Marine Structures 33 (2013) 56–70

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marstruc

Local joint flexibility element for offshoreplateforms structures

A. Akbar Golafshani a, Mehdi Kia b,*, Pejman Alanjari c

a Sharif University of Technology, Department of Civil and Environmental Engineering, Tehran, Iranb Structural Engineering, Sharif University of Technology, Tehran, Iranc Structural Engineering, K.N. Toosi University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 27 December 2011Received in revised form 13 April 2013Accepted 13 April 2013

Keywords:Joint flexibilityChordBraceTubular framed structureFlexibility matrix

* Corresponding author. Tel.: þ98 9111186949.E-mail addresses: golafshani@sharif.ir (A.A. Gola

alanjari@sina.kntu.ac.ir (P. Alanjari).

0951-8339/$ – see front matter � 2013 Elsevier Lthttp://dx.doi.org/10.1016/j.marstruc.2013.04.003

a b s t r a c t

A large number of offshore platforms of various types have beeninstalled in deep or shallow waters throughout the world. Thesestructures are mainly made of tubular members which are inter-connected by using tubular joints. In tubular frames, joints mayexhibit considerable flexibility in both elastic and plastic range ofresponse. The resulting flexibility may have marked effects on theoverall behavior of offshore platforms.This paper investigates the effects of joint flexibility on local andglobal behavior of tubular framed structures in linear range ofresponse. A new joint flexibility element is developed on the basisof flexibility matrix and implemented in a finite-element programto account for local joint flexibility effects in analytical models oftubular framed structures. The element formulation is consider-ably easy and straightforward in comparison with other existingtubular joint elements. It was concluded that developed flexiblejoint model produces accurate results comparing to sophisticatedmulti-axial finite element joint models.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In the past years, considerable investigative efforts have focused on understanding the real behaviorof offshore structures subjected to severe seismic loadings. Moreover, there is an increasing demand forreappraisal of existing installations of steel jacket structures. This perhaps could be due to the revised

fshani), mehdikia2519@gmail.com, M_kia@mehr.sharif.ir (M. Kia), Pejman_

d. All rights reserved.

A.A. Golafshani et al. / Marine Structures 33 (2013) 56–70 57

design recommendations based on a better knowledge of structural performance. As a consequence,variety of analytical models have been developed to simulate the response of offshore platforms.Welded steel tubular joints are the kind of connection used extensively in the construction of fixed,offshore steel structures. The behavior of these welded joints must be predicted to ensure safe andreliable analytical simulations of such structures. The use of tubular connections in circular hollowsections is not confined to offshore platforms and many structures employ these types of joints forinterconnecting their members such as large-span space frames and towers.

One of the earliest investigations on effects of local joint flexibility (LJF) on the response of offshoreinstallations dates back to 1980 when Boukamp et al. [1] conducted a research on analytical techniquesused to develop the joint flexibility (JF) model and to find some procedures to incorporate these effectsinto overall structural response. They provided joint analysis models which were assembled fromcomponent branch and chord substructures using a consistent multilevel substructure technique.Later, Ueda et al. [2] provided an improved joint model and equations for flexibility of tubular joints.The accuracy of their models was confirmed through comparisons with results of finite elementanalysis. They concluded that their models are capable of accurately representing the nonlinearbehavior of actual joints. Fessler and Spooner [3] and Fessler et al. [4] presented improved equations forflexibility coefficients in terms of joint parameters, in Y, X and gap-K joints. Buitrago et al. [5] alsoobtained new equations for flexibility coefficients and gave explicit formulas to determine the localjoint flexibilities for various joint types and geometries. Karamanos et al. [6] investigated the fatiguedesign of K-joint tubular girders. Skallerud et al. [7] performed experimental investigations on cyclic inelastic behavior of tubular joints and provided the researchers with valuable test data. Dier [8]described the recent developments that have taken place in offshore tubular joint technology. Mir-taheri et al. [9] investigated the comparative response of two analytical platformmodels and concludedthat LJF is of great significance in both elastic and plastic range of response. Static loading performanceof tubular joint inmulti-column composite bridge piers was studied by Lee et al. [10]. Lee and Parry [11]conducted a research on strength prediction for ring-stiffened DT-joints in offshore jacket structures.Holmås [12] describes a fully coupled finite element for local joint flexibility of tubular joints based onsolving the equations for elastic shell. Hellan [13] presents (among many topics) the different modelsfor joint flexibility and how the flexibility impacts the ultimate strength. Alanjari et al. [14] performedan experimental research on a small-scaled 2D platform and developed an analytical model whichmade use of uniaxial fiber elements to model a fracturing tubular joint. However, their model lackedjoint local biaxial and triaxial effects as they did not take LJF into account.

