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Design Considerations for Tapered Prestressed Concrete Poles

'I J. Bolander Jr.

Research Assistant PhD Candidate Department of Civil Engineering University of Michigan Ann Arbor, Michigan

Koz Sowlat

Associate The Datum/Moore Partnership Consulting Engineers Dallas, Texas

Antoine E. Naaman

Professor of Civil Engineering Department of Civil Engineering University of Michigan Ann Arbor, Michigan

restresse-d concrete poles are used for lighting, electric distribution and transmission lines, antenna masts, and other applications.' Compared to reinforced concrete members, these elements offer: (a) a wider range of elastic behavior, (b) better control of cracking and resistance to corrosion, (c) relatively high bending strength, and (d) ease of handling and installation.

44

P

Typical cross sections include circular and prismatic sections as well as Ishaped and circular hollow-cored sections. Prestressed concrete poles of circular or prismatic sections can be designed as prestressed concrete columns and are covered in previous investigations, 2'3 This study focuses on tapered prestressed circular hollow-cored and I-shaped poles for which additional de-

sign considerations are encountered. A hollow-cored circular section offers high torsional capacity, relatively low weight, and ample space for running wires and housing electric controls and accessories. The product lends itself to an automated manufacturing technique utilizing a spinning process that results in a smooth surface finish. An I-shaped section offers high mumerit resisting capacity for bending about its strong axis. At ultimate, it behaves identically to a rectangular section having the same outer dimensions and tendon configuration while offering considerable savings in both material and weight. Strength design of a prestressed concrete pole begins with determining external loads and calculating the axial load and moment due to these external loads at various section locations along the pole height. The pole is modeled as a cantilever beam rigidly fixed at its base. Since the section properties of a tapered pole change throughout the height, an iterative process is required to arrive at a design solution satisfying strength requirements. ', s Design charts may improve the efficiency of performing the iterative process. Load-moment interaction diagrams presented here have been developed based on a design procedure that accommodates the ACI 318-83 Building Code s provisions for fully prestressed concrete compression members. Section strength is accurately calculated by a strain compatibility procedure and an analytical representation of the nonlinear stress-strain relations of the prestressing tendons. As previously noted, nondimensionalized load-moment interaction diagrams for prestressed concrete columns of circular or prismatic cross sections have been addressed by other investigators. One objective of this paper is to extend this design concept to circular hollow-cored and I-shaped prestressed concrete tapered poles.

2a3

Synopsis

A strength design procedure based on the ACI 318-83 Building Code is developed for circular hollow -cored and I-shaped prestressed concrete poles and is implemented in a microcomputer program for an IBM PC. Load-moment interaction diagrams are automatically generated for any input values of the main variables. Three types of prestressing tendons with nonlinear stress-strain relations are considered. The effects of tendon configuration and other design parameters such as type of prestressing tendon, average prestress, concrete compressive strength and reinforcing index are evaluated. Nondimensionalized design charts for prototype conditions are presented.

ANALYTICAL FORMULATION

Design Assumptions and Approximations

The following assumptions are made: 1. Plane sections remain plane after loading. 2. Perfect bond exists between the steel and the concrete. 3. The tensile strength of the concrete is neglected. 4. The AC! 318-83 Building Code representation of the rectangular stress block in the concrete at ultimate load is valid. 5. The concrete compressive strain at failure is ee, = 0.003. 6. The design modulus of elasticityof concrete is E, = 33(y,)'' where E,, and f,' are in units of psi and y, is the unit weight of concrete in units ofpcf. 7. The stress-strain curve of the prestressing steel is represented by

45

PCI

JOURNAUJanuary-February 1988

A, = 2 (9 -- sin 9)r'

__f(1--

3

cosO)1.5

(B -- siii 6) r

A, = (B o - ;ih

,)r 2 -- 2(9; -- sin 9j)rl

3

(1 -- cnc -- (1 -- cos (0, -- sin 007 ,: -- (0, -- sin 8)t?

otes:

1. A. is the effective co!npre.,_a+n rote area (shaded regiion;s)

2. i is the distance from *tiec;aion centroid to the centroid of the effective compression zone area

Fig. 1. Possible cases of ultimate behavior for hallow-cored section.

