journal of constructional steel research volume 90 issue 2013 [doi 10.1016%2fj.jcsr.2013.07.024]...

7/25/2019 Journal of Constructional Steel Research Volume 90 Issue 2013 [Doi 10.1016%2Fj.jcsr.2013.07.024] Qian, Xudong;

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A loaddeformation formulation for CHS X- and K-joints in

push-over analyses

Xudong Qian , Yang Zhang, Yoo Sang Choo

Centre for Offshore Research and Engineering, Department of Civil and Environmental Engineering, National University of Singapore, 1 Engineering Drive 2, 117576, Singapore

a b s t r a c ta r t i c l e i n f o

Article history:

Received 23 February 2011

Accepted 20 July 2013Available online 25 August 2013

Keywords:

Joint formulation

Circular hollow section

Tubular joint

Pushover analysis

Phenomenological representation

This paper proposes a new loaddeformation formulation for circular hollow section (CHS) X- and K-joints to be

implemented in thepushoveranalysis of steel frames.The proposed formulation describes theloaddeformation

relationship of the CHSX- and K-joint through a simplefunction with the coefcients dependenton the ultimate

strength and the geometric parameters of the joint. The strength-dependent parameter follows the mean

strength equations in the latest IIW recommendations, while the geometric-dependent parameters derive

from the nite element results of the CHS X- and K-joints covering a practical geometric range. The proposed

joint formulation predicts closely the loaddeformation responses for planar CHS X- and K-joints measured in

the experiments. The non-dimensional loaddeformation formulation developed in the current study provides

a calibrated basis in the phenomenological representation of the nonlinear joint behavior in push-over analyses

of steel space frames. Theexperimental results from thelarge-scale 2-Dand 3-Dframetests validate theaccuracy

of the proposed formulation, which is implemented in a nonlinear pushover analysis as joint-spring elements.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

The extended service of steel offshore platforms beyond their initial

design life of 20 years has become a common practice due to economi-

cal considerations[1]. The reassessment of such platforms requires ad-

vanced nonlinear frame analyses which should include an accurate

representation of the local joint responses under overloading condi-

tions. The rigid joint assumption in the conventional frame analysis

often underestimates the deformation and over-estimates the ultimate

resistance of the critical joint. The rigid joint hypothesis therefore may

cause strong effects on the load-distribution and the sequence of the

component failure in the structure, leading to severe deviations in the

predicted frame behavior from the real structural response. However,

frame analyses with rigid-joint assumptions do not always provide con-

servative estimations on the ultimate strength of the structure, since

large deformation of the joint may mobilize adjacent redundant mem-

bers and leads to higher structural resistances than the prediction by

rigid-joint frame analyses. Hence, improved understandings on the ef-

fect of nonlinear joint behavior on the frame response become neces-

sary to develop a simple and calibrated engineering representation of

the nonlinear joint characteristics in pushover analyses.

The last three decades observe substantial experimental develop-

ments in the local joint exibility (LJF) for both uni-planar and multi-

planer tubular joints. Bouwkamp [2] investigates the effect of joint ex-

ibilities on the elastic response of offshore jacket structures where a

detailed shell element model describes the behavior of the joint-can

which is connected to the brace members modeled by beam elements.Efthymiou [3]develops the elastic exibility parametric formula for

T- and TY-joints based on the FE analysis using thin-shell elements

with theweld modeled by solid elements.Fessler [4,5] improves the for-

mula based on Araldite models for Y-, X-joint and multi-brace joints.

Chen[6]develops special elements to represent the joint exibility by

subdividing the brace end into nite strips, and derives parametric for-

mulae for the elastic stiffness of T-, Y- and K-joint. Kohoutek [7] investi-

gates the T-joint elastic stiffness based on the frequency measurement

by introducing a rigidity index into the stiffness matrix and calibrating

it through the natural frequency of the joint. Romeijn[8]investigates

the inuence of the joint geometry on the exibility of uni-planar and

multi-planar joints based on the nite element (FE) results. Holmas

[9,10]develops a joint shell element based on the small-displacement

theory and the load is represented by a series of concentrated loads

along the brace-to-chord intersection lines. The model yields a good

agreement with the parametric formulae for the elastic joint exibility

derived by Fessler[4,5]and Chen[6].

A natural extension of the work on the joint stiffness focuses on ex-

amining the inuence of the local jointexibility on the frame response

by incorporating the LJF into structural analyses. Sub-structural models

of thecritical joint using a detailed FE mesh [2,8,9,1115] provide an ac-

curate approach to incorporate the nonlinear joint deformation in the

frame analysis. However such methods become infeasible when applied

to a realistic steel offshore platform with multiple critical joints, which

require substantial computational resources in iterating the detailed

stress/strain/displacementelds in the 3-D local joint model.

Journal of Constructional Steel Research 90 (2013) 108119

Corresponding author. Tel.: +65 6516 6827; fax: +65 6779 1635.

E-mail address:[email protected](X. Qian).

0143-974X/$ see front matter 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.jcsr.2013.07.024

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Alternatively, the phenomenological representation of the joint be-

havior through nonlinear joint springs, as shown in Fig. 1a, provides a

convenient method for practicing engineers. Ueda [16,17] develops

the elasticperfectly-plastic springs for CHS T-, Y- and K-joints. Choo

et al. [18] implement the joint spring elements described by piece-

wise linear loaddeformation curves in the nonlinear frame analysis.

