Joint model for warping transmission

school of ciVil engineering

research rePort r922July 2011

issn 1833-2781

a neW Joint model for WarPing transmission in thin-Walled steel frames

shaBnam shayanKim Jr rasmuhao Zhang

ssen

SCHOOL OF CIVIL ENGINEERING

A NEW JOINT MODEL FOR WARPING TRANSMISSION IN THIN-WALLED STEEL FRAMES

RESEARCH REPORT R922

SHABNAM SHAYAN KIM JR RASMUSSEN HAO ZHANG

JULY 2011

ISSN 1833-2781

A New Joint Model For Non-Uniform Warping Transmission in Thin-Walled Steel Frames

School of Civil Engineering Research Report R922 Page 2 The University of Sydney

Copyright Notice

School of Civil Engineering, Research Report R922 A new joint model for warping transmission in thin-walled steel frames Shabnam Shayan ([email protected]) Kim JR Rasmussen ([email protected]) Hao Zhang ([email protected] July 2011

ISSN 1833-2781

This publication may be redistributed freely in its entirety and in its original form without the consent of the copyright owner.

Use of material contained in this publication in any other published works must be appropriately referenced, and, if necessary, permission sought from the author.

Published by: School of Civil Engineering The University of Sydney Sydney NSW 2006 Australia

This report and other Research Reports published by the School of Civil Engineering are available at http://sydney.edu.au/civil



ABSTRACT

Thin-walled steel moment frames are often used in construction and engineering practice. Since these frames are composed from open section members with low torsional stiffness, the members are likely to experience warping deformations. At joints, warping displacements in one member can be redistribute and produce warping and twist in all other attached members. This behaviour can be accurately captured at the expense of substantial computational effort by discretising the entire frame using shell finite elements. However, for one-dimensional beam finite elements, the development of a suitable model to incorporate with the effects of non-uniform warping restraint and transmission through the joint is a complex task.

A simple theory is developed in the study which considers the effect of warping continuity through the joint of thin-walled steel frames when using beam finite element analysis. The suggested model can readily be implemented into conventional types of analysis and without the need to modify the stiffness matrix or deal with shell finite element analysis which requires more time and computational efforts. Using a condensed stiffness matrix generated by the substructuring technique, warping springs are introduced to represent the condition of partial warping restraint at intersections between members. The general theory of static condensation, which is the basis of substructuring, is briefly described in this study. For validation purposes, the performance of the proposed model is demonstrated by a number of numerical examples. Excellent agreement is achieved comparing the result of beam finite elements using the suggested joint model and accurate shell finite element analysis.

KEYWORDS

Warping, thin-walled structures, warping spring, substructure, finite element analysis



TABLE OF CONTENTS

ABASTRACT………………………………………………………………………………………………………………3 KEYWORDS……………………………………………………………………………………………………………….3 TABLE OF CONTENTS…………………………………………………………………………………………………..4 1. INTRODUCTION……………………………………………………………………………………………………….52. BRIEF OVERVIEW OF BASIC CONCEPTS AND CURRENT MODLE………………………………………….93. NEW JOINT MODEL WITH WARPING CAPABILITY …………………………………………………………12

3.1. SUBSTRUCTURING AND STATIC CONDENSATION …………………………………………………123.2. JOINT MODEL ………………………………………………………………………………………………13

4. VERIFICATION AND ILLUSTRATIVE EXAMPLES …………………………………………………………….174.1. L-SHAPE PLANE FRAME …………………………………………………………………………………..184.2. PLANE PORTAL FRAME WITH AN OVERHANG MEMBER …………………………………………204.3. SPACE FRMAE …………………………………………………………………………………………….22

5. RESULT OBSERVATIONS AND REMARKS …………………………………………………………………..226. CONCLUSION ……………………………………………………………………………………………………247. REFERENCES ……………………………………………………………………………………………………248. APPENDIX 1: PERL SCRIPT TO GENERATE SUBSTRUCTURE …………………………………………..28



1. INTRODUCTION

Thin-walled steel moment frames are often used in steel construction industry. Since these frames are composed from open section members with low torsional stiffness, the members are likely to experience significant warping deformations. At joints, warping displacements in one member may redistribute and produce warping and twisting in other attached members. It means that when a member warps the other members must distort and their flanges must rotate. Thus, the resistance of the other members to distortion provides a level of restraint on the warping torsion of the loaded member.

However, despite the common use of such structures, the analysis method including the effect of torsional warping displacements and their transmission through the joints is still a major topic of interest in thin-walled steel research.

Past studies about warping effects on the behaviour of thin-walled structures may be divided into two distinct categorises: The effect of warping and end restraint (1) in an isolated member and (2) in a joint. In both cases the warping can be free, fully or partially prevented depends on joint configuration. These restraints need to be taken into account in analysis of thin-walled structures as the magnitude of warping stresses can be relatively large and cause failure.

Most of the early researches were focused on the response of a single member to warping deformations regardless of influence of loaded member to the others when they act as a whole structure. The first and most important contribution in this context have been made by Timoshenko [1,2] and followed by Wagner [3] and Vlasov [4, 5, 6] who studied non-uniform torsion of an I-beam which leads to a general theory of Thin-walled members. Later, many researches [7-13] introduced the first derivative of the rotation as the seventh degree of freedom to represent warping deformation. Toward this objective, the conventional 12×12 stiffness matrices were replaced by the new ones with warping considered as an additional degree of freedom. In that stage, the end warping restraint was assumed to be either completely free [1-6, 13-15] or fully prevented at both end of the member [16-18]. Next to it, another common assumption was to ignore warping in thin-walled member analysis [19, 20]. Later, the flexural-torsional behaviour of plane frames was studied by several investigators, but in most of the cases either the effect of warping was neglected [21-28] or extremely considered to be fully prevented [29-36]. Baigent and Hancock [37] investigated the behaviour of a special joint in which the webs of two C-profiles were joined by a flat plate. In that case, the members could warp freely and independently (see Figure 1) and there was no need to deal with the problem of transmission of warping through the joints. The assumed case was not a real and practical joint in most of the situations.

Several experimental works were conducted to find the warping restraint at member ends [38-41]. All experiments reported the difficulties in trying to restraint the warping and prove the fact that even very stiff warping restraints do not provide full torsional warping rigidity. Therefore the concept of continuous warping and partially restrained warping was introduced into research.

Figure 1: Joint configuration considered by Baigent and



Austin et.al [42] worked on elastic end warping restraint but no information was given to evaluate the degree of warping restraint. Trahair [13, 43] extended this idea and introduced the ratio between elastic flange and the fixed-end flange moments as the degree of warping restraint. The problem of these studies was high computational cost as both need to solve a set of deferential equations with various boundary conditions. So the methods could only be applied to very simple structures.

Ettouney and Kirby [44] proposed a warping restraint factor, which is the ratio between the bimoments of

the partially and fully restraint cases and has same concept as Yang and McGuire’s [45] warping indicator. For both studies, static condensation was used to eliminate undesired degrees of freedom. Although the basic idea of two methods was same, the Yang and McGuire’s procedure seems to be better representative of partial restraint warping between two members. Firstly, the elimination process proposed by Ettouney and Kirby, cannot predict the behaviour of a member correctly when the warping degree of freedom is partially restrained at both ends. The reason is that to obtain the bimoment of the element the influence of warping deformation of one end to the other is neglected. A correction to their method can be found in Ref.44. Secondly, the warping indicator proposed by Yang and McGuire [45] is dealt with warping deformation which is more physical and easier to measure than bimoment. Furthermore, the concept of warping spring was introduced by Yang and McGuire [45] and followed by many researches [61-62]. The model was a hypothetical warping spring applied as an internal spring at the joint.

As it was mentioned before, the warping deformation of a thin-walled open section member connected to

a joint can influence the warping of the other members, so for thin-walled frames transmission and continuity of warping through the joint is an interesting subject in literature.

The transmission of non-uniform torsion through the joint was investigated for two special connections by

Renton [60]. It was shown that the bimoment of two members at the common node are in equilibrium and equal warping occurs between adjacent elements with the same sign for first joint (a) and opposite sign for seconde joint (b) (see Figure 2).

