7 POSTBUCKLING BEHAVIOR OF WELDED BOX SECTION STEEL ... · postbuckling behavior of welded box section steel compression members using non-linear static Riks compared with a non-linear

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print),

ISSN 0976 – 6316(Online), Volume 6, Issue 4, April (2015), pp. 65-78 © IAEME

65

POSTBUCKLING BEHAVIOR OF WELDED BOX

SECTION STEEL COMPRESSION MEMBERS

Lilya Susanti1, Akira Kasai

2 Yuki Miyamoto

3

1Graduate School of Science and Technology (GSST), Kumamoto University,

2-39-1 Kurokami Chuo-Ku Kumamoto 860-8555, 2GSST, Kumamoto University, 2-39-1 Kurokami Chuo-Ku Kumamoto 860-8555, Japan,

3GSST, Kumamoto University, 2-39-1 Kurokami Chuo-Ku Kumamoto 860-8555, Japan,

ABSTRACT

Present paper observed the postbuckling behavior of welded box section bridge compression

members using static and response analysis. Parametric study using beam and shell varied in width-

thickness and slenderness ratio compared with full truss bridge beam model were used. Results

indicated that for design based purposes, beam element disregarding initial imperfection factors is

still suitable. But for analysis based purposes, which need the capability to perform real structure

behavior and to explore the postbuckling regime, shell is the best choice as it can perform more

detail compression members behavior and has more severe strength reduction in postbuckling

regime, especially.

Keywords: Compression Members, Postbuckling Regime, Static Analysis, Seismic Design

Methodology, Truss Bridge Structure

I INTRODUCTION

In the structure with based design purposes, prebuckling phase and maximum buckling load

have been considered as the most important parameters. But in the more complex analysis, especially

for steel column with thin-walled structures, postbuckling behavior is important to be investigated

since in the recent decades, numerical analysis using general finite element software and also

experimental study have been able to assess this phenomenon in more detail phases. It is found that

the prebuckling, stability and postbuckling behavior of a beam-column depend on the cross section

and material properties (area, inertia and elastic modulus), the magnitude of the end restraints and the

type and lack of symmetry about the beam-column mid span of the applied transverse loads and

initial crookedness [1]. Other studies have shown that the characteristics of postbuckling behavior

INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND

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ISSN 0976 – 6308 (Print)

ISSN 0976 – 6316(Online)

Volume 6, Issue 4, April (2015), pp. 65-78

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66

are significantly influenced by shell geometric parameter, stacking sequence, as well as initial

geometric imperfections [2,3]. In the recent achievement, compared to conventional shell design

methodology, recent research studies have turned to recognize postbuckling behavior as a desirable

response and shows a relatively more flexible response during initial loading and a lower critical

load. The postbuckling response with multiple bifurcation points is resulted due to changes in the

deformed geometry after each critical point [4]. After primary buckling, secondary buckling occurs

accompanied by successive reduction in the number of circumferential waves at every path jumping

[5].

Zang T. and Gu W. studied about the influences of asymmetric mesh and mesh density to the

secondary buckling mode and buckling load of composite laminated cylindrical shells [6]. Yamaki

N. et. al experimental study has found that as the cylinder is compressed beyond the primary

buckling, secondary buckling take place successively with diminishing wave numbers and that

postbuckling equilibrium load became significantly lower than those at buckling as geometric

parameter increases [7].

Calculation of the postcritical behavior and failure loads of shells has become a difficult

problem, even with finite element programs. Although verification of an important design by finite

element is imperative, the design is generally based on critical load solutions reduced by

imperfection sensitivity [8]. The postbuckling behavior of an axially compressed shell is also

exceedingly complicated due to an infinite number of closely spaced postbuckling branches and

bifurcation points [9]. The only possible affordable way is to perform computer simulations of the

structure with the help of a shell finite element code. But they should be able to approximate the

behavior that would be observed in an actual experiment [10].

