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Beam deflection per ACI 318-19: Bischoff Ie, minimum thickness, and long-term multiplier

Technical reference for structural engineers

Why deflection matters

ACI 318-19 checks beam deflections at service load level to prevent visible sagging, cracking of attached finishes, and functional impairment. The code gives two paths: satisfy the minimum-thickness table (Table 9.3.1.1) and skip the calculation, or compute deflections explicitly using the Bischoff effective moment of inertia (Table 24.2.3.5).

ACI 318-19 clause references

ACI 318-19 reference Topic
Cl 24.2.3Deflection calculation requirements
Table 24.2.3.5Bischoff Ie equation
Table 9.3.1.1Minimum beam thickness
Cl 24.2.4.1Long-term deflection multiplier
Table 24.2.2Deflection limits
Table 19.2.2.1bEc for normal-weight concrete
Cl 19.2.3.1Modulus of rupture fr

Minimum thickness (Table 9.3.1.1)

If the member depth h meets or exceeds minimum thickness, the explicit deflection calculation may be skipped:

Support condition hmin
Simply supportedl / 16
One end continuousl / 18.5
Both ends continuousl / 21
Cantileverl / 8

For steel grades other than Grade 60, multiply by:

$$h_{\min} = \frac{l}{\text{divisor}} \times \left(0.4 + \frac{f_y}{100{,}000}\right) \quad (f_y \text{ in psi})$$

Bischoff effective moment of inertia (Table 24.2.3.5)

When h < hmin or when an explicit check is desired, ACI 318-19 uses the Bischoff equation for the effective moment of inertia:

$$I_e = \frac{I_{cr}}{1 - \left(\dfrac{\frac{2}{3}\,M_{cr}}{M_a}\right)^{\!2} \left(1 - \dfrac{I_{cr}}{I_g}\right)} \;\leq\; I_g \quad \text{when } M_a > \tfrac{2}{3}\,M_{cr}$$

When Ma ≤ (2/3) Mcr, the section is uncracked and Ie = Ig.

The cracking moment is:

$$M_{cr} = \frac{f_r \cdot I_g}{y_t}$$

where the modulus of rupture is:

$$f_r = 7.5\,\lambda\,\sqrt{f'_c} \quad (\text{psi})$$

and Ec for normal-weight concrete (Table 19.2.2.1b):

$$E_c = 57{,}000\,\sqrt{f'_c} \quad (\text{psi})$$

Branson vs Bischoff: why it changed

ACI 318-14 used the Branson equation: Ie = (Mcr/Ma)3 Ig + [1 − (Mcr/Ma)3] Icr. This underestimated deflections for lightly reinforced and high-strength concrete members because the cubic interpolation overweighted Ig near the cracking threshold. The Bischoff formulation, based on integration of curvature along the member length, better predicts experimental behavior across all reinforcement ratios. The (2/3) Mcr threshold in ACI 318-19 accounts for the variability in actual cracking moment.

Immediate deflection

For uniformly loaded beams, immediate deflection is:

$$\Delta = C \cdot \frac{w \cdot l^4}{E_c \cdot I_e}$$

where C is the deflection coefficient for the support condition (5/384 for simple span, 1/384 for both ends continuous, etc.). Ie is computed separately for the dead-load moment and the total-load moment, giving ΔD and ΔD+L. The live-load deflection is the difference: ΔL = ΔD+L − ΔD.

Long-term deflection (Cl 24.2.4.1)

Sustained loads cause creep and shrinkage that increase deflection over time. The long-term multiplier is:

$$\lambda_\Delta = \frac{\xi}{1 + 50\,\rho'}$$

where ξ depends on load duration:

Duration ξ
3 months1.0
6 months1.2
12 months1.4
5 years or more2.0

Total long-term deflection:

$$\Delta_{LT} = \lambda_\Delta \cdot \Delta_{\text{sustained}} + \Delta_L$$

Deflection limits (Table 24.2.2)

Condition Limit
Immediate (flat roofs, no attachments)l / 180
Immediate (floors)l / 360
Long-term (damage likely)l / 480
Long-term (damage not likely)l / 240

What calcnote computes

  • Checks h vs hmin (Table 9.3.1.1) with fy adjustment
  • Computes Ig, Mcr, Icr for rectangular and T-beam sections
  • Applies Bischoff Ie for dead-load and total-load moments separately
  • Computes immediate and long-term deflections with the λΔ multiplier
  • Checks against all four Table 24.2.2 limits

Sources

  • ACI 318-19: Cl 24.2.3, Table 24.2.3.5, Table 9.3.1.1, Cl 24.2.4.1, Table 24.2.2.
  • Bischoff, P.H., "Reevaluation of Deflection Prediction for Concrete Beams Reinforced with Steel and FRP Bars," ASCE JSE, 2005.
  • Wight, Reinforced Concrete: Mechanics and Design, 7th ed.

Run a deflection check with your own beam geometry and loads: start a beam calculation or see a demo result.

What is the Bischoff equation for effective moment of inertia?

When the service moment Ma exceeds (2/3) Mcr, the effective moment of inertia is Ie = Icr / (1 − ((2/3) Mcr / Ma)2 (1 − Icr / Ig)), capped at Ig. When Ma is at or below (2/3) Mcr, the section is uncracked and Ie equals Ig. The (2/3) factor on Mcr is the key feature introduced in ACI 318-19 Table 24.2.3.5.

Why did ACI 318-19 replace the Branson equation with Bischoff?

The Branson equation underestimated deflections for lightly reinforced and high-strength concrete members because the cubic interpolation overweighted Ig near the cracking threshold. Bischoff's formulation, based on integration of curvature along the member length, better matches experimental results across the full range of reinforcement ratios and concrete strengths.

When can I skip the deflection calculation?

When the member depth h meets or exceeds the minimum thickness from Table 9.3.1.1 (e.g., l/16 for a simply supported beam with Grade 60 steel). If h is less than hmin, an explicit deflection check using the Bischoff Ie method is required.

How does ACI 318-19 account for long-term deflection?

The sustained-load deflection is multiplied by λΔ = ξ / (1 + 50 ρ'), where ξ depends on the load duration (1.0 at 3 months up to 2.0 at 5 years or more) and ρ' is the compression reinforcement ratio. The total long-term deflection adds this amplified sustained deflection to the immediate live-load deflection.