## PBEE toolbox

The PBEE toolbox combines a set of Matlab functions with OpenSees. It includes different functions for calculations of the moment-rotation relationship of the plastic hinges in columns and beams, functions for the generation of the tcl input code for OpenSees, functions for the post-processing of analysis results, and functions for structural performance assessment. Work with the PBEE toolbox is presented schematically in figure below. First the user has to define the structural data, which are defined in a Matlab environment. In addition to the structural data which are needed to define a structural model in OpenSees, some specific data (described elsewhere) are required in order to define the moment-rotation relationship of the plastic hinges. Once the structural data are prepared, the user calls functions in order to compute the moment-rotation envelopes in the plastic hinges, and after that the input files for OpenSees can be simply generated by calling functions for the generation of the tcl code. Depending on the type of analysis, the user has to prepare some data regarding the analysis, and runs the analysis in OpenSees through the Matlab internal memory. In general there is no need to work with OpenSees since all the results are stored in Matlab structures after the analysis is performed. The results are organized based on the so-called global results, which are, for example, the displacements at the center of mass, and local results, which include the forces and deformations at the plastic hinges. For some type of analyses, e.g. for incremental dynamic analysis, the PBEE toolbox provides an iterative process between the results of the non-linear dynamic analysis obtained by OpenSees, the requirements of the IDA as defined by the analysis data, and the automatic tcl code generation for OpenSees. Once the analysis is finished the user can define different limit states at the level of plastic hinge in order to link the damage in the plastic hinge to the global seismic response parameters. For example, if pushover analysis is performed, the user can simply link the damage at hinge level to the top displacement or base shear, or, based on the results of incremental dynamic analysis, can link the damage in the plastic hinges with the intensity measure. Such an approach enables graphical representation of the damage in the plastic hinges on the pushover or IDA curve. Additionally, the user can determine the target displacement and the damage to the structure according to the N2 method, and present the results visually using AD format. The damage to the plastic hinges can also be presented by plotting it on the structure plan.

The aim of the PBEE toolbox is to enable rapid definition of simple nonlinear structural models of RC frames. Such nonlinear models are permitted in different structural codes, where it is prescribed that the nonlinear behaviour can be modelled with concentrated plasticity. In this case the most time-consuming part of the work involves the determination of the properties of the plastic hinges. Since the PBEE toolbox automatically generates the properties of plastic hinges, based on data regarding material strength, reinforcement and section properties, the amount of work which is needed to prepare a structural model is reduced significantly. Although many different approaches exist for the definition of the moment-rotation relationship of plastic hinges, the PBEE toolbox combines the Eurocode 8 requirements for non-linear modelling and non-linear seismic analysis of buildings with some other approaches. However, the user can simply change the function which is used to determine the moment-rotation relationship of the plastic hinges, and still use other functions of the PBEE toolbox.

Different assumptions which, in general, follow the Eurocode 8 requirements, are used in order to establish as simple as possible and yet still adequate structural model. These assumptions are:

- The floor diaphragms are assumed to be rigid in their own planes, and the masses and moments of inertia of each floor are lumped at the corresponding centre of gravity.
- The beam and column flexural behaviour is modelled by one-component lumped plasticity elements, composed of an elastic beam and two inelastic rotational hinges (defined by a moment-rotation relationship). The element formulation is based on the assumption of an inflexion point at the midpoint of the element. For beams, the plastic hinge is used for one major axis bending only. For columns, two independent plastic hinges for bending about the two principal axes are used.
- The moment-rotation relationship before strength deterioration is modelled by a bi-linear or tri-linear relationship. Zero axial force and the axial load due to gravity loads are taken into account when determining the moment-rotation relationship for beams and columns, respectively. A linear negative post-capping stiffness is assumed after the maximum moment is achieved.
- Gravity load is represented by the uniformly distributed load on the beams and/or by concentrated loads at the top of the columns.

More details regarding the PBEE toolbox can be found in the following documents:

- PBEE toolbox Manual (pdf, 2.85 MB)
- PBEE examples of applications (pdf, 5.12 MB)

The PBEE toolbox is freely available as Matlab P-Code files (zip, 28.07 MB). Also M-files are available upon your request (contact).

## Web application

The aim of this application is to establish a seismic response database of simple system, e.g. single-degree-of-freedom system, which can be then used for multiple purposes. For example:

- Site-specific inelastic spectra, which can be used for determination of target displacement, within N2 method.
- Approximate IDA curves.
- Precedence list of ground motion record for nonlinear dynamic analysis. This approach can substantially reduce the number of required ground motion records for prediction of unbiased seismic response.
- Approximate dispersion measures for randomness and uncertainty in collapse capacity.

The parameters of the seismic response database of the SDOF system are divided into the input parameters and the output parameters. The input parameters are further classified as the structural input parameters and the loading input parameters. The structural input parameters describe the force-displacement relationship, period, damping, and the hysteretic behaviour of the SDOF system, whereas the loading input parameters consist of a seismic intensity measure and the ground motion record. The ground motion record is intentionally treated separately from the other input parameters since it was decided that the database has to be upgradable for at least this parameter. Therefore, SDOF-IDA curves for additional ground motion records can be easily added to the seismic response database. The output parameters, also called engineering demand parameters, are used for the description of the structural response of the model. The typical output parameters are the displacement and/or ductility demand.