### Design Procedure Assessment, DPA-01, Square Keys by Kotur Raghavan

**General**

Keys are among the simplest
mechanical elements. They are deployed to transmit power (torque) from a shaft
to a hub or vice versa. The hubs are normally an itegral part of a bigger
component such as a pulley, gearwheel or a turbine disk. The figure below is
representative of the functioning of keys.

Fig. 1

The applied torque from hub to
the shaft or the other way is entirely transmitted through the key by way of contacts.
There are four contact surfaces on the side flanks of the key. Two of them are
active depending on the mode of power transmission and also the direction of
rotation. In addition, the top and bottom surfaces also may come into play
because the key normally has snug fit with the other two components.

**Traditional Design Approach**

In the figure below, free body
diagrams of the key is shown. They correspond to the case of power transmission
from the hub to the shaft in the anti-clockwise direction or from the shaft to
the hub in the clockwise direction. The force P is the result of the applied
torque.

Fig. 2

The assumptions involved in
design are that the key will fail either in shear or in compression. These are
in line with contents of the book by Joseph Shigley and Charles Mischke *(Mechanical Engineering Design, Sixth
Edition*) and the book by V B Bhandari *(Design
of Machine Elements, Third Edition)*. Shear failure is along AB and
compression failure is at surfaces AC and BD. If W is the width of the key and
H the height, then the areas resisting shear and compression are respectively W
and H/2 per unit length of the key. If SA is the allowable stress in tension or
compression, the load capacities (corresponding to P) of the key in shear and
compression are given by

It is notable here that
the combined influence of the shear stress and compressive stresses is not
considered here because the affected areas are mutually exclusive. As we will
see later, the theme of this article revolves around this aspect.

**Finite Element Model and Analysis**

The state of stress can
be assumed to be uniform along the length of the key. Consequently a 2D plane
stress analysis is carried out. The mesh information is given in Fig. 3. There
are six contact pairs. Three of them connect the key to the hub and the other
three connect the key to the shaft.

Fig. 3

The shaft diameter is taken to be
400 mm and the dimensions of the key are 60*60. An allowable stress of 200 MPa
has also been assumed. For these data it the torque capacities have been
calculated to be **1386, 1200 and 907** kilo
newton-mm respectively for failure based on shear, compressive and combined
stresses.

For the purpose of analysis it is
assumed that the hub is transmitting torque to the shaft in the anti-clockwise
direction. The inner radius of the shaft is constrained. Torque is applied in
the form of equivalent tangential forces at the nodes on the outer periphery of
the hub. In order to avoid penetration pf shaft into the hub, the interface
nodes are radially coupled.

**Results for 1200 kN-mm torque**

The deformation patterns in the
assembly and the key alone are shown in Fig. 4. The patterns are along expected
lines.

Fig. 4

More important, however, are the
stresses experienced by the key. The figure below shows the contours of von
Mises stress.

Fig. 5

The contours are drawn for the
range of 0 to 200. The regions adjoining the red regions on the higher side
have stresses more than the allowable value of 200 MPa and hence are unsafe.
They are seen in grey colour. It is observable that the unsafe region is nearly
parallel to the load path and is spanning the entire width of the key. In order
to have a detailed look into the stresses the variations of stresses along the
line SS (Fig. 4) are looked into.

Fig. 6

Fig. 7

In figures 6 an7 the variations
of absolute compressive stress and absolute shear stress are shown. The
allowable limits for both these stress components are also indicated. The
stresses are within safe limits but for the high values at ends. These are
local effects due to singularities. An important observation from Fig. 6 is that
the compressive stress is dominant all along the path. This contradicts the
assumption made in the text books.

As both the compressive stress
and the shear stress are having significant magnitudes all along the load path,
their combined effect has made the key unsafe all along the load path.

Fig. 8

The variation of the von Mises
stress along the load path is shown in Fig. 8. The stresses all along are above
the allowable and this is in conformity with the contours seen in Fig. 5.

**Results for 907 kN-mm torque**

A torque magnitude of 907 kN-mm
corresponds the load limit based on the combined effect of the shear and
compressive stresses. Intuitively this is the appropriate criterion as it has
been demonstrated that shear and compressive stresses are present all along the
load path.

Fig. 9

The contours for von Mises stress
were computed as shown in Fig. 9. Grey colour regions are observed at the load
transfer location and at load resistance location. These are local peaks.
Importantly, however, the complete load path is seen to safe.

**Summary and Lessons Learnt**

· The main conclusion of this study is that the
design as per text book methodology (for a torque of 1200) is inadequate. It
has been shown that the torque capacity Is 907. An implication is that the originally
assigned factor of safety gets debited by a factor of 1.32. The normal industry
practice is to use a factor of safety of 1.5 for ductile materials. In such cases
the factor of safety effectively becomes 1.13.

·
It was stated in the earlier article that
normally only one stress component is normally considered in the traditional
design practice. In the case of keys, two stress components have come into
consideration. However, it is assumed that they act independently and on
regions which are mutually exclusive. The fact that they are dominant all along
the load path has been overlooked.

·
A recommendation thus is for introduction of
suitable modification in Machine Design books.

A notable aspect of the current study is that the analysis model does not involve any assumption. A more rigorous 3D model can be used. But the results are not expected to be different. The load path has been accounted for very realistically. This has been possible due to the use of contact elements.

That is nice! Is it possible to carry out a parametric analysis and generate a design document ( in the form of curves) as a supplement to the conventional design practice

ReplyDeleteYes. This present exercise of mine is for the ultimate goal of compilation i a book form. Thanks.

ReplyDeleteIt must be pointed out that your free-body diagram of the key is no in equilibrium, i.e., there are some forces missing which balance the unequilibrated couple in your diagram. Maybe if these were considered in the theoretical solution then closer agreement might be obtained?

ReplyDeleteThe free body diagram is representative of what one finds in Machine Design text books, It is covering the load application location and load resistance region. It is adequate, I feel, keeping in mind the scope and purpose of the article.

ReplyDeleteThe article appears to be picking a hole in the strength of materials solution from Shigley's text using FE as an alternative solution. Unfortunately, though, the FE model is not the same as the free-body diagram which itself is in error. The FE model has contact all the way round the key and you will find, if you look at reaction forces, there are normal forces on the top and bottom surfaces of the key and, miraculously, these equilibrate with the missing couple in the free-body diagram. As such, you are comparing solutions to different problems and it might therefore not be surprising that there is a difference. You could, for example, try running the FE model without the top and bottom contacts. You'll need to add friction to the remaining contacts in order to generate the equilibrating couple but a rather different solution will be obtained.

ReplyDeleteThe purpose of the work is just to highlight the effect of simplifying assumptions used in Machine Design textbooks. In the present case it has been shown that the assumptions happen to be non-conservative. The route used is a high-fidelity finite element model which involves no assumptions. I have a host of such studies. They will appear in this web site in due course.

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