1 Introduction

Steel frames are used widely in commercial buildings and pre-engineering buildings because of their significant advantages compared with reinforced concrete structures such as fastly construction, economical solution, sustainability, durability, eco-friendly product, etc. For hot-rolling steel, residual stresses are born in manufacturing procedures. Many researchers [1,2,3,4,5,6] employed the ECCS residual stress pattern [7], as shown in Fig. 2, for analyzing steel frames using W-sections. These plastic hinge method studies [8,9,10,11,12,13] considered indirectly residual stresses by using the CRC tangent modulus concept. Some researchers [1,2,3, 5, 6, 14] took into account residual stresses directly through meshing fibers on cross-sections.

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.

2 Nonlinear frame element

2.1 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 one-element modeling can precisely predict this effect. An incremental equilibrium equation for 2D frame elements is formulated as follows:

$$\left\{ {\begin{array}{*{20}c} {\Delta P} \\ {\Delta M_{I} } \\ {\Delta M_{J} } \\ \end{array} } \right\} = \frac{EI}{L}\left[ {\begin{array}{*{20}c} {A/I} & 0 & 0 \\ 0 & {S_{1} } & {S_{2} } \\ 0 & {S_{2} } & {S_{1} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta \delta } \\ {\Delta \theta_{I} } \\ {\Delta \theta_{J} } \\ \end{array} } \right\}$$
(1)

where \(S_{1}\) and \(S_{2}\) are stability functions found in the book of Chen và Lui [28].

2.2 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 both stability functions and fiber plastic hinges is formulated as follows:

$$\left\{ {\begin{array}{*{20}c} {\Delta P} \\ {\Delta M_{I} } \\ {\Delta M_{J} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {EA/L} & 0 & 0 \\ 0 & {K_{ii} } & {K_{ij} } \\ 0 & {K_{ij} } & {K_{jj} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta \delta } \\ {\Delta \theta_{I} } \\ {\Delta \theta_{J} } \\ \end{array} } \right\}$$
(2)
$$K_{ii} = \eta_{I} (S_{1} - \frac{{S_{2}^{2} }}{{S_{1} }}(1 - \eta_{J} ))\frac{EI}{L}$$
(3)
$$K_{ij} = \eta_{I} \eta_{J} S_{2} \frac{EI}{L}$$
(4)
$$K_{jj} = \eta_{J} (S_{1} - \frac{{S_{2}^{2} }}{{S_{1} }}(1 - \eta_{I} ))\frac{EI}{L}$$
(5)

where \(\eta_{I}\) và \(\eta_{J}\) 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:

$$\eta _{I} = \frac{{\sum\nolimits_{{i = 1}}^{{n}} {E_{{tIi}} \left( {A_{i} y_{i}^{2} + I_{i} } \right)} }}{{EI}}$$
(6)
$$\eta_{J} = \frac{{\sum\nolimits_{{i = 1}}^{{n}} {E_{tJi} \left( {A_{i} y_{i}^{2} + I_{i} } \right)} }}{EI}$$
(7)

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 cross-section; \(y_{i}\) is the coordinate of the center of ith fiber as illustrated in Fig. 1.

Fig. 1
figure 1

Discretion of cross-sections

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2.3 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:

$$\left\{ {\begin{array}{*{20}c} {\Delta N} \\ {\Delta M} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {x/L - 1} & {x/L} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta P} \\ {\Delta M_{I} } \\ {\Delta M_{J} } \\ \end{array} } \right\}$$
(8)

where x = 0 is considering at I end, and x = L is considering at L end.

The incremental sectional deformation vector is estimated by the flexibility matrix and the sectional force vector as

$$\left\{ {\begin{array}{*{20}c} {\Delta \varepsilon } \\ {\Delta \chi } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{n} {E_{i} A_{i} } } & { - \sum\limits_{i = 1}^{n} {E_{i} A_{i} y_{i} } } \\ { - \sum\limits_{i = 1}^{n} {E_{i} A_{i} y_{i} } } & {\sum\limits_{i = 1}^{n} {E_{i} \left( {A_{i} y_{i}^{2} + I_{i} } \right)} } \\ \end{array} } \right]^{ - 1} \left\{ {\begin{array}{*{20}c} {\Delta N} \\ {\Delta M} \\ \end{array} } \right\}$$
(9)

where \(\varepsilon\) and \(\chi\) 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:

$$\left\{ {\begin{array}{*{20}c} {\Delta e_{1} } \\ {\Delta e_{2} } \\ {...} \\ {\Delta e_{n} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} 1 & { - y_{1} } \\ 1 & { - y_{2} } \\ {...} & {...} \\ 1 & { - y_{n} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta \varepsilon } \\ {\Delta \chi } \\ \end{array} } \right\}$$
(10)

