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Finite element modelling and analysis of bolted joints of 3D tubular structures
Computers and Structures 82 (2004) 2173–2187 www.elsevier.com/locate/compstruc
Finite element modelling and analysis of bolted joints of 3D tubular structures La´szlo´ Gergely Vigh, La´szlo´ Dunai
*
Department of Structural Engineering, Budapest University of Technology and Economics, H-1111 Budapest, Bertalan Lajos u. 2, Hungary Received 3 February 2003; accepted 23 April 2004 Available online 13 August 2004
Abstract In the paper, research results on a special joint of a new type of steel cooling tower are presented. The structural members are tubular elements with high radius-to-thickness ratios. The 3D jointing and the bolted connections result in extremely complex joints that should be studied by experiments and advanced analysis. This paper focuses on the numerical modelling of two joint prototypes. The models are verified by test results. Virtual experiments are completed to study the elastic load-transfer as well as the ultimate behaviour. On these bases, the failure modes are determined and characterized, and the design procedure is improved. 2004 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. Keywords: Steel cooling tower; Tubular structure; Bolted endplate connection; Non-linear FEM; Shell buckling; Virtual experiment
1. Introduction In the framework of a governmental R&D project, a Hungarian ‘‘Steel Cooling Tower Consortium’’ aimed to develop a new type steel cooling tower [1]. The specialities of the structure can be described by three factors: Firstly, prefabricated tubular members with high radius-to-thickness ratio are applied (i.e. 60), as shown in Fig. 1a. Secondly, a special joint has been developed (Fig. 1b and c) to connect the structural elements. Itself the joint is made of gusset plates welded to tubular elements. Then the thin-walled horizontal and vertical bars are connected to the joint by bolted endplates. In case of the first prototype, the diagonal bars are connected to the joint directly to
*
Corresponding author. Fax: +36 1 463 1784. E-mail address: [email protected] (L. Dunai).
the gusset plates by simple pinned connections (Fig. 1b). Based on the results of the first laboratory test series, this connection has been reinforced, using rectangular hollow sections, as Fig. 1c illustrates. Although the presented joint configuration is highly advantageous with regard to easy and fast erection, the different connection types result in a complex interaction in the structural behaviour. The third special aspect is that the joint is non-planar due to the 3D shape of the structure, which even more complicates the structural behaviour. The complex behaviour leads to difficulties in the design. Although the separate failure modes of the different elements are well known, complicated interaction of the failure modes may appear, for which, there is no available design method. Consequently, major aim of the project is to investigate the complex elastic and ultimate behaviour, and to develop and verify an advanced design method. For this purpose, laboratory
0045-7949/$ - see front matter 2004 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.04.008
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Fig. 1. Steel cooling tower joint: 1,2––bolted endplates; 3a––pinned connection by earplates; 3b––pinned connection by hollow section; 4––gusset plate; 5––coverplate.
and virtual experiments are being completed on 1:2.5 scaled test specimens [2]. This paper concentrates on the numerical study of two prototypes of the joint: First it deals with the numerical modelling technique. The developed model is verified by test; numerical analyses and virtual experiments are fulfilled. The possible failure modes are separated and summarized. The results are applied in the preparation of the laboratory tests, in the improvement of the joint (development of the proto-
types) as well as in the development of the advanced design method.
2. Modelling technique 2.1. Modelling aspects The numerical analyses are completed on the base of the ANSYS [3] finite element program system. In the
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numerical model, 4-node quadrilateral shell elements (designated as SHELL 181 in ANSYS [4]) are applied, providing high efficiency and convenience. Additionally, this element is able to handle large displacements and strains; moreover, it is recommended in case of material and geometrical non-linearities [3]. The bolted endplate connection plays significant role in the elastic as well as in the ultimate behaviour of the joint. Due to the elongation of the bolts and the contact of the endplates, it has semi-rigid feature in elastic stage. Beside the global instability of the tubular member, the following failure modes of the bolted endplate connection and the connecting zone may occur: • yielding of the whole cross-section of the tubular member (‘‘I’’-type), • local buckling of the tubular member (‘‘II’’-type), • bolt failure: plastic deformation and fracture (‘‘A’’type), • bolt failure and plastic hinge arising at the intersection of the tubular wall and endplate (‘‘B’’-type), • plastic hinge at the tubular wall–endplate intersection and in the endplate at the location of a bolt (‘‘C’’type). The first two failure types are related to the tubular member, and independent from the connection configuration. The endplate thickness can influence the buckling mode and shape in extreme cases only (i.e. extremely thin endplate). The failure modes denoted as ‘‘A’’, ‘‘B’’ and ‘‘C’’ have great importance in the global joint behaviour; this is why the endplate connection is separately investigated by a local model. 2.2. Local model of the bolted connection The following cases are studied in the preliminary research program:
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• modelling as fully rigid connection; • model as semi-rigid connection with no bolt failure; • model as semi-rigid connection with possible bolt failure. The first model shown in Fig. 2a represents the theoretical case of rigid connection, e.g. relatively thick endplate and strong bolts. The endplates are modelled as one double-thickness member, and bolts are eliminated. The second one (Fig. 2b) separately models the endplates that are connected by linear tension-only spring elements (element type LINK10 [4]) acting for the bolts. This model can give satisfactory results in case of ‘‘C’’type failure (no bolt failure). Note that rigid compression-only elements (LINK10) are required to represent the contact phenomenon between the two endplates. Additionally, further links (coupled nodes) are used to avoid the lateral slide of endplates on each other. The element type COMBIN39 [2] with bilinear elasto-plastic spring characteristics acting for the real bolt characteristics is used in the third case, by which, the bolt failures can be considered (‘‘A’’ and ‘‘B’’-type failures), as well. Apart from this, the model has the same feature as the previous one (Fig. 2b). For the comparison of the models, the most unfavourable connection configuration of the actual structure is selected: D/t = 600/5, with 14 · M20 10.9 bolting. Parametric study is completed to find the effect of the endplate thickness on the structural behaviour. The numerical results show that the separate modelling of bolts is required even in case of relatively thick plates (Fig. 3a), because of the experienced semi-rigid behaviour. The rigid model results in a collapse caused by the interaction of full cross-section yielding and local plastic instability. Actually, this model symbolizes a theoretical case: This moment capacity cannot be achieved in real conditions. The differences are even higher in case of thinner endplates (Fig. 3b). It is also proved that non-linear spring characteristics cannot
Fig. 2. Finite element modelling ways of bolted endplate connection.
