Combined footing (two equal column loads): For an economical design of the longitudinal combined footing beam (supporting two equal column loads), the end projections are proportioned so that the bending moment under each column equals which reference bending moment?

Introduction / Context:
Combined footings for two columns are designed to achieve uniform soil pressure and economic use of concrete and steel. When the two column loads are equal and the footing is rectangular, the longitudinal strip acts like a beam on an elastic foundation. A practical proportioning guideline is to balance the bending moment under the columns with the maximum span moment to reduce peak steel demand and achieve a more uniform reinforcement layout.

Given Data / Assumptions:

• Two equal column loads supported by a single rectangular combined footing.
• Uniform allowable soil pressure targeted through appropriate footing dimensions.
• Longitudinal action idealized as a beam with overhangs (end projections).

Concept / Approach:
With equal column loads and uniform soil pressure, the shear and moment diagrams can be shaped by adjusting the end projections (cantilevers) beyond the outer columns. An economical proportion commonly adopted is to make the sagging bending moment at midspan essentially equal to the hogging (negative) moment at the column faces. This balances steel requirements and minimizes peak reinforcement disparity.

Step-by-Step Solution:
Assume uniform soil pressure under the footing area.Choose overhang lengths so that the resultant soil reaction under the overhangs counterbalances outer-span shears.Iterate until the hogging moment under each column approximately equals the largest sagging moment at midspan.This yields an efficient and symmetrical bending profile and reinforcement layout.

Verification / Alternative check:
Trial proportioning followed by soil-pressure and section-capacity checks confirms that reinforcement near the columns and at midspan can be kept similar, reducing congestion at the columns and avoiding unnecessarily deep sections at the span centre.

Why Other Options Are Wrong:

• Half the midspan moment: would leave the column zones under-reinforced compared to midspan.
• Twice the midspan moment: overconservative at columns, wasting material.
• None of these: not correct because matching column-face and midspan moments is a well-known economical criterion.

Common Pitfalls:

• Proportioning only by geometry without checking uniform soil pressure and resultant location.
• Ignoring shear near columns when maximizing overhangs.

Final Answer:
the bending moment at the centre (midspan) of the combined beam