DSpace at VNU: Second-order plastic-hinge analysis of planar steel frames using corotational beam-column element

Journal of Constructional Steel Research 121 (2016) 413–426

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Journal of Constructional Steel Research

Second-order plastic-hinge analysis of planar steel frames using
corotational beam-column element
Tinh-Nghiem Doan-Ngoc a,b, Xuan-Lam Dang a, Quoc-Thang Chu c, Richard J. Balling d, Cuong Ngo-Huu a,⁎
a

Faculty of Civil Engineering, University of Technology, VNU-HCM, 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Viet Nam
Department of Civil Engineering and Applied Mechanics, Ho Chi Minh City University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Viet Nam
Department of Civil Engineering, International University, VNU-HCM, Thu Duc District, Ho Chi Minh City, Viet Nam
d
Department of Civil and Environmental Engineering, Brigham Young University, Provo, UT 84602, United States
b
c

a r t i c l e

i n f o

Article history:
Received 9 November 2015
Received in revised form 15 February 2016
Accepted 11 March 2016
Available online 19 March 2016
Keywords:
Plastic-hinge
Corotational element

Nonlinear analysis
Steel frames

a b s t r a c t
A new beam-column element for nonlinear analysis of planar steel frames under static loads is presented in this
paper. The second-order effect between axial force and bending moment and the additional axial strain due to
the element bending are incorporated in the stiffness matrix formulation by using the approximate seventhorder polynomial function for the deflection solution of the governing differential equations of a beam-column
under end axial forces and bending moments in a corotational context. The refined plastic-hinge method is
used to model the material nonlinearity to avoid the further division of the beam-columns in modeling the structure. A Matlab computer program is developed based on the combined arc-length and minimum residual displacement methods and its results are proved to be reliable by modeling one or two proposed elements per
member in some numerical examples.
© 2016 Elsevier Ltd. All rights reserved.

1. Introduction
In the nonlinear analysis of steel structures, the beam-column
method has been considered as the simple and effective one in
modeling the second-order and inelastic effects and its results are
verified to be accurate enough for practical design application as
studied by Lui and Chen [1], Liew et al. [2], Chan and Chui [3],
Thai and Kim [4], Ngo-Huu and Kim [5], etc. However, the use of
the accurate stability functions obtained from the closed-form solution of the beam-column under end axial forces and bending moments can lead to some difficulties in derivation of the stiffness
matrix formulation, especially in corotational context. Chan and
Zhou [6] proposed the approximate fifth-order polynomial displacement function of the beam-column element and formulated
the element stiffness matrix considering the second-order effect
by principle of stationary total potential energy. The advantage of
using this polynomial function is its simplicity in formulation

⁎ Corresponding author.
E-mail address: (C. Ngo-Huu).

/>0143-974X/© 2016 Elsevier Ltd. All rights reserved.

while its accuracy is still maintained as the use of closed-form stability functions.
The corotational method has been widely used due to its efficiency
in deriving the formulation of geometrically nonlinear beam-column element for elastic analysis (Nguyen [7], Le et al. [8]) and inelastic analysis
(Balling and Lyon [9], Thai and Kim [10], Saritas and Koseoglu [11]). This
study proposes a new seventh-order polynomial displacement function
for the approximate solution of the governing differential equations to
formulate the element stiffness matrix considering the second-order effect following the beam-column theory in corotational context as presented by Balling and Lyon [9]. The bowing effect is integrated in the
formulation to consider the change in element length due to the bending of the element. The refined plastic-hinge method is used to simulate
the inelastic behavior of the steel material as lumped concept. To solve
the system of equilibrium nonlinear equations, the arc-length combined
with minimum residual displacement methods are employed due to
their robustness in nonlinear analysis application. A computer program
is developed using the Matlab programing language to automate the
analysis of nonlinear behavior of planar steel frames under static
loads. The obtained analysis results are compared to those of existing
studies to verify the reliability and effectiveness of the proposed
program.


