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Stiffness of joints in bolted connected cold-formed steel trusses
Journal of Constructional Steel Research 62 (2006) 240–249 www.elsevier.com/locate/jcsr
Stiffness of joints in bolted connected cold-formed steel trusses Raul Zaharia, Dan Dubina∗ Department of Steel Structures and Structural Mechanics, Faculty of Civil Engineering and Architecture, “Politehnica” University of Timisoara, I. Curea no.1, 1900 Timisoara, Romania Received 10 January 2005; accepted 1 July 2005
Abstract Web members in cold-formed steel trusses are usually assumed to have pinned connections at the ends, but the latest AISI Cold-Formed Steel Truss Design Standard allows for the joint stiffness to be considered in design. The paper summarizes experimental research performed for several years at the University of Timisoara, Romania, aimed at evaluating the real behaviour of bolted joints in cold-formed steel trusses. By means of tests on single lap joints and on truss sub-assemblies, a theoretical model for joint stiffness was proposed. The formula for the joint stiffness, from which the buckling length of web members was further determined, was also validated through a test on a full-scale truss. © 2005 Elsevier Ltd. All rights reserved. Keywords: Cold-formed steel trusses; Bolted connections; Joint stiffness; Buckling length; Experimental tests; Numerical analysis
1. Introduction Cold-formed steel framing demonstrates extensive development, even if is a relatively new system, due to some great advantages, such as high strength-to-weight ratio, reduced labor costs and fast erection due to the light weight of cold-formed members. The cold-formed steel trusses represent an economical option to the classical wood trusses used mainly in residential buildings, and to the hot-rolled trusses used for industrial applications. Several proprietary products have been developed, considering C, Z, hat, or more particular sections for chords and webs. The connections may be realized by welding, by using adhesives, with mechanical connectors as bolts or screws, or by some innovative mechanical connecting techniques such as press joining or rosette joining. The mechanical connections are among the most suitable, taking into account the production costs and rapidity of execution. Initiated by the more widespread use of the cold-formed steel in the residential construction market, systematic research to investigate the behaviour of cold-formed steel trusses was carried out at the University of Missouri-Rolla. ∗ Corresponding author. Tel.: +40 56 192970; fax: +40 56 192970.
E-mail address: [email protected] (D. Dubina). 0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2005.07.002
On the basis of full-scale tests on fink C-section truss assemblies, Harper [10] studied the buckling lengths for the top chord members, realized by a single C-section. Riemann [22] developed a computer analysis model and conducted full-scale truss tests in order to determine the capacity of compression web members, and suggested an interaction equation for the design of compression webs as beam–columns. Ibrahim et al. [11] made another series of tests on full-scale truss assemblies, considering the same system, i.e. single C-sections for chords and webs, connected by self-drilling screws. The authors proposed an interaction equation for checking unreinforced top chords subjected to axial compression, bending and web crippling. LaBoube and Yu [13] synthesized the above-presented research conducted at the University of Missouri-Rolla, which strongly influenced the design recommendations contained in the Standard for Cold-Formed Steel Truss Design issued recently by AISI [1]. This standard is intended to be a response to the problems that these particular systems raise for the designer, and applies to the design, quality assurance, installation and testing of cold-formed steel trusses used for load carrying purposes in buildings. As shown in the Standard Commentary [2], even if the structural analysis requirements are based on available information concerning the behaviour of cold-formed steel
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single C-section truss assemblies [13], these requirements do not preclude the use of more rigorous analysis or design assumptions, as determined by rational analysis and/or testing. As regards innovative mechanical connection techniques, two interesting research areas are worth mentioning here. Pedreschi et al. [17] demonstrated the efficiency of press joining, by making tests on single lap joints or groups of press joins and on full-scale pitched trusses made from coldformed Z-sections. Mäkeläinen and Kesti [15] studied the behaviour of the rosette joining system and its possibilities in roof-truss structures, by making tests on simple joints in shear or in cross-tension, and also by making tests on sub-assemblies. The authors concluded that the rosette joint has very good capacity to resist tensile forces and that the shear capacity seems to be sufficient for applications in lightweight steel trusses. The research on cold-formed steel trusses is generally focused on systems for residential roofs, having relatively reduced spans. For larger spans, efficient solutions can be achieved with higher resistance members, made for example with optimized cross-sections, like “pentagon” sections, studied by Blumel and Fontana [3]. The authors showed that the use of this particular cross-section with a large radius of gyration for both axes offers statical and constructional advantages for the chords. However, the low-cost design of the truss joints using a gusset plate welded onto the ridge of the cross-section can lead to important section deformations. The authors developed a calculation model for the local loadbearing behaviour of this particular type of cold-formed truss joint, and validated this model by means of numerical and experimental analysis on truss segments. The trusses built of cold-formed steel sections with bolted connections, made from built-up C-sections for chords and single C-sections for webs, represent another possible constructive system for residential buildings, also reliable for larger spans. In this system, the webs are connected to the chords by means of bolts placed on both flanges of the C-section of the web member. Fig. 1 shows two applications using such trusses. Fig. 1(a) shows the structure of a supplementary storey built for the Alcatel Company building in Timisoara, Romania, while Fig. 1(b) shows the trusses used to build the roof of a church in Bucharest, Romania. As regards the design of this type of cold-formed truss system, two problems arise: the stability behaviour of the compressed chord, taking into account that for these kinds of built-up members no design recommendations exists in the norms, and the real behaviour of the joints. Studies concerning the stability behaviour of the built-up C-profiles connected by bolted C-stitches were performed by Niazi [16]: based on the results obtained by Johnston [12], for hot-rolled columns in which the battens are attached to the chords by hinged connections, the compressed builtup C-section is supposed to work on an elastic foundation, provided by the roof purlins. The authors of the present
241
Fig. 1. Cold formed steel trusses.
