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Palos Verdes

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Stability

Static load on a raft foundation

Retaining wall

Retaining wall

Storyboard

In many cases, soil movements occur around a house, leaving slopes that can become unstable and require a retaining wall. In some instances, it's not so much a stability issue as it is about providing a finish and preventing future detachment that could hinder traffic on the adjacent surface.

>Model

ID:(112, 0)

Need for containment

Concept

>Top

In this case, on the southern side of the embankment, there is a slope that emerges from its creation, constructed with ($$), and contains the soil of the slope itself, which measures ($$).

To achieve a better finish and prevent progressive deterioration of the slope, the construction of a retaining wall measuring ($$) and the wall depth ($d$) is being considered.

ID:(733, 0)

Principle of instability

Concept

>Top

The retaining wall holds the soil that is at an angle greater than the angle of friction between soil and wall ($\varphi$), or the angle of repose, meaning the angle beneath which the soil rests without sliding. If the retaining wall were removed, it would be the segment of soil with the steepest angle that would slide:

The entire mechanics of soil containment boils down to the measures necessary to retain the soil that would slide without it, and which lies between the angle of friction between soil and wall ($\varphi$) and the wall slope angle from horizontal ($\alpha$).

ID:(752, 0)

Active force per length

Concept

>Top

On the element that can slide, a force, which we'll call the active force per length applied to the wall ($f_A$), acts. This force has components the horizontal force per floor length applied to the wall ($f_H$)

$ f_H = f_A \cos \varphi $

and the vertical force per floor length applied to the wall ($f_V$) with

$ f_V = f_A \sin \varphi $

which are projections in the angle of friction between soil and wall ($\varphi$):

Additionally, it's necessary to consider the total wall weight per length ($w$), as it contributes to stabilizing the retaining wall.

These components can cause displacement, settlement, and overturning of the retaining wall, leading to safety factors that must be met to ensure the design remains stable.

ID:(753, 0)

Construction of the retaining wall

Concept

>Top

A simple way to construct a retaining wall is using bricks measuring ($$) in length, reaching ($$) in height:

This type of wall relies on its own weight to effectively retain the soil it contains. Therefore, the mass of the wall is crucial.

ID:(735, 0)

Brick Alternative

Concept

>Top

One alternative is the Graublock GST-20 concrete blocks, which have a height of 19 cm, a width of 39 cm, and a depth of 19 cm.

Since the angle of the wall would be the wall slope angle from horizontal ($\alpha$) of 60°, the bricks should be stacked with an offset of 11 cm.

With an opening width of 12.6 cm, each edge would protrude by 3.2 cm, leaving an opening of 7.8 cm where plants can be planted. This wall would receive some sunlight (partially covered by the house) and would be more protected from the wind, creating a type of vertical garden.

To maintain a substantial mass and aesthetically utilize the wall, the spaces between the bricks can be filled with soil and planted with vegetation. In this case, each brick has a space of $6\times 10^{-3} m^3$, allowing for approximately 9 kg of soil to be added, increasing the mass of the brick to 26 kg.

ID:(734, 0)

Sliding

Concept

>Top

The total wall weight per length ($w$) and the vertical force per floor length applied to the wall ($f_V$) generate via the angle of internal friction of the soil ($\phi$) A force per length that resists slipping ($f_R$), which can be calculated by:

$ f_R = ( w + f_V )\tan \phi $

This can prevent the horizontal force per floor length applied to the wall ($f_H$) from displacing the retaining wall, destabilizing the system:

To ensure that the retaining wall does not move, it is required that the ratio between both forces, expressed in the slip safety factor ($SFS$), is for safety purposes greater than 2.0.

ID:(736, 0)

Support

Concept

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The total wall weight per length ($w$) and the vertical force per floor length applied to the wall ($f_V$) generate, through the wall depth ($d$), a force of a pressure below the base of the block ($\sigma_W$), which can be calculated by:

$ \sigma_W = \displaystyle\frac{1}{ d }( w + f_V )$

This force can overcome the bearing capacity ($\sigma_S$), leading to the retaining wall sinking:

To ensure that the retaining wall does not move, it is required that the relationship between both tensions, expressed in the safety factor for support ($FSB$), is for safety purposes greater than 2.0.

ID:(737, 0)

Overturn

Concept

>Top

ID:(738, 0)

Rollover stability

Concept

>Top

ID:(739, 0)

Mechanisms

Concept

>Top

The geometry of the retaining wall is reflected in the active pressure coefficient ($K_a$), which together with the wall height ($H$), defines the the active force per length applied to the wall ($f_A$) that could lead to the wall's destabilization. By considering the components the horizontal force per floor length applied to the wall ($f_H$) and the vertical force per floor length applied to the wall ($f_V$), we can estimate the risks of displacement (the slip safety factor ($SFS$)) and settling (the safety factor for support ($FSB$)), and through the torques the overturning moment per length ($m_O$) and the moment of resistance per length ($m_R$), the risk of overturning (the safety factor against overturning ($SFO$)).