In this study a JF element based on flexibility matrix is developed and formulated with the aid ofempirical Fessler [4] equations. Stiffness matrix derivation is discussed in detail through the use ofequilibrium of the uni-axial element without rigid body modes in the vicinity of chord and brace inter-section. The proposed element is subsequently implemented in nonlinear finite element (FE) programOpenSees [15] andverifiedusingmoregeneralmulti-purpose FEprogramswhichmakeuseofmulti-axialelements such as shell or solid elements. Finally, improved tubular framedstructuremodels aremadeandcompared against conventional rigid-joint models which are widely employed in engineering practice.

2. Tubular joint element

In general, a tubular joint comprises a number of independent chord/brace intersections and theultimate strength limit state of each intersection is to be checked against the design requirements.However, as mentioned earlier, JF has marked effects on overall deformation pattern of the structure,nominal stress distribution within the joints, buckling load of members as well as natural frequenciesand mode shapes of platform.

2.1. Analytical treatment of tubular joints

Conventionally, in structural analysis of offshore platforms, jacket structure is modeled by a plane orspace frame having tubular members rigidly interconnected to each other at nodal points. There existsome other techniques which can take into account LJF in a reasonable fashion such as the so-calledeffective length model. The model utilizes an effective length which is adopted to replace the real

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length of the brace so that local rotations and stress distributions of the joints are modified. That is, anoffset is defined between chord and brace joints and an equivalent element whose characteristics (i.e.type of the element and its cross section properties) are assigned a priori, is replaced to account forlocal deformations and rotations.

Mostly, the effective length method is treated in two separate approaches. Sometimes the offsetelement is assumed to be relatively rigid to account for a rigid tubular connection. This approach iscapable of considering offset moments which typically exist in the joints. The second method assignsan element similar to brace element to the offset distance to account for not only the offset momentsbut also the local flexibilities induced at the joint. In other words, the brace is extended to intersectwith chord central axis. The latter is called center-to-center model hereafter. However, it is difficult toselect an appropriate effective length to represent the exact value of the LJF. Moreover, beam-columnelements are mainly used in practice for representing the offset joint element which is not capable ofmodeling the exact multi-axial responses of the joints such as progressive ovalizing and local bucklingwhich are rather shell-like behaviors (Fig. 1).

Using three-dimensional shell and solid elements is another alternative to accurately simulate thebehavior of tubular joints; however, large computational effort which is made for these models makesit uninteresting for practical purposes.

2.2. Basic definitions

In an attempt to remedy the problems of center-to-center model, JF element has been proposed insome computer programs which contains joint information and characteristics based on empiricalrelationships [16] or solving the fundamental equations for shells [17]. The formulation and stiffnessmatrices of these elements differ from each other based on the computer programwhich is employed.For instance, Chen and Hu [18] proposed an LJF element based on equilibrium of external and internalforces and compatibility of displacements at element end nodes.

This section presents general formulation of the proposed JF element based on flexibility matrixwhich is developed and formulated with the aid of empirical Fessler [4] equations. These equations arebased on the fact that upon loading on a tubular joint (axial or flexural), chord wall would deformlocally in addition to overall consistent joint deformationwhich exists in a typical fixed connection. Theproposed formulation offers the advantage of simplicity over previous models.

The amount of LJF can be defined by the local deformation caused by unit external load in 3 di-rections namely axial deformation, in-plane and out-of-plane rotations as formulated by Eq. (1) [18]:

LJFAX ¼ fAX ¼ d

P; LJFIPB ¼ f IPB ¼

4I

MI; LJFOPB ¼ fOPB ¼ 4o

MO(1)

Fig. 1. Conventional analytical modeling of plane frame offshore jacket using rigid or center-to-center joints.

