Why are the units for the wobble coefficient, ‘K,’ in radians/unit length as defined in the FELT User Manual page 4-8? (NOTE: this also applies to friction loss calculations in ADAPT-PT and ADAPT-Builder.)
Units for friction coefficients in the formula for tendon stressing
– The formula commonly used for the calculation of stress loss due to friction is:
P(jack) = P(x)*e(µα + K*L)
In the above formula:
P(jack) = force at the stressing end;
P(x) = force at distance (x) from the stressing end; e = stands for exponential (“e” raised to the power of what follows in the parenthesis); µ = coefficient of angular friction; α = change in angle in radians; K = coefficient of wobble friction; and L = length of tendon from the stressing end to point x.
The following describes the background to the units of the coefficient K. However, before going into the explanation, it is important to be clear about the following two definitions. First, “friction” is defined as the force that can exist between two surfaces when the surfaces are compressed against one another. In engineering practice, it is assumed to be a characteristic of the two surfaces. Its value is given by:
Friction force: F = µ * N
Where µ is the coefficient of friction, and N is the axial (compression) force between the two surfaces. µ is dimensionless. N has dimension of force (lb, N etc). It is important to note that in engineering practice, it is assumed that the friction force in neither a function of the contact area nor length between the two surfaces. This is the type of force that the tendon friction formula is intended to address.
Second, “cohesion” is the bonding (gluing) force between two surfaces. Its value is given by:
Cohesion Force: C = γ * L
Where γ is the force per unit area, or unit length (if member is linear), and L is the area or length (for linear members).
In the stress loss calculation of tendons, cohesion is not accounted for. It is absent in the case of grouted tendons. In the general case, the cohesion between a strand and grease in unbonded tendons is assumed negligible compared with friction. If it were not negligible, it would have been calculated using a relationship of the following shape.
P(x) = P(jack) – γ*L
Where γ has the units of “force/unit length of tendon.”
With the forgoing made clear, we can now go back to the friction formula for tendon stressing. When a tendon path curves over an obstruction, the change in angle in tendon over the obstruction (such as a deviator) results in a compressive force along the contact between the tendon and the obstruction. This is the situation when a tendon path is not straight. When stressed, a tendon tends to straighten. In doing do, it pushes against the obstructions along its path. The extension of tendon at stressing end is resisted by the friction resulting from this compressive force. It is shown in most books on statics that drop in tendon force due to friction resulting from an angle change along the tendon is give by:
P(start) = P(end)*e(µ*θ)
Where θ is the accumulation of changes in angle along the length being considered.
Unbonded tendons, by design, are generally placed along a curved path, such as parabola. This is referred to as “design path,” or “intended path.” The accumulated change of angle between the two ends of tendon along this “design path” is known by design. It is denoted as α. Since the tendons are secured at selected points only along a design path, in practice the actual path of a flexible tendon will have small deviations from the design path. Also, other construction factors cause added departure of tendon path from its intended profile. The deviations from the design path are referred to as “wobble” of the tendon. The accumulation of angular change along the tendon length due to its wobble off the intended course is estimated and denoted as γ. Hence the accumulation of angular change becomes (α + γ). With wobble accounted for, the corrected friction loss relationship becomes:
P(start) = P(end)*e[µ*(α + γ)] Gamma is an estimated value and depends largely on construction practice and flexibility of tendon. It is best described as an estimated value per unit length of tendon. Hence,
P(start) = P(end)*e[µ*(α + (γ/L)*L] The following substitution is then made:
K = (µ*γ/L)
“K” is defined as coefficient of wobble friction. Since µ is dimensionless, γ is in radians, and L in units of length, the unit of K will be radians per unit length. With the above substitution, the friction formula reduces to:
P(jack) = P(x)*e(µ*α + K*L)
The preceding description is a stretched out effort in expressing a simple issue. The intent has been to avoid more involved formulations. The “cohesion” phenomenon was mentioned, in order to place it in the right perspective with respect to tendon friction loss, and to emphasize that it is not the phenomenon being generally considered in tendon stress loss.