the following general relationship:r.e

.fp s rx E e -

C+

t-

IN + ^ e {E )N 11 KJ PY

(1) Three types of prestressing steel are considered in this study. Design values for the parameters Q, K, and N of Eq. (1) were developed in Ref. 7. All symbols used in this paper are defined in a notation listing found at the end of this paper. Section Properties The rectangular stress block at ultimate of a circular hollow-cored section may act over either a segment of a circle 46

or an annulus (Fig. 1). Formulas for calculating the area of the effective compression zone and the distance of its geometric centroid from the center of the section are also presented in Fig. 1. The dimension variables assigned to a typical 1- shaped section are shown in Fig. 2. The rectangular stress block at ultimate of an I-shaped section (considering bending about either axis) may act upon any one of the cases shown in Fig. 3. Formulas for calculating the area of the effective compression zone and the distance of its geometric centroid from the center of the section are presented for each case in Fig. 3. Basic Equations for Analysis The analysis generally starts by selectinga strain diagram and then finding

f.

w' hs

cover

Indicates dimensions s matning constant over pole length

Fig. 2. Cross section configuration (I-shaped section).

the conditions leading to that strain diagram. For a given strain diagram (Fig. 4), two equations describing the equilibriurn conditions at the onset of failure can be written; one for the balance of forces and one for the balance of moments. Assuming the material and section properties are given, the strain in the concrete under effective prestress can be found using the following equation:'

f cr =

The stress in the ith layer of steel is calculated using the assumed stress-strain relationship for the steel [Eq. (1)]. The tensile force in each layer is given by:

T15 = (A 5a),(fy5 )1

(5)

The depth of the effective compression zone is given by: a = P I C(6) Assuming that the area occupied by the steel in the compression zone is negligible, the compressive force in the concrete is obtained from: C =0.85fAe(7) where A e is the area of the effective compression zone. The nominal axial force and nominal moment resistances can he obtained by writing equilibrium equations for forces and moments:

P,, =C -- T,,, (8)

Am

AnJ, -- (Am--Avs)Ec ARE,

(2)

where A,, = 1 i (A,,),. It is assumed that all tendons have the same effective prestress. At ultimate, the strain change in the ith steel layer becomes:

(

1 PA )i =

E fe+ E cv

r d, -- c I `

(3)

where d 1 can be smaller or larger than c. The strain in the ith layer of steel is then given by:

(E")i=E,+(AE").

PCI JOURNAL/January-February 1988

For hollow-cored shape: M,,=CT+ 1: Tip (d f --D,J2) (9)

47

(4)

x.

+_...................X

Ar=Ore&

}^ . ......_,....... + .............._..,,. 1{

A =bf+u(b_ u

tan o

= 1.

1 bf [2{h-f)+v(b-1ua)(2 -f - u )+ h tan 2

tan a 2

fh-I-u)

^.

X-................. + ........._... }C

A, = bf i-

+(2

A+w

)vtan o: + (u - vtan o)w

L

2 (h- f) +v z tanrr(2 f--v tan a)+uw(2-f-2)

«a A. = 2f3 ,f +

(}J1c)

2 tan u

r A. fricf(b- /rr)+(Brc) z tmra(b 23rc)1

A, = 2vf + r e ran u + h(J3j c - v)

Y ....._ ................ ....:..........y _

b z ^ rf(b -v)+u tancef2 2 )+h( tc-v.)( arc-v

2

)

Notes: 1. See Figure 2 for section dimensioning

2. A, is the effective compression zone area (slradec regions)

3- a is the distance from the section centroid to the centroid of the effective compression zone area 4. The following abbreviations have been used: · u =^c f

· L' =

(6 -- ii;)

Fig. 3. Possible cases of ultimate behavior for I-shaped section. 48

For I-shape:

114,,=Ci + 1T,(d 1 hl2) (10)

r

where i is the distance from the section centroid to the centroid of the area of the effective compression zone (i.e., the moment arm of the resultant concrete compressive force). The corresponding eccentricity of the applied load can be computed using: e=

Pn ·

·

PA

(11)

c

The above procedure can be repeated for various strain diagrams and leads to the determination of the load-moment interaction diagram for the pole section.

Load - Moment Interaction Diagram

IT

U.85f'.

