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

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

Second-order plastic-hinge analysis of planar steel frames using

corotational beam-column element

a

Faculty of Civil Engineering, University of Technology, VNU-HCM, 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Viet Nam

b 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

c

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

a b s t r a c t

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

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 seventh-order 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 struc-ture A Matlab computer program is developed based on the combined arc-length and minimum residual dis-placement 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

Keywords:

Plastic-hinge

Corotational element

Nonlinear analysis

Steel frames

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

so-lution of the beam-column under end axial forces and bending

mo-ments can lead to some difficulties in derivation of the stiffness

matrix formulation, especially in corotational context Chan and

Zhou[6]proposed the approximatefifth-order polynomial

dis-placement 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

while its accuracy is still maintained as the use of closed-form sta-bility functions

The corotational method has been widely used due to its efficiency

in deriving the formulation of geometrically nonlinear beam-column el-ement 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 ef-fect following the beam-column theory in corotational context as pre-sented by Balling and Lyon[9] The bowing effect is integrated in the formulation to consider the change in element length due to the bend-ing 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

⁎ Corresponding author.

E-mail address: ngohuucuong@hcmut.edu.vn (C Ngo-Huu).

http://dx.doi.org/10.1016/j.jcsr.2016.03.016

Contents lists available atScienceDirect Journal of Constructional Steel Research

2.1 Stability functions

Consider a simply supported planar beam-column element under end axial force and bending moments as presented inFig 1

The governing differential equations of the element using second-order Euler beam theory are

EI d

4

Δ xð Þ

dx4

!

2

Δ xð Þ

dx2

!

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

M1

M2

¼EI

L0

s11 s12

s21 s22

θ1

θ2

 

For compressive Fb0

s11¼ s22¼ λsinλ−λ2

cosλ

2−2 cos λ−λ sin λ

s12¼ s21¼ λ2

−λ sin λ

2−2 cos λ−λ sin λ

ð3Þ

whereλ ¼ L0

ffiffiffiffiffi

j Fj

EI

q

For tensile FN0

s11¼ s22¼ λ

2

coshλ−λ sinh λ

2−2 cosh λ þ λ sinh λ

s12¼ s21¼ λ sinh λ−λ2

2−2 cosh λ þ λ sinh λ:

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

The aicoefficients are determined from the compatibility and equilibrium conditions as follows

Δ xð Þx¼L

0

dΔ xð Þ

dx

x¼0

dΔ xð Þ

dx

x¼L 0

EI d

2

Δ xð Þ

dx2

!

x¼ L0

2

2

L0

x

ð Þx¼L0 2

EI d

3

Δ xð Þ

dx3

!

x¼0

ð Þ

¼ F dΔ xð Þ dx

x¼0

ð Þ

þðM1þ M2Þ

L0

ð11Þ

EI d

3

Δ xð Þ

dx3

!

x¼ L0

2

dΔ xdxð Þ

x¼ L0 2

ðM1Lþ M2Þ

0

ð12Þ

EI d

3

Δ xð Þ

dx3

!

x¼L 0

ð Þ

¼ F dΔ xð Þ dx

x¼L 0

ð Þ

þðM1þ M2Þ

L0

ð13Þ

M1¼ −EI d

2

Δ xð Þ

dx2

!

x¼0

ð Þ

ð14Þ

M2¼ EI d

2

Δ xð Þ

dx2

!

x¼L 0

ð Þ

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

The aicoefficients 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

s11¼ s22¼ − 5q3−1404q2þ 86400q−1209600

9 40ð −qÞ 840−11qð Þ

s12¼ s21¼ q3þ 252q2−25920q þ 1209600

18 40ð −qÞ 840−11qð Þ

ð16Þ

where q¼ λ2¼j Fj

EIL2

For tensile FN 0

s11¼ s22¼ 5q

3þ 1404q2þ 86400q þ 1209600

9 40ð þ qÞ 840 þ 11qð Þ

s12¼ s21¼ − q3−252q2−25920q−1209600

18 40ð þ qÞ 840 þ 11qð Þ :

