Probabilistic design of prestressed concrete poles

Building and Environment, Vol. 15, pp. 49 56

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© Pergamon Press Ltd. 1980. Printed in Great Britain

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~

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]

0360-1323/80/0301-0049 $02.00/0

Probabilistic Design of Prestressed Concrete Poles S. S. R A O * A probability based procedure is presented for the optimum design of prestressed concrete poles. The cube strength of concrete, the ultimate strength of steel, the jacking stress at transfer, the cross sectional dimensions of the pole, the lever arm at which the load acts and the magnitude of the load acting on the pole are treated as random variables. The results obtained by the probabilistic design procedure are compared with those given by the deterministic procedure. The probability of failure of the pole obtained from the analytical method is found to be in good agreement with the value predicted by Monte Carlo simulation. The effect of variation of parameters like probability of failure and variability of cube strength of concrete is also studied. The optimum cost of the pole is found to increase as the probability of failure decreases. For large values of variability of concrete strength and small values of probability of failure, the compressive stress carrying capacity of the top section of the pole is found to be critical at the optimum point.

of poles made of steel, prestressed concrete and reinforced concrete, and concluded that prestressed concrete poles were the most economical ones [1]. Intuitively also one can feel that, since concrete is weak in tension, it would be economical to use prestressing provided that concrete is of high quality. Normally in bending elements, the prestressing force is applied eccentrically so as to produce compression at fibres where working load would cause tension and very little compression or small tension in areas where the working load causes compression. But in the case of wind loading, the load is reversible and any particular zone may experience both tension and compression at different times. Thus one cannot have eccentric prestressing. However the pole can be given axial prestressing in this case. Another factor to be considered is that the top section will be subjected to large amount of compressive force due to the application of prestressing force. Thus there will be two design criteria to be satisfied in the problem: (i) the top section must be safe against the compressive force induced by the prestressing force, and (ii) the base section must be safe against the bending moment caused by the external load. The expression for the deterministic design according to the ultimate load design method are given by (Fig. 1):

NOMENCLATURE As

b D

d Fn~ Fu

f g~ H hi h2 kb L l M. N(xl,x2) P PI R

Ro so, ss sst

Vx t~ x

Z1

half of the total reinforcing steel area (for one direction of the load P) uniform breadth of the pole design vector effective depth at the base of pole load factor against dead load load factor against live load objective function constraint function total height of the pole depth at the top of pole depth at base of pole = d + cover thickness bond factor (taken as l since it is a bonded pretensioned construction) generalized load lever arm ultimate moment of resistance normally distributed random variable having a mean value of x~ and a standard deviation of x 2. transverse wind (working) load probability of failure generalized resistance reliability = l - Py cube strength of concrete ultimate strength of steel steel (jacking) stress at transfer coefficient of variation of x (tT~/~) standard deviation of x mean value of x upper limit of integration INTRODUCTION

0.68 bh j so, > 2 AsSstFdt

R U R A L electrification programs in any country require numerous transmission line poles. Since transmission poles are basically bending elements subjected to horizontal wind loads, a tapered shape will be ideal for them. The choice of structural material is based on factors such as available resources and functional requirements. Mhatre conducted a comparative study

(1)

(for safety of top section at transfer) (2)

M , > P" IF u

(for safety of bottom section) where

(°)

M , = k b'Asss d - ~

*Professor, Department of Mechanical Engineering, Indian Institute of Technology, Kanpur-208016, India.

and 49

,

13,

50

S. S. R a o

~-h,4 u

_] "

i

I

-la2~

1.2M

T

Fig. 1. A concrete pole.

As s s

a - 0.68 bs~,"

(4)

Tile design variables in this problem are b, h~, d and A~. These are to be chosen so as to satisfy the inequalities (1) and (2). The conventional design procedures, based on equations (1) and (2), have been followed by most of the designers until recently [2]. These procedures assume that the geometrical and material properties and loadings are deterministic. It has been established that the conventional design procedures based on factor of safety are not rational [3]. In this paper, the formulation and solution of probability based design of prestressed concrete poles is presented. The results obtained by the probabilistic design are compared with those given by the conventional deterministic design. The effect of variation of some of the parameters on the probability based design has also been studied.

