Concrete Slab on Grade Analysis Calculator (for Post or Wheel Loading)

For Slab Subjected to Interior Concentrated Post or Wheel Loading
Assuming Slab is Reinforced for Shrinkage and Temperature Only

Input Data

 Slab Thickness, t = in. Concrete Strength, f 'c = 3000350040004500500055006000 psi Conc. Unit Weight, wc = pcf Reinforcing Yield, fy = 40000500006000065000700007500080000 psi Subgrade Modulus, k = pci Concentrated Load, P = lbs. Contact Area, Ac = in.^2 Factor of Safety, FS = pci Dowel Bar Dia., db = 0.7511.25 in. Dowel Bar Spacing, s = in. Const. Joint Width, z = in. Joint Spacing, L = ft. Temperature Range, DT = deg. Increase for 2nd Wheel, i = % Result :

Check Slab Flexural Stress:
Effective Load Radius, a = in. a = SQRT(Ac/p)
Modulus of Elasticity, Ec = psi Ec = 33*wc^1.5*SQRT(f 'c)
Modulus of Rupture, MR = psi MR = 9*SQRT(f 'c)
Cracking Moment, Mr = ft-k/ft. Mr = MR*(12*t^2/6)/12000 (per 1' = 12" width)
Poisson's Ratio, m =   m = 0.15 (assumed for concrete)
Radius of Stiffness, Lr = in. Lr = (Ec*t^3/(12*(1-m^2)*k))^0.25
Equivalent Radius, b = in.
1 Load: fb1(actual) = psi fb1(actual) = 3*P*(1+m)/(2*p*t^2)*(LN(Lr/b)+0.6159) (Ref. 1)
2 Loads: fb2(actual) = psi fb2(actual) =
Fb(allow) = psi Fb(allow) = MR/FS

Check Slab Bearing Stress:
fp(actual) = psi fp(actual) = P/Ac
Fp(allow) = psi Fp(allow) = 4.2*MR

Check Slab Punching Shear Stress:
bo = in. bo = 4*SQRT(Ac) (assumed shear perimeter)
fv(actual) = psi fv(actual) = P/(t*(bo+4*t))
Fv(allow) = psi Fv(allow) = 0.27*MR

Shrinkage and Temperature Reinf.:(assuming subgrade drag method)
Friction Factor, F =   F = 1.5 (assumed friction factor between subgrade and slab)
Slab Weight, W = psf W = wc*(t/12)
Reinf. Allow. Stress, fs = psf fs = 0.75*fy
As = in.^2/ft. As = F*L*W/(2*fs)
Determine Estimated Crack Width:(assuming no use of stabilized or granular subbase)
Slab-base Frict. Adjust., C = C = 1.0 (assumed value for no subbase)
Thermal Expansion, a = in./in./deg a = 5.5x10^(-6) (assumed thermal expansion coefficient)
Shrinkage Coefficient, e = in./in. e = 3.5x10^(-4) (assumed coefficient of shrinkage)
Est. Crack Width, DL = in. DL = C*L*12*(a*DT+e)

Check Bearing Stress on Dowels at Construction Joints with Load Transfer: Min. req'd. slab thk. for single interior load: Min. req'd. slab thk. for single corner load: Min. req'd. slab thk. for single edge load (circular area): Min. req'd. slab thk. for single edge load (semi-circular area): Le = in. Le = 1.0*Lr = applicable dist. each side of critical dowel Effective Dowels, Ne = bars Ne = 1.0+2*S(1-d(n-1)*s/Le) (where: n = dowel #) Joint Load, Pt = lbs. Pt = 0.50*P (assumed load transferred across joint) Critical Dowel Load, Pc = lbs. Pc = Pt/Ne Mod. of Dowel Suppt., kc = psi kc = 1.5x10^6 (assumed for concrete) Mod. of Elasticity, Eb = psi Eb = 29x10^6 (assumed for steel dowels) Inertia/Dowel Bar, Ib = in.^4 Ib = p*db^4/64 Relative Bar Stiffness, b = b = (kc*db/(4*Eb*Ib))^(1/4) fd(actual) = psi fd(actual) = kc*(Pc*(2+b*z)/(4*b^3*Eb*Ib)) Fd(allow) = psi Fd(allow) = (4-db)/3*f 'c t(min) = in. Set Fb(allow) = 3*P*(1+m)/(2*p*t^2)*(LN(Lr/b)+0.6159) (Ref. 1) t(min) = in. Set Fb(allow) = 3*P/t^2*(1-(1.772*a/Lr)^(0.72)) (Ref. 1) t(min) = in. SSet Fb(allow) = 3*(1+m)*P/(p*(3+m)*t^2)*(LN(Ec*t^3/(100*k*a^4))+1.84-4*m/3+(1-m)/2+1.18*(1+2*m)*a/Lr) (Ref. 1) t(min) = in. Set Fb(allow) = 3*(1+m)*P/(p*(3+m)*t^2)*(LN(Ec*t^3/(100*k*a^4))+3.84-4*m/3+(1+2*m)*a/(2*Lr)) (Ref. 1)