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800
700
700
600
600
500
500
Moment [kNm]
Moment [kNm]
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400 300 200
400 300 200
rigid endplate lin. spring
100
rigid endplate lin. spring 10.9 NL-spring
100
10.9 NL-spring 0 0.000
0.005
0.010 0.015 Rotation [rad]
0.020
0 0.000
0.025
(a) thick endplate (30 mm)
0.005
0.010 0.015 Rotation [rad]
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0.025
(b) thin endplate (10 mm)
Fig. 3. Comparison of models.
800 700
500
C
B
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Moment capacity [kNm]
Moment [kNm]
600
14 x M20 10.9 bolts
300 200
700 690 680 670 660 650 640 630 620 610 600 10
15 20 25 30 Thickness of endplate [mm]
100 0 0.000
0.005
0.010
0.015
0.020
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Rotation [rad] rigid endplate, tk = 30
tk = 30
tk = 20
tk = 15
tk = 10
Fig. 4. Effect of endplate thickness on ultimate behaviour.
be eliminated in case of ‘‘C’’ and even in case of ‘‘B’’type failures. The results of the parametric study coincide with the expectations in terms of the failure mode tendency (Fig. 4): thick endplate goes with the ‘‘B’’-type failure, while the ‘‘C’’-type one appears with less than 15 mm of thickness (this failure mode is illustrated in Fig. 5. Among the possible and still practical cases, ‘‘A’’-type failure could
not be found. It is also observable that increasing the endplate thickness is not economical after reaching the ‘‘B’’-type failure, since the differences in ultimate capacity are relatively small. After all, with respect to the whole steel cooling tower joint, it is concluded that the bolts and contact of endplates should be modelled in the third discussed way.
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Fig. 5. Failure mode ‘‘C’’.
2.3. Global model of the joint Based on the preliminary studies, the shell-element model of the whole joint, shown in Fig. 6, is developed. The model geometry is derived from the actual 1:2.5 scaled test specimen. To avoid the disturbance caused by direct loading, the length of the tubular members is determined as about five times of the diameter. The possible mesh densities are determined by the given bolt allocation. For most of the analyses, the shown mesh is eligible; however, for plastic analyses, the inner joint parts are refined to follow the phenomena (e.g. plastic buckling). The ends of the bars are loaded: The horizontals can be tensioned, compressed and/or bended; the verticals are always compressed. The compressed or tensioned diagonals are not modelled at this level of the research: the pinned connection gives the opportunity to substitute them by their load only. At the modelled bars, uniform load distribution is achieved by means of relatively rigid plates at the loaded end.
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As the structural material, linear elastic or linear elastic––perfectly plastic material model is set, for the stability analysis and for the virtual experimenting, respectively. The material model comes with the following steel properties: YoungÕs modulus of E = 210 GPa, PoissonÕs ratio of m = 0.3, yield stress of ry = 240 MPa. It is noted that ANSYS requires true stresses vs. logarithmic strains as input data when large displacement and/or large strains are applied [3]. For the rigid elements (load-transfer plate and pressure-only springs), linear elastic material model is adjusted, too, but the YoungÕs modulus and their geometry are increased with orders of magnitude (i.e. E = 210,000 GPa, plate thickness as t = 60 mm, spring ‘‘cross-sectional’’ area as As = 106 mm2). As mentioned before, the bolt-representing springs are associated with bilinear spring characteristics derived from test results of bolts. In the analysis, large displacements are considered. For the instability investigations, the Block Lanczos method that uses the sparse matrix solver was applied [3,5]. During the virtual experimenting, force-controlled, arc length solution method was used [3]. The iteration was completed on the bases of the Newton–Raphson method. The convergence of the iterations was checked by the Euclidian norm of the unbalanced forces and moments; the applied convergence tolerance factor is 0.1%. 2.4. Model developing platform It is clearly seen that ‘‘manual’’ model building of the complicated joint consumes much time, resulting in inefficient research work. This fact motivated us to prepare an input interface shown in Fig. 7, using the programme language MATLAB [6]. The developed software creates the script-file that is directly readable by ANSYS and the entire numerical model can be built up in a few seconds.
Fig. 6. Finite element model of the joint.
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Fig. 7. MATLAB platform, input interface.
3. Numerical analyses––first prototype of the joint 3.1. Convergence test Beside the check in stage of the preliminary research, convergence study is also done in case of the whole joint, subjected to a general load case (bended horizontals, compressed column and diagonals). Three meshes are tested: the total element number of 2844 (2712 nodes), 4920 (4774 nodes) and 9428 (9229 nodes). Even between the first and the third models, the difference in reaction forces (at the bottom column) and in maximal deflections is less than 0.5% and 1.7%, respectively. Finally, the second meshing––which uses twice number of elements as bolts are located around the circum-
ference––is selected for the further elastic analyses, meaning the element number of 5364 and the node number of 5164 at the final configuration. The illustrated mesh is eligible for global analysis. Nevertheless, when analysing peak stresses or local phenomena, the corresponding parts have to be refined, as the results––discussed later in this paper––prove. 3.2. Stability analyses As the first step of the numerical analyses, the elastic stability phenomena are investigated. Based on the original geometry and an actual load case including dominantly bended horizontal, compressed vertical and diagonal members (Fig. 1a), parametric study is com-
Fig. 8. Buckling modes of gusset plates.
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Fig. 9. Buckling modes of tubular members.
right horizontal (24.32 kN)
right horizontal
column (40 kN) left horizontal (24.32 kN)
column (80 kN)
right diagonal (25.6 kN)
left diagonal (25.6 kN )
column
left horizontal right diagonal
left diagonal
column
(b) Loadcase “T”
(a) Loadcase “A” Fig. 10. Investigated load cases.
pleted by changing the thickness of gusset plates and tubular bars, in order to find the possible buckling modes in each member. The original geometry results in dominant failure modes of the gusset plates. This meets the expectations as the tubular bars are designed to avoid the local buckling. The first arising mode is related to the diagonals as Fig. 8a shows. The figure also introduces the further buckling modes of the gusset plates: due to the bending of horizontals (Fig. 8b) or due to the compression transferred from the columns (Fig. 8c and d). Evaluating the results, it is concluded that the buckling of the gusset plate with the applied parameters is a possible failure
mode that should be considered in the improvement of the joint. Changing the plate thicknesses leads to simple local buckling of the tubular members. However, the buckling shapes and locations are modified. In the compressed member, buckling appears on one side only (Fig. 9c). Similarly, the buckling position is changing along the axis of the bar in the horizontals (Fig. 9d). The reason is the interaction between the different members and their internal forces: The joint is forced to rotate around the longitudinal axis of the horizontal bars, causing additional bending in the columns and additional torsional moment in the horizontals.
