Design of
Unreinforced Slabs-on-Ground
Made Easy BY WAYNE W. WALKER AND JERRY A. HOLLAND
I
n our previous article “The First Commandment for Floor Slabs: Thou Shalt Not Curl Nor Crack...(Hopefully)” (Concrete International, January 1999), we discussed some important concepts concerning the design of slabs-onground such as curling, linear shrinkage, joint spacing, and serviceability issues. In this article we have used those concepts and others, plus a unique computer analysis, to develop design aids for slabs-onground that are unreinforced or have a small amount of reinforcing that is not included in the structural capacity. We have used these design aids for many years to design numerous successful industrial slabs. Design graphs presented here for joint spacings, uniform loads, rack loads,
and lift truck loads help make the design of unreinforced slabs much simpler and easier; furthermore, these design aids usually provide more economical slabs than those designed by more traditional methods.
What should the joint spacing be? There are a variety of slab design options a designer can choose from based on the project requirements. These options can be separated into two broad categories: slab designs that are based on the slab not cracking (uncracked option) and slab designs based on the slab cracking but remaining serviceable (cracked option). Typically, the uncracked slab design option would be a design using the concrete flexural tensile strength to resist the
loads, and the cracked option would be a slab reinforced to limit the cracks to an acceptable width. This article addresses slabs designed not to crack between joints, and an upcoming article in CI will address slabs designed to crack but remain serviceable. To minimize the size and number of cracks in an unreinforced slab, it is important to use the proper joint spacing to minimize the curling stresses, which are usually much larger than linear shrinkage stresses. The authors have observed numerous industrial slabs having a calculated curling stress of 200 psi (1.4 MPa) or less that have performed well. Based on a theoretical model that predicts our field-measured values, we derived equations using curvilinear regression that provide joint Concrete international
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Fig. 1: Recommended joint spacing for unreinforced slabs
curling and cracking. Application of the joint spacing criteria shown (solid lines) in Fig. 1 has resulted in many successful slabs that have remained much flatter and with far fewer cracks than typically seen. For joint spacings approaching 15 ft (4.6 m) or longer, with medium to high shrinkage potential concrete and heavy vehicular traffic, the slab designer should consider using dowels in the joints since the joints may no longer have adequate shear transfer by aggregate interlock. For aggregate interlock to be effective long-term, the joint spacing should theoretically be extremely close together (5 to 10 ft [1.5 to 3.0 m] for anything but low-shrinkage concrete), but in actuality, joints tend to open a little less than the theoretical prediction because of the restraints. If the designer cannot be sure of positive, long-term shear transfer at the joints through aggregate interlock, dowels should be used. It may be more economical to keep the joint spacing at 15 ft (4.6 m) for the thicker slabs and eliminate the dowels in the contraction joints; however, there will be more linear feet of joints to maintain. One other consideration concerning close joint spacing and joint maintenance is that, although there may be more joints, joint movement is less (both horizontally and vertically), which means there is less joint maintenance required per unit length of joint. These options should be discussed with the owner.
Fig. 2: Concentrated load on a slab-on-ground
Design for top or bottom cracking?
