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In recent years, the various mechanical weighing machines have been replaced by electromechanical industrial and commercial table-top versions. In modern types of weighing machines, an electrical signal that is directly proportional to the weight is provided for further processing by a microprocessor. The conversion from the mechanical quantity of mass or weight into an electrical signal is carried out by the piece of art termed the load cell (Karaus and Paul, 1992). The load cell is a force sensor that is used in weighing equipment. Most conventional load cells, for loads of 1000 kg or more, contain a spring element made from steel, which deforms under the load that is measured by sensor element, as shown in Figure 1.1. Usually, the sensor element consists of number of resistive strain gauges that are glued to the spring element. However, the accuracy of load cells is limited by the hysteresis and creep and to minimise these effects, expensive high-grade steels are required (Wiegerink et al., 2000)
Figure 1.1 Load cell concept of operation
Load cells are used in several industrial weighing applications. As the signal processing and control systems cannot operate correctly if they receive inaccurate input data, compensation of the imperfections of sensor response is one of the most important problems in sensor research. Influence of unwanted signals, non-ideal frequency response, parameter drift, nonlinearity, and cross sensitivity are the major defects in the primary sensors (Karaus and Paul, 1992; Piskorowski and Barcinski, 2008). Load cells have an oscillatory response which always needs time to settle down. Dynamic measurement refers to the ascertainment of the final value of a sensor signal while its output is still in oscillation. It is, therefore, necessary to determine the value of the measure and in the fastest time possible to speed up the process of measurement, which is of particular importance in some applications. One example of processing to the sensor output signal is filtering to achieve response correction (Piskorowski and Barcinski, 2008).
In this study, Finite Element Analysis (FEA) is conducted on a typical load cell. The stress and displacement of the load cell were modeled using the FE package. Moreover, manual calculations were performed and the results are compared with the model predictions.
The geometry of the load cell is relatively complex. It is therefore, was simplified to ease the construction and utilisation of the modelling techniques. The first phase in idealisation is to implement symmetry in modeling. Also, the upper and lower surfaces of the load cell are assumed horizontal and totally flat to ease modelling process. For the boundary conditions, the load cell is contacting fixed surface through its bottom surface i.e. the seating face. Therefore, the boundary conditions at this contact face are: no allowed any translation motion in x-direction and also in y-direction. Details of idealisation will be discussed in the latter sections.
2.1 Approximate stress calculation
As it is known, the Hook s law can be expressed as:
Thus, the normal stress under tension or compression is directly proportional to the relative elongation or shortening of the bar. The proportionally factor , which links the normal stress with the relative elongation, is called the modulus of elasticity of the material under tension (compression). The greater the modulus of elasticity of a material, the less the bar is stretched or compressed provided all other conditions remain unchanged. It should be borne in mind that Hook s law has been represented by a formula which sums up the experimental data only approximately; it cannot therefore be considered an accurate relation (Quek and Liu, 2003).
In order to manually evaluate the stress values, the positions of the neutral axis were firstly evaluated. For any rectangular cross sections, it is found that the neutral axis is to pass at the sections mid point. Therefore, it is considered that the mid section of the tested load cell takes the form of cantilever beam, which is subjected to normal force and accordingly a bending moment as shown in Figure 2.1. It was also considered as an assumption that the left hand side of the mid section of the load cell is restrained in all the degrees of freedom. It was also assumed that the normal force and the bending moment are acting on the right hand side of the simulated load cell s section.
Figure 2.1 representation of the section as cantilever beam
As the load is acted the result will be the bending moment which can be evaluated using the following expression.
The action of the bending moment is the expected deformation that will take place. For the clockwise affecting moments, the cross-sections located above the neutral axis will be subjected to tensile stresses whereas the cross-sections at the other side will experience compressive stresses. The area of the cross section can be evaluated from:
Given that b and h are the width and the height of the beam, the second inertial moment for the cell s cross section (i.e. rectangular shape) can be evaluated from:
The stress values at the area where the strain gauge is mounted are evaluated for the sections above the neutral axis (+ sign) and below the neutral axis (? sign) as follows:
Therefore, the stresses for the section above the neutral axis are evaluated at:
2.2 Approximate displacement calculation
By using equation (2.1, the strain can be evaluated as:
Given that the Poisson s ratio is expressed as the ratio of the transverse to axial elongations, therefore:
Same procedures can also be applied to evaluate the elongation in the z-direction, as similar value of the strain will be obtained in this direction.
