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Lateral-torsional buckling of tapered members with transverse loads

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Title:
Lateral-torsional buckling of tapered members with transverse loads
Creator:
Demos, Giorgios Paul
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
Physical Description:
xii, 96 leaves : ill. ; 29 cm.

Thesis/Dissertation Information

Degree:
Master's ( Master of Science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Civil Engineering, CU Denver
Degree Disciplines:
Civil engineering

Subjects

Subjects / Keywords:
Girders ( lcsh )
Buckling (Mechanics) ( lcsh )
Buckling (Mechanics) ( fast )
Girders ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (M.S.)--University of Colorado at Denver, 1994. Civil engineering
Bibliography:
Includes bibliographical references (leaves [95]-96).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Civil Engineering.
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Giorgios Paul Demos.

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Source Institution:
University of Colorado Denver
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Auraria Library
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
31159781 ( OCLC )
ocm31159781

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Full Text
LATERAL-TORSIONAL BUCKLING OF TAPERED MEMBERS
WITH TRANSVERSE LOADS
by
Giorgios Paul Demos
B.S., University of Colorado at Denver, 1992
A thesis submitted to the Facility of the Graduate School of the University of Colorado at Denver in partial fulfillment of the requirement for the degree of Master of Science Civil Engineering
1994


by Giorgios Paul Demos All rights reserved.


This thesis for the Master of Science
degree by
Giorgios Paul Demos has been approved for the Department of Civil Engineering by
VlX \I -1 a 1a
\ vlqS
Andreas S. Vlahinos
Date


V
used iu present-day practice. The loading consist of uniform bending, the end restraints applied to the bottom or to both flanges.
This abstract accurately represents the content of the candidates thesis. I recommend its publication.
Signed
\l ljJ^\

Andreas S. Vlahinos


Demos, Giorgios Paul Demos (M.S., Civil Engineering)
Lateral Torsional Buckling of Tapered Members With Transverse Loads Thesis directed by Assistant Professor Andreas S. Vlahinos
ABSTRACT
In recent years tapered beams have become increasingly popular in continuous frame construction due to their efficient utilization of structural material. Although analysis and design methods have been presented by many authors and three experimental programs have been carried out in recent years, further studies and design recommendations are needed.
In this thesis the mathematical formulation and the computer implementation of the elastic stability analysis of doubly symmetric tapered beams are presented. Solutions in the form of tables, charts and approximate formulas for the lateral buckling load are presented for a variety of beam geometry, end restraints and loading conditions. The critical loads corresponding to various buckling modes are plotted versus non-dimensionalized parameters in an effort to establish the parameter ranges at which local buckling or lateral-torsional buckling controls.
Since it is practically impossible to non-dimensionalize the many parameters in tapered sections, solutions were obtained for a range of tapered members


ACKNOWLEDGMENTS
I wish to express my gratitude and appreciation to my advisor, Professor Andreas
S. Vlahinos, for his guidance, direction, and assistance throughout this course of the work. I also would like to thank Professor James C-Y Gou, and JonathanT-H Wu for their willingness to serve on my thesis committee and in particular, their constructive criticisms.
This thesis is dedicated to my wonderful parents, who suffered a great deal after their immigration to the United States, in order to provide me with the best emotional, and financial support. Their love and patience has been exceptional, and their support has been immeasurable. This dedication is small token of my great appreciation for them. Thank you very much from the bottom of my heart. And I also would like to thank Dr. Andreas S. Vlahinos for his guidance, direction and his patience with me and good luck to his family and his future. And last but not least I want to thank Dr. Yang-Cheng Wang for his immeasurable help through-out this experience and best wishes.


CONTENTS
CHAPTER
I. INTRODUCTION ............................................ 1
II. MATHEMATICAL FORMULATION...................................3
11.1 Introduction..........................................3
11.11 Historical Review....................................3
11.111 Solution Procedure .............................. 5
III. COMPUTER IMPLEMENTATION ................................ 11
111.1 Problem Description.................................11
111.11 Problem Idealization............................. 12
111.111 Discretization....................................14
IV. NUMERICAL RESULTS........................................23
IV.I Section I .......................................... 23
IV.II Section II ........................................ 25
IV.III Section III........................................27


Vlll
IV.IV Section IV................................... 29
IV.V Section V ...................................... 31
IV.VI Section VI................................... 33
IV. VII Conclusion....................................35
V. CONCLUSIONS ......................................... 71
V. I Tables ...................................... 71
APPENDIX
A. INPUT FILE .......................................... 80
B. NOTATION ............................................ 85
C. MATLAB INPUT..........................................87
D. VERIFICATION PROBLEM..................................93
BIBLIOGRAPHY
95


FIGURES
Figure:
1. Geometry of linearly tapered I-beams ..............................15
2. Geometry of variable sections......................................16
3. Geometry of a tapered beam length (120 in) ...................... 17
4. Geometry of a tapered beam length (480 in) ........................ 18
5. End restraints without end plates...................................19
6. Plot of all finite elements for a typical case, without end plates .20
7. Plot of all the nodes and boundary conditions without end-plates.21
8. The buckling mode shape and the critical load of the verification
problem........................................................... 36
9. Plot of all the elements of the Finite Element model of
(section I)
37


X
10. Nodes and boundary conditions of the finite element model of
(section I) ............................................................ 38
11. Web buckling mode of (section I) ....................................... 39
12. lst lateral torsional buckling mode of (section I) ..................... 40
13. 2nd lateral torsional buckling mode of (section I).......................41
14. Plot of the critical load versus 7 of (section I)........................42
15. Plot elements of the Finite Element model in of (section II) ........... 43
16. Plot of nodes and boundary conditions of the finite element model of
(section II) ............................................................44
17. Web buckling mode of (section II) ...................................... 45
18. Critical load versus 7 of (section II) ................................. 46
19. Plot of all elements of the finite element model of (section III)........47
20. Nodes and boundary conditions of the finite element model of
(section III).......................................................... 48
21. Web buckling mode of (section III) ..................................... 49


XI
22. 1 et lateral torsional buckling mode of (section III)...................50
23. 2nd lateral torsional buckling mode of (section III) ................... 51
24. Plot of the critical load versus 7 of (section III) ..................... 52
25. Plot of all elements of the finite element model of (section IV)..........53 .
26. Nodes and boundary conditions of the finite element model of
(section IV) ............................................................ 54
27. Web buckling mode of (section IV) ....................................... 55
28. 1 at lateral torsional buckling mode (section IV) ...................... 56
29. 2nd web buckling mode of (section IV) ....................................57
30. Plot of the critical load versus 7 of (section IV) .......................58
31. Plot of all elements of the finite element model in of (section V)......59
32. Nodes and boundary conditions of the finite element model of
(section V) ..............................................................60
33. 1 at web buckling mode (section V)........................................61
34. 1 at lateral torsional buckling mode of (section V) ..................... 62


xn
35. 2nd web buckling mode of (section V) ................................... 63
36. Plot of the critical load versus 7 (section V).......................... 64
37. Plot of all elements of the finite element model in of (section VI) .... 65
38. Nodes and boundary conditions of the finite element model of
(section VI) ........................................................... 66
39. 13t local web buckling mode of (section VI) ............................ 67
40. 2nd local web buckling mode of (section VI) ............................ 68
41. Plot of the critical load versus 7 of (section VI) ..................... 69


TABLES
Table
1. Critical Loads and Mode Shapes for Various Sections and 7 .................... 22
2. Uniform Critical Loads for (L=120) ....................................... 73
3. Uniform Critical Loads for (L=180) ....................................... 74
4. Uniform Critical Loads for (L=240) ..................................... 75
5. Uniform Critical Loads for (L=300) ....................................... 76
6. Uniform Critical Loads for (L=360) ....................................... 77
7. Uniform Critical Loads for (L=420) ....................................... 78
8. Uniform Critical Loads for (L=480) ....................................... 79


CHAPTER I
INTRODUCTION
I.I Introduction
During the past several years there has been a sincere effort to reduce costs and therefore, cause the weight of immense structures to also reduce. Improved computational tools and high strength materials are the two primary reasons for efficient, lighter and stronger structures. The trend for producing lighter structures sometimes leads to slender elastic structures which can accept design loads but in which the stability safety factors are reduced due the fact that slender structures are more inclined to become unstable.
One of the ways to optimize structural components subjected to lateral loads is to generate variable cross sections. Ideally, if all sections must possess the minimum amount of required properties (i.e. moment of inertia) then individual sections will be unique for a particular loading conditions thus, consequently extremely expensive to construct. This shape optimization can be utilized effectively by the automotive and aerospace industry due to the fact that they have to manufacture the same components subjected to the same loading several thousand times. Furthermore, in civil engineering structures like high rise buildings, long span bridges, sports complexes, and industrial buildings, the structural members shape optimization is circumscribed to linearly varying cross


2
section like tapered beams. The constant cross section constructional components are the easiest to produce and are widely utilized. Tapered members are not as inexpensive to manufacture, but nevertheless, are primary to significant reserve in material expenditure. The bending potency requirements and design guidelines of tapered members are not significantly different than the ones of constants section members.
The immense majority of steel beams are incorporated in such a manner that their compression flanges are constrained opposing lateral buckling, since the upper flange of the beams utilized to support concrete floors are often incorporated in these floors; in those cases the lateral buckling of beams are not a major concern. In the event that the compression flange is without lateral support, it behaves like a compression member subjected to buckling, thus causing lateral buckling of the beam. The occurrence of lateral buckling of beams with constant cross sections subjected to lateral loads ha6 been investigated by several researchers for the past fifty years. An immense amount of analytical and experimental drudgery has been incorporated to design codes. No such design guidelines exist for the lateral buckling of tapered beams subjected to lateral loads. In this research an exertion was attempted to produce formulas two predict the critical uniform load which yields lateral buckling of tapered beams. The results were produced to incorporate the effects of several parameters such as section properties, beam lengths, and the slope of the flanges.


CHAPTER II
MATHEMATICAL FORMULATION
ILI Introduction
A structural member laterally loaded, bent about its major axis, may buckle laterally if the compression flange is unbraced. At the critical load, the compression flange tends to bend laterally and thus causing the remaining of the section to become restrain; thus resulting in lateral and torsional motion of the section. In other words, lateral buckling is a combination of twisting and lateral bending of the compression flange.
II.II Historical Review
The first theoretical work on the subject of lateral buckling beams with rectangular cross sections is by L. Prandt1 (1899) in his dissertation and A.G.M. Michelle2 in his paper Elastic Stability of Long Beams under Transverse Forces. In 1910 S. Timoshenko3 presented the equations for lateral buckling of beams with symmetric I sections. G. Winter4 in 1943 presented the equations for lateral stability of uniform section of unsymmetrical I-beams and trusses in bending. Lee5 first addressed the issue of lateral buckling of a tapered narrow rectangular
beam.


