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• Marc Mathieu

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8/22/2017

FOUNDATION ENGINEERING

Shallow Foundations- Bearing Capacity Prepared by: Engr. Marc Lin F. Abonales

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

INTRODUCTION The lowest past of a structure is generally referred to as the foundation. A properly designed foundation is one that transfers the load throughout the soil without overstressing the soil. Thus, geotechnical and structural engineers who design foundations must evaluate the bearing capacity of soils.

Can result in either excessive settlement or shear failure of the soil, both of which cause damage to the structure.

Its function is to transfer the load of the structure to the soil on which it is resting.

Shallow foundations – are those foundations that have a depth-of-embedment-to-width ratio of approximately less than four.

To perform satisfactorily, shallow foundations must have two main characteristics: 1. They have to be safe against overall shear failure in the soil that supports them. 2. They cannot undergo excessive displacement, or settlement.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

INTRODUCTION

TYPES OF FOUNDATION

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

GENERAL CONCEPTS Bearing capacity: The capacity of soil to support the loads applied to the ground. The bearing capacity of soil is the maximum average contact pressure between the foundation and the soil which should not produce shear failure in the soil. Ultimate bearing capacity: The load per unit area of the foundation at which shear failure in soil occurs. The maximum bearing capacity of soil at which the soil fails by shear.

Three modes of failure: a)

General Shear Failure

Most common type of shear failure; occurs in strong soils and rocks b) Local Shear Failure

Intermediate between general and punching shear failure c) Punching Shear Failure Occurs in very loose sands weak clays

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

GENERAL CONCEPTS Consider a strip (i.e., theoretically length is infinity) foundation resting on the surface of a dense sand or stiff cohesive soil, as shown in the figure, with a width of B. If load is gradually applied to the foundation, settlement will increase. The variation of the load per unit area on the foundation, q, with the foundation settlement is also shown in the figure. At a certain point – when the load per unit area equals qu – a sudden failure in the soil supporting the foundation will take place, and the failure surface in the soil will extend to the ground surface. This load per unit area, qu, is usually referred to as the ultimate bearing capacity of the foundation. When this type of sudden failure in soil takes place, it is called general shear failure.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

GENERAL CONCEPTS If the foundation under consideration rests on sand or clayey soil of medium compaction, an increase of load on the foundation will also be accompanied by an increase of settlement. However, in this case the failure surface in the soil will gradually extend outward from the foundation, as shown by the solid lines in the figure. When the load per unit area on the foundation equals qu(1), the foundation movement will be accompanied by sudden jerks. A considerable movement of the foundation is then required for the failure surface in soil to extend to the ground surface (as shown in broken lines in the figure). The load per unit area at which this happens is the ultimate bearing capacity, qu. Beyond this point, an increase of load will be accompanied by a large increase of foundation settlement. The load per unit area of the foundation, qu(1), is referred to as the first failure load. Note that a peak value of q is not realized in this type of failure, which is called local shear failure in soil.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

GENERAL CONCEPTS If the foundation is supported by a fairly loose soil, the load-settlement plot will be like the one in the figure. In this case, the failure surface in soil will not extend to the ground surface. Beyond the ultimate failure load, qu, the load-settlement plot will be steep and practically linear. This type of failure in soil is called punching shear failure.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

GENERAL CONCEPTS

General shear failure results in a clearly defined plastic yield slip surface beneath the footing and spreads out one or both sides, eventually to the ground surface. Failure is sudden and will often be accompanied by severe tilting. Generally associated with heaving. This type of failure occurs in dense sand or stiff clay.

Local shear failure results in considerable vertical displacement prior to the development of noticeable shear planes. These shear planes do not generally extend to the soil surface, but some adjacent bulging may be observed, but little tilting of the structure results. This shear failure occurs for loose sand and soft clay.

