BUCKLING ANALYSIS
CAPABILITIES
FOR USE IN THE DESIGN OF
LATTICE TRANSMISSION TOWERS
By
Mark Allen Celano
B.S., University of Colorado at Boulder, 1985
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Civil Engineering
1991
This thesis for the Master of Science
degree by
Mark Allen Celano
has been approved for the
Department of
Civil Engineering
by
Andreas S. Vlahinos
/ John R. Mays
Date
Celano, Mark Allen (M.S., Structural Engineering)
Buckling Analysis Capabilities for Use in the Design of
Lattice Transmission Towers
Thesis directed by Assistant Professor Andreas S. Vlahinos
Buckling analysis was done using two computer models to
determine the accuracy that can be achieved in calculating
the ultimate capacity of lattice transmission towers.
These capacities were compared to a fullscale tower test,
and to the current code requirements. The models were
based on a 345kV, selfsupporting, dead end, single circuit
lattice transmission tower. The models used were 1) spar
elements and 2) elastic beam elements. In the model using
the spar elements, it was determined that global buckling
is not an issue for the structure, and that subglobal
effects must be considered. Using the elastic beam element
model analysis, local buckling occurred. The location of
the local buckling was the same as in the fullscale tower
test. The results indicate that the theoretical tower
capacity given by the model can be predicted within 10 to
15 percent of the actual buckling capacity, and the
theoretical member capacity can be predicted with some
degree of accuracy.
The form and content of this abstract are approved. I
recommend its publication.
Signed
Andreas S. Vlahinos
IV
CONTENTS
Figures ................................... vi
Tables..................................... vii
CHAPTER
1. INTRODUCTION ............................. 1
Load Requirements ....................... 1
Tower Analysis........................... 5
2. TOWER TEST................................ 9
3. COMPUTER MODELING ....................... 19
Basic Assumptions....................... 19
The Spar Element Model.................. 20
The Elastic Beam Model.................. 22
4. CODE REQUIREMENTS........................ 28
Short Columns........................... 29
Slender Columns ........................ 31
Allowable Member Stress ................ 32
5. NUMERICAL RESULTS ....................... 33
6. CONCLUSION............................... 35
LIST OF REFERENCES........................... 39
APPENDIX
A. TOWER DRAWINGS.......................... 40
B. MATHEMATICAL METHODS .................... 54
C. COMPUTER MODEL
58
FIGURES
Figure
1.1. NESC Loading Zones....................... 2
1.2. Basic Wind Speed (MPH) ................. 4
1.3. Tower Outline ........................... 7
2.1. Load Case B3 Load Tree................ 11
2.2. Tower Failure in Face A,
Double Angle, Leg Bracing ................ 12
2.3. Load Case C2 Load Tree................ 13
2.4. Tower Failure in the Leg Post
Between Face A and Face B............... 14
2.5. Tower Strain Gage Locations, Face A . 16
2.6. Axial Force vs. Percent Load for
Strain Gage Locations 5 and 6,
Load Case B3.............................. 18
3.1. Spar Element Model...................... 21
3.2. Elastic Beam Model....................... 22
3.3. Load Case B3  Mode 1 (0.85)............ 24
3.4. Load Case B3  Mode 1 (0.85)............ 24
3.5. Load Case C2  Mode 1 (1.03)............ 25
3.6. Load Case C2  Mode 2 (1.12)............ 25
3.7. Load Case C2M Mode 1 (1.17)........... 26
4.1 StressStrain Curve Relationship,
Based on Residual Stress.................. 31
vi
TABLES
Table
1.1. NESC Loading............................... 3
2.1. Strain Gage Summary....................... 15
3.1. Axial Load (Kips), Deflection (inches) 23
3.2. Axial load (Kips)......................... 27
5.1. Member Capacity........................... 33
5.2. Tower Capacity Ratio ..................... 34
vii
CHAPTER 1
INTRODUCTION
Originally engineers designed wooden poles to support
iron wire that carried electric current. They struggled
with the same issues engineers face today: the cost of
transmission lines. For effective design of transmission
towers, one needs to consider the cost of a) material, b)
construction, and c) maintenance for the proposed
transmission line. These towers, for the most part, are
used in sparsely populated rural areas where large
deflection requirements and vibrations are not the critical
design factors. Tower costs control the design, thus an
effort to minimize the weight is desirable. Therefore
these towers are designed to their fullest capacities based
primarily on the economics of the transmission lines.
Load Requirements
The conductor, the ruling span and the geographic
location of the transmission line all play important roles
in the design load characteristics of the tower. Extensive
background information can be found for the loading of
transmission line structures in Guidelines for Transmission
Ling .Structural Loading [ 1 ], and the National Electric
Safety Code (NESC) [2]. The design loads are determined
by one code in particular, the NESC. The NESC gives two
specific loading requirements, "Combined Ice and Wind" and
"Extreme Wind," as mentioned in rule 250 of the NESC.
The combined ice and wind loading is broken down into
three geographic loading zones, or districts. The
geographic breakdown is shown in Figure 1.1 [2, pg. 257].
Figure 1.1 NESC Loading Zones
The wind, ice and temperature loads, shown in Table
1.1, are applied to the tower and conductor. The dead load
of the ice and the dead load of the conductor combine to
give the vertical component of the conductor load. The
2
vertical effect of the ice loading is calculated assuming
the ice to be attached uniformly (radially) around the
entire circumference of the conductor for the entire span.
The unit weight of the ice is 57 pcf. The wind load is
applied to the projected area of the bare conductor plus
ice (if any) resulting in the horizontal component of the
conductor load. The resultant load along the conductor is
the square root of the sum of the squares of the vertical
and horizontal components plus the NESC Constant shown in
Table 1.1.
Table 1.1 NESC Loading
NESC WIND ICE TEMP NESC CONSTANT
ZONE (PSF) (IN) (DEG. F) (LBS/FT)
Heavy 4.0 0.50 0.0 0.30
Medium 4.0 0.25 15.0 0.25
Light 9.0 0.00 30.0 0.05
The extreme wind loading is also given by the NESC.
Figure 1.2 [2, pg. 258] shows the wind pressures to be
applied to the conductor and tower, based on their
geographic location.