The joint industry project, led by a UK company [19,20], develops the

MSL formulation, which includes the interaction between the chord

and brace loadsin the joint response and the K-, X-, and Y-joint classi-

cations[21].

This study develops a nonlinear joint formulation which predicts

closely the loaddeformation responses for CHS X- and K-joints

subjected to brace axial compression. This proposed formulation de-

scribes the loaddeformation relationship for CHS X- and K-joints with

different geometric parameters covering a brace to chord diameter

ratio () from 0.3 to 1.0 and a chord radius to thickness ratio ( ) ratio

from 7 to 25. The proposed load

deformation relationship developsfrom loaddeformation results computed using calibrated FE analyses.

The parametric formulation,proposed in the currentjoint representation,

provides a convenient approach to characterize the loaddeformation

curve and eliminates the need for the elasticplastic, large-deformation

nite element analyses on CHS X- and K-joints. The nonlinear pushover

analysis, performed in the numerical tool USFOS (an acronym for

Ultimate Strength for Framed Offshore Structures) [22], provesthe valid-

ity of the proposed formulation by implementing the proposed formula-

tion via spring elements between the chord and brace members in 2-D

and 3-D space frames.

This paper starts with an introduction including a review on the re-

search of the joint exibility and the jointframe interaction. The next

section develops the joint phenomenological formulation to represent

the joint resistance with respect to the deformation due to the yielding

and plasticity mobilized under the remote brace loading. The study

compares the proposed joint formulation with calibratednite element

results. The following section presents the verication of the proposed

formulation using experimental results reported on 2D and 3D frames.

The last section summarizes the main conclusions drawn from the cur-

rent study.

2. Joint formulation

This section veries the accuracy of the elasticplastic, large defor-

mation FE analysis based on the experimental results for CHS X-joints

and K-joints. A loaddeformation relationship develops subsequently

from calibrated FE analyses, covering a practical geometric range for

X- and K-joints.

2.1. Validation ofnite element analysis

Table 1lists the geometric parameters for the CHS X- and K-joint

specimens reported by van der Vegte[23]and Kurobane et al. [24], re-

spectively. In Table 1, theratio indicates theratio of the bracediameter

over the chord diameter. Thevalue stands for the chord radius to the

chord wall thickness ratio. The parameter refers to the chord length

to the chord radius ratio and denotes the ratio of the brace wall thick-

ness over thechord wall thickness. The X-joint reported by vander Vegte

[23]experiences brace axial compression, as shown in Fig. 2a.Fig. 2b

shows the uni-axial true stress and true strain curve for the X-joint

Nomenclature

A geometric-dependent constant in the loaddeformation

formulation

B geometric-dependent constant in the loaddeformation

formulation

C constant in the loaddeformation formulation

P applied load

PE load at elastic limit

Pu peak load

P non-dimensional applied load

Pu non-dimensional peak load

d0 outer diameter of the chord

d1 outer diameter of the brace

fu material ultimate strength of the chord

fy material yield strength of the chord

g gap between two braces

g (non-dimensional) gap ratio (g/t0)

k0 initial stiffness of the joint

l0 length of the chord

t0 thickness of the chord

t1 thickness of the brace

chord length to half chord diameter ratio (= 2l0/d0)

brace diameter to chord diameter ratio (= d1/d0)

deformation parameter

E deformation at elastic limit

i limit deformation parameter in proposed joint

formulation

u deformation at the peak load

non-dimensional deformation

E non-dimensional deformation at elastic limit

u non-dimensional deformation at the peak load

chord radius to chord wall thickness ratio (= d0/2t0)

angle between the brace and the chord centerline

brace wall thickness over chord wall thickness ratio

(= t1/t0)

angle parameter in the proposed joint formulation

Chord

(Beam-column

element)

Brace

(Beam-column element)

Nodes for spring element

(a) (b)

Load

Deformation

Fig. 1.(a) Joint spring representation in the global frame analysis; and (b) load

deformation characteristics of the joint spring.

109X. Qian et al. / Journal of Constructional Steel Research 90 (2013) 108119


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material. The presence of three planes of symmetry allows the use of a

one-eighth model, as shown in Fig. 2c, which illustrates an FE meshbuilt from 20-node hexagonal elements in the preprocessor Patran

[25]. The elasticplastic, large-deformation analysis utilizes the general-

purposenite element package, ABAQUS[26].Fig. 2d demonstrates the

close agreement between the experimental loaddeformation response

of the CHS X-joint and that computed from the FE analysis, which pre-

dicts accurately the plastic deformation around the brace-to-chord inter-

section observed in the experiment.

Fig.3a illustrates theboundary conditions on thegapped K-joint test.

A test frame supports theends of the left brace and thechord by bearing

pins. The K-joint experiences axial load applied on the right brace, the

end of which is free to rotate in the plane of the K-joint. Fig. 3b shows

the uni-axial true stress and true strain relationship and Fig. 3c illus-

trates the typical FE mesh for the K-joint. Fig. 3d conrms the accuracy

of the FE analysis compared to the experimental loaddeformation

record.

