Slabs-On-Grade Subject to Concentrated Loads Elastoplastic Model by Victor D

Load-Carrying Capacity Employing an Concrete Slabs-On-Grade Subject To Concentrated Loads Elastoplastic Model By Victor D. Azzi, Ph.D., P.E., and Ralph H. Laird In earlier years, the determination of the allowable concentrated load applied to The scope of this article is limited to the design of industrial concrete floor an existing concrete floor slab system was slabs on grade for concentrated loads caused by the columns of free-standing solved according to the linear elastic ®theory work platforms, mezzanines, or mechanical support structures. The concrete floor of Westergaard. The elastic theory is cor- slab is presumed to be unreinforced. The structure supported by the floor slab is rect as long as the load is small. However, considered to be independent of the building structural system and, therefore, when the ultimate load-carrying capacity is is outside the scope of ACI 318. Further, the spacing of the structural columns required, the elastoplastic behavior of con- supporting these structural platforms is presumed to be sufficiently large, thereby crete should be taken into account. precluding any interactive effects of neighboring columns. The more recent research by Shentu and For generations, structural engineers, The evolution of SOG design and Al-Nasra indicate that there is a substantial and those who review their work, have behavior has allowed both designers difference in the results produced by been concerned that there was a lack of and building officials to have a better the methods presented in ACI when understanding of how concrete slabs-on- Copyrightunderstanding of, and confidence in, compared with those from an elastoplastic grade (SOG) behave, particularly under the load-carrying capacities of SOG analysis. Floor slabs on grade can carry the effects of discrete concentrated loads. subjected to large concentrated column significant additional load after the onset This has led to results that in many loads. These methods are applicable of initial cracking, and it is necessary to cases have been unrealistic and overly particularly to industrial warehouses take advantage of this additional load- conservative in their application. With and distribution centers where free- carrying capacity in design procedures for insufficient information, some have standing steel work platforms or engineering structures. speculated about concrete slab-on-grade mezzanines, typically carrying storage The inconsistency in designing exclusively behavior, including the surmising of and equipment loads on upper levels, in the elastic range is apparent. Most failures such as “punching shear,” and its cause large concentrated forces to act engineering publications acknowledge the dire consequences in applications where at discrete locations on the warehouse existence of shrinkage cracks in concrete none have been observed. floor on which the work platforms have floors. To use a design procedure, based on Earlier work done by Westergaard, and been installed. the analytical model of a floor presumed Ringo and Anderson, have long been the The work of Shentu and his colleagues, to be uncracked, for a floor slab known standard of practice. Standards by the through a comparison of their analytical to be cracked, is inappropriate. The long- ACI also addressed the design of slabs on predictions and test results, has demon- standing use of design methods that grade, and Face and Al-Nasra developed strated that the load-carrying capacity, as presume a crack-free slab, while simplifying a finite-element basis for the design well as settlement behavior, can be well the analytical model, has encouraged the magazineuse of methods that produce results that of SOG. More recently, a definitive predicted with good results. On that research paper by Shentu, Jiang, and basis, Shentu proceeded to develop a may not be applicable for the design of a S T R U C T Urealistic floor slab-on-grade.R The E design Hsu has brought about some rigorous “Simplified Analytical Method” that ign focus to the problem, and their analytical is the basis for this article. With this table presented later in this article is based on research results employing s and confirming testing results have method, and the resulting equations allowed a better understanding of the presented in that paper, the determina- an elastoplastic model of concrete e behavior and design of SOG. And even tion of the load-carrying capacity of a structural behavior. more recently, Higgins has introduced SOG is found to be simple, practical, the Shentu method as an approach to and reliable. Analysis

D a practical design method for slabs-on- Extensive investigations conducted by, The design of a floor slab-on-grade in- grade. The significance of this work is and on behalf of, the Storage Equipment volves the interaction of a concrete slab further demonstrated by the City of Los Manufacturers Association (SMA) have, and a soil support system. The concrete Angeles issuing an Information Bulletin for several years, been directed to develop is a material considered to be heteroge- stating that this approach, among others, an acceptable and reliable design method neous and statistically isotropic, becom- is an “Acceptable Design and Analysis for floor slabs on grade. This article is ing orthotropic with the development Method for Use of Slabs-on-Grade as intended to summarize their findings. of micro-cracks. Concrete strength in Foundation.” The approach developed Included here is a brief description of compression is significantly greater than here is further cited as an acceptable the design parameters for subgrade its strength in tension. Micro-cracks may design and analysis method in a recent properties, concrete tensile properties, form in the concrete even before load- Guidance Document, FEMA 460, by and the concept of the radius of relative ing. The soil system, in general, is also the Building Seismic Safety Council of stiffness and its relevance to this problem. heterogeneous; its characteristics and the Federal Emergency Management Also included is a representative design mechanical properties may vary within a Agency, which is focused on issues table, summarizing recommendations wide range. tructural related to seismic behavior of industrial resulting from this work for an example In addition to the concrete slab steel storage racks. six-inch slab. thickness, the following two properties discussions on design issues for structural engineers S are critical to the design of a floor slab- on-grade: subgrade strength and concrete tensile strength.

