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Journal of Constructional Steel Research 121 (2016) 413–426 Contents lists available at ScienceDirect Journal of Constructional Steel Research Second-order plastic-hinge analysis of planar steel frames using corotational beam-column element Tinh-Nghiem Doan-Ngoc a,b, Xuan-Lam Dang a, Quoc-Thang Chu c, Richard J Balling d, Cuong Ngo-Huu a,⁎ a Faculty of Civil Engineering, University of Technology, VNU-HCM, 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Viet Nam Department of Civil Engineering and Applied Mechanics, Ho Chi Minh City University of Technology and Education, Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Viet Nam Department of Civil Engineering, International University, VNU-HCM, Thu Duc District, Ho Chi Minh City, Viet Nam d Department of Civil and Environmental Engineering, Brigham Young University, Provo, UT 84602, United States b c a r t i c l e i n f o Article history: Received November 2015 Received in revised form 15 February 2016 Accepted 11 March 2016 Available online 19 March 2016 Keywords: Plastic-hinge Corotational element Nonlinear analysis Steel frames a b s t r a c t A new beam-column element for nonlinear analysis of planar steel frames under static loads is presented in this paper The second-order effect between axial force and bending moment and the additional axial strain due to the element bending are incorporated in the stiffness matrix formulation by using the approximate seventhorder polynomial function for the deflection solution of the governing differential equations of a beam-column under end axial forces and bending moments in a corotational context The refined plastic-hinge method is used to model the material nonlinearity to avoid the further division of the beam-columns in modeling the structure A Matlab computer program is developed based on the combined arc-length and minimum residual displacement methods and its results are proved to be reliable by modeling one or two proposed elements per member in some numerical examples © 2016 Elsevier Ltd All rights reserved Introduction In the nonlinear analysis of steel structures, the beam-column method has been considered as the simple and effective one in modeling the second-order and inelastic effects and its results are verified to be accurate enough for practical design application as studied by Lui and Chen [1], Liew et al [2], Chan and Chui [3], Thai and Kim [4], Ngo-Huu and Kim [5], etc However, the use of the accurate stability functions obtained from the closed-form solution of the beam-column under end axial forces and bending moments can lead to some difficulties in derivation of the stiffness matrix formulation, especially in corotational context Chan and Zhou [6] proposed the approximate fifth-order polynomial displacement function of the beam-column element and formulated the element stiffness matrix considering the second-order effect by principle of stationary total potential energy The advantage of using this polynomial function is its simplicity in formulation ⁎ Corresponding author E-mail address: ngohuucuong@hcmut.edu.vn (C Ngo-Huu) http://dx.doi.org/10.1016/j.jcsr.2016.03.016 0143-974X/© 2016 Elsevier Ltd All rights reserved while its accuracy is still maintained as the use of closed-form stability functions The corotational method has been widely used due to its efficiency in deriving the formulation of geometrically nonlinear beam-column element for elastic analysis (Nguyen [7], Le et al [8]) and inelastic analysis (Balling and Lyon [9], Thai and Kim [10], Saritas and Koseoglu [11]) This study proposes a new seventh-order polynomial displacement function for the approximate solution of the governing differential equations to formulate the element stiffness matrix considering the second-order effect following the beam-column theory in corotational context as presented by Balling and Lyon [9] The bowing effect is integrated in the formulation to consider the change