ELASTIC BUCKLING OF TAPERED MONOSYMMETRIC I-BEAMS

By Mark A. Bradford,' Associate Member, ASCE and Peter E. Cuk2

ABSTRACT: A finite-element method of analysis is presented for the elastic flexural-torsional buckling of non-prismatic I-section beam-col-umns. The formulation presents a general approach to the problem, in which the coupling of torsion and bending is simplified by adopting the web mid-height as an arbitrary axis of twist. By making this simplifi-cation, the formulation does not sutler from the restrictions of other solutions, such as the use of uniform elements and finite differences. Buckling stiffness and stability matrices are developed, and these may be readily included in existing finite-element programs. The method is shown to be in good agreement with the more complex finite-integral treatment, and its scope is demonstrated by application to the buckling of tapered cantilevers.

INTRODUCTION Modern techniques of fabricating plated steel members have resulted in

an increased emphasis on the use of nonprismatic tapered elements in steel frames. This is because the member is commonly used in situations where the maximum moment is reached only locally, so that economy can be gained by reducing the member section in the low moment regions.

When a tapered member of compact or semicompact cross section does not have adequate lateral support, its strength is governed by its resistance to flexural-torsional buckling. However, significant economies can still be achieved if the elastic critical load can be determined for the tapered member. While the stability of such members has interested researchers for several decades, there have been few general approaches to the problem contributed (Structural Stability Research Council 1976). This paper presents a simple finite-element approach for determining the elastic critical loads of monosymmetric I-section beam-columns that taper in flange width and thickness and in web depth. This approach is suitable for microcomputers of modest capacity.

Kitipornchai and Trahair (1972) presented a detailed review of research progress in the field prior to 1971. Since then, several contributions considering the flexural-torsional buckling of tapered members have ap-peared. Among these are papers by Lee, Morrell, and Ketter (1972), Home, Shakir-Khalil, and Akhtar (1979), Brown (1981), and Wekezer (1985) on simply supported beams, and by Nethercot (1973) on cantilevers. Design approximations have been proposed by Nethercot (1973a,b) and Taylor, Dwight, and Nethercot (1974) for flange-tapered doubly symmetric I-beams, and by Morrell and Lee (1974) for web tapered beams. Tapered

'Lect. in Civ. Engrg., The Univ. of New South Wales, Kensington, NSW 2033, Australia.

2Assoc. Dir., Wargon Chapman Partners, Sydney, NSW 2000, Australia. Note. Discussion open until October 1, 1988. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on February 27, 1987. This paper is part of the Journal of Structural Engineering, Vol. 114, No. 5, May, 1988. ASCE, ISSN 0733-9445/88/0005-0977/$1.00 + $.15 per page. Paper No. 22414.

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haunches in portal frames have been considered by Home and Morris (1977) and Nakane (1984). In addition, the flexural-torsional buckling of nonuniform columns has been considered by O'Rourke (1977) and Ku (1979), and that of beam-columns by Shiomi and Kurata (1984).

The early applications of the finite-element method to elastic flexural-torsional buckling used one-dimensional line elements of constant cross section (Barsoum and Gallagher 1970; Powell and Klinger 1970). These incorporated the classical assumptions of line-element analysis (Timosh-enko and Gere 1961; Vlasov 1961), including the coincidence of the axis of twist with the shear center axis, which is parallel to the centroidal axis. This assumption allows the decoupling of the out-of-plane bending and torsional resistances of a uniform monosymmetric beam. However, the finite-element model developed in this paper makes no such assumptions about the axis of twist.

Application of uniform elements to tapered monosymmetric beams causes difficulties because of the artificial discontinuities introduced in the centroidal and shear center axes at nodes. Moreover, the rate of conver-gence can be expected to be slow because of the comparatively crude model provided for a tapered element. It is therefore desirable to develop a tapered finite element that will provide an accurate and rapidly converg-ing method of representing a tapered member and that will not introduce any artificial discontinuities.

