Learn how to determine design a concrete slab-on-grade according to ACI 360R-10.
Introduction
A slab-on-grade, also referred to as slab-on-ground, is a slab supported by the ground whose main purpose is to support the applied loads by bearing on the ground.
The American and British Standards method for design is to compare "allowable stresses" against "actual stresses", where actual stresses are based upon characteristic loads with an overall Factor of Safety (FoS). The designer chooses the FoS to minimise the likelihood of serviceability failure, such as cracking and decrease to surface durability. In contrast, the Eurocode and Australian Standards is based upon limit state design with partial factors of safety on materials and loads.
The design checks to ACI 360R-22 are based upon ensuring:
$$ \text{actual stress}\le\dfrac{\text{allowable stress}}{\text{FoS}} $$
There are multiple failure modes for a slab-on-ground:
- Flexural failure of the slab, when the slab develops tension stresses in its soffit that exceed its flexural capacity
- Bearing failure of the slab, when the slab bearing stresses exceed its bearing strength
- Punching failure of the slab, when the slab shear stresses exceed its shear strength
- Bearing stress of dowels that causes the slab to fail, where the effectiveness of the dowel bars depend on the relative stiffness between the slab compared to its subgrade
Design Checks
The required checks are:
- Minimum required slab thickness
- Required shrinkage and temperature reinforcement
- Estimated crack width, to ensure they are below limits
- Slab flexural stress
- Slab bearing stress
- Slab punching shear stress
- Bearing stress on dowels, to ensure the dowels can adequately transfer loads
\( \bf{1} \) Minimum Required Slab Thickness
The minimum required thickness, \( t_{min} \) of a slab-on-grade is determined by evaluating the allowable flexural stress, \( F_{b\space(\text{allow})} \) under various loading conditions. According to ASI 360 R-10 Appendix 4.2, the \( F_{b\space(\text{allow})} \) is expressed as:
$$ F_{b\space(\text{allow})} = \frac{M_R}{\text{FoS}} $$
where:
\( \rlap{M_R}\hspace{2.5em} \): Modulus of rupture
\( \rlap{M_R}\hspace{2.5em} = 9 \sqrt{f'_c} \)
Below are the equations for \( F_{b\space(\text{allow})} \) for specific load scenarios. The equations take into account the subgrade modulus, \( k \) and therefore takes into account the strength of the founding soil.
The following formulas are derived from Load Testing of Instrumented Pavement Sections by the University of Minnesota, Department of Civil Engineering, submitted to Mn/DOT in March 2002.
\( \bf{1.1} \) Single Interior Load
For a concentrated load applied at the interior of the slab, the allowable flexural stress is calculated as:
$$ F_{b\space(\text{allow})} = \frac{3P(1+m)}{2\pi t^2} \left(\ln\left(\frac{L_r}{b}\right) + 0.6159\right) $$
where:
\( \rlap{P}\hspace{2.5em} \): Applied load
\( \rlap{m}\hspace{2.5em} \): Poisson's ratio of concrete (0.15)
\( \rlap{t}\hspace{2.5em} \): Slab thickness
\( \rlap{L_r}\hspace{2.5em} \): Radius of stiffness
\( \rlap{L_r}\hspace{2.5em} = \left(\frac{E_c \, t^3}{12 \, (1 - m^2) \, k}\right)^{0.25} \)
\( \rlap{b}\hspace{2.5em} \): Equivalent radius
\( \rlap{b}\hspace{2.5em} =
\begin{cases}
\sqrt{1.6 \, a^2 + t^2} - 0.675 \, t & \text{if } \sqrt{\frac{A_c}{\pi}} < 1.724 \, t, \\
\sqrt{\frac{A_c}{\pi}} & \text{otherwise.}
\end{cases}
\)
\( \rlap{A_c}\hspace{2.5em} \): Contact area
Rearranging for \( t \) gives:
$$ t = \sqrt{\frac{3P(1+m)}{2\pi F_{b\space(\text{allow})} \left(\ln\left(\frac{L_r}{b}\right) + 0.6159\right)}} $$
\( \bf{1.2} \) Single Corner Load
For a concentrated load applied at the corner of the slab, the allowable flexural stress is given by:
$$ F_{b\space(\text{allow})} = \frac{3P}{t^2} \left(1 - \left(1.772 \frac{a}{L_r}\right)^{0.72}\right) $$
where:
\( \rlap{a}\hspace{2.5em} \): Effective load radius