4
The analysis of lateral buckling of beams with constant cross sections was also presented by J.W. Clark and H.N. Hill6,7 in 1962. These authors presented the results of a series of investigations into the lateral buckling of channel sections and Z-sections that were loaded in the direction of the strong axis. They also introduced the torsion-bending factor which is defined as a property of the cross section. Torsion bending is the resistance arising from the restraint to warping of the cross section of the beams. The authors also developed a general formula for the elastic buckling strength of beams with coefficients depending on the loading and support conditions. Significant investigations of the elastic and inelastic buckling of various shapes have been reported by T.V. Galambos6 in 1963.
A method was presented for determining the inelastic lateral buckling problem for simply supported wide-flange beams subjected to equal end moments. His methods are based on the determination of the diminishing in the lateral and torsional stiffness caused by yielding. B.A. Boley9 established that for small tapering angles, less than 15 degrees, that the Bernoulli-Euler theory of various cross section yields adequate results.
In addition to the above-mentioned analytical studies, two test programs were started. The first experimental study were conducted by D.J. Butler and G.C. Anderson10,11 at Columbia University in 1966. In this program tapered I section beams and channels sections tapered in both the web and flanges were examined as cantilever beam-columns and the elastic stability loads and the brae-


5
ing requirements were discovered. The subsequent experimental study ended in 1974 and was conducted by Lee, Ketter, and Prawel1213 at the State University of New York at buffalo. In this program the inelastic stability of tapered I-shape beam columns were also investigated.
S. Kitipornchai and N. Trahiar14 in 1972 studied the elastic flexural-torsional buckling of I-beams. They concluded there elastic critical loads of Flange-Tapered beams decreased significantly as degree of taper increased; on the other hard, elastic critical loads of depth-tapered beams do not vary greatly as the degree of taper increases. This thesis is related to the insensitivity of the torsional stiffness to the degree of taper.
II.III Solution Procedure
Following is a brief derivation of the relevant equation using the potential energy formulation12,13,14. The expression of the elastic strain energy U is:
U = f Jvol
where ey is the Lagrangian strain tensor. Using Hookes law cr,j = C^e*/, where Cijki is the material property tensor, the above equation can be written as:
/ Cijkiekieijd(vol) Jvol
(*. J. 1 1)2,3;
(2-2)


6
The Lagrangian strain tensor can be expressed in Cartesian coordinates in terms of the displacement derivatives Ujj, as
e (2-3)
It is convenient to group the linear strain terms e,-^ = |(uij+Ujj) and the
nonlinear-strain terms Sij = |(ufc(i,iijt(j).Thus
ei
£,j + £
j
(2-4)
Substitution of the above equation into equation (2-2) results in
I (@ijkl£kl£ij "b 2C|jfc/£fc/£ij -(- Cijkl£kl£ij )d( Vol'j Jvol
(2-5)


7
This is the basic form of strain energy for nonlinear analysis. The total potential energy, denoted by Ut, is given by
Ut = U-V
(2-6)
where V is the potential energy of the external forces. Through the principal of stationary value of the total potential, one may derive the equilibrium equations and the proper end conditions. In the finite element formulation the displacements are approximated by the shape function matrix [N] times the nodal degrees of freedom vector {d}.
M = [N]{d}
For elastic stability analysis some simplifications can be made7. First
equation (5) can be rewritten as


8
I {@ijkl£kl£ij
Jvol
+ 2[Cijki(eki + £w)]ew Cijkieijiki}d(vol)
(2-7)
Discarding the higher order term and using Hookes law
U f (CijkiEki"I- 2 Z Jvol
(2-8)
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} = \{p)
(2-9)
The objective is to find the critical intensity Ac- at which buckling occurs.
If the load vector {P0} is applied, a state of stresses of,- can be found. Therefore


9
for any load vector P the state of stresses can be written as U = ^ [ (CijkiekiEij + 2\Gijeij)d{vol)
Z Jvol
(2 10)
Differentiation of the displacements and integration of the strain energy
results in
u =
if}
2
in+
if**}
(2-11)
where [iif]o is the linear elastic stiffness matrix and
n=/
/uoi
(2 12)
[JST]i is the initial-stress or geometric stiffness matrix. If the total potential 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


10
[m + TO* = 0 (2-13)
where $ is the buckling mode shape.Furthermore, the determinant of the above matrix must be zero
l[[jf]0 + A[jsrj1]| = o
(2 14)
The critical load is Per = AcrP- At this load level an adjacent equilibrium position exists, which means that a bifurcation point appears on the equilibrium
path. Equation 2-13 converts the buckling problem to the classical eigenvalue form which can be solved efficiently with the available eigensolving routines.


CHAPTER III
COMPUTER IMPLEMENTATION
III.I Problem Description
The elastic stability analysis of a simply supported steel tapered beam was considered. The objective of this study was to determine formulas in order to predict the uniformly distributed critical load of the beams in terms of sectional properties, beam length and degree of depth-tapering. The top flange was considered un-braced for the entire length. Figure 1 adumbrates the geometry of the linearly tapered beam. (L) represents the length, (do) represents the smallest depth of the beam and (di) the largest depth of the beam. The depth of the beam at a distance (x) from the lesser end is expressed as:
d = d, (l +^7) (3-1)
The tapering ratio 7 is a measure of the degree of depth-tapered and for a prismatic member thus yielding zero. Six various sections were considered. The flange thickness (tf), the web thickness (tw), the flange width (b), and the small depth (dQ), considered are adumbrates in table 1. In this study it was considered as a variable varying from (0 to 8) with increments of 1. The lengths considered were L= 120,180, 240, 300, 360, 420 and 480 inches, Figure 2. adumbrates all the


12
sections. Figure 3 adumbrates the top, side, front and isometric view of the case L=120 and 7 = 8. Figure 4 adumbrates the top, side, front and isometric view of the case L = 480, 7 = 8. The load considered was a uniformly distributed load at the top flange. The lower flange was restrained for the entire flange width in both (x) and (y) directions at the largest depth section, the lower flange was also restrained in the (y) direction at the smallest depth section as it is adumbrated in figure 5.
III.II Problem Idealization
The following assumptions were made:
The flange and web panels originally were entirely flat; in other words, all geometric imperfections are disregarded.
The material is isotropic, linearly elastic with modules of elasticity E=29 x 106 psi and Poissons ratio i/=0.3. No temperature effects were examined on the above parameters.
The loading uniform pressure was applied on the top flange and the effects were disregarded.
Bifurcation buckling is adequate enough to determine the critical load; in other words, the materials non-linear, the snap through buckling and im-


13
perfection sensitivity were not examined in the determination of the critical load.


14
IILIII Discretization
The tapered beam is descritized with 355 nodes and 328 four node shell elements, 12 geometry points, 20 lines, 9 surfaces, 21 loads (nodal), 25 constraints, 2 sets of thicknesses. This element contains both bending and membrane capabilities. Each node has six degrees of freedom: Translation in the nodal (X), (Y), and (Z) directions and rotations about the nodal (X), (Y), and (Z) axis. The largest element size is approximately a quarter of the flange width. The thickness in web and flange is constant (tw) and (tf) respectively. The aggregate numbers of the degrees of freedom is (500). Figure 6 adumbrates a plot of all elements for a typical case without end plates. For clarity all the elements have been decreased by (20degree of freedom of the bottom flange width of the section with the smallest depth are restrained. Figure 7 adumbrates a plot of all the nodes contained in the above described boundary conditions without end plates.


15
Figure 1. Geometry of linearly tapered I-beams


16
0.25
r .00 0
L
L-4.0D-J Section I
0.75
0.25
0.75
7f
- 0.25 f

12.00
Section IV
Section V
Section VI
Figure 2. Geometry of various sections


17
12.00
Figure 3. Geometry of a tapered beam length (120 in)


18
6.00
lr
!
54.00
~ I
J i
1 L- 12.00
Figure 4. Geometry of a tapered beam length (480 in)


19


Figure 6. Plot of all finite elements for a typical case, without end plates


21

Figure 7. Plot of all the nodes with the above described boundary conditions
without endplates


22
FACT=2
UZ
TOP
RSYS=0 DMX =1 SMX =1 PRES
NODAL
STEP=1
SOB =1
FACT=2
UZ
TOP
RSYS=0 DMX =1
8.
The buckling mode shape and the critical load of the verification
problem
.linn .222222 .333333 .444444 .555556 .666667 .777778 .888889
SOLUTION
.14


CHAPTER IV
NUMERICAL RESULTS
IV .I Section L
In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with the above dimension of section I (do)=6.00, b=4.00, (£/)=0.25, (£,)=().10. The length of section I was L=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various value of (7 = 0, 1, 2, 3, 4, 5, 6, 7,and 8). The largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates, they were connected to the top and bottom flange; the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; in order to illiminate the effect of the length on the value of the critical loads. That load was also nondimensionalized with respect to (^j) of the section. In section I the moment of inertia about the horizontal axis (strong axis) (Ix) is (17.9281 inches4) and the moment of inertia about the vertical axis (Iy) is (2.6671 inches4). The non reported critical loads are defined as:
(4-1)


24
(equation (4-1) where (pa) is the actual critical loads in pounds, L= length of the beam, E=the modules of elasticity and (Ix) the strong axis moment inertia at the smallest end. Figure (9) represents a plot of all the elements of the finite element model in (section I), figure (10) represents all the nodes and boundary conditions of the finite element model in (section I). In section I three buckling modes seemed to dominate the behavior of this beam, local web buckling, 1st lateral torsional buckling (single wave of the top flange and 2nd lateral torsional double wave of the top flange occurred.) Figure (11) represents web buckling mode in (section I), figure (12) represents the 1st lateral torsional buckling mode (section I), figure (13) represents 2nd lateral torsional buckling mode (section I), figure (14) represents a plot of the critical load versus7 in (section I). In all cases the local web buckling appears to be the smallest critical loads. To prevent local budding stiff plates were used, lateral torsional budding will be dominate. As 7 increases local web buckling and 1st lateral torsional buckling modes increase monotonecly, the 2nd lateral torsional buckling mode decreases monotonecly as 7 increases. A cubic polynomial coefficients for all three critical loads were generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use this polynomials to predict any of the critical loads for this section and any value of 7.


25
IV.II Section II
In this put of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with dimensions of section I (do)=6.00, b=12.00, (tf)=0.25, (tw)=0.10. The length of section II was L=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (7 = 0,1, 2, 3, 4, 5, 6, 7,and 8) the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates, they were connected to the top and bottom flange, the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reprted in pounds; in order to elliminate the effect of the length on the value of teh critical load. This load was also nondimensionalized with respect to (|^) of the section. In section II the moment of inertia about the horizontal axis (strong axis) (Ix) is (51.0155 inches4) and the moment of inertia about the vertical axis (Iy) is (72.005 inches4.) figure (15) represents a plot of all the elements of the finite element model in (section II), figure (16) represents all the nodes and boundary conditions of the finite element model in (section II). In this case all buckling modes were local. The lateral torsional does not appear in the 1st ten modes; figure (17) represented the web buckling modes in section (II). Figure (18) represents the plot of critical load


26
versus 7 in section (II). A cubic polynomial coefficients for all three critical loads was generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use these polynomials to predict any of the critical for this section and any value of 7.