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T.-N. Doan-Ngoc et al. / Journal of Constructional Steel Research 121 (2016) 413–426

Fig. 1. Simply-supported beam-column element.

2. Formulation
2.1. Stability functions
Consider a simply supported planar beam-column element under end axial force and bending moments as presented in Fig. 1.
The governing differential equations of the element using second-order Euler beam theory are

!
!
4
2
d ΔðxÞ
d ΔðxÞ
−F
¼ 0:
EI
dx4
dx2

ð1Þ

The closed-form solution to the differential equations leads to following end moment-end rotation relationship (Oran [12])
&

M1
M2

'
¼

'
θ1
:
θ2

ð2Þ

λ sin λ−λ2 cos λ
2−2 cos λ−λ sin λ
λ2 −λ sin λ
¼
2−2 cos λ−λ sin λ

ð3Þ

EI s11
L0 s21

s12
s22

!&

For compressive F b 0
s11 ¼ s22 ¼
s12 ¼ s21
where λ ¼ L0

qffiffiffiffiffi

j Fj
EI .

For tensile F N 0
λ2 cosh λ−λ sinh λ
2−2 cosh λ þ λ sinh λ
λ sinh λ−λ2

:
¼
2−2 cosh λ þ λ sinh λ

s11 ¼ s22 ¼
s12 ¼ s21

ð4Þ

For the simplicity in mathematical handling, instead of using the closed-form solution with above-mentioned complicated stability functions, the
deflection solution is assumed in following seventh-order polynomial function
ΔðxÞ ¼ a7 x7 þ a6 x6 þ a5 x5 þ a4 x4 þ a3 x3 þ a2 x2 þ a1 x þ a0 :

Fig. 2. Comparison of proposed and closed-form stability functions.

ð5Þ


T.-N. Doan-Ngoc et al. / Journal of Constructional Steel Research 121 (2016) 413–426

415

Fig. 3. Initial and displaced positions of the beam-column element.

The ai coefficients are determined from the compatibility and equilibrium conditions as follows
ΔðxÞðx¼0Þ ¼ 0

ð6Þ

ΔðxÞðx¼L0 Þ ¼ 0

ð7Þ






dΔðxÞ
¼ θ1
dx ðx¼0Þ

ð8Þ


dΔðxÞ
¼ θ2
dx ðx¼L0 Þ

ð9Þ

!
2
d ΔðxÞ
EI
À
dx2
!
3
d ΔðxÞ

EI
dx3

EI

EI

M 1 ¼ −EI

M 2 ¼ EI

À Á ðM1 þ M2 Þ ðxÞÀ L Á −M 1
Á ¼ FΔðxÞ x¼L20 þ
x¼ 20
L0

¼F
ðx¼0Þ

!
3
d ΔðxÞ
À
dx3
!
3
d ΔðxÞ
dx3

L

x¼ 20

L

x¼ 20



ðM1 þ M2 Þ
dΔðxÞ
þ
L0
dx ðx¼0Þ

Á¼F

ð10Þ

ð11Þ



dΔðxÞ
ðM1 þ M2 Þ
À LÁþ
dx
L0
x¼ 0

ð12Þ

2


¼F
ðx¼L0 Þ

!
2
d ΔðxÞ
dx2

!
2
d ΔðxÞ
dx2


dΔðxÞ
ðM 1 þ M 2 Þ
þ
dx ðx¼L0 Þ
L0

ð13Þ

ð14Þ
ðx¼0Þ

:

ð15Þ

ðx¼L0 Þ

Fig. 4. Cantilever with an end point load.


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Fig. 5. Large displacement analysis of cantilever with an end point load.

