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Optimum Design of Trussed Columns
6.5 Optimum Design of Trussed Columns Ferenc Orban University ofPecs, H-7624 Pecs, Hungary, e-mail:[email protected] Abstract Trussed columns for overhead lines for power transmission were designed about 40 years ago. The aim of this study is to elaborate the optimum design for these columns with different cross sections of bars. The weight was the objective function. Different constraints were taken into account eg. global buckling constraints for compressed members, displacement constraints for the column and frequency analysis. Mathematical problem-solving software was used to calculate the optimum values. Keywords: truss structure, optimum design, FEM analysis 1
Introduction
Columns for overhead lines for power transmission are often constructed with rectangular trussed steel section. Earlier the cross - section of struts were designed only by angle profile. The goal of this study is to investigate how economic these columns are, however, we take only the mass into account. Further examinations were carried out for these structures when we used CHS (circular hollow section). Supporting columns are proportioned to different loads and to their corresponding combinations. These are: • Dead load, • Wind load, • Uneven wiring internal forces. The load carrying capacity of the column can be characterized by the so called forces reduced to the peak. For optimum design two loads were chosen: wind load and force reduced to the peak. Both of them are constrained by buckling capacity of the bars. While carrying out an optimum design the displacement and eigenvalue frequency were taken into account. The general arrangement of trussed columns for overhead lines for power transmission can be seen in Figure 1. Our main goal is to determine dimensions of parts of the columns. Design of head structures is not part of this study as it has several electric and constructional limitations. 2 Optimum design of trussed columns for buckling The internal bar forces of truss structure are calculated according to pin joint analysis. The chord member force: 4
2-a
The dimensioning condition relates to buckling capacity of struts. Check for global buckling according to Eurocode 3 (2002):
302
Hollow
A
Sections
(2)
y
t
Figure 1. General arrangement of trussed columns.
Figure 2. Dimensions and loads of the trussed columns. X can be determined in function of slenderness. KL (3) L is the length of the compression member, r is the radius of gyration, K is effective length factor. For brace members the slenderness is limited.
DFE2008 Design, Fabrication and Economy of Welded Structures
r>
KL
^
=150
303
(4)
The effective length factor of the chords is 0.76. At the brace members K = 1
I, = 0 , 7 6
• L
Figure 3. Effective length of chords. In case of angle profile the cross-section is: A = 2-b-t b is the side length of the equal leg angle profile. The radius of gyration is b r = 5,17
(5)
(6)
We make an approach for the flexural buckling factor ^ = 1,122-0-574761 At CHS of the cross section: A = Dn-t D - middle diameter. The radius of gyration is r= D
if:
0,2
ft
(7) (8)
(9)
The flexural buckling factor is X = 1,00658 + 0,0181 Oil - 0,35661'
(10)
3 Optimum design for the part of column The type of the column under investigation is V20 - 2500, means that in this case the column is 20 m high and the force reduced to the peak is 25 kN (2500 kp). Forces reduction to the peak is based on the requirement of the moment of all the forces should be equal with the moment reduced to the foundation (Figure 5).
304
Hollow
Sections
Figure 4. Arrangement of columns.
Figure 5. Explanation of reduced force F -m F
(11) / The optimization was carried out when the load is peak force and the cross section is angle profile. (Table 1 and Table 2.) m is mass of the column with standardized values. Table 1. Optimized values for upper part of the column. a] mm
S kN
800 900 1000 1100 1200
228 202,7 182,5 166 152
c
Chord member L65x65xll L75x75x8 L75x75x8 L75x75x7 L70x70x7
l mm brace 943 1029 1118 1208 1300 r
brace section L35x35x4 L40x40x4 L45x45x5 L45x45x5 L50x50x5
m/kg 147 134 140,8 144 141
DFE2008 Design, Fabrication and Economy of Welded Structures
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Table 2. Optimized values for bottom part of columns a mm
S kN
2
c
1000 1100 1200 1300 1400
502,5 228,4 209,3 193,2 179,4
Chord member L80x80xl0 L75x75x8 L75x75x8 L75x75x8 L75x75x8