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where subscript ‘AX’, ‘IPB’ and ‘OPB’ denote axial, in-plane and out-of-plane bending, respectively. ‘d’,‘4I ’ and ‘4o’ are axial, in-plane and out-of-plane deformations which are caused by corresponding axialforces and bending moments. Fessler et al. [4] conducted experimental research on test specimens andproposed parametric equations bywhich it would be possible to evaluate local flexibility of certain typeof offshore tubular joints. For local flexibilities (LFs) provided in (1), the parametric equations are asfollows[4]:

LJFAX ¼ 1:95g2:15ð1� bÞ1:3sin2:194

ED(2)

LJFOPB ¼ 85:5g2:2exp�� 3:85b

�sin2:16

4

ED3 (3)

LJFIPB ¼ 134g1:73exp�� 4:52b

�sin1:22

4

ED3 (4)

In these equations, b ¼ d=D and g ¼ D=2T are typical joint characteristics.‘d’, ‘D’, ‘T ’, ‘E’and ‘4’arebrace and chord diameters, chord thickness, elastic modulus of the joint members and chord-braceintersection angle, respectively.

In most general case for a 3-dimensioanl element having 2 nodes, 9 LFs may be defined corre-sponding to 3 main flexibilities defined in Eq. (1). These LFs are shown in Eq. (2):

F ¼24 f11 f12 f13f21 f22 f23f31 f32 f33

35 (5)

According to reciprocal theorem, LJF matrix defined in Eq. (5) is symmetric. Since out-of-planebending is not coupled with axial force and in-plane bending (i.e. f13 ¼ f31 ¼ f23 ¼ f32 ¼ 0).Furthermore, entry ‘f12’ represents the local in-plane rotation at the cord-brace intersection which iscaused by a unit axial force acting on the brace. Since this value seems to be very small, it is assumed tobe zero in the flexibility matrix. As a consequence the matrix in (5) reduces to:

F ¼24 f11 0 0

0 f22 00 0 f33

35 (6)

In the case of planar problems, thismatrix reduces further to a 2� 2 diagonal matrix containing ‘f11’,‘f22’.

In a tubular framed structure model, a typical joint can be divided into 3 connected elementsnamely brace, joint and chord element as schematically shown in Fig. 3. The reference coordinatesystem (CS) for the entire elements is the LCS ‘x0’, ‘y0’, while ‘x’, ‘y’ denotes the GCS. The angle ‘q’ be-tween local and global CS and chord inclination angle are complementary. The tubular joint elementconnects chord and brace nodes even though it may not be necessarily aligned with brace direction.This offers the advantage of automatically calculating the offset moment which normally exist intubular framed structure joints given the fact that mutli-brace connections are not necessarily coin-cident. The element degrees of freedom are illustrated in Fig. 2 on the right. As can be seen the elementhas 6 degrees of freedom in order to be compatible with other surrounding beam-column elements.Fig. 3(a) shows brace and chord beam-column elements which are connected to the joint element. Itwas discussed earlier that flexibility matrix of the joint element provides axial and in-plane bendingdegrees of freedom in planar problems. However, in order for displacements of adjacent nodes in ananalytical model to be compatible, 3 degrees of freedom are defined for the tubular joint element atend nodes. The resulting element has 6 degrees of freedom including rigid body modes. These degreesof freedom are termed as initial degrees of freedom in this study. Due to the presence of these rigidbody motions in local and global coordinate system, the corresponding element stiffness matrix wouldbe singular. As a consequence, in general there would be no flexibility matrix associated with this

Fig. 2. Tubular joint element in the vicinity of chord-brace intersection.

Fig. 3. Idealization of a tubular joint; (a) tubular joint element with 2 degrees of freedom at each node, (b) representation of basicdegrees of freedom.

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system. Thereby, the element is formulated in another system inwhich rigid body modes are excluded(Fig. 3(b)). Rigid body modes can be incorporated with a simple geometric transformation. Theresulting element has 3 degrees of freedom: two perpendicular translations and one rotation relative tothe element longitudinal axis. These degrees of freedom are termed as basic degrees of freedomhereafter. Flexibility matrix is derived for this simple 2D-CS and extended to more general CS which isshown in Fig. 3(b) on the left. The former can be achieved explicitly using the so-called method of unitload application and computing the corresponding deformations. In other words, flexibility matrix of astructure represents the displacements developed in the structure when a unit force corresponding toone degree of freedom is introduced while no other nodal forces are applied. Applying a unit loadhorizontally on free end (i.e. degree of freedom #4), one can evaluate corresponding nodal displace-ments in all 3 directions. In this case, using Eq. (1), the displacements are ‘LJFv’, 0, 0 in three directionsshown in Fig. 4(b) on the right. It should be noted that ‘LJFv’ is obtained using simple decomposition of‘LJFAX ’ and its projection along the degree of freedom #4. That is:

LJFv ¼ LJFAXsin2f

(7)

The evaluation can be performed for other degrees of freedom and local flexibility matrix is ob-tained for basic degrees of freedom as follows:

24 LJFv

bLJFIPB

35 (8)

The entry in row 2 and column 2 is set to a small number which is denoted by ‘b’ in (8). It couldalso be seen that this matrix is diagonal, that is, applying a unit force in one direction does not induceany displacement in the directions of other degrees of freedom. This evaluation is resulted fromphysical interpretation of joint behavior in the vicinity of chord-brace intersection. In fact axialstiffness of the chord is too high to be altered by joint deformations and that is the primary reasonwhy it is neglected.

The next step is transformation between local and global coordinate system (i.e. from ‘x0y0’ to ‘xy’Fig. 2). It should be noted that in Fig. 3, LCS and GCS assumed to be coincident (i.e. no batter is assumedfor the legs of the platform). The transformation task could be conveniently carried out using simplerotation matrix in which ‘q’ is the rotation angle. Rotating local flexibility matrix yields the global onewhich is denoted by ‘F’ in the following Eq.:

F ¼ TTF 0T (9)

In which ‘T ’ and ‘F 0’ are rotation and LF matrices.

Fig. 4. Unit load application for obtaining transformation matrix ‘R’.

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2.3. Stiffness matrix derivation

The flexibility matrix which was obtained in (9) is to be employed for calculation of stiffness matrix.However, the global stiffness matrix contains both rigid and non-rigid degrees of freedom. Generally,the following 2 relationships hold between applied loads and corresponding displacements in con-ventional structural analysis:

�kFFDF þ kFCDC ¼ PFkCFDF þ kCCDC ¼ PC

(10)

The above equations relate applied forces in free and constrained degrees of freedom to corre-sponding displacements. These equations show how four stiffness sub-matrices constitute the globalstiffness matrix (indices ‘F’ and ‘C’ stands for free and constrained). Consequently, the problem reducesto evaluating four constituents of the global stiffness matrix. The first sub-matrix could be convenientlyobtained by inverting the flexibility matrix given in (9), that is:

kFF ¼ F�1 (11)

As for the other sub-matrices, a transformation matrix is required to relate the basic and initialdegrees of freedom. Fig. 5 illustrates the tubular joint element with three degrees of freedom whichhad been fixed earlier (Fig. 3(b)). A transformation matrix can be used to connect basic and initialdegrees of freedom again with the aid of unit load application. As can be observed, three unit loadshave been applied at one end while the corresponding reactions have been evaluated using simplestatic equilibrium. These reactions yield transformation matrix ‘R’ as follows:

R ¼24 �1 0 0

0 �1 0Lsina �Lcosa �1

35 (12)

Fig. 5. Rigid body movement of brace upon pure axial loading.

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In (12) and Fig. 5, ‘a’ denotes the angle between the direction of tubular joint element and the globalhorizontal axis ‘x’. Having this transformation matrix, it would be possible to obtain:

�PC ¼ RPFPF ¼ kFFDF

/PC ¼ RkFFDF ¼ RF�1DF (13)

It is clear that PC ¼ kCFDF , thereby substituting in (13) yields:

kCF ¼ RF�1 (14)

From symmetry it is concluded thatkCF ¼ kFC . As a consequence, the problem reduces to obtaining‘kCC ’ which can also be evaluated using transformation matrix ‘R’ and ‘kFF ’. The relation betweenconstrained and free applied forces are given in (13) asPC ¼ RPF . On the other hand,PF ¼ kFCDCandkFC ¼ F�1RT ¼ kFFRT , thereby it is possible to writePC ¼ RkFFRTDC . SincePC ¼ kCCDC ,it is concluded that:

kCC ¼ RkFFRT ¼ RF�1RT (15)

Using (11), (14) and (15), the global stiffness matrix can be explicitly formulated as:

K ¼�kFF kFCkCF kCC

�(16)

3. Element verification

The proposed stiffness matrix in (16) was used to introduce a new tubular joint element. Theelement is planar and has 6 degrees of freedom as shown in Fig. 3(b) on the left. Relevant coding wasperformed using object oriented programming framework and implemented in finite element programOpenSees [15]. Several planar problems are studied in order to validate the proposed element. A simplesingle-braced planar tubular joint model is to be modeled using four different analytical approaches.The conventional rigid connection, center-to-center joint model and a joint model containing thetubular joint element are conducted using uniaxial elements. The general purpose finite elementprogram ANSYS [19] was also employed to model a tubular joint model by using three-dimensionalshell elements which accounts for LJF through local deformation of the chord membrane. Twodifferent loading conditions are to be defined to verify the proposed model namely axial and flexuralloadings. A sensitivity analysis is to be performed subsequently to not only ensure the suitability of theelement, but also to evaluate the effect of chord length on the behavior of tubular joint element.

