Impacts of residual stress and shear deformation on 2D ...

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SN Applied Sciences (2021) 3:686 | https://doi.org/10.1007/s42452-021-04638-w

Research Article

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

Phu‑Cuong Nguyen1 · T. D. Tran1

Received: 5 October 2020 / Accepted: 5 May 2021

© The Author(s) 2021 OPEN

AbstractIn 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 deforma-tion should be considered in the advanced direct analysis and design procedures for steel-framed structures.

Article highlights

• A nonlinear 2D beam column element is developed suc-cessfully using onlyone element per member for modeling.

• Geometric nonlinearities, material plasticity, residual stress, and sheardeformation are investigated simultaneously.

• Both residual stress and shear deformation should be considered in the engineering design of steel frames.

Keywords Residual stress · Shear deformation · Steel frames · Nonlinear analysis · Fiber plastic hinge · Geometric nonlinearity

* Phu-Cuong Nguyen, [email protected]; [email protected] | 1Advanced Structural Engineering Laboratory, Faculty of Civil Engineering, Ho Chi Minh City Open University, Ho Chi Minh City, Vietnam.

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, sustain-ability, durability, eco-friendly product, etc. For hot-rolling steel, residual stresses are born in manufacturing proce-dures. Many researchers [1–6] employed the ECCS residual stress pattern [7], as shown in Fig. 2, for analyzing steel

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Research Article SN Applied Sciences (2021) 3:686 | https://doi.org/10.1007/s42452-021-04638-w

frames using W-sections. These plastic hinge method stud-ies [8–13] considered indirectly residual stresses by using the CRC tangent modulus concept. Some researchers [1–3, 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 predict-ing material inelasticity. For geometry nonlinearity, they employed Hermite interpolation functions [1, 14–17], high-order interpolation functions [18], stability functions [2, 9, 12, 13, 19, 20], or corotational approach [21–24], 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 defor-mation 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-interpola-tion functions combining with the flexibility method and monitoring some integration points along the member length [25–27], or it can be considered by using corota-tional formulations, but this method is relatively compli-cated. 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:

where S1 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 meth-ods consider indirectly residual stresses through proposed equations or formulations of internal forces. An incremen-tal equilibrium equation for 2D frame elements combining

(1)

⎧⎪⎨⎪⎩

ΔP

ΔMI

ΔMJ

⎫⎪⎬⎪⎭=

EI

L

⎡⎢⎢⎣

A∕I 0 0

0 S1S2

0 S2S1

⎤⎥⎥⎦

⎧⎪⎨⎪⎩

Δ�

Δ�IΔ�J

⎫⎪⎬⎪⎭

Fig. 1 Discretion of cross-sections

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SN Applied Sciences (2021) 3:686 | https://doi.org/10.1007/s42452-021-04638-w Research Article

both stability functions and fiber plastic hinges is formu-lated as follows:

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; EtIi and EtJi 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; Ai is the area of ith fiber; Ii is inertia moment of ith fiber around its centroid; I is inertia moment of frame cross-section; yi is the coordinate of the center of ith fiber as illustrated in Fig. 1.

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:

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

(2)

⎧⎪⎨⎪⎩

ΔP

ΔMI

ΔMJ

⎫⎪⎬⎪⎭=

⎡⎢⎢⎣

EA∕L 0 0

0 Kii Kij0 Kij Kjj

⎤⎥⎥⎦

⎧⎪⎨⎪⎩

Δ�

Δ�IΔ�J

⎫⎪⎬⎪⎭

(3)Kii = �I(S1 −S22

S1

(1 − �J))EI

L

(4)Kij = �I�JS2EI

L

(5)Kjj = �J(S1 −S22

S1

(1 − �I))EI

L

(6)�I =

∑n

i=1EtIi

�Aiy

2

i+ Ii

�EI

(7)�J =

∑n

i=1EtJi

�Aiy

2

i+ Ii

�EI

(8)

�ΔN

ΔM

�=

�1 0 0

0 x∕L − 1 x∕L

�⎧⎪⎨⎪⎩

ΔP

ΔMI

ΔMJ

⎫⎪⎬⎪⎭

The incremental sectional deformation vector is esti-mated by the flexibility matrix and the sectional force vec-tor as

where � and � are the sectional axial strain and the sec-tional curvature.

