TECHBriefs www.burnsmcd.com A Burns & McDonnell Publication 2011 No. 1

By Philip Terry, PEContact mechanics involves the study of forces transmitted from one surface/solid to another and the consequent stresses in those solids. Initially, when non-conforming solids come into contact, they touch at a point or along a line. Under load, the solids deform so that the contact area increases, but the contact areas can still be very small, causing stresses to be intense. Typically, structural engineers designing concrete slabs do not need to consider contact mechanics. However, in two recent projects, Burns & McDonnell structural engineers found the need to use principles from the area of contact mechanics to evaluate the bearing pressures of steel wheels on slab-on-grade concrete floors.

In both projects, the facility owners desired to move heavy items over their concrete floors. In designing the cart/dolly for moving the heavy items, the cart designers consulted manufacturers’ catalogs to select the wheels that would support the heavy loads, but they did not consider the interaction of the curved steel wheels and the concrete surface. They did not try to keep the maximum compressive stresses within the elastic ranges of both materials. Using contact mechanics, we evaluated the interaction between the steel wheel and the concrete and predicted possible harm to the concrete floors.

One might think that the concrete is very rigid and durable and that there is little or no need to consider the contact of the wheels on the concrete. This may be true for small loads and large plates or solids of flexible, conforming materials, but steel is a very rigid material compared to concrete.

From rudimentary structural mechanics in the elastic range, deflection is related to load by the value of Modulus of Elasticity (also known as Young’s Modulus), E, such that Deflection = Load x E. The Modulus of Elasticity varies with the material being considered. For this discussion, the typical value of E for steel is 29,000,000 psi (Figure 1) while for various strengths of concrete, E values are shown in Table 1. A material with a smaller value of E will compress more than a material with a larger value of E. Materials can be deformed beyond their ability to recover, beyond their elastic limits.

Consider a heavy load supported by a steel wheel and the steel wheel supported by concrete.If the areas of the steeland the concrete are the same and the elastic limitsof the materialshave not been exceeded,the concrete willdeformmore than the steel. Contact mechanics is used to determine the width ofthe small contactarea interface.

ContactMechanicsAnalyzing the Effect of Steel Wheels on Concrete Slabs

Figure 1: Stress calculations consider the relative elasticity of materials in contact under load conditions.

SteelE= 29 million psi

ConcreteE= 3.6 million psi

Defle

ctio

n/St

rain

Load/Stress

Modulus of Elasticity (E) in Elastic Range

Concrete f'c (psi) E concrete (psi)2Maximum Nominal

Bearing Strength (psi)3

3,000 3,122,000 5,100

4,000 3,605,000 6,800

5,000 4,030,000 8,500

1 f’c = concrete compressive strength, psi = pounds per square inch.2 E concrete = 57,000√(f’c) (from ACI 318)3 ACI 318-08, Section 10.14, Maximum Nominal Bearing Strength, B

N = 0.85 x f’c

x 2.0., where 2.0 is the maximum amplification factor permitted; accounting

for the effects of confinement provided by the surrounding concrete.

Table 1: Modulus of Elasticity (E) for various concrete compressive strengths.

TECHBriefs 2011 No. 1 2 Burns & McDonnell

Contact StressesUsing the cylinder analogy from Roark’s Formulas for Stress and Strain by Warren C. Young, the contact stresses can be calculated according to the formulas shown in Figure 3.

Calculation ExampleAs an example, consider a 6-inch diameter, steel wheel with a length of 3 inches and an unfactored load of 12,000 pounds normal to a concrete slab with a concrete strength of 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 given in Figure 3, the unfactored load per unit length, p, is 4,000 pounds per inch. KD is 6 inches. Using the typical E1, E2, v1 and v2 values for concrete and steel, the value of CE is 2.9768 e-7; the bearing contact width, b, is calculated to be 0.1352 inches and the contact area would be b x L, or 0.4057 square inches.

Under an unfactored/service load of 12,000 pounds, the contact pressure on the concrete and steel is 29,577 psi. The maximum compressive stress within the concrete would be 37,764 psi. Under strength design, the dead-load portion is factored by 1.2 and the live-load portion is factored by 1.6. Thus, in this example, the factored/ultimate design bearing pressure is 1.4 x 29,577 = 41,408 psi. American Concrete Institute (ACI) 318 (building code requirements for reinforced concrete) reduces the ultimate 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.

