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A simplified model for buckling mechanism in lattice structures
Pergamon
0045-7949(95)00057-7
TECHNICAL A SIMPLIFIED
Compurm & S/rurruresVol. 57, No. 4, pp. 745-750. 1995 Copyright r, 1995 Ekvier Science Ltd Printedin Great Britain.All rights reserved 004s7949/95 $9.50 + 0.00
NOTE
MODEL FOR BUCKLING LATTICE STRUCTURES
MECHANISM
IN
R. W. Hopkins, J. L. Meek and F. A. Al-bermani Department
of Civil Engineering,
The University
of Queensland,
St Lucia,
Qld 4072, Australia
(Receioed 2 1 April 1994) Abstract-This paper presents a simplified model of buckling mechanism which occurs in slender steel column elements under compressive axial loading. This model has been used to predict buckling capacities of certain structures, and comparisons have been made with test results. The method presented is not as accurate as the current finite element methods which use complete large displacement elasto-plastic analysis. However, it provides a reasonable upper bound load very economically. and with reasonable accuracy. It will be argued that it is pointless striving for a 1% accuracy when material parameters and load target values themselves can only be accurate to, say, 10%. The study has used software developed by Meek [3DBUCKLE_eigenvalue analysis program for buckling of steel columns, University of Queensland (1973)], which uses eigenvalue analysis by iteration to determine the Euler (theoretical) value of the buckling load of a column structure. This program was adapted to include the possibility that, during buckling, a reduced modulus section develops at the central portion of the critical column length. The reduction in the second moment of area of the section over a given length ensures that the revised estimate of the failure load is less than that predicted by the Euler formula for the undamaged column. The program was used to predict the failure loads of structures previously tested in other research programs, and the results show reasonable agreement with experiment. The paper shows the potential of the method as a simple design tool.
1. INTRODUCTION
given structure will have a stiffness, and this stiffness can be represented mathematically as a total stiffness matrix, KT. This matrix provides a relationship between the forces, R, applied to a structure, and the nodal deflections, r, such that
It was Euler [2]. who in the mid-1700s first expressed the buckling load of a column structure in terms of its geometric and material properties. He suggested the following relationship:
(1) where n describes the buckling mode or shape (i.e. n half sine waves), and K is the effective length factor. Though still a familiar equation today. the Euler formula can be readily proved unsuitable for the prediction of failure loads. This is because, whereas the “ideal” or Euler column is perfect, “real” columns contain slight imperfections such as out-of-straighnesses and small eccentricities of the applied load. The resulting “secondary effects” cause the system to become unstable before the expected buckling load has been reached. Used in conjunction with the compression yield line, the Euler curve represents an upper bound for the estimate of failure load (see Fig. 1). but for practical purposes cannot be used in design, except for very slender columns. Included in Fig. 1 are two standard multiple column curves (a, and b) recommended by the European Convention of Constructional Steelwork (ECCS) (31. These curves are more conservative than the theoretical curves, and, having been developed through extensive experimentation, were considered suitable as boundary curves for the development of the model.
2. EIGENVALUE
ANALYSIS
(2)
[&I = [&I + &I
(3)
where (KE] is the elastic stiffness matrix and is a function of material properties and the geometry of the system, and [KG] is the geometric stiffness matrix, which is a function of axial load and column length, and is non-positive definite for a compressive axial load. Thus, the effect of a compressive axial force will be to reduce the stiffness of a column. Consider, then, the situation where the axial load is so high that the stiffness is reduced to zero (that is, KE = -KG). At this point, according to eqn (2). for a finite load, R. the deflections. r, become indeterminate, and the system becomes unstable, that is. buckling has occurred. Working the other way, if the point can be found where the stiffness becomes zero, the buckling load can be inferred. Since the stiffness is represented by a matrix, it will actually be the matrix determinnnt which must go to zero. Also, since the stiffness is a function of the axial load, that is. the answer is required in the solution, the method of solution must be iterative. The determinant is calculated using the Gaussian elimination method of matrix manipulation. The basic function of 3DBUCKLE, then, is as follows:
PROGRAM
(1) calculate KE; (2) assume a critical
The original program 3DBUCKLE, was written in 1973 by Meek [l]. The basis of the eigenvalue method is that any CAS 57,&M
{RI = [Krl{r) where
745
buckling
load,
P_;
Technical
746 1.6
Note
\
I red
Dam
=
FAC
x
D
i 01
0
0.5
1
1.5
Non-d,mensionei~zed
Fig.
2
2.5
Slenderness
I
3
“Notch”
Rat10.h
1. Real and idealized relationships and slenderness
between critical load ratios.
(3) calculate KG using PC,; (4) K,=K,+K,; (5) calculate det 1KT I: (6) if det 1KTI is not equal to zero, adjust P,, and go to step 3. If det (KTi equals zero, then the last estimate of PC, was correct. For a simple pin-ended column, this program calculates a close approximation to the Euler buckling load, which is known to be an over-estimate. The question then arises: if the stiffness of the column was reduced over the central portion of the length, would a more realistic estimate of the critical buckling load be achieved? The answer is yes. How much the stiffness should be reduced and over what length is the purpose of this paper.