The load-moment interaction diagram is determined point by point through: (1) selecting a priori a linear strain diagram at ultimate, (2) computing the corresponding load-moment value for which equilibrium is satisfied, and (3) repeating the process for other strain diagrams. The procedure is described in detail in Re£ 2. To develop the design charts, several load-moment values were determined, including the point of pure compression, the point of zero tension, the balanced point, and the point of pure flexure. In this study, a curve was computer plotted and smoothed through these points and 60 other intermediate points.

M.A.

1)

( p,)

A iz

d,

Jm

(A,.)2

Fig. 4, Typical strain diagram and corresponding forces at the onset of failure in a column subjected to compression and bending (also applicable to other section shapes. including I-shape).

PARAMETRIC EVALUATION

A microcomputer program was developed for the IBM-1'C to generate load-moment interaction diagrams for prestressed concrete poles with: (1) circular hollow-cored sections, (2) rectangular sections, and (3) I-shaped sections for bending about either principal axis.

PCI JOURNALUanuary-February 1988

A flowchart outlining the basic operation of the computer program is presented in Fig. 5. This program was used first to evaluate the influence of various parameters on the load-moment interaction diagram and then to generate design charts for a practical range of those parameters found to have a nonnegligible influence on the load-moment interaction diagram.

49

TITLE PAGE ON SCREEN LOAD MOMENT INTERACTION DIAGP.PJ TYPE I , Nominal 2. Factored

NG

TYPE

SELECT CROSS-SECTION(i) TYPE 1. Rectangular 2. Circular 3. Circular Hollow-Core

4. I-Shane

q

SELECT SECTION(i} BENDING DIRECTION

INPUT STEEL VARIABLES FOR SECTIDN(i} Type Quantity Configuration

INPUT CONCRETE VAR [ABLES FOR SECTION(i) - Compressive Strength Unit Weiuht SELECT/INPUT PARAMETER VALUE 1 CHOOSE PROGRAM OPTIONS 1. Select Another Section 2, Vary Parameter of Previous Section 3. Compute LMID Values

FORj=1 TOi FOR k = 1 TO number of points

ASSUME STRAIN DISTRIBUTION(k) AT ULTIMATE

COMPUTE Nondimensional Load PPSU,k) - Nondimensional Moment MM(i,k) CONTINUE PROGRAM OUTPUT Plot Load Moment Interaction Diagrams Output Values on Line Prinior

Fig. 5. Flowchart showing basic operations of computer program.

nondimensionalized only if a reference tendon configuration is assumed, Two variables should be considered in deCircular Hollow-Cored Section -- veloping a reference tendon configuraLoad-moment interaction diagrams for tion--the number of tendons and their the circular hollow-cored section can he position within the concrete section, 50

Development of a Reference Tendon Configuration

0.40

fr = 7 ksi D1100 = 0.70 ffU =270ksi D5/D0=0.85 f 150 ksi A s/Ag = 0.0045

0.35

D Di

0.30

0.25

Q

a4

0.20

0.15

0.10

IDS

DS 0.05

0.00 0.00

0.10

0.20 (Mnr po

030

040

0.50

}*10

Fig. 6. Comparison of tendon configurations (circular hollow-cored section).

A common method of representing the location of the steel reinforcement in circular columns is the equivalent tube method. This method approximates the prestressing tendons by a prestressing steel tube with an area equal to that of the actual tendons. A minimum of four tendons is generally required in round columns to insure symmetric behavior. The interaction diagrams for two seePCI JOURNAU'January-February 1988

tions with equal prestressing ratios are compared in the following manner. One is modeled with four tendons and the other with an equivalent tube of equal area, as illustrated in Fig. 6. It is observed that the section modeled with four tendons (at 45 degrees) leads to slightly more conservative results. To investigate the effect of the positioning of the tendons, the interaction

51

U.

0. s5

f =7 ksi Di/Do = 0.70 fp, = 270 ksi D S/Do = 0.85 fp= ISO ksi A ps/Ag = 0.0045 ,

Do D^

0

so

o..

b

repo

:

0.