paper calibrated a finite element model suitable for predicting the ultimate load for such built-up elements, used for the compressed chords of cold-formed steel trusses, but also as columns in cold-formed steel framing [7]. The numerical model showed good results compared with the above procedure and experimental results. As regards the analysis of the web members, it must be emphasized that the use of two or more bolts for each flange of the C-section, in relation with the element slenderness, is supposed to modify the classical assumption of pinned connections, used in case of truss structures. Moreover, the eccentricity of joints could not be avoided, and this fact should also be considered into a global analysis, because it may require additional efforts. For these reasons, the analysis of trusses built of cold-formed steel sections with bolted connections should consider the real behaviour of the joints. This may lead to reduced buckling lengths of the web members, but at the same time, to supplementary bending moments in these elements. In chapter D3 “Analysis” of the latest AISI Cold-Formed Steel Truss Design Standard [1] it is shown that: “in lieu of a rigorous analysis to define joint flexibility, the following analysis model assumptions should be assumed: . . . (b) web members are assumed to have pinned connections at
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each end. Use of a specific joint stiffness other than the complete rotational freedom of a pin for a connection shall be permitted if the connection is designed for the forces resulting from a structural analysis with this specific joint stiffness”. This means that the Standard allows for joint stiffness to be considered in the design of a truss, even if no specific equations for calculating this parameter are given. This should be in fact difficult, considering the number of different cold-formed steel truss systems. In the case of coldformed trusses with bolted connections, a rigid behaviour of the joints is also not realistic, taking into account the deformability of the joint due to the bearing deformation produced by the bolt in the thin plate, associated with the hole elongation, bolt tilting and slippage due to the hole clearance. Generally, the research in the field of bolted connections in cold-formed steel framing is focused on determining their bearing resistance. For the first time, Zadanfarrokh and Bryan [25] analysed both experimentally and theoretically the flexibility of bolted connections in cold-formed steel sections, and gave a formula for the flexibility of a single lap bolted joint, but this approach was not included in any design recommendation. More recently, [24] investigated experimentally some particular column base connections and beam–column sub-frames, made from cold-formed steel sections, in different bolted connection configurations, in order to assess their strength and stiffness. The study identified different failure modes and concluded that the bolted moment connections were effective in transmitting moment between the connected sections, enabling effective moment framing in cold-formed steel structures. Another recent study concerning the stiffness of bolted connections in a steel portal framing system was made by Lim and Nethercot [14]. The authors described a finite element model that can be used to determine the stiffness of the individual bolt joint. Using this stiffness, a beam idealization of a coldformed steel bolted moment connection was determined, in order to predict the initial stiffness of the apex joints. The numerical and theoretical study was validated through tests on full-scale joints. The research presented in this paper summarizes the work performed for several years at the University of Timisoara, Romania, aimed at evaluating the real behaviour of joints in cold-formed steel trusses connected by bolts, and at proposing a theoretical model for the joint stiffness. The experimental programme was developed in three steps. First, the rotational rigidity of some truss connections was evaluated, by means of tests on typical T-joints. In the second step, a formula for the stiffness of a single lap bolted joint was determined, together with a theoretical model for the rotational stiffness of cold-formed steel truss bolted joints. Based on this model, an equation for determining the reduced buckling length of the web members was also proposed. The third step of the experimental programme
included a full-scale test of a cold-formed steel truss, in order to validate the theoretical formulations at structural level. 2. Experimental programme step 1: Tests on T-joint specimens This first step of the experimental programme has already been described in detail [5,6], and the results were included in the Database for Research on Cold-formed Steel Structure from University of Missouri-Rolla, USA [4]. From the experimental moment–rotation characteristic of the ten Tjoint specimens tested, it was concluded that all tested joints were of semi-rigid type with partial resistance, according to Eurocode 3 [21] criteria of joint classification. An important initial rotational slippage was observed for all specimens tested, but it was not considered in the evaluation of the joint rigidity, because the triangulated shape of the truss, which is geometrically and kinetically stable, and the presence of the axial forces in connected members prevent, or limit, at least, this phenomenon at structural level. The test of the cold-formed steel truss, in the third step of the experimental programme, was aimed also at validating this assumption. From this first part of the experimental programme it was also concluded that the rotational deformability is mainly due to the bearing work of the bolts in the thin plates, i.e. the elastic and plastic deformation of the bolt holes and bolt tilting. Consequently, the rotational rigidity of the joints may be determined by analysing the single lap bolted joint. 3. Experimental programme step 2: Tests on single lap joints Experimental studies carried out in order to calibrate a formula for the flexibility of single lap bolted joint for thin-walled cold-formed elements were already performed by Zadanfarrokh and Bryan [25]. The formula proposed by these authors gives the flexibility of a single lap joint, as a function of the thickness of the plates and the presence of the threaded portion of the bolt in the connection. The formula does not include the effect of bolt diameter, being determined for lap joints using M16 bolts, but makes a distinction between perfect fit and 2 mm clearance of the bolt hole. For the case of T-joints tested at Timisoara, for all ten specimens, M12 bolts with 1 mm clearance of the bolt hole, as specified in the Romanian code, were considered. Consequently, the second step of the experimental programme was aimed at calibrating a formula for the stiffness of single lap joints, considering the plate thickness as in the Zadanfarrokh and Bryan study, but also the bolt diameter, for the practical case of a threaded portion of the bolt in the connection and 1 mm hole clearance. The experimental programme considered three different thicknesses for the plates, between 1.85 and 3.75 mm, and five bolt diameters, between 8 and 16 mm. The mechanical characteristics of the steel plates are given in Table 1.
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Table 1 Mechanical characteristics (N, mm) t
fy
fu
1.85 3.15 3.75
279.8 276.8 258.5
402.1 392 375.5
Fig. 3. Typical load–extension characteristics.
Fig. 2. Test set-up.
The test set-up is shown in Fig. 2. The specimens were tested in a ZWICK universal testing machine, using an angular displacement transducer to record the extension readings. The loading rate of 1 kN/min and the plate dimensions were in accordance with the ones used in the Zadanfarrokh and Bryan experimental programme. This value of loading rate is also given in the European Recommendations [8]. Table 2 gives the experimental values of the specimens’ stiffnesses. For each thickness of plate, three different bolt diameters were used, and for each combination of plate thickness and bolt diameter, three tests were performed (a–c). Thus, a total of 27 experimental results are available. Typical load–extension characteristics for determining the initial stiffness of the lap joint, considering an identical set of parameters, are presented in Fig. 3. The formula for the stiffness of a single lap bolted joint was calibrated using Annex D of Eurocode 0: Basis of Structural Design [18]. This appendix describes a standard procedure for the determination of the characteristic values, design values and partial factor values for strength from tests that is in compliance with the basic safety assumptions outlined in Eurocode 1: Actions on Structures [19]. As shown by Tomà [23], in a pioneering study concerning the rigidity of screw connections in cold-formed elements, underestimating the rigidity of the connection in the elastic range will relieve the connection, but will put a supplementary bending moment into the element.
Overestimating the rigidity, on another hand, leads to a supplementary moment into the joint, but this can be resolved by an appropriate design of the connection. It is safe, for the stability and displacement analysis, to underestimate the rigidity. Therefore, the Annex D procedure for calibration of the formula for the stiffness of a single lap bolted joint, will follow the same steps as for the determination of the characteristic values, design values and partial factor values for a strength-type formula. The final characteristic value, obtained after the application of the standard procedure of Annex D, for the stiffness of a single lap bolted joint, is given by the following formula [27]: √ D (kN/mm) (1) K = 6.8 5 5 + − 1 t1 t2 with a partial safety factor γ R = 1.25. In the above equation, D is the nominal diameter of the bolt while t1 , t2 represent the thicknesses of joined plates. The range of validity of this formula is for bolts between 8 and 16 mm nominal diameter and thickness of plates between 2 and 4 mm, using 1 mm hole clearance and considering the threaded portion of the bolt in the connection. It can be noted that the partial safety factor for this formula is identical to the partial safety factor used in Eurocode 3 Part 1.3 [20] for the resistance of bolted connections. On the basis of Eq. (1), the stiffness of truss joints may be determined. For the displacement analysis, and for the computation of the buckling lengths of the web members, the design value must be applied, while for the connection design the characteristic one is suitable. 4. Computation models for rotational stiffness of truss joints The computation scheme for the rotational stiffness of a truss joint with two bolts on each flange of the C-section of the web member (four bolts in total) is presented in Fig. 4. The rotational stiffness of the joint, K node,t , can be expressed in terms of total bending moment, Mtot , and
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Fig. 4. Computation model for a four bolt joint. Table 2 Experimental values for single lap joint stiffness (kN/mm) Bolt
Plate thickness t (mm)
M8 M10 M12 M14 M16
1.85 a
b
4.237 5.102 7.353 – –
4.695 6.211 5.263 – –
c
3.15 a
b
c
3.75 a
b
c
4.348 5.025 5.236 – –
– 10.000 10.869 11.111 –
– 10.417 10.753 11.628 –
– 10.204 10.526 11.765 –
– – 9.259 14.286 16.667
– – 13.333 14.493 16.393
– – 13.699 14.925 15.385
Table 3 Comparison between experimental and theoretical values of joint stiffness K node,t /K node,exp
d K node,t /K node,exp
9 830
0.971 0.958
0.777 0.766
12 480 11 110
13 083
1.047 1.177
0.838 0.942
2.05
10 560 10 968
11 418
1.080 1.041
0.864 0.833
4.05
3
15 320 15 490
16 057
1.048 1.037
0.838 0.830
4.05
4.05
21 189 20 361
20 779
0.981 1.021
0.785 0.817
t1 (mm)
t2 (mm)
K node,exp (kN mm/rad)
1 3
3
2.05
10 130 10 270
2 4
3
3
5 8
4.05
6 9 7 10
Node
corresponding rotation, θ , as [27] 2(Fa) 2K da Mtot = = d = K a2 θ tan θ 0.5a √ 6.8a 2 D (kN mm/rad) = 5 5 + − 1 t1 t2
K node,t =
(2)
with the same partial safety factor as in Eq. (1), γ R = 1.25. The term a represents the distance between bolts.