For a tapered monosymmetric I-beam, Kitipornchai and Trahair (1975) showed that the shear center axis was not parallel to the centroidal axis. Moreover, any external minor axis moments acting in the plane of the nonparallel flanges also exert torques about the shear center axis; thus the minor axis bending and torsion actions cannot be separated. Hence the out-of-plane bending and torsion resistances of a tapered monosymmetric I-beam are interdependent and also cannot be separated.

The finite-element method developed in this paper is considered to be superior to the use of uniform elements, in that it develops a line element that correctly provides for nonuniformity and monosymmetry. This is achieved by abandoning the usual shear center and centroidal axis sys-tems in the development of the finite line element, thereby avoiding the complications of the differential equations that govern the behavior of nonuniform monosymmetric members (Kitipornchai and Trahair 1975). The element therefore uses a convenient and arbitrary x-, y-, and z-axis system passing through the midheight of the web as the reference system for the lateral displacements and twists. The stiffness and stability matrices are easily calculated by making this assumption of an arbitrary axis of twist.

Following development of the stiffness and stability matrices for buck-ling, the accuracy of the method is compared with the solutions of Kitipornchai and Trahair (1972, 1975), which are based on the finite-integral method, and the improved convergence and accuracy is shown by a comparison with a solution using uniform elements (Hancock and Trahair 1978). The scope of the finite-element method is then demonstrated by studying the flexural-torsional buckling of cantilever beams that taper in web depth and flange width.

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FINITE ELEMENT BUCKLING ANALYSIS

General The finite-element discretization used in this paper is similar to that

presented by Barsoum and Gallagher (1970) for prismatic members. For this method, the beam-column is divided into one-dimensional elements, and the stiffness equations for flexural-torsional buckling of the beam-column are developed.

The geometry and axis system of each element of the beam-column is shown in Fig. 1. The z-axis is positioned at the web midheight, and the y-z plane is the plane of symmetry. The initial actions producing out-of-plane instability Plt Ml,Vl,P2,M2, and V2 are applied initially in the plane of symmetry. These actions are increased proportionally by the load factor X until buckling occurs. During buckling, they move with the beam-column, but their planes of action remain parallel to the plane of symmetry.

Displacements The cross section consists of two unequal flanges connected rigidly at

their centerhnes to the web. The cross section does not undergo distortion during buckling. The lateral deflection it and twist (j> of such a cross section during buckling are shown in Fig. 2(a). These deformations can be written in matrix format as

{} = [M]{a} (1) where {it} = (u, (j>) T; [M] = a matrix of cubic interpolation polynomials; and {a} = (a j , . . . , a8)T = a vector of displacement coefficients.

The vector {a} may be related to the vector of nodal displacements {q} = (ux, u2, u'u u2, fa, 4>2, 4>i, '2)T (2) shown in Fig. 2(b) by suitable differentiation and substitution in Eq. 1,

la) (b)

FIG. 1. Geometry of Beam-Column Element: (a) Elevation; (b) Section A-A 979

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Erf (a)

FIG. 2. Element Deformations: (a) Deformations in Plane of Cross Section; (b) Nodal Displacements

so that

{

UT = 2 \ L [EIj(ufi2 + EIB(uBf + EIW{ ,>---). . , . , '..-u-.-; ( 7 )

where h = the distance between the flange centroids. When the web second moment of area /, is considered negligible compared with either flange value IT or IB , Eq. 5 may be written as

w /r = - | [(EIT + EIB)(u'f + (h')2(EIT + EIB)W)2 h2

+ j (EIT + EIBW)2 + 2h'(EIT - EIB)(u"V) + h(EIT - EIB)(u"V)

+ hh'{EIr + EIB)WV) + GJW)2} dz (8) where

J = JT + JB + Jn (9) The terms ", '. and " in Eq. 8 are determined by appropriate differentiation of Eq. 1, so that Eq. 8 can be represented as

UT = \ {qYVtM . (10) where [k] - the element stiffness matrix given in Appendix II. The integration with respect to z is carried out by four-point Gaussian quadra-ture (Zienkiewicz 1971).