\( \rlap{a}\hspace{2.5em} = \sqrt{\frac{A_c}{\pi}} \)
Rearranging for \( t \) gives:
$$ t = \sqrt{\frac{3P}{F_{b\space(\text{allow})} \left(1 - \left(1.772 \frac{a}{L_r}\right)^{0.72}\right)}} $$
\( \bf{1.3} \) Single Edge Load (Circular Area)
For a concentrated circular load area applied along the slab edge, the allowable flexural stress is:
$$
F_{b\space(\text{allow})} = \frac{3(1+m)P}{\pi(3+m)t^2} \left(\ln\left(\frac{E_c t^3}{100 k a^4}\right) + 1.84 - \frac{4m}{3} + \frac{(1-m)}{2} + 1.18(1+2m)\frac{a}{L_r}\right)
$$
where:
\( \rlap{E_c}\hspace{2.5em} \): Modulus of elasticity of concrete
\( \rlap{k}\hspace{2.5em} \): Subgrade modulus
Rearranging for \( t \) gives:
$$
t = \sqrt{\frac{3(1+m)P}{\pi(3+m)F_{b\space(\text{allow})}} \left(\ln\left(\frac{E_c t^3}{100 k a^4}\right) + 1.84 - \frac{4m}{3} + \frac{(1-m)}{2} + 1.18(1+2m)\frac{a}{L_r}\right)}
$$
\( \bf{1.4} \) Single Edge Load (Semi-Circular Area)
For a semi-circular concentrated load area applied along the slab edge, the allowable flexural stress is expressed as:
$$
F_{b\space(\text{allow})} = \frac{3(1+m)P}{\pi(3+m)t^2} \left(\ln\left(\frac{E_c t^3}{100 k a^4}\right) + 3.84 - \frac{4m}{3} + \frac{(1+2m)a}{2L_r}\right)
$$
Rearranging for \( t \) gives:
$$
t = \sqrt{\frac{3(1+m)P}{\pi(3+m)F_{b\space(\text{allow})}} \left(\ln\left(\frac{E_c t^3}{100 k a^4}\right) + 3.84 - \frac{4m}{3} + \frac{(1+2m)a}{2L_r}\right)}
$$
\( \bf{2} \) Required Shrinkage and Temperature Reinforcement
Shrinkage and temperature reinforcement is crucial in slab-on-grade design to control cracking due to temperature fluctuations and concrete shrinkage. The allowable stress, \( f_s \) in the reinforcement is:
$$ f_s = 0.75 \times f_y $$
The required area of reinforcement can be calculated with the formula:
$$ A_s = F\times L \times \frac{W}{2\times f_s} $$
where:
\( \rlap{F}\hspace{2.5em} \): Friction factor between subgrade and slab
\( \rlap{L}\hspace{2.5em} \): Joint spacing
\( \rlap{W}\hspace{2.5em} \): Slab weight
\( \bf{3} \) Estimated Crack Width
The formulas discussed in this section is derived from "Stresses and Strains in Rigid Pavements" (Lecture Notes 3) by Charles Nunoo, Ph.D., P.E.
Cracks in concrete occur due to thermal expansion and contraction, drying shrinkage, and other factors. The estimated crack width (\( \Delta L \)) can be calculated using the formula:
$$
\Delta L=C\times L\times 12\times(\alpha\Delta T+\varepsilon)
$$
where:
\( \rlap{C}\hspace{2.5em} \): Slab-base friction
\( \rlap{\alpha}\hspace{2.5em} \): Coefficient of thermal expansion of concrete
\( \rlap{\varepsilon}\hspace{2.5em} \): Coefficient of shrinkage
\( \bf{4} \) Slab Flexural Stress
The formulas discussed in this section is derived from "Load Testing of Instrumented Pavement Sections" by the University of Minnesota, Department of Civil Engineering, submitted to Mn/DOT in March 2002.
The flexural stress in a slab-on-grade is a parameter for assessing the slab's ability to resist bending under applied loads. The flexural stress is calculated using the following formulas:
$$
\begin{align}
f_{b1 \space\text{(actual)}} &= \frac{3P(1+\mu)}{2\pi t^2}\times \ln(\frac{L_r}{b} + 0.6159) \newline
f_{b2\space\text{(actual)}} &= f_{b1\space\text{(actual)}}\times (1+\frac{i}{100}) \newline
F_{b\space(\text{allow})} &= \frac{M_R}{\text{FoS}}
\end{align}
$$
where:
\( \rlap{f_{b1}}\hspace{2.5em} \): Flexural stress for 1 load
\( \rlap{f_{b2}}\hspace{2.5em} \): Flexural stress for 2 load
\( \rlap{\mu}\hspace{2.5em} \): Coefficient of friction between the slab and ground
\( \rlap{i}\hspace{2.5em} \): Increase for second load
The required check is therefore given by:
$$ f_{b \space\text{(actual)}} \leq F_{b\space\text{(allow)}} $$
\( \bf{5} \) Slab Bearing Stress
The formulas discussed in this section is derived from "Slab Thickness Design for Industrial Concrete Floors on Grade" (IS195.01D) by Robert G. Packard.