3.1. Axial and flexural loadings

The general configuration of the tubular joint discussed earlier is similar to the one in Fig. 2 with thechord-brace intersection anglef, which can take four different angles given in Table 1.

For each analytical joint model, six differentmodels weremade for all individual intersection anglesbased on the joint characteristics. The lengths of the brace and chord are 1 and 2 m respectively. Anaxial load was applied at the tip of the brace and corresponding displacement was monitored. Usingloads and displacements, it is possible to obtain the stiffnesses of the joint model. Table 1 gives axialstiffness ratios for the models confirming the superiority of the tubular joint model. The stiffnesses ofthe models have been divided by the stiffness of the model made of shell elements in each row. Thevalues shown in the table address the over-stiffness of the rigid-joint and center-to-centermodel whilethe results of model made of proposed tubular joint element reasonably agree with those of shell-element-based model. An interesting observation is made upon axial loading which is rigid move-ment of the brace perpendicular to the load application direction due to local deformations of chordwall (Fig. 6(a)&c). Regular rigid and center-to-center models are unable to capture this effect, however,tubular joint element can successfully account for this phenomenon through the rotation of theelement as depicted in Fig. 5(b). This can be further investigated through the use of local flexibility

Table 1Axial stiffness ratios of different joint models.

Chord-braceintersection angle(f)

Joint characteristics(b;g respectively)

Conventional rigidjoint model

Center-to-centerjoint model

Proposed tubularjoint element

Shell elementstiffness (KN/m)

30 0.4–15.625 1.92 1.37 0.94 7033290.4–20 2.12 1.51 0.91 4731400.5–15.625 1.82 1.32 0.91 8958590.5–20 1.98 1.43 0.88 6096290.6–15.625 1.67 1.23 0.90 11265400.6–20 1.79 1.31 0.86 777349

45 0.4–15.625 3.02 2.38 0.97 4211280.4–20 3.41 2.68 0.93 2775410.5–15.625 2.86 2.29 0.96 5294590.5–20 3.19 2.54 0.91 3536470.6–15.625 2.62 2.12 0.98 6596630.6–20 2.88 2.32 0.92 446556

60 0.4–15.625 3.99 3.31 0.98 3002690.4–20 4.57 3.78 0.93 1962770.5–15.625 3.71 3.12 0.96 3803560.5–20 4.20 3.51 0.90 2519310.6–15.625 3.36 2.85 0.98 4748550.6–20 3.78 3.20 0.92 316492

90 0.4–15.625 4.80 4.11 0.96 2364090.4–20 5.58 4.76 0.91 1526330.5–15.625 4.42 3.83 0.95 2988300.5–20 5.08 4.38 0.89 1956230.6–15.625 3.97 3.48 0.97 3731680.6–20 4.53 3.94 0.91 246751

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matrices of the tubular joint element. Fig. 6(a) shows a tubular joint assembly comprising chord,tubular joint and axially-loaded brace element. In an analogous fashion to Fig. 2(a), local and globalcoordinate systems have been given. One could decompose applied force ‘P’ into ‘P1’ and ‘P2’ in thedirection of local coordinate system ‘x0 � y0’. Correspondingly, induced deformation ‘d’ could bedecomposed into two perpendicular deformations ‘d1’ and ‘d2’. It was stated earlier that axial stiffnessof the chord is very high and consequently no axial deformation due to joint flexibility is assumed to

Fig. 6. Axially-loaded brace along with chord and tubular joint element; (a) decomposition of deformations in LCS, (b) decompo-sition of deformations in brace CS.