The stress–strain relationship of steel is assumed to be elastic perfectly plastic. Sectional internal forces are estimated as follows:

$$\left\{ {\begin{array}{*{20}c} N \\ M \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\sum\limits_{i = 1}^{m} {\sigma_{i} } A_{i} } \\ { - \sum\limits_{i = 1}^{m} {\sigma_{i} } A_{i} y_{i} } \\ \end{array} } \right\}$$
(11)

2.4 Residual stresses

By meshing the cross-section into many fibers, as illustrated in Fig. 1, assuming that residual stresses [7] are assigned directly to fibers in the beginning, as plotted in Fig. 2.

Fig. 2
figure 2

Assumed residual stresses [7]

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2.5 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

$$\left\{ {\begin{array}{*{20}c} {\Delta \theta_{IM} } \\ {\Delta \theta_{JM} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\frac{{K_{jj} }}{{K_{ii} K_{jj} - K_{ij}^{2} }}} & {\frac{{ - K_{ij} }}{{K_{ii} K_{jj} - K_{ij}^{2} }}} \\ {\frac{{ - K_{ij} }}{{K_{ii} K_{jj} - K_{ij}^{2} }}} & {\frac{{K_{ii} }}{{K_{ii} K_{jj} - K_{ij}^{2} }}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta M_{I} } \\ {\Delta M_{J} } \\ \end{array} } \right\}$$
(12)

where Kii, Kij, and Kjj are the factors of stiffness matrix in a 2D frame element. θIM and θJM are the slopes of the neutral axis under bending moments. The flexibility matrix corresponding to flexural shear deformation can be written as

$$\left\{ {\begin{array}{*{20}c} {\Delta \theta_{IS} } \\ {\Delta \theta_{JS} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\frac{1}{{GA_{s} L}}} & {\frac{1}{{GA_{s} L}}} \\ {\frac{1}{{GA_{s} L}}} & {\frac{1}{{GA_{s} L}}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta M_{I} } \\ {\Delta M_{J} } \\ \end{array} } \right\}$$
(13)

where G is the shear modulus, As 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

$$\left\{ {\begin{array}{*{20}c} {\Delta \theta_{I} } \\ {\Delta \theta_{J} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\Delta \theta_{IM} } \\ {\Delta \theta_{JM} } \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} {\Delta \theta_{IS} } \\ {\Delta \theta_{JS} } \\ \end{array} } \right\}$$
(14)

The force–displacement equation considering shear deformation is obtained by inverting the flexibility matrix

$$\left\{ {\begin{array}{*{20}c} {\Delta M_{I} } \\ {\Delta M_{J} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {D_{ii} } & {D_{ij} } \\ {D_{ij} } & {D_{jj} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta \theta_{I} } \\ {\Delta \theta_{J} } \\ \end{array} } \right\}$$
(15)

The incremental equilibrium equation can be written for 2D frame element considering shear deformation as

$$\left\{ {\begin{array}{*{20}c} {\Delta P} \\ {\Delta M_{I} } \\ {\Delta M_{J} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {E_{t} A/L} & 0 & 0 \\ 0 & {D_{ii} } & {D_{ij} } \\ 0 & {D_{ij} } & {D_{jj} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\Delta \delta } \\ {\Delta \theta_{I} } \\ {\Delta \theta_{J} } \\ \end{array} } \right\}$$
(16)

where

$$D_{ii} = \frac{{K_{ii} K_{jj} - K_{ij}^{2} + K_{ii} A_{s} GL}}{{K_{ii} + K_{jj} + 2K_{ij} + A_{s} GL}}$$
(17)
$$D_{ij} = \frac{{ - K_{ii} K_{jj} + K_{ij}^{2} + K_{ij} A_{s} GL}}{{K_{ii} + K_{jj} + 2K_{ij} + A_{s} GL}}$$
(18)
$$D_{jj} = \frac{{K_{ii} K_{jj} - K_{ij}^{2} + K_{jj} A_{s} GL}}{{K_{ii} + K_{jj} + 2K_{ij} + A_{s} GL}}$$
(19)

