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Local joint flexibility element for offshore plateforms structures
Marine Structures 33 (2013) 56–70
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Marine Structures journal homepage: www.elsevier.com/locate/ marstruc
Local joint flexibility element for offshore plateforms structures A. Akbar Golafshani a, Mehdi Kia b, *, Pejman Alanjari c a b c
Sharif University of Technology, Department of Civil and Environmental Engineering, Tehran, Iran Structural Engineering, Sharif University of Technology, Tehran, Iran Structural Engineering, K.N. Toosi University of Technology, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 December 2011 Received in revised form 13 April 2013 Accepted 13 April 2013
A large number of offshore platforms of various types have been installed in deep or shallow waters throughout the world. These structures are mainly made of tubular members which are interconnected by using tubular joints. In tubular frames, joints may exhibit considerable flexibility in both elastic and plastic range of response. The resulting flexibility may have marked effects on the overall behavior of offshore platforms. This paper investigates the effects of joint flexibility on local and global behavior of tubular framed structures in linear range of response. A new joint flexibility element is developed on the basis of flexibility matrix and implemented in a finite-element program to account for local joint flexibility effects in analytical models of tubular framed structures. The element formulation is considerably easy and straightforward in comparison with other existing tubular joint elements. It was concluded that developed flexible joint model produces accurate results comparing to sophisticated multi-axial finite element joint models. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Joint flexibility Chord Brace Tubular framed structure Flexibility matrix
1. Introduction In the past years, considerable investigative efforts have focused on understanding the real behavior of offshore structures subjected to severe seismic loadings. Moreover, there is an increasing demand for reappraisal of existing installations of steel jacket structures. This perhaps could be due to the revised
* Corresponding author. Tel.: þ98 9111186949. E-mail addresses: [email protected] (A.A. Golafshani), [email protected], [email protected] (M. Kia), Pejman_ [email protected] (P. Alanjari). 0951-8339/$ – see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.marstruc.2013.04.003
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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 and reliable analytical simulations of such structures. The use of tubular connections in circular hollow sections is not confined to offshore platforms and many structures employ these types of joints for interconnecting 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 offshore installations dates back to 1980 when Boukamp et al. [1] conducted a research on analytical techniques used to develop the joint flexibility (JF) model and to find some procedures to incorporate these effects into overall structural response. They provided joint analysis models which were assembled from component 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 element analysis. They concluded that their models are capable of accurately representing the nonlinear behavior of actual joints. Fessler and Spooner [3] and Fessler et al. [4] presented improved equations for flexibility coefficients in terms of joint parameters, in Y, X and gap-K joints. Buitrago et al. [5] also obtained new equations for flexibility coefficients and gave explicit formulas to determine the local joint flexibilities for various joint types and geometries. Karamanos et al. [6] investigated the fatigue design of K-joint tubular girders. Skallerud et al. [7] performed experimental investigations on cyclic in elastic 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. Mirtaheri et al. [9] investigated the comparative response of two analytical platform models and concluded that LJF is of great significance in both elastic and plastic range of response. Static loading performance of tubular joint in multi-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 on solving the equations for elastic shell. Hellan [13] presents (among many topics) the different models for joint flexibility and how the flexibility impacts the ultimate strength. Alanjari et al. [14] performed an experimental research on a small-scaled 2D platform and developed an analytical model which made use of uniaxial fiber elements to model a fracturing tubular joint. However, their model lacked joint 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 of empirical Fessler [4] equations. Stiffness matrix derivation is discussed in detail through the use of equilibrium of the uni-axial element without rigid body modes in the vicinity of chord and brace intersection. The proposed element is subsequently implemented in nonlinear finite element (FE) program OpenSees [15] and verified using more general multi-purpose FE programs which make use of multi-axial elements such as shell or solid elements. Finally, improved tubular framed structure models are made and compared 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 the ultimate 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 frequencies and 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 or space frame having tubular members rigidly interconnected to each other at nodal points. There exist some other techniques which can take into account LJF in a reasonable fashion such as the so-called effective 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, an offset 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 for local deformations and rotations. Mostly, the effective length method is treated in two separate approaches. Sometimes the offset element is assumed to be relatively rigid to account for a rigid tubular connection. This approach is capable of considering offset moments which typically exist in the joints. The second method assigns an element similar to brace element to the offset distance to account for not only the offset moments but also the local flexibilities induced at the joint. In other words, the brace is extended to intersect with chord central axis. The latter is called center-to-center model hereafter. However, it is difficult to select an appropriate effective length to represent the exact value of the LJF. Moreover, beam-column elements are mainly used in practice for representing the offset joint element which is not capable of modeling the exact multi-axial responses of the joints such as progressive ovalizing and local buckling which are rather shell-like behaviors (Fig. 1). Using three-dimensional shell and solid elements is another alternative to accurately simulate the behavior of tubular joints; however, large computational effort which is made for these models makes it 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 in some computer programs which contains joint information and characteristics based on empirical relationships [16] or solving the fundamental equations for shells [17]. The formulation and stiffness matrices of these elements differ from each other based on the computer program which is employed. For instance, Chen and Hu [18] proposed an LJF element based on equilibrium of external and internal forces and compatibility of displacements at element end nodes. This section presents general formulation of the proposed JF element based on flexibility matrix which is developed and formulated with the aid of empirical Fessler [4] equations. These equations are based on the fact that upon loading on a tubular joint (axial or flexural), chord wall would deform locally in addition to overall consistent joint deformation which exists in a typical fixed connection. The proposed 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 directions 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 4 ; LJFOPB ¼ fOPB ¼ o MI MO
Fig. 1. Conventional analytical modeling of plane frame offshore jacket using rigid or center-to-center joints.
(1)
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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 axial forces and bending moments. Fessler et al. [4] conducted experimental research on test specimens and proposed parametric equations by which it would be possible to evaluate local flexibility of certain type of offshore tubular joints. For local flexibilities (LFs) provided in (1), the parametric equations are as follows[4]: 1:3
LJFAX ¼
LJFOPB
1:95g2:15 ð1 bÞ ED
sin2:19 4
(2)
85:5g2:2 exp 3:85b sin2:16 4 ¼ ED3
(3)
134g1:73 exp 4:52b sin1:22 4 ED3
(4)
LJFIPB ¼
In these equations, b ¼ d=D and g ¼ D=2T are typical joint characteristics.‘d’, ‘D’, ‘T’, ‘E’and ‘4’are brace and chord diameters, chord thickness, elastic modulus of the joint members and chord-brace intersection angle, respectively. In most general case for a 3-dimensioanl element having 2 nodes, 9 LFs may be defined corresponding to 3 main flexibilities defined in Eq. (1). These LFs are shown in Eq. (2):
2
f11 F ¼ 4 f21 f31
f12 f22 f32
3 f13 f23 5 f33
(5)
According to reciprocal theorem, LJF matrix defined in Eq. (5) is symmetric. Since out-of-plane bending 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 is caused by a unit axial force acting on the brace. Since this value seems to be very small, it is assumed to be zero in the flexibility matrix. As a consequence the matrix in (5) reduces to:
2
f11 F ¼ 4 0 0
0 f22 0
3 0 0 5
(6)
f33
In the case of planar problems, this matrix 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 elements namely brace, joint and chord element as schematically shown in Fig. 3. The reference coordinate system (CS) for the entire elements is the LCS ‘x0 ’, ‘y0 ’, while ‘x’, ‘y’ denotes the GCS. The angle ‘q’ between local and global CS and chord inclination angle are complementary. The tubular joint element connects 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 in tubular framed structure joints given the fact that mutli-brace connections are not necessarily coincident. The element degrees of freedom are illustrated in Fig. 2 on the right. As can be seen the element has 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. It was discussed earlier that flexibility matrix of the joint element provides axial and in-plane bending degrees of freedom in planar problems. However, in order for displacements of adjacent nodes in an analytical model to be compatible, 3 degrees of freedom are defined for the tubular joint element at end nodes. The resulting element has 6 degrees of freedom including rigid body modes. These degrees of freedom are termed as initial degrees of freedom in this study. Due to the presence of these rigid body motions in local and global coordinate system, the corresponding element stiffness matrix would be singular. As a consequence, in general there would be no flexibility matrix associated with this
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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 basic degrees of freedom.