I mm brace 1118 1208 1300 1393 1480 r
brace section L40x40x4 L45x45x5 L45x45x5 L50x50x5 L55x55x5
m kg 160,6 167 154,6 157,6 167,3
Optimization was carried out when cross-section is CHS. Table 3. Optimized values for upper part of columns. a mm
SkN
D cm
1000 1100 1200 1300 1400
182,5 166 152 140,3 130,3
6,98 6,39 5,9 5,49 5,14
t
l mm brace 1118 1208 1300 1393 1480
brace section 035x3 035x3 035x3 035x3 035x3
m kg 115,7 112,4 107,4 105,8 106,7
I mm brace 1393 1486 1581 1676 1772
brace section 035x3 035x3 035x3 035x3 035x3
m kg 118,7 115,7 112,8 109,9 115
r
Table 4. Optimized values for bottom part of columns. a mm
S kN
D cm
1300 1400 1500 1600 1700
193,25 179,4 167,5 157 147,8
7,36 6,874 6,449 6,079 5,75
c
2
r
Standardized values for the upper part of column is 0 6 0 x 4 and the bottom part is 0 6 5 x 4 . The dimensions of the brace members are 0 3 5 x 3 . 4 Optimum design of the column for wind load The load case when the direction of the wind is perpendicular to the wire was examined. F is value of the wind load for wire, when the distance between columns is 150 m. F i is the wind load acting on the head structure. The wind load to column depends on the wind pressure the surface and the form of the cross-section. The form factor of the angle profile is 1,4 and CHS is 0,7 . wo
w
Table 5. Optimized values for angle profile {optimum is marked by bold letters). a 700 800 900 1000 2
b cm 6,66 6,0 5,64 5,3
m kg 181,2 176,1 174,9 184,2
Chord section L70x70x6 L60x60x6 L60x60x6 L55x55x6
m kg 194,6 177,08 181,9 179,5
306
Hollow
Sections
a
2
Figure 6. Wind forces to the column. Table 6. Optimized values for CHS D cm 4,23 3,848 3,55 3,31
A 800 900 1000 1100 2
m kg 124,1 120,7 124,6 121,2
Chord section 048x4 042x4 040x4 038x4
Mkg 151,2 142,3 145,0 142,9
5 Optimum design for displacement The displacement of truss structure can be calculated with the formula (12)
-Xr
Where: S\, internal force, S j internal force when acting force is 1 N, /,, length of bars, A\ cross-section of bars. At cantilever:
w, =—
(13) °* 150 Figure 4.b shows a simplification of the Figure 4.a structure to ease the calculation of the mass. According to Figure 4.b we got the following condition: 14.79 39.36 ^ . . w= - + - + 9,65<44ww (14) A •a A -a Ai ,A ,&re the cross-section of the chord members. The minimum cross-section area was determined from buckling condition. n
x
2
l
2
2
£
c
DFE2008 Design, Fabrication and Economy of Welded Structures
307
Table 7. Optimized values for angle profile. j A mm 1000 1026 1052,5
a mm 1600 1400 1300
2
A i mm 1000 1000 1000
a; mm 900 1000 1100
2
2
m kg 298,5 295,55 296,73
The standardized section for optimized value is L75x75x8 Table 8. Optimized values for CHS a mm 1800 1600 1500
2
Ai mm 800 800 800
a; mm 1000 1100 1200
2
2
A
2
mm 800 806 813
m kg 225,2 223,3 224,3
The standardized section for optimized value is 0 7 0 x 4 . 6 Design for natural frequency In general, the minimum eigenvalue frequency is determined. (15) The frequency examination was carried out by FEM. We make comparisons between values, FEM calculations and Eq 16. See Table 9. 3,52 L
[W
a,
\ m
2K
0
The trussed column was assumed to be a beam.
l = A-a\
(17)
m , mass of a 1 m long column. 0
«. = M e w
2
+8- /«, +0,5 ^ J 2
A
R A
(18)
/ 7
The column length was 20 m. According to FEM and Eq. (16) calculations, the eigenvalue frequency is increasing, if ai dimension is bigger but the chords cross-section area has less influence on the value of frequency. Table 9. Eigenfrequency values for column. a mm
Chord section
1500
L70x70x7 L75x75x7 L90x90x9 L70x70x7 L75x75x7 L90x90x9
2
1800
f,H COSMOS 3,33 3,52 3,79 3,74 3,8 4,1 z
m kg 1543 1587 1926 1714 1759 2098
fi theoretical 3,36 3,43 3,86 3,9 3,91 4,44
308 7
Hollow Sections Conclusion
According to optimum values we got smaller mass for CHS both for peak force and wind load, when the constraint is the buckling of strut. For displacement restriction the CHS gave smaller mass. The standards determine a minimum eigenvalue frequency value. It is possible to fulfd the frequency condition, if we increase the distance of legs then we have to look for a cross section area value to get prescribed frequency. References Eurocode 3 (2002) Design of steel structures. Part 1-1: General structural rules. Brussel, CEN. Farkas, J. & Jarmai, K. (1997) Analysis and optimum design of metal structures Balkema, Rotterdam-Brookfield. Farkas J. & Jarmai, K. (2003) Economic design of metal structures. Rotterdam, Millpress.
