ONE-STOREY TWO-BAY FRAMES (WITH MIDDLE COLUMN STIFFNESS DOUBLED)
corresponding to Lh/Lv=1, reveals that by doubling the bending stiffness of the middle column the total sway-buckling load is increased by approximately 33%, regardless of the relative stiffness of the horizontal bars. Another important observation is that the ratio of axial forces (inside to outside, P2/P1; ) is not affected appreciably by the doubling of the bending stiffness of the middle column. This ratio varies (increases) with increasing bending stiffness of the horizontal bars.
All of the above observations indicate that there exists an optimum distribution of bending stiffness, in multi-bay multi-storey orthogonal frames which are subject to sway buckling, for maximising their load carrying capacity.
5.5 CONCLUDING REMARKS
From the several studies performed on elastic orthogonal plane frameworks, some of which are reported herein, one may draw the following general conclusions.
(1) The effect of flexible joint connections (bolted, riveted and welded connections are flexible rather than rigid) on the frame response characteristics is negligibly small.
Hence, assuming rigid connections in analysing elastic plane frameworks will lead to accurate predictions.
(2) Eccentrically loaded two-bar frames lose stability through the existence of a limit point and do not experience bifurcational buckling. For these frames, the slenderness ratio of the bars has a small but finite effect on the critical load. Moreover, depending on the value for the slenderness ratio, there exists a critical eccentricity which divides the
response of the frame into two parts. On one side the response is characterised by stable equilibrium positions and on the other side it exhibits limit point instability (within the limitations of the theory, ).
(3) Unbraced multi-bay multi-storey frames (including portal frames) are subject to bifurcational (sway) buckling with stable post-buckling behaviour. Sway buckling takes place when the frame is structurally symmetric and the load is symmetric. Because of this, the frame is insensitive to geometric imperfections regardless of the type (load eccentricity, variation in geometry: length, stiffness, etc.). In many respects, the behaviour of these frames is similar to the behaviour of columns, especially cantilever columns.
(4) The effect of slenderness ratio on the nondimensionalised response characteristics of plane frameworks (except the two-bar frame) is negligibly small.
(5) Starting with a portal frame, addition of bays increases appreciably the total sway-buckling load, while addition of storeys has a very small effect.
(6) For multi-storey frames, distributing the load in various amounts among the different floors does not alter appreciably the total sway-buckling load. In all cases, the first storey vertical bars (columns) carry the total load.
REFERENCES
ACKROYD, M.H. (1979) Nonlinear inelastic stability of flexibly-connected plane steel frames.
PhD Thesis, University of Colorado.
ARGYRIS, J.H. and DUNNE, P.C. (1975) On the application of the natural mode technique to small strain large displacement problems. Proceedings of World Congress on Finite Element Methods in Structural Mechanics, Bournemouth, UK.
BAKER, J.F., HORNE, M.R. and RODERICK, J.W. (1949) The behavior of continuous stanchions. Proc. Roy. Soc. A., 198, 493.
BATHO, C. and ROWAN, H.C. (1934) Investigation on beam and stanchion connections. Second report of the Steel Structure Research Committee, HMSO, London.
BELLMAN, R. (1969) Perturbation Techniques in Mathematics, Physics and Engineering, Holt, Rinehart and Winston, New York.
BLEICH, F. (1919) Die Knickfestigkeit elasticher Stabverbindungen. Der Eisenbau, 10, 27.
BLEICH, F. (1952) Buckling Strength of Metal Structures, McGraw-Hill, New York.
BRITVEC, S.J. (1973) The Stability of Elastic Systems, Pergamon Press, New York.
BRITVEC, S.J. and CHILVER, H.A. (1963) Elastic buckling of rigidly jointed braced frames. J.
Eng. Mech. Div., ASCE, 89, EM6, 217.
BROWN, K.M. (1969) A quadratically convergent newton-like method based upon gaussian elimination. SIAM Journal on Numerical Analysis, 6(4), 560.