The lever arm l also becomes a random variable during erection and fitting. The area of steel is not taken as a random variable as reinforcing bars are manufactured in factory under quality control, and hence its variability will be negligible compared to that of other quantities. Thus there are eight random variables, viz., s~., s~, s,,, b, h 1, d, P and l in this problem. In this work, all the random variables are assumed to tollow normal distribution, and mean and standard deviations of functions of random variables are calculated by the partial derivative rule. Although the assumption of normal distribution is made for simplicity, the same procedure can be adopted even if the random variables follow non-normal distributions. To derive the expressions for mean and standard deviation of a function of random variables, consider a general function t~=qJ(vl,v 2 ..... v,) where Ul,l~2..... vn are normally distributed random variables with known mean values (6~,152..... /~,) and standard deviations (av,, %~,...,a,,). The Taylor's series expansion of the function 0 about the mean values of v I gives if/ = ~/(Vl' V2 ..... 12n) ~ I/J (t~l, f:2 .....

~_ ~0

+, , ~ (et,e2,.,e,I

(v,-~O.

/~n)

(5)

Thus ~ can be expressed as a linear combination of normally distributed random variables vi (since ~(vl, v2..... ~.) and all

~Ui (1~1, v2 .....

f'n)

are constants). Hence ~O also follows normal distribution and only the first two moments (mean and standard deviation) will completely specify its numerical characteristics. The mean and standard deviations of q* can be obtained from equation (5) as [4]. = ~/,071,z72..... ~,)

(6)

STATEMENT OF PROBABILISTIC DESIGN PROBLEM As many parameters of the design problem are random in nature, the problem is formulated within a reliability framework. The ultimate strength of prestressing wire and the cube strength of concrete have high mean values and like most of the recently developed high strength materials, they have a large variability about the mean strength. It would therefore be uneconomical to use mean values only (by neglecting their variable nature about mean values) in deterministic design with large factors of safety. Hence the problem of the design of concrete poles is considered according to probabilistic approaches. In this work, excepting the area of steel A~, all other quantities are taken as random variables. The jacking stress s~, is taken as a random variable because it is not possible to precisely control the jacking operation. The variables b, h 1, h z or d are random since formwork cannot be placed exactly according to the design specifications. The wind load P is random by nature.

In probability based design, the design requirements, analogous to equations (1) and (2) of deterministic design, are stated in the form [5]: P [ R - L >- 0] >=R o

(8)

where P [...] denotes the probability of occurrence of the event [...], R the generalized resistance, L the generalized load and R 0 the specified reliability. It is to be noted that the probabilities stated in equation (8) are notional and represent the uncertainties that arise only due to the variability of the parameters. Hence an extra random variable needs to be included in the analysis to represent the uncertainty in the way the equations used represent the poles when built. However, only the variability of the parameters is considered in this work for simplicity. The probabilistic design requirements can be stated in several ways

Probabilistic Design of Prestressed Concrete Poles m

2

as alternatives to equation (8). Two commonly used alternative statements are 1-6]

2

2

a l l = L 1 (V~,,)

a~ =(A~'gfl. PIR>_-1]>R o

51 (23)

l'5A~s--Z~ ~ . g~, -/) z V~, + (A~' g=a)2" V~

(9) + (0.75A: "g-2\ 2

\

and

P[L - R < 0] < Pf

(10)

~gc. * ) (V~ + VZ~.) 2

2

2

aL2 = L 2 ( V 1, + V 2 ).

where P: is the probability of failure, equal to 1 - R o. According to the first design criterion, the probability of realizing the prestressing force less than the compressive force carrying capacity of the top section of the pole must be greater than R o. Here the load and the resistance are given by

L 1 =2A,s~,

(11)

R 1=0.68bhls~,.

(12)

(24)

(25)

Here the standard deviation of a variable, say G , has been replaced by V~" ~ where V~ is the coefficient of variation and ~ is the mean value of the random variable x. It is to be noted that the expression for M~ is based on the assumption that the section is underreinforced. For the section to be under-reinforced, the following condition is to be satisfied.

and

Thus the criterion P ( R ~ - L 1 > 0 ) > R 0 can be restated in terms of a normalized variable as R1 - E l

(a2j+a2 )l/2 >=Z1

(13) d b _>-. -2

where

Ro=f

Another code specification [7] is that the area moment of inertia of the section in the breadth direction should be at least one fourth of that in the depth direction. This requirement leads to the condition

e-t*2/2) •dx.

(14)

Similarly, according to the second design criterion, the probability of realising the applied bending moment less than the moment of resistance of the pole must be greater than R o. Here the load and resistances are given by

L2=P'I

(15)

R2=M =A .sfl( 1 0.75A:~'] bds~. ]"

(16)

and

Hence the design criterion becomes

R 2- L z

(a~2 + a~2)1/2 > z,.