Some attempt can be found in the literature to define different warping magnitude for different members at

joint [57-59]. Bazant and El-Nimeiri [57] considered a unique warping deformation for all elements sharing a node and warping displacement continuity between adjacent elements was imposed to the model. A six DOF, zero length connection element was presented to model warping deformation discontinuity at the member joints by Blandford [58] and Chandramouli, et.al [59] (see Figure 3). Three moments ( , and ), three rotations ( , and ) and a linear moment-rotation relationship have been considered for this joint element.

Since multiple nodes are provided for different members in same location, it is possible to have different

warping displacements in members connected to the joint.

Figure 2: Joints details considered by Renton

(a (b



The warping and distortion at six rigid, angle joints composed from two same steel I-section members were

investigated by Vacharajittiphan and Trahair [46]. Four types of joints have been considered to account end warping restraint (see Figure 4) and the warping stiffness was numerically formulated but no explicit method or approaches was proposed. They came to the conclusion that warping and distortion are interdependent and depend on the joint configuration and details.

The four common joint types, introduced by Vacharajittiphan and Trahair, were then extensively studied

by Sharman [47], Krenk and Damkilde [48], Morrel, et.al [49], Massarira [50], Tong et al [51], Camotim, at al [52, 53] and Basaglia, et al [54-56] to determine the effect of warping transmission through the joints.

To investigate the behaviour of above joints, Krenk and Damkilde [48] considered the continuity conditions

for the flanges in all cases to express the distortion in terms of warping parameters of the two beams at the joint. They developed a simple method to formulate the two elastic-energy components associated with warping and distortion at beam ends. It was concluded that unstiffened joint has two independent warping parameters while the joints with two and three stiffeners impose equal magnitude warping in both connected beams. Before that, this result was also proved by Sharman [47]. The last joint with three pairs of stiffeners acts as full warping restraint.

Later, Massarira [50] made a more comprehensive study to present the energy components for the various

joints and proposed simple coefficients to approximate the joint warping effects. The warping stiffness coefficients ( ) were derived for several types of joints using finite element modelling. These coefficients were

Figure 3: Connection element proposed by Ref. 58

Unstiffen One pair of

Two pairs of Three pairs of

(a)

(b

Figure 4: (a) Angle joint under load (b) joint stiffening arrangements [Vacharajittiphan and Trahair, 1974]



compatible with the coefficients ( value) obtained using the energy components for warping and distortion by Krenk and Damkilde [48].

A new warping transmission model was presented by Tong, et al [51] for diagonal stiffener joint. This joint

was considered to have continuous warping through the joint by previous researchers [47-48, 60]. Bending deformation and twisting of the diagonal stiffener as well as restraining of the web in the joint panel zone to the twist of the diagonal stiffener were taken into account. More accurate results were obtained using proposed theory compare to the assumption of continuous warping at the joints.

One of the most important contributions in transferring warping through the joint was done by Camotin,

Basaglia and Silvestre [52-56]. A GBT-based finite element formulation, which includes local, distortional and global deformation modes, was proposed by Camotin, et.al [52, 53] and Basaglia, et.al [54, 55]. The numerical model can deal with the torsion warping transmission and local displacement compatibility at frame joints with various configurations. The result of model compared with shell element FE analysis using ANSYS and an excellent agreement was achieved.

A brief overview of the literature in transmission of warping through the joints shows that all developed

models need substantial numerical or computational effort. It is obvious that such approaches are not practical in common applications in industry, since designers need easier and faster methods. Due to the complexity of current models, the partial transmission of warping through the joint is ignored in most design cases. Even if a designer wanted to consider transmission of warping through joints, available commercial finite element softwares are limited to either completely prevent warping or allow warping to occur freely at joints when using beam finite element analysis (B-FEA). At this point in time, there is no FE software available that allows the seventh degree of freedom (warping) to be partially prevented. The only option to model warping accurately is using shell element which is not a desired method for complex structures due to its high computational cost.

A few models can be found in literature for partial transmission of warping at joints when using beam finite

element. Basaglia, et al [56] developed a simple kinematic model to simulate the torsion warping restraint and transmission at thin-walled frame joints in the context of beam finite element analysis. They imposed constraint condition between the warping degrees of freedom of the member end nodes which meeting at the joint. This can be done by the “linear constraint equations” ability which is available in most commercial FE softwares (e.g. ABAQUS and ANSYS). Despite its simplicity, this model is not a general method applicable to all types of joints rather than those presented in Basaglia, et al [56]. This model has been discussed in more details in this report. Another method in this context was reported by MacPhedran [63] using external springs to represent warping at the joint and was the basis idea of the proposed model in this report. There is no verification of model against shell element analysis. After lots of investigation by author, it was found that although the concept of using spring is correct, it was not applied correctly into the model and it cannot predict warping behaviour of the joint.

Since there are few accurate and appropriate methods to deal with warping transmission through the joint

using beam finite elements, the present research is concerned with developing a new method for modelling joints including warping effects, benefitting from the accuracy of 3D finite elements and from the computational efficiency of 1D elements. The model is general and can be applied to any kind of joints in thin-walled steel structures. Results from the beam finite elements using ABAQUS are validated in some illustrative examples against shell element finite element analysis and an excellent agreement was achieved.



2. BRIEF OVERVIEW OF BASIC CONCEPTS AND CURRENT MODLES

When a thin-walled open-section member is subjected to a transverse load, which is away from the shear centre, it causes a torque. Because of the open nature of the section, this torque leads the member to experience large out-of-plane warping displacements at the ends. The warping deformation may be defined as displacement of cross-section flanges in different directions (see Figure 5).

As the warping stresses can be large in some cases, the member may fail in non-uniform torsion.

Remarkable analytical models have been introduced by many researchers which are including the effect of warping and torsion in members. Warping conditions at the end of an isolated member can be divided into three main groups: completely free, fully or partially prevented. In some cases, a member is connected to a joint such that there is no restraint against warping. Such a condition applies, for example, when an I-section is welded only through its web and flanges are free to warp (Figure 6 (a)). A member can also be rigidly connected to a joint which effectively and sufficiently prevent member end warping deformations. This situation may occur when the flanges of an I-section are rigidly connected to a stiffened joint (Figure 6 (b)).

If a member is connected to a flexible joint, then the degree of restraint provided at the end of the member

is not sufficient to fully prevent warping. In this case the warping displacement is elastically restraint and bimoment is produced (Figure 7). This elastic restrain can be explained by following relationship in which is warping restraint stiffness, is bimoment and is warping degree of freedom.

Figure 6: joints configuration (a) warping free (b) warping fully t d

(a (b

Figure 5: Warping displacement of an I-



′ (1) Modelling warping effects are more complicated when it goes from studying an isolated member to a

complete three-dimensional frame. In general, the basic problem in modelling thin-walled structures is that warping of a member usually produces warping and distortion in all the other members connected to a joint. In the other words, the joint itself depends on its type and configuration can provide warping stiffness to the members end. Transmission of torsion warping through the joint is one of the major difficulties in the analysis of thin-walled frames. It can be performed precisely by use of shell or solid finite element models which can capture the full behaviour due to their shape and nature. (Figure 8)

The problem of these types of modelling is that it is time consuming, computational intensive and needs

lots of efforts to generate inputs and interpretation outputs. It is obvious that such an approach is not interesting in common applications in industry where the time is one of the most important aspects in all projects. In this context, beam finite element models can provide easier and faster way for designers to model and analyse the structures. In most new commercial structural softwares, the first derivative of rotation considered as seventh degree of freedom to capture warping in beam elements and there is no problem to fully fix or leave this degree of freedom free at the end of each member. This is what most of the designers do at the moment when considering warping is essential.

But in reality in most cases warping is partially restraint and transmission of warping displacement through

the joints needs to be taken into account. Using beam elements, beam-to-column joints are treated as non-dimensional points and they cannot capture any local effects including warping. Some efforts have been done to simulate this effect by Basaglia, et al [56] and a kinematic model for four specific types of joints is introduced (Figure 9). The model rely on the “linear constraint equation” ability of structural softwares which establish constraint conditions between the torsion warping degrees of freedom of the member end nodes.

