stress of about one-half ofhu .

3. The transition curve is identical in length and shape with the one in the tension region. This results in (see Fig. 5):

Therefore:

Therefore:

where Era = strain at end of elastic range in compression Eb = strain at buckling in compression f Pti = stress at strain of Era

(25)

(26)

Note that once the buckling strain, Eb , has been ex­ceeded, the tendon stress is assumed to remain constant at the buckling stress Sb until the ultimate compressive strain Ep;; is reached. The tendon stress becomes zero at Ep;; and beyond.

4. The linear elastic part of this branch has the same slope as that of the tensile region up to the point (Era ,f ra ). Therefore, the strain Era can be calculated if the linear slope is available and thus Eb can be determined.

These assumptions lead to:

For Era :5: Ep :5: 0 :

For Eb < Ep :5: Epa :

68

w -fp+-+Y

Ep

(27)

(28)

where

where

ForEp ~ Ep;;:

W = Epa Eb ifea - sb)

Eb- Epa

fp=O

(29)

(30)

(31)

(32)

For cyclic loading, curved loops can be used to trace the loading and unloading stress-strain paths. Relationships for cyclic stress-strain loops using the Ramberg-Osgood function have been derived by Thompson et al,17 assuming that the monotonic curve with origin adjusted to account for cyclic loading history forms an envelope curve for the cyclic stress-strain branches.

Stress Limit for Prestressing Steel in End Regions of Members as a Function of the Transfer and Development Lengths

The stress in a pretensioned tendon in the end region of a pile depends on its distance from the end of the p~le. and on the bond conditions. For a pile in flexure, the hm1t of tendon stress hum that can be developed at a section de­pends on a number of variables. These variables are t~e stress in the tendon due to prestress alone fs., the maxi­mum possible stress in the tendon at the flexural strength • the transfer length of the tendon 11 and the development Jps'

length ofthe tendon /d. In the analysis, it is assumed that the relationship be­

tween the limit of the tendon stress hum at any section and the distance from the end of the pile are linear, as shown in Fig. 6. At any section within the region of the develop-

fps

fp(im

Pretensioned tendon,

1 Not prestressed

I I

res tressed 1 I I I I I

Fig. 6. Variation in stress limit in a pretensioned steel tendon in end region of a pile.

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ment length ld, the stress in the tendonJ;,um will be less than J;,s since some slip of the tendon occurs. Hence, if the stress in the tendon obtained from the compatibility of strains with the adjacent concrete in a flexural strength analysis is greater than J;,um• only J;,um can be sustained. Tendons at sections within the pile beyond the development length are capable of attaining the stressJ;,s.

For tendons which are prestressed, the variation in ten­don stress will be assumed to be in three regions as shown in Fig. 6. The tendon stress at distance lx from the end of the pile is given as follows:

(33)

(35)

For tendons which are not prestressed, it will be as­sumed that the variation in tendon stress is in two regions as shown in Fig. 6. The tendon stress at distance lx from the end of the pile is given as follows:

(36)

(37)

According to the commentaries of the ACI Building Code'9 and the New Zealand Concrete Design Code,20 for 3- and 7 -wire strand the transfer length is given by:

It= [~i )db (38)

• 12.5mm (Jin J dia. tendon.

and the development length is given by:

2 fd = (J;,s - 3 fse ) db /7 (39)

where f., andJ;,s are in MPa (1 MPa = 145 psi) and db is the nominal diameter of the strand. As an approximation, It = 50 db and ld = 140 db will be used.

In recent years, several investigators have proposed changing the above equations for It and ld , which were de­termined by tests in the 1950s. For example, Martin and Scott2' have proposed more conservative equations. How­ever, in piles containing significant amounts of spiral con­fining reinforcement, the bond conditions are improved and Eqs. (38) and (39) should be conservative enough.

MOMENT-CURVATURE ANALYSIS General Assumptions

Monotonic moment-curvature relations for octagonal­shaped prestressed concrete piles subjected to combined axial load and flexure were obtained assuming that plane sections remain plane after bending and using the stress­strain relationships for concrete, longitudinal nonpre­stressed reinforcing steel and longitudinal prestressing steel described by Eqs. (1) to (39). Perfect bond between the longitudinal steel and the concrete was assumed, ex­cept within the development lengths of the steel in the end regions of the pile. In the end regions, when the available development length for the prestressed or nonprestressed longitudinal steel was smaller than the specified develop­ment length, some slip was considered to occur [as per Eqs. (33) to (39) for prestressing steel].

For the purpose of analysis, the cross section of the pile was divided into a number of discrete laminas (strips). To calculate a point on the moment-curvature curve, the ex­treme fiber concrete compressive strain, eT, was specified

o 20 mm (0.79 in) dia. deformed bar Concrete cover to tendons = 40mm (1.57in)

Core concrete

400mm (15.7in)

Section with only prestressed long itudina I steel

Section with both prestressed and non prestressed longitudinal steel

f ___ _ Cover concrete

Strain distribution

Concrete stress distribution

Fig. 7. Cross sections, strain and stress distribution for theoretical analysis of piles.

July-August 1990 69

(see Fig. 7). Then, an iterative procedure was used to de­termine the neutral axis depth at which the resultant inter­nal force was equal to the external axial load. The extreme fiber concrete compressive strain was varied over a range of strains to trace out the moment-curvature relationship until failure occurred.

The equations used in the moment-curvature analysis were:

where P = resultant internal axial force fc; = stress in ith concrete strip (positive if compres­

sion) Aci = area of ith concrete strip hi = stress injth prestressing tendon (positive if com­

pression) Apj area of jth prestressing tendon !sk = stress in kth nonprestressed longitudinal steel bar

(positive if compression) Ask = area of kth nonprestressed longitudinal steel bar nc = number of effective concrete strips np = number of prestressing tendons ns = number of nonprestressed longitudinal steel bars The neutral axis depth at which P is equal to the external

axial load Pe was calculated using the Newton-Raphson method. The corresponding section bending moment and curvature were given by:

where M = G> = Er = c = Yci =

YPi =

Ysk =

Er G>=-c (42)

bending moment curvature extreme fiber concrete compressive strain neutral axial depth distance from centroid of ith concrete strip to centroid of pile section (positive if it lies above centroid of pile section) distance from the centroid of the jth prestressing tendon to the centroid of pile section (positive if it lies above centroid of pile section) distance from centroid of kth nonprestressed lon­gitudinal steel bar to centroid of pile sec!"ion (positive if it lies above centroid of pile section)

Two Types of Monotonic Moment-Curvature Relations

The large number of variables means that the mon­otonic moment-curvature relations of spirally confined prestressed concrete piles can have a range of shapes. Fig. 8 shows two cases that are commonly found.

70

MOMENT.M

(/)

---..!.2) I --"I II obtained by I I using Method 0 I I

II II I I

¢u,2 ¢uJ

CURVATURE, ¢

Fig. 8. Moment-curvature relations showing ideal moments, yield moments, yield curvatures and ultimate curvatures.

The first moment-curvature curve in Fig. 8, indicated by the full line marked 1, represents a case where the com­pressive axial load level, the longitudinal reinforcement ratio and the volume of spiral reinforcement are all rela­tively large and the cover concrete thickness is relatively small. In this case, the moment of resistance does not re­duce significantly after spalling of the cover concrete and then increases considerably during the subsequent in­crease in curvature when the confinement takes effect. In this case, the maximum moment occurs in the region of curvature after the concrete cover spalls.

The second curve, indicated by the dashed line marked 2, represents a case where the compressive axial load level, the longitudinal reinforcement ratio and the volume of spiral reinforcement are all relatively low, and the cover concrete thickness is relatively large. In this case, the mo­ment of resistance reduces significantly after the spalling of the cover concrete and then increases only a small amount during the subsequent increase in curvature. In this case, the maximum moment occurs just before the concrete cover spalls.

Definition of Ideal Flexural Strength

The ideal flexural strength, M;, is defined as the maxi­mum moment of resistance reached in the moment-curva­ture curve. For a given pile, the peak moments before and after spalling of the concrete cover should be calculated and M; is taken as the greater of the two values.