spacing data to maintain curling stress at approximately 200 psi (1.4 MPa). Figure 1 shows the graph for this equation for two types of concrete: a concrete with typical shrinkage values and a concrete with low shrinkage values. Figure 1 also shows the joint spacing criteria of 36 and 24 times the slab thickness which has been referenced in the past. Using the commonly specified joint spacing of 36 times the slab thickness may cause a high curling stress, especially for the thicker slabs. Using a joint spacing of 36 times the slab thickness, along with some of the newer concrete materials that have higher shrinkage potential, has caused many slabs to exhibit
For a single concentrated load on a slab, the first cracks that will develop will be in the bottom of the slab in a radial direction, as shown in Fig. 2. As the load continues to increase, the next crack will be in the top of the slab in a partially or completely circumferential direction, as also shown in Fig. 2. The cracks in the bottom of the slab obviously cannot be seen and normally do not cause any serviceability problems. The cracks in the top of the
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“Using a joint spacing of 36 times the slab thickness, along with some of the newer concrete materials that have higher shrinkage potential, has caused many slabs to exhibit curling and cracking. Application of the joint spacing criteria shown (solid lines) in Fig. 1 has resulted in many successful slabs that have remained much flatter and with far fewer cracks than typically seen.” slab can be seen, cause serviceability problems, and raise aesthetic concerns. Thus, the design aids are based on minimization of the top cracks, except for conditions where lift trucks produce highly repetitive loads. (See the section on lift truck capacity.) The authors have developed a unique computer program that allows the slab to crack on the bottom and therefore lets the negative moment (moment that causes tension stress in the top) increase. As can be seen in Fig. 3, the negative moment for the cracked slab model is about half of the traditional moment (which assumes the slab is uncracked) that is typically used to design slabs. In other words, once the slab has cracked on the bottom, the second load path has much more capacity. This increase in capacity explains why we have observed, on many occasions, slabs that have supported much higher concentrated loads than that predicted by traditional analysis. In such cases it is thought that the slab with the high concentrated loads has cracked on the bottom (not visible) and the slab has increased capacity on the second load path. The slab’s actual capacity to support concentrated loads will most likely be greater than twice the capacity calculated by a traditional analysis and would depend on the number of bottom radial cracks. Flexural stresses occur in slabson-ground when either the uniform storage loads are not continuous (such as at aisle ways), the soil support beneath the slab is not uniform, or both. At aisle ways, the maximum tension stress in the top of the slab will occur when the uniform loads are located at the critical aisle width, as shown in
Fig. 4. Aisle widths that are less than or greater than the critical aisle width will have less flexural tension stress in the top.
Drag stress due to linear shrinkage Drag stresses (due to linear shrinkage and frictional restraints) that will occur for the joint spacings shown in Fig. 1 will be small if the slab is placed on a low friction base
with small, gradual undulations. If the undulations in the base become large and/or with abrupt changes, the drag stresses can easily become significant, and stress concentrations can develop that can lead to cracking. Therefore, it is important that the base under a slab be constructed as smooth and planar as practicably possible (to help minimize the drag stress due to the linear shrinkage) and maintain a uniform thickness.
Fig. 3: Moments for a 6 in. (150 mm) thick slab-on-ground with a 10,000 lb (45 kN) concentrated load
Fig. 4: Uniform loading on a slab-on-ground Concrete international
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What allowable flexural stress should be used?
Fig. 5: Rack plan
Fig. 6: Stresses perpendicular to the aisle for a 6 in. (150 mm) thick slab-on-ground that has cracked on the bottom of the slab at the rack base plate locations (red color indicates high tension stress in the top of the slab)
Modulus of subgrade reaction Design graphs for two values of modulus of subgrade reaction (K) are given for uniform loads and rack loads. The 30 pci (9000 kN/m3) value is for a soft soil with a wide area loading that will be subjected to some longterm consolidation. The 80 pci (22,000 kN/m3) value is for a firm soil with a wide area loading with minimal longterm consolidation. The modulus of subgrade reaction used for lift trucks is 100 pci (30,000 kN/m3), which is typical for many soils with temporary loads applied over small areas.
