IN-PLANE BUCKLING BEHAVIOR OF PITCHED ROOF STEEL FRAMES WITH SEMI-RIGID CONNECTIONS N. Silvestre, A. Mesquita D. Camotim, L. Silva INTRODUCTION Traditionally, the design and analysis of steel frames assumes either rigid or pinned connections. However, a large numer of experimental studies have shown that this hypothesis is not valid for the vast majority of the connections employed in practical situations [, ]. In fact, the most commonly used frame connections display a ehavior which lies somewhere etween rigid and pinned. Such connections are named semi-rigid and influence significantly the frame structural ehavior, namely the aspects related to in-plane uckling. This paper presents and discusses some results concerning the in-plane elastic staility and non-linear uckling ehavior of unraced one ay pitched roof frames with semi-rigid connections (figure ). Initially, typical connection configurations are displayed and a methodology to evaluate their stiffness values is descried and illustrated. Then, earlier 3 I r, L r I r, L r α I c, L c Semi-rigid connection (a) I c, L c H Sd L () V Sd /L H Sd (c) Figure : Pitched roof frame with semi-rigid connections (a) geometry () loading (c) staility model N c N r N c N r R I=I c/i r R L=L r/l c γ=n r/n c N r N c N r N c Civil Engineering Department, Technical University of Lison, Instituto Superior Técnico, Av. Rovisco Pais, 096 Lisoa Codex, Portugal Civil Engineering Department, University of Coimra, 3049 Coimra Codex, Portugal
results related to the in-plane staility of pitched roof frames [3] are extended to incorporate the influence of the connections semi-rigidity. In particular, analytical expressions are developed to estimate the relevant ifurcation loads. Finally, the paper deals with the consideration of second-order effects (P- type). It is shown that an indirect amplification method, previously proposed and validated in the context of rigid frames [4], remains valid for semi-rigid frames. CONNECTIONS IN PITCHED ROOF FRAMES Introduction. The need to transmit high ending moments and the tight constraints related to the column and rafter layout make the commonly used pitched roof frame connections quite standard. Figure (a)-() shows typical apex and eaves connection configurations. The presence of extended end-plates (towards the tension zone - dominant load case) and haunches increases the olts lever arm and, therefore, improves oth the steel section capacity and the olt force distriution. As the column axial forces are usually small, most column ase connections are unstiffened, typically consisting of a thick plate welded to the column section and anchored to a rigid concrete lock, y two or more olts (figure (c)). Concerning the column ase stiffness, one should mention that it highly depends on the geotechnical conditions and frame span-to-height ratio. As a result, one may have column ase connections ranging from nominally pinned to nominally fixed. (a) () (c) Figure : Typical connection configurations (a) apex () eaves (c) column ases In order to determine the in-plane elastic uckling ehavior of semirigid frames, it is essential to assess the moment-rotation characteristics of every connection, namely the initial stiffness values K j,init (figure 3). It is well known that an accurate evaluation of a connection stiffness is
a rather difficult prolem, requiring lengthy and complex finite element calculations. In order to avoid such procedures, a simpler methodology, known in the literature as component method [5] and providing the ackground for Annex J of Eurocode 3 (EC3) [6], is employed here. The joints are replaced y mechanical models, made of extensional springs and rigid links, which adequately incorporate all the relevant aspects concerning the connections deformaility (figures 4 and 5). M K j,init M j,rd M j,sd φ Ed φ Xd φ Cd φ Figure 3: Connection moment -rotation curve and initial stiffness K j,init k eq k eff, φ k 3 k 4 k 5 k 6 k k M k eff, Figure 4: Mechanical model for the eaves and apex connections Eaves and Apex Connection Stiffness. The mechanical model shown in figure 4 illustrates the application of the component method to the eaves and apex connections. All the different connection components contriuting to its overall deformaility are displayed. Eaves and apex connections are acted y ending moment M, axial force N and shear force V and their stiffness stems from a several individual component ehaviors comined. It should e outright mentioned that only the connection M-φ ehavior is dealt with here. Moreover, the (weak) influence of the deformations due to axial and shear forces is neglected.
