Global structural modelling of space frame structures with ... · EC3-1-1, three joint models are...

EUROSTEEL 2017, September 13–15, 2017, Copenhagen, Denmark

© Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin · CE/papers (2017)

Globalstructuralmodellingofspaceframestructureswithconsiderationofjointbehaviour

An integral analysis approach

Tobias Mähr*,a, Miguel Espinhab, René Zieglerb aFrick+Gattinger AG, Vaduz, Liechtenstein

[email protected] b Waagner-Biro Stahlbau AG, Vienna, Austria

[email protected], [email protected]

ABSTRACT

The effects of the behaviour of joints on the distribution of internal forces and moments within a structure and on the overall deformations of the structure are generally neglected [1]. However, where such effects are significant, they should be taken into account. Provisions are given in Eurocode 3, Part 1-8 [2] for analysis, classification and modelling of joints with respect to rotational degrees of freedom. Provisions for translational degrees of freedom are missing. In shear connections slip may occur in case slip resistance is smaller than design load. This holds true for shear connections category A and B according to Eurocode 3, Part 1-8 [2]. In structures with predominant normal forces such slip effects may accumulate and considerably affect the internal forces as well as the deflections. Structural modelling and calculation of slip effects is a complicated task in structural mechanics. In this paper, concepts are presented how to consider slip effects in joints when dealing with large space frame structures. Stick-slip-models are outlined and implemented in 3d structural models. Calculation techniques are discussed relating to the additional degrees of freedom (statically over-determined systems) and the non-linear behaviour. The importance and benefit of this integral analysis approach, where the effects of the joints are considered in the global modelling, are illustrated by the example of the Louvre Abu Dhabi Dome. The dome measures 185m in diameter and consists of approx. 11’000 steel members, most of which having shear connections. The results show that slip in joints must not be neglected in the global structural analysis. The quality and accuracy of the calculated results are verified by deflection measurements on site. Keywords: stick-slip modelling, joint behaviour, hybrid bolted-welded structures, Dome of the Louvre Abu Dhabi

1 INTRODUCTION

Space frame structures are generally modelled using 3d-beam elements (2 nodes, 6 or 7 degrees of freedom (DOF) per node), truss elements (2 nodes, 3 DOF per node) or a combination of both. In EC3-1-1, three joint models are defined to identify whether the effects of joint behaviour on the analysis need be taken into account [1]:

simple (or “pinned”): the joint does not transmit bending moments continuous (rigid and full-strength): bending moments are fully transmitted (the behaviour of

the joint may be assumed to have no effect on the analysis) semi-continuous (semi-rigid, partial-strength): the behaviour of the joint needs to be taken

into account in the analysis The joints should be classified according to their rotational stiffness. Detailed provisions are given in EC3 how to model and analyse the rotational stiffness of joints and their components. Thus, the behaviour of the rotational degrees of freedom (3 per node in beam elements) are covered in-depth,


but the remaining DOF are not treated. In this paper, the translational behaviour of joints and its effects on the global structural analysis is treated. All findings are based on practical experience gained during design and execution of the Louvre Abu Dhabi Dome. The results show that the slip in bearing type bolted connections become significant in large space frame structures and affect both the global deflection and internal forces.

2 STATE OF THE ART: EUROCODE PROVISIONS

As mentioned in chapter 1, three joint models are defined in Eurocode 3 to identify the rotational behaviour of joints. In EC3-1-1, 5.1.2 Joint modelling it says: “The effects of the behaviour of the joints on the distribution of internal forces and moments within a structure, and on the overall deformations of the structure, may generally be neglected, but where such effects are significant (such as in the case of semi-continuous joints) they should be taken into account, see EN 1993-1-8.” Consulting EC3-1-8, further provisions can be found relating to the rotational stiffness of joints, but only few statements on translational stiffness. A comment on lattice girders states the following, EN 1993-1-8, 5.1.5: “The distribution of axial forces in lattice girder may be determined on the assumption that the members are connected by pinned joints.” In EC3-1-1 there is a chapter titled “5.2.1 Effects of deformed geometry of the structure”. Article (6) hints at translational displacements caused by slip in joints: “The effects on the global analysis of the slip in bolt holes and similar deformations of connection devices like studs and anchor bolts on action effects should be taken into account, where relevant and significant.” There are no further explanations how treat this problem in detail.