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Design graphs are based on the allowable concrete flexural tension stress to give the designer flexibility in selecting the method in which the concrete modulus of rupture is determined and which factor of safety should be used. There are several published empirical equations that give the modulus of rupture based on the concrete compressive strength. Equations based on these data have considerable scatter because the modulus of rupture is sensitive to other concrete design mixture parameters and components, such as the aggregate shape and texture. For similar compressive strengths, the authors have seen modulus of rupture values vary from less than 450 psi (3.1 MPa) to more than 750 psi (5.2 MPa). Due to the variability of the modulus of rupture values for different concrete mixture designs, when we are not familiar with the local concrete materials, we require flexural beam tests prior to construction of large, critical slabs. When beam tests are specified, compression cylinders typically are made of the same concrete to correlate between the flexural and the compressive strengths. After the relationship is found to be consistent, only compression cylinders are tested. Flexural beam tests are highly sensitive to sampling, storage, transportation, and testing; thus, only highly qualified testing companies and certified technicians should be used. The factors of safety used range from 1.25 to 2.0. Lower factors of safety can be applied when there are the following conditions: good quality control of the concrete mixture design; dependable geotechnical information with proper base control; adequate field testing; good construction supervision; well-known applied loads; and required flexural beam tests. Higher factors of safety are used where there is less information on the conditions mentioned above or highly repetitive dynamic loads that would cause fatigue of the concrete, such as frequent lift truck traffic.
Rack load considerations The rack plan dimensions used for the design aid are as shown in Fig. 5. This type of rack is commonly used for double-deep pallet storage. The analysis used was as noted above for concentrated loads, allowing the slab bottom to crack under the post. As can be seen in the moment contours in Fig. 6 for a center rack post load of 10,000 lbf (45 kN) (note: no curling or drag stresses are added to the moment contours for this graph), the high moments occur away from the post loads, unlike traditional analysis based on uncracked section properties that predict the maximum stress directly under the post load. Also, a traditional analysis for these soil values and rack plan dimensions would provide a higher rack post load capacity than for a single post load, thereby indicating the surrounding rack
Fig. 7: Slab thickness required for 3000 lb (13 kN) and 4000 lb (18 kN) capacity lift trucks
Fig. 9: Allowable rack post loads
post loads act to reduce the positive moment under the center rack post. Since the surrounding rack posts may not be loaded to their full capacity, the single post load capacity would be the controlling factor for traditional analysis. For typical slab thicknesses and soil properties, this difference in load capacity would range between 10 to 20%. The maximum post load is controlled by one of the following: allowable flexural stress; base plate bearing stress; punching shear; or beam shear.
Lift truck considerations
Fig. 8: Allowable uniform loads
Typical lift trucks with rated capacities of 3000 lbf (13 kN) and 4000 lbf (18 kN) and solid tires were chosen because these are the most commonly used. The analytical model used for the design aid (Fig. 7) is based on no cracking on the slab bottom in order to ensure that fatigue is not a significant issue. However, for lift truck loading with a low to moderate number of load repetitions, the slab could be designed for bottom cracking but no top cracking (similar to the rack loading above). Concrete international
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Conclusions There are many variables that should be considered in the design of slabs-on-ground. Here we have provided a sophisticated computer analysis program developed to address these variables with graphs (Fig. 7, 8, and 9) that will make designing unreinforced and lightly reinforced slabs easier and provide more economy for various slab applications. Many slabs have been designed and successfully constructed using these design aids, but readers are cautioned to make sure they understand all contributing design factors before using these design aids or seek technical assistance from appropriately competent designers. Look for an upcoming article in Concrete International on the design of reinforced slabs based on crack width control. Selected for reader interest by the editors.
er is a senior structural engineer at ACI Member Wa y ne W. Walk alker Lockwood Greene Engineers, Atlanta, Ga. He has been a speaker at ACI seminars and is a member of ACI Committee 360, Design of Concrete Slabs on Ground. He has also published other papers and has developed many computer programs to analyze and design slabs on grade.
Jerr Jerryy A A.. Ho Holl l and and,, FACI, is a structural engineering consultant with Lockwood Greene Engineers and Architects in Atlanta, Ga. He has 33 years of experience in design, construction, and troubleshooting concrete materials and structures. Holland is past chairman of ACI Committee 360, Design of Slabs on Ground, and is currently a member of that committee; as well as 223, Shrinkage-Compensating Concrete; 301, Specifications for Concrete; 302, Construction of Concrete Floors; and 350, Environmental Engineering Concrete Structures.
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