### 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

where L is the length of the key. Lower of these two values will form the design basis. It can be seen that load capacity in compression is lower in the case of square cross section (W = H).

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.

### Comments

1. 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

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

3. It 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?

4. The 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.

5. The 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.

6. The 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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