3. TDB
3.1. The plasticity
PROGRAM
mechanism
It is recognized
that, during buckling, the central portion of the column length will undergo a sudden and severe change in strain profile. For a column with a medium slenderness ratio (for example 60 < L/r < 120). that profile might look as shown in Fig. 2(a). This is associated with a sudden increase in the curvature of the column. The material on the concave side will undergo an increase in compression, while that on the convex side will undergo a reduction in compression, or, as in this case, may actually go into tension. More significantly,
Fig. 3.
“Notch”
i strain
Ten!5
ProfIle
I Fig. 2.
Assumed
stress and strain
profiles.
of column
it can be seen from the diagram that part of the section in compression has actually yielded. For the sake of simplicity, it can be argued that any part of the section which has yielded does not contribute to the second moment of area of the section. Since the elastic stiffness matrix, 4, is a function of the second moment of area, the required reduction in stiffness, discussed at the conclusion of Section 2. can be achieved by recognizing this reduction in the second moment of area, Ired. That is, the column has effectively become “notched” (see Fig. 3). The objective of this paper, then, is to attempt to model this notching or plasticity mechanism, and somehow incorporate the model into the main program. The eigenvalue analysis will then be performed on a “weakened” column, and hopefully provide a more realistic estimate of the buckling load of the column. The program is set up such that the critical column is divided into five sub-lengths. There are two reasons for this: (1) so that the geometric details of the central sub-length can be altered individually; (2) when using the eigenvalue method the more sublengths used. the more accurate the analysis. The following simplifying imposed on the model:
(a)
at centre
assumptions
must
now
be
(1) the relationship between the stress and strain is idealized. as shown in Fig. 4; (2) the compressive yield strain is reached at a distance x from the tension face; (3) as indicated in Fig. 4. for all strains exceeding the yield strain, the stress is constant, and equal to the yield stress, (TV;this section provides the only contribution to the axial force. C2; (4) the stress profile is presumed to be such that the forces T and C 1 are equal and opposite and therefore cancel (see Fig. 2): (5) the section which has yielded does not contribute to the second moment of area: (6) there is no yielding in tension. Having made these simplifying assumptions, it can be seen that the true stress and strain profiles do not need to be known. It is possible that, for extremely high values of slenderness, the convex side will actually yield in tension, causing
Technical
Note
747
entered into the final program in equation form. Thus, the damage is, in effect, pre-determined. A more detailed description of the procedure is as follows. 3.3. Determination of the FAC curves
I/
E
=
Elastic
Modulus
stra Fig. 4.
Real and idealized
n
stress-strain
relationship.
a further reduction in second moment of area. Alternatively, it is likely that, for extremely low values of slenderness, there will be no part of the section in tension. The given strain and stress profiles shown in Fig. 2 would then not apply, and separate models would be required to account for these possibilities. This paper, however, is only concerned with the general case shown in Fig. 2. It is anticipated, however, that, although for these extreme cases the model is not technically correct, the partly empirical nature (see note below) of the method will ensure that a good approximation to the critical buckling load may still be obtained for many practical cases. Note: because the lack of information concerning the moment precludes a rigorous strain based analysis, it is accepted that the method will not be purely analytical. That is, since exhaustive experimentation which has been carried out in the field had led to a reasonable agreement as to the true shape of the non-dimensionalized load vs slenderness curve, it was decided that this be used as the boundary curve against which to set up the TDB program. More specifically, the ECCS curves a, and h were used (see Fig. 1). 3.2. FAC-damage
length ,factor
Section 3.1 discussed the plasticity mechanism which occurs at the mid-length of the buckled column, and can be modelled as a reduction in the second moment of area of the section. It is now necessary to look at the length of column over which this reduction is to take place. Figure 3 introduces a damage length, DAM, which is the product of damage length factor, FAC. and the depth of the column, D. It was considered appropriate that the length be non-dimensionalized so as to keep it universal for all values of the variable D. It has been mentioned previously that this method will not be entirely analytical. but partly empirical as well. In the final version of the TDB program, the FAC factor will be calculated directly from the known non-dimensionalized slenderness ratio of the column, the relationship between the two quantities having been predetermined using the procedure described in the next section. Note: non-dimensionalized slenderness ratio, 1. is defined as follows:
The following procedure was followed for several column types (UCs, UBs and equal angles), for buckling failure about several axes. Firstly the program 3DBUCKLE is altered to accept a variable FAC value which is used within the program to calculate the lengths of the sub-sections of the column. Then, for a range of slenderness ratios, the value of FAC is determined, by iteration, as follows: For 1 = 0.5-2.8, in increments of 0.1;
