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BYANJALI A(1211108)JEYAGOMATHI(1211136)FINITE ELEMENT METHOD-ELEMENTS OF ELASTICITY1 FINITE ELEMENT METHOD (FEM)FEM is a numerical procedure for solving physical problems governed by differential equation or energy equation.In simple terms, FEM is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other.It provides approximate solution.The domain of physical problems is discretized into the finite elements.The elements are connected at points called nodes.The assemblage of elements is called finite element mesh.

APPLICATIONSEQUATION OF EQUILIBRIUMx, y, z normal stresses in x,y,z directionxy, yz, zx-shear stressFx,Fy,Fz-body forces acting in x,y,z direction xy= xy; xz= zx, yz= zy complementary plane

STRAIN DISPLACEMENT RELATIONWhen the deformation are small products & square of first derivative are negligible compared with derivatives themselves

u,v,w-displacement component of a point in x,y,z directionx,y, z-normal strain in x,y,z directionx,y,z-shear strain

The above equation can be written in a matrix form as given below x /x 0 0 y 0 /y 0 u z = 0 0 /z v xy /y /x 0 w yz 0 /z /y zx /z 0 /x STRESS-STRAIN RELATIONS-Poisson ratioE-Youngs modulusx,y, z-normal strain in x,y,z directionx,y,z-shear strain

The above equation can be written in matrix form x 1 x x - 1 (sym) y x = 1/E - - 1 z xy 0 0 0 2(1+) xy yz 0 0 0 0 2(1+) yz zx 0 0 0 0 0 2(1+) zxFor a linear elastic material, the stress-strain relation comes from generalized Hookes law.For isotropic material, two material properties are youngs modulus(E) & poisson ratio() The inverse relation of stresses in terms of strain components can be expressed as x 1- x y 1- (sym) z = 1- z xy 0 0 0 (1-2)/2 x z 0 0 0 0 (1-2)/2 z zx 0 0 0 0 0 (1-2)/2 zx

Where,

{} = [C]{}[C] -constitutive matrix

PLANE STRESS

PLANE STRESSPLANE STRAIN

PLANE STRAIN

POTENTIAL ENERGY IN ELASTIC BODIESPRINCIPLE OF MINIMUM POTENTIAL ENERGYIf a deformable body are structural system is in equilibrium where total potential energy of the system as a stationary value. =0 Total P.E=U+H =U+HPrinciple energy of external force H is equal but opposite to total virtual Work done by external force. H= - He = u- H By Principle of virtual work, u=H =0

RAYLEIGH RITZ METHODA beam AB of span l simply supported at ends & carries a concentrated load W at the centre. Determine the deflection at the midspan by using Rayleigh Ritz method. w For a simply supported beam, the fourier series => approximate function Where a1, a2 are ritz parameters =U-H =potential energy ; H=work done by external force Strain energy (U) of the beam due to bending

Substitute dy/dx in U

Work done by external force, H=Wymax@ x=l/2 , ymax=a1-a2 [substitute x in y equation] H=W(a1-a2)

By principle of minimum potential energy =0 ( /a1 )=0 ; (/ a2)=0

ymax = a1-a2

Deflection