## 12. Rod Snap Problem

This simple problem consisting of two rods illustrates the snap phenomenon and its solution with B2000++. The FE model is show below (note that the two rods initially meet at a right angle). The rod dimensions are B=H=10 and the sections are circular with a radius of 0.5641896, .e. an area of 1.0. The material is linear isotropic with a modulus of elasticity E=1000. The rods are restrained at nodes 1 and 3, node 2 being free to move in the x- and y-directions. Two loading cases are considered:

• Essential boundary condition case: A displacement v applied in the y-direction of node 2.

• Natural boundary condition case: A force F applied in the y-direction of node 2. Figure 45. Rod snap problem: Model.

When trying to solve the problem for the natural boundary condition case, i.e. the case with a force F applied in the y-direction of node 2, the Newton-type solution procedure will fail, because the procedure will not overcome the situation where v=h. Arch-length methods are then indicated.

The essential boundary condition case, i.e. the case with displacement v applied in the y-direction of node 2, is the most elementary one, because the Newton method will always work. Besides, an analytical expression for the reaction force Ry as a function of the displacement v can then be developed:

{ h = 1.414 l 0 S = AE l 0 l = ( v 2 2 Hv + l 0 2 ) 1 / 2 R y = 2 S ( l l 0 ) ( h v ) 2 l

With a full Newton method the iteration converges with one step per displacement increment. The reaction force Ry as a function of the load factor is displayed in the figure below: Figure 46. Rod snap problem: Reaction Force Ry as a function of the load factor