Using Galerkin's method, derive the stiffness matrix for the following beam element, which has an additional node in the center (higher-order element). Let the displacements be v(x) = a1 x^5 + a2 x^4 + a3 x^3 + a4 x^2 + a5 x + a6 (two degrees of freedom per node).
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Derive the element stiffness matrix for the bar element.
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