2.2. Proposed joint formulation

This research work targets at developing an accurate nonlinear rela-

tionship for the loaddeformation responses of CHS X- and K-joints, in-

cluding both the elastic and the elasticplastic responses. The expected

function, which needs to describe such a relationship, shall entail the

following characteristics. The formulation should provide highly accu-

rate estimates on the joint response for a wide range of practical geo-

metric parameters. The function should be at least C2 continuous,

implying that the change in the joint stiffness as the load increases

should be continuous before any unstable failure occurs. The basic

form of the function should remain universal for different types of joints

under various loading conditions. In addition, the function should adopt

a simplest possible form for curve tting and subsequent engineering

applications.

Fig. 4 shows the typical loaddeformation curve for an X-joint under

the brace axial compression and that for a gapped K-joint under bal-

anced brace axial loads. Except for very thick-walled chords[27], the

X-joint under brace axial compression often exhibits a peak load as

the deformation increases. The joint resistance decreases graduallyafter the peak load as the plastic deformation propagates in the chord

wall. In contrast, the gapped K-joint under balanced axial loads sustains

monotonically increasing loads until the joint resistance is limited by

the ductility of the material, or extensive plastic deformations in the

chord.

Coupling the physical response of the X- and K-joints with the re-

quirements on the expected loaddeformation function, the proposed

load deformation formula follows,

P f Pu

h 1

where Pand Puare the non-dimensional load and ultimate load, respec-

tively, or,

P Psin fyt

20

2

PuPu sin

fyt20

3

wherefy denotes the yield strength of the chord material, t0 refersto the

thickness of the chord member, and measures the intersection angle

between the brace and the chord. The parameter in Eq. (1) represents

the non-dimensional deformation of the joint,

d0

4

Table 1

Geometric parameters of the X- and K-joint specimens in the reported test.

Joint d0(mm) g

X-joint 408.0 0.6 20 12 1.0

K-joint 216.4 0.76 13.7 16 0.67 60 3.8

(a) (b)

(c) (d)

400

600

800

200

00 10 20 30 40

2

0/ yP f t

0

5

10

15

0 0.25 0.50 0.75

Test

FE

0/d

t0 d0

d1

t1

l0

Reaction

Load

331MPa

435MPa

= 205 GPa

= 0.3

y

u

f

f

E

No. of nodes: 28,000

No. of elements: 5,300

(MPa)

CrownSaddle

(%)

=

=

Fig. 2. A CHSX-joint test: (a) test set-up; (b) uni-axial true stress andtrue strain curve;(c) FE mesh; and(d) comparisonof theload

deformation curve between thetest andFE analysis.

110 X. Qian et al. / Journal of Constructional Steel Research 90 (2013) 108119


4/12

whered0denotes the chord diameter. In Eq.(1), f Pu

is a linear func-

tion ofPuand h

is a logarithmic function of,

P CPu 1A ln 1 B

1=ffiffiffiA

ph i2 : 5

In Eq. (5), CPu refers to theextreme value of thefunction.The deriv-

ative of the loadPfollows,

dP

d 2AC ln 1 B

e

1ffiffiA

p

" # B

1 B

: 6

ThecoefcientA determines the decreasing rate of the joint strength

beyond the peak load. A large value ofAin Eq.(5)creates a sharp vari-

ation in the joint resistance as the deformation increases, while a small

value ofA generates a smooth loaddeformation relationship. The value

ofA, therefore, exhibits strong dependence on the geometric parame-

ters of the joint. The coefcient Btogether withAdetermines the initial

stiffness of the curve. The joint displacement at the peak load derives

from Eq.(5)by settingP CPu, or by setting Eq.(6)to zero,

uud0

e

1=ffiffiffi

Ap1

B

: 7

The proposed joint formulation includes four independent parame-ters: Pu ,A,B and C. The current approach employs the mean strength

equations in IIW[28]forPu, which follows,

Pu 3:16 1 10:7

0:15 8

Pu 2 161:6

0:3

1 11:2 g=t0 0:8

: 9

Eq.(8)denes non-dimensional ultimate strength X-joints under

brace axial compression and Eq. (9)calculates that for K-joints under

(a) (b)

(c) (d)

Loading

Pin support400

600

800

200

00 10 20 30

0

10

20

30

0 0.25 0.50 0.75 1

0/d

2

0sin / yP f t

480MPa

532MPa

= 205 GPa

= 0.3

y

y

f

f

E

No. of nodes: 47,000

No. of elements: 9,000

CrownSaddle

(MPa)

(%)

Test

FE

=

=

Fig. 3. A CHSK-jointtest:(a) loading andboundaryconditions;(b) uni-axialtrue stress andtruestraincurve; (c)FE mesh;and (d)comparisonof theloaddeformation curvebetweenthe

test and FE analysis.

0/d

2

0/ yP f t

d0 = 406 mm

= 0.6

=15

2

0sin / yP f t

0/d

d0 = 406 mm

= 0.9

= 15

= 60o

(a) (b)

4

8

12

16

00 0.04 0.08 0.12 0.16

10

30

40

50

0

20

0 0.02 0.04 0.06 0.08

Fig. 4.Typical load

deformation curves for: (a) an X-joint under axial brace compression; and (b) a K-joint under balanced axial brace loading.



