transmission tower limit analysis and design

IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 4, April 1981

TRANSMISSION TOWER LIMIT ANALYSIS AND DESIGNBY LINEAR PROGRAMMING

Jun W. Lee, Member, IEEEStructural Research, Inc.

Madison, Wisconsin

Abstract: A method is presented for the limitanalysis of indeterminate space trusses utilizing thestatic equilibrium equation and linear programmingtechniques. Special emphasis is given to obtainingcollapse loads for self-supporting lattice transmis-sion towers. A piece-wise step-by-step linear solu-tion employing the stiffness matrixhas been the usualway of handling this problem. The static equilibriumequation has several advantages over the stiffnessmatrix equation for limit analysis: a direct solu-tion can be obtained in one step; tension onlymemberscan be given zero compression capacities; and spatiallyunstable joints can be stabilized by assigning zerocapacity members to them. A very efficient computerprogram was written by rewriting the linear program-ming equality constraints in a condensed form and ex-ploiting its sparsity (zero terms). Example problemswith the computer program are discussed to demonstratethe analysis method. Potential application of thegeneral theory to the design problem may have sub-stantial advantages over current methods.

INTRODUCTION

In dealing with indeterminate lattice transmissiontowers, presently in the transmission line industrythe "Stiffness Method" (also called "DisplacementMethod") is generally used for the analysis. Themethod involves the distribution of joint loads intovarious members according to their relative individualstiffness, in view of geometric compatibility as wellas force equilibrium at various joints. In con-junction with this method, the -first yield conditionis normally used as the criteria for the determinationof the load carrying capacity of the structure.

For determinate structures, the first yield con-dition is the ultimate capacity of the structure.However, for structures with multiple degrees of in-determinacy, the collapse load is normally higher thanthe load causing the first yield condition. With therecent concern in failure containment as well as up-grading of existing lines to meet higher conductorloads, the ultimate capacity of a transmission towerhas become more important in transmission tower design.Limit analysis is such an analytic procedure for ob-taining the ultimate load of a structure at collapse.

Limit analysis techniques were established byearlier researchers by using the piece-wise linear aswell as mathematic programming techniques either in-dividually, or in combination. The techniques involvedthe replacement of yielded members by a set of equiva-lent forces and the subsequent revision of structuregeometry and its stiffness matrix in a stepwise manner(10). In this procedure, each step involved the in-

80 SM 681-7 A paper recommended and approved by theIEEE Transmission & Distribution Committee of theIEEE Power Engineering Society for presentation atthe IEEE PES Summer Meeting, Minneapolis, Minnesota,July 13-18, 1980.Manuscript submitted February 11,1980; made ayailable for printing May 9, 1980

H. Gordon Jensen, Member, IEEEBonneville Power Administration

Portland, Oregon

version of the stiffness matrix and the completesolution of the structural system.

In this paper,a distinct approach is proposed forthe limit analysis and design problems. A linearprogramming formulation based on the basic, limitanalysis theorems will be presented.

Although the use of linear programming in a struc-tural analysis procedure is not commonly used, thedirect applicability of the linear programming al-gorithm to limit analysis will become obvious in con-

junction with the Upper and Lower Bound Theorems whichwill be presented later in this paper. The linear pro-gramming technique used in this paper will be theSimplex Method (11), which is a maximization procedureon an objective function with a number of inequalityconstraints. In general, a linear programming problemcan be represented by Equation (1):

Max. W = b x + bx + ... b x1 22 mm

Subject To: c2 = a21x1 + a22x2 + an nx + k->0

C2 = 211 + 222 +- 2mmn 2-

c =a x +a x + ...a x + k > 0m mll1 m2 2 mn n m-

X1 > 0, x2 > 0, ... x > 0 (1)

Where W is the objective function, and c1... c arethe underlying constraints. The Simplex Method findsthe solution by going through vertices defined by theintersection of the n hyperplanes: c.1= 0, c.2= 0,...c. =0 and x. =0, x =0, ... x. =0, where

ip sij 2j 3q W'p + q m=n The soilution is3reached when W is at itsmaximum and none of the constraints are violated. Iffor xl - 0, x2 = 0,. x = 0 none of the constraints2 ~~~n.are violated, the progranuning problem is said to beprimal feasible. Primal feasible problems can besolved efficiently through the use of well establishedpivotal techniques (11, 12). In this paper, it will beshown that the limit analysis problem formulatedthrough linearprogrammingis a primalfeasible problem.

