BUCKLING OF SPLICED COLUMNS IN STEEL FRAMES

Transcription

1 EUROSTEEL 2011, August 31 - September 2, 2011, Budapest, Hungary BUCKLG OF SPLCED COLUMS STEEL FRAMES Pedro D. Simão a,b, Ana M. Girão Coelho b,c and Frans S. K. Bijlaard c a University of Coimbra, Dept. of Civil Engineering, Portugal b nstitute of Computers and Systems Engineering of Coimbra (ESC-Coimbra), Portugal c Delft University of Technology, Dept. of Structural and Building Engineering, The etherlands TRODUCTO Steel frame building design practice is based on the behaviour of individual members. Column design, in particular, accounts for the effect of the interaction among neighbouring framing members, but it completely disregards construction details such as column splices. This practice is questionable as the splice may adversely affect the overall behaviour in terms of stiffness and strength [1,2]. Columns in buildings are predominantly axially loaded and thus fail by crushing or by buckling. Most practical columns fail by buckling. The European code of practice for the design of steel structures E 1993 [3] adopts a family of column curves for different cross-section types and derived from the Rondal-Maquoi formula [4]. This research project proposes a way of supplementing this formula for spliced columns. The current paper starts with a stability analysis of a framed spliced column using the classical equilibrium approach. The possibility of columns with stepped cross-section is also taken into account. This is followed by a study of the imperfection sensitivity to the linearly evaluated critical load. Linear elasticity is assumed in both analyses. The initial imperfections are applied by using the first eigenmode from linear buckling analysis. A discussion on the variation of the load carrying capacity with the level of imperfection on a practical spliced column is also presented. 1 ELASTC BUCKLG OF SPLCED COLUMS 1.1 Differential equation of equilibrium The fourth-order equilibrium equations for an initially straight spliced column loaded axially by a compressive load are derived using the equilibrium method. The force retains its direction as the column deflects. n this classical approach, the problem is reduced to an eigen-boundaryvalue problem and the critical conditions are the eigenvalues. The column in Fig. 1a consists of two independent members, and, of length L = αl and L = (1- α)l, 0 α 1, respectively, connected by a spring at point C. Each member has a constant bending stiffness E and E. The mathematical formulation of this problem is given below. Fig. 1b shows the free-body diagram of an infinitesimal segment of this column. The general fourth-order differential equilibrium equation is written as follows: where w μ w + μ w (1) V 2 is the lateral displacement, is given by: μ μ μ μ = or = (2) E E The general solution of this equation is given below: Member : w = A1sin μx+ A2cos μx+ A3x+ A4 Member : w = B sin μ x+ B cos μ x+ B x+ B where A i, B i are constants (i = 1, 2, 3, 4). This solution must satisfy the boundary conditions that are given by: (3)

2 x, u x, u dz dm M + K θb dq Q+ L K Δb L L C K θc K θa M Q a) Frame spliced column system b) Column segment Fig. 1. nitially perfect column model at x w E w = K w θa at the splice location E w = K x θc w w = L x E w = E w w x = L x = w x = L x x = L ( E w w ) = ( E w w ) x = L Ì x (4) at x = L E w = Kθbw E w w = K w Δb From Eqs. (3) and (4), the following eight linear homogeneous algebraic equations in the eight constants A i and B i are obtained: A1 0 Kθaμ Kθa A 2 0 c3,1 c3,2 Kθc 0 Kθcμ 0 Kθc 0 A 3 0 c4,1 c4, A4 0 = (5) sin ( μl) cos( μl) L B B c7,5 c7,6 Kθb 0 B c8,5 c8,6 c8,7 KΔ b B 4 0 C [ ]