Element Stability Matrix The element stability matrix [g] may be obtained by considering the

work done during buckling. Fig. 3(a) shows the x-, y-, z-axis system that is conveniently taken as now being centroidal, so that principal axis cen-troidal properties may be used. The angle of twist is taken at the web midheight 0, and the coordinate y represents the coordinate of the web midheight relative to the centroidal axis system.

During buckling, each longitudinal fiber 8A deflects laterally and be-comes inclined to its original position. The web midheight 0 displaces laterally u to a position 0, , and the general point Q in the cross section displaces laterally u to gi and then rotates through an angle about 0, to the position Q2. The coordinates of Q are (x, y), while those for Q2 can be

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(midheight)

(centroid)

o(o,-y) a,(u,-y)

*-x

Q2 (x+u-(y+y)0,y+x0)

I u-ly+y)0

(a)

(Pre-buckling Displacement)

4 ^ TvB U=0=O U,0*O

(Pre-buckling (Buckled Position) Position)

(b)

FIG. 3. Buckling Displacements: (a) Axis System and Longitudinal Fiber Displace-ment; (b) Buckling Displacement v

obtained by consideration of Fig. 3(a) as [x + u (y + y), y + x4>]. The displacements 8 te and 89), of Q are therefore given by

e* = u - (y + y)4> \ = x The angles of inclination of fiber 8A through Q2 are given by

e - ^ 6^ ~ dz

d8,

u'-(y + y)V

'&1 dz = x'

( ID (12)

(13)

(14)

on noting that the reference axis is now centroidal. When a fiber of length 8z rotates through an angle 0, there is an axial,

shortening 8A given by

8A ='8z Sz cos 8 (15)

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or

8A= ( | e 2 j 8^ (16) on neglecting fourth-order terms in 9.

The work done by the axial load XP can be found by calculating the work done by the axial stresses on the fiber 8A, so that

XP - Vi

2 i o2 SVP = - 8AI 2 % + 2%)dz ( ly) Integrating over the whole member and making use of the centroidal axis properties

f ydA = 0 (18)

f x2 dA = Iy (19)

f y2dA = Ix (20)

Eq. 17 may be written as

VP = \ P XP [{u'f - lyiSV + (r20 + ?)W)2] dz... (21) Jo

where r% = (Ix + Iy)/A, which represents the work done by the axial load P during buckling.

The applied major axis moment \MX causes the monosymmetric section to twist and, when coupled with shears, causes an additional deflection vB in the plane of symmetry during buckling, as shown in Fig. 3(b). The moment creates an axial stress given by

' *Mxy X2) 8z (23)

Integrating Eq. 23 produces

VM = - ^ P \M,p.?(')2 dz + [L \Mxu'V dz (24)

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Curvature u7cosiJ

Finally, the effect of off-axis vertical load is given by

Vfi= - ^ W M ? (30) i

where the load W, is applied at a section / which twists cf),- during buckling, and where d,- = the height of application of this load above 0.

Thus the total work done by the axial forces, shear forces, and bending moment is found by combining Eqs. 21, 24, and 29 to give

VT = x \ fL \P[(u')2 - 2yn'(|> + (ri + fW)2] - \ P \MX\$>W? Jo Jo

+ 2, and 4>' in Eq. 31 are determined by appropriate differentiation of Eq. 1, so that Eq. 31 may be written as

VT = j iqMeM (32) where [g] = the element stability matrix given in Appendix III. As for the stiffness matrix, the integration with respect to z is carried out by four-point Gaussian quadrature (Zienkiewicz 1971).

