Slab on grade
Point loads on a plain (unreinforced or shrinkage-reinforced) concrete slab on ground: rack posts on base plates or lift-truck wheels, with an optional uniform storage load beside an aisle. The design follows the approach of ACI 360R and the PCA method (Packard): the working flexural tensile stress from Westergaard's theory of a slab on a dense-liquid (Winkler) subgrade must not exceed the modulus of rupture divided by a factor of safety. Under a rack post the calculation also checks two-way shear and bearing of the plain concrete to ACI 318-19 Chapter 14.
This is a working-stress (allowable stress) design: the loads are unfactored and the factor of safety is yours, so the LRFD / ASD switch is hidden. Only the plain-concrete shear and bearing checks use a load factor, which you enter.
Inputs of the slab on grade #
- Concentrated load: a rack post on a base plate (plate x along the joint, plate y) or a wheel load (tire contact pressure q, single or dual tires with their centre spacing). P is the working load of one post or one wheel. The contact area is the plate area or P / q; it is treated as a circle of equal area (radius a). Dual tires are replaced by one circle (Huang, Pavement Analysis and Design, 2nd Ed., 4.1.3) carrying both loads.
- Other loads nearby: the back-to-back post of a double rack, the next posts along the rack, the other wheel of the axle: their positions (x, y from the design load) and loads. They are superposed at the interior position (see Checks).
- Load positions: interior (away from joints), edge (at a joint or free edge), corner (joint intersection). Check every position where the load can stand.
- Joint load transfer: the reduction of the edge and corner stresses for load transfer across the joint. 0 % is a free edge or an open joint; up to about 25 % is used for doweled joints or tight sawcut joints with aggregate interlock (ACI 360R, UFC 3-320-06). It is never assumed.
- Edge and corner formulas: Westergaard's 1948 edge formula for a circular load with the Ioannides et al. (1985) corner formula (the default: both agree with finite-element results), or Westergaard's 1926 edge and corner formulas, which ACI 360R and the PCA papers quote. The interior formula is Westergaard's 1926 formula in both.
- Joint spacing x, y: the panel size, for the drawing and for the check that the panel is large enough for the theory.
- Slab and subgrade: thickness h, f'c, the modulus of rupture (9√f'c, the PCA value used with its charts; 7.5√f'c, ACI 318-19 19.2.3.1; or a tested flexural strength), Ec (57,000√f'c, ACI 318-19 19.2.2.1, or entered: the PCA and Corps of Engineers charts use 4,000 ksi), Poisson's ratio (0.15 usual; the Corps uses 0.20), the modulus of subgrade reaction k (from plate load tests or the geotechnical report; usual values by soil type in UFC 3-320-06 Table 2-2) and the factor of safety on the modulus of rupture (lift-truck wheels 1.7 to 2.0, rack posts 1.7 to 2.0 in ACI 360R and 2.0 to 5.0 in UFC 3-320-06 Table 3-7; more for unlimited repetitions or impact).
- Shear and bearing (rack posts): on or off; the load factor on P (1.6 for storage live load, ASCE 7-22 2.3.1); whether h is reduced by 2 in for shear because the slab is cast against soil (ACI 318-19 14.5.1.7).
- Uniform storage load: the load w on both sides of an aisle, the aisle layout (variable: the critical aisle width governs; fixed: the aisle width entered, valid only while the layout does not change) and its factor of safety (2.0 usual, 1.4 the lower limit in UFC 3-320-06 Table 3-7).
Auto-design (editor toolbar) varies the slab thickness (1/2-in. steps); the loads, the base plate, the subgrade, the concrete and the factors of safety are kept. Objective: the thinnest slab. See Auto-design.