(17)

The mean values and standard deviations of the loads and resistances can be calculated as follows (using equations (6) and (7): /~1 = 0.68 6~lScu

(18)

L, =2A, .g,,

(19)

/~2=A . g f l ( 1 0.75A~.gs)

I2o)

L 2 =P-/-

(21)

~1 =R~ (v~. + v~, + v~)

(22)

)

(27)

If R 0 is specified, the value o~ z 1 can be obtained from standard normal tables and the design variables can be determined so as to satisfy the reliability based constraints, equations (13), (17), (26) and (27).

N U M E R I C A L EXAMPLE To illustrate the probability based design procedure, a numerical, example is considered with the following data: total length of the p o l e = H = N (8.8,0)m, lever arm=l=N (7,0.07)m, wind load=P=N (180,18)kg, n o m ~ a l grade of concrete M-350 with Scu=N (367.33, 42.61)-kg/cm 2, nominal ultimate steel s t r e s s = s s = N (16,200, 1215)kg/cm 2, jacking stress=s~,=N (10,500, 105) kg/cm 2, Ro=0.99999 , V A = 0 and Vh =Vb=Va =0.01. Here N(xl,x2) represents a normally distributed random variable with mean x 1 and standard deviation x 2. Corresponding to the specified reliability of Ro=0.99999 (probability of failure is 10-5), the value of z 1 can be obtained as 4.26 from standard normal tables. There are four design variables and basically two reliability equations, namely, equations (13) and (17), in this problem. The inequality (27) can be incorporated by taking b=d/2, but the inequality (26) cannot be taken with equality sign as it would make the section balanced whereas the aim is to use less amount of steel. However, from practical considerations, a value of 15cm is selected for hx. By taking d=2b, the unknowns b and A~ can be found by solving the two nonlinear equations (13) and (17). The values thus obtained will be acceptable if they satisfy the inequality (26). The inequalities (13) and (17), when written in

S. S. Rao

52 equality form, give

zation problem as follows:

2 =ZI(0"R, 2 2+~[1 (/~1-L1)

)

(28)

find the vector of mean values of the~ design variables

J

and

(R2 - L2)2 = zlZ(a2. +

a~.).

(29)

For the given data, /~1 =3740~, a2~ = 19.05 x 104/92, L 1 =2300A~, a~1=52,900A~z, z~=18.2 and hence equation (28) gives the quadratic equation (A~) 2 - 1 6 . 3 ( ~ ) + 50.0 = 0

which minimizes the criterion or objective function f(D) and satisfies the constraints

from which the value of b/A, can be found as 12.18. With the values of b=12.18A~, d = 2 4 . 3 6 A s, /~2 =(39.4A~ - 4.39As) x 104, L 2 = 12.6 x 10 4, 622 =(881.5A~-388A3+69.4A2)x106 and cr~ =161 x 106, equation (29) gives the following fourth order polynomial equation for As:

gj(D)<0, j = 1,2 ..... m.

(30)

Although the actual cost of the pole is comprised of materials' cost, formwork cost and labour costs, only the materials cost is considered for minimization in this work. Thus the objective function can be stated as

12.52A~-2.49A~-8.87A~ + A~+ l.16=O.

f(D)=c~'H'b(h~+d+2.5)+G'H'2"As

This equation can be solved to obtain As=0.77cm 2 and hence /~=9.37cm and d=18.70cm. With the values of design variables found, the right side of the inequality (26) can be evaluated as

where c c and cs indicate the costs of concrete and steel per unit volume. Since H is a constant, the objective function can be rewritten as f (D) =162(61 + 63 + 2.5) + c r "fi4

0.236~ g2)= 0.936cm2. As this value is greater than the actual steel area of 0.77cm 2, the section remains under-reinforced. Thus the results of the probabilistic design can be given as h 1= b= d= h2 = As =

(15.00, 0.15) cm; (9.37, 0.0937) cm; (18.74,0.1874)cm; (20.24, 0.2024) cm with (2.5, 0.025) cm cover; (0.77, 0.00) cm 1.

In actual specification, these quantities may have to be rounded-off, for example, as b=(9.5,0.095)drm, d = (19.0, 0.190) cm, h 2 = (20.5,0.205) cm, and A~ =0.78cm 2 (11 nos. 3 m m ~b wires). With these roundoff values, the reliability of the concrete pole becomes 0.999997888 at the top section (first design criterion) and 0.9999744 at the base section (second design criterion) of the pole.