According to N.M. Hawkins in “The Bearing Strength of Concrete Loaded Through Rigid Plates” (Magazine of Concrete Research, 1968) the initial cracking in the concrete slab under a loaded rigid plate is a vertical crack under the loaded plate (Figure 4, Label 1). At maximum 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.

b=1.6 √ p x KD x C

E (see Figure 2)

Where:

p = load per unit length = P/L

KD = D

2, (since the support material is flat, not curved)

CE = (1 – v

12)/E

1 + (1 – v

22)/E

2

E= See Figure 1 and Table 1.

Poisson’s Ratio for concrete, v1, is approximately 0.2 and

for steel, v2, is approximately 0.3. The subscript number

indicates the material. The maximum compressive stress, g

c, is 0.798 √p⁄(K

D x C

E ) and the maximum shear stress, ,

is 1/3 x oc .

Figure 3: Formulas for calculation of contact stress.

Load

Plate

Concreteslab

1 Crackinitiation

2 Conical wedge

Figure 4: Initial cracking of concrete slab under a loaded rigid plate.

Cylinder material 2

L(Length)

P(Total Load)

D(Diameter)

b(Contact width)

Support Material 1

Figure 2: This illustration of Young’s Cylinder analogy defines

the variables for calculating contact stresses.

Burns & McDonnell 3 TECHBriefs 2011 No. 1

ConclusionThe ultimate design bearing pressure of 41,408 psi is considerably higher than 4,420 psi; therefore it is expected that the steel wheel will cause the concrete to fail locally in bearing/crushing. Since the load will roll across the floor, there will be numerous failure locations. If redesign of the cart is not possible, then the concrete should be armored to distribute the load and protect the surface from wear.

The maximum nominal bearing strength values shown in Table 1 on page 1 include the maximum 2.0 amplification factor, which accounts for the permitted compressive stress increases due to the triaxial compressive stress condition of confined concrete. According to ACI 318, “When the supporting area is wider that the loaded area on all sides (a typical condition for slab loading, except at slab edges), the surrounding concrete confines the bearing area, resulting in an increase in bearing strength. No minimum depth is given for a support material. The minimum depth of support will be controlled by the shear requirements.”

Or as Warren Young states in Roark’s Formulas for Stress and Strain (1968), “... because of the facts that the stress is highly localized and triaxial, the actual stress intensity can be very high without producing apparent damage.”

In their article “Bearing Capacity of Concrete Blocks” (Journal of the American Concrete Institute, 1960) Au Tung and Donald Baird conducted a series of tests of loads bearing on equal-sided cubes and on blocks with a depth of half the sides. For the cubes, they observed the vertical cracks starting near the top of the cube and progressing downward and the formation of an inverted cone/pyramid under the plates that was forced downward to split the cube. For the blocks (with reduced depth), they did not

observe a clear-cut cone or pyramid. Instead, there was a vertical core with a narrow section at mid-depth (in some cases, appearing to form mirrored pyramids). Cracks appeared at the bottom of the block first and then progressed upward. They also observed that the failure load was consistently and slightly higher than the cube results. William Shelson, observing amplification factors larger than 2.0 for confined areas (higher ratios of footing area to loaded area) stated in his article “Bearing Capacity of Concrete” (Journal of the American Concrete Institute, 1957), “For comparatively shallow blocks on steel platens, the region directly beneath the loaded area is subject to a generally uniform compressive stress throughout the total depth. Conditions favorable to wedge [cone/pyramid] formation are not present, and splitting is of the block or penetration of the wedge is retarded.”

This article has not included the effects of load impact, lateral forces, non-smooth (rough surfaces), elevated slabs, aggregate types and sizes, concrete mix designs, wear, durability and fatigue from cyclical loading, soil interaction, punching shear, pre-existing cracks, joints and slab edges or the beneficial effects of reinforcement. In his 2006 paper “Concentrically Loaded Circular Steel Plates Bearing on Plain Concrete” (ASCE Journal of Structural Engineering, 2006) Edgard Escobar-Sandoval indicates that under some conditions, the amplification factor permitted in the ACI equation for nominal bearing strength “is not conservative and appears to over-predict the ultimate load for most concrete strengths,” and that in other conditions it is conservative. The conservative conditions tend to be where the loaded area is much smaller than the overall concrete area, as in a load on a slab. Thus, considering the current level of knowledge about bearing, it would not be conservative to use a larger amplification factor than permitted by ACI.

For more information, please e-mail: pterry@burnsmcd.com.

Philip Terry, PE, is an associate structural engineer in the Burns & McDonnell Aviation Group. He has more than 32 years of structural engineering experience.