In eaves connections one identifies the following relevant component ehaviors (haunch and rafter flange -we assumed undeformale in compression): (i) unstiffened column we shear k, (ii) unstiffened column we compression k, (iii) column we tension k 3, (iv) column flange flexure k 4, (v) end-plate flexure k 5 and (vi) olt tension k 6. Then, the initial stiffness value K j,init may e evaluated using the equations K k j,init = E zeq kc () = + + () c k k keq n keff,r h r k r eq = = (3) zeq keff,rhr z = r eq (4) k h r eff,r r k eff,r =, (5) k where h r is the distance etween olt-row r and the connection center of compression, and the equivalent stiffness coefficient k eq takes into account the comined influence of the different component ehaviors which are grouped in the various olt-rows in tension. The effective stiffness coefficient for olt-row r, k eff,r, takes into account the stiffness coefficients of the corresponding i relevant components (k 3 -k 6 in this case) and the distance z eq represents an equivalent lever arm. A similar reasoning may e applied to apex connections. The haunch and rafter flanges-wes are assumed undeformale in compression (small pitches) and the component ehaviors are: (i) extended end-plate flexure k 5 and (ii) olt tension k 6. Equations ()-(5) also remain valid. i i, r
Column Base Connection Stiffness. Due to the sustantial influence of the axial force, the same column ase may show rather different rotational characteristics. Two phenomena are involved, namely the deformaility of the links (i) etween the steel column and the concrete ase and (ii) etween the concrete foundation and the soil [7]. In the first case, the rotation ehavior is represented y a M-φ curve, which depends on the connection ending moment-axial force ratio. In the second case, however, the M-φ curve (rotation of the concrete lock within the soil) must take into account the soil settlement due to the column compression. Since no availale information exists concerning the concrete-to-soil link, this paper deals solely with the deformaility taking place etween the steel column and the concrete foundation. Figure 5 shows the corresponding mechanical model, which accounts for the interaction etween ending moment and axial force. The model components include (i) extensional springs to simulate the column cross-section deformation (tension and compression), (ii) extensional springs to simulate the anchor olts and ase plate deformation (tension), (iii) extensional springs to simulate the concrete under the ase plate (compression) and (iv) rotational springs to simulate the plastic deformation associated to an eventual development of a yield line in the extended part of the plate (connection compressive zone). It should also e mentioned that the use of this mechanical model involves the performance of an iterative numerical procedure. N Column deformation Plate to concrete contact M Yield line Anchors Undeformed position Figure 5: Mechanical model of the column ase connection
Illustrative Numerical Results. In order to illustrate the application of the component method, the initial stiffness values of typical pitched roof frame connections were evaluated. The results otained are shown in tales (a) (apex olts without haunch and 3 olts with haunch), () (eaves) and (c) (column ase - and 4 olts [7]) and correspond to IPE 80-330 rafters and a HEB60 column. Rafters Tale : Connections initial stiffness values (a) apex () eaves (c) column ase Bolt rows (a) K j,init (knm/rad) Bolt rows K j,init (knm/rad) IPE 80 M 6 6380 3 M 6 4694 IPE 00 M 6 770 3 M 6 59954 IPE 0 M 6 79 3 M 6 70378 IPE 40 M 6 35509 3 M 6 87774 IPE 70 M 6 45545 3 M 6 894 IPE 300 M 6 6003 3 M 6 3893 IPE 330 M 6 7564 3 M 6 73496 () Column / Rafter Bolt rows K j,init (knm/rad) HEB60 / IPE 80 3 M 6 7963 HEB60 / IPE 00 3 M 6 9865 HEB60 / IPE 0 3 M 6 30 HEB60 / IPE 40 3 M 6 559 HEB60 / IPE 70 3 M 6 954 HEB60 / IPE 300 3 M 6 4407 HEB60 / IPE 330 3 M 6 8808 (c) Column Bolts K j,init (knm/rad) HEB60 M 6 750-5000 HEB60 4 M 6 3500-8000
