8 Summary and Conclusion

Transcription

1 8 Summary and Conclusion Summary and Conclusion 8.1. Generalised Mechanical Model Based on the general principles of the component method, a generalised mechanical model was proposed, in the present thesis, to estimate the endplate joint behaviour when both bending moments and axial forces are present. This mechanical model is able to deal with three basic requirements for the joint performance: strength, stiffness and deformation capacity. Application and validation of this mechanical model, using experimental tests executed by Lima et al. (2004) on six extended endplate joints, was performed and led to accurate prediction of the experiment s key variables. The utilization of this generalised mechanical model is simple and provides an accurate approach to estimate the bending moment versus rotation curve for any axial force level acting on the joint. Additionally, bending moment versus axial force interaction diagrams can also be obtained by using the proposed mechanical model. The tri-linear characterisation of the joint components suggested in this work, is shown to be capable of reasonable approximations for the momentrotation curve construction. However, further experimental examination and numerical analysis using different ranges of joints to check the validity and application of the proposed strain hardening coefficients beyond the scope of studied joints in this work is still desirable. The approach proposed for evaluation of lever arm d, by taking into account the change of the joint compressive centre position according to the axial force levels and bending moment applied to the joint, is directly responsible for a satisfactory estimation for the joint initial stiffness, even before yielding of the first weakest component was reached. Parametric and sensitivity investigations demonstrate the application scope of the proposed mechanical model. Various scenarios involving the key

2 8 Summary and Conclusion 165 parameters that influence on the joint structural behaviour were considered and the main conclusions are: - The prediction of the bending moment versus axial force interaction diagram using the proposed mechanical model demonstrated to be in agreement with the experimental points of the extended endplate joint tested by Lima et al. (2004). Additionally, the mechanical model was able to capture an important characteristic observed in the experimental tests performed by Lima et al. (2004) where for certain compressive force levels it was possible to obtain a joint resistance bending moment larger than that one without axial forces. - The use of different beam profiles strongly affects the joint response under axial forces and bending moments. The joint ultimate bending moment resistance is reduced in alignment with a profile reduction, whilst larger profiles increase the ultimate bending moment resistance. The joint initial stiffness is slightly reduced by downsizing the beam profiles for compressive forces and slightly increased for tensile forces. From the analytical moment-axial load interaction diagram, at different beam profiles, it was observed that the joint tensile resistance is inversely proportional to the downsizing of the beam profile. This fact occurs due to the reduction of the lever arm defined by the distance from the load application line to the midpoint between the first and second bolt-rows. However, this was not identified for larger beam profile sizes, where others factors may become more relevant than the lever arm and, consequently, the joint tensile resistance might be larger than smaller profiles. - The influence of the studied column profile types on the joint response was not as pronounced, as expected, as the previous investigated beam profile cases. The increase in the column profile sizes does not significantly affect the joint characteristic curve. On the other hand, the use of smaller profile causes a pronounced reduction in the joint ultimate bending moment when coupled with increasing compressive forces. The joint initial stiffness presents a slightly reduction by downsizing the column profiles for the compressive forces. However, for the others cases the initial stiffness remains almost unchanged. The analytical moment-axial load interaction diagram, at different column profiles, depicts an increasing joint compressive resistance when column profile sizes were increased. On the other hand, for tensile forces applied to the joint, the results were very similar for the whole set of the investigated column profiles

3 8 Summary and Conclusion 166 demonstrating the small influence of the column profile variations in the joint tensile resistance. - The endplate thickness influence over the joint structural behaviour was more significant, as expected, than the previously investigated cases referred to the beam and column profile variations, causing large variations for the joint ultimate bending moment resistance mainly for decreasing endplate thickness. It is interesting to note that this is in line with the experimental observations depicted by Lima (2003). The joint initial stiffness is strongly dependent on the endplate thickness, mainly for endplate thickness smaller than the reference 15 mm endplate thickness when tensile forces are acting on the joint. This fact was also noted in the analytical moment-axial load interaction diagram at different endplate thicknesses, where the joint tensile resistance was reduced with a simultaneous decrease of the endplate thickness. - The bolts, similar to the endplate thickness variations, had a significant effect on the joint response, as it was again expected. However, decreasing the bolt sizes caused a larger joint ultimate bending moment resistance variation than when the bolt sizes were increased. The joint initial stiffness is strongly dependent on the bolt type, similarly to the finding observed in the investigation of the endplate thicknesses. Cases involving bolts smaller than the reference M 20 bolt, in general, present significant variation in the joint initial stiffness, fact that did not happen for bolts larger than the reference M 20 bolt. Associated also with the reduction in the bolt sizes is the associated reduction of the joint tensile resistance as presented in the analytical moment-axial load interaction diagram at different bolt sizes. In general, from the parametric investigations, it is possible to note that the axial force significantly affects the joint structural behaviour. The effect of the axial force might be more pronounced or not when coupled with variations in the joint basic components arising from, for instance, different profile sizes, endplate thickness and bolts. Some axial force levels may be also beneficial for the joint ultimate bending moment as identified in the analytical bending moment versus axial load interaction diagram for the majority of the investigated variations. Based on the investigation, it was also possible to conclude that the positive contribution of the axial force in the maximum joint ultimate bending moment resistances was more significant with a joint stiffness decrease. The joints that had

