Pole Analysis

7/27/2019 Pole Analysis

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Journal of Structural Engineering

Vol.

38 No.6

February-March 2012 pp. 507-518

No.38-41

quivalent pole concept for tapered power poles

Sriram Kalaga*

[8J Email: [email protected]

*Allgeier Martin Associates, Inc., Missouri 64834, USA.

Received: 04 August 2010; Accepted: 30 January 2011

An Equivalent Pole concept is introduced to analyze tapered

power

poles. Using stiffness and strength criteria diameters

of q u i v ~ l n t

const nt section poles

are

derived

for

wood

and

steel poles

by comparing

deflections

and

stresses with those

of t pered poles. Axial, flexural and torsion loading were considered. The derivations

are

validated

for

wood and steel

poles

using exact

computer

analyses.

Both

qualitative

and quantitative

inferences

were drawn and

suggestions

for further

extensions

are

made.

KEYWORDS:

Transmission poles; steel; wood; stiffness; strength; finite elements.

structural response of ransmission poles is usually

by the behavior of the tapered element under

of

wire, wind, ice and other loads.

t fiber-reinforced composite (FRC) poles tare also

employed successfullyas transmission structures

1

.

of these poles involve non-linear finite

critical buckling capacities

of

guyed, tapered steel

es (8- and 12-sided), are hard to find; solutions for

literature

2

3

, are not

A brief literature survey shows significant basic

to

the mid

4

.

Past investigations covered topics such

as

5

, formulation

of

explicit FE stiffness

8

, torsion

9

, combined non-linearity

1

and

of steel poles

,

among others.

dli e

to multiple integrations for varying area and moment

of inertia

12

13

With specific reference to buckling of

guyed poles, most research dealt with wide-flange,

box and other cross sections

3

but not dodecagonal

(12-sided) steel poles commonly used in high-voltage

transmission applications. Banerjee et al

7

presented

buckling solutions for hollow tapered beam-columns,

but the procedure is part of a complex Bernoulli-Euler.

stiffness analysis procedure. The ASCE guidelines

14

for steel poles simply give an expression for allowable

compressive stress based on limiting width/thickness

wit) ratios, but this refers to local buckling rather than

overall pole buckling.

To

the extent the author knows, there is little

information available on the application of equivalency

concepts - using both strength and stiffness - to the

analysis of transmission poles. This study is a small

step in that direction. The aim of this paper is to present

the concept of an Equivalent Pole (EP) which can

be used to convert tapered poles into constant section

elements. The EP can then be used to develop simple

analytical models covering various load patterns. The

proposed process is validated

on

poles made of steel

JOURNAL OF STRUCTURAL ENGINEERING 507.

Vol. 38, No.6, FEBRUARY- MARCH 2012


2/12

(hollow) and wood (solid). Possible extensions

of

the

idea

are proposed.

EQUIVALENT POLE CONCEPT

Figures

1

and

2

show a typical

tapered

transmission

pole

of length 'L' and cross sections associated with

different materials. Conventional

FE

pole modeling

usually involves a piece-wise linear approach where

the system is considered as made up ofseveral elements

of

equal length,

each

with a constant cross section15.

Alternatively, the entire pole can be transformed into

one single element of

constant

cross section (Fig. 3).

The idea is illustrated here by proposing the concept of

an

'Equivalent Pole' whose strength an stiffness are

approximately the same as that of the original tapered

system.

Fig. 1 Typical transmission pole

GI

Steel

Fig. 2 Pole cross sections

508 JOURNAL OF

STRUCTURAL

ENGINEERING


daq

1'7 ''

,f ..

.

I

:

_

....

.:

..

L

=:>

: ~ ~ ; ~ ;

.:

\i

;I

I . .

. I

.

. L

:1

'i/

I '

_; .... ; .

