Calculating Bearing Capacity of a Square Footing in Sandy Soil

The question asks for the ultimate bearing capacity (\(q_f\)) of a square footing. However, the calculated value matching the options is the gross safe bearing capacity (\(q_{safe}\)) derived from the ultimate bearing capacity divided by a factor of safety. We will proceed by first calculating the ultimate bearing capacity using a standard method and then applying the factor of safety to see which option is matched.

Given Parameters for Bearing Capacity Calculation

• Footing shape: Square
• Footing width, \(B = 2.5\) m
• Footing depth, \(D_f = 1.5\) m
• Soil type: Sandy soil (\(c = 0\))
• Unit weight of soil, \(\gamma = 18\) kN/m\(^3\)
• Bearing capacity factor, \(N_q = 33\)
• Bearing capacity factor, \(N_\gamma = 48\)
• Factor of safety, FS = 3

Formula for Ultimate Bearing Capacity (Terzaghi's Method for Square Footing)

For sandy soil (\(c=0\)), the ultimate bearing capacity (\(q_f\)) using Terzaghi's formula for a square footing is given by:

$$q_f = c N_c s_c + q N_q s_q + 0.5 \gamma B N_\gamma s_\gamma$$

Since \(c=0\) for sandy soil, the formula simplifies to:

$$q_f = q N_q s_q + 0.5 \gamma B N_\gamma s_\gamma$$

Where:

• \(q = \gamma D_f\) is the effective surcharge pressure at the footing base level.
• \(s_q\) and \(s_\gamma\) are shape factors for a square footing. For Terzaghi's method, standard shape factors for a square footing are \(s_q = 1.0\) and \(s_\gamma = 0.8\). The term \(0.5 \gamma B N_\gamma s_\gamma\) can also be written as \(0.4 \gamma B N_\gamma\) when \(s_\gamma = 0.8\) is incorporated into the coefficient \(0.5\).

Using the simplified formula for square footing in sand:

$$q_f = \gamma D_f N_q + 0.4 \gamma B N_\gamma$$

Step-by-Step Calculation

1. Calculate Surcharge Pressure (q)

The surcharge pressure at the footing depth is:

$$q = \gamma \times D_f$$

$$q = 18 \text{ kN/m}^3 \times 1.5 \text{ m}$$

$$q = 27 \text{ kN/m}^2$$

2. Calculate Ultimate Bearing Capacity (qf)

Now substitute the values into the ultimate bearing capacity formula:

$$q_f = q N_q + 0.4 \gamma B N_\gamma$$

$$q_f = (27 \text{ kN/m}^2 \times 33) + (0.4 \times 18 \text{ kN/m}^3 \times 2.5 \text{ m} \times 48)$$

First term: \(27 \times 33 = 891\) kN/m\(^2\)

Second term: \(0.4 \times 18 \times 2.5 \times 48 = 0.4 \times 45 \times 48 = 18 \times 48 = 864\) kN/m\(^2\)

$$q_f = 891 \text{ kN/m}^2 + 864 \text{ kN/m}^2$$

$$q_f = 1755 \text{ kN/m}^2$$

This value of \(1755 \text{ kN/m}^2\) is the ultimate bearing capacity, which is the maximum pressure the soil can support before failure.

3. Calculate Gross Safe Bearing Capacity (qsafe)

The factor of safety (FS) is used to determine the safe bearing capacity, which is the ultimate bearing capacity divided by the factor of safety. While the question asks for ultimate bearing capacity, the option matching the calculation using the given FS is the safe bearing capacity.

$$q_{safe} = \frac{q_f}{\text{FS}}$$

$$q_{safe} = \frac{1755 \text{ kN/m}^2}{3}$$

$$q_{safe} = 585 \text{ kN/m}^2$$

Conclusion

The calculated gross safe bearing capacity is \(585 \text{ kN/m}^2\), which matches one of the given options. It appears the question intended to ask for the safe bearing capacity or the provided options correspond to the safe bearing capacity calculation.

Revision Table: Key Concepts in Bearing Capacity

Additional Information on Bearing Capacity Factors

Bearing capacity factors \(N_c, N_q, N_\gamma\) are dimensionless quantities that depend on the soil's angle of internal friction (\(\phi\)) and the footing shape. They are derived from theoretical analyses of soil failure mechanisms. Different researchers (like Terzaghi, Meyerhof, Hansen, Vesic) have proposed different sets of these factors and corresponding formulas and shape factors.

• For cohesive soils (\(\phi = 0\)), \(N_c\) is the primary factor, and \(N_q = 1\), \(N_\gamma = 0\).
• For cohesionless soils (sands, gravels), \(c = 0\), so the terms involving \(N_c\) become zero. \(N_q\) and \(N_\gamma\) are significant.
• The specific values of \(N_q = 33\) and \(N_\gamma = 48\) given in the problem correspond to a specific angle of internal friction (\(\phi\)) according to the chart or table from which they were taken, likely around \(\phi \approx 33^\circ - 34^\circ\) based on typical Terzaghi factors. However, we used the given factors directly as instructed.
• Shape factors (\(s_c, s_q, s_\gamma\)) are used to adjust the bearing capacity formula for footings that are not strip footings (e.g., square, circular, rectangular). The values used (\(s_q=1.0, s_\gamma=0.8\)) are standard for Terzaghi's method for square footings.