1. Introduction
Steel tubular structural sections allow a higher load capacity for axial force, and torsion as well as for combined effects. Recommendations for the use of circular hollow sections are generally based on ^{CIDECT (2008)}. Many researches have been performed in order to develop new techniques and propose solutions for the proper use of these sections as seen in works done by ^{Munse and Chesson (1963)}, ^{Beke and Kvocak (2008)}, Martinez-Saucedo and Packer (2009), ^{Freitas and Requena (2009)}, and others.
Square hollow sections (SHS), rectangular hollow sections (RHS), and circular hollow sections (CHS) are the tubular sections found in the market. This article deals with steel CHS submitted to tensile force with bolted connections. The connection is studied for use in a truss system. Differently from the tubular sections with flange studied by ^{McGuire (1968)}, shown in Figure 1(a), the continuity of the profile is by a sleeve connection, which presents a much more elegant appearance and allows the use of standardized elements, as shown in Figure 1 (b). This new type of connection has been studied by ^{Vieira et al. (2011)}, ^{Silva (2012)} and ^{Amparo (2014)}, ^{Amparo et al. (2014)}.
The sleeve connection is constituted by two tubes of the same diameter (outer tubes) connected by another tube with smaller diameter and bolts (inner tube), Figure 1 (c). This type of connections transfers the loading from the tube by its contact with the bolt. This characteristic shows the presence of the shear lag coefficient indicated by the high stress concentration on the region of the cross-section area with holes. This work shows one study about the influence of the shear lag coefficient in CHS with sleeve connection.
2. Materials and methods
To represent the connection, the prototypes are formed by an outer tube and an inner tube (sleeve) with aligned and crossed bolts. The prototypes were tested under tension in a controlled servo-hydraulic press with 200kN of capacity. An LVDT (Linear Variational Displacement Transducer) and strain gauges were used for the test instrumentation. Figure 2 shows the general scheme of the experiment.The displacement rate used was set to 0.4 mm/min.
The prototypes have geometric property variations: thickness and diameter. Therefore they were separated into 3 groups, as shown in Table 1. Each group had a variation in the number of bolts. Group A has prototypes with staggered bolts and groups B and C are prototypes with crossed bolts. X and Y is the steel resistance of outer and inner tube, respectively.
Group | Prototypes identification | Prototypes Numbers | Bolts Numbers | Outer tube | Inner tube | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Diameter (mm) | Thickness (mm) | Steel Resistance | Diameter (mm) | Thickness (mm) | Steel Resistance | ||||||
yielding | ultimate | yielding | ultimate | ||||||||
f_{y} (MPa) | f_{u} (MPa) | f_{y} (MPa) | f_{u} (MPa) | ||||||||
A | LA-3-X3Y2 | 1 | 3 | 73.0 | 5.5 | 399.5 | 539.5 | 60.3 | 5.5 | 381.0 | 479.0 |
LA-4-X3Y2 | 2 | 4 | 73.0 | 5.5 | 399.5 | 539.5 | 60.3 | 5.5 | 381.0 | 479.0 | |
B | CB-3-X2Y3 | 4 | 3 | 76.1 | 3.6 | 386.0 | 545.0 | 60.3 | 3.6 | 424.0 | 535.0 |
CB-4-X2Y3 | 3 | 4 | 76.1 | 3.6 | 386.0 | 545.0 | 60.3 | 3.6 | 424.0 | 535.0 | |
C | CC-5-X2Y3 | 2 | 5 | 88.9 | 5.5 | 375.0 | 474.0 | 73.0 | 5.5 | 399.5 | 539.5 |
After experimental analysis, it was observed that in some prototypes, the bolts showed a flexural mode. In some cases the limit state of bolt bending is dominant, and not the yielding of the gross section nor the fracture through the effective net area, group A. More details are shown in ^{Amparo et al. (2014)} and ^{Amparo (2014)}, including one expression for the bolt bending failure.
The gross area and net area were calculated considering the tubular section as a rectangular prismatic bar (rectangular plate). The net area is calculated according to the scheme shown in Figure 3 (a) and (b); the figure also shows the possible fracture lines for the two configurations of the connection. The cross-sections analyzed are identifiedas 1-1, 2-2 or 3-3. This adaptation can be found in ^{Amparo (2014)}.
The effective net area (A_{e}) is calculated according to Equation 1:
The value considered for the shear lag coefficient (C_{t}) was equal to 1.0 for the calculation of theoretical values.
Table 2 shows the expressions used to calculate the failure modes and the theoretical values adapted according to Brazilian code ABNT NBR:2008, using Ae as shown in Equation 1. The failure modes werethe yielding of the gross section (Y.G.S. - N_{t}), the fracture through the effective net area (F.N.A. - N_{t}), and the shear failure of bolt (S.B. - F_{v}).
