Thin-Walled Structures 60 (2012) 98–104 Contents lists available at SciVerse ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Second-order plastic-hinge analysis of space semi-rigid steel frames Cuong Ngo-Huu a,1, Phu-Cuong Nguyen b, Seung-Eock Kim b,n a b Faculty of Civil Engineering, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam Department of Civil and Environmental Engineering, Sejong University, 98 Gunja-dong, Gwangjin-gu, 143-747 Seoul, South Korea a r t i c l e i n f o a b s t r a c t Article history: Received 27 October 2011 Received in revised form 28 June 2012 Accepted 28 June 2012 Available online August 2012 This paper presents a numerical procedure based on the beam–column method for nonlinear static analysis of space semi-rigid steel frames The second-order effects are considered by the use of stability functions obtained from the exact solution of beam–columns under end forces and the inelastic behavior of material are presented by the lumped inelastic concept using yield surface An independent zero-length element comprising of six translational and rotational springs, of which rotational spring stiffness uses the Kishi–Chen power model, is used to simulate the rigidity of the steel beam-to-column connection The generalized displacement control method is applied to solve the nonlinear equilibrium equations in an incremental–iterative scheme The nonlinear response results are compared with those of existing studies to verify the accuracy of the proposed numerical procedure & 2012 Elsevier Ltd All rights reserved Keywords: Second-order effects Stability functions Semi-rigid connection Plastic-hinge Beam–columns Steel frames Introduction The beam-to-column connections of steel structures are conventionally considered as fully rigid or ideally hinged types for simplicity in structural analysis However, the behavior of actual connections, especially bolted ones, is more complicated in comparison with two above-mentioned simple extreme cases and has been studied by many researchers in the efforts of deriving appropriate moment–rotation relation model from experimental data (Frye and Morris [1], Chen and Kishi [2], Azizinamini and Radziminski [3]) The connection rigidity significantly affects the strength and displacement response of steel frames and therefore many studies have been done to address these Two basic ‘‘line-element’’ methods of finite element and beam–column have been used for second-order inelastic analysis of steel semi-rigid frame structures The finite element method is able to accurately predict the nonlinear behavior of steel structures because it easily considers the coupling effects among axial, flexural and torsional deformations and simulates the spread of plasticity along the length and across the cross section as the actual behavior of the steel member However, it is generally recognized to be computationally expensive because numerous elements and monitoring points/bers are required in modeling n Corresponding author Tel.: ỵ82 3408 3291; fax: ỵ82 3408 3332 E-mail address: sekim@sejong.ac.kr (S.-E Kim) Formerly Visiting Scholar, Department of Civil and Environmental Engineering, Sejong University, 98 Gunja-dong, Gwangjin-gu, 143-747 Seoul, South Korea 0263-8231/$ - see front matter & 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.tws.2012.06.019 Foley and Vinnakota [4,5] developed a nonlinear finite element for use in second-order inelastic ultimate load analysis of steel frames under static loading Amongst the features of these studies were that it was the first time the full nonlinear load-deflection response including unloading behavior of large-scale rigid and semi-rigid steel frames was computed using the spread of plasticity model and shifting of the location of the elastic neutral axis was directly considered in the stiffness matrix The beam– column member is modeled by an appropriate way of eliminating its further subdivision and then it becomes the simple and effective way in representing the nonlinear effects with least elements in modeling Generally, the second-order effects are considered by the use of stability functions obtained from the closed-form solution of the beam–columns