Sensitivity Analysis of Perforated Plate using Design of Experiments for Weight Optimization

Sensitivity Analysis of Perforated Plate using Design of Experiments for Weight Optimization

Sujith Kumar C

1. (Design) Mechanical Engineering Department Vishwakarma institute of technology

   Pune, India

   Prof. Dr. M.V.Walame Mechanical Engineering Department Vishwakarma institute of technology Pune, India

   Abstract – In todays world, speed to market and cost are major drivers for every product development process in an organization. Utilization of tools like Design of experiments, Topology optimization combined with finite element analysis will help in reducing the complexity, cost and product life cycle of structural components without compromising the structural strength. Perforated plates have found variety of applications in structural members and various studies have been conducted on perforated plate with lateral load and effect of cut out shapes on the strength. As the cutouts in such plates induces stress concentration, it is important to study the effect of parameters like plate thickness, hole diameter, spacing of perforation on the strength of these plates. The objective is to study the effect of mainly three parameters, hole diameter, plate thickness and hole pattern on stress. Design of experiment methodology is applied to list out all possible combinations for these parameters. FEA results for all 144 combinations is analyzed and experimental verification is done for 3 combinations. From the results of DOE and Finite element analysis, a generalized equation was established correlating effects of hole diameter, plate thickness, hole pattern for sensitivity analysis. It is found that plate thickness has major effect on stress and deformation followed by hole spacing in direction perpendicular to bolting face. The effect of hole diameter is minimal.

   KeywordsPerforated Plate; Design Of Experiments; Stress Analysis; Sheetmetal; Optimization

   1. INTRODUCTION

      Structural optimization [1] has gained high prominence in recent years, thereby increasing need for lightweight structures without compromising the structural efficiency. One such structural member is perforated sheet metal plates. These plates are easily manufactured and are widely used in many engineering applications such as platforms in various agricultural and earth moving machines, offshore platforms, ship decks and hulls, box sections of bridge girders and air craft industries. There is an often need for cut-outs in plates for services, dirt drain, inspections, maintenance and majority of times to reduce weight of structure.

      The presence of holes on the plate changes the stress distribution and cause reduction in its strength [2] [3]. Hence proper combination of hole size, plate thickness, hole spacing is crucial in maintaining the strength of the plate.

      In this study a typical perforated plate to be used in stairs of an agricultural machinery is studied. Design of Experiment is used to identify the important interactions of plate thickness, hole size and hole spacing on perforated plate, effect of them on strength and specimen selection for performing detailed FEA and experiment.

   2. LITERATURE SURVEY

      Various studies have been carried out related perforated plate with lateral loads as well as axial loads.

      Jae-Hoon Kang[4] investigated Exact solutions for stresses, strains, and displacements of a perforated rectangular plate by a central circular hole subjected to linearly varying in- plane normal stresses on two opposite edges by two- dimensional theory of elasticity using the Airy stress function.

      M. Aydin Komur [2] investigated the elasto-plastic buckling behavior of simply supported square and rectangular thin steel plates having elliptic cut-outs.

      Jinho Woo [5] investigated stress concentration of perforated plates with cur out, orientation of cut out and bluntness.

      EmanueleMaiorana [6] analyzed linear buckling of square and rectangular plates with circular and rectangular holes in various positions subjected to axial compression and bending moment.

      D.B.Kawadkar [7] studied the stress concentration in plate with various cutouts and bluntness with different cutout orientation.

      J. Rezaeepazhand [8] conducted analytical stress analysis of plates with different central cutout. Particular emphasis was placed on flat square plates subjected to a uni-axial tension load.

      Venkatachalam G. [9] investigated the influence of holes on the flexural strength of Aluminium 8090 alloy sheets. They carried out various experiments to study the influence of hole size, % of open area and hole arrangement pattern.

      O.R. Nandagopan [10] investigated perforated plate with lining to determine the static deflection of the plate.

      Very less references is found for effects of perforated plate parameters when vertical loaded.

      Application of design of experiments on sheet metal plate optimization.

   3. DESIGN OF EXPERIMENTS AND PARAMETERS OF

      PERFORATED PLATE

      Experiment is an integral part of optimization. Experiments are performed in all fields and are used to study the performance of processes and systems [11].

      One of the issue in the conventional CAD based design development is that the number of iterations required for finalizing the design in which each iterations output of finite element analysis. In addition the time taken for actual testing considering all variables. The impact is time consuming and high process cost.

      DOE techniques provide guidance to choose the experiments to be performed in an efficient way [12] and hence the technique is utilized in this study to minimize the experimental specimens to three from the available combinations.

      The objectives of the experiment include: determining the variable that has highest influence on response, determining where to set the influential controllable variables so that the response is almost always near the desired optimal value, which in turn will result in minimizing the variations in response and the effect of uncontrollable variables.

      Perforated sheet metal plates has manually or mechanically punched holes where the hole shapes can be round, square, triangle, diamond, oval, hexagonal etc. It is generally advisable to have the hole size larger than material thickness.