STRUCTURE magazine 18 April 2008 Table 1: 6-inch slab. Allowable load in Kips Soil 1.5 Rad Rel Concrete PSI 10-inch baseplate 12-inch baseplate 14-inch baseplate 16-inch baseplate 50 pci 52 inch 3000 36 37 38 39 4000 41 43 44 45 100 pci 44 inch 3000 41 44 46 48 ® 4000 48 50 52 55 200 pci 37 inch 3000 52 56 60 65 4000 60 65 70 75

Example Consider a 6-inch thick unreinforced slab made of 4000 psi concrete; sitting on a soil whose modulus of subgrade reaction, ks, is 100 pci; and with column loads being applied through 14-inch base plates. Using Table 1, the allowable concentrated load, Pa, is determined to be 52 kips; and the columns should be no closer than 2x44 inches, or 88Copyright inches, or 7.33 feet.

Soil-bearing capacity, soil compressibility, carrying capacities, Pa, for various concrete slabs has a radius of three times the radius of and modulus of subgrade reaction are on grade for a variety of parametric values, based relative stiffness. The radius of relative properties of the soil system that require on the application of the following relationship stiffness, b, is expressed as the fourth root understanding. Soil-bearing capacity is a developed by Shentu: of the result found by dividing the concrete measure of soil shear failure. This value is plate stiffness by the modulus of subgrade 4 2 determined by using various standardized Pn = 1.72 [(ksR1/Ec)10 + 3.60] ftd reaction as follows: soil tests. Soil compressibility is a measure of (Equation 1) 3 2 0.25 long-term settlement in a soil under load. and b = [Ecd /(12 (1 – µ ) ks)] This value is normally determined using soil Pa = Pn / FS (Equation 3) consolidation tests. (Equation 1a) where Modulus of subgrade reaction is the propor- or, alternatively, solving for the thickness, d, and b = radius of relative stiffness, in inches. tionality constant, ks, in a Winkler subgrade. introducing a Factor of Safety (FS) yields: Ec = modulus of elasticity of concrete, taken here as 4,000,000 psi. Its value depends upon the kind of soil, the 4 0.5 degree of compaction, and the moisture con- d = [(FSxPa)/(1.72 [(ksR1/Ec)10 +3.60] ftβ)] d = slab thickness, in inches. tent. The modulus of subgrade reaction has (Equation 2) µ = Poisson’s ratio, taken here as 0.15. where units of pci; it is the pressure in psi per inch ks = modulus of subgrade reaction, in pci. of soil deformation. The procedure for determagazine- Pn = nominal load-carrying capacity of the Additionally, the table shows a value, in mining k is outlined in ASTM D 1196. slab on grade, in pounds. s inches, which represents a distance of 1.5 ForS the general relationshipT betweenR the soil PUa = allowable load-carryingC capacityT of the U R E times the radius of relative curvature for the classifications and the modulus of subgrade slab on grade, in pounds. slab/soil system. From a practical point of reaction, see their depiction in Figure 3.3.5 ks = modulus of subgrade reaction, in pci. view, the radius of relative stiffness is used of ACI R-92. Essentially, soils that have high R1 = one-half the width or diameter of the to determine the distance from the point compressibility and low subgrade strength column base plate, in inches. of an applied concentrated load to a point will have a design k value of about 50 pci. Ec = modulus of elasticity of concrete, in psi. s where the load has virtually no effect on the Natural soils of higher subgrade strength ft = tensile strength in flexure of concrete, in slab stress. A load that is within a distance have a design k value of about 200 pci. psi. s of 1.5 times the radius of relative stiffness The tensile strength of concrete is usually d = slab thickness, in inches. from another load may have an influence determined by using the split cylinder test in FS = factor of safety, here taken as 3.0. on the slab stresses. Essentially, the loads accordance with ASTM C 496. The tensile β = load reduction factor, 1.0 for d < 7.0 shown in Table 1 assume that the load strength is a more variable property than inches; 0.85 for d  7.0 inches. under consideration is the only load within the compressive strength; it is about 10 to In the analysis on which this article is based, the distance shown on that Table. 15 percent of the compressive strength. tables were developed for slab thicknesses 0.5 The tensile strength is between 6(fc) and from four to eight inches; however, the loads 0.5 Factor of Safety 7(fc) for normal stone concrete. for the seven-inch and eight-inch thick floor The tensile strength in flexure is the modulus slabs were reduced by fifteen percent, using The primary focus of this article is the analysis of rupture (ASTM C 78). The modulus of a load-reduction factor, β, to compensate for and design of concrete floor slabs-on-grade, 0.5 rupture is generally accepted as 7.5(fc) apparent deviation of the results of Equation 1 in warehouses or industrial-type buildings, for normal concrete. The magnitude of the from the finite-element results presented in the on which free-standing work platforms or compressive strength for concrete is generally Shentu paper. mezzanine structures are built. These structures available for use by the design engineer. The earlier work of Packard (12), Pickett and are normally designed for heavy storage floor or The values presented in Table 1 are for unre- Ray (13) and, more recently, by Spears and deck loads of 125 psf or more. Further, these inforced concrete slabs of six-inch thickness. Panarese (14), and further detailed in ACI free-standing structures are independent of the The tabulated values represent the results 360R-92 (4), treated the area of influence of building structure and, therefore, the floor slabs for the determination of the allowable load- a single concentrated load. The slab analyzed are outside the scope of ACI 318.