Flexural strength before spalling of concrete cover -As a good approximation, M; for the curve marked 2 in Fig. 8 can be calculated by satisfying the requirements of strain compatibility and equilibrium and assuming a rec­tangular concrete compressive stress block with a mean stress of 0.85/: and an extreme fiber concrete compressive strain of 0.003 as recommended by the ACJ19 and SANZ20

codes. The measured stress-strain curves of the prestress­ing tendons and nonprestressed longitudinal reinforce­ment are also used and it is assumed that no slip of longitudinal reinforcement occurs. A strength reduction

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factor of <1> = 1.0 is also assumed. This ideal flexural strength will be referred to as MA-

Flexural strength after spalling of concrete cover -As a good approximation, M; for the curve marked 1 in

Fig. 8 can be calculated by satisfying the requirements of strain compatibility and equilibrium and using a concrete compressive stress distribution for the spirally confined core concrete obtained from the stress-strain curve pro­posed by Mander et al.t4 The concrete compressive strain at the extreme fiber of the core concrete E,m can be esti­mated as that at 0.8 1c; on the falling branch of the stress­strain curve, where 1c; is the compressive strength of spirally confined concrete. The measured stress-strain curves of the prestressing tendons and nonprestressed lon­gitudinal reinforcement are also used. Allowance is made for any slip of longitudinal steel which may occur in end regions. Again, a strength reduction factor of <1> = 1 is as­sumed. This ideal flexural strength will be referred to as Mv.

The magnitude of E,m to be used to calculate M v depends

2.0

1. 8

ffc 1.6

7T 1.4

1.2

1.0 0

2.0

1.8

ffc 1.6

f~ 1.4

1.2

1.0 0

1.8

f~, 1.6

f~ 1.4

1.2

1.0 0

f~ (fviPa) 40 35 30

0.02 0.04 Ecm

50 40 f~ = 55 145/ 35 30 fviPa

I R=41

0.02 0.04 0.06E

I R=5l

0.02 0.04 0.06

em

Ecm

Fig. 9. Concrete compression strains Ecm corresponding to a compressive stress of 0.8 f:C on falling branch of stress-strain curve for confined concrete.

July-August 1990

on many variables such as E,0 ,f:, R,J,. and the section con­figuration. Values of E,m found corresponding to a con­

crete compressive stress of 0.8.fc; on the falling branch of

the stress-strain curve for spirally confined concretew are shown in Fig. 9. It is evident that E,m is high iff; is low, R is high and.fc; I J:is high.

It should be noted that E,m as defined here is an approxi­mation for the concrete compressive strain at the extreme fiber of the concrete core at the peak moment of the mo­ment-curvature relation after spalling of the concrete cover. In fact, M0 calculated with E,m as defined here will not coincide exactly with the peak moment. For example, the theoretical monotonic moment-curvature relations cal­culated for six prestressed concrete piles with 400 mm (15.7 in.) octagonal cross sections gave peak moments which were on average 4 percent higher than the moment M0 calculated at E,m as defined above. 10•4 However, simpli­fying assumptions for the maximum concrete compres­sion strain at the extreme fiber of the core will always result in a moment which is equal to or less than the actual peak moment of the moment -curvature relation.

Definition of Yield Curvature and Yield Moment

The defmition of the yield curvature, <J>Y, is illustrated on the moment-curvature relations in Fig. 8. Following the custom in New Zealand,10.6 <J>Y is defined as the curvature at the point of intersection of the horizontal line passing through the ideal flexural strength M; and a straight line joining the origin with the point (<J>;, My) on the moment­curvature relation, where My is the yield moment. There­fore:

(43)

This definition for the yield curvature, <J>Y' is that for an equivalent elasto-plastic moment-curvature relation which has the elastic stiffness measured as the secant value at the yield moment My.

This definition for <J>Y is very dependent on the definition for the yield moment My. Several definitions for My were considered.w It was concluded10 that assuming My = 0.75M; in Eq. (43) results in the most consistent definition of yield curvature. That assumption is used in this study.

Definition of Ultimate Curvatures

The ultimate curvature, <J>u, is defined as the curvature attained by the pile when the performance of the cross sec­tion is considered to have reached an unsatisfactory condi­tion. The ultimate curvature is assumed to have been attained if one of the following limit conditions applies:

(a) The moment capacity has dropped to 80 percent of the peak moment of the moment-curvature relation be­yond the point of peak moment.

(b) The total strain in the prestressing tendon closest to the extreme tension fiber has exceeded 90 percent of the ultimate tensile strain of the tendon, Epu·

71

(c) The total strain in the prestressing tendon closest to the extreme compression fiber has exceeded 70 percent of the ultimate compressive strain of the tendon, E/Ju.

(d) The strain in the nonprestressed longitudinal steel (if present) closest to the extreme tension fiber has ex­ceeded 90 percent of the ultimate tensile strain for the steel.

(e) The strain in the nonprestressed longitudinal steel (if present) closest to the extreme compression fiber has exceeded a maximum permissible compression strain.

(f) The strain in the spiral reinforcement has exceeded the fracture strain of that steel.

Limit condition (a) avoids excessive reduction in flex­ural strength. Limit conditions (b) and (c) appear reason­able in view of experimental results obtained from pile and pile-pile cap connection tests. 10.4 Also, Muguruma et al.J found that fracture of prestressing steel could govern the available ultimate curvature of piles if the fracture strain of that steel was relatively small. Limit conditions (d) and (e) are normally not critical. Limit condition (f) re­flects the condition when the strain energy capacity of the spiral steel is no longer adequate to accommodate the ex­pansion of the core concrete due to compression. Fracture of the spiral steel will occur when the strain energy ab­sorbed as a result of confinement has exceeded the strain energy capacity of the spiral reinforcement.'4 This limit condition will not be considered further in this study since fracture of spiral steel does not occur if that steel is ade­quately ductile.

Types of Moment-Curvature Analyses

In this study moment-curvature analyses for only monotonic loading were used, whereas earthquakes cause cyclic loading. However, reasonable agreement has been obtained between the theoretical monotonic moment­curvature relations and the measured experimental moment-curvature points at the peaks of the loading cycles obtained from tests on prestressed concrete piles and pile-pile cap connections subjected to simulated severe seismic (cyclic) loading.10.4 In those tests the exter­nal axial compressive load applied to the piles was 0.2 t: Ag, where Ag is the gross area of the pile cross sec­tion. The degradation of flexural strength which occurs with cyclic loading may be greater when the external axial load level is higher.

Moment-curvature theory for structural concrete mem­bers subjected to cyclic loading has been derived by many investigators in the past. Such theory has been based on analytical stress-strain relations for the steel and concrete subjected to cyclic loading. For example, Blakeley et al.,'6 Thompson et al., 17 Whittaker et at.zz and Pam et al.JO have obtained reasonable agreement between theoretically derived and experimentally measured cyclic moment­curvature responses of prestressed and partially pre­stressed concrete members.

As an example, Fig. 10 shows a comparison between the experimental and theoretical lateral load versus lateral displacement hysteresis loops for a prestressed concrete pile obtained by Pam et al.JO The theoretical displacements

72

jUNIT Pt.j

"450

.:::

D FLrst\'!soblecrushmgol

]_ -~

1 Sp~ral bar fractured

(a) Experimental curves

-450

(b) Theoretical curves

Fig. 1 0. Lateral load-lateral displacement hysteresis loops for pile unit P4.

in Fig. 1 Ob were obtained by integrating the theoretical curvatures to obtain displacements. The details of this prestressed concrete pile (Unit P4) are given elsewhere. lOA

Unit P4 had a 400 mm (15.7 in.) octagonal cross section, was pretensioned by ten 12.5 mm (1!2 in.) diameter 7-wire strands, and had an applied external axial load level of 0.3 t: Ag while subjected to cyclic lateral loading. The pile was well confined by spiral reinforcement with Ps = 1.75 percent and/yh = 630 MPa (91.4 ksi). It is evident that the pile withstood a considerable number of cycles in the post­elastic range with little loss of strength and stiffness.

The agreement between the experimental and theoreti­calloops of Figs. 1 Oa and lOb is reasonable. The reduction in the laterat load capacity with increasing displacement is mainly due to the P d effect of the axial loading rather than to a reduction in the flexural strength. On the experimental curves of Fig. lOa are superimposed inclined lines which correspond to the lateral loads obtained when the theoreti­cal flexural strengths M0 (with R = 3) and MA are reached at the critical section of the pile, allowing for the Pd ef­fect.

As concluded previously,w.4 it will be assumed that monotonic moment-curvature theory will give a reason­able prediction of the envelope curve for cyclically loaded piles.

