This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 2 1 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 1 u u + where u denotes the displacement field, u represents an equilibrium configuration on the primary path, and 1 u is a small increment. Now let 1 1 u u u w w w → + → + (2) where the arrows are read “be replaced by.” The variables ( , u w ) represent a configuration on the primary equilibrium path, the incremental displacements ( 1 1 , u w ) are infinitesimally small, and both ( , 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 , u w alone is equal to zero because , u w satisfy Eqs. (1). 1 2 Second and higherorder terms in 1 1 , u w may be omitted because of the smallness of the incremental displacements. The resulting equations are ( 29 ( 29 1 1 2 1 1 1 1 1 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 , u w appearing as constants. For the column, the primary equilibrium path represents undeflected configurations. Thus, w = for all values of x , and Eqs. (3) reduces to 1 1 1 iv u EIw EAu w ′′ = ′ ′′ = (4) For u the equilibrium equation for the column in the undeflected form yields...
View
Full
Document
This note was uploaded on 09/24/2011 for the course CIVL 7690 taught by Professor Staff during the Summer '10 term at Auburn University.
 Summer '10
 Staff

Click to edit the document details