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3.3. Virtual experimenting 3.3.1. Elastic behaviour The force-flow of the joint and the interaction of the members is further investigated by extended experimental studies and linear elastic analyses. Hereafter, the load cases shown in Fig. 10 are discussed in details. Firstly, a general load case is applied, representing the actual load ratios of the structure: compressed and tensioned horizontals, compressed column, and tensioned and compressed diagonals (Fig. 10a). Fig. 11 shows the von Mises stress distribution in the joint. It is clearly observable that the intersection of the columns and the gusset plates can be a critical point, especially in
the case when the horizontal displacements of the joint cause extra bending of the column. Similar tendency can be observed in the case of compressed column (Figs. 10b and 12). Peak stresses caused by the complex geometry arise in the bar and the gusset plates at their intersections. Based on the analysed cases, good agreement is found between the numerical model and the tests, in the comparison level of reaction forces and displacements. For instance, at load case ‘‘A’’, at the load level of 40 kN in the column and 24 kN in the horizontals, the bottom reaction forces are exactly the same (37.96 kN), while the central displacements are 2.2% greater than in the test (5.03 and 4.92 mm, respectively). Generally, the
Fig. 11. Stress distribution due to the general load case ‘‘A’’.
Fig. 12. Stress distribution due to the compressed column (load case ‘‘T’’).
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Fig. 13. Comparison of test and analysis results (load case ‘‘T’’).
load preload
load
preload
preload
preload
Fig. 14. Load case ‘‘Y’’.
model also follows the tendency of the complicated global stress distribution, as shown in Figs. 12 and 13. However, although the peak stresses are clearly observable, for their detailed analyses the mesh should be refined in the vicinity of geometrical intersections in order to get more realistic values. 3.3.2. Ultimate behaviour Various load cases are applied and virtual experiments are completed by non-linear analyses in order to separate the possible failure modes of the joint. One typical failure mode is discussed hereafter. In order to get the pure failure of the horizontals, non-axial forces are applied at their ends, meaning bend-
ing and compression, as illustrated in Fig. 14. Considering the actual experimental conditions stabilizing the joint setup, the diagonals and the column are preloaded (tension and compression, respectively) by a relatively small force (Fig. 14). In this way, the collapse occurs due to local plastic buckling in the horizontal member, inside the joint, as it can be seen in Fig. 15. Fig. 16a proves that the tubular member has not reached its plastic capacity. Based on the evaluation of strains and stresses, a buckling load could be defined (Figs. 15 and 16b). It is also found that the endplate connection does not fail before the instability phenomenon appears. Consequently, the capacity of the endplate connection as well as the capacity of the horizontals is not utilized. In order to decrease the dominance of the observed failure and improve the load bearing capacity of the structure, the plate thickness inside the joint should be increased to take the capacity closer to the ones of the bolted connection and the tubular members outside the joint (uniform resistance design). In a separate analysis, it is found that thickening by 1 mm results in approximately 10% increase in the load capacity. It is important to highlight that the same collapse phenomenon is established in the virtual experiment as in the real test (compare Figs. 15 and 17a). The ultimate load is close to the test result (compare 127.9 to 132.9 kN, meaning a difference of 3.8%). Moreover, the joint rigidity meets the reality (Fig. 17b); the maximum displacement is 21.02 and 22.27 mm in the analysis and in the experiment, respectively. The only difference is the buckling location: in the test, it is close to the ending gusset plate. This difference may come from the geometrical imperfections of the test specimen.
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Fig. 15. Failure in horizontal tubular members.
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45 40
30
yielding start
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Moment [kNm]
35
20
1
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Rotation at the end of the horizontals [-]
(a) moment vs. rotation curve (at the end of horizontals)
50
Strain [x10-3]
(b) stress vs. strain relationship nodes at the buckling position
Fig. 16. Failure in horizontal tubular member––details.
Separate analyses prove that weakening the bolted connections in the horizontals (even if applying the half bolt number of the designed one) does not influence significantly the shown ultimate behaviour.
gusset plate and the horizontals, as shown in Fig. 1c. Additionally, the radius of the lower column is increased to one of the other tubular members. Further real and virtual experiments are completed on this specimen, in the same configuration as discussed previously.
4. Numerical analyses––second prototype of the joint 4.2. Stability analyses 4.1. Joint improvement During the test series of the first prototype of the joint, it was found that the pinned connection of the diagonals is not eligible: the lateral movements of the bars keep bending the gusset plate in highly unfavourable way. This is why the connection has been reinforced by rectangular hollow section welded to the
As expected, the applied reinforcement does not influence significantly the elastic stability phenomena. However, the first buckling mode corresponding to the pinned connection––due to the reinforcement–– disappears. In the other cases, the same buckling modes are found; the critical loads are hardly modified.
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Fig. 17. Comparison to the experimental results.
4.3. Virtual experimenting 4.3.1. Elastic behaviour Similarly to the previous test series, various elastic load cases are investigated. Detailed numerical analysis is completed in a general load case shown in Fig. 18: the horizontals are compressed and bended, the column and the diagonals are compressed. The computer results meet the test outcomes, in term of displacements, reaction forces (bottom reaction of 2.12 kN at the load level of 20 kN at the top), as well as the tendency of the stress distribution. As an example, the stress distribution in a column cross-section close to the bolted plate is compared in Fig. 19. The test as well as the FE analysis proves that the actual configuration of the pinned connections is satisfactory.
4.3.2. Ultimate behaviour in case of compressed column In the last experimentally investigated load case with simply supported horizontals and eliminated diagonals, only the column was loaded. At a relatively low load level (193.85 kN), the lower column failed by local plastic buckling (Fig. 21a), which forced us to complete a detailed analysis. During the previous experiments that are not discussed here, a couple of imperfections that surely influenced the last test were created, such as • curvature of the column as global imperfection (Fig. 20a); • local distortion as local imperfection of the lower column, caused by an accidental crash of the column and the diagonal (Fig. 20c);
20 kN
12.16 kN
12.80 kN 12.16 kN 12.80 kN
Fig. 18. Load case ‘‘A’’.
• failure of the horizontal local plastic buckling (Fig. 20b). Parametric study is completed to investigate the effect of the above imperfections and the radius-to-thickness ratio of the column to find out whether the experimented failure may occur in practice. Fig. 21 proves that the test failure mode is found by the analysis. However, in case of perfect specimen, an ultimate load of 242.5 kN is obtained, which is 22% higher than the test one. Note that if the plate buckling is excluded, the load capacity of the perfect specimen is theoretically 358.9 kN; the difference from the above analytical result well represents the influence of the local plastic buckling. Fig. 22 shows the ultimate load vs. global imperfection relation, while Fig. 23 illustrates the corresponding
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buckled shapes. In case of the more practical imperfections (L/70–L/35), the change in ultimate load can be neglected. Although an extremely big imperfection (i.e. 95 mm; L/15) would result in the experimental ultimate load, the comparison of the buckled shapes proves that not a large global imperfection reduced the load capacity. Note that, with regard to the test experiences, an imperfection of 40 mm can be considered as the most realistic.