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system. Thereby, the element is formulated in another system in which rigid body modes are excluded (Fig. 3(b)). Rigid body modes can be incorporated with a simple geometric transformation. The resulting element has 3 degrees of freedom: two perpendicular translations and one rotation relative to the element longitudinal axis. These degrees of freedom are termed as basic degrees of freedom hereafter. Flexibility matrix is derived for this simple 2D-CS and extended to more general CS which is shown in Fig. 3(b) on the left. The former can be achieved explicitly using the so-called method of unit load application and computing the corresponding deformations. In other words, flexibility matrix of a structure represents the displacements developed in the structure when a unit force corresponding to one degree of freedom is introduced while no other nodal forces are applied. Applying a unit load horizontally on free end (i.e. degree of freedom #4), one can evaluate corresponding nodal displacements in all 3 directions. In this case, using Eq. (1), the displacements are ‘LJFv ’, 0, 0 in three directions shown 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
LJFAX
(7)
sin2 f
The evaluation can be performed for other degrees of freedom and local flexibility matrix is obtained for basic degrees of freedom as follows:
2
3 LJFv 4 b 5 LJFIPB
(8)
The entry in row 2 and column 2 is set to a small number which is denoted by ‘b’ in (8). It could also be seen that this matrix is diagonal, that is, applying a unit force in one direction does not induce any displacement in the directions of other degrees of freedom. This evaluation is resulted from physical interpretation of joint behavior in the vicinity of chord-brace intersection. In fact axial stiffness of the chord is too high to be altered by joint deformations and that is the primary reason why 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 assumed for the legs of the platform). The transformation task could be conveniently carried out using simple rotation matrix in which ‘q’ is the rotation angle. Rotating local flexibility matrix yields the global one which is denoted by ‘F’ in the following Eq.:
F ¼ T T F0T
(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 conventional structural analysis:
kFF DF þ kFC DC ¼ PF kCF DF þ kCC DC ¼ PC
(10)
The above equations relate applied forces in free and constrained degrees of freedom to corresponding displacements. These equations show how four stiffness sub-matrices constitute the global stiffness matrix (indices ‘F’ and ‘C’ stands for free and constrained). Consequently, the problem reduces to evaluating four constituents of the global stiffness matrix. The first sub-matrix could be conveniently obtained 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 initial degrees of freedom. Fig. 5 illustrates the tubular joint element with three degrees of freedom which had been fixed earlier (Fig. 3(b)). A transformation matrix can be used to connect basic and initial degrees of freedom again with the aid of unit load application. As can be observed, three unit loads have been applied at one end while the corresponding reactions have been evaluated using simple static equilibrium. These reactions yield transformation matrix ‘R’ as follows:
2
1 R ¼ 4 0 Lsina
0 1 Lcosa
3 0 0 5 1
Fig. 5. Rigid body movement of brace upon pure axial loading.
(12)
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In (12) and Fig. 5, ‘a’ denotes the angle between the direction of tubular joint element and the global horizontal axis ‘x’. Having this transformation matrix, it would be possible to obtain:
PC ¼ RPF /PC ¼ RkFF DF ¼ RF 1 DF PF ¼ kFF DF
(13)
It is clear that PC ¼ kCF DF , 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 between constrained and free applied forces are given in (13) asPC ¼ RPF . On the other hand, PF ¼ kFC DC andkFC ¼ F 1 RT ¼ kFF RT , thereby it is possible to writePC ¼ RkFF RT DC . SincePC ¼ kCC DC , it is concluded that:
kCC ¼ RkFF RT ¼ RF 1 RT
(15)
Using (11), (14) and (15), the global stiffness matrix can be explicitly formulated as:
K ¼
kFF kCF
kFC kCC
(16)
3. Element verification The proposed stiffness matrix in (16) was used to introduce a new tubular joint element. The element is planar and has 6 degrees of freedom as shown in Fig. 3(b) on the left. Relevant coding was performed using object oriented programming framework and implemented in finite element program OpenSees [15]. Several planar problems are studied in order to validate the proposed element. A simple single-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 the tubular joint element are conducted using uniaxial elements. The general purpose finite element program ANSYS [19] was also employed to model a tubular joint model by using three-dimensional shell elements which accounts for LJF through local deformation of the chord membrane. Two different loading conditions are to be defined to verify the proposed model namely axial and flexural loadings. A sensitivity analysis is to be performed subsequently to not only ensure the suitability of the element, 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 the chord-brace intersection anglef, which can take four different angles given in Table 1. For each analytical joint model, six different models were made for all individual intersection angles based on the joint characteristics. The lengths of the brace and chord are 1 and 2 m respectively. An axial load was applied at the tip of the brace and corresponding displacement was monitored. Using loads and displacements, it is possible to obtain the stiffnesses of the joint model. Table 1 gives axial stiffness ratios for the models confirming the superiority of the tubular joint model. The stiffnesses of the models have been divided by the stiffness of the model made of shell elements in each row. The values shown in the table address the over-stiffness of the rigid-joint and center-to-center model while the results of model made of proposed tubular joint element reasonably agree with those of shellelement-based model. An interesting observation is made upon axial loading which is rigid movement of the brace perpendicular to the load application direction due to local deformations of chord wall (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 the element as depicted in Fig. 5(b). This can be further investigated through the use of local flexibility
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Table 1 Axial stiffness ratios of different joint models. Chord-brace intersection angle (f)
Joint characteristics (b; g respectively)
Conventional rigid Center-to-center Proposed tubular joint model joint model joint element
Shell element stiffness (KN/m)
30
0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20
1.92 2.12 1.82 1.98 1.67 1.79 3.02 3.41 2.86 3.19 2.62 2.88 3.99 4.57 3.71 4.20 3.36 3.78 4.80 5.58 4.42 5.08 3.97 4.53
703329 473140 895859 609629 1126540 777349 421128 277541 529459 353647 659663 446556 300269 196277 380356 251931 474855 316492 236409 152633 298830 195623 373168 246751
45
60
90
1.37 1.51 1.32 1.43 1.23 1.31 2.38 2.68 2.29 2.54 2.12 2.32 3.31 3.78 3.12 3.51 2.85 3.20 4.11 4.76 3.83 4.38 3.48 3.94
0.94 0.91 0.91 0.88 0.90 0.86 0.97 0.93 0.96 0.91 0.98 0.92 0.98 0.93 0.96 0.90 0.98 0.92 0.96 0.91 0.95 0.89 0.97 0.91
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 global coordinate systems have been given. One could decompose applied force ‘P’ into ‘P1 ’ and ‘P2 ’ in the direction of local coordinate system ‘x0 y0 ’. Correspondingly, induced deformation ‘d’ could be decomposed into two perpendicular deformations ‘d1 ’ and ‘d2 ’. It was stated earlier that axial stiffness of 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) decomposition 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 behavior which is found to be of significance. Using local flexibility matrix provided in (8), force-displacement equations can be written for node ‘j’ which is common between the joint and the brace element:
2 3 32 3 LJFv P1 d1 4 b 54 P2 5 ¼ 4 d2 5 0 LJFIPB q 2
(17)
It yields:
8 <
q¼0 d2 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 coordinate system (i.e. ‘xb yb ’) attached to it (Fig. 6(b)). Node ‘j’ is shared by the tubular joint element and the brace beam-column element on the chord wall, thereby the only existing local deformation ‘d1 ’ is also induced at node ‘j’ for the beam element. Projecting this deformation in the directions of beam local coordinate 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 to write:
