EUROSTEEL 2017, September 13–15, 2017, Copenhagen, Denmark
An efficient approach to limiting the elastic range in advanced analysis of steel structures via RPHM Ígor José Mendes Lemes, Rafael Cesário Barros, Everton André Pimentel Batelo, Ricardo Azoubel da Mota Silveira* Federal University of Ouro Preto, School of Mines, Department of Civil Engineering, Ouro Preto, MG, Brazil
[email protected],
[email protected],
[email protected],
[email protected]
ABSTRACT Over the years many research on the nonlinear behavior of steel structures has been made. The Refined Plastic Hinge Method (RPHM) is a fast and accurate alternative to simulate, in concentrated form, the plasticity of the structural systems materials. In recent studies, there was a loss of accuracy in the analysis of steel structures under minor axis bending, in the conventional method of RPHM where pseudo-springs are used at the ends of the elements to simulate the flexural stiffness degradation. In this approach, the pseudo-springs at the finite elements ends have infinite stiffness in elastic range, with gradual loss of stiffness in elastoplastic stage and a null stiffness in plastic range. For the methodology adjust to this issue, the fictitious springs were replaced by the tangent modulus of elasticity of the material and presented as a viable solution. In this paper, we propose the study of the interaction curves behavior for application in RPHM with pseudo-springs, considering the bending around major and minor axes of inertia, with a greater focus to RPHM using the tangent Young's modulus. The Strain Compatibility Method (SCM) is used, which through a nonlinear procedure, the momentcurvature relationship is defined allowing to obtain the exact point where the cross section starts the yield process. Thus, it is possible to trace the initial yield curve realistically minimizing simplifications. The residual stresses models of AISC-LRFD e ECCS for I sections will be used. The results will be compared with numerical and experimental data available in the literature. Keywords: RPHM, pseudo-springs, initial yield curve, residual stresses 1
INTRODUCTION
Over the years researchers have directed their research into the nonlinear material study of steel structures [1-5]. These studies were responsible for the development of numerical formulations capable of accurately simulating the behavior of these structures. The Refined Plastic Hinge Method (RPHM) emerged as a fast and accurate option for structural simulation outside the laboratories [25]. The classical RPHM approach consists of the use of a hybrid finite element, basically composed of a beam-column element with pseudo-springs elements at its ends [2-5]. Thus, the degradation of the flexural stiffness of the cross section occurs only in the spring elements, characterizing the concentrated plasticity. This degradation occurs within the axial-bending moment diagram. For this, a curve that limits the elastic range and another curve to limit the bearing capacity must be used [5]. The use of these springs generates limitations to the methodology, especially in the simulations of reinforced concrete elements [6-7], and steel elements under minor-axis bending [8-9]. The use of pseudo-springs to simulate the degradation of flexural stiffness suggests a linear elastic regime until the initial yield curve is reached. It then becomes an incompatibility in the simulation of structures with concrete in its composition, since the material under compression has non-linear behavior [7]. For steel structures, Zubydan [8] proposed equations for the variation of the tangent modulus of elasticity as external loads increased. Gonçalves et al. [9] applied these equations to the RPHM by removing the pseudo-springs, so the simulation of the flexural stiffness degradation was no longer done by the classical methodology approach. © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin · ce/papers 1 (2017), No. 2 & 3 https://doi.org/10.1002/cepa.177
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Lemes et al. [10] made comparisons between the use of pseudo-springs and the coupled RPHM approach with the Strain Compatibility Method (SCM). The RPHM / SCM coupling suggests a more realistic simulation of the problem since the behavior of the material is treated explicitly with its constitutive relationship. In one of the simulations, the authors focused on the minor-axis bending of steel structures and verified, in all cases, a more rigid behavior of the structure when the degradation is done by pseudo-springs. This stimulated the study of the limitation of the elastic range of steel structures. Then, Lemes et al. [11] studied the behavior of steel I sections and verified the requirement for revision of the initial yield curves. The symmetry in the four quadrants of the axial force-bending moment diagram, usually applied in the RPHM, remained for the residual stress model of the ECCS [12] when the section is subjected a major-axis bending. In the minor-axis this symmetry does not occur. For this, a significant variation in the form of the initial yield curve for sections with h/b ≤ 1,2 and h/b > 1,2 was also observed. The residual stress model of the AISC-LRFD [13] was also studied. The objective of the present paper is to recover the classical RPHM approach and simulate any problems in steel structures, continuing the previous studies [10-11]. For this, the residual stress models of the ECCS [12] and the AISC-LRFD [13] are again used, however, with global structural analyzes. Structural systems such as isolated steel beams, columns with minor-axis bending and a simple portal frame will be simulated for validation of the proposed numerical formulations. 2