Concerning the apex and eaves connections, it is oserved that, depending on the rafter, K j,init varies from 6380 to 73496 knm/rad (apex) and from 7963 to 355 knm/rad (eaves). The column ase results show the sustantial influence of the axial force on the stiffness. In fact, K j,init varies from 750 to 5000 knm/rad ( olts) and from 3500 to 8000 knm/rad (4 olts) when N changes from 00 to 000 kn. FRAME STABILITY BEHAVIOR Introduction. It is assumed that the frame loading consists only of compressive forces acting on the columns and rafters, related y the parameter γ (figure (c)). It was shown efore [3] that the in-plane elastic staility ehavior of rigid frames is conditioned y two uckling modes, one anti-symmetric (ASM) and the other symmetric (SM), oth involving lateral (sway) displacements at the column top s (figure 6(a)). ASM Ν r Ν ASM r.0 Ν SM r.0 ASM SM α 3> α > α >0 SM (a) γ Ν ASM c.0 () 0 α α Ν SM c.0 α 3 Ν c Figure 6: (a) Buckling modes () Variation of (N c -N r ) with α and γ An extensive parametric study unveiled the essential aspects of the inplane elastic staility of one ay symmetric pitched roof rigid frames (fixed and pinned-ase), which are shown in figure 6() [8]. Concerning the critical load λ cr (scaled to the columns Euler load P Ec ), the variation of the comination (N c -N r ) at ifurcation with the inclination α and the loading ratio γ, for given R I and R L, was shown to display the following characteristics (N ASM c.0, N SM c.0, N ASM r.0 and N SM ifurcation loads for γ=0, ): (i) The ASM ehavior (solid line) does not depend on the value of α. r.0 -
(ii) The SM ehavior (dashed lines) depends on the value of α, as the frame staility is controlled essentially y the rafters. (iii) For low γ values, λ cr = (no dependence on α). (iv) For high γ values, λ cr = or λ cr =λ SM (dependence on α). (v) The variation of λ cr with γ involves either oth the ASM and SM (α=0 ; α ; α ) or only the ASM (α=α 3 ). Semi-Rigid Connections. When performing a linear staility analysis, one looks for equilirium configurations in the vicinity of the fundamental path. Therefore, in the case of semi-rigid frames, such an analysis only requires the knowledge of the connections initial stiffness values K j,init (the shape of the full M-φ curve is not involved). Let us now consider the pitched roof frame depicted in figure, with symmetrically located semi-rigid connections in all joints (figure (a)). The initial stiffness values of the three connection types are designated as K (column ase), K (eaves) and K 3 (apex) and, out of convenience, one defines the non dimensional parameters S, S and S 3 (S j =K j L c /EI c ). The frame staility ehavior continues to e conditioned y two uckling modes (ASM and SM), with shapes similar to the ones shown in figure 6(a), and it still exhiits the characteristics displayed in figure 6(). The conclusions drawn with respect to rigid frames remain qualitatively valid in the presence of semi-rigid connections. As the frame overall stiffness is reduced, the ifurcation loads and λ SM oviously decrease and, moreover, the nature of the critical mode may vary. The aim of this study is to estimate the order of magnitude of these changes and also to appraise the relative importance of each connection type. Naming the ifurcation loads of the rigid (S j = ) and