Punching shear failure occurs in very loose sands and soft clays and there is little or no development of planes of shear failure in the underlying soil. Slip surfaces are generally restricted to vertical planes adjacent to the footing, and the soil may be dragged down at the surface in this region.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

GENERAL CONCEPTS

Load settlement curves for different shear

From the curves the different types of shear failures can be predicted : • For general shear failure there is a pronounced peak after which load decreases with increase in settlement. The load at the peak gives the ultimate stress or load. • For local shear failure there is no pronounced peak like general shear failure and hence the ultimate load is calculated for a particular settlement. • For punching shear failure the load goes on increasing with increasing settlement and hence there is no peak resistance.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

GENERAL CONCEPTS

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY Terzaghi (1943) was the first to present a comprehensive theory for evaluating of rough shallow foundations. According to this theory, a foundation is shallow if the depth, Df, of the foundation is less than or equal to the width of the foundation. Later investigators, however, have suggested that foundations with Df equal to 3 to 4 times the width of the foundation may be defined as shallow foundation. Terzaghi suggested that for a continuous, or strip foundation (that is, the width-to-length ratio of the foundation approaches 0), the failure surface in soil at ultimate load may be assumed to be similar to that shown below (This is the case of general shear failure). The effect of soil above the bottom of the foundation may also be assumed to be replaced by an equivalent surcharge, q = γDf (where: γ = unit weight of the soil).

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY Assumptions for Terzaghi's Method: • Depth of foundation is less than or equal to its width • No sliding occurs between foundation and soil (rough foundation) • Soil beneath foundation is homogeneous semi infinite mass • Mohr-Coulomb model for soil • General shear failure mode is the governing mode (but not the only mode) • No soil consolidation occurs • Foundation is very rigid relative to the soil • Soil above bottom of foundation has no shear strength; is only a surcharge load against the overturning load • Applied load is compressive and applied vertically to the centroid of the foundation • No applied moments present

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY The failure zone under the foundation can be separated into three parts: 1. The triangular zone ACD immediately under the foundation. (Active zone) 2. The radial shear zones ADF and CDE, with the curves DE and DF being arcs of a logarithmic spiral. (Transition zone) 3. Two triangular Rankine passive zones AFH and CEG. (Passive zone) The angles CAD and ACD are assumed to be equal to the soil friction angle (that is, α = Φ’). Note that, with the replacement of the soil above the bottom of the foundation by an equivalent surcharge q, the shear resistance of the soil along the failure surfaces GI and HJ was neglected.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY Using the equilibrium analysis, Terzaghi expressed the ultimate bearing capacity in the form (For foundations that exhibit the general shear failure mode in soils)

𝟏 𝒒𝒖 = 𝒄′ 𝑵𝒄 + 𝒒𝑵𝒒 + 𝜸𝑩𝑵𝜸 (for continuous or strip foundation) 𝟐 𝒒𝒖 = 𝟏. 𝟑𝒄′ 𝑵𝒄 + 𝒒𝑵𝒒 + 𝟎. 𝟒𝜸𝑩𝑵𝜸 (for square foundation) 𝒒𝒖 = 𝟏. 𝟑𝒄′ 𝑵𝒄 + 𝒒𝑵𝒒 + 𝟎. 𝟑𝜸𝑩𝑵𝜸

(for circular foundation)

Where: B = width for square footing = diameter for circular footing c' = cohesion of soil γ = unit weight of soil q = γDf Nc, Nq, Nγ = bearing capacity factors that are nondimensional and are only functions of the soil friction angle, Φ’

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY

𝒒𝒖 = 𝒒𝒄 + 𝒒𝒒 + 𝒒𝜸 𝟏 𝒒𝒖 = 𝒄 𝑵𝒄 + 𝒒𝑵𝒒 + 𝜸𝑩𝑵𝜸 𝟐 ′

Cohesion term

Above foundation level

Below foundation level

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY The bearing capacity factors Nc, Nq, and Nγ are defined by:

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY For foundations that exhibit the local shear failure mode in soils, Terzaghi suggested the following modifications:

𝐜′ =

𝟐 ′ 𝐜 𝟑

𝟐 ′ 𝟏 (for continuous or strip foundation) 𝒄 𝑵′𝒄 + 𝒒𝑵′𝒒 + 𝜸𝑩𝑵′𝜸 𝟑 𝟐 𝒒𝒖 = 𝟎. 𝟖𝟔𝟕𝒄′ 𝑵′𝒄 + 𝒒𝑵′𝒒 + 𝟎. 𝟒𝜸𝑩𝑵′𝜸 (for square foundation)