CLASSICAL REFINED PLASTIC HINGE METHOD
The computer program CS-ASA [5] has been under development since 2009 and has already been established for the nonlinear analysis of steel structures. This program uses the Refined Plastic Hinge Method (RPHM) [2] to simulate the concentrated plasticity in the nodal points. To model the structures considered in this work, the study adopts the following assumptions: the elements are initially straight, prismatic, and warpage may be ignored the structure is perfectly locked on to the orthogonal axis (2D problem) effects of localized instability are neglected large displacements and rigid body rotations are allowed; shear strain effects are neglected. In the modeling, the beam-column finite element used is shown in Fig. 1. This element is defined by the nodes i and j. Also illustrated are the internal forces, Mi, Mj and P, as well as their respective degrees of freedom i, j and , referenced to the corrotational system. Null length pseudo-springs are inserted, at the ends of the element, to simulate the degradation of flexural stiffness by parameter S. material nonlinearity
Mi , i i
S si
Mj , j S sj
j
P,
Fig. 1. Hybrid finite element with pseudo-springs
The S parameter is set within three domains (Fig. 1). When in the elastic state, the value of S is assumed to be 1016. Such a state is limited by the beginning of initial yield curve. Upon reaching the plastic state (a situation where the internal forces reach the full yield curve), the flexural stiffness is completely degraded. In this case, S is taken to be numerically equal to 10-10. Between the initial and full yield curves, the loss of stiffness is considered to occur gradually, according to the following equation: S=
EI L
M pr - M M - M er
© Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017)
(1)
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where L is the length of the finite element; Mpr is the full yield bending moment and Mer the initial yield bending moment, which will be defined in Section 3; and EI is the flexural stiffness of the steel section; and is a constant that will be studied in the present work. N
Full yield curve Initial yield curve
Elastoplastic Stage
Plastic stage
Elastic Stage M
Fig. 2. Hybrid finite element with pseudo-springs
For the element shown in Fig. 2, the force-displacement relationship is expressed by [2]: Δδ 0 0 ΔP EA / L 2 Si -Si (k jj +S j ) / β (Si S j kij ) / β Δθi or Δfco Kco Δuco ΔMi = 0 ΔM (Si S j k ji ) / β S j -S 2j (kii +Si ) / β Δθ j j 0
(2)
with = (Si + kii)(Sj + kjj) - kjikij. The subscripts i and j indicate the nodes of the element and co the subscript to the coordinate system used (corrotational); EA is the axial stiffness of the steel crosssection; P and M are, respectively, the incremental axial force and bending moment; and are the increments of axial strain and nodal rotation. The terms kii, kij, kji and kjj are obtained according to the Yang and Kuo’s formulation [14]. When the section plastify, any increase in incremental load causes the internal forces to extrapolate the full yield curve, therefore, violating the cross-section bearing capacity. The formulation used here avoiding this violation is a strategy known as Return Mapping [5]. In this strategy, the element axial force remains constant and returns the internal bending moment to the full yield curve. In other words, Eq. (2) is modified so that in these situations the bearing capacity of the elements is not violated. 3
STRAIN COMPATIBILITY METHOD
The bending moments Mpr and Mer, used in Eq. (1), are obtained in the context of the Strain Compatibility Method (SCM). Compared to the simplified design codes procedures [1, 2], a more realistic approach is produced by the coupling of the deformed shape of section and the constitutive relationships of the materials comprising it. To satisfactorily obtain the strain field in the section, the cross-section discretization is necessary. In the discretization, the strength may be sufficiently evaluated with two-dimensional layers, but this procedure is sensitive to the degree of refinement imposed by the analyst. 3.1 Steel behavior The steel behavior is described by an elastic-perfectly-plastic constitutive relationship (shown in Fig. 3), as described below:
-fy Ea fy
se u y se y y
(3)
se y u
The ECCS [12] uses the residual stress model shown in Fig. 3a. In this model the residual stresses are arranged in all the plates of the cross section by means of a bilinear distribution. Furthermore, © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017)
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the values of r depend directly on the relationship between the height h and the width b of the cross section. The AISC-LFRD follows the proposal of Galambos and Ketter [13] that describes the residual stresses in a similar way at the flanges, but with different maximum values of traction and compression. In the web the distribution is constant as shown in Fig. 3b.