Slab bearing stress is the compressive stress distributed over the contact area between the applied load and the slab. The bearing stress is calculated using the following formulas:
$$
\begin{align}
&f_{p\space\text{(actual)}} = \frac{P}{A_c} \newline
&F_{p\space\text{(allow)}} = \frac{4.2M_R}{\text{FoS}}
\end{align}
$$
The required check is therefore given by:
$$ f_{p \space\text{(actual)}} \leq F_{p\space\text{(allow)}} $$
\( \bf{6} \) Slab Punching Shear Stress
The formulas discussed in this section is derived from "Slab Thickness Design for Industrial Concrete Floors on Grade" (IS195.01D) by Robert G. Packard.
Calculating the punching shear stresses ensure the slab can resist shear failure around the load area. The formulas to evaluate punching shear stress are:
$$
\begin{align}
&f_{v\space\text{(actual)}} = \frac{P}{t\times(b_o + 4t)} \newline
&F_{v\space\text{(allow)}}=\frac{0.27M_R}{\text{FoS}}
\end{align}
$$
where:
\( \rlap{ b_o }\hspace{2.5em} \): Load perimeter
The required check is therefore given by:
$$ f_{v \space\text{(actual)}} \leq F_{v\space\text{(allow)}} $$
\( \bf{7} \) Bearing Stress on Dowels
The formulas referenced in this section is derived from "Dowel Bar Optimization: Phases I and II - Final Report" by Max L. Porter (Iowa State University, 2001).
The bearing stress on dowels is a parameter to ensure that the dowels connecting structural elements can adequately transfer loads without failure.
$$
\begin{align}
&f_{d\space\text{(actual)}} = k_c \times \dfrac{P_c \times (2 + \beta z)}{4\beta^3 E_b I_b} \newline
&F_{d\space\text{(allow)}} = \frac{4-d_b}{3} \times f'_c
\end{align}
$$
\( \rlap{ k_c }\hspace{2.5em} \): Modulus of dowel support, assumed to be \(1.5\times 10^6 \) for steel dowels
\( \rlap{ P_c }\hspace{2.5em} \): Critical dowel load, taken as \( P_c = \frac{0.5P}{N_e} \)
\( \rlap{ N_e }\hspace{2.5em} \): Effective number of dowels
\( \rlap{ \beta }\hspace{2.5em} \): Relative bar stiffness, taken as \( \beta = \left( \frac{k_c d_b}{4 E_b I_b} \right)^{\frac{1}{4}} \)
\( \rlap{ z }\hspace{2.5em} \): Constant joint width
\( \rlap{ E_b }\hspace{2.5em} \): Modulus of elasticity, assumed to be \(29\times 10^6 \) for concrete
\( \rlap{ I_b }\hspace{2.5em} \): Inertia of dowel, taken as \(I_b = \frac{\pi d_b^4}{64} \)
\( \rlap{ d_b }\hspace{2.5em} \): Dowel diameter
The required check is therefore given by:
$$ f_{d \space\text{(actual)}} \leq F_{d\space\text{(allow)}} $$
Recommended Values for Calculations
Factor of Safety (FoS)
ACI recommends the FoS values as per the table below.
Concentrated Load
Wheel loadings are converted into a static concentrated load for the purpose of analysing a slab-on-ground. To replicate the load of a truck, ACI 360R-10 provides representative axle loads and wheel spacings for various lift truck capacities.
Modulus of Subgrade
The modulus of subgrade (k) is also known as the modulus of soil reaction of Winkler foundation. It is a spring constant that assumes a linear response between load and deformation from the subgrade, although in reality the relationship is non-linear. The load-deformation relationship depends on factors including the density, moisture content and prior loading of the soil and the width of the loaded area.
Nonetheless, tests performed on site called "plate load field tests" are used to estimate k for the purposes of slab-on-ground design.
Recommended modulus of subgrade values are provided below.
Dowel and Joint Parameters
Dowels are used to transfer shear force over construction joints in slabs. This calculator assumes use of plain bars. The following are recommended values for the dowel diameter, db and the joint spacing, L.
