Geotechnical Engineering 1

1. Shallow Foundations

1) General

* Shallow Foundations : Foundations that transmit structural loads to the near- surface soils. (Spread footing foundation + Mat foundation)

Df/B ≤1 (by Terzaghi) → Later D f/B ≤3∼4

Df

B

* Requirements to satisfactory foundations i) Safe against shear failure (bearing capacity failure). ii) Should not undergo excessive displacements. (settlements ←differential settlements) iii) Consideration on the any future influences which could adversely affect its performance. (frost action, scouring of pier foundations of bridge, … )

* Types of shallow foundation
Spread footings - Spread footing foundations: An enlargement at the bottom of a column or bearing wall that spreads the applied structural loads over a sufficiently large soil area. i) Square spread footings : Supporting a single centrally-supported column. ii) Rectangular spread footings : In cases that obstructions prevent

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construction of a square footing with a sufficiently large base area and large moment loads are present. iii) Circular spread footings : Supporting a single centrally-supported column, but less common than square footing. (flagpoles). iv) Continuous spread footings(Strip footings) : Used to support bearing walls. v) Combined footing : When columns are located too close together for each to give its own footing. vi) Strap footing with a grade beam : Provides the necessary moment resistance in the exterior footing with eccentric load and a more rigid foundation system.

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Raft (mat) foundation

- Mat foundations (Raft foundations): A very large spread footing that usually encompasses the entire footprint of the structure.

- The advantages of the mat foundation over individual spread footings i) Spreads the structure load over a larger area, thus reduces bearing pressure. ii) Provides much more structural rigidity and thus reduces the potential for excessive differential settlements. iii) Is easier to water proof. iv) Has a greater weight and thus is able to resist greater uplift pressure.

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2) Bearing Pressure

* Bearing Pressure : the contact forces per unit area along the bottom of the footing.

* The actual distribution of the bearing pressure depends on ; - Structural rigidity of the footing. - Stress-strain properties of the soil. - Eccentricity of the applied load. - Magnitude of the applied moment. - Roughness of the bottom of the footing.

* Perfectly flexible footings

Bend but maintain a uniform bearing pressure.

* Perfectly rigid footings

Settle uniformly but have variations in the bearing pressure. Close to behaviors of the real footing. (especially for spread footing but not for raft foundation)

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For bearing capacity and settlement analysis, we assume the uniform pressure with rigid footings. (Error is not significant for spread footing.)

* The footing subjected to eccentric and/or moment loads. The bearing pressure is biased toward one side.

< Distribution of bearing pressure along the base of spread footing subjected to eccentric and/or moment loads : (a) actual distribution (b) simplified distribution >

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* Presumptive Bearing Pressures

- Presumptive bearing pressures are allowable bearing pressures based on experience and expressed as a function of soil type. (An attempt to control excessive settlements.) Give quick reference values of foundation design. (Tales 6.1) Useful for small structure at sites with good soils (Column Loads < 200kN).

- Presumptive bearing pressures vary considerably, due to differing degree of conservatism and reflecting the subsurface conditions in the area where the code is used .

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3) Failure modes

 General shear failure →  Local shear failure →  Punching shear failure →

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 Failure modes →a function of relative density and relative depth(D f /B)

Bo=B for circular (diameter) or square ft . Bo = 2BL/(B+L) for rectangular ft.

 Ultimate load occurs at 4∼10% of B for general shear failure 15 ∼25% of B for local-punching shear failure

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4) General Bearing Capacity Equation

qu = ultimate bearing capacity (stress)

qa = allowable bearing capacity = qu / F.S.

* Failure zones for strip footing.

