Long-Term Deflection Multiplier (ACI 318-19 24.2.4)
Why bottom-only beams sag and doubly-reinforced ones hold: a concrete beam keeps deflecting for years from creep and shrinkage. ACI 24.2.4.1.1 multiplier lambda = xi / (1 + 50 rho'), where xi is 2.0 at 5+ years and rho' is the compression-steel ratio. A bottom-only beam (rho' = 0) with a 0.4 in immediate deflection gets the full lambda = 2.0 -> 0.80 in additional -> 1.20 in total, sagging triple its snapshot value; add compression bars so rho' = 0.01 and lambda drops to 1.33 -> 0.93 in total, 22% less. This long-term total is what the L/240 and L/480 limits are actually checked against. A design aid; the engineer of record governs.
Formula and source
xi = 2.0 (>=60 mo) / 1.4 (>=12) / 1.2 (>=6) / 1.0 (else); lambda = xi / (1 + 50 rho'); additional = lambda x immediate; total = immediate + additional.
The ACI 318-19 24.2.4.1.1 additional time-dependent (creep and shrinkage) deflection multiplier lambda = xi / (1 + 50 rho'), with the 24.2.4.1.3 time factor xi, by name.
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