semi-rigid (at least one S j ) frames as λ and λ S, tale shows, for a frame with a commonly used geometry (R I =.5, R L =3,α=6 ) and γ values produced y a single uniformly distriuted vertical span load (figures ()-(c)), the variation of the ratios S /, λ SM S /λ SM and S /λ SM S with the S j values. Assuming a test frame with HEB60 columns, the connection stiffness numerical results presented earlier correspond to the S j ranges: 0.9 S 9., 9. S 37.3 and 8.8 S 3 98.9. On the asis of these results, the following realistic S j values were considered: S =0,, 5, ; S =0, and S 3 =0,. It is possile to oserve that:
Tale : Staility results of semi-rigid frames: variation with S j S γ S λ SM S λ SM λ SM S S S S 3 =0 S 3 = S 3 =0 S 3 = S 3 =0 S 3 = S 3 =0 S 3 = 0 5 0 0.879 0.876 0.544 0.545.37.355 0.440 0.43 0.89 0.889 0.555 0.555.358.386 0.439 0.430 0.078.075.08.030.6.85 0.950 0.933.085.08.04.05.95.9 0.90 0.903 0.339.335.07.074.04.044.3.04.338.335.08.083.058.079.096.077 0.563.559 0.98 0.984 0.949 0.967.0.09.557.553 0.998.000 0.98.000.09.074 (i) As the results for S 3 =0 and S 3 = are practically identical, an apex connection with S 3 =0 may e treated as rigid. (ii) As the results for S =0 and S = differ only slightly (maximum difference of 3%, for λ SM S /λ SM ), eaves connections with S =0 may e treated as almost rigid. (iii) As S varies from 0 to : (iii.) γ increases sustantially (higher rafter compression) and, as a result, the λ S /λ ratios reach values higher than (i.e., semi-rigid column ases may improve the frame staility). (iii.) λ SM S /λ SM steadily decreases 30%. (iii.3) S / increases 00% for 0 S 5 ( 90% for 0 S ) and then decreases 0% (5 S < ). (iii.4) As a result of (iii.-3), S /λ SM S varies, for S =S 3 =, etween 0.430 (S =0) or 0.903 (S =) and.077 (S =5). This means that the critical mode changes from AS to S. In order to assess the influence of the column and rafter geometry, the variation of S / *, λ SM /λ SM * and /λ SM with R I and R L is now investigated. Designating the reference frame (R I =.5, R L =3,α=6 ) ifurcation loads as λ *, tale 3 shows, for two of the previous connection stiffness cominations and similarly varying γ values, the variation of the three ifurcation ratios.
Tale 3: Staility results of semi-rigid frames: variation with R I and R L S =S =S 3 = S = S =0 S 3 =0 R I R L γ λ SM λ SM λ SM γ λ SM λ SM λ SM 0.956 3.73 4.6 0.975 0.675.340 3.854 0.577 3.459.436.496.035.07.356.435 0.898 4.945 0.670 0.754 0.958.370 0.76 0.74 0.967.00.75.830.048 0.7.876.79 0.639.5 3.553.000.000.074.078.000.000 0.950 4.083 0.454 0.504 0.973.440 0.55 0.488.004.05.03.3.4 0.736.597.77 0.697 3.634 0.76 0.745.00.5 0.806 0.763.004 4.0 0.340 0.377 0.973.497 0.403 0.368.039 The following set of geometrical parameter values was considered: R I =,.5, ; R L =, 3, 4. One oserves that: (i) The influence of R L clearly surpasses that of R I. (ii) As R I varies from to : (ii.) γ increases 9-3% in a fairly proportional fashion. (ii.) / * decreases 40-50% (S = ) and 30-45% (S =). (ii.3) λ SM /λ SM * decreases 45-50%. (ii.4) /λ SM increases -4% (S = ) and 7-0% (S =). The variation magnitude decreases when R L varies from to 4. (iii) As R L varies from to 4: (iii.) γ increases 00% in a fairly proportional fashion. (iii.) / * decreases 85% (S = ) and 70-75% (S =). (iii.3) λ SM /λsm * decreases 8-83%. (iii.4) /λ SM increases 50-70% (S =) and varies 8-5% without a consistent pattern (S = ). Overall, the /λ SM values range from 0.58 (S =; R I =; R L =) to. (S = ; R I =; R L =). Although no clear pattern can e identified concerning the variation of the critical mode nature, the results show that the SM is more relevant in fixed-ase frames.
