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Engineering, Technology & Applied Science Research Vol. 10, No. 4, 2020, 5974-5978 5974 
 

www.etasr.com Nguyen et al.: Nonlinear Inelastic Analysis of 2D Steel Frames  

 

Nonlinear Inelastic Analysis of 2D Steel Frames  
An Improvement of the Plastic Hinge Method 

 

Phu-Cuong Nguyen 

Faculty of Civil Engineering 

Ho Chi Minh City Open University 

Ho Chi Minh City, Vietnam 
cuong.pn@ou.edu.vn 

Binh Le-Van
 

Faculty of Civil Engineering 

Ho Chi Minh City Open University 

Ho Chi Minh City, Vietnam 
binh.lv@ou.edu.vn 

Son Dong Tam Vo Thanh 

Faculty of Civil Engineering 

Ho Chi Minh City Open University 

Ho Chi Minh City, Vietnam 
son.dtvt@ou.edu.vn 

 

 

Abstract—In this study, a new method for nonlinear analysis of 

2D steel frames, by improving the conventional plastic hinge 

method, is presented. The beam-column element is established 

and formulated in detail using a fiber plastic hinge approach. 

Residual stresses of I-shape sections are declared at the two ends 

through fibers. Gradual yielding by residual stresses along the 

member length due to axial force is accounted for by the tangent 

elastic modulus concept. The P-δ effect is captured by stability 

functions, whereas the P-∆ effect is estimated by the geometric 

stiffness matrix. A nonlinear algorithm is established for solving 
nonlinear problems. The present study predicts the strength and 

behavior of 2D steel frames as efficiently and accurately as the 
plastic zone method did. 

Keywords-fiber plastic hinge; nonlinear algorithm; residual 

stress; stability functions; steel frames   

I. INTRODUCTION  

Nowadays, direct design for steel frames is permitted by the 
modern design codes. A direct design including the effects of 
geometric nonlinearity, inelasticity of materials, imperfections, 
residual stress, etc. is accounted directly and simultaneously in 
advanced analysis. There are usually two methods for advanced 
analysis: plastic hinge methods [1-15] and plastic zone methods 
[3, 16-25]. Authors in [14, 15, 22, 23, 26] invented a spring 
element for accounting stiffness of beam-to-column 
connections for nonlinear behavior analysis of steel frames 
with flexible connections. Recently, authors in [27-29] tried to 
investigate the behavior of steel frames with the effects of 
connections. Author in [30] investigated the effect of the 
Iranian standard no. 2800 on the elastic and inelastic behavior 
of dual steel systems by using the nonlinear pushover analysis 
of commercial software Sap2000. Up to now, in spite of the 
developments in computer science and technology, plastic zone 
methods are still expensive for the daily engineering design of 
steel frames. Plastic hinge methods are more simple, efficient 
in computational time, and with acceptable accuracy, so they 
are suitable for practical design. Plastic hinge methods have 
been studied widely from 1980 to 2000 [1-8]. Authors in [1, 3] 
used Hermite interpolation functions to predict the 
displacements of beam-column elements. Plastic hinges were 
assumed to concentrate at the two elemental ends. For 
considering geometric nonlinearity, the beam-column elements 
were divided into many short-elements. Residual stresses and 
imperfections were not accounted for in direct analysis. 

Authors in [6, 7] used a five-order interpolation function for 
considering the second-order effects of beam-column elements. 
The plasticity of cross-sections at the two ends of the element 
is modeled by two springs using the section assemblage 
concept. The tangent stiffness matrix of a structural system is 
established by integrating the stiffness of beam-column 
elements and the stiffness of springs. Authors in [10] derived a 
finite element formulation for a beam-column element that had 
an arbitrarily plastic hinge along the element length. Authors in 
[9] proposed a second-order inelastic large-deflection analysis 
method using only one element per member, including three 
plastic hinges in one member. 

