Flexural resistance of a singly reinforced concrete beam: If At is the area of tension steel, f is the design stress in that steel at the limiting state, d is the effective depth, and y is the depth to the resultant compressive force in concrete, what is the expression for the moment of resistance M?

Introduction / Context:
In reinforced concrete flexure, internal compression in concrete and tension in steel form a couple. The moment of resistance equals one of the forces times the lever arm between the lines of action. For singly reinforced beams, the tension side is governed by steel stress at the design limit state.

Given Data / Assumptions:

• At = area of tension reinforcement.
• f = design stress in tension steel at the limit state.
• d = effective depth to tension steel centroid.
• y = depth to the resultant compressive force in concrete from the extreme compression fiber (so lever arm is approximately d - y).

Concept / Approach:
Equilibrium requires compressive force C = tensile force T. With steel governing: T = At * f. The internal moment is T multiplied by the lever arm z = d - y, giving M = At * f * (d - y).

Step-by-Step Solution:
Compute tensile force: T = At * f.Determine lever arm: z = d - y.Moment of resistance: M = T * z = At * f * (d - y).

Verification / Alternative check:
Using compressive block: C = 0.36 * fck * b * x (code-based) with its centroid at y; matching C = T gives the same lever arm to evaluate M, consistent with M = T * z.

Why Other Options Are Wrong:

• M = At * (d - y): missing the steel stress f.
• M = At * f * (d + y): lever arm is not d + y.
• M = (At / f) * (d - y): dimensionally inconsistent.
• None of these: incorrect because a correct expression is available.

Common Pitfalls:
Confusing y with neutral axis depth or using effective depth incorrectly; mixing working-stress and limit-state symbols can also cause errors.

Final Answer:
M = At * f * (d - y).