Negative Reactions – Insights and Applications by Kotur Raghavan
General
In an earlier article dealing with tower bolts, it was observed that deployment of more number of set screws turn out to be counter-productive.
(https://www.fembestpractices.com/2020/11/tower-bolts-2-review-of-assumptions-by.html ).
The reason was that negative reaction forces induce positive reactions of higher magnitude in order to satisfy the equilibrium needs. It was also observed that the simplified model, in which displacement constraints are applied to simulate the socket behaviour, the some of the screws are effectively in compression. This condition, however, physically not possible. In this article we will first take a look into the phenomenon of negative reactions. Based on the understanding a revised model of the tower bolt will be arrived at.
Example Problem, A Table with Six Legs
Fig.1
A table as shown in the figure
above is taken up as a case study problem. Two-dimensional shell and
one-dimensional link (or spar) elements are used as shown. This is analysed for two different load
cases. In the first case, a load (1000 newtons, downward) is centrally applied.
In the second case the load is applied close to the midpoint of one of the
shorter edges.
Fig. 2
For the case of symmetric loading
the deflection pattern and the reaction forces are shown in the figure 2 above.
All the reactions are positive. This indicates that all the legs are supporting
the load.
Fig. 3
For the case of eccentrically
applied load, the corresponding results are given in the figure 3 above. The
nodes 13 and 16 are having negative reactions. The associated legs, which are farthest
from the load application location, are trying to leave the ground. They cannot
do so because of the applied displacement constraint.
Nonlinear Analysis
The same problem is now solved by
introducing nonlinearity. The nonlinearity is imposed by assigning the
condition that the links can take only compressive loads. Because the elements
behave differently in tension and compression, the problem becomes nonlinear.
The results for this analysis are given in the Fig. 4 below.
Fig. 4
The legs at 13 and 16 are free
now. Consequently the reactions forces at these nodes are zero. The applied
load is supported by the other four support nodes – 11, 12, 14 and 15.
This problem shows the right way
of dealing with negative reaction issues. The important fact is that the legs
are not built into the ground and hence it is inappropriate to apply
displacement constraint. We also note here that in the centrally laded case, it
was alright to constrain the nodes. The reason is that all the legs supported
the applied load.
What has been simulated is the
so-called ‘support nonlinearity’. In addition to the option used in the current
case study, there are two additional options available for simulating support
nonlinearity. They are ‘tension-only’ link elements and contacts. The former is
often referred to as CABLE element.
A Revisit to the Tower Bolt Problem
In light of the foregoing
discussion and the lessons learnt, the tower bolt structural analysis model
will now be slightly modified. In the model it was assumed that the bolts are
integral appendages to the door. The projecting portion of the bolt was
constrained at the set screw locations. In the revised model these constraints
will be done away with. Instead, the bolt will be grounded through link
elements. In the problem dealing with a table the legs were modelled using
compression-only links. In the tower bolt problem the screws will be simulated
using ‘tension-only’ link elements.
Fig. 5
The changes which have been introduced
are indicated in the figure above. All other modelling details remain unchanged.
Earlier, two different configurations, having six screws and twelve screws were
studied and compared. For both the cases comparison are made between the
earlier model and the present model.
Comparative results are provided in the figures below.
Fig. 6
Fig. 7
In the above figures nodes with number
difference of twenty define the end nodes of the links. For example 1 is
connected to 21, 11 to 31 etc. Zero reaction nodes are not supporting the
applied loads. We see no inconsistencies with the revised model results. It can
be seen that the positive reactions are identically same in the six-screw and
twelve-screw configurations. This means that the additional screws
corresponding to 11 to 16 are supporting no load. However, they will serve as
second line of defence.
The inference is that use of
tension-only links is acceptable and is recommended for use.
Deformations
In the model, it is to be noted
that the plate which is simulation the bolt is floating in the frame region but
for the tension links. Under this condition, the deformation contours are as
shown in the figure 8 below.
Fig. 8
The frame portion of the bolt is
undergoing both positive (away from the frame) and negative deformation. In
reality, however, the bolt will not have full freedom. It is in close contact
with the socket and the frame. The socket also is in contact with the frame. It
may be expected that the bolt will touch the socket away from the wall and
touch the frame near the wall. These aspects are not accounted for in the
model. They can be accounted for only in full 3D model including contacts. Load
paths and load sharing are likely to be different in the high fidelity model.
That will be a future work.
Key Takeaways
1. Learning
the use of nonlinear elements which have bilinear behaviour.
2. Understanding
the phenomenon of negative reaction – symptoms and remedies.
3. Development
of a more realistic structures model for the tower bolt involving tension-only
elements.
4. Recognition
of the need for an even more rigorous model which includes 3D elements and
contacts.
_ve reactions are best comprehended for column supports. Using that to interpret our problem has provided the clue.
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