chp.8 pile foundations lecture 2

8.4 STATIC CAPACITY OF AXIALLY LOADED SINGLE PILE

1.Point bearing capacity, Qp: The ultimate resistance per unit area developed at the pile tip qp may be expressed by an equation similar in form to that given by the general bearing capacity of shallow foundations:

qu = qp: ultimate pile tip resistance/unit area Nc*,Nq*,Nγ* : bearing capacity factors including the necessary shape

and depth factors D: width or diameter of pile However D is relatively small so that γD Nγ* can be neglected, hence

The point bearing of piles is:

Ap: area of pile tip; qp : unit point resistance; q’: effective vertical overburden stress at the pile tip level

QU = QP+QS

A- Meyerhof’s Method SAND: The point bearing capacity qp, of a pile in sand generally increases with the depth of embedment in the bearing stratum and reaches a maximum value at an embedment ratio of Lb/D = (Lb/D)cr. Note that in a homogeneous soil Lb = L; However, where a pile has penetrated into a bearing stratum ,Lb< L. Beyond(Lb/D)cr ,the value of qp remains constant (qp=ql) For piles in Sand, C =0 simplifies the equation to:

qp=ql

L/D =Lb/D

The variation of Nq* with soil friction angle Φ is shown in figure(8.4.1), however Qp shouldn’t exceed the limiting value that is:

Fig.8.4.1

Soil friction ,Φ

B- Vesic’s Method:

Qp = Ap.qp = Ap(C Nc* + σ’0 Nσ*)

Where σ’0 =mean effective normal ground stress at the level of the pile point = ( 1+2 K0)q’/3 K0 = earth pressure coefficient at rest = 1-sin Φ Nσ* = bearing capacity factor

= f(Irr)

In case of saturated clay in undrained conditions(Φ=0)

For saturated clay, no volume changes,Δ = 0, Ir =Irr

Vesic

C- Coyle and Castello’s Method

Coyle and castello

2- Frictional Resistance, Qs The frictional or skin resistance of a pile may be written as:

where p = perimeter of the pile section ∆L = incremental pile length over which p and f are taken constant f = unit friction resistance at any depth z

A- Sand 1- Critical depth Method The unit frictional resistance at any depth for a pile is

where K = earth pressure coefficient σ’o = effective vertical stress at the depth under consideration δ = soil–pile friction angle The effective vertical stress σ’o increases with pile depth to a maximum limit at a depth of 15 to 20 pile diameters and remains constant thereafter,

A conservative estimate is to assume that L = 15D The values of δ = 0.5 Φ to 0.8 Φ Qs = Qs1+Qs2

Where Qs1 from z=0 to 15 D (favg is considered) Qs2 from z=15D to L

2- Coyle and Castello:

B-Clay Several methods are available for obtaining the unit frictional resistance of piles in clay. Three of the presently accepted procedures are described briefly.

1. The α Method:

According to the α method, the unit skin resistance in clayey soils

can be represented by the equation:

Where α = empirical adhesion factor, cu =undrained cohesion

2. The β Method

the unit frictional resistance for the pile can be determined on the basis of the effective stress parameters of the clay in a remolded state (c =0).

and

• K = 1 - sin ΦR (for normally consolidated clays) • K = (1 - sin ΦR) √OCR (for over consolidated clays) where OCR = σ’p/σ’v over consolidation ratio

where σ’o = mean vertical effective stress β = K tan ΦR ; ΦR= drained friction angle of remolded clay K = earth pressure coefficient

3. The λ Method: the average unit skin resistance is:

where σ’o :mean effective vertical stress for the entire embedment length cu : mean undrained shear strength ( Φ = 0 concept) The value of λ changes with the depth of pile penetration Thus, the total frictional resistance may be calculated as Qs = pLfav

• the mean value of cu is (cu1(L1) + cu2(L2 ) . . . )/L • the mean effective stress is

The λ method

α method λ method

8.5 Allowable Pile Capacity

The ultimate total load carrying capacity of a pile is determined by: Qu = Qp+Qs

• Then Q all = Qu/FS where FS = factor of safety = 2.5 to 4

8.6 Elastic Settlement of Piles

The elastic settlement of a pile under a vertical working load, Qw, is determined by three factors:

S = S1+S2+S3 where S = total pile settlement S1 = settlement of pile shaft S2= settlement of pile caused by the load at the pile point S3= settlement of pile caused by the load transmitted along the pile

shaft

1

1- determination of S1

Ap: area of the pile cross section L : length of the pile Ep: modulus of elasticity of the pile material The magnitude of ξ depends on the nature of the unit friction (skin) resistance distribution along the pile shaft. ξ ≈ 0.6 is acceptable.

where Qwp : load carried at the pile point under working load condition Qws : load carried by frictional (skin) resistance under working load

condition

2- determination of S2

2

where D : width or diameter of the pile qwp : point load per unit area at the pile point= Qwp/Ap Es: modulus of elasticity of soil at or below the pile point μ: Poisson’s ratio of soil

Iwp : influence factor = αr (fig.8.6.1)

Vesic also proposed a semi empirical method to obtain the magnitude of the settlement

Where: qp : ultimate point resistance of the pile Cp :an empirical coefficient in the following table

2

Fig.8.6.1

3-determination of S3:

3

where p : perimeter of the pile L : embedded length of the pile Iws : influence factor

Or Vesic

3

where Cs an empirical constant (=0.93 + 0.16√L/D )Cp The values of Cp may be estimated from Table

8.7 Laterally Loaded Piles

Consider pile of length L subjected to a lateral force Qg and a moment Mg at the ground surface (z=0) as shown in the next figure. According to Winkler’s model, the soil can be replaced by a series of infinitely close independent elastic springs:

From the fundamentals of mechanics of materials: 1-Granular soils: The subgrade modulus for granular soils at a depth z is defined as:

Kz = nh.z

Where nh: constant of modulus of horizontal subgrade reaction

a-Pile deflection at any depth

b- slope of pile at any depth

c-Moment at any depth

d- shear force at any depth

e- soil reaction at any depth

Remark: • When L ≥ 5T, the pile is considered “Long pile” • When L ≤ 2T, the pile is considered “Rigid pile”

Representative values of Nh

Note that Z = /T

2- cohesive soils For cohesive soil the subgrade reaction may be assumed to be approximately constant with depth according to vesic:

a-Pile deflection at any depth

b- pile moment at any depth

and