fy
Ea 1
y
u
a)
b)
c)
Fig. 3. a) Steel stress-strain; b) Residual stresses: ECCS [12] model; c) Residual stresses: AISC-LRFD [13] model
3.2 Moment-curvature relationship Once the cross-section is discretized, in using the SCM coupled with the Newton-Raphson Method, there are two relevant variables: the area of the fibers and their respective positions. This second is referred to as the plastic centroid (PC), so as to minimize convergence problems [15]. Figure 4 illustrates the distribution of strains in the steel section from a combination of axial force and bending moment. Note that the total axial strain at the ith fiber, i, is given by a linear function. Therefore: f Fiber i
PC
PC
i
Ai
yi
0 Initial shape
Deformed shape
Fig. 4. Linear strain field around x axis
i 0 ri fyi
(4)
where yi is the distance between the of the analyzed fiber and the cross-section plastic centroids (PCs), 0 is the axial strain in the section CP, ri is the residual strain, and f its curvature. For the matrix notation, the study adopted the following: the variables 0 and f are the positions of the strain vector X = [0 f]T. It is necessary to adjust the vector X until the deformed shape of the section is consistent with the active external forces. This adjustment is made through the iterative process described below. Numerically, it can be said that the balance of the section is obtained when the following equation is satisfied: F X fext fint 0
where the external forces vector is depicted as fext = [N M]T.
© Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017)
(5)
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However, the internal force vector is given by classical integral expressions for the axial force, Nint, and bending moment, Mint. Once areas, Ai, and positions, yi, of each fiber are known, the integral becomes the sum described as:
nfib i Ai i 1 fint n fib i Ai yi i 1
(6)
with nfib being the number of fibers used in the discretization of the cross-section. Fibers stresses, , are dependent on the deformed shape of the cross-section, and are therefore functions of 0 and f. While it is appropriate to initiate the process with X = 0, convergence is achieved only in the first iteration if external forces are null. Thus, for the next iteration, k+1, the strain vector is given by: Xk+1 Xk F' Xk F Xk -1
(7)
where F' is the tangent stiffness matrix of the cross-section or the Jacobian matrix of the nonlinear problem stated in Eq. (5), that is:
Nint F 0 F' X M int 0
Nint f M int f
(8)
The convergence criterion adopted in this work is based on the ratio of the Euclidean norms of the unbalanced force vector, F, and the external forces vector, fext. Thus, this ratio should be less than a tolerance, assumed here to be 10-5. Figure 5 details the flow chart of the process for obtaining the moment-curvature relationship. Input: geometry and material data
Cross section discretization in fibers
Evaluate the plastic centroid (PC)
Translation of reference system for the PC
Update 0 and f (Eq. 7)
Evaluate the axial strain in the layers (i) (Eq. 4)
Initialize 0 e f
Assemble fint (Eq. 6)
Assemble F’ (Eq. 8)
no
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yes
Output: internal forces (fint)
End
Fig. 5. Nonlinear local problem solver – moment-curvature relationship
When, for a given axial force, the maximum bending moment of the moment-curvature is reached, there is a total plastification of the section. It is defined such that a pair of forces is a point on the full yield curve. © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017)
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The initial yield curve also is obtained from the moment-curvature relation. When the first fiber of the section presents axial strain, , greater than the yield strain of the steel, y, the fiber begins to degrade and consequently the section loses stiffness. The moment responsible for this fact is considered the initial yield bending moment. 4
NUMERICAL ANALYSIS