In bifurcation buckling, the deflection curves of imperfect structures exhibit a limit point with

snap-through rather than bifurcation. Since some imperfection is always present, bifurcation

buckling is merely an abstraction, albeit a very useful one [8]. Some of shell models may experience

large bending deformation resulting in buckling due to transverse loading and the shells may exhibit

snap-through and snap-back type postbuckling behavior [11]. After the primary buckling, the

deformed shell jumped to a stable bifurcation path with a non-axisymmetric shape, along with the

dropping of the load. Subsequently, the shell repeated consecutive snap-through buckling (secondary

buckling) as the number of waves decreased one by one in the circumferential direction as the

compression progressed [5]. Mode jumping or snap-through and snap-back during the post buckling

response leads to sudden and high-rate deformations due to generally smaller changes in the

controlling load ordisplacement input to the system [12].

Kundu C. K. And Sinha P. K. have observed that the thicker shell exhibits a snap-through

behavior while both snap-through and snap-back behavior are observed for the thinner one [11]. It

has also been found that thin-walled structure with small imperfection has higher possibility in the

occurrence of snap-back and snap-through behavior [13]. Jamal M. et. al studied the influences of

thickness and localized imperfection in the appearance probability of snap-back behavior [14].

However, the prediction and control over this behavior are comparably more difficult due to higher

imperfection sensitivity in thin-walled shells [4].

Besides static analysis, post buckling behavior has also been also important to be investigated

using dynamic response analysis. For short time load duration, dynamic buckling strength is higher

than static buckling strength. But as the load duration increases, dynamic buckling strength decreases

quickly and becomes much smaller than the static strength [15].

A previous study by Wullschleger L. and Meyer-Piening H. R has reported the different

approaches, including linear, non-linear and dynamic non-linear FE analysis results and discussed

the related effects and potential difficulties in the buckling behavior of geometrically imperfect

cylindrical shells [16]. Kubiak T. has studied about the dynamic buckling of thin-walled composite

plates with varying width wise material properties [17].



67

Considering many previous study results that have been listed above and according to the

previous studies that have clearly discussed about the elastic buckling regime and ultimate buckling

strength of welded box section steel compression members [18,19], present paper observed the

postbuckling behavior of welded box section steel compression members using non-linear static Riks

compared with a non-linear dynamic method. In more detail discussion, snap-back and snap-through

behavior as a result of internal instability of the compression members was explored including the

influence of slenderness and width-thickness ratio of the structures that may lead snap-back and

snap-through phenomenon.

Present paper gives many advantages for both engineers and researchers because the study is

designed as close as real methodology that is usually applied in the practical fields. First analysis was

conducted by analyzing parametric beam and shell models using static loading condition to observe

the differences of postbuckling behavior by varying width-thickness ratio and slenderness.

Parametric study was also aimed to observe the snap-back and snap-through and also multiple

bifurcation point phenomena. Since the strength demand is always conducted in dynamic or static

equivalent methods, therefore, the second analysis was performed in dynamic loading condition

using a full truss bridge model. In final analysis, static and dynamic postbuckling behavior was

compared in order to verify which one resulted the most severe buckling load. It also verified

whether the beam type that was widely used in based design purpose has sufficient capability in

performing real structure behavior.

II NUMERICAL PROCEDURE

2.1 Beam and shell finite element models

The important parameters to analyze column with thin-walled structures are slenderness ( λ )

and width-thickness ratio(RR). Present paper defined the slenderness and width-thickness ratio as

follows (Eqs. (1) and (2)) where σy is yield stress, EYoung’s modulus, Lis column height, ris radius

of gyration, t is plate thickness, υis Poisson’s ratio, n number of panels. Twenty box section beam

and shell models varied in slenderness and width-thickness ratio as shown in the Table 1 were

employed. Slenderness and width-thickness ratio presented in this paper were designed in the range

that can display both local and global buckling behavior.

