## 4 Computational and Mathematical Aspects
### Structural Modeling Techniques for Masonry Structures
#### 4.2.4 Modeling Masonry Using Equivalent Frame Method  
The most widely used method for the modeling of real cases masonry structures is the simplified analysis method (SAM). The SAM is based on the macro-element approach that has been first proposed and developed by [PERSON] and [PERSON] (1998) [133] to overcome the shortcomings of the previously proposed method of \"story-mechanism\" by [PERSON] and [PERSON] (1990) [134]. The SAM stems from the concept of framed buildings analysis using beam and beam-columns elements. In fact, in the SAM to simulate the in-plane behavior of a masonry wall panel, the constituent piers and spandrel beams are substituted by the equivalent beam-columns elements with appropriate mechanical properties. Therefore, sometimes the SAM is called the equivalent frame method (EFM) which is a familiar method and has been widely used to analyze RC coupled shear walls in the literature (Figure 5).  
The SAM has been used successfully by [PERSON] and [PERSON] (1998) [133] to analyze two multi-story masonry walls subjected to lateral loads. The SAM was then developed by [PERSON] (2000) [135] to be used for seismic assessment of practical and historic masonry buildings making use of displacement spectra and of the substitute-structure approach. [PERSON] et al. (2002) [136] investigated the effectiveness of the SAM or EFM for the design and/or assessment of masonry buildings. [PERSON] (2006) [137] based on the concept of the EFM proposed an approach for the pushover analysis of URM buildings. This approach is similar to the SAM of [PERSON] and [PERSON] (1998) [133], however, it uses an analytical approach and a Mohr-Coulomb failure criteria for flexural and shear strengths, respectively. Furthermore, the shear stiffness and ultimate shear deformation capacity of wall elements are estimated from experimental tests results ([PERSON], 2006, [137]). [PERSON] et al. (2008) [138] investigated the capability of a commercial software for pushover analysis of masonry buildings using the EFM. This study showed that the EFM can be used only for pushover analysis of masonry walls of usual and regular geometry ([PERSON] et al., 2008, [138]). This is because the software does not take into account the effects of axial load variation in the piers' shear strength capacity during the analysis process.  
The main advantage of the SAM is that it is based on simple strength criteria of masonry piers and spandrel beams corresponding to the well-known flexural and shear failure modes ([PERSON] and [PERSON], [133]). In addition, it requires low computational effort to simulate both linear and nonlinear lateral load behavior of practical large masonry wall panels. Detailed state-of-the-art reports can be found in [139; 140; 141; 142; 143; 144; 145; 146; 147; 148; 149; 150; 151; 152; 153; 154].  
Figure 5: A schematic representation of equivalent frame model for planar walls with openings ([PERSON], 2000, [134]).  
In light of the above and despite the plethora of mathematical models that have been proposed for the modeling of masonry structures, for real masonry structures, it is preferable to use simplified macro-models such as modeling the masonry as a one-phase material (macro-modeling approach) while the use of more sophisticated micro-models is suggested for the modeling of special parts of the masonry structural model. Detailed works on the advantages and disadvantages among different mathematical models about masonry structures can be found in [40; 100; 155; 156].