_i__Li

Fig. 3 Equivalent pole

/

For a given pole class and height, the base diameter

(and ground line diameter) and taper are fixed. For

example, Class 1 wood poles have a tip diameter of

8.60 inches (21.8 em) and a taper

of

0.12 in/ft (3

em/

m), which gives a base diameter of 15.7 inches (40

em). For steel poles, the taper is slightly larger at 0.16

in/ft (4 cm/m). Class 1 steel poles have a top diameter

ranging from 7.25 inches (18.4 mm) to 10 inches (25.4

mm), depending on the manufacturer.

For stiffness, deflections and/or rotations under

various loadings (Fig. 4) are evaluated. The load cases

cover axial loading

a),

bending b,

c

an jl

d)

and torsion

e). The strength criteria considered

her.e

are buckling,

bending and torsion:. The diameter of the equivalent

pole, deq. which satisfies both stiffness and strength

conditions, is the parameter governing equivalency.

N

p

r.t

T

M

ll)

{b

d)

e)

Fig. 4 Loadings considered for equivalency


3/12

Loadings on transmission structures involve dead

the

expressions

for these stresses are more or less

loads, ice loads, wind pressure and wire tensions,

identical.

depending on the type of structure. Most tangent

Numerical

values

of equivalent

diameters

are

suspension) transmission poles i.e.) those primarily

calculated

for

wood and

steel

poles

of various

loaded by transverse forces are governed by flexure. )heights. In each height class, the

m ximum

value is

They are also directly embedded into the ground or

determined. These are plotted for pole heights ranging

fixed to a concrete pier; so the boundary conditions

from 45 13.5

m)

to 90 (27 m) in Fig. 5 and Fig.

6.

are similar to that of a cantilever i.e.) fixed-free

All equations are assembled and solved with a special

conditions.

computer program

19

Tables 1-a and

1-h show the configurations and

16.00

equations associated with the stiffness and strength

criteria, for wood poles. Similarly Tables 2-a and 2-b

15.50

show the configurations and equations associated with

' '

15.00

.s

......

steel poles. These expressions are readily available in

14.50

-

' '

. . . . ~

literature

16

-

18

B

14.00

---

n each load category, the theoretical deflections or

....

slopes) of the original tapered system are compared

....

13.50

0

r

with those

of

the equivalent system; the value of deq

13.00

is computed from the equality. Typical computation

12.50

for selected loadings is shown in tbe Appendix. The

12.00

process is repeated for the

s t r e t t ~ t h

category. Tables 3

45

50

55

60 65

70

75

80 85 90

and 4 show the expressions obtained for deq in each

Pole Height

ft)

case. It can be seen that diameters for cases involving

bending and torsion are identical since the form of

Fig. 5

Equivalent diameters for wood poles

,' :

..

TABLE 1-A

EQUIVALENCY CONCEPT FOR SOLID WOOD) POLES

.

Oridnal Tapered Solid beam

1.

[

: : r N

2.

[

I

tp

3.

[

C)M

4.

f I I I I I I IIW

[ :::1

5.

E

~ T

r = Ai/Aa)-

I =

d,jda) =

l

tp

=

I

+ +

32)13 f

{ =

diJda

DEFLECTIONS

Equation for Deflection or Slope

Equivalent Constant Section

Equation for Deflection or Slope

at Free End

at

Free End

Beam

Col.

1)

Col. 2)

l

=

NL

I

EAa [In (l+r)lr]

I

1 N

l

=NL

I EAeq

l

=

P3l3Ela [dt/daP

I

r

l

=

P3 I 3Eleq

8

=

MLI1.075 E

0

[di/da ]1.587

I

~ p

O=ML/

Eleq

fiiiiiiiiW

l

= wL4 I 7.872

E

0

[db

I d

0

] 3282

I

I

t l

= wL4f Eleq

321/J

TL :Jr Gda4

I I

T

8 32

TL n G

deq4

JOURNAL OF STRUCTURAL ENGINEERING 509

Vol.

38, No.6,

FEBRUARY- MARCH 2012


4/12

TABLE 1-B

EQUNALENCY

CONCEPT FOR SOLID (WOOD} POLES

STRENGTHS

Oridnal Tapered Solid beam

Equation for Strength

Equivalent Constant Section Beam


Col. (3)

Col. (4)

6.

