# Elastic Modulus

## Plate Load Test

The settlement of a rigid circular or square plate resting on the surface of an isotropic elastic material may be reasonably obtained from:

where

E_{s} = modulus of elasticity 6 = settlement

q = applied plate stress = load/area В = diameter of plate о = Poisson’s ratio or

For tests performed rapidly in saturated clays, the value of о = 0.5 and Equation 9.7 gives:

Typically, the results of PLTs and footing load tests show that at bearing stresses up to about one-third of the ultimate capacity, the load-displacement behavior is approximately linear. Therefore, the elastic modulus should be obtained within this region of applied stress. Additionally, it may be advantageous to perform at least one unload-reload cycle in this region in order to more accurately obtain the reload modulus.

In sands, the value of Poisson’s ratio may typically vary from about 0.1 to 0.33. For a value of u = 0.3 and for surface tests, Equation 9.8 gives:

Equations *9.7-9.9* are applicable to PLTs performed on the surface. If the plate is embedded below the surface, then the surface equations must be modified by a depth reduction factor, p_{0}. For a rigid plate, Equation 9.9 becomes:

where

go = depth reduction factor

Different values of g_{0} have been suggested by Burland (1970), Pells & Turner (1979), Donald et al. (1980), and Pells (1983). For surface tests, g_{0} = 1.0. For PLTs performed at depths greater than about four times the plate diameter, the value of p_{0} may be taken as 0.87 (Burland 1970). The values of g_{0} as shown in Figure 9.13 are for PLTs performed in excavations that are the same size as the plate. If the excavation is larger than the plate, the reduction is less. Pells (1983) has suggested one solution for such a reduction as shown in Figure 9.14.

## Screw Plate Test

Selvadurai et al. (1980) summarized a variety of numerical results derived from various theoretical models of the SPLT. In clays, the test is usually performed rapidly enough that full drainage does not occur. In this case, the *in situ* undrained elastic modulus, E_{u}, may be obtained from:

where

q = applied plate stress r = radius of the screw plate 6 = plate settlement К = undrained modulus factor

*Figure 9.13* Modulus reduction factor for embedded plate having the same diameter as the shaft. (After Pells 1983.)

*Figure 9.14* Modulus reduction factor for the embedded plate having larger diameter as the shaft. (After Pells 1983.)

Theoretical values of К range from 0.525 to 0.750 (Bergado & Huan 1987); however, a more realistic range, which is more applicable to the conditions of the SPLT, is К = 0.60 to

0.75.

Selvadurai & Nicholas (1979) recommended that the undrained elastic modulus from the SPLTs be defined as a secant modulus and taken at the point where a secant slope passes through a point on the bearing stress vs. settlement curve corresponding to one-half of the ultimate bearing stress. Using a value of К = 0.66 gives:

For the drained loading of clays, a drained value of Poisson’s ratio = 0.20 may be more reasonable, which gives an expression for the drained elastic modulus as:

The value of S_{100} is the settlement taken at the end of the test corresponding to 100% consolidation.

# Shear Modulus

Marsland & Powell (1990, 1991) suggested that the shear modulus be calculated from the PLTs as:

where

G = secant shear modulus

Д5 = change in displacement for a given change in stress, Aq В = plate diameter о = Poisson’s ratio

Since the results of PLTs are nonlinear, the modulus decreases with increasing applied stress. Therefore, it is necessary to define the stress range over which the moduli are determined.

# Undrained Shear Strength of Clays

The interpretation of PLT and SPLT results for estimating the undrained shear strength of clays requires the interpretation of the load vs. settlement curve to give the ultimate bearing stress, q_{uU}. In the absence of an obvious plunging failure condition, several methods have been suggested to evaluate the ultimate bearing stress from PLTs.

One method of interpreting the ultimate stress is to take the bearing pressure at some fixed relative plate displacement as the ultimate bearing pressure. For example, Skempton (1951) showed that for remolded clay, the ultimate bearing capacity of shallow foundations in clays is achieved at a displacement on the order of 3%-5% of the plate diameter. Powell & Quaterman (1986) used a bearing stress producing a settlement of 15% of the plate diameter for SPLTs. Most footing tests that are taken to large displacements indicate that the bearing capacity may be conveniently defined as the bearing stress producing a settlement of 10% of the footing width. The author suggests that this be applied to PLTs as well. Results from a 0.3-m (1 ft)-diameter PLT conducted in a stiff clay are shown in Figure 9.15. The 10% criterion indicates an ultimate load of about 19,750 lbs (q_{ult} = 25,950 lbs/fU).