M I N I M U M COST D E S I G N In this section, the design problem is cast as an optimization problem. The cost function is minimized subject to the four constraints given by equations (13), (17), (26) and (27), and two side constraints. Since an infinite number of feasible solutions satisfy the constraints of equations (13), (17), (26) and (27), a criterion like cost can be used to judge the superiority of one design over another. Out of all the feasible designs, if the one corresponding to the least cost is required, the problem has to be posed as an optimi-

131)

(32)

where c,=(cost of steel/cost of concrete). The constraint functions can be stated as follows: R1 - Z l (a21 + o . 2 t ) l / 2 ~-Z1

gl(D)

g2( D ) =

R 2 -

L 2

[~2 ,_,,2 ~1/2 ~-Zl I"~R2 | ~L2 !

g3(D)=As

0.236bdgcu

(33)

(34)

(35)

g4(D) = d / b - 2.0

(36)

gs(D)=b/10- 1

(37)

g6(D) = h~/15 - 1.

(38)

Here the first four constraints have been derived earlier and the last two constraints are practical constraints on the m i n i m u m size, below which all the reinforcing wires cannot be placed properly. For the solution of the constrained minimization problem, the interior penalty function method or the sequential unconstrained minimization technique (SUMT) is used. In this method, the constraints are appended to the objective function to form a new function 4~ as [8] m

=f-r ~ l j=l gj

(39)

and the unconstrained minima of q5 are found several

Probabilistic Design of Prestressed Concrete Poles times with decreasing values of r till the actual constrained minimum of f is reached. The termination criterion used in SUMT was that whenever the change in the function value is less than 1~o for two consecutive values of r and at the same time the change in all the design variables is less than 0.005, the process is stopped. Otherwise, the process will be terminated if the number of iterations exceed a certain value or if the value of the penalty parameter becomes smaller than a certain value. For the unconstrained minimization of ~b, the variable metric method due to Davidon, Fletcher and Powell has been used. Here also the same convergence criterion on design variables and function value has been used. For linear minimization that arises within the unconstrained minimization, the golden section search scheme has been used with suitable convergence ~criterion [8].

2'52 344

2,2,6

_e ~. 328 ~o 320

V=¢.=

312 ._E

_ j ll.5O% - 12 . oo|,

I

Fig. 2. Minimum cost vs probability of failure.

20.O

o•19.0 .c_ o 180

g

a

V$cu=O-1507 V$¢u'O'"6 ]

v, I 10-4

I iO-S

10-6

Decreasing probability of failure

Fig. 3. h I vs probability of failure.

h, r2 OO1 b = 15oo

11.8

125"001 , corresponding cost = 553.75. / /

IL4f

L l.~j

,,°1t

The results indicate the following behaviour: 1. Figure 2 shows that although the optimum cost of

For V~c==0.05, 0.075 and 0.116, the optimum cost approaches a constant value below a value of Ps =10 -5" 2. Normally a higher optimum cost is expected (for any specified value of reliability) whenever the variability of strength increases. This behaviour is evident even in Fig. 2 up to a value of V~c =0.1t6. However, for ~c =0.15, the optimum cost is less than

/

1/

v'°°'°°"l

t5.o 14.010_a

the pole increases with decreasing probability of failure for all values of V~, the optimum cost increases at a faster rate in the case of V~,==0.15"

i0-e

iO-5

Decreasing probability of foilure

As this starting point violated the constraint on the probability of failure at top section at transfer for V~, =0.15 with P I = 10 -5 and 10 -6, the following starting point has been used in these cases.

A,

I

i0 -4

L 1.165j

d

......

~./"I--

288

o ~ 16.0 corresponding cost = 445.15.

~

296

~ 17..0

~'20.0001

.

v,¢==o.oT~.-~S~/

~ 2,04

PARAMETRIC STUDY

hi

0

s~ ~ -

2810-3

The effects of variation of the reliability and the variability of the cube strength of concrete on the minimum cost design are studied. Four values of the probability of failure (10 -3, 10 -4, 10 -5 and 10 -6) and four values of the coefficient of variation of concrete (0.050, 0.075, 0.116 and 0.150) were used and the results obtained are shown in Figs 2 to 6. Excepting for two cases (when P s = 1 0 - 5 and 10 -6 with V~. =0.150), the following starting point was used for optimization:

53

2

/1 v,°:°-'°°7 I / .... I

I ~'o~'°"~-L /

10,2 . . . . . . .

rV,o."~°~ I

~ 9.8 94 f 9DIO-a

10-4

10-5

iO-e

Decreasing probability of failure

Fig. 4. b vs probability of failure.

