Galerkin - Galerkins Method The requirement that the total...

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Galerkin’s Method The requirement that the total potential energy of a column has a stationary value is shown in the following equation: ( 29 ( 29 ( 29 0 0 0 0 iv EIy Py ydx EIy y EIy y δ δ δ ′′ ′′ ′′′ + + - = l l l (1) where y δ is virtual displacement. Assume that it is possible to approximate the deflection of the column by a series of independent functions, ( 29 i g x , multiplied by an undetermined coefficients, i a . ( 29 ( 29 ( 29 1 1 2 2 approx n n y a g x a g x . . . . . . a g x + + + B (2) If each ( 29 i g x satisfies the G.B.C. and N.B.C., then the second term in Eq (1) vanishes when substitutes approx y to y . Also the coefficients, i a , must be chosen such that approx y will satisfy the first term. Let the operator be 4 2 4 2 d d Q EI P dx dx = + (3) and ( 29 1 n i i i a g x φ = = (4) From Eqs (3) and (4), the first term of Eq (1) becomes ( 29 0 0 Q dx φ δφ = l (5) Since φ is a function of n parameters, i a , 1 2 1 2 1 1 2 2 1 n n n n n i i i a a . . . . a a a a g a g a . . . . . g a g a φ φ φ δφ δ δ δ δ δ δ δ = = + + + = + + + = (6) 1
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( 29 ( 29 0 1 0
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