ð17Þ

Fig 2shows 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, s21and s22with respect to q are as follows

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

For F≤ 0

ds11

dq

¼ ds22

dq

¼ −5 11q4−2560q3þ 270144q2−13547520q þ 270950400

9 40ð −qÞ2

840−11q

ds12

dq

¼ ds21

dq

¼ 11q4−2560q3þ 63360q2−9676800q þ 677376000

18 40ð −qÞ2

840−11q

For FN 0

ds11

dq

¼ ds22

dq

¼5 11q4þ 2560q3þ 270144q2þ 13547520q þ 270950400

9 40ð þ qÞ2

840þ 11q

ds12

dq

¼ ds21

dq

¼ − 11q4þ 2560q3þ 63360q2þ 9676800q þ 677376000

18 40ð þ qÞ2

840þ 11q

As q goes to zero, Eqs.(16) through (21)become to

ds11

dq

¼ ds22

dq

¼ −152 ds12

dq

¼ ds21

dq

¼ 1

ds11

dq

¼ ds22

dq

¼ 2 15

ds12

dq

¼ ds21

dq

¼ −1

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 7 Deflections at free end of column.

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

F¼EA

L0

ZL 0

0

dδ

dxdxþ1

2

ZL 0

0

dΔ dx

dx

0

@

1

A ¼ EA

L0ΔL0þEA 2L0

ZL 0

0

dΔ dx

 2

For the closed-form solution, the axial force can be presented by stability functions as

F¼AE

L0ΔL0−EA 12 ds11

dq

θ2þ ds12

dq

θ1θ2þ1 2

ds22

dq

θ2

F≤ 0

F¼AE

L0ΔL0þ EA 1

2

ds11

dq

θ2þ ds12

dq

θ1θ2þ1 2

ds22

dq

θ2

FN0

The axial force of the proposed approach is also derived from above relations

2.3 Plastic hinge

Letη1andη2(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

¼EI

L0

s1p s2p

s2p s3p

θ1

θ2

 

ð27Þ

Fig 9 Load-deflection curves of William toggle frames.

L0ΔL0 EA 1

2

ds1p

dq

θ2þ ds2p

dq

θ1θ2þ1 2

ds3p

dq

θ2

The ± symbol in Eq.(28)is assigned as“+” when F N 0 and “–” when F ≤ 0; the modified stability functions s1p, s2pand s3pare determined from the stability functions and inelastic ratios as proposed by Chan and Chui[3]as

s1p¼ η1 s11−s212

s11

1−η2

s2p¼ η1η2s12 s3p¼ η2 s22−s221

s22

1−η1

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

ds1p

dq

¼ η1

ds11

dq

− 2s11s12

ds12

dq

−s2 12

ds11

dq

s2 11

1−η2

2

6

4

3 7

ds2p

dq

¼ η1η2

ds12

dq

ð31Þ

ds3p

dq

¼ η2

ds22

dq

− 2s22s21

ds21

dq

−s2 21

ds22

dq

s2 22

1−η1

2

6

4

3 7

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

Table 1

Buckling loads of pinned-ended column.

