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Floor Slab Design

Introduction
PlastiSpan® insulation has provided designers and building owners with long-term thermal performance for over 45 years as a component in residential, commercial and industrial floor systems. Structural slab design is governed by the types and magnitude of loads on it which generally include wheel loads from forklifts or delivery vehicles, point loads from the legs of storage racks or distributed loads from product stored on the floor.

Review
Selection of a sub-slab insulation product is often based upon its ability to respond to compressive loads transferred from the slab, without full and accurate determination of the load distribution characteristics of the slab.  Table 1 below provides sample calculations based upon this method for structural concrete slab design using two of the load types noted above.

Table 1 - Examples of Typical Loads on Concrete Slabs

Example 1 - Forklift Wheel Load   Example 2 - Point Load (Storage Racks)
Wheel Load - F kN 35   Point Load - F kN 50  
lbf 7,875 lbf 11,250
Wheel Contact Area m 0.203 x 0.203 Base Plate Contact Area m 0.152 x 0.152
in 8.0 8.0 in 6.0 6.0
Stress Distribution Angle 45   Stress Distribution Angle 45  
Slab Thickness m 0.152 Slab Thickness m 0.152  
in 6.0 in 6.0
Loaded Area m2 0.26 Loaded Area m2 0.21  
in2 398 in2 322
EPS Compressive Stress kPa 136 EPS Compressive Stress kPa 240  
psi 20 psi 35


The calculations above are based upon the assumption that loads distributed over a contact area on a concrete floor slab area can be "assumed" to be distributed by and through the slab to a largely hypothetical insulation bearing area. The insulation load exposure calculated on this basis would dictate use of a high density, high strength insulation material increasing cost unnecessarily.

Design Considerations
An accepted design procedure to use for structural slab design with these types of loads is the theory of plates on elastic foundations. When a concrete slab is constructed over a compressible or elastic subgrade such as soil or rigid insulation, load distribution and transfer to the sub-slab insulation is controlled by the slab itself and its response to loads. Floor loads will cause slab deflection as a function of both the concrete slab properties and the compressibility of the materials beneath it.

In order to use this method, designers use the insulation or subgrade response factor referred to as the modulus of subgrade reaction (k) or, in other cases, foundation modulus, k-modulus, k-value, etc. The use of k-values in the design of structural slabs as discussed in PCA Concrete Information1 reflects the response of the insulation and subgrade under temporary (elastic) conditions when small deflections occur. Modulus of subgrade reaction values expressed in units of NM/m3 or lbs/in3 (pci) for various PlastiSpan insulation types and product thickness are listed below.

Table 2 - PlastiSpan Insulation Modulus of Subgrade Reaction (k)

PlastiSpan Insulation Types Units PlastiSpan Insulation Thickness - mm (in)
25(1") 50(2") 75(3") 100(4")
PlastiSpan HD Insulation NM/m3 176 147 111 92
pci 650 540 410 340
PlastiSpan 20 Insulation NM/m3 198 163 125 103
pci 730 600 460 380
PlastiSpan 25 Insulation NM/m3 255 212 160 133
pci 940 780 590 490
PlastiSpan 30
Insulation
NM/m3 299 247 187 157
pci 1100 910 690 580
PlastiSpan 40
Insulation
NM/m3 346 285 217 182
pci 1275 1050 800 670


For floor slab designs incorporating multiple insulation layers and a subgrade material, k can be found by adding k values for each layer as follows: 1/kT = 1/k1 + 1/k2 + …1/kn

Floor slab deflection establishes magnitude of unit load transferred to the subgrade material, in this case thermal insulation.  Based on slab-on-grade design theory using the theory of plates on elastic foundations2, slab deflection (W) is determined by load exposure, slab strength characteristics and subgrade (insulation) response to load transfer using the equation below.

 

where:             
W         = slab deflection
P          = applied load
K          = modulus of subgrade reaction
D          = Eh3/12(1-μ2)  
where:             
E          = modulus of elasticity of concrete
h          = slab thickness, in.
μ          = Poisson’s ratio of concrete

The following sample calculations are provided to illustrate a process to review the effect of the load on the concrete slabs.   Slab deflection and compressive load are calculated using elastic foundation design analysis based upon the combined characteristics of the insulation and a subgrade material.

