large deformation of cantilever

hello Dear friend

I am mohammad jalili. I am working on a paper that solves large displacement of a cantilever by total lagrangian approach. because of satisfying field consistency symbolic method is used. beam element have three nodes. two end nodes have three degree of freedom and middle node has one degree of freedom along axix of beam. middle node is added for increasing order of axial interpolation. external load are not allowed at middle node. after applying total lagrangian method a nonlinear stiffness equation obtained. and we shoud solve this equation for unknown displacements by newton raphson method. my question is about middle degree of freedon. as you know it is possible that we remove middle degree of freedom by static condensation method or solve enlarged system of equilibrium equations. I can not gain a interactive response that apply static condensation method during newton raphson method. on the other hand as I told degree of freedom of two node is different with middle node. both end nodes have three and middle node has one degree of freedom. please guide me for the method of assembling this form of elements. I attach paper for furthur information.

thanks in advance.