The incremental sectional fiber strain vector is defined by the linear geometric matrix and the incremental sec-tional deformation vector:

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

2.4 Residual stresses

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

(9)�

Δ�

Δ�

�=

⎡⎢⎢⎢⎣

n∑i=1

EiAi −n∑i=1

EiAiyi

−n∑i=1

EiAiyi

n∑i=1

Ei�Aiy

2

i+ Ii

�⎤⎥⎥⎥⎦

−1

�ΔN

ΔM

�

(10)

⎧⎪⎨⎪⎩

Δe1

Δe2

...

Δen

⎫⎪⎬⎪⎭=

⎡⎢⎢⎢⎣

1 −y1

1 −y2

... ...

1 −yn

⎤⎥⎥⎥⎦

�Δ�

Δ�

�

(11)

�N

M

�=

⎧⎪⎨⎪⎩

m∑i=1

�iAi

−m∑i=1

�iAiyi

⎫⎪⎬⎪⎭

Fig. 2 Assumed residual stresses [7]

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assigned directly to fibers in the beginning, as plotted in Fig. 2.

2.5 Shear deformation

The factors of bending moments in the element stiffness matrix are developed for calculating the additional flex-ural shear effects in a frame member. The flexural flexibil-ity matrix can be obtained by inverting the flexural stiff-ness matrix, and the incremental equilibrium equation of moments and slopes is written as

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 corre-sponding to flexural shear deformation can be written as

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

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)

�Δ�IMΔ�JM

�=

⎡⎢⎢⎣

Kjj

KiiKjj−K2

ij

−Kij

KiiKjj−K2

ij−Kij

KiiKjj−K2

ij

Kii

KiiKjj−K2

ij

⎤⎥⎥⎦

�ΔMI

ΔMJ

�

(13)

{Δ�ISΔ�JS

}=

[1

GAsL

1

GAsL1

GAsL

1

GAsL

]{ΔMI

ΔMJ

}

(14)

{Δ�IΔ�J

}=

{Δ�IMΔ�JM

}+

{Δ�ISΔ�JS

}

(15)

{ΔMI

ΔMJ

}=

[Dii Dij

Dij Djj

]{Δ�IΔ�J

}

(16)

⎧⎪⎨⎪⎩

ΔP

ΔMI

ΔMJ

⎫⎪⎬⎪⎭=

⎡⎢⎢⎣

EtA∕L 0 0

0 Dii Dij

0 Dij Djj

⎤⎥⎥⎦

⎧⎪⎨⎪⎩

Δ�

Δ�IΔ�J

⎫⎪⎬⎪⎭

(17)Dii =KiiKjj − K2

ij+ KiiAsGL

Kii + Kjj + 2Kij + AsGL

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:

where the transformation matrix [T ]3×6 for the frame ele-

ment is calculated as

and the geometry stiffness matrix is established as

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 prob-lems 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 pro-posed program can accurately predict the nonlinear behav-ior and ultimate strength of 2D steel frames considering

(18)Dij =−KiiKjj + K2

ij+ KijAsGL


(19)Djj =KiiKjj − K2

ij+ KjjAsGL


(20)[K ]T = [T ]T3×6

[Ke]3×3

[T ]3×6 +[Kg]6×6

(21)[T ]3×6 =

⎡⎢⎢⎣

−1 0 0 1 0 0

0 1∕L 1 0 −1∕L 0

0 1∕L 0 0 −1∕L 1

⎤⎥⎥⎦

(22)�Kg�6×6

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0MI+MJ

L20 0 −

MI+MJ

L20

MI+MJ

L2P

L0 −

MI+MJ

L2−

P

L0

0 0 0 0 0 0

0 −MI+MJ

L20 0

MI+MJ

L20

−MI+MJ

L2−

P

L0

MI+MJ

L2P

L0

0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(23)[K ij−1

]{ΔDi

j

}= 𝜆i

j

{P̂}+{Rij−1

}

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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.

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 nonlin-ear inelastic analysis of steel frames. Initial imperfections are assumed for columns through the initial out-of-plump-ness of � = 1∕400 . Elastic modulus is E = 205000MPa . The yield stress of steel is �y = 235MPa . Vogel [14] developed a plastic zone method for analyzing this frame, while this

study is using a fiber plastic hinge method with one-ele-ment modeling.