Introduction
Columns for overhead lines for power transmission are often constructed with rectangular trussed steel section. Earlier the cross - section of struts were designed only by angle profile. The goal of this study is to investigate how economic these columns are, however, we take only the mass into account. Further examinations were carried out for these structures when we used CHS (circular hollow section). Supporting columns are proportioned to different loads and to their corresponding combinations. These are: • Dead load, • Wind load, • Uneven wiring internal forces. The load carrying capacity of the column can be characterized by the so called forces reduced to the peak. For optimum design two loads were chosen: wind load and force reduced to the peak. Both of them are constrained by buckling capacity of the bars. While carrying out an optimum design the displacement and eigenvalue frequency were taken into account. The general arrangement of trussed columns for overhead lines for power transmission can be seen in Figure 1. Our main goal is to determine dimensions of parts of the columns. Design of head structures is not part of this study as it has several electric and constructional limitations. 2 Optimum design of trussed columns for buckling The internal bar forces of truss structure are calculated according to pin joint analysis. The chord member force: 4
2-a
The dimensioning condition relates to buckling capacity of struts. Check for global buckling according to Eurocode 3 (2002):
302
Hollow
A
Sections
(2)
y
t
Figure 1. General arrangement of trussed columns.
Figure 2. Dimensions and loads of the trussed columns. X can be determined in function of slenderness. KL (3) L is the length of the compression member, r is the radius of gyration, K is effective length factor. For brace members the slenderness is limited.
DFE2008 Design, Fabrication and Economy of Welded Structures
r>
KL
^
=150
303
(4)
The effective length factor of the chords is 0.76. At the brace members K = 1
I, = 0 , 7 6
• L
Figure 3. Effective length of chords. In case of angle profile the cross-section is: A = 2-b-t b is the side length of the equal leg angle profile. The radius of gyration is b r = 5,17
(5)
(6)
We make an approach for the flexural buckling factor ^ = 1,122-0-574761 At CHS of the cross section: A = Dn-t D - middle diameter. The radius of gyration is r= D
if:
0,2
ft
(7) (8)
(9)
The flexural buckling factor is X = 1,00658 + 0,0181 Oil - 0,35661'
(10)
3 Optimum design for the part of column The type of the column under investigation is V20 - 2500, means that in this case the column is 20 m high and the force reduced to the peak is 25 kN (2500 kp). Forces reduction to the peak is based on the requirement of the moment of all the forces should be equal with the moment reduced to the foundation (Figure 5).
304
Hollow
Sections
Figure 4. Arrangement of columns.
Figure 5. Explanation of reduced force F -m F
(11) / The optimization was carried out when the load is peak force and the cross section is angle profile. (Table 1 and Table 2.) m is mass of the column with standardized values. Table 1. Optimized values for upper part of the column. a] mm
S kN
800 900 1000 1100 1200
228 202,7 182,5 166 152
c
Chord member L65x65xll L75x75x8 L75x75x8 L75x75x7 L70x70x7
l mm brace 943 1029 1118 1208 1300 r
brace section L35x35x4 L40x40x4 L45x45x5 L45x45x5 L50x50x5
m/kg 147 134 140,8 144 141
DFE2008 Design, Fabrication and Economy of Welded Structures
305
Table 2. Optimized values for bottom part of columns a mm
S kN
2
c
1000 1100 1200 1300 1400
502,5 228,4 209,3 193,2 179,4
Chord member L80x80xl0 L75x75x8 L75x75x8 L75x75x8 L75x75x8
I mm brace 1118 1208 1300 1393 1480 r
brace section L40x40x4 L45x45x5 L45x45x5 L50x50x5 L55x55x5
m kg 160,6 167 154,6 157,6 167,3
Optimization was carried out when cross-section is CHS. Table 3. Optimized values for upper part of columns. a mm
SkN
D cm
1000 1100 1200 1300 1400
182,5 166 152 140,3 130,3
6,98 6,39 5,9 5,49 5,14
t
l mm brace 1118 1208 1300 1393 1480
brace section 035x3 035x3 035x3 035x3 035x3