      The parameter that effects the stress and deformation of a perforated plate are: Plate thickness, Hole size, Pattern (Linear or scattered), Pitch (distance between the hole centers), Open area (total area of holes divided by total area of sheet), Margins (blank area along the edges of the sheet), Material property, Manufacturing process.

   4. GEOMETRY AND MATERIAL PROPERTIES

      Geometry of the specimen for study is shown in Fig 1 Assumptions (based on practical application):

      • Plates size 250mm x 250 mm

      • Hole pattern is square

      • Bolted on two sides

   5. FINITE ELEMENT MODELING

      Ansys 16.2 is used to carry out finite element analysis and design of experiments [16].

      1. Finite element modeling:

         Meshing element type is 4 node quad shell 181 Mesh size is 1mm

         Node population count 186932 Element population count 92331

         Design load of 2724 N applied centrally on a surface of 145mm diameter

         Load Behavior is taken as Rigid which resembles the practical case.

         Boundary condition- blotted on two faces with three bolts and hence the holes surfaces were fixed.

         Figure 2 shows the distribution of stress in perforated plate.

         Fig. 2. FEA model set up

         Strain gauge location is picked on the basis of vector principal strain directionshown in figure 3.The maximum strain location happens to be at the point below the load which is impractical to gauge. So the next high strain area (close to bolted side) was chosen as gauge location for correlation.

         Material properties:

         Fig. 1. Geometry

         The material specifications of the specimen is listed in table 1 [13].

         Load:

         TABLE I. MATERIAL DATA

         Material	MS IS 513 CR2
         Yield Stress	240Mpa
         Density	7.85e-6 kg/mm3
         Modulous of elasticity	210GPA
         Poisonss ratio	0.3

         Fig. 3. Vector principal strain direction

      2. DOE set up:

      The parameters and their range considered in the study are shown below

      Input parameters: Plate thickness-2, 2.5, 3, 4 mm, Hole spacing in X direction-40, 70, 90 mm, Hole spacing in Y direction-40, 70, 90 mm, Hole diameter-12,16,20,24mm

      General weight for large operator 114.1Kg [14] with a bag of seed 25Kg.

      Design load considered is 278 Kg (2724N) (including factor of safety-2) [15].

      Output Parameter: Mass (Kg), Maximum equivalent stress (Mpa), Strain (µstrain), Deformation (mm).

      Input/output parameters and resulting design point example is shown in figure 4 and 5.

      Fig. 4. Desing of experiment in Ansys

      Fig. 5. Desing of experiment in Ansys Fig. 6.

      The data from DOE was analyzed (144 design points) and 3 specimens were picked to perform experiment for validation as per below table.

      TABLE II. TEST SPECIMEN

      	THICKNESS (mm)	HOLE SPACING (mm)		HOLE DIA (mm)	NO OF HOLES
      		X	Y
      SPECIMEN 1	3	45	45	12	25
      SPECIMEN 2	2	45	45	24	25
      SPECIMEN 3	2.5	90	90	16	9

      Specimen 1- The combination with lowest stress and lowest mass was picked.

      Specimens 2- Lowest mass picked. All combinations were failing for equivalent stress. It was added to correlate the fea results

      Specimen 3-This plate was already available. Hence used for correlation.

      The DOE data was utilized to establish an equation and the values of coefficients was found. This was for interpolation between the limits of the parameters studied. A sensitivity analysis of the variables was studied using this equation.

      Equation for deformation:

      = 1.57-5.195 (t) +0.0 (u)-0.0012(v) +0.08(d) +5.914(t2)

      -0.01(u2) -0.0(v2) +0.023(d2) +0.021(tu) +0.014(tv)-0.08(td)

      +0.03(uv)-0.14(ud)-0.02(vd)-2.27t3

      Equation for stress:

      =2.154-7.036 (t) +0.535(u) +0.4167(v) +0.343(d) +7.007(t2)

      -0.28(u2) -0.31(v2) -0.025(d2) -0.004(tu) +0.09(tv)-0.16(td)

      -0.03(uv)-0.14(ud)-0.02(vd))-2.4t3

      Where:

      1. Plate thickness

      2. Hole spacing in X direction (perpendicular tol bolting face) v- Hole spacing in Y direction

      d- Hole Diameter -Deformation

      – Max equivalent stress.

   6. EXPERIMENTATION

      1. SPECIMEN AND GAUGE LOCATION

         Figure 6 shows the specimens and the gauge locations.

         Two gauges were laid, one on the side near the bolt on faces as per FEA vector principal direction and the other underneath the load.

         Fig. 7. Specimen and gauge location

      2. EXPERIMENTAL SET UP AND PROCEDURE

      The specimen was held between the side plates (fixture) as shown in the figure 6. The load is gradually varied from 50kg to 400 kg in increments of 50 using a hydraulic actuator.

      A load cell and a load plate (145mm diameter plate to replicate FEA model) is placed on the actuator. The gauge readings are recorded through eDAQ. Figure 7 shows the output format.

