End plate joints A–C with corner bolts and three different end plate thicknesses (tpn = 10, 15 and 20 mm) under arbitrarily inclined bending are analysed by the
rake. These three joints are associated with a total of five conducted tests as explained in section 3.1 (A: TE1, TE7, B: TE2, C: TE3, TE8). The rotational resistances were estimated using the actual geometrical dimensions and material properties of the joints (section 2).
The initially vertical point load P (|P| = P) is supposed to act at mid span of the splice beam in such a way that its line of action goes through the centre of symmetry of the joint (and tubes) as shown in Figure 11. The angle between the weak axis direction and the line of action of load P is denoted by α. Then the direction of the resultant moment vector MR is indicated by the same angle α with respect to the strong
axis direction. The absolute value of the resultant moment MR can be expressed by the
components My and Mx associated with weak and strong axis bending, respectively, as
|MR| = MR=Mx2 + My2. (4)
Figure 11. Load P and associated bending moment MR in the joint.
The tension component is the predominant component for joint response in all the cases considered here. Corner mechanism 8 with prying (Figure 9) gives the minimum resistances amongst all yield mechanisms in question. Otherwise the resistances were determined as far as possible in accordance with Table 6.2 of EN1993-1-8. The resistance values of equivalent T-stubs (defined here per bolt) are given in Table 6. For joints A and B (tpn=10 or 15 mm), mode I (=complete yielding
of the flange) gives the lowest resistance values, whereas the lowest resistance value for joint C with the thickest end plate (tpn=20 mm) is given by mode II (= bolt
failure with yield of the flange). According to the standard, “the possible modes of failure of the flange of an equivalent T-stub may be assumed to be similar to those expected to occur in the basic component that it represents”. These modes are compatible with the observations made during tests TE1–TE3 (Figures 5 to 8). However, for joint B, the minimum resistance values given by mode I (method 2) and by mode II are quite close (dashed box in Table 6). If the plate thickness were
Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012 315 316 Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012
15.1 mm, these modes could give equal minimum resistances. In general, the resistance formula used with method 2 is more relevant with the thinner end plates. Based on partly intuitional reasoning, method 1 should be used with joint B whose associated failure mode is near the borderline between mode I (method 2) and mode II. This is compatible with the observations made during test TE2 whose failure mode was clearly consistent with mode I (flange yields completely under prying). The resistance values exploited in the rake in bold in Table 6.
Table 6. Resistances of tension components. Resistances FtRd [kN]
Joint and its
orientation A (TE1,TE7) tpn = 10 mm B (TE2) tpn = 15 mm C (TE3,TE8) tpn = 20 mm method 1 61.8 109 186 mode I method 2 77.5 137 233 mode II 125 138 160 mode III 250 250 250
After the assembly of the rake, the rotational resistances of the considered joints could be determined for arbitrarily inclined bending moment. The results are presented in Figure 12 as moment-moment interaction curves (Mx–My) for joints A–C.
Calculations were done only for selected values of angle α belonging to the first quarter of the coordinate system (α=0°, 10°, 20°,...,´90°). Based on the double symmetry of the joint, the interaction curves were extended to other quarters.
In the moment-moment interaction curve, the distance from origin (4) gives the resistance value (MRd,α) of resultant bending moment in the arbitrary direction.
The shapes of the interaction curves are elongated. Ratio MS/MW between resistances
in strong axis bending (α = 90°) and in weak axis bending (α = 0°) is about 1.5 for all joints A–C. For comparison, the Mx-My curves determined by method 1 for joint
A and by method 2 for joint B are also drawn in Figure 12 as dashed lines. The test points are indicated by red circles. Calculated resistance values of MRd,α are given in
Table 7 (decisive resultant value in bold). Ratio MRd,α/Mtest is given in the lowest row of
the table. They indicate that resistances can be evaluated quite well. In every case, ratio MRd,α/Mtest is,roughlyappropriate(=onthesamelevel)comparedtotypicalapproximations
achievedbythecomponentmethodinstrongaxisbendingcases.However,theselection of method 1 or 2 associated with the mode I is not a foregone conclusion. This can be investigated, for example, by parametric study based on FE simulations.