27
IY.III Section III
In this part of the analysis a finite element model was constructed to c calculate the buckling loads of a linear tapered beam with dimensions of section 1= (do)=6.00, b=12.00, (tf)=0.25, (tw)=0.10. The length of section III was L=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (7 = 0,1, 2, 3, 4, 5, 6, 7,and 8); the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates; they were connected to the top and bottom flange. The reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; in order to ilBminate the effect of the length on the value of the critical loads. This load was also nondimensionalized with respect to (j^j) of the section. In section III the moment of inertia about the horizontal axis (strong axis) (Ix) is (43.52345 inches4) and the moment of inertia about the vertical axis (Iy) is (8.0006 inches4.) Figure (19) represents a plot of all the elements of the finite element model in (section III), figure (20) represents all the nodes and boundary conditions of the finite element model in (section III).
Once again three buckling modes seemed to dominate the behavior of
this beam, local web buckling, Ist lateral torsional buckling (single wave of the


28
top flange and 2nd lateral torsional double wave to the top flange occurred.) Figure (21) represents web buckling mode in (section III), figure (22) represents the l*t lateral torsional buckling mode in (section III), figure (23) represents the 2nd lateral torsional buckling mode in (sectionlll); figure (24) represents a lot of the critical load versus 7 in (section III.) for 7 less than (1), local web buckling appears to be the smallest critical load. For 7 greater than (1), the lateral torsional buckling controls. As 7 increases local web buckling and 2nd lateral torsional mode also increases monotonecly, the 1st lateral torsional mode seems to remain constant for all 7 values. A cubic polynomial coefficients for all there critical loads were generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use these polynomials to predict any of the critical loads for this section and any value of 7.


29
rV.IV Section IV
In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with dimensions of section I (do)=12.00, b=6.00, (tf)=0.75, (tw)=0.25. The length of section IV was L=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (7 = 0,1,2,3,4,5,6,7,8); the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates; they were connected to top and bottom flange; the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; one observation that was noticed was that length did not play a role in the critical loads. This load was also nondimensionalized with respect to (|rf) of the section. In section IV the moment of inertia about the horizontal axis (strong axis) (Ix) is (126.773 inches4) and the moment of inertia about the vertical axis (Iy) is (216.006 inches4.) Figure (25) represents a plot of all the elements of the finite element model in (section IV); figure (26) represents all the nodes and boundary conditions of the finite element model in (section IV).
Once again two buckling modes seemed to dominate the behavior of this
beam: local web buckling, 1 t lateral torsional buckling (single wave of the top


30
flange and 2nd lateral torsional double wave of the top flange occurred.) Figure (27) represents web buckling mode in (section IV); figure (28) represents the Yt lateral torsional buckling mode in (section Iv), figure (29) represents the 2nd lateral torsional buckling mode in (section IV), figure (30) represents a plot of the critical load versus 7 in (section VI). For7 less than (3.25), local web buckling appears to be the smallest critical load. For 7 greater than (3.25), the lateral torsional buckling controls. As 7 increases local web buckling and 2nd lateral torsional mode also increases monotonecly; the Yt lateral torsional mode seems to remain constant for all 7 'values. This type of behavior is similar to the case in section (III) and thus, leads to the following very important conclusion: The eifect of the degree of taper 7 is negligibly small on the lateral torsional buckling load of tapered beams; thus one may use the formulas of constant section beam to predict the approximate value of the lateral torsional buckling load. A cubic polynomial coefficients for all three critical loads were generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use these polynomials to predict any of the critical loads for this section and any value of 7.


31
rV.V Section V
In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with dimensions of section I (do)=12.00, b=6.00, (tf)=0.25, (tw)=0.10. The length of section IV was L=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (7 = 0,1,2,3,4,5,6,7,8); the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates; they were connected to the top and bottom flange; the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; in order to illiminate the effect of the length on the value of the critical loads. This load was also
m O
non dimensions,, iaed with respect to (j^j) of the section. In section V the moment of inertia about the horizontal axis (strong axis) (Ix) is (116.236 inches4) and the moment of inertia about the vertical axis (Iy) is (9.00096 inches4.) Figure (31) represents a plot of all the elements of the finite element model in (section V); figure (32) represents all the nodes and boundary conditions of the finite element model in (section V).
Once again two buckling modes seemed to dominate the behavior of this
beam; local web buckling, 1st lateral torsional buckling (single wave of the top


32
flange and 2nd lateral torsional double wave of the top flange occurred.) Figure (33) represents web buckling mode in (section V); figure (34) represents the 1 at lateral torsional buckling mode in (section V), figure (35) represents the 2nd lateral torsional buckling mode in (section V), figure (36) represents a plot of the critical load versus 7 in (section V). For 7 less than (6.75), local web buckling appears to be the smallest critical load. For 7 greater than (6.75), the 2nd local web buckling mode controls. The 1 t local web buckling and l*t lateral torsional mode increases monotonecly as 7 increase. The 2nd local web critical load decreases as 7 increases. A cubic polynomial coefficients for all three critical loads were generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use these polynomials to predict any of the critical for this section and any value of 7.


33
rV.VI Section VI
In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with dimensions of section 1= (do)=12.00, b=6.00, (tf)=0.75, (tw)=0.25. The length of section IV was L=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (7 =0,1,2,3,4,5,6,7,8); the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates; they were connected to the top and bottom flange; the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; in order to illiminate the effect of the length on the value of the critical load. This load was also nondimensionalized with respect to (|^) of the section. In section IV the moment of inertia about the horizontal axis (strong axis) (Ix) is (288.841 inches4) and the moment of inertia about the vertical axis (Iy) is (170.668 inches4.) Figure (37) represents a plot of all the elements of the finite element model in (section VI); figure (38) represents all the nodes and boundary conditions of the finite element model in (section VI).
Once again two buckling modes seemed to dominate the behavior of this
beam: local web buckling, 1 *t lateral torsional buckling (single wave of the top


34
flange and 2nd lateral torsional double wave of the top flange occurred.) Figure (39) represents 1 at local web buckling, figure (40) represents 2nd local web buckling in (section VI). Figure (41) represents a plot of the critical load versus 7 in
ii
b
(section VI). For 7 less than (5), the the 1£ local web buckling load appears to
I
be the smallest critical load. For 7 greater than (5), the 2nd local web buckling dominates. The 18t local web buckling increase monotonecly as 7 increase. The 2nd local web buckling load decreases as 7 increases as in all previous cases. The code that generated the coefficients in the table was matlab, and is represented in appendix a. One any of the critical for this section and any value of 7.


35
IV.VII Conclusion
The value of the critical loads and corresponding mode shape for all sections and 7 are summarized in table (1). The (*) represents that the corresponding mode did not appear in the first ten modes. The most important conclusion can be summarized below:
The effect of the degree of taper (7) is negligibly small on the lateral torsion buckling load of tapered beams; thus, one may use the formulas of constant section beam to predict the approximate value of the lateral torsion buckling load.
The first local buckling mode seems to controls in most cases and increase as 7 increases.
The second local buckling mode seems to controls only in cases in high 7 and small thickness. The corresponding critical load decrease as 7 increases.


Figure 9. Plot of all the elements of the Finite Element model of (section I)


38
r
V-.
v
Figure 10. Nodes and boundary conditions of the finite element model of
(section I)


39
cr-1563.44299. L-360, g-8, do-6, b-4, tf-0.25, tw-0.1
Figure 11. Web buckling mode of (section I)


40
2 Pcr-1907.087S4, L-360, g-8, do-6, b-4, t£-0.2S, tw-O.l
Figure 12. 1 "t lateral torsional buckling mode of (section I)


41
3 Pcr-1933.04399, L-360, g-B, do-6, b-4, tf-0.25, tw-0.1
Figure 13. 2nd lateral torsional buckling mode of (section I)


42
Critical Loads Versus Gamma for Local and Lateral Torsional Buckling
Figure 14. Plot of the critical load versus 7 of (section I)


43
Figure 15. Plot elements of the Finite Element model in of (section II)


44
V
<*


Figure 16. Plot of nodes and boundary conditions of the finite element model of
(section II)


45
:r-3055.97155, L-360, g-B, do-6, b-12, tf-0.25, tw-0.1
Figure 17. Web buckling mode of (section II)


Critical Loads Versus Gamma for Local and Lateral Torsional Buckling
Figure 18. Critical load versus 7 of (section II)


47
Figure 19. Plot of all elements of the finite element model of (section III)


48
Figure 20. Nodes and boundary conditions of the finite element model of
(section III)


2 Pcr-27980.3082, L-360, g-B, do-6, b-4, tf-0.75, tw-0.25
Figure 21. Web buckling mode of (section III)


50
Per-12497. 7696, L-360, g-B, do-6, b-4, tf-0.75, tw-0.25
Figure 22. 18t lateral torsional buckling mode of (section III)


51
4 Per-36924.6197, L-360, g-8, do-6, b-4, tf-0.75, tw-0.25
Figure 23. 2nd lateral torsional buckling mode of (section III)


(t rr\ / 7
52
Critical Loads Versus Gamma for Local and Lateral Torsional Buckling
Figure 24. Plot of the critical load versus 7 of (section III)


53
Figure 25. Plot of all elements of the finite element model of (section IV)


54
Figure 26. Nodes and boundary conditions of the finite element model of
(section IV)


55
2 Pcr-68300.8266, L-360, g-8, do-6, b-12, tf-0.75, tw-0.25
Figure 27. Web buckling mode of (section IV)


56
cr-46576.9522, L-360, g-8, do-6, b-12, t£-0.75, tw-0.25
Figure 28. 1 "t lateral torsional buckling mode (section IV)


57
3 Pcr-69364.0824, L-360, g-8, do-6, b-12, tf-0.75, tw-0.25
Figure 29. 2nd web buckling mode of (section IV)


Critical Loads Versus Gamma for Local and Lateral Torsional Buckling
Figure 30. Plot of the critical load versus 7 of (section IV)



2 Pcr-3629.82207, L-360, g-8, do-12, b-6, tf-0.25.
tv-0.X
Figure 31. Plot of all elements of the finite element model in of (section V)


60
Figure 32. Nodes and boundary conditions of the finite element model of
(section V)


61

3 Pcr-4106.01647, L-360, g-0, do-12, b-6, tf-0.25, tw-0.1
Figure 33. 1 at web buckling mode (section V)


3 Pcr-4106.01647, L-360, g-8, do-12, b-6, tf-0.25, tW-0.1
Figure 34. l*t lateral torsional buckling mode of (section V)


63
cr-3060.61157, L-360, g-8, do-12, b-6, tf-0.25, tw-0.1
Figure 35. 2nd web buckling mode of (section V)


(ua) /
64
Critical Loads Versus Gamma for Local and Lateral Torsional Budding
Figure 36. Plot of the critical load versus 7 (section V)


65

Figure 37. Plot of all elements of the finite element model in of (section VI)


66
I
Figure 38. Nodes and boundary conditions of the finite element model of
(section VI)


67
2 Pcr-4444.17938, L-360, g-B, do-12, b-16, tf-0.25, tw-0.1
Figure 39. lt local web buckling mode of (section VI)


68
cr-4014.74397,
L-360, g-8, do-12, b-16, tf-0.25.
tw-0.1
Figure 40. 2nd local web buckling mode of (section VI)


69
Critical Loads Versus Gamma for Local and Lateral Torsional Buckling
Figure 41. Plot of the critical load versus 7 of (section VI)


CHAPTER V
CONCLUSION
V.I The effect of length and gamma on the critical loads
In order to investigate the effect of beam length on the critical loads for various values of gamma consider a beam with section VI which had the smallest end. A Unite element model was generated having the length L as element model was generated having the length L as a parameter, the length L as parameter, had values of Length = 120,180,240,300,360,420 and 480 inches and (7= 0,1,2,3,4,5,6,7 and 8). The load was uniformed pressure on the top flange. The boundary conditions were the same as the ones in the previous chapter. In this case no end plates were considered at the end supports; the first ten mode shapes were also computed. Due to the lack of end plates and the thickness of the web and flange, small local buckling caused the critical mode to always appear.