Stiffness and Stability Matrices for Beam-Column By considering equilibrium and compatibility at the nodes of the

beam-column, the global matrices [K] and [G] can be assembled from [k] and [g], respectively. The finite-element buckling analysis for the complete beam-column can then be represented in matrix form by

([K) -K[G]){Q} = {0} (33) where {Q} = the vector of (global) out-of-plane displacements. The values of X which yield a nontrivial solution for {Q} in Eq. 33 are the eigenvalues, while the corresponding values of {Q} are the eigenvectors. In this paper, the eigenvalue problem is solved using the routines set out by Hancock (1984) for banded matrices.

ACCURACY OF SOLUTION

Doubly Symmetric Tapered Beam The finite-element method was used to calculate the flexural-torsional

buckling loads of a simply supported, doubly symmetric I-beam with a central concentrated load on the top flange. The three cases considered were independent uniform tapers in web depth, flange width, and flange thickness. For this beam, the largest cross section (which was at midspan) had the dimensions h = 72.8 mm, BT~ BB = 31.6 mm, TT = TB = 3.11 mm

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1000

BOO

S" 600

g

400

200

0 02 04 06 0-8 TO

Taper constants h , B , T

FIG. 5. Critical Loads of Doubly Symmetric Tapered l-Beam

and / = 2.13 mm, while the material properties were E = 65,160 MPa and G = 25,650 MPa.

These cases were considered by Kitipornchai and Trahair (1972), and their numerical solutions are compared in Fig. 5 with those of the finite-element method, using four elements over the half-span. In Fig. 5, the beam tapers from dimensions h, BT, TT, and t at midspan to ahh, uBBT, aTTT, and t at both ends. It can be seen from Fig. 5 that the agreement between the two solutions is very good, with the maximum discrepancy being about 1% for depth and thickness tapered beams when the degree of tapering is large.

Monosymmetric Width Tapered Beam The accuracy of the finite-element model for beams with tapered flanges

was further assessed by considering the monosymmetric nonuniform I-beam shown in Fig. 6. The maximum dimensions and material properties were taken as those given in the previous subsection.

Numerical solutions for this problem were presented by Kitipornchai and Trahair (1975). When the top flange is narrow at midspan [Fig. 6(b)], the solutions produced using the finite-element model are the same as those of Kitipornchai and Trahair, but when the top flange is wide at midspan [Fig. 6(a)], significant discrepancies exist between the finite-element solutions and those of Kitipornchai and Trahair. An independent analysis was therefore made of this case by using a stepped, monosym-

Kitipornchai and Trahair (1972) This study

Depth Tapered Width Tapered Thickness Tapered

J l_ _J I

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1000

600

600

a a 2 a

% 400

1

200

0 0-2 0-4 06 08 10

Flange taper constant KB

FIG. 6. Critical Loads of Tapered Monosymmetric l-Beams

metric uniform-element approximation (Hancock and Trahair 1978). The closeness of this finite-element approximation to the results of this paper suggests that there may be some errors in the finite-integral solutions of Kitipornchai and Trahair.

In both Figs. 6(a and b), four elements were used over the half-span for the tapered finite-element idealization. However, twelve elements were used over the half-span for the uniform finite-element (Hancock and Trahair 1978) representation of Fig. 6(a): Demonstration of Convergence

The rate of convergence of the finite-element method of this paper was investigated for a doubly symmetric cantilever that had a web taper ah of 0.167. The cantilever was loaded by a concentrated load on the top flange at its end. This problem was also considered by Brown (1981), and the geometry of the cantilever is given in his paper. Fig. 7 plots the dimen-sionless buckling load WJ2/V(EIyGJ)0 against the number of elements, where the subscript

0 refers to the values at the deepest section. It can be seen that convergence of the tapered element model is very rapid, with the solution remaining constant to four significant figures with four or more elements.