There are many types of transmission. tower structurescurrently in. use: lattice structures, pole structures,frame structures, as well as truss-frame type struc-tures. Although the general concept presented hereinis applicable to tower structures in general, thescope of this paper is directed towards lattice struc-tures. In considering the limit analysis solution, itis assumed that the members are prismatic, and exhibitan ideal elasto-plastic load-deflection behavior.

GENERAL FORMULATION - ANALYSIS

The approach to limit analysis presented in thispaper is based on two classical theorems, namely, theUpper Bound and the Lower Bound Theorems (7). They arestated here without proof as follows:

1. The Upper Bound Theorem: For a given frame/truss subject to a set of loads P, the value P whichis found to correspond to any assumed mechanism mustbe either greater than or equal to the collapse loadP.u 2. The Lower Bound Theorem: For a given frame/

truss and loading, if there exists any distribution offorces throughout the frame which is both safe and

(C) 1981 IEEE

1999

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statically admissible (satisfying static equilibrium)with a set of loads P, the value of P must be less thanor equal to the collapse load Pu

Consider a truss with joint loadings as shown inFigure 1. Assuming a load vector field "P: as shownin Figure 2 and a force vector field "F" representingmember forces as shown in Figure 3, we can, from con-sidering equilibrium of members and joints, arrive atthe following equilibrium equation:

{P} = [A]{F}

lP -v.

T4P

(2)

3P

4P

Figure 1: Imposed Loads

where pk = {p}, the load matrix. The force matrix Fand the scaler p are all variables, and the columnmatrices P, Mt and M are known quantities. Ingeneral, Equation (7) represents ageneral mathematicalprogramming formulation for limit analysis followingthe equilibrium approach.

Defining the following variables:NF = Number of Forces (members)NP = Number of Degrees of FreedomNI = NF - NP = Degree of Indeterminacy

then, two points are observed for the above linear pro-gramming tableau:

1) The tableau dimension is (2NF+NP+l) X (NF+2).2) The tableau is always primal feasible.

Considering the duality of the above linear programmingproblem, it can be shown that the dual formulation ofEquation (7) represents a formulation derived from theLower Bound Theorem (5). Consequently, the solutionof the linear programming problem as stated in Equation(7) takes into account simultaneously both the UpperBound and the Lower Bound Theorems.

In an indeterminate structure where NP # NF, thestatic matrix (A) expressed in Equation (2) is a rec-tangular matrix. It is possible to partition thestatic matrix such that:

[Al = [A*:a] (8)

PI=(pi P, P .. )T{P}=' P2 3 f NP

Figure 2: The Assumed Load Field

Af F4F5 10

FF3 ZY F7

1?K F2

F8

F9

where A* is a square matrix of size NP X NP and [a] isrectangular matrix of NP X NI. Similarly, we can alsopartition the force matrix,the tension capacity matrixand the compression capacity matrix as follows:

F* M* M*{F} = {...};{M } = {...}.;{M } = {..C..} (9)f m mt C

It can be shown that a non-singular A* submatrixexists for all stable structures. Physically, thepartitioning of the A matrix is merely the selectionof a set of redundant members such that the remainingstructure is a stable one. One systematic method ofselecting the correct redundant members is given in thereordering algorithm of the sparse routine, which willbe discussed later.

Assuming A* being non-singular, then its inverseexists. Using Equations (8), (9) and (2), F* can be ex-pressed in terms of the redundant member forces f asfollows:

-F --F*= -A* af +pA* k (10)

{F} = (F1, F2, F3, ... FNF)

Figure 3: The Assumed Force Field

If the tension and compression capacities of the mem-bers are (corresponding to the assumed force field)

Mt = (Mt Mt2 . )T (3)t ti t2' ""

tNF

= (M ,M M )Tc cil c2f...cNF (4)

then the necessary and sufficient conditions for stableequilibrium or equilibrium at bifurcation are thatEquation (2) be satisfied, and that

F< Mt (5)

F > -M (6).- c

Thus, if we have a load matrix with a common loadfactor p, we can, following the Upper Bound Theorem,formulate the equilibrium approach as follows:

Max. p

Subject To: (i) AF - pk = 0

(ii) -Mc < F < Mt (7)

Using the above equation, the primary formulation ofthe limit analysis problem presented in Equation (8)becomes:

Max. p

Subject To:

(i) -A* af + pA* i + M > 0c-+ f + m > 0

c

(ii) +A*- af -pA* 1k +M*>Ot -

- f + m > 0 (11)

The numberof constraints in Equation'(11) is 2NF, andthe number of variables is (NI + 1) where NI is thedegree of indeterminacy. Consequently, the size ofthe linear programming tableau becomes (2NF + 1) X(NI + 2). Since both Mt and M are positive vectors,

.t cthe tableau remains primal feasible.