Model cross section is shown in Fig. 1(a) where B is flange width D is web width. Assembled

in ABAQUS general-purpose finite element software [22], geometrical non-linearity and large

displacement theory was applied. Timoshenko wire beam B31(Fig. 1(b)) and conventional reduced

integration S4R elements (Fig. 1(c)) were used to assemble beam and shell models. Generally, both

models are divided into three parts. Center part that is used to assess the buckling behavior of the

compression members consisted of smaller meshing pattern than both outside parts. Maximum

global initial displacement (δ0) as L/1000 was employed in all models as an initial imperfection

factor. SM490 steel grade was used with material non-linearity in stress-strain relationship as shown

through Eq. (3) where ξ is steel grade constant, Est is modulus of elasticity in hardening region, εstis

strain in hardening region, εyis yield strain while material properties is shown in Table 2 respectively.

r

L

E

yσ

πλ

1= (1)

( )22

2

4

112

nEt

bR

y

Rπ

υσ −= (2)



68

1exp11

+

−−−=

y

st

y

st

y E

E

ε

ε

ε

εξ

ξσ

σ (3)

Table 1: Geometrical properties of beam and shell FE models

Column λ RR L (mm) B(mm) Column λ RR L (mm) B(mm)

1

2

3

4

5

6

7

8

9

10

0.2

0.5

0.8

1.1

1.4

0.2

0.5

0.8

1.1

1.4

0.2

0.2

0.2

0.2

0.2

0.5

0.5

0.5

0.5

0.5

670.2

1675.6

2681.0

3686.3

4691.7

1782.4

4455.9

7129.5

9803.0

12476.6

115

115

115

115

115

287.5

287.5

287.5

287.5

287.5

11

12

13

14

15

16

17

18

19

20

0.2

0.5

0.8

1.1

1.4

0.2

0.5

0.8

1.1

1.4

0.7

0.7

0.7

0.7

0.7

0.8

0.8

0.8

0.8

0.8

2524.5

6311.3

10098.1

13884.9

17671.6

2896.0

7240.0

11584.1

15928.1

20272.1

402.4

402.4

402.4

402.4

402.4

459

459

459

459

459

d

b

1t

2t

B

D

(a) cross section (b) beam model (c) shell model

Figs. 1:beam and shell finite element models

Table 2 Material properties of SM490 steel grade

Property Value

E(GPa)

yσ (MPa)

uσ (MPa)

υ

stEE

stεε

ξ

200

315

490

0.3

30

7

0.06



69

Pin-roll simply supported end restraints were employed in the both sides of the compression

member as well as the concentrated compression loads taken place (Fig. 2(a)) where δx , δy, δzandθx ,

θy ,θz are displacement and rotation in x, y and z direction while δ0 is maximum initial displacement.

Although there are three types of analysis that can be used to assess the buckling behavior of

compression members such as Eigen value analysis, non-linear Riks method and response analysis,

present work only compared two methods: non-linear Riks and response methods. In the first step,

beam and shell FE models varied in slenderness and width-thickness ratio were analyzed using Riks

(arch-length) method.. Residual stress was also considered as another initial imperfection factor as a

result of the welding process in the steel columns. Maximum tensile residual stress as σy as well as

compression residual stress as 0.25 σy was used for both models in ABAQUS software (Fig. 2(b)).

The residual stress pattern was applied according to previous research by Imamura et. al. [23].

0δδ

0===== yxzyx θθδδδ 0==== yxzy θθδδ

yσ

yσ)25.0(−

yσ)25.0(−

yσ)25.0(−

yσ)25.0(−

yσ

yσ

yσ

yσ

yσ

yσ yσ

(a) boundary and loading condition (b) residual stress

Figs. 2:boundary condition and residual stress distribution

2.2 Truss bridge FE model

The next step in the present study was analyzing full truss bridge model using the response

method. All members in the present truss bridge FE model were assembled using B31 Timoshenko

beam in ABAQUS software following the engineer requirement in the practical fields. Large

meshing patterns were used to simplify the running time process. Each span of the main

superstructure is only divided into 5 to 20 segments depend on the analysis requirement. The main

part that were used to verify the buckling phenomenon consists of smaller meshing patterns.