Figure 5 shows the result comparison of this study, Vogel [14], and ABAQUS [30]. It can be seen that the pre-dicted 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 cap-ture 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

1. Find vector of total external force and internal force

2. Form unbalanced force vector

Yes

Convergence?

No

GD

C it

erat

ive

algo

rithm

No

1. Calculate strain and stress of fibers2. Calculate charateristic of sections

j = j

+ 1

Nex

t inc

rem

ent,

i = i

+ 1

START: Input data

1. Form incremental load vector2. Form tangent stiffness matrix

"Solve equilibrium equations"

1. Find displacements at two ends of members2. Find sectional force vectors

YesNext load step?

FINISH: Output data

Find incremental displacements

Fig. 3 Flowchart of the proposed program

Fig. 4 Portal frame

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

Figures 6, 7, 8, and 9 show load–deflection relation-ships at node A for four different cases: Case 1 considers

both shear deformation and residual stress; Case 2 con-siders only shear deformation; Case 3 considers only residual stress; Case 4 does not consider both shear deformation and residual stress. Table 1 shows differ-ences 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

Fig. 5 Load–deflection rela-tionship at A of portal frame

0.00.10.20.30.40.50.60.70.80.91.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Loa

d co

effic

ient

Horizontal deflection at A (mm)

This studyABAQUSThis study w/o She DefVogel

Fig. 6 Load–deflection rela-tionship at A of portal frame with different effects

0.00.10.20.30.40.50.60.70.80.91.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Loa

d co

effic

ient


This studyThis study w/o She DefThis study w/o Res StrThis study w/o (Res Str + She Def)VogelABAQUS

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stresses do not affect global structural stiffness in the beginning, but shear deformation has significant influ-ences during the incremental iterative solution proce-dure. 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 addi-tion, 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

Fig. 7 Load-vertical deflec-tion relationship at A of portal frame

0.00.10.20.30.40.50.60.70.80.91.0

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Loa

d co

effic

ient

Vertical deflection at A (mm)

This studyThis study w/o She DefThis study w/o Res StrThis study w/o (Res Str + She Def)

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

0.80

0.85

0.90

0.95

1.00

1.05

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4L

oad

coef

ficie

ntVertical deflection at A (mm)


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does not affect the vertical deflection at node A or axial shortening of members.

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 imperfec-tions of columns are assumed to be out-of-plumpness of � = 1∕450 . Elastic modulus is E = 20500MPa . Yield stress is �y = 235MPa.

Figure 11 shows the comparison of the load-horizon-tal deflection relationship predicted by this study and Vogel [14]. Those without the effects of shear deforma-tion 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 dif-ferent 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 con-sider both shear deformation and residual stress. Tables 2

Fig. 9 Load-rotation relation-ship at A of portal frame

0.00.10.20.30.40.50.60.70.80.91.0

0.0000 0.0004 0.0008 0.0012 0.0016 0.0020

Loa

d co

effic

ient

Rotation at A (rad)


Table 1 Deflection at limit load at point A for portal frame

Case 1 considers both shear deformation and residual stress; Case 2 considers only residual stress; Case 3 considers only shear deformation; Case 4 does not consider both shear deformation and residual stress

Case Load coefficient Error (%) Horizontal deflection (mm)

Error (%) Vertical deflec-tion (mm)

Error (%) Rotation (rad) Error (%)

Case 1 1.007 – 19.92 – −6.18 – 0.00116 –Case 2 1.014 0.70 17.93 −9.97 −6.22 0.59 0.00098 −15.63Case 3 1.022 1.49 11.77 −40.91 −5.08 −17.84 0.00101 −12.94Case 4 1.027 1.99 10.97 −44.95 −5.16 −16.58 0.00086 −26.05

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Fig. 10 Six-story frame

Fig. 11 Load-horizontal deflection relationship at A of six-story frame

0.00.10.20.30.40.50.60.70.80.91.01.1

0 30 60 90 120 150 180 210 240 270 300 330

Load

coe

ffici

ent


This study w/o She Def

Vogel

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and 3 show differences of limit load coefficient and deflec-tion 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 deflec-tion at node A and B or axial shortening of members, while

Fig. 12 Load-horizontal deflection relationship at A of six-story framewith different effects

0.00.10.20.30.40.50.60.70.80.91.01.1

0 30 60 90 120 150 180 210 240 270 300 330

Load

coe

ffici

ent


This studyThis study w/o She DefThis study w/o Res StrThis study w/o (Res Str + She Def)Vogel

Fig. 13 Load-horizontal deflection relationship at A of six-story frame with different effects

0.500.550.600.650.700.750.800.850.900.951.001.051.10

30 60 90 120 150 180 210 240 270

Load

coe

ffici

ent


This studyThis study w/o She DefThis study w/o Res StrThis study w/o (Res Str + She Def)Vogel

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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.

