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A vertical pile has its axis coinciding with the plumb line while a batter pile has its axis
inclined to the plumb line. Vertical piles are usually used to resist dead and live loads, uplift due
to swelling and frost expansion of soil, and forces due to hydrostatic pressure beneath the base of
a structure. Batter piles are commonly used to resist inclined or large lateral loads.
A short pile is relatively rigid and rotates as one unit under lateral loads while a long pile
is relatively flexible and acts like a beam under lateral loads. The criteria for the classification of
short and long piles is given in Figure (1). The design length of a pile mainly depends on the profile
of the subsoil and the type and magnitude of loading.
There is an extensive amount of available literature on axially and laterally loaded piles.
Generally, the vertical capacity of a pile is dictated by the ultimate bearing capacity of the soil.
The ultimate bearing capacity in turn may be defined as the maximum load which the pile can
support without undergoing significant settlements. The ultimate bearing capacity of vertical piles
under axial loads in sands is generally evaluated using soil properties such as its density and the
angle of shear resistance (Terzaghi 1943, Meyerhof 1951, Vesic 1963). The ultimate lateral load
of vertical short rigid piles is generally computed based on lateral earth pressure theories (Brinch
Hansen 1961, Broms 1964, Petrasovits et al. 1972, Meyerhof et al. 1981), and the ultimate lateral
load of a long flexible pile can be evaluated using the theory of elasticity (Rowe 1955, Matlock
and Reese 1962, Broms 1964, Poulos 1971).
Literature on the ultimate capacity of vertical rigid piles subjected to inclined loads is
somewhat limited. One of the present methods to compute the ultimate capacity under inclined
loads is to use an interaction equation (Meyerhof 1981). A brief review of the existing theories of
vertical and lateral ultimate capacity of short rigid piles is presented here.
Vertical piles under axial loads
The ultimate bearing capacity (Q) of a vertical pile under an axial load (Figure l) is
generally expressed as the sum of point resistance force (Qp) and total shaft resistance (Qs) as
follows: (Equation 1).
The point resistance pressure (qp) and the average unit shaft resistance (qs) are functions
of several parameters but mainly depend on the type of soil, the density of soil, the angle of friction,
and the physical properties of the pile. For practical purposes, Equation (1) is formulated on the
premise that the two components (qp) and (qs) are independent of each other. In fact, for piles
driven into cohesionless soils there is some interdependence between the two components (Kezdi
l957), but this small influence is generally neglected (Broms l956). The magnitudes of the two
components (Qp) and (Qs) in cohesionless soil may be intuitively expected to be proportional to
the embedded depth, but according to laboratory and field test results, the proportionality cannot
be satisfied beyond the critical depth below which the ultimate load remains relatively constant
(Kerisel l964, Vesic l963, Vesic l964, Tavenas l970). The relative magnitudes of (Qp) and (Qs)
depend on the type of soil and the method of installation of the pile.
Based on the method of placement, vertical piles may be classified into two broad
categories. A pile driven into the soil is classified as a displacement pile. A pile which is placed
by removing an equal volume of the soil is generally called non-displacement pile (sands) or a
bored pile (clays). The capacity of the pile is predominantly the end bearing resistance for a nondisplacement (bored) pile, while it is the sum of the end bearing and side frictional resistance for
a displacement (driven) pile.
The ultimate bearing capacity of a pile (Q) can be estimated by several methods and the
most commonly used are: (1) based upon bearing capacity theories, (2) from the results of in-situ
tests, and (3) prototype pile load tests.
The first two methods which are relevant to this thesis will be reviewed in the following
Estimation of Q based on bearing capacity theories:
Point resistance force, Qp:
Most of the present solutions for the point resistance force of pile foundations are derived
using Prandtl's (1920) and Reissner's (1924) general bearing capacity theories based on the
assumption of weightless material, and Ohde' (1938) theory considering the weight of the material.
The resulting point resistance pressure is expressed by the following general equation: (Equation
For deep foundations in cohesionless soils, the first term will be zero and the third term is
negligibly small in comparison to the second term. Hence, Equation 2 can be simplified as:
When it is necessary to consider the weight of the pile, the net point resistance pressure
(Qpn) of a pile can be determined based on the assumption that the unit weight of pile material is
equal to that of soil. (Equation 4).
Equations (3) and (4) indicate that the bearing capacity of a pile varies with the bearing
capacity factor (Nq), which depends on the deformation characteristics of the soil. Vesic (1967,
1977) has summarized the various theoretical approaches to simulate the failure mechanism of soil
as shown in Figure (2). The corresponding (Nq) values in sand as suggested by various
investigators are reproduced in Figure (3) and Table (1).
It is reported (Norlund 1963, Broms 1955, Vesic 1964, 1967) that in practice, the (Nq)
values of Berezantzev are found to correlate well with the measured value. However, Coyle and
Castello (1979) suggested that Terzaghi’s (Nq) values for general shear failure were found to fit
their experimental results.