𝒒𝒖 =

𝒒𝒖 = 𝟎. 𝟖𝟔𝟕𝒄′ 𝑵𝒄 + 𝒒𝑵𝒒 + 𝟎. 𝟑𝜸𝑩𝑵′𝜸

(for circular foundation)

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY N’c, N’q, and N’γ, the modified bearing capacity factors, can be calculated by using the bearing capacity factor equations (for Nc, Nq, and Nγ, respectively) by replacing

𝛟′ = 𝐭𝐚𝐧−𝟏

𝟐 ′ 𝛟 𝟑

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

TERZAGHI’S ULTIMATE BEARING CAPACITY THEORY Terzaghi’s bearing capacity equations have now been modified to take into account the effects of the foundation shape (B/L), depth of embedment (Df) and the load inclination. Many design engineers, however, still use Terzaghi’s equation, which provides fairly good results considering the uncertainty of the soil conditions at various sites.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

FACTOR OF SAFETY Calculating the gross allowable loadbearing capacity of shallow foundations requires the application of a factor of safety (FS) to the gross ultimate bearing capacity, or

𝐪𝐚𝐥𝐥 =

𝐪𝐮 𝐅𝐒

The net ultimate bearing capacity is defined as the ultimate pressure per unit area of the foundation that can be supported by the soil in excess of the pressure caused by the surrounding soil at the foundation level. If the difference between the unit weight of concrete used in the foundation and the unit weight of soil surrounding is assumed to be negligible, then

However, some practicing engineers prefer to use a factor of safety such that

The factor of safety as defined by the preceding equation should be at least 3 in all cases.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 1.

PROBLEM 2.

Based from Problem 1, assume that local shear failure occurs in the soil.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

THE GENERAL BEARING CAPACITY EQUATION The ultimate bearing capacity equations based from Terzaghi’s equations are for continuous, square, and circular foundations only; they do not address the case of rectangular foundations (0 < B/L < 1). Also, The equations do no take into account the shearing resistance along the failure surface in soil above the bottom of the foundation. In addition, the load on the foundation may be inclined. To account for all these shortcomings, Meyerhof (1963) suggested the following form of the general bearing capacity equation:

qu = c’NcFcsFcdFci + qNqFqsFqdFqi + ½ γBNγFγsFγdFγi Where: c‘ = cohesion q = effective stress at the level of the bottom of the foundation γ = unit weight of soil B = width of foundation ( = diameter for a circular foundation) Fcs, Fqs, Fγs = shape factors Fcd, Fqd, Fγd = depth factors Fci, Fqi, Fγi = load inclination factors Nc, Nq, Nγ = bearing capacity factors

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

THE GENERAL BEARING CAPACITY EQUATION Bearing Capacity Factors Based on laboratory and field studies of bearing capacity, the basic nature of the failure surface in soil suggested by Terzaghi now appears to be correct (Vesic, 1973). However, the angle α shown in is closer to 45 + ϕ’/2 than to ϕ’, as was originally assumed by Terzaghi. With α = 45 + ϕ’/2, the relations for Nc and Nq can be derived as

The equation for Nc was originally derived by Prandtl (1921), and the relation for Nq was presented by Reissner (1924). Caquot and Kerisel (1953) and Vesic (1973) gave the relation for Nγ as

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

THE GENERAL BEARING CAPACITY EQUATION Bearing Capacity Factors The table shows the variation of the preceding bearing capacity factors with soil friction angles.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

THE GENERAL BEARING CAPACITY EQUATION Shape, Depth, Inclination Factors

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

THE GENERAL BEARING CAPACITY EQUATION Shape, Depth, Inclination Factors

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

THE GENERAL BEARING CAPACITY EQUATION Shape, Depth, Inclination Factors

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 3. Solve problem 1 using the general bearing capacity equation. PROBLEM 4.