5 Case study

In the last two examples, we proved the proposed meth-od’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

Fig. 14 Load-vertical deflec-tion relationship at A of six-story frame

0.00.10.20.30.40.50.60.70.80.91.01.1

-24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

Loa

d co

effic

ient


This study


This study w/o Res Str

This study w/o (Res Str + She Def)

Fig. 15 Load-vertical deflec-tion relationship at A of six-story frame

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

-20 -18 -16 -14 -12 -10 -8 -6 -4Lo

ad c

oeffi

cien

tVertical deflection at A (mm)

This study




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loadings is studied, as shown in Fig. 25. Elastic modulus is E = 20500MPa . Yield stress is �y = 235MPa . 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-sec-tions 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 F2= 240 kN . Wind loadings of

F3= 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

Fig. 16 Load-rotation relation-ship at A of six-story frame

0.00.10.20.30.40.50.60.70.80.91.01.1

-0.024 -0.021 -0.018 -0.015 -0.012 -0.009 -0.006 -0.003 0.000

Loa

d co

effic

ient

Rotation at A (rad)

This study




Fig. 17 Load-rotation relation-ship at A of six-story frame

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

-0.018 -0.016 -0.014 -0.012 -0.010 -0.008 -0.006 -0.004L

oad

coef

ficie

ntRotation at A (rad)

This study




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into sixty-six fibers for accuracy and considering residual stresses by inputting initial stress values for each fiber.

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 stiff-ness. While residual stresses impact substantially on the structure’s behavior and stiffness as fiber plastic hinges

Fig. 18 Load-horizontal deflection relationship at B of six-story frame

0.00.10.20.30.40.50.60.70.80.91.01.1

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Loa

d co

effic

ient

Horizontal deflection at B (mm)

This study




Fig. 19 Load-vertical deflec-tion relationship at B of six-story frame

0.00.10.20.30.40.50.60.70.80.91.01.1

-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0L

oad

coef

ficie

nt

Vertical deflection at B (mm)

This study




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Fig. 20 Load-vertical deflec-tion relationship at B of six-story frame

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

-5.4 -5.1 -4.8 -4.5 -4.2 -3.9 -3.6 -3.3 -3.0 -2.7 -2.4

Loa

d co

effic

ient


This study




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

0.00.10.20.30.40.50.60.70.80.91.01.1

0.000 0.002 0.004 0.006 0.008 0.010 0.012

Loa

d co

effic

ient

Rotation at B (rad)

This study




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Fig. 22 Load-rotation relation-ship at B of six-story frame

0.900

0.930

0.960

0.990

1.020

1.050

1.080

1.110

0.000 0.002 0.004 0.006 0.008 0.010 0.012

Loa

d co

effic

ient

Rotation at B (rad)

This study




Table 2 Deflection at limit load at point A for six-story frame





Case 1 1.113 – 306.82 – −13.87 – −0.01241 –Case 2 1.116 0.30 303.08 −1.22 −14.28 2.96 −0.01352 8.90Case 3 1.116 0.29 317.50 3.48 −15.00 8.14 −0.01444 16.35Case 4 1.119 0.58 303.61 −1.05 −15.01 8.25 −0.01471 18.49

Table 3 Deflection at limit load at point B for six-story frame





Case 1 1.113 – 52.17 – −5.19 – 0.00365 –Case 2 1.116 0.30 48.03 −7.93 −5.19 −0.12 0.00338 −7.24Case 3 1.116 0.29 51.89 −0.52 −5.11 −1.54 0.00429 17.66Case 4 1.119 0.58 47.74 −8.48 −5.12 −1.30 0.00389 6.82

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start yielding. The load-carrying capacity of the frame is no much change as residual stresses and shear deforma-tion are considered or not.