The conductor also plays an important role in the
design load characteristics of the tower. The conductor
size is determined by the current carrying capacity of the
line. The composition of the conductor is typically a
3
Figure 1.2 Basic Wind Speed (MPH)
multistrand cable composed of aluminum and steel core
strands. The aluminum strands carry most of the electrical
current while the steel core reinforcement carries the
majority of the tension load. The conductor is strung to
a given percentage of its ultimate strength, as required by
rule 261H2 of NESC. Given this tension, the design
engineer can establish a table of sag vs. ruling span data
for the corresponding loading zone.
4
Through experience and through the use of
sophisticated computer programs, the engineer determines
the location of the towers, the ruling span, and
corresponding sag to produce the most economical
transmission line. The terrain and sag of the conductor
dictate the height of the tower that must be used.
Based on experience, geographic location, and
conductor type and size, the design loads are found. These
two basic design loads and others are multiplied by
appropriate load factors, given by NESC rule 261A1, to
achieve the ultimate design loads. The tower members and
their connections are designed to resist these ultimate
design loads. The stresses for the well designed tower,
approach the elastic limit, for tension loaded members, and
approach buckling capacity, for compression loaded members.
Tower Analysis
Tower analysis today is done mainly by the use of
computer programs. The most common kind of analysis used
is a FirstOrder Linear Elastic analysis using an ideal
3D space truss. The model gives axial loads in members,
and joint displacements. The moment in the members are
ignored in the model. Moments are caused by two primary
sources 1) member connectivity (framing) eccentricities and
5
2) the continuation of member such as legs and beam chords.
The code used to check these members is the ASCE Manuals
and Reports on Engineering Practice No. 52. Guide for
Design of Steel Transmission Towers (ASCE Manual No. 52^ .
2nd Edition [3]. The method used by the code accounts for
connectivity eccentricities by modifying the slenderness
ratio of the member, and thus reducing the load capacity.
SecondOrder Elastic analysis is not practical to use
for selfsupporting lattice towers. The tower in general
is too rigid, resulting in small deflections as shown in
the static analysis below. Thus PDelta effects, or
geometric nonlinearities, are of little significance.
Given the ultimate design loads and the desire to
achieve the most economical towers, buckling analysis
modeling will be considered. The buckling analysis will be
compared to a fullscale test and to the current code (the
design code most widely used for the design of these types
of structures), ASCE Manual No. 52 [3].
More specifically we will compare the loads carried
by the tower shown in Figure 1.3, for two loading
conditions. The comparison will be made on the buckling
capacity of the tower and the buckling capacity for two
members of the tower. The first member is the leg post
located between face A and B. This member is an
6
7
L 6 x 6 x 1/2, Fy=50 Ksi, and has an unsupported length of
6.81' in the X and Z direction. The second member is the
leg brace located on face A, This member is a double angle
L 5 x 3 1/2 x 5/16, Fy=36 Ksi, with unsupported lengths of
6.967 in the X and 34.8 in the Y direction. The
reference directions for the unsupported lengths are the
same as given in ASCE Manual No.52 [3].
8
CHAPTER 2
TOWER TEST
A full scale tower test was conducted as reported in
the Structural Loading Test of a Western Area Power
Administration 345kV. SelfSupporting. Double Dead End,
Single Circuit Lattice Transmission Tower fTvpe 36Y) [4].
The test was conducted at the Transmission Line Mechanical
Research Center (TLMRC) and sponsored by the Electric Power
Research Institute, Inc. (EPRI). The tower details and
erection drawings can be found in Appendix A. The overall
height was 115'0" and the tower base was 34'2 3/4"
square. Several load cases were used on this structure to
verify the tower capacity. Strain gages were utilized to
record axial forces in various members.
The apparatus utilized to load the tower was two
reaction frames, variablespeed winches, and an automated
computer controlled loading system. The steel reaction
frames support cables and rigging that apply loads to the
test structure.
The load cases of interest are load cases B3 and C2
as shown in Figures 2.1 and 2.3 respectively. Both load
cases caused buckling to be experienced in the tower.
Case B3 was completed first and caused the tower to buckle
in the double angle leg bracing in face A as shown in
Figure 2.2. This member was not replaced prior to the
conducting of load case C2. Load case C2 caused the tower
to buckle in the leg post between face A and face B as
shown in Figure 2.4.
Figure 2.1 is the schematic of the loads applied to the
tower in load case B3 at failure. In Figure 2.2 the arrow
addresses the failure of the double angle leg bracing in
face A for load case B3.
10
Figure 2.1 Load Case B3 Load Tree
11
Figure 2.2 Tower Failure in Face A, Double Angle, Leg
Bracing
12
Figure 2.3 is the schematic of the loads applied to
the tower in load case C2 at failure. In Figure 2.4 the
arrow addresses the failure of the leg post between face A
and face B for load case C2.
13
14
Strain gages were strategically placed on tower
members to monitor the loads during the tests. Table 2.1
gives the axial loads recorded, and Figure 2.5 illustrates
the locations of the recording devices (strain gages) for
each load case. In case B3, the strain gages, located at
reference points 5 and A (on the outside angle of the
double angle member) and 6 and B (on the inside angle) are
located on the leg brace member that failed. The gages
were located oneandonehalf panel points above the actual
failure location. Thus, the recorded results only reflect
the actual axial load at the failure point. In case C2,
the gages designated E and 19 are located on the buckled
leg member between faces A and B. These gages also reflect
Table 2.1 Strain Gage Summary
AXIAL LOAD (LB)
MEMBER TYPE STRAIN GAGE LOAD CASE B3 AVG LOAD CASE C2 AVG
BRACE 5 12044 12538 5297 5728
A 13032 6160
6 39510 38738 7897 7175
BRACE B 37966 6454
LEG 19 172560 173300 207360 207325
LEG E 173040 207290
15
16
the load that actually occurred in the failed member.
These gages are located two panel points above the critical
area.
Table 2.1 gives a summary of the axial loads that were
recorded in the leg post and the leg brace. The average
value is listed for the strain gages located on the same
member. Note that the total axial load carried by the
double angle leg brace is the total of the two averages
shown in Table 2.1.
Figure 2.6:
shows the individual and combined axial forces, as a
function of percent load, at locations A and B for the same
test. Both plots are virtually identical, and suggest that
the outside angle (locations 5 and A) stopped carrying
additional compressive load between 80 and 85 percent load.