M

M

Figure 8: Warping moments and deformation of joint [50]

Figure 7: Flange moments of an I-section to produce bimoment



The general form of linear constraint equation which must be imposed in the case of two connected

members is: 0 (2) where a and b are the beam and column ends respectively, are the coefficients depends on joint

configuration and ′ is warping degree of freedom. The values and mechanical characteristic of four different joint types of Figure 9 have been summarised in Table 1.

Although this model contributed a lot to solve the problem of transmission warping through the joint when

using beam elements, it is only applicable to four specific types of joints and can’t model all possible cases with partially restraint warping. For example, fully prevented warping assumption for diagonal/box stiffened joints may be really conservative if the stiffener has small thickness. For these types of joints, comparing the result of frame analysis using beam and shell elements shows even more than 8% error in some cases. Also this method cannot extend into 3D or space frames.

Therefore the research motivation for the proposed method originates from the need to have more general

and precise model for transferring warping through the joints.

Joint Type Warping Transmission coefficients

Unstiffened Complete and direct 1 and 1

Diagonal-stiffened Complete and direct 1 and 1

Box-stiffened Complete and inverse 1

Diagonal/Box stiffened Null 0

Figure 9: Joint configuration connection channel members: (a) unstiffened with flange continuity, (b) diagonal stiffened, (c) box stiffened, (d)

diagonal/box stiffened joints

Table 1: Summary of mechanical characteristic for four specific joint typesreported in Ref.56



3. NEW JOINT MODEL WITH WARPING CAPABILITY Due to the lack of a mechanism to provide warping stiffness in the joints when using beam finite elements

in commercial softwares, a joint model was developed with the help of substructuring technique and linear springs. The joint accept warping deformation from loaded member and redistribute it to all connected beams and columns. In fact, the suggested joint acts as a flexible interface between members and provides a partially warping restraint by means of springs. The basic for the model is assemblage of ABAQUS’ shell elements (S4R) and substructuring analysis.

As an overall overview of the joint model, the joint itself is first modelled as a substructure using shell

elements, only warping degrees of freedom are retained and all others will be eliminated using ”static condensation” . The condensed stiffness matrix for warping degrees of freedom is generated. The stiffness matrix components which are the warping stiffness were used as springs to model joints in frames analysis. It is worthy to mention that the connection model in some cases is not as generic as the joint using shell element due to some simplification imposed into model. First, it is assumed that the shared flange of beam and column in the joint must have same width. In the other word, the model cannot be applied when joint connected members have different size. Second, the model limited to have same rotation in both flanges and cannot simulate the situation when two opposite flanges have different support conditions. An example of this case is where the bottom flange of a beam is rigidly connected to a column but the top flange is free. In this circumstance, the upper flange may experience more rotation than the lower one and this effect is neglected in proposed model.

3.1. SUBSTRUCTURING AND STATIC CONDENSATION

Substructring is a technique commonly used to overcome the difficulty of working with large dimensional

problems. In principle a whole structure is subdivided into smaller parts and each part will be analysed separately. The basic idea of substructuring analysis is that only certain degrees of freedom are retained while others are eliminated by static condensation. This methodology offer many advantages in finite element analysis:

A substantial reduction in analysis time is achieved due to only modelling the joints using 3D shell finite elements rather that the entire frame. In spite of the fact that joints are repeated from the frame model, the substructure stiffness matrix needs to be computed once for each type of joint with similar geometry. By having a script to generate the substructure, there is no need to make the joint manually each time when the geometry or configuration changed so the stiffness matrix calculated automatically.

The formulation of a substructure element, which is used by ABAQUS [64] to generate the condensed stiffness matrix of the joint, is outlined below.

According to the standard and conventional finite element method, the global stiffness matrix K and the

global force matrix F are obtained by assembling of all elements stiffness matrices. The general form of global stiffness matrix can be described as:

(3) where K is the n × n global stiffness matrix, and are n ×1 node-displacement and load matrix

respectively while n is total degrees of freedom. In substructure analysis, Equation (3) is modified to ∗ ∗ (4) where ∗ and ∗ are stiffness and force matrix respectively after static condensation. To get the above

equation, in first step the displacement vector is divided into degrees of freedom which are retained ( ,) and eliminated ( )

(5)

In partitioned matrix form, the equation between forces and displacements can be written as



(6)

in which , , and are displacements and forces of retained and eliminated DOFs respectively and

K’s are stiffness sub-matrices. Solving the second set of Equation (6) with respect to the eliminated DOFs the expression of Equation (7) may be obtained:

(7)

The condensed equilibrium equation can be written by substituting Equation (7) into the first set of Equation

(6):

(8) After application of static condensation, only the retained degrees of freedom are presented in stiffness

equation (Equation 8). It means that after a substructure is defined and analysed, its stiffness and load vectors are condensed to the specified retained nodes. Thus, each joint can be represented with an equivalent stiffness matrix ( ∗).

3.2. JOINT MODEL

ABAQUS [64] is used in this study for creating numerical models. As mentioned before the joint model is

developed to enable the designers to use beam finite element analysis while the transmission of warping at joints is also captured correctly. So, the beam finite element (B31OS) is used which has 7 DOFs. The seventh DOF is warping and can be free or fully prevented in ABAQUS. To partially prevent warping, this new method is introduced which is based on substructuring technique. First the joint is modelled by S4R shell elements as a substructure in the same way that any structural model may be created. The S4R shell elements in ABAQUS can support all six degrees of freedom in the translational and rotational directions. But because there is no compatibility between the number of degrees of freedom in beam elements (7 DOFs) and shell elements (6 DOFs), no direct method was found to connect beam elements warping degree of freedom to joint model created by shell elements. Therefore it was decided to convert an exact 3D model of the joint to a 2D simple system using linear springs to reduce complexity of the joint element.

After the joint is modelled using shell elements the next step is to map the warping deformations into joint

by a series of constraint equations which tie the longitudinal displacement of one corner node to the longitudinal displacements of other corner nodes on the same face so as to represent a warping displacement in the direction of the attached beam (Figure 10(a)). For sections composed of three plate elements (e.g. I-sections), this requires three constraint equations for each face. The *EQUATION command in ABAQUS can be used to set the longitudinal displacements of corner nodes on each face of the joint to be equal and opposite of each other. For example for front face of the box joint presented in Figure 10 (a) a general set of equations will be

0 (9) 0 (10) 0 (11) where and are the coefficient to make the displacements inverse

1 and is the longitudinal displacement (degree of freedom one) at node i.



By imposing warping deformations into the joint, for each connected face one degree of freedom, which is chosen as the longitudinal displacement of a corner node, is retained (Figure 11). The number of degrees of freedom retained in the substructuring process depends on the number of adjoining beams and columns. A warping deformation of the beam will cause a compatible deformation at each corner of the connected face of the joint, as shown in Figure 10(b). Each faces of the joint can provide a degree of freedom which represents an attached beam element’s warping degree of freedom. Figure 10(a) shows the warping displacement required to model warping degree of freedom of beam element that is attached to that face. A warping deformation from the beam will cause deformation at each corner of the joint plate of the connected face and the joint deform like Figure 10 (b).

Currently, substructure modelling is not supported by ABAQUS/CAE, thus a PERL script was written to

generate and analyse the joint element (Appendix 1). (PERL is a common programming language that can be run in most computer operating systems). After determining the retained DOFs, a “*SUBSTRUCTURE GENERATE” step is then considered to performed static condensation of the substructure and the reduced stiffness matrix is produced by use of “* SUBSTRUCTURE MATRIX OUTPUT” command. Once the substructure is generated, it can be employed to the model as an element. The remained DOFs of the substructure define its connectivity and can directly be bound to the adjacent beam and column nodes in FEA. The problem of with this approach is that the substructure cannot be used with Eigenvalue buckling analysis in ABAQUS as the program does not generate the geometric stiffness matrix for the element. To create a more general model, it was decided to get the stiffness matrix and applying springs to model the joint rather than using substructure directly as an element. It is also possible to utilise the stiffness matrix as a “user element”. This approach would be more complex and requires writing a subroutine outside ABAQUS. Although both approaches provide same results, it was decided to model the joint using springs since all the stiffness values are positive.