(1) read member size from data file; (2) calculate total area, A,,,,,, second moment of area, I, and the radius of gyration of the gross section, r; (3) read P/P, from ECCS curve (note: P = known buckling load = C2 in Fig. 2); (4) calculate P = P/Py*Py = C2; area; (5) calculate Acomp= P/P,*total (6) calculate x using Acomp and known section geometry (x described in Fig. 2). (7) calculate Ired using x; using Iti for middle (8) set up data file for 3DBUCKLE sub-element of column; (9) Finally, vary FAC until the eigenvalue analysis produces an estimate of critical buckling load, P,,, equal to P calculated above. This value of FAC is then plotted onto the curve showing FAC vs 1. The FAC curves having now been developed for several column types buckling about several axes, it is possible to set up the final version of the 3DBUCKLE program, TDB. 3.4. TDB program Whereas during the development of the FAC curves a manual iteration of FAC was performed, for the TDB program the iteration will be over the variable X, FAC being calculated directly within the program. Thus, the procedure use to calculate an estimate of the critical buckling load of a column will be as follows:
(1) input x (guessed); area of yielded section); (2) calculate A (3) calculate PC”2 ar I; Fomp*oy ; (4) calculate I, ; input Ired into properties file; calculate 1 (using details given); calculate FAC using d; calculate lengths of five sub-sections; perform eigenvalue analysis to give estimate of critical buckling load, P,,; (10) if Pa, = P,, then exit, else, adjust x and go to Step 2. (5) (6) (7) (8) (9)
”
ElastIc
where leR= effective length = L for pin-ended column; r = radius of gyration (gross section). Briefly. for a given column type, the reduced second moment of inertia having been determined, the value of FAC will be varied until such time as the buckling load calculated by eigenvalue analysis agrees with the value given by ECCS curve. This value of FAC will be recorded, for a range of slenderness ratios, and for several possible column types, to form the FAC curves. These curves are then
Section yielded in compression
Portion
\\
C A con0 1 ”
: \ Caiculate mcment port
Fig. 5.
Buckling
mechanism
of
reduced
area
IO”,
second
using
about
v-v
elastic axis
for equal angle about
x-axis.
Technical
748
Note
I
I
Table 2. Comparison of various predictions to actual failure load
PC,
(
Fig. 6.
Arrangment
]
represents
of multi-bay
a
cross-braced
OF
COMPARISON
STRUCTURES OF
frame.
Case no. Al BI
TDB ANGLE ASCE
120 127 194
0.98 1.04 I.59
4.2. K-brace of a lattice tower LeMaster [5] performed an analysis on a K-brace system, using a finite element package. ANGLE. This system had already been tested to failure at the EPRI Transmission Line Mechanical Research Facility (TLMRF). The frame set-up was as follows: The critical member was a 3.5 x 2.5 x 114 angle (in). Since the TDB program requires dimensions in millimetres, this was approximated as a 74 x 74 x 6 angle. A TDB analysis was performed and the result presented below, along with the prediction of the ANGLE program, and an estimate produced in accordance with the American Society of Civil Engineers (ASCE) standard procedure. These are compared to the TLMRF experimental result of 122 kN. Both the TDB and ANGLE programs can be seen to give excellent predictions, far more accurate than the standard
AND
Cr / t I ca I
RESULTS
Fig. 7.
I. Comparison of TDB with Finch’s experiment
K-brace
Member
system.
c
Ii
cross-braced frcmtes
Finch [4], in 1992 performed a series of experiments on some multi-bay cross braced frames, two of which are presented here. A schematic arrangement of the frame is as shown in Fig. 6. The chords were formed by halving a Table
Ratio
200UB30 section to form what were labelled as IOOBTlS T-sections. The verticals were pairs of 50 x 50 x 5 angles, while the diagonals were single 40 x 40 x 5 angles. Each crossed pair of diagonals was connected by a zero torsion “bolt”. as described above, while all other connections were welded. The total load was applied equally at nodes 2 and 4. In each case, the structure failed via the buckling of the diagonal between nodes 2 and I I. The results obtained were as shown in Table I. As can be seen from Table I, the TDB method has predicted the failure load to within 4% accuracy for the first case, and I% for the second.
This section presents a description of various structures for which the buckling loads have been estimated using the TDB program. These results are shown along with the results of actual tests performed in the field by other researchers. 4.1. Multi-hay
(kN)
bolt"
TDB will perform Steps 229 automatically, while the input of .Y (step 1) and the comparison of P_ and P,, (Step 10) are performed by the operator. Thus, a partly empirical, partly analytical method has been developed which is able to produce estimates of the critical buckling load of pin-ended columns which agree with the ECCS experimental curves. This would seem a pointless exercise, in itself, when it is considered that these estimates could simply be read directly off the curves. The strength of this method, however, does not concern pin-ended columns, but rather columns which belong to a sub-assembly, in a situation where neighbouring elements can provide some measure of rotation restraint at the connecting points. which, in turn. can enhance the buckling load of the critical column by some unknown degree. It is usual, in design. to consider that the end of a column is fully hinged, or fully fixed, and there are relatively simple methods to account for these situations by assigning effective length factors. These methods. however, are unable to accurately quantify a situation where a neighbouring member can provide “partial” fixity. That is. even though a critical member is fully fixed to a neighbouring member. if this other member is of limited robustness, the node can still rotate a little, and this will affect the performance of the column in question. The eigenvalue method, on the other hand, can take this effect into account, and it is this fact which makes the TDB program useful. In fact, the eigenvalue method automatically assumes that all connections are rigid. In the situation where a hinged connection is required. this can be modelled by inserting an extra member, say IOmm long. with a zero torsional constant. but second moment of area equal to that of the connecting member. This allows the member end to rotate. while staying in position. These short members are referred to as “bolts”.