d4b ¼ d1b þ
PL ; d6b ¼ d3b ¼ 0; d5b ¼ d2b ¼ d1 cos4 EA
(19)
In which ‘E’ and ‘A’ are elastic modulus and cross section area of the beam-column element. The above equation simply justifies the physical interpretation of rigid body motion of brace due to the chord wall deformations. The proposed element was shown to be capable of incorporating these local deformations into the structural displacement calculations through the use of a simple assumption in the 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, a flexural moment was applied to the structure, at the tip of the brace and corresponding rotation was monitored and consequently flexural stiffnesses of the models with different joint characteristics were obtained (Table 2). The loadings started with 1 m-long brace model and relevant stiffnesses were recorded. It can be observed in Table 2 that though the model made of tubular joint element demonstrates superior agreement with shell model, some discrepancies can be seen when intersection angle is 30 . This problem stems from two major reasons which are firstly Fessler’s Equations inadequacy for small chord-brace intersection angles and secondly low contribution of joint element in overall flexural stiffness of the structure having 1 m-long brace. The former is rarely observed in practice since most of offshore jackets are constructed using wider brace-to-chord intersection angles while the latter can be overcome using shorter brace in analytical models. As a consequence, for other models which have 30 intersection angle, brace beam element was modeled shorter (50 cm), to emphasize the role of tubular joint element in overall flexural stiffness of the structures. Moreover it is seen for high amount ofg, results agree much better with those of model made of shell elements emphasizing 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 the chord 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 on response of offshore structures. In an analogous way to previous section, an axial force along the brace axis is applied to the up end of the brace. Four different values of the chord length are selected to be modeled using shell, rigid and flexibility models. The dimensions of the structure are chord length ¼ 500 mm, chord diameter ¼ 16 mm, brace length ¼ 1.0 m, brace diameter ¼ 250 mm and brace thickness ¼ 14 mm.
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Table 2 Flexural stiffness ratios of different joint models. Chord-brace intersection angle (f)
Joint characteristics (b; g respectively)
Conventional rigid joint model
Center-to-center Proposed tubular joint model joint element
Shell element stiffness (KN.m/RAD)
30
0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20
1.11 1.25 1.14 1.26 1.17 1.27 1.74 2.00 1.90 2.13 2.01 2.23 2.05 2.33 2.24 2.52 2.37 2.63 2.27 2.66 2.49 2.83 2.64 2.94
0.74 0.83 0.77 0.84 0.78 0.85 1.03 1.18 1.12 1.26 1.20 1.33 1.30 1.48 1.44 1.61 1.52 1.69 1.52 1.78 1.68 1.90 1.78 1.99
63899 43370 126162 86254 217522 149403 81238 53686 150294 100621 246806 166114 69182 46158 127053 85063 210045 140800 62321 40439 114231 75562 188371 126088
45
60
90
0.86 0.93 0.83 0.87 0.82 0.85 0.92 0.98 0.88 0.91 0.88 0.90 0.91 0.99 0.90 0.93 0.90 0.91 0.96 1.02 0.90 0.93 0.89 0.91
These dimensions are similar to those of examples given by Chen and Hu [18]. Table 3 gives three different stiffness ratios through which it can be concluded that as the chord length increases, joint flexibility effect on axial stiffness of the structure deteriorates. In this table, F.M., S.M. and C.R.M. denote flexibility, shell and conventional rigid models respectively as explained earlier. In the second column the validity of proposed tubular joint element in modeling the real behavior of offshore joints are further emphasized while in the third column the ratio of flexibility model to conventional rigid model are provided. For a very long chord the difference between two models can be neglected while for shorter chord lengths which are frequently encountered in practice, this difference are quite significant. 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 mathematical framework. 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 were formerly tested by Honarvar et al. [20] and Gates et al. [21] were selected for this research. These structures are 2D experimental platforms which were subjected to cyclic static loadings. OpenSees program [15] was selected for modeling and analyses of the structures.
Table 3 Effect 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 5 10 50
0.912 0.942 0.982 0.996
0.289 0.657 0.932 0.999
0.258 0.579 0.896 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]. Progressively increasing cyclic loading was applied to the platform to gain insight into hysteretic behavior of the frame. In conventional design of offshore platforms, lateral elastic stiffness of the jacket is of considerable significance since it controls the overall drift ratio and the displacements of individual members and appurtenances upon application of laterally-exerted loads such as wind, wave and currents. A false estimation of elastic lateral stiffness not only results in erroneous displacement calculations of nodes and members, but also local forces in the members are obtained with a certain amount of error. The latter directly influences the design calculations of the members and might have considerable impact on overall cost of the platform construction. A robust and sophisticated model is then required to obtain the most accurate results in push-over analysis. For the purpose of this research initial elastic stiffness of the platform was extracted from cyclic loading response of the structure and compared against elastic lateral stiffness of three analytical models namely conventional rigid model (i.e. model whose joints use rigid offset element to consider LJF), center-to-center model (i.e. model whose braces have been extended to intersect with chord centerline) and flexibility model (i.e. model which utilizes tubular joint element to consider LJF). Table 4 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 not considerably tall, JF has affected the platform response to lateral loading. Brace to chord diameter ratio seems to a more important parameter in LJF of the tubular framed structures rather than any other parameters.
Fig. 7. Sample planar offshore frames; (a) Honarvar et al. frame [20] (units in mm), (b) Gates et al. frame [21] (units in m).
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Table 4 Comparison 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. and C.R.M.
4.2. Gates et al. [21] Gates et al. [21] conducted an analytical model of a small four-pile production facility platform using computer program DYNAS [22]. Fig. 8(b) illustrates this platform along with its dimensions. Member sizes are also given in Table 5. Similar to the previous section, three analytical models were created to investigate the effect of LJF on the global behavior of offshore frames. Moreover, a model made of three-dimensional shell elements capable of considering local chord and brace deformations was provided as a reference for uniaxial-element-based models. Piles were not incorporated into the models since the focus of the study 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 and fundamental 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 rather similar behavior to the model made of shell elements. A simple triangular progressively-increasing load pattern is applied on all the platform models in linear range of response and results are given in Table 6. As can be seen, C-C.M. and C.R.M. can not simulate the lateral response of the platform accurately comparing to the S.M. due to rigidities in their connections. On the other hand, F.M. has successfully predicted the overall elastic stiffness of the structure. The internal forces induced within the members at the end of elastic response may be extracted to investigate the differences resulted from different kinds of analytical modeling. Table 7 gives the force Table 5 Member sizes of the platform (Gates et al. [21]). Member name
Diameter (cm)
Wall thickness (cm)
Chord Lower braces Upper braces Deck braces Horizontal braces
192.75 76.20 60.96 91.44 45.72
4.75 1.58 1.27 1.92 0.9525
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Table 6 Lateral stiffness values and fundamental periods of vibration of the models.