3 where c i,j are coefficients that are defined as follows: c3,1 = sin ( μl) μkθccos( μl) c3,2 = cos( μl) + μkθc sin ( μl) c4,1 = sin ( μl) c4,2 = cos( μl) c7,5 = sin μ L μ Kθbcos μ L c7,6 = cos μ L + μ Kθb sin μ L c8,5 = KΔ b sin ( μ L ) c8,6 = KΔ b cos( μ L ) c = KΔ L 8,7 b A nontrivial solution exists if any of the eight constants is not equal to zero. This happens if the determinant of matrix [C] vanishes. The expansion of this determinant leads to the characteristic equation. The smallest positive route yields the buckling load cr and the shape of the deflection curve Eq. (3). mperfect columns ow consider the initial imperfect spliced column represented in Fig. 2a. Before the load is applied, the column is assumed to be slightly bowed and to be stress-free. The shape functions for the geometric imperfections associated with out-of-straightness of the column are given by: Member : w = δ A sin μ x+ A cos μ x+ A x+ A 0, 0 0,1 cr, 0,2 cr, 0,3 0,4 Member : w = δ B sin μ x+ B cos μ x+ B x+ B 0, 0 0,1 cr, 0,2 cr, 0,3 0,4 where δ 0 is the amplitude of the shape function, A 0,i, B 0,i are the first eigenvector coefficients (i = 1, 2, 3, 4), μ cr,, μ cr, are obtained from Eq. (2) at critical buckling ( = cr ). The total deflection w T (x) is obtained by superimposing the lateral deflection w(x) to the imperfection w 0 (x), for each member: wt x = w0 x + w x (8) The equilibrium equation is then written as: V 2 2 w + μ w = μ w 0 (9) where μ is given by Eq. (2). The complementary solution w c has the form of Eq. (3). The particular solution w p for this equation is obtained by using the method of undetermined coefficients. This leads to: Member : w = δ C sin μ x+ C cos μ x p, 0 1 cr, 2 cr, Member : w = δ D sin μ x+ D cos μ x where C i, D i are the following constants (i = 1, 2): p, 0 1 cr, 2 cr, C = A C = A 1 0,1 2 0,2 cr cr D = B D = B 1 0,1 2 0,2 cr cr Use of the boundary conditions, Eqs. (4) and the procedure described in section 1.1 gives the total deflection for the imperfect column. 2 MPERFECTO SESTVTY STUDY 2.1 Examples The variation of the relative basic amplitudes of the imperfection δ 0 is considered to investigate the imperfection sensitivity of a non-uniform spliced column. The dimensions and properties of the column used is this study are as follows: (6) (7) (10) (11)

4 x, u x, u dz dm M + L K Δb K θb w 0,ΙΙ,max w 0 (x) w(x) dq Q+ L Δ 0 S K θc L K θa w 0,Ι,max M Q a) Geometric imperfections b) Column segment Fig. 2. nitially imperfect column model Column length L = 4 m, Member cross-section Member : HE240B ( = mm 4 ), Member : HE200B ( = mm 4 ), Splice location α.5 (L = 2 m) and α.125 (L.5 m), Splice stiffness K θc = E /L and K θc = 10E /L, Young modulus E = 210 k/mm 2, End conditions Sway prevented case Clamped ends Pinned ends Sway case Fixed-guided ends Pinned-guided ends. First, a linear elastic eigenvalue analysis is conducted to obtain the buckling load cr for each column. Results are set out in Table 1 and agree well with analytical results obtained by means of an energy-based formulation of the problem [2,5]. For imperfection sensitivity studies, the selected amplitudes for the initial out-of-straightness were chosen as a percentage of the column length: 2%, 1%, %, 5% and 0.1%. 2.2 nfluence of imperfections on the column load carrying capacity The effect of the amplitude of the imperfection δ 0 on the maximum load of the column is depicted in Fig. 3. The diagrams are represented by the axial load ( 0.95 cr ) versus the maximum lateral displacement w max with a variation of δ 0. n each case, the graph is initially linear. As the load increases, the increase in displacement becomes disproportionately larger. ncrease in the amplitude of the imperfection reduces the pre-buckling stiffness and the carrying capacity of the column. This type of behaviour is independent from the column boundary conditions and the splice stiffness. This study also shows that: the magnitude of the column initial out-of-straightness has an unfavourable effect on the load carrying that may result in a very significant deterioration of the load bearing capacity. The maximum load is thus function of the imperfection, the maximum lateral displacement becomes unbounded when the applied load approaches the critical load, the equilibrium curves seem to converge on to the path / cr = 1, the maximum lateral displacement is proportional to a factor δ 0 /( cr ), irrespective of the column end restraints (Fig. 4).

