# FEA of steel cable conveyer belt splices.

Finite element analysis has existed for many years. In simple terms, it is a numerically intensive analysis of an arbitrary geometry. The user subdivides complex two dimensional shapes into small rectangular or triangular sections, or three dimensional shapes into rectangular prismatic or pyramidal sections, that are then analyzed as a continuous structure by combining large numbers of basic equations. The analysis produces displacements, strains and stresses that result from a loading condition applied to the geometry. A simple example will be used to answer the often-asked question: Why is FEA needed?

Let's assume an analyst is designing a metal handle for a briefcase and he or she wishes to analyze it, to determine if it will support the weight of the case. The model seems reasonably simple, but to calculate the stresses and displacements using conventional methods would take days and still be only an approximation. Analyzing the handle with finite element techniques would only take an hour or two and produce a complete picture of the structure. To show a typical FEA, this handle was analyzed with the basic dimensions and loads shown in figure 1. Because hand-calculating the results would be difficult and time consuming, an analyst would choose to use FEA.

To begin the FEA, the handle is broken into the small rectangles or finite elements as shown in figure 2. This model is a series of points or nodes in space represented by x, y and z coordinates from a 0, 0, 0 origin, which is chosen by the analyst. The finite elements are four nodes arranged in a rectangle and represented by four lines. Each node is used in multiple finite elements and will transfer the reactions to loads from one element to the next based on the material's stiffness. Although the analyst sees the model as two dimensional with no thickness, the software is told a thickness to allow for the calculation of the correct stiffness. The loads are applied to the nodes as concentrated forces in the x or y directions. To avoid instability, some of the nodes are fixed so that they cannot move in one or more of the two primary directions.

At this point, the software is executed in the computer using the mathematical model. Several types of results are produced for the analyst's review. The first is the amount of movement or displacement that occurs from the loads. Displacements are then used to compute strains, which are numeric expressions relating deformed shapes to undeformed shapes. In other words, if you have a 100 mm long object and it is stretched to a total length of 200 mm, the object experiences 100% strain. The strain is calculated at every finite element.

Stress is calculated from the strain in the model and is the amount of load that exists on a specific area, often shown in Newtons per millimeter squared. The question becomes: What can be done with this information?

Assuming the analyst knows the strain or stress limits of the material, a determination of failure can be made with respect to the loads used in the analysis. For example, let's assume that the handle can withstand a maximum strain of .01% and the predicted strain based on the analysis is .000966. In this case, the part can easily survive the load it is expected to support, because the handle's material can withstand more strain than the analysis predicts.

Experimental

Although FEA has been used to analyze metal structures for years, analysis of elastomeric parts has been rare. Elastomers have two major complexities that do not affect metals. The first is the fact that they are theoretically liquids and virtually incompressible. This means, if you have 100 cubic millimeters of an elastomer, it can change shape, but cannot be reduced in volume, unlike a metal which can lose volume. This leads to an interesting consideration that can be demonstrated with a simple FEA model. Figure 3 shows both the geometry and finite element model of a steel chamber with a pocket filled with an elastomer. The model appears two dimensional to the analyst, but the software is told that it is a solid cylinder with a center axis against the left edge of the finite elements. A finite element analysis is done by loading the top of the elastomer with an even pressure of 206 MPa and fixing the bottom of the steel cylinder. When the analyst looks at the results in figure 4, he or she can see that all the stress appears in the steel portion of the structure. However, in figure 5, all the strains are in the elastomeric portion. This is easily explained by the elastomer being an incompressible liquid which transfers the applied pressure to the steel causing high stresses. The high strains in the elastomer were produced by the material's natural attempt to flow away from the load, causing localized large distortions as the steel fails and collapses. This example shows that strains are the controlling factor in elastomeric analysis.

The incompressibility of elastomers which causes high distortions or strains also creates the second complexity in the analysis of these materials. As an elastomer distorts to flow away from the load, the molecular structure actually modifies itself. This causes the material to have a varying stiffness, depending on the distortion or strain. This unique characteristic is known as non-linear behavior. Figure 6 shows a graph of the straight sloped line representing the load versus displacement plot of a typical metal. In metal analysis, an analyst can perform an FEA at any load that will not fail the metal and determine the stress, strain and displacements. The results can then be ratioed up or down the sloped line, producing results at any other load below the failure point of the metal. Figure 7 shows the complex curve of an elastomer. The non-linearity of the curve is produced by the elastomer's stiffness variation which requires the analyst to do non-linear FEA. This involves a series of smallstepped analyses that allow the software to accurately advance along the elastomer's material stiffness curve. The analyst tells the software the maximum load that is to be used and the number of steps or increments to be analyzed to achieve this load. If the maximum load is 100 Newtons and the steps are in 10 Newton increments, the software will do an initial analysis at 10 Newtons and establish the strain levels in each finite element. It will then modify the material stiffness in each element to suit its particular strain level and then proceed to a load of 20 Newtons. At each step, the material stiffnesses are modified for every finite element. The process is repeated for 30 Newtons, 40 Newtons and so on until the analysis reaches 100 Newtons, at this point, every element in the structure will have a unique stiffness and set of results. The more steps taken, the more accurately the material stiffness curve will be followed, but the longer the analysis will take. So the analyst must have the experience to optimize the accuracy while minimizing computer execution time.