Fundamental period (s) Lateral stiffness (KN/m)
C.R.M.
C-C.M.
F.M.
S.M.
0.745 92028
0.774 85045
0.847 71230
0.837 73500
Table 7 Force 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 Moment Axial load Moment Axial load Moment
1.278 1.493 1.263 0.858 0.832 0.891
1.176 1.131 1.165 0.866 1.08 1.107
0.952 0.961 0.970 0.931 1.030 1.052
4560.72 95.45422 13556.9 7012.688 2.53809 54.7461
Lower chord Horizontal brace
and moment ratios (i.e. model forces divided by those of shell model) for all the models as well as shell model forces and moments induced within its members at the end of elastic range of response. As can be seen in this table, F.M. can adequately predict local forces induced within the members comparing to S.M. whereas, C-C.M. and C.R.M. variably over-predict or under-predict the forces and moments existing in the structure. This suggests the greater impact of LJF on the design procedure which is markedly dependent upon internal forces within the members in elastic analyses. Dynamic response of the flexibility model can be verified through time-history analysis in which an arbitrary seismic loading is exerted to the model peak displacements are monitored to gain insight into structural dynamic response. 1979 El Centro earthquake ground motion was selected and applied to the models and peak horizontal displacements were monitored and presented in Fig. 8. On the right-hand side, 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. even in linear range of response. It can be concluded that F.M. is able to better predict the dynamic response of the tubular framed structure upon earthquake loading. 5. Conclusion A new local joint flexibility element based on flexibility matrix and Fessler [4] empirical equations was proposed in this paper. The element is derived through the elimination of rigid body modes and the assumption of negligible axial deformation of the chord resulting from joint flexibility. Based on the verification studies performed on the models in which the proposed element had been used, the following conclusions can be drawn: The proposed stiffness matrix derivation is easy and straightforward comparing to other existing tubular joint elements and can be incorporated in any FE program which makes use of typical uniaxial beam-column element for modeling of tubular framed structures. The proposed element can be used for modeling of tubular framed structures conveniently without the prior information about the joint behavior. In order to incorporate the element into any analytical model, brace and chord geometric properties at their intersection should be introduced to the model. In other words, the element is capable of incorporating the chord wall deformations into global stiffness matrix of the structure using simple geometric characteristics of the relevant connecting tubular members. Brace to chord diameter ratio and chord length seem to be the most important factors in the extent of LJF effect on offshore platforms. Conventionally in structural analysis of tubular framed structures, brace and chord are connected to 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 inefficiencies however, flexibility model which contains proposed tubular joint element, shows better agreement with the result of the analyses of sophisticated 3-dimensional models of tubular joints. Special care should be taken when designing a tubular framed structure using conventional linear limit-state design procedure. Internal forces predicted by flexibility model have shown to be more reliable for design purposes. References [1] Bouwkamp JG, Hollings JP, Maison BF, Row DG. Effects of joint flexibility on the response of offshore towers. In: Proc. OTC, paper 3901, 1980. [2] Ueda Y, Rashed SMH, Nakacho K. “An improved joint model and equations for flexibility of tubular joints. J Offshore Mech Arct Eng 1990;vol. 112:157–68. [3] Fessler H, Spooner H. Faulkner D, Cowling M, Frieze P, editors. Experimental determination of stiffness of tubular joints. London: Appl. science publishers; 1981. [4] Fessler H, Mockford PB, Webster JJ. Parametric equations for the flexibility matrix of single brace tubular joint in offshore structures. Proc Inst Civil Engs 1986. Part 2. [5] Buitrago J, Healy BE, Chang TY. Local joint flexibility of tubular joints. Glasgow: Offshore Mech. Arct. Eng. Conf., OMAE; 1993. [6] Karamanos SA, Romeijn A, Wardenier J. On the fatigue design of K-Joint tubular girders. Int J Offshore Polar Eng 2000; 10(1). [7] Skallerud BH, Eide O, Amdahl J, Johansen A. On the cyclic capacity of tubular joints subjected to extreme cyclic loads. 14th Int. Conf. on offshore Mech. And Arctic Eng., OMAE’ 95. Copenhagen: June 19–22 1995. [8] Dier A. Tubular joint technology for offshore structures. J Steel Struct 2005. [9] Mirtaheri M, Zakeri H, Alanjari P, Assareh M. Effect of joint flexibility on overall behavior of jacket type offshore platforms. Am J Eng Appl Sci 2009;2(1):25–30. [10] Lee J, Hino S, Toshiaki O, Seo S. Static loading performance of tubular joint in multi-column composite bridge piers, 62. Memoirs of the faculty of eng., Kyushu Univ; 2002. No. 3. [11] Lee M, Parry A. Strength prediction for ring-stiffened DT-joints in offshore jacket structures. J Eng Struct 2004. http://dx. doi.org/10.1016/j.engstruct.2004.11.004. [12] Holmås T. Implementation of tubular joint flexibility in global frame analysis. Dr. ing thesis. Trondheim Norway: Norwegian Institute of Technology; 1987. [13] Hellan Ø. Nonlinear pushover and cyclic analysis in ultimate limit state design and reassessment of tubular steel offshore structures. Dr. ing thesis. Trondheim Norway: Norwegian Institute of Technology; 1995. [14] Alanjari P, Asgarian B, Honarvar MR, Bahari MR. On the energy dissipation of jacket type offshore platforms with different pile_leg interactions. J Appl Ocean Res 2009. http://dx.doi.org/10.1016/j.apor.2009.07.002. [15] Mazzoni S, McKenna F, Scott M, Fenves G. OpenSees command language manual 2006. [16] Release 5SACS User’s manual, collapse analysis 2001. [17] Skallerud B. USFOS shell element theory manual. Sintef Group; 1998-03-30. [18] Chen B, Hu Y. Fatigue in offshore structures: local flexibility of tubular joints of offshore platforms. In: Dover WD, Madhava R, editorsVolume 1. Rotterdam/Brookfield: A.A. Balkema; 1996. [19] ANSYS. User’s manual, version 10. (PA, 153421300): ANSYS Inc.; Houston: 2005. [20] Honarvar MR, Bahari MR, Asgarian B, Alanjari P. Cyclic inelastic behavior and analytical modeling of pile_leg interaction in jacket type offshore platform. J Appl Ocean Res 2008. http://dx.doi.org/10.1016/j.apor.2008.02.001. [21] Gates W, Marshall W, Mahin S. Analytical methods for determining the ultimate earthquake resistance of fixed offshore structures 1979. OTC paper (2751). [22] Kanaan AE, Powell GH. General purpose computer program for the inelastic dynamic response of plane structures. Univ. California; 1975. Berkeley Report no. EERC 75–5.
