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
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. The bulletin provides calculation examples to illustrate use of PlastiSpan insulation for structural floor slab applications.
Structural slab design is governed by the types and magnitude of loads on the concrete slab which may include wheel loads from forklifts or delivery vehicles, point loads from the legs of storage racks or distributed loads from product stored on the concrete slab. Often, selection of a sub-slab insulation product for structural slab-on-grade applications is based upon ability to sustain compressive loads transferred through the concrete slab, without full and accurate determination of the load distribution characteristics of the concrete slab.
In one design methodology, design calculations 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 wheel contact area or base plate contact area through the concrete slab to a largely hypothetical bearing area on the top surface of the insulation. As illustrated in Table 1, the load exposure for the insulation (EPS compressive stress) calculated on this basis could dictate use of a high density, high compressive resistance insulation material increasing cost unnecessarily.
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 | 45 | |||||
lb_{f} | 7,875 | lb_{f} | 10,125 | |||||||
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 | m^{2} | 0.26 | Loaded Area | m^{2} | 0.21 | |||||
in^{2} | 398 | in^{2} | 322 | |||||||
EPS Compressive Load Exposure | kPa | 136 | EPS Compressive Load Exposure | kPa | 240 | |||||
psi | 20 | psi | 31 |
Another accepted design procedure to use for structural slab design with these types of loads is the theory of plates on elastic foundations. Using the theory of plates on elastic foundations design procedure, 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 concrete 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 Information reflects the response of the insulation and subgrade under temporary (elastic) conditions when small deflections occur.
Disclaimer Regarding Example Calculations
This bulletin provides examples of calculations to illustrate applying the theory of plates on elastic foundations procedure to concrete slab design based upon a hypothetical wheel load with a defined wheel contact area or point load supported on a defined base plate contact area. It must be stressed that in all cases, design calculations and details for specific applications must be prepared by a registered design professional to verify compliance with applicable codes for the jurisdiction in which the project is to be constructed.
Limitations of Use:
- It is not the intent of this bulletin to provide comprehensive design guidance. Concrete slab deflection for each
application must be calculated by the design professional responsible for concrete slab design. - Design approaches expressed in this bulletin are related specifically to the distribution of floor loads to subgrade
insulation. Final slab design is generally controlled by flexural stresses to which the slab is exposed under long term
or short term rolling load conditions. - Relationships used to establish slab deflection assume load intensities on the subgrade insulation do not exceed
the elastic limit of the insulation. - Such factors as the bearing capacity and compressibility of subsoil and/or subgrade slabs below the insulation must
be considered in the design of slab/subgrade insulation composites. - These design considerations are applicable to concrete slabs-on-grade (insulation serving as grade) exposed to
storage loads, storage rack post loads and vehicle axle or wheel loads causing limited slab deflection. High intensity
column or wall loading on floor slabs requires further consideration. - Soils, concrete, steel and subgrade insulation exhibit creep or cold flow under long-term load exposures. Such
long-term load exposures must be considered in slab design in order to prevent objectionable slab settlement.
Calculation Examples Using Theory of Plates on Elastic Foundation Procedure
The following calculation examples use the hypothetical loads for the two load types provided in Table 1 to illustrate the theory of plates on elastic foundation procedure. Floor slab deflection establishes magnitude of unit load transferred to the subgrade material, in this case thermal insulation, based on slab-on-grade design using the theory of plates on elastic foundations . Slab deflection (W) is determined by load exposure, concrete slab strength and subgrade response (insulation and soil) to load transfer using the equation:
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
Based upon the theory of plates on elastic foundation procedure, slab deflection and insulation compressive load are calculated using elastic foundation design analysis based upon the combined characteristics of the insulation and a subgrade material.
_______________________
^{1}Portland Cement Association, Concrete Information, Packard, Robert G., Slab Thickness Design for Industrial Concrete Floors on Grade, 1996.
^{2}Timoshenko, S. and Woinowsky-Kreiger, S., Theory of Plates and Shells, McGraw-Hill, 1959.
Assumptions for Theory of Plates on Elastic Foundation Calculations:
- Concrete strength () = 28 MPa (4000 psi)
- Concrete thickness (h) as noted in table 1
- Poisson's ratio for concrete = 0.15
- Insulation thickness = 76 mm (3")
- Subgrade k-value (ks) = 100 MN/m3 (368 pci)
- k-value Insulation and soil = 1/kT = 1/ki + 1/ks
E-modulus of Concrete (Ec):
In SI units:
= = 24,870 MPa (3.605 x 106 psi)
Compressive resistance at 1% strain, the industry accepted allowable stress for live and dead loads, is provided in Table 2 for a variety of PlastiSpan insulation types.
Table 2 - PlastiSpan Insulation Compressive Resistance @ 1% Strain
Units | PlastiSpan
HD Insulation |
PlastiSpan
25 Insulation |
PlastiSpan
30 Insulation |
PlastiSpan
40 Insulation |
PlastiSpan
60 Insulation |
---|---|---|---|---|---|
kPa | 45 | 60 | 75 | 103 | 124 |
psi | 6.50 | 8.7 | 10.9 | 15.0 | 18.0 |
Modulus of subgrade reaction values (k_{i}) expressed in units of MN/m^{3} or lbs/in^{3} (pci) for various PlastiSpan insulation types and thickness are provided in Table 3.
