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www.burnsmcd.com A Burns & McDonnell Publication 2011 No. 1 Contact Mechanics Analyzing the Effect of Steel Wheels on Concrete Slabs By Philip Terry, PE From rudimentary structural mechanics in the Contact mechanics involves the study of forces elastic range, deflection is related to load by transmitted from one surface/solid to another the value of Modulus of Elasticity (also known and the consequent stresses in those solids. as Young’s Modulus), E, such that Deflection Initially, when non-conforming solids come = Load x E. The Modulus of Elasticity varies into contact, they touch at a point or along with the material being considered. For this a line. Under load, the solids deform so that discussion, the typical value of E for steel is the contact area increases, but the contact 29,000,000 psi (Figure 1) while for various areas can still be very small, causing stresses strengths of concrete, E values are shown in to be intense. Typically, structural engineers Table 1. A material with designing concrete slabs do not need to a smaller value of E will consider contact mechanics. However, in two compress more than a recent projects, Burns & McDonnell structural material with a larger engineers found the need to use principles from value of E. Materials can the area of contact mechanics to evaluate the be deformed beyond their Steel bearing pressures of steel wheels on slab-on- ability to recover, beyond E= 29 million psi grade concrete floors. their elastic limits.

Concrete Consider a heavy load Deflection/Strain In both projects, the facility owners desired to E= 3.6 million psi move heavy items over their concrete floors. supported by a steel In designing the cart/dolly for moving the wheel and the steel wheel Load/Stress heavy items, the cart designers consulted supported by concrete. If the areas of the steel manufacturers’ catalogs to select the wheels Modulus of Elasticity (E) in Elastic Range that would support the heavy loads, but they did and the concrete are the not consider the interaction of the curved steel same and the elastic limits wheels and the concrete surface. They did not of the materials Figure 1: Stress calculations consider the relative elasticity of try to keep the maximum compressive stresses have not been exceeded, materials in contact under load conditions. within the elastic ranges of both materials. the concrete will Using contact mechanics, we evaluated the deform interaction between the steel wheel and the more than the Maximum Nominal concrete and predicted possible harm to the steel. Contact Concrete f'c (psi) E concrete (psi)2 Bearing Strength (psi)3 concrete floors. mechanics is used to 3,000 3,122,000 5,100 One might think that the concrete is very rigid determine the 4,000 3,605,000 6,800 and durable and that there is little or no need width of 5,000 4,030,000 8,500 to consider the contact of the wheels on the the small 1 f’c = concrete compressive strength, psi = pounds per square inch. concrete. This may be true for small loads and contact 2 E concrete = 57,000√(f’c) (from ACI 318) large plates or solids of flexible, conforming area interface. 3 ACI 318-08, Section 10.14, Maximum Nominal Bearing Strength, BN = 0.85 x f’c materials, but steel is a very rigid material x 2.0., where 2.0 is the maximum amplification factor permitted; accounting compared to concrete. for the effects of confinement provided by the surrounding concrete.

Table 1: Modulus of Elasticity (E) for various concrete compressive strengths. Contact Stresses Using the cylinder analogy from Roark’s P Cylinder material 2 (Total Load) Formulas for Stress and Strain by Warren C. Young, the contact stresses can be calculated according to the formulas shown in Figure 3.

L Calculation Example D (Length) As an example, consider a 6-inch diameter, (Diameter) steel wheel with a length of 3 inches and an unfactored load of 12,000 pounds normal to b a concrete slab with a concrete strength of (Contact width) f ’c = 4,000 psi. Half of the load is due to dead loads and half of the load is due to live loads. Employing the formulas for contact stresses Support Material 1 given in Figure 3, the unfactored load per unit

length, p, is 4,000 pounds per inch. KD is 6 inches. Using the typical E , E , v and v values Figure 2: This illustration of Young s Cylinder analogy defines 1 2 1 2 ’ for concrete and steel, the value of C is 2.9768 the variables for calculating contact stresses. E e-7; the bearing contact width, b, is calculated to be 0.1352 inches and the contact area would be

b=1.6 √ p x KD x CE (see Figure 2) b x L, or 0.4057 square inches. Where: p = load per unit length = P/L Under an unfactored/service load of 12,000

KD = D2, (since the support material is flat, not curved) pounds, the contact pressure on the concrete 2 2 CE = (1 – v1 )/E1 + (1 – v2 )/E2 and steel is 29,577 psi. The maximum E= See Figure 1 and Table 1. compressive stress within the concrete would be 37,764 psi. Under strength design, the dead-