An alternative approach is to use a model that reasonably describes the pressure- settlement results and predicts an ultimate bearing pressure. This is a form of curve fitting, most commonly done using a transformed hyperbolic model. If the plate settlement/plate

*Figure 9.15* Results of PLTs in stiff clay.

stress is plotted as a function of settlement, the inverse slope of the linear portion of the curve gives the ultimate bearing stress. This model has previously been used to describe the load-displacement behavior of PLTs and footing tests (e.g., Chin 1983; Wrench *&C *Nowatzki 1986; Wiseman Sc Zeitlan 1994; Thomas 1994). Figure 9.16 shows the PLT data from Figure 9.15 using this transformed model, indicating an ultimate load of 41,800 lbs. (q_{ult} = 52,900 lbs/ft^{2}).

Kay & Parry (1982) used the hyperbolic model to extrapolating the load-displacement curve to obtain the ultimate plate capacity without actually plotting the data. By measuring the plate displacement at two points on the stress-displacement curve, they estimated the ultimate plate capacity as:

q_{u]t} = ultimate stress

q_{t} 5 = plate stress at a displacement of 1.5% of the plate diameter

q_{2} = plate stress at a displacement of 2% of the plate diameter

*Figure 9.16* Transformed axes for hyperbolic estimation of ultimate plate bearing stress.

Using the interpreted ultimate bearing stress, the undrained shear strength may be estimated from a traditional bearing capacity equation, e.g., Skempton (1951). For PLTs performed on the surface:

where

s_{u} = undrained shear strength

q_{u]t} = interpreted ultimate bearing stress

N_{c} = shallow bearing capacity factor

The bearing capacity factor, N_{c}, for a surface footing is approximately 6.0 for a square or round plate.

For embedded plate tests or SPLTs beyond a relative embedment of about five times the plate width (diameter), the undrained shear strength is obtained from:

where

s_{u} = undrained shear strength

q_{u]t} = interpreted ultimate bearing stress

o_{v} = total overburden stress at the depth of the test

N_{c} = deep bearing capacity factor

Equation 9.16 is also sometimes given as Equation 9.17 for SPLTs.

The bearing capacity factor for deep plate tests has traditionally been taken as N_{c} = 9, from the work of Skempton (1951); however, more rigorous analyses suggest that N_{c} varies from about 5.69 to 11.35 (e.g., Selvadurai et al. 1980; Bergado & Fluan 1987). Several studies have shown that the undrained shear strength obtained from the SPLT or PLT is compared well with results from laboratory or other *in situ* tests.

# Coefficient of Consolidation

Janbu & Senneset (1973) suggested that the results of SPLTs could be used to estimate the *in situ* coefficient of consolidation by using the time-settlement measurements. It was suggested that since the drainage is predominantly radial, the results provide an estimate of the horizontal coefficient of consolidation, c_{h}. In this procedure, the settlement is plotted vs. the square root of time for each load increment, as shown in Figure 9.17; a straight line is fitted to the initial portion of the curve and is projected back to the axis to give the corrected theoretical zero point. From this point, a second straight-line offset from the first line and having a slope 1.3:1 is constructed. This line intersects the curve at a point that approximately represents 90% primary consolidation, with the time corresponding to *t _{90}. *The value of c

_{h}is obtained from:

where

c_{h} = coefficient of radial consolidation

T_{90} = theoretical time factor for 90% consolidation

*Figure 9.17* Graphical estimation of coefficient of consolidation from screw plate.

R = plate radius

t_{90} = interpreted time for 90% consolidation

Kay & Avalle (1982) observed that in many cases, the test data show a more S-shaped curve. They felt that the radial drainage model was not appropriate and suggested that interpretations of the coefficient of consolidation be made using three-dimensional isotropic consolidation. The suggested construction procedure presented by Kay & Avalle (1982) using the settlement vs. square root of time plot is as follows:

- 1. Ignore the initial data points and extrapolate the reverse curve portion of the plot to zero on the time axis. This represents the point of zero percent drained settlement, t
_{0}. - 2. Draw a straight line from t
_{0}tangent to the settlement vs. square root of time plot. - 3. Construct a line from t
_{0}with a slope of 1.28:1 flatter to the first line. This line intersects the curve at a point representing 70% consolidation, t_{70}, and gives t_{70}.

The coefficient of consolidation is given as:

This procedure is illustrated in Figure 9.17. Since the test results may be evaluated for each loading increment, the results give an interpretation of the coefficient of consolidation over a range of applied stresses. This means that the results may be plotted as a coefficient of consolidation vs. stress and may be used to estimate values for any stress level (Figure 9.18).

*Figure 9.18* Graphical estimation of coefficient of consolidation from screw plate.