in element length due to the bending of the element The refined plastic-hinge method is used to simulate the inelastic behavior of the steel material as lumped concept To solve the system of equilibrium nonlinear equations, the arc-length combined with minimum residual displacement methods are employed due to their robustness in nonlinear analysis application A computer program is developed using the Matlab programing language to automate the analysis of nonlinear behavior of planar steel frames under static loads The obtained analysis results are compared to those of existing studies to verify the reliability and effectiveness of the proposed program 414 T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 Fig Simply-supported beam-column element Formulation 2.1 Stability functions Consider a simply supported planar beam-column element under end axial force and bending moments as presented in Fig The governing differential equations of the element using second-order Euler beam theory are ! ! d ΔðxÞ d ΔðxÞ −F ¼ 0: EI dx4 dx2 ð1Þ The closed-form solution to the differential equations leads to following end moment-end rotation relationship (Oran [12]) & M1 M2 ' ¼ ' θ1 : θ2 ð2Þ λ sin λ−λ2 cos λ 2−2 cos λ−λ sin λ λ2 −λ sin λ ¼ 2−2 cos λ−λ sin λ ð3Þ EI s11 L0 s21 s12 s22 !& For compressive F b s11 ¼ s22 ¼ s12 ¼ s21 where λ ¼ L0 qffiffiffiffiffi j Fj EI For tensile F N λ2 cosh λ−λ sinh 22 cosh ỵ sinh sinh : ẳ 22 cosh ỵ sinh s11 ẳ s22 ẳ s12 ẳ s21 4ị For the simplicity in mathematical handling, instead of using the closed-form solution with above-mentioned complicated stability functions, the deflection solution is assumed in following seventh-order polynomial function xị ẳ a7 x7 ỵ a6 x6 ỵ a5 x5 ỵ a4 x4 ỵ a3 x3 ỵ a2 x2 ỵ a1 x ỵ a0 : Fig Comparison of proposed and closed-form stability functions ð5Þ T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 415 Fig Initial and displaced positions of the beam-column element The coefficients are determined from the compatibility and equilibrium conditions as follows xịxẳ0ị ẳ 6ị xịxẳL0 ị ẳ 7ị dxị ¼ θ1 dx ðx¼0Þ ð8Þ dΔðxÞ ¼ θ2 dx xẳL0 ị 9ị ! d xị EI dx2 ! d ΔðxÞ EI dx3 EI EI M ẳ EI M ẳ EI M1 ỵ M2 ị xị L M ẳ Fxị xẳL20 ỵ xẳ 20 L0 ẳF xẳ0ị ! d xị dx3 ! d xị dx3 L xẳ 20 L xẳ 20 M1 ỵ M2 ị dxị ỵ L0 dx xẳ0ị ẳF 10ị 11ị dxị M1 ỵ M2 ị Lỵ dx L0 xẳ 12ị ẳF xẳL0 ị ! d ΔðxÞ dx2 ! d ΔðxÞ dx2 dΔðxÞ ðM ỵ M ị ỵ dx xẳL0 ị L0 13ị 14ị xẳ0ị : 15ị xẳL0 ị Fig Cantilever with an end point load 416 T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 Fig Large displacement analysis of cantilever with an end point load The coefficients are solved from Eqs (6) through (13) and the end moment-end rotation relationship is identical as Eq (2) with following sij functions For compressive F ≤ 5q 1404q2 ỵ 86400q1209600 s11 ẳ s22 ẳ 940qị84011qị q ỵ 252q2 25920q ỵ 1209600 s12 ẳ s21 ẳ 1840qị84011qị 16ị where q ¼ λ2 ¼ jEIFj L20 For tensile F N 5q3 ỵ 1404q2 ỵ 86400q ỵ 1209600 940 2ỵ qị840 ỵ 11qị q 252q 25920q1209600 : ẳ 1840 ỵ qị840 ỵ 11qị s11 ¼ s22 ¼ s12 ¼ s21 ð17Þ Fig shows a comparison of the proposed and closed-form stability functions and it can be seen that all curves are almost identical The differentiations of the proposed stability functions s11 , s12 , s21 and s22 with respect to q are as follows Fig Beam-column with end point loads T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 417 Fig Deflections at free end of column For F ≤ ds11 dq ds12 dq ẳ 11q4 2560q3 ỵ 270144q2 13547520q ỵ 270950400 ds22 ẳ dq 940qị2 84011qị2 18ị ẳ 11q4 2560q3 ỵ 63360q2 9676800q ỵ 677376000 ds21 ẳ dq 1840qị2 84011qị2 19ị ẳ 11q4 ỵ 2560q3 ỵ 270144q2 ỵ 13547520q ỵ 270950400 ds22 ẳ dq 940 ỵ qị2 840 ỵ 11qị2 20ị ẳ 11q4 ỵ 2560q3 þ 63360q2 þ 9676800q þ 677376000 ds21 ¼− dq 18ð40 þ qÞ2 ð840 þ 11qÞ2 ð21Þ For F N ds11 dq ds12 dq As q goes to zero, Eqs (16) through (21) become to s11 ¼ s22 ¼ ds11 dq ds11 dq s12 ¼ s21 ¼ ¼ ds22 ¼− 15 dq ¼ ds22 ẳ 15 dq 22ị ds12 ds21 ¼ ¼ 30 dq dq ð F ≤ 0Þ ð23Þ ds12 ds21 ¼ ¼− 30 dq dq ð F N0Þ: ð24Þ These results are identical to