4. ANALYSIS
Method
Cr
t
t
ca
,/’
I
lement
,,’
results
TDB (kN)
Finch (kN)
Ratio (&)
I87 187
194 185
0.96 1.01
431
37
Fig. 8.
kN
Inverted
tower
leg.
Technical
Fig. 9.
Note
Full scale transmission
ASCE method. It is acknowledged. however, that these are comparatively simple structures, and that the eigenvalue method has not been applied to a more involved structure, such as transmission tower. The next examples will deal with this. 4.3. Inverted tower leg In 1991. Knight and Santhakumar [6] tested, to failure, the tower leg shown in Fig. 8. The leg was inverted simply to allow easier application of the load. During testing, the lower third of the main angle buckled inwards about the minor principal axis, under an axial load of 431.7 kN (with secondary loads as indicated in Fig. 8). The TDB analysis predicted a buckling load of 492 kN, which represents a 13% over estimate. This is obviously not as good a result as for previous examples, yet is still considered reasonable. It is believed that the accuracy of the plasticity model may have been reduced for a column of such low slenderness. Recall that Section 3.1 discussed the possibility that a different model might be preferable for the extreme slenderness ratios.
749
tower
4.4. Full scale transn~ission tower In 1986, Meuller er al. [7] performed a full scale load test on the transmission tower shown in Fig. 9. The load was applied in the positive u-direction. at nodes 15 and 16, and the member connecting nodes 68 and 92 failed in the second mode at a total applied load of 19.5 kips. This represents a member load of 85 kips, or 380 kN. It was decided for simplicity that, since the effects which neighbouring members have on a critical member decrease with distance from the member, it was not necessary to include all the members of the tower in the input file for TDB. The simplified structure used in the TDB analysis looked as shown in Fig. 10. The TDB analysis gave a member buckling load of 441 kN, which represents an over-estimate of 16%. Notes. (I) For the above examples, the ratio between the total applied load and the critical member load was found using the MicroSTRAN [8] structural analysis package. (2) An uncertainty is introduced into the TDB analysis by the lack of knowledge of the end connection details. The buckling load of a member is highly dependent on the
Technical
750
.,i
Note known to great accuracy, there is a limitation on how accurate a prediction can be. If a series of identical experiments produces a scatter of IO%, a claim to be accurate to 5% would be meaningless. Further refinements to the model, however, are still considered necessary. The program may be adjusted to include the alternative plasticity models for the higher and lower extremes of slenderness. It would also be sensible to look into the possibility of improving the way the end connections are modelled, since this is known to have a significant effect on the performance of a column.
REFERENCES
Fig. 10. Simplified
tower structure
for TDB analysis.
method of connection to neighbouring members, and in these cases some assumptions have been made which may have introduced some error. 5. CONCLUSIONS AND RECOMMENDATIONS
The TDB analyses performed on the structures described in the previous section have shown the potential of the modified eigenvalue method for the prediction of critical buckling loads. Most predictions have been accurate to less that 15%. It is accepted that this method will not be as accurate as the more involved finite element methods, however it does offer reasonable estimation. It is also argued that, since a given member’s material properties and details of imperfections cannot themselves be
1. J. L. Meek, 3DBUCKLE-Eigenvalue analysis program for buckling of steel columns, University of Queensland (1973). 2. L. Euler, Sur la forces des colonnes, Acad. R. Sci. Belles Letr. Berlin, Mem. 13, 252, (1759). English translation by J. A. Van den Broek, Am. J. Phys. 15, 309, (1947). 3. European Convention for Constructional Steelwork (ECCS), In Second International Colloquium on Slabilitl Introductory Report (Stabilit~~ Manual), 2nd Edn, Subchapter 3.1.5, Angles, pp. 98103, and Sub-chapter 9.2. Angles in Lattice Transmission Towers (1976). 4. D. L. Finch, Double diagonal cross bracing, Submission for Ph.D., University of Queensland (1992). 5. R. A. LeMaster, Structural development studies at the EPRI transmission line Mechanical Research Facility, Interim Report 1, EPRI EL-4756, Electric Power Research Institute, prepared by Sverdrup Technology Inc.. Tullahama, TN (1986). 6. G. M. S. Knight and A. R. Santhakumar, Joint effects of behaviour of transmission towers, J. S/rucr. Engng AXE. 119, 6988712 (1993). 7. W. H. Meuller, S. L. Prickett and L. Kempner, Nonlinear and full scale test of a transmission tower. Portland State University (1986). 8. MicroSTRAN Version 5.21. Structural analysis program (1993).
