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In The Name O f ALLAH

In The Name O f ALLAH. Introduction To Finite Elements In Engineering By: T.R.Chandruputla The Finite Element Method By: Zienkiewicz and Taylor Concepts And Applications Of Finite Element Analysis By: R.D.Cook Applied Finite Element Analysis for Engineers By: Frank L. Stasa. 1/25.

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In The Name O f ALLAH

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  1. In The Name Of ALLAH

  2. Introduction To Finite Elements In Engineering • By: T.R.Chandruputla • The Finite Element Method • By: Zienkiewicz and Taylor • Concepts And Applications Of Finite Element Analysis • By: R.D.Cook • Applied Finite Element Analysis for Engineers • By: Frank L. Stasa 1/25

  3. Fundamental Concepts Chapter 1

  4. Body Force ( distributed force per unit volume) Traction Force ( distributed force per unit Area) Point load( load acting at a point) Loads

  5. Deformation of a pint Stress Vector Strains Strain _ Displacements Stresses, Strains and displacements

  6. eQUILIBRIUM Equations for internal differential volume

  7. eQUILIBRIUM Equations for Boudary differential volume Where is a unit normal vector to a surface

  8. Stress _ strain relation

  9. Stress _ strain relation Plane Stress Plane Strain Axisymmetric 3D problems can be converted to 2D problems by assumsion of :

  10. Plane stress • A thin planar body subject to in_plane loading. • When one dimension of a body is very smaller than the other two dimensions.

  11. Plane strain • A long body of uniform cross section , subjected to transverse loading along its length. • When one dimension of a body is very longer than the other two dimensions.

  12. Axisymmetric bodies • When The body can be generated by 360 degree revolution of an area around the axis.

  13. Temperature effects Plane Stress : Plane Strain : Axisymmetric :

  14. Total Potential Energy = Strain Energy + Work Potential Strain Energy : area under the curve of stress-strain S.E. = Work Potential : work of external forces with reverse sign W.P.= Potential energy

  15. For conservative systems of all the kinematically admissible displacement field, those corresponding to equilibrium extremise the total potential energy. Where qi are the degrees of freedom Principle of minimum potential energy

  16. Example

  17. Derivative of a matrix with respect to a scalar More Mathematics

  18. Integral of a matrix with respect to a scalar More Mathematics

  19. Derivative of a scalar function with respect to a vector More Mathematics Define N and a Consider the scalar function

  20. More Mathematics

  21. Square Matrix Symmetric Matrix Diagonal Matrix Identity (Unit) Matrix Transpose (AT) Inverse (A-1) Positive Definite (XT.A.X > 0 ) More Mathematics

  22. Half Band Width (nbw) • When all of nonzero elements of a square matrix are contained within a band, this matrix is called banded. • For symmetric matrices half of this band is Half Band Width (nbw). More Mathematics nbw

  23. 5. The integral as a sum of other integrals • The follows directly from the definition of integration that an integral may be evaluated by dividing the domain  into M non_overlapping subdomain e and summing the integrals over each subdomain, or More Mathematics

  24. Thanks for your attention از توجه شما متشکریم 25/25

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