under end forces and the inelastic behavior of material is presented by the lumped inelastic concept using yield surface This method is called the ‘‘practical advanced method’’ because it can consider all key factors influencing the ultimate strength of the steel frame structures in an effective way and then is suitable for adoption in design practice (Hsieh and Deierlein [6], Chen et al [7], Chan and Chui [8], Kim and Choi [9], Choi and Kim [10], Sekulovic and Nefovska-Danilovic [11]) Recently, Chiorean [12] proposed the beam–column method for nonlinear inelastic analysis of space semi-rigid steel frames The nonlinear inelastic force–strain relationship and stability functions are used in representing the material nonlinear and second-order effects, respectively The advantage of this study is that it is able to trace the spread-ofplasticity along the member length and only one beam–column element per frame member is necessary in modeling However, it C Ngo-Huu et al / Thin-Walled Structures 60 (2012) 98–104 seems that the shape parameters a and n of the Ramberg–Osgood model and a and n of the proposed modified Albermani model for force–strain relationship of the cross-section, which considerably affect the inelastic behavior of the steel frames, are not consistently used In above-mentioned studies, the connections are usually modeled as rotational springs attached at the beam ends for representing the moment–rotation relations of the beam-tocolumn connections A static condensation algorithm is then applied to eliminate the degrees of freedom between the springs and beam ends so that the condensed stiffness matrix of the beam with semi-rigid connections has the usual dimension ready for assemblage (Lui and Chen [13]) In this paper, an independent zero-length connection element with six different translational and rotational springs connecting two identical nodes is developed to simulate the beam-to-column connections This is an efficient way because the modification of beam–column stiffness matrix considering the semi-rigid connections is not necessary and the connection is ready to integrate with any beam–column model The Kishi–Chen power model is used for rotational springs Although the translational springs are able to model linear semi-rigid connections with constant stiffness, it is just used herein as rigid springs The second-order effects of the steel members are considered by the use of stability functions obtained from the exact solution of beam–columns under end forces and the inelastic behavior of material are presented by the lumped inelastic concept using Orbison or LRFD yield surface The generalized displacement control method is applied to solve the nonlinear equilibrium equations in an incremental–iterative scheme The nonlinear response results are compared with those of existing studies to verify the accuracy of the proposed numerical procedure Element formulation 2.1 Beam-to-column connection element An independent zero-length element with three translational and three rotational springs is developed to simulate the rigidity of the steel beam-to-column connection The coupling effects between springs in a connection are neglected This multi-spring element connects two nodes having identical coordinates Each translational spring has the linear stiffness for translational degrees of freedom while the rotational one can have linear or nonlinear stiffness for rotational degrees of freedom In this study, the nonlinear rotational spring uses the Kishi–Chen power model (Chen and Kishi [2]) as follows: Mẳ Rki yr ẵ1 ỵ yr =y0 Án 1=n ð1Þ where M and yr are moment and rotation of the connection, n is shape parameter, and Rki is initial connection stiffness Rki ¼ Mu y0 fDF S g ẳ ẵK S fDU S g lin stiffness values for linear spring ki lin rot ki ð3Þ where [KS] is the tangent diagonal stiffness matrix comprising of tangent stiffness components for each spring The tangent rot and nonlinear spring ki ki ẳ Rlin k,i are 4aị ẳ Rrot kt,i ẳ Rrot ki,i n ẵ1 ỵ yr =y0 ỵ 1=n À ð4bÞ The linear spring can simulate the rigid or hinge spring by assigning a very large or small stiffness value Because the purpose of this study is mainly to verify the bending moment