0045-7949(95)00057-7
TECHNICAL A SIMPLIFIED
Compurm & S/rurruresVol. 57, No. 4, pp. 745-750. 1995 Copyright r, 1995 Ekvier Science Ltd Printedin Great Britain.All rights reserved 004s7949/95 $9.50 + 0.00
NOTE
MODEL FOR BUCKLING LATTICE STRUCTURES
MECHANISM
IN
R. W. Hopkins, J. L. Meek and F. A. Al-bermani Department
of Civil Engineering,
The University
of Queensland,
St Lucia,
Qld 4072, Australia
(Receioed 2 1 April 1994) Abstract-This paper presents a simplified model of buckling mechanism which occurs in slender steel column elements under compressive axial loading. This model has been used to predict buckling capacities of certain structures, and comparisons have been made with test results. The method presented is not as accurate as the current finite element methods which use complete large displacement elasto-plastic analysis. However, it provides a reasonable upper bound load very economically. and with reasonable accuracy. It will be argued that it is pointless striving for a 1% accuracy when material parameters and load target values themselves can only be accurate to, say, 10%. The study has used software developed by Meek [3DBUCKLE_eigenvalue analysis program for buckling of steel columns, University of Queensland (1973)], which uses eigenvalue analysis by iteration to determine the Euler (theoretical) value of the buckling load of a column structure. This program was adapted to include the possibility that, during buckling, a reduced modulus section develops at the central portion of the critical column length. The reduction in the second moment of area of the section over a given length ensures that the revised estimate of the failure load is less than that predicted by the Euler formula for the undamaged column. The program was used to predict the failure loads of structures previously tested in other research programs, and the results show reasonable agreement with experiment. The paper shows the potential of the method as a simple design tool.
1. INTRODUCTION
given structure will have a stiffness, and this stiffness can be represented mathematically as a total stiffness matrix, KT. This matrix provides a relationship between the forces, R, applied to a structure, and the nodal deflections, r, such that
It was Euler [2]. who in the mid-1700s first expressed the buckling load of a column structure in terms of its geometric and material properties. He suggested the following relationship:
(1) where n describes the buckling mode or shape (i.e. n half sine waves), and K is the effective length factor. Though still a familiar equation today. the Euler formula can be readily proved unsuitable for the prediction of failure loads. This is because, whereas the “ideal” or Euler column is perfect, “real” columns contain slight imperfections such as out-of-straighnesses and small eccentricities of the applied load. The resulting “secondary effects” cause the system to become unstable before the expected buckling load has been reached. Used in conjunction with the compression yield line, the Euler curve represents an upper bound for the estimate of failure load (see Fig. 1). but for practical purposes cannot be used in design, except for very slender columns. Included in Fig. 1 are two standard multiple column curves (a, and b) recommended by the European Convention of Constructional Steelwork (ECCS) (31. These curves are more conservative than the theoretical curves, and, having been developed through extensive experimentation, were considered suitable as boundary curves for the development of the model.
2. EIGENVALUE
ANALYSIS
(2)
[&I = [&I + &I
(3)
where (KE] is the elastic stiffness matrix and is a function of material properties and the geometry of the system, and [KG] is the geometric stiffness matrix, which is a function of axial load and column length, and is non-positive definite for a compressive axial load. Thus, the effect of a compressive axial force will be to reduce the stiffness of a column. Consider, then, the situation where the axial load is so high that the stiffness is reduced to zero (that is, KE = -KG). At this point, according to eqn (2). for a finite load, R. the deflections. r, become indeterminate, and the system becomes unstable, that is. buckling has occurred. Working the other way, if the point can be found where the stiffness becomes zero, the buckling load can be inferred. Since the stiffness is represented by a matrix, it will actually be the matrix determinnnt which must go to zero. Also, since the stiffness is a function of the axial load, that is. the answer is required in the solution, the method of solution must be iterative. The determinant is calculated using the Gaussian elimination method of matrix manipulation. The basic function of 3DBUCKLE, then, is as follows:
PROGRAM
(1) calculate KE; (2) assume a critical
The original program 3DBUCKLE, was written in 1973 by Meek [l]. The basis of the eigenvalue method is that any CAS 57,&M
{RI = [Krl{r) where
745
buckling
load,
P_;
Technical
746 1.6
Note
\
I red
Dam
=
FAC
x
D
i 01
0
0.5
1
1.5
Non-d,mensionei~zed
Fig.
2
2.5
Slenderness
I
3
“Notch”
Rat10.h
1. Real and idealized relationships and slenderness
between critical load ratios.
(3) calculate KG using PC,; (4) K,=K,+K,; (5) calculate det 1KT I: (6) if det 1KTI is not equal to zero, adjust P,, and go to step 3. If det (KTi equals zero, then the last estimate of PC, was correct. For a simple pin-ended column, this program calculates a close approximation to the Euler buckling load, which is known to be an over-estimate. The question then arises: if the stiffness of the column was reduced over the central portion of the length, would a more realistic estimate of the critical buckling load be achieved? The answer is yes. How much the stiffness should be reduced and over what length is the purpose of this paper.
