Macroelement formulation - eesd-epfl/OpenSees GitHub Wiki
The macroelement is formulated as a one-dimensional element defined by three nodes in the three-dimensional space. It consists of an assemblage of two panels subjected to shear deformations separated by three nonlinear sections accounting for axial deformations.
The shear response is controlled by a non-linear interface located at mid-height of the element, in which all shear deformations of the two panels are lumped. The shear model is coupled, in terms of forces, to the axial response as it accounts for the axial load that acts on the mid-section. For describing the shear response, any nd Opensees material can be used that couples the shear to the axial load. As not many were available when the macro-element was developed, some have been implemented; see information here.
In order to decouple shear deformations from the axial ones and avoid iterative element determination procedures, a zero-dilatancy behaviour of the shear interface is postulated.
The flexural response is described by the three sections, that can model coupled in-plane and out-of-plane rocking. The uplift related to the rocking response can be properly described by suitable sectional models.
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The element is defined by three nodes, located at the extremities and at mid-height. These nodes are standard three-dimensional nodes with degrees of freedom that describe three displacements and three rotations in space. The internal node e defines the three displacements of the extremities of, respectively, block A and B. The local degrees of freedom become:
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From the vector of local displacements the 11 displacements defining the shear deformations of the panels and the displacements at the zero-length sections (ubasic) can be obtained through compatibility equations. To avoid the complexity of the co-rotational formulation, still accounting for second-order effects for moderate displacements, a p-Δ formulation can be obtained through a second-order Taylor-series expansion of exact compatibility equations.
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It allows capturing in a simplified manner the geometrical nonlinear effects, in particular the second-order effects that govern the out-of-plane response of a masonry wall. It is applicable in moderate displacements range, as the formulation is not exact; normally it is however adequate up to the loss of out-of-plane stability of masonry walls with typical aspect ratios. It requires the update of the compatibility matrix of each element at every displacement update, but does not introduce further complications.
Through proper sectional models one can express sectional forces and stiffness, and express them with reference to the local reference system.