A

B

C

D

E

F

G

H

2

1 3

Figure 10: (a) Displacements to generate a single warping degree of freedom, (b) Joint deformations due

t li d i

(a) (b)



For the particular corner joint consisting of one beam and one column, the condensed stiffness matrix has

four components: one for the warping degree of freedom of the vertical face connected to the beam ( , another one for the warping degree of freedom of the horizontal face attached to the column ( and two off-diagonal components to represent interaction between those warping degrees of freedom ( and ). The general form of the stiffness matrix for the substructure after condensation is:

where (12)

This can be thought of stiffness matrix created by applying unit displacement to the exposed degrees of freedom one-by-one while the other degrees of freedom are fixed and have no displacement. As the retained degrees of freedom are longitudinal displacements, their multiplication by the stiffness matrix produces forces while the corresponding terms for the warping DOFs are bimoments. Thus the relation between forces and longitudinal displacements (Equation 13) needs to be converted to a relation between bimoments (B) and warping deformations ( ′) (Equation 14) in which is the spring stiffness matrix.

(13) (14) Bearing in mind that a warping deformation may be produced by applying four axial forces to the flange tips (see Figure 11), the bimoment matrix may be expressed by Equation 15: (15) where is the flange width and the section height. The warping displacement matrix ( ) can also be defined as: ⁄ (16) in which is the sectorial coordinate of the flange tips. Substituting Equation 13 and 14 into Equation 15 and 16 results in:

(17)

where ’s are stiffness terms.. Once the converted stiffness matrix is formulated, it can be applied to the beam finite element model. Ordinarily, this would be achieved using spring elements. The terms “force” and “displacement” are used throughout the description of spring elements. When the spring is associated with displacement degree of freedom, these variables are the forces and relative displacement in the spring. If the springs are associated with rotational degrees of freedom, these variables will then be the moment transmitted

2

1 3

H

E

Figure 11: Schematic configuration to show retained degrees of freedom for a



by the spring and the relative rotation across the spring [64]. However, in ABAQUS [64], spring elements can only be linked to translations and rotations (i.e. the first six degrees of freedom), while there is no direct way of relating spring elements to the warping degree of freedom. Therefore, an indirect method of connecting springs to the seventh DOF is proposed. The model consists of using a combination of a spring and a linear constraint equation for each member connected to the joint. For the particular corner joint shown in Figure 11, two springs and two constraint equations are needed. The relation between warping and bimoment in the corner joint takes the form,

(18)

where ( , ) and , are warping and bimoments respectively of beams and columns. To obtain the required linear constraint equations, the first diagonal stiffness term is considered as a fixed value ( and the other terms are written as a fraction of that term. For example, for the present case is used as the fixed value and the stiffness matrix is changed to

(19)

Substituting the new matrix into Equation 18 and rearrange it, gives:

(20)

where represents stiffness, and are bimoments and the terms in round brackets are warping displacements which must be tied to the linear displacements of the springs by using constraint equations in ABAQUS [64] (see Equation 21). In Equation 13, the terms and are the warping degrees of freedom of the beam and column respectively, while and are linear spring displacements in DOF directions 1 and 2 respectively. It is obvious that the stiffness of all springs should be same and equal to (see Figure 12).

(21)

Keeping in mind that the frame should be modelled in such a way that separate nodes are used for the adjoining members at the joint and that these nodes are located at the perimeter of the joint, then, new “dummy” nodes are created near the ends of the adjoining elements and springs are attached to these nodes (Figure 12). The SPRING2 element of the ABAQUS library [64] is used to represent the warping stiffness. While the seventh DOF (warping) is partially transferred using the presented spring model, the other six DOFs are transmitted directly and rigidly between members connected to the joint. That is, the joint is considered to be rigid with respect to rotation and translation (Figure 12).

Figure 12: joint spring model

2

1

Dummy

Constraint

SprinRigid link



In this report three different joint types are studied (Figure 13) but the concept is general and can be extended to all other types of joints. The first two joints (Figure 13 (a) and (b)) were chosen as they are common configurations in structural building while the last one (Figure 13 (c)) can represent full warping restraint and it is a good case to compare with conventional approaches. All joint types are capable to be used in multi-stories frames.

4. VERIFICATION AND ILLUSTRATIVE EXAMPLES

In order to evaluate the performance and capabilities of the proposed joint few examples are presented in this section. Three thin-walled steel frames has been analysed using one-dimensional beam elements in ABAQUS (B-FEA) and then compared with the same frames using 3D shell elements (S-FEA) to demonstrate the validity and accuracy. It is shown that the results have excellent agreement with the kinematic model proposed by Basaglia et.al [56] (which is called Basaglia model in this report for ease of reference) as well as the shell elements. All the frame are made of steel I-sections with Elastic modulus (E) equal to 200GPa and Poisson’s ratio equal to 0.3. The cross section geometry for all frame members has been presented in Figure 14. To avoid any local deformation all sections considered to be compact. All beam-column joints are laterally restraint and cannot move out of plane. The column bases are fully fixed and the joints are laterally restrained to prevent out-of-plane displacements. For this study, three dimensional beam elements with 7 DOFs per node (B31OS) and S8R shell elements are used. The shell finite element analyses are based on discretisations of the flanges and webs into 4 and 6 elements respectively while the length of each element in the beam analyses is taken as 25mm.

section Section height

h (mm) Flange width

b (mm) Web thickness

(mm) Flange thickness

(mm) Sectorial coordinate( )( )

150UB 14 150 75 5 7 2681 I 300×8 308 150 8 8 11250

Figure 14: Cross-sections geometries

Figure 13: Joint configurations for d d l

(a) (b) (c)



The first study illustrates the buckling and elastic behaviour of the L-shape steel frame, which has only one beam and one column meeting at the joint. The second frame is a plane portal frame with an overhanging member in which there is a joint with three connected members. The third one is a space frame which was chosen to see the performance of the joint in 3D model. These examples show the model is general and does not depend on the number and configuration of attached members to the joint.

Three different joint types have been considered for all frames which are box-stiffened, box-stiffened with

one diagonal stiffener and box-stiffened with two diagonal stiffeners (see Figure 13). The stiffener plate has the thickness same as the other plates formatting all four faces. It is worthy to note that the loading conditions have been chosen in way to ensure producing significant warping deformation at the joints.

4.1 L-SHAPE PLANE FRAME

The first study illustrates the buckling behaviour of a L-shape steel frame which has only one beam and

one column meeting at the joint. Figure 15 shows the frame geometries and loading using beam and shell elements as well as the critical buckling shape of both cases. Both beam and column are made by 150UB14 and a point load P applied to the top flange at the end of the beam. The column is fixed at the base and the out-of-plane displacements are prevented at centre node of the joints for shell element frame and at top of the columns for frames using beam elements (It means that the beam-column joints cannot move along X and Z). It can be obviously seen that shell elements involve more analysis time and computational expenses.

It is worthy to note that the spring system in joint modelling is only in charge of transferring warping and it

doesn’t deal with other six degrees of freedom. In beam finite element frames the joint considered to act as rigid joint regarding to lineal and rotational deformations. Toward this objective, a rigid link is used between member’s end nodes to provide same bending and displacement of beam and column at the joint. Figure 16 provides the arrangement of springs and the rigid link in ABAQUS for proposed model.

(a)

(b)

(c)

P

8000 mm

4000 mm

P

Y

X Z

Fixed

(a(b

(c(d

Figure 15: L-shape plane frame (a) configuration using beam elements, (b) configuration using shell elements, (c) buckling shape using beam element, (d) buckling shape using shell elements



Table 2 shows the spring stiffness as well as bimoments of attached members to the joint. The spring

stiffness increases by adding stiffener into the joint. The warping transmission is complete and inverse for box joint and almost prevented for box joint with one and two diagonal stiffeners, as it can be also found in literature (Basaglia et al [56]). The critical loads of the frame buckling analysis with different joints using beam and shell finite element analysis as well as Basaglia’s model [56] has been summarized in Table 3.

