Abstract
The ill-condition of stiffness matrix at the unstable region for example at the strain-softening region, the load control method will not be valid to give the solution therefore the displacement control method is essential to use. The stiffness matrix is changed at different steps by used load control and solution is found using any nonlinear solution method of Newton’s family. Therefore, Newton’s method will not give a solution to the unstable region. The iteration method and solution process may also stop when the diagonal element of the stiffness matrix becomes negative or zero if Newton's method is adopted. These difficulties need to be overcome and a non-positive definite stiffness matrix is retained and used for iteration in the arc-length method. A scientific review of literature has been made to pinpoint the difficulties faced in the prediction of complete load-displacement response of structures and techniques available to overcome the same. Displacement control methods have been reviewed with emphases on the arc length methods which can be integrated with the traditional finite element method.
Highlights
The complete nonlinear solution path of structure including limit points and beyond of this point is need to check and track
Newton-Raphson method is called Load controlled method, it is unsuccessful near the limit point for geometrically and material nonlinearity problem, the displacement control method is required for the solution beyond the ultimate point of the load-displacement relationship
Most of the inelastic analysis of structure bring singular vale of the diagonal element of stiffness matrix and unstable region will be appear due to the changing in the sign from positive to zero or negative value
Summary
The complete nonlinear solution path of structure including limit points and beyond of this point is need to check and track. Newton-Raphson method is called Load controlled method, it is unsuccessful near the limit point for geometrically and material nonlinearity problem, the displacement control method is required for the solution beyond the ultimate point of the load-displacement relationship. The conventional method of the nonlinear solution consists of the use of an increment loading when procedure combined with incremental iterative Newton-Raphson solution lead to out of balance force will be vanish. The Newton-Raphson technique fails in the neighborhood of critical points as the stiffness matrix approaches the singularity. The displacement control solution producers like Crisfield’s arc-length method, Riks’ arc-length method and modified Riks’ arc-length method can be used to obtain the complete inelastic response of structure. The strain-softening is a poplar case for negative stiffness and unstable condition (Fig. 1) if traditional nonlinear solution techniques of Newton's family are used in the inelastic analysis of structures
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