4 8 Summary and Conclusion 167 their dimensions reduced when compared to the joint reference dimensions presented a beneficial contribution in terms of the maximum ultimate bending moment. On the other hand, for upper joint dimensions the maximum ultimate bending moments was reached without axial forces. First order approximations for the trigonometric expressions were used throughout the generalised mechanical model formulation. Figure 104 presents the error due to these approximations versus joint rotations. According to Nethercot & Zandonini (1989), rotations beyond 0.05 radians had little practical significance. Based on this, this 0.05-radian rotation was adopted for the joint final rotation. For this rotation value it is possible to observe in Figure 104 an error of %. This indicated that the developed equations in this work were accurate for the usual problems involving beam-to-column joints. Error = abs(sin(q)-q)x100 (%) q (rad) Figure First-order approximations error magnitudes versus joint rotation Alternative Methodology A consistent and alternative methodology to determine any moment versus rotation curve from experimental tests, including the axial versus bending moment interaction, was also presented. This method extends the application range of available data so as to produce moment-rotation characteristics that implicitly make proper allowance for the presence of significant levels of either tension or compression at the beam. This methodology can also be applied to results obtained analytically, empirically, mechanically, and numerically. Due to its simplicity and to the fact that its basis is M- curves that already consider the moment versus axial force

5 8 Summary and Conclusion 168 interaction, it can be easily incorporated into a nonlinear semi-rigid joint finite element formulation. It is also important to observe that the use of the proposed methodology does not change the basic formulation of the non-linear joint finite element, only requiring a rotational stiffness update procedure. This proposed method is a simple and accurate way of introducing semirigid joint experimental test data into structural analysis, through M- curves. Application and validation of the proposed methodology to obtain M- curves, for different axial force levels, were performed against experimental tests executed by Simões da Silva et al. (2004) and Guisse et al. (1996) on eight flush endplate and on twelve column base joints, respectively. Finally, it may be suggested that an alternative, though accurate, method to determine M- curves for endplate and baseplate joints, considering the bending moment versus axial force interactions, can be made with a simple linear interpolation between two reference M- curves providing a straightforward procedure to obtain M- curves for any axial force level Design Considerations Using the Eurocode 3:1-8 (2005) component method, it is possible to evaluate the rotational stiffness and moment capacity of semi-rigid joints when subject to pure bending. However, this component method is not yet able to calculate these properties when, in addition to the applied moment, an axial force is also present. Eurocode 3:1-8 (2005) suggests that the axial load may be disregarded in the analysis when its value is less than 5% of the beam s design axial plastic resistance (N pl,rd ), but provides no information for cases involving larger axial forces. However, if the applied axial force exceeds the 5% limit, a conservative approach may be used: M M j, Ed j, Rd N N j, Ed j, Rd 1.0 (8.1) where M j,ed is the design value of the joint internal moment, M j,rd is the joint moment design resistance, N j,ed is the design value of the joint internal axial force and N j,rd is the joint axial force design resistance.

