2016 Fall ME 7210 Elasticity and Plasticity Final Project
- 1. 2016 Fall ME 7210 Elasticity and Plasticity Final Project Strip stress over half space Northeastern University Department of Mechanical Engineering Mutian Fan
- 2. Abstract This project discusses the theory of analysis of stress in a uniform loaded strip stress over a half space. Based on textbook Elasticity 3rd edition by Martin Sadd on page 196-198, the equations to calculate stress field in x and y direction and xy shear stress are given. A MATLAB routine is scripted to calculate and plot the theoretical stress field over 3 straight line under the loaded area on the half space. Then Solidworks FEA analysis is employed to test out theory. Also, correct boundary condition and mesh size are discussed in this report.
- 3. Theory The principal theory behind the strip stress over a half space comes from superposition of single point load solution. (pg 196-197) The equations are shown in figure 1 and figure 2 below. Figure 1 Figure 2 For practical, I only calculated and plotted 3 lines of stress that are directly under the boundaries and at the middle of loaded area. I implemented a MATLAB routine and plot the stress vs relative distance. Please refer to appendix A for MATLAB routine. For all these 3 lines, I calculated xx normal ,yy normal, and xy shear stress. They are shown in the appendix.
- 4. Finite element analysis The part is created as a 100*100 mm square with 1mm thickness plate and a loaded area of 20 mm on the top. It is as shown in the figure 9. For FEA, the boundary conditions have to be set up properly, I used different boundary conditions to test and compare the theory plot to determine the correct boundary conditions. I used only fixtures on bottom and sides without roller support, roller support with only bottom and sides support with same dimension of the half space and mesh size to ensure the accuracy of the effect of the boundary condition. After mesh and simulation, X,Y normal stress and xy shear stress are showed and probe function are used to plot the stress field along the lines. Because there are 9 plot for each condition and we have 36 plots in total, I am not able to compare them all here. In this paragraph, I will only use one set of plot to compare and explain which is the correct boundary condition. Figure 3 Theory x1 stress plot Figure 4 x1 stress with fixtures only Figure 5 x1 stress with bottom fixture and roller slide support
- 5. It is quite obvious that in figure 5 and 6 the general trend of the plot is not right compare to the theoretical plot. The major difference between plot 4 and 7 is that the plot 4 has a “wiggle” at the end of the plot. It is been pointed out in the zoomed out version in the following. Figure 4 x1 with fixtures only zoomed out Other plots show same pattern, hence it is safe to conclude that the correct boundary conditions are bottom and sides fixtures and roller slides on both sides just as shown in the following SolidWork sketch. Figure 6 x1 stress with side fixture and roller slide support Figure 7 x1 stress with full boundary condition include all fixtures and roller slide support Line 1 Line 2
- 6. Figure 9 SolidWorks Model with full boundary conditions Next, the distance of loaded are to the edge must be determined to be far enough so the edge end effect won’t affect the stress field in the center. In order to do that, the loaded area in the FEA simulation remain 20 mm while size of the part(square) grows from 100 mm to 600 mm with 100 mm increment. Each individual case has generated 9 plots so the results can be compared to determine the end effects from the edge. The following plots are from 500 mm and 600 mm size parts. It is quite obvious the shape of the plots is nearly same and it is safe to conclude that 500 mm square is big enough for this particular topic i.e. (ratio of the loaded area and actual size of the edge has to be more than 1:25). Also, for the demonstration purpose, the 100 mm size parts are also included to show the effect of increase the size. X normal stress along line 1 Figure 10 X normal stress along line 1 500*500 mm part Figure11X normal stress along line 1 600*600 mm part Line 3
- 7. Figure 12 X normal stress along line 1 100*100 mm part Y normal stress along line 2 Figure 14 Y normal stress along line 2 600*600 mm part Figure 13 Y normal stress along line 1 500*500 mm part Figure 15 Y normal stress along line 1 100*100 mm part
- 8. XY shear stress along line 3 Figure 17 xy shear stress along line 3 100*100 mm part Finally mesh size is refined as small as possible for 500*500 mm part so that the FEA results plot are close to the plots from theory. The mesh size for the region under the loaded area is refined and global mesh size remained relatively bigger (7.4mm) to reduce the time for processing. The mesh size started from 5mm to 0.8 mm. Due to the limitation of the computing power, finer results took very long time and system crushes after a few attempts. Following plots are the best result and theory plots are added to compare. Figure 16 x normal stress along line 1 vs theory Figure 15 xy shear stress along line 3 500*500 mm part Figure 16 xy shear stress along line 3 600*600 mm part
- 9. Figure 17 y normal stress along line 1 vs theory Figure 18 xy shear stress along line 1 vs theory Conclusion and Discussion By comparing the theory and actual FEA simulation, it can be concluded that FEA generates reasonable plots with boundary conditions to simulate the theory of a half space as plane stress. The boundary conditions are fixtures on bottom and sides with roller support on both plane surfaces. The x-normal stress have the best result in very refined mesh, the plots from FEA and theory are almost same. The y normal and xy shear stress have some edge end effect, this could also because the mesh is not refined enough. It is worth mentioning that the stress field of xy shear stress in line 2 is almost 0 in the theory and FEA plots generates very small stress in the magnitude of 1-2 pa. But in the plots they seem very drastic because of the range of the axises. The improvements for any future progress on this topic are refined mesh and study of the edge end effect in the FEA.
- 10. Appendix A Matlab Code %ME 7210 final project %equation from text book sadd 3rd edition 8.4.50 %half space uniform loading clear clc p=500; %uniform loading 500 %% t11d=linspace(0.01,84) t11=degtorad(t11d); t12=degtorad(90); sx1=-p/(2*pi())*(2*(t12-t11)+(sin(2*t12)-sin(2*t11))) sy1=-p/(2*pi())*(2*(t12-t11)-(sin(2*t12)-sin(2*t11))) sxy1=p/(2*pi())*(cos(2*t12)-cos(2*t11)) d=tan(t11)*20 plot(d,sxy1) %manualy change for different stress title('xy shear stress along line1') xlabel('distance in mm') ylabel('stress in pa') %% clear clc p=500; t21d=linspace(0.01,84); t22d=180-t21d; t21=degtorad(t21d); t22=degtorad(t22d); d=(tan(t21)*10)/100; sx2=-p/(2*pi())*(2*(t22-t21)+(sin(2*t22)-sin(2*t21)))
- 11. sy2=-p/(2*pi())*(2*(t22-t21)-(sin(2*t22)-sin(2*t21))) sxy2=p/(2*pi())*(cos(2*t22)-cos(2*t21)) plot(d,sy2) %manualy change for different stress title('y normal stress along line2') xlabel('distance in mm') ylabel('stress in pa') %% clear clc p=500 t32d=linspace(179,100) t31=degtorad(90); t32=degtorad(t32d); d=tan(3.14159-t32)*20/100 sx3=-p/(2*pi())*(2*(t32-t31)+(sin(2*t32)-sin(2*t31))) sy3=-p/(2*pi())*(2*(t32-t31)-(sin(2*t32)-sin(2*t31))) sxy3=p/(2*pi())*(cos(2*t32)-cos(2*t31)) plot(d,(sxy3)) %manualy change for different stress title('xy shear stress along line3') xlabel('distance in mm') ylabel('stress in pa')
- 12. Appendix B Theory plots