m kg 115,7 112,4 107,4 105,8 106,7
I mm brace 1393 1486 1581 1676 1772
brace section 035x3 035x3 035x3 035x3 035x3
m kg 118,7 115,7 112,8 109,9 115
r
Table 4. Optimized values for bottom part of columns. a mm
S kN
D cm
1300 1400 1500 1600 1700
193,25 179,4 167,5 157 147,8
7,36 6,874 6,449 6,079 5,75
c
2
r
Standardized values for the upper part of column is 0 6 0 x 4 and the bottom part is 0 6 5 x 4 . The dimensions of the brace members are 0 3 5 x 3 . 4 Optimum design of the column for wind load The load case when the direction of the wind is perpendicular to the wire was examined. F is value of the wind load for wire, when the distance between columns is 150 m. F i is the wind load acting on the head structure. The wind load to column depends on the wind pressure the surface and the form of the cross-section. The form factor of the angle profile is 1,4 and CHS is 0,7 . wo
w
Table 5. Optimized values for angle profile {optimum is marked by bold letters). a 700 800 900 1000 2
b cm 6,66 6,0 5,64 5,3
m kg 181,2 176,1 174,9 184,2
Chord section L70x70x6 L60x60x6 L60x60x6 L55x55x6
m kg 194,6 177,08 181,9 179,5
306
Hollow
Sections
a
2
Figure 6. Wind forces to the column. Table 6. Optimized values for CHS D cm 4,23 3,848 3,55 3,31
A 800 900 1000 1100 2
m kg 124,1 120,7 124,6 121,2
Chord section 048x4 042x4 040x4 038x4
Mkg 151,2 142,3 145,0 142,9
5 Optimum design for displacement The displacement of truss structure can be calculated with the formula (12)
-Xr
Where: S\, internal force, S j internal force when acting force is 1 N, /,, length of bars, A\ cross-section of bars. At cantilever:
w, =—
(13) °* 150 Figure 4.b shows a simplification of the Figure 4.a structure to ease the calculation of the mass. According to Figure 4.b we got the following condition: 14.79 39.36 ^ . . w= - + - + 9,65<44ww (14) A •a A -a Ai ,A ,&re the cross-section of the chord members. The minimum cross-section area was determined from buckling condition. n
x
2
l
2
2
£
c
DFE2008 Design, Fabrication and Economy of Welded Structures
307
Table 7. Optimized values for angle profile. j A mm 1000 1026 1052,5
a mm 1600 1400 1300
2
A i mm 1000 1000 1000
a; mm 900 1000 1100
2
2
m kg 298,5 295,55 296,73
The standardized section for optimized value is L75x75x8 Table 8. Optimized values for CHS a mm 1800 1600 1500
2
Ai mm 800 800 800
a; mm 1000 1100 1200
2
2
A
2
mm 800 806 813
m kg 225,2 223,3 224,3
The standardized section for optimized value is 0 7 0 x 4 . 6 Design for natural frequency In general, the minimum eigenvalue frequency is determined. (15) The frequency examination was carried out by FEM. We make comparisons between values, FEM calculations and Eq 16. See Table 9. 3,52 L
[W
a,
\ m
2K
0
The trussed column was assumed to be a beam.
l = A-a\
(17)
m , mass of a 1 m long column. 0
«. = M e w
2
+8- /«, +0,5 ^ J 2
A
R A
(18)
/ 7
The column length was 20 m. According to FEM and Eq. (16) calculations, the eigenvalue frequency is increasing, if ai dimension is bigger but the chords cross-section area has less influence on the value of frequency. Table 9. Eigenfrequency values for column. a mm
Chord section
1500
L70x70x7 L75x75x7 L90x90x9 L70x70x7 L75x75x7 L90x90x9
2
1800
f,H COSMOS 3,33 3,52 3,79 3,74 3,8 4,1 z
m kg 1543 1587 1926 1714 1759 2098
fi theoretical 3,36 3,43 3,86 3,9 3,91 4,44
308 7
Hollow Sections Conclusion
According to optimum values we got smaller mass for CHS both for peak force and wind load, when the constraint is the buckling of strut. For displacement restriction the CHS gave smaller mass. The standards determine a minimum eigenvalue frequency value. It is possible to fulfd the frequency condition, if we increase the distance of legs then we have to look for a cross section area value to get prescribed frequency. References Eurocode 3 (2002) Design of steel structures. Part 1-1: General structural rules. Brussel, CEN. Farkas, J. & Jarmai, K. (1997) Analysis and optimum design of metal structures Balkema, Rotterdam-Brookfield. Farkas J. & Jarmai, K. (2003) Economic design of metal structures. Rotterdam, Millpress.