The applied load on a shallow square foundation makes an angle of 15° with the vertical. Given: B = 6 ft, Df = 3 ft, γ = 115 lb/ft3, ϕ’ = 25°, and c’ = 500 lb/ft2. Use FS = 4 and determine the gross allowable load using the general bearing capacity equation.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

MODIFICATION OF BEARING CAPACITY EQUATIONS FOR WATER TABLE The preceding equations give the ultimate bearing capacity, based on the assumptions that the water table is located well below the foundation. However, if the water table is close to the foundation, some modifications of the bearing capacity equations will be necessary.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

MODIFICATION OF BEARING CAPACITY EQUATIONS FOR WATER TABLE The preceding equations give the ultimate bearing capacity, based on the assumptions that the water table is located well below the foundation. However, if the water table is close to the foundation, some modifications of the bearing capacity equations will be necessary.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 5. A column foundation is 3 m x 3 m in plan. Given: Df = 2 m, ϕ’ = 25°, c’ = 70 kPa. If FS = 3, determine the net allowable load the foundation could carry. a. Using Terzaghi’s Bearing Capacity Equation b. Using General Bearing Capacity Equation

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 6.

Use the Terzaghi’s Equation and General Bearing Capacity Equation

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Shallow Foundations-Bearing Capacity

EFFECT OF SOIL COMPRESSIBILITY Some Terzaghi’s bearing capacity equations which apply to the case of general shear failure were modified to Terzaghi’s bearing capacity equations in case of local shear failure to take into account the change of failure mode in soil. The change of failure mode is due to soil compressibility, to account for which Vesic (1973) proposed the following modifications of the General Bearing Capacity Equation that was proposed by Meyerhof:

qu = c’NcFcsFcdFcc + qNqFqsFqdFqc + ½ γBNγFγsFγdFγc Where: Fcc, Fqc, and Fγc are soil compressibility factors. The soil compressibility factors were derived by Vesic (1973) by analogy to the expansion of cavities.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EFFECT OF SOIL COMPRESSIBILITY According to that theory, in order to calculate Fcc, Fqc, and Fγc, the following steps should be taken: Step 1. Calculate the rigidity index, Ir, of the soil at a depth approximately B/2 below the bottom of the foundation, or

𝐈𝐫 = Where: Gs = shear modulus of the soil =

𝐆𝐬 𝐜 ′ + 𝐪′ 𝐭𝐚𝐧∅′

𝐄𝐬 𝟐(𝟏+𝛍𝐬 )

Es = Modulus of Elasticity µs = Poisson’s ratio q = effective overburden pressure at a depth of Df + B/2 Step 2. The critical rigidity index, Ir(cr), can be expressed as 𝐈𝐫(𝐜𝐫) =

𝟏 𝐞𝐱𝐩 𝟐

𝟑. 𝟑𝟎 − 𝟎. 𝟒𝟓

𝐁 ∅′ 𝐜𝐨𝐭 𝟒𝟓 − 𝐋 𝟐

The variations of Ir(cr) with B/L are given in the next table.

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Shallow Foundations-Bearing Capacity

EFFECT OF SOIL COMPRESSIBILITY

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EFFECT OF SOIL COMPRESSIBILITY According to that theory, in order to calculate Fcc, Fqc, and Fγc, the following steps should be taken: Step 3. If Ir ≥ Ir(cr), then Fcc, Fqc, and Fγc = 1 However, if Ir < Ir(cr), then 𝐅𝛄𝐜 = 𝐅𝐪𝐜 = 𝐞𝐱𝐩

−𝟒. 𝟒 + 𝟎. 𝟔

𝐁 (𝟑. 𝟎𝟕𝐬𝐢𝐧∅′ )(𝐥𝐨𝐠𝟐𝐈𝐫 ) 𝐭𝐚𝐧∅′ + 𝐋 𝟏 + 𝐬𝐢𝐧∅′

For Φ = 0, 𝐅𝐜𝐜 = 𝟎. 𝟑𝟐 + 𝟎. 𝟏𝟐 For Φ’ > 0,

𝐁 + 𝟎. 𝟔𝟎𝐥𝐨𝐠𝐈𝐫 𝐋

𝟏 − 𝐅𝐪𝐜 𝐍𝐜 𝐭𝐚𝐧∅′ Note: The bearing capacity factors to be used is the same table used in the general bearing capacity equations. 𝐅𝐜𝐜 = 𝐅𝐪𝐜 −

