Seismic Analysis of RC Walls with Unequal Lengths (Analisi Sismica di pareti in CA di diversa lunghezza)

Lumped, or concentrated, plasticity models normally are elastic beam elements with inelastic springs located at the member ends. Usually the springs have zero-length and for that reason they are also called "point hinges". The stiffness matrix of the entire element is computed from the stiffness of the single parts that compose the member.
Lumped inelasticity models have been widely used in earthquake engineering applications because their simple formulation allows running very fast analyses. The main disadvantage of this modeling approach is that it oversimplifies certain aspects of the modeling, as for example the hysteretic behavior of the structure. Thus it can result in limited applicability (Elnashai & Di Sarno, 2008).
In the next analyses, when the Lumped Plasticity (LP) approach is applied, the following modeling assumptions are made:

- The structural elements are represented by sticks with section properties concentrated on the centre line of the section, coupled with plastic hinges placed at the bottom-end nodes of the structure.

- The sticks consist in elastic beam-column frame elements that represent the behavior of the structure before the attainment of the yielding.

- The plastic hinge, which controls the post-yielding phase, is modeled as a nonlinear link element. That is an inelastic rigid-plastic rotational spring which behaves differently in the elastic and the plastic phase:

- In the elastic phase, the stiffness of the hinge is very high, so basically the flexibility of the combined system is the flexibility of the elastic element; conversely,

- In the inelastic phase, after reaching the yield moment, the flexibility of the system depends mainly on the capacity of the spring.

- The link element connects two initially coincident structural nodes, and requires the definition of an independent force-displacement (or momentrotation) response curve for each of its local six degrees-of-freedom (F1, F2, F3, M1, M2, M3). In a 2D analysis case, a rotational spring describes the moment-rotation relationship for a rotation about the axis perpendicular to the structural plan.

- For most analyses carried out within this study, the moment-rotation model utilized is the simplified Takeda hysteretic model. It is characterized by several parameters: Ki is the initial stiffness, thus it is the slope of the first branch, My is the yielding strength, ris the post yield to initial stiffness ratio, so basically it defines the post yield stiffness and β is the stiffness degradation factor, which is taken 0.5 as suggested by (Priestley, Calvi, & Kowalsky, 2007).