The combined load plot in both figures suggests that the
double angle unit loaded up in a near linear manner despite
the independent and opposite loading experienced by the
individual angles. During this test, the above mentioned
member was very visible bowing outward, away from the
center of the tower. An outward bowing of a double angle
member would subject the "inside" angle to additional
compressive force while the "outside" member experienced
some tension forces. Thus, (Figure 2.6] seems to agree
with visual observation during the test [4, pg. 511 &
512].
17
A FACE
Figure 2.6 Axial Force vs. Percent Load for Strain Gage
Locations 5 and 6, Load Case B3
18
CHAPTER 3
COMPUTER MODELING
Two computer models were used for the comparison. The
spar element model and the elastic beam element model. A
brief description of the mathematical methods used can be
found in Appendix B. Both models were analyzed using first
order elastic, and buckling analysis methods. ANSYS,
Swanson Analysis Systems, Inc., [5] general purpose finite
element program was used for the analysis. A full
description of the input data for each model can be found
in Appendix C.
The loads applied to the tower computer models were
the same as described for the Tower Test, and as shown in
Figures 2.1 and 2.3.
Basic Assumptions
1. The material is elastic.
2. There are the same compressive stressstrain properties
throughout the section.
3. No initial internal stresses exist such as those due to
cooling after rolling.
4. The members are perfectly straight and prismatic.
5. The load resultant acts through the centroidal axis of
the member.
6. The small deflection theory is applicable.
7. Distortion of the crosssection does not occur during
bending.
8. No local type of instability will occur.
The Spar Element Model
The spar element model is shown in Figure 3.1. This
represents the conventional 3D space truss model being
used most commonly for FirstOrder Linear Elastic analysis.
The model contains 50 nodes and 170 elements (two node 3D
space, 150 D.O.F.).
The linear elastic analysis produced a maximum deflection
of 6.9" for load case B2, and 7.4" for load case C2.
Keeping in mind that the total tower height is 115' or
1380" the total deflection in the horizontal is
approximately 1/200. Thus verifying the earlier conclusion
that geometric nonlinear effects are of little
significance, and that SecondOrder Elastic analysis is not
required.
20
The results from the
Figure 3.1 Spar Element
Model
buckling analysis of the 3D
spar element model resulted in
a buckling/load factor of 38.6
for load case B3 and a factor
of 41.8 for load case C2. The
buckling/load factor is
defined below:
Pcr=(Buckling/Load Factor) P
Per (critical) is the
load on the structure that
causes buckling, and P is the
load applied to the structure.
The critical buckling load predicted by this model is
to the order of 40 times the "actual" (capacity found by
the fullscale test) load. These results indicate that
overall (global) buckling of this type of tower is not the
issue at hand and that a more complex model would be needed
to consider the local (subglobal) buckling effects. For
this, the elastic beam element model was used, as described
below.
21
The Elastic Beam Model
The elastic beam element
model is shown in Figure 3.2.
This model contains 346 nodes
and 570 elements (two node
3D space, 2076 D.O.F.).
All joints in the model
are rigid. The model contains
all the "redundant" bracing
not required in the simple
3D spar from the waist of the
tower down. The leg post
sections between each
"redundant" member were broken
into four elastic beam
elements in order to achieve
the most accurate buckling
load factor possible.
The linear elastic analysis produced a maximum
deflection very close (within onetenth of an inch) to the
spar element model. The deflections were 6.8" for load
case B2, and 7.4" for load case C2. This implies that we
have not tampered with the overall stiffness of the
structure.
Figure 3.2. Elastic Beam
Model
22
Table 3.1 relates the leg post and leg brace members'
axial loads for both load cases, and the maximum
deflections for each load case.
Table 3.1 Axial Load (Kips), Deflections (inches)
3D SPAR 3D ELASTIC BEAM
LOAD CASE B3 C2 B3 C2
LEG POST 115.59 135.63 113.02 131.52
LEG BRACE 87.83 28.25 87.93 29.91
MAX DEFL. 6.9 7.4 6.8 7.4
As can be seen in the Table 3.1 the axial loads
between the two analysis types are of little difference.
The buckling analysis for the 3D elastic beam element
resulted in a buckling/load factor of .85 for load case B3.
The primary buckling mode is shown in Figures 3.3 and 3.4.
This mode shape is very similar to the buckling mode shape
that occurred in the full scale tower test as is shown in
Figure 2.2.
The buckling analysis for the 3D elastic beam element
for load case C2 gave a buckling/load factor of 1.03. This
result did not occur under the same model shape as the
fullscale test. The model as shown in Figure 3.5 buckled
23
Figure 3.3 Load Case B3 
Mode 1 (0.85)
along the leg brace in face A for the leg post between
faces A and D. The second mode of buckling occurred at a
factor of 1.12 in the leg brace on face B for the leg post
between faces A and B as shown in Figure 3.6.
Referring back to the tower test results, Table 2.1,
and the axial loads shown in Table 3.1 for load case C2 we
see that the model's leg brace is carrying considerably
24
more load than in the fullscale tower test. This
indicates that the member may have been damaged to some
extent under the buckling load encountered under load case
B3. This load case was conducted first under the full
scale test. Remodeling the structure using the same
configuration as used in the elastic beam model with part
of the leg brace removed, the axial load of the leg brace
member was lower than in the actual tower test.
25
Load case C2M refers to
the modified 3D elastic beam
model described above. The
axial loads for this analysis
are listed in Table 3.2. For
load case C2M (modified) the
model resulted in a
buckling/load factor of 1.17.
The buckling mode can be
seen in Figure 3.7. The
buckling occurred on the leg
brace on face B for the leg
post between faces A and B.
This result indicates that
failure is very probably
predicted in this area of the
tower. If the leg brace was
significantly damaged or significant bolt slippage occurred
for this member, the tower would have buckled as it did
under load case C2M. If the leg brace was slightly damaged
or minor bolt slippage occurred, then the tower would have
buckled in the range of 1.12 (load case C2, mode 2) to 1.17
(load case C2M, mode 1), for an average of 1.15.
Figure 3.7 Load Case C2M
 Mode 1 (1.17)
26
Table 3.2 Axial Load (Kips)
Tower Test 3D ELASTIC BEAM
LOAD CASE C2 C2(1.12) C2M C2M*(1.17)
LEG LEG POST BRACE 207.33 12.90 147.31 33.53 144.28 1.71 168.52 2.00
27
CHAPTER 4
CODE REQUIREMENTS
The following equations and their corresponding
numbers were extracted from the ASCE Manuals and Reports
on Engineering Practice No. 52. Guide for Design of Steel
Transmission Towersf 2nd Edition [3]. Equation 4.61 is
valid only for short columns buckling in the inelastic
range and equation 4.62 is valid for only slender columns
buckling in the elastic range.