3 LOCAL EFFECTS OF SLIP IN BOLT HOLES ON JOINTS

3.1 Relevant connection types Bolted connections which are defined as shear connections in EC3-1-8 implicate a certain amount of slip. In categories B and C, slip-resistance is provided at serviceability (B) or at ultimate limit state (C). The possible amount of slip depends on the design of the connection (e.g. geometry, number of buttstraps etc.) and the borehole tolerances. These tolerances are defined in EN 1090-2 [3]. When shear connections category C are used, no slip will occur at all. However, using category C involves the design and control of the friction surfaces and the bolt preloading. This causes higher expenses and complicates the installation procedure. Therefore, category C connections are often inefficient and avoided whenever possible.

3.2 Kinematic behaviour of connections with slip In Fig. 1, the basic kinematics of a splice joint is shown. The total possible slip depends on several parameters:

- borehole tolerances - hole pattern - initial position of the bolts and splice plate before loading

The initial position of the bolts and splice plate is random and has to be estimated for each system individually. The installation procedure may influence the position of the connection elements prior to loading.


longitudinal

transversal

rotattional

Fig 1. Longitudinal, transversal and rotational degrees of freedom of splice joints.

4 GLOBAL ANALYSIS OF THE LOUVRE-ABU-DHABI DOME IN CONSIDERATION OF JOINT BEHAVIOUR

4.1 Project overview The city of Abu Dhabi has the ambition to build a museum complex in the spirit of openness and dialogue. Jean Nouvel has designed a museum which blends into the traditional landscape of the United Arab Emirates. The museum is on the transition from land to sea, the buildings are islands in the water connected by bridges and pontoons. The shallow dome spans 180m and is the cohesive for the archipelago of buildings. The obvious and precise overall shape of the dome shows a complex topology of load bearing members in detail. These members together with the multiple layers of woven outer and inner cladding create a shadow. At the same time seemingly random openings allow rays of light to enter the dome and touch the buildings and the seawater underneath. The ensemble of buildings and the dome create a sanctuary for the valuable art and the visitors.

4.2 General description of the structure The supporting steel structure of the dome consists of a trussed space frame, point supported on four concrete piers located on diametrically opposite sides, with a span of approximately 180m. The steel elements with hollow cross-sections are joined mainly using bolted connections and constitute the supporting truss system. The truss system itself consists of a top and bottom chord, connected with posts and diagonals, all acting mainly in tension and compression. Due to the uniform topology of the space frame the structural behaviour and load path to the support are not revealed, see Fig. 2.


Fig. 2. Structural model of the steel space frame of the dome

The space frame is embraced by a ring beam which gathers the forces and transfers them to the four support points. The members transferring high internal forces such as members connected to the ring beam or close to one of the supports are executed with welded connections. The structural system of the spaced frame has a total of approximately 11.000 members.

4.3 Structural modelling

4.3.1 Bolted connections

The majority of the nodes of the space frame structure consist of joints where the connecting members are bolted to a solid round node bar by means of double cover splice connection. The connection type was chosen in alignment with the architectural intent. Jean Nouvel wanted to show how the members are connected and transfer loads. In the design process it was described as a "mechanical connection".

Fig.3. Typical bolted node connection

This connection topology is characterized by having two shearing planes and two bolt rows on each side with variable number of lines, which transfer the normal forces through bolt shear and bearing.

4.3.2 Stick-slip considerations

In order to reflect the stick-slip effect in the global analysis, a non-linear force-displacement diagram comprising three phases is defined at the member ends in axial direction of the member, s. Fig. 4. The first phase is a ‘stick’ phase, in which no movement occurs as the friction force itself is sufficient to transfer the axial force. If the axial force in the member becomes higher than the friction limit, than


the ‘slip’ phase is reached and the connection starts to experience slip. During this phase, the axial force in the member remains constant. The slippage movement continues until the bolt is in contact with the splice plate, so that the axial forces can be transferred via bearing (phase 3). This model has been defined as the ‘stick-slip’ model.