CHILVER, A.H. (1956) Buckling of a simple portal frame. Journal of Mechanics and Physics of Solids, 5, 18.
CHWALLA, E. (1938) Die Stabilitat lotrecht belasteter Rechteckrahmen. Der Bauingenieur, 19, 69.
DEFALCO, F. and MARINO, F.J. (1966) Column stability in type 2 construction. AISC Engineering Journal, 3(2), 67.
FRYE, J.M. and MORRIS, G.A. (1975) Analysis of flexibly connected steel frames. Canadian Journal of Civil Engng, 2, 280.
GOLDBERG, J.E. (1960) Buckling of one-storey frames and buildings. Journal of the Structural Division, ASCE, 86, ST10, 53.
HALLDORSSON, O.P. and WANG, G.K. (1968) Stability analysis of frameworks by matrix methods. Journal of the Structural Division, ASCE, 94, ST7, 1745.
HORNE, M.R. (1962) The effect of finite deformations in elastic stability of plane frames. Proc.
Roy. Soc. A., 266, 47.
HORNE, M.R. and MERCHANT, W. (1965) The Stability of Frames, Pergamon Press, London.
HUDDLESTON, J.V. (1967) Nonlinear buckling and snap-over of a two-member frame.
International Journal of Solids and Structures, 3(6), 1023.
JOHNSTON, B.G. (1976) Guide to Stability Design Criteria for Metal Compression Members, (3rd edition) John Wiley and Sons, Inc., New York.
KOITER, W.T. (1945) On the stability of elastic equilibrium. (English Translation NASA TT-2 F-10833), Thesis, Polytechnic Institute, Delft, The Netherlands.
KOITER, W.T. (1966) Postbuckling analysis of a simple two-bar frame. Recent Progress in Applied Mechanics, Almquist and Wiksell, Stockholm, p. 337.
KOUNADIS, A.N., GIRI, J. and SIMITSES, G.J. (1977) Nonlinear stability analysis of an eccentrically loaded two-bar frame. Journal of Applied Mechanics, 44(4), 701.
LEE, S.-L., MANUEL, F.S. and Rossow, E.C. (1968) Large deflections and stability of elastic frames. Journal of the Engineering Mechanics Division, ASCE, 94, EM2, 521.
LIVESLEY, R.K. (1956) Application of electronic digital computers to some problems of structural analysis, Structural Engineer, 34(6), 161.
LOTHERS, J.E. (1960) Advanced Design in Structural Steel, Prentice-Hall, Inc., New Jersey.
LUI, M.E. and CHEN, W.F. (1983) End restraint and column design using LRFD. Engineering Journal, AISC, First Quarter, 29.
MASUR, E.F., CHANG, I.C. and DONNELL, L.H. (1961) Stability of frames in the presence of primary bending moments. Journal of the Engineering Mechanics Division, ASCE, 87, EM4, 19.
McMINN, S.J. (1962) Matrices for Structural Analysis, John Wiley and Sons, Inc., New York.
MERCHANT, W. (1954) The failure load of rigid-jointed frameworks as influenced by stability.
Structural Engineer, 32(7), 185.
MERCHANT, W. (1955) Critical loads of tall building frames. Structural Engineer, 33(3), 84.
MONCARZ, P.D. and GERSTLE, K.H. (1981) Steel frames with nonlinear connections. Journal of the Structural Division, ASCE, 107, ST8, 1427.
MÜLLER-BRESLAU, H. (1908) Die Graphische Statik der Bau-Konstruktionen (Vol. II), Kroner, Berlin.
NELDER, J.A. and MEAD, R. (1964) A simplex method of function minimization. Computer Journal, 7, 308.
OBRECHT, H., WUNDERLICH, W. and SCHRODTER, V. (1982) Large deflections and stability of thin-walled beam structures. Stability in the Mechanics of Continua (Ed. by F.H.Schroeder), Springer-Verlag, Berlin, 165.