CURVE PITTING COEFICIENTS FOR CRITICAL LOADS
Pcr(Y)=dy3+CY2+by+a
d C b a
Sec I -0.00006167508418 0.00106996248196 0.02328497017797 0.15322762626263
Sec I -0.00041843350168 0.00622837193362 -0.00570030206830 0.31930697979798
Sec I -0.00000220370370 0.00025586111111 -0.00515758597884 0.74842169047619
Sec IX -0.00005973905724 0.00092830375180 0.01774929894180 0.15152301010101
Sec II -0.00045675084175 0.00793322330447 0.16677549723425 1.16750695959596
Sec III -0.00104570117845 0.02013005375180 -0.12214888287638 1.46282461616162
Sec III -0.00275279629630 0.04046075360750 -0.05811292700818 2.94536995959596
Sec III -0.00090196885522 0.01133516847042 0.11669223977874 1.13368107070707
Sec IV 0.00073663973064 -0.01036882611833 0.03851385305435 1.56986452525253
Sec IV -0.00000771548822 0.00007210642136 0.01332185149110 0.04533932323232
Sec IV 0 0 0.01888750000000 0.12320480000000
Sec V 0.00010566666667 -0.00017450000000 -0.03159316666667 0.32357600000000
Sec V 0 -0.00049512500000 0.01268691071429 0.04396853571429
Sec V -0.00017933333333 0.00491050000000 -0.05208416666667 0.25425500000000


Table 1. Critical Loads and Mode Shapes for Various Sections and l
0.75 L 0.75 JL 17 I i
0.75 J. 0.75 0 0.10 1 .5 0.10 1 2
i 6.DO L. .10 0.10 -J-0 25 II r 0.25
n 1 2 t n .1 L 4.0D-'-action I. 1) t . i ll 0J 1- a.o J Sick Ion VI
action Sac Ion II Sac Ion IV Sactlon V
' fiiekUig Local Vab III | 2al la( local let 2i4 Local lab Local fab lat | 2aA lal Lacal fab Ultral-Teraiasal lat Ucal Vab Ultra! Twaiml 2*4 Lacal fab III Ucal Vab 2al Ucal Vab
UUral-faralaaal Lalaral- aralaaal
0 0.153 0.321 * 0.132 1.770 1.403 2.944 1.134 1.301 0.040 0.116 0.043
1 0.177 0.318 0.170 0.259 1.338 1.307 2.928 1.618 1.280 0.009 0.143 0.000
OJ 0.203 0.329 0.190 0.239 1.028 1.289 1.083 1.407 1.610 0.073 0.188 0.087
3 0.231 0.349 0.733 0.212 0.239 1.729 1.231 3.077 1.004 1.004 0.080 0.108 a 0.070 a
4 0.200 0.371 0.732 0.234 0.200 1.930 1.232 3.204 1.723 1.800 0.090 0.191 * 0.086
5 0.209 0.395 0.729 0.2M 0.209 2.149 1.220 3.331 1.804 1.098 0.113 * o.m 0.094
CO 0.318 0.417 0.720 0.278 0.239 3.333 1.227 3.495 2.040 1.002 0.128 0.101 0.080
7 0.347 0.439 0.724 0.301 0.209 2.003 1.233 3.008 2.203 1.080 0.139 a 0.130 0.006
CO 0.377 0.480 0.722 0.323 0.280 2.777 1.241 3.888 1.327 1.087 0.103 0.114 0.000


Table 2. Uniform Critical Loads for L=120
X L = 120 in ^cr
0 1.770
1 0.846
2 0.515
3 0.362
4 0.277
5 0.223
6 0.188
7 IN 0.162
8 N 0.143


Table 3. Uniform Critical Loads for L=180
L = : 180 in 9cr
0 0.899
1 = 0.169
2 0.379
3 0.268
4 - % 0.206
5 0.166
6 0.139
7 [\ 0.120
8 0.105


76
Table 4. Uniform Critical Loads for L=240
X L = 240 in qcr
0 0.427
1 i 0.432
2 0.287
3 0.204
4 0.157
5 0.127
6 0.107
7 0.092
8 r\. 0.081


77
Table 5. Uniform Critical Loads for L=300
L = 300 in qCr
0 0.249
1 ir j 0.251
2 i= 0.226
3 0.161
4 0.124
5 0.101
6 0.270
7 0.247
8 1.770


Table 6. Uniform Critical Loads for L=360


79
I
Table 7. Uniform Critical Loads for L=420
L = 420 in 9cr
0 0.426
1 . 0.328
2 0.280
3 0.109
4 0.118
5 0.068
6 0.120
7 0.217
8 0.300


Table 8. Uniform Critical Loads for L=480


APPENDIX A
INPUT DATA FILE


81
INPUT DATA FUE
Tapered Beam analysis.(Done ansys 5.0)
Location:
CCUDNVR:disk$user5:[thesis.gdemos.thesis]gthesis.dat Date created: Oct., 16, 1993
Preprocessor
/prep7
! PARAMETERS: ( For the beam properties)
! 1 = length of the tapered beam
! b = the width of the flange (top and bottom flanges
! have the same width)
! tf = the thickness of the flange
! tw = the thickness of the web
! gamma = the ratio of the height between the both end
! do = the height of the small end of the beam
! dl = the height of the top end
! dl = (l+gamma)*do
1=30*12 do=12.0 b=16.0 tf=.25 tw=.10 gamma=8
dl=(1+gamma)*do
! PARAMETERS: (For ANSYS)
! r = real (cross-section dimension)
! et = element type
! k = keypoint (create keypoints)
! a = area (define area)
! lesize = line size


82
! asesh area mesh (automatically mesh area) mp = material properties
et,1,63
mp, ex, 1,30e6
mp,nuxy,l,0.3
mp,dens,1,0.0007346
r,l,tf
r,2,tw
! Geometry of beam
k,l,-b/2,,-dl/2
k,2,,,-dl/2
k,3,b/2,,-dl/2
k,4,-b/2,,dl/2
k,6,,,dl/2
k,6,b/2,,dl/2
k,7,-b/2,l,-do/2
k,8,,l,-do/2
k,9,b/2,l,-do/2
k,10,-b/2,l,do/2
k,ll,0,l,do/2
k,12,b/2,l,do/2
! Area defined
a,1,2,8,7 a,2,3,9,6 a,4,5,11,10 a,5,6,12,11 a,5,2,8,11
Element size defined
lesize,l,,,2 lesize,5,,,2 lesize,8,,,2 lesize,12,,,2 lesize,3,,,2 lesize,7,,,2


83
lesize,10,,,2 lesize,14,,,2 lesize,15f,,6 lesize,16,,,6 lesize,all,,,20
! Elements being defined
real,1 amesh,1,4,1 real,2 amesh, 5
! +++ end plates +++++
A,3,6,5,2 A,1,2,5,4 A,9,12,11,8 A,7,8,11,10 LESIZE,ALL,,,6 amesh,6,9,1 +++++++++++++++++++ save f ini
The First Solution Pass
/solu
Static Analysis by Default
pstres,on
nsel,s,loc,y,0,0
nsel,r,loc,z,-61/2,-dl/2
d,all,uz
d,all,ux
d,all,uy
nsel,all
nsel,s,loc,y,1,1
nsel,r,loc,z,-do/2,-do/2
d,all,uz
d,all,ux
nsel,all


84
LSEL,,P50X lplo NSLL,,1
F,ALL,FZ,-1/20 F,107,FZ,-1/40 F,109,FZ,-1/40 nsel.all
! esel.s.elem, 81,160 sfe,all,1,pres,,-l esel.all save


APPENDIX B NOTATION


NOTATION
= Material property tensor.
= Nodal degrees of freedom vector.
= Concentrated load eccentricity.
= Lagrangian strain tensor.
= Modulus of Elasticity.
= Geometric stiffness matrix.
= Shape function matrix.
= axial displacement component along element.
= in-plane 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.


APPENDIX C
MATLAB INPUT FILE FOR CURVE FITTING OF BUCKLING LOADS
% newr.m
g=[ 0 1 2 3 4 5 6 7 8]'; sl=[0.153493 0.321444 0
0.177107 0.315661 0
0.203372 0.328673 0
0.231324 0.348914 0.735177 0.259758 0.371323 0.731803 0.288914 0.394697 0.728671 0.31783 0.417347 0.726261 0.347227 0.439275 0.724092 0.376624 0.459516 0.722406];
s2=[0.151832 0 0
0.169614 0.259037 0
0.190107 0.259291 0
0.211785 0.259121 0
0.233887 0.259037 0
0.256073 0.258783 0
0.278259 0.258698 0
0.30053 0.258698 0
0.322547 0.258698 0];
s3=[l. 169842 1.464933 2.944259
1.337686 1.356545 2.925996 1.528259 1.28905 2.982671 1.728858 1.250836 3.027436 1.934617 1.232077 3.203518 2.143156 1.225724 3.330766 2.353382 1.227014 3.454638 2.564899 1.232672 3.568287 2.777210 1.240513 3.665063];
s4=[l. 13356 1.561188 0
1.259507 1.616154 0
1.407127 1.610054 0
1.563505 1.604125 0
1.723222 1.600308 0
1.883928 1.594958 0
2.044599 1.591652 0
2.204555 1.589097 0
2.32747 1.587188 0];
s5=[0.045379 0.11763 0


coef=[cl;c2;c3];
% 222222222222222222222222222222222222222222222222222222222222
% section II
al=s2(:,l);
cl=polyfit(g,al,m);
aa 1 =polyval(c 1 ,bigg);
%
pi ot (bigg, aa 1 ,g, a 1,'+') yIabeI('Pcr*L**2 / (E*I)') xlabel('Gamma ( dl/dO -1)')
title('Critical Loads Versus Gamma for Local and Lateral Torsional Buckling') text(1.25,0.25,'Web Local Buckling') text(6.5,. 15,'Section II') grid
meta sec2 pause
coef=[coef;cl];
% 333333333333333333333333333333333333333333333333333333333333
% section III
al=s3(:,l);
a2=s3(:,2);
cl=polyfit(g,al,m);
aal=polyval(cl ,bigg);
c2=polyfit(g,a2,m);
aa2=polyval(c2,bigg);
a3=s3(:,3);
c3=polyfit(g,a3,m);
aa3=polyval(c3 ,bigg);
%
plot(bigg, aa 1 ,bigg,aa2,g,a 1 ,'+',g, a2, '+',bigg,aa3 ,g,a3,'+') ylabel('Pcr*L**2 / (E*I)') xlabel('Gamma (dl/dO -1 )')
title('Critical Loads Versus Gamma for Local and Lateral Torsional Buckling')
text(l,3.5,'2nd Mode Lateral-Torsional Buckling')
text(3,1.4,'1st Mode Lateral-Torsional Buckling')
text(2,2,'Web Local Buckling')
text(6,3.75,'Section III')
grid
pause
meta sec3
coef=[coef;cl;c2;c3];
% 444444444444444444444444444444444444444444444444444444444444


Full Text

PAGE 1

LATERAL-TORSIONAL BUCKLING OF TAPERED MEMBERS WITH TRANSVERSE LOADS by Giorgios Paul Demos B.S., University of Colorado at Denver, 1992 A thesis submitted to the Faculty of the Graduate School of the University of Colorado at Denver in partial fulfillment of the requirement for the degree of Master of Science Civil Engineering 1994

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@ by Giorgios Paul Demos All rights reserved.