2.6 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:

$$\left[ K \right]_{T} = \left[ T \right]_{3 \times 6}^{T} \left[ {K_{e} } \right]_{3 \times 3} \left[ T \right]_{3 \times 6} + \left[ {K_{g} } \right]_{6 \times 6}$$
(20)

where the transformation matrix \(\left[ T \right]_{3 \times 6}\) for the frame element is calculated as

$$\left[ T \right]_{3 \times 6} = \left[ {\begin{array}{*{20}c} { - 1} & 0 & 0 & 1 & 0 & 0 \\ 0 & {1/L} & 1 & 0 & { - 1/L} & 0 \\ 0 & {1/L} & 0 & 0 & { - 1/L} & 1 \\ \end{array} } \right]$$
(21)

and the geometry stiffness matrix is established as

$$\left[ {K_{g} } \right]_{6 \times 6} = \left[ {\begin{array}{*{20}c} 0 & {\frac{{M_{I} + M_{J} }}{{L^{2} }}} & 0 & 0 & { - \frac{{M_{I} + M_{J} }}{{L^{2} }}} & 0 \\ {\frac{{M_{I} + M_{J} }}{{L^{2} }}} & \frac{P}{L} & 0 & { - \frac{{M_{I} + M_{J} }}{{L^{2} }}} & { - \frac{P}{L}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & { - \frac{{M_{I} + M_{J} }}{{L^{2} }}} & 0 & 0 & {\frac{{M_{I} + M_{J} }}{{L^{2} }}} & 0 \\ { - \frac{{M_{I} + M_{J} }}{{L^{2} }}} & { - \frac{P}{L}} & 0 & {\frac{{M_{I} + M_{J} }}{{L^{2} }}} & \frac{P}{L} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]$$
(22)

3 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:

$$\left[ {K_{j - 1}^{i} } \right]\left\{ {\Delta D_{j}^{i} } \right\} = \lambda_{j}^{i} \left\{ {\hat{P}} \right\} + \left\{ {R_{j - 1}^{i} } \right\}$$
(23)

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 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.

Fig. 3
figure 3

Flowchart of the proposed program

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4 Numerical examples

4.1 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 \(\psi = 1/400\). Elastic modulus is \(E = 205000 \, MPa\). The yield stress of steel is \(\sigma_{y} = 235 \, MPa\). 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.

Fig. 4
figure 4

Portal frame

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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 shell elements in the nonlinear analysis while the proposed program uses only one beam-column line element per member. Moreover, Kim and Lee [30] developed a complicated Fortran subroutine for assigning initial residual stresses for shell elements.

Fig. 5
figure 5

Load–deflection relationship at A of portal frame

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Figures 6, 7, 8, and 9 show load–deflection relationships at node A 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. 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 stresses do not affect global structural stiffness in the beginning, but shear deformation has significant influences during the incremental iterative solution procedure. Residual stresses definitely affect the behavior and strength of this frame at the load coefficient of 0.6 because, from this time, some fibers start yielding. From Fig. 6 and Table 1, horizontal deflection at node A of the frame in case 3 and 4 without considering residual stresses are much smaller than those of case 1 and 2 with the effect of residual stresses. The post-collapse load curves of horizontal deflection at node A for four cases are matched together, as plotted in Fig. 6. In addition, from Figs. 8 and 9, the post-collapse load curves of vertical deflection and rotation at node A for cases 1 and 3 are overlapped, and those for cases 2 and 4 are also overlapped. As shown in Figs. 7 and 8, shear deformation does not affect the vertical deflection at node A or axial shortening of members.