In 2014, Liu et al. [13] also proposed an arbitrarily-located 
plastic hinge element for direct analysis of planar steel frames. 
Their method was directly developed from the initial out-of-
straight element. King et al. [5] proposed a second-order 
inelastic analysis method for steel frames. This method 
employed stability functions for predicting the second-order 
effects accurately. Gradual yielding at plastic hinges accounted 
for using LRFD’s interactive equations. Gradual yielding by 
residual stresses along the member length due to axial force 
was calculated by the CRC tangent modulus concept. One 
element was used for modeling. In 2002, Ziemian and McGuire 
[8] improved the result of plastic hinge method using modified 
tangent modulus formulation. The result was nearly identical 
with the sophisticated plastic zone method in some examples, 
but the method should be verified even more with different 
problems. The plastic hinge method of [5] is effective and 
saves computational time because it uses only one element per 
member. However, residual stresses at two plastic hinges are 
not considered in the analysis, and in some problems, the result 
has a significant error when compared with the ‘exact’ 
solutions using plastic zone methods. 

This study tries to develop a new plastic hinge method 
which can capture the nonlinear behavior of steel frames 
accurately. The proposed method employs the fiber 
discretization of a cross-section. Stability functions and the 
geometric stiffness matrix are utilized for accounting for the 
second-order effects. A nonlinear static algorithm 
implementing the generalized displacement method is 
established for solving nonlinear problems. By some examples, 
the present method is proved to be reliable and straightforward 
for tracing the nonlinear behavior of 2D steel frames. 

Corresponding author: Phu-Cuong Nguyen



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www.etasr.com Nguyen et al.: Nonlinear Inelastic Analysis of 2D Steel Frames  

 

II. BEHAVIOR OF THE BEAM-COLUMN ELEMENT 

A. P δ−  Effect 

Stability functions studied in [31] were utilized for 
predicting the P δ−  effect. With one element for the member, 
it is efficient to economize sources and analysis time. The 
force-displacement relationship using the incremental form of a 
2D beam-column element can be written as: 

1 2

2 1

/ 0 0

0

0

I I

J J

P A I
EI

M S S
L

M S S

δ

θ

θ

∆ ∆

∆ = ∆

∆ ∆

    
    
    
        

    (1) 

where P∆ , IM∆ , and JM∆  are the axial force and moments, 

δ∆ , Iθ∆ , and Jθ∆  are the axial movement and rotations, A is 
the sectional area, I is the moment of inertia around the z axis, 
L is the elemental length, E  is the Young’s modulus of the 

steel, and 1S  and 2S  are stability functions. 

B. Fiber Plastic Hinge 

Figure 1 illustrates the fiber plastic hinge method. In this 
method, two ends, I and J, of the element are monitored 
regarding the behavior of stress and strain of fibers. The force-
displacement relation of a 2D element considering both the 
P δ−  effect and plasticity can be formulated as in (2): 

2
2

1 2
1

2
2

2 1
1

/ 0 0

0 ( (1 ))

0 ( (1 ))

t
I I J I J I

J J

I J J I

A I
P

E I S
M S S

L S
M

S
S S

S

δ
η η η η θ

θ

η η η η

 
 
 

∆ ∆    
    ∆ = − − ∆       ∆ ∆    

 
− − 

  
    

(2) 

where Iη  and Jη  are scalar parameters accounting for gradual 
yielding of fiber hinges.  

 

 
Fig. 1.  Fiber plastic hinge method. 

They are estimated as: 

( )2
1

n

tIi i i i

i
I

t

E A y I

E I
η =

+

=
∑

    (3) 

( )2
1

n

tJi i i i

i
J

t

E A y I

E I
η =

+

=
∑

    (4) 

where n  is the sum of fibers on the sections at I and J, tIiE  and 

tJiE  are the tangent modulus of the 
th
i  fiber at I and J, iA  is 

the area of the 
th
i  fiber, iI  is the moment of inertia of the 

th
i  

fiber, iy  is the center coordinate of the 
th
i  fiber, shown in 

Figure 1, and tE  is the tangent modulus of an element. 

C. Residual Stresses 

From [5], the CRC tangent modulus tE  is applied with the 

aim to consider the effects of residual stresses along the length. 
This effect is similar to the spread of plasticity on the length 
due to axial force, formulated as: 

tE E=   for   0.5 yP P≤     (5) 

4 1t
y y

P P
E E

P P

 
= − 

 
 

  for   0.5 yP P>     (6) 

where Py is squash load. ECCS residual stress [2] is admitted as 
initial condition to fibers, as shown in Figure 2. 