0.1

0.0

450

0.0

u ^^

^.IV

u -zU

030

0.40

0.50

(Mn/f^D03)* 10

Fig. 7. Comparison of tendon configurations (circular hollow-cored section).

diagrams of two sections are compared; one containing four tendons aligned with the bending axis of the section and the other containing four tendons at a 45 degree angle to the bending axis. As illustrated in Fig. 7, the latter also leads to slightly more conservative results. Hence, for practical safe design, a reference tendon configuration may be taken as four tendons positioned at a 45 52

degree angle to the principal axis of the section. The area of each prestressing tendon is taken as one-fourth the total area of the prestressing steel. I-Shaped Section--It is the practice of commercial producers of prestressed concrete I-shaped poles to hold certain dimensions constant over the length of the pale while the pole height and width are varied. The following

parametric study for the I-shaped section assumes that the flange thickness, the web thickness, the flange taper angle, and the concrete cover remain fixed over the pole length. Fig. 3 shows the various dimension parameters and the prestressing steel configuration related to the I-shaped section.

Effects of Design Parameters

Concrete poles are generally designed for loading conditions corresponding to the low axial load region of the interaction diagrams. Therefore, the variables influencing this region are of particular interest. Circular Hollow-Cored Section The effects of the following parameters on the load-moment interaction diagram are considered: (1) concrete compressive strength, (2) prestressing steel reinforcement ratio, (3) effective prestress in the tendons, (4) type of prestressing steel, (5) the diameter of the circle passing through the tendons' centerline D8 , and (6) the ratio of the inside diameter to the outside diameter of the section D 1 /Dg. The results of this parametric evaluation can be summarized as follows: 1. The compressive strength of the concrete has a significant influence at high axial loads. However, this effect diminishes considerably at lower axial loads (Fig. 8). 2. The amount of prestressing reinforcement significantly affects strength. This effect is more pronounced in the low axial load region of the interaction diagrams (Fig. 9). 3. The effective prestress in the tendons does not have a significant effect on the load-moment interaction diagram, especially in the low axial load region (Fig. 10). 4. The ultimate prestressing index defined here as:

(L)pv APe.ff,11

AAf,

(12)

is useful in comparing the effects of different types of steel. In the low axial load region of the interaction diagrams, different types of steel lead to similar results provided the ultimate prestressing index remains constant (Fig. 11). 5. The ratio of the diameter of the circle passing through the tendons' centerline to the section's outer diameter, D,/D O , generally does not have a significant effect on the strength (Fig. 12). However, in cases where a high reinforcing index is used, the D, ID, ratio becomes significant and therefore should be included as a design variable (Fig. 14). For the case presented in Fig. 13, one obtains more than a 10 percent increase in nominal moment resistance in the pure flexure region of the load-moment interaction diagram by increasing the D, ID, ratio from 0.70 to 0.90. 6. The ratio of the inside diameter to the outside diameter of the section, DID,, has a marked elect on the resulting strength (Fig. 13). I-Shaped Cross Section -- The effects of the following parameters on the load-moment interaction diagram for strong axis bending were considered: (1) concrete compressive strength, (2) prestressing steel reinforcement ratio, (3) effective prestress in the tendons, (4) type of prestressing steel, (5) prestressing steel layer height to overall section height ratio, h,/h, and (6) the flange thickness to overall section height ratio flh. The parametric study on the first four of these design variables provided results in agreement with those obtained for the circular hollow-cored section, In particular, Conclusions 1 through 4 listed above are valid here. Consequently, only the last two of the above listed design parameters need be discussed. 1. For lower to moderate reinforcing ratios, the ratio of the prestressing steel layer height to overall section height, hg llt, does not have a significant elect on normalized strength (Fig. 15). How53

PC] JOURNAUJanuary-February 1988

YV

DIDa=0.70 fpu = 270 ksi D S /DO - 0.85 s

6U

f = I50 ksi A p ^AS = 0.0060

Da

4 80

· D^

·

4 .06

U) N

eksi 3 20

0

"0

O,

9 40

0 4s .

90

·

0

ks.

60.5.

0.

o.