The ai coefficients are solved from Eqs. (6) through (13) and the end moment-end rotation relationship is identical as Eq. (2) with following sij
functions.
For compressive F ≤ 0
À 3
Á
5q −1404q2 þ 86400q−1209600
s11 ¼ s22 ¼ −
9ð40−qÞð840−11qÞ Á
À 3
q þ 252q2 −25920q þ 1209600
s12 ¼ s21 ¼
18ð40−qÞð840−11qÞ

ð16Þ

where q ¼ λ2 ¼ jEIFj L20 .
For tensile F N 0
À

Á
5q3 þ 1404q2 þ 86400q þ 1209600
À 3 9ð40 2þ qÞð840 þ 11qÞ
Á
q −252q −25920q−1209600
:
¼−
18ð40 þ qÞð840 þ 11qÞ

s11 ¼ s22 ¼
s12 ¼ s21

ð17Þ

Fig. 2 shows a comparison of the proposed and closed-form stability functions and it can be seen that all curves are almost identical.
The differentiations of the proposed stability functions s11 , s12 , s21 and s22 with respect to q are as follows.

Fig. 6. Beam-column with end point loads.


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417

Fig. 7. Deflections at free end of column.

For F ≤ 0




ds11
dq

ds12
dq

¼

À
Á


5 11q4 −2560q3 þ 270144q2 −13547520q þ 270950400
ds22
¼−
dq
9ð40−qÞ2 ð840−11qÞ2

ð18Þ

¼

Á

 À

11q4 −2560q3 þ 63360q2 −9676800q þ 677376000
ds21
¼
dq
18ð40−qÞ2 ð840−11qÞ2

ð19Þ

¼

À
Á


5 11q4 þ 2560q3 þ 270144q2 þ 13547520q þ 270950400
ds22
¼
dq
9ð40 þ qÞ2 ð840 þ 11qÞ2

ð20Þ

¼

À
Á


11q4 þ 2560q3 þ 63360q2 þ 9676800q þ 677376000
ds21

¼−
dq
18ð40 þ qÞ2 ð840 þ 11qÞ2

ð21Þ





For F N 0




ds11
dq

ds12
dq





As q goes to zero, Eqs. (16) through (21) become to
s11 ¼ s22 ¼ 4




ds11
dq
ds11
dq



s12 ¼ s21 ¼ 2

¼



ds22
2
¼−
15
dq

¼



ds22
2
¼
15
dq



ð22Þ


 

ds12
ds21
1
¼
¼
30
dq
dq

ð F ≤ 0Þ

ð23Þ


 

ds12
ds21
1
¼
¼−
30
dq

dq

ð F N0Þ:

ð24Þ

These results are identical to those obtained by using the approximate deflection function as common Hermite third-order polynomial function
for the beam element.

Fig. 8. William's toggle frame.


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T.-N. Doan-Ngoc et al. / Journal of Constructional Steel Research 121 (2016) 413–426

Fig. 9. Load-deflection curves of William toggle frames.

2.2. Axial strain attributed to element bending
Axial force considering the axial strain attributed to element bending because of end rotations is shown as follows
0L
1
ZL0  2
Z0
ZL0  2
EA @ dδ
1
dΔ
EA
EA

dΔ
dx þ
dxA ¼
ΔL0 þ
dx:
F¼
L0
dx
2
dx
L0
2L0
dx
0

0

ð25Þ

0

For the closed-form solution, the axial force can be presented by stability functions as
AE
ΔL0 −EA
L0
AE
ΔL0 þ EA
F¼
L0
F¼






 !
1 ds11 2
ds12
1 ds22 2
θ1 þ
θ1 θ2 þ
θ
ð F ≤ 0Þ
2  dq 
2  dq  2 !
 dq 
:
1 ds11 2
ds12
1 ds22 2
θ1 þ
θ1 θ2 þ
θ2
ð F N0Þ
2 dq
2 dq
dq

ð26Þ

The axial force of the proposed approach is also derived from above relations.
2.3. Plastic hinge
Let η1 and η2 (0≤η1, η2 ≤ 1) be the inelastic ratios of the end sections of the beam-column element in which the values of one and zero indicate
fully elastic and plastic-hinge states, respectively, and a value between zero and one indicates the partially plastic state of the section. Eqs. (2) and
(26) are modified to account for the presence of the plastic hinges as follows
&

M1
M2

'
¼

EI s1p
L0 s2p

s2p
s3p

!&

θ1
θ2

'
ð27Þ

Fig. 10. Pinned-ended column under axially compressed load.