5/12

balanced brace axial loading. The use of Eqs. (8) and (9)reduces the

number of undetermined coefcient to three:A, B and C. The value ofco-

efcientsA andB should remain in reasonable ranges to avoid a nega-

tive value of the dependent variable P, which requires a positive value

of theg

function, or,

A ln 1 B 1= ffiffiffiAph i2b 1 or e 2ffiffiAp N1 B: 10The deformation at the peak load in Eq. (7)should remain as a pos-

itive value, which requires,

A N 0 and B N0: 11

2.3. X-joint formulation

Thecurrent study determines the value ofA, B and Cfor CHS X-joints

through a regression analysis of the results obtained from 30 elastic

plastic, large-deformation analysis, covering a ratio from 0.3 to 1.0

and aratio from 7 to 25.

The loaddeformation characteristics of the X-joint depend signi-

cantly on the ratio. Based on the plastic hinge model proposed by

Togo[29], plastic hinges form at the saddle point and the mid-depth

point of thechord crosssection when an X-joint reaches its peak capac-

ity. For a joint with a small ratio, the chord wall around the brace-to-

chordintersection area undergoes membrane,bending and shearing ac-

tions. As theratio increases, the two braces become closer in locations

and the load transfers from one brace to the other predominantly via

the membrane action in the chord wall material between the two

braces. To reect this change, the parametricstudy includes sixratios:

0.3, 0.6, 0.9, 0.93, 0.96 and 1.0 for the X-joint.

For thin-wall joints with a large value, the transverse shear across

the chord wall thickness is negligible based on thethin-shell theory. The

joint strength depends primarily on the interaction between the bend-

ing and axial stresses acting on the chord wall. Large deformations of

the chord wall create strong variationsin the magnitudes of these bend-

ingstressesand axial stresses,causing a pronounced changein theresis-tance of the joint. This sharp change in the joint resistance with

increasing joint deformations yields a relatively largeA value. As the

ratio approaches 1.0, the membrane action becomes dominant, which

leads to a much higher joint capacity than that of a joint with the

same chord size under dominant bending actions in the chord wall.

Therefore, the loaddeformation curve for X-joints with a large ratio

shows a smooth variation, corresponding to a smallAvalue in Eq.(5).

For thick-walled joints with a small value, the transverse shear

across the chord wall contributes to the joint strength. The large bend-

ing and shear stiffness of the chord wall limits the deformation in the

chord wall and leads subsequently to a small variation in the joint resis-

tance with increasing joint deformations, as compared to thin-walled

joints. This smooth variation in the loaddeformation relationship for

the thick-walled joint leads to a relatively small value ofA.The termf Pu

CPu inEq. (5) characterizes a reference load level inthe loaddeformation relationship. The non-dimensional ultimate

strength Pu incorporates the geometric dependence of the load resis-

tance, while the parameter Cincludes theeffect of joint types andloading

conditions. TheCvalue for CHS X-joints under brace axial compression,

which often exhibits a peak in the loaddeformation curve, equals to 1.0.

The curve-tting procedure to evaluate the coefcientsA and B con-

sists oftwo steps.Therststep determines thevaluesofA and B foreach

discrete loaddeformation curve obtained from thenite element anal-

ysis.The geometric-dependent formulationsofA and B then derivefrom

a nonlinear regression procedure [30]using the values ofA and B for all

joints included in the parametric study.Table 2lists the corresponding

formulation for A and B, which demonstrates a close agreement with

the discrete values obtained using the FE results, as shown by the

small standard-deviation values.Fig. 5compares the loaddeformation

curvespredicted by theproposed joint formulation and those computed

from the FE analysis for four typical CHS X-joints. The proposed load

deformation formulation agrees well with the loaddeformation rela-

tionships computed from the large-deformation, elasticplastic FE

analysis.

This study compares the predictions of the critical joint deformation

at thepeakload and the initial joint stiffness derived from the proposed

joint formulation with the reported studies[18,31]to ensure that the

proposed formulation provides reliable estimations on these important

parameters. Lu's deformation limit, which corresponds to a joint defor-

mation equal to 3% of the chord diameter [31], has become a widely rec-

ognized deformation parameter to dene the ultimate strength of

tubular joints. The initial stiffness formulation, reported by Choo et al.

[18] based on an extensive numerical study, estimates the joint stiffness

as,

k0 PE=E cPu=E 0:8Pu=E 12

where the loadPEcorresponds to the limit of elasticity and assumes a

value of 0.8Pu based on the FE analysis [18]. The initial joint stiffness

(k0), therefore, equals,

k0 0:8Pu=E 0:8BPu= e 1ffiffiffiffiffiffiffiffiffi

0:2=Ap

1

13

whereEdenotes the displacement at 0.8Pu.Table 3shows the agree-ment in the uand k0 values obtained from the proposed joint formula-

tion in comparison with Lu's deformation and the k0 results reported by

Chooet al.[18].

