190 C. Haojun and W. Jiqing assuming the approximate deflection, y=Ssin('rrz/2H). 4. DISCUSSION Consideration of Eqn. 3.26 leads to (1) If the stiffness of beams are equal to zero, that is, EbIb=0, then Kb=12Eblb/hl=0 and Eqn. 3.26 becomes (qH)cr = Fc _ :rt "2 E cI ~ 8.299Ec1~ 0.29 ~ - 0.297 x ~" H 5- = H 2 (4.1) The exact value for Kb=0 is 7.837Ej~ (qH)cr = H 2 (4.2) Comparison of Eqn. (4.1) with Eqn. (4.2) leads to the error less than 6%. (2) If the stiffness of beams tends to infinite, that is, EbIb ~ o0, then Kb-~ oo and Eqn. 3.27 becomes (qH)~r = (F 0 + Fc)= 8.299(E~/~ + EcI r) 0.297 H 2 (4.3) The exact value for Kb > oo is (qH)~ r = 7.837(Ecl c + Eclr) H 2 (4.4) (3) Eqn. (3.27) may be also written in the form (qH)cr = Fc + K b + KbF ~ / F o rO+X /Fo) If it is taken that 3,=0.315, a more accurate value of the critical load may be obtained. (4.5) References 1 Zalka, K.A. and Armmer, G.S.T. (1992). Stability of Large Structure, Butterworth Heinmann Ltd. 2 Bao Shihua and Fang Ehua. (1994), Structural Design of Tall Building. Qinhua Publishing Housing, Beijing. 3 Chen Haojun. (1996), Stability Analysis of Multistory Framework under Vertical Loading. Proceedings of International Conference on Advances in Steel Structures, 257-262. 4 Timoshenko, S.E and Gere, S.M. (1958), Theory of Elastic Stability, Chinese Science Publishing House. Space Structures This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank STUDIES ON THE METHODS OF STABILITY FUNCTION AND FINITE ELEMENT FOR SECOND - ORDER ANALYSIS OF FRAME STRUCTURES S. L. CHAN and J.X. GU Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, CHINA ABSTRACT Imperfect beam-column element for second-order analysis of two- and three-dimensional frames is derived in this paper. Initial imperfection of element is restricted to a curvature in the form of a single sinusoidal half-wave. Force deformation equations and tangent stiffness matrix in Eulerian local coordinate system have been obtained using stability function method, as an extension of Oran's equations for straight element. Comparison is made between the present element and the cubic Hermit element by two numerical examples. The obtained results show accuracy and practicality of presented beam-column element. KEYWORDS Steel Frame, Structural analysis, Initial Imperfection, Finite element methods, Stability function method, Second-order analysis, Geometric nonlineality INTRODUCTION Second-order nonlinear analysis of steel frame has been studied extensively over the past few decades and is referred in modern design codes of practice such as the American Load and Resistance Factor Design (LRFD) specification (1986), the Australian Standard 4100 (1990) and the British Standard 5950 (1990). The finite element method and the method of stability function are the two main approaches. The simplest and most typical stiffness matrix method of analysis is to extend the cubic Hermite element to the nonlinear case by inclusion of the geometric stiffness to the linear stiffness matrix to form the tangent stiffness matrix. This approach has been used by many researchers( for example, Barosum and Gallagher[ 1970], Meek and Tan[ 1984], and Chan and Kitipornchai[ 1987] ) and have been quite successful. However, the result by using a single element for each member was noted to 193 194 S.L. Chan and J.X. Gu be somewhat different from the more accurate equilibrium curve obtained by using more elements. Using a single cubic element for each member has been demonstrated by So and Chan (1991) to contain an error of more than 20% for the simple case of a column with both ends pinned. The cause is due to the displacement function independent of the axial force, thus violating the equilibrium condition along the element. Albermani and Kitipornchai (1990) proposed an improved analysis technique to allow less elements to be used via the addition of some terms for large displacement effects. Izzuddin (1991) suggested the use of a higher order element for highly non- linear analysis of frame. The element, although reported to be more accurate than the cubic Hermite element, does not consider the inter-dependence of the axial force and the element displacement. Recently, Chan and Zhou (1994, 1995) developed a pointwise equilibrating polynomial (PEP) element for slender frames. Their element includes initial imperfection and good results were obtained for second-order analysis using a single element to model each member. As an exact solution of the beam-column, the method of Stability function has been widely studied (Livesley and Chandler [ 1956], Oran [ 1973], and Chen and Lui[ 1987]). The method develops the element matrix by solving the differential equilibrium equation of a beam-column under the action of axial load. Unlike the finite element approach, which assumes a displacement shape function. The accuracy of the analysis using stability function is affected only by the numerical truncating error. Although it has the disadvantage of inconsistency in stiffness expression and numerical problem when the axial force is close to zero, it enables only one beam-column element per member to capture the second-order effect. Satisfactory accuracy can generally be achieved without resorting to a fine discretization. Therefore, it can be used for analysis of structures accurately and economically. McConnel (1992) proposed force deformation equations for initially curved laterally loaded beam column, but his element was only for compression and the tangent stiffness matrix was not derived. This paper presents an exact beam-column element allowing for second-order effect due to axial force and initial imperfection. Force deformation equations and tangent stiffness matrix in Eulerian local coordinate system have been obtained using the stability function method, as an extension of Oran's (1973) equations for straight element. Whilst all codes require the consideration of initial imperfection and the equivalent notational force is difficult to quantify, straight element may not be useful in practice. The correctness and effectiveness of presented beam-column element are demonstrated by several numerical examples. ASSUMPTIONS The present theory is based on the assumption of Timoshenko's beam-column. The cross-section of the element is doubly symmetric and the material is linearly elastic. The applied loads are conservative and nodal. Shear deformations and warping effects are neglected. Small strain but arbitrarily large deflections is considered. The initial shape of the element is assumed to in a half sine curve as follows, 9 7/'x V0 Vmo sm ~ (1) L in which v0 is the lateral initial imperfection, Vmo is the magnitude of imperfection at mid-span, x is the distance along the element longitudinal axis and L is the element length. Stability Function and Finite Element for Second-Order Analysis FORCE DEFORMATION EQUATIONS 195 For a given axial compressive force, the equilibrium equation along the element length can be expressed as, EI ~=- dzV1 P(vo + VI) + M1 + M2 x- M~ (2) dx 2 L in which EI is the usual flexural rigidity of the element, M1 and M2 are the nodal moments and vl is the lateral displacement induced by loads. Making use of the boundary conditions that when x=0 and x=L, v 1=0, we have, [jr] M~ sin(c~-kx) L-x M2 sinkx x + q . :,rx Vl V s~n~z~ L KL~ L 1-~q Vm~ (3) Superimposing the deflection to the initial imperfection, we have the final offset of the element centroidal axis from the axis joining the two ends of the element as, V VI + VO L-x I Esin xl+• sin 'X P ' sin~) -~ - sin~ L 1-q vm~ (4) in which, ~-~ P PL 2 k = P 9 r q Per ~ EI ' ' 5 (5,6,7) Pcr is the buckling axial force parameter given by Pcr- 27 "2 EI L 2 Differentiation Eqn. 3 with respect to x, and expressing the rotations at two ends as the nodal d Vl . dvl rotations as, ~ x=0 = 01 ~ x=L = 02, we have, Eli Vmo ] MI= L C, Ol +c202 +cOl L ~ (8) EII (VII Mz= L c281 +clO2-c0 (9) Axial strain can be expressed in terms of the nodal shortening, u and the bowing due to initial imperfection and deflection as u 1 r.vo [ ] = ~ + ;L 1 dv L 2 l_dx ] 2]'L ~X dx (10) 196 S.L. Chan and J.X. Gu Vmo - P = EA s = EA - bl (8~ + 82 )2_ bE (81- 82 )2_ bus-Z- (8~- 82)- bvv (11) In Eqn. 8, 9 and 11, Cl, C2 and co are stability functions and bl, b2, bvs and bvv are curvature functions. They are required to be derived for the case of positive, zero and negative values of axial force parameter, q. in which, Cl, c2, bl, bE are correspondent to the terms by Oran (1973). The term co, can be expressed in terms of q, as, za/r r - - for compression, q>0 (12) c~ (1-q)(1-cosr ' c 0 = 0, for no axial force, q=0 (13) Co =- nqgsinhg/ , for tension, q<0 (14) (1 - q)(cosh ~- 1) bvs and bvv, can be expressed in terms of q, el, C2 and co as follows, bvs= cl-c~ + c2co (15) n"(1 - q)2 2(c~ + c2)(c~ - c2) BVV = 2 z2q(2-)q) + 2Co )2 + C2Co )2 (16) 4(1 - q n'(1 - q 2(Cl + c2)(cl - c2 the axial force parameter, q, can be written as, ~2Fu )2 )2 VmO ] q=~-L~-b~(Ol+02 -b2(0,-02 -bvs ~(O~-O2)-bvv(~) 2 (17) in which ~. is the slendemess ratio given by A= L/~I//A . TANGENT STIFFNESS MATRIX To complete the procedure for the Newton-Raphson type of incremental-iterative method, the tangent stiffness is required to formulate for the prediction of displacement increment subjected to an incremental force. Defining [Fi] and [ui] as the basic nodal variables at two ends of an element, we have, [F] = [M,, M2, p]T (18) [U] -" [01, 02, U] T (19) The tangent stiffness equation for the incremental forces and displacements can then be written as, Stability Function and Finite Element for Second-Order Analysis [AF] =[ke][Au] in which the element tangent stiffness matrix in local coordinate system is obtained from, OFi OFi Oq kij = +~~ Ouj c3q Ouj Operating Eqn. 21, we have the following entries for the tangent stiffness matrix, 0q_ Gl 0q_ G2 0q 1 9 o _ ' 2 ' 001 x2H 002 X H 0u LH in which, 197 (20) (21) (22,23,24) t t t VmO t Vm0 Gl = Cl 0~ + c2 02 + Co ~; G2 = c2'01 + Cl'02-co L L (25, 26) 2 =7~ +b1,(01+02)2 , )2 ,Vmo/o , VmO 2 H ~ + bz (0l- 02 "+ bvs "-~\U1- 02) "+ b,,v ( L ) (27) The resulting tangent stiffness matrix about a principal axis can be determined as, EI [ke ] = L G12 G1G 2 GI C l + 2 H C2 + 2 H rt rt LH G1G z G2 2 G 2 C 2 + C 1 + rc2H ~:2H LH 2 G l G 2 71; LH LH L2H _ (28) Eqn.28 can be very easily extended to three-dimensional space by repeating the process for the other principal axis. The element-stiffness matrix, [ke] in Eqn. 28 will than be 6 by 6. The tangent stiffness in the local coordinate system can be evaluated as, [kE] = [T][ke ][T ]T + [N] (29) in which [T] is the internal to external transformation matrix relating the six independent internal force and moments to the external 12 forces and moments, [ke] is the element tangent stiffness matrix derived above and [N] is the matrix allowing for work done due to nodal displacements and initial stress (Ho and Chan, 1991). The complete element stiffness matrix in global coordinate system, [kG], can finally be determined as, [kG] = [L][kE][L ]T (30) in which [L] is the standard local to global transformation matrix (Gere and Weaver,1965). 198 NUMERICAL EXAMPLES S.L. Chan and J.X. Gu The derived element stiffness matrix is incorporated into the computer program NAF-NIDA (Chan, 1999). In smaller scope where the axial force is close to zero, the stability functions are expressed by interpolation in order to avoid numerical instability. The bucking behavior of column with both ends fixed is used to verify newly presented element. Fig. 1 shows the response of the column due to axial load and different value of initial imperfection. The load versus axial shortening curves for column obtained by the present single element and by 8 cubic straight elements are very close. The error arises since a smoothly curved member is replaced by eight segments. 