Impacts of residual stress and shear deformation on 2D steel frames using fiber plastic hinge element: nonlinear behavior and strength

In this study, impacts of residual stress and shear deformation are investigated on 2D steel frames using a new fiber plastic hinge method. Geometry and material nonlinearities, residual stress, shear deformation, imperfections are considered simultaneously in the nonlinear analysis. The proposed method is efficient in computational efforts since the one-element modeling is used for the nonlinear analysis by employing stability functions for capturing the P-small delta phenomenon. The P-large delta phenomenon is considered using the geometric stiffness matrix. Plastic hinges are assumed to be formed at two ends of members. Cross-sections at two ends of members are divided into many fibers. The ECCS residual stress pattern is assigned directly through fibers as initial stress conditions. A finite element program is coded using the Fortran programming language for predicting the nonlinear behavior and ultimate strength of planar steel frames. The behavior and load-carrying capacity of steel frames are predicted precisely and efficiently using the nonlinear inelastic analysis. A case study of a large-scale planar steel frame is investigated for the frame's behavior and strength under the effects of residual stresses and shear deformation. Through numerical examples, we recommend that both residual stress and shear deformation should be considered in the advanced direct analysis and design procedures for steel-framed structures. A nonlinear 2D beam column element is developed successfully using only one element per member for modeling. Geometric nonlinearities, material plasticity, residual stress, and shear deformation are investigated simultaneously. Both residual stress and shear deformation should be considered in the engineering design of steel frames. A nonlinear 2D beam column element is developed successfully using only one element per member for modeling. Geometric nonlinearities, material plasticity, residual stress, and shear deformation are investigated simultaneously. Both residual stress and shear deformation should be considered in the engineering design of steel frames.

Many studies developed a lot of nonlinear inelastic analysis methods for steel frames. They used plastic hinge methods or distributed plasticity methods for predicting material inelasticity. For geometry nonlinearity, they employed Hermite interpolation functions [1,[14][15][16][17], high-order interpolation functions [18], stability functions [2,9,12,13,19,20], or corotational approach [21][22][23][24], in which stability functions is more efficient than other approaches since they use only one-element modeling for capturing second-order effects precisely. So this study will employ stability functions for developing a new method for analyzing the nonlinear inelastic behavior of steel frames. Recently, da Silva et al. [4] consider shear deformation using Timoshenko's beam theory in the nonlinear inelastic analysis of steel frames. Moreover, no researchers are discussing in detail the effects of residual stresses and shear deformation on the nonlinear inelastic behavior and strength of steel frames. That is the reason why we do this study.
This study presents a new method named the fiber plastic hinge method for analyzing the nonlinear inelastic behavior of 2D steel frames, considering both the effects of residual stress and shear deformation. With the primary purpose of this study, we investigate the impacts of shear deformation and residual stress on the nonlinear inelastic behavior and load-carrying capacity of 2D steel frames in detail. Next sections, we present the proposed analysis method, numerical examples, discussion, and withdraw important conclusions for advanced direct analysis and design procedures of steel frames.

P-small delta phenomenon
P-small delta phenomenon is the impact of axial force on bending moments (e.i. bending moments are increased by the axial force), leading to instability of beam-column elements. P-small delta phenomenon can be considered by using the geometric stiffness matrix combining with meshing elements into many short elements, or it can be considered by using high-order displacement-interpolation functions combining with the flexibility method and monitoring some integration points along the member length [25][26][27], or it can be considered by using corotational formulations, but this method is relatively complicated. To reduce computer resources and computational efforts, this study employed stability functions [28] to consider the P-small delta phenomenon since only oneelement modeling can precisely predict this effect. An incremental equilibrium equation for 2D frame elements is formulated as follows: where S 1 and S 2 are stability functions found in the book of Chen và Lui [28].

Fiber plastic hinges
In this study, two fiber plastic hinges are monitored at two ends of 2D frame elements. Cross-sections of these hinges have meshed into many fibers, as illustrated in Fig. 1. This method is more effective than traditional plastic hinge methods since it can directly consider residual stresses as initial conditions, whereas traditional plastic hinge methods consider indirectly residual stresses through proposed equations or formulations of internal forces. An incremental equilibrium equation for 2D frame elements combining where I và J are factors considering the plasticity of fiber hinges at two ends of frame elements, these factors are value from 0 to 1, the value of 0 is fully elastic, the value of 1 is fully plastic, proposed as: where n is the number of meshed fibers at two ends of frame elements; E tIi and E tJi are current modulus of ith fiber at ends I and J, if the strain of fiber is larger than plastic strain, current modulus of i fiber will be assigned to be equal to 0, e.i. contributed stiffness of fiber is equal to 0; A i is the area of ith fiber; I i is inertia moment of ith fiber around its centroid; I is inertia moment of frame crosssection; y i is the coordinate of the center of ith fiber as illustrated in Fig. 1.