K node,t (kN mm/rad)
Table 3 shows the comparison between the experimental values of the rotational stiffness for the T-joints obtained in the first step of the experimental programme (K node,exp ) and the theoretical values obtained with the proposed relation (K node,t ). It may be observed that there is a good correlation between the experimental results and the characteristic values of the joint rotational stiffness. The average reported ratio between the theoretical characteristic values and the experimental ones is 1.036 and the correlation coefficient
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245
is ρ = 0.982. Considering the design theoretical value of d the formula (K node,t ), affected by the partial safety factor, it may be observed that all the theoretical values are in the safe range. A similar model may be used for determining the rotational stiffness for a six bolt truss joint. Considering that the centre of rotation of the bolt group is in the axis of the middle bolt, as shown in Fig. 5, the following equation may be determined [26]: 2(F × 2a) 4K da Mtot = = d = 4K a 2 θ tan θ a √ 2 27.2a D (kN mm/rad). = 5 5 + − 1 t1 t2
K node,t =
(3)
Fig. 5. Computation model for a six bolt joint.
5. Buckling length of truss web members The rotational stiffness of joints may be used to determine the buckling lengths of the web members. Trusses may be classified as fixed-node-type structures. The computation model for the buckling length of truss web members L b,web is then as for an element with fixed nodes for lateral displacement and elastic rotational springs on both ends. For this model, the buckling length should be determined considering the following equation: L b,web = µL web
(4)
where L web is the length of the web member measured between centrelines of connections and µ is computed with Eq. (5): µ = 0.5 + 0.14(η1 + η2 ) + 0.055(η1 + η2 )2 .
(5)
For the case of cold-formed steel trusses with bolted joints, the coefficients η1 and η2 may be computed as follows: K web K web η2 = η1 = K web + K node,1 K web + K node,2 E Iweb with K web = (6) L web where Iweb is the second moment of area of the web member and K node,1 , K node,2 are the rotational rigidities of the two joints connecting the web member on the chords, computed with Eq. (2) or (3), a function of the number of bolts. Note that for the computation of the buckling length, the design values of the rotational rigidities K node,i should be considered. 6. Experimental programme step 3: Test on the truss structure In order to demonstrate that the initial rotational slippage from the moment–rotation characteristics of T-joints determined in the first step of the experimental programme is not present in the truss, and to validate at
Fig. 6. Specimen dimensions. Table 4 Cross-section characteristics (mm) Profile
h
b1
b2
c
t
C100/2 C120/2
100 120
40 40
45 45
20 20
1.91 1.91
structural level the theoretical formulas presented above, a full-scale test of a truss was performed. The dimensions of the experimental model are presented in Fig. 6. All joints used six M12 8.8 grade bolts. The section characteristics are presented in Table 4 (LINDAB® profiles) and the mechanical proprieties of steel are presented in Table 5. Fig. 7 presents the experimental arrangement. The load was introduced by means of a 50 ton actuator, in displacement control, imposing a rate of 2.5 mm/min. Fig. 8 presents the instrumentation of the test. In order to measure the joint rotations, two inclinometers (R1 –R2 ) were
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Table 5 Material characteristics (N, mm) Profile
fy
fu
εu (%)
C100/2 C120/2
367.2 354
542 493.4
19 14
Fig. 7. Experimental arrangement.
Fig. 8. Instrumentation.
placed on the web of the C-section of each web member, in the axis of the lower chord joints. For determining the slippage on the connections, along the axial forces in the web members, four LVDT displacement transducers (I1 –I4 ) were placed along the axis of web members, near the joints. Four potentiometric displacement transducers (P1 –P4 ) were also used for global structural displacement control. The load increased until the failure of the structure, produced by the flexural instability of the compressed web member, occurred in the plane of the truss, as shown
in Fig. 9(a). A local buckling of the lower chord C-section webs, due to the shear of the panel between the joints, was observed (Fig. 9(b)), before reaching the ultimate load. This phenomenon increases the deformability of the joint. Fig. 9(c) presents the bolt hole plastic deformations, for the compressed web member. It may be observed that the hole of the middle bolt suffers only elongations along the axis of the member, which confirms the computation model from Fig. 5, in which the middle bolt was considered the centre of rotation for the six bolt connection. Fig. 10 presents the evolution of the displacements reported by the LVDT transducers I3 and I4 , along the axis of the compressed web member. One may observe the typical behaviour of a thin-walled bolted connection subjected to shear. After reaching the slippage force (corresponding, at bolt level, to approximately 200 daN) the initial slippage extension is extended until the hole clearance is reached; the size of this extension is a function of the initial position of the elements in the structure. Fig. 11 shows the evolution of the rotations in web member connections. Corresponding to the load range in which the axial slippage occurs, only small rotations are observed. Until the structure ‘shakes down’, the presence of the axial forces and the triangulated shape of the truss prevent the development of significant rotational slippage in connections. Consequently, the initial rotational slippage observed for the T-joints is not present in the structure, and the rotational stiffness evaluated without considering the initial slippage represents the real initial stiffness of the connection in the truss. For quantitative validation of the proposed theoretical formulas, the tested truss was numerically analysed by means of PEP-micro programme [9]. PEP-micro is a specialized programme for the non-linear analysis of steel structures with semi-rigid joints. The static scheme of the structure is presented in Fig. 12. For the stability verification of a structure, Eurocode 3 allows for a second order analysis with initial sinusoidal imperfection of the elements. The amplitude of those initial imperfections is a function of the buckling curve for the corresponding cross-section of the element. For lipped channels, according to Eurocode 3 Part 1.3, the corresponding buckling curve is B, with an initial equivalent imperfection e0 = l/380. The ultimate load of the element is then the load corresponding to the reach of the yield stress in the extreme fibre of the cross-section, taking into account the second order effects. A step-by-step second order analysis was performed, with a load step corresponding to 1% of the ultimate load. The connection eccentricities were taken into account by introducing supplementary rigid elements on the edges of the diagonals, of length L exc in Fig. 12. The structure was analysed considering the classical pinned assumption for joint behaviour and also the rotational stiffness, computed by means of Eq. (3), for a six bolt joint. The effect of the axial stiffness of the connections on the direction of the axial force in web members
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247
Fig. 9. Structure after test.
Fig. 10. Axial displacements of the web member.