The frame critical loads obtained from B-FEA and S-FEA for three different joint configurations are

summarized in Table 2. The maximum discrepancy using Basaglia model is 4.56% corresponding to joint (b) and decreases to 1.8% using proposed joint model which indicates good accuracy. At the same time, the new joint model can predict the critical buckling load of a frame with joint (c) very accurate while the Basaglia method doesn’t enable to model this type of joint. It is worth noticing that the joint configuration influences the critical buckling load and that by adding two stiffeners to the joint, the buckling load is increased by 12% compared to the frame with box-joint. Figure 17 further illustrate an excellent agreement between the load-deflection curves of the L-shape frame with box-joint using beam and shell element first order elastic analyses.

Spring stiffness ( ) Beam Bimoment ( . ) Column Bimoment ( . ) Joint (a) 1.09E+13 5510 4473

Joint (b) 1.71E+13 11875 1081

Joint (c) 2.36E+13 12179 1340

Joint configuration Proposed spring model

Basaglia’s model

Shell FEA Error (%) (spring model)

Error (%) (Basaglia model )

Joint (a) 3.97 3.95 3.92 1.26 0.76

Joint (b) 4.47 4.60 4.39 1.8 4.56

Joint (c) 4.52 NA 4.52 0 NA

Beam

Column

Rigid link to transfer the first 6 DOF

Figure 16: Joint configuration in ABAQUS

Table 3: Critical buckling loads ( )

Table 2: spring stiffness and bimoments



4.2. PLANE PORTAL FRAME WITH AN OVERHANGING MEMBER

A portal frame with an overhanging member has been studied. In addition to the corner joint considered in the previous example, the frame features an intermediate joint which connects three members. This example has been chosen to show that the method is general and does not depend on the number of adjoining members. All conditions such as cross sections geometry and support conditions are same as example 4.1 and frame configuration can be seen in Figure 18.

For those frames analysed with beam elements and springs joint model, two spring configurations needed.

The spring’s arrangement for the joint including three members can be found in Figure (19) while the model for joints made by two elements was introduced before in Figure 12. The terms , and represent beams and column respectively and is spring stiffness.

0

2

4

6

8

10

12

0 50 100 150 200 250

Loa

d (k

N)

Deflection (mm)

L-shape frame with box-joint

B‐FEA

S‐FEA

Figure 17: First order elastic analysis

Figure 18. Configuration and dimension of plane portal frame with an overhanging member

8000 mm

4000 mm

3857mm

P

(a)

(b)

(c)

Fixed

P

Y

X Z



The stiffness matrix in Equation 23 was obtained from analysis of box-joint as substructure with three

retained degrees of freedom, two for horizontal beams and one for vertical column.

440568163044251363

163044440568251363

251363251363523643

(23)

Applying Equation 17, the spring stiffness matrix can be obtained from substructure stiffness matrix

(see Equation 24). The value of b (flange width), h (section height) and (sectorial coordinate) can be found in Figure 14.

1.2668 134.688 12

7.2276 12

4.688 121.2668 137.2276 12

7.2276 127.2276 121.5057 13

(24)

As mentioned before the first component in spring stiffness matrix is considered as (spring stiffness)

and the others are rewritten as a portion of that value (Equation 25).

0.370.57

0.37

0.57

0.570.571.188

(25)

Then, the general equilibrium equation between bimoments and warping deformations may be written as

Equation 26.

(26)

By Rearranging Equation 26 and substituting the stiffness matrix from Equation 25, the following set of

equations can be derived

0.37 0.57

0.37 0.57

0.57 0.57 1.188

(27)

These equations then applied in Abaqus as linear constraint equations with an appropriate format such:

2

1

Figure 19: Spring configuration



0.37 0.57

0.37 0.57

0.57 0.57 1.188 (28)

The following table shows the result of all frame analysis using different joints:

Based on the result in Table 4, the largest error is 2.3% using spring joint model which relates to the box

joint with one diagonal stiffener and increase to 5.54% using basaglia kinematic mode. Similar to the previous example, changing the joint configuration from a box joint to box joints with one and two diagonal stiffener increases the critical buckling load by 11.5% and 12% respectively. 4.3. SPACE FRAME

For this example, the space frame presented in Figure 20 has been analysed using beam and shell finite

elements to show that the proposed method can be extended to 3D models. The frame is a symmetric space frame which is made from two portal frames connected through three transverse beams (see Figure 20). Five equal point loads P are applied at the top of the columns and the mid-span of the transverse beam at the centre. While the beam-column joints exhibit web continuity, there is flange continuity in the beam-to-beam ones [56]. The frame features two types of joint: (a) beam-column joint and (b) beam-beam joint. Warping through the joints can be modelled by the use of three springs and three constraint equations in the beam finite element analysis. The out-of-plane displacement is restraint at all joints and the displacement along X and Z is prevented. All column bases are fully fixed and a roller support is used at beam mid-span to prevent the displacement along X. When shell finite element analysis is applied the lateral supports are located at the centre of the joints or at cross-section mid-web point for transverse beam. For beam element the all these displacement restraints are located at top of the columns. For beam finite element analysis the frame is discretized to 72 elements corresponding to1008 degrees of freedom while the discretization for shell element frames cause more than 52000 DOFs. Figure 20 shows the frame configuration, critical buckling shape and joint types for both B-FEA and S-FEA. The critical load using B-FEA is 350.30 kN while it decreases to 342.83 kN for S-FEA and shows the discrepancy of 2.13%. The beams incur significant flexural-torsional buckling deformations which cause pronounced warping of the joints, so the performance of the joint model is well tested by this example. 5. RESULT OBSERVATIONS AND REMARKS

The observation and comparison of all frames buckling loads, lead to the following remarks: 1. There is an excellent agreement between the buckling results of three considered finite element

analysis. A more precise view can show that for box-stiffened joints the results of spring model and Basaglia kinematic approach are close or identical in some cases but when it goes to box-stiffened with one diagonal stiffener the difference increases. All the current models for transmission warping as well as Basaglia, et.al [56] considered fully prevented warping for diagonal-stiffened joint. This assumption is correct when the diagonal stiffener is thick enough to prevent warping. It can be proved in example 4.2 when the difference between buckling result of frame using shell element and Basaglia model for this type of joint reaches 5.54%.

2. Despite one case related to the result of L-shape frame buckling analysis with box joint, all the errors

are smaller for spring model compare with Basaglia model which proves more accurate approach. Next to that,

Joint configuration Proposed spring model

Basaglia model

Shell FEA Error (%) (spring model)

Error (%) (Basaglia’s model )

Joint (a) 7.28 7.32 7.25 0.41 0.95

Joint (b) 8.19 8.47 8 2.3 5.54

Joint (c) 8.27 NA 8.25 0.24 NA

Table 4: Critical buckling loads



the suggested method can considerably reduce the number of degrees of freedom and computational expenses in comparison with shell finite element analysis.

3. Studying different types of joints show that the critical buckling load relates to the joint configuration and

progressively increases when the joint becomes stiffer. The roll of stiffener is to prevent distortion of the cross-section and increase the resistance to warping which cause more critical buckling load.

4. It was mentioned before in Table 1 that Basaglia, et.al [56] considered fully prevented warping for box-

stiffened joints with one diagonal stiffener. The results of analysis show that the critical buckling load for warping prevented case is matched more with the result of box-joint with two diagonal stiffener rather that the connection with one stiffener. It is proved that the assumption of prevented warping in diagonal/box-stiffened joint is not completely correct when the thickness of the stiffener plate is small and the propped model can predict the behaviour of this type of joint more precisely.

5. The presented connection model is general and can easily be used for all arbitrary joint configurations

by means of substructuring techniques. It was also showed that the method does not applicable only corner joint as most of the methods do and can be used for all joints in plane frames regardless of the number of connected members.