3. TDB
3.1. The plasticity
PROGRAM
mechanism
It is recognized
that, during buckling, the central portion of the column length will undergo a sudden and severe change in strain profile. For a column with a medium slenderness ratio (for example 60 < L/r < 120). that profile might look as shown in Fig. 2(a). This is associated with a sudden increase in the curvature of the column. The material on the concave side will undergo an increase in compression, while that on the convex side will undergo a reduction in compression, or, as in this case, may actually go into tension. More significantly,
Fig. 3.
“Notch”
i strain
Ten!5
ProfIle
I Fig. 2.
Assumed
stress and strain
profiles.
of column
it can be seen from the diagram that part of the section in compression has actually yielded. For the sake of simplicity, it can be argued that any part of the section which has yielded does not contribute to the second moment of area of the section. Since the elastic stiffness matrix, 4, is a function of the second moment of area, the required reduction in stiffness, discussed at the conclusion of Section 2. can be achieved by recognizing this reduction in the second moment of area, Ired. That is, the column has effectively become “notched” (see Fig. 3). The objective of this paper, then, is to attempt to model this notching or plasticity mechanism, and somehow incorporate the model into the main program. The eigenvalue analysis will then be performed on a “weakened” column, and hopefully provide a more realistic estimate of the buckling load of the column. The program is set up such that the critical column is divided into five sub-lengths. There are two reasons for this: (1) so that the geometric details of the central sub-length can be altered individually; (2) when using the eigenvalue method the more sublengths used. the more accurate the analysis. The following simplifying imposed on the model:
(a)
at centre
assumptions
must
now
be
(1) the relationship between the stress and strain is idealized. as shown in Fig. 4; (2) the compressive yield strain is reached at a distance x from the tension face; (3) as indicated in Fig. 4. for all strains exceeding the yield strain, the stress is constant, and equal to the yield stress, (TV;this section provides the only contribution to the axial force. C2; (4) the stress profile is presumed to be such that the forces T and C 1 are equal and opposite and therefore cancel (see Fig. 2): (5) the section which has yielded does not contribute to the second moment of area: (6) there is no yielding in tension. Having made these simplifying assumptions, it can be seen that the true stress and strain profiles do not need to be known. It is possible that, for extremely high values of slenderness, the convex side will actually yield in tension, causing
Technical
Note
747
entered into the final program in equation form. Thus, the damage is, in effect, pre-determined. A more detailed description of the procedure is as follows. 3.3. Determination of the FAC curves
I/
E
=
Elastic
Modulus
stra Fig. 4.
Real and idealized
n
stress-strain
relationship.
a further reduction in second moment of area. Alternatively, it is likely that, for extremely low values of slenderness, there will be no part of the section in tension. The given strain and stress profiles shown in Fig. 2 would then not apply, and separate models would be required to account for these possibilities. This paper, however, is only concerned with the general case shown in Fig. 2. It is anticipated, however, that, although for these extreme cases the model is not technically correct, the partly empirical nature (see note below) of the method will ensure that a good approximation to the critical buckling load may still be obtained for many practical cases. Note: because the lack of information concerning the moment precludes a rigorous strain based analysis, it is accepted that the method will not be purely analytical. That is, since exhaustive experimentation which has been carried out in the field had led to a reasonable agreement as to the true shape of the non-dimensionalized load vs slenderness curve, it was decided that this be used as the boundary curve against which to set up the TDB program. More specifically, the ECCS curves a, and h were used (see Fig. 1). 3.2. FAC-damage
length ,factor
Section 3.1 discussed the plasticity mechanism which occurs at the mid-length of the buckled column, and can be modelled as a reduction in the second moment of area of the section. It is now necessary to look at the length of column over which this reduction is to take place. Figure 3 introduces a damage length, DAM, which is the product of damage length factor, FAC. and the depth of the column, D. It was considered appropriate that the length be non-dimensionalized so as to keep it universal for all values of the variable D. It has been mentioned previously that this method will not be entirely analytical. but partly empirical as well. In the final version of the TDB program, the FAC factor will be calculated directly from the known non-dimensionalized slenderness ratio of the column, the relationship between the two quantities having been predetermined using the procedure described in the next section. Note: non-dimensionalized slenderness ratio, 1. is defined as follows:
The following procedure was followed for several column types (UCs, UBs and equal angles), for buckling failure about several axes. Firstly the program 3DBUCKLE is altered to accept a variable FAC value which is used within the program to calculate the lengths of the sub-sections of the column. Then, for a range of slenderness ratios, the value of FAC is determined, by iteration, as follows: For 1 = 0.5-2.8, in increments of 0.1;
