Successful use of a mathematical analytical model can provide information on the type, extent and location of damage and unsafe zones and safety levels.
The results of the numerical analysis of church are shown as originally designed that some of the surface subsidence occurred during its construction on thick layers of soft clay (6 m) and because of the drainage or dewatering project in 2000. Displacement developed on the surface, above the maximum value of about 122.85 mm, is the total consolidation settlement and failure of local shearing of the soil due to the in-plane loading. Figure 8 shows the geometry and FE discretization of the 3D model and the deformed mesh and the calculated vertical displacement Uy of the saturated silty clay soil with the structure loading and appears distributed under the superstructure of the church in Figs. 9 and 10, the extreme value Uy is 122.85 mm. The volumetric strain εv distribution of the subsoil is shown in, with a maximum value of 4.50%. The initial effective compressive stresses was determined at the middle of the clay layer depth, where σv = 60 kN/m2. The effective mean stresses P on the subsoil is shown below the structure with a maximum value of 192.85 kN/m2, as shown in Fig. 11. The average effective stress is also calculated and the maximum value is 192.85 kN/m2.
The active pore pressure Pactive distribution in the ground, where the maximum value of Pactive is 240 kN/m2 as shown in Fig. 12.
Settlement calculations determined from the empirical study give the same settlement value, Eq. 1 [25];
$$\Delta {\text{H}} = \frac{\text{CcH}}{{1 + {\text{e}}_{ 0} }}\log \left( {\frac{{{\text{P}}_{0} + \Delta {\text{P}}}}{{{\text{P}}_{0} }}} \right)$$
(1)
ΔΗ is the consolidation adjustment, Cc is the compression index, H is the height of the clay soil layer, e0 is the initial vacuum ratio, P0 is the initial effective vertical stress in the middle of the mud layer depth, ΔP is the change in the vertical effective stress [26].
$$\begin{aligned} \Delta {\rm H}\;{\text{is the consolidation settlement of the soft clay layer}}\, & = \,{{\left( { 5. 5\, \times \,0.3} \right)} \mathord{\left/ {\vphantom {{\left( { 5.5\, \times \,0.3} \right)} {\left( { 1\, + \, 1.84} \right)}}} \right. \kern-0pt} {\left( { 1\, + \, 1.84} \right)}}\, \times \,\left( {{\text{log1}}0{ (60} + 60 )/ 60} \right)\, \\ & = \, 165\,{\text{mm}}. \\ \end{aligned}$$
where the thickness of layer H is 5.5 m, the compression index Cc = 0.3, the initial stress in the depth of the clay layer is P0 = 60 kN/m2. The initial void ratio is 1.84.
The subsidence and settlement of the subsoil may affect seriously the superstructure of the church. The main reasons were the consolidation of subsoil clay layers. Also the shear failure characteristics of subsoil may lead to the ground subsidence. The results of the numerical analysis indicated that the distribution of the Uy vertical displacement of the church superstructure, with a maximum value of 67.62 mm, and minimum Uy value is 67.62 mm as shown in Fig. 13. While the maximum value of horizontal displacement is 8.91 mm, represent the distribution of the horizontal displacement Ux of the structure of the Abo Serga church, the maximum value of Ux is 8.91 mm and the minimum value of Ux is 2.83 mm.
Figure 14 illustrates the normal axial force N1 on the superstructure of the church; the maximum/minimum principal stresses, the maximum value is 784.85 kN/m and the minimum value is 380.88 kN/m.
The shear force Q1 on the superstructure of the church, is with a maximum value of 305.38 kN/m, and the minimum value is 208.96 kN/m as shown in Fig. 15 which represent the maximum/minimum principal stresses. Figure 16 shows the maximum/minimum principal stresses, the bending moments of M1 on the superstructure of the church, with a maximum value of 104.88 kN/m and the minimum value is 80.32 kN/m. bending moment 105 kN per length unit is relatively high value, it may be the main cause of the out-plane deformation and the cracks patterns on the main facades of Abu Serga church, and it observed and recorded from the field and experimental studies.
The computed static surface ground displacements under Abu Serga church are in high values: maximum total vertical displacements is 122.85 mm, which is not acceptable or permissable. Many researches like [27,28,29,30,31,32] discussed the permissable maximum settlement for the shallow foundations in clay soils; and indicated that, for the loading bearing walls, the permissible maximum settlement is 60 mm in case of isolated footings, and 125 mm in case of raft foundations.
The maximum normal axial force N1 on the structure is 784.85 kN/m is very close to the uniaial compressive strength of the original brick (1400 kN/m2). Moreover shear force Q1 is on the superstructure of the church, with a maximum value of 305.38 kN/m, which is also close to the measured shear strength of the rock material (600 kN/m2). The results also indicated that the overstress state is beyond the elastic regime. With a global factor of safety (FoS = strength of component/load on component) equal to 1.78 (< 2) the Abu Serga church should not be considered as safe under static conditions, an FoS of 2 means that a component will fail at twice the design load. In conclusion the detailed analysis of the Abu Serga church proved that these important monuments present low safety factors for both static loading and soil consolidation settlement. Consequently a well-focused strengthening and retrofitting program is deemed necessary. It is deemed necessary to upgrade the safety reserves due to the special nature of the structure.
From the results of the numerical modeling, indicated that the structural deficiencies in the superstructure of the oldest Cairo church, mainly the diagonal, shear and vertical cracks and other distortions within the plane, are mainly induced by the differential settlement of the full saturated clay subsoil consolidation.
The technical assessment revealed that almost all level masonry structural walls presented a brittle mode of failure and more than that, from the first level were of “weak and soft stories” type.