transference in the major axis of the beam, the translational and torsional degrees of freedom and the rotational degree of freedom corresponding to the rotation about the minor axis of the beam of the connection are all modeled by rigid springs in all numerical examples 2.2 Beam–column element stiffness matrix To capture the effect of axial force acting through the lateral displacement of the beam–column element (P–d effect), the stability functions reported by Chen and Lui [14] are used to minimize modeling and solution time Generally only one element per member is needed to accurately capture the P–d effect The material nonlinearity includes gradual yielding of steel beam–column member under axial force and bending moments considering the effects of residual stresses and initial geometric imperfection The incremental force–displacement equation of space beam–column element can be expressed as 38 E A=L 0 Dd > DP > t > > > > > > > > 7> > > 60 > > > kiiy kijy DyyA > > > > 7> DMyA > > > > > > > > > 7> > > > = < < DM = k k ijy jjy yB DyyB 5ị ẳ6 k 0 k DyzA > > > ijz 7> iiz > > > > > > DM zA > 7> > > > > > > DyzB > > kijz kjjz 0 > > > > > > 5> > > > > DM zB > ; > : ; : Df > GJ=L 0 0 DT where E and G are elastic and shear modulus of material; A and L are area and length of beam–column element; J is torsional constant; DP, DMyA, DMyB, DMzA, DMzB, and DT are incremental axial force, end moments with respect to y and z axes, and torsion respectively; Dd, DyyA, DyyB, DyzA, DyzB, and Df are the incremental axial displacement, joint rotations, and angle of twist; and ! Et Iy S2 ð6aÞ kiiy ẳ ZA S1 1ZB ị S1 L kijy ¼ ZA ZB S2 kjjy ¼ ZB Et Iy L ð6cÞ ! S24 Et Iz ð1ÀZB Þ S3 L 6dị kiiz ẳ ZA S3 kijz ẳ ZA ZB S4 kjjz ẳ ZB 6bị ! S22 Et Iy S1 À ð1ÀZA Þ S1 L ð2Þ where Mu is ultimate moment capacity and y0 is a reference plastic rotation The relation between the incremental force vector {DFS} and displacement vector {DUS} of the multi-spring element corresponding to six degrees of freedom between two connected nodes is as follows: 99 Et Iz L ! S24 Et Iz S3 À ð1ÀZA Þ S3 L ð6eÞ ð6fÞ where In is moment of inertia with respect to n axes (n ¼y,z); S1n and S2n are stability functions with respect to n axis (n ¼y,z), and 100 C Ngo-Huu et al / Thin-Walled Structures 60 (2012) 98–104 are expressed as n L sinðkn LÞÀðkn LÞ cosðkn LÞ > < k2À2 if P o coskn Lịkn L sinkn Lị S1n ẳ k Lị2 coshðk LÞÀk L sinhðk LÞ n n n n > if P : 22 coshkn Lị ỵ kn L sinhkn Lị S2n ẳ > < kn Lị2 Àkn L sinðkn LÞ 2À2 cosðkn LÞÀkn L sinðkn LÞ 7aị fDij g ẳ fDij1 g ỵ fDDij g 7bị if P for P r0:39P y Et ¼ À2:7243E ð8aÞ P P ln Py Py for P 0:39P y 8bị Z ẳ 1:0 for a r0:5 9aị Z ẳ 4a1aị for a 0:5 9bị where a can be expressed in yield surfaces proposed by AISCLRFD [15] as 9 ð10aÞ 2 my ỵ mz 9 10bị a ẳ p ỵ my ỵ mz for p Z my ỵ my aẳ p ỵ my ỵmz for p o or by Orbison et al [16] as a ¼ 1:15p2 þ m2z þ m4y þ 3:67p2 m2z þ 3:0p6 m2y þ 4:65m4z m2y where p ¼ P=Py , (weak-axis) mz ¼ Mz =M pz (strong-axis), my ¼ M y =M py An incremental–iterative algorithm using the generalized displacement control method proposed by Yang and Shieh [17] is applied to trace the equilibrium path of the frame structure under static loading Its significant feature is the general numerical stability and efficiency in solving the nonlinear problem The incremental equilibrium equation of structure can be rewritten for the jth iteration of the ith incremental step as i ^ ½K ijÀ1 fDDij g ẳ lj fPg ỵfRij1 g 12ị fDDij g where is the tangent stiffness matrix, is the ^ is the reference load vector, displacement increment vector, fPg i fRijÀ1 g is the unbalanced force vector, and lj is the load increment parameter For convenience, Eq (12) can be decomposed into the following equations: ^ ^ i g ẳ fPg ẵK ij1 fDD j 13ị i ẵK ij1 fDDj g ẳ fRij1 g i i 14ị i ^ g ỵ fDD g fDDij g ẳ lj fDD j j 17ị i The load increment parameter lj is an unknown It is determined from a constraint condition For the first iterative step i (j ¼1), the load increment parameter lj is determined based on the Generalized Stiffness Parameter (GSP) as pffiffiffiffiffiffiffiffiffiffiffi ð18Þ li1 ¼ l11 jGSPj where l1 is an initial value of load increment parameter, and the GSP is defined as T GSP ¼ ^ g