(1) read member size from data file; (2) calculate total area, A,,,,,, second moment of area, I, and the radius of gyration of the gross section, r; (3) read P/P, from ECCS curve (note: P = known buckling load = C2 in Fig. 2); (4) calculate P = P/Py*Py = C2; area; (5) calculate Acomp= P/P,*total (6) calculate x using Acomp and known section geometry (x described in Fig. 2). (7) calculate Ired using x; using Iti for middle (8) set up data file for 3DBUCKLE sub-element of column; (9) Finally, vary FAC until the eigenvalue analysis produces an estimate of critical buckling load, P,,, equal to P calculated above. This value of FAC is then plotted onto the curve showing FAC vs 1. The FAC curves having now been developed for several column types buckling about several axes, it is possible to set up the final version of the 3DBUCKLE program, TDB. 3.4. TDB program Whereas during the development of the FAC curves a manual iteration of FAC was performed, for the TDB program the iteration will be over the variable X, FAC being calculated directly within the program. Thus, the procedure use to calculate an estimate of the critical buckling load of a column will be as follows:
(1) input x (guessed); area of yielded section); (2) calculate A (3) calculate PC”2 ar I; Fomp*oy ; (4) calculate I, ; input Ired into properties file; calculate 1 (using details given); calculate FAC using d; calculate lengths of five sub-sections; perform eigenvalue analysis to give estimate of critical buckling load, P,,; (10) if Pa, = P,, then exit, else, adjust x and go to Step 2. (5) (6) (7) (8) (9)
”
ElastIc
where leR= effective length = L for pin-ended column; r = radius of gyration (gross section). Briefly. for a given column type, the reduced second moment of inertia having been determined, the value of FAC will be varied until such time as the buckling load calculated by eigenvalue analysis agrees with the value given by ECCS curve. This value of FAC will be recorded, for a range of slenderness ratios, and for several possible column types, to form the FAC curves. These curves are then
Section yielded in compression
Portion
\\
C A con0 1 ”
: \ Caiculate mcment port
Fig. 5.
Buckling
mechanism
of
reduced
area
IO”,
second
using
about
v-v
elastic axis
for equal angle about
x-axis.
Technical
748
Note
I
I
Table 2. Comparison of various predictions to actual failure load
PC,
(
Fig. 6.
Arrangment
]
represents
of multi-bay
a
cross-braced
OF
COMPARISON
STRUCTURES OF
frame.
Case no. Al BI
TDB ANGLE ASCE
120 127 194
0.98 1.04 I.59
4.2. K-brace of a lattice tower LeMaster [5] performed an analysis on a K-brace system, using a finite element package. ANGLE. This system had already been tested to failure at the EPRI Transmission Line Mechanical Research Facility (TLMRF). The frame set-up was as follows: The critical member was a 3.5 x 2.5 x 114 angle (in). Since the TDB program requires dimensions in millimetres, this was approximated as a 74 x 74 x 6 angle. A TDB analysis was performed and the result presented below, along with the prediction of the ANGLE program, and an estimate produced in accordance with the American Society of Civil Engineers (ASCE) standard procedure. These are compared to the TLMRF experimental result of 122 kN. Both the TDB and ANGLE programs can be seen to give excellent predictions, far more accurate than the standard
AND
Cr / t I ca I
RESULTS
Fig. 7.
I. Comparison of TDB with Finch’s experiment
K-brace
Member
system.
c
Ii
cross-braced frcmtes
Finch [4], in 1992 performed a series of experiments on some multi-bay cross braced frames, two of which are presented here. A schematic arrangement of the frame is as shown in Fig. 6. The chords were formed by halving a Table
Ratio
200UB30 section to form what were labelled as IOOBTlS T-sections. The verticals were pairs of 50 x 50 x 5 angles, while the diagonals were single 40 x 40 x 5 angles. Each crossed pair of diagonals was connected by a zero torsion “bolt”. as described above, while all other connections were welded. The total load was applied equally at nodes 2 and 4. In each case, the structure failed via the buckling of the diagonal between nodes 2 and I I. The results obtained were as shown in Table I. As can be seen from Table I, the TDB method has predicted the failure load to within 4% accuracy for the first case, and I% for the second.
This section presents a description of various structures for which the buckling loads have been estimated using the TDB program. These results are shown along with the results of actual tests performed in the field by other researchers. 4.1. Multi-hay
(kN)
bolt"
TDB will perform Steps 229 automatically, while the input of .Y (step 1) and the comparison of P_ and P,, (Step 10) are performed by the operator. Thus, a partly empirical, partly analytical method has been developed which is able to produce estimates of the critical buckling load of pin-ended columns which agree with the ECCS experimental curves. This would seem a pointless exercise, in itself, when it is considered that these estimates could simply be read directly off the curves. The strength of this method, however, does not concern pin-ended columns, but rather columns which belong to a sub-assembly, in a situation where neighbouring elements can provide some measure of rotation restraint at the connecting points. which, in turn. can enhance the buckling load of the critical column by some unknown degree. It is usual, in design. to consider that the end of a column is fully hinged, or fully fixed, and there are relatively simple methods to account for these situations by assigning effective length factors. These methods. however, are unable to accurately quantify a situation where a neighbouring member can provide “partial” fixity. That is. even though a critical member is fully fixed to a neighbouring member. if this other member is of limited robustness, the node can still rotate a little, and this will affect the performance of the column in question. The eigenvalue method, on the other hand, can take this effect into account, and it is this fact which makes the TDB program useful. In fact, the eigenvalue method automatically assumes that all connections are rigid. In the situation where a hinged connection is required. this can be modelled by inserting an extra member, say IOmm long. with a zero torsional constant. but second moment of area equal to that of the connecting member. This allows the member end to rotate. while staying in position. These short members are referred to as “bolts”.