Allowable compression:
For members with concentric load at both ends of the
unsupported panel, for 0 < 1/r < 120:
Fa=[ 12$ [ (kl/r)/Cc]2 ]Fy
kl/r < Cc (4.61)
Fa=286,000/(kl/r)2
Cc=7r/(2E/Fy)
kl/r > Cc (4.62)
(4.63)
kl/r = 1/r
(4.75)
For members partially restrained against rotation at
both ends of the unsupported panel, for 120 < kl/r < 250:
kl/r = 46.2 + 0.615 1/r (4.710)
A Crosssectional area (square inches)
E Young/s modulus (29,000 KSI)
Et Tangent modulus (KSI)
Fa Allowable compressive stress (KSI)
Fy Minimum guaranteed yield stress (KSI)
k Effective length coefficient
1 Unbraced length (inches)
P Axial load on member (Kips)
r Radius of gyration (inches)
Rs Residual stress (KSI)
Short Columns
Equation (4.61) is valid for short columns (kl/r <
Cc) which buckle inelastically. This equation is the "CRC
[Column Research Council] ColumnStrength Curve," based on
computed critical load curves for rolled steel Hshaped
sections [6] (averaged about the strong and weak axes of
buckling). The "CRC ColumnStrength Curve," initially
published in 1960, considers the effects of residual
stress. The effects of these stresses are the primary
factor in the determination of the column buckling strength
for Hshaped members.
29
Residual stresses are caused by two primary sources:
1) the cooling of hot rolled sections and 2) the effects
of welding. Residual stresses (Rs) average 0.3Fy for
structural carbon steels [7, pg. 319].
The maximum residual stress in the crosssection
defines the effective proportional limit for the column.
At point A, in Figure 4.1, the stressstrain curve becomes
nonlinear; caused by the yielding of some fibers in the
section (P/A + Rs > Fy ). Thus, the effective A*E for the
total section begins to drop off defining the curved
portion of the graph shown in Figure 4.1. The residual
stresses of the crosssection have been analytically
modeled as linear and/or parabolic in their distribution.
These distributions were used to generate the effective
stressstrain relationship. From this, the tangent modulus
theory can be applied for a range of values of (kl/r).
Fcr=7r2Et/(kl/r)2 [7, pg. 301]
The analytical solution produces a curve that
corresponds very closely to the "CRC ColumnStrength
Curve."
30
For equal angles the residual stress in the cross
section probably will have less effect than in Hshaped
sections where there is a deferential in the thickness
between the flange and the web. One needs to consider
residual stresses, the member crookedness, the
eccentricities of the loading and possible material
defects, such as variation in thickness, when modeling
members in this slenderness range.
Sjender column?
Equation 4.62 is Euler's equation [8], valid in the
elastic range. The modulus of elasticity (E) was taken as
29,000 KSI. This equation is valid in the slenderness
range of (kl/r) > Cc. The column is assumed initially
31
All fibers in the
straight and concentrically loaded,
member are stressed below their elastic limit until
reaching the buckling load.
Allowable Member Stress
Leg Brace L 5x3 1/2x5/16? Fy=36 KSI ? lx=6.96/ ; ly=34.8'
rx=1.03 in. ; ry=2.44in.; s=l/2 in.; A=5.12 in
Control: ly/ry=171; Fa=12.5 KSI from equation (4.710), and
(4.62)
Leg Post L 6x6xl/2 ? Fy=50 KSI ? lz=6.81/ ; lx=6.81/
rz=1.18 in. ; A=5.75 in
Control: lz/rz=69.3? Fa=39.5 KSI from equation (4.75), and
(4.61)
32
CHAPTER 5
NUMERICAL RESULTS
Table 5.1 compares the member capacities, in Kips,
for the double angle leg bracing in face A, and the leg
post between face A and face B.
Table 5.1 Member Capacity
MEMBER TOWER ELASTIC ASCE
CAPACITY TEST BEAM MANUAL 52
LEG BRACE
LOAD B3 51.28 74.74 63.5
LEG POST
LOAD C2 207.33 168.52* 227.3
* The leg post between face A and face B was not the
primary mode of buckling that occurred for the elasticbeam
model under load case C2. The member capacity is based on
the load applied under the primary mode shape.
The tower capacity ratio given in Table 5.2 is based
on the tower test buckling at 1 (buckling/load factor =
1.0). The ratios given for the ASCE Manual 52 provide
numerical results for both the leg post between face A and
face B and the double angle leg brace in face A, based on
the axial loads found from the 3D spar model.
Table 5.2 Tower Capacity Ratio
TOWER
CAPACITY
TOWER
TEST
ELASTIC
BEAM
ASCE
MANUAL 52
LOAD B3 
LEG POST
LEG BRACE
1.00
85
1.97
.72
LOAD C2 
LEG POST
LEG BRACE
1.00
1.15*
1.68
2.25
* The buckling/load factor for the elasticbeam model
was averaged between load case C2 and C2M. The factor is
based on the buckling of the leg brace, not the leg post
which buckled during the tower test, and limits the
structure under the design code.
34
CHAPTER 6
CONCLUSION
The buckling analysis with the 3D elasticbeam model
predicts very closely the buckling mode shape, and capacity
of both the members and the tower. The 3D sparelement
model, on the other hand, is not effective for this type of
analysis.
In load case B3 the 3D elasticbeam computer model
predicted the same buckling mode shape that occurred in the
full scale test. The computer model underestimated the
tower capacity by 15% of the actual. The design manual
underestimates the tower capacity by 28%. The model,
therefore, would allow an additional 13% more capacity than
the design manual. The member that buckles under this load
is the double angle leg brace on face A. This member is
slender (1/r = 171) and buckles in the elastic range (Euler
buckling). The model can predict the effective column
length for members more accurately than the design manual
can. The manual uses k=.88, whereas the model corresponds
to an effective length coefficient of k=.81 for the leg
brace member. The model reduces the effective length due
to the action of the redundant members along the axis
transverse to the buckling axis.
