Prestress losses by creep and shrinkage (working-stress viewpoint): If C is the creep coefficient of concrete, f is the original concrete stress at tendon level, m is modular ratio, E is the Young’s modulus of steel, and e is the shrinkage strain of concrete, what is the combined loss/effect in steel stress due to creep and shrinkage?

Introduction / Context:
In prestressed concrete, time-dependent effects like creep and shrinkage of concrete reduce the effective prestress in steel. Analytical formulae express the change in steel stress in terms of concrete parameters and modular ratio. Understanding the signs and relative contributions of creep and shrinkage is fundamental in estimating long-term losses.

Given Data / Assumptions:

• C = creep coefficient of concrete referred to the initial concrete stress at tendon level.
• f = initial concrete stress (compression) at tendon level due to prestress and loading.
• m = modular ratio = Es / Ec.
• E = Young’s modulus of steel; e = free shrinkage strain of concrete (shortening).
• Working-stress treatment, linear elastic superposition, no secondary effects considered.

Concept / Approach:
Creep under sustained compression increases concrete strain beyond the instantaneous elastic value, causing steel to relax (loss of steel stress). Shrinkage is a free uniform contraction of concrete; compatibility through bond imposes shortening on steel, again reducing steel stress. Sign convention: additional concrete shortening implies a reduction in steel stress (negative contribution for shrinkage term).

Step-by-Step Solution:
Instantaneous steel stress linked to concrete stress: m f (compatibility/transformations).Creep addition to concrete strain proportional to (C - 1), hence steel stress change term involves (C - 1) m f.Shrinkage causes a direct steel stress change of magnitude E * e but as a loss (opposes initial prestress), so with negative sign.Combined expression: Δσs = (C - 1) m f - e E.

Verification / Alternative check:
Both mechanisms reduce effective steel stress: creep term depends on existing concrete stress (through m f), and shrinkage term depends on the free strain e multiplied by Es. The signs therefore align with a net loss in steel stress.

Why Other Options Are Wrong:

• (1 - C) m f ± e E: Reverses creep sign; would imply gain if C > 1.
• (C - 1) m f + e E: Assigns a positive sign to shrinkage; contradicts loss nature.

Common Pitfalls:
Mixing up whether the expression represents “loss” (drop in steel stress) or “remaining stress”; always check sign conventions. Also, ensure e refers to concrete shrinkage strain (shortening), not thermal expansion.

Final Answer:
(C - 1) m f - e E