OLESEN, J.F. and BYSKOV, E. (1982) Accurate determination of asymptotic postbuckling stresses by the finite element method. Computers and Structures, 15(2), 157.
PRAGER, W. (1936) Elastic stability of plane frameworks. Journal Aeronaut. Sci., 3, 388.
QASHU, R.K. and DADEPPO, D.A. (1983) Large deflection and stability of rigid frames. Journal of the Engineering Mechanics Division, ASCE, 109 , EM3, 765.
RATHBUN, J.E. (1936) Elastic properties of riveted connections. Transactions of American Society of Civil Engineers, 101, 524.
REINHOLDT, W.C. (1974) Methods for Solving Nonlinear Equations, Heyden, London.
ROORDA, J. (1965) Stability of structures with small imperfections. Journal of the Engineering Mechanics Division, ASCE, 91, EM1, 87.
SAAFAN, S.A. (1963) Nonlinear behavior of structural plane frames. Journal of the Structural Division, ASCE, 89, ST4, 557.
SEWELL, M.J. (1965) The static perturbation technique in buckling problems. J. Mech. Phys.
Solids, 13, 247.
SIMITSES, G.J. (1976) An Introduction to the Elastic Stability of Structures, Prentice-Hall, Inc., New Jersey.
SIMITSES, G.J. and GIRI, J. (1982) Non-linear analysis of unbraced frames of variable geometry.
International Journal of Non-Linear Mechanics, 17(1), 47.
SIMITSES, G.J., and KOUNADIS, A.N. (1978) Buckling of imperfect rigid-jointed frames.Journal of the Engineering Mechanics Division, ASCE, 104, EM3, 569.
SIMITSES, G.J. and VLAHINOS, A.S. (1982) Stability analysis of a semi-rigidly connected simple frame. Journal of Constructional Steel Research, 2(3), 29.
SIMITSES, G.J., KOUNADIS, A.N. and GIRI, J. (1977) Nonlinear buckling analysis of imperfection sensitive simple frames. Proceedings, International Colloquium on Structural Stability under Static and Dynamic Loads, Washington, DC, May 17–19, 1977. Published by ASCE, 1978, p. 158.
SIMITSES, G.J., GIRI, J. and KOUNADIS, A.N. (1981) Nonlinear analysis of portal frames.
International Journal of Numerical Methods in Engineweering, 17, 123.
SOUROCHNIKOFF, B. (1949) Wind stresses in semi-rigid connections of steel framework.
Transactions of American Society of Civil Engineers, 114, 382.
SWITZKY, H. and WANG, C. (1969) Design and analysis of frames for stability. Journal of the Structral Division, ASCE, 95, ST4, 695.
VLAHINOS, A.S. (1983) Nonlinear stability analysis of elastic frames, PhD Thesis, Georgia Institute of Technology, Atlanta.
WEAVER, W., JR. and GERE, J.M. (1980) Matrix Analysis of Framed Structures, D.Van Nostrand Co., London, 417.
WILLIAMS, F.W. (1964) An approach to the non-linear behavior of the members of a rigid jointed plane framework with finite deflections. Quarterly Journal of Mechanics and Applied
Mathematics, 17, 451.
ZIMMERMANN, H. (1909) Die Knickfestigkeit des geraden Stabes mit mehreren Feldern.
Sitzungsberichte der preussischen Akademie der Wissenschaften, 180.
ZIMMERMANN, H. (1910) Die Knickfestigkeit der Druckgurte offener Brucken, W. Ernst und Sohn, Berlin.
ZIMMERMANN, H. (1925) Die Knickfestigkeit der Stabverbindungen, W.Ernst und Sohn, Berlin.
ZWEIG, A. and KAHN, A. (1968) Buckling analysis of one-storey frames. Journal of the Structural Division, ASCE, 94, ST9, 2107.