Fiber behavior
Fiber behavior needs to be defined for estimating the state of fiber plastic hinges. Based on the force interpolation function matrix, we can calculate the incremental sectional force vector at two ends as follows: where x = 0 is considering at I end, and x = L is considering at L end. (2) The incremental sectional deformation vector is estimated by the flexibility matrix and the sectional force vector as where and are the sectional axial strain and the sectional curvature.
The incremental sectional fiber strain vector is defined by the linear geometric matrix and the incremental sectional deformation vector: The stress-strain relationship of steel is assumed to be elastic perfectly plastic. Sectional internal forces are estimated as follows:

Residual stresses
By meshing the cross-section into many fibers, as illustrated in Fig. 1, assuming that residual stresses [7] are

Shear deformation
The factors of bending moments in the element stiffness matrix are developed for calculating the additional flexural shear effects in a frame member. The flexural flexibility matrix can be obtained by inverting the flexural stiffness matrix, and the incremental equilibrium equation of moments and slopes is written as where K ii , K ij , and K jj are the factors of stiffness matrix in a 2D frame element. θ IM and θ JM are the slopes of the neutral axis under bending moments. The flexibility matrix corresponding to flexural shear deformation can be written as where G is the shear modulus, A s is the area subjected to shear, L is the length of a frame element. The total rotation at the I and J ends is summed by Eqs. (12) and (13) as The force-displacement equation considering shear deformation is obtained by inverting the flexibility matrix The incremental equilibrium equation can be written for 2D frame element considering shear deformation as where (12)

P-large delta phenomenon
The tangent stiffness matrix of the element, including the P-large delta phenomenon by using the geometry stiffness matrix, is summed as follows: where the transformation matrix [T ] 3×6 for the frame element is calculated as and the geometry stiffness matrix is established as

Nonlinear algorithm and analysis program
The general displacement control method [29] is used for developing the nonlinear static solution procedure. This method owns numerical stability and can solve the problems with many snap-back and snap-through points. The incremental equilibrium equation for solving nonlinear static problems of 2D steel frames is written as follows: The detail formulation can be found in the original article of Yang and Shieh [29] Based on the proposed formulation, an analysis program is coded using the Fortran programming language. The proposed program can accurately predict the nonlinear behavior and ultimate strength of 2D steel frames considering Vol.:(0123456789) SN Applied Sciences (2021) 3:686 | https://doi.org/10.1007/s42452-021-04638-w Research Article geometric nonlinearity, plasticity, residual stress, shear deformation using beam-column line elements. Figure 3 illustrates the flowchart of the nonlinear inelastic analysis procedure of the proposed program.

Portal frame
Vogel [14] invented a portal steel frame as plotted in Fig. 4. This frame is used as a benchmark problem for the nonlinear inelastic analysis of steel frames. Initial imperfections are assumed for columns through the initial out-of-plumpness of = 1∕400 . Elastic modulus is E = 205000 MPa . The yield stress of steel is y = 235 MPa . Vogel [14] developed a plastic zone method for analyzing this frame, while this study is using a fiber plastic hinge method with one-element modeling. Figure 5 shows the result comparison of this study, Vogel [14], and ABAQUS [30]. It can be seen that the predicted result is almost identical to the study of Vogel. It is noted that Vogel's result (red solid line) and this study (w/o shear deformation, blue dash line) do not consider the effects of shear deformation and use the residual stress pattern [7]. In the case of considering both residual stress and shear deformation, ABAQUS's result [30] (red dot line with green squares) and the proposed program (blue solid line) are identical in the range of load coefficient from 0.0 to 6.0 since this load range is in the elastic regime of steel material. When the load coefficient increases by more than 6.0, the frame behavior is change due to plasticity mainly. In this case, the limit load coefficients are 1.030 of ABAQUS and 1.007 of the proposed program. Less than 2.23% error is obtained comparing with ABAQUS's result. Without the effect of shear deformation, the limit load coefficient is obtained by Vogel's method to be 1.022 while the limit load coefficient is obtained by this study to be 1.014. Less than 0.78% error is obtained comparing with Vogel's result. It can be seen that this fiber plastic hinge method can capture the nonlinear behavior and ultimate strength of 2D steel frames as the sophisticated plastic zone method but more effective in the computational effort because the ABAQUS modeling of Kim and Lee [30] uses 8952 S4R5    Table 1 shows differences in limit load coefficient and deflection at node A generated by this study with four mentioned-above cases. From Figs. 6, 7 and 9, it can be seen that residual