On the other hand, the problem was analyzed using a stepped, uniform-element approximation (Hancock and Trahair 1978). The comparison of convergence using uniform elements with that of the tapered element approach in Fig. 7 clearly demonstrates the efficiency of the method of this

This study Kitipornchai and Trahair (1975) Hancock and Trahair (1978)

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2 3 4 5 6

Number of elements

FIG. 7. Convergence of Buckling Loads for Cantilever

paper. It is of interest to note that while the two finite-element methods are reasonably close when a large number of elements are used, neither solution is consistent with the value of the dimensionless buckling load of 5.32 given by Brown for this problem.

DESIGN OF TAPERED CANTILEVERS

The finite-element method of this paper has been used to obtain dimensionless elastic critical loads for tip-loaded cantilevers. The two cases considered were flange tapering (Fig. 8) and web tapering (Fig. 9). In these figures, the dimensionless buckling load WcI2/V(EIyGJ)0 is plotted against the beam parameter

K = Eh GJ (34)

where EIm = the warping rigidity, and the subscript 0 = the rigidities at the largest section. Both top flange loading (2d/ahh = -1) and centroidal loading (2a/ahh = 0) were considered. The buckling differential equations that apply to this problem as derived by Kitipornchai and Trahair (1972) suggest that K, a, and the taper constants aB and a,, are the independent parameters for the buckling load.

Fig. 8 shows that the buckling load for a cantilever with flange tapering decreases markedly as the degree of tape the effective

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'. ; ELEV

"T j-u

Beam Parameter K= ^ / (E I W /QJ !

FIG. 8. Buckling Loads for Flange-Tapered Cantilevers

10.

a

-

-

h _ , ELEV.

PLAN.

.-/. ,. 2a /K h h = 1

J0rl'^'

i-a

ZZZzz

T y 10

' / 075 - ^ / ^ 0-50

^ 0 7 5

0 1 2 3

Beam Parameter K = tL/lEIw/GJ)

FIG. 9. Buckling Loads for Web-Tapered Cantilevers 989

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Beam Parameter K= fVlEIw/OJ)

FIG. 10. Effect of Load Height on Buckling Load

rigidities are decreased most in the region where the buckling deformations are largest. The effect is greater for centroidal loading than for top flange loading. On the other hand, the effect of web tapering on the critical load for centroidal loading (Fig. 9) is much less than for flange tapering.

The influence of the height of application of the load on the cantilever tip is shown in Fig. 10, in which the ratio

(Wc)2a/a,,h = - 1 (Wc)2a/ahl, = o

is plotted against the beam parameter K. It can be seen that the ratio of the critical load for top flange loading to that for centroidal loading increases generally as the tapers aB and ah decrease. This indicates that the detrimental effect of placing the load above the shear center decreases as the tapering becomes more severe. This is particularly so for web tapered beams, since the height of application of the load decreases as ah decreases.

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CONCLUSIONS

A finite-element method of analysis has been developed that is capable of making accurate predictions of the elastic buckling loads of tapered, monosymmetric I-section beam-columns. The element is unique in that it uses an arbitrary axis system passing through the midheight of the web as the reference system for lateral displacements and twists. By abandoning the usual shear center and centroidal axis representations, the compli-cations of the differential equations that govern the behavior of nonuniform monosymmetric members are avoided.

The accuracy of the method is shown by a comparison with independent solutions, and its rapid convergence when compared with a finite-element representation that uses uniform elements is shown. Finally the scope of the finite-element method is demonstrated by obtaining dimensionless buckling loads for tip-loaded, tapered cantilevers.

The tapered element may therefore be employed to study parametrically the stability of tapered beams, columns, and beam-columns. Because of the relatively simple formulation of the stiffness and stability matrices and its rapid convergence, the method is particularly suitable for microcom-puter applications.