Based on separate studies, it is found that the local accidental imperfection hardly modifies the ultimate behaviour; e.g. the load capacity is reduced by 0.4% only. Further investigations prove the important role of the horizontal failure as imperfection. It is observed that horizontals––even though they are not loaded––bear relatively high stresses due to the global curvature (Fig. 24a). Although applying a further imperfection to represent the plastic buckling of horizontal (Fig. 20b) causes
Fig. 19. Comparison of test and analysis results (load case ‘‘A’’).
Fig. 20. Imperfections.
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Fig. 21. Failure due to vertical compression.
negligible changes in the ultimate behaviour (Fig. 24b), a theoretical case eliminating the horizontals––as if they are not able to carry any load––results in a drastically
250 240
Ultimate load at top
Ultimate load [kN]
230 220 210 200
Imperfection [mm]
190
Ult. load Reaction (bottom) (top) [kN] [kN]
0
242.50
231.69
20 (L/70)
242.39
230.40
170
30 (L/47)
238.95
226.01
40 (L/35)
235.73
221.06
160
95 (L/15)
197.18
182.12
180
Reaction at bottom
150
0 20 40 60 80 100 Global imperfection (lateral deflection of joint center) [mm]
Fig. 22. Effect of global imperfection.
reduced capacity of 166.52 kN (Fig. 24b) that is even smaller than the test value (193.85 kN). Consequently, large value of imperfection should represent the previous failure. Additionally, it is concluded that the vertical ultimate load is highly influenced by this local failure and the global imperfection. The parametric study on the radius-to-thickness ratio (Figs. 25 and 26) certify that the buckling mode does not change at higher value of R/t (Fig. 26b–e) and that the plastic instability becomes less dominant at increasing thickness, the stress distribution tends to the fully plastic one (Fig. 26a). It is also found that increasing the wall thickness inside the joint only efficiently improves the capacity. After all, it can be stated that the low level of ultimate load experienced in the test was resulted by a misadventures interaction of different––and practically big–– imperfections: global curvature and horizontal failure. According to these findings it can be concluded that the ultimate load in practical cases is surely higher than it was in the test.
Fig. 23. Deformed shape and stress distribution at different global imperfections.
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Fig. 24. Role of horizontals.
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400
300 250 200
t [mm]
R/t [-]
120
3
40.0
359.7
345.9
120
2
60.0
241.0
229.6
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170.6
163.1
120
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120.0
104.9
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120
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68.7
65.8
Load (compression) [kN]
Ultimate load [kN]
350
Ult. load Reaction (top) (bottom) [kN] [kN]
R [mm]
150 100
350 300 250 200
0
0 100
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80 120
50
50
60
100
50 0
40
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-2
171.4
0
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4
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8
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Axial displacement of the top [mm]
R/t of the column [-]
Fig. 25. Effect of the radius-to-thickness ratio.
Fig. 26. Deformed shape and stress distribution at different R/t.
5. Summary and conclusions The research work presented in this paper can be summarized as follows:
Firstly, shell-element numerical model for the bolted endplate connection is developed. Based on the separate studies on the elastic and ultimate behaviour, its structural role is evaluated.
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The modelling way of the whole steel cooling tower joint is examined. Input interface is programmed in MATLAB, by which, the presented models of the two types of joint are created. Stability analysis and virtual experimenting (including elastic and ultimate analyses) are completed on the first prototype of the joint. By these, the possible failure modes are scanned and separated. The analyses representing various load cases draw general idea about the structural behaviour, show how the joint transfers the forces in the different cases. It is found that elastic local instability phenomena do not appear in the practical cases, especially when concentrating on the original geometry. The collapse always occurs in plastic stage, due to yielding and/or plastic instability. The load-bearing capacity relation of the joint and the connected members is evaluated; the critical points of the structure are investigated. It can be stated that the inner parts of the tubular members should be thickened in order to achieve more economical structure. To determine the required plate thicknesses, further parametric studies are needed. The bolted endplate configuration, based on the preliminary studies, is considered eligible; moreover, it can be weakened. As proved, its load bearing capacity approaches the one of the theoretically rigid connection. The real experiments showed that the connection zone of the diagonals should be studied by the numerical model. In case of the second prototype, a general elastic load case is investigated and particular attention is paid for the ultimate behaviour due to vertical loading. The elastic analysis as well as the test proves that the improved joint can safety transfer the forces, without undesirable phenomena. By wide-range parametric studies of the compressed column, the effect of different imperfections on the ultimate behaviour is investigated. The experimented failure type is found. Based on the results, it is concluded that the early collapse observed in the test cannot occur in practice. It is also stated that increasing the plate thickness of the tubular members inside the joint can efficiently improve the load bearing capacity.
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The presented analyses highly supported and guided the preparation of the real tests, by defining the details that are worthy to examine in the reality. The developed numerical model, after further calibrations by the laboratory tests, is going to be applicable in the further virtual experiments. The completed comparison of the results of the real experiments and the numerical models promises that the virtual experiments can substitute the costly real ones. This directly supports the design of the cooling tower. By the help of the complete MATLAB platform, further parametric studies can be done, which can lead to the improvement of the joint and the development of the advanced design method.
Acknowledgment The research work is conducted under the financial support of the NKFP 3/023/2001 R&D project.
References [1] Dunai L, Horva´th L, Kaltenbach L, To´th J, Vigh LG. Analysis of steel cooling tower joint/Research report #1 [in Hungarian]. Budapest: Budapest University of Technology and Economics, Department of Structural Engineering; 2001. [2] Dunai L, Horva´th L, Kaltenbach L, To´th J. Experimental study of steel cooling tower joint/Research report #2.1 [in Hungarian]. Budapest: Budapest University of Technology and Economics, Department of Structural Engineering; 2002. [3] ANSYS Structural Analysis Guide. Online Documentation ANSYS Inc.; 2001. [4] ANSYS Elements Reference. Online Documentation ANSYS Inc.; 2001. [5] Rajakumar C, Rogers CR. The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem. Int J Numer Meth Eng 1992;32:1009–26. [6] The MathWorks Home. Online Documentation MathWorks Inc.; 2001.