In load case C2 the 3D elasticBeam model did not
predict the same buckling mode shape as occurred in the
fullscale test. The fullscale test buckled the leg post,
a member with an 1/r of 69.3 and a kl/r of even less. This
member buckles in the inelastic range. The model follows
the Euler's buckling equation, or elastic buckling, and
therefore is not valid in the inelastic range unless the
tangent modulus theory is applied. Thus, the computer
model will overpredict the capacity for this member, and
the capacity of the tower. The 3D ElasticBeam model
exhibited buckling in the leg brace under this load case.
Again, the leg brace is a slender member and therefore
controlled over the short leg post in buckling. The tower
capacity for this load case was overpredicted by 15% of the
actual, thus, verifying the earlier assumption that an
inelastic analysis would be required to ensure more
accurate results. This analysis would need to include the
stressstrain relationship including any residual stresses,
eccentricities of member connections, and if possible
member crookedness and material defects. The later two
items would be very difficult to model and may have very
little effect on the results.
The loading of members for the fullscale tower test,
in particular the member that may have been damaged by the
36
initial loading of load case B3, the leg brace, exhibited
different load bearing characteristics than in the model.
The member may have been damaged or considerable bolt
slippage in this area may have taken place. Bolt slippage
is another item that needs to be addressed to help
understand the complete characteristics of the structure.
Linear Elastic Buckling analysis, with the use of
complex 3D elasticbeam models, can be accurate when
applied in the elastic range of buckling (kl/r > Cc). Most
structures contain members in the short column class.
Thus, this would limit the use of this kind of analysis or
require a more sophisticated nonlinear elastic buckling
approach.
FirstOrder Linear Elastic analysis, the most common
analysis type used today, is simple, accurate and
inexpensive. The 3D sparelement model contains 50 nodes
and 170 elements vs. the more complex 3D elasticbeam with
346 nodes and 570 elements. Thus, 229 more nodes and 520
more elements must be input by the design engineer. The
computer, on the other hand, has 150 D.O.F. for the 3D
spar vs. the 2076 D.O.F. for the 3D elasticbeam model.
This corresponds roughly to a process time for the 3D
elasticbeam model of 200 times the time required to solve
the 3D sparelement model. The 3D sparelement model
37
used in the analysis predicts member loads, and tower
deflections very well, as shown in Table 3.1. The member
loads for the 3D spar model were within 5% of the axial
loads given by the 3D elasticbeam computer model. The
tower deflections for the 3D sparelement model, and the
3D elasticbeam model were essentially the same (within
1.5%).
Extensive research has gone into the creating of the
design manual fASCE Manual No. 521. Good correlations
between the elastic and inelastic column curves have been
proven in research. The practical design engineer should
use, with confidence, the design manual ASCE Manual No. 52.
but take special care in the calculations of the effective
lengths for the lattice tower members. Doing this will
ensure a simple design.
38
LIST OF REFERENCES
1. Guildlines for Transmission Line Structural Loading.
(1984). Commit, on Electr. Transm. Struct, of Commit, on
Anal, and Design of Struct, of Struct. Div. ASCE. New York:
ASCE.
2. National Electric Safety Code. (1987). Amer. Nat. Stand.
C2, Inst, of Electr. and Electron. Eng., Inc., New York:
Author
3. ASCE Manuals and Reports on Enar. Practice No. 52. Guide
for Design of Steel Transmission Towers 2nd Edition.
(1988). Task Commit, on Updating Manual No. 52 of Struct.
Div. ASCE. New York: ASCE.
4. Structural Loading Test of a Western Area Power
Administration 345kV. SelfSupporting. Double Dead End.
Single Circuit Lattice Transmission Tower (Type 36YK
(1988). S. E. Powell. Haslet, TX: Sverdrup Technology, Inc.
5. ANSYS. (1988). Swanson Analysis Systems, Inc. Houston,
PA
6. Structural Stability Research Council. Guide to
Stability Design Criteria for Metal Structures. (1976).
Bruce G. Johnston, ed., 3rd ed. New York: John Wiley &
Sons, Inc.
7. Steel Structures. Design and Behavior. Emphasizing Load
and Resistance Factor Design. (1990). Charles G. Salmon,
John E. Johnson, 3rd ed. New York: Harper & Row,
Publishers.
8. L. Euler. De Curvis Elasticis. Additamentum I, Methodus
Inveniendi Lineas Cuvas Maximi Minimive Proprietate
Gaudentes. Lausanne and Geneva, 1744 (pp. 267268); and
"Sir le Forces des Colonnes," Memoires de l'Academie Royale
des Science et Belles Lettres, Vol. 13, Berlin, 1759;
English translation of the letter by J. A. Van den Broek,
"Euler's Classic Paper 'On the Strength of Columns',"
American Journal of Physics, 15 (JanuaryFebruary 1947),
309318.
9. Manufacturer Drawing No. 864MD85175. 176. 180. 182
through 190. and 201. (1979). U.S. Department of Interior,
Bureau of Reclamations.
Appendix A
Tower Drawings
[9]
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Appendix B
Mathematical Methods
MATHEMATICAL FORMULATION
Ciju
{d}
e'i
E
m o
[N]
P
vi
u2
U
Ut
V
Â£'j
Â£ij
X
a'i
[*]
NOTATION
= Material property tensor.
= Nodal degrees of freedom vector.
= Concentrated load eccentricity.
= Lagrangian strain tensor.
= Modulus of Elasticity.
= Linear elastic stiffness matrix.
= Geometric stiffness matrix.
= Shape function matrix.
= Concentrated load.
= axial displacement component along element.
= inplane normal displacement component along element.
= Elastic strain energy.
= Total potential energy.
= Potential energy of external forces.
= Linear strain tensor.
= Nonlinear strain tensor.
= Load intensity factor.
= Stress tensor.
= Buckling mode shape.