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exist (i.e.d2 ¼ 0). This assumption is resulted from physical interpretation of the chord wall behaviorwhich is found to be of significance. Using local flexibility matrix provided in (8), force-displacementequations can be written for node ‘j’ which is common between the joint and the brace element:

24 LJFv

bLJFIPB

3524 P1P20

35 ¼

24 d1d2q

35 (17)

It yields:8<:

q ¼ 0d2 ¼ bP2; bz0/d2 ¼ 0

d1 ¼ LJFv � P1(18)

On the other hand, brace beam-column element has 6 degrees of freedom and a local coordinatesystem (i.e. ‘xb � yb’) attached to it (Fig. 6(b)). Node ‘j’ is shared by the tubular joint element and thebrace beam-column element on the chord wall, thereby the only existing local deformation ‘d1’ is alsoinduced at node ‘j’ for the beam element. Projecting this deformation in the directions of beam localcoordinate system introduces two local deformations ‘d1b’ and ‘d2b’ as schematically shown in Fig. 6(b).Considering the equilibrium of a planar 6-degree-of-freedom beam element, it would be possible towrite:

d4b ¼ ��d1b þ

PLEA

�; d6b ¼ d3b ¼ 0; d5b ¼ d2b ¼ d1cos4 (19)

In which ‘E’ and ‘A’ are elastic modulus and cross section area of the beam-column element. Theabove equation simply justifies the physical interpretation of rigid body motion of brace due to thechord wall deformations. The proposed element was shown to be capable of incorporating these localdeformations into the structural displacement calculations through the use of a simple assumption inthe local flexibility matrix of the tubular joint element.

In order to verify the ability of the element to model the flexural behavior of an offshore joint, aflexural moment was applied to the structure, at the tip of the brace and corresponding rotation wasmonitored and consequently flexural stiffnesses of the models with different joint characteristics wereobtained (Table 2). The loadings started with 1 m-long brace model and relevant stiffnesses wererecorded. It can be observed in Table 2 that though the model made of tubular joint element dem-onstrates superior agreement with shell model, some discrepancies can be seen when intersectionangle is 30�. This problem stems from two major reasons which are firstly Fessler’s Equations in-adequacy for small chord-brace intersection angles and secondly low contribution of joint element inoverall flexural stiffness of the structure having 1 m-long brace. The former is rarely observed inpractice since most of offshore jackets are constructed using wider brace-to-chord intersection angleswhile the latter can be overcome using shorter brace in analytical models. As a consequence, for othermodels which have 30� intersection angle, brace beam element was modeled shorter (50 cm), toemphasize the role of tubular joint element in overall flexural stiffness of the structures. Moreover it isseen for high amount ofg, results agree much better with those of model made of shell elementsemphasizing the effect of chord diameter on LJF in offshore platforms.

3.2. Effect of chord length on LJF

The chord length was considered to be 2 m for all models in the previous section. However, as thechord length increases, the effect of JF on overall behavior of tubular framed structure decreases.Generally, chord length and diameter are two important factors which dominate the effect of JF onresponse of offshore structures.

In an analogous way to previous section, an axial force along the brace axis is applied to the up endof the brace. Four different values of the chord length are selected to be modeled using shell, rigid andflexibility models. The dimensions of the structure are chord length ¼ 500 mm, chorddiameter ¼ 16 mm, brace length ¼ 1.0 m, brace diameter ¼ 250 mm and brace thickness ¼ 14 mm.

Table 2Flexural stiffness ratios of different joint models.

Chord-braceintersection angle(f)

Joint characteristics(b;g respectively)

Conventionalrigid joint model

Center-to-centerjoint model

Proposed tubularjoint element

Shell elementstiffness (KN.m/RAD)

30 0.4–15.625 1.11 0.74 0.86 638990.4–20 1.25 0.83 0.93 433700.5–15.625 1.14 0.77 0.83 1261620.5–20 1.26 0.84 0.87 862540.6–15.625 1.17 0.78 0.82 2175220.6–20 1.27 0.85 0.85 149403

45 0.4–15.625 1.74 1.03 0.92 812380.4–20 2.00 1.18 0.98 536860.5–15.625 1.90 1.12 0.88 1502940.5–20 2.13 1.26 0.91 1006210.6–15.625 2.01 1.20 0.88 2468060.6–20 2.23 1.33 0.90 166114

60 0.4–15.625 2.05 1.30 0.91 691820.4–20 2.33 1.48 0.99 461580.5–15.625 2.24 1.44 0.90 1270530.5–20 2.52 1.61 0.93 850630.6–15.625 2.37 1.52 0.90 2100450.6–20 2.63 1.69 0.91 140800

90 0.4–15.625 2.27 1.52 0.96 623210.4–20 2.66 1.78 1.02 404390.5–15.625 2.49 1.68 0.90 1142310.5–20 2.83 1.90 0.93 755620.6–15.625 2.64 1.78 0.89 1883710.6–20 2.94 1.99 0.91 126088