D. Fiber State 

The cross-section ends of the element are divided into 
several fibers n for considering gradual yielding of two fiber 
plastic hinges at I and J, as shown in Figure 1. Fibers are 
monitored and their behavior (stress, strain) is updated. If the 
fiber is yielded, its elastic modulus is equal to zero. Axial strain 

ε∆  and curvature χ∆  of cross-section, and section forces are 
written as: 

Sectional force vector: 

{ }TN M∆ ∆     (7) 

Sectional deformation vector: 

{ }Tε χ∆ ∆     (8) 

The sectional deformation vector is estimated as: 

( )

1

1 1

2

1 1

n n

i i i i i

i i

n n

i i i i i i i

i i

E A E A y
N

M
E A y E A y I

ε

χ

−

= =

= =

−
∆ ∆

=
∆ ∆

− +

 
 

    
       

 
  

∑ ∑

∑ ∑
 (9) 



Engineering, Technology & Applied Science Research Vol. 10, No. 4, 2020, 5974-5978 5976 
 

www.etasr.com Nguyen et al.: Nonlinear Inelastic Analysis of 2D Steel Frames  

 

   
Fig. 2.  ECCS residual stress pattern. 

E. P − ∆  Effect 
The elemental tangent stiffness matrix is written as: 

[ ] [ ] [ ]
3 5 3 3 3 5 5 5

T

e gT
K T K T K

× × × ×
 = +         (10) 

where the transformation matrix [ ]
3 5

T
×
 of the element is 

formulated as: 

[ ]
3 5

1 0 0 0 0

0 1 0 1/ 1/

0 0 1 1/ 1/

T L L

L L
×

− 
 = − 
 − 

    (11) 

and gK    is the geometric stiffness matrix: 

2 2

5 5

2

2

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0

0 0

I J I J

g

I J

I J

M M M M

L L

K
M M P P

L LL

M M P P

L LL

×

+ +
−

=
+

−

+
− −

 
 
 
 
 
     
 
 
 
 
 

    (12) 

III. NONLINEAR SOLUTION 

The nonlinear solution algorithm invented by Yang and 
Shieh [32] is developed to find a solution to the structural 
system. Yang and Shieh’s method is one of the most efficient 
and stable numerical methods. It can easily capture the 
problems with several critical points. The equilibrium equation 
of steel frames is: 

{ } { } { }1 1ˆi i i ij j j jK D P Rλ− −  ∆ = +      (10) 

where 1
i
jK −    is the tangent stiffness matrix, { }

i
jD∆  is the 

displacement vector, { }P̂  is the consultation load vector, 
{ }1ijR −  is the residual force vector, ijλ  is the load coefficient. 

IV. EXAMPLES AND DISCUSSION 

A. Column Flexural Buckling 

Figure 3 illustrates a simply supported steel column under 
axial compression. Young’s modulus is E=200000Mpa and 
Poisson’s ratio of steel is v=0.3. A horizontal load is put in the 
middle of the column for considering geometry imperfection. 

 

 
Fig. 3.  Simply supported column. 

   
Fig. 4.  Load-deflection curves of the column. 

The load-deflection curves of the column captured by the 
present study and ABAQUS are plotted in Figure 4. By one 
element per column, the presented method can predict precisely 
the behavior and strength of the column, but ABAQUS 
overestimates the strength by 10.6%. ABAQUS needs more 
than five elements for obtained accuracy. This example 
illustrates the accuracy of the presented study in predicting the 
second-order effect. 

B. Portal Steel Frame 

Vogel [3] invented the portal steel frame for benchmark 
second-order inelastic methods. Nguyen and Kim [25] 
proposed a plastic zone method for analyzing this frame. The 
frame configuration is described in Figure 5. The elastic 
modulus is E=205000Mpa, the yield stress of the steel is 
σy=235MPa. The cross-sections are HEA340 and HEB300. 
Authors in [3, 25] used 50 elements for columns and 40 
elements for the beam for analyzing the frame, while this 
present program uses one element for beam-column members, 
and I-shape cross-sections have been meshed into 24 fibers for 
flanges and 18 fibers for the web. The load-deflection curve of 
the present study closely matches with Vogel’s result, as 
plotted in Figure 6. Nguyen and Kim’s result is lower than 
Vogel’s result: -2.05% error is obtained when compared with 

P

0.004P

W
8
x
3
1 2
.5
 m

2
.5
 m



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www.etasr.com Nguyen et al.: Nonlinear Inelastic Analysis of 2D Steel Frames  

 

Vogel’s result. The collapse load coefficient of the different 
applied methods is listed in Table I. Less than 0.8% error of the 
present study is achieved when comparing with Vogel’s result. 
Analyzing this problem on a computer, with a configuration of 
Intel Core i7-7500 4CPUs 2.70GHz and 16GB RAM, the 
analyzing time is only 15s. This shows the accuracy and 
computation speed of the proposed method. 