Checks performed #
| Check | Basis |
|---|---|
| Radius of relative stiffness | ℓ = [Ec h³ / (12 (1 − ν²) k)]^¼ (Westergaard 1926, Eq. 1) |
| Equivalent radius | b = √(1.6a² + h²) − 0.675h when a < 1.724h, else a (Westergaard 1926, Eq. 8) |
| Interior stress | σi = 3(1 + ν)P / (2πh²) [ln(ℓ/b) + 0.6159] (Westergaard 1926; ACI 360R form 0.316 P/h² [4 log(ℓ/b) + 1.069]) |
| Loads nearby (interior) | Moments of each nearby load from the infinite-plate solution: Mr = P/(2π)[kei″ + ν kei′/x], Mt = P/(2π)[kei′/x + ν kei″], x = r/ℓ (Hertz, Westergaard 1926 pp. 104-106); added in x, y and shear, and the largest principal stress is checked |
| Edge stress | Default: Westergaard 1948 circular load, σe = 3(1 + ν)P / (π(3 + ν)h²) [ln(Eh³/(100ka⁴)) + 1.84 − 4ν/3 + (1 − ν)/2 + 1.18(1 + 2ν)a/ℓ] (Ioannides et al. 1985); or 1926: 0.572 P/h² [4 log(ℓ/b) + 0.359]; times (1 − load transfer) |
| Corner stress | Default: σc = 3P/h² [1 − (c/ℓ)^0.72], c = √(contact area) (Ioannides et al. 1985); or 1926: 3P/h² [1 − (a√2/ℓ)^0.6]; times (1 − load transfer) |
| Allowable flexural stress | MR / FS (ACI 360R, PCA) |
| Uniform load beside an aisle | Beam on elastic foundation (Hetényi; Rice 1957, PCA): λ = [k/(4EcI)]^¼, I = h³/12, aisle-centre moment M = w/(2λ²) e^(−λc) sin(λc) (UFC 3-320-06 Eq. 5-5); variable layout at the critical aisle λc = π/4 (2.209ℓ), which is UFC Eq. 5-4, w = 257.9 (MR/FS) √(kh/Ec) psf |
| Two-way shear under the base plate | ACI 318-19 Table 14.5.5.1(b), (c): Vn = min(4/3 + 8/(3β), 2.66) λ√f'c bo h, perimeter at h/2 (14.4.3.4.1) for the smallest perimeter of the positions checked (interior: four sides; edge: three; corner: two), h − 2 in when chosen (14.5.1.7), φ = 0.60 (Table 21.2.1(i)) |
| Bearing under the base plate | ACI 318-19 Table 14.5.6.1: 0.85 f'c A1 × min(√(A2/A1), 2), A2 of the 1:2 frustum within h, cut by the joints at an edge or corner, φ = 0.60 |
| Panel size (advisory) | Smaller joint spacing at least 3.5ℓ (interior), 5ℓ (edge), 4ℓ (corner) for Westergaard's solutions to hold (Ioannides et al. 1985) |
| Nearby loads at an edge or corner (advisory) | Shown when a nearby load is within 3ℓ and an edge or corner position is checked: they are not superposed there |
Limitations #
- Westergaard's theory: an elastic slab of uniform thickness in full contact with a dense-liquid subgrade, large compared with ℓ. Curling and warping stresses, loss of support at joints and settlement are not included.
- Nearby loads are superposed at the interior only. No free published method gives the influence of a nearby load on an edge or corner stress; the edge and corner stresses are those of the design load alone (an advisory says so). ACI 360R and the PCA charts treat the back-to-back post at the interior in the same way.
- The PCA design charts (Packard 1976, ACI 360R Appendix A) are graphs and are not reproduced; the closed-form Westergaard solutions they rest on are. Fatigue is covered by the factor of safety you choose, not by Miner's rule. The PCA "variable layout" table for uniform loads (UFC Table 5-4) is not implemented: the critical aisle width (UFC Eq. 5-4, PCA Table 5-5) governs the variable layout here.
- Reinforced, post-tensioned, shrinkage-compensating and fiber-reinforced slabs, wall and line loads, and the subgrade reaction inside the shear perimeter (neglected, conservative) are not covered.
- The shear and bearing checks are made only for rack posts (tire pressures do not govern).
Verification examples #
Verification status: 50 values compared in 28 examples (26 published, 2 hand calculations from the cited equations): 48 within tolerance, 2 recorded deviations. Code basis: partial. Only some provisions were read in the target edition; the rest rests on older editions, examples or summaries.