The numerical strategies presented in previous sections are used now to aim at a nonlinear analysis of steel structural systems. Evaluated examples are divided into three groups: isolated beams, columns under minor axis bending and simple steel portal frame. The results are compared with those reported in the literature, obtained numerically. 4.1 Isolated steel beam Liew et al. [16] made a numerical simulation of an isolated steel beam under major-axis bending. It is a steel beam with section WF 8x48 and length (L) equal to 3 m. A concentrated vertical load is applied to L/3 of the left end, as shown in Fig. 6. In this figure, the finite element mesh used in the analyzes (3 elements and 4 nodes) is also shown. The steel used is ASTM A7 with fy = 235 MPa and E = 205 GPa. The both ECCS [12] and AISC-LRFD [13] residual stresses models are used. For this example the h parameter is taken 6 [2]. P
L/3
2L/3
2
1
3
4
u
Fig. 6. Steel beam
The numerical simulation of the steel beam demonstrates the good accurately with the numerical results present in the literature, as seen in Fig. 7. A more rigid structure behavior is highlighted when the initial yield curve (IYC) was obtained with the AISC-LRFD residual stress model. This occurs because the cross section has a square shape h/b = 1.04, so the maximum ECCS [12] residual stresses (equal to 0.5fy) are higher than the maximum AISC-LRFD [13] residual stresses (equal to 0.3fy). For this structure, the bearing capacity was not influenced by the adopted residual stress model. It is verified both in the full yield curve (FYC), the same for both models, as for the critical load applied to the structure. 10 P
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Fig. 7. a) Equilibrium paths of steel beam b) WF 8x48 Full and initial yield curves (FYC and IYC) © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017)
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4.2 Simple columns under minor axis bending A cantilever steel column as shown in Fig. 8 is simulated in this section. This structural element was proposed by Zubydan [8] and also studied by Gonçalves et al. [9]. It is a W21x160 steel crosssection, with fy equal to 250 MPa and Young's modulus taken as 200 GPa, under minor axis bending. H is a incremental load and P is a fixed vertical load, three different analyses are performed: considering P = 0,2Py, P = 0,4Py and P = 0,6Py. Several analyses are done by varying the parameter . P H
4
P
u
H 3
2
1
Fig. 8. Cantilever steel column under minor axis bending
Figure 8 shows the equilibrium paths and yield curves (YC). It is seen in the YC a similar behavior to that observed by Lemes [11]. This behavior of the Initial Yield Curve (IYC) is associated to the fact that in both models of residual stresses [12-13], at the ends of the flanges present compressive residual stresses, determinants in the analysis of steel sections under minor axis bending. The value of H was varied and the curves are plotted in Fig. 10. It has been found, for this example, that as the fixed axial load P is larger, the smaller the value of . In the analyzes with higher P values, the curves closest to the Zubydan [8] response present = 1 (Figs 10e-10f). When P is equal to 0.4Py, better results are obtained when = 1 and for P = 0.2Py when = 6. These values were taken as reference in the curves obtained using the residual stress model of the ECCS [12], even model used by Zubydan [8]. Thus, these values were taken as comparable and then plotted in Fig. 9 together with the results of the literature. Thus, it can be said that with the adjustment of the elastic range, highlighting the IYC, there is a need to revise Eq. (1). The parameter S can be improved, since it has been proposed for the degradation of the flexural stiffness of steel elements major axis bending. For this, a parametric study is recommended that is not the objective of the present study. It is worth mentioning that the results lead to a new level in the inelastic analysis of steel structures under minor axis bending via the classical Refined Plastic Hinge Method (RPHM). Present analysis - ECCS [12] Present analysis - AISC [13] Zubydan [8]
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Fig. 10. Equilibrium paths a) P/Py = 0,2 - ECCS [12] b) P/Py = 0,2 - AISC [13] c) P/Py = 0,4 - ECCS [12] d) P/Py = 0,4 - AISC [13] e) P/Py = 0,6 - ECCS [12] f) P/Py = 0,6 - AISC [13] © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017)
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CONCLUSIONS