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419

Table 1
Buckling loads of pinned-ended column.
λc

L (mm)

P/Py
Residual stress ignored

1141.97
2283.95
3425.92
4567.84
6851.90
9135.78
11419.73
13703.67
15987.62
18271.56
20555.51
22839.45

F¼

0.25

0.50
0.75
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00

Residual stress considered

Euler

Ngo-Huu & Kim

Proposed

Diff. (%)

CRC

Ngo-Huu & Kim

Proposed

Diff. (%)

16
4
1.7778
1.0000
0.4444
0.2500
0.1600
0.1111
0.0816
0.0625
0.0494
0.0400

0.9870
0.9870
0.9870
0.9870
0.4450
0.2500
0.1600
0.1120
0.0820
0.0630
0.0500
0.0400

1.0000
1.0000
1.0000
0.9973

0.4433
0.2494
0.1597
0.1110
0.0816
0.0625
0.0494
0.0400

–
–
–
0.27
0.25
0.24
0.19
0.09
0.00
0.00
0.00
0.00

0.9844
0.9375
0.8594
0.7500
0.4444
0.2500
0.1600
0.1111

0.0816
0.0625
0.0500
0.0400

0.987
0.936
0.861
0.76
0.445
0.25
0.16
0.112
0.082
0.063
0.0496
0.0402

0.9843
0.9373
0.8590
0.7494
0.4433
0.2494
0.1597
0.1110
0.0816
0.0625
0.0494
0.0400

0.01
0.02
0.05
0.08
0.25
0.24
0.19
0.09
0.00
0.00
1.20
0.00






 !
ds2p
AE
1 ds1p 2
1 ds3p 2
θ1 þ
θ1 θ2 þ
θ2 :
ΔL0 Æ EA
L0
2 dq

2 dq
dq

ð28Þ

The ± symbol in Eq. (28) is assigned as “+” when F N 0 and “–” when F ≤ 0; the modified stability functions s1p, s2p and s3p are determined from the
stability functions and inelastic ratios as proposed by Chan and Chui [3] as

s1p ¼ η1 s11 −

Á
s212 À
1−η2
s11

!
s2p ¼ η1 η2 s12

s3p ¼ η2 s22 −

!
Á
s221 À
1−η1 :
s22

ð29Þ

The differentiations of the modified stability functions with respect to q are as follows






ds1p
dq

ds2p
dq

ds3p
dq







2
3
ds12
ds11

 2s11 s12
−s212
6 ds11
Á7
dq

dq À
¼ η1 6
1−η2 7
4 dq −
5
s211


¼ η1 η2





ds12
dq

ð30Þ

ð31Þ





2
3
ds21
ds22


 2s22 s21
−s221
6 ds22
Á7
dq
dq À
¼ η2 6
1−η1 7
4 dq −
5:
s222

ð32Þ

Clearly, the mathematic handling of the modified stability functions and their differentiations with the use of the approximate seventh-order deflection function is much simplified.

Fig. 11. Strength curve of pinned-ended column.


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T.-N. Doan-Ngoc et al. / Journal of Constructional Steel Research 121 (2016) 413–426

Fig. 12. One-bay two-storey frame with pinned support.