fDD ^ g fDD 1 T ^ iÀ1 g fDD ^i g fDD 1 ð19Þ i For the iterative step (j Z2), the load increment parameter lj is calculated as T lij ¼ À ^ iÀ1 g fDDi g fDD j iÀ1 T i ð20Þ ^ g fDD ^ g fDD j ^ iÀ1 g is the displacement increment generated by the where fDD ^ at the first iteration of the previous (i À 1) reference load fPg ^ i gand fDDi g denote the displacement incremental step, and fDD j j increments generated by the reference load and unbalanced force vectors, respectively, at the jth iteration of the ith incremental step, as defined in Eqs (13) and (14) 11ị Nonlinear analysis algorithm ẵK ij1 ^ fP ij g ẳ fPij1 g ỵ lj fPg The terms ZA and ZB in Eq (2) are scalar parameters that allow for gradual inelastic stiffness reduction of the element associated with plastification at ends A and B These terms are equal to 1.0 when the element is elastic, and zero when a plastic hinge is formed The parameter Z is assumed to vary according to the parabolic function as The total applied load vector at the jth iteration of the ith ^ as incremental step relates to the reference load vector fPg i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kn ¼ jP j=EIn ; Et is the AISC-LRFD column tangent modulus (Chen et al [7]) considering the effects of residual stresses and initial geometric imperfection and is defined as Et ¼ E ð16Þ fPij g if P o0 kn L sinðkn Lịkn Lị2 > : 22 coshk n Lị ỵ kn L sinhðkn LÞ Once the displacement increment vector fDDij g is determined, the total displacement vector fDij g of the structure at the end of jth iteration can be accumulated as ð15Þ Verification A computer program written in Fortran programming language is developed based on the proposed numerical procedure to predict the second-order inelastic response of space semi-rigid steel frames under static loading It is verified for validity using the results from existing studies through four following numerical examples 4.1 Vogel portal frame Vogel [18] presented this portal rigid frame as the European calibration frame for static inelastic analysis The initial out-ofplumb straightness and the ESSC residual stress model were assumed for the frame and its members (Fig 1) The rigid beam-to-column connections were replaced by semi-rigid ones to study the second-order inelastic behavior considering the connection rigidity by Chen and Kim [19] using the plastic-hinge method The values of three parameters for Kishi–Chen power model of these semi-rigid connections are: Rki ¼280,000 kipUin/ rad, Mu ¼1,250 kipUin, and n ¼0.98 The results of the proposed program using Orbison yield surface and existing studies in predicting the second-order inelastic response of frames with rigid, semi-rigid, and hinged beam-to-column connections shown in Fig 2, in which NASF is a nonlinear finite element program for second-order spread-of-plasticity analysis of semi-rigid planar steel frames (Nguyen [20]) It can be seen that the nonlinear load–displacement curves agree well C Ngo-Huu et al / Thin-Walled Structures 60 (2012) 98–104 Δ P 4.2 Stelmack two-storey frame P H This one-bay two-storey flexibly connected steel frame was tested by Stelmack [21] and is selected as a benchmark frame in the present study (Fig 3) The A36 W5 Â 16 hot-rolled profile was used for all frame members The connections used in the frame were bolted top and seat angles connections of L4 Â Â 1/2 made of A36 with bolt fasteners of A325 3/4 in D, and its experimental moment–rotation relationship is shown in Fig Gravity loads was first applied at third points of the beam of the first floor, and then a lateral load was applied as the second loading sequence The three parameters of the Kishi–Chen power model are determined by a curve-fitting as follows: Rki ¼40,000 kipUin/rad, Mu ¼220 kipUin, and n ¼0.91 The moment–rotation relationship of the connection by the experiment and curve-fitting data result in a good agreement as shown in Fig The lateral load– displacement curves obtained by the proposed program using 5.0 m HEB300 HEB300 HEA340 ψ = 1/400 ψ 4.0 m Fig Vogel portal frame with semi-rigid connections Fig Moment–rotation behavior of Stelmack two-storey frame Fig Load–roof displacement response of Vogel portal frame (PZ: plastic-zone