Fig. 6
figure 6

Load–deflection relationship at A of portal frame with different effects

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Fig. 7
figure 7

Load-vertical deflection relationship at A of portal frame

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Fig. 8
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Load-vertical deflection relationship at A of portal frame

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Fig. 9
figure 9

Load-rotation relationship at A of portal frame

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Table 1 Deflection at limit load at point A for portal frame
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4.2 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 \(\psi = 1/450\). Elastic modulus is \(E = 20500 \, MPa\). Yield stress is \(\sigma_{y} = 235 \, MPa\).

Fig. 10
figure 10

Six-story frame

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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.

Fig. 11
figure 11

Load-horizontal deflection relationship at A of six-story frame

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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 and 3 show differences of limit load coefficient and deflection at nodes A and B generated by this study with four mentioned-above cases. From Figs. 12 and 18, it can be seen that residual stresses do not affect global structural stiffness in the beginning, but shear deformation has significant influences during the incremental iterative solution procedure. Residual stresses definitely affect the behavior and strength of this frame at the load coefficient of 0.6 because, from this time, some fibers start yielding. From Figs. 12 to 22, Tables 2 and 3, deflection at nodes A and B of the frame in four different cases are not much different such as the behavior of the portal frame in the one example. The rotation at nodes A and B in four cases is not much different. As shown in Figs. 14, 15, 19 and 20, shear deformation does not affect on the vertical deflection at node A and B or axial shortening of members, while 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 six-story frame is collapsed when both beams and columns are yielding, as shown in Fig. 24.

Fig. 12
figure 12

Load-horizontal deflection relationship at A of six-story framewith different effects

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Fig. 13
figure 13

Load-horizontal deflection relationship at A of six-story frame with different effects

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Fig. 14
figure 14

Load-vertical deflection relationship at A of six-story frame

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Fig. 15
figure 15

Load-vertical deflection relationship at A of six-story frame

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Fig. 16
figure 16

Load-rotation relationship at A of six-story frame

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Fig. 17
figure 17

Load-rotation relationship at A of six-story frame

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Fig. 18
figure 18

Load-horizontal deflection relationship at B of six-story frame

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Fig. 19
figure 19

Load-vertical deflection relationship at B of six-story frame

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Fig. 20
figure 20

Load-vertical deflection relationship at B of six-story frame

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Fig. 21
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Load-rotation relationship at B of six-story frame

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Fig. 22
figure 22

Load-rotation relationship at B of six-story frame

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Table 2 Deflection at limit load at point A for six-story frame
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Table 3 Deflection at limit load at point B for six-story frame
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Fig. 23
figure 23

Plasticity of portal frame at limit load [1]

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Fig. 24
figure 24

Plasticity of six-story frame at limit load [1]

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5 Case study

In the last two examples, we proved the proposed method's accuracy and reliability in predicting the second-order 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 loadings is studied, as shown in Fig. 25. Elastic modulus is \(E = 20500 \, MPa\). Yield stress is \(\sigma_{y} = 235 \, MPa\). The beam-span 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 seven-to-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-to-three 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 into sixty-six fibers for accuracy and considering residual stresses by inputting initial stress values for each fiber.

Fig. 25
figure 25

Five-bay nine-story frame

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Figures 26, 27, 28, 29, 30, and 31 show comparisons of the load–deflection relationships at A and B of the frame under nonlinear static analysis considering shear deformation and residual stresses and without these effects. Based on 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 start yielding. The load-carrying capacity of the frame is no much change as residual stresses and shear deformation are considered or not.

Fig. 26
figure 26

Load–deflection relationship at A of nine-story frame with different effects

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Fig. 27
figure 27

Load-vertical deflection relationship at A of nine-story frame

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Fig. 28
figure 28

Load-rotation relationship at A of nine-story frame

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Fig. 29
figure 29

Load–deflection relationship at B of nine-story frame

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Fig. 30
figure 30

Load-vertical deflection relationship at B of nine-story frame

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Fig. 31
figure 31

Load-rotation relationship at B of nine-story frame

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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.

Table 4 Deflection at limit load at point A for nine-story frame
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Table 5 Deflection at limit load at point B for nine-story frame
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6 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.

  • The proposed formulation for the effects of shear deformation does not affect the axial shortening of frame members.

  • Both residual stresses and shear deformation should be considered in the advanced direct analysis and design procedures for steel frames.

  • In engineering design, designers should check the effects of residual stresses and shear deformation on service conditions of buildings