. .

Per=

[diJdaJ2-61

;r2

/,/42)

I

1 N

Per -;r2 E eq14L2

: [

: : J N

7.

a=32

l : ~ r d x

:

tp

a=

32M/

r

deq3

. i

dp

I

'

[

i

:

:X

8.

..

a=4wL2/;r dx3

a = ~

M/;r deq3

I

i

~

[

l

Q M

X

9.

a 4 w L 2 : ~ r d i

I I I I

I I

I

IIW

a

4wL2/;rdeq3

I

I

I I 1 I I IIW

I

I

[

J

x

10.

f;, 16T /;rdx3

cr

'fmax= l6T/;rdeq3

'

T

'

:x

All bending and shesr stresses refer to

rnid- ,lpan.

d:x

= lh.

(1 +

3)/da {

=db/ da

TABLE2-A

EQUNALENCY CONCEPT FOR HOLLOW (STEEL) POLES

DEFLECTIONS

..

Equation for Deflection or Slope Equation for Deflection or Slope


Oridnal Tapered Hollow Beam

at Free End

Hollow Beam

Col. (5)

1.

[

:::.1-N

A

NL

I

EAa [In l+r)lr]

11-N

2.

Jp

:

r

'

A=rJ PL3f2E C t [rbl

raP

i

:X

3.

[

QM

(}

=.

[ML/2ECt]*

[ ra

+

rb)/ ra

2

f'M

b2]

'

X

4.

I I If I I

2

w

IIIIIIIIIW

A = ~ w4 I 2E C t [rb-ra]4

II

'X

5.

~

() = [TLI GJa]* 1 J

:x

~ T

C

=

Cross-sectional constant related to shape = 3.29 ( 12-sided steel pole)

r= ATJAa) 1 = (riJr

0

) -1 f} = [2ln (ri/ra)]- [(rb- ra) I rb]* [3-

r

,/rb)]

=

3ra

[ - In

(ri/r

0

] [r

0

-

rb] +

ri/6 rb2)

+

lh.]

+

rb 1

=

(1

+ +

3

2

)/3

{33,{3 =diJda

510 JOURNAL OF STRUCTURAL ENGINEERING

Vol. 38. No.6. FEBRUARY- MARCH 2012

at Free End

Col. (6)

A

NL/ EAeq

A = P3 /3Eleq

O=MLI

Eleq

A=

V:,4/SEI

.

eq

(J=TLI GJeq


5/12

TABLE2-B

EQUIVALENCY CONCEPT

FOR

HOLLOW (STREEL) POLES

STRENGTHS

Oridnal Tapered Hollow

beam


Col. (7)

6.

Per= ([di/daJ261 n2 E lj4L2)

[

l+-N

7.

[

i

Jp

a=PLI2Sx

:X

8.

[

:

S \ M

a MI Sx

X

9.

1

I

I

I

I IIW

a

4 wL2

I n dx3

[

J

'x

10.

=0

r = 16 TIn d 3

X

:X

All bending and shes r stresses refer to mid-span.

r = (Ai/Aa) l = (ri/ra) l

TABLE3

EQUIVALENT DIAMETER FOR SOLID (WOOD)

POLES

Load

Expression for Equivalent Dilm:_).eter deq

Case

Stiffness Criteria

Strength Ch.teria

1

[r/ln l+r)]O.SO

da

[

db2.67 dal.33]0.25

2

[db3

da]0.25

Y:z (1+,8) da

3

[1.075 db .581 d}.413]0.25

Y:z (1+,8) da

4

[0.984 db3.282

da0.718]0.25

Y:z

(

1+,8)

da

5

[3,83/l + 8 + ,82]0.25 da

Y:z

(1+,8) da

TABLE4

EQUIVALENT

DIAMETER

FOR HOLLOW

(STEEL) POLES

Load

Expression for Equivalent Diameter deq

Case

Stiffness Criteria

Strength Criteria

1

[rlln (I+ r)] da

[db2.61

dal.33]0.33

2

[(5.34 1]) (rb- ra)3]0.33

Y:z

(1+,8) J

a

3

2.52*[ra2 rb2 ra +

rb]0.33

Y (1+,8)

da

4

[(l/8;) (db-

da)4]0.33

Y (1+,8) da

5

[3,83/l + 8 + ,82]0.33 da

Y:z

(1+,8)

da

It

is observed that for wood poles, the maximum

equivalent diameter from deflection point of view

corresponded to the case with uniform load whereas


Equation for Stre

Hollow

Beam

Col. (8)