3. Finite Element Model
3.1 Model justification
The geometry of the load cell is illustrated in Figure 3.1 and the dimensions are listed in Table 3.1. Three dimensional proper FE model has been created using the commercial SolidWorks package. The load cell has a simple construction with a uniform thickness throughout. The load can be applied via rods screwed into the M10 threads through two holes at the two ends so that the load can be either tensile or compressive.
Figure 3.1 (a) 2-D projection of load cell model and (b) basic geometry
Table 3.1 Dimensions and properties of the load cell
Wherever there is symmetry in the problem it should be made use. By doing so, lot of memory requirement is reduced or in other words more elements can be used with the use of a refined mesh for the same processing time. When symmetry is to be used, it is worth to note that at the right angles to the line of symmetry the displacement is zero (Belyaev, 1979; Rao, 2010). For the load cell simulation in this study, planar symmetry is used, see Figure 3.2.
Figure 3.2 Views of planar symmetry as applied to the load cell
In the FEA, stiffness matrix of size 1000 1000 or even more is not uncommon. Hence, memory requirement for storing stiffness matrix would be very high. If the user tries to implement the Gaussian elimination straight, he will end up with the problem of memory shortage. So, to reduce memory requirement, according to Belyaev (1979) and Rao (2010), the following techniques are used to store the stiffness matrices:
* Use of symmetry and banded nature
* Partitioning of matrix (frontal solution).
* Skyline storage.
3.3 Stress rising effect
In the development of the basic stress equations for tension, compression, bending, and torsion, it was assumed that no geometric irregularities occurred in the member under consideration. But it is quite difficult to design a machine without permitting some changes in the cross sections of the members. Rotating shafts must have shoulders designed on them so that the bearings can be properly seated and so that they will take thrust loads; and the shafts must have key slots machined into them for securing pulleys and gears. A bolt has a head on one end and screw threads on the other end, both of which account for abrupt changes in the cross section. Other parts require holes, oil grooves, and notches of various kinds. Any discontinuity in a machine part alters the stress distribution in the neighborhood of the discontinuity so that the elementary stress equations no longer describe the stress state in the part at these locations. Such discontinuities are called stress raisers, and the regions in which they occur are called areas of stress concentration.
The distribution of elastic stress across a section of a member may be uniform as in a bar in tension, linear as a beam in bending, or even rapid and curvaceous as in a sharply curved beam. Stress concentrations can arise from some irregularity not inherent in the member, such as tool marks, holes, notches, grooves, or threads. The nominal stress is said to exist if the member is free of the stress raiser. This definition is not always honored, so check the definition on the stress-concentration chart or table you are using.
A theoretical, or geometric, stress-concentration factor or is used to relate the actual maximum stress at the discontinuity to the nominal stress. The factors are defined by Belyaev (1979) as:
where is used for normal stresses and for shear stresses. The nominal stress or is more difficult to define. Generally, it is the stress calculated by using the elementary stress equations and the net area, or net cross section. But sometimes the gross cross section is used instead, and so it is always wise to double check your source of or before calculating the maximum stress. The subscript in means that this stress-concentration factor depends on the geometry of the part, see Figure 3.3. So, the material has no effect on and this is the reason it is called theoretical stress-concentration factor.
Figure 3.3 Stress concentration factor versus dimensions
The analysis of geometric shapes to determine stress-concentration factors is a difficult problem, and not many solutions can be found. Most stress-concentration factors are found by using experimental techniques. Though the finite-element method has been used, the fact that the elements are indeed finite prevents finding the true maximum stress. Experimental approaches generally used include photo-elasticity, grid methods, brittle-coating methods, and electrical strain-gauge methods. Of course, the grid and strain-gauge methods both suffer from the same drawback as the finite-element method (Budynas and Nisbett, 2007).
In this study and for the load cell, the simulation demonstrated that the stress is concentrated at two main regions represented at A and B. Stresses are aso concentrated at the threaded holes, as demonstrated in Figure 3.4. As shown, there is a considerably sharp rise of the stress at these locations because the strain gauges at situated at the middle section. Also, this section is of considerably small area compared with the other load cell s cross sections.
Figure 3.4 Areas of concern for stress concentration in the load cell
3.4 Restraints justification
With the aim of calculating the stress and strain in the middle section of the load cell, the appropriate restraint is used. As we know, the line of action of the applied load, at the upper seat hole, is through a M10 screw. Meanwhile, screw of same size is used to fix the load cell at its bottom base. Accordingly, for the idealisation purposes, it can be said that all the degrees of freedoms (DOFs) are restrained at the location of the hole at the bottom surface, see Figure 3.5.