Solution procedure
Following is a brief derivation of the relevant equation using the poten
tial energy formulation5,6,7. The expression of the elastic strain energy U is:
U = Jvo[oijd(eij)d(vol) (1)
where e,y is the Lagrangian strain tensor. Using the Hookes law aij =
Cijki^kt, where Cijki is the material property tensor, the above equation
can be written as:
U = \ fvol Cijkieueijd^ol) (i,j, M = 1,2,3) (2)
55
The Lagrangian strain tensor can be expressed in Cartesian coordinates
in terms of the displacement derivatives u,j, as
e>7 = 2^' + +
(3)
It is convenient to group the linear strain terms = j(u{j + xtyt;) and
the nonlinearstrain terms = 1 (utility).Thus
7 Â£*j "b 7"
(4)
Substitution of the above equation into equation (2) results in
= 2 Jvol(C'iuÂ£uÂ£ij + lCijkiÂ£kiÂ£ij + CijkiÂ£ktÂ£ij)d(vol) (5)
This is the basic form of strain energy for nonlinear analysis. The total
potential energy, denoted by t/j, is given by
Ut = U V (6)
where V is the potential energy of the external forces. Through the prin
cipal of stationary value of the total potential, one may derive the equi
librium equations and the proper end conditions. In the finite element
formulation the displacements axe approximated by the shape function
matrix [IV] times the nodal degrees of freedom vector {d}.
M = M{d}
For elastic stability analysis some simplifications can be made7. First
equation (5) can be rewritten as
U = J^{CijklÂ£klÂ£ij + 2[Ciju(Â£kl + CijkiÂ£ijÂ£kl}d(vol)
(7)
Discarding the higher order term and using Hookes law'
[^{CijklÂ£klÂ£ij 1 2(7ijEij')d(vol)
(8)
56
Since proportional loading is assumed, the applied force vector {P}
may be represented by an initial load vector {P0} times A a scalar load
intensity parameter:
{P} = A{P0}
(9)
The objective is to find the critical intensity A. at which buckling oc
curs. If the load vector {P0} is applied, a state of stresses afy can be
found. Therefore for any load vector P the state of stresses can be written
as
U = \ jjPiiHCueii + 2\
Differentiation of the displacements and integration of the strain energy
results in
u=^Vi0{d}+yfmiW (ii)
where [K]o is the linear elastic stiffness matrix and
[K\i = Jvol*iAAvo1) (12)
[K]i is the initialstress or geometric stiffness matrix. If the total poten
tial has a relative minimum at an equilibrium position (stationary value),
then the equilibrium position is stable and the second variation is positive
definite. If the second variation is negative definite, the static equilibrium
is unstable thus when a critical condition for instability is that the second
variation is zero
[[K}0 + \[K)1}$ = 0
(13)
where $ is the buckling mode shape.Furthermore, the determinant of the
above matrix must be zero
[[jr]0 + A(2T]I] = 0
(14)
The critical load is PCT = A^.P0. At this load level an adjacent equi
librium position exists, which means that a bifurcation point appears on
the equilibrium path. Equation 13 converts the buckling problem to the
classical eigenvalue form which can be solved efficiently with the available
eigensolving routines.
57
Appendix C
Computer Model
59
ELASTIC BEAM MODEL
ansys SCommand to begin ANSYS Program
/inter,no SCommand to Indicate Interactive Run
/prep7 SGeneral Analysis Data Generator
/input,key
SFile Containing Key Point Data
no. XCord YCord ZCord
k, 1 , 0 , 252 , 1020
k, 10 , 44.52 , 252 , 732
k, 14 , 44.52 , 126 , 732
k, 18 , 44.52 , 0 , 732
k, 30 , 55.68 , 252 , 660
k, 34 , 55.68 , 126 , 660
k, 38 , 55.68 , 0 , 660
k, 40 , 0 , 402 , 660
k, 50 , 69.72 , 220.8 , 564
k, 55 , 83.76 , 189.6 , 468
k, 60 , 104.4 , 0 , 326.4
k, 70 , 117.12 , 117.12 , 240
k, 80 , 151.2 , 151.2 , 0
k, 85 , 151.2 , 0 , 0
k, 87 , 0 , 151.2 , 0
k, 150 , 204 , 204 , 360
k, 2 , 0 , 252 , 1020
k, 11 , 44.52 , 252 , 732
k, 15 , 44.52 , 120 , 732
k, 19 , 44.4 , 0 , 732
k, 31 , 55.68 , 252 , 660
k, 35 , 55.68 , 120 , 660
k, 39 , 55.2 , 0 , 660
k, 41 , 0 , 396 , 660
k, 51 , 69.72 , 216 , 564
k, 56 , 83.76 , 180 , 468
k, 61 , 104.4 , 0 , 326.4
k, 71 , 117.12 , 116.4 , 240
k, 81 , 151.2 , 144 , 0
k, 86 , 144 , 0 , 0
k, 88 , 0 , 144 , 0
k, 151 , 204 , 204 , 360
k, 12 , 44.4 , 252 , 732
k, 16 , 44.4 , 120 , 732
k, 32 , 55.2 , 252 , 660
k, 36 , 55.2 , 120 , 660
k, 52 , 69.6 , 216 , 564
k, 57 , 82.8 , 180 , 468
k, 72 , 116.4 , 116.4 , 240
k, 82 , 144 , 144 , 0
k, 152 , 204 , 204 , 360
k, 13 , 44.4 , 252 , 732
k, 17 , 44.4 , 126 , 732
k, 33 , 55.2 , 252 , 660
