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Slopes have the issue that the soil can slide if the forces generated by its own weight exceed the soil's cohesion. Since cohesion can vary due to external factors, there is a possibility that a mass may lose stability and shift, making it essential to understand its vulnerability and the likelihood of future destabilization.

>Model

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Iframe

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Description

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Para modelar la estabilidad de un terreno asumimos un fondo rocoso con una pendiente dada y una capa de suelo homogénea que se puede deslizar sobre esta.

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Image

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La sección que estamos estudiando tiene un ancho \Delta.y un largo L:

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Image

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En primera instancia podemos considerar que la masa genera una fuerza gravitacional que trata de deslizar el suelo por la pendiente. Por otro lado la componente vertical al fondo rocoso genera el roce necesario para mantener la masa en su lugar:

De no existir agua ambas fuerzas son proporcionales a la masa por lo que finalmente solo dependerá del coeficiente de roce si la capa es estable.

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Image

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De existir agua en el suelo esta contribuye en varias formas para desestabilizar la capa de suelo. Una primera forma es creando una fuerza de sustentación que reduce la fuerza normal y con ello el roce que sujeta el suelo en el lugar:

Este comportamiento corresponde a lo que se podría llamar en el limite la tendencia a que el suelo flote.

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Image

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La segunda contribución del agua tiende, en la medida que el agua este adecuadamente dosificada, a estabilizar el suelo. Si solo figura como humedad relativa alta se forman meniscos de agua entre los granos que ejercen fuerzas cohesivas. Sin embargo si la capa de suelo es inundada dicha sección pierde esta cohesión y es el resto sobre el nivel del agua que debe soportar el peso de la masa:

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Concept

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The cohesion of the material ($c$) and the angle of internal friction of the soil ($\phi$) depend on the soil composition (the mass fraction of sand in the sample ($g_a$), the mass fraction of silt in the sample ($g_i$), the mass fraction of clay in the sample ($g_c$)) and water content (the mass fraction of water in the sample ($g_w$)).

Based on measurements, phenomenological models can be developed to describe these properties:

Cohesion Model

Cohesion the cohesion of the material ($c$) is expressed using the equation:

Where the constants the inherent cohesion of dry material ($c_0$), the degree of cohesion induced by fine particles ($k$), and the sensitivity of cohesion to water ($m$) take the following typical values:

• the inherent cohesion of dry material ($c_0$):

• the degree of cohesion induced by fine particles ($k$): 20 - 200 kPa
• the sensitivity of cohesion to water ($m$): 5 - 20 kPa

Internal Friction Angle Model

The internal friction angle the angle of internal friction of the soil ($\phi$) is described using the equation:

Where the constants the internal friction angle of the base soil ($\phi_0$), the friction angle sensitivity to clay ($k_c$), the friction angle sensitivity to sand ($k_a$), and the friction angle sensitivity to water ($k_w$) take the following values:

• the internal friction angle of the base soil ($\phi_0$):

• the friction angle sensitivity to clay ($k_c$): 5° - 10°
• the friction angle sensitivity to sand ($k_a$): 3° - 8°
• the friction angle sensitivity to water ($k_w$): 5° - 15°

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Parameters

$\phi$

phi

Angle of internal friction of the soil

rad

$c$

c

Cohesion of the material

Pa

$k$

k

Degree of cohesion induced by fine particles

Pa

$k_c$

k_c

Friction angle sensitivity to clay

rad

$k_a$

k_a

Friction angle sensitivity to sand

rad

$k_w$

k_w

Friction angle sensitivity to water

rad

$g$

g

Gravitational Acceleration

m/s^2

$c_0$

c_0

Inherent cohesion of dry material

Pa

$\phi_0$

phi_0

Internal friction angle of the base soil

rad

$\sigma$

sigma

Normal tension

Pa

$s$

s

Saturation

-

$SF$

SF

Security factor

-

$m$

m

Sensitivity of cohesion to water

Pa

$\rho_s$

rho_s

Solid Density

kg/m^3

$\gamma_s$

gamma_s

Unit weight of soil

N/m^3

$\gamma_w$

gamma_w

Unit weight of water

N/m^3

$\rho_w$

rho_w

Water density

kg/m^3

Variables

$H$

H

Layer height

m

$g_c$

g_c

Mass fraction of clay in the sample

-

$g_a$

g_a

Mass fraction of sand in the sample

-

$g_i$

g_i

Mass fraction of silt in the sample

-

$g_w$

g_w

Mass fraction of water in the sample

-

$\tau$

tau

Shear stress

Pa

$\theta$

theta

Slope angle of the hillside

$p_v$

p_v

Water pressure in pores

Pa

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(16105, 0)

Equation

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The the security factor ($SF$) represents the ratio of the stress that prevents sliding. It is calculated based on the cohesion of the material ($c$), adjusted by the normal tension ($\sigma$), reduced by the water pressure in pores ($p_v$), and weighted using the tangent of the angle of internal friction of the soil ($\phi$) and the normal tension ($\sigma$), as expressed in the following equation:

$SF = \displaystyle\frac{ c + ( \sigma - p_v )\tan \phi }{ \tau }$

Conditions

Variables

$\phi$

Angle of internal friction of the soil

$rad$

10528

$c$

Cohesion of the material

$Pa$

10527

$\sigma$

Normal tension

$Pa$

10510

$SF$

Security factor

$-$

10526

$\tau$

Shear stress

$Pa$

10512

$p_v$

Water pressure in pores

$Pa$

10511

Relation

ID:(16112, 0)

Equation

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The shear stress ($\tau$) is calculated from unit weight of soil ($\gamma_s$), combined with the layer height ($H$), and weighted by the sine of the slope angle of the hillside ($\theta$), as shown in the following formula:

$\tau = \gamma_s H \sin \theta$

Conditions

Variables

$H$

Layer height

$m$

8239

$\tau$

Shear stress

$Pa$

10512

$\theta$

Slope angle of the hillside

$rad$

4953

$\gamma_s$

Unit weight of soil

$N/m^3$

10508

Relation

ID:(16111, 0)

Equation

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unit weight of water ($\gamma_w$) of water is determined from the water density ($\rho_w$) and the gravitational Acceleration ($g$), using the following formula:

$\gamma_w = \rho_w g$

Conditions

Variables

$g$

Gravitational Acceleration

9.8

$m/s^2$

5310

$\gamma_w$

Unit weight of water

$N/m^3$

10509

$\rho_w$

Water density

$kg/m^3$

6000

Relation

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Equation

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unit weight of soil ($\gamma_s$) of a body is calculated using the solid Density ($\rho_s$) and the gravitational Acceleration ($g$), as expressed in the following formula:

$\gamma_s = \rho_s g$

Conditions

Variables

$g$

Gravitational Acceleration

9.8

$m/s^2$

5310

$\rho_s$

Solid Density

$kg/m^3$

4944

$\gamma_s$

Unit weight of soil

$N/m^3$

10508

Relation

ID:(16107, 0)

Equation

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The water pressure in pores ($p_v$) generated by water in the pores is calculated using the saturation ($s$), unit weight of water ($\gamma_w$), and the layer height ($H$), as shown in the following formula:

$p_v = s \gamma_w H$

Conditions

Variables

$H$

Layer height

$m$

8239

$s$

Saturation

$-$

10529

$\gamma_w$

Unit weight of water

$N/m^3$

10509

$p_v$

Water pressure in pores

$Pa$

10511

Relation

ID:(16110, 0)

Equation

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The normal tension ($\sigma$) is the stress that counteracts sliding, calculated using unit weight of soil ($\gamma_s$), the layer height ($H$), and the slope angle of the hillside ($\theta$), as shown in the following formula:

$\sigma = \gamma_s H \cos \theta$

Conditions

Variables

$H$

Layer height

$m$

8239

$\sigma$

Normal tension

$Pa$

10510

$\theta$

Slope angle of the hillside

$rad$

4953

$\gamma_s$

Unit weight of soil

$N/m^3$

10508

Relation

ID:(16109, 0)

Equation

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The cohesion of the material ($c$) can be estimated using the inherent cohesion of dry material ($c_0$), the degree of cohesion induced by fine particles ($k$), the sensitivity of cohesion to water ($m$), the mass fraction of clay in the sample ($g_c$), the mass fraction of silt in the sample ($g_i$), and the mass fraction of water in the sample ($g_w$), with the following formula:

$c = c_0 + k ( g_i + g_c ) - m g_w$

Conditions

Variables

$c$

Cohesion of the material

$Pa$

10527

$k$

Degree of cohesion induced by fine particles

$Pa$

10532

$c_0$

Inherent cohesion of dry material

$Pa$

10531

$g_c$

Mass fraction of clay in the sample

$-$

10099

$g_i$

Mass fraction of silt in the sample

$-$

10098

$g_w$

Mass fraction of water in the sample

$-$

10530

$m$

Sensitivity of cohesion to water

$Pa$

10535

Relation

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Equation

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The angle of internal friction of the soil ($\phi$) can be estimated using the internal friction angle of the base soil ($\phi_0$), the friction angle sensitivity to clay ($k_c$), the friction angle sensitivity to sand ($k_a$), the friction angle sensitivity to water ($k_w$), the mass fraction of clay in the sample ($g_c$), the mass fraction of sand in the sample ($g_a$), and the mass fraction of water in the sample ($g_w$), with the following formula:

$\phi = \phi_0 + k_a g_a - k_c g_c - k_w g_w$

Conditions

Variables

$\phi$

Angle of internal friction of the soil

$rad$

10528

$k_c$

Friction angle sensitivity to clay

$rad$

10538

$k_a$

Friction angle sensitivity to sand

$rad$

10537

$k_w$

Friction angle sensitivity to water

$rad$

10536

$\phi_0$

Internal friction angle of the base soil

$rad$

10534

$g_c$

Mass fraction of clay in the sample

$-$

10099

$g_a$

Mass fraction of sand in the sample

$-$

5797

$g_w$

Mass fraction of water in the sample

$-$

10530

Relation

ID:(16124, 0)