In the present work was studied a methodology for the definition of the curves of plasticization start and applications in global structures analysis. For this, an implementation of the fiber model was made in cross sections and its local analysis is done via Strain Compatibility Method (SCM). This approach is original by the explicit introduction of residual stresses in the fibers and global analysis coupled with the classic Refined Plastic Hinge Method (RPHM), in which pseudo-springs are used to simulate flexural stiffness degradation of cross section. It was verified good convergence in the results, especially in the isolated steel beam under major axis bending, both for the residual stress model of the ECCS [12] and for the AISC [13]. In the study of the cantilever steel column proposed by Zubydan [8], it was verified, in this case, of bending minor axis bending, that the structural behavior is sensitive to the parameter . Therefore, a simplified analysis was done. It was concluded that when the axial load increases the value of decreases. It was verified the need to review the equation describing the parameter S, emphasizing the minor axis bending. For this, it is intended in the future, to carry out a parametric study of the behavior of pseudo-springs and to adjust this parameter. 6
ACKNOWLEDGMENT
The authors would like to thank CAPES, CNPq, Fapemig, Fundação Gorceix and PROPEC/UFOP for their support during the preparation of this work. REFERENCES [1] Ziemian R.D., McGuire W. “Modified tangent modulus approach, a contribution to plastic hinge analysis”. Journal of Structural Engineering 128 No. 10, pp. 1301-1307, 2002 [2] Chan S.L., Chui P.P.T. “Non-linear static and cyclic analysis of steel frames with semi-rigid connections”. Oxford: Elsevier, 2000 [3] Liew J., White D., Chen W, “Second-order refined plastic hinge analysis for frame design: part 1”. Journal of Structural Division 119, No. 11, pp. 3196-3216, 1993 [4] Liew J., White D., Chen W, “Second-order refined plastic hinge analysis for frame design: part 2”. Journal of Structural Division 119, No. 11, pp. 3217-3237, 1993 [5] Silva A.R.D. “Sistema computacional para análises avançada estática e dinâmica de estruturas metálicas” PhD Thesis, PROPEC/UFOP, Ouro Preto-MG-Brazil, 2009 (in portuguese) [6] Liu S.W., Liu Y.P., Chan S.L. “Advanced analysis of hybrid steel and concrete frames part 2: Refined plastic hinge and advanced analysis”. Journal of Constructional Steel Research 70, pp.337-349, 2012 [7] Lemes Í.J.M., Silva, A.R.D., Silveira, R.A.M., Rocha P.A.S., “Acomplamento MCD/MRPR para Análise de Estruturas Metálicas, de Concreto e Mistas”, Proceedings of the XXXVI Iberian Latin American Congress on Computational Methods in Engineering, Rio de Janeiro-RJ-Brazil, CILAMCE 2015 (in portuguese) [8] Zubydan A.H. “Inelastic Second Order Analysis of Steel Frame Elements Flexed About Minor Axis”. Engineering Structures 33, pp. 1240-1250, 2011 [9] Gonçalves G.A., Silva A.R.D., Silveira R.A.M. “Avaliação do comportamento inelástico de colunas e pórticos metálicos com flexão em torno do eixo de menor inércia”. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería 32, No. 1, pp. 13-21, 2016 [10] Lemes Í.J.M., Silva, A.R.D., Silveira, R.A.M., Rocha P.A.S., “Estudo Numérico Comparativo de Metodologias para a Degradação da Rigidez à Flexão no Contexto do Método da Rótula Plástica Refinado”, Proceedings of the XXXVII Iberian Latin American Congress on Computational Methods in Engineering, Brasília-DF-Brazil, CILAMCE 2016 (in portuguese) [11] Lemes Í.J.M., Silva, J.L, Batelo, E.A.P., Silveira, R.A.M., “Influência dos modelos de tensões residuais prescritos em códigos normativos no comportmento de perfis I”, Proceedings of the XXXVII Iberian Latin American Congress on Computational Methods in Engineering, Brasília-DF-Brazil, CILAMCE 2016 (in portuguese) © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017)
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[12] ECCS, Ultimate Limit State Calculation of Sway Frames with Rigid Joints, Pub. no. 33, European Convention for Constructional Steelwork, 1983. [13] Galambos, T.V., Ketter, R.L. “Columns under Combined Bending and Thrust”. Journal of the Engineering Mechanics Division 85, pp. 1-30, 1959 [14] Yang Y.B., Kuo S.B. “Theory & Analysis of Nonlinear Framed Structures”. Singapore: Prentice Hall, 1994 [15] Chen S., Teng J.G., Chan S.L. “Design of biaxially loaded short composite columns of arbitrary section”. Journal of Structural Engineering 127, No. 6, pp.678-685, 2001 [16] Liew J., White D., Chen W. “Second-order refined plastic hinge analysis of frames”. Purdue University, West Lafayette, 1992
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