For X-joints under brace axial compression, a re-development of the

joint strength occurs at a large deformation level due to the direct con-

tact of the compression braces, as observed in the BOMEL 2D and 3D

frame tests[32,33].Fig. 6a shows the large deformation of the chord,

which leads to the contact of the two braces through the inner surface

of the chord. The direct contact of the two brace members leads to a

re-gained joint strength equal to the axial yield strength of the brace

member at a joint deformation of = 0.5d0. The initialization of the

strength re-development depends on theratio, as shown inFig. 6b,

which denes i to be thedisplacement corresponding to the initialcon-

tact of the two braces,

i 0:5d0t0 sin cos1

: 14

The value oficorresponds to the distance between the inner sur-

faces of thechord member near thesaddle point, measured alonga ver-

tical axis corresponding to the mid-thickness of the brace wall, as

shown inFig. 6c.Fig. 6d shows the close agreement between Eq. (14)

and theivalues obtained from the nite element analysis, which pro-

hibits self penetration of the chord inner surface in the contact algo-

rithm. The proposed joint formulation includes this redevelopment of

joint strength for CHS X-joints under brace axial compression through

a bilinear model in the loaddeformation relationship, as shown by

the dashed line inFig. 6b.

Table 2

Coefcient in the proposed formulation for X-joints.

Coefcient Formulation Proposed/FE

Mean Standard deviation No. of data

A 0:275:2 2:527:54:3 1.00 0.08 30

B (2333.24 40+ 820)0.6 1.01 0.09 30

C 1.00 1.00 0.02 30



6/12

2.4. K-joint formulation

The loaddeformation curve for the K-joint under balanced axial

loads follows the response of the compression brace [18]. Based on

the typical loaddeformation curve for a gapped K-joint, which con-

tinues to sustain increasing loads beyond the Lu's deformation limit

[31], the K-joint formulation also follows Eq.(5).The strength of the K-joint depends on the membrane, shear and

bending resistance of the chord wall around the brace-to-chord inter-

section. In addition, thegap in the chord between the two braces trans-

fers theload from onebrace to theother andexperiences bending, shear

and membrane actions at large deformations. Similar to the X-joint, the

transverse shear in a thick-walled chord of a K-joint also contributes to

the joint strength. The shear contribution leads to a smooth loaddefor-

mation curve for thethick-walled K-jointwith a small ratio.Therefore,

theA value, which implies the rate of changein the loadlevel, decreases

as decreases.

The determination of the coefcientsA, B and C follows the same

procedure as that for the X-joint. The parametric FE analysis covers a

ratio from 0.3 to 1.0 and a ratio from 7 to 25. The numerical analysis

xes the gap between the two braces to be twice the wall thickness ofthe chord. The boundary conditions for the K-joint follows that shown

inFig. 7a, which provides a conservative representation of framing ef-

fect on the K-joint[34].

Table 4 illustrates the formulation for A, B and C based on the

nonlinear regression analysis, which leads to a close agreement with

the values determined from the FE analysis, as reected by the mean

and standard deviation values in the same table. Fig. 7b and c sketches

the loaddeformation curves predicted by the proposed joint formula-

tion and those computed from the FE analysis for two typical K-joints.

Theproposed joint formulation based on theIIW equation [28] providesa close agreement with the FE results.

3. Validation of the proposed formulation in pushover analyses

The current study implements the proposed joint formulation in the

frame analysis performed using the nonlinear frame analysis tool,

USFOS[22]. The verication study utilizes experimental results from

large-scale 2D and 3D frame tests[32,33,35]. The element formulation

in USFOS employs the exact solution of the governing equation for

beam-columns subjected to end-forces, which enables the modeling

of each physical member by one element. The plastic hinges at the

mid-span and at the end of the beam-column element simulate the ma-

terial nonlinearity[22].

3.1. BOMEL 2D frames

Boltet al.[33]report an experimental study of a series of 2D large-

scale frame tests under the scope of a joint industry project. The frame

test consists of 6 double-bay X-frames and 4 single-bay K-frames. The

current study compares the results of three X-braced frames, namely

Frame I, Frame II and Frame III, as shown in Fig. 8. The design of

X-frames follows practical congurations representative of offshore

jacket structures. Frame I has strong joint-cans at both the top and the

bottom bays, together with a horizontal member in the middle of the

top and the bottom bays, while frame II includes a weak joint-can at

the top bay to investigate the load shedding and redistribution. Frame

III remains the same as Frame I, except that the mid-horizontal member

(a) (b)

(c) (d)

2

0/ yP f t

d0 = 406 mm

= 0.6

= 10

Proposed

FE

0 0.03 0.06 0.09 0.12 0.15

0/d

2

0/ yP f t

d0 = 406 mm

= 0.9

= 10

0/d

0 0.03 0.06 0.09 0.150.12

5

10

20

25

0

15

2

0/ yP f t

5

10

15

0

d0 = 406 mm

= 0.6

= 20

Proposed

FE

0/d

2

0/ yP f t

5

10

15

0

d0 = 406 mm

= 0.9

= 20

0 0.03 0.06 0.09 0.12 0.15

0/d

Proposed

FE

Proposed

FE

0 0.03 0.06 0.09 0.12 0.15

5

10

15

0

Fig. 5.Comparison between the proposed loaddeformation formulation and FE results for CHS X-joints with: (a) = 0.6, = 10; (b) = 0.9, = 10; (c) = 0.6,= 20; and

(d)= 0.9,= 20.

Table 3

Comparisons of the critical deformation at the peak load and the joint stiffness with

reference studies for X-joint.