::T

0.00

0 2)

n an

VV

I,UI}

M0/D03 (ksi)

Fig. 8. Influence of concrete compressive strength on the loadmoment interaction diagram (circular hollow-cored section).

ever, when considering higher reinforcement ratios, the ha /h ratio has a nonnegligible effect on the strength in the lower regions of the load-moment interaction curve and therefore must be considered as a design variable (see Fig. 16). 2. Consideration of the flange thick54

ness to section height ratio, fh, as a design variable is presented here as one solution to the problems arising due to nonuniform sealing of the cross section over the pole length. The f/h ratio has a significant effect on strength over the entire load-moment curve when the reinforcing ratio p is held constant (Fig.

0.4

I)/1)= 0.70 fc = 7 ksI 270 ksi D51D0 = 0.85 f f= 150 ksi

0 35

0.30

0.25 0 0.20

QI

05-

0.0015

0.10-

0,0030 00045 0.0060 1 APS/A9

0.0 5

0.0075 0,0090

0.0U 0.00

0.02

0.04

0.06

0.08

0.10

Mnlf,Do

Fig. 9. Influence of prestressed reinforcing ratio on the load-moment interaction diagram (circular hollow-cored section).

17). This may be explained as follows: Since several of the cross-sectional dimensions are held constant, the area of the cross section at any given location along the pole length is no longer linearly related to the product of the given section's width and height (the nondimensionalizing parameters). For

PCI JOURNAUJanuary-February 1988

example, as one approaches the pole tip, the flange portion of the cross section will take up a greater percentage of total section area. It is the relative increase in the prestressing steel area necessary to maintain the reinforcing ratio p constant which accounts for the increase iu flexural resistance observed.

55

o., v f, = 7 ksi Di1D0 = 0.70 fP , = 270 ksi D s/Do = 0.85 A P S = 0.0045 ,/A

5

0.1 100 k5

025

fPe

125 ksi 150 ksi 175 ksi

0.20

0.15

0.10

0.05

0.00

0.10

0.20

0.30

0,40

0.50

(Mn/f^Do 3 )* 10

Fig. 10. Influence of effective prestress on the load-moment interaction diagram (circular hollow-cored section).

DESIGN CHARTS

Both nominal load-moment interaction diagrams and factored load-moment interaction diagrams have been developed as design charts. g· 1 ° Nominal load-moment diagrams were used in the preceding parametric investigations and 56

may be regarded as a design aid lacking any code safety constraints. The factored load-moment interaction diagrams are simply the nominal load-moment interaction curves modified for design purposes by the ACI factors a and 6 (Ref: 6), These factors are as follows: 1, For continuous spiral lateral rein-

0 40

op,, AP. fPU I(A X fd =0.25

fe = 5

ksi D1 /D 0 = 0.70

D5/D 0 = 0-85 D0

0.30

^s D^

025

fpu=l60ksi f pe ;, 72 ksi

e-I 0 0.20

fpu=235 ksi fpe = l25ksi f pu =270"'

G.a

0.15

fpe = 150ksi

0.j 0

,r ·

0 0 000

002

004

0.06

0.08

01

Mn/f^D0

Fig. 11. Influence of ultimate prestressing index on the load-moment interaction diagram (circular hollow-cored section).

18. In accordance with the AC! provisions, the point of pure flexure is multi= 0.75 = 0.$5; plied by the less severe reduction factor, 0 = 0.90, related to pure bending. This 2. For hoop-type lateral reinforcement: point is connected to an AC! defined A = 0-80; 4> = 0.70 transition point by a straight line repreExamples of factored load-moment senting a linear transition from ¢ = 0.90 interaction diagrams are given in Fig. back to 0 = 0.70 or 4> = 0.75. forcement:

PCI JOURNAL/January-February 19BB 57

0.4

D/Do = 0.60 f, = 7 ksi fpu = 270 ksi fpc 150 ksi Ap5 lAg = 0.0060

0.35

0.30

0.25

nr

Q

ap

0

0.20

0.90 0.80 D S / Do

0.15

0.70

0.10

005

0.0C

] 00

00?