The point resistance pressure (qp) has been normally found to increase up to a certain depth
beyond which any increase in (D) does not result in significant increase of (qp). This depth has
been normally designated as the critical depth. Kerisel (1964) and Meyerhof (1976) reported that
the value of (Nq) in sand increases with depth and reaches its maximum value at less than half of
the critical depth. While Berzantsev et al. (1961) and Drugunoglu & Mitchell (1973) found that
(q) decreases with increasing (D/B) ratio, Vesic (1977) concluded that (Nq) is a constant,
independent of the depth.
In addition to the depth, (Nq) depends on many factors such as density of the soil,
overburden pressure, shape of the pile and method of installation. For driven piles the change of
density of soil due to driving a pile has to be taken into account to evaluate the ultimate load
capacity of the pile. However, as mentioned earlier, the available theories are based on the
assumption that the soil density during pile driving is not changed. In fact, the density index
increases for driven piles in sand except in very dense sand, and therefore the angle of internal
friction (¢2) after driving the piles is larger than the initial internal angle of friction (¢1). The
relationship between (¢1) and (¢2) in sand has been suggested as follows (Kishida and Meyerhof
19(5): Equation (5).
Equation (5) implies that there is no change in density index for soils with an internal
friction angle of 40°.
Based on the failure mechanism shown in Figure (2), Vesic (1977) has given the following
equation for (qp): Equation (6).
The point resistance force Qp can thus be computed as the product of (qp) and the area of
the pile base (Ap).
Total shaft resistance, Qs
For deep foundations the total shaft resistance Qs' can be defined as the resistance to the
sliding of a rigid body relative to the surrounding soil and is generally expressed by two
components: (1) adhesion, and (2) friction, dependent on normal stresses.
The unit shaft resistance of driven piles (qs) at any depth (Z) below the ground surface can
be calculated from Mohr-Coulomb's theory of rupture as follows: (Equation 7).
For piles in cohesionless soils the value of ca is zero. Equation (7) can be rewritten
integrating along the embedded pile length for the total shaft resistance (Qs) as follows (Dorr 1922,
Meyerhof 1951, Norlund 1963). (Equation 8).
The magnitude of coefficient (Ks) in Equation (8) depends mainly on the initial relative
density, the displacement volume of the pile, the shape of pile, and the method of pile installation.
However, for practical purposes the averaged values of (Ks) can be taken for piles driven into
cohesionless soil. The coefficient (Ks) for driven steel piles has been suggested as 1.0 for dense
sand and 0.5 for loose sand regardless of pile type and roughness of the pile surface (Meyerhof
1951, Broms 1966, Coyle and Castello 1979).
The angle of friction (¢) between the soil and the shaft has been suggested based on
experimental data as 0.54 (¢) for smooth steel piles and 0.76 (¢) for rusted steel piles where (¢)
represents the angle of internal friction for the soil (Pontyondy, 1961). Other researchers have
given the skin friction angle (¢) as 20 degrees for steel piles assuming that the value of (¢) is
independent of the density index of the soil surrounding the pile (Broms 1966, Craig 1978,
Estimation of Q based on in-situ tests
The vertical capacity of piles can be also estimated based on in-situ tests such as the
standard penetration test (SPT) and the cone penetrometer test (CPT). Both types of tests are
routinely done as part of site investigations.
The standard penetration test can be used to determine the ultimate bearing capacity of
piles in cohesionless soils. This ultimate bearing capacity (Q) in sands has been expressed as
(Meyerhof 1956, 1976): (Equation 9).
It should be noted that the accuracy of the above estimate depends on the reliability of the
blow count (N). As is common knowledge, the standard penetration test is not generally used for
cohesive soils and Equation 9 is valid only for cohesionless materials.
If the ratio of depth to diameter of the pile is less than 10, the point resistance pressure (qp)
can be expressed as (Meyerhof 1956): (Equation 10).
The cone penetrometer test in cohesive and cohesionless soils has been correlated with the
ultimate bearing capacity of piles. The ultimate bearing capacity (Q) of piles in cohesionless soil
has been given as (Meyerhof 1956): (Equation 11).
Equation 11 has been derived on the assumption that the point resistance pressure of the
pile is equal to the average static cone point resistance (qc) over a depth of 4 pile diameters above
and one pile diameter below the anticipated depth of the pile tip (Meyerhof 1956, Menzenbaeh
1961). It is also assumed that the unit shaft resistance is equal to 0.5% of the average static cone
point resistance (Meyerhof 1956, 1976).
If the depth of foundation is less than 10 times the pile diameter, the point resistance
pressure (qp) can be expressed as: (Equation 12).
Subsequent work shows good correlations for pile diameters less than 50 cm (Kerisel
1961). However, the total shaft resistance on concrete piles was found to be greater than that given
by Equation (13) (Mohan et a1. 1963). Tomlinson (1977) has given a slightly different approach
in which the values of (Ks) and (¢) can be estimated from cone penetrometer tests. The suggested
values are given in Table (2), and from these, the ultimate bearing capacity (Q) can be estimated
as: (Equation 13).