Contents lists available at SciVerse ScienceDirect
Marine Structures journal homepage: www.elsevier.com/locate/ marstruc
Local joint flexibility element for offshore plateforms structures A. Akbar Golafshani a, Mehdi Kia b, *, Pejman Alanjari c a b c
Sharif University of Technology, Department of Civil and Environmental Engineering, Tehran, Iran Structural Engineering, Sharif University of Technology, Tehran, Iran Structural Engineering, K.N. Toosi University of Technology, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 December 2011 Received in revised form 13 April 2013 Accepted 13 April 2013
A large number of offshore platforms of various types have been installed in deep or shallow waters throughout the world. These structures are mainly made of tubular members which are interconnected by using tubular joints. In tubular frames, joints may exhibit considerable flexibility in both elastic and plastic range of response. The resulting flexibility may have marked effects on the overall behavior of offshore platforms. This paper investigates the effects of joint flexibility on local and global behavior of tubular framed structures in linear range of response. A new joint flexibility element is developed on the basis of flexibility matrix and implemented in a finite-element program to account for local joint flexibility effects in analytical models of tubular framed structures. The element formulation is considerably easy and straightforward in comparison with other existing tubular joint elements. It was concluded that developed flexible joint model produces accurate results comparing to sophisticated multi-axial finite element joint models. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Joint flexibility Chord Brace Tubular framed structure Flexibility matrix
1. Introduction In the past years, considerable investigative efforts have focused on understanding the real behavior of offshore structures subjected to severe seismic loadings. Moreover, there is an increasing demand for reappraisal of existing installations of steel jacket structures. This perhaps could be due to the revised
* Corresponding author. Tel.: þ98 9111186949. E-mail addresses: [email protected] (A.A. Golafshani), [email protected], [email protected] (M. Kia), Pejman_ [email protected] (P. Alanjari). 0951-8339/$ – see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.marstruc.2013.04.003
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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 and reliable analytical simulations of such structures. The use of tubular connections in circular hollow sections is not confined to offshore platforms and many structures employ these types of joints for interconnecting 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 offshore installations dates back to 1980 when Boukamp et al. [1] conducted a research on analytical techniques used to develop the joint flexibility (JF) model and to find some procedures to incorporate these effects into overall structural response. They provided joint analysis models which were assembled from component 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 element analysis. They concluded that their models are capable of accurately representing the nonlinear behavior of actual joints. Fessler and Spooner [3] and Fessler et al. [4] presented improved equations for flexibility coefficients in terms of joint parameters, in Y, X and gap-K joints. Buitrago et al. [5] also obtained new equations for flexibility coefficients and gave explicit formulas to determine the local joint flexibilities for various joint types and geometries. Karamanos et al. [6] investigated the fatigue design of K-joint tubular girders. Skallerud et al. [7] performed experimental investigations on cyclic in elastic 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. Mirtaheri et al. [9] investigated the comparative response of two analytical platform models and concluded that LJF is of great significance in both elastic and plastic range of response. Static loading performance of tubular joint in multi-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 on solving the equations for elastic shell. Hellan [13] presents (among many topics) the different models for joint flexibility and how the flexibility impacts the ultimate strength. Alanjari et al. [14] performed an experimental research on a small-scaled 2D platform and developed an analytical model which made use of uniaxial fiber elements to model a fracturing tubular joint. However, their model lacked joint 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 of empirical Fessler [4] equations. Stiffness matrix derivation is discussed in detail through the use of equilibrium of the uni-axial element without rigid body modes in the vicinity of chord and brace intersection. The proposed element is subsequently implemented in nonlinear finite element (FE) program OpenSees [15] and verified using more general multi-purpose FE programs which make use of multi-axial elements such as shell or solid elements. Finally, improved tubular framed structure models are made and compared 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 the ultimate 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 frequencies and 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 or space frame having tubular members rigidly interconnected to each other at nodal points. There exist some other techniques which can take into account LJF in a reasonable fashion such as the so-called effective 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, an offset 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 for local deformations and rotations. Mostly, the effective length method is treated in two separate approaches. Sometimes the offset element is assumed to be relatively rigid to account for a rigid tubular connection. This approach is capable of considering offset moments which typically exist in the joints. The second method assigns an element similar to brace element to the offset distance to account for not only the offset moments but also the local flexibilities induced at the joint. In other words, the brace is extended to intersect with chord central axis. The latter is called center-to-center model hereafter. However, it is difficult to select an appropriate effective length to represent the exact value of the LJF. Moreover, beam-column elements are mainly used in practice for representing the offset joint element which is not capable of modeling the exact multi-axial responses of the joints such as progressive ovalizing and local buckling which are rather shell-like behaviors (Fig. 1). Using three-dimensional shell and solid elements is another alternative to accurately simulate the behavior of tubular joints; however, large computational effort which is made for these models makes it 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 in some computer programs which contains joint information and characteristics based on empirical relationships [16] or solving the fundamental equations for shells [17]. The formulation and stiffness matrices of these elements differ from each other based on the computer program which is employed. For instance, Chen and Hu [18] proposed an LJF element based on equilibrium of external and internal forces and compatibility of displacements at element end nodes. This section presents general formulation of the proposed JF element based on flexibility matrix which is developed and formulated with the aid of empirical Fessler [4] equations. These equations are based on the fact that upon loading on a tubular joint (axial or flexural), chord wall would deform locally in addition to overall consistent joint deformation which exists in a typical fixed connection. The proposed 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 directions 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 4 ; LJFOPB ¼ fOPB ¼ o MI MO
Fig. 1. Conventional analytical modeling of plane frame offshore jacket using rigid or center-to-center joints.
(1)
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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 axial forces and bending moments. Fessler et al. [4] conducted experimental research on test specimens and proposed parametric equations by which it would be possible to evaluate local flexibility of certain type of offshore tubular joints. For local flexibilities (LFs) provided in (1), the parametric equations are as follows[4]: 1:3
LJFAX ¼
LJFOPB
1:95g2:15 ð1 bÞ ED
sin2:19 4
(2)
85:5g2:2 exp 3:85b sin2:16 4 ¼ ED3
(3)
134g1:73 exp 4:52b sin1:22 4 ED3
(4)
LJFIPB ¼
In these equations, b ¼ d=D and g ¼ D=2T are typical joint characteristics.‘d’, ‘D’, ‘T’, ‘E’and ‘4’are brace and chord diameters, chord thickness, elastic modulus of the joint members and chord-brace intersection angle, respectively. In most general case for a 3-dimensioanl element having 2 nodes, 9 LFs may be defined corresponding to 3 main flexibilities defined in Eq. (1). These LFs are shown in Eq. (2):
2
f11 F ¼ 4 f21 f31
f12 f22 f32
3 f13 f23 5 f33
(5)
According to reciprocal theorem, LJF matrix defined in Eq. (5) is symmetric. Since out-of-plane bending 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 is caused by a unit axial force acting on the brace. Since this value seems to be very small, it is assumed to be zero in the flexibility matrix. As a consequence the matrix in (5) reduces to:
2
f11 F ¼ 4 0 0
0 f22 0
3 0 0 5
(6)
f33
In the case of planar problems, this matrix 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 elements namely brace, joint and chord element as schematically shown in Fig. 3. The reference coordinate system (CS) for the entire elements is the LCS ‘x0 ’, ‘y0 ’, while ‘x’, ‘y’ denotes the GCS. The angle ‘q’ between local and global CS and chord inclination angle are complementary. The tubular joint element connects 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 in tubular framed structure joints given the fact that mutli-brace connections are not necessarily coincident. The element degrees of freedom are illustrated in Fig. 2 on the right. As can be seen the element has 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. It was discussed earlier that flexibility matrix of the joint element provides axial and in-plane bending degrees of freedom in planar problems. However, in order for displacements of adjacent nodes in an analytical model to be compatible, 3 degrees of freedom are defined for the tubular joint element at end nodes. The resulting element has 6 degrees of freedom including rigid body modes. These degrees of freedom are termed as initial degrees of freedom in this study. Due to the presence of these rigid body motions in local and global coordinate system, the corresponding element stiffness matrix would be singular. As a consequence, in general there would be no flexibility matrix associated with this
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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 basic degrees of freedom.