Dynamic load with the implicit integration operator in ABAQUS/Standard was employed to evaluate

the buckling loads. The boundary conditions used in the present bridge system were simplified in

such a way that they can provide the bearing supports between superstructure and abutments. Simple

pins on one side and rollers on another side were used. All members were rigidly connected. All

initial imperfection factors such as initial displacement and residual stress were disregarded. Multipe

point constrains were employed to impose constraints between members. Geometrical property of

main steel members of this truss bridge model is shown in Table 3 and the layout of the full truss

bridge model with the 2-direction of earthquake motion in the ABAQUS software is drawn in the

Fig. 3 respectively where B is total width, D is the total depth and t is thickness.



70

Table 3: geometrical properties of main steel members Bridge Part Section Type Dimension(mm)

Main girder,main truss and overside beam

Under side beam

Undercross and overcross beam

Box section

I section

I section

B = 320

D = 400

t = 15

B upper flange = 300

B lower flange = 350

D = 843

t web = 12

t upper flange = 21

t lower flange = 15

Bflanges = 200

D = 220

tweb = 8

tflanges = 10

Fig. 3: full truss bridge model

The truss bridge model was assembled using the actual dimensions of a railway truss bridge

in Japan. SM490 steel grade with modulus elasticity (E) = 200 GPa, yield stress (σy) = 315 MPa and

Poisson’s ratio (υ) = 0.3 were used in the present analysis. Stress-strain curve input data was drawn

as a bi-linear stress-strain relationship with modulus elasticity slope in the strain hardening regime is

defined as E/100. Train load is necessary to be taken into consideration. Distributed gravity load and

train load (TL) as 35 kN/m along bridge axis direction was employed. In the superstructure, train

load was modeled as point mass (PM) with total point mass as TL x L where L is the total length of

truss bridge. The point masses were located in 8 certain location according to the bridge design

requirement (Fig. 3). Therefore, each point mass has the load as PM = m x g where m is load applied

at the point mass and g is gravity acceleration.

Natural frequency of the structure has to be defined in order to get mass coefficient and

stiffness matrix, which is necessary to calculate the damping matrix. Rayleigh damping type was

chosen in present analysis. Natural frequency is defined using Eigen value analysis by displaying

severe deformed modes the structure. Ground motion input data were taken from the recent severe

earthquakes in Japan which were known as Hyogo prefecture earthquake (Kobe earthquake) in

1995, Tokachi earthquake (2004) and Tohoku earthquake (2011). Present study also employed El-

Centro (1940) and Taft (1952) earthquake ground motions that have been widely chosen as ground

motion input data for design based purposes. All acceleration response spectra of earthquakes ground

motion data are shown through Figs. 4. East-West (EW) earthquake was applied to the direction 3

transverse axis of bridge structure as well as North-South (NS) earthquake taken place in the

direction 1 longitudinal to the bridge axis.



71

Figs. 4:acceleration response spectra

First part conducted parametric analysis of beam and shell models to observe the differences

of postbuckling behavior in various width-thickness and slenderness ratio. Parametric study is also

aimed to observe the snap-back and snap-through and also multi-bifurcation point phenomena. The

first ultimate strength analysis was conducted in static Riks methods. In the design based purposes

generally, strength demand is always conducted in dynamic response analysis or static equivalent

methods. Therefore, to illustrate the actual design procedure, the second analysis was performed in

dynamic pattern using the full truss bridge model. Most of the bridge structures in the actual field

have a good capability in carrying the earthquake load due to the design specifications have required

the bridge structures have to be able to carry severe earthquake motions. Therefore, to gain the

maximum buckling strength, the ground motions input data were magnified until the bridge totally

collapse, so that the prebuckling and postbuckling regime can be clearly seen. In the final analysis,

static and dynamic postbuckling behavior was compared each other in order to verify which one

resulted the most severe buckling load and strength reduction.