Finite element modelling and analysis of bolted joints of 3D tubular structures La´szlo´ Gergely Vigh, La´szlo´ Dunai
*
Department of Structural Engineering, Budapest University of Technology and Economics, H-1111 Budapest, Bertalan Lajos u. 2, Hungary Received 3 February 2003; accepted 23 April 2004 Available online 13 August 2004
Abstract In the paper, research results on a special joint of a new type of steel cooling tower are presented. The structural members are tubular elements with high radius-to-thickness ratios. The 3D jointing and the bolted connections result in extremely complex joints that should be studied by experiments and advanced analysis. This paper focuses on the numerical modelling of two joint prototypes. The models are verified by test results. Virtual experiments are completed to study the elastic load-transfer as well as the ultimate behaviour. On these bases, the failure modes are determined and characterized, and the design procedure is improved. 2004 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. Keywords: Steel cooling tower; Tubular structure; Bolted endplate connection; Non-linear FEM; Shell buckling; Virtual experiment
1. Introduction In the framework of a governmental R&D project, a Hungarian ‘‘Steel Cooling Tower Consortium’’ aimed to develop a new type steel cooling tower [1]. The specialities of the structure can be described by three factors: Firstly, prefabricated tubular members with high radius-to-thickness ratio are applied (i.e. 60), as shown in Fig. 1a. Secondly, a special joint has been developed (Fig. 1b and c) to connect the structural elements. Itself the joint is made of gusset plates welded to tubular elements. Then the thin-walled horizontal and vertical bars are connected to the joint by bolted endplates. In case of the first prototype, the diagonal bars are connected to the joint directly to
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Corresponding author. Fax: +36 1 463 1784. E-mail address: [email protected] (L. Dunai).
the gusset plates by simple pinned connections (Fig. 1b). Based on the results of the first laboratory test series, this connection has been reinforced, using rectangular hollow sections, as Fig. 1c illustrates. Although the presented joint configuration is highly advantageous with regard to easy and fast erection, the different connection types result in a complex interaction in the structural behaviour. The third special aspect is that the joint is non-planar due to the 3D shape of the structure, which even more complicates the structural behaviour. The complex behaviour leads to difficulties in the design. Although the separate failure modes of the different elements are well known, complicated interaction of the failure modes may appear, for which, there is no available design method. Consequently, major aim of the project is to investigate the complex elastic and ultimate behaviour, and to develop and verify an advanced design method. For this purpose, laboratory
0045-7949/$ - see front matter 2004 Civil-Comp Ltd. and Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.04.008
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Fig. 1. Steel cooling tower joint: 1,2––bolted endplates; 3a––pinned connection by earplates; 3b––pinned connection by hollow section; 4––gusset plate; 5––coverplate.
and virtual experiments are being completed on 1:2.5 scaled test specimens [2]. This paper concentrates on the numerical study of two prototypes of the joint: First it deals with the numerical modelling technique. The developed model is verified by test; numerical analyses and virtual experiments are fulfilled. The possible failure modes are separated and summarized. The results are applied in the preparation of the laboratory tests, in the improvement of the joint (development of the proto-
types) as well as in the development of the advanced design method.
2. Modelling technique 2.1. Modelling aspects The numerical analyses are completed on the base of the ANSYS [3] finite element program system. In the
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numerical model, 4-node quadrilateral shell elements (designated as SHELL 181 in ANSYS [4]) are applied, providing high efficiency and convenience. Additionally, this element is able to handle large displacements and strains; moreover, it is recommended in case of material and geometrical non-linearities [3]. The bolted endplate connection plays significant role in the elastic as well as in the ultimate behaviour of the joint. Due to the elongation of the bolts and the contact of the endplates, it has semi-rigid feature in elastic stage. Beside the global instability of the tubular member, the following failure modes of the bolted endplate connection and the connecting zone may occur: • yielding of the whole cross-section of the tubular member (‘‘I’’-type), • local buckling of the tubular member (‘‘II’’-type), • bolt failure: plastic deformation and fracture (‘‘A’’type), • bolt failure and plastic hinge arising at the intersection of the tubular wall and endplate (‘‘B’’-type), • plastic hinge at the tubular wall–endplate intersection and in the endplate at the location of a bolt (‘‘C’’type). The first two failure types are related to the tubular member, and independent from the connection configuration. The endplate thickness can influence the buckling mode and shape in extreme cases only (i.e. extremely thin endplate). The failure modes denoted as ‘‘A’’, ‘‘B’’ and ‘‘C’’ have great importance in the global joint behaviour; this is why the endplate connection is separately investigated by a local model. 2.2. Local model of the bolted connection The following cases are studied in the preliminary research program:
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• modelling as fully rigid connection; • model as semi-rigid connection with no bolt failure; • model as semi-rigid connection with possible bolt failure. The first model shown in Fig. 2a represents the theoretical case of rigid connection, e.g. relatively thick endplate and strong bolts. The endplates are modelled as one double-thickness member, and bolts are eliminated. The second one (Fig. 2b) separately models the endplates that are connected by linear tension-only spring elements (element type LINK10 [4]) acting for the bolts. This model can give satisfactory results in case of ‘‘C’’type failure (no bolt failure). Note that rigid compression-only elements (LINK10) are required to represent the contact phenomenon between the two endplates. Additionally, further links (coupled nodes) are used to avoid the lateral slide of endplates on each other. The element type COMBIN39 [2] with bilinear elasto-plastic spring characteristics acting for the real bolt characteristics is used in the third case, by which, the bolt failures can be considered (‘‘A’’ and ‘‘B’’-type failures), as well. Apart from this, the model has the same feature as the previous one (Fig. 2b). For the comparison of the models, the most unfavourable connection configuration of the actual structure is selected: D/t = 600/5, with 14 · M20 10.9 bolting. Parametric study is completed to find the effect of the endplate thickness on the structural behaviour. The numerical results show that the separate modelling of bolts is required even in case of relatively thick plates (Fig. 3a), because of the experienced semi-rigid behaviour. The rigid model results in a collapse caused by the interaction of full cross-section yielding and local plastic instability. Actually, this model symbolizes a theoretical case: This moment capacity cannot be achieved in real conditions. The differences are even higher in case of thinner endplates (Fig. 3b). It is also proved that non-linear spring characteristics cannot
Fig. 2. Finite element modelling ways of bolted endplate connection.
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800
700
700
600
600
500
500
Moment [kNm]
Moment [kNm]
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400 300 200
400 300 200
rigid endplate lin. spring
100
rigid endplate lin. spring 10.9 NL-spring
100
10.9 NL-spring 0 0.000
0.005
0.010 0.015 Rotation [rad]
0.020
0 0.000
0.025
(a) thick endplate (30 mm)
0.005
0.010 0.015 Rotation [rad]
0.020
0.025
(b) thin endplate (10 mm)
Fig. 3. Comparison of models.