Figure 20: Configuration and dimension of Space

P

P

P

P

P

Fixed

(a)

Fixed

(a)

(b)



6. CONCLUSION This report outlined a new and more accurate joint model into beam finite element analysis considering

the effect of warping and its transmission between the connected members. The suggested model can be implemented simply into analysis and there is no need to deal with modifying traditional stiffness matrix or challenging with shell finite element analysis which needs more time and computational efforts. In this method, the joint itself modelled as an assemblage of shell elements and analysed separately as a substructure. A smaller stiffness matrix which only includes warping degrees of freedom was achieved by the help of static condensation method. These stiffness components applied as spring associated with a set of linear constraint equations to model a joint with the capability to prevent warping partially. The introduction of warping springs provide a simple and more accurate estimate of the joint warping stiffness in one-dimensional beam finite element analysis as there is no direct way to do that in common commercial finite element softwares. The model is general and applicable to any arbitrary joint types. Three different frames have been analysed using beam elements to evaluate the performance and capabilities of the proposed joint model. All examples were done in FEM package ABAQUS which provides spring and ‘linear constraint equation’ facilities. It can be seen that an excellent agreement has been achieved comparing the result of critical buckling loads of these frames with the exact values using shell finite element analysis. Finally, it is worthy to note that due to simple way of modelling, this approach can be used in all common applications in industry where designers look for fast and easy methods and avoid any implication in design methodologies. 7. REFERENCES [1] Timoshenko, S. P. (1905-6). Bull. Polyt. Inst. St. Petersburg. [2] Timoshenko, S. P. and Gere, J. M. (1961). “Theory of Elastic Stability”, McGraw-Hill, New York. [3] Wagner, H. (1936). “Verdrehung und Knickung von Offenen Profilen (Torsion and Buckling of Open Sections)”, 25th Anniversary Publication, Technische Hochschule, Danzig, 1904-1929, Translated as Technical Memorandum No. 87, National Committee for Aeronautics. [4] Vlasov, V. Z. (1939). “Twisting, stability and vibrations o f thin-walled members “, (in Russian), Appl. Math. Mech. (PMM), 3(1). [5] Vlasov ,V. Z. (1959). “Thin-walled elastic members”, (in Russian), 2nd ed. Moscow: GIFML. [6] Vlasov, V. Z. (1961). “Thin Walled Elastic Beams”, Jerusalem: Israel Program for Scientific Translation. [7] Krahula, J. L. (1967). “Analysis of bent and twisted bars using the finite element method”, AIAA Jnl5 (6), 1194-1197. [8] Krajcinovic, D. (1969). “A consistent discrete elements technique for thin-walled assemblages”, International Journal of Solids and Structures, 5(7), 639-662. [9] Yoo, C.H. (1980). “Bimoment contribution to stability of thin-walled assemblages”, Computers & Structures 11(5), 465-471. [10] Rajasekaran, S. and Murray, D. W. (1973). “Finite element solutions of inelastic beam equations”, Journal of the Structural Division, ASCE, 99(12), 2423-2438. [11] Barsoum, R.S. and Gallagher, R.H. (1970). “Finite element analysis of torsional and torsional–flexural stability problems”, International Journal for Numerical Methods in Engineering 2(3), 335-352. [12] Epstein, M. and Murray, D.W. (1976). “Three-dimensional large deformation analysis of thin walled beams”, International Journal of Solids and Structures 12(12), 867-876. [13] Trahair, N.S. (1966). “Elastic stability of I-beam elements in rigid-jointed structures”, Journal of the Institution of Engineers, 38, 171-180. [14] Kitipornchai,S. and Trahair, N. S. (1971). “Elastic lateral buckling of stepped I-Beams”, Journal of the Structural Division, 97(10), 2535-2548.



[15] Tebedge, N. and Tall, L. (1973). “Linear stability analysis of beam- columns”, Journal of the Structural Division, ASCE, 99 (STI2), 24-39. [16] Razzaq, Z. and Galambos, T.V. (1979). “Biaxial bending of beams with or without torsion”, Journal of the Structural Division, ASCE, 105 (ST11), 2145-2162. [17] Dinno, K. S. and Merchant, W. (1965). ” A procedure for calculating the plastic collapse of I-sections under bending and torsion”, Journal of Structural Engineering ,43, 219-221. [18] Chaudhary, A.B. (1982). “Generalized stiffness matrix for thin-walled beams”, Journal of the Structural Division, ASCE, 108(3), 559-577. [19] Argyris, J.H., Hilbert, O., Malejannakis, G.A. and Scharpf, D.W. (1979). “ On the geometrical stiffness of a beam in space-a consistent approach”, Computer Methods in Applied Mechanics and Engineering, 20, 105-131. [20] Bathe, K. J. and Bolourchi, S. (1979). “Large displacement analysis of three-dimensional beam structures”, International Journal for Numerical Methods in Engineering, 14(7), 961-986. [21] Kim, S.E., Kim, Y. and Choi, S.H. (2001). “Nonlinear analysis of 3-D steel frames”,Thin-Walled Structures, 39(6), 445-461. [22] Chen, W.F., Kim, S.E. and Choi, S.H. (2001). “Practical second-order inelastic analysis for threedimensional steel frames”, Steel Structures, 1, 213–223. [23] Kim, S.E., Park, M.H. and Choi, S.H. (2001). ”Direct design of three-dimensional frames using practical advanced analysis”, Engineering Structures, 23(11), 1491-1502. [24] Teh, L.H. and Clark, M.J. (1999). “Plastic-zone analysis of 3D steel frames using beam elements”, Journal of Structural Engineering, 125 (11), 1328–1337. [25] Jiang, X.M., Chen, H. and Liew, J.Y.R. (2002). “Spread-of-plasticity analysis of three-dimensional steel frames”, Journal of Constructional Steel Research, 58(2), 193-212. [26] Liew, J.Y.R., Chen, H., Shanmugam, N.E. and Chen, W.F. (2000). “Improved nonlinear plastic hinge analysis of space frame structures”, Engineering Structures, 22(10), 1324-1338. [27] Kim, S.E., and Choi, S.H. (2005). “Practical second-order inelastic analysis for three-dimensional steel frames subjected to distributed load”, Thin-Walled Structures 43(1), 135-160. [28] Orbiso, J.G. (1982). “Nonlinear Static Analysis of Three-Dimensional Steel Frames”, a thesis presented to the faculty of the graduate school of Cornell University in partial fulfillment for the degree of Doctor of Philosophy, Ithaca, New York. [29] Hartmann, A. J. and Munse, W. H. (1966). “Flexural-torsional buckling of planar frames”, Journal of the Structural Division, ASCE, 92 (EM2), 37-59. [30] Renton, J.D. (1962). ”Stability of space frames by computer”, Journal of the Structural Division, ASCE, 88 (ST4), 81–103. [31] Renton, J.D. (1967). “Buckling of frames composed of thin-walled members”, Thin-walled Structures (edited by AH CHILVER), Chatto and Windns, London. [32] Trahair, N. S. (1969). “Elastic Stability of continuous beams”, Journal of the Structural Division, ASCE, 95 (ST 6), 1295-1312. [33] Vacharajittiphan, P. and Trahair, N. S. (1975). “Analysis of lateral buckling in plane frames”, Journal of the Structural Division, ASCE, 101(7), 1497-1516.