method, PH: plastic-hinge method) H W5x16 W5x16 P = 10.7 kN (2.4 k) P 1.7 m (66 in) C-1/2 W5x16 C-1/2 Δ 2H W5x16 0.9 m (36 in) C-3/4 0.9 m (36 in) E = 29,000 ksi fy = 36 ksi 1.7 m (66 in) W5x16 0.9 m (36 in) W5x16 C-3/4 2.7 m (108 in) Fig Stelmack two-storey frame 101 Fig Load–displacement response of Stelmack two-storey frame 102 C Ngo-Huu et al / Thin-Walled Structures 60 (2012) 98–104 Δ P 4.3 Liew portal frame P H Liew et al [22] performed a series of tests on a variety of portal frames and their joints in order to provide test results for calibration of analysis and design methodology of semi-rigid steel frames The SRF3 portal semi-rigid frame under non-proportional loading shown in Fig is used in this research to validate the reliability of the proposed program in predicting the nonlinear behavior of steel frames The vertical load P and horizontal load H are first applied proportionally until reaching the values of P¼612 kN and H¼29 kN The horizontal load H is then increased until the frame collapses while the gravity loads are kept B2 3.025 m C2 C2 E = 200 GPa 1.4 x 1.86 k/ft Δt 2.800 m 144 in C2: 209.6x205.2x9.3x14.2; fy= 336 MPa Fig Liew SRF3 portal frame W8x31 B2: 256.0x146.4x6.4x10.9; fy= 345 MPa C-1/2 W16x31 W8x31 C-1/2 ψ ψ 1.4 x 2.70 k/ft Δb Rki (kN m/rad) n Column-to-base Beam-to-column 430 300 321,000 25,000 0.265 0.3 C-3/4 W18x35 ψ = 1/400 E = 29,000 ksi fy = 36 ksi C-3/4 W8x31 Mu (kN m) W8x31 Connection 144 in Table Curve-fitting connection parameters of Liew SRF3 portal frame ψ 288 in Fig Two-storey semi-rigid frame Table Connection parameters of Chen two-storey semi-rigid frame Connection type Mu (kips in.) Rki (kips in./rad) n C-1/2 C-3/4 841 1,773 205,924 954,014 1.27 0.81 Fig Moment–rotation relations of SRF3 portal frame Fig Load–displacement response of Liew SRF3 portal frame Orbison yield surface and experimental work compare well in Fig As a result, the proposed analysis is adequate in predicting the behavior and strength of semi-rigid connections Fig 10 Load–displacement response of two-storey semi-rigid frame (PH: plastichinge method) C Ngo-Huu et al / Thin-Walled Structures 60 (2012) 98–104 103 structure is modeled, analyzed and designed using their advanced analysis program with the consideration of the connection flexibility (Fig 9) The three parameters of Kishi–Chen power model of the connections C-1/2 and C-3/4 were computed by Chen et al [7] based on the steel properties and geometrical variables of the connections and are presented in Table Using same AISC-LRFD yield surface, the proposed load-deflection curves for the top and bottom floors under gravity loads match well with those of Chen et al [7] in the beginning but abruptly stop while the behavior of Chen frame proves to be more ductile (Fig 10) The reason is that the frame analyzed by the proposed analysis collapsed as soon as the beam mechanism forms in all beams constant Based on the test data on moment–rotation relations of column-to-base and beam-to-column joints [22], two groups of three parameters of Kishi–Chen power model are obtained by the curve-fitting technique for those joints (Table 1) and the comparison of test and proposed curve-fitting curves is shown in Fig It can be seen in Fig that the initial curve, the ultimate load value and the ductility of test and proposed results correlates well 4.4 Chen two-storey frame This one-bay two-storey steel frame was presented by Chen et al [7] as a design example to show how an actual steel Y W12x53 W12x87 W12x26 W12x53 7.315 m (288 in) W12x26 W12x26 W12x26 X 7.315 m (288 in) 7.315 m (288 in) Z W12x87 W12x87 W10x60 W12x120 W10x60 W12x120 W12x87 W12x87 H = x 3.658 m = 21.948 m (864 in) W10x60 W10x60 A Y X Fig 11 Orbison six-storey frame (a) Plan view and (b) perspective view 104 C Ngo-Huu et al / Thin-Walled Structures 60 (2012) 98–104 connection element comprising of six translational and rotational springs in order to connect six degrees of freedom of two identical nodes is developed to simulate the rigidity of the steel beamto-column connection The Kishi–Chen power model is used for rotational spring to model the bending moment transference The generalized displacement control method is applied to solve the