800 700
500
C
B
400
Moment capacity [kNm]
Moment [kNm]
600
14 x M20 10.9 bolts
300 200
700 690 680 670 660 650 640 630 620 610 600 10
15 20 25 30 Thickness of endplate [mm]
100 0 0.000
0.005
0.010
0.015
0.020
0.025
Rotation [rad] rigid endplate, tk = 30
tk = 30
tk = 20
tk = 15
tk = 10
Fig. 4. Effect of endplate thickness on ultimate behaviour.
be eliminated in case of ‘‘C’’ and even in case of ‘‘B’’type failures. The results of the parametric study coincide with the expectations in terms of the failure mode tendency (Fig. 4): thick endplate goes with the ‘‘B’’-type failure, while the ‘‘C’’-type one appears with less than 15 mm of thickness (this failure mode is illustrated in Fig. 5. Among the possible and still practical cases, ‘‘A’’-type failure could
not be found. It is also observable that increasing the endplate thickness is not economical after reaching the ‘‘B’’-type failure, since the differences in ultimate capacity are relatively small. After all, with respect to the whole steel cooling tower joint, it is concluded that the bolts and contact of endplates should be modelled in the third discussed way.
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Fig. 5. Failure mode ‘‘C’’.
2.3. Global model of the joint Based on the preliminary studies, the shell-element model of the whole joint, shown in Fig. 6, is developed. The model geometry is derived from the actual 1:2.5 scaled test specimen. To avoid the disturbance caused by direct loading, the length of the tubular members is determined as about five times of the diameter. The possible mesh densities are determined by the given bolt allocation. For most of the analyses, the shown mesh is eligible; however, for plastic analyses, the inner joint parts are refined to follow the phenomena (e.g. plastic buckling). The ends of the bars are loaded: The horizontals can be tensioned, compressed and/or bended; the verticals are always compressed. The compressed or tensioned diagonals are not modelled at this level of the research: the pinned connection gives the opportunity to substitute them by their load only. At the modelled bars, uniform load distribution is achieved by means of relatively rigid plates at the loaded end.
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As the structural material, linear elastic or linear elastic––perfectly plastic material model is set, for the stability analysis and for the virtual experimenting, respectively. The material model comes with the following steel properties: YoungÕs modulus of E = 210 GPa, PoissonÕs ratio of m = 0.3, yield stress of ry = 240 MPa. It is noted that ANSYS requires true stresses vs. logarithmic strains as input data when large displacement and/or large strains are applied [3]. For the rigid elements (load-transfer plate and pressure-only springs), linear elastic material model is adjusted, too, but the YoungÕs modulus and their geometry are increased with orders of magnitude (i.e. E = 210,000 GPa, plate thickness as t = 60 mm, spring ‘‘cross-sectional’’ area as As = 106 mm2). As mentioned before, the bolt-representing springs are associated with bilinear spring characteristics derived from test results of bolts. In the analysis, large displacements are considered. For the instability investigations, the Block Lanczos method that uses the sparse matrix solver was applied [3,5]. During the virtual experimenting, force-controlled, arc length solution method was used [3]. The iteration was completed on the bases of the Newton–Raphson method. The convergence of the iterations was checked by the Euclidian norm of the unbalanced forces and moments; the applied convergence tolerance factor is 0.1%. 2.4. Model developing platform It is clearly seen that ‘‘manual’’ model building of the complicated joint consumes much time, resulting in inefficient research work. This fact motivated us to prepare an input interface shown in Fig. 7, using the programme language MATLAB [6]. The developed software creates the script-file that is directly readable by ANSYS and the entire numerical model can be built up in a few seconds.
Fig. 6. Finite element model of the joint.
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Fig. 7. MATLAB platform, input interface.
3. Numerical analyses––first prototype of the joint 3.1. Convergence test Beside the check in stage of the preliminary research, convergence study is also done in case of the whole joint, subjected to a general load case (bended horizontals, compressed column and diagonals). Three meshes are tested: the total element number of 2844 (2712 nodes), 4920 (4774 nodes) and 9428 (9229 nodes). Even between the first and the third models, the difference in reaction forces (at the bottom column) and in maximal deflections is less than 0.5% and 1.7%, respectively. Finally, the second meshing––which uses twice number of elements as bolts are located around the circum-
ference––is selected for the further elastic analyses, meaning the element number of 5364 and the node number of 5164 at the final configuration. The illustrated mesh is eligible for global analysis. Nevertheless, when analysing peak stresses or local phenomena, the corresponding parts have to be refined, as the results––discussed later in this paper––prove. 3.2. Stability analyses As the first step of the numerical analyses, the elastic stability phenomena are investigated. Based on the original geometry and an actual load case including dominantly bended horizontal, compressed vertical and diagonal members (Fig. 1a), parametric study is com-
Fig. 8. Buckling modes of gusset plates.
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Fig. 9. Buckling modes of tubular members.
right horizontal (24.32 kN)
right horizontal
column (40 kN) left horizontal (24.32 kN)
column (80 kN)
right diagonal (25.6 kN)
left diagonal (25.6 kN )
column
left horizontal right diagonal
left diagonal
column
(b) Loadcase “T”
(a) Loadcase “A” Fig. 10. Investigated load cases.
pleted by changing the thickness of gusset plates and tubular bars, in order to find the possible buckling modes in each member. The original geometry results in dominant failure modes of the gusset plates. This meets the expectations as the tubular bars are designed to avoid the local buckling. The first arising mode is related to the diagonals as Fig. 8a shows. The figure also introduces the further buckling modes of the gusset plates: due to the bending of horizontals (Fig. 8b) or due to the compression transferred from the columns (Fig. 8c and d). Evaluating the results, it is concluded that the buckling of the gusset plate with the applied parameters is a possible failure
mode that should be considered in the improvement of the joint. Changing the plate thicknesses leads to simple local buckling of the tubular members. However, the buckling shapes and locations are modified. In the compressed member, buckling appears on one side only (Fig. 9c). Similarly, the buckling position is changing along the axis of the bar in the horizontals (Fig. 9d). The reason is the interaction between the different members and their internal forces: The joint is forced to rotate around the longitudinal axis of the horizontal bars, causing additional bending in the columns and additional torsional moment in the horizontals.
