Bond (development) length: For a reinforcing bar of diameter d subjected to allowable tensile stress ft in concrete with allowable bond stress fb, what is the required development/bond length Lb?

Introduction / Context:
The bond length (development length) is the minimum embedment required so that the stress in a reinforcing bar can be safely transferred to surrounding concrete through bond. It is a fundamental detailing parameter in reinforced concrete design.

Given Data / Assumptions:

• Bar diameter = d.
• Allowable tensile stress in steel = ft.
• Allowable bond stress in concrete = fb.
• Straight bar in tension; standard bond conditions.

Concept / Approach:
Equating the tensile force in steel to the bond resistance along the embedded length gives the basic formula. Tensile force = area * stress = (π d^2 / 4) * ft. Bond resistance = perimeter * length * bond stress = (π d) * Lb * fb. Solving for Lb yields the classical expression.

Step-by-Step Solution:
Tension in steel: T = (π d^2 / 4) * ft.Bond capacity: R = (π d) * Lb * fb.Equate T = R and simplify: (π d^2 / 4) * ft = (π d) * Lb * fb.Cancel π d and solve: Lb = (d * ft) / (4 * fb).

Verification / Alternative check:
The derived expression matches standard RC design texts for straight tension bars under allowable stress design assumptions.

Why Other Options Are Wrong:

• Forms with 2fb or 3fb in denominator mis-state the equilibrium constants.
• Inverse or squared-diameter forms do not come from basic force equilibrium of bar area vs. bond perimeter.

Common Pitfalls:
Forgetting to multiply perimeter by length, or using area times bond stress; mixing design stress formats (working vs. ultimate) without adjusting fb values.

Final Answer:
Lb = (d * ft) / (4 * fb)