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RESULTS OBTAINED FROM MONOTONIC MOMENT-CURVATURE

ANALYSES

Pile Section and Material Properties

Piles with the section properties shown in Fig. 7 were analyzed. The pile cross section was 400 mm (15.7 in.) oc­tagonal and the cover to the spiral reinforcement was 30 mm (1.18 in.). The piles contained ten 12.5 mm (1,.-2 in.) diameter 7-wire strands and sometimes ten 20 mm (0.79 in.) diameter nonprestressed longitudinal bars distributed between the tendons.

For the concrete fco = 0.002, R = 3, Espall = 0.006 and!/= 0.11: were assumed in the stress-strain Eqs. (1) to (10). It was also assumed that the compressive strength of a stan­dard concrete cylinder t: and the compressive strength of the concrete in the pile are identical.

For the nonprestressed reinforcing steel, it was assumed thatJ;, = 275 MPa (40 ksi) andEs= 204,000 MPa (29,600 ksi) in the stress-strain Eqs. (11) to (16). In addition, for the branch curve in tension it was assumed that Esh = 0.022, Esh = 4900 MPa (711 ksi), Esu = 0.200 andfsu = 420 MPa (60.9 ksi). For the branch curve in compression, it was as­sumed that fsh = -0.012, Esh = 6870 MPa (996 ksi), fsu = -0.070 andfsu = -400 MPa (58.0 ksi). These values for the parameters of the branch curves were established by Zahn23 for Grade 275 reinforcement.

For the prestressing steel the values of the parameters assumed in the analysis and in stress-strain Eqs. (17) to (32) are listed in Table 1. The basic properties listed were based on actual stress-strain curves for Japanese 7-wire strand used in pile tests.Io,4

Influence of Axial Load Level on Available Flexural Strength and Curvature Ductility Factor for Unconfined and Confined Piles

It has been demonstrated experimentallyi0.4 that ductile behavior during severe cyclic loading is achieved by pre­stressed concrete piles containing the following volu­metric ratio of spiral confining reinforcement in the plastic hinge regions:

Ps =0.45 (A_l) t:{o.5 + 1.25

( ~· +&, )} lAc J;,h $ lf;4g Jc (44)

or

whichever is greater, where Ps = volume of spiral steel to volume of concrete core measured to outside of spiral, Ag = gross area of pile cross section, Ac = area of concrete core of pile cross section measured to outside of spiral,/;= concrete compressive cylinder strength, /.;h = yield

July-August 1990

Table 1. Values assumed for parameters for 12.5 mm (11.2 in.) diameter 7-wire steel strand.

Stress (MPa) Strain Other parameters* (MPa)

/pa = 1242 Epa = 0.0064 Ep = 194000 fpa= -642* Ejiii = -0.0033* u =· 6122

/pb = 1500 Epb = 0.016 W= -2.752 -Sb = -900 Eb = -0.0129* W= -1.144

/pu = 1800 Epu = 0.065 y = 1672 fse = 1080 Ejjii = -0.065 y = -988.7

Note . .fse effecbve stress m prestressmg steel after all losses. Vanables marked* were obtained by calculation from the corresponding equations. Conversion factor: 1 MPa = 145 psi

strength of spiral steel, P, =external compressive load on pile, G> =strength reduction factor, and/cp =compressive stress in concrete due to prestress. Eqs. (44) and (45) are in accordance with the provisions for reinforced concrete in the New Zealand concrete design codezo except that the axial load due to prestress is added to the external axial load level. Eqs. (44) and (45) indicate that the quantity of spiral reinforcement increases as the axial load level in­creases.

Fig. 11 illustrates the theoretical variation in the avail­able ideal flexural strength M; and the curvature ductility factor G>u IG>Y' with axial load level, for piles with confined and unconfined cross sections. The properties of the piles are shown in the figure. The confined pile has a level of confmement expressed by /,.If:= 0.12, where/,.= effective lateral confinement pressure as given by Eqs. (7) and (8). For the pile properties assumed,/,. It:= 0.12 is equivalent to 185, 137, 108 and 90 percent of that required by Eq. ( 44) for P, It: Ag = 0, 0.2, 0.4 and 0.6, respectively, assum­ing G>= 1.0.

In Fig. 10, two curves for unconfined piles are pre­sented. One curve (M A) is calculated by satisfying the re­quirements of strain compatibility and equilibrium and using the concrete compressive stress block and extreme

•12-Smm r;m) d1a. tendon Prop_erlies ·- I(; ~ 45MPo 16.530psil

R ~ 3 fcp ~ 7.6MPo(1.100pS1) Pst : 0 ~~SD~E~ 8~ ~~

"'t~ <"0)~

F/, t'Ag

tx/td ~ 1.0

1.0r----

-- con:med. } Mander ef a/. ffr/fc: 0.12) stress-strain

_---Unconfined relatron for 0.8 (ff/fC=OI concrete

-·-Uncont;ned,ACI concrete

7 I I I I 1 o.4 I I

I o.2 I

I I

1kNm~0.737

kip II

40 30 20 10 0 75 150 225 300

1\J/~y M,lkNml

Fig. 11. Variation in available flexural strength and curvature ductility factor for confined and unconfined prestressed concrete piles with various axial load levels.

73

fiber concrete compressive strain recommended by the ACJI9 and SANZ codes,2o the adopted stress-strain curves for the prestressing steel assuming no slip of steel, and a strength reduction factor <1> = 1.0. The second curve was calculated in the same manner, except that the concrete compressive stress block was obtained using the stress­strain curve for unconfined concrete due to Mander et al. 14

with a concrete compressive strain of Ecm at the outside of the tendon closest to the extreme compression fiber, where Ecm is the strain at a stress of 0.81: on the falling branch of the stress-strain curve. On this basis, the con­crete compressive strain at the extreme fiber varied from 0.00436 for zero external compressive load (only bend­ing) to 0.00297 for maximum external compressive load (zero bending).

As expected, Fig. 11 shows a large increase in flexural strength and curvature ductility factor due to confinement. The flexural strength increase is particularly high at large axial load levels, due to the greater dependence of the flex­ural strength on the contribution from the compressive force in the concrete at large axial loads. The ACI Code method is seen to give a reasonable indication of the flex­ural strength for unconfined piles.

Fig. 11 also shows that for a constant quantity of spiral reinforcement (as in the pile section analyzed), the avail­able curvature ductility factor, <l>u I<I>Y' reduces with in­crease in the level of external axial load. Hence, to obtain a more uniform available curvature ductility factor, more spiral reinforcement should be provided at high axial load levels, and less at low axial load levels, as recommended by Eqs. (44) and (45).

Rather than using in design semi-empirical equations, such as Eqs. ( 44) and ( 45), a more sophisticated design ap­proach would make the required quantity of the spiral re­inforcement a function of the external axial load level and the curvature ductility factor required for a particular pile. Also, advantage could be taken in design of the increase in flexural strength due to confmement.

e 2t.o <: -'1:

.... 160

~ ~

80

•12.5mm !fin) dia. tendon

f~ = t.SMPa (6530psi} R =3 Pst = 0

lx/(d = 1.0 fr/fr! = 0.09 (0

t· .~ ::·:=:·_· - ·::.__:::::~ .. ,--::=:::.-=::.· :::==:-.::.

i 1kNm= 0.737 . /,.-------------- kip ft

''Legend. ';{J,~~~-09 fcp/f~ Fiotal/f~Ag 1m = 3.28 ft.

0.2 0 0.2 0 0.2 0.2

0.1 0.1 0.2 o.t. o o.t. o.2 0.2 o.t. •

o.ot. o.o8 0.12 o.16 o.2o o.2t.-

CURVATURE,t/J lrad/m}

Fig. 12. Influence of levels of prestress and external axial compressive load on moment-curvature behavior of a confined prestressed concrete pile.

74

Influence of Prestress and External Axial Load Level on Moment-Curvature Behavior of Confined Piles

It is of interest to investigate the influence of the relative amounts of axial prestress and external compressive load on the moment-curvature behavior of prestressed concrete piles. Fig. 12 shows theoretical moment-curvature curves which were computed for pile cross sections having either zero prestress and some external axial load, or some pre­stress and some external axial load, or full prestress and no external axial load. The properties of the piles are given in the figure.

