Abstract
Tracing load-displacement paths in structural mechanics problems is complicated in the presence of critical points of instability where conventional load- or displacement control fails. To deal with this, arc-length methods have been developed since the 1970s, where control is taken over increments of load at these critical points, to allow full transit of the load-displacement path. However, despite their wide use and incorporation into commercial finite element software, conventional arc-length methods still struggle to cope with non-zero displacement constraints. In this paper we present a new displacement-controlled arc-length method that overcomes these shortcomings through a novel scheme of constraints on displacements and reaction forces. The new method is presented in a variety of serving suggestions, and is validated here on six very challenging problems involving truss and continuum finite elements. Despite this paper’s focus on structural mechanics, the new procedure can be applied to any problems that involve nonhomogeneous Dirichlet constraints and challenging equilibrium paths.
Highlights
In structural mechanics, the modelling of complex non-linear load-displacement paths poses difficulties in passing critical points, associated with instabilities found in practical engineering problems, such as buckling
This allows physical problems to be modelled, and their associated equilibrium paths to be traced, that are not possible with existing arc-length methods that rely on the specification of nodal forces
The graphical difference between spherical and cylindrical constraints are shown in Fig. 4, where the constraint equations are plotted in the three-dimensional axes of the prescribed displacements, the free displacements and the scaled reaction forces, and it can be observed that the intersection between the equilibrium path and the constraint is more severe in the spherical case
Summary
The modelling of complex non-linear load-displacement paths poses difficulties in passing critical points, associated with instabilities found in practical engineering problems, such as buckling. The only boundary conditions that can be applied when adopting Riks solvers are tractions or body forces which, when integrated over the boundary of volume of the domain, manifest themselves as equivalent nodal forces This limits the physical problems that can be modelled using arc-length methods to those where it is appropriate to load the analysed structure via nodal forces, non-zero displacement constraints are often more appropriate especially when analysing the response of experimental tests where rigid constraints are imposed on the boundary of loaded specimens. This allows physical problems to be modelled, and their associated equilibrium paths to be traced, that are not possible with existing arc-length methods that rely on the specification of nodal forces. It is worth emphasising that the current displacement-controlled arc-length method differs from the technique under the same name presented in Verhoosel et al [39], as will be explained below
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