Approximate Analytical Expressions. In order to avoid the need to perform linear staility analyses, easy -to-use analytical expressions to estimate and λ SM were previously developed, in the context of rigid frames [3]. The ifurcation loads are otained from the equation / RR L C γ λ = λ +, (6) c.0 r.0 ρ ρ where R=R I R L, ρ c.0 and ρ r.0 are related to the ifurcation loads when only the columns or rafters are compressed (expressions for their estimation are also availale) and λ is scaled to the columns Euler load P Ec. The form of equation (6) stems from the fact that all the curves shown in figure 6() are almost elliptic, C λ eing an adjustment factor (0.9 C λ.). An extensive parametric study [8] showed the formulas to e valid and to lead to accurate results for all commonly uilt frames. Similar analytical expressions, accounting for semi-rigid connections, are now presented. It was found that the form of equation (6) may e retained, provided that C λ, ρ c.0 and ρ r.0 are evaluated y the formulas given in tale 4 (R H =R L sinα), which contain the stiffness values S j. Making S j or 0, they are also valid for rigid or pinned connections. Tale 4: Approximate analytical expressions to evaluate and λ SM ASM SM.. S ( ) ( ) S R+ 3 + 3 S + S ρ c.0 = SS ( R + 3) + 6( S + S ) +. 5( 3+ RS ) S Cλ = + 09 + S C = λ ρ r.0 = SS S3 SS ρ r 0. = SS ( R + 6) + 6( S + S) (.4R + 6) + 6( S + S ) ( 6 + H + 4) + ( 48 + 6) + ( + )( 6 + 48) ( H + ) + ( 48 + ) + ( + )( + 48) SS S3 R R SS3 R S S3 S ρ c.0 = SS S3 R R SS3 R S S3 S SS S3( 8R + RH + ) + SS 3( 4R + 8) + ( S + S3)( 8S + 4) ( 8R R + 4) + S S ( 4R + 6) + ( S + S )( 6S + 48) + 5 ( + SS ) H 3 3 R
In order to assess the accuracy and validity of the proposed formulas, the exact and approximate ifurcation loads were compared for all the 3 frames considered in tales -3. The results yielded y the formulas were found to e quite accurate and mostly conservative (53 out of 64). Concerning the errors, one has 4% ε MAS +% and 6% ε MS +%. SECOND-ORDER EFFECTS Let us now turn our attention to the non-linear uckling ehavior of semi-rigid unraced pitched roof frames acted y horizontal and vertical loads (figures (a)-()), namely the consideration of secondorder effects of the P- type. As symmetric vertical distriuted loads acting on symmetric pitched roof frames lead to significant displacements at the column tops, three first-order displacement(u)/moment(m) components may e identified: (i) non-sway (NS) and (ii) symmetric sway (SS), oth due to the vertical load, and (iii) anti-symmetric sway (AS), due to the horizontal load. The similarity etween (i) the first-order deformed configurations and ui AS and (ii) the S and AS uckling mode shapes led to the formulation, in the context of rigid pitched roof frames, of an indirect amplification method to incorporate the P- effects [4]. Two amplifications are involved and the second-order displacements and moments are calculated y means of u I SS u II ap M II ap I uss Sd I uas ASM I VSd V = uns + + V (7) SM V I VSd I VSd = I MNS + M SS + MAS V SM V, (8) ASM where V Sd, V ASM e V SM are the design and ifurcation values of the vertical load. The relative magnitude of the two amplified terms depends on the comined values of the ratios (i) (V SM /V ASM ) and (ii) (uss/u I AS) I or (M SS/M I AS). I The method was shown to yield accurate results for a wide range of fixed and pinned-ase frames [8].