Parameters Results from Reference Proposed/reference

Mean Standard

deviation

No. of

data

u/d0 0.03 (Lu's deformation limit[31]) 1.10 0.20 30

k0 P d0sinfy t201185 0:8

20:150:4 [18] 1.09 0.15 30



7/12

is removed. Each frame connects to a test rig through pin connections at

the bottoms of thetwo vertical legs, with out-of-plane pinsupports pro-

vided at sixprimary legjoints. Thetest arrangement applies a horizontal

load at the top of theframe until thecritical joints and members deform

signicantly, causing pronounced reductions in the frame resistance.

The current study includes three types of joint formulation for each

frame analysis: 1) the rigid joint assumption, 2) MSL joint formulation

[22],and 3) the proposed joint formation. Fig. 9a compares the numer-

ical analysis and the test results for Frame I. In Frame I, the buckling of

the top-bay compression brace (shown in Fig. 8a) dominates the

(a) (b)

(c) (d)

0.1

0.2

0

0.3

0.4

0.5

i by ABAQUS

0 0(0.5 ) sini d t

0 0.2 0.4 0.6 1.00.8 1.2

0/

i d

0 0= (

=

0.5 ) sini

d t

Brace yielding

capacity

P

00.5d

Eq. (5)

Bilinear model

1 1 1

0 0

0.5 0.5cos ( )

0.5

d t

d t

0 0(0.5 )sini d t0 00.5d t

=

=

=

Fig. 6.(a) Contact of two compression braces under a large deformation level for X-joint; (b) schematic load deformation relationship for the X-joint with strength re-development;

(c) deformation level at the initial contact of the two compression braces; and (d) comparison ofiobtained from FE analyses and Eq. (14).

(a) (b)

(c)

2

0sin /

yP f t

8

16

24

32

0

0/d

0.015 0.030 0.045 0.0500

d0 = 406 mm

= 0.6

= 15

Proposed

FE

15

30

45

60

00.02 0.04 0.06 0.080

20

sin / yP f t

0/d

d0 = 406 mm

= 0.9

= 20

Proposed

FE

Fig. 7. (a) Load andboundaryconditions forFE analyses of CHSK-joints; (b) theproposed loaddeformation formulation for the CHS K-joint with= 0.6, = 15; and(c) theproposed

load

deformation formulation for the CHS K-joint with = 0.9,= 20.



8/12

frame strength. All three analyses predict this failure mechanism. The

proposed formulation leads to a slightly better prediction on the ulti-

mate strength of the test frame than the MSL joint formulation.

Fig. 9b compares the global response for Frame II, in which the top-

bay X-joint is the critical structural component. This X-joint with= 1

underbrace axial compression softens gradually due to increased plastic

deformations in thechord wall beyond thepeakload. The contact of the

two braces at a further deformation redevelops the joint strength suf-

cient to cause buckling of the compression brace in the top bay. The

global load applied on the frame thus increases until the buckling of

that compression brace occurs. The MSL formulation shows the soften-

ing of the X-joint in line with the experimental observation. However,

the MSL formulation imposes a deformation limit on the X-joint and

leads to the termination of the analysis before the unstable brace buck-

ling takes place. The proposed joint formulation predicts both the soft-

ening of the CHS joint due to plastic deformations in the chord wall

and the re-strengthening of the X-joint at a large deformation level.

The predicted frame response using the proposed joint formulation

thus reects correctly the buckling failure of the test frame, albeit that

this brace buckling occurs at a lower load level than that observed in

the test.

Without the horizontal member, Frame III shows a similar ultimate

strength level compared to Frame I, as shown inFig. 9c. The absence

of the horizontal membergenerates signicant load re-distributions be-

yond the top bay brace buckling. This forces the buckling of the bottom

bay brace to occur at a small global displacement level. The frame anal-

ysis based on the proposed joint formulation provides a better

prediction on the global frame response than thatusing the MSLformu-

lation and that based on the rigid joint assumption.

3.2. Kurobane's 2D frames

Kurobane and Ogawa [35] summarize the cyclic tests on 15 2D

frames, with six frame congurations investigated. The current study

veries the K-joint formulation based on three typical frames, as

shown in Fig. 10. All three frames shown in Fig. 10 experience a vertical

load at the right end of the frame, while the chord ends on the left are

xed viaange connections to a reaction wall. Thetest measures the ro-

tation as the deection of the frame divided by the length of the truss.

Fig. 11 compares the numerical prediction of the frame response

based on different joint formulations with the experimental results.

Similar to the BOMEL 2D frames, the numerical study includes three

types of joint formulation in the frame analysis. Each frame analysis in-

cludes the joint formulation for all K-joints in the frame. All three analy-

ses (the rigid joint, the MSL formulation and the proposed formulation)

predict accurately thefailure mode of Frame A, whichis governed by the

member buckling (Figs. 10a and 11a). The proposed formulation agrees

with the test results on both the frame stiffness and the ultimate frame

strength. The MSL formulation predicts a more exible frame response

than thetest results. A detailed examinationreveals that theMSL formu-

lation predicts a much lower joint stiffness than does the proposed joint

formulation. The latter agrees with the joint stiffness obtained from a

separate FE analysis for the K-joint in Frame A.

Table 4

Coefcient in the proposed formulation for K-joints.