The moment-curvature relations of Fig. 12 illustrate that for pile sections having the same total (external plus prestress) axial load, the magnitude ofthe moment capac­ities is affected mainly by the component of external axial load. That is, the increase in moment capacity at large curvatures due to increase in external axial load is greater than that due to increase in prestressing force. This is be­cause the effect of external axial load is constant and al­ways present throughout the loading, whereas the effect of prestress reduces when the tendons are in the compression zone. This has also been observed in the experimental tests reported elsewhere.10.4 It is also observed from Fig. 12 that, for piles that have a high total axial load, there­placement of the external axial load by prestress results in piles having larger available ultimate curvatures.

Combinations of Variables Analyzed to Obtain Design Tables and Charts

The pile cross section with the previously described material properties were analyzed to obtain design tables and charts for combinations of the following variables, which were considered to be useful for the practical design of prestressed concrete piles and their connections to pile caps:

1. Effective lateral confining pressure level: /,.It: =

0.03, 0.06, 0.09 and 0.12. The value for/,. is given by Eqs. (7) and (8) and depends on the diameter, spacing and yield strength ofthe spiral bar.

2. Concrete compressive strength: t: = 30, 45 and 60 MPa (4350, 6530 and 8700 psi). A typical value for pre­stressed concrete piles is 45 MPa (6530 psi).

3. Compressive stress in concrete due to prestress:fcp = 0 and 7.6 MPa ( 1100 psi). All pile sections considered had ten 12.5 mm (V2 in.) diameter seven-wire strands. For a pile, fcP = 0 represents the case at a pile-pile cap connec­tion when the concrete at the end of the pile has been broken back to expose the tendons which are therefore embedded in a nonprestressed condition in the pile cap. For a pile, fcP = 7.6 MPa (1100 psi) represents ap­proximately the case within a pile where the effective stress in the tendons after losses in prestress is 0.6 h,u, where h,u is the ultimate tensile strength of the tendon.

4. The ratio of the available anchorage length to the re­quired development length of tendon: l)ld = 0.4, 0.6, 0.8 and 1.0. The lx /ld value will be less than 1.0 when the criti-

PCIJOURNAL

cal section of the pile is at a distance lx from the end of the tendon which is less than the development length of the tendon ld = 140 db, where db= tendon diameter. This may occur at the pile to pile-cap interface where the available development length of the tendon, prestressed or non­prestressed, is short due to space limitations in the pile cap.

5. Content of nonprestressed longitudinal reinforce­ment: Psr = 0 and 0.0237. The former case represents a cross section without nonprestressed longitudinal steel. The latter case is a section which contains ten 20 mm (0. 79 in.) diameter bars distributed between the tendons. This latter case, if combined with.fcP = 0, simulates a cross sec­tion at the pile-pile cap interface of a broken back pile which contains reinforcing steel as starter bars.

6. External compression axial load level: P, I j;A8 = 0,

0.2, 0.4 and 0.6. These load levels result from the gravity and earthquake loading on the structure above.

l~eal Flexural Strength, M1

A simple method to accurately determine the real flexural strength of a prestressed concrete pile taking into account the enhanced compressive stress-strain properties of spirally confined concrete, the measured stress-strain properties of the prestressing tendons and the non­prestressed longitudinal reinforcement, and allowing for possible slip of the tendons, would be to use a modifica­tion factor to modify the flexural strength calculated by the ACJt9 and SANZ20 code approaches. On this basis:

(46)

where Mi = ideal flexural strength as previously defined,

equals MA or M 0 , whichever is larger

K = flexural strength modification factor which depends on the pile variables

M A = flexural strength calculated by satisfying the re­quirements of strain compatibility and equi­librium and using the concrete compressive stress block and extreme fiber strain recommended in the ACJt9 and SANzzo codes, the adopted stress­strain curves for the prestressing steel and non­prestressed steel assuming no slip of that steel, and a strength reduction factor of 4> = 1.0.

K

2.2

2.0

1.8

1.6

1.4

1.2

f.' c

•12.5mm rfin) dia. tendon

= 45MPa (6530psi} fcp = 7.6MPa (1100psi} R = 3 Pst = 0

lx/ld = 1.0 .06

0.8 L.__.....__ _ _.__~---'-----''------'

0 0.1 0.2 0.3 0.4 0.5 0.6

Pe/fCAg

Fig. 13. Flexural strength modification factor Kfor a prestressed concrete pile with various levels of axial load and confinement.

Table 2. Flexural strength modification factor Kfor 400 mm (15.7 in.) octagonal piles having fcp = 7.6 MPa (11 00 psi) and Pst= 0.

Effective lateral confining stress level/,//;

Pe J: 0.12 0.09 -(MPa) f/Ag

lxlla lxlla

1.0 0.8 0.6 0.4 1.0 0.8

30 1.24 1.19 1.00* 1.00* 1.18 1.15 0 45 1.13 1.08 1.00* 1.00* 1.08 1.06

60 1.08 1.02 1.00* 1.00* 1.03 1.00

30 1.31 1.31 1.23 1.18 1.25 1.25 0.2 45 1.15 1.15 1.09 1.05 1.10 1.10

60 1.07 1.07 1.03 1.00* 1.02 1.02

30 1.67 1.67 1.63 !.55 !.56 1.56 0.4 45 1.36 1.36 1.33 1.28 1.26 1.26

60 1.22 1.22 1.21 1.17 1.13 1.13

30 3.74 3.74 3.70 3.60 3.27 3.27 0.6 45 2.17 2.17 2.16 2.11 1.89 1.89

60 1.76 1.76 1.75 1.72 1.51 1.51

Note: Factors marked • are obtained from MA; the remainder are obtained from Mo.

Conversion factor: I MPa = 145 psi.

July-August 1990

0.06

lxlla

0.6 0.4 1.0 0.8 0.6

1.04 1.00* 1.09 1.09 1.00* 1.00* 1.00* 1.01 1.01 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

1.19 1.14 1.15 1.15 1.11 1.05 1.01 1.01 1.01 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

!.53 1.46 1.35 1.35 1.34 1.24 1.20 1.09 1.09 1.09 1.12 1.08 1.00* 1.00* 1.00*

3.26 3.18 2.54 2.54 2.54 1.89 1.85 1.44 1.44 1.44 1.51 1.49 1.11 1.11 1.11

O.Q3

lxlla

0.4 1.0 0.8 0.6 0.4

1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

1.06 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

1.29 1.05 1.05 1.05 1.02 1.05 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

2.50 1.42 1.42 1.42 1.42 1.42 1.00* 1.00* 1.00* 1.00* 1.11 1.00* 1.00* 1.00* 1.00*

75

Table 3. Flexural strength modification factor Kfor 400 mm (15.7 in.) octagonal piles having fcp=O and Pst= 0.

Effective lateral confining stress levelf,./f:

Pe f/ 0.12 0.09 f/Ag (MPa)

lxlld lxlld

1.0 0.8 0.6 0.4 1.0 0.8

30 1.32 1.19 1.00* 1.00* 1.27 1.17 0 45 1.27 1.00* 1.00* 1.00* 1.22 1.12

60 1.23 1.00* 1.00* 1.00* 1.18 1.07

30 1.38 1.34 1.18 1.03 1.33 1.31 0.2 45 1.31 1.28 1.15 1.02 1.26 1.25

60 1.25 1.23 1.12 1.01 1.21 1.19

30 1.38 1.37 1.28 1.16 1.29 1.29 0.4 45 1.27 1.27 1.21 1.12 1.19 1.19

60 1.19 1.19 1.14 1.08 1.11 1.11

30 1.44 1.44 1.39 1.32 1.30 1.30 0.6 45 1.34 1.34 1.32 1.27 1.19 1.19

60 1.27 1.27 1.26 1.22 1.11 1.11

Note: Factors marked* are obtained from MA; the remainder are obtained from Mo.

Conversion factor: I MPa = 145 psi.