STRUCTURE magazine 19 April 2008 Table 1: 4-inch Slab Allowable load in Kips Soil 1.5 Rad Rel Concrete PSI 10” baseplate 12” Baseplate 14” Baseplate 16” Baseplate 50 pci 39 inch 3000 16 17 17 17 4000 19 19 20 20 100 pci 33 inch 3000 19 19 20 21 4000 21 22 23 24 ® 200 pci 27 inch 3000 23 25 27 29 4000 27 29 31 33

Table 2: 5-inch Slab Allowable load in Kips Soil 1.5 Rad Rel Concrete PSI 10” baseplate 12” Baseplate 14” Baseplate 16” Baseplate 50 pci 46 inch 3000 25 26 26 27 4000Copyright29 30 31 31 100 pci 38 inch 3000 29 30 32 33 4000 33 35 36 38 200 pci 32 inch 3000 36 39 42 45 4000 41 45 48 52

Table 3: 6-inch Slab Allowable load in Kips Soil 1.5 Rad Rel Concrete PSI 10” baseplate 12” Baseplate 14” Baseplate 16” Baseplate 50 pci 52 inch 3000 36 37 38 39 4000 41 43 44 45 100 pci 44 inch 3000 41 44 46 48 4000 48 50 52 55 200 pci 37 inch magazine3000 52 56 60 65 S T R U4000 60C 65T U70 75R E Table 4: 7-inch Slab Allowable load in Kips Soil 1.5 Rad Rel Concrete PSI 10” baseplate 12” Baseplate 14” Baseplate 16” Baseplate 50 pci 59 inch 3000 41 43 44 45 4000 48 49 51 52 100 pci 49 inch 3000 48 50 53 55 4000 55 58 61 63 200 pci 42 inch 3000 60 65 70 75 4000 69 75 80 86

When selecting a factor of safety (FS), the will stored goods become unstable or of three was chosen, pending any further following factors should be considered: dislodged, and will roofs leak due to the research results on the effects of control joints a) Will the design load be applied to the floor-slab settlement? and the effects of other possibly-neighboring entire deck surface simultaneously? d) Will slab failure lead to costly repair? loads on the overall behavior and load- The likelihood of the design load being carrying capacity of the floor-slab system. applied over the entire deck surface may Good engineering judgment should be Further, as stated earlier, the loads for seven- be unlikely. exercised in the selection of any factor of inch and eight-inch thick floor slabs have been b) Will any slab failure lead to a safety. The tables developed in this study, such reduced by approximately fifteen percent to catastrophic result? as the example presented here, in general use compensate for the apparent deviation of the c) Will excessive settlement under load cause a factor of safety of three versus the predicted results of Equation 1 from the finite-element problems of function or inconvenience, nominal load, Pn, of Equation (1). While this results presented in the Shentu paper.▪ e.g., will windows break, will doors stick, is considered to be very conservative, a factor

STRUCTURE magazine 20 April 2008 Table 5: 8-inch Slab Allowable load in Kips Soil 1.5 Rad Rel Concrete PSI 10” baseplate 12” Baseplate 14” Baseplate 16” Baseplate 50 pci 65 inch 3000 55 56 57 59 4000 62 64 66 68 100 pci 55 inch 3000 62 65 69 72 4000 72 75 79 83 ® 200 pci 46 inch 3000 78 85 91 97 4000 90 98 105 112