Fig.4. Stick-slip model used for the connections

The stick-slip parameters are defined based on the configuration and geometry of the connections and depend on the number of bolts, number of contact surfaces and friction coefficient, and on the pre-stress value of the bolts, s. Fig. 4. The stick parameter (i.e., friction force) per bolt FR,b is calculated acc. EN1993-1-8 s. 3.9.1 [2]:

CpSbR FnkF ,, (1)

The maximum amount of possible slip depends on the available tolerance. In case of a bolt with nominal diameter d and hole diameter d0, the total hole clearance is c = d0-d, which means a possible slip of ±c/2 mm assuming the bolt is centred in the hole. The actual maximum slip of each bolt in a single connection depends on the initial position of the bolt in respect to the hole. It has been assumed that in average the overlapping holes are aligned with each other and with the bolt axis after installation. Assuming the most unfavourable position of the bolts, and since we have a connection plate on each of the two sides, we have for each connection a maximum possible slip of 4c, which is the sum of the maximum possible relative displacement due to slippage within the connection and in respect to the node. If the bolts are centred in the hole, the possible slip reduces to 2c. In order to minimize the effects due to slippage in the connections, the bolted connections have been executed with hole clearance reduced by 1mm in respect to the nominal hole clearance as defined in

Friction

Slip

1

2

3

(symmetric)

uslip

FR,b


section 6.6.1 of the EN 1090-2. Taking a M20 bolt as an example, instead of the normed hole size of 22mm only 21mm was executed. A sensitivity study has been performed using the stick-slip modelling approach, for which different stick (friction parameters) and different slip properties (different possible slip deformation) have been selected and varied, using different combinations. From the different models tested, the three models listed in Table 1 have been selected for comparison purposes.

Table 1. Overview of different considered models

No. Model type Description

1 Elastic without stick-slip

2 Average stick-slip with average slip (bolt installed in the centre of the hole)

3 Maximum possible stick-slip with maximum possible slip (bolt installed on opposite sides of thehole, in the most unfavourable position)

Model type no. 3 represents a situation which is very unlikely to occur, for which all the bolts in the structure are located in the most unfavourable position. However, it provides an estimate of the upper bound limit for the deflections.

4.4 Deflections of the Dome The deformed shape of the dome under self-weight is shown qualitatively below (deflections scaled for better visualisation).

Fig.5. Dome deformed shape (qualitative representation)

The largest vertical deflections are located at the outer ring beam in between the supports. Another important deflection of reference is the deflection in the middle of the dome. The maximum deflections under self-weight are evaluated at these locations for each of the three different model types considered, s. Table 2.

Table 2. Maximum dome vertical deflections under dead loads in mm, for different locations and different models

Isometrie

In X-Richtung


Model type Max. deflection at outer

ring beam [mm] Max. deflection at dome

centre [mm]

1. Elastic deflection 291 232

2. Average stick-slip 695 794

3. Max. possible stick-slip 1021 1095

In order to compare the calculated values with the actual deflections, the level of the outer ring beam has been selected as a reference and its level has been continuously surveyed on site during the dome installation. The level is plotted in a diagram where the y-axis is the absolute vertical level in mm and the x-axis gives the position along the circular development of the outer ring beam, starting at the pier 4 according to the sketch included in the upper right-hand corner in the diagram below, in which the dome area is shown divided in four quadrants. The uppermost curve represents the planned geometry as the structure is built on the installation scaffold. This is the initial pre-cambered shape of the dome. The computed curves 1 to 3 represent the theoretical level of the outer ring beam under total dead load, using 1) elastic, 2) average stick-slip and 3) maximum possible stick-slip conditions, respectively. Furthermore, two curves are obtained based on actual measured values obtained from site surveys. The two chosen surveys are taken at different stages of construction. Survey 1 was at a stage where the dome was partially covered with cladding.

Fig.6. Outer ring beam level diagram

Survey 2 was after full installation of all components of the dome. The valleys of the diagram occur at the locations between piers which are the locations with the highest deflections. The difference between the maximum stick-slip level and the elastic level shows the possible range of deflections, being the elastic level a lower bound value and the maximum stick-slip an upper bound. The average stick-slip curve develops between both, as expected.