004

n nF.

ors

n Ir

M/f"D 3 Q c o Fig. 12. Influence of D5 ID9 ratio on the load-moment interaction diagram for moderate prestressing ratios (circular hollow-cored section). The transition point is defined in DESIGN PROCEDURE FOR terms of an axial load equal to 0.10f,. A9. TAPERED POLES Any point within the boundaries set by the factored load-moment interaction Load-moment interaction diagrams diagram represents a feasible design for can be used as a design aid to expedite a given section, design parameters, and the design of prestressed concrete colloading. In Fig. 18 both the ACI defined umns and poles. A typical factored demaximum allowable compression value sign chart is presented in Fig. 18. The and the transition point are indicated by design procedure suggested here should lead to a conservative yet economically pointmarks. 58

N

O

Q

0.00

0.02

0.04

0.06

0.08

0.10

Mnl fDo

Fig. 13. Influence of DID,, ratio on the load-moment interaction diagram (circular hollow-cored section).

attractive design. The design steps are: 1. The following design variables are initially selected: (a) Concrete compressive strength (b) Type of prestressing steel (c) Pole tip dimensions (d) Pole taper slopes (e) Concrete cover 2. Several locations along the pole height are chosen as design sections. It

PCI JOURNAL/January-February 1988

is recommended to section the pole at 10 ft (3.05 m) intervals along its upper 60 ft (18.3 m) of length and at 5 ft (1.52 m) intervals over any remaining length of the structure.' Note that it may be necessary to choose additional design sections along the pole length to check possible critical locations not accounted for by the above arbitrary length divisions.

59

rn

Q

FP

I

:...: . . : ..:

,

.;,

.... , . , , ; ...BPS= 181 INA2 .' ,, ,,,,,,,,,,,,,,,,, R I1 /^ 111 INA2 } ^C = . .. . aO : 0 1 014 i

}SI FPE=150 )BSI FC:7.00 ES I

0160

..

. , ...

..; , .....

1 i.^^i..

FPU:270 ES I

^i^

:..

.FP-15 f^ ESI} /{E 1'^f^0

FC :7.09

ESI

'APS=0189 INA2 :AG:222 6 INAI RHO: 1 040 .. ...,.. ..

0145

^..

..

..

...I

, . ti.,

0

I

0130

1115

A

I,

I.

I

tb 20

18

;

20

Fig. 14. Influence of D, ID, ratio on the load-moment interaction diagram for higher prestressing ratios (circular hollow-cored section)-

Fig. 15. Effect of steel layer height on the load-moment interaction diagram for lower reinforcing ratios (I-shape section).

C)

C-

PP B 140 FPU:270 SI ,.. , ,,:FPE:150 ){SI FC:? , .. . ..: . :...:... RPS:2 00 IBS I 67 IN"2

P 86 , . ' ,;. FP=270 HSI : FPE :I5O HSI F=7,00 HSI 'APS:X X IH"2

,AC:XXX,X INA2

0

C M z

C CD

a C

CD CD

0^4 0130

.

;. : ^^

,,,.,

,

: RN 4= I^ ^14^

I

CO

W

0.20

0.3

011 0110 ' ' `

1

1 1

lI

^

i

)I

003

0I6

0.09

MM

I

m

Fig. 16. Effect of steel layer height on the load-moment interaction diagram for higher reinforcing ratios (I-shape section).

Fig. 17. Effect of flange thickness on the load-moment interaction diagram holding the prestressing steel ratio constant (I-shape section).

rn

PHI=PP 0.3 9r

7 .......`

............................. ..

si

PHI*PP 0.3 e

f^,u = 270 ksi f^0 = 150 ksi fc 7.00ksi

1Pe -- 150 ksi

ff = 7.00 ksi

0.20

..

..

...;...

...

;.......'

0.0013 ... ; .....

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

......i ..............'..............:............

0.04 0.a5 PHIsmrn

..

0.0093..

0187

6.a1

e.02

0.a3

e.1

0.02 20.03

0.04

0.05 PHI * l

(a) Rho - 0,0013, 0.0033,0.0053,0.0073,0.0093

{b) Rhos 0.0043, 0.0117, 0,0140, 0.0163, 0.0187

+ These Rho values correspond to an effective prestress in the concrete of 200, 500, 800, 1100, 1400, 1750, 2100, 2450, and 2800 psi, respectively.

Fig. 18. Factored load-moment interaction diagrams for circular hollow-cored section.