Fig. 12. Static scheme of the experimental model.
for the corresponding number of bolts in the connection, i.e. multiplied by six: √ D E Aech = K axial = 6 × 6.8 (7) 10 L ech t −1
Fig. 11. Web member joint rotations.
was also considered. The PEP-micro programme is not able to model this axial stiffness, and consequently an equivalent finite element to simulate this behaviour of the connections was considered. The equivalent cross-section area of this finite element may be determined by equalizing the expression for the axial stiffness of the bar element having a length L ech equal to the distance between the centre of rotation of the bolts group and the last bolt, with the axial stiffness of the connection, K axial , determined using Eq. (1)
Fig. 13 presents the comparison between the experimental load–displacement characteristic and the results of the numerical analysis. In Fig. 13(a) are represented the numerical load–displacement characteristics, considering the rotational stiffness (K node ), together or not with the axial stiffness of the joint (K axial ). Fig. 13(b) shows also the response of the numerical model for the complete rotational freedom of web member connections (pinned), considering the axial stiffness of the joints. One may observe, in the experimental load–displacement curve, an initial structural slippage, at the level of the force corresponding to the connection slippage along the axis of the web members. Neglecting this phenomenon, not considered in the numerical analysis, the structural rigidity
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Table 6 Results of the numerical and experimental analysis K node without K axial (1)
K node with K axial (2)
Pinned with K axial (3)
Experiment (4)
(1)/(4)
(2)/(4)
(3)/(4)
Ultimate load (daN) 6650
7665
5790
7820
0.85
0.98
0.74
Displacement (mm) 3.3
9.9
7.5
15.8
0.21
0.63
0.47
Structural rigidity (daN/mm) 2015
774
772
734
2.74
1.05
0.99
the other hand, the rotational stiffnesses of the joints have a very slight influence on the structural rigidity compared with the classical assumption of pinned joints, due to the triangulated shape of the truss, but affect in a significant way the resistance of the structure. Considering the results of the numerical and experimental analysis at structural level, it may be concluded that the theoretical formulas proposed may be successfully applied to represent the joint stiffness in cold-formed steel trusses with bolted joints. 7. Conclusions
Fig. 13. Comparison between experimental and numerical analysis.
obtained numerically, taking into account both axial (K axial ) and rotational (K node ) stiffness of the connections, is very close to the experimental one. Table 6 presents the results of the numerical analysis, in comparison with the experimental values. It may be observed that the analysis considering both axial and rotational stiffness of the connections gives differences around only 2% for the ultimate load and 37% for the corresponding displacement. The difference at displacement level is due to the initial axial slippage in the joints and to the bolt plastic bearing appearing at high force levels, phenomena not considered in the numerical analysis. However, the comparison between the initial numerical and experimental structural rigidities, after the consumption of the slippage, gives differences of only 5%. It may be observed that the axial stiffness of the joints plays an important role in the structural rigidity, and it affects also, but to a lower extent, the ultimate load. On
Cold-formed steel trusses are much used, especially in residential construction, but such systems are also reliable for larger span applications. Trusses made from built-up Csections for chords and single C-sections for webs, with bolted joints, represent a suitable solution for both situations, taking into account the production costs and the rapidity of execution. Despite the extensive use of cold-formed steel truss systems and of cold-formed steel framing in general, and despite the fact that new standards appeared in recent years to cover this domain, there is still a lack of information considering different behavioural aspects of this particular type of structural system. In the case of steel trusses, current design practice assumes that web members have pinned connections, but the latest revision of the AISI Cold-Formed Steel Truss Design Standard allows for joint stiffness to be considered in the design of a truss. However, the standard did not offer formulas for calculating this parameter. In order to determine a theoretical model for the joint stiffness of bolt connected cold-formed steel trusses, the authors developed an experimental programme. In the first step, the stiffness of some cold-formed steel truss bolted joints was evaluated, by means of tests on typical T-joints. It was emphasized that the joint deformability is mainly due to the bearing work of the bolts, and consequently the rotational rigidity of the connection may be determined by analysing the single lap bolted joint. In order to determine the stiffness of a single lap bolted joint, a second experimental programme was performed, and a formula for the characteristic and design stiffness was calibrated. Using this formula, a computational model
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for the rotational stiffness of truss joints was established. The theoretical model for a four bolt joint demonstrates a good correlation in comparison with the experimental results on T-joints. Considering this model, a formula for the buckling length of truss diagonals was further determined. In the last part of the experimental programme, one test on a full-scale truss demonstrated that the initial rotational slippages observed in the T-joint tests are not present at structural level, so these slippages are not influencing the initial rotational stiffness of the truss joints. The numerical analysis of the truss tested, considering the axial and rotational stiffness of the joints, computed with the corresponding formulas for a six bolt joint, demonstrates a good correlation with the experimental results. Consequently, the formulas proposed by the authors may be successfully applied to represent the joint stiffness in cold-formed steel trusses with bolted joints. For design purposes, when computing the displacements of the truss or the buckling lengths of the web members, the design values of the joint stiffness computed with the formulas proposed in this paper have to be applied, while for the connection design the characteristic values are suitable. The study revealed that the response of the truss is influenced not only by the rotational stiffness, but also by the axial stiffness of joints, in the direction of the axial forces of web members. References [1] AISI/COFS. Standard for cold-formed steel framing — Truss design, Revision of AISI/COFS/TRUSS 2000. American Iron and Steel Institute, Committee on Framing Standards (COFS); 2001 [1st printing June 2002]. [2] AISI. Commentary on the standard for cold-formed steel framing — Truss design. American Iron and Steel Institute. Committee on Framing Standards (COFS); 2001 [1st printing June 2002]. [3] Blumel S, Fontana M. Load-bearing and deformation behaviour of truss joints using thin-walled pentagon cross-sections. Thin Wall Struct 2004;42(2):295–307. [4] Database. Database for research on cold-formed steel structures project title: Behaviour of bolted connections in cold-formed steel plane trusses — Center for Cold-formed Steel Structures. University of Missouri-Rolla M 065401-0249 USA; 1996. [5] Dubina D, Zaharia R. Experimental evidence of semi-rigid behaviour of some cold-formed steel truss bolted joints. In: International conference on experimental model research and testing of thin-walled steel structures. 1997. [6] Dubina D, Zaharia R. Cold-formed steel trusses with semi-rigid joints. Thin Wall Struct 1998;29(1–4):237–87. [7] Dubina D, Zaharia R, Ungureanu V. Behaviour of built-up columns made of C sections connected with bolted stitches. In: International colloquium on stability and ductility of steel structures. 2002.