By comparing the actual surveyed values with the theoretical, it can be observed that both measured values lie within the expected range. It is also interesting to observe that the survey indicates that the actual deflection is not only higher than the value calculated using an elastic model but higher than the results of the average stick-slip model as well. By interpreting these results, one could infer that the actual amount of total slippage which occurred was higher than the one predicted by the average stick-slip model. This result can be explained by the bolts being (in average) in a rather unfavourable position that the one assumed. Another reason might be lower friction properties or lower bolt pre-tensioning. The lack of symmetry of the survey is probably due to the installation procedure, which followed an unsymmetrical pattern. The diagram in Fig. 6 allows to conclude that in the particular case of the Louvre Abu Dhabi Dome the influence of the stick-slip effects on the deflections must be considered in order to obtain an accurate estimate of the deflections, which is confirmed by the surveyed values.

4.5 Distribution of internal forces of the Dome Analyses of the load distribution are particularly important in case of hybrid bolted and welded structures which are part of the supporting space frame of the dome. A comparison between the internal force distribution shows in general only moderate changes between an elastic model and a model using stick-slip properties for most of the members in dome centre.

(a) Normal forces without translational stick-slip effects.

max N: 19300kNmin N: -19278kN

(b) Normal forces with translational stick-slip effects.

max N: 25004kNmin N: -22482kN

Fig.7. Comparison of member normal forces near support without consideration of stick-slip effects (a) and with stick-slip effects (b).


However, there are certain areas of the structure where a more significant change in internal force distribution can be observed. These are located in areas close to the supports or to the inner ring beam, where the connection types usually change from bolted to welded and therefore the hybrid characteristics of the structure become more relevant. Depending on the connection configuration and on the pre-tensioning of the bolts, as soon as the friction force is reached the bolted connections start to slip and, therefore, keep the load for any load increment until the gaps are closed. During this process the global load path changes and any load increments are redistributed to members which can take additional loads, e.g.:

(a) members with bolted connections for which the friction limit has not been reached yet (b) members with bolted connections where the slip length is consumed (c) members with welded connections

The forces which are re-distributed to neighbouring bolted members of type (a) can be transferred until the friction limit reached, and then will also be re-distributed to other members, which results in an iterative process where the structural system may be changing repeatedly as the load keeps being incremented, until it reaches equilibrium. Finally, if in average the internal forces on the members tend to exceed the friction limits of most of the connections, then the system will achieve an equilibrium situation for which it would be expected that the welded connections to become more onerous as these will need to carry a bigger portion of the redistributed loads. In Fig. 7, an example is given which illustrates possible load re-distribution in the structural system of the Louvre Dome. The internal normal forces near one of the support points are evaluated for dead load. Part (a) of the figure shows the results without consideration of stick-slip effects whereas in part (b) the results with stick-slip effects are shown. The members near the bearing are welded and therefore the resulting normal forces are significantly higher when stick-slip effects are taken into account. The same is true for the outer ring-beam (shown in the bottom part of the figures) which is completely welded. Due to the rigidity of the ring-beam additional normal forces are attracted by load re-distribution.

5 CONCLUSION

The effects on the global analysis of the slip in bolt holes are significant for the structure of the Louvre Abu Dhabi Dome. A simple stick-slip model is developed for the bolted joints and integrated in the global structural design. The parameters for the stick-slip model are derived both from design codes and site surveys. This heuristic method leads to satisfying results, but further research is necessary to establish more general rules and methods for this problem.

6 SUMMARY AND ACKNOWLEDGMENT

Acknowledgements are addressed to structural engineers Brian Cole, Andy Pottinger and Chris Shrubshall (all Buro Happold) and Amer Ellewi (ASO JV - Arabtec, San Jose and Oger Joint Venture)

REFERENCES

[1] EN 1993-1-1, Eurocode 3: Design of steel structures – Part 1-1: General rules and rules for buildings.

[2] EN 1993-1-8, Eurocode 3: Design of steel structures – Part 1-8: Design of joints.

[3] EN 1090-2, Execution of steel structures and aluminium structures – Part 2: Technical requirements for steel structures.

[4] Imbert, F., Frost, K., Fischer, A., Witt, A., Tourre, V. and Koren, B.; Concurrent Geometric, Structural and Environmental Design. In: Hesselgren, Sharma, Wallner, Baldassini, Bompas, Rauynaud (Hrsg.) Advances in Architectural Geometry 2012

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