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This thesis for the Master of Science degree by Giorgios Paul Demos has been approved for the Department of Civil Engineering by Andreas S. Vlahinos Date

PAGE 4

v used in present-day practice. The loading consist of uniform bending, the end restraints applied to the bottom or to both flanges. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. Signed Andreas S. Vlahinos

PAGE 5

Demos, Giorgios Paul Demos (M.S., Civil Engineering) Lateral Torsional Buckling of Tapered Members With Transverse Loads Thesis directed by Assistant Professor Andreas S. Vlahinos ABSTRACT In recent years tapered beams have become increasingly popular in contin uous frame construction due to their efficient utilization of structural material. Although analysis and design methods have been presented by many authors and three experimental programs have been carried out in recent years, further studies and design recommendations are needed. In this thesis the mathematical formulation and the computer implementation of the elastic stability analysis of doubly symmetric tapered beams are presented. Solutions in the form of tables, charts and approximate formulas for the lateral buckling load are presented for a variety of beam geometry, end restraints and loading conditions. The critical loads corresponding to various buckling modes are plotted versus non-dimensionalized parameters in an effort to establish the parameter ranges at which local buckling or lateral-torsional buckling controls. Since it is practically impossible to non-dimensionalize the many parameters in tapered sections, solutions were obtained for a range of tapered members

PAGE 6

ACKNOWLEDGMENTS I wish to express my gratitude and appreciation to my advisor, Professor Andreas S. Vlahinos, for his guidance, direction, and assistance throughout this course of the work. I also would like to thank Professor James C-Y Gou, and JonathanT-H Wu for their willingness to serve on my thesis committee and in particular, their constructive criticisms. This thesis is dedicated to my wonderful parents, who suffered a great deal after their immigration to the United States, in order to provide me with the best emotional, and financial support. Their love and patience has been exceptional, and their support has been immeasurable. This dedication is small token of my great appreciation for them. Thank you very much from the bottom of m.y heart. And I also would like to thank Dr. Andreas S. Vlahinos for his guidance, direction and his patience with me and good luck to his family and his future. And last but not least I want to thank Dr. Yang-Cheng Wang for his immeasurable help through-out this experience and best wishes.

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CONTENTS CHAPTER I. INTRODUCTION . . . . . . . . . . . . 1 II. MATHEMATICAL FORMULATION ................................. 3 II.I Introduction . . . . . . . . . . . . . 3 II.II Historical Review . . . . . . . . . . . . 3 II.III Solution Procedure ............................................. 5 III. COMPUTER IMPLEMENTATION . . . . . . . . 11 III.I Problem Description . . . . . . . . . . . 11 III.II Problem Idealization . . . . . . . . . . 12 III.III Discretization ................................................. 14 IV. NUMERICAL RESULTS ............................................ 23 IV .I Section I . . . . . . . . . . . . . 23 IV .II Section II . . . . . . . . . . . 25 IV .III Section III . . . . . . . . . . . . . 27

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Vlll IV .IV Section IV . . . . . . . . . . . . . 29 IV.V Section V ..................................................... 31 IV. VI Section VI .......... . . . . . . . . . . 33 IV.VII Conclusion ................................................... 35 V. CONCLUSIONS .................................................... 71 V.I Tables . . . . . . . . . . . . . . 71 APPENDIX A. INPUT FILE . . . . . . . . . . . . . 80 B. NOTATION . . . . . . . . . . . . . . 85 C. MATLAB INPUT ................................................... 87 D. VERIFICATION PROBLEM ........................................ 93 BIBLIOGRAPHY . . . . . . . . . . . . . . 95

PAGE 9

FIGURES Figure: 1. Geometry of linearly tapered I-beams ................................ 15 2. Geometry of variable sections .. .. .. .. .. .. .. .. .. .. .. .. 16 3. Geometry of a tapered beam length (120 in) ......................... 17 4. Geometry of a tapered beam length ( 480 in) .. .. .. .. .. .. .. .. .. .. .. .. 18 5. End restraints without end plates .................................... 19 6. Plot of all finite elements for a typical case, without end plates ....... 20 7. Plot of all the nodes and boundary conditions without end-plates ..... 21 8. The buckling mode shape and the critical load of the verification problem .............................................................. 36 9. Plot of all the elements of the Finite Element model of (section I) . . . . . . . . . . . . . . 3 7

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X 10. Nodes and boundary conditions of the finite element model of (section I) . . . . . . . . . . . . . . 38 11. Web buckling mode of (section I) .................................... 39 12. l8t lateral torsional buckling mode of (section I) ..................... 40 13. 2"d lateral torsional buckling mode of (section I) ..................... 41 14. Plot of the critical load versus "Y of (section I) ........................ 42 15. Plot elements of the Finite Element model in of (section II) .......... 43 16. Plot of nodes and boundary conditions of the finite element model of (section II) . . . . . . . . . . . . . . 44 17. Web buckling mode of (section II) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 45 18. Critical load versus"'( of (section II) ................................. 46 19. Plot of all elements of the finite element model of (section III) ........ 47 20. Nodes and boundary conditions of the finite element model of (section III) . . . . . . . . . . . . . . 48 21. Web buckling mode of (section III) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 49

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XI 22. l't lateral torsional buckling mode of (section III) . . . . . 50 23. 2nd lateral torsional buckling mode of (section III) ................... 51 24. Plot of the critical load versus 'Y of (section III) . . . . . 52 25. Plot of all elements of the finite element model of (section IV) . . 53 26. Nodes and boundary conditions of the finite element model of (section IV) . . . . . . . . . . . . . . 54 27. Web buckling mode of (section IV) .. .. .. .. .. .. .. .. .. .. .. .. .. .. 55 28. Pt lateral torsional buckling mode (section IV) ...................... 56 29. 2nd web buckling mode of (section IV) ............................... 57 30. Plot of the critical load versus 'Y of (section IV) ...................... 58 31. Plot of all elements of the finite element model in of (section V) . 59 32. Nodes and boundary conditions of the finite element model of (section V) . . . . . . . . . . . . . . 60 33. IBt web buckling mode (section V) ................................... 61 34. l't lateral torsional buckling mode of (section V) .................... 62

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Xll 35. 2"d web buckling mode of (section V) ............................... 63 36. Plot of the critical load versus "( (section V) .. .. .. .. .. .. .. .. .. .. .. .. 64 37. Plot of all elements of the finite element model in of (section VI) ..... 65 38. Nodes and boundary conditions of the finite element model of (section VI) . . . . . . . . . . . . . . 66 39. Pt local web buckling mode of (section VI) .......................... 67 40. 2"d local web buckling mode of (section VI) ......................... 68 41. Plot of the critical load versus "( of (section VI) .. .. .. .. .. .. .. .. .. .. 69

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TABLES Table 1. Critical Loads and Mode Shapes for Various Sections and 'Y 22 2. Uniform Critical Loads for (L=120) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 73 3. Uniform Critical Loads for (L=180) ................................. 74 4. Uniform Critical Loads for (L=240) ................................. 75 5. Uniform Critical Loads for (L=300) ................................. 76 6. Uniform Critical Loads for (L=360) ................................. 77 7. Uniform Critical Loads for (L=420) ................................. 78 8. Uniform Critical Loads for (L=480) ................................. 79

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I.I Introduction CHAPTER I INTRODUCTION During the past several years there has been a sincere effort to reduce costs and therefore, cause the weight of immense structures to also reduce. Improved computational tools and high strength materials are the two primary reasons for efficient, lighter and stronger structures. The trend for producing lighter structures sometimes leads to slender elastic structures which can accept design loads but in which the stability safety factors are reduced due the fact that slender structures are more inclined to become unstable. One of the ways to optimize structural components subjected to lateral loads is to generate variable cross sections. Ideally, if all sections must pos sess the minimum amount of required properties (i.e. moment of inertia) then individual sections will be unique for a particular loading conditions thus, con sequently extremely expensive to construct. This shape optimization can be utilized effectively by the automotive and aerospace industry due to the fact that they have to manufacture the same components subjected to the same loading several thousand times. Furthermore, in civil engineering structures like high rise buildings, long span bridges, sports complexes, and industrial buildings, the structural members shape optimization is circumscribed to linearly varying cross

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2 section like tapered beams. The constant cross section constructional compo nents are the easiest to produce and a.re widely utilized. Tapered members are not as inexpensive to manufacture, but nevertheless, are primary to significant reserve in material expenditure. The bending potency requirements and design guidelines of tapered members are not significantly different than the ones of constants section members. The immense majority of steel beams are incorporated in such a manner that their compression flanges are constrained opposing lateral buckling, since the upper flange of the beams utilized to support concrete floors are often in corporated in these floors; in those cases the lateral buckling of beams are not a major concern. In the event that the compression flange is without lateral sup port, it behaves like a compression member subjected to buckling, thus causing lateral buckling of the beam. The occurrence of lateral buckling of beams with constant cross sections subjected to lateral loads has been investigated by sev eral researchers for the past fifty years. An immense amount of analytical and experimental drudgery has been incorporated to design codes. No such design guidelines exist for the lateral buckling of tapered beams subjected to lateral loads. In this research an exertion was attempted to produce formulas two pre dict the critical uniform load which yields lateral buckling of tapered beams. The results were produced to incorporate the effects of several parameters such as section properties, beam lengths, and the slope of the flanges.

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CHAPTER II MATHEMATICAL FORMULATION 11.1 Introduction A structural member laterally loaded, bent about its major axis, may buckle laterally if the compression flange is unbraced. At the critical load, the compression flange tends to bend laterally and thus causing the remaining of the section to become restrain; thus resulting in lateral and torsional motion of the section. In other words, lateral buckling is a combination of twisting and lateral bending of the compression flange. 11.11 Historical Review The first theoretical work on the subject of lateral buckling beams with rectangular cross sections is by L. Prandt1 (1899) in his dissertation and A.G.M. Michelle2 in his paper "Elastic Stability of Long Beams under Transverse Forces." In 1910 S. Timoshenko3 presented the equations for lateral buckling of beams with symmetric I sections. G. Winter4 in 1943 presented the equations for lateral stability of uniform section of unsymmetrical 1-beams and trusses in bending. Lee5 first addressed the issue of lateral buckling of a tapered narrow rectangular beam.

