How to Calculate the Bearing Capacity of Soils

How to Calculate the Bearing Capacity of Soils
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The ​bearing capacity of soil​ is given by the equation


in which ​Qa​ is the allowable bearing capacity (in kN/m2 or lb/ft2), ​Qu​ is the ultimate bearing capacity (in kN/m2 or lb/ft2) and FS is the safety factor. The ultimate bearing capacity ​Qu​ is the theoretical limit of the bearing capacity.

Much like how the Leaning Tower of Pisa leans due to the deformation of soil, engineers use these calculations when determining the weight of buildings and houses. As engineers and researchers lay down foundation, they need to make sure their projects are ideal for the ground that supports it. Bearing capacity is one method of measuring this strength. Researchers can calculate the bearing capacity of soil by determining the limit of contact pressure between the soil and the material placed on it.

These calculations and measurements are performed on projects involving bridge foundations, retaining walls, dams and pipelines that run underground. They rely on the physics of soil by studying the nature of the differences caused by pore water pressure of the material underlying the foundation and the inter-granular effective stress between the soil particles themselves. They also depend upon fluid mechanics of the spaces between soil particles. This accounts for cracking, seepage and the shear strength of the soil itself.

The following sections go into greater detail on these calculations and their uses.

Formula for Bearing Capacity of Soil

Shallow foundations include strip footings, square footings and circular footings. The depth is usually 3 meters and allow for cheaper, more feasible and more easily transferable results.

Terzaghi Ultimate Bearing Capacity Theory​ dictates that you can calculate the ultimate bearing capacity for shallow continuous foundations ​Qu with


in which ​c​ is the cohesion of soil (in kN/m2 or lb/ft2), ​g​ is the effective unit weight of soil (in kN/m3 or lb/ft3), ​D​ is the depth of footing (in m or ft) and B is the width of the footing (in m or ft).

For shallow square foundations, the equation is ​Qu with


and, for shallow circular foundations, the equation is


In some variations, the g is replaced with ​γ​.

The other variables depend upon other calculations. ​Nq​ is

N_q=\frac{e^{2\pi (0.75-\phi '/360)\tan{\phi '}}}{2\cos{(2(45+\phi '/2))}}

Nc​ is 5.14 for ​ф'=0​ and

N_C=\frac{N_q-1}{\tan{\phi '}}

for all other values of ф', ​Ng​ is:

N_g=\tan{\phi '}\frac{K_{pg}/\cos{2\phi '}-1}{2}

Kpg​ is obtained from graphing the quantities and determining which value of ​Kpg​ accounts for the trends observed. Some use ​Ng = 2(Nq+1)tanф'/(1+.4sin4​​ф')​ as an approximation without needing to calculate ​K​​pg.

There can be situations in which the soil shows signs of local ​shear failure​. This means the soil strength cannot show enough strength for the foundation because the resistance between the particles in the material isn't great enough. In these situations, the square foundation's ultimate bearing capacity is ​Qu = .867c Nc + g D Nq + 0.4 g B Ng ,​ the continuous foundation's i​s​ Qu = 2/3c Nc + g D Nq + 0.5 g B Ng and the circular foundation's is ​Qu= .867c Nc+ g D Nq+ 0.3 g B N​​g​.

Methods of Determining Bearing Capacity of Soil

Deep foundations include pier foundations and caissons. The equation for calculating ultimate bearing capacity of this type of soil is is ​Qu = Qp + Qf ​in which ​Qu​ is the ultimate bearing capacity (in kN/m2 or lb/ft2), ​Qp​ is the theoretical bearing capacity for the tip of the foundation (in kN/m2 or lb/ft2) and ​Qf is the theoretical bearing capacity due to shaft friction between the shaft and soil. This gives you another formula for bearing capacity of soil

You can calculate the theoretical end bearing (tip) capacity foundation ​Qp as ​Qp = Apqp​ in which ​Qp​ is the theoretical bearing capacity for the end bearing (in kN/m2 or lb/ft2) and ​Ap​ is the effective area of the tip (in m2 or ft2).

The theoretical unit tip-bearing capacity of cohesion-less silt soils ​qp​ is ​qDNq​ and, for cohesive soils, ​9c,​ (both in kN/m2 or lb/ft2). ​Dc​ is the critical depth for piles in loose silts or sands (in m or ft). This should be ​10B​ for loose silts and sands, ​15B​ for moderate density silts and sands and ​20B​ for very dense silts and sands.

For the skin (shaft) friction capacity of pile foundation, the theoretical bearing capacity ​Qf​ is ​Afqf for a single homogeneous soil layer and ​pSqfL​ for more than one layer of soil. In these equations, ​Af ​is the effective surface area of the pile shaft, ​qf​ is ​kstan(d)​, the theoretical unit friction capacity for cohesion-less soils (in kN/m2 or lb/ft) in which ​k​ is the lateral earth pressure, ​s​ is the effective overburden pressure and ​d​ is the external friction angle (in degrees). ​S​ is the summation of differing soil layers (i.e. ​a1​ + ​a2​ + .... + ​an​).

