## 4 Design of a Reinforced-Concrete Conical Foundation
### Calculation of Bearing Capacity of a Reinforced Cone-Top Conical Foundation  
To calculate the bearing capacity of the conical pile foundation, the estimated load from a 2-story building measuring 12 by 24 m with a column grid of 6 by 6 m was considered, as well as the placement of cone-top pile foundations under load-bearing walls every 3 m. The estimated design load on each cone-top-pile block of the foundation was calculated to be 177.55 kN. The diameter of the base was selected based on calculations considering the loads on the structure, and was taken as 1200 mm. Calculation of bearing capacity at the cone base was made according to the geometric characteristics in Figure 14b. According to [PERSON]'s bearing capacity formula for c-\(\varphi\) soil:  
\[q_{u}=c\cdot N_{c}+\gamma\cdot D\cdot N_{q}+0.5\cdot\gamma\cdot B\cdot N_{ \gamma}, \tag{2}\]  
\(q_{u}\)--Ultimate bearing capacity kN/m\({}^{3}\);  
\(c\cdot N_{c}\)--Contribution of soil cohesion to bearing capacity;  
\(\gamma\cdot D\cdot N_{q}\)--Contribution from surcharge (vertical stress due to overlying soil at base depth);  
0.5\(\cdot\gamma\cdot B\cdot N_{\gamma}\)--Contribution from unit weight of the soil acting directly under the base;  
\(D\)--Depth of the footing, 1.61 m;  
\(B\)--Width of the footing, the same as pile shaft diameter, 0.26 m;  
\(\gamma\)--Unit weight of the soil, 21.2 kN/m\({}^{3}\);  
\(N_{c}\), \(N_{q}\), \(N_{\gamma}\)--Bearing capacity factors, which are dimensionless values that depend on the soil's angle of internal friction (\(\varphi\)) and the shape of the foundation. \[N_{c}=25.13\text{, }\text{Nq}=12.72\text{, }N_{\gamma}=7.44\text{, }q=\gamma\cdot\text{depth}\approx 34.2 \text{ kPa}\]  
\[q_{u}=39\cdot 25.13+21.2\cdot 1.61\cdot 12.7+0.5\cdot 21.2\cdot 0.26\cdot 7.44=143 4.91\text{ kPa}\]  
Note on net bearing capacity:  
\[q_{net}=q_{u}-\gamma\cdot D \tag{3}\]  
\[q_{net}=1434.91-(21.2\times 1.61)\approx 1434.91-34.13=1400.78\text{ kPa}\]  
Figure 14: Example of reinforcement of conical foundations: (**a**) conical block with an opening angle of 90\({}^{\circ}\); (**b**) cone-top foundation with a pile-stand. Allowable bearing capacity with FS = 3:  
\[q_{allow,\ gross}=\frac{q_{u}}{3}=\frac{1434.91}{3}=478.30\;\mathrm{kPa} \tag{4}\]  
Base area: \(A_{b}=\frac{\pi}{4}0.85^{2}=0.5685\;\mathrm{m}^{2}\). Applied pressure: \(q_{applied}=177.55\;\mathrm{kN/0.5685\;m}^{2}\approx 312.3\;\mathrm{kPa}\). 312.3 kPa < 478.30 kPa safety is ensured. The cone base is safe under the applied load of 177.55 kN with a factor of safety of 3. Allowable reserve capacity: \(P_{max}=q_{allow, gross}\cdot A=478.3\cdot 0.5685=271.58\;\mathrm{kN}\). Side friction resistance for the pile shaft Qs is calculated for using the \(\alpha\)-method:  
\[\alpha=0.5\;(\mathrm{typical\;for\;tiff\;clays\;and\;c-\varphi\; soils})\]  
\[Qs=\alpha\cdot c\cdot As=0.5\cdot 39\cdot 0.98\approx 19.11\;\mathrm{kN} \tag{5}\]  
where the pile shaft friction area is \(A_{\mathrm{s}}=\pi\cdot\mathrm{D_{pile}\cdot h_{pile}}\approx\)3.14 \(\cdot\) 0.26 \(\cdot\) 1.2 = 0.98 m\({}^{2}\). Total resistance = 271.8 kPa * 0.98 m\({}^{2}\) + 19.11 kN = 285.47 kN capacity > 177.55 kN applied. Side friction-resistance safety is ensured. No extrusion is expected under this load, given the depth, cohesion, and effective surface contact. It is assumed that the cone is a frustum, and so its lateral surface area is  
\[A_{cone}=\pi\;\frac{D_{top}+D_{bottom}}{2}\rfloor \tag{6}\]  
\[N_{c}=25.13\text{, }\text{Nq}=12.72\text{, }N_{\gamma}=7.44\text{, }q=\gamma\cdot\text{depth}\approx 34.2 \text{ kPa}\]0  
where \(D_{top}\)--diameter of the top (smaller) base of the frustum;  
\(D_{bottom}\)--diameter of the bottom (larger) base of the frustum;  
\(h_{c}^{\prime}\)--vertical height of the frustum (not slant height);  
\(l\)--slant height of the frustum;  
\(A_{cone}\)--lateral surface area of the conical frustum.