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New macro element for equivalent frame modelling of masonry buildings

This wiki provides information on modelling unreinforced masonry buildings using the open-source software OpenSees. The buildings are modelled using an equivalent frame approach and modelling piers, spandrels and gable walls by means of the newly developed macro-element. The novelty of this macro-element lies in capturing next to the inelastic in-plane response of the masonry element also its out-of-plane response.

This method aims at capturing all the main features of the dynamic response of a masonry building, through a simple modelling technique that allows performing multiple complex dynamic analyses with relatively little numerical cost. The use of equivalent frame models, for their characteristics and computational demand, is applicable both in the research community and among practitioners.

To maximise the versatility of the element and investigate various modelling assumptions on the seismic response of a masonry building, the macro-element offers several options with regard to the following points:

• Sectional model describing axial and bi-axial flexural response: The sectional behaviour can be described by a fibre section and uni-axial material models (see OpenSees manual for details) or by the computationally more efficient analytically integrated section that was developed explicitly for the macro-element.
• Shear model: Any nD-material model that couples the axial to the shear response can be used. Several new material models (see below) have been implemented.
• Mass matrix: Choice between consistent and lumped mass matrix; special lumped mass matrix for triangular gable walls is implemented.
• Drift capacity models: the drift models for flexural and shear failure can be a user-defined function of the applied axial load ratio and can include a dependency on the shear span; after the drift capacity is attained, the user can impose that the element retains a fraction of its capacity.
• Response at axial load failure: after loss of lateral capacity, the loss of axial load bearing capacity can be controlled by similar drift model, if the user needs to conduct a collapse analysis.
• Damping models: viscous damping models in OpenSees are available. i.e. Rayleigh damping proportional to the mass matrix and to the initial, or current (or last committed) tangent stiffness matrix.

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Deformed shape of an example building with elements failing in-plane and out-of-plane.

In addition to the macro-element and the corresponding nD-models, the following models have been implemented in order to increase the modelling options for masonry buildings:

• Unseating of beams: a frictional law to be applied through zero-length elements for floor-to-wall connections, allows modelling sliding of the floor with respect to the wall, and possible pounding or unseating of beams.
• Timber floor stiffness: simplified modelling of the in-plane stiffness of deformable floors through an elastic orthotropic membrane.

Next to providing the necessary documentation as well as the input code and an executable of OpenSees that includes the newly developed element, this wiki shares models of masonry subassemblages and entire buildings, which make use of the new macro-element. This wiki presents only the new features that were introduced in the software OpenSees by the Earthquake Engineering and Structural Dynamics (EESD) of EPFL; the general user guide is referred to for any further information.

Theory

• Macroelement formulation
• Sectional response
• Shear response
• Gambarotta–Lagomarsino model
• Ibarra–Medina–Krawinkler model
• Mass matrix
• Drift model

Modelling commands

• Node definition
• Material/section definition
• Macroelement definition
• Self weight and element loads

Output commands

• Element recorders
• Sectional recorders
• Shear interface recorders

Additional elements and material models for equivalent frame modelling of masonry buildings

• Zero-length element definition
• No-crushing material for wall-to-wall connections
• Frictional material for floor-to-wall connection
• Orthotropic elastic membrane for floors
• nD material for unit-to-unit connections in aggregates

Preprocessing

• Plug-in for Rhino - coming soon

Postprocessing

• Matlab and Python files for post-processing of results - coming soon

Downloads

• Installing OpenSees on a new machine
• Libraries download
• Supplemental materials

Examples library

	Analytical section and fiber section response Moment-curvature analysis, for different moment orientations, of a section analytically integrated or a fiber section with the same material model. Model used to produce Figure 5 in Vanin et al. (2019).
	Single macro-element subjected to in-plane loading The model represents cyclic shear-compression tests on masonry walls. The model consists of one macro-element, subjected to a constant axial load and cyclic in-plane lateral displacements. a. Cantilever; b. Double-bending: model used to produce Figures 6 and 7 in Vanin et al. (2019).
	Single macro-element subjected to static and dynamic out-of-plane loading a. Static out-of-plane analyses, with concentrated and distributed loads (case a, case b, case c, case d). The model was used to produce Figure 8 in Vanin et al. (2019); b. Dynamic free-vibrations of a cantilever wall with distributed load. The model was used to produce Figures 9 and 10 in Vanin et al. (2019).
	One way rocking Model consisting of three macro-elements and zero-length elements modelling the wall-to-wall connection, subjected to dynamic free-vibrations from a deformed configuration. Input files and matlab files for postprocessing are provided in the .zip folder.