APPENDIX I. DISPLACEMENT MATRICES

[A/] Matrix u\_\L Li Lg Lg 0 0 0 0 / " [o o o o i a1 e W (36)

z (37)

[C]"1 Matrix

W = [C,,]-1 [0]

to] [ A r 1 {?}

[0,,]-'=[

Nonzero terms in symmetric [k*]: 4EL

^ 3 = ^ - , . (41)

/C3*,4 = - ^ (42)

2EI,h' kU = n- (43)

4EIJt'i 2EImh kh = ir^+^r w

6EImh'e t 6EI,M kt& = JJ + Li (45)

36/ L kU = jP- (46)

kt^g-1 (47)

\2EImh'i2 6EImhi kh = Z ^ + ^ (48)

lSEImh'e , l*EImh? kfa = JJI + j i (49)

, ^ Elph'2 GJ kte = -jj- + JJ (50)

2EIph% Elphh' 2GJJ kh = JT. +

Li + jTT (51)

, lEIph'2? -hElphh'i 3GJ? ks= PLi + "Li +^r- (52)

_ 4EIph'\2 AEIphh'i EIph2 4GJg kh- pLi + pLi + ~ l r + - [ r (53)

6EIph'2e ^ 9EIphh'e 3EIPh2Z, 6Gjg kU = JT + J} + [A + -JT- (54)

9EIph'2^ lSEIphh'l-3 9EIph2^2 9GJ^

Elp=EIT + EIB (56) 992

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EI, = EIT-EIB (57) z Z = l (58)

APPENDIX III. ELEMENT STABILITY MATRIX

-(I L 1 [gl = [CrT[ I lg*l dz\[Crl (59) Nonzero terms in symmetric [g*]:

gi,2=P (60) gh = 2Pi (6i) gt4 = 3Pe (62) gt6=-Si (63) gb= - 25 ,5 (64) 8ti= - 35*,2 (65) 3*3 = 4P2 (66) gf,4 = 6P& (67) gts=--lM . . . . . . . . . . . ..(68) gf,6 = - 25,5 - 2MC (69) 3*7 = - 45,52 - 2Mi2 (70) S3*8 = - 65,53 -2Me .....'. (71) gtA = 9Pi? ......... (72) gts= -6M (73) gt6 = - 35,52 - 6M2 . . . . . . (74) gfr = - 65,53 -6M53 . . . ; . . . . (75) gU = - 95,54 - m (76) ge = S2 , (77) g7 = 2S2ii... :.....:,...... ..(78) gs = 3S2e . . . . : . . . . . . ; (79) gh = 4s2e -.;; (so)

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gf,s~6S2e (81) gts = 9S2? (82)

Mx M = -j- (83)

Py Sx = -[ (84)

c [(>j + y2)P-B*Mxl 02 = T5 (->)

4 (86) APPENDIX IV. MONOSYMMETRY PARAMETER (3*

M = TJ{h-y)

i V R T , ( ,.2 , ih-yft

- j ^ - + BB TB(h - yf + -^

-y BTTT

,-,2 ?< 12 + BTTTy + + 2y (87)

APPENDIX V. REFERENCES

Barsoum, R. S., and Gallagher, R. H. (1970). "Finite element analysis of torsional and torsional-flexural stability problems." Int. J. Numer. Methods Eng., 2, 335-352.

Brown, T. G. (1981). "Lateral torsional buckling of tapered I-beams." J, Struct. Div., ASCE, 107(ST4), 689-697.

Cuk, P. E. (1984). "Flexural-torsional buckling in frame structures," thesis presented to the University of Sydney, at Sydney, Australia, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Hancock, G. J. (1984). "Structural buckling and vibration analyses on microcom-puters." Civ. Engrg. Trans., Institution of Engineers, Australia, CE26, 327-332.

Hancock, G. J., and Trahair, N. S. (1978). "Finite element analysis of the lateral buckling of continuously restrained beam-columns." Civ. Engrg. Trans., Insti-tution of Engineers, Australia, CE20, 120-127.

Home, M. R., and Morris, L. J. (1977). "The design against lateral stability of haunched members restrained at intervals along the tension flange." Proc. of Second International Colloquium on Stability, Washington, D.C., 618-629.

Home, M. R., Shakir-Khalil, H., and Akhtar, S. (1979). "Stability of tapered and haunched members." Proc, Inst. Civ. Engrg., London, U.K., 67, Part 2, 677-694.