For silts, this theoretical capacity is ​cA+​ ​kstan(d)​ in which ​cA​ is the adhesion. It is equal to ​c,​ the cohesion of soil for rough concrete, rusty steel and corrugated metal. For smooth concrete, the value is ​.8c​ to ​c​, and, for clean steel, it is ​.5c​ to ​.9c​. ​p​ is the perimeter of the pile cross section (in m or ft). ​L​ is the effective length of the pile (in m or ft).

For cohesive soils, ​qf= aSu in which a is the adhesion factor, measured as ​1-.1(Suc)2​ for ​Suc​ less than 48 kN/m2 where ​Suc = 2c​ is the unconfined compression strength (in kN/m2 or lb/ft2). For ​Suc​ greater than this value, ​a = [0.9 + 0.3(Suc - 1)]/Suc​.

What is the Factor of Safety? 

The safety factor ranges from 1 to 5 for various uses. This factor can account for magnitude of damages, relative change in the chances a project may fail, the soil data itself, tolerance construction and accuracy of design methods of analysis.

For instances of shear failure, the safety factor varies from 1.2 to 2.5. For dams and fills, the safety factor ranges from 1.2 to 1.6. For retaining walls, it's 1.5 to 2.0, for shear sheet piling, it's 1.2 to 1.6, for braced excavations, it's 1.2 to 1.5, for shear spread footings, the factor is 2 to 3, for mat footings it is 1.7 to 2.5. By contrast, instances of seepage failure, as materials seep through small holes in pipes or other materials, the safety factor ranges from 1.5 to 2.5 for uplift and 3 to 5 for piping.

Engineers also use rules of thumb for the safety factor as 1.5 for retaining walls that are overturned with granular backfill, 2.0 for cohesive backfill, 1.5 for walls with active earth pressure and 2.0 for those with passive earth pressures. These safety factors help engineers avoid shear and seepage failures as well as the soil may move as a result of the load bearings on it.

Practical Calculations of Bearing Capacity

Armed with the test results, engineers calculate how much load the soil can safely bear. Beginning with the weight required to shear the soil, they add a safety factor so the structure never applies enough weight to deform the soil. They can adjust the footprint and depth of a foundation to stay within that value. Alternatively, they can compress the soil to increase its strength, by, for instance, using a roller to compact loose fill material for a roadbed.

Methods of determining bearing capacity of soil involve the maximum pressure that the foundation can exert onto the soil such that the acceptable safety factor against shear failure is below the foundation and the acceptable total and differential settlement are met.

The ultimate bearing capacity is the minimum pressure that would cause the shear failure of the supporting soil immediately below and adjacent to the foundation. They take into account the shear strength, density, permeability, internal friction and other factors when building structures on soil.

Engineers use their best judgement with these methods of determining bearing capacity of soil when performing many of these measurements and calculations. The effective length requires the engineer making a choice about where to start and stop measuring. As one method, the engineer may choose to use the pile depth and subtract any disturbed surface soils or mixtures of soils. The engineer may also choose to measure it as the length of a pile segment in a single soil layer of soil that consists of many layers.

What Causes Soils to Become Stressed?

Engineers need to account for soils as mixtures of individuals particles that move around with respect to one another. These units of soils can be studied to understand the physics behind these movements when determining the weight, force and other quantities with respect to the buildings and projects engineers build upon them.

Shear failure can result from the stresses applied to soil that cause the particles to resist one another and disperse in ways that are detrimental to building. For this reason, engineers must be careful in choosing designs and soils with appropriate shear strengths.

The ​Mohr Circle​ can visualize the shear stresses on the planes relevant to building projects. The Mohr Circle of Stresses is used in geological research of soil testing. It involves using cylinder-shaped samples of soils such that the radial and axial stresses act on the layers of soils, calculated using planes. Researchers then use these calculations to determine the bearing capacity of soils in foundations.

Classifying Soils by Composition

Researchers in physics and engineering can classify soils, sands and gravels by their size and chemical constituents. Engineers measure the specific surface area of these constituents as the ratio of the surface area of particles to the mass of the particles as one method of classifying them.

Quartz is the most common component of silt and sand and mica and feldspar are other common components. Clay minerals like montmorillonite, illite and kaolinite make up sheets or structures that are plate-like with large surface areas. These minerals have specific surface ares from 10 to 1,000 square meters per gram of solid.

This large surface area allows for chemical, electromagnetic and van der Waals interactions. These minerals can be very sensitive to the amount of fluid that may pass through their pores. Engineers and geophysicists can determine the types of clays present in various projects to calculate the effects of these forces to account for them in their equations.

Soils with high-activity clays can be very unstable because they are very sensitive to fluid. They swell in the presence of water and shrink in its absence. These forces can cause cracks in the physical foundation of buildings. On the other hand, materials that are low-activity clays that are formed under more stable activity can be much more easy to work with.

Soil Bearing Capacity Chart has a list of soil bearing capacity values you can use as a soil bearing capacity chart.

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