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These dimensions are similar to those of examples given by Chen and Hu [18]. Table 3 gives threedifferent stiffness ratios through which it can be concluded that as the chord length increases, jointflexibility effect on axial stiffness of the structure deteriorates. In this table, F.M., S.M. and C.R.M. denoteflexibility, shell and conventional rigid models respectively as explained earlier. In the second columnthe validity of proposed tubular joint element in modeling the real behavior of offshore joints arefurther emphasized while in the third column the ratio of flexibility model to conventional rigid modelare provided. For a very long chord the difference between two models can be neglected while forshorter chord lengths which are frequently encountered in practice, this difference are quite signifi-cant. Furthermore, this table compares the results of proposed model with those of Chen and Hu [18]which proves the fair agreement of the two approaches in formulating the JF in a mathematicalframework.

4. Effect of LJF on behavior of tubular framed structures

In order to test the proposed element in planar offshore frames, two sample jackets which wereformerly tested by Honarvar et al. [20] and Gates et al. [21] were selected for this research. Thesestructures are 2D experimental platforms which were subjected to cyclic static loadings. OpenSeesprogram [15] was selected for modeling and analyses of the structures.

Table 3Effect of chord length on local joint flexibility of different models.

Chord length (m) F:M:=S:M: F:M:=C:R:M: F:M:=C:R:M:(Chen & Hu)[18]

2.4 0.912 0.289 0.2585 0.942 0.657 0.57910 0.982 0.932 0.89650 0.996 0.999 0.999

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4.1. Honarvar et al. [20]

Fig. 7(a) shows a 2D small-scaled offshore jacket frame tested by Honarvar et al. [20]. Progressivelyincreasing cyclic loadingwas applied to theplatform to gain insight intohysteretic behaviorof the frame.

In conventional design of offshore platforms, lateral elastic stiffness of the jacket is of considerablesignificance since it controls the overall drift ratio and the displacements of individual members andappurtenances upon application of laterally-exerted loads such as wind, wave and currents. A falseestimation of elastic lateral stiffness not only results in erroneous displacement calculations of nodesand members, but also local forces in the members are obtained with a certain amount of error. Thelatter directly influences the design calculations of the members and might have considerable impacton overall cost of the platform construction. A robust and sophisticated model is then required toobtain the most accurate results in push-over analysis.

For the purpose of this research initial elastic stiffness of the platform was extracted from cyclicloading response of the structure and compared against elastic lateral stiffness of three analyticalmodels namely conventional rigid model (i.e. model whose joints use rigid offset element to considerLJF), center-to-center model (i.e. model whose braces have been extended to intersect with chordcenterline) and flexibility model (i.e. model which utilizes tubular joint element to consider LJF). Table4 gives the four elastic stiffness values of the platforms and pronounces the superior ability of the F.M.to simulate the real response of the experimental test frame. It is seen that although the platform is notconsiderably tall, JF has affected the platform response to lateral loading. Brace to chord diameter ratioseems to a more important parameter in LJF of the tubular framed structures rather than any otherparameters.

Fig. 7. Sample planar offshore frames; (a) Honarvar et al. frame [20] (units in mm), (b) Gates et al. frame [21] (units in m).

Table 4Comparison between experimental test and analytical platform models.

Platform model Experimental test F.M. C-C.M. C.R.M.

Lateral stiffness (N/mm) 586.3 588.6 605.247 615.17

Fig. 8. Time-history analysis of the structures; (a) comparison between S.M. and F.M., (b) Comparison between S.M., C-C.M. andC.R.M.

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4.2. Gates et al. [21]

Gates et al. [21] conducted an analytical model of a small four-pile production facility platform usingcomputer program DYNAS [22]. Fig. 8(b) illustrates this platform along with its dimensions. Membersizes are also given in Table 5.

Similar to the previous section, three analytical models were created to investigate the effect of LJFon the global behavior of offshore frames. Moreover, a model made of three-dimensional shell ele-ments capable of considering local chord and brace deformations was provided as a reference foruniaxial-element-based models. Piles were not incorporated into the models since the focus of thestudy is more on LJF of the jacket structure rather than flexibilities produced by pile–soil interaction.Furthermore, rigid elements were utilized to represent the rigid deck at the top of the platform.