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system. Thereby, the element is formulated in another system in which rigid body modes are excluded (Fig. 3(b)). Rigid body modes can be incorporated with a simple geometric transformation. The resulting element has 3 degrees of freedom: two perpendicular translations and one rotation relative to the element longitudinal axis. These degrees of freedom are termed as basic degrees of freedom hereafter. Flexibility matrix is derived for this simple 2D-CS and extended to more general CS which is shown in Fig. 3(b) on the left. The former can be achieved explicitly using the so-called method of unit load application and computing the corresponding deformations. In other words, flexibility matrix of a structure represents the displacements developed in the structure when a unit force corresponding to one degree of freedom is introduced while no other nodal forces are applied. Applying a unit load horizontally on free end (i.e. degree of freedom #4), one can evaluate corresponding nodal displacements in all 3 directions. In this case, using Eq. (1), the displacements are ‘LJFv ’, 0, 0 in three directions shown 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
LJFAX
(7)
sin2 f
The evaluation can be performed for other degrees of freedom and local flexibility matrix is obtained for basic degrees of freedom as follows:
2
3 LJFv 4 b 5 LJFIPB
(8)
The entry in row 2 and column 2 is set to a small number which is denoted by ‘b’ in (8). It could also be seen that this matrix is diagonal, that is, applying a unit force in one direction does not induce any displacement in the directions of other degrees of freedom. This evaluation is resulted from physical interpretation of joint behavior in the vicinity of chord-brace intersection. In fact axial stiffness of the chord is too high to be altered by joint deformations and that is the primary reason why 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 assumed for the legs of the platform). The transformation task could be conveniently carried out using simple rotation matrix in which ‘q’ is the rotation angle. Rotating local flexibility matrix yields the global one which is denoted by ‘F’ in the following Eq.:
F ¼ T T F0T
(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 conventional structural analysis:
kFF DF þ kFC DC ¼ PF kCF DF þ kCC DC ¼ PC
(10)
The above equations relate applied forces in free and constrained degrees of freedom to corresponding displacements. These equations show how four stiffness sub-matrices constitute the global stiffness matrix (indices ‘F’ and ‘C’ stands for free and constrained). Consequently, the problem reduces to evaluating four constituents of the global stiffness matrix. The first sub-matrix could be conveniently obtained 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 initial degrees of freedom. Fig. 5 illustrates the tubular joint element with three degrees of freedom which had been fixed earlier (Fig. 3(b)). A transformation matrix can be used to connect basic and initial degrees of freedom again with the aid of unit load application. As can be observed, three unit loads have been applied at one end while the corresponding reactions have been evaluated using simple static equilibrium. These reactions yield transformation matrix ‘R’ as follows:
2
1 R ¼ 4 0 Lsina
0 1 Lcosa
3 0 0 5 1
Fig. 5. Rigid body movement of brace upon pure axial loading.
(12)
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In (12) and Fig. 5, ‘a’ denotes the angle between the direction of tubular joint element and the global horizontal axis ‘x’. Having this transformation matrix, it would be possible to obtain:
PC ¼ RPF /PC ¼ RkFF DF ¼ RF 1 DF PF ¼ kFF DF
(13)
It is clear that PC ¼ kCF DF , 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 between constrained and free applied forces are given in (13) asPC ¼ RPF . On the other hand, PF ¼ kFC DC andkFC ¼ F 1 RT ¼ kFF RT , thereby it is possible to writePC ¼ RkFF RT DC . SincePC ¼ kCC DC , it is concluded that:
kCC ¼ RkFF RT ¼ RF 1 RT
(15)
Using (11), (14) and (15), the global stiffness matrix can be explicitly formulated as:
K ¼
kFF kCF
kFC kCC
(16)
3. Element verification The proposed stiffness matrix in (16) was used to introduce a new tubular joint element. The element is planar and has 6 degrees of freedom as shown in Fig. 3(b) on the left. Relevant coding was performed using object oriented programming framework and implemented in finite element program OpenSees [15]. Several planar problems are studied in order to validate the proposed element. A simple single-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 the tubular joint element are conducted using uniaxial elements. The general purpose finite element program ANSYS [19] was also employed to model a tubular joint model by using three-dimensional shell elements which accounts for LJF through local deformation of the chord membrane. Two different loading conditions are to be defined to verify the proposed model namely axial and flexural loadings. A sensitivity analysis is to be performed subsequently to not only ensure the suitability of the element, 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 the chord-brace intersection anglef, which can take four different angles given in Table 1. For each analytical joint model, six different models were made for all individual intersection angles based on the joint characteristics. The lengths of the brace and chord are 1 and 2 m respectively. An axial load was applied at the tip of the brace and corresponding displacement was monitored. Using loads and displacements, it is possible to obtain the stiffnesses of the joint model. Table 1 gives axial stiffness ratios for the models confirming the superiority of the tubular joint model. The stiffnesses of the models have been divided by the stiffness of the model made of shell elements in each row. The values shown in the table address the over-stiffness of the rigid-joint and center-to-center model while the results of model made of proposed tubular joint element reasonably agree with those of shellelement-based model. An interesting observation is made upon axial loading which is rigid movement of the brace perpendicular to the load application direction due to local deformations of chord wall (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 the element as depicted in Fig. 5(b). This can be further investigated through the use of local flexibility
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Table 1 Axial stiffness ratios of different joint models. Chord-brace intersection angle (f)
Joint characteristics (b; g respectively)
Conventional rigid Center-to-center Proposed tubular joint model joint model joint element
Shell element stiffness (KN/m)
30
0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20
1.92 2.12 1.82 1.98 1.67 1.79 3.02 3.41 2.86 3.19 2.62 2.88 3.99 4.57 3.71 4.20 3.36 3.78 4.80 5.58 4.42 5.08 3.97 4.53
703329 473140 895859 609629 1126540 777349 421128 277541 529459 353647 659663 446556 300269 196277 380356 251931 474855 316492 236409 152633 298830 195623 373168 246751
45
60
90
1.37 1.51 1.32 1.43 1.23 1.31 2.38 2.68 2.29 2.54 2.12 2.32 3.31 3.78 3.12 3.51 2.85 3.20 4.11 4.76 3.83 4.38 3.48 3.94
0.94 0.91 0.91 0.88 0.90 0.86 0.97 0.93 0.96 0.91 0.98 0.92 0.98 0.93 0.96 0.90 0.98 0.92 0.96 0.91 0.95 0.89 0.97 0.91
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 global coordinate systems have been given. One could decompose applied force ‘P’ into ‘P1 ’ and ‘P2 ’ in the direction of local coordinate system ‘x0 y0 ’. Correspondingly, induced deformation ‘d’ could be decomposed into two perpendicular deformations ‘d1 ’ and ‘d2 ’. It was stated earlier that axial stiffness of 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) decomposition 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 behavior which is found to be of significance. Using local flexibility matrix provided in (8), force-displacement equations can be written for node ‘j’ which is common between the joint and the brace element:
2 3 32 3 LJFv P1 d1 4 b 54 P2 5 ¼ 4 d2 5 0 LJFIPB q 2
(17)
It yields:
8 <
q¼0 d2 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 coordinate system (i.e. ‘xb yb ’) attached to it (Fig. 6(b)). Node ‘j’ is shared by the tubular joint element and the brace beam-column element on the chord wall, thereby the only existing local deformation ‘d1 ’ is also induced at node ‘j’ for the beam element. Projecting this deformation in the directions of beam local coordinate 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 to write:
d4b ¼ d1b þ
PL ; d6b ¼ d3b ¼ 0; d5b ¼ d2b ¼ d1 cos4 EA
(19)
In which ‘E’ and ‘A’ are elastic modulus and cross section area of the beam-column element. The above equation simply justifies the physical interpretation of rigid body motion of brace due to the chord wall deformations. The proposed element was shown to be capable of incorporating these local deformations into the structural displacement calculations through the use of a simple assumption in the 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, a flexural moment was applied to the structure, at the tip of the brace and corresponding rotation was monitored and consequently flexural stiffnesses of the models with different joint characteristics were obtained (Table 2). The loadings started with 1 m-long brace model and relevant stiffnesses were recorded. It can be observed in Table 2 that though the model made of tubular joint element demonstrates superior agreement with shell model, some discrepancies can be seen when intersection angle is 30 . This problem stems from two major reasons which are firstly Fessler’s Equations inadequacy for small chord-brace intersection angles and secondly low contribution of joint element in overall flexural stiffness of the structure having 1 m-long brace. The former is rarely observed in practice since most of offshore jackets are constructed using wider brace-to-chord intersection angles while the latter can be overcome using shorter brace in analytical models. As a consequence, for other models which have 30 intersection angle, brace beam element was modeled shorter (50 cm), to emphasize the role of tubular joint element in overall flexural stiffness of the structures. Moreover it is seen for high amount ofg, results agree much better with those of model made of shell elements emphasizing 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 the chord 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 on response of offshore structures. In an analogous way to previous section, an axial force along the brace axis is applied to the up end of the brace. Four different values of the chord length are selected to be modeled using shell, rigid and flexibility models. The dimensions of the structure are chord length ¼ 500 mm, chord diameter ¼ 16 mm, brace length ¼ 1.0 m, brace diameter ¼ 250 mm and brace thickness ¼ 14 mm.
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Table 2 Flexural stiffness ratios of different joint models. Chord-brace intersection angle (f)
Joint characteristics (b; g respectively)
Conventional rigid joint model
Center-to-center Proposed tubular joint model joint element
Shell element stiffness (KN.m/RAD)
30
0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20 0.4–15.625 0.4–20 0.5–15.625 0.5–20 0.6–15.625 0.6–20
1.11 1.25 1.14 1.26 1.17 1.27 1.74 2.00 1.90 2.13 2.01 2.23 2.05 2.33 2.24 2.52 2.37 2.63 2.27 2.66 2.49 2.83 2.64 2.94
0.74 0.83 0.77 0.84 0.78 0.85 1.03 1.18 1.12 1.26 1.20 1.33 1.30 1.48 1.44 1.61 1.52 1.69 1.52 1.78 1.68 1.90 1.78 1.99
63899 43370 126162 86254 217522 149403 81238 53686 150294 100621 246806 166114 69182 46158 127053 85063 210045 140800 62321 40439 114231 75562 188371 126088
45
60
90
0.86 0.93 0.83 0.87 0.82 0.85 0.92 0.98 0.88 0.91 0.88 0.90 0.91 0.99 0.90 0.93 0.90 0.91 0.96 1.02 0.90 0.93 0.89 0.91
These dimensions are similar to those of examples given by Chen and Hu [18]. Table 3 gives three different stiffness ratios through which it can be concluded that as the chord length increases, joint flexibility effect on axial stiffness of the structure deteriorates. In this table, F.M., S.M. and C.R.M. denote flexibility, shell and conventional rigid models respectively as explained earlier. In the second column the validity of proposed tubular joint element in modeling the real behavior of offshore joints are further emphasized while in the third column the ratio of flexibility model to conventional rigid model are provided. For a very long chord the difference between two models can be neglected while for shorter chord lengths which are frequently encountered in practice, this difference are quite significant. 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 mathematical framework. 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 were formerly tested by Honarvar et al. [20] and Gates et al. [21] were selected for this research. These structures are 2D experimental platforms which were subjected to cyclic static loadings. OpenSees program [15] was selected for modeling and analyses of the structures.
Table 3 Effect 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 5 10 50
0.912 0.942 0.982 0.996
0.289 0.657 0.932 0.999
0.258 0.579 0.896 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]. Progressively increasing cyclic loading was applied to the platform to gain insight into hysteretic behavior of the frame. In conventional design of offshore platforms, lateral elastic stiffness of the jacket is of considerable significance since it controls the overall drift ratio and the displacements of individual members and appurtenances upon application of laterally-exerted loads such as wind, wave and currents. A false estimation of elastic lateral stiffness not only results in erroneous displacement calculations of nodes and members, but also local forces in the members are obtained with a certain amount of error. The latter directly influences the design calculations of the members and might have considerable impact on overall cost of the platform construction. A robust and sophisticated model is then required to obtain the most accurate results in push-over analysis. For the purpose of this research initial elastic stiffness of the platform was extracted from cyclic loading response of the structure and compared against elastic lateral stiffness of three analytical models namely conventional rigid model (i.e. model whose joints use rigid offset element to consider LJF), center-to-center model (i.e. model whose braces have been extended to intersect with chord centerline) and flexibility model (i.e. model which utilizes tubular joint element to consider LJF). Table 4 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 not considerably tall, JF has affected the platform response to lateral loading. Brace to chord diameter ratio seems to a more important parameter in LJF of the tubular framed structures rather than any other parameters.
Fig. 7. Sample planar offshore frames; (a) Honarvar et al. frame [20] (units in mm), (b) Gates et al. frame [21] (units in m).
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Table 4 Comparison 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. and C.R.M.
4.2. Gates et al. [21] Gates et al. [21] conducted an analytical model of a small four-pile production facility platform using computer program DYNAS [22]. Fig. 8(b) illustrates this platform along with its dimensions. Member sizes are also given in Table 5. Similar to the previous section, three analytical models were created to investigate the effect of LJF on the global behavior of offshore frames. Moreover, a model made of three-dimensional shell elements capable of considering local chord and brace deformations was provided as a reference for uniaxial-element-based models. Piles were not incorporated into the models since the focus of the study 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 and fundamental 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 rather similar behavior to the model made of shell elements. A simple triangular progressively-increasing load pattern is applied on all the platform models in linear range of response and results are given in Table 6. As can be seen, C-C.M. and C.R.M. can not simulate the lateral response of the platform accurately comparing to the S.M. due to rigidities in their connections. On the other hand, F.M. has successfully predicted the overall elastic stiffness of the structure. The internal forces induced within the members at the end of elastic response may be extracted to investigate the differences resulted from different kinds of analytical modeling. Table 7 gives the force Table 5 Member sizes of the platform (Gates et al. [21]). Member name
Diameter (cm)
Wall thickness (cm)
Chord Lower braces Upper braces Deck braces Horizontal braces
192.75 76.20 60.96 91.44 45.72
4.75 1.58 1.27 1.92 0.9525
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Table 6 Lateral stiffness values and fundamental periods of vibration of the models.
