8 Summary and Conclusion
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8
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.
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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
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
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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
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.
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- 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
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 0.0021%.
This indicated that the developed equations in this work were accurate for the
Error = abs(sin(q)-q)x100 (%)
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usual problems involving beam-to-column joints.
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
0.00
0.01
0.02
0.03
0.04
0.05
q (rad)
Figure 104 - First-order approximations error magnitudes versus joint rotation.
8.2.
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
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
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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.
8.3.
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 (Npl,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 j ,Ed
M j ,Rd

N j , Ed
N j , Rd
 1 .0
(8.1)
where Mj,Ed is the design value of the joint internal moment, Mj,Rd is the joint
moment design resistance, Nj,Ed is the design value of the joint internal axial force
and Nj,Rd is the joint axial force design resistance.
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, My) defines the
joint initial stiffness corresponding to the attainment of the weakest component
yield while the second point (u, Mu) is obtained when the weakest component
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reaches its ultimate strength. The third point (f, Mf) 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.
8.4.
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
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
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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.
8.5.
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 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
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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.