0.6

1.02 1.00* 1.00*

1.16 1.13 1.09

1.22 1.14 1.08

1.28 1.19 1.11

0.06 0.03

lxlld lxlld

0.4 1.0 0.8 0.6 0.4 1.0 0.8 0.6 0.4

1.00* 1.20 1.14 1.00 1.00* 1.14 1.11 1.00* 1.00* 1.00* 1.16 1.09 1.00* 1.00* 1.09 1.06 1.00* 1.00* 1.00* 1.12 1.05 1.00* 1.00* 1.05 1.02 1.00* 1.00*

1.00 1.26 1.25 1.12 1.00* 1.16 1.16 1.09 1.00* 1.00* 1.20 1.20 1.09 1.00* 1.11 1.11 1.05 1.00* 1.00* 1.14 1.14 1.06 1.00* 1.05 1.05 1.01 1.00*

1.12 1.17 1.17 1.12 1.05 1.03 1.03 1.03 1.00* 1.08 1.08 1.08 1.06 1.00 1.00* 1.00* 1.00* 1.00* 1.03 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

1.22 1.08 1.08 1.08 1.07 1.00* 1.00* 1.00* 1.00* 1.16 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.10 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

Table 4. Flexural strength modification factor Kfor 400 mm octagonal piles having fcp = 7.6 MPa (1100 psi) and Pst = 0.0237.

Effective lateral confining stress level/,.//:

P, f( 0.12 0.09 f/Ag (MPa)

lxlld lxlld

1.0 0.8 0.6 0.4 1.0 0.8

30 1.22 1.21 1.15 1.14 1.16 1.16 0 45 1.15 1.13 1.06 1.05 1.09 1.08

60 1.09 1.07 1.01 1.00 1.03 1.03

30 1.23 1.23 1.22 1.19 1.22 1.22 0.2 45 1.17 1.17 1.13 l.lO 1.12 1.12

60 1.11 1.11 1.08 1.05 1.07 1.07

30 1.50 1.50 1.47 1.43 1.42 1.42 0.4 45 1.35 1.35 1.33 1.29 1.28 1.28

60 1.27 1.27 1.26 1.22 1.19 1.19

30 1.99 1.99 1.97 1.93 1.81 1.81 0.6 45 1.72 1.72 1.71 1.68 1.54 1.54

60 1.57 1.57 1.57 1.54 1.38 1.38

Note: Factors marked • are obtained from MA; the remainder are obtained from Mo.

Conversion factor: I MPa = 145 psi.

0.6

1.09 1.02 1.00*

1.18 1.09 1.04

1.40 1.26 1.18

1.80 1.54 1.38

Tables 2 through 5 list the values forK found for com­binations of variables considered to be useful for the prac­tical design of 400 mm (15.7 in.) octagonal prestressed concrete piles and their connections to pile caps.

To transform the K values which are listed in Tables 2 through 5 to graphs would result in a large number of fig­ures, namely 48 graphs containing 192 curves in total. A typical example of a set of curves is shown in Fig. 13. This figure was obtained from these tables and illustrates the modification factor, K, for various transverse confining steel contents, as represented by f,. If; ratios, and external axial load levels for a particular case.

From Tables 2 through 5, it is evident that the trends are K to increase with increase in lateral confining stress level fr If:, increase in the ix lid ratio, decrease in the concrete

76

0.06 0.03

lx!ld lxlld

0.4 1.0 0.8 0.6 0.4 1.0 0.8 0.6 0.4

1.09 1.09 1.09 1.04 1.02 1.02 1.02 1.00* 1.00* 1.00 1.02 1.02 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

1.15 1.13 1.13 1.11 1.08 1.02 1.02 1.01 1.00* 1.07 1.04 1.04 1.02 1.00 1.00* 1.00* 1.00* 1.00* 1.02 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

1.36 1.29 1.29 1.28 1.25 1.09 1.09 1.09 1.08 1.23 1.15 1.15 1.14 1.12 1.00* 1.00* 1.00* 1.00* 1.15 1.06 1.06 1.06 1.03 1.00* 1.00* 1.00* 1.00*

1.76 1.52 1.52 1.52 1.50 1.13 l.l3 l.l3 l.l2 1.52 1.26 1.26 1.26 1.24 1.00* 1.00* 1.00* 1.00* 1.37 1.10 l.lO 1.10 1.08 1.00* 1.00* 1.00* 1.00*

cylinder strength/;, and increase in the axial load level. In those cases marked with an asterisk, the K value is governed by the ACJt9 and SANZ2o code value for mo­ment (M0 5 MA), since the moment capacity after spalling of the concrete cover did not rise above the value before spalling.

Tables 2 through 5 emphasize the effect of neutral axis depth on the K value. A high axial load ratio results in a large neutral axis depth and hence a greater contribution of the concrete compressive force to the moment capacity. Hence, the K factors generally increase with increase in axial load ratio for a given lateral confining stress level. Also, the values forK decrease with decrease in ix lid ratio, but this reduction is not so great when the axial load ratio is high; this is because a high axial compressive load leads

PCIJOURNAL

Table 5. Flexural strength modification factor Kfor400 mm (15.7 in.) octagonal piles having fcp = 0 and Pst= 0.0237.

Effective lateral confining stress level/,.///

Pe f( 0.12 0.09

f(Ag (MPa)

lx!ld lxlld

1.0 0.8 0.6 0.4 1.0 0.8

30 1.31 1.26 1.15 1.09 1.25 1.22 0 45 1.27 1.21 1.10 1.07 1.21 1.16

60 1.22 1.17 1.08 1.06 1.17 1.13

30 1.29 1.27 1.18 1.08 1.25 1.24 0.2 45 1.24 1.23 1.14 1.06 1.21 1.20

60 1.21 1.19 1.12 1.04 1.17 1.16

30 1.28 1.27 1.22 1.15 1.21 1.21 0.4 45 1.24 1.24 1.19 1.14 1.18 1.18

60 1.22 1.22 1.18 1.14 1.15 1.15

30 1.38 1.38 1.34 1.28 1.28 1.28 0.6 45 1.34 1.35 1.32 1.28 1.23 1.23

60 1.29 1.29 1.28 1.25 1.16 1.16

Note: Factors marked* are obtained from MA; the remainder are obtained from Mo. Conversion factor: I MPa = 145 psi.

to a large neutral axis depth and hence smaller tensile strains in the concrete adjacent to the tendons, making slip less likely.

Available Ultimate Curvature and Curvature Ductility Factor

Figs. 14 through 17 display the available ultimate curva­ture <l>u and curvature ductility factor <l>u !<l>y for combina­tions of variables considered to be useful for practical cases of 400 mm (15.7 in.) octagonal piles with the range of P e If: A8 = 0 to 0.6, except that only J: = 45 MPa ( 6530 psi) and lx /ld = 1.0 are considered.

0.4

Failure criteria : From -,onward: M~O.BMmax " Tendon failure due to compression o Tendon failure due to tension

1m = 3.28ft Proeerties:

0.6

1.11 1.06 1.04

1.14 1.12 1.09

1.16 1.14 1.13

1.26 1.22 1.16

f! =45MPa(6530psi} fcp = 7. 6MPa ( 1100psi} R = 3 Pst= 0

tx/td = 1.0

0.2

fr/f(: 0.12

0.1 0.09

0.06

0 0.03

0 0.1 0.2 0.3 0.4 o.s 0.6

0.06 0.03

lxlld lxlld

0.4 1.0 0.8 0.6 0.4 1.0 0.8 0.6 0.4

1.05 1.19 1.16 1.05 1.00* 1.15 1.14 1.01 1.00* 1.03 1.14 1.14 1.01 1.00* 1.09 1.08 1.00* 1.00* 1.02 1.10 1.08 1.00* 1.00* 1.05 1.04 1.00* 1.00*

1.05 1.18 1.18 1.12 1.02 1.12 1.12 1.07 1.00 1.03 1.15 1.15 1.09 1.01 1.08 1.08 1.05 1.00* 1.02 1.11 1.11 1.06 1.00* 1.04 1.04 1.02 1.00*

1.10 1.13 .1.13 1.10 1.04 1.04 1.04 1.04 1.00 1.09 1.09 1.09 1.07 1.03 1.00* 1.00* 1.00* 1.00* 1.09 1.05 1.05 1.04 1.02 1.00* 1.00* 1.00* 1.00*

1.21 1.13 1.13 1.13 1.12 1.02 1.02 1.02 1.02 1.20 1.08 1.08 1.08 1.08 1.00* 1.00* 1.00* 1.00* 1.16 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00* 1.00*

The limiting failure criteria, as previously outlined, are indicated on the curves. It is evident that, generally, the criterion M ~ 0.8 M; governed. The tensile strains in the tendon, £P' and in the nonprestressed longitudinal rein­forcement, Es, located closest to the extreme tensile fiber, are also shown on the figures. The steel strain generally governed at the upper end of each curve. The strains in the tendons at failure were assumed to be 0.0455 in compres­sion and 0.0585 in tension. Failure of tendons can drasti­cally change the slope of the curves in the figures. Failure of the nonprestressed reinforcement never occurred before the prestressed tendons. It is evident from the figures that the available curvature increases with increase in lateral

50

40

!& 1/Jy 30

20

10

0 0 0.1 0.2 0.3

•12.5mm f.jin) dia. tendon

(0 fr/f(: 0.12 0.09 0.06

0.03

0.4 o.s 0.6

Fig. 14. Available ultimate curvature and curvature ductility factor for prestressed concrete piles with fcp = 7.6 MPa (11 00 psi) and without nonprestressed longitudinal reinforcement.

