Beam deflection verification: Bischoff I_e and long-term deflection

This page walks through the complete ACI 318-19 deflection check for test case D1: a 12×20 in simply supported beam spanning 20 ft, with 3-#8 bars, w_D = 1.0 klf, w_L = 0.8 klf, 5-year duration. The full Bischoff I_e chain is traced from E_c through long-term deflection. Every intermediate value is pinned by a regression test. This is one of 30 cases in the beam verification report.

Case definition

Section	12 × 20 in, d = 17.625 in
Span	L = 20 ft, simply supported
Reinforcement	3-#8 (A_s = 2.37 in²)
Materials	f′c = 4,000 psi, Grade 60
Loads	w_D = 1.0 klf, w_L = 0.8 klf
Duration	5 years, no compression steel (ρ′ = 0), no sustained live load

Step 1: Material properties

Modulus of elasticity (ACI 318-19 §19.2.2) and modulus of rupture (§19.2.3):

E_c = 57\sqrt{f'_c} = 57\sqrt{4{,}000} = 3{,}605\ \text{ksi}

f_r = 7.5\,\frac{\sqrt{f'_c}}{1{,}000} = 7.5\,\frac{\sqrt{4{,}000}}{1{,}000} = 0.4743\ \text{ksi}

Modular ratio n = E_s / E_c = 29,000 / 3,605 = 8.044.

Step 2: Gross and cracked moment of inertia

Gross moment of inertia and cracking moment (ACI 318-19 §24.2.3.5):

I_g = \frac{bh^3}{12} = \frac{12 \times 20^3}{12} = 8{,}000\ \text{in}^4

M_{cr} = \frac{f_r \, I_g}{y_t} = \frac{0.4743 \times 8{,}000}{10.0} = 379.47\ \text{kip-in}

Cracked neutral axis depth from the transformed-section quadratic, with nA_s = 8.044 × 2.37 = 19.064 in²:

\frac{b \cdot (kd)^2}{2} = nA_s(d - kd) \quad \Rightarrow \quad kd = 6.062\ \text{in}

I_{cr} = \frac{b(kd)^3}{3} + nA_s(d - kd)^2 = \frac{12 \times 6.062^3}{3} + 19.064 \times 11.563^2 = 3{,}440.1\ \text{in}^4

Step 3: Bischoff effective I_e

ACI 318-19 Table 24.2.3.5 uses the Bischoff formulation instead of the older Branson equation. For moment M_a at the critical section:

I_e = \frac{I_{cr}}{1 - \left(\frac{(2/3)\,M_{cr}}{M_a}\right)^2 \left(1 - \frac{I_{cr}}{I_g}\right)}

Under dead load only: M_D = w_DL²/8 = 1.0 × 20²/8 = 50 k-ft = 600 kip-in. Since M_D = 600 > M_cr = 379.47, the section is cracked:

I_{e,D} = \frac{3{,}440.1}{1 - ((2/3) \times 379.47/600)^2 \times (1 - 3{,}440.1/8{,}000)} = 3{,}828.0\ \text{in}^4

Under total load: M_total = (w_D + w_L)L²/8 = 1.8 × 20²/8 = 90 k-ft = 1,080 kip-in:

I_{e,total} = \frac{3{,}440.1}{1 - ((2/3) \times 379.47/1{,}080)^2 \times (1 - 3{,}440.1/8{,}000)} = 3{,}551.2\ \text{in}^4

Step 4: Immediate and long-term deflection

Immediate deflection under dead load (simply supported, uniform load, w_D converted from k/ft to k/in):

\delta_D = \frac{5 w_D L^4}{384 E_c I_{e,D}} = \frac{5 \times (1.0/12) \times 240^4}{384 \times 3{,}605 \times 3{,}828.0} = 0.261\ \text{in}

Live load deflection δ_L = 0.225 in (computed from the difference between total-load and dead-load deflections using their respective I_e values).

Long-term multiplier (ACI 318-19 §24.2.4.1) for 5-year duration (ξ = 2.0) with no compression steel (ρ′ = 0):

\lambda_\Delta = \frac{\xi}{1 + 50\rho'} = \frac{2.0}{1 + 0} = 2.0

\delta_{LT} = \delta_D \cdot \lambda_\Delta + \delta_L = 0.261 \times 2.0 + 0.225 = 0.747\ \text{in}

Limit checks (Table 24.2.2)

ACI Table 24.2.2 specifies limits for different deflection components: immediate live-load deflection for damage-sensitive limits, and long-term total deflection for partition/attachment limits.

Limit	Component	Threshold	Actual	Status
L/180 (flat roofs, no damage)	δ_L	1.333 in	0.225 in	PASS
L/360 (floors, no damage)	δ_L	0.667 in	0.225 in	PASS
L/480 (floors, brittle partitions)	δ_LT	0.500 in	0.747 in	FAIL
L/240 (roofs/floors, nonstructural)	δ_LT	1.000 in	0.747 in	PASS

Overall: FAIL (L/480 exceeded). The 12×20 beam with these loads and 5-year creep satisfies immediate live-load limits but exceeds the long-term deflection limit for floors supporting brittle partitions.

Pinned values

These values are locked by regression tests. A drift beyond tolerance fails the build before it can deploy.

Quantity	Value	ACI reference
E_c	3,605 ksi	Cl 19.2.2
f_r	0.4743 ksi	Cl 19.2.3
I_g	8,000 in⁴
M_cr	379.47 kip-in	Cl 24.2.3.5
n	8.044
kd	6.062 in
I_cr	3,440.1 in⁴
I_e (dead)	3,828.0 in⁴	Table 24.2.3.5
I_e (total)	3,551.2 in⁴	Table 24.2.3.5
δ_D	0.261 in
δ_L	0.225 in
λ_Δ	2.0	Cl 24.2.4.1
δ_LT	0.747 in