Coefcient Formulation Proposed/FE

Mean Standard deviation No. of data

A 0:07521:5 26:4 e 0:001720:0141:47 =1000 1.00 0.04 16B 2267e1.9 1.00 0.05 16

C 1.13 1.00 0.03 16

5944

1524

6096

6096

1524

168OD7.1WT

168OD9.5WT

168OD6.3WT

168OD9.5WT

356ODx12.7W

T

5944

1524

6096

6096

1524

168OD7.1WT

168OD5.1WT

168OD6.3WT

168OD9.5WT

356ODx12.7W

T

5944

1524

6096

6096

1524

168OD7.1WT

168OD9.5WT

168OD6.3WT

168OD9.5WT

356ODx12.7W

T

All units: (mm)

168OD4.5WT 168OD4.5WT 168OD4.5WT

Buckled

brace

Load Load Load

(a) (b) (c)

Buckled

brace

Buckled

brace

Buckled

brace

All members

168OD4.5WT

All members

168OD4.5WT

All members

168OD4.5WT

Fig. 8.Conguration of BOMEL 2D frames: (a) Frame I; (b) Frame II; and (c) Frame III.



9/12

Frame B fails by the out-of-plane buckling and local buckling of the

brace (Figs. 10b and 11b). The three analyses show similar strength pre-

dictions as the test, as shown inFig. 11b. The MSL formulation provides

a lower prediction on the frame stiffness than the test frame.

Fig. 11c compares the numerical prediction and the experimental re-

cord on the global loaddisplacement response for frame T, which is

governed by the buckling of the brace shown inFig. 10c. The rigid joint

formulation predicts a sequence of member buckling due to the stiffjoint response. The MSL formulation predicts a weak joint and the

frame exhibits a much lower strength and a much lower stiffness than

the test results. The proposed joint formulation predicts closely the soft-

ening of the joint and agrees well with the peak strengthof the test frame.

3.3. BOMEL 3D frames

Bolt and Billington [36] report thelarge-scale3D frame testsshown in

Fig. 12. The double-bay test frame consists of six vertical legs[37]. Asshown in Fig. 12a, thestructurepresents a hybrid of bracing conguration

(a) (b)

(c)

0.05 0.10 0.15 0.200

Global load (kN)

Global displacement (m)

200

400

600

800

0

1000Global load (kN)


Test

MSL

Rigid

Proposed

400

800

0

1200Global load (kN)

0.05 0.10 0.150 0.20 0.25Global displacement (m)

0.05 0.10 0.150 0.20

200

400

600

800

0

1000

Test

MSL

Rigid

Proposed

Test

MSL

Rigid

Proposed

Brace buckling

Frame III

Frame IIFrame I

Brace buckling

Joint yielding

Brace buckling (top bay)

Brace buckling (bottom bay)

Fig. 9.Comparison of the global loaddeformation response between numerical analysis and experimental records for: (a) Frame I; (b) Frame II; and (c) Frame III.

(a) (b)

(c)

3572

1500

60.5OD3.8WT

60.6OD2.2WT

165.5OD5.7WT

2418

1000

60.4OD

3.8WT

60.6OD2.2WT

165.4OD5.7WT

4054

60.5OD

2.1WT

139.9OD4.1WT

1250

All units: (mm)

Buckled braceBuckled brace

Buckled brace

Load

Load

Load

60.5OD

2.1WT60.5OD

2.1WT

Fig. 10.Conguration of Kurobane's 2D frames: (1) Frame A; (b) Frame B; and (c) Frame T.



10/12

typical for offshore jacket structures. The two longitudinal panels in the

horizontal plane (designated as Panel A and Panel B) are X-braced. In

Panel A (the bottom panel in Fig. 12a) the X-joints have thick joint-

cans. In Panel B (the top panel inFig. 12a), the two level I X-joints do

not include joint-cans and the through chords run in opposite directions.

The transverse panels C and D are K-braced with intermediate diamond

bracing in between the two panels. In Panel C, neither of the gapped K-

joints has a joint-can. The distant transverse panel E is X-braced but

without a horizontal member in the middle height. The entire structure

is mounted in a self-reacting frame made of I- and H-sections, as

illustrated in Fig. 12a. The bottomof the self-reacting frame, which is par-

allel to Panel A in Fig. 12a, sitson a strongoor. The entire testing proce-

dure includes three load cases, as shown inFig. 12cd, in which the self-

reacting frame is removed.

The testing of the 3D frame includes three load cases. In Load Case I,

the front K-braced panel along Panel C is loaded vertically upwards, as

(a) (b)

(c)

0.02 0.040

Global load (kN)

Global rotation (Radian)

50

100

0

150

0.01 0.020

Global load (kN)


Global load (kN)

0.100 0.20


50

150

0

200

100Test

MSL

Rigid

Proposed

Test

MSL

Rigid

Proposed

Test

MSL

Rigid

Proposed

50

100

0

150

Frame A

Frame T

Frame B

Brace buckling

Out-of-plane buckling

Brace buckling

Fig. 11.Comparison of the global loaddeformation response between the numerical analysis and experimental records for: (a) Frame A; (b) Frame B; and (c) Frame T.

(a) (b)

(c)

Buckled brace

Buckled

braceLoad

Load

Crack

Crack

CrackBuckled

brace

Load

Panel EPanel D

Panel CPanel B

Panel A

Level II

Level I

Panel CPanel E

Panel A

Panel E

Load Case I

Load Case II

Load Case III(d)

Fig. 12.Conguration of BOMEL 3D frame test: (a) test model; (b) Load Case I; (c) Load Case II; and (4) Load Case III.