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[8] ECCS. TC7 European recommendations for steel construction: The design and testing of connections in steel sheeting and sections, Publication No. 21; 1983. [9] Galea Y, Bureau A. PEP–micro: Analyse plastique au second ordre de structures plannes a barres. Paris (France): Centre Technique Industriel de la Construction Metallique; 1992. [10] Harper MM. Cold-formed steel in residential roof trusses. Thesis presented to the Faculty of the University of Missouri-Rolla in partial fulfillment for the degree master of Science. Missouri-Rolla, USA; 1995. [11] Ibrahim TM, LaBoube RA, Yu WW. Behaviour of cold-formed steel roof trusses under concentrated loads. J Constr Steel Res 1998; 46(1–3):176–7. [12] Johnston BG. Spaced steel columns. J Struct Div, ASCE 1971; 97(ST5):1465–79. [13] LaBoube RA, Yu WW. Recent research and developments in coldformed steel framing. Thin Wall Struct 1998;32(1–3):19–39. [14] Lim JPB, Nethercot DA. Stiffness prediction for bolted momentconnections between cold-formed steel members. J Constr Steel Res 2004;60(1):85–107. [15] Mäkeläinen P, Kesti J. Advanced method for light-weight steel joining. J Constr Steel Res 1999;49(2):107–16. [16] Niazi A. Contribution a l’etude de la stabilite des structures composees de profils a parois minces et section ouverte de type C. These presentee en vue de l’obtention du grade scientifique de Docteur en Sciences Appliquees. Universite de Liege; 1993. [17] Pedreschi RF, Sinha BP, Davies R. Advanced connection technique for cold-formed steel structures. J Struct Eng 1997;123(2):138–44. [18] prEN1990. Eurocode 0, Basis of structural design. Brussels (Belgium): European Committee for Standardisation; 2001. [19] prEN1991. Eurocode 1: Actions on structures. Brussels (Belgium): European Committee for Standardisation; 2001. [20] prEN1993-1-3. Eurocode 3, Design of steel structures, Part 1.3 General rules, Supplementary rules for cold formed thin gauge members and sheeting. Brussels (Belgium): European Committee for Standardisation; 2003. [21] prEN1993-1-8. Eurocode 3, Design of steel structures, Part 1.8, ‘Design of steel joints’. Brussels (Belgium): European Committee for Standardisation; 2003. [22] Riemann JA. The behaviour of compression web members in coldformed steel truss assemblies. Thesis presented to the Faculty of the University of Missouri-Rolla in partial fulfillment for the degree master of Science. Missouri-Rolla, USA; 1996. [23] Tomà AW. Connections in cold-rolled sections. Research report BI76-78, TNO Delft; 1976. [24] Wong MF, Chung KF. Structural behaviour of bolted moment connections in cold-formed steel beam–column sub-frames. J Constr Steel Res 2002;58(2):253–74. [25] Zadanfarrokh F, Bryan ER. Testing and design of bolted connections in cold-formed steel sections. In: Eleventh international specialty conference on cold-formed steel structures, University of MissouriRolla, St. Louis, Missouri, U.S.A, 1992. [26] Zaharia R. Cold-formed steel trusses with bolted connections. Ph.D. thesis. Romania: The ‘Politehnica’ University of Timisoara; 2000 [in Romanian]. [27] Zaharia R, Dubina D. Behaviour of cold-formed steel truss bolted joints. In: The IV’th international workshop on connections in steel structures. 2000.
Stiffness of joints in bolted connected cold-formed steel trusses Raul Zaharia, Dan Dubina∗ Department of Steel Structures and Structural Mechanics, Faculty of Civil Engineering and Architecture, “Politehnica” University of Timisoara, I. Curea no.1, 1900 Timisoara, Romania Received 10 January 2005; accepted 1 July 2005
Abstract Web members in cold-formed steel trusses are usually assumed to have pinned connections at the ends, but the latest AISI Cold-Formed Steel Truss Design Standard allows for the joint stiffness to be considered in design. The paper summarizes experimental research performed for several years at the University of Timisoara, Romania, aimed at evaluating the real behaviour of bolted joints in cold-formed steel trusses. By means of tests on single lap joints and on truss sub-assemblies, a theoretical model for joint stiffness was proposed. The formula for the joint stiffness, from which the buckling length of web members was further determined, was also validated through a test on a full-scale truss. © 2005 Elsevier Ltd. All rights reserved. Keywords: Cold-formed steel trusses; Bolted connections; Joint stiffness; Buckling length; Experimental tests; Numerical analysis
1. Introduction Cold-formed steel framing demonstrates extensive development, even if is a relatively new system, due to some great advantages, such as high strength-to-weight ratio, reduced labor costs and fast erection due to the light weight of cold-formed members. The cold-formed steel trusses represent an economical option to the classical wood trusses used mainly in residential buildings, and to the hot-rolled trusses used for industrial applications. Several proprietary products have been developed, considering C, Z, hat, or more particular sections for chords and webs. The connections may be realized by welding, by using adhesives, with mechanical connectors as bolts or screws, or by some innovative mechanical connecting techniques such as press joining or rosette joining. The mechanical connections are among the most suitable, taking into account the production costs and rapidity of execution. Initiated by the more widespread use of the cold-formed steel in the residential construction market, systematic research to investigate the behaviour of cold-formed steel trusses was carried out at the University of Missouri-Rolla. ∗ Corresponding author. Tel.: +40 56 192970; fax: +40 56 192970.
E-mail address: [email protected] (D. Dubina). 0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2005.07.002
On the basis of full-scale tests on fink C-section truss assemblies, Harper [10] studied the buckling lengths for the top chord members, realized by a single C-section. Riemann [22] developed a computer analysis model and conducted full-scale truss tests in order to determine the capacity of compression web members, and suggested an interaction equation for the design of compression webs as beam–columns. Ibrahim et al. [11] made another series of tests on full-scale truss assemblies, considering the same system, i.e. single C-sections for chords and webs, connected by self-drilling screws. The authors proposed an interaction equation for checking unreinforced top chords subjected to axial compression, bending and web crippling. LaBoube and Yu [13] synthesized the above-presented research conducted at the University of Missouri-Rolla, which strongly influenced the design recommendations contained in the Standard for Cold-Formed Steel Truss Design issued recently by AISI [1]. This standard is intended to be a response to the problems that these particular systems raise for the designer, and applies to the design, quality assurance, installation and testing of cold-formed steel trusses used for load carrying purposes in buildings. As shown in the Standard Commentary [2], even if the structural analysis requirements are based on available information concerning the behaviour of cold-formed steel
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single C-section truss assemblies [13], these requirements do not preclude the use of more rigorous analysis or design assumptions, as determined by rational analysis and/or testing. As regards innovative mechanical connection techniques, two interesting research areas are worth mentioning here. Pedreschi et al. [17] demonstrated the efficiency of press joining, by making tests on single lap joints or groups of press joins and on full-scale pitched trusses made from coldformed Z-sections. Mäkeläinen and Kesti [15] studied the behaviour of the rosette joining system and its possibilities in roof-truss structures, by making tests on simple joints in shear or in cross-tension, and also by making tests on sub-assemblies. The authors concluded that the rosette joint has very good capacity to resist tensile forces and that the shear capacity seems to be sufficient for applications in lightweight steel trusses. The research on cold-formed steel trusses is generally focused on systems for residential roofs, having relatively reduced spans. For larger spans, efficient solutions can be achieved with higher resistance members, made for example with optimized cross-sections, like “pentagon” sections, studied by Blumel and Fontana [3]. The authors showed that the use of this particular cross-section with a large radius of gyration for both axes offers statical and constructional advantages for the chords. However, the low-cost design of the truss joints using a gusset plate welded onto the ridge of the cross-section can lead to important section deformations. The authors developed a calculation model for the local loadbearing behaviour of this particular type of cold-formed truss joint, and validated this model by means of numerical and experimental analysis on truss segments. The trusses built of cold-formed steel sections with bolted connections, made from built-up C-sections for chords and single C-sections for webs, represent another possible constructive system for residential buildings, also reliable for larger spans. In this system, the webs are connected to the chords by means of bolts placed on both flanges of the C-section of the web member. Fig. 1 shows two applications using such trusses. Fig. 1(a) shows the structure of a supplementary storey built for the Alcatel Company building in Timisoara, Romania, while Fig. 1(b) shows the trusses used to build the roof of a church in Bucharest, Romania. As regards the design of this type of cold-formed truss system, two problems arise: the stability behaviour of the compressed chord, taking into account that for these kinds of built-up members no design recommendations exists in the norms, and the real behaviour of the joints. Studies concerning the stability behaviour of the built-up C-profiles connected by bolted C-stitches were performed by Niazi [16]: based on the results obtained by Johnston [12], for hot-rolled columns in which the battens are attached to the chords by hinged connections, the compressed builtup C-section is supposed to work on an elastic foundation, provided by the roof purlins. The authors of the present
241
Fig. 1. Cold formed steel trusses.
