R EVIEW OF L ITERATURE
2.3.1 Macromodels
Alternatively, infills may be modelled by means of detailed finite element micromodelling. Two ways of modelling infills were observed in the literature: either by simplistic macromodelling approach or by the more complicated micromodelling approach. A brief review of the different analytical models pertaining to the current study and their limitations are reviewed in this section.
masonry infilled RC frames. Based on large experimental study on full-scale infilled steel frames tested under lateral loading, Flanagan and Bennett (1999) proposed a piecewise linear strut area (Aw) as shown in Eq. (2.20).
d h w
m s
λ θ 2
2 sin 95 .
=0 (2.19)
θ λ
π cos C
Aw = tm (2.20)
where C is an empirical constant which varied with the in-plane drift displacement and is an indicator of the limit state of infill. Puglisi et al. (2009) proposed a new modified diagonal strut model based on the theory of plasticity (plastic concentrator) to overcome the drawback of equivalent strut model under cyclic or earthquake loadings. The modified model considers a concentrator (plastic/elastic) linked between the two diagonal strut bars, which transfer the effects from one strut to other. From the analysis, it was found that the conventional strut models need to be modified by considering coupling between the bars in order to observe the more realistic behavior under lateral loading. To study the nonlinear hysteretic behavior of masonry infill in RC frames, Rodrigues et al. (2010) upgraded the equivalent diagonal compression strut model to consider the interaction of masonry in both directions (Fig. 2.5).
Fig. 2.5. Macromodel for the simulation of infill and force displacement monotonic curve (Rodrigues et al. 2010).
The masonry panel hysteretic procedure considers rules for loading, reloading, unloading and also degradation of stiffness and strength along with pinching effect. It was reported that the proposed model was able to represent the response in terms of
displacements, global shear-drift at each storey, and cumulative dissipated energy.
Among the various parameters influencing the behavior of equivalent strut, the effect of vertical load acting on the frames is well recognized as discussed in the previous sections (2.2, 2.3, and 2.4) but not quantified. Addressing this effect, Campione et al. (2015) modified the equivalent strut taking into account the stiffening effect produced by the vertical loads on the infill in the initial state. A family of curves to modify the width of the strut was proposed for different vertical loads and for different Poisson’s ratio by imposing the equivalence between the frame containing the infill and the frame containing the diagonal strut. Along the same lines, Asteris et al. (2016) modified the equivalent pin-jointed diagonal compressive strut by a stiffness reduction factor which considers the effects of both openings and vertical loads. The effect of vertical loads on the stiffness of the dimensionless strut width is prominent in case of fully infilled frames and negligible for panels with larger openings. From the study, it was also concluded that the proposed mathematical macromodel may be used as a reliable and useful tool for the determination of the equivalent compressive strut width, since it accounts for a large number of parameters, which are not generally accounted by the available models.
2.3.1.2 Multiple-strut Models
To overcome the limitations of single-strut models, multiple-strut models were proposed to simulate the effect of infill under lateral loading more accurately. In this regard, Thiruvengadam (1985), proposed multiple-strut model to evaluate the realistic natural frequencies and modes of vibration using several diagonal and vertical struts in each direction. The model has been adopted by various researchers as the model accounts effect of openings (Singh and Das 2006; FEMA 2000).
Chrysostomou et al. (2002) considering infill hysteretic stiffness and strength degradation simulated the response of infilled frames using six struts with three struts in each diagonal direction [Fig. 2.6(a)]. Only three struts were activated at any time of analysis, whenever tensile force exists the struts are switched to other loading direction.
The infill model allows the interaction between infill and the frame, and takes into account the strength and stiffness degradation of the infill. El-Dakhakhni et al. (2003) simulated the effect of infill by three diagonal struts in each direction to estimate the stiffness and the lateral load capacity of concrete masonry infilled frames. The masonry
model was developed based on the orthotropic properties of masonry infill. It was observed that the method can be easily implemented in finite element models and design of three dimensional masonry infilled concrete frames.
(a) (b)
Fig. 2.6. Multiple-strut models: (a) six-strut idealization of infill wall (Chrysostomou et al.
2002); and (b) double strut model with shear spring for infill panel (Crisafulli and Carr 2007).
To appropriately, simulate the response of infilled frame in terms of lateral stiffness and strength a new macromodel consisting of two parallel struts with a shear spring [Fig. 2.6(b)] was proposed by Crisafulli and Carr (2007). The major inefficacy was the model not able to predict the distribution bending moment and shear forces in the frame elements. Uva et al. (2012) performed sensitivity analysis varying width of the strut (ws), the constitutive force-displacement law of the panel and number of struts adopted.
The equivalent strut with larger width (ws) has shown higher capacity but exhibited brittle behavior, whereas, vice-versa behavior observed for lower values of ws. The constitutive force-displacement relation of the masonry panel has a significant dependence on the type of failure mechanisms of the panel. The authors reported that only the multi-strut model can represent the brittle failure mechanism occurring at the nodes due to the presence of the masonry infill.
2.3.1.3 Improved Macromodels to Capture Local Shear Failure of Columns
It has been already discussed in the previous section that the single strut models were not be able to reproduce the realistic shear forces and bending moments in the surrounding frame elements. Under-estimation of shear demand on the columns may lead to their brittle shear failure which is not desirable. In the recent years, the basic strut macromodels were improved to predict the shear distribution in frame critical sections.
D’Ayala et al. (2009) analysed single storey infilled RC frames to evaluate the shear
U αL
αh h
L Φ=U/h
θ
capacity of the columns due to the detrimental effect of infill using two finite element programmes and validated the results with similar experiments (Al-chaar et al. 2002).
Even though the analytical model predicted the shear failure of columns, the applicability of the model in practical engineering problems is limited and computationally intensive as the contact elements were found to be difficult to calibrate.
Celarec and Dolšek (2013) carried out an iterative pushover based procedure involving model adaptation by modifying the moment-rotation and force-displacement relationships of the columns and diagonal strut, respectively. Their procedure was found to be approximate due to several limitations and simplifications, especially, the limitations concerning the shear strength model and the proportion of force transferred from infill to column. Cavaleri and Di Trapani (2015) proposed a predictive tool, to estimate the shear forces at the ends of the beam and column as a fraction of axial load experienced by the equivalent diagonal strut. Fiore et al. (2016) tried to predict the brittle shear collapse mechanism of columns due to the infill-frame interaction using the double- strut model (Fig. 2.7) proposed previously by Fiore et al. (2012). However, the major difficulty of the double-strut model is the placement of the equivalent non-parallel struts and the shear force and bending moment distribution due to the application of the concentrated force near the stress resultants.
(a) (b)
Fig. 2.7. Macromodels of infill panels: (a) single-strut model; and (b) double-strut model (Fiore et al. 2016).