those obtained by using the approximate deflection function as common Hermite third-order polynomial function for the beam element Fig William's toggle frame 418 T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 Fig Load-deflection curves of William toggle frames 2.2 Axial strain attributed to element bending Axial force considering the axial strain attributed to element bending because of end rotations is shown as follows 0L ZL0 2 Z0 ZL0 2 EA @ dδ dΔ EA EA d dx ỵ dxA ẳ L0 ỵ dx: Fẳ L0 dx dx L0 2L0 dx 0 ð25Þ For the closed-form solution, the axial force can be presented by stability functions as AE ΔL0 −EA L0 AE ΔL0 þ EA F¼ L0 F¼ ! ds11 ds12 ds22 θ1 þ θ1 θ2 þ θ ð F ≤ 0Þ dq dq ! dq : ds11 ds12 ds22 ỵ ỵ F N0ị dq dq dq ð26Þ The axial force of the proposed approach is also derived from above relations 2.3 Plastic hinge Let η1 and η2 (0≤η1, η2 ≤ 1) be the inelastic ratios of the end sections of the beam-column element in which the values of one and zero indicate fully elastic and plastic-hinge states, respectively, and a value between zero and one indicates the partially plastic state of the section Eqs (2) and (26) are modified to account for the presence of the plastic hinges as follows & M1 M2 ' ¼ EI s1p L0 s2p s2p s3p !& θ1 θ2 ' ð27Þ Fig 10 Pinned-ended column under axially compressed load T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 419 Table Buckling loads of pinned-ended column λc L (mm) P/Py Residual stress ignored 1141.97 2283.95 3425.92 4567.84 6851.90 9135.78 11419.73 13703.67 15987.62 18271.56 20555.51 22839.45 F¼ 0.25 0.50 0.75 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Residual stress considered Euler Ngo-Huu & Kim Proposed Diff (%) CRC Ngo-Huu & Kim Proposed Diff (%) 16 1.7778 1.0000 0.4444 0.2500 0.1600 0.1111 0.0816 0.0625 0.0494 0.0400 0.9870 0.9870 0.9870 0.9870 0.4450 0.2500 0.1600 0.1120 0.0820 0.0630 0.0500 0.0400 1.0000 1.0000 1.0000 0.9973 0.4433 0.2494 0.1597 0.1110 0.0816 0.0625 0.0494 0.0400 – – – 0.27 0.25 0.24 0.19 0.09 0.00 0.00 0.00 0.00 0.9844 0.9375 0.8594 0.7500 0.4444 0.2500 0.1600 0.1111 0.0816 0.0625 0.0500 0.0400 0.987 0.936 0.861 0.76 0.445 0.25 0.16 0.112 0.082 0.063 0.0496 0.0402 0.9843 0.9373 0.8590 0.7494 0.4433 0.2494 0.1597 0.1110 0.0816 0.0625 0.0494 0.0400 0.01 0.02 0.05 0.08 0.25 0.24 0.19 0.09 0.00 0.00 1.20 0.00 ! ds2p AE ds1p ds3p ỵ ỵ : L0 ặ EA L0 dq dq dq 28ị The ± symbol in Eq (28) is assigned as “+” when F N and “–” when F ≤ 0; the modified stability functions s1p, s2p and s3p are determined from the stability functions and inelastic ratios as proposed by Chan and Chui [3] as s1p ¼ η1 s11 − Á s212 À 1−η2 s11 ! s2p ¼ η1 η2 s12 s3p ¼ η2 s22 − ! Á s221 À 1−η1 : s22 ð29Þ The differentiations of the modified stability functions with respect to q are as follows ds1p dq ds2p dq ds3p dq ds12 ds11 2s11 s12 −s212 ds11 Á7 dq dq À ¼ η1 1−η2 dq − s211 ¼ η1 η2 ds12 dq ð30Þ ð31Þ ds21 ds22 2s22 s21 −s221 ds22 Á7 dq dq À ¼ η2 1−η1 dq − 5: s222 ð32Þ Clearly, the mathematic handling of the modified stability functions and their differentiations with the use of the approximate seventh-order deflection function is much simplified Fig 11 Strength curve of pinned-ended column 420 T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 Fig 12 One-bay two-storey frame with pinned support The refined plastic-hinge method presented by Liew et al [2] is used in this research The inelastic ratio η =4β(1− β) is determined through inelastic parameter β, where β is calculated based on the following strength curve of Orbison [13] β ¼ 1:15 2 2 2 2 P M P M ỵ ỵ 3:67 : Py Mp Py Mp ð33Þ To consider the gradual plasticity due to the effect of the axial force on the presence of residual stresses in the section the tangent modulus Et proposed by the Column Research Council is used as follows P Et ≤ 0:5 for ¼1 Py E : Et P P P ¼4 1− N0:5 for Py Py E Py ð34Þ 2.4 Corotational element stiffness matrix Consider the change in geometry of the beam-column element shown in Fig Fig 13 Elastic load-deflection curves