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Shallow Foundations-Bearing Capacity

EFFECT OF SOIL COMPRESSIBILITY

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 7. For a shallow foundation, B = 0.6 m, L = 1.2 m, and Df = 0.6 m. the known soil characteristics are as follows: Soil: • Φ’ = 25° • c’ = 48 kPa • γ = 18 kN/m3 • Modulus of Elasticity, Es = 620 kPa • Poisson’s Ratio, μs = 0.3

Calculate the ultimate bearing capacity.

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Shallow Foundations-Bearing Capacity

ECCENTRICALLY LOADED FOUNDATIONS In several instances, as with the base of a retaining wall, foundations are subjected to moments in addition to the vertical load, as shown in the figure.

In such cases, the distribution of pressure by the foundation on the soil is not uniform. The nominal distribution of pressure is

The exact distribution of pressure is difficult to estimate.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ECCENTRICALLY LOADED FOUNDATIONS

The figure (b) shows a force system equivalent to that shown in the figure (a).

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Shallow Foundations-Bearing Capacity

ECCENTRICALLY LOADED FOUNDATIONS The distance, e = M/Q, is the eccentricity. The nominal distribution of pressure is

Case 2: e = B/6

Case 1: e < B/6

qmin = 0 Case 3: e > B/6

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ECCENTRICALLY LOADED FOUNDATIONS The figure below shows the nature of failure surface in soil for a surface strip foundation subjected to an eccentric load. The factor of safety for such type of loading against bearing capacity failure can be evaluated as

𝐅𝐒 =

𝐐𝐮𝐥𝐭 𝐐

Where: Qult = ultimate load-carrying capacity

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Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – ONE-WAY ECCENTRICITY A. Effective Area Method (Meyerhof, 1953)

Step 2. Use the general bearing capacity equation for the ultimate bearing capacity:

In 1953, Meyerhof proposed a theory that is generally referred to as the effective area method. The following is a step-by-step procedure for determining the ultimate load that the soil can support and the factor of safety against bearing capacity failure: Step 1. Determine the effective dimensions of the foundation: B’ = effective width = B – 2e L’ = effective length = L

q'u = c’NcFcsFcdFci + qNqFqsFqdFqi + ½ γB’NγFγsFγdFγi

To evaluate Fcs, Fqs, and Fγs, use the effective length and effective width dimensions instead of L and B, respectively. To determine Fcd,Fqd,and Fγd, do not replace B with B’. Step 3. The total ultimate load that the foundation can sustain is Qult = q’u(B’)(L’) = q’u(A’)

Note that if the eccentricity were in the direction of the length of the foundation, the value of L’ would be equal to L – 2e. The value of B’ would equal B. The smaller of the two dimensions (i.e., L’ and B’) is the effective width of the foundation.

Where: A’ = effective area Step 4. The factor of safety against bearing capacity failure is 𝐐𝐮𝐥𝐭 𝐅𝐒 = 𝐐

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – ONE-WAY ECCENTRICITY B. Prakash and Saran Theory Prakash and Saran (1971) analyzed the problem of ultimate bearing capacity of eccentrically and vertically loaded continuous (strip) foundations by using the onesided failure surface in soil. According to this theory, the ultimate load per unit length of a continuous foundation can be estimated as

Prakash and Saran (1971) also recommended the following for the shape factors:

•

Fcs(e) = 1.2 – 0.025

•

Fqs(e) = 1

•

Fγs(e) = 𝟏. 𝟎 +

𝟐𝐞 𝐁

𝐋 𝐁

(with a minimum of 1.0)