Figure 3.5 Schematics of the first problem Idealisation step
In the second step of the problem idealisation, it was assumed that by tightening the screw in the bottom face hole of the load cell will cause all the degrees of freedom to be restrained. Accordingly, this condition can cause decreased simulation lead time and enhance the results, see Figure 3.6.
Figure 3.6 Schematics of the second problem Idealisation step
As it is clear, different restraint conditions produce variants of boundary conditions. Finally, in the third idealisation, it is assumed that the load cell can rotate about its y-axis to bring the results as close as possible to reality, see Figure 3.7.
Figure 3.7 Schematics of the third problem Idealisation step
3.5 Load justification
In this section, justifying the applied load is considered throughout the hole of the upper seat. In the first step of the idealisation process, it was assumed that the load is to be applied to affect on the edges of the hole. Therefore, the tension stress transfer to the middle section of load cell where the measurement of stresses and strains are needed, see Figure 3.8.
Figure 3.8 First idealisation step required for the load justification
The applied force transfers to whole the upper section, there, this points that considering a uniform distributed load in upper section might be a proper assumption. Therefore, to apply the consequent idealisation, uniformly distributed load was allowed to takes an affect directly on the upper section. In the first idealisation, the magnitude of point load was assumed to be 300 N. Therefore, the magnitude of the uniformly distributed load (UDL) is found to be 2.3 N/m2, which is equal to the magnitude of point load, see figure 3.9.
Figure 3.9 Application of the uniformly distributed load
In the third idealisation, the applied load is assumed to act by means of the M10 screw and throughout the whole upper hole, see Figure 3.10. This assumption is very close to reality and may present very good results which are in good agreement with the hand calculation of stress and strain.
Figure 3.10 Applied load act by M10 screw throughout the upper hole
3.6 Element type
The largest commercial finite element packages, which have facilities to solve stress and a variety of field problems, might easily have more than one hundred different finite element available for the user. The selection of which element to use by given problem is not as difficult it might first appear, first, the type of problem to be analysis, secondly, the chosen dimensionality of the module restricts range .Before choosing the element type; the engineer should try to predict what is taking place in the problem to be examined. Figure 3.11 shows a typical range of element.
Figure 3.11 Typical ranges of elements
4. Discussion of Results
4.1 Aspect ratio
The finest accuracy values can be guaranteed with the use of elements meshed using uniform perfect tetrahedral as solid mesh, which has equal length edges. For a general geometry, it is impossible to create a mesh of perfect tetrahedral elements. Due to small edges, curved geometry, thin features, and sharp corners, some of the generated elements can have some of their edges much longer than others. When the edges of an element become much different in length, the accuracy of the results deteriorates. It should be noted that the shape of mesh is critical to analysis as higher density improves solution at the cost of increased computational time. The simple geometry require fewer elements, more complexity requires increased density and the mesh shape is related to the loads and the boundary conditions.
The aspect ratio of a perfect tetrahedral element is used as the basis for calculating aspect ratios of other elements. The aspect ratio of an element is defined as the ratio between the longest edge and the shortest normal dropped from a vertex to the opposite face normalized with respect to a perfect tetrahedral (Belyaev, 1979; Rao, 2010). By definition, the aspect ratio of a perfect tetrahedral element is 1.0. The aspect ratio check assumes straight edges connecting the four corner nodes. The aspect ratio check, see Figure 4.1 is automatically used by the program to check the quality of the mesh.
Figure 4.1 Aspect ratio checks
4.2 Jacobian check
The elements with the parabolic nature can be effectively used with the curved geometry shapes. It is therefore expected to result in more accurate predictions compared with the linear elements even if they are of similar size. In this case, the elemental nodes (on the middle side) of the boundary corners can be situated on the model s real geometry. However, these placements of nodes can cause distorted elements with crossing by edges, in boundaries of very sharp curvature. Accordingly, the Jacobian of such distorted element would be of negative values, which can cause cancelled software operation of analysis. Selected points situated within each model element can be used to perform the Jacobian checks. The software package allows the user to select the Jacobian check limits i.e. using 4, 16, or 29 nodal Gaussian points.
The Jacobian ratio of a parabolic tetrahedral element, with all mid-side nodes located exactly at the middle of the straight edges, is 1.0. This ratio increases with the curvatures of the edges. At a point inside the element, this ratio provides a measure of the degree of local elemental distortion. The software calculates the Jacobian ratio at the selected number of Gaussian points for each tetrahedral element, see Figure 4.2. Based on stochastic studies, it is generally seen that a Jacobian ratio of forty or less is acceptable. The software adjusts the locations of the mid-side nodes of distorted elements automatically to make sure that all elements pass the Jacobian check (Belyaev, 1979; Rao, 2010).