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4 The analysis of lateral buckling of beams with constant cross sections was also presented by J.W. Clark and H.N. Hill6 7 in 1962. These authors presented the results of a series of investigations into the lateral buckling of channel sections and Z-sections that were loaded in the direction of the strong axis. They also introduced the torsion-bending factor which is defined as a property of the cross section. Torsion bending is the resistance arising from the restraint to warping of the cross section of the beams. The authors also developed a general formula for the elastic buckling strength of beams with coefficients depending on the loading and support conditions. Significant investigations of the elastic and inelastic buckling of various shapes have been reported by T.V. Galambos8 in 1963. A method was presented for determining the inelastic lateral buckling problem for simply supported wide-:O.ange beams subjected to equal end moments. His methods are based on the determination of the diminishing in the lateral and torsional stiffness caused by yielding. B.A. Boley9 established that for small tapering angles, less than 15 degrees, that the Bernoulli-Euler theory of various cross yields adequate results. In addition to the above-mentioned analytical studies, two test programs were started. The first experimental study were conducted by D.J. Butler and G.C. Anderson10 11 at Columbia University in 1966. In this program tapered I section beams and channels sections tapered in both the web and :O.anges were examined as cantilever beam-columns and the elastic stability loads and the brae-

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5 ing requirements were discovered. The subsequent experimental study ended in 1974 and was conducted by Lee, Ketter, and PraweP2 13 at the State University of New York at buffalo. In this program the inelastic stability of tapered !-shape beam columns were also investigated. S. Kitipornchai and N. Trahiar14 in 1972 studied the elastic flexuraltorsional buckling of I-beams. They concluded there elastic critical loads of FlangeTapered beams decreased significantly as degree of taper increased; on the other hard, elastic critical loads of depth-tapered beams do not vary greatly as the degree of taper increases. This thesis is related to the insensitivity of the torsional stiffness to the degree of taper. 11.111 Solution Procedure Following is a brief derivation of the relevant equation using the potential energy formulation1213 14 The expression of the elastic strain energy U is: U = f CTijd( eij )d( vol) luol (2-1) where eij is the Lagrangian strain tensor. Using Hooke's law CTij = Cijklekh where Cijkl is the material property tensor, the above equation can be written as: (i,j, k, l = 1, 2, 3) (2 -2)

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6 The Lagrangian strain tensor can be expressed in Cartesian coordinates in terms of the displacement derivatives Ui,j, as 1 e .. = -(u +u+ukUk ) ., 2 ,,. ,, (2 -3) It is convenient to group the linear strain terms ei; = i(ui,j +u;,i) and the nonlinear-strain terms ei,j = !(uk,iUk,;).Thus (2 -4) Substitution of the above equation into equation (2-2) results in (2 -5)

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7 This is the basic form of strain energy for nonlinear analysis. The total potential energy, denoted by Ut, is given by Ut=U-V (2 -6) where Vis the potential energy of the external forces. Through the principal of stationary value of the total potential, one may derive the equilibrium equations and the proper end conditions. In the :finite element formulation the displace ments are approximated by the shape function matrix [ N] times the nodal degrees of freedom vector { d}. {u} = [N]{d} For elastic stability analysis some simplifications can be made7 First equation (5) can be rewritten as

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8 {2 -7) Discarding the higher order term and using Hooke's law {2 -8) 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: {2 -9) The objective is to find the critical intensity Acr at which buckling occurs. If the load vector {P0 } is applied, a state of stresses u?; can be found. Therefore

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9 for any load vector P the state of stresses can be written as CTij = Thus the strain energy can be written as (210) Differentiation of the displacements and integration of the strain energy results in U = + {diT (2-11) where [K]o is the linear elastic stiffness matrix and (2 12) [ K]I is the initial-stress or geometric stiffness matrix. If the total potential 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

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10 (2-13) where is the buckling mode shape.Furthermore, the determinant of the above matrix must be zero I[[K]o + .,\[Kh]l = 0 (2-14) The critical load is Per = .,\crP0 At this load level an adjacent equilibrium position exists, which means that a bifurcation point appears on the equilibrium path. Equation 2-13 converts the buckling problem to the classical eigenvalue form which can be solved efficiently with the available eigensolving routines.

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CHAPTER III COMPUTER IMPLEMENTATION III.I Problem Description The elastic stability analysis of a simply supported steel tapered beam was considered. The objective of this study was to determine formulas in order to predict the uniformly distributed critical load of the beams in terms of sec tional properties, beam length and degree of depth-tapering. The top flange was considered un-braced for the entire length. Figure 1 adumbrates the geometry of the linearly tapered beam. (L) represents the length, (do) represents the smallest depth of the beam and ( d1) the largest depth of the beam. The depth of the beam at a distance (x) from the lesser end is expressed as: (3-1) The tapering ratio "Y is a measure of the degree of depth-tapered and for a pris matic member thus yielding zero. Six various sections were considered. The flange thickness (tr), the web thickness (tw), the flange width (b), and the small depth ( d0), considered are adumbrates in table 1. In this study it was considered as a variable varying from (0 to 8) with increments of 1. The lengths considered were L= 120, 180, 240, 300, 360, 420 and 480 inches, Figure 2. adumbrates all the

PAGE 25

12 sections. Figure 3 adumbrates the top, side, front and isometric view of the case 1=120 and '"Y = 8. Figure 4 adumbrates the top, side, front and isometric view of the case L = 480, '"Y = 8. The load considered was a uniformly distributed load at the top :flange. The lower flange was restrained for the entire flange width in both (x) and (y) directions at the largest depth section, the lower :flange was also restrained in the (y) direction at the smallest depth section as it is adumbrated in figure 5. 111.11 Problem Idealization The following assumptions were made: The :flange and web panels originally were entirely flat; in other words, all geometric imperfections are disregarded. The material is isotropic, linearly elastic with modules of elasticity E=29 x 106 psi and Poisson's ratio v=0.3. No temperature effects were examined on the above parameters. The loading uniform pressure was applied on the top :flange and the effects were disregarded. Bifurcation buckling is adequate enough to determine the critical load; in other words, the material's non-linear, the snap through buckling and im-

PAGE 26

13 perfection sensitivity were not examined in the determination of the critical load.

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14 III.III Discretization The tapered beam is descritized with 355 nodes and 328 four node shell elements, 12 geometry points, 20 lines, 9 surfaces, 21loads (nodal), 25 constraints, 2 sets of thicknesses. This element contains both bending and membrane capa bilities. Each node has six degrees of freedom: Translation in the nodal (X), (Y), and (Z) directions and rotations about the nodal (X), (Y), and (Z) axis. The largest element size is approximately a quarter of the flange width. The thickness in web and flange is constant ( tw) and ( tr) respectively. The aggregate numbers of the degrees of freedom is (500). Figure 6 adumbrates a plot of all elements for a typical case without end plates. For clarity all the elements have been de creased by (20degree of freedom of the bottom flange width of the section with the smallest depth are restrained. Figure 7 adumbrates a plot of all the nodes contained in the above described boundary conditions without end plates.

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15 tf tr L 7, dr-Il:i _t Etilf dO b.....J '1. J L L Figure 1. Geometry of linearly tapered !-beams

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Section I 0.75 L4.11JJ Section III l 12..00 0.10-t-I.-.-6.00 --l Section V 0.25 0.10 1----12.00 ----l Section II 0.75 0.25 1----12.11l------' Section IV 0.10 .25 1------16.00----...l Sect ion VI Figure 2. Geometry of various sections. 16

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17 120.00 ... 6.00 t l f 54.00 J 12.00 Figure 3. Geometry of a tapered beam length (120 in)

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18 54.00 6fc:==::======:Jj _j L 12.00 Figure 4. Geometry of a tapered beam length ( 480 in)

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p Jgure 5 E lld restrai nts lV.i t.h out end plates 19

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20 Figure 6. Plot of all finite elements for a typical case, without end plates

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21 ., .. : . ..... Figure 7. Plot of all the nodes with the above described boundary conditions without endplates

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FACT=2.l4 uz TOP RSYS=O DMX =l SMX =l PRES 22 III!IBII Ei!iili'Eil 1111!1 -0 O.llllll .. 0.222222 0.333333 0.444444 0.555556 0.666667 0.777778 0.888889 1 NODAL SOLUTION STEP=l SOB =l FACT=2.14 uz TOP RSYS=O DMX =l 8. The buckling mode shape and the critical load of the verification problem

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CHAPTER IV NUMERICAL RESULTS IV .I Section I. In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with the above dimension of section I (do)=6.00, b=4.00, (tJ )=0.25, (tw)=0.10. The length of section I was 1=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various value of (-y = 0, 1, 2, 3, 4, 5, 6, 7,and 8). The largest end of the model was supported on the lower :flange and its entire length with pined supports. Both ends of the model contained plates, they were connected to the top and bottom :flange; the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top :flange. The critical loads were reported in pounds; in order to illiminate the effect of the length on the value of the critical loads. That load was also nondimensionalized with respect to ( of the section. In section I the moment of inertia about the horizontal axis (strong axis) (Ix) is (17.9281 inches4 ) and the moment of inertia about the vertical axis (Iy) is (2.6671 inches4). The non reported critical loads are defined as: (4 -1)

PAGE 37

24 (equation (4-1) where (Pa) is the actual critical loads in pounds, L= length of the beam, E=the modules of elasticity and (Ix) the strong axis moment inertia at the smallest end. Figure (9) represents a plot of all the elements of the finite element model in (section I), figure (10) represents all the nodes and boundary conditions of the finite element model in (section I). In section I three buckling modes seemed to dominate the behavior of this beam, local web buckling, 1st lateral torsional buckling (single wave of the top flange and 2nd lateral torsional double wave of the top flange occurred.) Figure (11) represents web buckling mode in (section I), figure (12) represents the 1st lateral torsional buckling mode (section I), figure (13) represents 2nd lateral torsional buckling mode (section I), figure (14) represents a plot of the critical load versus"'( in (section I). In all cases the local web buckling appears to be the smallest critical loads. To prevent local buckling stiff plates were used, lateral torsional buckling will be dominate. As "'( increases local web buckling and 1st lateral torsional buckling modes increase monotonecly, the 2nd lateral torsional buckling mode decreases monotonecly as"'( increases. A cubic polynomial coefficients for all three critical loads were generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use this polynomials to predict any of the critical loads for this section and any value of "'(.

PAGE 38

25 IV .II Section II In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with dimensions of section I (do)=6.00, b=12.00, (tr)=0.25, (tw)=O.lO. The length of section II was 1=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (-y = 0, 1, 2, 3, 4, 5, 6, 7,and 8) the largest end of the model was supported on the lower :flange and its entire length with pined supports. Both ends of the model contained plates, they were connected to the top and bottom :flange, the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top :flange. The critical loads were reprted in pounds; in order to elliminate the effect of the length on the value of teh critical load. This load was also nondimensionalized with respect to ( of the section. In section II the moment ofinertia about the horizontal axis (strong axis) (Ix) is (51.0155 inches4 ) and the moment of inertia about the vertical axis (Iy) is (72.005 inches4.) figure (15) represents a plot of all the elements of the finite element model in (section II), figure (16) represents all the nodes and boundary conditions of the finite element model in (section II). In this case all buckling modes were local. The lateral torsional does not appear in the 1st ten modes; figure (17) represented the web buckling modes in section (II). Figure (18) represents the plot of critical load

PAGE 39

26 versus ; in section (II). A cubic polynomial coefficients for all three critical loads was generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use these polynomials to predict any of the critical for this section and any value of;.