11/12

shown in Fig. 12b. Fig. 13a shows the comparison of three analyses with

the experimental results. The weld-toe crack near the tension brace in

the K-joint (Fig. 12a) initiates a slight decrease in the frame strength.

However, this crack does not grow extensively under increasing loads.

Instead, the diamond brace in level I redistributes the load to Panel D

and the K-brace in Panel D buckles at the peak frame load.

None of the three types of joint formulation includes a representa-

tion on the fracture failure in tubular joints. Both the MSL and the pro-

posed formulation predict the weakening of the joint under plasticdeformation as well as the subsequent brace buckling in the frame.

The proposed formulation provides a close prediction on the ultimate

strength of the test frame.

In Load Case II, the X-braced Panel E experiences a vertical load ap-

plied in an upward direction as shown in Fig. 12c. The weakening of

the X-joint under plastic deformation in the chord wall leads to a ductile

frame response. Similar to Frame II in the BOMEL 2D frame (see Fig. 8b),

the large deformation of the joint enables contact of the two compres-

sion braces through the inner surface of the chord. This contact leads

to the redevelopment of the joint strength and causes the buckling of

the compression brace. Both the MSL joint formulation and the pro-

posed joint formulation predict the weakening of the joint, as shown

inFig. 13b. Similar to Frame II (inFig. 8b), the deformation limit in the

MSLjoint formulation terminates the frame analysis at a small deforma-

tion level, insufcient to mobilize the subsequent brace buckling. The

proposed joint formulation shows a good agreement with the test re-

sults for Load Case II. The rigid joint formulation estimates a relatively

smaller frame capacity than the test by forcing the compression brace

to buckle at a very small global deformation level.

In Load Case III, a horizontal load is applied along Panel A to thebot-

tom X-braced panel. After all the compressionbraces in Panel A buckles,

the horizontal braces redistribute the load to Panel B and leads to the

crack in two joints shown inFig. 12d. The test stops after the K-brace

in Panel C buckles. Similar to Load Case I, crack initiation is not captured,

which contributes to the difference between the proposed formulation

and test results.Fig. 13c shows that the proposed formulation predicts

the frame ultimate strength accurately.

4. Summary and conclusions

The current study develops a new loaddeformationformulation for

CHS X- and K-joints to describe their nonlinear loaddeformation be-

havior in the global pushover analysis. The current study focuses on

the loaddeformation response of X- and K-joints subjected only to

brace axial loads. The reference ultimate strength in the proposed

joint formulationfollows the latestIIW recommendations [27]. The pro-

posed joint formulation develops through regression analyses of the FEresults, which are validated against reported experimental results. The

verication study of the proposed formulation on CHS X- and K-joints

in the pushover analysis utilizes 2D BOMEL [32,33] and Kurobane

frame tests [35] as well as 3D BOMEL [36] frame experiments. The

study summarized above supports the following conclusions:

(1) The proposed joint formulation provides a convenient approach

to estimate the loaddeformation relationship for CHS X- and K-

joints. The parametric formulation based on the joint geometry

and loading conditions eliminates the need for the elasticplastic,

large-deformationnite element analyses on CHS X- and K-joints.

The verication based on the reported experimental study proves

the accuracy of the proposed formulation.

(2) The comparison between the frame analyses with various joint

formulations and the experimental data demonstrates the signi-

cance of the nonlinear loaddeformation joint behavior in the

frame response, especially for simple 2-D frames with low redun-

dancy. The rigid joint assumption leads to completely different fail-

ure modes in a frame with weak joints. The proposed formulation,

implemented as joint-spring elements in the frame analysis, pro-

vides close predictions on both the failure modes and the ultimate

strength for 2-D and 3-D tested frames.

(3) The proposed formulations describe the loaddeformation

behavior of the axially loaded CHS X- and K-joints without fracture

failure. The incorporation of the fracture failure as reliable phenom-

enological representations in the frame analysis is the focus of a sep-

arate research effort[38].

(a) (b)

(c)Global load (kN)

Global load (kN) Global load (kN)

Test

MSL

Rigid

Proposed

0.05 0.150 0.10

300

600

0

1200

900

0.100 0.300.20

250

1000

0

1250

750

500

1000

0

2000

3000

0.15 0.250 0.05 0.10 0.20

Test

MSL

Rigid

Proposed

Test

MSL

Rigid

Proposed




Load Case IILoad Case I

Load Case III

Fig. 13.Comparison of the global loaddeformation response between numerical analysis and experimental records for BOMEL 3D test: (a) Load Case I; (b) Load Case II; and (c) Load

Case III.



12/12

Acknowledgment

We acknowledge the support of Lloyd's Register Foundation towards

funding the research and development program in the Centre for Off-

shore Research & Engineering in National University of Singapore. The

research scholarship provided by the National University of Singapore

is also gratefully acknowledged. The authors would like to extend

their appreciation to Professor Peter Marshall for providing the very

useful suggestions on the research.

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journal of constructional steel research volume 90 issue 2013 [doi 10.1016%2fj.jcsr.2013.07.024]...

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