Civil Engineering Department, BITS Pilani Rajasthan
Civil Engineering Department, BITS Pilani Rajasthan
This paper presents a powerful analytical tool based on the limit analysis methodology to estimate the stability of the stone masonry structures, especially structures such as arches, vaults and domes. The approach provided is largely 2-dimensional, but with some approximation it can also be applied to complex 3-dimensional problems. Further, the strengthening of unstable masonry arches is studied with the help of composites such as FRPs, which show excellent properties of high tensile strength and stiffness. The analysis is based on the stress results obtained through the linear finite element analysis in ABAQUS and thrust line analysis to evaluate the stability of unstable masonry arches. Furthermore, some recommendations have been provided to expand this work in the future along with the limitations of the present work.
Masonry structures are a major part of our architectural heritage. The most important components of these structures are arches, buttresses and vaults. Though such structures have existed for years, little study has been performed regarding their stability. This may be because of the uncertainties in the material properties, difficulty in assessing the conditions without being able to check the interior of the structure and consequently, difficulty in obtaining measurements.
Generally, masonry structures fail due to instability rather than lack of material strength. As the primary load on a stone masonry arch is the self-weight, an analysis of the equilibrium of the structure is suitable for such scenarios. Further, limit analysis method provides a quantitative understanding of this approach. Three assumptions are made in the study as a part of the methodology- masonry has no tensile capacity; masonry can resist infinite compression (i.e., the structure will not fail due to compressive stresses); sliding does not take place between the masonry stones.
The masonry structures have deteriorated due to ageing and have developed cracks due to an increase in lateral load, change in occupancy, environmental changes, soil settlements or dynamic forces such as earthquakes, etc. Hence, there is a need to analyze their stability, and undertake suitable strengthening and retrofitting to preserve our heritage for a long time.
Further, the use of fibre-reinforced polymer (FRP) has increased recently in the repair and rehabilitation of stone masonry and other important structures. They provide many benefits such as ease of application, non-evasiveness, easy removal, high tensile strength and stiffness. FRPs are lightweight and most importantly, provide a very cost-effective solution to the problem of instability. Moreover, they do not alter the natural behaviour of the existing structure as they do not add any substantial mass to the system. FRP sheets are bonded to the surface of the structures at the intrados or extrados in the form of strips of complete sheets, depending on the level of stability required for a particular structure and accessibility conditions.
An arch deforms under the self-loading through a three-hinge stable mechanism. Thus, limit analysis is a powerful tool to estimate the range of lower bound solutions for the equilibrium of compression-only structures. It allows analysis of the stability of a structure and visualization of possible collapse mechanisms.
Further, a thrust line is a line joining the resultant of the compressive forces in the arch. An arch is stable if the thrust line lies completely inside the arch section and within the middle third to avoid tension. Wherever the thrust line touches the boundary of the arch or is not contained within the arch section, there is a possibility of the formation of a hinge; if sufficient hinges form, it provides a collapse mechanism and the arch fails. Therefore, three arches of varying thicknesses have been considered to evaluate their stability. The arches have a radius of 5 m, and thickness of 0.4 m, 1 m and 1.2 m, respectively.
The stresses in the arches due to their self-weight have been obtained by linear elastic finite element analysis in ABAQUS. The maximum and minimum principal stresses are obtained. Further, their moment resisting capacity and axial force is calculated using the expressions mentioned below:
M = (σ1– σ2)t2/12 (1)
P = (σ1+ σ2)t/2 (2)
M = Moment
P = Axial Force
σ1 = Maximum principal stress
σ2 = Minimum principal stress
t = Thickness of the arch
The distance of the thrust from the centreline (eccentricity) of the radial arch section is calculated as:
e = M/P (3)
Further, the values of e/t at various locations of the arch where the stresses have been obtained by the finite element analysis are estimated. If the value of e/t lies between -0.5 to 0.5, then the arch is stable; this is because it implies that the thrust line lies completely inside the radial section of the arch and no extra hinges would be formed.
The results of the finite element analysis are shown in Fig. 1, where all the three arches considered for the study have been compared. Further, the e/t ratio obtained by carrying out the limit analysis on these arches has been shown with the help of some graphs in Fig. 2.
The analysis shows that the e/t ratio for the first and second arches exceeds the range of [-0.5,0.5] at some regions; hence, these arches are unstable, while the third arch of thickness 1.2 m is stable.
|a) 0.4 m||b) 1 m||c) 1.2 m|
|Fig. 1: Finite Element Analysis of Arches of Various Thicknesses|
The unstable arches require strengthening to increase their life span. FRPs are used for this purpose because of the large number of benefits provided by them. Further, they are found in different types depending on the type of fibre used, such as glass fibre reinforced polymer (GFRP), carbon fibre reinforced polymer (CFRP), aramid fibre reinforced polymer (AFRP), etc.
In the study, FRP strengthening is undertaken and the finite element analysis combined with limit analysis is performed again to find out the stability of the arch once it is strengthened with FRP.
The results of the limit analysis of the three arches (Fig. 4) of varying thicknesses show that the stability of an arch changes with the change in geometry. The results show that the thin arches are not stable (additional hinge forms to produce a collapse mechanism, which is evident from the values of e/t which do not lie in the specified range), while the thick arch of 1.2 m thickness is stable. Moreover, after strengthening of the unstable thin arch of 0.4 m thickness with FRP at the intrados and extrados, it is seen that the arch remains stable. This shows that considerable stability can be achieved by reinforcing either at the intrados or extrados; so, the strengthening can be done according to the convenience and structural difficulties.
|(a) 0.4 m||(b) 1 m||(c) 1.2 m|
|Fig. 2: Range of Values of e/t for Arches of Thickness|
|a) Intrados||b) Extrados|
|Fig. 3: FRP Strengthening|
|a) Intrados||b) Extrados|
|Fig. 4: Range of Values of e/t for Strengthening|
Hence, FRP strengthening is a powerful technique to preserve our rich architectural heritage. FRPs are anisotropic in nature and the best mechanical properties are obtained in the direction of application of the fibres. Further, the limit analysis provides a common framework for engineers and architects to study historic structures completely, which is a great necessity in the present scenario.
Limitations And Future Work
The model presented here is a 2D analysis. However, in real-world scenarios, many problems would be difficult to present and model with a 2D approach. So, this analysis must be extended to 3D approach. Also, in this study only self-weight has been considered while evaluating the stability; however, the model can be extended to include live loads. Further, the limit analysis provides an overall approach for determining the stability of masonry structures; it can be expanded to incorporate some local effects in the structure as well. Also, as earthquakes are a major reason for the deterioration of these historic structures, dynamic analysis should be performed in case of an earthquake to assess the performance and stability of the structure, and to ensure that FRP strengthening helps to make the structure stable and resistant against seismic loading.
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