It is interesting to know how FEA functions, but can it accurately predict the behavior of a part made of an elastomer? Here are two examples to determine how accurate this type of analysis is for elastomers.

The first example is the simple dumbbell sample shown in figure 8. This sample was pulled to break and the load versus displacement curve was recorded. An FEA model was then created to duplicate the actual test.

Figure 9 shows the series of FEA-predicted displacements from no load to 146.3 Newtons. As can be seen, the model is executed over several load steps to accurately predict its final displacement. Figure 10 shows a graph of the actual test results versus the FEA results. In this case, the FEA was able to reproduce the test very accurately. The small differences in the curves are similar to the variations seen in replicate dumbbell tensile tests.

Since it is obvious a simple part can be successfully analyzed, can a significantly more complex structure also be simulated? To determine this, a five cable elastomeric pull block was created. This sample consists of four steel cables extending from one end of an elastomeric block and a single center cable extending from the other end. The finite element model must three-dimensionally duplicate the elastomer and the flexibility of the cables. Cables are unique steel structures that have high tensile strength along their lengths, but are flexible in all other directions. These characteristics can be duplicated by using orthotropic material matrix theory. In other words, by using a complex set of calculations from known cable information, the flexibility and strength of a steel cable can be duplicated with a mathematical matrix. Figure 11 shows the three dimensional FEA model of the sample which was created and executed within the computer. The sample's FEA-predicted displaced shapes and maximum strains are shown in figure 12. The resulting load versus displacement curve was plotted and compared to actual pull test results. The two curves shown in figure 13 are a comparison of actual test results versus FEA predicted results. The resulting curves are strikingly close for such a complex structure considering similar result variations are seen in duplicate testing of such blocks. It appears from this example that even complex structures with multiple materials can be accurately simulated with finite element analysis.

Results and discussion

Since it can be demonstrated that FEA will accurately predict actual results, the technology can be used on product development. To demonstrate one of the many applications, let us look at a steel cable conveyor belt splice analysis. A splice of this nature is extremely complex, typically involving top and bottom cover layers of elastomer and a layer of flexible steel cables surrounded by a second elastomer. All of these features require accurate material properties and modelling. There are many types of splices, but for this demonstration a single-stage splice was used. Using FEA, multiple complex three-dimensional models can be created to check various splice variables. To show this capability, it is assumed that the analyst wishes to determine the splice's optimum length. This can be done by creating five 3-D FEA models with lengths of 900, 1,200, 1,500, 1,800 and 2,100 mm. Figure 14 shows the 900 mm model and cut away views of the imbedded steel cables. Belts are typically designed based on a 100% rated tension which is the maximum load the belt is expected to see during its normal fully loaded operation. The 150% rated tension is the maximum load the belt is expected to see during a fully loaded start up operation. The analysis is done by tensioning the various splices up to 150% of the belts rated tension by equally loading each belt cable. After running the five models, the maximum green strains in the elastomer were reviewed to determine the optimum splice length. In general terms the maximum green strains are the non-directional large displacement maximum strains that are calculated on the distorted elastomer at each load step. In all the models, the maximum green strains occur in the area where the steel cables are bent to become part of the splice.

The maximum green strain of 93.8% at the cable bends is shown in the 900 mm splice. As the splices become longer, as observed in the 1200 mm splice, the maximum green strain decreases to 74.3%. The 1,500 mm splice continues the downward trend with a strain of 65.1%. The 1,800 mm splice's maximum green strain level starts to level off at 54.4% and the 2,100 mm splice verifies the formation of a valley with a maximum green strain of 47.7%. Figure 15 compares the maximum green strains over a series of loads up to 150% of the belts rated tension. Each of the various length splices are shown as individual non-linear curves. The top curve represents the 900 mm splice and the bottom the 2,100 mm splice. It can be seen that the curves are slowly coming together as the splice lengths increases.

The 1,800 mm and 2,100 mm are almost on top of each other showing that additional length is becoming meaningless. Figure 16 summarizes the FEA data for all five splices at three tension ratings, by plotting maximum green strain versus splice length. This graph shows the strain levelling out at 2,000 mm length at all three tension ratings. An experienced analyst can see that to minimize elastomeric strain in the belt splice, it is not necessary to make the splice longer than 2,000 mm. This demonstrates how FEA can provide the conveyor belt designer with information that allows the optimization of the splice designs. Although this example used a simple single stage splice design, the FEA technique has been used with equal success on very complex steel cable belt splices.

Conclusion

Even though finite element analysis of elastomerics is considerably more complex than for other materials such as metals, due to the uniqueness of the material and the large computational needs of the software, it can be performed successfully. Finite element analysis can be a useful tool for conveyor belt and slice design as well as many other elastomeric parts.
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