The refined plastic-hinge method presented by Liew et al. [2] is used in this research. The inelastic ratio η =4β(1− β) is determined through inelastic parameter β, where β is calculated based on the following strength curve of Orbison [13]

β ¼ 1:15

 2  2

 2  2
P
M
P
M
þ
þ 3:67
:
Py
Mp
Py
Mp

ð33Þ

To consider the gradual plasticity due to the effect of the axial force on the presence of residual stresses in the section the tangent modulus Et proposed by the Column Research Council is used as follows
P
Et
≤ 0:5
for
¼1
Py
E
 

:
Et
P
P
P

¼4
1−
N0:5
for
Py
Py
E
Py

ð34Þ

2.4. Corotational element stiffness matrix
Consider the change in geometry of the beam-column element shown in Fig. 3.

Fig. 13. Elastic load-deflection curves.


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421

Fig. 14. Inelastic load-deflection curves.

The original length L0 and the deformed length L of the element:
L0 ¼

L¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðxB −xA Þ2 þ ðzB −zA Þ2

ð35Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðxB þ u4 −xA −u1 Þ2 þ ðzB þ u5 −zA −u2 Þ2 :

ð36Þ

The nodal rotations of the element:
θ1 ¼ u3 −ðα−α 0 Þ

ð37Þ

θ2 ¼ u6 −ðα−α 0 Þ

ð38Þ

where
sin α ¼

z þ u −z −u 
B
5
2
A
L

ð39Þ

cos α ¼

x þ u −x −u 
B
4
1
A
L

ð40Þ

Fig. 15. Two-bay four-storey frame.


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Fig. 16. Lateral deflection of top right joint.

α 0 ¼ sin−1




zB −zA
:
L0

ð41Þ

The differentiations of L, θ1, θ2, sinα, cosα, F, M1 and M2 with respect to nodal displacements ui(i = 1, ... , 6) are as follows
&

&

∂L
∂u

'

∂θ1
∂u

¼ f − cos α
&

'
¼

−

sin α
L

− sin α

0

cos α

cos α
L

1

sin α
L

sin α

−

cos α
L

0 gT

ð42Þ
'T
ð43Þ

0

Fig. 17. One-bay four-storey frame.


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423

Table 2
Limit load factor of proposed program and the others of four-storey frame.
Value of lateral load (H)

Ratio of limit inelastic load

0.10P
0.24P
0.50P

&

&

&







∂θ2
∂u

&

'
¼

∂ sin α
∂u

¼

∂M2
∂ui

&

'



∂M1
∂ui

sin α
L

&

'

∂ cos α
∂u

∂F
∂ui

−

¼

¼

cos α
L

sin α cos α
L
sin2 α
−
L

Kassimali

Yoo and Choi

Proposed

Difference with Yoo and Choi's results (%)

1.687
1.502
1.075

1.660
1.479
1.062

1.656
1.465
1.045

−0.24
−0.95
−1.60

sin α
L

0

cos2 α
−
L
sin α cos α
L

−

cos α
L

'T
ð44Þ

1

0

sin α cos α
−
L

0

sin2 α
L

cos2 α
L

sin α cos α
−
L

'T
ð45Þ

0
'T
0

 

  
 !
∂s1p

∂s2p
∂s2p
∂s3p
EA ∂L
∂θ1
∂θ2
Æ EA
þ
θ1 þ
θ2
θ1 þ
θ2
L0 ∂ui
∂q
∂q
∂ui
∂q
∂q
∂ui



ð46Þ

ð47Þ

¼





 
∂s1p
∂s2p
EI
∂θ1
∂θ2
∂F
s1p
þ s2p
Æ L0
θ1 þ
θ2
L0
∂ui
∂ui
∂q
∂q
∂ui

ð48Þ

¼




 
∂s2p
∂s3p

EI
∂θ1
∂θ2
∂F
s2p
þ s3p
Æ L0
:
θ1 þ
θ2
L0
∂ui
∂ui
∂q
∂q
∂ui

ð49Þ



The ± symbol in above equations are assigned as “+” when F N 0 and “–” when F ≤ 0.
The nodal element resistance vector in local coordinate system is

fzg ¼ −F

ðM 1 þ M 2 Þ
L

M1

F

−

ðM1 þ M2 Þ
L

!T
M2

:

ð50Þ

The nodal element resistance vector in global coordinate system is fZg ¼ ½T T Šfzg, where [T] is the transformation matrix of planar element.
The local and global element tangent stiffness matrix ½kT Š and [KT], respectively, are determined as follows
3
0 2 h Ti
1
!
!
h i
∂ T
∂fZ g
∂fzg
T
T
TA
4

5
@
k T ¼ ½T Š
½T Š ¼ ½T Š
½T Š
fzg ½T Š þ
∂fug
∂fug
∂fug
h i
½K T Š ¼ ½T ŠT kT ½T Š:

ð51Þ

ð52Þ

Fig. 18. Load-displacement curve of the four-storey frame.


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T.-N. Doan-Ngoc et al. / Journal of Constructional Steel Research 121 (2016) 413–426

Applying the results of above differentiations with respect to nodal variables ui(i =1, ... ,6), the local element stiffness matrix ½kT Š ¼ ½kG Š þ ½kθ Š is
derived with following ½kG Š and ½kθ Š matrices
2

A
6 I
6

6
6
6
6
h i EI 6
6
kG ¼ 6
L0 6
6
6
6
6
6
4

À
G1 þ

0
s1p þ 2s2p þ s3p

Á

À

L2

0
s1p þ s2p
L

s1p

Á

−

3

A
I

0

À
−G1 −

0
s1p þ 2s2p þ s3p

L2 Á
À
− s1p þ s2p
L

0
A
I

À
G1 þ

0
s1p þ 2s2p þ s3p

2

−T 1

0

L0 T 1 ðT 1 þ T 2 Þ
L2

ðT 1 þ T 2 Þ
−T 3 þ
L

L0 T 21

T1
0

Á

L2

sym:
ðT 1 þ T 2 Þ
T3−
6 0

L
6
6
6
T
4
6
6
6
h i
6
kθ ¼ EA6
6
6
6
6
6
6
6
4
sym:

Á

−T 3 þ

ðT 1 þ T 2 Þ
L

−T 4

−

L0 T 1 ðT 1 þ T 2 Þ

L2
ðT 1 þ T 2 Þ
T3−
L
T4

0

Á 7
7
7
7
7
7
7
7
s2p
7
7
7
7
0
À
Á7
7
− s2p þ s3p 7

5
L
s3p
À

s2p þ s3p
L

ð53Þ

3
−T 2

7
7
L0 T 2 ðT 1 þ T 2 Þ 7
7
7
L2
7
7
7
L0 T 1 T 2
7
7
7
7
T2
7
7

L0 T 2 ðT 1 þ T 2 Þ 7
7
−
2
5
L
2
L0 T 2

ð54Þ

0
where G1 ¼ AI ðL−L
L Þ and



∂s1p
∂s2p
θ1 þ
θ2
T1 ¼ Æ
∂q
∂q

ð55Þ



∂s2p

∂s3p
T2 ¼ Æ
θ1 þ
θ2
∂q
∂q

ð56Þ

T3 ¼

T4 ¼

ðM 1 þ M 2 Þ

ð57Þ

EAL2
T 1 θ1 þ T 2 θ2 L0 ðT 1 þ T 2 Þ2
þ
:
2L
L2

ð58Þ

The “±” symbol in Eqs. (55) and (56) are assigned as “+” when F N 0 and “–” when F ≤ 0.
2.5. Nonlinear solution algorithm
The arc-length method combined with minimum residual displacement method proposed by Chan and Zhou [6] is used as nonlinear solution algorithm in this research to solve the nonlinear equation system. The incremental equilibrium equation is presented as follows:
È