Figure 4.2 Jacobian ratio checks
4.3 Connectivity of elements and mesh grading
To achieve an accurate result we need to check the connectivity of all elements so precisely. Any discontinuity may result in large error in stress or strain or displacement calculation in purposed area. With the aim of this, after checking all the area of the load cell, no dis-connectivity was observed. Also mesh grading illustrated in Figure 4.3.
Figure 4.3 Mesh grading checks
In areas of the model where there are high stress gradients it is normally necessary to use more elements to obtain a high quality solution. Often this will happen automatically when an automatic mesh generator is used. This is because the mesh generator uses the segments (e.g. arcs, straight lines, surfaces) of the solid model as a starting point for the mesh. Since the high stress gradients will be around geometry that changes within a short distance, these seeding features will be small. However, it may be necessary to control mesh quality either to force smaller elements where they have not been automatically generated or to allow larger elements where the analysis does not need to be accurate.
4.4 Displacement and stress discontinuity
The plot representing displacement variations can be utilised for displacement discontinuity checks (Barrans, 2010). This can solely takes place at the elements connected incorrectly. It also takes place for the improperly defined geometries so slivers and small gaps can exist as a blackboard. Checking the displacement of load cell visually shows that there is no displacement discontinuity, see Figure 4.4.
Figure 4.4 Displacement discontinuity checks
After the nodal displacements evaluation, the code continued to evaluate, for each element, the strain and stress values, separately. The stress was evaluated at specific element points, which are intentionally placed to enable having accurate outcomes and they are termed Gaussian or quadrature. After calculating the stresses at these points, the code calculated the nodal stresses for each element by extrapolation. For an exact solution, all elements should give identical stress values at their common nodes. While the displacement field obtained by FEA was continuous, stress field was discontinuous from an element to another. Different elements give stress values that are generally different at a common node. The code calculated the nodal stress, see Figure 4.5, at common node by averaging the values at the contributing elements (Belyaev, 1979; Rao, 2010).
Figure 4.5 Nodal stresses evaluation
4.5 Sensible displaced shape
Figure 4.6 shows, and as predicted, the most sensible displaced section is the middle section of the load cell.
Figure 4.6 most sensible displaced section
4.6 Approximate stress and displacement
As shown in Figures 4.7 through 4.9, the results of the simulation are in good agreement with the hand calculation of stress and strain.
Figure 4.7 Manually evaluated stresses are as the marked value (4.92 x107N/m2)
Figure 4.8 Manually evaluated strain values are about the marked value (1.46 x10-4)
Figure 4.9Justification of stresses matching
4.7 Stress discontinuity
In order to evaluate the stress discontinuity, three values are requires for the principle stress, which are the maximum, mid and the minimum value. The dark spots represent the places at which there is stress discontinuity, see Figure 4.10.
Figure 4.10 Discontinuity in the values of stresses in the adjacent elements.
Stress discontinuity evaluation
The values of the principle stresses at different shown in figure 4.11 were evaluated and then used to calculate the stress discontinuity. The stress values and displacement are also shown.
Figure 4.11 Values of the stresses in the adjacent elements
The stress discontinuity at each node is evaluated from:
Stress discontinuity (%) = =
Stress discontinuity (%) = 17.12 %
It should be noted that the nearly zero displacements at the two nodes used in the calculations proved the right choice of constrains of the complete fixation of the seating face.
4.8 Convergence study displacement and stress
Figures 4.12 and 4.13 show the stress and displacement convergence diagrams. These figures demonstrated the convergence with continue solution using the software as plotted against the loop numbers.
Figure 4.12 Stress convergence diagram
Figure 4.13 Displacement convergence diagram
Moreover, Table 4.1 shows the convergence results for Von-Mises stress values at different nodes. Also the presentation of these stresses against the number of elements is given in Figure 4.14.
Table 4.1 Stress convergence at different nodes
Figure 4.14 Von Mises stress versus elemental number
Also, Table 4.2 shows the convergence results for the displacement values at different nodes. Also the presentation of these displacement values versus the number of elements is given in Figure 4.15.
Table 4.2 Displacement convergence at different nodes
Figure 4.15 Displacement stress versus elemental number
Load cell unit has been modeled using the finite element software. As well, hand calculations were performed to evaluate the values of the stresses and displacement. The load cell was first idealised so as to ease the modelling processing. The model was built and the predicted results showed that the displacement was higher at the mid sections of the load cell. The predicted results when compared with the manual calculations showed good agreement for the stress and displacement.