Horizontal deflection at A (mm)
This study ABAQUS This study w/o She Def Vogel

Horizontal deflection at A (mm)
This study

Six-story frame
The six-story frame, as shown in Fig. 10, was also chosen by Vogel [14] as one of the benchmark problems for the nonlinear inelastic analysis of steel frames. Initial imperfections of columns are assumed to be out-of-plumpness of = 1∕450 . Elastic modulus is E = 20500 MPa . Yield stress is y = 235 MPa. Figure 11 shows the comparison of the load-horizontal deflection relationship predicted by this study and Vogel [14]. Those without the effects of shear deformation are in agreement well. It is noted that Vogel used a plastic zone method with a lot of beam-column elements in modeling, while this study used a fiber plastic hinge method using one-element modeling. The limit load coefficient is obtained by Vogel's approach to be 1.111. The limit load coefficient is achieved by this study to be 1.116, without the effect of shear deformation. Less than 0.45% error is obtained comparing with Vogel's result. It can be seen that this fiber plastic hinge method can capture the nonlinear behavior and ultimate strength of 2D steel frames as the sophisticated plastic zone method but more effective in computational effort. Figures 12,13,14,15,16,17,18,19,20,21, and 22 show load-deflection relationships at node A and B for four different cases: Case 1 considers both shear deformation and residual stress; Case 2 considers only shear deformation; Case 3 considers only residual stress; Case 4 does not consider both shear deformation and residual stress. Tables 2

Rotation at A (rad)
This study   residual stresses make the influences on the behavior and axial shortening of columns in load coefficient range from 0.7 to 1.1. There are differences between the portal frame and the six-story frame in the behavior curves because the portal frame is collapsed by yielding along the length of columns, mainly as shown in Fig. 23. In contrast, the sixstory frame is collapsed when both beams and columns are yielding, as shown in Fig. 24.

Case study
In the last two examples, we proved the proposed method's accuracy and reliability in predicting the secondorder inelastic analysis of steel frames. In this case study, we analyze the impacts of residual stresses and shear deformation for a large-scale steel frame. A five-bay nine-story steel frame with geometry, cross-sections, and Load coefficient

Vertical deflection at A (mm)
This study  Fig. 25. Elastic modulus is E = 20500 MPa . Yield stress is y = 235 MPa . The beamspan length of the frame is 5.0 m. The story height of the frame is 3.5 m. Cross-sections of exterior columns from one-to-three stories, from four-to-six stories, from sevento-nine stories are HEB240, HEB220, HEB200. Cross-sections of interior columns from one-to-three stories, from four-to-six stories, from seven-to-nine stories are HEB260, HEB240, HEB220. Cross-sections of beams from one-tothree stories, from four-to-six stories, from seven-to-nine stories are IPE400, IPE360, IPE330. Applied loadings at exterior column tops are F 1 = 120 kN . Applied loadings at interior column tops are F 2 = 240 kN . Wind loadings of F 3 = 12 kN and F 4 = 24 kN are put at positions, as shown in Fig. 25. We use one element per member for modeling all beam-columns for the frame. All cross-sections are divided

Rotation at A (rad)
This study  Figs. 26 and 29, it can be concluded that shear deformation significantly affects the structure's stiffness. While residual stresses impact substantially on the structure's behavior and stiffness as fiber plastic hinges    Tables 4 and 5 list deflection values at the limit load of the frame for nodes A and B in four cases of considering or not considering shear deformation and residual stresses. It can be seen that in the case 1 of considering both shear deformation and residual stresses, the horizontal deflection of the frame is biggest. This recommends that structural designers check the buildings' service conditions with the effects of shear deformation and residual stresses simultaneously.

Conclusion
A fiber plastic hinge method for nonlinear inelastic analysis of 2D steel frames considering both shear deformation and residual stress is developed successfully. Geometric nonlinearity is considered by using stability functions and the geometric stiffness matrix. Material nonlinearity is simulated by a proposed fiber plastic hinge model. Residual stress is directly considered by dividing several small fibers on the cross-sections. The proposed procedure can predict precisely and effectively the behavior and load-carrying capacity of 2D steel frames under static loadings by using advanced nonlinear analysis as complicated plastic zone methods or commercial general finite element software (ABAQUS, ANSYS, etc.) done. Local buckling, lateral-torsional buckling, panel zone, etc., are not considered in this study. In the next work, the effects of shear deformation and residual stress on the nonlinear inelastic dynamic analysis of steel frames will be investigated and evaluated. Some original conclusions are withdrawn from this study: • Residual stresses definitely affect the behavior and strength of steel-framed structures when some monitoring fibers start yielding. • Shear deformation significantly affects the global and local structural stiffness during the analysis procedure. • The influence of shear deformation on transverse deflection is considerable in the nonlinear analysis of steel frames.   IPE400   IPE400   IPE400   IPE360   IPE360   IPE360   IPE330   IPE330   IPE330   IPE400   IPE400   IPE400   IPE360   IPE360   IPE360   IPE330   IPE330   IPE330   IPE400   IPE400   IPE400   IPE360   IPE360   IPE360   IPE330   IPE330   IPE330   IPE400   IPE400   IPE400   IPE360   IPE360   IPE360   IPE330   IPE330   IPE330   = 24