5 Sway prevented columns Table 1. Buckling load Case α KθcL E cr (k) Case α KθcL E cr (k) Clamped ends Pinned ends Sway columns Fixedguided ends Pinnedguided ends Load ratio / cr Perfect column mp 0.1% mp 5% mp % mp 1% mp 2% Load ratio / cr Perfect column mp 0.1% mp 5% mp % mp 1% mp 2% Sway prevented columns Fig. 3. Equilibrium paths with initial curvature Sway columns δ 0/( cr - ) mp 0.1% mp 5% mp % mp 1% mp 2% δ 0/( cr - ) mp 0.1% mp 5% mp % mp 1% mp 2% Sway prevented columns Sway columns Fig. 4. Representative relationship δ 0 /( cr ) vs. w max /L 3 COCLUDG REMARKS AD FURTHER WORK The work reported in this paper has focused on the elastic stability analysis of non-uniform stepped columns with initial geometric imperfections. The formulation of the problem was based on the exact solution of the governing equations for buckling. The initial imperfection is applied by using the first mode shape from linear buckling analysis and its influence on the load-carrying capacity is studied compared to the perfect column. The introduction of the imperfection in the form of the first eigenmode will not necessarily provide a lower limit to the pre-buckling stiffness and collapse load.

6 Analyses would need to be carried out based on imperfections in the form of linear combination of the column buckling modes to ensure that the results are conservative and can safely be used in design calculations [6]. The effect of the level of a geometric imperfection was examined in the case study of a practical framed spliced column and adopting different relative basic values of the imperfection. Results indicate that spliced columns can be highly sensitive to initial imperfections in the geometry. Small imperfections in these structures are however inevitable and may result in a very significant deterioration of their load bearing capacity. The margin between the maximum strength and the design load should be decided by complementary elastic-plastic imperfection sensitivity studies that the authors are undertaking. The investigated configuration was rather limited to a particular case; investigation with other configuration cases would be necessary to obtain more general conclusions. Designers and steel fabricators would potentially be interested in this issue and the authors are currently working on this topic. Additional imperfections such as column segment misalignment and imperfect contact between the cut surfaces at the column splice deserve further investigation. Except for work by Lindner [6] and Popov and Stephen [7], who investigated contact splices, data on the behaviour of such structural members are scarce. This work affords some basis to produce design guidance on framed spliced columns. Design codes such as E 1993 [3] account for the out-of-straightness effects explicitly by using the column maximum strength as the basic criterion. The authors attempt to extend this concept and to set up sound design criteria regarding (i) the requirements for stiffness and strength of column splices and (ii) the generalization of the column buckling curves adopted in E 1993 [3] for spliced columns by means of appropriate forms of the generalized imperfection factors. REFERECES [1] Snijder, HH, Hoenderkamp, JCD, nfluence of end plate splices on the load carrying capacity of columns, Journal of Constructional Steel Research, Vol. 64, pp , [2] Girão Coelho, AM, Simão, PD, Bijlaard, FSK, Stability design criteria for steel column splices, Journal of Constructional Steel Research, Vol. 66, pp , [3] E Design of steel structures, Part 1-1: General rules and rules for buildings, CE, European Committee for Standardization, [4] Rondal, J, Maquoi, R, Single equation for SSRC column strength curves, Journal of the Structural Division, American Society of Civil Engineers, 105(ST1), pp , [5] Simão, PD, Girão Coelho, AM, Bijlaard, FSK, nfluence of splices on the stability behaviour of columns and frames, Proc. ntl. Colloquium on Stability and Ductility of Steel Structures (SDSS Rio 2010), (s.: E Batista, P Vellasco, L de Lima), pp , Rio de Janeiro, [6] Gonçalves, R, Camotim, D, On the incorporation of equivalent member imperfections in the in-plane design of steel frames, Journal of Constructional Steel Research, Vol. 61, pp , [7] Lindner, J, Old and new solutions for contact splices in columns, Journal of Constructional Steel Research, Vol. 64, pp , [8] Popov, EP, Stephen RM Capacity of columns with splice imperfections, Engineering Journal, American nstitute of Steel Construction, 1 st quarter, pp , 1977.

7 To cite this paper: Simão, P.D., Girão Coelho, A.M., Bijlaard, F.S.K. (2011), Buckling of spliced columns in steel frames, in: L. Dunai et al. (s.), Proceedings of the 6th European Conference on Steel Structures (Eurosteel 2011), Budapest, Hungary,