An efficient computerprogram was developed to auto-mate the execution of the foregoing algorithm with thesparse matrix techniques.

An analysis of the ultimate capacity of a singlecircuit 345 KV tower under wind loading was performed


2001

using the computer program. The tower has an overallheight of approximately 105 ft (32m) above ground. Thegeneral configuration is shown in Figure 4. The dia-meter of the ground wire and phase conductor are

.385 inches (9.8 mm) and 1.737 inches ( 44.1 mm)respectively. Span is 1000 ft (305m) and wind direction isstrictly transverse. Only wind forces on the conduc-tors and shield wires are considered.

Table I presents the loading on the tower. Theultimate load factor and ultimate loads are tabulatedin Table II. The labeled members as shown in Figure 4are the yielded members.

TABLE I

Load Factors Applied' at Joints (Kips)- (448 N)

JOINTS

1

221

222529

X-ACTION

0.350.351.001.000.500.50

Y-ACTION

-0.12-0.12-0.64-0.64-0. 32-0. 32

Z-ACTION

0.000.000.000.000.000.00

y

Figure 4: 345 KV Tower

TABLE II

Ultimate Load Capacity (Kips)

JOINT X-ACTION Y-ACTION Z-ACTION

1 1.8152 1.815

21 5.18522 5.18525 2.59229 2.592

-0.622-0.622-3. 318-3. 318-1.659-1.659

0.00*0.000.000.000.000.00

Ultimate Load Factor = 5.18 kips (23,040 N)Equivalent Wind Load = 120 MPH (208 KM/H)

It is interesting to note that the total executiontime of this program is 69 seconds on a CDC 6500computer, with 174 degrees of freedom.

DISCUSSION

The linear programming formulation of the limitanalysis is based on the Upper Bound and Lower BoundTheorems presented earlier in the paper. It is impor-tant to note that the theorems pertain to the correctcollapse loads, but not necessarily the correct inter-nal force field. In the case that there is a one to

one correspondence between the collapse load factorand the internal force field, the correctness of themember forces is inferred through the uniqueness ofthe solution.

However, there may be situations in which thereexists more than one set of internal forces which are

statically admissible, and at the same time, give thesame collapse load factor. Physically, there are two

situations where this could occur: the inclusion oflocked-in internal forces in a substructure, and theexistence of an indeterminate substructure at collapse.The latter occurs when the failure of a substructureeffected the collapse of the load carrying capacity ofthe structure. In such a case, the forces in the re-

maining indeterminate substructure can be uniquelydetermined through elastic considerations taking intoaccount the proper stiffness of each individual memberas well as the initial locked-in forces created throughmisalignment induced during the fabrication and erec-

tion process. For the linear programming solution,since the algorithm considers the vertices of the con-

straints, the member forces will be given such thatsome of the members are at eithex compression or ten-sion yield.

For asubstructure at collapse, the linear program-

ming routine takes into consideration statical'ly admis-sible locked-in forces within the substructure.Therefore, the resulting forces from the linear pro-

gramming solution may include some pre-existingforces and consequently, may not compare directly withthe results derived fromthe piece-wise elastic method.However, for comparison purposes, it is possible toinclude a minimization routine to insure that thelocked-in forces are removed in all or most of themembers (13). Such an algorithm was included in theaforementioned computer program.

Following the same logistic development, a similarprocedure can be formulated for the limit design pro-

blem.

GENERAL FORMUIATION - DESIGN

For the design problem, it is common that severalloading conditions may have to be considered. De-noting the NLC number of loadings as Pl, P2, P3 .pNLC, and {L}, a matrix expressing the length of eachmember, the limit design problem can be written as:

Min. w = {L}T [Mt1/ftSubject To: I. For Loading Condition #1

1 1(1) p = AF

(2) -M < F < + Mc - - t

II. For Loading Condition #22 2

(1) p = AF2

(2) -M < F < Mc - - t

N. For Loading Condition #NLC

(1) pNLC =AFNLC(2) -M <F <M (1?)

where f is the tensile allowable stress, F is themember force matrix corresponding to the ith loadingcondition, and Mt, M are the tension and compression

ccapacity matrices respectively.


It should be noted that in the design problem, thematrices p' and A are known values, while thematrices McI Mt, F1 are all variables to be determinedfrom the optimization.

It is possible to perform thebasic transformationon the design optimization as presented earlier in theanalysis problem. Following notations used in theprevious section.