nonlinear equilibrium equations in an incremental–iterative scheme Through some numerical examples, it is verified that the proposed program can accurately predict the combined effects of geometric and material nonlinearities together with the connection rigidity for space steel frames Acknowledgments Fig 12 Load–displacement response at Y direction of Orbison six-storey frame This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No 2011-0030847) and the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No 20104010100520) 4.5 Orbison six-storey frame References Fig 11 shows a six-storey space rigid frame analyzed previously by Orbison et al [16] using the plastic hinge approach and recently by Chiorean [12] the beam–column method considering the spread-of-plasticity along the member length A36 steel with the yield stress of 250 MPa and Young’s modulus of 206,850 MPa is used for all members Uniform floor load of 9.6 kN/m2 is converted into equivalent concentrated loads on the top of the columns Wind loads are simulated by point loads of 53.376 kN in the Y-direction at every beam–column joints One proposed element per member is used to model this structure Aside from the rigid frame case, two more cases of the frames with linear and nonlinear semi-rigid connections are proposed and investigated by Chiorean [12] The bolted top and seat angles connections were assumed for all beam-to-column connections of the frame and their parameters using Kishi–Chen power model are as follows: (1) for W12 Â 53 and W12 Â 87 beams framing about the major-axis of columns: the fixity factor g ¼0.86, ultimate moment Mu ¼300 kN m, and shape factor n ¼1.57; (2) for W12 Â 26 beams framing about the weak-axis of columns: the fixity factor g ¼0.86, ultimate moment Mu ¼200 kN m, and shape factor n¼ 0.86 From the above fixity factor value, the initial connection stiffness is calculated by the following formulation [12]: Rki ẳ 4EI0 3g L 41gị ð21Þ where I0 is the moment of inertia of the beam cross-section The load–displacement curves at point A at the roof predicted by the proposed program using Orbison yield surface for rigid, linear semi-rigid, and nonlinear semi-rigid frames compare well with those of Chiorean [12] (Fig 12) It can be concluded that the proposed program is reliable in predicting the nonlinear inelastic behavior of space steel semi-rigid frames Conclusions A simple and efficient numerical procedure using beam–column method is proposed for second-order inelastic analysis of space semirigid steel frames under static loading An independent zero-length [1] Frye MJ, Morris GA Analysis of flexibly connected steel frames Canadian Journal of Civil Engineering 1975;2(3):280–91 [2] Chen WF, Kishi N Semirigid steel beam-to-column connections: data base and modeling Journal of Structural Engineering 1989;115(1):105–19 [3] Azizinamini A, Radziminski JB Static and cyclic performance of semirigid steel beam-to-column connections Journal of Structural Engineering 1989;115(12):2979–99 [4] Foley CM, Vinnakota S Inelastic behavior of multistory partially restrained steel frames Part I Journal of Structural Engineering 1999;125(8):854–61 [5] Foley CM, Vinnakota S Inelastic behavior of multistory partially restrained steel frames Part II Journal of Structural Engineering 1999;125(8):862–9 [6] Hsieh S-H, Deierlein GG Nonlinear analysis of three-dimensional steel frames with semi-rigid connections Computers and Structures 1991;41(5): 995–1009 [7] Chen WF, Goto Y, Liew JYR Stability design of semi-rigid frames John Wiley and Sons; 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Nonlinear static and cyclic analysis of steel frames with semi-rigid connections Elsevier; 2000 [9] Kim SE, Choi SH Practical advanced analysis for semi-rigid space frames International Journal of Solids... behavior of space steel semi-rigid frames Conclusions A simple and efficient numerical procedure using beam–column method is proposed for second-order inelastic analysis of space semirigid steel frames. .. calibration of analysis and design methodology of semi-rigid steel frames The SRF3 portal semi-rigid frame under non-proportional loading shown in Fig is used in this research to validate the