[34] Razzaq, Z. and Naim, M. M. (1980). “Elastic instability of unbraced space frames”, Journal of the Structural Division, ASCE, 106(ST7), 1389-1400. [35] Chu, K.H. and Rampetsreiter, R.H. (1972). “Large deflection buckling of space frames”, Journal of the Structural Division, ASCE, 98(12), 2701–2722. [36] Morino, S. (1970). “Analysis of Space Frames”, Ph.D. dissertation, Lehigh University, University Microfilms, Ann Arbur, Michigan. [37] Baigent, A.H. and Hancock, G.J. (1982). “Structural analysis of assemblages of thin-walled members”, Engineering Structures, 4(3), 207-216. [38] Dinno, K.S. and Gill, S.S. (1964). “The plastic torsion of I-sections with warping restraint”, International Journal of Mechanical Sciences 6(1), 27-43. [39] Ojalvo, M. and Chambers, R. S. (1977). “Effect of warping restraints on I-beam buckling”, Journal of the Structural Division, ASCE, 103(ST12), 2351–2360. [40] Heins, C.P. and Potocko, R.A. (1979). ” Torsional stiffening of I-Girder Webs”, Journal of the Structural Division, ASCE, 105(ST8), 1689–1698. [41] Beerman, H.J. (1980). “Warping torsion in commercial vehicle frames, taking into consideration flexible joints”, International Journal of Vehicle Design, 1(5), 397-414. [42] Austin, W.S., Yegian, S. and Tung, T.P. (1957). “Lateral buckling of elastically end restrained I-beams”, Transactions, ASCE, 122, 374–388. [43] Trahair, N.S. (1968). “Elastic stability of propped cantilevers”, Civil Eng Trans (Institution of Engineering, Australia), CE10 (1), 94–100. [44] Ettouney, M.M. and Kirby, J.B. (1981). “Warping restraint in three-dimensional frames”, Journal of the Structural Division, ASCE, 107(ST8), 1643–1656. [45] Yang, Y.B. and McGuire, W. (1984). ”A procedure for analysing space frames with partial warping restraint”, International Journal for Numerical Methods in Engineering, 20, 1377–1390. [46] Vacharajittiphan, P. and Trahair, N.S. (1974). “Warping and distortion at I-section Joints”, Journal of the Structural Division, ASCE, 100(ST3), 547-564. [47] Sharman, P.G. (1985). “Analysis of structures with thin-walled open sections”, International Journal of Mechanical Science, 27(10), 665–667. [48] Krenk, S. and Damkilde, L. (1991). “Warping of joints in I-beam assemblages”, Journal of Engineering Mechanics, 117(11), 2457–2474. [49] Morrell, P.J.B., Riddington, J.R., Ali, F.A. and Hamid, H.A. (1996). “Influence of joint detail on the flexural/torsional interaction of thin-walled structures”, Thin-Walled Structures, 24(2), 97-111. [50] Masarira, A. (2002). “The effect of joints on the stability behaviour of steel frame beams”, Journal of Constructional Steel Research, 58(10), 1375-1390. [51] Tong, G.S., Yan, X.X. and Zhang, L. (2005). “Warping and bimoment transmission through diagonally stiffened beam-to-column joints”, Journal of Constructional Steel Research, 61(6), 749-763. [52] Camotim D., Basaglia C. and Silvestre N.( 2008). “GBT buckling analysis on thin-walled steel frames”, Proceedings of the fifth International Conference on Coupled Instabilities in Metal Structures CIMS2008, Rasmussen K. and Wilkinson T. (Eds.), University Publishing Service, The University of Sydney, 1-18.



[53] Camotim, D., Basaglia, C. and Silvestre, N. (2010) “GBT buckling analysis of thin-walled steel frames: A state-of-the-art report”, Thin-Walled Structures, 48(10-11), 726-743. [54] Basaglia, C., Camotim, D. and Silvestre, N. (2009). “GBT-based local, distortional and global buckling analysis of thin-walled steel frames”, Thin-Walled Structures, 47(11), 1246-1264. [55] Basaglia, C., Camotim, D. and Silvestre, N. (2008). “Global buckling analysis of plane and space thin-walled frames in the context of GBT”, Thin-Walled Structures, 46(1), 79-101. [56] Basaglia, C., Camotim, D. and Silvestre, N. (2010). “Kinematic models to simulate the torsion warping transmission at thin-walled steel frame joints”, Steel & Composite Structures- Proceeding of the 4th International Conference. [57] Bazant, Z.P. and El Nimeiri, M. (1973). “Large-deflection spatial buckling of thin walled beams and frames”, Journal of Engineering Mechanics Division, 99(EM6), 1259–1281. [58] Blandford, G.E. (1990). “Thin-walled space frame analysis with geometric and flexible connection nonlinearities”, Computers & Structures, 35(5), 609-617. [59] Chandramouli, S., Wang, S.T. and Blandford, G.E. (1994). “Stability response of flexibly connected cold-formed steel space frames”, Thin-Walled Structures, 18(4), 333-346. [60] Renton, J.D. (1974). “On the transmission of non-uniform torsion through joints”, Report No. 1086/74, Department of Engineering Science, Oxford University . [61] Gotluru, B.P., Schafer, B.W. and Pek ِ◌z, T. (2000). “Torsion in thin-walled cold-formed steel beams”, Thin-Walled Structures, 37(2), 127-145. [62] Mohammed, Z.A. and Frank, E.W. (1996). “Torsion constant for matrix analysis of structure including warping effect”, International Journal of Solids and Structures, 33(3), 361–374. [63] MacPhedran, I.J. (2009). “Frame stability considering member interaction and compatibility of warping deformation”, a thesis presented to the Faculty of Graduate Studies and Research of University of Alberta in partial fulfillment of the requirements for the degree of Doctor of Philosophy, Edmonton, Alberta. [64] ABAQUS/STANDARD (2009). User’s manual version 6.8, Hibbit, USA: Karlsson & Sorensen.



8. APPENDIX 1: PERL SCRIPT TO GENERATE SUBSTRUCTURE As it was mentioned before currently ABAQUS/CAE does not support modelling the substructure. It means that it is not possible to generate model manually in visual user interface and the analysis can be done only by creating a text file as an input file. Because it is difficult to form this text file by hand every time when the dimensions change, a PERL script was made to do it automatically. The following program is an example for a corner box joint with diagonal stiffener using 150UB14 cross-section. It is obvious that this code can be modified easily to generate any other joint types. #!/usr/local/bin/perl open (MYFILE, '> diagonal_stiffener _150_ UB _14.txt'); # # Define substructure element for box joint # # Element number for library $ElemeNum = 200; # Required inputs for "box" type joint # Centre to centre depth of beam, thickness of horizontal plates $depthb = 150 - 7; $thik1 = 7; # Centre to centre width of column, thickness of vertical plates $depthc = 150 - 7; $thik2 = 7; # web thickness $webt = 5; # full width of joint $width = 75; # # Width by number of elements $nelem = 16; # # This part writes the substructure element(s) # print MYFILE"*HEADING\n This is a superstructure for a box type joint\n"; # # define geometry for joint # # The centre of the element is nominally at 0,0,0 # $nelemup = int($depthb/$width * $nelem); if (($nelemup % 2) == 1) {$nelemup++;} $nelemfr = int($depthc/$width * $nelem); if (($nelemfr % 2) == 1) {$nelemfr++;} $nodenum = 1; # Generate all nodes for joint $dx = $depthc/$nelemfr; $dy = $depthb/$nelemup; $dz = $width/$nelem; print MYFILE"*NODE\n"; for ($h=0; $h<=$nelemfr; $h++) { $x = ($h - $nelemfr/2)*$dx; for ($i=0; $i<=$nelemup; $i++) { $y = $dy*($i - $nelemup/2); for ($j=0; $j<=$nelem; $j++) { $z = $dz*($nelem/2 - $j); print MYFILE $nodenum,",", $x,",", $y,",", $z,"\n"; # Flag corner nodes if ($h == 0) { if ((($j==0) || ($j == $nelem)) && (($i==0) || ($i == $nelemup))) { if (($j==0) && ($i==0)) { $nodea = $nodenum; } elsif (($j==0) && ($i==$nelemup)) { $nodec = $nodenum; } elsif (($j==$nelem) && ($i==0)) {