Let: A = [A*:a] M =c m'c

F={----} Mt = {-m--} (13)f t

Then, the constraints for loading condition i becorne:

-M* < -A* af + A* p < +M*c- t

i-m < f < +mc - - t (14)

Changing the minimization problem to an optimizationproblem, the general formulation becomes:

Max.

S.T.

-w = -{L} {M }/ft t

I. Loading Condition #1

-M* < -A* 1af + A*- p < M*c-

-m < fi < mc- - t

II. Loading Condition #2

-M* < -A*- 1af2 '+ A* lp2 < M*,.c - 2t

-m < f < mc - - t

iSince the force variables Fj could become negative,which violates an automatic implication of the tableau,the force variables Fi were first reduced out of thetableau before the istandard optimization procedureswere used. It should be noted also that the initialtab-leau from the above design formulation will alwaysbe dual feasible.

Limit design solution to the example problem:

r 0.6 0 -0.6 -0.81-0.8 -1 -0.8 -0.6J

[-1]33 -1] [A*1a] = [-1.6 1.667]

-[ 0 8.3331[A* P] = 10 -0.6671

N. Loading Condition #NLC

-M* < -A* af + A* PN < M*c - NLC -t

-nt < f < m (15)

Example: This 4-bar truss example problem is takendirectly from Reference (10), basically for comparisonof the limit design optimization and the con-ventionalelastic design optimization. Unlike the limit design,the elastic optimization procedure is an iterative pro-cedure, with each iteration involving the solution ofa linear programming problem. In the design problempresented by Wang, after one iteration, the objectivevalue becomes 4.837 in2-ft (951 mm2-m).

The geometric configuration and loading conditionsof the 4-bar truss is shown in Figure 5.

L. I.a L.3.6' 3.6'

V 2.8 ;

2

113,~ ~ ~ ~

c~~~~~~~~I

(1'=0. 305m)

NLC

1 2

P 1_10 T-62

Initial Tableau:

-F1 -F -F23 4 3

cl11

c12

c1c2

cl1

cl11

c211

c22

21

c23

1c24

4

c2c12

cl2c2c3cl4

-2c1c22

2c23

22c4-w

Figure 5: Example Problem

,It is interesting to note that the objectivefunction as given in the optimal tableau is 3.720 in2-ft (732' mm2_m) as compared to 4.837 in2-ft (951 mm2_M)given by Wang in using the elastic concept..

In this example problem, the notation clj repre-

sents the compressive constraint of the jth member forthe ith loading condition. Consequently, they can bewritten as:

f. F. + M > 0c cj -

f = F + Mt; > 0tj j

(16)

-F -M -M4 tl t2

0.3 .24 0.3 0.4

2002

t3 t4

-1.0 -1.3 -.75

1.6 1.7 -.75 10.0

-1.0 -.75

-1.0 -.75

1.0 1.3 -1.0

-1.6 -1. 7 -1.0 -10.0

1.0 -1.0

1.0 -1.0

-1. 0 -1.3 -.75 8. 33

1.6 1.7 -.75 -.667

-1.0 -.75

-1. 0 -.75

1.0 1.3 -1.0 -8.33

-1.6 -1.7 -1.0 0.667

1.0 -1.0

1.0 -1.0


Intermediate Tableau:

c2 c2 c223 4 3c22c24 Mtl Mt2 Mt3 t4

1.0 1.3 -.75 -1.0 -1.3

-1.6 -1.7 -.75 1.6 1.7 10.0

1.0 -1.8

1.0 -1.8

-1.0 -1.3 -1.0 1.0 1.3

1.6 1.7 -1.0 -1.6 -1.7 -10.0

1.0 1.3 -.75 -1.0 -1.3 8.33

-1.6 -1.7 -.75 1.6 1.7 -.667

1.0 -1.8

1.0 -1.8

-1.0 -1.3 -1.0 1.0 1.3 -8.33

1.6 1.7 -1.0 -1.6 -1.7 .667

0.3 0.24 0.3 0.4

2003

To limit the scope of this preliminary study, steelangle shapes listed in the AISC Steel Manual were con-sidered. A-36 steel with ayield strength of 36 ksi wasassumed and both ecual leg angles as well as unequal legangles were included. There were two studies done: Thefirst involved the computation of the capacities of allthe angle sections for a given effective length. Theresulting capacities were then plotted against their re-spective area for each of the lengths considered. Thesecond study involved the selection of the two mosteconomical angle sections for a given length and in-cremental load. The designed load and the resultingareas of the selected members were plotted in a graph.Again, A-36 steel was assumed with a yield strength of36 ksi and both equal leg and unequal leg members wereconsidered in this study.