paper calibrated a finite element model suitable for predicting the ultimate load for such built-up elements, used for the compressed chords of cold-formed steel trusses, but also as columns in cold-formed steel framing [7]. The numerical model showed good results compared with the above procedure and experimental results. As regards the analysis of the web members, it must be emphasized that the use of two or more bolts for each flange of the C-section, in relation with the element slenderness, is supposed to modify the classical assumption of pinned connections, used in case of truss structures. Moreover, the eccentricity of joints could not be avoided, and this fact should also be considered into a global analysis, because it may require additional efforts. For these reasons, the analysis of trusses built of cold-formed steel sections with bolted connections should consider the real behaviour of the joints. This may lead to reduced buckling lengths of the web members, but at the same time, to supplementary bending moments in these elements. In chapter D3 “Analysis” of the latest AISI Cold-Formed Steel Truss Design Standard [1] it is shown that: “in lieu of a rigorous analysis to define joint flexibility, the following analysis model assumptions should be assumed: . . . (b) web members are assumed to have pinned connections at
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each end. Use of a specific joint stiffness other than the complete rotational freedom of a pin for a connection shall be permitted if the connection is designed for the forces resulting from a structural analysis with this specific joint stiffness”. This means that the Standard allows for joint stiffness to be considered in the design of a truss, even if no specific equations for calculating this parameter are given. This should be in fact difficult, considering the number of different cold-formed steel truss systems. In the case of coldformed trusses with bolted connections, a rigid behaviour of the joints is also not realistic, taking into account the deformability of the joint due to the bearing deformation produced by the bolt in the thin plate, associated with the hole elongation, bolt tilting and slippage due to the hole clearance. Generally, the research in the field of bolted connections in cold-formed steel framing is focused on determining their bearing resistance. For the first time, Zadanfarrokh and Bryan [25] analysed both experimentally and theoretically the flexibility of bolted connections in cold-formed steel sections, and gave a formula for the flexibility of a single lap bolted joint, but this approach was not included in any design recommendation. More recently, [24] investigated experimentally some particular column base connections and beam–column sub-frames, made from cold-formed steel sections, in different bolted connection configurations, in order to assess their strength and stiffness. The study identified different failure modes and concluded that the bolted moment connections were effective in transmitting moment between the connected sections, enabling effective moment framing in cold-formed steel structures. Another recent study concerning the stiffness of bolted connections in a steel portal framing system was made by Lim and Nethercot [14]. The authors described a finite element model that can be used to determine the stiffness of the individual bolt joint. Using this stiffness, a beam idealization of a coldformed steel bolted moment connection was determined, in order to predict the initial stiffness of the apex joints. The numerical and theoretical study was validated through tests on full-scale joints. The research presented in this paper summarizes the work performed for several years at the University of Timisoara, Romania, aimed at evaluating the real behaviour of joints in cold-formed steel trusses connected by bolts, and at proposing a theoretical model for the joint stiffness. The experimental programme was developed in three steps. First, the rotational rigidity of some truss connections was evaluated, by means of tests on typical T-joints. In the second step, a formula for the stiffness of a single lap bolted joint was determined, together with a theoretical model for the rotational stiffness of cold-formed steel truss bolted joints. Based on this model, an equation for determining the reduced buckling length of the web members was also proposed. The third step of the experimental programme
included a full-scale test of a cold-formed steel truss, in order to validate the theoretical formulations at structural level. 2. Experimental programme step 1: Tests on T-joint specimens This first step of the experimental programme has already been described in detail [5,6], and the results were included in the Database for Research on Cold-formed Steel Structure from University of Missouri-Rolla, USA [4]. From the experimental moment–rotation characteristic of the ten Tjoint specimens tested, it was concluded that all tested joints were of semi-rigid type with partial resistance, according to Eurocode 3 [21] criteria of joint classification. An important initial rotational slippage was observed for all specimens tested, but it was not considered in the evaluation of the joint rigidity, because the triangulated shape of the truss, which is geometrically and kinetically stable, and the presence of the axial forces in connected members prevent, or limit, at least, this phenomenon at structural level. The test of the cold-formed steel truss, in the third step of the experimental programme, was aimed also at validating this assumption. From this first part of the experimental programme it was also concluded that the rotational deformability is mainly due to the bearing work of the bolts in the thin plates, i.e. the elastic and plastic deformation of the bolt holes and bolt tilting. Consequently, the rotational rigidity of the joints may be determined by analysing the single lap bolted joint. 3. Experimental programme step 2: Tests on single lap joints Experimental studies carried out in order to calibrate a formula for the flexibility of single lap bolted joint for thin-walled cold-formed elements were already performed by Zadanfarrokh and Bryan [25]. The formula proposed by these authors gives the flexibility of a single lap joint, as a function of the thickness of the plates and the presence of the threaded portion of the bolt in the connection. The formula does not include the effect of bolt diameter, being determined for lap joints using M16 bolts, but makes a distinction between perfect fit and 2 mm clearance of the bolt hole. For the case of T-joints tested at Timisoara, for all ten specimens, M12 bolts with 1 mm clearance of the bolt hole, as specified in the Romanian code, were considered. Consequently, the second step of the experimental programme was aimed at calibrating a formula for the stiffness of single lap joints, considering the plate thickness as in the Zadanfarrokh and Bryan study, but also the bolt diameter, for the practical case of a threaded portion of the bolt in the connection and 1 mm hole clearance. The experimental programme considered three different thicknesses for the plates, between 1.85 and 3.75 mm, and five bolt diameters, between 8 and 16 mm. The mechanical characteristics of the steel plates are given in Table 1.
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Table 1 Mechanical characteristics (N, mm) t
fy
fu
1.85 3.15 3.75
279.8 276.8 258.5
402.1 392 375.5
Fig. 3. Typical load–extension characteristics.
Fig. 2. Test set-up.
The test set-up is shown in Fig. 2. The specimens were tested in a ZWICK universal testing machine, using an angular displacement transducer to record the extension readings. The loading rate of 1 kN/min and the plate dimensions were in accordance with the ones used in the Zadanfarrokh and Bryan experimental programme. This value of loading rate is also given in the European Recommendations [8]. Table 2 gives the experimental values of the specimens’ stiffnesses. For each thickness of plate, three different bolt diameters were used, and for each combination of plate thickness and bolt diameter, three tests were performed (a–c). Thus, a total of 27 experimental results are available. Typical load–extension characteristics for determining the initial stiffness of the lap joint, considering an identical set of parameters, are presented in Fig. 3. The formula for the stiffness of a single lap bolted joint was calibrated using Annex D of Eurocode 0: Basis of Structural Design [18]. This appendix describes a standard procedure for the determination of the characteristic values, design values and partial factor values for strength from tests that is in compliance with the basic safety assumptions outlined in Eurocode 1: Actions on Structures [19]. As shown by Tomà [23], in a pioneering study concerning the rigidity of screw connections in cold-formed elements, underestimating the rigidity of the connection in the elastic range will relieve the connection, but will put a supplementary bending moment into the element.