4. ANALYSIS
Method
Cr
t
t
ca
,/’
I
lement
,,’
results
TDB (kN)
Finch (kN)
Ratio (&)
I87 187
194 185
0.96 1.01
431
37
Fig. 8.
kN
Inverted
tower
leg.
Technical
Fig. 9.
Note
Full scale transmission
ASCE method. It is acknowledged. however, that these are comparatively simple structures, and that the eigenvalue method has not been applied to a more involved structure, such as transmission tower. The next examples will deal with this. 4.3. Inverted tower leg In 1991. Knight and Santhakumar [6] tested, to failure, the tower leg shown in Fig. 8. The leg was inverted simply to allow easier application of the load. During testing, the lower third of the main angle buckled inwards about the minor principal axis, under an axial load of 431.7 kN (with secondary loads as indicated in Fig. 8). The TDB analysis predicted a buckling load of 492 kN, which represents a 13% over estimate. This is obviously not as good a result as for previous examples, yet is still considered reasonable. It is believed that the accuracy of the plasticity model may have been reduced for a column of such low slenderness. Recall that Section 3.1 discussed the possibility that a different model might be preferable for the extreme slenderness ratios.
749
tower
4.4. Full scale transn~ission tower In 1986, Meuller er al. [7] performed a full scale load test on the transmission tower shown in Fig. 9. The load was applied in the positive u-direction. at nodes 15 and 16, and the member connecting nodes 68 and 92 failed in the second mode at a total applied load of 19.5 kips. This represents a member load of 85 kips, or 380 kN. It was decided for simplicity that, since the effects which neighbouring members have on a critical member decrease with distance from the member, it was not necessary to include all the members of the tower in the input file for TDB. The simplified structure used in the TDB analysis looked as shown in Fig. 10. The TDB analysis gave a member buckling load of 441 kN, which represents an over-estimate of 16%. Notes. (I) For the above examples, the ratio between the total applied load and the critical member load was found using the MicroSTRAN [8] structural analysis package. (2) An uncertainty is introduced into the TDB analysis by the lack of knowledge of the end connection details. The buckling load of a member is highly dependent on the
Technical
750
.,i
Note known to great accuracy, there is a limitation on how accurate a prediction can be. If a series of identical experiments produces a scatter of IO%, a claim to be accurate to 5% would be meaningless. Further refinements to the model, however, are still considered necessary. The program may be adjusted to include the alternative plasticity models for the higher and lower extremes of slenderness. It would also be sensible to look into the possibility of improving the way the end connections are modelled, since this is known to have a significant effect on the performance of a column.
REFERENCES
Fig. 10. Simplified
tower structure
for TDB analysis.
method of connection to neighbouring members, and in these cases some assumptions have been made which may have introduced some error. 5. CONCLUSIONS AND RECOMMENDATIONS
The TDB analyses performed on the structures described in the previous section have shown the potential of the modified eigenvalue method for the prediction of critical buckling loads. Most predictions have been accurate to less that 15%. It is accepted that this method will not be as accurate as the more involved finite element methods, however it does offer reasonable estimation. It is also argued that, since a given member’s material properties and details of imperfections cannot themselves be
1. J. L. Meek, 3DBUCKLE-Eigenvalue analysis program for buckling of steel columns, University of Queensland (1973). 2. L. Euler, Sur la forces des colonnes, Acad. R. Sci. Belles Letr. Berlin, Mem. 13, 252, (1759). English translation by J. A. Van den Broek, Am. J. Phys. 15, 309, (1947). 3. European Convention for Constructional Steelwork (ECCS), In Second International Colloquium on Slabilitl Introductory Report (Stabilit~~ Manual), 2nd Edn, Subchapter 3.1.5, Angles, pp. 98103, and Sub-chapter 9.2. Angles in Lattice Transmission Towers (1976). 4. D. L. Finch, Double diagonal cross bracing, Submission for Ph.D., University of Queensland (1992). 5. R. A. LeMaster, Structural development studies at the EPRI transmission line Mechanical Research Facility, Interim Report 1, EPRI EL-4756, Electric Power Research Institute, prepared by Sverdrup Technology Inc.. Tullahama, TN (1986). 6. G. M. S. Knight and A. R. Santhakumar, Joint effects of behaviour of transmission towers, J. S/rucr. Engng AXE. 119, 6988712 (1993). 7. W. H. Meuller, S. L. Prickett and L. Kempner, Nonlinear and full scale test of a transmission tower. Portland State University (1986). 8. MicroSTRAN Version 5.21. Structural analysis program (1993).