Figure 6 is a typical computer plot from the firststudy. They represent the available capacities of allthe equal and unequal members as listed in the AISCManual (1973 edition). In computing the member cap-acities, the AISC formulae were used with the factorof safety terms ommitted to reflectthe ultimate con-ditions. In this typical plot, it is important to notethe linearirty of the upper envelope since the enveloperepresents approximately the economical selections fora given loading.

At Optimal:

Mem. M M f1 fl f2 f2t c t c t c

1 5.481 4.111 1.672 7.920

F1 F2

0 9 592 3-Rn q -A4

2 3.095 2.929 0 6.834 0 6.822 3.905 3.905M

3 3.809 2.857 0 6.666 6.666 0 3.809 -2.857 P

R

4 0 0 0 0 0 0 0 0 ES

Member Force: S

Loading Condition #1 N

F, = 3.809 F2 = 3.905 C

F3 = 3.809 F4 = p

Loading Condition #2

F = 5.481 F =3905 T~~~~~2

F3 = -2.857

Member Capacity:

M t = 5.481

Mt2 = 3.905

Mt3 = 3.809

Mt4 °

F4=0

M = 4.111

M = 2.929c2

M 2 = 2.857t2

M = 0c4

ANGLE MEMBER PROPERTIES

One of the more difficult questions in the limitdesign problem is in the appropriate methods with whichthe compression members are designed. In the previousexample problem,an arbitrary allowable stress value isused. In reality, the allowable or ultimate compressivestress is a function of the radius of gyration of theavailable sections, as well as the effective length ofthe members. Yet another variable is the fact that onlya finite number of angles are available for a towerdesigner to select from. Consequently, it was felt thata study of the capacities of the currently availablesteel angle shapes is appropriate at this point for thedesign program.