In addition to push-over analysis, modal analysis was also performed on all analytical models andfundamental periods of vibration of platforms were obtained and presented in Table 6.

As can be seen, conventional rigid model is the stiffest whereas, flexibility model exhibits rathersimilar behavior to the model made of shell elements.

A simple triangular progressively-increasing load pattern is applied on all the platform models inlinear range of response and results are given in Table 6. As can be seen, C-C.M. and C.R.M. can notsimulate the lateral response of the platform accurately comparing to the S.M. due to rigidities in theirconnections. On the other hand, F.M. has successfully predicted the overall elastic stiffness of thestructure.

The internal forces induced within the members at the end of elastic response may be extracted toinvestigate the differences resulted from different kinds of analytical modeling. Table 7 gives the force

Table 5Member sizes of the platform (Gates et al. [21]).

Member name Diameter (cm) Wall thickness (cm)

Chord 192.75 4.75Lower braces 76.20 1.58Upper braces 60.96 1.27Deck braces 91.44 1.92Horizontal braces 45.72 0.9525

Table 6Lateral stiffness values and fundamental periods of vibration of the models.

C.R.M. C-C.M. F.M. S.M.

Fundamental period (s) 0.745 0.774 0.847 0.837Lateral stiffness (KN/m) 92028 85045 71230 73500

Table 7Force and moment ratios for analytical models.

Member Force component at one end C.R.M. C-C.M. F.M. S.M. force component (KN&KN-m)

Lower brace Axial load 1.278 1.176 0.952 �4560.72Moment 1.493 1.131 0.961 95.45422

Lower chord Axial load 1.263 1.165 0.970 �13556.9Moment 0.858 0.866 0.931 7012.688

Horizontal brace Axial load 0.832 1.08 1.030 �2.53809Moment 0.891 1.107 1.052 �54.7461

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and moment ratios (i.e. model forces divided by those of shell model) for all the models as well as shellmodel forces and moments induced within its members at the end of elastic range of response. As canbe seen in this table, F.M. can adequately predict local forces inducedwithin themembers comparing toS.M. whereas, C-C.M. and C.R.M. variably over-predict or under-predict the forces and momentsexisting in the structure. This suggests the greater impact of LJF on the design procedure which ismarkedly dependent upon internal forces within the members in elastic analyses.

Dynamic response of the flexibility model can be verified through time-history analysis inwhich anarbitrary seismic loading is exerted to themodel peak displacements aremonitored to gain insight intostructural dynamic response.1979 El Centro earthquake groundmotionwas selected and applied to themodels and peak horizontal displacements were monitored and presented in Fig. 8. On the right-handside, the excellent agreement between F.M. and S.M. displacements can be seen whereas, Fig. 8(b)pronounces the differences between displacements of C-C.M. and C.R.M. with those of the S.M. evenin linear range of response. It can be concluded that F.M. is able to better predict the dynamic responseof the tubular framed structure upon earthquake loading.

5. Conclusion

A new local joint flexibility element based on flexibility matrix and Fessler [4] empirical equationswas proposed in this paper. The element is derived through the elimination of rigid body modes andthe assumption of negligible axial deformation of the chord resulting from joint flexibility. Based on theverification studies performed on the models in which the proposed element had been used, thefollowing conclusions can be drawn:

� The proposed stiffness matrix derivation is easy and straightforward comparing to other existingtubular joint elements and can be incorporated in any FE program which makes use of typicaluniaxial beam-column element for modeling of tubular framed structures.

� The proposed element can be used formodeling of tubular framed structures convenientlywithoutthe prior information about the joint behavior. In order to incorporate the element into anyanalytical model, brace and chord geometric properties at their intersection should be introducedto the model. In other words, the element is capable of incorporating the chord wall deformationsinto global stiffness matrix of the structure using simple geometric characteristics of the relevantconnecting tubular members.

� Brace to chord diameter ratio and chord length seem to be themost important factors in the extentof LJF effect on offshore platforms.

� Conventionally in structural analysis of tubular framed structures, brace and chord are connectedto each other using fixed connections. These models have shown to exhibit unreliable predictions

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even in linear range of response. Center-to-center models have relatively resolved their in-efficiencies however, flexibility model which contains proposed tubular joint element, showsbetter agreement with the result of the analyses of sophisticated 3-dimensional models of tubularjoints. Special care should be takenwhen designing a tubular framed structure using conventionallinear limit-state design procedure. Internal forces predicted by flexibility model have shown to bemore reliable for design purposes.

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