PAGE 40

27 IV .III Section III In this part of the analysis a finite element model was constructed to c cal culate the buckling loads of a linear tapered beam with dimensions of section I= (do)=6.00, b=12.00, (tr)=0.25, (tw)=0.10. The length of section III was 1=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (-y = 0, 1, 2, 3, 4, 5, 6, 7,and 8); the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates; they were connected to the top and bottom flange. The reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; in order to illiminate the effect of the length on the value of the critical loads. This load was also nondimensionalized with respect to ( of the section. In section III the moment of inertia about the horizontal axis (strong axis) (Ix) is ( 43.52345 inches4 ) and the moment of inertia about the vertical axis (ly) is (8.0006 inches4.) Figure (19) represents a plot of all the elements of the finite element model in (section III), figure (20) represents a.ll the nodes and boundary conditions of the finite element model in (section III). Once again three buckling modes seemed to dominate the behavior of this beam, local web buckling, 1 8t lateral torsional buckling (single wave of the

PAGE 41

28 top :Bange and 2nd lateral torsional double wave to the top :Bange occurred.) Figure (21) represents web buckling mode in (section III), figure (22) represents the l8t lateral torsional buckling mode in (section III), figure (23) represents the 2nd lateral torsional buckling mode in (sectioniii); figure (24) represents a lot of the critical load versus 'Y in (section III.) for 'Y less than (1), local web buckling appears to be the smallest critical load. For 'Y greater than (1), the lateral torsional buckling controls. As 'Y increases local web buckling and 2nd lateral torsional mode also increases monotonecly, the 1st lateral torsional mode seems to remain constant for all 'Y values. A cubic polynomial coefficients for all there critical loads were generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use these polynomials to predict any of the critical loads for this section and any value of 'Y

PAGE 42

29 IV .IV Section IV In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with dimensions of section I (d0)=12.00, b=6.00, (tr)=0.75, (tw)=0.25. The length of section IV was 1=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (1 = 0,1,2,3,4,5,6,7,8); the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates; they were connected to top and bottom flange; the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; one observation that was noticed was that length did not play a role in the critical loads. This load was also nondimensionalized with respect to ( of the section. In section IV the moment of inertia about the horizontal axis (strong axis) (Ix) is (126.773 inches4) and the moment of inertia about the vertical axis (ly) is (216.006 inches4.) Figure (25) represents a plot of all the elements of the finite element model in (section IV); figure (26) represents all the nodes and boundary conditions of the finite element model in (section IV). Once again two buckling modes seemed to dominate the behavior of this beam: local web buckling, 1 "t lateral torsional buckling (single wave of the top

PAGE 43

30 flange and 2"d lateral torsional double wave of the top flange occurred.) Figure (27) represents web buckling mode in (section IV); figure (28) represents the Pt lateral torsional buckling mode in (section Iv), figure (29) represents the 2"d lateral torsional buckling mode in (section IV), figure (30) represents a plot of the critical load versus 'Yin (section VI). For"( less than (3.25), local web buckling appears to be the smallest critical load. For 'Y greater than (3.25), the lateral torsional buckling controls. As 'Y increases local web buckling and 2"d lateral torsional mode also increases monotonecly; the l8t lateral torsional mode seems to remain constant for all 'Y values. This type of behavior is similar to the case in section (III) and thus, leads to the following very important conclusion: The effect of the degree of taper 'Y is negligibly small on the lateral torsional buckling load of tapered beams; thus one may use the formulas of constant section beam to predict the approximate value of the lateral torsional buckling load. A cubic polynomial coefficients for all three critical loads were generated in table (1 ). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use these polynomials to predict any of the critical loads for this section and any value of 'Y.

PAGE 44

31 IV. V Section V In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with dimensions of section I (do)=12.00, b=6.00, (tr)=0.25, (tw)=O.lO. The length of section IV was 1=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (1 = 0,1,2,3,4,5,6,7,8); the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates; they were connected to the top and bottom flange; the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; in order to illiminate the effect of the length on the value of the critical loads. This load was also nondimensionalized with respect to ( of the section. In section V the moment ofinertia about the horizontal axis (strong axis) (Ix) is (116.236 inches4 ) and the moment of inertia about the vertical axis (Iy) is (9.00096 inches4.) Figure (31) represents a plot of all the elements of the finite element model in (section V); figure (32) represents all the nodes and boundary conditions of the finite element model in (section V). Once again two buckling modes seemed to dominate the behavior of this beam: local web buckling, 1st lateral torsional buckling (single wave of the top

PAGE 45

32 flange and 2nd lateral torsional double wave of the top flange occurred.) Figure (33) represents web buckling mode in (section V); figure (34) represents the 18t lateral torsional buckling mode in (section V), figure (35) represents the 2nd lateral torsional buckling mode in (section V), figure (36) represents a plot of the critical load versus 'Y in (section V). For 'Y less than (6.75), local web buckling appears to be the smallest critical load. For 'Y greater than (6.75), the 2nd local web buckling mode controls. The 1 8t local web buckling and 1 8t lateral torsional mode increases monotonecly as 'Y increase. The 2nd local web critical load decreases as 'Y increases. A cubic polynomial coefficients for all three critical loads were generated in table (1). The code that generated the coefficients in the table was matlab, and is represented in appendix a. One may use these polynomials to predict any of the critical for this section and any value of 'Y.

PAGE 46

33 IV. VI Section VI In this part of the analysis a finite element model was constructed to calculate the buckling loads of a linear tapered beam with dimensions of section I= (do)=12.00, b=6.00, (tr)=0.75, (tw)=0.25. The length of section IV was 1=360 inches; the smaller end of the beam was supported with rollers. Several models were implemented for various values of (-y =0,1,2,3,4,5,6,7,8); the largest end of the model was supported on the lower flange and its entire length with pined supports. Both ends of the model contained plates; they were connected to the top and bottom flange; the reason for this was to prevent local buckling of the supports. This model (beam) was subjected to a uniformly distributed load on the top flange. The critical loads were reported in pounds; in order to illiminate the effect of the length on the value of the critical load. This load was also nondimensionalized with respect to ( of the section. In section IV the moment of inertia about the horizontal axis (strong axis) (Ix) is (288.841 inches4 ) and the moment of inertia about the vertical axis (Iy) is (170.668 inches4.) Figure (37) represents a plot of all the elements of the finite element model in (section VI); figure (38) represents all the nodes and boundary conditions of the finite element model in (section VI). Once again two buckling modes seemed to dominate the behavior of this beam: local web buckling, 111t lateral torsional buckling (single wave of the top

PAGE 47

34 flange and 2nd lateral torsional double wave of the top flange occurred.) Figure (39) represents Pt local web buckling, figure ( 40) represents 2nd local web buckling in (section VI). Figure (41) represents a. plot of the critical load versus "Yin II ,, (section VI). For "Y less than (5), the the l't local web buckling load appears to ,, be the smallest critical load. For "Y greater than (5), the 2nd local web buckling dominates. The l't local web buckling increase monotonecly as "( increase. The 2nd local web buckling load decreases as "Y increases as in all previous cases. The code that generated the coefficients in the table was matlab, and is represented in appendix a. One any of the critical for this section and any value of "Y

PAGE 48

35 IV.VII Conclusion The value of the critical loads and corresponding mode shape for all sec tions and 1 are summarized in table (1). The(*) represents that the correspond ing mode did not appear in the first ten modes. The most important conclusion can be summarized below: The effect of the degree of taper ( 1) is negligibly small on the lateral torsion buckling load of tapered beams; thus, one may use the formulas of constant section beam to predict the approximate value of the lateral torsion buckling load. The first local buckling mode seems to controls in most cases and increase as 1 increases. The second local buckling mode seems to controls only in cases in high 1 and small thickness. The corresponding critical load decrease as 1 increases.

PAGE 49

37 Figure 9. Plot of all the elements of the Finite Element model of (section I)

PAGE 50

. :!' Jl .. .. .. r J r .. .. r tJ .. fl JJ .. r 1!! < k l ... .... Figure 10. Nodes and boundary conditions of the finite element model of (section I) 38

PAGE 51

39 I 3 t cr-1563.44289, L, g, do, b, tf-0.25, twO.l Figure 11. Web buckling mode of (section I)

PAGE 52

40 .-----rm 2 Pcr-1907.08754, L, g, do, b, t.25, tw-0.1 Figure 12. l"t lateral torsional buckling mode of (section I)

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41 3 Pcr-1933.04399, L, g, do-6, b, tf.25, twO.l Figure 13. 2nd lateral torsional buckling mode of (section I)

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Critical Loads Versus Gamma for Local and Lateral Torsional Buckling ------!--. t. l 0.7 --. I !-2nd M9de Laterai-Tqrsional Bucklipg ; 1 0.6 .... 0.5 .. _,_ ----.. ___ --______ _J _:_. i -+ -----+--. -. _.;. __ ------+--------... -..,.------. 0 0 25 ___ ..!_ __ -I 6.00 0.10 ; ---------42 I L 4 00, I --f"' 1 i t 1st Molle LateralTorsional BucWg ; I 0.4 0.3 . -. ..... tI ....... ------+""... I -------. -----------r----------i-Web Locaj Buckling ---+------. 1 2 3 4 s 6 Gamma ( dVdO -1 ) Section I l 7 Figure 14. Plot of the critical load versus 1 of (section I) 8

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43 Figure 15. Plot elements of the Finite Element model in of (section II)

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r r l .. l'l . r r.r ... .. 1 1 l l l .. rll l .lG 44 Figure 16. Plot of nodes and boundary conditions ,of the finite element model of (section II)

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45 :rJ055.97155, L, gB, do-6, b, tf.25, twO.l Figure 17. Web buckling mode of (section II)

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' Critical Loads Versus Gamma for Local and Lateral Torsional Buckling 0.34 .----..-------.------.-----..--------.---'------,.-----.,-----, I I .. ----------+--------+---------'-----------j o.;rz .. I 0.3 -----------------j--------I I --------;--------------' -,------r---------t--i Web Locaj Buckling -:---r-----------------""7""! I I ... !. ------_____ .J. ___ I i I 0.24 ... -+ .... -----+-----1 I 0.22 ..... +---------L----, I i i i ---l-----+---------.. --r--: : l 1.: 0.25 0.10 12.00-----...11 D 0.14ol_ ___ .J..1 ___ __,2 ____ 3.L..,_ ___ .J..4 ___ __,5 ____ 6J...._ ___ .J..7 ___ ---!8 Gamma ( dVdO -1 ) Figure 18. Critical load versus "'' bf (section II) 46

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47 Figure 19. Plot of all elements of the finite element model of (section III)

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. r r l .. [. y
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49 2 PcrBO.JOB2, LJ60, gB, do, b, tf.75, tw.25 Figure 21. Web buckling mode of(section III)

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50 ,---,, L, gB, do, b, tf.75, tw.25 Figure 22. 18 t lateral torsional buckling mode of (section III)

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51 -4 Pcr.6l97, L, do, b, tf.75, tw.25 Figure 23. 2" d lateral torsional buckling mode of (section III)

