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adjacentequil - Adjacent Equilibrium Columns The...

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Adjacent Equilibrium Columns The differential equation governing equilibrium on the secondary path for the column is linear because the factor N happens to be constant. ( 29 ( 29 2 2 1 0 2 1 0 2 iv N EA EA u w EIw EA u w w ε = = + = ′′ - + = (1) If the primary path is, say, a straight line through the origin, as for a column, the adjacent equilibrium configurations occur at a bifurcation point. The linear equations necessary for this process may be derived from the nonlinear Eqs. (1) by use of a perturbation technique in which u is replaced by 0 1 u u + where u denotes the displacement field, 0 u represents an equilibrium configuration on the primary path, and 1 u is a small increment. Now let 0 1 0 1 u u u w w w + + (2) where the arrows are read “be replaced by.” The variables ( 0 0 , u w ) represent a configuration on the primary equilibrium path, the incremental displacements ( 1 1 , u w ) are infinitesimally small, and both ( 0 0 , u w ) and ( , u w ) are equilibrium configurations. Then, in the equations obtained by introducing Eqs. (2) into Eqs. (1), one sees that 1 In each equation the sum of all terms containing 0 0 , u w alone is equal to zero because 0 0 , u w satisfy Eqs. (1). 1
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2 Second- and higher-order terms in 1 1 , u w may be omitted because of the smallness of the incremental displacements. The resulting equations are ( 29 ( 29 1 0 1 2 1 0 0 1 0 1 0 1 0 1 0 2 iv u w w EIw EA u w w w u w w + = ′′ ′′ - + + + = (3) Equations (3) are linear in the unknowns 1 1 , u w as desired, the variables 0 0 , u w appearing as constants. For the column, the primary equilibrium path represents undeflected configurations. Thus, 0 0 w = for all values of x , and Eqs. (3) reduces to 1 1 0 1 0 0 iv u EIw EAu w ′′ = ′′ - = (4) For 0 u the equilibrium equation for the column in the undeflected form yields 0 P u x EA = - (5) Substituting Eq. (5) into the second of Eqs. (4), gives
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