Understanding Bending Moment in Square Footings

This question asks us to determine the maximum bending moment acting on a square footing that supports a square column. The footing is subjected to a net upward pressure from the soil. Calculating this bending moment is crucial in foundation design to ensure the footing is strong enough to resist the forces.

Analyzing the Footing as a Cantilever

A square footing supporting a centrally located square column can be analyzed by considering strips of the footing acting as cantilevers projecting outwards from the faces of the column. The maximum bending moment occurs at the face of the column, where the cantilever is "fixed".

• The footing has a side length denoted by $b$.
• The square column has a side length denoted by $a$.
• The net upward pressure on the footing is given as $p$.

Calculating the Cantilever Length

For a square footing centered under a square column, the projection of the footing beyond each face of the column is equal. The total length of the footing is $b$, and the column width is $a$. The remaining length $(b - a)$ is distributed equally on both sides of the column.

Therefore, the cantilever length, denoted as $L$, is:

$\qquad L = \frac{b - a}{2}$

Determining Load and Bending Moment

We consider a strip of the footing with a width equal to the full side length of the footing, $b$, running perpendicular to the column face. This strip behaves like a cantilever beam subjected to a uniformly distributed load (UDL) from the upward soil pressure $p$.

The intensity of the UDL on this strip, per unit length along the cantilever direction, is the pressure multiplied by the strip width:

$\qquad w = p \times b$

The maximum bending moment ($B.M.$) for a cantilever beam subjected to a UDL $w$ over a length $L$ occurs at the fixed end (the column face) and is given by the formula:

$\qquad B.M. = \frac{wL^2}{2}$

Substitute the values of $w$ and $L$ into the formula:

$\qquad B.M. = \frac{(p \times b) \times \left(\frac{b - a}{2}\right)^2}{2}$

Now, let's simplify the expression:

$\qquad B.M. = \frac{pb \times \frac{(b - a)^2}{4}}{2}$

$\qquad B.M. = \frac{pb(b - a)^2}{4 \times 2}$

$\qquad B.M. = \frac{pb(b - a)^2}{8}$

Comparing with Options

Let's compare our derived formula with the given options:

• A. ${\rm{B}}.{\rm{M}} = \frac{{{\rm{pb}}\left( {{\rm{c}} - {\rm{a}}} \right)}}{4}$ (Incorrect, 'c' is undefined)
• B. ${\rm{B}}.{\rm{M}} = \frac{{{\rm{pb}}{{\left( {{\rm{b}} - {\rm{a}}} \right)}^2}}}{4}$ (Incorrect, factor of 1/2 is missing)
• C. ${\rm{B}}.{\rm{M}} = \frac{{{\rm{pb}}{{\left( {{\rm{b}} - {\rm{a}}} \right)}^2}}}{8}$ (Matches our derived formula)
• D. ${\rm{B}}.{\rm{B}}.{\rm{M}} = \frac{{{\rm{pb}}\left( {{\rm{b}} + {\rm{a}}} \right)}}{8}$ (Incorrect, the term $(b+a)$ is not relevant to the cantilever length)

The derived formula for the maximum bending moment matches option C.

Conclusion

The maximum bending moment in the square footing acting as a cantilever from the column face is given by the formula $\frac{pb(b - a)^2}{8}$. This calculation assumes that the pressure distribution is uniform and that the critical section for bending is at the face of the column.

Revision Table: Square Footing Bending Moment

Additional Information: Footing Design Considerations

Calculating the maximum bending moment is just one step in the overall design of a square footing. Other important considerations include:

• Shear Strength: Checking for one-way shear and two-way (punching) shear around the column. This ensures the concrete won't fail due to diagonal tension or punching action.
• Bearing Capacity: Ensuring the soil can safely support the total load transmitted by the column through the footing.
• Settlement: Predicting and limiting the settlement of the footing to acceptable levels.
• Reinforcement: Determining the amount and placement of steel reinforcement required to resist the calculated bending moments and shear forces. The bending moment calculated here dictates the amount of steel needed in the bottom mat of the footing in both directions.
• Concrete Strength: Specifying the appropriate concrete compressive strength.

The formula $\frac{pb(b - a)^2}{8}$ provides the maximum bending moment per unit width if $b$ was 1 unit. However, the calculation performed here considers the total load on a strip of width $b$, resulting in the total bending moment across the full width of the column face.