420.82 +

374.121+

327.42 +

280.71 +

Over 5 pts0 5 ptsX 4 pts* 3 pts+ 2 pts* 1 pt

~~~~~~*

+ * *

234.01 + **+ * *

+ *

+ * *

* #

187.31 + *

+4.

*~~~~*

140.60 **

9390+ X++ +++

+ + *

93.90 * *

+4.**++ 4.4X*

*O*47.20 + *X++

#

*#*4.+0M#'4

#-#

0.50 +- +-+----+- +----+----4_---C.0 3.34 6.68 10.02

AREA

Figure 6: Angle Section Capacity (KL = 6')

c11

1cl1

cl1

cl14c1c21

c22

2c1

2c12

2c13

2c14

2c21

2c22

-w

**

*

**

* *

- * *

+ *


2004

Figure 7 is a typical plot of angle members selec-ted for a given load on the most economical basis interms of weight. This plot formed a basis for the re-lationship of a given ultimate design load and theavailable member area. Noting that all of the datapoints can be approximated by straight lines, linearregression analyses were performed on all the availablegraphs. Both the resulting slope, B1, and the inter-cepts, Bo, are summarized in Table 3.

TABLE III

Intercept and Slope Values from Regression Analysisof Load-Area Relationships of Angles

Effective AngleLength(ft)

4

6

8

10

12

14

16

Intercept SlopeB B1

- 0.267 32.182

-10.458 31.577

-19.527 30.209

-27.471 28.224

-34.735 25.943

-36.695 23.032

-30.334 19.066

CorrelationFactor

R

0.994

0. 994

0.992

0.988

0.983

0.976

0.965

420.82 +

+ Over 5 pts+ 0 5 pts

C 374.12 + X 4 pts0 + # 3 ptsM + + 2 pts *R + * 1pt *E 327.42 + *S + **S + **I + +0 + * *N 280.71 + * *

4. *$C +A + **p + +A 234.01 + +C + **I + **T +Y + **

187.31 + **+ ~**

+ **+ **+ * *

140.60 + **4. **+ ++ **+ +

93.90 + ++ +

+ **+ +

47.20 + ++ ++ 4.+ **

C.C 3.34 6.68 10.02ARE A

The intercept and slope values summarized in TableIII can be approximated with the fQllowing empiricalequations, expressing the quantities in terms of themember effective length L:

(L - 39)1.2Bo = -0.8591 - 34.9145 Sin 31.128

B1 = 32.1990 - 0.0773 (L - 3.00)2

(17)

(18)

where the angle expressed within the sine function ofEquation (17) is in radians and the length L is expressedin feet.

The relationsship bewteen the design ultimate loadand the available angle area can then be approximatedby the following equation:

M =B + Ac 0 1 (19)

Equation (19) expresses continuously, the availableangle area with respect to a given ultimate design com-pressive load. In reality, only a finite number ofangles are available in a discrete fashion. The com-

puted area A from a given load through the use ofEquation (19) is therefore termed the 'TheoreticalDesign Area".

For tension capacity, the 85% rule for net sectionarea is used. Denoting Fy as the yield stress, thetension capacity Mt can be written in terms of anglearea A:

Mt = 0.85 F A

which for A-36 steel:

(20)

Mt = 30.6 A (21)

Substituting A into Equation (19) yields the relationshipbetween M and M:

c t

M =B + BiMtc 0 30.6(22)

Using Equation (22}1 the general formulation given inEqcuation (15) becomes;

TMax. -w - L Mt /30.6t

S.T. I. Loading Condition #1

-[B + lmt ] < -A*- af1 + A p30.6

< M ~ [B + 1 t ]< f < m-t' 0 ~30.6

II. Loading Condition #2B M * -l 2 -12

-[B0+ lt] < -A* af + A*- p30.6

i1t 2<M* [B + ] < f < m- t 0 30.6 t

N. Loading Condition #NLC

BM -l1 NLC-[B0 + 1 t] < -A* af +

30.6-l NLC B1m

A* P < M ;[B + i- t 0 306

Figure 7: Available Economical Angle Sections

(KL = 6 ft)

< fNLC <_ - t (23)

Example: The plane tower shown in Figure .4 was de-signed using both the elastic concept as well as the


2005

limit design concept with the aid of the computer. In

both cases, the "Theoretical Design Area" was used forcomparison purposes. The plane tower is assumed tohave symmetry about its vertical axis.

The result of the elastic design is shown in TableIV where the total weight function is 461 in2-ft (90,676mm2-m). The result of the limit design is shown inTable V, where the total weight function is 420 in2-ft(82,612 mm2-m).

It is interesting to observe that a "theoretical"weight savings of 9.8% is effected through the use ofthe limit design concept.

LOADING

P CONDITIONS

1 2

p1 0.5 16

p2 -55 -2

P3 0.5 16

P4 -55 -2

10'

8o

Fig'ure 8. Plane Tower Example

TABLE IV

Plane Tower, Elastic Design Results

MEMBER CAPACITY MEMBER FORCEAREA

NUMBER TEN COMP LC#1 LC#2

1 4.66 142.60 70.20 -41.01 67.042 3.00 91.80 23.72 -14.70 -18.043 3.00 91.80 23.72 -13.60 17.014 0.70 21.42 6.40 10.90 0.405 2.03 62.12 54.42 -51.65 40.956 1.35 41.31 16.47 -3.87 -15.98

7 1.35 41.31 16.47 -2.88 15.738 0.62 18.97 10.18 11.89 0.439 1.60 48.96 41.05 -39.84 14.53

10 1.50 45.90 23.45 -21.44 -23.38

11 1.60 48.96 41.05 -40.84 -17.46

12 1.28 39.17 30.97 -30.78 -1.12

13 1.74 53.24 45.46 -45.44 -1.65

14 1.18 36.11 13.89 -13.52 -0.4915 1.74 53.24 45.46 -45.44 -1.65

16 2.12 64.87 57.43 64.56 2.3517 2.55 78.03 54.82 77.78 2.8318 2.06 63.04 55.54 -55.50 -18.0019 2.55 78.03 54.82 77.78 2.8320 2.06 63.04 55.54 -54.50 14.0021 1.18 36.11 13.89 -13.52 -0.4922 1.50 45.90 23.45 -20.03 21.8723 2.03 62.12 54.42 -54.33 -44.8124 4.66 142.60 70.20 -45.30 -70.18

2 ~~~2w = 461 ft-in (90,676 mm -im)

TABLE V

Plane Tower, Limit Design Results

-MEMBER AA CAPACITY MEMBER FORCE