Poisson’s Ratio for concrete, v1, is approximately 0.2 and load portion is factored by 1.2 and the live-load

for steel, v2, is approximately 0.3. The subscript number portion is factored by 1.6. Thus, in this example, indicates the material. The maximum compressive stress, the factored/ultimate design bearing pressure g , is 0.798 √p⁄(K x C ) and the maximum shear stress, , c D E is 1.4 x 29,577 = 41,408 psi. American Concrete is 1/3 x o . c Institute (ACI) 318 (building code requirements for reinforced concrete) reduces the ultimate Figure 3: Formulas for calculation of contact stress. bearing pressure permitted on the concrete by a strength reduction factor, o, of 0.65, thus, the permitted bearing strength becomes 0.65 x 6,800 = 4,420 psi.

Load According to N.M. Hawkins in “The Bearing Plate Strength of Concrete Loaded Through Rigid Plates” (Magazine of Concrete Research, 1968) 2 Conical Concrete wedge slab the initial cracking in the concrete slab under a loaded rigid plate is a vertical crack under the 1 Crack loaded plate (Figure 4, Label 1). At maximum initiation load, there is a conical wedge beneath the plate (Figure 4, Label 2) and radial cracks appear on the concrete surface. Failure occurs when the inverted cone is pushed downward sufficiently to split the concrete toward the sides. Figure 4: Initial cracking of concrete slab under a loaded rigid plate.

TECHBriefs 2011 No. 1 2 Burns & McDonnell Conclusion observe a clear-cut cone or pyramid. Instead, The ultimate design bearing pressure of 41,408 there was a vertical core with a narrow section psi is considerably higher than 4,420 psi; at mid-depth (in some cases, appearing to form therefore it is expected that the steel wheel will mirrored pyramids). Cracks appeared at the cause the concrete to fail locally in bearing/ bottom of the block first and then progressed crushing. Since the load will roll across the upward. They also observed that the failure floor, there will be numerous failure locations. load was consistently and slightly higher than If redesign of the cart is not possible, then the the cube results. William Shelson, observing concrete should be armored to distribute the amplification factors larger than 2.0 for confined load and protect the surface from wear. areas (higher ratios of footing area to loaded area) stated in his article “Bearing Capacity of The maximum nominal bearing strength Concrete” (Journal of the American Concrete values shown in Table 1 on page 1 include Institute, 1957), “For comparatively shallow the maximum 2.0 amplification factor, which blocks on steel platens, the region directly accounts for the permitted compressive stress beneath the loaded area is subject to a generally increases due to the triaxial compressive stress uniform compressive stress throughout the condition of confined concrete. According to total depth. Conditions favorable to wedge ACI 318, “When the supporting area is wider [cone/pyramid] formation are not present, and that the loaded area on all sides (a typical splitting is of the block or penetration of the condition for slab loading, except at slab wedge is retarded.” edges), the surrounding concrete confines the bearing area, resulting in an increase in This article has not included the effects bearing strength. No minimum depth is given of load impact, lateral forces, non-smooth for a support material. The minimum depth (rough surfaces), elevated slabs, aggregate of support will be controlled by the shear types and sizes, concrete mix designs, wear, requirements.” durability and fatigue from cyclical loading, soil interaction, punching shear, pre-existing Or as Warren Young states in Roark’s Formulas cracks, joints and slab edges or the beneficial for Stress and Strain (1968), “... because of the effects of reinforcement. In his 2006 paper facts that the stress is highly localized and “Concentrically Loaded Circular Steel Plates triaxial, the actual stress intensity can be very Bearing on Plain Concrete” (ASCE Journal of high without producing apparent damage.” Structural Engineering, 2006) Edgard Escobar- Sandoval indicates that under some conditions, In their article “Bearing Capacity of Concrete the amplification factor permitted in the ACI Blocks” (Journal of the American Concrete equation for nominal bearing strength “is not Institute, 1960) Au Tung and Donald Baird conservative and appears to over-predict the conducted a series of tests of loads bearing on ultimate load for most concrete strengths,” and equal-sided cubes and on blocks with a depth of that in other conditions it is conservative. The half the sides. For the cubes, they observed the conservative conditions tend to be where the vertical cracks starting near the top of the cube loaded area is much smaller than the overall and progressing downward and the formation concrete area, as in a load on a slab. Thus, of an inverted cone/pyramid under the plates considering the current level of knowledge Philip Terry, PE, is an that was forced downward to split the cube. For about bearing, it would not be conservative to associate structural engineer the blocks (with reduced depth), they did not use a larger amplification factor than permitted in the Burns & McDonnell Aviation by ACI. Group. He has more than 32 years of structural engineering experience.

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