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3.3. Virtual experimenting 3.3.1. Elastic behaviour The force-flow of the joint and the interaction of the members is further investigated by extended experimental studies and linear elastic analyses. Hereafter, the load cases shown in Fig. 10 are discussed in details. Firstly, a general load case is applied, representing the actual load ratios of the structure: compressed and tensioned horizontals, compressed column, and tensioned and compressed diagonals (Fig. 10a). Fig. 11 shows the von Mises stress distribution in the joint. It is clearly observable that the intersection of the columns and the gusset plates can be a critical point, especially in
the case when the horizontal displacements of the joint cause extra bending of the column. Similar tendency can be observed in the case of compressed column (Figs. 10b and 12). Peak stresses caused by the complex geometry arise in the bar and the gusset plates at their intersections. Based on the analysed cases, good agreement is found between the numerical model and the tests, in the comparison level of reaction forces and displacements. For instance, at load case ‘‘A’’, at the load level of 40 kN in the column and 24 kN in the horizontals, the bottom reaction forces are exactly the same (37.96 kN), while the central displacements are 2.2% greater than in the test (5.03 and 4.92 mm, respectively). Generally, the
Fig. 11. Stress distribution due to the general load case ‘‘A’’.
Fig. 12. Stress distribution due to the compressed column (load case ‘‘T’’).
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Fig. 13. Comparison of test and analysis results (load case ‘‘T’’).
load preload
load
preload
preload
preload
Fig. 14. Load case ‘‘Y’’.
model also follows the tendency of the complicated global stress distribution, as shown in Figs. 12 and 13. However, although the peak stresses are clearly observable, for their detailed analyses the mesh should be refined in the vicinity of geometrical intersections in order to get more realistic values. 3.3.2. Ultimate behaviour Various load cases are applied and virtual experiments are completed by non-linear analyses in order to separate the possible failure modes of the joint. One typical failure mode is discussed hereafter. In order to get the pure failure of the horizontals, non-axial forces are applied at their ends, meaning bend-
ing and compression, as illustrated in Fig. 14. Considering the actual experimental conditions stabilizing the joint setup, the diagonals and the column are preloaded (tension and compression, respectively) by a relatively small force (Fig. 14). In this way, the collapse occurs due to local plastic buckling in the horizontal member, inside the joint, as it can be seen in Fig. 15. Fig. 16a proves that the tubular member has not reached its plastic capacity. Based on the evaluation of strains and stresses, a buckling load could be defined (Figs. 15 and 16b). It is also found that the endplate connection does not fail before the instability phenomenon appears. Consequently, the capacity of the endplate connection as well as the capacity of the horizontals is not utilized. In order to decrease the dominance of the observed failure and improve the load bearing capacity of the structure, the plate thickness inside the joint should be increased to take the capacity closer to the ones of the bolted connection and the tubular members outside the joint (uniform resistance design). In a separate analysis, it is found that thickening by 1 mm results in approximately 10% increase in the load capacity. It is important to highlight that the same collapse phenomenon is established in the virtual experiment as in the real test (compare Figs. 15 and 17a). The ultimate load is close to the test result (compare 127.9 to 132.9 kN, meaning a difference of 3.8%). Moreover, the joint rigidity meets the reality (Fig. 17b); the maximum displacement is 21.02 and 22.27 mm in the analysis and in the experiment, respectively. The only difference is the buckling location: in the test, it is close to the ending gusset plate. This difference may come from the geometrical imperfections of the test specimen.
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Fig. 15. Failure in horizontal tubular members.
50
-300
45 40
30
yielding start
25
-200
Stress [MPa]
Moment [kNm]
35
20
1
-250
local plastic buckling
2
-150 -100
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15 -50
10 5
5
-0.005
0 0.000 -5
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-5
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Rotation at the end of the horizontals [-]
(a) moment vs. rotation curve (at the end of horizontals)
50
Strain [x10-3]
(b) stress vs. strain relationship nodes at the buckling position
Fig. 16. Failure in horizontal tubular member––details.
Separate analyses prove that weakening the bolted connections in the horizontals (even if applying the half bolt number of the designed one) does not influence significantly the shown ultimate behaviour.
gusset plate and the horizontals, as shown in Fig. 1c. Additionally, the radius of the lower column is increased to one of the other tubular members. Further real and virtual experiments are completed on this specimen, in the same configuration as discussed previously.
4. Numerical analyses––second prototype of the joint 4.2. Stability analyses 4.1. Joint improvement During the test series of the first prototype of the joint, it was found that the pinned connection of the diagonals is not eligible: the lateral movements of the bars keep bending the gusset plate in highly unfavourable way. This is why the connection has been reinforced by rectangular hollow section welded to the
As expected, the applied reinforcement does not influence significantly the elastic stability phenomena. However, the first buckling mode corresponding to the pinned connection––due to the reinforcement–– disappears. In the other cases, the same buckling modes are found; the critical loads are hardly modified.
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Fig. 17. Comparison to the experimental results.
4.3. Virtual experimenting 4.3.1. Elastic behaviour Similarly to the previous test series, various elastic load cases are investigated. Detailed numerical analysis is completed in a general load case shown in Fig. 18: the horizontals are compressed and bended, the column and the diagonals are compressed. The computer results meet the test outcomes, in term of displacements, reaction forces (bottom reaction of 2.12 kN at the load level of 20 kN at the top), as well as the tendency of the stress distribution. As an example, the stress distribution in a column cross-section close to the bolted plate is compared in Fig. 19. The test as well as the FE analysis proves that the actual configuration of the pinned connections is satisfactory.
4.3.2. Ultimate behaviour in case of compressed column In the last experimentally investigated load case with simply supported horizontals and eliminated diagonals, only the column was loaded. At a relatively low load level (193.85 kN), the lower column failed by local plastic buckling (Fig. 21a), which forced us to complete a detailed analysis. During the previous experiments that are not discussed here, a couple of imperfections that surely influenced the last test were created, such as • curvature of the column as global imperfection (Fig. 20a); • local distortion as local imperfection of the lower column, caused by an accidental crash of the column and the diagonal (Fig. 20c);
20 kN
12.16 kN
12.80 kN 12.16 kN 12.80 kN
Fig. 18. Load case ‘‘A’’.
• failure of the horizontal local plastic buckling (Fig. 20b). Parametric study is completed to investigate the effect of the above imperfections and the radius-to-thickness ratio of the column to find out whether the experimented failure may occur in practice. Fig. 21 proves that the test failure mode is found by the analysis. However, in case of perfect specimen, an ultimate load of 242.5 kN is obtained, which is 22% higher than the test one. Note that if the plate buckling is excluded, the load capacity of the perfect specimen is theoretically 358.9 kN; the difference from the above analytical result well represents the influence of the local plastic buckling. Fig. 22 shows the ultimate load vs. global imperfection relation, while Fig. 23 illustrates the corresponding
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buckled shapes. In case of the more practical imperfections (L/70–L/35), the change in ultimate load can be neglected. Although an extremely big imperfection (i.e. 95 mm; L/15) would result in the experimental ultimate load, the comparison of the buckled shapes proves that not a large global imperfection reduced the load capacity. Note that, with regard to the test experiences, an imperfection of 40 mm can be considered as the most realistic.