III RESULTS AND DISCUSSION

3.1 Parametric study in static loading condition

Beam and shell models were numerically analyzed, and the results are compared each other

as shown in the Figs. 5. This first step is aimed to assess the influences of slenderness and width-

thickness ratio to the postbuckling behavior of the compression members. The differences of beam

and shell in the postbuckling regime, including the existence of snap-back and snap-through

behavior as a result of numerically internal instability were explored. Figs. 5 show the normalized

load and longitudinal displacement of beam and shell models in various slenderness and width-

thickness ratio.

In the postbuckling regime generally, curves of beam model indicated smoother reduction of

load-axial displacement relationship compared with shell models. Especially in the width-thickness

ratio as 0.7, beam and shell curves started to show different postbuckling behavior. Shells generally

have more significant load-axial displacement reduction than beam models (Figs. 5(k) – (q)). In the

high slenderness ratio range, most of the shell running process have stopped due to numerical

instability problems. Most of their running steps have stopped after reaching the maximum buckling

load (Figs. 5(d), (e), (h), (i), (j), (r), (s) and (t)). Even in some curves such as Figs. 5(c) and (g), load-

longitudinal displacement curves of shell model stopped in the maximum buckling point so that the

postbuckling behavior of these curves cannot be explored. But generally, as slenderness increases,

the gap become smaller.

0

1

2

3

4

5

6

7

8

0.01 0.1 1

Acc

/g

Natural Period T(s)

KOBE

TOKACHI

TOHOKU

EL CENTRO

TAFT



72

(a) RR=0.2 – λ=0.2 (b) RR=0.2 – λ=0.5 (c) RR=0.2 – λ=0.8 (d) RR=0.2 – λ=1.1

(e) RR=0.2 – λ=1.4 (f) RR=0.5 – λ=0.2 (g) RR=0.5 – λ=0.5 (h) RR=0.5 – λ=0.8

(i) RR=0.5 – λ=1.1 (j) RR=0.5 – λ=1.4 (k) RR=0.7 – λ=0.2 (l) RR=0.7 – λ=0.5

(m) RR=0.7 – λ=0.8 (n) RR=0.7 – λ=1.1 (o) RR=0.7 – λ=1.4 (p) RR=0.8 – λ=0.2

(q) RR=0.8 – λ=0.5 (r) RR=0.8 – λ=0.8 (s) RR=0.8 – λ=1.1 (t) RR=0.8 – λ=1.4

Figs. 5:load-axial displacement relationship

The shell model in Fig. 5(q), smooth snap-back behavior is smoothly seen. In the other

curves, smooth snap-through behavior was also found especially in the beam model curves as shown

in the Figs. 5(a), (f), (k), and (p) which all of these figures have the similar slenderness ratio as 0.2.

Snap-back behavior is usually occurred in the finite element analysis which use large displacement

theory and displacement-control method as well as snap-through exist in the analysis with force-

control method. However, according to the results of present study analysis where arch-length

control with large with large-displacement theory was employed in ABAQUS software, both snap-

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6P

/Py

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shellbeam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shellbeam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

P/P

y

δ/δy

shell beam



73

back and snap-through behavior can still be found in some graphs of beam and shell models,

although in the smoother appearance. Snap-back and snap-through behavior are usually found in

buckling analysis of frames, rings and shells. In exploring postbuckling regime, generally there are

more problems found in the shell than beam models. The snap-back phenomenon in Fig. 5(q) is

occurred when the axial displacement value suddenly decreased after the load undergo its ultimate

buckling point.

As it has been mentioned in the introduction chapter that the use of shell models can perform

a more flexible response such as multiple bifurcation points due to the changes in the deformed

geometry after each critical point [4], indeed present shell analysis showed double bifurcation points,

which can be seen especially in Figs. 5(m), (n) and (o). It is noted that all shell figures with double

bifurcation points have the similar width-thickness ratio as 0.7. Width-thickness ratio is often

associated with the local buckling phenomenon in the compression members. As it has well known

from some specifications such as Japan Specification for Highway Bridges 2002 Part II – Steel

Bridges [20] and American Institute for Steel Constructions 2005 [21] that local buckling is occurred

starting at the width-thickness ratio as 0.65 – 0.7. However, it needs more investigation to conclude

that the double bifurcation points resulted in this analysis is caused by the local buckling effect.