k, 37 , 55.2 , 126 , 660
k, 53 , 69.6 , 220.8 , 564
k, 58 , 82.8 , 189.6 , 468
k, 73 , 116.4 , 117.12 , 240
k, 83 , 144 , 151.2 , 0
60
k, 153 , 204 , 204 , 360
ex,l,29e3 SAssumed Modulus of Elasticity
/input,properties
area, Izz . iyy t t t f Ixx, SHEARZ, SHEARY
r, 1 1.688 0.812 , 3.208 mn 0.035 > 0.797 0.803
r, 2 1.688 0.812 , 3.208 i i i i i 0.035 0.797 0.803
r, 3 4.359 6.201 ,24.573 i t i t 0.204 y 2.057 2.069
r, 3 4.359 6.201 ,24.573 y t t 0.204 y 2.057 2.069
r, 4 4.359 6.201 ,24.573 t f r t 0.204 y 2.057 2.069
r, 5 5.75 8.071 ,31.745 y y t t 0.479 2.716 2.744
r, 6 5.75 8.071 ,31.745 y t t 0.479 y 2.716 2.744
r, 10 2.859 1.774 , 6.944 y y y t y 0.134 y 1.351 1.369
r, 11 2.859 1.774 , 6.944 y y y y 0.134 y 1.351 1.369
r, 12 2.402 1.503 , 5.925 r y y y 0.078 y 1.134 1.145
r, 13 1.938 1.225 , 4.854 f i f t 0.04 y 0.914 0.92
r, 14 1.09 0.387 , 1.536 1 t t t 1 0.013 0.514 0.517
r, 15 2.402 1.503 , 5.925 y t f r 0.078 y 1.134 1.145
r, 16 1.688 0.812 , 3.208 y t y t 0.035 0.797 0.803
r, 17 1.438 0.504 , 1.984 y t t t 0.03 0.679 0.686
r, 18 2.402 1.503 , 5.925 t t i t 0.078 1.134 1.145
r, 19 1.688 0.812 , 3.208 r t > t 0.035 0.797 0.803
r, 19 1.688 0.812 , 3.208 i t t t 0.035 0.797 0.803
r, 20 0.809 0.148 , 0.652 t t 0.009 0.385 0.309
r, 21 3.027 2.99 , 11.85 t i t 0.098 1.429 1.437
r, 22 3.875 10.219 , 5.247 mil 0.081 1 1.667
r, 23 3.375 5.538 , 5.213 t t t t f 0.07 1 1.333
r, 24 1.938 1.225 , 4.854 i i m 0.04 0.914 0.92
r, 25 3.875 10.219 , 5.247 } t t t 0.081 1 1.667
r, 28 1.09 0.387 , 1.536 t t t * 0.013 0.514 0.517
r, 29 0.902 0.221 , 0.872 i t i t t 0.011 0.426 0.43
r, 30 1.438 0.504 , 1.984 t t t t 0.03 0.679 0.686
r, 31 1.813 0.976 , 4.03 t t 0.038 0.857 0.752
r, 32 1.813 0.976 , 4.03 t t t t 0.038 0.857 0.752
r, 33 1.938 1.225 , 4.854 f t t t 0.04 0.914 0.92
r, 34 1.438 0.504 , 1.984 t t i t t 0.03 0.679 0.686
r, 35 1.438 0.504 , 1.984 i r t 0.03 0.679 0.686
r, 36 3.309 2.037 , 7.911 r t t t 0.211 1.565 1.594
r, 37 2.063 1.223 , 6.392 t t y 0.043 1 0.697
r, 38 0.621 0.064 , 0.303 y f t f 0.007 0.299 0.225
r, 39 0.621 0.064 , 0.303 t y t y y 0.007 0.299 0.225
r, 41 1.688 0.716 , 3.409 y y t y 0.035 0.81 0.606
r, 40 5.117 13.21 ,10.179 y y y y 0.167 1.458 2.083
r, 42 0.621 0.064 , 0.303 y y y y y 0.007 0.299 0.225
/input,element
et,l,4 $See Documentation Section (JSTIF=4)
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1, 1 , 10
1, 1 , 13
1, 2 , 11
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elsi ,,1
lmesh,all
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1, 40 , 13
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elsi ,.l
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1, 31 , 52
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elsi ,,1
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elsi ,,1
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1, 55 , 73
1, 58 , 70
1, 56 , 72
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1, 30 , 37
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1, 32 , 35
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elsi ,,1
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1, 37 , 38
1, 36 , 38
1, 35 , 39
elsi .,1
lmesh, ,all
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elsi ,,1
lmesh, ,all
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65
1, 87 , 80
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e,326,123
66
e,123,325
e,325,119
e,119,324
e,324,115
e,115,323
e,323,47
e,107,342
e,342,103
e,103,341
e,341,99
e,99,340
e,340,95
e,95,339
e,339,35
e,107,322
e,322,103
e,103,321
e,321,99
e,99,320
e,320,95
e,95,319
e,319,35
e,87,334
e,334,83
e,83,333
e,333,79
e,79,332
e,332,75
e,75,331
e,331,23
e,87,318
e,318,83
e,83,317
e,317,79
e,79,316
e,316,75
e,75,315
e,315,23
e,63,209
e,209,67
e,67,205
e,205,283
e,283,67
e,63,232
e,232,67
e,67,228
e,228,297
e,297,67
e,51,198
e,198,55
e,55,194
e,194,276
e,276,55
e,51,255
67
e,255,55
e,55,251
e,251,311
e,311,55
e,39,244
e,244,43
e,43,240
e,240,304
e,304,43
e,39,186
e,186,43
e,43,182
e,182,269
e,269,43
e,27,221
e,221,31
e,31,217
e,217,290
e,290,31
e,27,175
e,175,31
e,31,171
e,171,262
e,262,31
d,71,all
d,91,all
d,111,all
d,131,all
/input,c2
f,l,fz,4.6
f,4,Â£z,6.8
Â£,151,fz,19.6
f,157,Â£z,19.5
f,158,fz,25.9
f,i,fy,3.6
f,A,fy,7.2
Â£,151,fy,11.2
f,157,fy,16.7
f,158,fy,27.8
f,l,Â£x,11.7
f,151,Â£x,36.1
f, 157,Â£x,36
modmsh,detach
edele,417
vsort,z $Reorder Elements Based on a Sort in the Zdir.
