Element Iterative Strategy

In SeismoStruct, all analyses are treated as potentially nonlinear, and therefore an incremental iterative solution is needed.

Force-based Element Type / Force-based Plastic-Hinge Elements Type
Individual force-based frame elements require a number of iterations to be carried in order for internal equilibrium to be reached [e.g. Spacone et al. 1996; Neuenhofer and Filippou 1997]. The maximum number of such element loop iterations, together with the corresponding (force) convergence criterion or tolerance, can be defined herein:

• Element Loop Convergence Tolerance. The default value is 1e-5 (users may need to relax it to e.g. 1e-4, in case of convergence difficulties)
• Element Loop Maximum Iterations (elm_ite). The default value is 300 (although this is already a very large value (typically not more than 30 iterations are required to reach convergence), users may need to increase it to 1000 in cases of persistent elm_ite error messages)

Whilst running an analysis, elm_inv and elm_ite flag messages may be shown in the analysis log, meaning respectively that the element stiffness matrix could not be inverted or that the maximum allowed number of element loop iterations has been reached. In both cases, the global load increment is subdivided, as described in Appendix A, unless the ‘Do not allow element unbalanced forces in case of elm_ite’ option discussed below has been deactivated by the user.

Users are also given the possibility of allowing the element forces to be output and passed on to the global internal forces vector upon reaching the maximum iterations, even if convergence is not achieved. This non-default option may facilitate the convergence of the analysis at global/structure level, since it avoids the subdivision of the load increment (note that the element unbalanced forces are then to be balanced in the subsequent iterations).

Displacement-based Plastic-hinge Element Type
Since the element consists of a series of three sub-elements (two links at the member edges and an elastic frame element in the middle) an iterative procedure is required, in order to achieve internal equilibrium.

The parameters required for the element iterative strategy are the maximum and the minimum iterations allowed, and the value for the convergence norm. It is noted that a relative small value is given as default for the maximum number of iterations, as it has been observed that typically convergence is achieved within a limited number of iterations. Hence, if convergence is not achieved relatively early, it is highly probable that no convergence will be achieved.

Finally, an additional setting is provided for employing a smoother descending branch in the moment-rotation curve of the plastic hinge, in order to have a smoother and more stable numerically curve. This setting is applied only when users choose the automatic definition of the modelling parameters by the program, but not when they define the a, b and c hinge modelling parameters.

Masonry Element Type
Since the element consists of a force-based element type employed in modelling mainly the bending behaviour of the masonry member (herein called the ‘internal sub-element’) with two links at the two edges that are employed to simulate the shear behaviour of the member (herein referred to as the ‘external links’ or the ‘link sub-elements’), two internal iterative procedures are required, in order to achieve equilibrium on the element level: one for the internal force-based sub-element, and the second for the assemblage of the three sub-elements, links and frame.

As a result, parameters for both iterative procedures should be provided. The parameters for the internal force-based sub-element are the same with the typical force-based elements, and have the same default values. The parameters for the external loop of the entire element are the maximum and the minimum iterations and the value for the convergence norm. It is noted that a relatively small value is given as default for the maximum number of iterations, as it has been observed that typically convergence is achieved within just a limited number of iterations.

Elastic with Plastic-hinges Element Type & Rack with Plastic-hinges Element Type
Since these elements consist of a series of three sub-elements (two links at the member edges and an elastic frame element in the middle) an iterative procedure is required, in order to achieve internal equilibrium.

The parameters required for the element iterative strategy are the maximum and the minimum iterations allowed, and the value for the convergence norm. It is noted that a relatively small value is given as default for the maximum number of iterations, as it has been observed that typically convergence is achieved within a limited number of iterations. Hence, if convergence is not achieved relatively early, it is highly probable that no convergence will be achieved.