(a) shell model

RR=0.2 – λ=0.5

(b) shell model

RR=0.7 – λ=0.5

(c) beam model

RR=0.2 – λ=0.5

beam model

RR=0.7 – λ=0.5

Figs. 6:beam and shell contour plots

The differences between beam and shell models deformed shape can be verified using Figs. 6. In the

width-thickness ratio as 0.2, as can be seen in Fig. 6(a), shell model only performed pure global

buckling phenomenon. Similar appearance was also occurred in other shell models which have a

width-thickness ratio smaller than 0.7. In Fig. 6(b), starting at the width-thickness ratio as 0.7, shell

models perform both local and global buckling. Therefore, snap-back behavior did not affect the

deformation shape of the models. Figs 6(c) and (d) show beam models deformed shape. Local

buckling absolutely cannot be performed in this type of models. Therefore, beam type is only

suitable for based design purposes. Maximum axial deformation is shown in each legend. Regions

with red color indicate largest deformation than the others. The maximum deformation value at the

legends of beam and shell models could not be compared each other since these values indicate the

deformation point at the end iterations in ABAQUS software.

3.2 Response Analysis Present part is aimed to explore the dynamic postbuckling behavior of the compression

member using a full truss bridge model in ABAQUS nonlinear software. Five Earthquake ground



74

motions data from Kobe, Tokachi, Tohoku, El centro and also Taft earthquake were employed as

ground motion input data. Since the present analysis is aimed to investigate dynamic postbuckling

behavior of the compression members, the bridge structure has to reach its maximum buckling point.

Most of the bridge structures are designed to carry the severe earthquake motions. Therefore, to

make the bridge compression members exceed the buckling strength, earthquake data were

magnified until the compression members enter the postbuckling regime. In the present analysis

obviously, to reach the postbuckling regime, Kobe, Tokachi and Tohoku earthquake ground motions

were magnified three times, eight times of El Centro earthquake and also eleven times of Taft

earthquake . Results of the analysis are shown in the Figs 7.

In the Figs 7, it was observed that all models have passed their ultimate buckling point so that

postbuckling behavior can be explored. The bridge compression members that were analyzed in here

have box section profiles with the width-thickness ratio as 0.366 and slenderness ratio as 0.637.

According to previous static Riks analysis, these parameters did not produce the local buckling

behavior. Moreover, present dynamic analysis was carried out in wire beam elements so that the

local buckling behavior cannot be performed. Otherwise, in the static Riks analysis, similar width-

thickness and slenderness ratio resulted normalized ultimate stress around 0.7. The average of

normalized ultimate stress for all models from response analysis is 0.965. It might be occurred due to

the present dynamic analysis disregarded initial imperfection factors, including initial displacement

and residual stress.In here, the significant influence of initial imperfection effect can be clearly seen.

(a) Kobe earthquake (b) Tokachi earthquake

(c) Tohoku earthquake (d) El Centro earthquake

(e) Taft earthquake

Figs. 7:average stress-strain relationship

-1

-0.5

0

0.5

1

-2 -1 0 1 2σav

g/σ

y

εavg/εy

-1

-0.5

0

0.5

1

-2 -1 0 1 2σavg/σ

y

εavg/εy

-1

-0.5

0

0.5

1

-2 -1 0 1 2σav

g/σ

y

εavg/εy

-1

-0.5

0

0.5

1

-2 -1 0 1 2

σa

vg/σ

y

εavg/εy

-1

-0.5

0

0.5

1

-2 -1 0 1 2σav

g/σ

y

εavg/εy



75

All models in the Figs. 7 performed multiple bifurcation points due to the typical loading

motions in the response analysis. In the postbuckling regime especially, average stress decreases as

the average strain increases. The reduction shape of average stress in this regime was difficult to be

measured since it was affected by earthquake intensity. According to the acceleration response

spectra in the Figs. 4, when the earthquake intensity decrease significantly after the peak

acceleration, the stress intensity also decreased significantly following the earthquake magnitude.