Modes of failure | Equation | Prototype | Modes of failure | ||
---|---|---|---|---|---|
Y.G.S. | N_{t}=A_{g} f_{y} | Y.G.S. | F.N.A. | S.B. | |
F.N.A. | N_{t}=A_{e} f_{u} | N_{t} (kN) | N_{t}(kN) | F_{s} (kN) | |
S.B. | F_{v}=0,4 A_{b}f_{ub} | ||||
LA-3-X3Y2 | 360.8 | 378.2 | 313.5 | ||
LA-4-X3Y2 | 360.8 | 378.2 | 418.0 | ||
CB-3-X2Y3 | 271.9 | 259.3 | 313.5 | ||
CB-4-X2Y3 | 271.9 | 259.3 | 418.0 | ||
f_{ub} is the ultimate resistance of the bolt. | CC-5-X2Y3 | 466.0 | 498.0 | 522.5 |
3. Results and discussion
Table 3 shows the experimental maximum load capacity and the average load of the experimental maximum load capacity for all of the prototypes of each group.
Prototype | Experimental maximum load capacity (kN) | Average load (kN) | |||
---|---|---|---|---|---|
LA-3-X3Y2 | 425.7 | 440.4 | |||
LA-4-X3Y2 | 448.7 | 446.9 | |||
CB-3-X2Y3 | 247.1 | 248.6 | 241.6 | 250.8 | 244.5 |
CB-4-X2Y3 | 244.2 | 232.8 | 246.4 | ||
CC-5-X2Y3 | 400.7 | 425.3 | 413.0 |
The Figure 4 shows the curves of tension load (P) versus displacement for 3 prototypes of the group A. It was possible to observe that the load capacity of these prototypes was above the theoretical yielding gross section and the fracture through the effective net area (Table 2). The theoretical values of group A use the shear lag coefficient C_{t} equal 1.0.
The average load corresponding to the fracture of the prototypes of the group A was 440.4kN (Table 3), 16.45% higher than the expected theoretical value (Table 2). Changes in the experimental curve slopes characterize the different failure modes. In this group, the bending in bolts was the first failure mode of the connection, observed experimentally.
In the prototypes of the groups B and C, which have a sleeve connection with crossed bolts, the experimental load failure mode in the net area had a lower value than the theoretical load, as shown in Figure 5(a) and (b). Figure 5 shows the curves of tension load (P) versus displacement for the prototypes of groups B and C, respectively.
The prototypes of the group B had the average load value corresponding to the fracture equal to 244.5kN (Table 3). This value is 10.08% lower than the expected theoretical value (Table 2). Meanwhile, for the prototypes of group C, the average load value corresponding to the fracture was equal to 413.0kN (Table 3), 17.07% lower than the expected theoretical value.
The prototypes of groups B and C had a different behaviour compared with the prototypes of group A; all experimental curves of tension load (P) versus displacement were below the theoretical values. In these prototypes, the value 1.0 was used as the shear lag coefficient. So, it is necessary to adopt a value that reduces the net area of the effective net area, improving the results.
The international standards do not provide a shear lag coefficient for this type of connection given by the eq. (2). This equation was proposed by AISC (2005) and ABNT NBR 8800:2008 for a slotted tube connected to a gusset plate, (Figure 6), adapted to sleeve connections. Table 4 shows the experimental shear lag coefficient determined by dividing the average value for the experimental load corresponding to the fracture by the net area and the ultimate stress of the inner tube (tube where the fracture occurred).
Prototypes | Number of bolts | Experimental C_{t} | C_{t} by NBR |
---|---|---|---|
CB | 3 | 0.909 | 0.760 |
4 | 0.887 | 0.840 | |
CC | 5 | 0.829 | 0.855 |
where: C_{t} is the shear lag coefficient, e_{c} is the connection eccentricity, l_{c} is the length of connection.
The prototypes with crossed bolts do not have a geometric configuration equal to the model presented by ABNT NBR 8800:2008 (Figure 3 (b) and Figure 6). Consequently, the stress distribution is different in these prototypes. This fact could explain the differences shown in Table 4 between the experimental C_{t} and the C_{t} by NBR. However, the use of the equation proposed by ^{AISC (2005)} and ^{ABNT NBR 8800 (2008)} provides conservative results and therefore is reliable.
4. Conclusions
The goal of this work was perform theoretical and experimental evaluations of the influence of the shear lag coefficient on the new connection known as sleeve connection. This connection is constituted by steel hollow sections, and the prototypes were formed by two tubes, the outer tube and the inner tube, connected with aligned bolts or crossed bolts in 90º. The inner tube represented the sleeve.
The theoretical analysis of the modes of failure was based according to ABNT NBR 8800:2008, and the consideration of the shear lag coefficient (Ct) was based on the equation proposed by ^{AISC (2005)} and NBR.
The methodology used in the experiments was adequate to evaluate the new connection. For group A, the experimental results were good and showed high capacity load resistance, while the flexural failure mode was dominant, and the C_{t}=1.0. It was observed that the fracture through the net area, in the prototypes with crossed bolts, needs to consider a coefficient that reduces this net area to an effective net area, since the fractures occurred at levels below the load stipulated as standard.
The international and national standards do not yet include the sleeve connections. The results and the analyses proved the need and suitability of a formulation for the calculation of the shear lag coefficient for crossed bolts. Thus, an equation based in AISC/NBR to calculate the C_{t}, was proposed. The results showed good correlation with the experimental results.