Table 3 - 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 | MN/m^{3} | 176 | 147 | 111 | 92 |
pci | 650 | 540 | 410 | 340 | |
PlastiSpan 25 Insulation | MN/m^{3} | 255 | 212 | 160 | 133 |
pci | 940 | 780 | 590 | 490 | |
PlastiSpan 30 Insulation | MN/m^{3} | 299 | 247 | 187 | 157 |
pci | 1100 | 910 | 690 | 580 | |
PlastiSpan
40 Insulation |
MN/m^{3} | 346 | 285 | 217 | 182 |
pci | 1275 | 1050 | 800 | 670 | |
PlastiSpan
60 Insulation |
MN/m^{3} | 434 | 358 | 271 | 228 |
pci | 1600 | 1320 | 1000 | 840 |
Step 1: Calculate the modulus of subgrade reaction for 76 mm (3") PlastiSpan insulation plus subgrade material. 1/k_{T}= 1/k_{i} + 1/k_{s}
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
Table 4 - K-value - PlastiSpan Insulation Plus Subgrade Material (k_{T})
k_{T} | PlastiSpan
HD Insulation |
PlastiSpan
25 Insulation |
PlastiSpan
30 Insulation |
PlastiSpan
40 Insulation |
PlastiSpan
60 Insulation |
---|---|---|---|---|---|
MN/m^{3} | 53 | 62 | 65 | 68 | 73 |
pci | 194 | 227 | 240 | 252 | 269 |
Step 2: Calculate slab deflection (W) due to load.
Table 5 - Slab Deflection (W)
Insulation Type | PlastiSpan
HD Insulation |
PlastiSpan
25 Insulation |
PlastiSpan
30 Insulation |
PlastiSpan
40 Insulation |
PlastiSpan
60 Insulation |
---|---|---|---|---|---|
Example 1 - 152 mm (6") Concrete Slab Deflection (W) Under Wheel Load | |||||
mm | 0.22033 | 0.20381 | 0.19806 | 0.19326 | 0.18707 |
in. | 0.00867 | 0.00802 | 0.00780 | 0.00761 | 0.00737 |
Example 2 - 152 mm (6") Concrete Slab Deflection (W) Under Point Load | |||||
mm | 0.28328 | 0.26204 | 0.25464 | 0.24848 | 0.24052 |
in. | 0.01115 | 0.01032 | 0.01003 | 0.00978 | 0.00947 |
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 from Table 3: F = K_{i}W.
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 | 32 | 4.57 |
PlastiSpan 25 Insulation | 33 | 4.73 | 42 | 6.09 |
PlastiSpan 30 Insulation | 37 | 5.38 | 48 | 6.92 |
PlastiSpan 40 Insulation | 42 | 6.09 | 54 | 7.83 |
PlastiSpan 60 Insulation | 41 | 5.89 | 52 | 7.58 |
Based upon the above calculation examples, the compressive load transferred to the PlastiSpan insulation is within the allowable stress range provided in Table 2 for all insulation types.
Step 4: Check concrete slab bending stress (f_{b}).
Where ƒ_{b} = Concrete bending stress, h = slab thickness and a = radius of load contact
; where Ac = load contact area
Table 7 - Concrete Bending Stress (ƒ_{b})
Design Loads | Wheel Load – Load Factor =1.5 | Point load – Load Factor =1.25 | ||
---|---|---|---|---|
kN | lb_{f} | kN | lb_{f} | |
35 | 7,875 | 45 | 10,125 | |
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 | 2.88 | 417 | 3.70 | 536 |
PlastiSpan 25 Insulation | 2.83 | 410 | 3.35 | 486 |
PlastiSpan 30 Insulation | 2.81 | 407 | 3.33 | 484 |
PlastiSpan 40 Insulation | 2.79 | 405 | 3.32 | 481 |
PlastiSpan 60 Insulation | 2.77 | 402 | 3.30 | 478 |
Concrete Tensile Strength ():
MPa in SI Units:
Bending stress should not exceed the concrete tensile strength = 3.41 MPa (495 psi).
Selection of the PlastiSpan insulation type based upon concrete slab design using the theory of plates on elastic foundation would result in:
- Based upon the hypothetical wheel load for calculation example 1 from Table 1, PlastiSpan HD insulation
would satisfy the design requirements. - Based upon the hypothetical point load for calculation example 2 from table 1, PlastiSpan 25 insulation
would satisfy the design requirements.
Calculation Example Summary:
- Under "assumed" or hypothetical load distribution patterns, the EPS compressive resistance requirement for the
load exposure from the two load types typical for structural slabs would dictate use of a high strength
thermal insulation. - Using elastic foundation design theory, a much lower compressive load would be transferred to the sub-slab
insulation based upon slab deflection under the two hypothetical load types allowing the use of a more
cost-effective PlastiSpan insulation alternative.