S. S. Rao

54 230

probability of failure at the optimum point for V~° =0.05~0.116. For V~ =0.15, all these variables can be seen to increase with a decrease in the value of

225

Pf. E ¢J

220

The reinforcing area (As) at the optinmm design has been found to increase nearly linc~l~ with an increase in reliability of the pole for l~ =0.05 0.116. However, for V~,,=0.15, the optimum value of As first increases as P/ changes from 10 -3 to 10 4 and then decreases in the range P: = 10 -4 to 10 -~'.

21.5 Vscu" o .E

21.0

20,5

/

I,I ,o.o

C O M P A R I S O N WITH DETERMINISTIC DESIGN

19,5

19.0 10-3

I 10-4

I

10-5

10-6

Decreasing probabiiify of failure

Fig. 5. d vs P: at optimum design.

The deterministic design of the pole is considered as per I.S. Code specifications for comparing with the probabilistic design. The ultimate load method is used by taking the load factors for dead and live loads as 1.5 and 2.5 respectively. The design requirements can be stated as follows: 0.68 bh 1s,., >=2A:srF el

(40)

M , > P . I "Fu

(41)

0.8~

E (J

0.8(:

where Vscu~ 0.116

076 o

.E

~-

Vscu=O.075 , ~ f "

v,~:o~

0.72

_

_

M"=As'ss'dI1

/y'~_~ ./v,~,-o.,5o

"6 0.64 "1I

10-4

(42)

The mean (nominal) values of the parameters are used in the computation. Thus s,.,=350kg/cm 2, ss =16.000kg/cm 2, s~=10.500kg/cm 2, P = 1 8 0 k g and / = 7 m . By assuming h 1 = 15cm and b=d/2, strict equalities of equations (40) and (41) give

0.68

0.60 10-3

0L75-As-s~]'bJ'dsc,

_

I

10-5

10-6

Decreasing probability of failure

Fig. 6. A~ vs P/at optimum design.

that of V~, =0.116 for P : = I 0 --3 and 10 4. This must be due to the existence of local minima in the nonlinear programming problem. Another important point to be noted regarding the results of Fig. 2 is that for V~=0.15 with P y = 1 0 s and 10 6, the compressive stress carrying capacity of the top section of the pole has become critical while it was not so in all the other cases. The resistance of the bottom section, which is under-reinforced, is basically governed by steel, whereas at the top section, the concrete has to take the compressive stress. This shows that with a high variability of concrete strength and high values of desired reliability, the top section actually governs the design and hence the abrupt change noticed in the optimum cost in this case can be justified. 3. Figures 3 6 show the variation of design variables with respect to the probability of failure of the concrete pole. The depth of the pole at the top section (h~), the breadth of the pole (b) and the effective depth at the base of the pole (d) can be seen to be essentially constant for all values of

b=9.65A~(d=19.30A,),

and

A ~ = l . 1 3 c m 2.

Thus the deterministic design gives b = 1 0 . 9 c m , d = 2 1 . 8 c m and if 6 nos. 5 q~ wires are provided. A~ =1.165cm 2. The right hand side of inequality (26) gives

0.236(.bds""]= 1.38 \ s, / which is greater than the actual steel area used. Thus the section of the pole is under-reinforced. It is not possible to directly compare the deterministic design with the probability based design since the approaches are totally different. In deterministic design, all the quantities are fixed and a structure, which will fail when the loads are increased by a certain factor of safety, is designed. In probability based design, some of the quantities influencing the design are taken as random variables and the structure is designed in such a way that the probability of failure of the structure under working load remains less than a certain preassigned quantity. Thus there is very little relationship between the load factor and the probability of failure. However the probability of failure of the pole corresponding to the deterministic design at working load

Probabilistic Design of Prestressed Concrete Poles (with the mean values and coefficients of variation of random variables as given earlier) can be found. Thus for the top section of the pole, /~1 =43 x 103, L1 =26.8 × 103, 0"21=25.1 X 106, tTL, 2 =0.072 x 106 and hence

55

ximately same as those calculated earlier, the probability of failure can be computed by first finding Za as

R-L

zl

43.0 - 26.8 5.02

z , - ( , ~ + 0.~),/2 .