 

 
Fig. 5.  Portal steel frame. 

TABLE I.  COLLAPSE LOAD COEFFICIENT FOR PORTAL FRAME 

Method Collapse load coefficient Error (%) 

[3] 1.022 – 

[25] 1.001 -2.05 

Present study 1.014 -0.78 

 

 
Fig. 6.  Load-deflection curves of portal frame. 

C. Six-story Steel Frame 

The six-story steel frame plotted in Figure 7 was firstly 
analyzed by Vogel in [3]. Vogel used both plastic zone and 
plastic hinge methods, Chan and Chui [7] used the refined 
plastic hinge method, Nguyen and Kim in [22] used a fiber 
beam-column method and in [25] the plastic zone method. All 
the columns are inclined with an angle of 

ψ = 1/450.  Properties of steel are E=205000MPa and 
σy=235MPa. In the present study, five elements for beams and 
one for columns were used for modeling. Figure 8 and Table II 
show the results predicted by various methods. In this frame, 
the predicted load-deflection curve of various methods is not 
much different. The load-deflection curve of the present study 
is almost identical to Vogel’s plastic zone method. The critical 
strength predicted by the proposed method (1.116) has less than 
0.45% error when compared with Vogel’s result (1.111) using 

the plastic zone method. Analyzing this problem on the same 
Intel computer, it takes only 53s, showing that the proposed 
program is accurate and efficient in predicting the nonlinear 
behavior and the strength of 2D steel frames. 

 

 
Fig. 7.  Six-story steel frame. 

 
Fig. 8.  Load-deflection curves of a six-story frame. 

TABLE II.  COLLAPSE LOAD COEFFICIENT FOR A SIX-STORY FRAME 

Method Collapse load coefficient Error (%) 

[3] 1.111 – 

[25] 1.100 -0.99 

Present study 1.116 +0.45 
 

V. CONCLUSION 

A second-order inelastic analysis program based on the 
finite element method for 2D steel frames has been developed 

successfully. The effects of P-δ, P-∆, inelasticity of materials, 
residual stresses, and imperfections have been accounted for 
the nonlinear analysis by the generalized displacement method. 
The proposed method is simple, accurate, and efficient in 
predicting the strength and behavior of steel frames. The 

4.0 m

5
.0
 m

2800 kN

35 kN ∆

H
E
B
3
0
0

2800 kN

H
E
B
3
0
0

HEA340

ψ = 1/400 ψ

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25

L
o
a
d
 c
o
e
ff
ic
ie
n
t

Horizontal deflection (mm)

Vogel

Nguyen and Kim

Present study

6.0 m6.0 m

49.1 kN/m

49.1 kN/m

49.1 kN/m

49.1 kN/m

49.1 kN/m

31.7 kN/m

IPE400

IPE360

IPE330

IPE300

IPE300

IPE240

H
E
B
2
2
0

H
E
B
2
2
0

H
E
B
2
2
0

H
E
B
2
2
0

H
E
B
1
6
0

H
E
B
1
6
0

H
E
B
2
6
0

H
E
B
2
6
0

H
E
B
2
4
0

H
E
B
2
4
0

H
E
B
2
0
0

H
E
B
2
0
0

6
 x
 3
.7
5
 m
 =
 2
2
.5
 m

∆

= 1/450ψ

= 10.23 kNF 1

= 22.44 kNF 2

F 2

F 2

F 2

F 2

ψ ψ

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150 200 250 300 350

L
o
a
d
 c
o
e
ff
ic
ie
n
t

Horizontal deflection (mm)

Vogel

Nguyen and Kim

Present study



Engineering, Technology & Applied Science Research Vol. 10, No. 4, 2020, 5974-5978 5978 
 

www.etasr.com Nguyen et al.: Nonlinear Inelastic Analysis of 2D Steel Frames  

 

proposed method can be integrated into commercial software 
for daily engineering design using advanced analysis. 

ACKNOWLEDGMENT 

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 Ho Chi Minh City 
Open University and colleagues for supporting this project. 

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