Based on separate studies, it is found that the local accidental imperfection hardly modifies the ultimate behaviour; e.g. the load capacity is reduced by 0.4% only. Further investigations prove the important role of the horizontal failure as imperfection. It is observed that horizontals––even though they are not loaded––bear relatively high stresses due to the global curvature (Fig. 24a). Although applying a further imperfection to represent the plastic buckling of horizontal (Fig. 20b) causes
Fig. 19. Comparison of test and analysis results (load case ‘‘A’’).
Fig. 20. Imperfections.
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Fig. 21. Failure due to vertical compression.
negligible changes in the ultimate behaviour (Fig. 24b), a theoretical case eliminating the horizontals––as if they are not able to carry any load––results in a drastically
250 240
Ultimate load at top
Ultimate load [kN]
230 220 210 200
Imperfection [mm]
190
Ult. load Reaction (bottom) (top) [kN] [kN]
0
242.50
231.69
20 (L/70)
242.39
230.40
170
30 (L/47)
238.95
226.01
40 (L/35)
235.73
221.06
160
95 (L/15)
197.18
182.12
180
Reaction at bottom
150
0 20 40 60 80 100 Global imperfection (lateral deflection of joint center) [mm]
Fig. 22. Effect of global imperfection.
reduced capacity of 166.52 kN (Fig. 24b) that is even smaller than the test value (193.85 kN). Consequently, large value of imperfection should represent the previous failure. Additionally, it is concluded that the vertical ultimate load is highly influenced by this local failure and the global imperfection. The parametric study on the radius-to-thickness ratio (Figs. 25 and 26) certify that the buckling mode does not change at higher value of R/t (Fig. 26b–e) and that the plastic instability becomes less dominant at increasing thickness, the stress distribution tends to the fully plastic one (Fig. 26a). It is also found that increasing the wall thickness inside the joint only efficiently improves the capacity. After all, it can be stated that the low level of ultimate load experienced in the test was resulted by a misadventures interaction of different––and practically big–– imperfections: global curvature and horizontal failure. According to these findings it can be concluded that the ultimate load in practical cases is surely higher than it was in the test.
Fig. 23. Deformed shape and stress distribution at different global imperfections.
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Fig. 24. Role of horizontals.
400
400
300 250 200
t [mm]
R/t [-]
120
3
40.0
359.7
345.9
120
2
60.0
241.0
229.6
120
1.5
80.0
170.6
163.1
120
1
120.0
104.9
100.8
120
0.7
171.4
68.7
65.8
Load (compression) [kN]
Ultimate load [kN]
350
Ult. load Reaction (top) (bottom) [kN] [kN]
R [mm]
150 100
350 300 250 200
0
0 100
150
200
80 120
50
50
60
100
50 0
40
150
-2
171.4
0
2
4
6
8
10
Axial displacement of the top [mm]
R/t of the column [-]
Fig. 25. Effect of the radius-to-thickness ratio.
Fig. 26. Deformed shape and stress distribution at different R/t.
5. Summary and conclusions The research work presented in this paper can be summarized as follows:
Firstly, shell-element numerical model for the bolted endplate connection is developed. Based on the separate studies on the elastic and ultimate behaviour, its structural role is evaluated.
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The modelling way of the whole steel cooling tower joint is examined. Input interface is programmed in MATLAB, by which, the presented models of the two types of joint are created. Stability analysis and virtual experimenting (including elastic and ultimate analyses) are completed on the first prototype of the joint. By these, the possible failure modes are scanned and separated. The analyses representing various load cases draw general idea about the structural behaviour, show how the joint transfers the forces in the different cases. It is found that elastic local instability phenomena do not appear in the practical cases, especially when concentrating on the original geometry. The collapse always occurs in plastic stage, due to yielding and/or plastic instability. The load-bearing capacity relation of the joint and the connected members is evaluated; the critical points of the structure are investigated. It can be stated that the inner parts of the tubular members should be thickened in order to achieve more economical structure. To determine the required plate thicknesses, further parametric studies are needed. The bolted endplate configuration, based on the preliminary studies, is considered eligible; moreover, it can be weakened. As proved, its load bearing capacity approaches the one of the theoretically rigid connection. The real experiments showed that the connection zone of the diagonals should be studied by the numerical model. In case of the second prototype, a general elastic load case is investigated and particular attention is paid for the ultimate behaviour due to vertical loading. The elastic analysis as well as the test proves that the improved joint can safety transfer the forces, without undesirable phenomena. By wide-range parametric studies of the compressed column, the effect of different imperfections on the ultimate behaviour is investigated. The experimented failure type is found. Based on the results, it is concluded that the early collapse observed in the test cannot occur in practice. It is also stated that increasing the plate thickness of the tubular members inside the joint can efficiently improve the load bearing capacity.
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The presented analyses highly supported and guided the preparation of the real tests, by defining the details that are worthy to examine in the reality. The developed numerical model, after further calibrations by the laboratory tests, is going to be applicable in the further virtual experiments. The completed comparison of the results of the real experiments and the numerical models promises that the virtual experiments can substitute the costly real ones. This directly supports the design of the cooling tower. By the help of the complete MATLAB platform, further parametric studies can be done, which can lead to the improvement of the joint and the development of the advanced design method.
Acknowledgment The research work is conducted under the financial support of the NKFP 3/023/2001 R&D project.
References [1] Dunai L, Horva´th L, Kaltenbach L, To´th J, Vigh LG. Analysis of steel cooling tower joint/Research report #1 [in Hungarian]. Budapest: Budapest University of Technology and Economics, Department of Structural Engineering; 2001. [2] Dunai L, Horva´th L, Kaltenbach L, To´th J. Experimental study of steel cooling tower joint/Research report #2.1 [in Hungarian]. Budapest: Budapest University of Technology and Economics, Department of Structural Engineering; 2002. [3] ANSYS Structural Analysis Guide. Online Documentation ANSYS Inc.; 2001. [4] ANSYS Elements Reference. Online Documentation ANSYS Inc.; 2001. [5] Rajakumar C, Rogers CR. The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem. Int J Numer Meth Eng 1992;32:1009–26. [6] The MathWorks Home. Online Documentation MathWorks Inc.; 2001.