Table 7. Resistances estimated by the rake (decisive values in bold)
Resistances MRd,α [kNm]
A: tpn = 10 mm B: tpn = 15 mm C: tpn = 20 mm
Joint and its orientation
0° (TE7) 35° (TE1) 35° (TE2) 0° (TE8) 35° (TE3)
method 1 22.9 26.7 47.0 — —
mode I
method 2 28.8 33.5 59.1 — —
mode II — — — 59.3 69.3
MRd,α/Mtest = 0.73 0.67 0.75 0.66 0.70
Figure 12. Mx-My interaction curves of joints A–C.
In addition to resistance, also (initial) stiffness can be estimated by a rake. The active components determine the rotational stiffness of a joint. The compression components of joints A–C are very stiff compared to their tension components. The selected stiffness of the compression component was 103 times the stiffness of the tension component. Larger multipliers (from 104 to 106) were tried out, too, but their influence on the solution was negligible and, moreover, caused some numerical problems. There are no formulas to evaluate the stiffness (or the resistance) of the tension components associated with corner bolts in EN1993-1-8. As a first approximation, the formulas given for normally positioned bolts in the standard were exploited (Table 6.11 of EN1993-1-8) with the difference that the equivalent length leff at issuewas
replaced by the value determined for the corner mechanism. This procedure, however, produces clearly too high calculated rotational stiffness values for all joints. This can be seen from Table 8 in the case of tests TE1–TE3, TE7 and TE8. The low accuracy of the estimates stands out especially with the thicker end plates when ratio Sini,α/Sini,test has far too high values.
Table 8. Rotational stiffnesses of joints. Stiffnesses Sini,α [kNm/mrad]
A: tpn = 10 mm B: tpn = 15 mm C: tpn = 20 mm
Joint and its orientation
0°(TE7) 35°(TE1) 35°(TE2) 0°(TE8) 35°(TE3)
mode I 11.7 14.9 32 — —
mode II — — — 43.1 55.9
Sini,α/Sini,test = 4.0 5.5 7.0 9.6 13.6
5. CONCLUSIONS
The rotational resistances and stiffnesses of the three joints A–C were analysed by a 3D mechanical model called the rake model. The method represents an enlargement of the component method presented in standard EN1993-1-8. The motive for conducting the tests described in this article (and the larger series of tests they
Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012 317 318 Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012
belong) was to develop the 3D model. The research is still going on (2012) and only preliminary results were presented.
Based on these few examples it appears that the component method integrated with the principles of plasticity theory (limit state analysis based on plastic mechanisms) for bolted end plate joints could be applied easily to the 3D analysis of end plate joints. On the other hand, the poor quality of the estimates of initial stiffness is obvious. The main reason for this is that an end plate in bending accompanied by a corner bolt is much more flexible than an end plate in bending with normally positioned bolts (test TE11 compared to the others, section 3.3). The used stiffness formulas found from the standard are based on the analogy between flange behaviour and one- dimensional beam analysis (i.e. their applicabilty is limited only to “normally positioned” bolts). The stiffness estimates for flanges with corner bolts definitely need to be derived separately. Because of the (unavoidable!) two-dimensional character of the flange bending problem with the corner bolts, more appropriate analytical formulas are, however, cumbersome to formulate. Perhaps, a more attractive approach is to determine proper multipliers for “associated one dimensional formulas” by the aid of numerical analysis (FEA) supported by tests.
Numerical analysis plays an important role in addition to the experimental study in the ongoing research of 3D modelling of end plate joints. Numerical analysis makes it possible to investigate issues difficult to reach by tests only. Parametric considerations are especially valuable there. Numerical analysis methods usually offer a more economical way to study complicated structural problems than experimental ones.