## Saturday, October 1, 2011

### Terms explanation

When designing a retaining wall we need to evaluate and understand  the following terms:

• cohesive strength(c) -of the retained material wich can be explained as the interior own strength(shear     strength)of a  material besides his own weight;
• unit weight(y)-of a material wich is defined as the weight per unit volume of a material;
• angle of  friction((φ-phi)-wich is the maximum angle of a certain material that will remain stable and will not begin sliding.
Also in order to proper design a retaining wall we need to take in consideration the following earth pressures categories:
• active earth pressure
• passive earth pressure
• earth pressure at rest
In addition it is possible to account for the following effects having on the earth pressure magnitude:
• influence of water pressure
• influence of broken terrain
• friction between soil and back of structure
• influence of earth wedge at cantilever jumps
• influence of earthquake
When specifying rocks it is also necessary to input both the cohesion of rock c and the angle of internal friction of rock (φ). These values can be obtained either from the geological survey or from the table of recommended values.

Recommended values for typical calculations:
Soil Type                        Ø(φ , °)      γ(KN/m3)                c(kPa)
Gravel                               30                20                          0
Sand                                20                20                          0
Clay                                 15               19                          10
Concrete                                30               24                          200
Armed concrete                              45               25                          400
Geocelular protection                       20               18                           10

## Active and passive pressure

The active state occurs when a soil mass is allowed to relax or move outward to the point of reaching the limiting strength of the soil; that is, he soil is at the failure condition in extension. Thus it is the minimum lateral soil pressure that may be exerted. Conversely, the passive state occurs when a soil mass is externally forced to the limiting strength (that is, failure) of the soil in compression. It is the maximum lateral soil pressure that may be exerted. Thus active and passive pressures define the minimum and maximum possible pressures respectively that may be exerted in a horizontal plane.

### Rankine theory

Rankine's theory, developed in 1857, is a stress field solution that predicts active and passive earth pressure. It assumes that the soil is cohesionless, the wall is frictionless, the soil-wall interface is vertical, the failure surface on which the soil moves is planar, and the resultant force is angled parallel to the backfill surface. The equations for active and passive lateral earth pressure coefficients are given below. Note that φ' is the angle of shearing resistance of the soil and the backfill is inclined at angle β to the horizontal:
$K_a = \cos\beta \frac{\cos \beta - \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}{\cos \beta + \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}$  ; Ka=active coeficient for earth type
$K_p = \cos\beta \frac{\cos \beta + \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}{\cos \beta - \left(\cos ^2 \beta - \cos ^2 \phi \right)^{1/2}}$ ;  Kp=passive coeficient for earth type
For the case where β is 0, the above equations simplify to
$K_a = \tan ^2 \left( 45 - \frac{\phi}{2} \right) \$
$K_p = \tan ^2 \left( 45 + \frac{\phi}{2} \right) \$

### Coulomb theory

Coulomb (1776)  first studied the problem of lateral earth pressures on retaining structures. He used limit equilibrium theory, which considers the failing soil block as a free body in order to determine the limiting horizontal earth pressure. The limiting horizontal pressures at failure in extension or compression are used to determine the Ka and Kp respectively. Since the problem is indeterminate, a number of potential failure surfaces must be analysed to identify the critical failure surface (i.e. the surface that produces the maximum or minimum thrust on the wall). Mayniel (1908) later extended Coulomb's equations to account for wall friction, symbolized by δ. Müller-Breslau (1906) further generalized Mayniel's equations for a non-horizontal backfill and a non-vertical soil-wall interface (represented by angle θ from the vertical).
$K_a = \frac{ \cos ^2 \left( \phi - \theta \right)}{\cos ^2 \theta \cos \left( \delta + \theta \right) \left( 1 + \sqrt{ \frac{ \sin \left( \delta + \phi \right) \sin \left( \phi - \beta \right)}{\cos \left( \delta + \theta \right) \cos \left( \beta - \theta \right)}} \ \right) ^2}$

$K_p = \frac{ \cos ^2 \left( \phi + \theta \right)}{\cos ^2 \theta \cos \left( \delta - \theta \right) \left( 1 - \sqrt{ \frac{ \sin \left( \delta + \phi \right) \sin \left( \phi + \beta \right)}{\cos \left( \delta - \theta \right) \cos \left( \beta - \theta \right)}} \ \right) ^2}$
For an in-depth analysis on how to proper estimate your geotechnical in site conditions click here ! For an automated calculation of active,passive and at-rest pressures you can try a free application that you can acces by clicking   here .