− 𝟎. 𝟔𝟖

𝐁 + 𝐋

𝟎. 𝟒𝟑 −

𝟑 𝟐

𝐞 𝐁

𝐁 𝟐 𝐋

Qult = B[c’Nc(e) + qNq(e) + ½ γBNγ(e)] Where: Nc(e), Nq(e), Nγ(e) = bearing capacity factors under eccentric loading. For rectangular foundations, the ultimate load can be given as Qult = BL[c’Nc(e)Fcs(e) + qNq(e)Fqs(e) + ½ γBNγ(e)Fγs(e)] Where: Fcs(e), Fqs(e), and Fγs(e) = shape factors

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Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – ONE-WAY ECCENTRICITY The variations of Nc(e), Nq(e) and Nγ(e) with soil friction angle Φ’ are given in the following figures:

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – ONE-WAY ECCENTRICITY C. Reduction Factor Method (For Granular Soil) Purkayastha and Char (1977) carried out stability analysis of eccentrically loaded continuous foundations (Fqs = 1, Fγs = 1) supported by a layer of sand using the method of slices. Based on that analysis, they proposed 𝐑𝐤 = 𝟏 −

𝐪𝐮(𝐞𝐜𝐜𝐞𝐧𝐭𝐫𝐢𝐜) 𝐪𝐮(𝐜𝐞𝐧𝐭𝐫𝐢𝐜)

Where: Rk = reduction factor qu(eccentric) = ultimate bearing capacity of eccentrically loaded continuous foundations qu(centric) = ultimate bearing capacity of centrally loaded continuous foundation

The magnitude of Rk can be expressed as

𝑹𝒌 = 𝒂

𝒆 𝑩

𝒌

Where a and k are functions of the embedment ratio Df/B. Hence,

qu(eccentric) = qu(centric) (1 – Rk) = qu(centric) 𝟏 − 𝒂

𝒆 𝒌 𝑩

Where

qu(centric) = qNqFqd + ½ γBNγFγd The relationships for Fqd and Fγd are the same used in general bearing capacity equations. The ultimate load per unit length of the foundation can then be given as Qu = Bqu(eccentric)

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Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – ONE-WAY ECCENTRICITY C. Reduction Factor Method (For Granular Soil)

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 8. A continuous foundation is shown in the figure below. If the load eccentricity is 0.2 m, determine the ultimate load, Qult, per unit length of the foundation. Use: a. Meyerhof’s effective area method b. Prakash and Saran Theory c. Reduction Factor Method

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Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY Consider a situation in which a foundation is subjected to a vertical ultimate load Qult and a moment M, as shown in figures (a) and (b). For this case, the components of the moment M about the x- and y-axes can be determined as Mx and My, respectively, as shown figure (c). This condition is equivalent to a load Qult placed eccentrically on the foundation with x = eB and y = eL, as shown in figure (d).

Analysis of foundation with two-way eccentricity

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY Therefore, 𝐞𝐁 =

𝐌𝐲 𝐐𝐮𝐥𝐭

If Qult is needed, it can be obtained from

𝐚𝐧𝐝

𝐞𝐋 =

𝐌𝐱 𝐐𝐮𝐥𝐭

Qult = q’uA’

Where: q’u = c’NcFcsFcdFci + qNqFqsFqdFqi + ½ γB’NγFγsFγdFγi A’ = effective area = B’L’ As before, to evaluate Fcs, Fqs, and Fγs (the same formulas used in the general bearing capacity equation), use the effective length L’ and effective width B’ instead of L and B, respectively. To calculate Fcd, Fqd, and Fγd (the same formulas used in the general bearing capacity equation), do not replace B with B’.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY In determining the effective area, A’, effective width, B’, and effective length, L’, five possible cases may arise (Higher and Anders, 1985). Case I. eL/L ≥ 1/6 and eB/B ≥ 1/6. The effective area for this condition is shown in the figure, or A’ = ½ B1L1 Where: 𝟑𝐞𝐁 𝐁𝟏 = 𝐁 𝟏. 𝟓 − 𝐁 𝟑𝐞𝐋 𝐋𝟏 = 𝐋 𝟏. 𝟓 − 𝐋 The effective length L’ is the larger of the two dimensions B1 and L1. So the effective width is 𝐀′ 𝐁′ = 𝐋′