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 deflec-tion of the frame is biggest. This recommends that struc-tural designers check the buildings’ service conditions with the effects of shear deformation and residual stresses simultaneously.

6 Conclusion

A fiber plastic hinge method for nonlinear inelastic analysis of 2D steel frames considering both shear defor-mation 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 effec-tively 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 buck-ling, 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 analy-sis 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 moni-toring 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.

Fig. 23 Plasticity of portal frame at limit load [1]

Fig. 24 Plasticity of six-story frame at limit load [1]

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5.0 m

HEB

240

HEB

240

HEB

240

HEB

220

HEB

220

HEB

220

HEB

260

HEB

260

HEB

260

HEB

240

HEB

240

HEB

240

F4

F4

HEB

200

HEB

220

HEB

200

HEB

220

= 24 kNF4

HEB

200

HEB

220

= 12 kNF3

= 120 kNF1

= 100 kNF1

F1

F1

F1

F1

F1

F1

F1= 120 kN

5.0 m

HEB

260

HEB

260

HEB

260

HEB

240

HEB

240

HEB

240

HEB

220

HEB

220

HEB

220

= 240 kNF2

F2= 240 kN

5.0 m

HEB

260

HEB

260

HEB

260

HEB

240

HEB

240

HEB

240

HEB

220

HEB

220

HEB

220

5.0 m

HEB

260

HEB

260

HEB

260

HEB

240

HEB

240

HEB

240

HEB

220

HEB

220

HEB

220

5.0 m

IPE400

IPE400

IPE400

IPE360

IPE360

IPE360

IPE330

IPE330

IPE330

HEB

240

HEB

240

HEB

240

HEB

220

HEB

220

HEB

220

HEB

200

HEB

200

HEB

200

= 240 kNF2 = 240 kNF2 = 240 kNF2 = 120 kNF1

9 x

3.5

m =

31.

5 m

F1

F1

F1

F1

F1

F1

F1

F1F2= 240 kN F2= 240 kN F2= 240 kN

F2 F2 F2 F2

F2 F2 F2 F2

F2 F2 F F2

F2 F2 F2 F2

F2 F2 F2 F2

2

F2 F2 F F22

F2 F2 F F22

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 kNF4

= 24 kNF4

F4

F4

F4

A

B

Fig. 25 Five-bay nine-story frame

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Fig. 26 Load–deflection relationship at A of nine-story frame with different effects

0.0

0.10.2

0.30.4

0.50.6

0.70.8

0.91.01.1

0 12 24 36 48 60 72 84 96 108 120

Loa

d co

effic

ient


This study




Fig. 27 Load-vertical deflec-tion relationship at A of nine-story frame

0.0

0.10.2

0.30.4

0.50.6

0.70.8

0.91.01.1

-24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

Load

coe

ffici

ent


This study




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Fig. 28 Load-rotation relation-ship at A of nine-story frame

0.0

0.10.2

0.30.4

0.50.6

0.70.8

0.91.01.1

-0.0010 -0.0008 -0.0006 -0.0004 -0.0002 0.0000

Loa

d co

effic

ient

Rotation at A (rad)

This study




Fig. 29 Load–deflection relationship at B of nine-story frame

0.0

0.10.2

0.30.4

0.50.6

0.70.8

0.91.01.1

0 8 16 24 32 40 48 56 64 72

Loa

d co

effic

ient

Horizontal deflection at B (mm)

This study




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Fig. 30 Load-vertical deflec-tion relationship at B of nine-story frame

0.0

0.10.2

0.30.4

0.50.6

0.70.8

0.91.01.1

-18 -16 -14 -12 -10 -8 -6 -4 -2 0

Loa

d co

effic

ient


This study




Fig. 31 Load-rotation relation-ship at B of nine-story frame

0.0

0.10.2

0.30.4

0.50.6

0.70.8

0.91.01.1

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

Loa

d co

effic

ient

Rotation at B (rad)

This study




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• 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

Acknowledgements The authors gratefully acknowledge the financial support granted by the Scientific Research Fund of the Ministry of Education and Training (MOET), Vietnam. (No. B2019–MBS–01). The authors would also like to thank colleagues at Ho Chi Minh City Open University for supporting this work.

Declarations`

Conflict of interest The authors declare that they have no conflict of interest.

Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adap-tation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted

use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.

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