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Critical !.Dads Versu5 Gamma for !.Deal and lJlteral Torsional Buckling __ i I I I .,. -. ---T j_' ______---1 2.5 -----------2 ------. -----: ------+---. I -------r-1 I I ----;--I i t I __;_ -------11-------1 I i l.Dcal stickling f.. -------! j i i I I +' 0.75 0.25 i I i i 1.5 ----------------. ruckling i----L 41Xl J -------t------J _________ l ______________ --+--------t-----j_:_'_--1 I ; i I 10 2 3 4 Gamma ( dVdO 1 ) J Figure 24. Plot of the critical load versus -y of (section III) 52

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53 Figure 25. Plot of all elements of the finite element model of (section IV)

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54 _/ .. .. .. rrrt .. .. rr .. r ... l 1 ... I . .. Figure 26; Nodes and boundary conditions of the finite element model of (section IV)

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55 2 Pcr.8266, LJ60, gB, do, b, tf.75, tw.25 Figure 27. Web buckling mode of (section IV)

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56 .. cr.9522, L, g, do, b, tf.75, tw.25 Figure 28. l8t lateral torsional buckling mode (section IV)

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. 57 \ 3 Pcr.0824, L, g, do, bl2, tf.75, tw.25 Figure 29. 2"d web buckling mode of IV)

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N -E Critical Loads Versus Gamma for Local and Lateral Torsional Buckling 2 1.8 ---. --------------+--..... _, ___ -----------l Web Loajl Buckling i I I I 1.6 J.. .. : .. :::::: ____ 1st 1.4 -,t: L_ ___j-12.111 Sectiod IV 1 2 3 4 5 6 7 Gamma ( dl/dO -1 ) Figure 30. Plot of the critical load versus 1 of (section IV) 8 58

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59 iiM $>$ !!1!1-i!ES -#!! .Q.iij. ... ... f j .Z iii!! t tilt F:tt L. 2 Pcr.82207, L, gB, dol2, b, tf-0.25, twO.l Figure 31. Plot of all elements of the finite element model in of {section V)

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. i I rrrr r. ;z ; y
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61 a ea J1 l-t-tktt!t i !@@( 3 Pcr.01647, L, g, do, b, tf.25, tw.1 Figure 33. l8t web buckling mode (section V)

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62 f1 5 I U s e: 7 I 3 Pcr.01647, LJ60, gB, do, b, tf.25, tw.1 Figure 34. l6t lateral torsional buckling mode of {section V)

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63 cr.61157, L, g, do, b, tf-0.25, twO.l Figure 35. 2nd web buckling mode of (section V)

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64 Critical Loads Versus Gamma for Local and Lateral Torsional Buckling 0.35 r----,.-------,-------.------,-------,-----r----,-------, ... 0.3 ; -----------0.25 -----__ Web Lodl Buckling -----.. : ____________________ .. -------------;-0.2 ----. --, -... ---------r "" _..-----1st I ----_ _j ____ __j ___ I I -...... i -..... -------; ___ :::_.:::_+o:----. I 0.1 r--"1""-a+l l---: -------.25 ; 11.1110.10-1-Section v 1..-a.lll--' 1 2 3 4 s 6 1 8 Gamma ( dildO -1 ) Figure 36. Plot of the critical load versus 'Y (section V)

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65 Figure 37. Plot of all elements of the finite element model in of (section VI)

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66 Figure 38. Nodes and boundary conditions of the finite element model of {section VI)

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67 2 Pcr-4444.17938, L, gB, do, b, tf.25, tw.1 Figure 39. l't local web buckling mode of (section VI)

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68 ., ... cr.74397, L, gB, do, b, tf.25, twO.l Figure 40. 2nd local web buckling mode of (section VI)

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0.2 .0.18 0.16 0.14 :::-0.12 -. -.. N 0.1 -. -l .. 0.06 0.04 --0.02 --00 .. -1-.. .. 69 Critical Loads Versus Gamma for Local and Literal Torsional Buckling -----I I I Web 1.ocaj Buckling I i 2 3 4 i .. __________ i I -----r----""'1"' ........... -+--t ... .25 I 11.111 1 Section VI I '-----15.111----...1 s 6 7 Gamma ( dVdO -1 ) 8 Figure 41. Plot of the critical load versus 1 of (section VI)

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CHAPTER V CONCLUSION V .I The effect of length and gamma on the critical loads In order to investigate the effect of beam length on the critical loads for various values of gamma consider a beam with section VI which had the smallest end. A finite element model was generated having the length L as element model was generated having the length L as a parameter, the length L as parameter, had values of Length= 120,180,240,300,360,420 and 480 inches and ('r= 0,1,2,3,4,5,6,7 and 8). The load was uniformed pressure on the top flange. The boundary conditions were the same as the ones in the previous chapter. In this case no end plates were considered at the end supports; the first ten mode shapes were also computed. Due to the lack of end plates and the thickness of the web and flange, small local buckling caused the critical mode to always appear.

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CURVE FITTING COEFICIENTS FOR CRITICAL LOADS Pcr(Y)=dy3+cy2+by+a d c b a Sec I -0.00006167508418 0.00106996248196 0.02328497017797 0.15322762626263 Sec I -0.00041843350168 0.00622837193362 -0.00570030206830 0.31930697979798 Sec I -0.00000220370370 0.00025586111111 -0.00515758597884 0.74842169047619 Sec II -0.00005973905724 0.00092830375180 0.01774929894180 0.15152301010101 Sec II -0.00045675084175 0.00793322330447 0.16677549723425 1.16750695959596 Sec III -0.00104570117845 0.02013005375180 -0.12214888287638 1.46282461616162 Sec III -0.00275279629630 0.04046075360750 -0.05811292700818 2.94536995959596 Sec III -0.00090196885522 0.01133516847042 0.11669223977874 1.13368107070707 sec IV 0.00073663973064 -0.01036882611833 0.03851385305435 1. 56986452 5252 53 Sec IV -0.00000771548822 0.00007210642136 0.01332185149110 0.04533932323232 Sec IV 0 0 0.01888750000000 0.12320480000000 Sec V 0.00010566666667 -0.00017450000000 -0.03159316666667 0.32357600000000 Sec v 0 -0.00049512500000 0.01268691071429 0.04396853571429 Sec V -0.00017933333333 0.00491050000000 -0.05208416666667 0.25425500000000

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Table 1. Critical Loads and Mode Shapes for Various Sections and K I I .L 1 0 .25 0 .25 0 .7'5 0 .75 [!" I :E :E 11.11 0 .10-1-0 .10-lo-Luo J [__12.00_J L___:_IJ,g) I L--a. lll---1 I .ID Sctllon I Stctlon II Soctlon Ill hcllon IV Slcllon Y S.tllon Yl Leu I ltl :z .. Lou I lol Zd Ill lotonl h. .... ,., ... .... ... ltl Loul "' Zd Locol It' ,,, LlhrI-Tenl111l Ill Local Loloroi-Toroloaal ...... ..... Loco I Iii Local ld a. Locol 0 0.1)3 0 .321 0.1)2 1 .770 l.tl!l I.IU 1 .134 1.581 o.ocs 0 .111 "' 1 o.l'n 0 311 0 .170 0 251 1.331 1.31? 1.128 1.818 1.280 D .ell D.IU D .DU 2 0 .203 0.328 0.110 1.523 1.281 1.183 l.cn 1.810 o.on ..... 0.017 3 0.231 O .SCI o.m 0.112 0.251 1.7Zt 1.251 s.cm 1.184 1.106 0.011 0.111 o.m 4 0.210 0 .371 0.732 O.Zlt 0.201 1.031 1.232 l.ZOC 1.m 1.100 0.000 0.111 0.011 5 O.ZII 0.31& o.m O.ZIWI 1.201 2.143 1.1!21 3.:131 1.114 ''" O.IU O.I'Pt O.Oit 6 0.311 0 .117 o.ne o.ne 0.151 a.w 1 .11%7 s .tss I Otl 1.102 0 .121 0.151 o .oeo 7 o.sn O.UI o nc 0.:101 O .ZOI I.MS 1.233 s aee 1.285 1.1111 0.131 0.1:10 o.oat 8 o.3n 0.110 0.722 0.323 O.ZBI z.m I.ZU 3.1BI 1.:12? .. ..., 0.183 D.IU 0.0011 ----i

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74 Table 2. Uniform Critical Loads for 1=120 0 L = 120 lll qcr 0 1. 770 1 r ""::I 0.846 2 r----. 0.515 3 r---, 0.362 4 0.277 5 D 0.223 0.188 7 0.162 8 0.143

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75 Table 3. Uniform Critical Loads for 1=180 (S L = 180 Ill qcr 0 0.899 1 0.169 2 r-----, 0.379 3 r-------. 0.268 4 'r-----., 0.206 5 0.166 6 0.139 7 0.120 I 8 0.105

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76 Table 4. Uniform Critical Loads for 1=240 0 L = 240 lll qcr 0 0.427 1 I 0.432 2 r 0.287 3 r----.. 0.204 4 r-------. 0.157 5 0127 6 0.107 7 0.092 8 0.081

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77 Table 5. Uniform Critical Loads for 1=300 0 L = 300 lll qcr 0 0.249 1 0.251 2 I 0.226 3 r---. 0.161 4 0.124 5 0. 101 6 0.270 7 0.247 8 1.770

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78 Table 6. Uniform Critical Loads for L=360 lS L = 360 qcr Ill 0 0.219 1 0.164 2 I 0.165 3 r "-::::1 0.167 4 r-------..., 0.168 5 r--:------_, 0.170 6 0.069 7 0.176 8 0.179

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79 Table 7. Uniform Critical Loads for 1=420 Q L = 420 lll qcr 0 0.426 1 0.328 2 I 0.280 3 r 0.109 4 [ ""::::1 0.118 5 r-------_, 0.068 6 0.120 7 0.217 8 0.300

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Table 8. Uniform Critical Loads for L=480 '6 L = 480 lll qcr 0 0.377 1 0.317 2 I 0.464 3 I 0.488 4 I :::J 0.258 ., I t 5 0.344 --. 6 0.445 7 0.461 8 0.389

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APPENDIX A INPUT DATA FILE

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INPUT DATA FDE ============================================================= Tapered Beam analysis.(Done ansys 5.0) Location: CCUDNVR:disk$user5: [thesis.gdemos.thesis]gthesis.dat Date created: Oct., 16, 1993 ============================================================= ============================================================= Preprocessor ============================================================= /prep7 ============================================================= PARAMETERS: ( For the beam properties) l = length of the tapered beam b = the width of the flange (top and bottom flanges have the same width) tf = the thickness of the flange tv = the thickness of the web gamma = the ratio of the height between the both end do = the height of the small end of the beam dl = the height of the top end dl = (1+gamma)do ============================================================= 1=30 do=12.0 b=16.0 tf=.25 tv=.10 gamma=8 dl=(1+gamma)do ============================================================= PARAMETERS: (For ANSYS) r = real (cross-section dimension) et = element type k = keypoint (create keypoints) a = area (define area) lesize = line size 81

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amesh = area mesh (automatically mesh area) mp = material properties ============================================================= et,1,63 mp,ex,1,30e6 mp,nuxy,1,0.3 mp,dens,1,0.0007346 r,1,t r,2,tv ============================================================= Geometry o beam ============================================================= k,1,-b/2 -dl/2 k,2,,,-dl/2 k,3,b/2,,-dl/2 k,4,-b/2, ,dl/2 k, 5 ,dl/2 k,6. b/2. ,dl/2 k