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ANALYSIS OF BEAM BY USING FEM

Presented by Dr R SURYA KIRAN Phd , Associate Professor DEPARTMENT OF MECHANICAL ENGINEERING VISAKHA INSTITUTE OF ENGINEERING & TECHNOLOGY. ANALYSIS OF BEAM BY USING FEM. For Beam Analysis.

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ANALYSIS OF BEAM BY USING FEM

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  1. Presented by Dr R SURYA KIRAN Phd, Associate Professor DEPARTMENT OF MECHANICAL ENGINEERING VISAKHA INSTITUTE OF ENGINEERING & TECHNOLOGY ANALYSIS OF BEAM BY USING FEM

  2. For BeamAnalysis

  3. In finite element method, the structure to be analyzed is subdivided into a mesh of finite- sized elements of simple shape, and then the whole structure is solved with quiteeasiness. • Finite SizedElement RectangularBody CircularPlate

  4. The rectangular panel in the rectangular body and triangular panel in the circular • plate are referred to an‘element’. • There’re one-, two- and three-dimensional elements. • The accuracy of the solution depends upon the number of the finite elements; the more there’re, the greater theaccuracy.

  5. If a uniaxial bar is part of a structure then it’s usually modeled by a spring element if and only if the bar is allowed to move freely due to the displacement of the whole structure. (One dimensionalelement) • Bar Uniaxial bar ofthe • structure Structure Springelement

  6. Here goes the examples of two- and three- dimensional finite sizedelements. Triangle Rectangle Hexahedron

  7. The points of attachment of the element to other parts of the structure are callednodes. • The displacement at any node due tothe deformation of structure is known as the nodaldisplacement. • Node

  8. Simple trusses can be solved by just using the equilibrium equations. But for the complex shapes and frameworks like a circular plate, equilibrium equations can no longer be applied as the plate is an elastic continuum not the beams or bars as the case of normal trusses. • Hence, metal plate is divided into finite subdivisions (elements) and each element is treated as the beam or bar. And now stress distribution at any part can be determined accurately.

  9. By The Help OfFEM

  10. Consider a simple bar made up of uniform material with length L and the cross- sectional area A. The young modulus of the material isE. • L • Since any bar is modeled as spring in FEM thuswe’ve: k x2 F1 x1 F2

  11. Let us suppose that the value of spring constant is k. Now, we’ll evaluate the value of k in terms of the properties (length, area, etc.) of thebar: • We knowthat: • i.e. Also: i.e. And i.e.

  12. Now substituting the values of x and F is the base equation of k, we’llhave: But Hence, we maywrite:

  13. According to the diagram, the force atnode • x1 can be written in theform: • Where x1– x2is actually the nodal displacement between two nodes.Further: • Similarly:

  14. Now further simplificationgives: • These two equations for F1 and F2 can also be written as, in Matrixform: • Or:

  15. Here Ke is known as the Stiffness Matrix. So a uniform material framework of bars, the value of the stiffness matrix would remain the same for all the elements of bars in the FEMstructure.

  16. k1 k2 F1 F2 F3 x1 x2 x3 • Similarly for two different materials bars joined together, we maywrite: • ;

  17. With The Help Of FEM Analysis ForBars

  18. Three dissimilar materials are friction welded together and placed between rigid end supports. If forces of 50 kN and 100 kN are applied as indicated, calculate the movement of the interfaces between the materials and the forces exerted on the endsupport. Steel Rigidsupport 50kN Brass Aluminium 100kN

  19. k1 k2 k3 F1 F2 F3 F4 x1 x2 x3 x4

  20. The system may be represented as the system of three springs. Hence, the spring are shown. Values of spring constant can be determinedas:

  21. From the extension of FEM, we can write the force-nodal equations for this systemas: • Solving this system and adding similar equationsyields:

  22. Now: • From these equations we can easily determine the unknowns, but we’ll have to apply the boundary conditions first.

  23. At point 1 and 4, the structure is fixed, and hence no displacement can be produced here. Thus, we’ll saythat: • And also, from the given data, we knowthat:

  24. Now, simply putting these values in the equations, weget: • And: • And, that was therequired.

  25. Complex StructureAnalysis

  26. Complex structures which contain the material continuum, are subdivided into the elements and are analyzed on the computers. Software packages are available for the determination of the Stiffness matrix of thosestructures. • Some software packages also allow virtual subdivision on the computer as well i.e. computer automatically analyzes the shape, and gives the stress-strain values at any point of thestructure.

  27. ComplexStructure FEMStructure

  28. components. The value of the inclined angle is always known and then the components areevaluated. • Three-dimensional structures involve three dimensional elements i.e. elements with three dimensions (length, width,thickness).

  29. FEM has become very familiar in subdivision of continuum. It gives reliable and accurate results if the number of elements are kept greater. • Modern computer technology had helped this analysis to be very easy and less time consuming. • Large structures under loadings are now easily solved and stresses on each and every part are now beingdetermined.

  30. Sheikh HarisZia 08-ME-39 IbrahimAzhar 08-ME-53 MuhammadHaris 08-ME-69

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