NUMBER TEN COMP LC#l LC#21 5.59 171.00 91.10 -53.53 51.532 1.35 41.40 0.00 0.00 0.00

3 1.35 41.40 0.00 1.09 35.00

4 1.23 37.60 22.70 1.71 -18.83

5 2.03 62.20 54.50 -51.82 32.60

6 0.92 28.00 3.68 - 3.52 - 3.68

7 0.92 28.00 3.68 - 2.53 28.00

8 0.86 26.40 17.90 4.58 -17.86

9 1.82 55.60 47.80 -46.80 6.13

10 1.10 33.70 11.50 -11.50 -11.50

11 1.82 55.60 47.80 -47.80 -25.83

12 2.04 62.50 55.00 -54.83 - 9.84

13 2.28 69.90 62.60 -62.47 - 1.99

14 0.35 10.80 0.00 10.65 0.00

15 2.28 69.90 62.60 -62.47 - 1.99

16 1.55 47.40 39.40 47.40 2.00

17 2.54 77.80 54.60 77.69 2.82

18 2.06 63.00 55.50 -55.43 -17.98

19 2.54 77.80 54.60 77.69 2.82

20 2.06 63.00 55.50 -54.43 13.9821 0.35 10.80 0.00 10.-65 0.0022 1.10 33.70 11.50 -10.09 33.70

23 2.03 62.20 54.50 -54.50 -53.0524 5.59 171.00 91.10 -57.81 -85.51

w = 420 ft-in2 (82,612 mm-2M)

CONCLUSIONS

A general formulation for both the limit analysisand design problems were developed in this paper.Although this paper represents only a fraction of the

effort in realizing the full potential of the linear

programming techniques in limit analysis and design, it

was felt that the feasibility of the above techniqueshas been demonstrated.


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Additional research is required to enhance thefuture use of limit design for the transmission struc-tures,especially in the retrofitting of existing trans-mission for higher electrical loads as well as in thedesign of new transmission towers.

ACKNOWLEDGE!MENT

This paper is a result of a research project sup-ported by Bonneville Power Administration. The authorsgratefully acknowledge the consideration and supportprovided by Bonneville Power Administration. Theauthors further acknowledge the contributions of Mr.John Osteraas and Mr. Bowen Shih who helped develop andde-bug the computer program.

REFERENCES

1) Cohn, M. Z., Franchi, A. "Structural PlasticityComputer System: STRUPL", ASCE Journal StructuralDiv., April, 1979.

2) Jensen, H. G., "Designing Self-Supporting Trans-mission Towers With the Digital Computer", IEEEPower Industry Computer Applications Conf. Rec.,May, 1967, pp. 303-319.

3) Jensen, H. G., "Efficient Matrix TechniquesApplied to Transmission Tower Design", Pro-ceedings, IEEE, Vol. 55, No. 11, November, 1967,pp. 1997-2000.

4) Jensen, H. G., and Parks, G. A., "Efficient Solu-tions for Linear Matrix Equations", ASCE Journal,Structural Deivision, January, 1970.

5) Lee, J. W., "Limit Analysis by Linear ProgrammingMethod", Independent Study Report, University ofWisconsin, CEE Department-, 1970.

6) Marjerrison, M. M., "Electric Transmission TowerDesign", Journal of the Power Division, ASCE,94, No. P01, Proc. Paper 5932, May, 1968, pp. 1-23.

7) Neal, B. G., The Plastic Methods of StructuralAnalysis. Chapman & Hall, 1963.

8) Tinney, W. G. and Walker, J. W., "Direct Solu-tions of Sparse Network Equations by OptimallyOrdered Triangular Factorization", Power IndustryComputer Applications Conf. Rec., Sponsored byIEEE Power Group, February, 1967; revised August9, 1967.

9) Turner, A. E. and Wood, D. L., "Computer DesignsTransmission Towers", Electrical World, January22, 1962, pp. 32-34.

10) Wang, C. K., Computer Methods in Advanced Struc-tural Analysis. Intext Educational Publishers,New York, New York, 1973.

11) Zukhovitskiy, S. I. and Avdeyeva, L. I., Linearand Convex Programming. W. B. Saunders Company,Philadelphia and London, 1966.

12) Hadley, G., Linear Programming. Addison-WesleyPublishing Co., Inc., 1963.

13) Lee, J. W. and Jensen, H. G.,"Limit Truss Analysisby Linear Programming Techniques", Paper pre-sented in ASCE Convention in Portland, OR, April14, 1980.

14) Wang, C. K. Matrix Methods of Structural AnalysisSecond Edition, International Textbook Company,1970.

BIOGRAPHY

Jun W. Lee obtained his Ph.D. Degree in StructuralEngineering at the University of Wisconsin, Madison.He was with EngineeringlResearch Consultants, Inc. un-til 1978. Presently, he is President of StructuralResearch, Inc. of Madison, Wisconsin.

Dr. Lee has executed numerous projects in the areaof Structural Engineering. His experience includesstructural design of various structures, static anddynamic analysis of structures, experimental and the-oretical research in Structural and Materials Engineeringas well as field insturmentations and full-scale struc-tural testing. In addition, Dr. Lee has presented andpublished numerous papers and reports in the area ofStructural and Materials Engineering.

Dr. Lee is a Member of IEEE as well as an AssociateMember of ASCE. He is also a member of the HonorSocieties Sigma Xi, Tau Beta Pi, Chi Epsilon and Pi TauSigma. He is presently registered as a ProfessionalEngineer in the State of Wisconsin.

H. Gordon Jensen was born in Portland, Oregon onApril 13, 1931. He received the B.S. Degree in CivilEngineering from Oregon State University, Corvallis, in1955.

Since 1962 he has been with the U. S. Departmentof Energy, Bonneville Power Administration (BPA), Port land,Oregon, hwere he has been responsible for transmissioncomputer applications and design of transmission lines.He is currently Senior Structural Engineering Advisorfor Bonneville Power Administration.

