RAM Concept ’s treatment of the effect of cross section strains on ultimate unbonded tendon stresses is loosely based on a paper by Naaman, Burns, French, Gable and Mattock [Naaman, A. E. et. al, "Stresses in Unbonded Prestressing Tendons at Ultimate: Recommendation", ACI Structural Journal, V. 99, No. 4, July-August 2002, pp. 518-529]. In the paper the authors, who are members of the Subcommittee of Stresses in Unbonded Tendons of Joint ASCE-ACI committee 423, Prestressed Concrete, recommend code modifications for ACI 318.

The paper provides an equation for estimating tendon stresses at ultimate bending strength of a cross section. The proposed equation is shown to have a correlation with test results that is 2.5 times better than the ACI equations 18-4 and 18-5. The equation is:

fps = fse + Ωu Ep εcu (dp/c – 1)(L1/L2) ≤ 0.80 fpu

where

fps	=	tendon stress at ultimate bending strength
fse	=	effective prestress in prestressed reinforcement
Ep	=	elastic modulus of prestressed reinforcement
εcu	=	failure strain of concrete (typically assumed as 0.003)
dp	=	distance from extreme compression fiber to centroid of prestressed reinforcement.
c	=	depth of neutral axis at ultimate strength
L	=	span under consideration
L1	=	sum of lengths of loaded spans
L2	=	total length of tendon between anchorages
Ωu	=	K(dp/L) where K = 3 for uniform or third point loadings and 1.5 for midspan loading
fpu	=	specified tensile strength of prestressed tendons

It can be shown that:

Δεp ≈ εcu (dp/c - 1)

where

Δεp

=

change in strain in concrete adjacent to the tendon from effective prestress level to ultimate bending

With this substitution (and the one for Ω_u ) the equation becomes:

f_ps = f_se + K(d_p/L) E_p Δε_p (L₁/L₂) ≤ 0.80 f_pu

L can both realistically and conservatively be assumed to equal L₁ as it is unlikely for two spans to simultaneously have large inelastic deformations. This simplifies the equation further to:

f_ps = f_se + E_p (Kd_p/ L₂)Δε_p≤ 0.80 f_pu

It is obvious that in the above equations that (Kd_p /L₂) is a strain reduction factor that accounts for the distribution of the localized strain over the length of the tendon. The numerator is a consideration of the length of the yielding (high strain) region, while the denominator is a consideration of the length over which this strain is distributed.