L (mm) λ c P/P y

Residual stress ignored Residual stress considered Euler Ngo-Huu & Kim Proposed Diff (%) CRC Ngo-Huu & Kim Proposed Diff (%) 1141.97 0.25 16 0.9870 1.0000 – 0.9844 0.987 0.9843 0.01 2283.95 0.50 4 0.9870 1.0000 – 0.9375 0.936 0.9373 0.02 3425.92 0.75 1.7778 0.9870 1.0000 – 0.8594 0.861 0.8590 0.05 4567.84 1.00 1.0000 0.9870 0.9973 0.27 0.7500 0.76 0.7494 0.08 6851.90 1.50 0.4444 0.4450 0.4433 0.25 0.4444 0.445 0.4433 0.25 9135.78 2.00 0.2500 0.2500 0.2494 0.24 0.2500 0.25 0.2494 0.24 11419.73 2.50 0.1600 0.1600 0.1597 0.19 0.1600 0.16 0.1597 0.19 13703.67 3.00 0.1111 0.1120 0.1110 0.09 0.1111 0.112 0.1110 0.09 15987.62 3.50 0.0816 0.0820 0.0816 0.00 0.0816 0.082 0.0816 0.00 18271.56 4.00 0.0625 0.0630 0.0625 0.00 0.0625 0.063 0.0625 0.00 20555.51 4.50 0.0494 0.0500 0.0494 0.00 0.0500 0.0496 0.0494 1.20 22839.45 5.00 0.0400 0.0400 0.0400 0.00 0.0400 0.0402 0.0400 0.00

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

β ¼ 1:15 PP

y

 2

Mp

þ 3:67 PP

y

 2

M

Mp

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 pro-posed by the Column Research Council is used as follows

Et

E ¼ 1

Et

E ¼ 4 P

Py

 

1−PP

y

P

Py ≤ 0:5 for P

2.4 Corotational element stiffness matrix

Consider the change in geometry of the beam-column element shown inFig 3

Fig 13 Elastic load-deflection curves.

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

The original length L0and the deformed length L of the element:

L0¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xB−xA

ð Þ2þ zðB−zAÞ2

q

ð35Þ

L¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

xBþ u4−xA−u1

ð Þ2þ zðBþ u5−zA−u2Þ2

q

The nodal rotations of the element:

where

sinα ¼ zBþ u5−zA−u2

L

ð39Þ

cosα ¼ xBþ u4−xA−u1

L

ð40Þ

Fig 14 Inelastic load-deflection curves.

α0¼ sin−1 zB−zA

L0

The differentiations of L,θ1,θ2, sinα, cosα, F, M1and M2with respect to nodal displacements ui(i = 1, , 6) are as follows

∂L

∂u

 

∂θ1

∂u

¼ −sinLα cosLα 1 sinα

L −cosLα 0

ð43Þ

Fig 16 Lateral deflection of top right joint.

∂u

¼ −sinα

L

cosα

sinα

ð44Þ

∂ sin α

∂u

¼ sinα cos α

L −cosL2α 0 −sinα cos αL cosL2α 0

ð45Þ

∂ cos α

∂u

¼ −sinL2α sinα cos αL 0 sin

2

α

L −sinα cos αL 0

ð46Þ

∂F

∂ui

¼EA

L0

∂L

∂ui

 EA ∂s1p

∂q θ1þ∂s2p

∂q θ2

∂θ1

∂ui

þ ∂s2p

∂q θ1þ∂s3p

∂q θ2

∂θ2

∂ui

ð47Þ

∂M1

∂ui

¼EI

L0

s1p∂θ1

∂ui

þ s2p∂θ2

∂ui

 L0 ∂s1p

∂q θ1þ∂s2p

∂q θ2

∂F

∂ui

ð48Þ

∂M2

∂ui

¼EI

L0

s2p∂θ1

∂ui

þ s3p∂θ2

∂ui

 L0 ∂s2p

∂q θ1þ∂s3p

∂q θ2

∂F

∂ui

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

z

f g ¼ −F ðM1þ M2Þ

L M1 F −ðM1þ M2Þ

The nodal element resistance vector in global coordinate system isfZg ¼ ½TTfzg, where [T] is the transformation matrix of planar element The local and global element tangent stiffness matrix½kT and [KT], respectively, are determined as follows

kT

h i

¼ T½  ∂f gZ

∂ uf g

T

½ T¼ ½ T ∂ Th iT

∂ uf gf gz

2 4

3

5 T½ Tþ ∂f gz

∂ uf g

T

½ T

0

@

1

KT

½  ¼ T½ T

kT

h i

T

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

Kassimali Yoo and Choi Proposed Difference with Yoo and Choi's results (%)