4.0 0.001 rib3.0 1 present element o" ,-" 2.0 8 cubic elements o o o ~ P ~ ~[ "~P .~ 1.0 E=le7, 1=0.8333, A=I, L=100 (units: Ib, inch) 0.0 I t i 0.0 0.5 1.0 1.5 2.0 Axial shortening, u (in) Figure 1. Buckling Analysis of fixed-fixed Column The second example is a 90 member hexagonal shallow dome, and its dimensions and properties are shown in Fig.2. Members with initial imperfections of various magnitudes and in the direction of the deflection caused by the external loads are assumed and the dome is analyzed. Their load deflection curves for these imperfections are plotted in Fig.2. In all cases, only a single element is used to model a member. It can be seen in the Figure that the cubic element over estimates considerably the buckling load of the structure than the presented element due to member under high axial force. Another observation gained from the analysis is that imperfection affects the buckling loads of the structure. When a member has a larger initial imperfection, the buckling load of the complete structure is reduced significantly. This observation cannot be found in bifurcation type of analysis. CONCLUSION Methods for analyzing large deflections and stability of frame structures in the past have been based on either the finite element approach or the stability function. The cubic finite element is inaccurate when a single element is used to model member under high axial load. The stability function is assumed straight in previous work. The exact stiffness matrix of an imperfect member under large Stability Function and Finite Element for Second-Order Analysis 199 29.0~ 86.9 ~~/~ ,58., ~~~. 624 7200 Unit: mm 9 Loaded node E = 1.95e5 N/mm 2 Iy = 1.44e4 mm 4 G = 0.80e5 N/mm 2 Iz = 1.44e4 mm 4 A = 142.3 mm 2 J = 2.89e4 1TIn'I 4 1500 Z "-" 1000 EL -6 O ._1 500 Vmo/L = -*- 0.0 cubic element 0.0 / '*- .001 present element .005 J -*- .010" I I I I 0 -10 -20 -30 -40 -50 Displacement, v (mm) Figure 2. Buckling Analysis of Hexagonal Shallow Dome axial force is derived in this paper and incorporated into a second-order analysis computer program NIDA for analysis of skeletal structures. The element is accurate even when the axial force is four times the Euler's buckling load, which refers to the extreme case of the buckling of a column with both ends fixed in direction and in rotation. Whilst all practical member possess initial imperfection, the derived element will be of great practical use. ACKNOWLEDGEMENT The authors are thankful to the financial support by The Research Grant Council, Hong Kong SAR Government under the project "Analysis and design of steel frames allowing for beam warping and lateral-torsional buckling (B-Q233)" REFERENCES AISC, (1986). Load and Resistance Factor Design Specification for Structural Steel Buildings, AISC, Chicago, IL, U.S.A. A1-Bermani and Kitipornchai, S. (1990). Nonlinear analysis of thin-walled structures using least element/member, Journal of Structural Engineering, ASCE, 116:1, 215-234. Australian Standards (1990). AS4100-1990, Steel Structures, Standards Association of Australia, Sydney. Barosum, R.S., and Gallagher, R.H. (1970). Finite element analysis of torsional-flexural stability problems. International Journal for Numerical Methods in Engineering, 2, 335-52. BS5950 (1990). Structural Use of steelwork in building, British Standard Institutions. . 1TIn'I 4 1500 Z " ;-& quot; 1000 EL -6 O ._1 500 Vmo/L = -* - 0.0 cubic element 0.0 / ' *- .001 present element .005 J -* - .010" I I I I 0 -1 0 -2 0 -3 0 -4 0 -5 0. centroidal axis from the axis joining the two ends of the element as, V VI + VO L-x I Esin xl+• sin 'X P ' sin~) -~ - sin~ L 1-q vm~ (4) in which, ~-~ P PL 2 k = P 9 r q Per. z2q( 2-) q) + 2Co )2 + C2Co )2 (16) 4(1 - q n'(1 - q 2(Cl + c2)(cl - c2 the axial force parameter, q, can be written as, ~2Fu )2 )2 VmO ] q=~-L~-b~(Ol+02 -b2(0 ,-0 2 -bvs ~(O~-O2)-bvv(~)