6 8 Summary and Conclusion 169 Aiming to overcome this limitation in this existing code related to the component method and based on the results obtained in this work the following design considerations are suggested, as an extension of the current procedures of Eurocode 3:1-8 (2005) accounting for the full interaction of the bending moment and axial forces: - Rotational stiffness: the generalised mechanical model, developed in this word, is suggested to estimate the rotational joint stiffness, considering the influence of the interaction between bending moment and axial loading. The bending moment versus rotation curve can be readily predicted by evaluating three main points of the moment-rotation curve: the first point ( y, M y ) defines the joint initial stiffness corresponding to the attainment of the weakest component yield while the second point ( u, M u ) is obtained when the weakest component reaches its ultimate strength. The third point ( f, M f ) depends on the joint assumed final rotational capacity for the moment-rotation curve, which is adopted to be equal to 0.05 radians. - Strength interaction: the proposed mechanical model can be straightforwardly used to build bending moment versus axial force interaction diagrams, where the proposed analytical model is subjected to different levels of axial load. This is followed by increasing bending moment until the joint ultimate capacity is reached. - Deformation capacity: the joint deformation capacity is controlled by the ductility of its constituent components. In this way a tri-linear characterisation of the joint basic components is suggested in this thesis. The bending moment versus axial load interaction diagram, constructed by using the ideas development in this work, can be used to determine the joint resistance subjected to any combination of bending moments and axial loads, supplying an efficient and complete tool for structural joint designs Main Contributions and Developments of the Present Investigation This section summarises the main contributions and developments of the present investigation: - A generalised component-based mechanical model was proposed to estimate the endplate joint behaviour when both bending moments and axial

7 8 Summary and Conclusion 170 forces are present. It must be underlined the simplicity of the mechanical model utilization, given by analytical equations developed in this thesis, and its accurate prediction of the moment-rotation curves and moment-axial load interaction diagrams. However, the most important and unprecedented contribution could be related to the ability that this model has in representing the changes of the joint compressive centre position according to the axial load levels and bending moments applied to the joint. - A tri-linear characterisation of the joint basic components was suggested in this work, highlighting the novelty of the strain hardening coefficients proposed for endplate joints that are used to estimate the plastic and ultimate stiffness of the joint basic components. - The use of the proposed component-based mechanical model as an extension of the current procedures of Eurocode 3:1-8 (2005) accounting for the full interaction of the bending moment and axial forces and dealing with three basic requirements for the joint performance: strength interaction, stiffness and deformation capacity. - A consistent and alternative methodology to determine any moment versus rotation curve from experimental tests or results obtained analytically, empirically, mechanically and numerically, including the moment-axial load interaction, was also presented. From this alternative methodology it may be underlined its straightforward implementation into nonlinear semi-rigid joint finite element formulation. However, the most important observation referred to this alternative methodology is that the prediction of M- curves for endplate and baseplate joints, considering the bending moment versus axial force interactions, can be made with a simple linear interpolation between two reference M- curves Future Research Recommendations This research work has focused on the development of a component-based mechanical model to describe the beam-to-column joint behaviour including the full interaction of the bending moment and axial forces. This model is based on a general idea that permits the model to represent any kind of joint. Moreover, this model offers practical improvements over current procedure of Eurocode 3:1-8 (2005), because it considers the influence of the axial force effect in the joint

8 8 Summary and Conclusion 171 behaviour and allows modifications of the compressive centre position even before reaching the first component yield, i.e. in the linear-elastic regime, enabling accurate predictions of the moment-rotation curve. The research topics that have been identified in the process of developing and applying the proposed mechanical model include the following issues: - Tri-linear characterisation of the joint components: further experimental examination and numerical analysis using different ranges of joints to check the validity and application of the proposed strain hardening coefficients is still desirable. - Composite joints: a mechanical model for composite joints may be formulated from the proposed mechanical model by accounting for the contribution of the reinforcing bars. A row of reinforcing bars in tension might be similarly treated as a bolt-row in tension in a steel joint while the interaction slab-connectors-beam could be considered by adding a new vertical spring described by the force-displacement characteristic of this slab-connectors-beam system. - Lever arm position: further investigation about the lever arm d, which considers the change of the joint compressive centre position according to the axial force levels and bending moment applied to the joint. It would be enviable to aim on determining a single equation for both tensile and compressive forces and also to prove mathematically if the suggested lever arm position evaluation accurately represents the variations in the joint compressive centre position as a function of the joint loads. - Experimental investigations: few experiments considering the interaction bending moment and axial force have been reported in the literature. Additionally, the available experiments are associated with a small number of axial force levels and associated bending moment versus rotation curves. There is, therefore, the need of further tests associated with various axial force magnitudes and different joint layouts. In conclusion, although there is clearly scope for further improvements, it is believed that the proposed mechanical model offers an effective tool for assessment of structural joints, considering the axial-moment interaction.