Mr. Jensen is a Registered Civil Engineer in theState of California.


Discussion

Hong-To Lam, (South Carolina Public Service Authority, MoncksCorner, SC): I would like to congratulate the authors who introducedthe limit concept to analyze the latticed transmission line structures. Aswe all know by assuming pin-connected members, we only consideraxial loads in a latticed transmission tower, i.e. tension or compression,(in reality, because of the joint detailing, gusset plate arrangement andstress introduced during erection, the shear center will not usually coin-cide with the center of gravity of the loads), thus, the size of stiffnessmatrix can be reduced considerably.

If we look into the tension or compression failure moods, we will findthat the member fails in tension if its fiber stress reaches yield stress.There are many members in a latticed transmission line structure. Cer-tain members' fiber stress reaches yield point while other members'fiber stress is probably well under the yield stress of the material. Byapplying limit approach to find collapse load under which all members'fiber stress reaches yield point of the material, as long as the membersare made from ductile steel. On the other hand, failure in compression,as the classical Euler's formula n2EI/P for pin-connected member in-dicates that it is independent of yield stress of the material. For a shortmember, i2EI/P may reach or exceed the yield while a long membermay buckle before its fiber stress reaches the yield. In short, compres-sion failure is a stability problem. Based on the preceeding, I would likethe authors to comment on the following:

(1) What is the validity and applicability of the limit approach toanalyze a tension compression system structure?

(2) Is this method also applicable to the other structure material suchas aluminum, etc., whose characteristics are not the same as steelwhich is ductile?

Manuscript received August 4, 1980.

681-7a

Jun W. Lee and Gordon Jensen: The discussion for this paper focusmostly on the applicability of the limit analysis concept to transmissiontower analysis and design. The concerns arose from the recognition ofsuch factors as joint eccentricity, member continuity, etc. Another con-cern is the behavior of members under compression.

There are two separate tasks involved in determining the capacity of atransmission line (regardless of the analysis concept or failure defini-tion). The capacities of the individual members must be determined,which is then used with the overall analysis of the tower in determiningthe tower capacity. The limit analysis procedure presented in the paperdeals only with the overall tower analysis. No attempt was made inrec'ommending a procedure for member capacity computation. Thepresence of joint eccentricity, etc., represents the need for refinement ofmember capacity computation techniques in conjunction with overallanalysis techniques, which should be a future topic of research inadvancing the understanding and predicting of transmission towerbehavior.

The basic assumption of the limit analysis procedure presented in thepaper is that the ideal elasto-plastic behavior exists in both tension andcompression members. Elasto-plastic behavior is well known in tensionmembers made of ductile materials. For compression members, failurenormally occurs in buckling. Therefore, the load-deflection behavior ofcompression members undergoing the buckling process must be inves-tigated. It should be noted that even for cases where the load-deflectioncurves are not ideally elasto-plastic, the calculated tower capacity using

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the proposed method would still be conservative if the assumed membe:capacities do not exceed the actual member capacity through thenecessary displacement.To study the buckling phenomena in angle members, the Bonneville

Power Administration is currently sponsoring a research project atPortland State University which involves the experimental determina-tion of load-deflection characteristics. One such preliminary curve isshown in Figure 1. Assuming that the required displacement to effectredistribution of axial forces is 0.2" (5.08 mm), then the elasto-plasticcurve for this particular column can be idealized as shown, having anultimate capacity of 9.7 instead of the actual 9.8 kips.

12

10

4

o

8

6

4

2

.2 .4 .6 .8 1.0

AXIAL DISPLACEMENT (IN) (25.4 MM)

FIGURE 1

In summary, both the limit analytic and design procedure presentedin this paper represent the theoretical bases for the application of suchtechniques. Practical implementation will be realized with additionalresearch into the basic member behaviors, especially pertaining to com-pression members. Some of this research is currently under considera-tion.

Manuscript received September 15, 1980.


transmission tower limit analysis and design

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