3.22.

This gives a probability of failure of 0.64 x 10 -3. For the bottom section of the pole, /~2=36.3 x 104, /,2 = 12.6 x 104, 0.25= 569.5 x 106, 0"22= 161 x 106 and hence

zx

36.3 - 12.6 ----8.8. 2.7

This corresponds to a probability of failure of about 10-s. From these results, it can be seen that for the given values of mean and coefficient of variation of strength and load, the probability of failure at working load at bottom is very low compared to the probability specified in probabilistic design. Further, the probabilities of failure at top and bottom sections are totally different for the deterministic design whereas they are specified to be equal in probabilistic design. Thus although the structure has a large safety against the failure of the base section, the probability of failure is high at the top during the transfer of prestress and hence the overall probability of failure of the pole will be high (the total probability of failure can be taken as the sum of the two probabilities as an upper bound). Hence in a good design, normally these two probabilities should be kept the same. However, it is to be noted that the specification of these two target probabilities depends upon the nature of the limit state and the consequences of failure [9]. For example, in some applications, the failure of the component does not lead to the failure of the entire system. Even if the entire system fails, it may not result in any catastrophy. Under these circumstances, the target probability of failure may be specified to be a relatively large quantity. On the other hand, there are applications like aircraft and nuclear plants where the consequences of failure are serious and hence the target probability of failure in the design of such systems should be very low. In the probabilistic design procedure presented in this paper, the probabilities of failure of the top section and base are specified to be same. By proportioning the design variables (essentially the area of steel), the two probabilities have been made almost equal at the final design. The probability of failure corresponding to a deterministic ultimate load (= working load × load factor) can also be computed. For this, assume a suitable load factor and find the values of resistance and load used in ultimate load design (which can be interpreted as mean values of resistance and load). Then by assuming the resistance and load as normally distributed random variables with mean values same as those used in ultimate load design and standard deviations appro-

Thus for the top section of the pole, /~1 =43 x 10 3, L 1 =40.2×103, a~,=25.1×106 , aLZl=105, z1=(43.0 --40.2)/5.02=0.556 and hence P:=0.288. For the bottom section of the pole,/~z =36.3 × 104, L2 =31.5 × 104, a ~ = 569 x 106, a~2 = 1000 x 106, z 1= (36.3 - 31.5)/3.88 =1.235 and P:=0.109. An upper bound on the total probability of failure can be obtained as 0.288 +0.109 = 0.397. Thus the P: of the pole designed at its theoretical ultimate load is very low compared to what it should have been (nearly one) has the ultimate load design really meant ultimate load design. It is to be noted that the probability of failure at ultimate load design would be nearly equal to one if all the variables are deterministic, that is, if the standard deviations of load and resistance are zero. Due to the existence of finite values of a R and aL, and the assumption of normal distribution, theoretically it is possible to realize infinite values of resistance and load (with different probabilities). Hence the probability of failure at ultimate load under realistic conditions of nonzero values of a R and aL came out to be 0.397 instead of 1.00. This shows that the ultimate load design, though a better approach compared to elastic design, is not really rational. This is basically because that all the deterministic designs are made with only mean values and some fixed code-assigned load factor (or safety factor, which it now seems has little physical meaning) and by totally neglecting the variabilities of strengths and loads. To verify Pf predicted at the ultimate load design of the pole, the Monte Carlo simulation method is also used. Five hundred poles have been simulated and probabilities of failures of 0.266 for the top section and 0.128 for the bottom section, which give the upper bound on the total probability of failure as 0.394, have been obtained. It can be seen that the results of Monte Carlo method are in good agreement with those calculated theoretically.

CONCLUSION 1. The minimum cost of the pole has been found to increase as the probability of failure decreases. This is to be expected since, under any prescribed design conditions, a more reliable concrete pole costs more. 2. The present method permits the specification of different probabilities of failure for the compressive force (prestress) carrying capacity of concrete at top section and moment of resistance of bottom section of the pole.

56

S. S. Rao

3. The optimum cost of the pole has been found to increase with an increase in the value of coefficient of variation of concrete strength except for Vsc° =0.15. 4. The probability of failure of the pole predicted by the Monte Carlo simulation method has been

found to be in good agreement with the one given by the analytical method. 5. The probabilistic design procedure presented in this paper is expected to be more realistic and rational since the variabilities of random design parameters are considered in the problem formulation.

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