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### Introduction finite element method by A.Vinoth Jebaraj

• 1. Dr.A.Vinoth Jebaraj, SMEC VIT University, Vellore
• 3. So, On what basis we have to design a machine component?
• 4. Methods to solve any Engineering problem Experimental Analytical Numerical Time consuming & needs experimental setup Atleast 3 to 5 prototypes must be tested Applicable only if physical model is available Approximate solution Applicable if physical model is not available Real life complicated problems 100% accurate result Applicable only for simple problems = y ? Is this equation is correct for the above beam?
• 5. Area = l × b Area = ? Error Solution
• 6. FEA is a numerical method to find the location and magnitude of max stress and deflection in a structure. Solid Plate - Theoretical solution is possible Plate with Holes – No theoretical solution available Load Load
• 7. Challenge lies in representing the exact geometry of the structure, especially, the curves. Coarser mesh Fine mesh Regions where geometry is complex (curves, notches, holes, etc.) require increased number of elements to accurately represent the shape.
• 8. Atomic Structure Finite Element model Infinite to Finite Degrees of Freedom ? Why do we carry out MESHING? Machine component
• 9. Types of Finite elements 1D (line) element 2D (plane) element 3D solid element Truss, beam, spring, pipe etc. Membrane, plate, shell etc. 3D fields
• 10. Traditional Design cycle Vs. FEA FE Model & BC’sFinite Element ModelCAD Model Max Stress Max Displacement Simple Bracket FEA  Replacement for costly and Time consuming Testing Pre-processing or modeling the structure Post processing
• 11. Stresses vs. Resisting Area’s (Fundamentals of stress analysis) For Direct loading or Axial loading For transverse loading For tangential loading or twisting Where I and J  Resistance properties of cross sectional area I  Area moment of inertia of the cross section about the axes lying on the section (i.e. xx and yy) J  Polar moment of inertia about the axis perpendicular to the section
• 12. Plane of Bending X – Plane Y - Plane Z - Plane Under what basis Ixx, Iyy and Izz have to be selected in bending equation? Bending Bending Twisting
• 14. Planar Assumptions  All real world structures are three dimensional.  For planar to be valid both the geometry and the loads must be constant across the thickness. When using plane strain, we assume that the depth is infinite. Thus the effects from end conditions may be ignored.
• 15. Plane Stress  All stresses act on the one plane – normally the XY plane.  Due to Poisson effect there will be strain in the Z direction. But We assume that there is no stress in the Z – direction.  σx, τxz, τyz will all be zero.  All strains act on the one plane – normally the XY plane. And hence there is no strain in the z-direction.  σz will not equal to zero. Stress induced to prevent displacement in z – direction.  εx, εxz, εyz will all be zero. Plane Strain
• 16.  A thin planar structure with constant thickness and loading within the plane of the structure (xy plane).  A long structure with uniform cross section and transverse loading along its length (z – direction).
• 17. Stiffness Axial stiffness = ; Bending stiffness = ; Torsional stiffness =
• 18.