Assumptions for Calculations:

  1. Concrete strength = 28 MPa (4000 psi)
  2. Concrete thickness (h) as noted in table
  3. Poisson's ratio for concrete = 0.15
  4. Insulation thickness = 76 mm (3")
  5. Subgrade k-value (ks) = 100 NM/m3 (368 pci)
  6. k-value Insulation and soil = 1/kT = 1/ki + 1/ks
  7. E-modulus of Concrete (Ec):

  = 24,870 MPa (3.605 x 106 psi)

Step 1: Calculate the modulus of subgrade reaction for 76 mm (3") insulation plus subgrade material.
1/kT = 1/ki + 1/ks

Table 3 - K-value - PlastiSpan Insulation Plus Subgrade Material (KT)

kT PlastiSpan HD
Insulation
PlastiSpan 20
Insulation
PlastiSpan 25
Insulation
PlastiSpan 30
Insulation
PlastiSpan 40
Insulation
NM/m3 53 56 62 65 68
pci 194 204 227 240 252

 

Step 2: Calculate slab deflection (W) due to load.


Table 4 - Example 1 – Slab Deflection (W) Under Wheel Load

152 mm (6") Concrete Slab
mm 0.22033 0.21459 0.20381 0.19806 0.19326
in. 0.00867 0.00845 0.00802 0.00780 0.00761

Table 5 - Example 2 – Slab Deflection (W) Under Point Load

152 mm (6") Concrete Slab
mm 0.31475 0.30655 0.29116 0.28294 0.27609
in. 0.01239 0.01207 0.01146 0.01114 0.01087


Step 3: Check compressive stress (F) in 76 mm (3") thick EPS insulation. The above slab deflection (W) will transfer load to the insulation material at intensity directly related to the insulation k-value – F = KiW.

Table 6 - EPS Compressive Stress (F)

PlastiSpan Insulation Types Example 1 - Wheel Load Example 2 - Point Load
152 mm (6") Slab 152 mm (6") Slab
kPa psi kPa psi
PlastiSpan HD Insulation 25 3.56 35 5.08
PlastiSpan 20 Insulation 27 3.89 38 5.55
PlastiSpan 25 Insulation 33 4.73 47 6.76
PlastiSpan 30 Insulation 37 5.38 53 7.69
PlastiSpan 40 Insulation 42 6.09 60 8.70

 

Table 7 below provides compressive resistance at 1% strain which should be used for determination allowable compressive stress for the various PlastiSpan insulation types based upon long term duration load exposure. 

Table 7 - PlastiSpan Insulation Compressive Resistance @ 1% Strain

Units
PlastiSpan HD Insulation PlastiSpan 20 Insulation PlastiSpan 25 Insulation PlastiSpan 30 Insulation PlastiSpan 40 Insulation
mm 45 50 65 76 100
in. 6.50 7.30 9.40 11.00 14.50

 

In the above examples, the compressive load transferred to the PlastiSpan insulation is within the allowable stress range for all insulation types. 

Step 4: Check bending stress (fb) in concrete

 

Where ƒb = Concrete bending stress, h = slab thickness and a = radius of load contact

Table 8 - Concrete Bending Stress (ƒb)

Design Loads Wheel Load – Load Factor =1.5 Point load – Load Factor =1.25
kN lbf kN lbf
35 7,875 50 11,250
Radius of Contact (a) 115 mm (4.5") 86 mm (3.4")
PlastiSpan Insulspan Type 152 mm (6") Slab 152 mm (6") Slab
MPa psi MPa psi
PlastiSpan HD Insulation 2.65 385 3.51 509
PlastiSpan 20 Insulation 2.62 380 3.46 502
PlastiSpan 25 Insulation 2.54 369 3.37 489
PlastiSpan 30 Insulation 2.49 361 3.31 481
PlastiSpan 40 Insulation 2.45 355 3.26 473



Concrete Tensile Strength (ƒT):

Bending stress should not exceed the concrete tensile strength = 3.41 MPa (495 psi).

Based upon the above analysis, PlastiSpan HD insulation or higher density would be adequate for the wheel load application.  PlastiSpan 25 insulation or higher density would be adequate for the point load application.

Summary:

Limitations of Use:

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1Portland Cement Association, Concrete Information, Packard, Robert G., Slab Thickness Design for Industrial Concrete Floors on Grade, 1996.

2Timoshenko, S. and Woinowsky-Kreiger, S., Theory of Plates and Shells, McGraw-Hill, 1959.