3. For each of the above design sections, the factored design load and moment are calculated (loadings applied during manufacturing and transportation should also be considered here). Increase the moment values to account for slenderness effects as prescribed by the ACT 318-83 Building Code where required. fi'°'° These two values are then nondimensionalized according to section type in the following manner: (a) Circular hollow-cored section:

PP =P^1(f,Do) MM M u l(f, D^

(h) I-shaped section:

PP P,,I(f'bh)

MM = M,,1(f,bh2) where PP and MM are the nondimensionaI load and moment, respectively, The cross-sectional area and values accounting for the effect of tapering over the pole length (see Design Step 4) should also be computed at this time. The above calculations can be quickly performed using a programmable hand calculator or a small computer program. 4. For each section location a design chart (load-moment interaction diagram) is either computer generated or selected from a set of previously prepared design charts developed for the concrete compressive strength to be used. The design chart chosen for a given section should obey the following restrictions: (a) Circular hollow-cored section:

(D

E

/D)

CHART

}

(D IDo).cNCTWON

1

(D^ ID

O)CHART =

(D1 / D0 )

SECTION

(b) 1-shaped section:

(flh )CILRT = (f/h)SECTIOx (h, /h) CHART 1

( he l h ) SECTION

The closer to equality the above relations are, the more accurate the design chart will be in estimating the section's behavior. Note that the second relation in each of the above two cases becomes much

PC) JOURNALJanuary-February 1988

less significant when considering lower levels of prestressing. 5. Reinforcing ratios, p, for each section location are obtained by plotting a given section's nondimensionalized load and moment on the corresponding design chart and then linearly interpolating horizontally (a conservative approximation) between the two bounding p values on the chart. Design charts focusing on the lower axial load region of the load-moment interaction diagram could be used to improve the accuracy of this procedure. 6. The number of prestressing tendons required at a given section is determined by multiplying this p value by the corresponding gross cross-sectional area of the section, dividing through by the area of one tendon, and then rounding up to the next integer value. Check that the number of tendons so determined satisfies the design requiremonts set forth by the PCI Committee on Prestressed Concrete Poles in Ref. 5 and any other constraints made necessary due to special applications. 7. If a satisfactory design is not achievable for each section location, one or more of the input variables selected in Step 1 are adjusted and the procedure is repeated. In general, one would increase the outer dimension(s) of the pole tip for cases where one or more of the section locations fail to meet strength requirements. A numerical example illustrating the above design procedure for a 130 ft (39,7 m) concrete hollow-cored pole under extreme wind conditions is presented in Ref. 9. It should be noted that computer programs exist for direct design of circular hollow-cored poles circumventing the need to interpolate between values on a design chart.' Furthermore, a comprehensive design is more readily obtainable using a complete computer solution. It is expected that the use of design charts will diminish as such progtams become more widely accepted and available.

63

For most applications, it would he advantageous to develop design charts Nondimensionalized load-moment concentrating around the pure flexural interaction diagrams were developed in region of the load-moment interaction this study to predict the nominal resis- diagrams. tance of circular hollow-cored and IA microcomputer program was develshaped tapered prestressed concrete oped in this study to generate loadcolumns and poles. The results are pre- moment interaction diagrams for colsented in forms suitable for both analyumns and poles of various cross-secsis and design. Through application of tional shapes. The program runs on the ACI defined safety factors A and 0, IBM-PC systems with graphics capabilnominal load-moment interaction dia- ities. Details of the program and its opgrams are translated into factored design eration are found in Ref. 9. In addition, charts in accordance with the ACI Ref. 9 contains a numerical example il318-&3 Building Code specifications for lustrating the use of load-moment interfully prestressed concrete members, action diagrams as a design tool, The use of load-moment interaction Note that Refs. 9 and 10 are available diagrams as a design aid should provide from PCI Headquarters at cost of reproa quick and economical solution for the duction. design of prestressed concrete columns and poles when a complete computer solution is not available. To improve the ACKNOWLEDGMENT accuracy of the results, design charts developed for actual usage should be This study was supported in part by a plotted or drafted in a manner more re- PCI Student Fellowship Award to the fined than those presented in this study. University of M ichigan,

CONCLUDING REMARKS

64

REFERENCES

1. Rodgers, T. E., "Prestressed Concrete Poles: The State of the Art," PC! JOURNAL, V. 29, No. 5, September-October 1984, pp. 52-91. 2. Naaman, A. E., Prestressed Concrete Analysis and Design, McGraw Hill Book Company, New York, N.Y., 1982, 670 pp. 3, Salmons, J. R., and McLaughlin, D. G., "Design Charts for Proportioning Rectangular Prestressed Columns and Poles," PCI JOURNAL, V. 27, No, 1, January-February 1982, pp. 120-143.