É
fΔu þ ΔλΔug ¼ ½K T Š−1 ΔP þ ΔλΔP

ð59Þ

where {ΔP}= {P − Z} is applied incremental load vector for the first iteration or the unbalanced force vector in second iteration onward; {Δu} = corresponding displacement increment due to this force; fΔPg = a force vector parallel to the applied load vector; fΔug = conjugate displacement
solved; Δλ = a load corrector factor for imposition of the constraint condition.
For the first iterative step
arc length
Δλ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
fΔugT fΔug
From the second step, Δλi is determined from the following condition
h
i
T
∂ðequilibrium error Þ ∂ fΔλi Δu þ Δui g fΔλi Δu þ Δui g
¼
¼ 0:
∂Δλi
∂Δλi

ð60Þ

ð61Þ

Simplifying the Eq. (61), we have:
Δλi ¼

fΔugTi fΔug
fΔugT fΔug

ði ≥ 2Þ:

ð62Þ


T.-N. Doan-Ngoc et al. / Journal of Constructional Steel Research 121 (2016) 413–426

425

3. Numerical examples

3.5. Single-bay two-storey frame

A computer program using the above-mentioned nonlinear solution
algorithm is developed using Matlab for nonlinear inelastic analysis of
planar steel frames subjected to static loads. The analysis results are
compared with results from the literature to validate accuracy on the
following numerical examples.

The single-bay two-storey frame with pinned supports as shown in
Fig. 12 was analyzed by Lui and Chen [18] and later by Chan and Chui
[3]. Each beam and column member of the frame was respectively
modeled by two and one elements by Chan and Chui while the frame
member is modeled by one element herein. The analysis results presented in Fig. 13 and Fig. 14 show that the elastic and inelastic loaddeflection curves of the frame are almost identical. The elastic and inelastic limit loads of Chan and Chui are 746 kips and 417 kips while
the corresponding results of proposed program are 732 kips and
421 kips with the respective differences of 1.9% and 1.0%.

3.1. Cantilever beam with an end point load
The cantilever beam with geometric and material properties shown

in Fig. 4 is used for nonlinear elastic analysis in this research. This example was firstly introduced by Bisshopp and Drucker [14] with exact solution and then it has been analyzed by many researchers by modeling
with two or more elements per member for accuracy comparison. The
cantilever beam is modeled by two proposed elements herein and by
two and twenty BEAM3 elements by ANSYS for verification purpose.
The results of Bisshopp and Drucker, the proposed method, and
ANSYS program are shown in Fig. 5. It can be seen that the obtained results from proposed method are acceptable in accuracy in comparison
with the closed-form solution of Bisshopp and Drucker and the ANSYS
result with 20 beams in modeling.

3.2. Cantilever column with end eccentric axial point load
The cantilever column subjected to eccentric axial load at its free end
shown in Fig. 6 was firstly introduced by Wood and Zienkiewicz [15] by
using five paralinear elements and was recently analyzed by Nguyen [7]
by using three corotational elements. The column is modeled by two
proposed elements.
The nonlinear elastic analysis results shown in Fig. 7 prove that the
displacement responses of proposed method have good agreement
compared to those of existing studies.

3.3. William's toggle frame
The William toggle frame shown in Fig. 8 was analyzed by many researchers to verify their analysis methods in predicting large displacement behavior at the system level. For existing studies, the member
was generally modeled from two or more elements for nonlinear elastic
analysis. The William toggle frame was analyzed by Chan and Chui [3]
with two different pinned and fixed support conditions. Only one proposed element is used herein for modeling each frame member. Fig. 9
shows that a good agreement is seen between the proposed result and
existing ones.