T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 421 Fig 14 Inelastic load-deflection curves The original length L0 and the deformed length L of the element: L0 ẳ Lẳ q xB xA ị2 ỵ zB zA ị2 35ị q xB ỵ u4 xA u1 ị2 ỵ zB ỵ u5 zA u2 ị2 : 36ị The nodal rotations of the element: ẳ u3 ị 37ị ẳ u6 ị 38ị where sin ẳ z ỵ u z u B A L 39ị cos ẳ x ỵ u −x −u B A L ð40Þ Fig 15 Two-bay four-storey frame 422 T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 Fig 16 Lateral deflection of top right joint α ¼ sin−1 zB −zA : L0 ð41Þ The differentiations of L, θ1, θ2, sinα, cosα, F, M1 and M2 with respect to nodal displacements ui(i = 1, , 6) are as follows & & ∂L ∂u ' ∂θ1 ∂u ¼ f − cos α & ' ¼ − sin α L − sin α cos α cos α L sin α L sin α − cos α L gT ð42Þ 'T ð43Þ Fig 17 One-bay four-storey frame T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 423 Table Limit load factor of proposed program and the others of four-storey frame Value of lateral load (H) Ratio of limit inelastic load 0.10P 0.24P 0.50P & & & ∂θ2 ∂u & ' ¼ ∂ sin α ∂u ¼ ∂M2 ∂ui & ' ∂M1 ∂ui sin α L & ' ∂ cos α ∂u ∂F ∂ui − ¼ ¼ cos α L sin α cos α L sin2 α − L Kassimali Yoo and Choi Proposed Difference with Yoo and Choi's results (%) 1.687 1.502 1.075 1.660 1.479 1.062 1.656 1.465 1.045 −0.24 −0.95 −1.60 sin α L cos2 α − L sin α cos α L − cos α L 'T ð44Þ sin α cos α − L sin2 α L cos2 α L sin α cos α − L 'T ð45Þ 'T ! ∂s1p s2p s2p s3p EA L ặ EA ỵ ỵ ỵ L0 ui q q ui q q ui 46ị 47ị ẳ ∂s1p ∂s2p EI ∂θ1 ∂θ2 ∂F s1p þ s2p Ỉ L0 θ1 þ θ2 L0 ∂ui ∂ui q q ui 48ị ẳ s2p s3p EI F s2p ỵ s3p ặ L0 : ỵ L0 ui ui q q ∂ui ð49Þ The ± symbol in above equations are assigned as “+” when F N and “–” when F ≤ The nodal element resistance vector in local coordinate system is fzg ẳ F M ỵ M ị L M1 F M1 ỵ M2 ị L !T M2 : ð50Þ The nodal element resistance vector in global coordinate system is fZg ẳ ẵT T fzg, where [T] is the transformation matrix of planar element The local and global element tangent stiffness matrix ½kT and [KT], respectively, are determined as follows h Ti ! ! h i ∂ T ∂fZ g ∂fzg T T TA @ k T ¼ ½T ½T ¼ ½T ½T fzg ẵT ỵ fug fug fug h i ẵK T ẳ ẵT T kT ẵT : 51ị 52ị Fig 18 Load-displacement curve of the four-storey frame 424 T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 Applying the results of above differentiations with respect to nodal variables ui(i =1, ,6), the local element stiffness matrix ẵkT ẳ ẵkG ỵ ẵk is derived with following ½kG and ½kθ matrices A I 6 6 h i EI 6 kG ¼ L0 6 6 6 G1 ỵ s1p ỵ 2s2p ỵ s3p L2 s1p ỵ s2p L s1p Á − A I À −G1 s1p ỵ 2s2p ỵ s3p L2 s1p ỵ s2p L A I G1 þ s1p þ 2s2p þ s3p −T L0 T T ỵ T ị L2 T ỵ T ị T ỵ L L0 T 21 T1 Á L2 sym: ðT ỵ T ị T3 L 6 T 6 h i kθ ¼ EA6 6 6 6 sym: T ỵ T ỵ T Þ L −T − L0 T ðT ỵ T ị L2 T ỵ T Þ T3− L T4 Á 7 7 7 7 s2p 7 7 7 s2p ỵ s3p L s3p s2p ỵ s3p L 53ị T 7 L0 T T ỵ T Þ 7 L2 7 L0 T T 7 7 T2 7 L0 T T ỵ T ị 7 − L L0 T 54ị where G1 ẳ AI LL L ị and s1p s2p ỵ T1 ẳ ặ q q 55ị s2p s3p T2 ẳ ặ ỵ q q 56ị T3 ẳ T4 ẳ M ỵ M ị 57ị EAL2 T 1 ỵ T 2 L0 T ỵ T ị2 ỵ : 2L L2 58ị The symbol in Eqs (55) and (56) are assigned as “+” when F N and “–” when F ≤ 2.5 Nonlinear solution algorithm The arc-length method combined with minimum residual displacement method proposed by Chan and Zhou [6] is used as nonlinear solution algorithm in this research to solve the nonlinear equation system The incremental equilibrium equation is presented as follows: ẩ ẫ fu ỵ ug ẳ ẵK T P ỵ P 59ị where {P}= {P Z} is applied incremental load vector for the first iteration or the unbalanced force vector in second iteration onward; {Δu} = corresponding displacement increment due to this force; fΔPg = a force vector parallel to the applied load vector; fΔug = conjugate displacement solved; Δλ = a load corrector factor for imposition of the constraint condition For the first iterative step arc length Δλ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : fΔugT fΔug From the second step, Δλi is determined from the following condition h i T equilibrium error ị fi u ỵ ui g fi u ỵ ui g ẳ ẳ 0: i i 60ị 61ị Simplifying