4. PCI Design Handbook -- Precast Prestressed Concrete, Second Edition, Pre-

stressed Concrete Institute, Chicago, Illinois, 1978. 5. PCI Committee on Prestressed Concrete Poles, "Guide for Design of Prestressed Concrete Poles," PCI JOURNAL, V. 27, No. 3, May-J une 1983, pp. 22-87. 6. ACI Committee 318, "Building Code Requirements for Reinforced Concrete (ACI 318-83)," American Concrete Institute, Detroit, Michigan, 1983. 7 Menegotto, M., and Pinto, P. E., "Method of Analysis for Cyclically Loaded R. C. Plane Frames Including Changes in Geometry and Non-Elastic Behavior of Elements Under Combined

Normal Force and Bending," IASBE Preliminary Report for the Symposium on Resistance and Ultimate Deformability of Structures Acted on by Well Defined Loads, Lisbon, Portugal, 1973, pp. 15-22. 8. Naarnan, A. E., "An Approximate Nonlinear Design Procedure for Partially Prestressed Concrete Beams," Computers and Structures, V. 17, No, 2,1983, pp. 287-293. 9. Bolander, J. Jr., and Naaman A. E., "Load-Moment Interaction Diagrams as Design Aids for Prestressed Concrete Columns and Poles," University of Michigan Department of Civil Engineering Report No. 85-1, February, 1985, 91 pp. 10. Sowlat, K., and Naaman, A. E., "Design Aids for Hollow-Cored Prestressed Concrete Poles," University of Michigan Report No. 84-1, September, 1984, 70 pp. 11. Nathan, N. D., "Slenderness of Prestressed Concrete Columns," PCI JOURNAL, V. 28, No. 2, March-April 1983, pp. 50-77. 12. Nathan, N. D., "Rational Analysis and Design of Prestressed Concrete Beam Columns and Wall Panels," PCI JOURNAL, V. 30, No. 3, May-June 1985, pp. 82-133.

NOTE: Discussion of this article is invited. Please submit your comments to PCI Headquarters by October 1, 1988.

PCI JOURNAL7January-February 1988 65

APPENDIX - NOTATION

= effective compression area of = effective prestress concrete fP, = stress in prestressing steel Aa = gross area of concrete section fp,, = yield strength of prestressing A,^ = nominal area of concrete secsteel tion fnu = ultimate strength of pre(Ava)j = area of prestressing steel in stressing steel layeri ft = uniaxial compressive strength (1. = effective depth of compresof concrete sion zone h = section height a = flange taper angle (I-shape) h, = distance between centerlines b = flange width (I-shape) of opposing tendon groups = reduction factor of compres(I-shape) sion area as per ACI Code M,, = nominal moment capacity of C = total effective compressive section force M. = ultimate moment capacity of C = distance of neutral axis from section outside edge of section MM = nondimensionalized ultimate Di = inside diameter of section moment capacity of section (circular shape) P,, = nominal axial load capacity of Do = outside diameter of section section (circular shape) Pa= ultimate axial load capacity of D, = diameter of circle passing section through tendon centerlines PP = nondimensionalized ultimate (circular section) axial load capacity of section distance of prestressing steel ri= inside radius of section (cirlayer i from outside edge of cular shape) section r, = outside radius of section (cirE, = modulus of elasticity of concular shape) crete p = prestressing steel reinforcing = modulus of elasticity of preratio stressing steel Ti,e = tensile force in prestressing e eccentricity of applied load steellayeri E,e = strain in concrete under ef- yc = unit weight of concrete fective prestress w = web thickness (I-shape) = ultimate compressive strain of w,,P= effective prestressing index Ecu concrete ri = ultimate prestressing index E" = strain in prestressing steel z = moment arm of the resultant under effective prestress concrete compressive force of E, = strain in prestressing steel section ep,, = yield strain of prestressing K,Q,N = empirically determined pasteel rameters in prestressing steel = flange thickness (I-shape) f stress-strain relation

Ap

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