      Fig. 8. Fixture and experimental set up

      Fig. 9. eDAQ output file

   7. RESULTS AND COMPARISONS

      1. FEA VS EXPERIMENTAL RESULTS

         FEA of 3 specimens was carried with 8 steps load ranging from 50 Kg to 400 Kg with increment of 50.

         Maximum equivalent stress from FEA for the design load 2724N are:

         Specimen 1: 176 Mpa

         Specimen 2: 458 Mpa

         Specimen 3: 274 Mpa

         From the maximum stress values it is found that specimen 1 meets the acceptance criteria.

         Strain comparison for gauge1 between FEA and experiment for all 3 specimens are shown in figures 9, 10, 11. Gauge 2 was ignored as the value was negligible.

         Fig. 10. Specimen 1 fea vs experiment

         Fig. 11. Specimen 2 fea vs experiment

         Fig. 12. Specimen 3 fea vs experiment

      2. DOE DATA ANALYSIS

         Sensitivity of hole size, plate thickness and hole spacing to stress is plotted in figure 12.

         Deformation is mainly sensitive to thickness.

         Fig. 13. Stress sensitivity

         Figure 13 shows the deflection fit that was done to compare the deformation values from FEA with the values from the equation. It is eveident from the graph that both values are close.

         Fig. 14. Deflection fit between actual (FEA) and predicted (curve fir)

         It is evident from figure 14 and 15 that as the hole diameter increases, there is increase in deformation and stress.

         Fig. 15. Deformation sensitivity

         Fig. 16. Stress sensitivity

         The behavior of stress in comparison with X spacing for different Y spacing is ploted in figure 16.

         Fig. 17. Stress sensitivity

   8. CONCLUSION AND REMARKS

• Experimental Validation of the specimens was done and the variation when compared to between FEA is around 11%.

Considering 11% variation between FEA and experiment, specimen 1 is the solution.

Form the analysis of FEA data for all the combinations, it was found that:

• Deformation and stress are mostly effected by thickness of the plate.

• The next parameter effecting stress is X spacing.

  Detailed analysis revealed that:

• Stress and deformation increases as the hole diameter is increased but the variation is minimum as compared to thickness and X spacing change.

• It is also seen that as X spacing is increased from 45mm to 70mm, stress increases but later with 90mm it falls(Y spacing is kept same as X). This is because at 70mm, the holes fall right below the outer edge of the load creating stress concentration and in addition X direction is the load path due to the boundary condition. In Y direction stress does not increase significantly even if the hole falls under the outer edge of load as it is not the load path.

REFERENCES

1. Singiresu S. Rao, Engineering Optimization: Theory and Practice,

   Wiley, Fourth Edition, 2009

2. M. Aydin Komur,Elasto-plastic buckling analysis for perforated steel plates subject to uniform compression, Mechanics Research Communications 38 (2011) 117122O.

3. G.N. Savin, Stress Concentration around Holes, Pergamon Press, New York (1961)

4. Jae-Hoon Kang, Exact Solutions of Stresses, Strains, and Displacements of a Perforated Rectangular Plate by a Central Circular Hole Subjected to Linearly Varying In-Plane Normal Stresses on Two Opposite Edges, International Journal of Mechanical Sciences, 2014

5. Jinho Woo1 and Won-Bae Na1, Effect of Cutout Orientation on Stress Concentration of Perforated Plates with Various Cutouts and Bluntness International Journal of Ocean System Engineering 1(2) (2011) 95-101

6. EmanueleMaiorana,CarloPellegrino, ClaudioModena, Elastic stability of plates with circular and rectanguar holes subjected to axial compression and bending moment, Thin-Walled Structures 47 (2009) 241255

7. D.B.Kawadkar, Dr.D.V.Bhope, S.D. Khamankar

   Evaluation of Stress Concentration in Plate with Cutout and its Experimental Verification, International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.566-571

8. J. Rezaeepazhand, and M. Jafari, Stress Analysisof Perforated Composite Plates, Composite Structures, 71 (3-4) (2005) 463-468.

9. VenkatachalamG,Influence Of Perforations On Flexural Strength Of Aluminium 8090 Alloy Sheets, ARPN Journal of Engineering and Applied Sciences, VOL. 8, NO. 4, April 2013

10. R. Nandagopan, S. Ranjith Kumar, M.R. Rajesh, K. Manoharan, and

    C.G. Nandakumar, Nonlinear Behaviour of Perforated Plate with Lining Defence Science Journal, Vol. 62, No. 4, July 2012, pp. 248- 251

11. M. Cavazzuti, Optimization Methods: From Theory to Design, DOI: 10.1007/978-3-642-31187-1_2, Springer-Verlag Berlin Heidelberg 2013

12. Design and Analysis of experiments by Douglas Montgomery

13. http://bis.org.in/sf/mtd/MTD4(4670).pdf

14. ISO 3411 physical dimensions of operators

15. ISO 4254-1:2013 – Agricultural machinery – Safety -Part 1

16. Ansys mechanicals user guide