Prestressing losses along a curved tendon — if k is the wobble (wave) friction coefficient, μ is the coefficient of friction, R is the radius of curvature, and x is distance from an anchorage, what is the tension ratio T(x)/T₀?

Introduction / Context:
In post-tensioned concrete, tendon force reduces from the jacking end due to curvature (friction proportional to μ times total angular change) and unintended wobble or misalignment (modeled by k per unit length). Estimating the remaining force at a distance x is necessary to verify stresses and ultimate capacity along the member.

Given Data / Assumptions:

• μ: coefficient of friction between tendon and duct.
• k: wobble coefficient (per unit length).
• Constant radius R → angular change over length x is approximately x / R.
• T₀: jacking force at the anchorage; T(x): force at distance x.

Concept / Approach:

The standard friction-loss model yields the exponential decay expression: T(x) = T₀ * exp(−(μ * θ + k * x)), where θ is the total angular change between the jacking end and section at x. For constant radius, θ = x / R, giving T(x)/T₀ = exp(−(μ * x / R + k * x)).

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

For straight tendon (R → ∞), term μ * x / R → 0, leaving T/T₀ = exp(−k x), matching the special case with wobble only.

Why Other Options Are Wrong:

Options A or B consider only one loss mechanism; D is a linear approximation, not used for design; E has sign error for the curvature term.

Common Pitfalls (misconceptions, mistakes):

Using degrees for θ without converting to radians; ignoring that k is per unit length; assuming μ effects vanish on small curvatures without checking magnitude.

Final Answer:

exp(−(μ * x / R + k * x))