July-August 1990 77

0.4

-..... 0.3 E

] ~0.2

0.1

Failure criteria: From -,onward: Ms O.BMmax o Tendon failure due to compression o Tendon failure due to tension

1m= 3.28 ft ProR_erties :

f~ = 45MPa(6530ps/} fcp = 0 R :3 Pst= 0 t/(d = 1.0

fr/f(; 0.12

0.09

0.06 0.03

OL----.J...._-...J..-_-1-----'------'-----' 0 0.1 0.2 0.3 0.4 0.5 0.6

Pe /ff:Ag

25

20

15

10

5

•12.5mm rlin) dia. tendon

(() 0.09

0.06

0.03

0,__ _ _._ _ __.. __ ..__ _ _._ _ ____._~

0 0.1 0.2 0.3 0.4 o.s 0.6

f//fCAg

Fig. 15. Available ultimate curvature and curvature ductility factor for prestressed concrete piles with fcp = 0 and without non prestressed longitudinal reinforcement.

Failure criteria: Framt onward: MsO.BMmax

o Tendon failure due to compression

0 4 o Tendon failure due to tension

• Proeerties

0.3

0.1 Es--­Ep--1m = 3.28ft.

f~ = 45MPa (6530psil fcp= 7. 6MPa (1100psi/ R =3

Pst= 0.0237 {/fd = 1.0

fr/f(: 0.12

0.09

0.06

0.1 0.2 0.3 0.4 0.5 0.6

f>e/fC Ag

~ ~y

50

40

30

20

10 Es ---Ep--

0 0 0.1

• 12.5mm rfin) dia. tendon o 20mm (0.791h dia. deformed bar

f,-/f(; 0.12 0.09 0.06

----0.03

0.2 0.3 o. 4 o.s 0.6

f>e /fC Ag

Fig. 16. Availabl9 ultimate curvature and curvature ductility factor for prestressed concrete piles w1th Tcp = 7.6 MPa (11 00 psi) and Pst = 0.0237.

confining stress ratio and with decrease in axial load level ratio, providing that none of the tendons fracture.

From Figs. 14 and 15, it can be seen that the available ultimate curvatures of piles having full prestress and zero prestress are about the same, with the latter case slightly more ductile. However, when considering the available curvature ductility factor, piles having full prestress have much higher available ductility factors than those having zero prestress. The difference between the curvature duc­tility factors for piles having full prestress and those having zero prestress is due mainly to the difference in their yield curvatures <I>Y as defined. Piles with full prestress generally have a smaller yield curvature. The

78

same phenomena can be observed in Figs. 16 and 17 for piles having nonprestressed longitudinal steel and prestressing steel with full prestress and zero prestress.

A comparison of Figs. 14 through 17 shows that adding nonprestressed longitudinal reinforcement to prestressed concrete piles results in a significant increase in the avail­able ultimate curvatures. It can be seen that the slopes of the ultimate curvature curves in Figs. 16 and 17 are steeper than those in Figs. 14 and 15, respectively, and the in­crease is higher for piles containing high lateral confine­ment levels. However, this is not always the case for curvature ductility factor curves. Some increase in the curvature ductility factor due to the presence of nonpre-

PCIJOURNAL

• 12.5mm t1in)dia tendon Failure criteria : o 20 mm 10.79 in dia. deformed bar From t onward: MsO.BMmax

PrfJP..erties: o Tendon failure due to compression o Tendon failure due to tension

0.4 f~ = 45MPa (6530psi) fcp = 0 25

0.3

0.1 Es--­Ep--

1m:3.28ft.

R :3 Pst =0.0237

lx/ld = 1.0

fr/f(; 0.12

0.09

0.06 0.03

0'----'-----'-----'---"-----'-----' 0 0.1 0.2 0.3 0.4 0.5 0.6

20

tllu 15 ¢y

10

5

Es--­Ep--

fr/fc 0.12

0.09

0.06

0.03

OL---L----'-----L---L--~--~ o 0.1 a2 o.J o.4 o.5 o.6

Fig. 17. Available ultimate curvature and curvature ductility factor for prestressed concrete piles with fcp = 0 and p51 = 0.0237.

stressed longitudinal reinforcement is apparent, especially at low axial load levels, except for piles having high lateral confinement levels. For piles having full prestress and high lateral confinement levels, the presence of non­prestressed longitudinal reinforcement results in a reduc­tion in the curvature ductility factor. However, for piles having zero prestress and a high lateral confinement level if, If:= 0.12), the presence of nonprestressed longitudinal reinforcement improves the curvature ductility factor even at high axial load levels.

CONCLUSIONS

As a result of the theoretical moment-curvature studies conducted on pretensioned prestressed concrete piles having a 400 mm (15.7 in.) octagonal cross section and containing various quantities of spiral reinforcement, it was concluded that:

1. The flexural strength and available ultimate curva­ture can be increased appreciably as a result of confine­ment from spiral reinforcement.

2. The flexural strength obtained from the theoretical moment-curvature method for prestressed concrete piles with spiral reinforcement ranged from 1.00 to 3.74 times flexural strength calculated using the ACI (and New Zealand) Code concrete compressive stress block proper­ties. This increased flexural strength was achieved at high concrete compressive strains after the concrete cover had spalled.

3. The increase in flexural strength was particularly great when the external compressive axial load level was high, and when the quantity of spiral reinforcement was large.

4. Design tables which enable this flexural strength en­hancement to be estimated can be determined for pile sec­tions with various prestress levels, nonprestressed longitudinal reinforcement contents, lateral confining

July-August 1990

stress levels, concrete strength, distances from the pile end to take into account the development of tendons, and ex­ternal axial load levels.

5. A simple procedure to accurately predict the flexural strength of prestressed concrete piles, M;, taking into ac­count the improved compressive stress-strain properties of spirally reinforced concrete, the stress-strain properties of the steel, and allowing for the development of tendons at sections of the pile near the ends, is to use a modifica­tion factor, K. On this basis M; = KMA, where MA is the flexural strength calculated by the ACI and New Zealand code approach. Tables for the coefficient K can be prepared as a function of the above variables.

6. The available ultimate curvature of a prestressed concrete pile, defined as that curvature when the moment capacity has reduced to 80 percent of the peak moment of the moment-curvature relation, or when failure of the prestressing strand has occurred, increases with increase in the quantity of spiral reinforcement. For a constant quantity of spiral reinforcement, the available ultimate curvature decreases with increase in the level of external axial load.

7. Design charts can be prepared to determine the quan­tity of spiral reinforcement necessary to achieve a speci­fied ultimate curvature as a function of the axial load level and the properties of the pile.

8. The presence of nonprestressed longitudinal rein­forcement in a prestr~ssed concrete pile increases the available ultimate curvature. However, well confined pre­stressed concrete piles are adequately ductile without the presence of nonprestressed longitudinal reinforcement.

9. It is necessary for prestressing tendons to have a fracture strain which is sufficiently large to avoid the ul­timate curvature of the pile being limited by tendon fail­ure, particularly at low axial load levels. In the analysis, the failure strains assumed for the tendons were 4.6 per­cent in compression and 5.9 percent in tension. Tendon

79

failure occurred in some cases at high curvatures. 10. It is hoped that the results of this study will stimu­

late further work in this area, leading to validation of the assumptions in this investigation and the development of simplified design methods and code revisions which would permit the rational design approach.

ACKNOWLEDGMENT

The work described in this paper was carried out with generous financial assistance provided by the National Roads Board of New Zealand and the University of Canterbury.

REFERENCES

1. Banerjee, S., Stanton, J. F., and Hawkins, N. M., "Seismic Per­formance of Precast Prestressed Concrete Piles," Journal of Structural Engineering, American Society of Civil Engineers, V. 113, No.2, February 1987, pp. 381-396.