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY In determining the effective area, A’, effective width, B’, and effective length, L’, five possible cases may arise (Higher and Anders, 1985). Case II. eL/L < 0.5 and 0 < eB/B < 1/6. The effective area for this case, shown in the figure, is A’ = ½ (L1 + L2)B

The magnitude of L1 and L2 can be determined from the next slide. The effective width is 𝐀′ 𝐋𝟏 𝐨𝐫 𝐋𝟐 (𝐰𝐡𝐢𝐜𝐡𝐞𝐯𝐞𝐫 𝐢𝐬 𝐥𝐚𝐫𝐠𝐞𝐫) The effective length is 𝐁′ =

L’ = L1 or L2 (whichever is larger)

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY

For Case II.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

For Case II.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY In determining the effective area, A’, effective width, B’, and effective length, L’, five possible cases may arise (Higher and Anders, 1985). Case III. eL/L < 1/6 and 0 < eB/B < 0.5. The effective area for this case, shown in the figure, is A’ = ½ (B1 + B2)L The effective width is B’ = A’/L The effective length is L’ = L

The magnitudes of B1 and B2 can be determined from the next slide.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY

For Case III.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY In determining the effective area, A’, effective width, B’, and effective length, L’, five possible cases may arise (Higher and Anders, 1985). Case IV. eL/L < 1/6 and eB/B < 1/6. The effective area for this case, shown in the figure, is A’ = L2B + ½ (B + B2)(L – L2) The ratio B2/B, and thus B2, can be determined by using the eL/L curves that slope upward. Similarly, the ratio L2/L, and thus L2, can be determined by using the eL/L curves that slope downward. (Refer to the figure found in the next slide.) The effective width is B’ = A’/L The effective length is L’ = L

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY

For Case IV.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

For Case IV.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

ULTIMATE BEARING CAPACITY UNDER ECCENTRIC LOADING – TWO-WAY ECCENTRICITY In determining the effective area, A’, effective width, B’, and effective length, L’, five possible cases may arise (Higher and Anders, 1985). Case V. In the case of circular foundations under eccentric loading, the eccentricity is always one-way. The effective area, A’, and the effective width, B’, for a circular foundation are given in a nondimensional form in the table found in the next slide. Once A’ and B’ are determined, the effective length can be obtained as

𝐋′ =

𝐀′ 𝐁′

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

For Case V.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 9. A square foundation is shown in the figure with eL = 0.3 m and eB = 0.15 m. Assume two-way eccentricity, and determine the ultimate load, Qult.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 10. Consider the foundation in problem 9 with the following changes: eL = 0.18 m eB = 0.12 m For the soil, γ = 16.5 kN/m3 Φ’ = 25° c’ = 25 kPa

Determine the ultimate load, Qult.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

BEARING CAPACITY OF A CONTINUOUS FOUNDATION SUBJECTED TO ECCENTRIC INCLINED LOADING The problem of ultimate bearing capacity of a continuous foundation subjected to an eccentric inclined load was studied by Saran and Agarwal (1991). If a continuous foundation is located at a depth Df below the ground surface and is subjected to an eccentric load (load eccentricity = e) inclined at an angle β to the vertical, the ultimate capacity can be expressed as

𝑸𝒖𝒍𝒕 = 𝑩 𝒄′ 𝑵𝒄

𝒆𝒊

+ 𝒒𝑵𝒒

𝒆𝒊

𝟏 + 𝜸𝑩𝑵𝜸 𝟐

𝒆𝒊

Where: Nc(ei), Nq(ei), and Nγ(ei) = bearing capacity factors. The variations of the bearing capacity factors with e/B, Φ’, and β derived by Saran and Agarwal are given in figures in the next slides.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

BEARING CAPACITY OF A CONTINUOUS FOUNDATION SUBJECTED TO ECCENTRIC INCLINED LOADING

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

BEARING CAPACITY OF A CONTINUOUS FOUNDATION SUBJECTED TO ECCENTRIC INCLINED LOADING

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

BEARING CAPACITY OF A CONTINUOUS FOUNDATION SUBJECTED TO ECCENTRIC INCLINED LOADING

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE PROBLEM 11. A continuous foundation is shown in the figure. Estimate the ultimate load, Qult, per unit length of the foundation.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