Overestimating the rigidity, on another hand, leads to a supplementary moment into the joint, but this can be resolved by an appropriate design of the connection. It is safe, for the stability and displacement analysis, to underestimate the rigidity. Therefore, the Annex D procedure for calibration of the formula for the stiffness of a single lap bolted joint, will follow the same steps as for the determination of the characteristic values, design values and partial factor values for a strength-type formula. The final characteristic value, obtained after the application of the standard procedure of Annex D, for the stiffness of a single lap bolted joint, is given by the following formula [27]: √ D (kN/mm) (1) K = 6.8 5 5 + − 1 t1 t2 with a partial safety factor γ R = 1.25. In the above equation, D is the nominal diameter of the bolt while t1 , t2 represent the thicknesses of joined plates. The range of validity of this formula is for bolts between 8 and 16 mm nominal diameter and thickness of plates between 2 and 4 mm, using 1 mm hole clearance and considering the threaded portion of the bolt in the connection. It can be noted that the partial safety factor for this formula is identical to the partial safety factor used in Eurocode 3 Part 1.3 [20] for the resistance of bolted connections. On the basis of Eq. (1), the stiffness of truss joints may be determined. For the displacement analysis, and for the computation of the buckling lengths of the web members, the design value must be applied, while for the connection design the characteristic one is suitable. 4. Computation models for rotational stiffness of truss joints The computation scheme for the rotational stiffness of a truss joint with two bolts on each flange of the C-section of the web member (four bolts in total) is presented in Fig. 4. The rotational stiffness of the joint, K node,t , can be expressed in terms of total bending moment, Mtot , and
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Fig. 4. Computation model for a four bolt joint. Table 2 Experimental values for single lap joint stiffness (kN/mm) Bolt
Plate thickness t (mm)
M8 M10 M12 M14 M16
1.85 a
b
4.237 5.102 7.353 – –
4.695 6.211 5.263 – –
c
3.15 a
b
c
3.75 a
b
c
4.348 5.025 5.236 – –
– 10.000 10.869 11.111 –
– 10.417 10.753 11.628 –
– 10.204 10.526 11.765 –
– – 9.259 14.286 16.667
– – 13.333 14.493 16.393
– – 13.699 14.925 15.385
Table 3 Comparison between experimental and theoretical values of joint stiffness K node,t /K node,exp
d K node,t /K node,exp
9 830
0.971 0.958
0.777 0.766
12 480 11 110
13 083
1.047 1.177
0.838 0.942
2.05
10 560 10 968
11 418
1.080 1.041
0.864 0.833
4.05
3
15 320 15 490
16 057
1.048 1.037
0.838 0.830
4.05
4.05
21 189 20 361
20 779
0.981 1.021
0.785 0.817
t1 (mm)
t2 (mm)
K node,exp (kN mm/rad)
1 3
3
2.05
10 130 10 270
2 4
3
3
5 8
4.05
6 9 7 10
Node
corresponding rotation, θ , as [27] 2(Fa) 2K da Mtot = = d = K a2 θ tan θ 0.5a √ 6.8a 2 D (kN mm/rad) = 5 5 + − 1 t1 t2
K node,t =
(2)
with the same partial safety factor as in Eq. (1), γ R = 1.25. The term a represents the distance between bolts.
K node,t (kN mm/rad)
Table 3 shows the comparison between the experimental values of the rotational stiffness for the T-joints obtained in the first step of the experimental programme (K node,exp ) and the theoretical values obtained with the proposed relation (K node,t ). It may be observed that there is a good correlation between the experimental results and the characteristic values of the joint rotational stiffness. The average reported ratio between the theoretical characteristic values and the experimental ones is 1.036 and the correlation coefficient
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245
is ρ = 0.982. Considering the design theoretical value of d the formula (K node,t ), affected by the partial safety factor, it may be observed that all the theoretical values are in the safe range. A similar model may be used for determining the rotational stiffness for a six bolt truss joint. Considering that the centre of rotation of the bolt group is in the axis of the middle bolt, as shown in Fig. 5, the following equation may be determined [26]: 2(F × 2a) 4K da Mtot = = d = 4K a 2 θ tan θ a √ 2 27.2a D (kN mm/rad). = 5 5 + − 1 t1 t2
K node,t =
(3)
Fig. 5. Computation model for a six bolt joint.
5. Buckling length of truss web members The rotational stiffness of joints may be used to determine the buckling lengths of the web members. Trusses may be classified as fixed-node-type structures. The computation model for the buckling length of truss web members L b,web is then as for an element with fixed nodes for lateral displacement and elastic rotational springs on both ends. For this model, the buckling length should be determined considering the following equation: L b,web = µL web
(4)
where L web is the length of the web member measured between centrelines of connections and µ is computed with Eq. (5): µ = 0.5 + 0.14(η1 + η2 ) + 0.055(η1 + η2 )2 .
(5)
For the case of cold-formed steel trusses with bolted joints, the coefficients η1 and η2 may be computed as follows: K web K web η2 = η1 = K web + K node,1 K web + K node,2 E Iweb with K web = (6) L web where Iweb is the second moment of area of the web member and K node,1 , K node,2 are the rotational rigidities of the two joints connecting the web member on the chords, computed with Eq. (2) or (3), a function of the number of bolts. Note that for the computation of the buckling length, the design values of the rotational rigidities K node,i should be considered. 6. Experimental programme step 3: Test on the truss structure In order to demonstrate that the initial rotational slippage from the moment–rotation characteristics of T-joints determined in the first step of the experimental programme is not present in the truss, and to validate at
Fig. 6. Specimen dimensions. Table 4 Cross-section characteristics (mm) Profile
h
b1
b2
c
t
C100/2 C120/2
100 120
40 40
45 45
20 20
1.91 1.91
structural level the theoretical formulas presented above, a full-scale test of a truss was performed. The dimensions of the experimental model are presented in Fig. 6. All joints used six M12 8.8 grade bolts. The section characteristics are presented in Table 4 (LINDAB® profiles) and the mechanical proprieties of steel are presented in Table 5. Fig. 7 presents the experimental arrangement. The load was introduced by means of a 50 ton actuator, in displacement control, imposing a rate of 2.5 mm/min. Fig. 8 presents the instrumentation of the test. In order to measure the joint rotations, two inclinometers (R1 –R2 ) were
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Table 5 Material characteristics (N, mm) Profile
fy
fu
εu (%)
C100/2 C120/2
367.2 354
542 493.4
19 14
Fig. 7. Experimental arrangement.
Fig. 8. Instrumentation.
placed on the web of the C-section of each web member, in the axis of the lower chord joints. For determining the slippage on the connections, along the axial forces in the web members, four LVDT displacement transducers (I1 –I4 ) were placed along the axis of web members, near the joints. Four potentiometric displacement transducers (P1 –P4 ) were also used for global structural displacement control. The load increased until the failure of the structure, produced by the flexural instability of the compressed web member, occurred in the plane of the truss, as shown
in Fig. 9(a). A local buckling of the lower chord C-section webs, due to the shear of the panel between the joints, was observed (Fig. 9(b)), before reaching the ultimate load. This phenomenon increases the deformability of the joint. Fig. 9(c) presents the bolt hole plastic deformations, for the compressed web member. It may be observed that the hole of the middle bolt suffers only elongations along the axis of the member, which confirms the computation model from Fig. 5, in which the middle bolt was considered the centre of rotation for the six bolt connection. Fig. 10 presents the evolution of the displacements reported by the LVDT transducers I3 and I4 , along the axis of the compressed web member. One may observe the typical behaviour of a thin-walled bolted connection subjected to shear. After reaching the slippage force (corresponding, at bolt level, to approximately 200 daN) the initial slippage extension is extended until the hole clearance is reached; the size of this extension is a function of the initial position of the elements in the structure. Fig. 11 shows the evolution of the rotations in web member connections. Corresponding to the load range in which the axial slippage occurs, only small rotations are observed. Until the structure ‘shakes down’, the presence of the axial forces and the triangulated shape of the truss prevent the development of significant rotational slippage in connections. Consequently, the initial rotational slippage observed for the T-joints is not present in the structure, and the rotational stiffness evaluated without considering the initial slippage represents the real initial stiffness of the connection in the truss. For quantitative validation of the proposed theoretical formulas, the tested truss was numerically analysed by means of PEP-micro programme [9]. PEP-micro is a specialized programme for the non-linear analysis of steel structures with semi-rigid joints. The static scheme of the structure is presented in Fig. 12. For the stability verification of a structure, Eurocode 3 allows for a second order analysis with initial sinusoidal imperfection of the elements. The amplitude of those initial imperfections is a function of the buckling curve for the corresponding cross-section of the element. For lipped channels, according to Eurocode 3 Part 1.3, the corresponding buckling curve is B, with an initial equivalent imperfection e0 = l/380. The ultimate load of the element is then the load corresponding to the reach of the yield stress in the extreme fibre of the cross-section, taking into account the second order effects. A step-by-step second order analysis was performed, with a load step corresponding to 1% of the ultimate load. The connection eccentricities were taken into account by introducing supplementary rigid elements on the edges of the diagonals, of length L exc in Fig. 12. The structure was analysed considering the classical pinned assumption for joint behaviour and also the rotational stiffness, computed by means of Eq. (3), for a six bolt joint. The effect of the axial stiffness of the connections on the direction of the axial force in web members
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247
Fig. 9. Structure after test.