But since the design based purposes only explore the prebuckling until maximum buckling regime,

beam element is still suitable for this purposes. For similar width-thickness ratio and slenderness

parameters, shell models in static Riks analysis resulted bigger stress reduction. Hence, most severe

stress reduction is resulted using shell models generally. Fig. 8 performs the buckling performance of

the compression members in the truss bridge system from ABAQUS software. It was shown that the

most severe main girder is located near the edge support, especially. This part has been collapse first

before the other parts reach the maximum buckling point. Hence the most severe part in the truss

bridge system under dynamic loading is the main girder nearest to the bearing supports.

Fig. 8: Buckling behavior in the truss bridge members

Regarding some numerical phenomena in the postbuckling regime, such as a snap-back and

double bifurcation points in shell model and snap-through in the beam model, a recommendation to

avoid those phenomena according to the present study results is presented through the flowchart in

Fig. 9. In here, first part is purposed to select the use of beam or shell model by verifying slenderness

and width and thickness ratio to be spared from snap-back or snap-through phenomenon. Next part is

choosing static or dynamic analysis according to bridges condition. For beam element especially,

buckling occurrence has to be verified with the available design criteria whether it actually has been

buckled or not. But for shell models, it can be directly decide whether it has buckled or not so that

the postbuckling behavior can be explored. Exploring the postbuckling regime is the main concern of

the future study.



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Fig. 9: Flowchart for designing a truss bridge structure

IV CONCLUSION

Present paper is aimed to observe the differences of static and dynamic postbuckling behavior

by varying width-thickness ratio and slenderness. Parametric study in static Riks analysis is aimed to

observe some postbuckling phenomenons, including snap-back and snap-through and also multiple

bifurcation points. Then, response analysis using full a truss bridge model was conducted. In final

analysis, static and dynamic postbuckling behavior was compared each other in order to verify which

one results the most severe buckling load. It also verified whether the beam type that is widely used

in based design purposes has sufficient capability in performing real structure behavior. According to

present analysis results, it can be conclude that:

i. In the postbuckling regime generally, curves of beam models showed smaller strength reduction

compared with shells. It can be said that generally shell models resulted more severe strength

reduction than beam models.



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ii. Snap-back was occurred in the shell models as well as snap-through behavior mostly exists in

the beam models. But both snap-back and snap-through were not related to the width-thickness

or slenderness ratio as they only related with the loading condition where snap-back is usually

occurred in the finite element analysis which used large displacement theory and displacement-

control method as well as snap-through occurred in the analysis with force-control.

iii. Some of the shell model curves showed double bifurcation points phenomenon. It might be

occurred as a result of local buckling behavior that can only be performed using shell models.

But it needs more analysis to prove this relation.

iv. The reduction trend of average stress in the postbuckling regime resulted from dynamic response

analysis was significantly affected by the earthquake acceleration response spectra. Hence, it

was difficult to predict the stress reduction in the postbuckling regime accurately. With similar

width-thickness and slenderness ratio, shell models in static analysis resulted more severe stress

reduction.

v. For exploring prebuckling until ultimate buckling regime based purposes, the use of beam

elements is still suitable because most specifications have provided the design criteria that can

overcome this problem. Otherwise, in prebuckling regime until ultimate buckling point, beam

and shell resulted good agreement each other. But for analysis based purposes, which need the

capability to perform actual structure behavior and to explore the postbuckling regime, it is

needed to consider the width-thickness ratio and slenderness influences in order to avoid some

numerical postbuckling phenomena such as snap-back and snap-through in selecting beam or

shell model. But overall, shell is the best choice to be considered as it can perform more detail

compression member behavior and result more severe strength reduction in postbuckling

regime, especially.

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7 POSTBUCKLING BEHAVIOR OF WELDED BOX SECTION STEEL ... · postbuckling behavior of welded box section steel compression members using non-linear static Riks compared with a non-linear