Published examples and tables (backend/tests/test_verify_slab_on_grade.py), all from free documents: Westergaard's 1926 paper (HRB Proceedings 5, Tables II to IV and the four-wheel example), Huang's Examples 4.1 to 4.4 as reproduced in a free university lecture, the Pavement Interactive calculation example, the finite-element results of Ioannides, Thompson and Barenberg (1984, AFOSR-TR-85-0083) and UFC 3-320-06 (2025) with the PCA tables it reprints.
| Example | What it checks | Published | ILX Calcs |
|---|---|---|---|
| Westergaard 1926, h = 6 in, k = 100 pci, a = 2 / 4 / 6 in | Corner (Table II) | 619 / 509 / 420 psi | 619 / 509 / 420 |
| same | Interior (Table III) | 461 / 395 / 335 psi | 461 / 395 / 335 |
| same | Edge (Table IV) | 721 / 601 / 493 psi | 721 / 601 / 493 |
| Westergaard 1926, h = 10 in, a = 2 / 6 in | Corner; edge | 239 / 182; 261 / 212 psi | 239 / 182; 261 / 212 |
| Westergaard 1926, four 10-kip wheels at 66 in (h = 7 in, k = 50 pci) | One wheel; two wheels; four wheels | 279; 301; 282 psi | 279; 301; 282 |
| Huang 4.1, 4.2 (h = 10 in, k = 100 pci, a = 6 in) | ℓ; b; interior; corner (1926) | 42.97 in; 5.804 in; 143.7; 186.6 psi | 42.97; 5.80; 143.7; 186.6 |
| Huang 4.3 | Edge, Westergaard 1948 circle | 279.4 psi | 279.3 |
| Huang 4.4, duals 14 in apart | a; corner; interior; edge | 7.85 in; 166.8; 130.8; 244.2 psi | 7.85; 166.8; 130.8; 244.1 |
| Pavement Interactive, h = 9 in, k = 200 pci, 9,000 lb, a = 5.9 in | ℓ; interior; edge; corner (1926) | 35.3 in; 150; 226; 193 psi | 35.3; 150; 226; 193 |
| Ioannides 1984, finite elements, corner, runs C001, C002, C004, C005, C007 | Corner stress | 307.5, 270.9, 282.1, 259.9, 159.9 psi | 310.0, 271.0, 282.6, 257.8, 160.5 |
| Ioannides 1984, runs C003, C006 (deviation: formula fit) | Corner stress | 242.7, 563.6 psi | 237.3, 554.9 |
| UFC 3-320-06 Table 5-3 (k = 100 pci, Ec = 4,000 ksi, FS = 2) | Allowable uniform load, h = 7 / 6 / 20 in | 1109 / 868 / 2018 psf | 1109 / 869 / 2018 |
| UFC 3-320-06 Example B-1.1 (k = 150 pci) | Allowable uniform load | 1358 psf | 1358 |
| PCA Table 5-5 (in UFC 3-320-06) | Critical aisle width; allowable load (three rows) | 5.4 / 5.6 / 5.6 ft; 950 / 610 / 2065 psf | 5.39 / 5.59 / 5.63; 948 / 612 / 2063 |
The Ioannides corner formula is a fit to finite-element results; two of the seven runs are 1.5 and 2.2 % below them (deviation SOG-IOANNIDES-FIT on the Verification page).
Hand calculations in closed form:
| Example | What it checks | Hand | ILX Calcs |
|---|---|---|---|
| SOG-1 (the defaults) | Two-way shear φVn, edge perimeter with h − 2 in | 18.17 kips | 18.17 |
| SOG-1 | Bearing φBn = 0.6 × 0.85 f'c A1 × 2 | 146.9 kips | 146.9 |
| SOG-2 | Uniform load beside a fixed 4 ft aisle | 1247 psf | 1247 |
The behaviour tests also check the Kelvin-function moments against Westergaard's printed coefficients (Mr/P = −0.0211 and Mt/P = 0.0181 at 1.813ℓ; −0.0186 and 0.0058 at 2.564ℓ). ACI 360R-10 and its worked examples (Appendix A), the PCA IS195 paper and Ringo and Anderson's book are not on file; their charts are graphical and are not used.