Flexural (average) bond stress in a reinforced concrete beam section: If jd is the lever arm and ΣO is the total perimeter of tension reinforcement at the section, express the bond stress τ_bd in terms of the applied shear/longitudinal force Q.

Introduction / Context:
Flexural bond stress represents the average shear stress developed along the steel–concrete interface that allows transfer of longitudinal force from concrete to reinforcing bars. It is an essential quantity in checking anchorage and development length.

Given Data / Assumptions:

• Beam carries shear force V; the corresponding change in tension force over a short length is related to Q (longitudinal force increment).
• ΣO is total bar perimeter in tension zone at the section considered.
• Lever arm between compression and tension resultants is jd.

Concept / Approach:
Equilibrium requires the increment of tensile force in steel over a short length to balance the shear flow along the interface. The average bond stress τ_bd times total interface perimeter ΣO equals the rate of change of steel force with length. For a short segment at peak shear, the relationship simplifies to τ_bd = Q / (ΣO * jd), where Q is the internal shear (linked to external shear V) developing the change in flexural force across the lever arm jd.

Step-by-Step Solution:
Relate shear to change in tension: dT/dx ≈ V / jd.Bond equilibrium: τ_bd * ΣO = dT/dx.Thus τ_bd = (V / jd) / ΣO = V / (ΣO * jd) = Q / (ΣO * jd).

Verification / Alternative check:
The formula is dimensionally consistent: Q has force units; ΣO has length; jd has length; τ has force/area. Using total perimeter consolidates multiple bars into a single average value.

Why Other Options Are Wrong:

• Multiplying (option b) or inverting (option c) violates equilibrium.
• Replacing jd with d (option d) ignores the actual lever arm and overestimates bond stress.
• Using π*d^2 (option e) erroneously introduces bar area rather than total perimeter.

Common Pitfalls:
Confusing average bond stress with local peak bond; using bar area instead of perimeter; overlooking that jd, not d, governs the tensile/compressive couple.

Final Answer:
τ_bd = Q / (ΣO * jd)