Ⅰ : Active Rankine Zone Ⅱ : Radial Zone Ⅲ : Passive Rankine Zone

α = φ' for Terzaghi = 45 + φ 2/' ⇒ realistic value

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* Bearing Capacity Equation

 Presented by Terzaghi. 1 q = cN + qN + γBN u c q 2 γ

Cohesion term Friction term Surcharge term

γ : unit weight of soil c : cohesion of soil

q = γDf

Nc, N q, N r => Bearing capacity factors = f( φ’)  based on α = φ' (Terzaghi, Table 3.1 (p.129))  based on α = 45 + φ 2/' (Vesic, Table 3.4(p138))

* Assumptions 1. 2. 3. 4. 5. 6. 7. 8.

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* General Bearing Capacity Equations (Bearing Capacity Factors and Other Influential Factors)

- To consider the influences of the shape of foundations, embedded depth, and inclined load , the following form of equation has been suggested.

1 q = cN F F F + qN F F F + γBN F F F u c cs cd ci q qs qd qi 2 γ γs γd γi

i) Nc , Nq , Nr

Nc by Prandtl (1921)

Nq by Reissner (1924)

Nr by Caquot and Kerisel (1953) and Vesic (1973)

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ii) Shape factor (De Beer (1970))

B N q Fcs =1 + L N c B F =1 + tanφ ’ qs L B F =1 − 4.0 γs L where L = length of the foundation (L > B)

iii) Depth factor (Hansen(1970))

For D f / B ≤1, For D f / B >1

−1 Fcd =1 + 4.0 D f / B Fcd =1 + 4.0 tan (D f / B) D  D  2 f 2 −1  f  Fqd =1 + 2 tanφ ()1' − sinφ' Fqd =1 + 2 tanφ ()1' − sinφ' tan   B  B 

Fγd = 1 Fγd = 1

iv) Inclination Factor (Hanna and Meyerhof(1981)) β F = F = 1( − ) 2 ci qi 90

β F = 1( − )2 γi φ' where β = inclination of the load on the foundation with respect to the vertical.

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- For inclined load, we must check sliding failure in addition to bearing capacity.

P Pv

PH

Passive Resistance s = interface strength (Ignored)

2 For clays, s = αsu (generally α ≤ ,0.1 but α = 0.1 for su ≤ 500lb / ft )

For sands, s = Pv /(Area) tanφint (φint = 3/2 φ)

PH ≤ s × (Area /) F.S.

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* Effect of water (influence on q and γγγ of bearing capacity equation)

I) D 1 < D f qN q = [D1γ t + D2 (γ sat − γ w )]N q 1 1 γBN = {}γ − γ BN 2 γ 2 sat w γ

II) 0 ≤ d ≤ B qN q = γ t D f N q 1 1  d  γBNγ = γ sat − γ w + []γ t − (γ sat − γ w ) BNγ 2 2  B 

γt : Total unit weight of soil

γsat : Saturated unit weight of soil

γw : Unit weight of water

III) d ≥ B No effect

Note) No seepage force

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* Effect of Compressibility

For compressible soils ⇒ local shear failure Ex) Loose sands ⇒ low relative density

- Shear (triaxial) test response and load-settlement curve for loose soil with those for dense sand

P σ1-σ3

General failure dense q q(u) (u) loose

S εa

- Terzaghi Recommendations Use c*, φ* (reduced strength parameter) in bearing capacity equation ;

2 c* = c' 3 2 tan φ* = ()tan φ' 3

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* Net ultimate bearing capacity

Q

D f D γ q =γ f

qu

(assuming γ conc = γ soil )

qu = Q / A + γD f

∴qnet = Q / A = qu − γD f

* Factor of Safety

i) qall = qu / F.S., qall(net) = (qu − q /) F.S.

ii) FS shear ⇒ cd = c / FS shear , φd = φ / FS shear

⇒ qall = cd N c Fcs Fcd Fci + qN q Fqs Fqd Fqi + 2/1 γBNγ Fγs Fγd Fγi ,

qall(net) = qall − q

F.S. = 3-4, FS shear = 4.1 − 6.1

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* Case history : Ultimate Bearing Capacity in Saturated Clay

- Comparison between measured and estimated bearing capacities on 5 different sizes of the foundations.

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