$nodeb = $nodenum; } else { $noded = $nodenum; } } } elsif ($h == $nelemfr) { if ((($j==0) || ($j == $nelem)) && (($i==0) || ($i == $nelemup))) { if (($j==0) && ($i==0)) { $nodee = $nodenum; } elsif (($j==0) && ($i==$nelemup)) { $nodeg = $nodenum; } elsif (($j==$nelem) && ($i==0)) { $nodef = $nodenum; } else { $nodeh = $nodenum; } } } elsif (($x == 0) && ($y == 0) && ($z == 0)) { $cntrnode = $nodenum; } $nodenum++; } } } # Nodes for "back" plate print MYFILE"*NSET, NSET=NODEA\n ".$nodea."\n"; print MYFILE"*NSET, NSET=NODEB\n ".$nodeb."\n"; print MYFILE"*NSET, NSET=NODEC\n ".$nodec."\n"; print MYFILE"*NSET, NSET=NODED\n ".$noded."\n"; print MYFILE"*NSET, NSET=FACEBA\n ".(($nelemup/2)*($nelem+1)+$nelem/2+1)."\n"; # Nodes for "front" plate print MYFILE"*NSET, NSET=NODEE\n ".$nodee."\n"; print MYFILE"*NSET, NSET=NODEF\n ".$nodef."\n"; print MYFILE"*NSET, NSET=NODEG\n ".$nodeg."\n"; print MYFILE"*NSET, NSET=NODEH\n ".$nodeh."\n"; print MYFILE"*NSET, NSET=FACEFR\n ".(($nelemup/2)*($nelem+1)+$nelem/2+1+($nelem+1)*($nelemup+1)*$nelemfr)."\n"; print MYFILE"*NSET, NSET=FACETP\n ".($cntrnode+($nelemup/2)*($nelem+1))."\n"; print MYFILE"*NSET, NSET=FACEBO\n ".($cntrnode- ($nelemup/2)*($nelem+1))."\n"; # Node for "centre" print MYFILE"*NSET, NSET=CENTRE\n ".$cntrnode."\n"; # # set up elements # print MYFILE"*ELEMENT, TYPE=S4R, ELSET=BACK\n"; $elem=1; for ($n=0; $n<$nelemup; $n++) { for ($m=1; $m<=$nelem; $m++) { print MYFILE $elem.",".($n*($nelem+1)+$m).",".($n*($nelem+1)+$m+1).",". (($n+1)*($nelem+1)+$m+1).",".(($n+1)*($nelem+1)+$m)."\n"; $elem++; } } # print MYFILE "*ELEMENT, TYPE=S4R, ELSET=FRONT\n"; $noffset = ($nelem+1)*($nelemup+1)*$nelemfr; for ($n=0; $n<$nelemup; $n++) { for ($m=1; $m<=$nelem; $m++) { print MYFILE $elem.",".($n*($nelem+1)+$m+$noffset).",".($n*($nelem+1)+$m+1+$noffset).",". (($n+1)*($nelem+1)+$m+1+$noffset).",".(($n+1)*($nelem+1)+$m +$noffset)."\n";



$elem++; } } # print MYFILE"*ELEMENT, TYPE=S4R, ELSET=TOP\n"; $noffset = ($nelem+1)*($nelemup); for ($n=0; $n<$nelemfr; $n++) { for ($m=1; $m<=$nelem; $m++) { print MYFILE $elem.",". ($noffset+$m+$n*($nelem+1)*($nelemup+1)).",". ($noffset+$m+1+$n*($nelem+1)*($nelemup+1)).",". ($noffset+$m+1+($n+1)*($nelem+1)*($nelemup+1)).",". ($noffset+$m+($n+1)*($nelem+1)*($nelemup+1))."\n"; $elem++; } } # print MYFILE"*ELEMENT, TYPE=S4R, ELSET=BOTTOM\n"; $noffset = 0; for ($n=0; $n<$nelemfr; $n++) { for ($m=1; $m<=$nelem; $m++) { print MYFILE $elem.",". ($noffset+$m+$n*($nelem+1)*($nelemup+1)).",". ($noffset+$m+1+$n*($nelem+1)*($nelemup+1)).",". ($noffset+$m+1+($n+1)*($nelem+1)*($nelemup+1)).",". ($noffset+$m+($n+1)*($nelem+1)*($nelemup+1))."\n"; $elem++; } } # print MYFILE"*ELEMENT, TYPE=S4R, ELSET=WEB\n"; $noffset = int($nelem / 2)+1; for ($n=0; $n<$nelemfr; $n++) { for ($m=0; $m<$nelemup; $m++) { print MYFILE$elem.",". ($noffset + $m*($nelem+1) + $n*($nelem+1)*($nelemup+1)).",". ($noffset + ($m+1)*($nelem+1) + $n*($nelem+1)*($nelemup+1)).",". ($noffset + ($m+1)*($nelem+1) + ($n+1)*($nelem+1)*($nelemup+1)).",". ($noffset + ($m)*($nelem+1) + ($n+1)*($nelem+1)*($nelemup+1))."\n"; $elem++; } } # print MYFILE"*ELEMENT, TYPE=S4R, ELSET=STIFFENER\n"; for ($n=$nelemup; $n>0; $n--) { for ($m=1; $m<=$nelem; $m++) { print MYFILE $elem.",". ($n*($nelem+1)+$m+($nelemup-$n)*($nelemup+1)*($nelem+1)).",". ($n*($nelem+1)+$m+1+($nelemup-$n)*($nelemup+1)*($nelem+1)).",". (($n-1)*($nelem+1)+$m+1+($nelemup-$n+1)*($nelemup+1)*($nelem+1)).",". (($n-1)*($nelem+1)+$m+($nelemup-$n+1)*($nelemup+1)*($nelem+1))."\n"; $elem++; } } # print MYFILE"*SHELL SECTION, ELSET=BACK, MATERIAL=STEEL1\n"; print MYFILE$thik2,"\n"; print MYFILE"*SHELL SECTION, ELSET=FRONT, MATERIAL=STEEL1\n"; print MYFILE$thik2,"\n"; print MYFILE"*SHELL SECTION, ELSET=TOP, MATERIAL=STEEL1\n"; print MYFILE$thik1,"\n"; print MYFILE"*SHELL SECTION, ELSET=BOTTOM, MATERIAL=STEEL1\n"; print MYFILE$thik1,"\n";



print MYFILE"*SHELL SECTION, ELSET=WEB, MATERIAL=STEEL1\n"; print MYFILE$webt,"\n"; print MYFILE"*SHELL SECTION, ELSET=STIFFENER, MATERIAL=STEEL1\n"; print MYFILE$thik2,"\n"; # print MYFILE"*MATERIAL, NAME=STEEL1\n"; print MYFILE"*ELASTIC\n 200E3, 0.3\n"; print MYFILE"*DENSITY\n 7.7E-9\n"; # # Constraints # print MYFILE"*EQUATION\n"; print MYFILE"** Back face\n"; print MYFILE"2\n"; print MYFILE"NODEC, 1, 1, NODEA, 1, 1\n"; print MYFILE"2\n"; print MYFILE"NODEA, 1, 1, NODEB, 1, 1\n"; print MYFILE"2\n"; print MYFILE"NODEB, 1, 1, NODED, 1, 1\n"; print MYFILE"** Front face\n"; print MYFILE"2\n"; print MYFILE"NODEG, 1, 1, NODEE, 1, 1\n"; print MYFILE"2\n"; print MYFILE"NODEE, 1, 1, NODEF, 1, 1\n"; print MYFILE"2\n"; print MYFILE"NODEF, 1, 1, NODEH, 1, 1\n"; print MYFILE"** Top face\n"; print MYFILE"2\n"; print MYFILE"NODEC, 2, 1, NODED, 2, 1\n"; print MYFILE"2\n"; print MYFILE"NODED, 2, 1, NODEH, 2, 1\n"; print MYFILE"2\n"; print MYFILE"NODEH, 2, 1, NODEG, 2, 1\n"; print MYFILE"** Bottom face\n"; print MYFILE"2\n"; print MYFILE"NODEA, 2, 1, NODEB, 2, 1\n"; print MYFILE"2\n"; print MYFILE"NODEB, 2, 1, NODEF, 2, 1\n"; print MYFILE"2\n"; print MYFILE"NODEF, 2, 1, NODEE, 2, 1\n"; # # Generate element # print MYFILE"*STEP\n"; # Note that type is of format Zn where 0<n<10000 printf MYFILE"*SUBSTRUCTURE GENERATE, TYPE=Z%d, OVERWRITE, RECOVERY MATRIX=YES\n",$ElemeNum; print MYFILE"*RETAINED NODAL DOFS, SORTED=NO\n"; # Lines to generate dof's - NodeNumber, dof_start, dof_end print MYFILE" ".$nodeh.","."1"."\n"; print MYFILE" ".$nodee.","."2"."\n"; # End of definition print MYFILE"*SUBSTRUCTURE MATRIX OUTPUT, OUTPUT FILE=USER DEFINED, FILE NAME= UB_150_14_one_stiffener, STIFFNESS=YES\n"; print MYFILE"*END STEP\n"; close (MYFILE);

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Joint model for warping transmission