Provided that the connections display a linear ehavior (M j.sd inside the linear portion of the M-φ curve - figure 3), the fundamentals of the aove method apply also for semi-rigid frames. It is, therefore, natural to expect similarly accurate results in this case. This statement is confirmed y the results shown in tale 5, which consist of the ratios etween the approximate and exact maximum second-order moments and displacements (top of right column) and correspond to V Sd =0.5V cr and H Sd =0.05V Sd. It is oserved that, for all the 0 frames considered (S =0,, 5, ; S =S 3 = ; R I =,.5, ; R L =, 3, 4), the approximate results are rather accurate (ε max 4%) and always conservative. Tale 5: Relation etween approximate and exact second-order results Map II II /Mex uii ap /uii ex R I R L S =0 S = S =5 S = S =0 S = S =5 S =.5 3.0.0.0.0.0.0.0.0.5.00.00.0.00.0.0.0.0.5 4.03.0.0.0.04.03.0.03 3.0.00.0.0.0.0.0.03 3.0.0.0.0.0.0.0.0 CONCLUDING REMARKS Some results concerning the in-plane elastic staility and non-linear uckling ehavior of unraced one ay pitched roof frames with semirigid connections were presented and discussed. First, typical connection configurations were shown and a simple methodology, proposed in the revised version of Annex J of EC3, was used to evaluate their initial stiffness values. While most commonly employed apex and eaves connections were, naturally, found to e almost rigid, the column ases displayed a wide stiffness value range. Next, it was investigated how the connections semi-rigidity influences the frame elastic in-plane staility. Different geometries were dealt with and it was found that the commonly used apex and eaves connections may e treated as rigid and that the column ase stiffness strongly
affects the frame staility ehavior. In particular, even small stiffness changes may alter the nature of the critical uckling mode. In order to enale an easy estimate of the two relevant frame uckling loads, approximate formulas were developed, which yield rather accurate and mostly conservative results. Finally, it was shown that, for linear semi-rigid connections, an existing indirect amplification method accurately takes into account the secondorder P- effects. The incorporation of the uckling load formulas into this method will certainly lead to an efficient design tool. One last word to mention that the authors are presently working on the uckling ehavior of semi-rigid frames with non-linear connections. REFERENCES [] Chen, W.F., Goto, Y. and Liew, J. Staility Design of Semi-Rigid Frames, John Wiley, 996. [] Semi-Rigid Connections in Steel Frames, Council on Tall Buildings and Uran Haitat, McGraw-Hill, 993. [3] Silvestre, N., Camotim, D. and Corrêa, M. - On the Design and Safety Checking of Unraced Pitched Roof Steel Frames, Journal of Constructional Steel Research, Vol. 46, nº-3, 998, 38-39 (full paper in the CD-ROM paper # 88). [4] Silvestre, N. and Camotim, D. Second-Order Effects in Pitched Roof Steel Frames, Proceedings of the SSRC Annual Technical Session and Meeting, Toronto, 997, 85-98. [5] Weynand, K., Jaspart, J-P. and Steenhuis, M. - The Stiffness Model of the Revised Annex J of Eurocode 3, Connections in Steel Structures III - Proceedings of the 3 rd International Workshop on Connections, Eds. R. Bjorhovde, A. Colson, R. Zandonini, Pergamon, 996, 44-45. [6] Revised Annex J of Eurocode 3, Document CEN/TC 50/SC 3 - N 49 E, CEN, 994. [7] Guisse, S., Vandegans, D. and Jaspart, J-P. - Application of the Component Method to Column Bases Experimentation and Development of a Mechanical Model for Characterization, Report MT 95, CRIF, Liège, 996. [8] Silvestre, N. Staility and Second-Order Effects in Pitched Roof Steel Frames, M.A.Sc. Thesis (in portuguese), 997.