I

1 - N Per= n

2

E Ieq I

;

tp

I

a

PL/2

:

I

~

a M/ Seq

I I I I I I

I IIW

I

i

I

a

4 wL2 In

f

T

r=

l 6Tind

it referred

to

axial compressive load for

pole. Equivalent diameters determined frc

perspectives came from bending stress for

W

and axial compressive stress for steel poles.

19

18

..-._ 17

3 16

:)

0 15

0 14

13

12

L: :

v

/

v

/

/

45 50 55

60

65

70

75

80

Pole Height (ft)

Fig. 6 Equivalent diameters for steel poles

For example, the maximum equivalent

di deq

=

[da ] ~

Derivation of Equivalent Diameter for Case 2 Hollow

Pole)

Equating col. (1) and 2) from Table 2-a:

t = I PL

3

2 E C t [rb- raP=PL

3

13 Eleq

or

2 3.29) t

[rb-

raP 7 = 3Ieq =3 0.411

delt)

=> 1.233

del=

6.58

[rb- ra]

3

1

7

or

del= 5.34 [rb- ra]

3

1 7

=> deq

= [5.34

[rb-

raJ31q]

3

where;

J

=

[2

In

r ~ r a ) ] -

[ rb-

ra)lrb]

*

[

rafrb,)]

Nomenclature

f3

=

d ~ d a )

J , ~

parameters as defined in Table 2-a, b

tjJ

= parameter as defined in Table 1-a

a

Bending Stress

7:

=

Shear Stress

A a

Area at top= 1t

di/4,

Ab =Area at bottom

=

1t dil4

w o o d )

A a

=

Area at

top=

3.22 da

t,

Ab =Area at bottom

= 3.22 db t steel)

EI

Flexural Stiffness

da

Diameter at Pole Top

db

=

Diameter at Pole Bottom ground line)

deq

Diameter ofEquivalent Pole

dx Diameter at Pole at Mid Span or Height

E Modulus

of

Elasticity

Fb

Maximum Bending Stress

Fy

Yield Stress of Steel

G

Shear Modulus

I a

Moment ofinertia at Pole Top= 1t da

2

164

wood)

leq

Moment of Inertia

of

Equivalent Pole =

1t

da

2

164 wood)

I a

Moment of Inertia at top = 0.411 da

3

t

steel)

leq

Moment of Inertia of Equivalent Pole =

0.411 deq3 t steel)

Ja

Polar Moment of Inertia at Pole

Top= 2*1a

=

n d/4132

wood)

Ja

Polar Moment oflnertia at Pole

Top=

2*Ia

= 0.822 da3 t steel)

L

Length of Pole

M Moment

N

Axial Load

JOURNAL OF STRUCTURAL ENGINEERING 517

Vol. 38, No.( ,

FEBRUARY- MARCH

2012


12/12

p

Lateral Load

r

parameter as defined in Table 1-a

Ya

Pole Radius at Top

rb

Pole Radius at Bottom (ground line)

t Thickness

of

Steel Pole

Sa

Section 'Modulus at Pole Top n da

3

132

(wood)

Seq

Section Modulus

of

Equivalent Pole n

dell3 (wood)

Sa

Section Modulus at top 0.822

da

2

t (steel)

Seq

Section Modulus of Equivalent Pole

0.822

deit

(steel)

T Torsion

w uniform load on beam

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1.

2.

3.

4.

Technical Overview, Shakespeare Composite

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(Discussion

on

this article must reach the editor

before

May

31,

2012

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