Fig. 10. Axial displacements of the web member.
Fig. 12. Static scheme of the experimental model.
for the corresponding number of bolts in the connection, i.e. multiplied by six: √ D E Aech = K axial = 6 × 6.8 (7) 10 L ech t −1
Fig. 11. Web member joint rotations.
was also considered. The PEP-micro programme is not able to model this axial stiffness, and consequently an equivalent finite element to simulate this behaviour of the connections was considered. The equivalent cross-section area of this finite element may be determined by equalizing the expression for the axial stiffness of the bar element having a length L ech equal to the distance between the centre of rotation of the bolts group and the last bolt, with the axial stiffness of the connection, K axial , determined using Eq. (1)
Fig. 13 presents the comparison between the experimental load–displacement characteristic and the results of the numerical analysis. In Fig. 13(a) are represented the numerical load–displacement characteristics, considering the rotational stiffness (K node ), together or not with the axial stiffness of the joint (K axial ). Fig. 13(b) shows also the response of the numerical model for the complete rotational freedom of web member connections (pinned), considering the axial stiffness of the joints. One may observe, in the experimental load–displacement curve, an initial structural slippage, at the level of the force corresponding to the connection slippage along the axis of the web members. Neglecting this phenomenon, not considered in the numerical analysis, the structural rigidity
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Table 6 Results of the numerical and experimental analysis K node without K axial (1)
K node with K axial (2)
Pinned with K axial (3)
Experiment (4)
(1)/(4)
(2)/(4)
(3)/(4)
Ultimate load (daN) 6650
7665
5790
7820
0.85
0.98
0.74
Displacement (mm) 3.3
9.9
7.5
15.8
0.21
0.63
0.47
Structural rigidity (daN/mm) 2015
774
772
734
2.74
1.05
0.99
the other hand, the rotational stiffnesses of the joints have a very slight influence on the structural rigidity compared with the classical assumption of pinned joints, due to the triangulated shape of the truss, but affect in a significant way the resistance of the structure. Considering the results of the numerical and experimental analysis at structural level, it may be concluded that the theoretical formulas proposed may be successfully applied to represent the joint stiffness in cold-formed steel trusses with bolted joints. 7. Conclusions
Fig. 13. Comparison between experimental and numerical analysis.
obtained numerically, taking into account both axial (K axial ) and rotational (K node ) stiffness of the connections, is very close to the experimental one. Table 6 presents the results of the numerical analysis, in comparison with the experimental values. It may be observed that the analysis considering both axial and rotational stiffness of the connections gives differences around only 2% for the ultimate load and 37% for the corresponding displacement. The difference at displacement level is due to the initial axial slippage in the joints and to the bolt plastic bearing appearing at high force levels, phenomena not considered in the numerical analysis. However, the comparison between the initial numerical and experimental structural rigidities, after the consumption of the slippage, gives differences of only 5%. It may be observed that the axial stiffness of the joints plays an important role in the structural rigidity, and it affects also, but to a lower extent, the ultimate load. On
Cold-formed steel trusses are much used, especially in residential construction, but such systems are also reliable for larger span applications. Trusses made from built-up Csections for chords and single C-sections for webs, with bolted joints, represent a suitable solution for both situations, taking into account the production costs and the rapidity of execution. Despite the extensive use of cold-formed steel truss systems and of cold-formed steel framing in general, and despite the fact that new standards appeared in recent years to cover this domain, there is still a lack of information considering different behavioural aspects of this particular type of structural system. In the case of steel trusses, current design practice assumes that web members have pinned connections, but the latest revision of the AISI Cold-Formed Steel Truss Design Standard allows for joint stiffness to be considered in the design of a truss. However, the standard did not offer formulas for calculating this parameter. In order to determine a theoretical model for the joint stiffness of bolt connected cold-formed steel trusses, the authors developed an experimental programme. In the first step, the stiffness of some cold-formed steel truss bolted joints was evaluated, by means of tests on typical T-joints. It was emphasized that the joint deformability is mainly due to the bearing work of the bolts, and consequently the rotational rigidity of the connection may be determined by analysing the single lap bolted joint. In order to determine the stiffness of a single lap bolted joint, a second experimental programme was performed, and a formula for the characteristic and design stiffness was calibrated. Using this formula, a computational model
R. Zaharia, D. Dubina / Journal of Constructional Steel Research 62 (2006) 240–249
for the rotational stiffness of truss joints was established. The theoretical model for a four bolt joint demonstrates a good correlation in comparison with the experimental results on T-joints. Considering this model, a formula for the buckling length of truss diagonals was further determined. In the last part of the experimental programme, one test on a full-scale truss demonstrated that the initial rotational slippages observed in the T-joint tests are not present at structural level, so these slippages are not influencing the initial rotational stiffness of the truss joints. The numerical analysis of the truss tested, considering the axial and rotational stiffness of the joints, computed with the corresponding formulas for a six bolt joint, demonstrates a good correlation with the experimental results. Consequently, the formulas proposed by the authors may be successfully applied to represent the joint stiffness in cold-formed steel trusses with bolted joints. For design purposes, when computing the displacements of the truss or the buckling lengths of the web members, the design values of the joint stiffness computed with the formulas proposed in this paper have to be applied, while for the connection design the characteristic values are suitable. The study revealed that the response of the truss is influenced not only by the rotational stiffness, but also by the axial stiffness of joints, in the direction of the axial forces of web members. References [1] AISI/COFS. Standard for cold-formed steel framing — Truss design, Revision of AISI/COFS/TRUSS 2000. American Iron and Steel Institute, Committee on Framing Standards (COFS); 2001 [1st printing June 2002]. [2] AISI. Commentary on the standard for cold-formed steel framing — Truss design. American Iron and Steel Institute. Committee on Framing Standards (COFS); 2001 [1st printing June 2002]. [3] Blumel S, Fontana M. Load-bearing and deformation behaviour of truss joints using thin-walled pentagon cross-sections. Thin Wall Struct 2004;42(2):295–307. [4] Database. Database for research on cold-formed steel structures project title: Behaviour of bolted connections in cold-formed steel plane trusses — Center for Cold-formed Steel Structures. University of Missouri-Rolla M 065401-0249 USA; 1996. [5] Dubina D, Zaharia R. Experimental evidence of semi-rigid behaviour of some cold-formed steel truss bolted joints. In: International conference on experimental model research and testing of thin-walled steel structures. 1997. [6] Dubina D, Zaharia R. Cold-formed steel trusses with semi-rigid joints. Thin Wall Struct 1998;29(1–4):237–87. [7] Dubina D, Zaharia R, Ungureanu V. Behaviour of built-up columns made of C sections connected with bolted stitches. In: International colloquium on stability and ductility of steel structures. 2002.
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