the Eq (61), we have: i ẳ fΔugTi fΔug fΔugT fΔug ði ≥ 2Þ: ð62Þ T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 425 Numerical examples 3.5 Single-bay two-storey frame A computer program using the above-mentioned nonlinear solution algorithm is developed using Matlab for nonlinear inelastic analysis of planar steel frames subjected to static loads The analysis results are compared with results from the literature to validate accuracy on the following numerical examples The single-bay two-storey frame with pinned supports as shown in Fig 12 was analyzed by Lui and Chen [18] and later by Chan and Chui [3] Each beam and column member of the frame was respectively modeled by two and one elements by Chan and Chui while the frame member is modeled by one element herein The analysis results presented in Fig 13 and Fig 14 show that the elastic and inelastic loaddeflection curves of the frame are almost identical The elastic and inelastic limit loads of Chan and Chui are 746 kips and 417 kips while the corresponding results of proposed program are 732 kips and 421 kips with the respective differences of 1.9% and 1.0% 3.1 Cantilever beam with an end point load The cantilever beam with geometric and material properties shown in Fig is used for nonlinear elastic analysis in this research This example was firstly introduced by Bisshopp and Drucker [14] with exact solution and then it has been analyzed by many researchers by modeling with two or more elements per member for accuracy comparison The cantilever beam is modeled by two proposed elements herein and by two and twenty BEAM3 elements by ANSYS for verification purpose The results of Bisshopp and Drucker, the proposed method, and ANSYS program are shown in Fig It can be seen that the obtained results from proposed method are acceptable in accuracy in comparison with the closed-form solution of Bisshopp and Drucker and the ANSYS result with 20 beams in modeling 3.2 Cantilever column with end eccentric axial point load The cantilever column subjected to eccentric axial load at its free end shown in Fig was firstly introduced by Wood and Zienkiewicz [15] by using five paralinear elements and was recently analyzed by Nguyen [7] by using three corotational elements The column is modeled by two proposed elements The nonlinear elastic analysis results shown in Fig prove that the displacement responses of proposed method have good agreement compared to those of existing studies 3.3 William's toggle frame The William toggle frame shown in Fig was analyzed by many researchers to verify their analysis methods in predicting large displacement behavior at the system level For existing studies, the member was generally modeled from two or more elements for nonlinear elastic analysis The William toggle frame was analyzed by Chan and Chui [3] with two different pinned and fixed support conditions Only one proposed element is used herein for modeling each frame member Fig shows that a good agreement is seen between the proposed result and existing ones 3.4 Pinned-ended column The pinned-ended column under top axial force shown in Fig 10 was analyzed by Ngo-Huu and Kim [16] by the fiber hinge method in which the column was divided into three elements, one middle elastic element using the conventional stability function and two end fiber-hinge elements Table presents the buckling load results obtained by the proposed program, the Euler's theoretical exact solution, CRC column curve (Chen and Lui [17]), and Ngo-Huu and Kim's fiber hinge element with a large range of the column length A comparison of results from the proposed program and Ngo-Huu and Kim's analysis result is also presented and the maximum difference of about 1.2% is found The strength curves corresponding to the slenderness parameter about the weak axis are shown in Fig 11 and it can be seen that the curves are almost identical This example demonstrates the capacity of the proposed program in predicting the elastic and inelastic buckling loads