2. Gerwick, Jr., Ben C., "Seismic Design of Prestressed Concrete Piles," Proceedings of the 9th Congress of the Federation Inter­nationale de la Precontrainte, V. 2, Stockholm, 1982, pp. 60-69.

3. Muguruma, H., Watanabe, F., and Nishiyama, M., "Improving the Flexural Ductility ofPretensioned High Strength Spun Con­crete Piles by Lateral Confining of Concrete," Pacific Con­ference on Earthquake Engineering, Proceedings, V. 1, W aira­kei, New Zealand, August 1987, pp. 385-396.

4. Pam Hoat Joen, and Park, R., "Simulated Seismic Load Tests on Prestressed Concrete Piles and Pile-Pile Cap Connections," to be published in the PCI JOURNAL.

5. Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley and Sons, New York, 1975, p. 769.

6. Priestley, M. J. N., and Park, R., "Strength and Ductility of Con­crete Bridge Columns Under Seismic Loadings," Structural Journal, American Concrete Institute, V. 84, No. 1, January­February 1987, pp. 61-76.

7. Zahn, F. A., Park, R., Priestley, M. J. N., and Chapman, H. E., "Development of Design Procedures for the Flexural Strength and Ductility of Reinforced Concrete Bridge Columns," Bul­letin of New Zealand National Society for Earthquake En­gineering, V. 19, No.3, September 1986, pp. 200-212.

8. Park, R., and Falconer, T. J., "Ductility of Prestressed Concrete Piles Subjected to Simulated Seismic Loading," PCI JOUR­NAL, V. 28, No.5, September-October 1983, pp. 112-144.

9. Park, R., Priestley, M. J. N., Falconer, T. J., and Pam HoatJoen, "Detailing of Prestressed Concrete Piles for Ductility," Bulletin of New Zealand National Society for Earthquake Engineering, V. 17, No.4, December 1984, pp. 251-271.

10. Pam HoatJoen, Park, R., and Priestley, M. J. N., "Seismic Per­formance of Prestressed Concrete Piles and Pile-Pile Cap Con­nections," Research Report 88-3, Department of Civil Engi­neering, University of Canterbury, 1988, p. 320.

11. Iyengar, K. S. R., Desayi, P., and Reddy, K. N., "Stress-Strain Characteristics of Concrete Confined in Steel Binders," Maga­zine of Concrete Research, V. 22, No. 72, September 1970, pp. 173-184.

12. Park, R., and Leslie, P. D., "Curvature Ductility of Circular Reinforced Concrete Columns Confined by the ACI Spiral," 6th Australasian Conference on the Mechanics of Structures and Materials, Proceedings V. 1, Christchurch, New Zealand,

80

August 1977, pp. 342-349. 13. Shah, S. P., Fafitis, A., and Arnold, R., "Cyclic Loading of

Spirally Reinforced Concrete," Journal of Structural Division, American Society of Civil Engineers, V. 109, No. ST7, July 1975,pp. 1695-1710.

14. Mander, J. B., Priestley, M. J. N., and Park, R., "Theoretical Stress-Strain Model for Confmed Concrete," Journal of Struc­tural Engineering, American Society of Civil Engineers, V. 114, No.8, August 1988, pp. 1805-1826.

15. Mander, J. B., Priestley, M. J. N ., and Park, R., "Seismic Design of Bridge Piers," Research Report 84-2, Department of Civil Engineering, University of Canterbury, 1984, p. 483.

16. Blakeley, R. W. G., and Park, R., "Prestressed Concrete Sec­tions With Cyclic Flexure," Journal of the Structural Division, American Society of Civil Engineers, V. 99, No. ST8, August 1973,pp.1717-1742.

17. Thompson, K. J., and Park, R., "Moment-Curvature Behavior of Cyclically Loaded Structural Concrete Members," Proceed­ings oflnstitution of Civil Engineers, Part 2, V. 69, June 1980, pp. 317-341.

18. Menegotto, M., and Pinto, P. E., "Method of Analysis for Cycli­cally Loaded Reinforced Concrete Plane Frames, Including Changes in Geometry and Non-Elastic Behavior of Elements Under Combined Normal Force and Bending," International Association for Bridge and Structural Engineering, Preliminary Report for Symposium on Resistance and Ultimate Defor­mability of Structures Acted on by Well-Defined Repeated Loads, Lisbon, Portugal, 1973, pp. 15-22.

19. ACI Committee 318, "Building Code Requirements for Rein­forced Concrete, ACI 318-83," American Concrete Institute, Detroit, MI, 1983.

20. "New Zealand Standard Code of Practice for the Design of Con­crete Structures, NZS 3101, Part 1:1982," Standards Associa­tion of New Zealand, Wellington, 1982. (Commentary: NZS 3101, Part 2: 1982.)

21. Martin, L. D., and Scott, N. L., "Development of Prestressing Strand in Pretensioned Members," ACI Journal, Proceedings, V. 73, No.8, August 1976, pp. 453-456.

22. Whittaker, D., Park, R., and Carr, A. J., "Seismic Performance of Offshore Concrete Gravity Platforms," Research Report 88-1, Department of Civil Engineering, University of Canterbury, 1988, p. 215.

23. Zahn, F. A., "Design of Reinforced Concrete Bridge Columns for Strength and Ductility," Research Report 86-7, Department of Civil Engineering, University of Canterbury, 1986, p. 402.

PCIJOURNAL

APPENDIX A- NOTATION

Ac = area of concrete core of section of pile measured to outside of peripheral spiral

Ac; = area of the ith concrete strip Ag = gross area of section of pile Ap = area of one prestressing steel tendon Apj = area of jth prestressing steel tendon Ask = area of kth nonprestressed longitudinal steel bar Asp = area of spiral bar c = depth of neutral axis measured from extreme

compression fiber of pile section db = diameter of nonprestressed longitudinal reinfor-

cement or prestressed tendon ds = diameter of spiral (center to center) Ec = modulus of elasticity of concrete Ep = modulus of elasticity of prestressing steel Es = modulus of elasticity of nonprestressed steel Esec = secant modulus of confined concrete at peak of

stress-strain relation (=.fc~IEcc) Esh = strain hardening modulus for nonprestressed

steel measured at strain of Esh

.fc = concrete stress f/ = compressive cylinder strength of concrete .fc~ = compressive strength of confined concrete /c; = stress in ith concrete strip fcp = compressive stress in concrete due to prestress J;, = prestressing steel stress J;,a = stress at strain of Epa for prestressing steel (elastic

limit stress) f pa = stress at strain of Epa for prestressing steel J;,b = stress at strain of Epb for prestressing steel fpj = stress in jth prestressing tendon fpum = stress limit in prestressing steel as a function of

development and transfer lengths J;,s = maximum stress in prestressing steel at section

where flexural strength is reached J;,u = ultimate strength of prestressing steel /r = effective lateral confining stress on concrete from

spiral reinforcement /s = stress in nonprestressed longitudinal steel /se = effective stress in prestressing steel after all loss-

es /sk = stress in the kth nonprestressed longitudinal steel

bar /su = ultimate strength of nonprestressed steel fl = modulus of rupture for concrete h = yield strength of nonprestressed longitudinal

steel reinforcement hh = yield strength of spiral steel reinforcement K = flexural strength modification factor ke = confinement effectiveness coefficient ld = development length of prestressed tendon or non-

prestressed longitudinal reinforcement /, transfer length of prestressed tendon lx = distance from section under consideration to end

of pile where tendon stress is zero M = moment

July-August 1990

MA = flexural strength calculated by strain compatibil­ity-equilibrium method using ACI (and New Zealand) Code concrete compressive stress block and extreme fiber concrete compressive strain, and assuming no slip of tendons occurs

Mv = flexural strength calculated by strain compatibil­ity-equilibrium method using compressive stress­strain curve for spirally confined concrete and ex­treme fiber concrete compressive strain of core concrete after spalling of cover concrete, and as­suming slip of tendons occurs if development length is inadequate