h i

¼ EA

0 T3−ðT1þ T2Þ

T4

L0T1ðT1þ T2Þ

L2 −T3þðT1þ T2Þ

L0T2ðT1þ T2Þ

L2

L0T2 T1 −L0T1ðT1þ T2Þ

0 T3−ðT1þ T2Þ

T4 −L0T2ðT1þ T2Þ

L2

2

6

6

6

6

6

6

6

6

4

3 7 7 7 7 7 7 7 7 5

ð54Þ

where G1¼A

IðL−L 0

L Þ and

T1¼  ∂s1p

∂q θ1þ∂s2p

∂q θ2

ð55Þ

T2¼  ∂s2p

∂q θ1þ∂s3p

∂q θ2

ð56Þ

T3¼ðM1þ M2Þ

T4¼T1θ1þ T2θ2

2L þL0ðT1þ T2Þ2

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 al-gorithm in this research to solve the nonlinear equation system The incremental equilibrium equation is presented as follows:

Δu þ ΔλΔu

where {ΔP}={P−Z} is applied incremental load vector for the first iteration or the unbalanced force vector in second iteration onward; {Δu} = cor-responding 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 thefirst iterative step

Δλ1¼ arc lengthffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Δu

f gT

Δu

From the second step,Δλiis determined from the following condition

∂ equilibrium errorð Þ

∂Δλi

¼∂ Δλf iΔu þ ΔuigT

ΔλiΔu þ Δui

Simplifying the Eq.(61), we have:

Δλi¼ Δf ug

T

ifΔug

Δu

f gT

Δu

3 Numerical examples

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

3.1 Cantilever beam with an end point load

The cantilever beam with geometric and material properties shown

inFig 4is used for nonlinear elastic analysis in this research This

exam-ple wasfirstly introduced by Bisshopp and Drucker[14]with exact

solu-tion 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 inFig 5 It can be seen that the obtained

re-sults 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 inFig 6wasfirstly introduced by Wood and Zienkiewicz[15]by

usingfive 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 inFig 7prove 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 inFig 8was analyzed by many

re-searchers to verify their analysis methods in predicting large

displace-ment 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 andfixed support conditions Only one

pro-posed 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 inFig 10was

analyzed by Ngo-Huu and Kim[16]by thefiber hinge method in which

the column was divided into three elements, one middle elastic element

using the conventional stability function and two endfiber-hinge

ele-ments.Table 1presents the buckling load results obtained by the

pro-posed program, the Euler's theoretical exact solution, CRC column

curve (Chen and Lui[17]), and Ngo-Huu and Kim'sfiber 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 inFig 11and it can be seen that the curves are

al-most identical This example demonstrates the capacity of the proposed

program in predicting the elastic and inelastic buckling loads of the

column

3.5 Single-bay two-storey frame The single-bay two-storey frame with pinned supports as shown in

Fig 12was 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

present-ed inFig 13 andFig 14show that the elastic and inelastic

load-deflection curves of the frame are almost identical The elastic and in-elastic 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%

3.6 Two-bay four-storey frame The two-bay four-storey frame shown inFig 15was 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 inFig 16 It can be seen that the curves are rela-tively 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 inFig 17was 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 mem-ber, respectively The analysis results of the proposed element and Kassimali and Yoo and Choi are shown inTable 2andFig 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 de flec-tion 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 addi-tional axial strain caused by the bending of the element and the inelastic simulation by refined plastic-hinges lumped at both ends A Matlab pro-gram is developed based on the arc-length combined with minimum re-sidual displacement methods to solve the system of nonlinear equilibrium equations step by step The analysis results of numerical ex-amples 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

s , s , s , s Elastic stability functions of beam-column element