of the column 3.6 Two-bay four-storey frame The two-bay four-storey frame shown in Fig 15 was analyzed by Kukreti and Zhou [19] by using the refined plastic-hinge method with LRFD bilinear plastic strength curve It is modeled herein by one element per column and two elements per beam The load-lateral deflection curves of the frame from the proposed program and Kukreti and Zhou's analysis are presented in Fig 16 It can be seen that the curves are relatively matched The limit load ratio of Kukreti and Zhou's analysis is 1.831 while that of the program is 1.834, which are different by only about 0.16% 3.7 One-bay four-storey frame The one-bay four-storey frame shown in Fig 17 was analyzed by Kassimali [20] using rigid-hinge method and later by Yoo and Choi (2008) [21] using inelastic buckling method based on bilinear and linear strength curves in order to compare the nonlinear behavior and ultimate loads with varying lateral loads of H = 0.1P, 0.24P, and 0.5P One and two proposed elements are used to model each column and beam member, respectively The analysis results of the proposed element and Kassimali and Yoo and Choi are shown in Table and Fig 18 It can be seen that the proposed element predicts the ultimate strength of the frame very well but there are the slight differences in load-lateral deflection curves Conclusion The seventh-order polynomial function is assumed as the deflection solution for the governing second-order differential equations of the beam-column member under end axial forces and bending moments and it is applied in the formulation of element stiffness in the corotational context The corotational element also integrates the additional axial strain caused by the bending of the element and the inelastic simulation by refined plastic-hinges lumped at both ends A Matlab program is developed based on the arc-length combined with minimum residual displacement methods to solve the system of nonlinear equilibrium equations step by step The analysis results of numerical examples prove that and the developed program from the proposed corotational element is capable of accurately predicting the nonlinear behavior of structural members and frames under the static loads Notations The followings notations are used throughout the paper A Sectional area of beam-column member E , Et Elastic and tangent modulus I Moment of inertia of the element section L0 , L Initial and deformed lengths of the element ΔL = L − L0 Change in element length Py , Mp Squash load and plastic moment of the cross-section s11 , s12 , s21 , s22 Elastic stability functions of beam-column element 426 T.-N Doan-Ngoc et al / Journal of Constructional Steel Research 121 (2016) 413–426 s1p , s2p , s3p Inelastic stability function of beam-column element considering end flexibility F , M1 , M2 Axial force and end bending moments ½kT ; ½K T Local and global element tangent stiffness matrices ½kG , ½kθ Local geometric and higher-order geometric tangent stiffness matrices [T] Local-global transformation matrix of planar element fzg; fZg Local and global nodal element resistance vectors {P}, {ΔP} Global total and incremental load vectors {u}, {Δu} Global total and incremental displacement vectors β Inelastic parameter λ Load factor Δ(x) Defection function of beam-column element η1 , η2 Inelastic ratios at left and right end sections θ1 , θ2 Rotations at element left and right ends Acknowledgments 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The corotational element also integrates the additional axial strain caused by the bending of the element and the inelastic simulation... developed using Matlab for nonlinear inelastic analysis of planar steel frames subjected to static loads The analysis results are compared with results from the literature to validate accuracy... Kim, Second-order inelastic dynamic analysis of steel frames using fiber hinge method, J Constr Steel Res 67 (2011) 1485–1494 [5] C Ngo-Huu, S.E Kim, Second-order plastic-hinge analysis of space