M; = ideal flexural strength = MA or Mv whichever is greater

My = yield moment, assumed to be 0.75 M; nc = number of effective concrete strips np = number of prestressing steel tendons ns = number of nonprestressed longitudinal steel bars P = resultant internal axial force, or strain hardening

power for nonprestressed steel P e = external axial load in compression due to gravity

and seismic loading R = ratio of increase in concrete strain at peak stress

to increase in concrete strength at peak stress due to confinement

r = parameter in stress-strain equation for com-pressed concrete

s' = clear spacing of spiral reinforcement s = center to center spacing of spiral reinforcement Sb = compressive buckling stress of prestressing steel U = slope of upper linear (plastic) branch of stress-

strain curve of prestressing steel W = coefficient in stress-strain relationship for

prestressing steel in tension region W = coefficient in stress-strain relationship for

prestressing steel in compression region X = EciEcc Y = coefficient in stress-strain relationship for

prestressing steel in tension region Y = coefficient in stress-strain relationship for

prestressing steel in compression region Yci = distance from the centroid of ith concrete strip to

the centroid of pile section ypj = distance from the centroid ofjth prestressing steel

tendon to the centroid of pile section Ysk = distance from the centroid of kth nonprestressed

longitudinal steel bar to the centroid of pile sec­tion

Eb = compressive buckling strain of prestressing steel tendon

Ec = concrete strain Ecc = compressive strain at maximum stressf/c of con­

fined concrete stress-strain curve Ecm = compressive strain at 0.8 /c~ on falling branch of

confined concrete stress-strain curve Eco = compressive strain at maximum stress f/ of un-

81

confined concrete stress-strain curve Pee = ratio of total area of longitudinal steel (pre-Ep = prestressing steel strain stressed tendons plus nonprestressed reinforce-Epa = tensile strain at elastic limit for prestressing steel ment) to area of concrete core (measured to cen-Epa = compressive strain at elastic limit for prestressing terline of spiral reinforcement)

steel Ps = volumetric ratio of spiral reinforcement= ratio of Epb = strain at beginning of upper linear (plastic) volume of spiral reinforcement to volume of con-

branch for prestressing steel crete core measured either to outside of spiral Epu = ultimate tensile strain of prestressing steel reinforcement or to centerline of spiral reinforce-Ejiii = ultimate compressive strain of prestressing steel ment Es = nonprestressed longitudinal steel strain Pst = ratio of area of nonprestressed longitudinal rein-Esh = steel strain at commencement of strain hardening forcement to gross area of pile Espall = strain at which cover concrete is considered to <1> = section curvature, or strength reduction factor

have spalled and cease to carry any load <jly = yield curvature Esu = ultimate strain of nonprestressed steel <J>; = curvature corresponding to My Er = concrete strain at extreme compressive fiber <J>u = ultimate curvature Ey = yield strain of nonprestressed steel

APPENDIX B- DESIGN EXAMPLE

The prestressed concrete pile shown in Fig. A1 has a 400 mm (15.7 in.) octagonal cross section. The end of the pile is embedded in a cast-in-place reinforced concrete pile cap. The compressive strength of the concrete of the pile is 45 MPa (6530 psi). The pile is prestressed by ten 12.5 mm (1;2 in.) diameter seven-wire strands placed as

Note.· Pile and pile cap reinforcement is not shown.

t 400mm

(15.7in)

i

Precast prestressed concrete pile

10mm (0.39in) dia. ira/ bars

Ten 12.5mm fj in) dia. Tendons

Concrete cover to tendons =40mm (1.57in)

SECTION A-A

Fig. A 1. Example of the calculation of the flexural strength and ductility of a spirally confined prestressed concrete pile.

82

shown in Fig. A 1. The minimum concrete cover to the strand is 40 mm (1.57 in.). The stress in the strand after all losses is ..fs, = 1080 MPa (157 ksi). The strand has an ultimate tensile strength offpu = 1800 MPa (261 ksi). The spiral reinforcement bar is 10 mm (0.39 in.) in diameter and has a yield strength of/yh = 430 MPa (62.4 ksi).

The design external axial load in compression on the pile due to gravity and seismic loading is P, = 2387 kN (536 kips).

Calculate the flexural strength of the pile at the critical section at the face of the pile cap if the center to center spacing of the spiral reinforcement s and the embedment length of pile lx are as follows: (a)s = 50mm(1.97in.)andlx=either700mrn(27.6in.)

or 1750 mm (68.9 in.) (b) s = 200 mm (7.87 in.) and lx =either 700 mm (27.6

in.) or 1750 mm (68.9 in.) Also, calculate the available ultimate curvature and

curvature ductility factor for that critical section of the pile for the above values of s if lx = 1750 mm (60.9 mm).

Solution

Gross area of pile, Ag = 132,550 mm2 (205.5 in. 2)

Area of each tendon, Ap = 93.1 mm2 (0.144 in.Z) Area of 10 mm diameter spiral bar, Asp= 78.5 mm2 (0.122 in.2

)

Therefore: 10 X 93.1 X 1080 fcp = lOAp fs,IA g = -=-=-..:...c.::...=.:..::.c...:.-=-::..=.::_

132,550 = 7.59MPa (1100 psi)

For the prestressing strand, the development length is ld = 140db = 140 x 12.5 = 1750 mm (68.9 in.).

For an embedment length of pile in pile cap of lx = 700 mrn (27.6 in.), lxlld = 700/1750 = 0.4.

For an embedment length of pile in pile cap of lx = 1750 mm (68.9 in.), lxlld = 1.0.

PCIJOURNAL

Also, P,IJ: Ag = 2,387,000/(45 x 132,550) = 0.40. The flexural strength calculated by the strain

compatibility-equilibrium method using the ACI rectangular concrete stress block and an extreme fiber compressive strain of 0.003, and assuming that the tendons do not slip, can be obtained from Fig. 11 as MA = 277 k-Nm (168 kip-ft).

From Eqs. (7) and (8), the effective lateral confining stress on the concrete from the spiral reinforcement is:

f, = 0 5 ~-" ( 1 - 0.5 (s' I d.) ) r • PsJyh 1 -Pee

where for the pile: Ps = 4Asp1 ds s = 4 X 78.5/(330s) = 0.952/s Pee = lOApl Ag = 10 X 93.1/132,550 = 0.00702 Pst = 0

(a) If s = 50 mm (1.97 in.)

f, = ( 0.5 X 0.952 X 430 ) ( 1 - (0.5 X 401330) ) r 50 1-0.007

= 3.87 MPa (561 psi)

Therefore, f,./f:= 3.87 I 45 = 0.086 Now the flexural strength isM;= KMA.

For embedment length lx = 700 mm (27.6 in.): From Table 2, for Pelt: Ag = 0.4,J; = 45 MPa (6530

psi), lxlld = 0.4 and linearly interpolating betweenf,./j; = 0.06 and 0.09:

0.026 K = 1.05 + 0.15 X 0.03 = 1.18

Therefore, M; = 1.18 x 227 = 268 k-Nm (198 kip-ft). For embedment length lx = 1750 mm (68.9 in.):

July-August 1990

From Table 2, for variables as above except putting lxlfd= 1.0:

0.026 K = 1.09 + 0.17 X 0.0

3 = 1.24

Therefore, M; = 1.24 x 227 = 281 k-Nm (207 kip-ft). Also from Fig. 14, for lx = 1750 mm (68.9 in.) the

available ultimate curvature is <Pu = 0.16 radians/m (0.0041 radians/in.) and the available curvature ductility factor is <Pulcpy = 19.

(b) Ifs = 200 mm (7.87 in.)

f, = ( 0.5 X 0.952 X 430 ) ( 1 - (0.5 X 1901330) ) r 200 1 - 0.007

= 0.734 MPa (106 psi)

Therefore,f,.l/:= 0.734/45 = 0.0163. For embedment length lx = 700 mm (27 .6 in.):

From Table 2, for P, It: I Ag = OA,J: = 45 MPa (6530 psi), lxlld = 0.4 andf,. It:= 0.0163, it is seen that K = 1.00.

Hence, the flexural strength is MA. Therefore, M; = 227 k-Nm (168 kip-ft).

For embedment length lx = 1750 mm (68.9 in.): From Table 2 for variables as above except putting

lxl ld = 1.0, it is seen that K = 1.00. Therefore, M; = 227 k-Nm (168 kip-ft). Also from Fig. 14, <Pu = 0.02 radians/m (0.00051

radians/in.) and <Pul cpy = 2.

Note: The spiral spacing of 0.5 h, where h is the diameter of the section of the octagonal pile, results in a relatively small available ductility of the pile at the flexural strength. However, a spiral spacing of0.125 h has resulted in a very ductile pile.

83