References:
1. "Load Testing of Instumented Pavement Sections - Improved Techniques for Appling the Finite Element
Method to Strain Predition in PCC Pavement Structures" - by University of Minnesota, Department of Civil
Engineering (submitted to MN/DOT, March 24, 2002)
2. "Dowel Bar Opimization: Phases I and II - Final Report" - by Max L. Porter (Iowa State University, 2001)
3. "Design of Slabs-on-Ground" - ACI 360R-06 - by American Concrete Institute (2006)
4. "Slab Thickness Design for Industrial Concrete Floors on Grade" (IS195.01D) - by Robert G. Packard
(Portland Cement Association, 1976)
5. "Streses and Stains in Rigid Pavements" (Lecture Notes 3) - by Charles Nunoo, Ph.D., P.E.
(Florida International University, Miami FL - Fall 2002)

Note : there will be a few situations where certain combinations of the Concentrated
Load, P, Subgrade Modulus, k, and Contact Area, Ac, result in a #N/A error message
and thus no solution for the minimum slab thickness, t(min), for one or more of the
equations listed above. For those cases, the user would then manually iterate the
input slab thickness to determine the minimum value if desired.

Subgrade Soil Types and Approximate Subgrade Modulus (k) Values

Type of Soil Support Provided k Values Range (pci)

Fine-grained soils in which
silt and clay-size particles
predominate
Low50 - 120

Sands and sand-gravel
mixtures with moderate
amounts of silt and clay
Medium 130 - 170

Sands and sand-gravel
mixtures relatively free
of plastic fines
High 180 - 220

Cement-treated subbases Very high 250 - 400
Representative Axle Loads and Wheel Spacings for Various Lift Truck Capacities
Truck Rated Capacity (lbs.) Total Axle Load (lbs.) Wheel Spacing (in.)
2,000 5,600-7,200 24-32
3,000 7,800-9,400 26-34
4,000 9,800-11,600 30-36
5,000 11,600-13,800 30-36
6,000 13,600-15,500 30-36
7,000 15,300-18,100 34-37
8,000 16,700-20,400 34-38
10,000 20,200-23,800 37-45
12,000 23,800-27,500 38-40
15,000 30,000-35,300 34-43
20,000 39,700-43,700 36-53

Note: Axle loads are given for trucks handling the rated loads at 24 in.
from load center to face of fork with mast vertical.
Data for Construction Joint Dowels for Load Transfer
Slab Depth Dowel Dia., db Total Dowel Length Dowel Spacing (c/c), s
5" - 6" 3/4" 16" 12"
7" - 8" 1" 18" 12"
9" - 11" 1-1/4" 18" 12"
Slab Thickness           Joint Spacing (ft.)
< 3/4" Aggregate> 3/4" Aggregate Slump < 4"
5" 10 13 15
6" 12 15 18
7" 14 18 21
8" 16 20 24
9" 18 23 27
10" 20 25 30
Radius of Stiffness, "Lr", is a measure of the stiffness of the slab relative to the foundation (subgrade).
It is a linear dimension and represents mathematically the 4th root of
the ratio of the stiffness of the slab to the stiffness of the foundation.
Subbase friction adjustment factor, 'C', is as follows: C = 0.65 for stabilized subbase C = 0.80 for granular subbase C = 1.00 for no subbase
Values of Portlant Cement Concrete Coefficient of Shrinkage (e)
Concrete Strength,           Modulus of Rupture,           Srinkage Coefficient,
f 'c (psi)MR (psi)e (in./in.)
3000 4930.00046
3500532 0.00040
4000569 0.00035
4500 604 0.00030
5000 636 0.00026
5500667 0.00023
6000 697 0.00020
Note: Indirect tensile strength = Modulus of Rupture (MR) = 9*SQRT(f 'c)
Recommendations for input of Increase for 2nd Wheel (or Post), 'i':
1. For 6" to 10" thick slabs on grade with 'k' values between 100 pci and 200
pci, the increase in stress, 'i', due to a 2nd wheel (or post) load as a
percentage of stress for a single wheel (or post) load is approximately
15% to 20% for a wheel (or post) spacing of 3' to 4'.

2. For wheel (or post) spacings of 5' to 15', the increase in stress is
approximately 0% to 5%.

3. For a single post load, input a value of i = 0%.

4. For situations outside of the above criteria and/or for a more in depth
analysis and evaluation of the effects of a 2nd wheel (or post) load, please
refer to the "BOEF.xls" (Beam On Elastic Foundation) spreadsheet workbook.
Note: direct solution for t(min) has been rounded up to nearest 1/4" thickness.
Note: direct solution for t(min) has been rounded up to nearest 1/4" thickness.
Note: direct solution for t(min) has been rounded up to nearest 1/4" thickness.
Note: direct solution for t(min) has been rounded up to nearest 1/4" thickness.

Disclaimer: This calculator is not intended to be used for the design of actual structures, but only for schematic (preliminary) understanding of structural design principals. For the design of an actual structure, a competent professional should be consulted.

‘Calculations courtesy of Alex Tomanovich, PE ’