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Structural Engineering, TMS 402 Reinforced CMU Masonry Walls & Buckling Slenderness

CMU Masonry Wall Prism Strength & Slenderness Formula

Calculate Masonry Wall Slenderness Ratio h/r, TMS 402 Piecewise Parabolic-to-Euler Buckling Reduction Factor, Nominal Axial Capacity Pn (kN), and LRFD Strength phi*Pn (kN).

Cross h/r = 99 on a tall warehouse CMU block wall and TMS 402 switches from parabolic crushing to inverse-square Euler buckling!

Reinforced concrete masonry unit (`CMU`) warehouse and gymnasium bearing walls are governed by **Specified Masonry Prism Compressive Strength `f’_m`** (MPa, tested per ASTM C1314 on mortared/grouted block prisms), Effective Net Cross-Sectional Area `A_n` (`mm²/m` of wall length), Unbraced Clear Height `h` (mm), and Section Radius of Gyration `r = SQRT(I_n / A_n)` (mm, `~58` mm for a fully grouted `190` mm nominal `8-inch` CMU wall). Under **TMS 402 / ACI 530 Strength Design**, **Slenderness Ratio `h / r`** triggers a sharp two-branch piecewise buckling reduction factor `R_s`: for **`h / r <= 99`**, parabolic inelastic crushing governs **`R_s = 1 - ((h / r) / 140)^2`**; for slender tall walls **`h / r > 99`**, elastic Euler column buckling governs **`R_s = (70 / (h / r))^2`**! Multiplying by `0.80` eccentricity-tie factor and `0.80 * f’_m * A_n` yields **Nominal Axial Strength `P_n = (0.80 * (0.80 * f’_m * A_n) * R_s) / 1,000`** (`kN`) and **LRFD Design Strength `phi * P_n`** (`phi = 0.90`)!

Verify on a `190` mm Fully Grouted CMU Bearing Wall (`1` m strip)—Prism_Strength `f’_m` = 13.8 MPa (`2,000` psi, B2), Grouted_Net_Area `A_n` = 190,000 mm²/m (B3), Unbraced_Wall_Height `h` = 4,200 mm (B4), and Radius_of_Gyration `r` = 58.0 mm (B5): `h / r` equals `72.41 <= 99` (E2), Parabolic Slenderness Factor equals `0.7325` (E3), Nominal `P_n` reaches `1,229.1` kN/m (E4), and LRFD `phi * P_n` equals `1,106.2` kN/m (E5)! Note on external AI review workflows: if your structural team tests scoped OAuth workbook inspection through the Arte Sheets ChatGPT plugin `0.1.0` (currently `In review` / `Not published`), remember that `read_range` (`<= 200` cells) inspects raw formulas and stored values on cloud collaboration workbooks without recalculating results—always verify both branches (`h / r <= 99` and `> 99`) directly in the live sheet.

TMS 402 Reinforced CMU Wall KPI (fm=13.8 MPa, An=190,000 mm², r=58 mm, h=4.2 vs 6.2 m)FormulaVerified Output
Wall Slenderness Ratio (h / r)Unbraced_Height_h_mm / Radius_Gyration_r_mm72.41 (106.90 > 99 Euler @ h=6,200 mm!)
TMS 402 Slenderness Factor RsIF(h/r<=99, 1-(h/(140*r))^2, (70/(h/r))^2)0.7325 (0.4288 @ h=6,200 mm!)
Nominal Axial Capacity Pn (kN/m)(0.80 * (0.80 * fm * An) * Rs) / 10001,229.1 kN/m (719.6 kN/m @ h=6,200 mm)
LRFD Design Capacity phi*Pn (kN/m)0.90 * Nominal_Pn_kN1,106.2 kN/m (647.6 kN/m = -41.5% drop!)

TMS 402 Piecewise Slenderness Transition (h/r = 99) & Axial LRFD Formula

Validate `Prism_Strength f’_m` in `[5, 45]` MPa, `Net_Area A_n > 10,000` mm², `Height h >= 1,000` mm, and `Radius_of_Gyration r > 15` mm!

=LET(fm_p, B2, an_m, B3, hw_m, B4, rg_m, B5, hr_r, hw_m / rg_m, pc_k, (0.8 * (0.8 * fm_p * an_m)) / 1000, IF(OR(fm_p < 5, fm_p > 45, an_m <= 10000, hw_m < 1000, rg_m <= 15), "Invalid Masonry Input", ROUND(0.9 * pc_k * IF(hr_r <= 99, 1 - ((hr_r / 140) ^ 2), (70 / hr_r) ^ 2), 1)))

High-Bay 6.2 m Big-Box Retail Wall Euler Buckling Transition Check (Set Wall Height B4 = 6,200 mm)

Raise clear roof-ledger height from `4,200` to `6,200` mm (`B4 = 6200`) without thickening the `190` mm block core: `h / r` E2 crosses `99` to `106.90`, switching E3 onto the inverse-square Euler curve (`R_s = 0.4288`) and cutting LRFD Design Strength E5 by `-41.5%` to `647.6` kN/m—signaling an upgrade to `240` or `290` mm (`10-` or `12-inch`) CMU pilasters!

=IF(ISTEXT(E5), E5, LET(hr_r, B4 / B5, ROUND(IF(hr_r <= 99, 1 - ((hr_r / 140) ^ 2), (70 / hr_r) ^ 2), 4)))

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