Kitipornchai, S., and Trahair, N. S. (1972). "Elastic stability of tapered I-beams." J. Struct. Div., ASCE, 98(ST3), 713-728.

Kitipornchai, S., and Trahair, N. S. (1975). "Elastic behavior of tapered mono-symmetric I-beams." J. Struct. Div., ASCE, 101 (ST8), 1661-1678.

Ku, A. B. (1979). "Buckling of non-uniform columns." Proc. of Third Engineering Mechanics Division Specialty Conference, ASCE, University of Texas, Austin, Tex., 240-243.

Lee, G. C , Morrell, M. L., and Ketter, R. L. (1972). "Design of tapered 994

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members." Bulletin No. 173, Welding Research Council. Morrell, M. L., and Lee, G. C. (1974). "Allowable stress for web tapered beams."

Bulletin No. 192, Welding Research Council. Nakane, K. (1984). "The design for instability of non-uniform beams." Proc, 9th

Australasian Conference on Mechanics of Structures and Materials, Sydney, Australia, 18-22.

Nethercot, D. A. (1973). "Lateral buckling of tapered beams." Publications, IABSE, 33-11, 173-192.

Nethercot, D. A. (1973). "The effective length of cantilevers as governed by lateral buckling." Struct. Engr., 57(5), 161-168.

0'Rourke,M. (1977). "Buckling loads for non-uniform columns." Comput. Struct., 7(6), 717-720.

Powell, G., and Klinger, R. (1970). "Elastic lateral buckling of steel beams." J. Struct. Div., ASCE, 96(ST9), 1919-1932.

Shiomi, H., and Kurata, M. (1984). "Strength formula for tapered beam-columns." J. Struct. Engrg., ASCE, 110(7), 1630-1643.

Structural Stability Research Council. (1976). Guide to stability design criteria for metal structures. John Wiley and Sons, New York, N.Y.

Taylor, J. C , Dwight, J. B., and Nethercot, D. A. (1974). "Buckling of beams and struts: proposals for a new British code." Proc, Conference on Metal Structures and the Practicing Engineer, Melbourne, Australia, 27-31.

Timoshenko, S. P., and Gere, J. M. (1961). Theory of elastic stability. McGraw Hill, New York, N.Y.

Vlasov, V. Z. (1961). Thin walled elastic beams. 2nd ed., Israel Program for Scientific Translation, Jerusalem, Israel.

Wekezer, J. W. (1985). "Instability of thin walled bars." J. Engrg. Mech., ASCE, 111(7), 923-935.

Zienkiewicz, O. C. (1971). The finite element method in engineering science. McGraw Hill, New York, N.Y.

APPENDIX VI. NOTATION

The following symbols are used in this paper:

A a [C]

E,G ITJB

hJy L J

JT JB >JW K

[K],[G] [kite!

L I

M Mx P {Q}

M rl

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

area of section; height of application of load above zero; matrix in Eq. 3; elastic and shear moduli; minor axis second moments of area of top and bottom flange; major and minor axis second moments of area; warping section constant; Jj- + JB + Jw ; torsion constants for top flange, bottom flange, and web; beam parameter in Eq. 34; global stiffness and stability matrices; element stiffness and stability matrices; length of element; length of beam; interpolation matrix; bending moment; axial force; global degrees-of-freedom; element degrees-of-freedom in Fig. 2; (/, + Iy)IA\

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uT,vT {} ,4>

Vy w

x,y,z y

ah ,aB ,aT {0} PJf

e X

=

=

=

=

=

=

=

=

=

=

=

=

total strain energy and work done; (uAf; buckling deformations in Fig. 2; shear force; transverse load; Cartesian axis system; coordinate of midheight of web relative to centroid depth, width, and thickness tapers; interpolation polynomials; monosymmetry parameter; load height parameter in Eq. 35; and buckling load factor.

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