Fundamental period (s) Lateral stiffness (KN/m)
C.R.M.
C-C.M.
F.M.
S.M.
0.745 92028
0.774 85045
0.847 71230
0.837 73500
Table 7 Force 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 Moment Axial load Moment Axial load Moment
1.278 1.493 1.263 0.858 0.832 0.891
1.176 1.131 1.165 0.866 1.08 1.107
0.952 0.961 0.970 0.931 1.030 1.052
4560.72 95.45422 13556.9 7012.688 2.53809 54.7461
Lower chord Horizontal brace
and moment ratios (i.e. model forces divided by those of shell model) for all the models as well as shell model forces and moments induced within its members at the end of elastic range of response. As can be seen in this table, F.M. can adequately predict local forces induced within the members comparing to S.M. whereas, C-C.M. and C.R.M. variably over-predict or under-predict the forces and moments existing in the structure. This suggests the greater impact of LJF on the design procedure which is markedly dependent upon internal forces within the members in elastic analyses. Dynamic response of the flexibility model can be verified through time-history analysis in which an arbitrary seismic loading is exerted to the model peak displacements are monitored to gain insight into structural dynamic response. 1979 El Centro earthquake ground motion was selected and applied to the models and peak horizontal displacements were monitored and presented in Fig. 8. On the right-hand side, 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. even in linear range of response. It can be concluded that F.M. is able to better predict the dynamic response of the tubular framed structure upon earthquake loading. 5. Conclusion A new local joint flexibility element based on flexibility matrix and Fessler [4] empirical equations was proposed in this paper. The element is derived through the elimination of rigid body modes and the assumption of negligible axial deformation of the chord resulting from joint flexibility. Based on the verification studies performed on the models in which the proposed element had been used, the following conclusions can be drawn: The proposed stiffness matrix derivation is easy and straightforward comparing to other existing tubular joint elements and can be incorporated in any FE program which makes use of typical uniaxial beam-column element for modeling of tubular framed structures. The proposed element can be used for modeling of tubular framed structures conveniently without the prior information about the joint behavior. In order to incorporate the element into any analytical model, brace and chord geometric properties at their intersection should be introduced to the model. In other words, the element is capable of incorporating the chord wall deformations into global stiffness matrix of the structure using simple geometric characteristics of the relevant connecting tubular members. Brace to chord diameter ratio and chord length seem to be the most important factors in the extent of LJF effect on offshore platforms. Conventionally in structural analysis of tubular framed structures, brace and chord are connected to 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 inefficiencies however, flexibility model which contains proposed tubular joint element, shows better agreement with the result of the analyses of sophisticated 3-dimensional models of tubular joints. Special care should be taken when designing a tubular framed structure using conventional linear limit-state design procedure. Internal forces predicted by flexibility model have shown to be more reliable for design purposes. References [1] Bouwkamp JG, Hollings JP, Maison BF, Row DG. Effects of joint flexibility on the response of offshore towers. In: Proc. OTC, paper 3901, 1980. [2] Ueda Y, Rashed SMH, Nakacho K. “An improved joint model and equations for flexibility of tubular joints. J Offshore Mech Arct Eng 1990;vol. 112:157–68. [3] Fessler H, Spooner H. Faulkner D, Cowling M, Frieze P, editors. Experimental determination of stiffness of tubular joints. London: Appl. science publishers; 1981. [4] Fessler H, Mockford PB, Webster JJ. Parametric equations for the flexibility matrix of single brace tubular joint in offshore structures. Proc Inst Civil Engs 1986. Part 2. [5] Buitrago J, Healy BE, Chang TY. Local joint flexibility of tubular joints. Glasgow: Offshore Mech. Arct. Eng. Conf., OMAE; 1993. [6] Karamanos SA, Romeijn A, Wardenier J. On the fatigue design of K-Joint tubular girders. Int J Offshore Polar Eng 2000; 10(1). [7] Skallerud BH, Eide O, Amdahl J, Johansen A. On the cyclic capacity of tubular joints subjected to extreme cyclic loads. 14th Int. Conf. on offshore Mech. And Arctic Eng., OMAE’ 95. Copenhagen: June 19–22 1995. [8] Dier A. Tubular joint technology for offshore structures. J Steel Struct 2005. [9] Mirtaheri M, Zakeri H, Alanjari P, Assareh M. Effect of joint flexibility on overall behavior of jacket type offshore platforms. Am J Eng Appl Sci 2009;2(1):25–30. [10] Lee J, Hino S, Toshiaki O, Seo S. Static loading performance of tubular joint in multi-column composite bridge piers, 62. Memoirs of the faculty of eng., Kyushu Univ; 2002. No. 3. [11] Lee M, Parry A. Strength prediction for ring-stiffened DT-joints in offshore jacket structures. J Eng Struct 2004. http://dx. doi.org/10.1016/j.engstruct.2004.11.004. [12] Holmås T. Implementation of tubular joint flexibility in global frame analysis. Dr. ing thesis. Trondheim Norway: Norwegian Institute of Technology; 1987. [13] Hellan Ø. Nonlinear pushover and cyclic analysis in ultimate limit state design and reassessment of tubular steel offshore structures. Dr. ing thesis. Trondheim Norway: Norwegian Institute of Technology; 1995. [14] Alanjari P, Asgarian B, Honarvar MR, Bahari MR. On the energy dissipation of jacket type offshore platforms with different pile_leg interactions. J Appl Ocean Res 2009. http://dx.doi.org/10.1016/j.apor.2009.07.002. [15] Mazzoni S, McKenna F, Scott M, Fenves G. OpenSees command language manual 2006. [16] Release 5SACS User’s manual, collapse analysis 2001. [17] Skallerud B. USFOS shell element theory manual. Sintef Group; 1998-03-30. [18] Chen B, Hu Y. Fatigue in offshore structures: local flexibility of tubular joints of offshore platforms. In: Dover WD, Madhava R, editorsVolume 1. Rotterdam/Brookfield: A.A. Balkema; 1996. [19] ANSYS. User’s manual, version 10. (PA, 153421300): ANSYS Inc.; Houston: 2005. [20] Honarvar MR, Bahari MR, Asgarian B, Alanjari P. Cyclic inelastic behavior and analytical modeling of pile_leg interaction in jacket type offshore platform. J Appl Ocean Res 2008. http://dx.doi.org/10.1016/j.apor.2008.02.001. [21] Gates W, Marshall W, Mahin S. Analytical methods for determining the ultimate earthquake resistance of fixed offshore structures 1979. OTC paper (2751). [22] Kanaan AE, Powell GH. General purpose computer program for the inelastic dynamic response of plane structures. Univ. California; 1975. Berkeley Report no. EERC 75–5.