afvrite $Vrites Analysis Data File to File27
68
finish $Exit Prep7
/input,27 SSolution Phase
finish
/buck,,3,3,, ,,1 SActivates the Eigenvalue Buckling Analysis
iter,1,1,1
end $Exits Buckling Solution Phase
finish $Exits ANSYS
69
SPAR ELEMENT
$ ansys
/inter
/prep7
et,l,8
ex,l,29e3
/input,no!2
N, 1 , 0 , 252 , 1020
N, 10 , 44.52 , 252 , 732
N, 14 , 44.52 , 126 , 732
N, 18 , 44.52 , 0 , 732
N, 30 , 55.68 , 252 , 660
N, 34 , 55.68 , 126 , 660
N, 38 , 55.68 , 0 , 660
N, 40 , 0 , 402 , 660
N, 50 , 69.72 , 220.8 , 564
N, 55 , 83.76 , 189.6 , 468
N, 60 , 104.4 , 0 , 326.4
N, 70 , 117.12 , 117.12 , 240
N, 80 , 151.2 , 151.2 , 0
N, 85 , 151.2 , 0 , 0
N, 87 , 0 , 151.2 , 0
N, 150 , 204 , 204 , 360
N, 2 , 0 , 252 , 1020
N, 11 , 44.52 , 252 , 732
N, 15 , 44.52 , 120 , 732
N, 19 , 44.4 , 0 , 732
N, 31 , 55.68 , 252 , 660
N, 35 , 55.68 , 120 , 660
N, 39 , 55.2 , 0 , 660
N, 41 , 0 , 396 , 660
N, 51 , 69.72 , 216 , 564
N, 56 , 83.76 , 180 , 468
N, 61 , 104.4 , 0 , 326.4
N, 71 , 117.12 , 116.4 , 240
N, 81 , 151.2 , 144 , 0
N, 86 , 144 , 0 , 0
N, 88 , 0 , 144 , 0
N, 151 , 204 , 204 , 360
N, 12 , 44.4 , 252 , 732
N, 16 , 44.4 , 120 , 732
N, 32 , 55.2 , 252 , 660
N, 36 , 55.2 , 120 , 660
N, 52 , 69.6 , 216 , 564
N, 57 , 82.8 , 180 , 468
N, 72 , 116.4 , 116.4 , 240
N, 82 , 144 , 144 , 0
N, 152 , 204 , 204 , 360
N, 13 , 44.4 , 252 , 732
N, 17 , 44.4 , 126 , 732
N, 33 , 55.2 , 252 , 660
N, 37 , 55.2 , 126 , 660
N, 53 , 69.6 , 220.8 , 564
N, 58 , 82.8 , 189.6 , 468
N, 73 , 116.4 , 117.12 , 240
MODEL
70
N, 83 ,
N, 153 ,
144 ,
204 ,
151.2
204
0
360
/input,elem
real
e, 1
e, 1
e, 2
e, 2 real
e, 10
e, 11
e, 12
e, 13 real
e, 30
e, 31
e, 32
e. 33
e, 50
e, 51
e, 52
e, 53 real
e, 55
e, 56
e, 57
e, 58 real
e, 70
e, 71
e, 72
e, 73 real
e, 80
e, 81
e, 82
e, 83 real
e, 38
e, 38
e, 39
e, 39 real
e, 30
e, 31
e, 32
e, 33 real
e, 40
e, 40
e, 41
e, 41 real
e, 40
e, 40
e, 41
e, 41 real
1
10
13
11
12
2
30
31
32
33
3
50
51
52
53
55
56
57
58
4
70
71
72
73
5
80
81
82
83
6
150
151
152
153
10
34
35
36
37
11
34
35
36
37
12
30
33
31
32
13
10
13
11
12
14
71
e, 1 , 14
e, 1 , 17
e, 2 , 15
e, 2 , 16
real, 15
e, 10 , 14
e, 11 , 15
e, 12 , 16
e, 13 , 17
real, 16
e, 14 , 30
e, 15 , 31
e, 16 , 32
e, 17 , 33
e, 10 , 34
e, 11 . 35
e, 12 , 36
e, 13 , 37
real, 17
e, 14 , 34
e, 15 , 35
e, 16 , 36
e, 17 , 37
real, 18
e, 18 , 14
e, 18 , 15
e, 19 , 16
e, 19 , 17
real, 19
e, 18 , 34
e, 18 , 35
e, 19 , 36
e, 19 , 37
e, 38 , 14
e, 38 , 15
e, 39 , 16
e, 39 , 17
real, 20
e, 18 , 38
e, 19 , 39
real, 21
e, 34 , 55
e, 35 , 56
e, 36 , 57
e, 37 , 58
real, 22
e, 60 , 55
e, 60 , 56
e, 61 , 57
e. 61 , 58
real, 23
e, 60 , 70
e, 60 , 71
e, 61 , 72
e, 61 , 73
real, 24
e, 70 , 71
e, 73 , 72
real, 24
e, 70 , 73
72
e, 71 , 72
real, 25
e, 85 , 70
e, 85 , 71
e, 86 , 72
e, 86 , 73
e, 87 , 70
e, 87 , 73
e, 88 , 71
e, 88 , 72
real, 28
e, 10 , 13
e, 11 , 12
real, 29
e, 10 , 33
e, 13 , 30
e, 11 . 32
e, 12 , 31
real, 30
e, 30 , 33
e, 31 , 32
real, 31
e, 30 , 53
e, 33 , 50
e, 31 , 52
e, 32 , 51
real, 32
e, 50 , 58
e. 53 , 55
e, 51 , 57
e, 52 , 56
real, 33
e, 55 , 73
e, 58 , 70
e, 56 , 72
e, 57 , 71
real, 34
e, 30 , 37
e, 33 , 34
e, 32 , 35
e, 31 , 36
real, 35
e, 34 , 39
e, 37 , 38
e, 36 , 38
e, 35 , 39
real, 36
e, 38 , 39
real, 37
e, 70 , 72
e, 71 , 73
real, 38
e, 10 , 17
e, 13 , 14
e, 12 , 15
e, 11 , 16
real, 39
e, 14 , 19
e, 17 , 18
e, 16 , 18
73
e, 15 , 19
real, 41
e, 85 , 80
e, 85 , 81
e, 86 , 82
e, 86 , 83
e, 87 , 80
e, 87 , 83
e, 88 , 81
e, 88 , 82
real, 40
e, 85 , 150
e, 85 , 151
e, 86 , 152
e, 86 , 153
e, 87 , 150
e, 87 , 153
e, 88 , 151
e, 88 , 152
real, 42
e, 34 , 50
e, 35 , 51
e, 36 , 52
e, 37 , 53
e, 60 , 72
e, 60 , 73
e, 61 , 70
e, 61 , 71
e, 85 , 87
e, 85 , 88
e, 86 , 87
e, 86 , 88
e, 85 , 86
/input,pro
<3,150,all,0,,153,1
/input,c2
f,l,fz,4.6
f,2,fz,6.8
f,38,fz,19.6
f,40,fz,19.5
f,41,fz,25.9
f,1,fy,3.6
f,2,fy,7.2
f,38,fy,11.2
f,40,fy,16.7
f,41,fy,27.8
f,1,fx,11.7
f,38,fx,36.1
f,40,fx,36
/input,b3
f,1,fz,6.435
f,2,fz,4.356
f,38,fz,18.258
f,40,fz,24.402
74
f,
f,
Â£,
f,
f,
f,
f,
f,
Â£,
f,
f,
afvrite
finish
/input,27
finish
,fz,18.121
fy,6.844
fy,3.352
,fy,10.462
,fy.31.387
,fy,10.373
fx,11.035
,fx,33.751
,fx,33.594
,fx,4.2465
,fx,4.2465
41
1,:
2,:
38
40
41
2,:
38
41
80
83
/eof