Victor D. Azzi, Ph.D., P.E. is a consulting structural engineer who, among other areas of his practice, serves as a consultant to various groups of the material handling industry. He was a long-time Professor of Engineering at the University of New Hampshire, from which he is now retired, and continues to serve on committees of the BSSC-NEHRP, ASCE 7, and AISI dealing with the seismic behavior of non-building structures. Victor can be reached at [email protected]. Ralph H. Laird, served as president of Wildeck, Inc. fromCopyright 2002 until his retirement in 2008. He was a registered P.E. until that retirement. Because of the apparent lack of design information for floor slab-on-grade, he worked with the MHI (Material Handling Insitute) to assist in the development of a more comprehensive and realistic approach to the SOG design.

References and Bibliography (1) Westergaard, H. M., “New Formulas for Stresses in Concrete Pavements of Airfield,” Transactions of the ASCE, 113(2340), pp. 425-444, 1948. (2) Ringo, Boyd C. and Robert B. Andersen, Designing Floor Slabs on Grade, Second Edition, Aberdeen Group, Addison, Illinois, 1996. (3) Ringo, Boyd C., Design, Construction, and Performance of Slabs-on-Grade for an Industry,” Journal of the American Concrete Institute, pp. 594-602, November 1978. (4) “Design of Slabs on Grade,” ACI 360 R-92, ACI Committee Report 360, 1992. (5) “Guide for Concrete Floor and Slab Construction,” ACI 302 R-89, ACI Committee Report 302, 1989. (6) Face, Edward W., “Slab-on-Grade Column Load Limit Analysis,” Face Company Report on Research conducted for the Storage Equipment Manufacturers Association, Material Handling Industry, March 1993. (7) Al-Nasra, Moayyad M., “Materially Nonlinear Finite Element Analysis of Trusses and Slabs-on-Grade Problems,” Doctoral Dissertation, Old Dominion University, Norfolk, Virginia, 1992. (8) Shentu, Longmei, Dahua Jiang, andmagazine Cheng-Tzu Thomas Hsu, “Load-Carrying Capacity for Concrete Slabs on Grade,” Journal of Structural Engineering, ASCE, pp. 95-103, January 1997. (9)S Higgins, PeterT S., “Industrial R Building Slabs Uon Grade – StrengthC Design, ServiceabilityT Considerations,” U SEAOSC R Slabs on GradeE Design Seminar, November 2003. (10) City of Los Angeles, “Acceptable Design and Analysis Methods for Use of Slabs-on-Grade as Foundations,” Department of Building and Safety, Information Bulletin/ Public-Building Code, Reference (A) 2 3 4 5 6 7 8 9 10 15 20 25 30 40 50 60 70 80 90 L.A.M.C.91.1806, Document No. P/ (BCS 314) CALIFORNIA BEARING RATIO “CBR”, PERCENT BC 2002-100, effective May 10, 2004. (11) “Seismic Considerations for Steel (B) 50 100 150 200 300 400 500 600 700 800 900 1000 11001200

3 Storage Racks Located in Areas (BCS 315) MODIFIED MODULUS OF SOIL REACTION “K12” LBS./IN. (12” DIAM. PLATE) Accessible to the Public,” Building Seismic Safety Council, Federal 25 50 75 100 150 200 250 300 350 400 450 500 550 600 Emergency Management Agency, (C) 3 FEMA 460, September 2005. STANDARD MODULUS OF SOIL REACTION “K”, LBS./IN. (30” DIAM. PLATE) (12) Packard, Robert G., Slab Thickness (D) G - GRAVEL GW Design for Industrial Concrete Floors on S - SAND GM M - SILT GP Grade, IS195.01D, Portland Cement C - CLAY GU W - WELL GRADED GC Association, Skokie, Illinois, 1976. P - POORLY GRADED (13) Pickett, Gerald, and Gordon K. U - UNIFORMLY GRADED SW L - LOW TO MEDIUM COMPRESSIBILITY SM Ray, “Influence Charts for Concrete H - HIGH COMPRESSIBILITY SP Pavements,” Transactions of the ASCE, O - ORGANIC SU Vol. 116, 1951. SC (14) Spears, Ralph, and William C. Panarese, CL CL ML ML LEGEND Concrete Floors on Ground, EB075.02D. OL OL Portland Cement Association, Skokie, COMPACTED DENSITIES Illinois, second edition, 1983; revised MH MH NATURAL DENSITIES CH CH 1990. OH OH Note: comparison of soil type to “K”, particularly in the “L” and “H” Groups, should generally be made in the lower range of the soil type. Figure 1 STRUCTURE magazine April 2008