3.4. Pinned-ended column
The pinned-ended column under top axial force shown in Fig. 10 was
analyzed by Ngo-Huu and Kim [16] by the fiber hinge method in which

the column was divided into three elements, one middle elastic element
using the conventional stability function and two end fiber-hinge elements. Table 1 presents the buckling load results obtained by the proposed program, the Euler's theoretical exact solution, CRC column
curve (Chen and Lui [17]), and Ngo-Huu and Kim's fiber hinge element
with a large range of the column length. A comparison of results from
the proposed program and Ngo-Huu and Kim's analysis result is also
presented and the maximum difference of about 1.2% is found. The
strength curves corresponding to the slenderness parameter about the
weak axis are shown in Fig. 11 and it can be seen that the curves are almost identical. This example demonstrates the capacity of the proposed
program in predicting the elastic and inelastic buckling loads of the
column.

3.6. Two-bay four-storey frame
The two-bay four-storey frame shown in Fig. 15 was analyzed by
Kukreti and Zhou [19] by using the refined plastic-hinge method with
LRFD bilinear plastic strength curve. It is modeled herein by one element
per column and two elements per beam. The load-lateral deflection
curves of the frame from the proposed program and Kukreti and Zhou's
analysis are presented in Fig. 16. It can be seen that the curves are relatively matched. The limit load ratio of Kukreti and Zhou's analysis is
1.831 while that of the program is 1.834, which are different by only
about 0.16%.
3.7. One-bay four-storey frame
The one-bay four-storey frame shown in Fig. 17 was analyzed by
Kassimali [20] using rigid-hinge method and later by Yoo and Choi
(2008) [21] using inelastic buckling method based on bilinear and linear
strength curves in order to compare the nonlinear behavior and ultimate
loads with varying lateral loads of H = 0.1P, 0.24P, and 0.5P. One and
two proposed elements are used to model each column and beam member, respectively. The analysis results of the proposed element and
Kassimali and Yoo and Choi are shown in Table 2 and Fig. 18. It can be
seen that the proposed element predicts the ultimate strength of the
frame very well but there are the slight differences in load-lateral deflection curves.

4. Conclusion
The seventh-order polynomial function is assumed as the deflection
solution for the governing second-order differential equations of the
beam-column member under end axial forces and bending moments
and it is applied in the formulation of element stiffness in the
corotational context. The corotational element also integrates the additional axial strain caused by the bending of the element and the inelastic
simulation by refined plastic-hinges lumped at both ends. A Matlab program is developed based on the arc-length combined with minimum residual displacement methods to solve the system of nonlinear
equilibrium equations step by step. The analysis results of numerical examples prove that and the developed program from the proposed
corotational element is capable of accurately predicting the nonlinear
behavior of structural members and frames under the static loads.
Notations
The followings notations are used throughout the paper.
A
Sectional area of beam-column member
E , Et
Elastic and tangent modulus
I
Moment of inertia of the element section
L0 , L
Initial and deformed lengths of the element
ΔL = L − L0 Change in element length
Py , Mp Squash load and plastic moment of the cross-section
s11 , s12 , s21 , s22 Elastic stability functions of beam-column element


426

T.-N. Doan-Ngoc et al. / Journal of Constructional Steel Research 121 (2016) 413–426

s1p , s2p , s3p Inelastic stability function of beam-column element considering end flexibility

F , M1 , M2 Axial force and end bending moments
½kT Š; ½K T Š Local and global element tangent stiffness matrices
½kG Š, ½kθ Š Local geometric and higher-order geometric tangent stiffness
matrices
[T]
Local-global transformation matrix of planar element
fzg; fZg Local and global nodal element resistance vectors
{P}, {ΔP} Global total and incremental load vectors
{u}, {Δu} Global total and incremental displacement vectors
β
Inelastic parameter
λ
Load factor
Δ(x)
Defection function of beam-column element
η1 , η2
Inelastic ratios at left and right end sections
θ1 , θ2
Rotations at element left and right ends
Acknowledgments
This research is funded by Vietnam National Foundation for Science
and Technology Development (NAFOSTED) under grant number
107.02-2012.28.
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