MAT FOUNDATIONS – COMMON TYPES Mat foundations are shallow foundations which is sometimes referred to as a raft foundation. It is a combined footing that may cover the entire area under a structure supporting several columns and walls. Mat foundations are sometimes preferred for soils that have low-bearing capacities but that will have to support high column and/or wall loads. Under some conditions, spread footings would have to cover more than half the building area, and mat foundations might be more economical. Several types of mat foundations are currently used. Some of the common types are shown schematically in the figures and include the following: 1. Flat plate (Figure a). The mat is of uniform thickness. 2. Flat plate thickened under columns. (Figure b) 3. Beams and slab (Figure c). The beams run both ways, and the columns are located at the intersection of the beams. 4. Flat plates with pedestals (Figure d). 5. Slab with basement walls as a part of the mat (Figure e). The walls act as stiffeners for the mat.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

MAT FOUNDATIONS – COMMON TYPES Mats may be supported by piles. The piles help in reducing the settlement of a structure built over highly compressible soil. Where the water table is high, mats are often placed over piles to control buoyancy. The figure shows the difference between the depth Df and the width B of isolated foundations and mat foundation. The next figure shows a mat foundation under construction.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

BEARING CAPACITY OF MAT FOUNDATIONS The gross ultimate bearing capacity A suitable factor of safety should be used to of a mat foundation can be determined by the calculate the net allowable bearing capacity: same equation used for spread footings, or • For mats on clay, the factor of safety should not be less than 3 under dead load and maximum live load. However, under the most extreme conditions, the factor of safety should be at least 1.75 to 2. The proper values of the bearing • For mats constructed over sand, a factor of safety capacity factors and the shape, depth, and of 3 should normally used. load inclination factors are the same used in • Under most working conditions, the factor of safety against bearing capacity failure of mats on the general bearing capacity equation. The sand is very large. term B in the equation is the smallest qu = c’NcFcsFcdFci + qNqFqsFqdFqi + ½ γBNγFγsFγdFγi

dimension of the mat. The net ultimate bearing capacity is qnet(u) = qu – q

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE

PROBLEM 12. Determine the net ultimate bearing capacity of a mat foundation measuring 12 m x 8 m on a saturated clay with cu = 80 kPa, Φ = 0, and Df = 2 m.

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

COMPENSATED FOUNDATIONS The settlement of a mat foundation can be reduced by decreasing the net pressure increase on soil and by increasing the depth of embedment, Df. This increase is particularly important for mats on soft clays, where large consolidation settlements are expected. In this design, the deeper basement is made below the higher portion of the superstructure, so that the net pressure increase in soil at any depth is relatively uniform. From the figure shown, the net average applied pressure on soil can be given as 𝐪=

𝐐 − 𝛄𝐃𝐟 𝐀

Where: Q = dead load of the structure and live load A = area of the mat

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

COMPENSATED FOUNDATIONS For no increase of the net soil pressure on soil below a mat foundation, q should be zero. Thus, 𝐐 𝐃𝐟 = 𝐀𝛄 This relation for Df is usually referred to as the depth of embedment of a fully compensated foundation. (The fully compensated mat foundation is one in which the net increase in soil pressure below the mat is zero.) The factor of safety against bearing capacity failure for partially compensated foundation (that is, Df < Q/Aγ) may be given as 𝐅𝐒 =

𝐪𝐧𝐞𝐭(𝐮) 𝐪𝐧𝐞𝐭(𝐮) = 𝐐 𝐪 𝐀 − 𝛄𝐃𝐟

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FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE

PROBLEM 13. The mat has dimensions of 40 m x 20 m, and the live load and dead load on the mat are 200 MN. the mat is placed over a layer of soft clay that has a unit weight of 17.5 kN/m3. Find the Df for a fully compensated foundations.

FOUNDATION ENGINEERING

Shallow Foundations-Bearing Capacity

EXAMPLE

PROBLEM 14. Refer to Problem 13. For the clay, cu = 60 kN/m3. If the required factor of safety against bearing capacity failure is 3, determine the depth of the foundation.

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