Ménard in combination with Brinch Hansen

The Brinch Hansen method also assumes the immediate development of full plastic soil pressures. Some software programs apply Brinch Hansen in combination with the soil stiffness for piles according to the Ménard method in order to model the bilinear soil behaviour.

To understand the principles of the modulus of subgrade reaction according to Ménard, a brief description of the Ménard pressuremeter test and the underlying theory is given below (Roctest).

Figure 3-5 shows the basic principle of the Ménard pressuremeter test. A cylindrical probe is penetrated into the soil and by pressing gas or fluid into the probe it expands laterally. If the volume is then plotted against pressure in the probe, the graph shown in figure 3-5 is derived. This relationship between deformation and stresses can be analysed if several assumptions are made. Given the length of the probe, the instrument exerts a radial and uniform field of stresses. The deformation of soil or rock comprises a pseudo-elastic and a plastic phase. When determination is done by a volumetric mean, the medium is considered to be isotropic in the test zone. That is, the limit pressure PL, the pressuremeter modulus E and the creep pressure PF are relevant. The pressuremeter modulus is based on the Lame equation giving the radial increment of a radial cavity in function of the pressure in an elastic medium. The formula that gives the shear modulus G is:

G = V \cdot (\partial P / \partial V)

(3.13)

Where:
$V$	= volume of the cavity
$P$	= pressure in the cavity
$\partial P / \partial V$	= slope of the pressuremeter curve in its linear pseudo-elastic part, taken for the volume V M , located in the middle of the segment V O - V F
$V_{O}$	= volume corresponding to the pressure of recompression of the walls on the borehole, which is more or less the 'at rest' pressure of the soil
$V_{F}$	= reep pressure

In an elastic medium the relation between shear modulus G and Young's modulus E is:

E = G \cdot (1 + v)

(3.14)

Where:
$v$	= Poisson's ratio, in the case of pressuremeter modulus E M , the Poisson's ratio 0.33.

If V_C is the 'at rest' volume of the probe, E_M can be calculated with:

E_{M} = 2.66 \cdot (V_{C} + V_{M}) \frac{\partial P}{\partial V}

(3.15)

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Figure 3-5 Menard pressuremeter test.

Cone penetration test (CPT) correlations were established between point resistance of a CPT and pressuremeter modulus; see table 3-2. Given the pressuremeter modulus, the modulus of subgrade reaction was then calculated with the following equations (Ménard, 1971) [3.18]:

$\frac{1}{k_{h}} = \frac{1}{3 E_{M}} [1.3 R_{0} {(2.65 \frac{R}{R_{0}})}^{α} + α R] (f o r R < R_{0})$	(3.16)
$\frac{1}{k_{h}} = \frac{2 R}{E_{M}} \frac{4 {(2.65)}^{α} + 3 α}{18} (f o r R < R_{0})$	(3.17)

Where:
$E_{M}$	= pressiometer modulus [kN/m 2 ]
$R_{0}$	= 0.3 [m]
$R$	= half diameter of pile [m]
$α$	= rheological coefficient [-], table 3-3
$K_{h}$	= modulus of horizontal subgrade reaction [kN/m 3 ]

Table 3-2 Relation cone resistance and pressuremeter modulus for different types of soil.

Soil	E M [kPa]
Peat	(3–4)q c
Clay	(2–3) q c
Loam	(1–2) q c
Sand	(0.7–1) q c
Gravel	(0.5–0.7) q c

Table 3-3 Rheological coefficients for different types of soil.

Soil	Peat	Clay	Loam	Sand	Gravel
Over consolidated	-	1	2/3	1/2	1/3
Normally consolidated	1	2/3	1/2	1/3	1/4
Decomposed, weathered	-	1/2	1/2	1/3	1/4

For more background information on Ménard, see [3.18]. For the sake of comprehensiveness, it must be mentioned that the DIN 4085 [3.2] method for ultimate soil resistance can also be combined with the soil stiffness according to Ménard.

Cautionary comments by the authors:
This method is known to show more pile deformation than compared to the other methods and field tests due to some limitations. These limitations should be considered when Ménard with Brinch Hansen is used:

Little to no stress dependency. The soil stiffness is stress dependent. At equal qc-value the soil stiffness at depth is higher when compared to the soil stiffness near surface level. This might underestimate the stiffness near the bottom of the pile, since the Ménard stiffness usually based on qc-values.
The Ménard soil stiffness is linear where in reality soil behaves non-linear. At small deformation the stiffness can be very high. When calculating the deflection of a dolphin, the deformation of the pile at the bottom is usually very small, hence the stiffness should be relatively high. The method Menard Brinch Hansen will not compensate for this.
Over-estimation of the pile deflection due to a too low soil reaction at the bottom is magnified with increased diameter of piles. With increasing diameter, the bending stiffness of the pile will increase more than the strength and this causes more rotation of the pile in oppose to bending. The result is more dependency of the overall pile deformation to the soil stiffness at the bottom of the pile.

The beforementioned limitations are of a theoretical nature and should therefore always be taken into account regardless of the dolphin type, be it a mooring or a breasting dolphin. However, when designing a breasting pile the design might end up to be unsafe, since the energy absorption is too high, and the calculated bending moment too low causing the pile to structurally fail. When designing a mooring dolphin deformations are less relevant and hence this overestimation might result in slightly conservative pile dimensions.

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Ménard in combination with Brinch Hansen