Knowledge

Code

Concept

Mechanisms

ID:(754, 0)

Active pressure coefficient

Equation

>Top, >Model

$ K_a = \displaystyle\frac{\sin^2( \alpha + \phi )}{\sin^2 \alpha \sin( \alpha - \varphi )\left[1+\sqrt{\displaystyle\frac{\sin( \phi + \varphi )\sin( \phi - \beta )}{\sin( \alpha - \varphi )\sin( \alpha + \beta )}}\right]^2}$

$K_a$

Active pressure coefficient

$-$

$\beta$

Angle of backfill slope

$rad$

$\varphi$

Angle of friction between soil and wall

$rad$

$\phi$

Angle of internal friction of the soil

$rad$

$\alpha$

Wall slope angle from horizontal

$rad$

ID:(740, 0)

Active force per length applied to the wall

Equation

>Top, >Model

$ f_A = \displaystyle\frac{1}{2} \rho_S g K_a H ^2$

$f_A$

Active force per length applied to the wall

$N/m$

$K_a$

Active pressure coefficient

$-$

$g$

Gravitational acceleration

9.8

$m/s^2$

$\rho_S$

Soil density

$kg/m^3$

$H$

Wall height

$m$

ID:(741, 0)

Total weight per wall length

Equation

>Top, >Model

$ w = \rho_W g H d $

$g$

Gravitational acceleration

9.8

$m/s^2$

$w$

Total wall weight per length

$N/m$

$\rho_W$

Wall density

$kg/m^3$

$d$

Wall depth

$m$

$H$

Wall height

$m$

ID:(742, 0)

Vertical soil force per length applied to the wall

Equation

>Top, >Model

Since the active force per length applied to the wall ($f_A$) has a the angle of friction between soil and wall ($\varphi$) with respect to the horizontal, its vertical projection is the vertical force per floor length applied to the wall ($f_V$) according to

$ f_V = f_A \sin \varphi $

$f_A$

Active force per length applied to the wall

$N/m$

$\varphi$

Angle of friction between soil and wall

$rad$

$f_V$

Vertical force per floor length applied to the wall

$N/m$

ID:(743, 0)

Horizontal soil force per length applied to the wall

Equation

>Top, >Model

Since the active force per length applied to the wall ($f_A$) has a the angle of friction between soil and wall ($\varphi$) angle with respect to the horizontal, its horizontal projection is the horizontal force per floor length applied to the wall ($f_H$) according to

$ f_H = f_A \cos \varphi $

$f_A$

Active force per length applied to the wall

$N/m$

$\varphi$

Angle of friction between soil and wall

$rad$

$f_H$

Horizontal force per floor length applied to the wall

$N/m$

ID:(744, 0)

Force per length that resists slipping

Equation

>Top, >Model

$ f_R = ( w + f_V )\tan \phi $

$\phi$

Angle of internal friction of the soil

$rad$

$f_R$

Force per length that resists slipping

$N/m$

$w$

Total wall weight per length

$N/m$

$f_V$

Vertical force per floor length applied to the wall

$N/m$

ID:(745, 0)

Slip safety factor

Equation

>Top, >Model

To determine the risk of the retaining wall displacing due to the force exerted by the retained soil, the slip safety factor ($SFS$) is determined by dividing the force per length that resists slipping ($f_R$) by the horizontal force per floor length applied to the wall ($f_H$):

$ SFS = \displaystyle\frac{ f_R }{ f_H }$

$f_R$

Force per length that resists slipping

$N/m$

$f_H$

Horizontal force per floor length applied to the wall

$N/m$

$SFS$

Slip safety factor

$-$

If the slip safety factor ($SFS$) exceeds unity, it indicates a situation where the force per length that resists slipping ($f_R$) surpasses the horizontal force per floor length applied to the wall ($f_H$), meaning the wall should not displace. Since the values are estimated, it may yield a factor greater than unity, even when the force per length that resists slipping ($f_R$) is actually less than the horizontal force per floor length applied to the wall ($f_H$). For safety, a factor greater than 2.0 is required to consider that the wall will not displace.

ID:(746, 0)

Overturning moment per length

Equation

>Top, >Model

"The distance between the point of application of the horizontal force per floor length applied to the wall ($f_H$) and the pivot point is equal to $H/3$ due to the force triangle in which it increases with the depth in the soil behind the retaining wall.

Since the torque per length is equal to the overturning moment per length ($m_O$), calculated from the horizontal force per floor length applied to the wall ($f_H$) by the vertical arm corresponding to the wall height ($H$) thirds, we have:

$ m_O = \displaystyle\frac{1}{3} H f_H $

$f_H$

Horizontal force per floor length applied to the wall

$N/m$

$m_O$

Overturning moment per length

$N$

$H$

Wall height

$m$

ID:(747, 0)

Moment of resistance to overturning

Equation

>Top, >Model

For the torque stabilizing the retaining wall, both the total wall weight per length ($w$) and the vertical force per floor length applied to the wall ($f_V$) are considered. The necessary arms can be derived from the geometry by referring to the following diagram:

Here, the arm of the total wall weight per length ($w$) consists of half the length of the wall depth ($d$) and the distance from the center of mass of the retaining wall, calculated from the wall height ($H$) and the wall slope angle from horizontal ($\alpha$):

$\displaystyle\frac{1}{2}d + \displaystyle\frac{1}{2}H \text{cot}\alpha$

For the case of the vertical force per floor length applied to the wall ($f_V$), the wall depth ($d$) must be added to the horizontal distance from the point of application of the active force per length applied to the wall ($f_A$), associated with the wall height ($H$) and the angle of friction between soil and wall ($\varphi$), through:

$d + \displaystyle\frac{1}{3}H \text{cot}\varphi$

This yields:

$ m_R = \displaystyle\frac{1}{2}\left( d + H \text{cot} \alpha \right) w + \left( d + \displaystyle\frac{1}{3} H \text{cot} \varphi \right) f_V $

$\varphi$

Angle of friction between soil and wall

$rad$

$m_R$

Moment of resistance per length

$N$

$w$

Total wall weight per length

$N/m$

$f_V$

Vertical force per floor length applied to the wall

$N/m$

$d$

Wall depth

$m$

$H$

Wall height

$m$

$\alpha$

Wall slope angle from horizontal

$rad$

ID:(748, 0)

Safety factor against overturning

Equation

>Top, >Model

$ SFO = \displaystyle\frac{ m_R }{ m_O }$

$m_R$

Moment of resistance per length

$N$

$m_O$

Overturning moment per length

$N$

$SFO$

Safety factor against overturning

$-$

ID:(749, 0)

Pressure applied to the ground below the base of the block

Equation

>Top, >Model

The pressure beneath the block arises from the total wall weight per length ($w$) and the vertical force per floor length applied to the wall ($f_V$), which must be divided by the area along the wall, resulting in the wall depth ($d$). Therefore, it is concluded that the pressure below the base of the block ($\sigma_W$) is

$ \sigma_W = \displaystyle\frac{1}{ d }( w + f_V )$

$\sigma_W$

Pressure below the base of the block

$Pa$

$w$

Total wall weight per length

$N/m$

$f_V$

Vertical force per floor length applied to the wall

$N/m$

$d$

Wall depth

$m$

ID:(750, 0)

Safety factor for support

Equation

>Top, >Model

To determine whether there is a risk of the retaining wall sinking into the ground it supports, the pressure below the base of the block ($\sigma_W$) must be compared with the bearing capacity ($\sigma_S$), with the former being less than the latter.

Thus, the safety factor for support ($FSB$) can be defined as the ratio between the bearing capacity ($\sigma_S$) and the pressure below the base of the block ($\sigma_W$):

$ FSB = \displaystyle\frac{ \sigma_S }{ \sigma_W }$

$\sigma_S$

Bearing capacity

$Pa$

$\sigma_W$

Pressure below the base of the block

$Pa$

$FSB$

Safety factor for support

$-$

ensuring that for safety reasons, this factor is equal to or greater than 2.0.

ID:(751, 0)

Model

Concept

>Top

Parameters

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$\beta$

beta

Angle of backfill slope

rad

$\varphi$

vphi

Angle of friction between soil and wall

rad

$\phi$

phi

Angle of internal friction of the soil

rad

$\sigma_S$

sigma_S

Bearing capacity

$g$

Gravitational acceleration

m/s^2

$\rho_S$

rho_S

Soil density

kg/m3

Variables

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$f_A$

f_A

Active force per length applied to the wall

N/m

$K_a$

K_a

Active pressure coefficient

$f_R$

f_R

Force per length that resists slipping

N/m

$f_H$

f_H

Horizontal force per floor length applied to the wall

N/m

$m_R$

m_R

Moment of resistance per length

$m_O$

m_O

Overturning moment per length

$\sigma_W$

sigma_W

Pressure below the base of the block

$SFO$

SFO

Safety factor against overturning

$FSB$

FSB

Safety factor for support

$SFS$

SFS

Slip safety factor

$w$

Total wall weight per length

N/m

$f_V$

f_V

Vertical force per floor length applied to the wall

N/m

$\rho_W$

rho_W

Wall density

kg/m3

$d$

Wall depth

$H$

Wall height

$\alpha$

alpha

Wall slope angle from horizontal

rad

Calculations

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Calculations

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Equation

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Equations

Equation

$ FSB = \displaystyle\frac{ \sigma_S }{ \sigma_W }$

FSB = sigma_S / sigma_W

$ f_A = \displaystyle\frac{1}{2} \rho_S g K_a H ^2$

f_A = rho_S * g * K_a * H ^2/2

$ f_H = f_A \cos \varphi $

f_H = f_A * cos( vphi )

$ f_R = ( w + f_V )\tan \phi $

f_R = ( w + f_V )* tan( phi )

$ f_V = f_A \sin \varphi $

f_V = f_A * sin( vphi )

K_a = sin( alpha + phi )^2/(sin( alpha )^2*sin( alpha - vphi )*[1+sqrt((sin( phi + vphi )*sin( phi - beta ))/(sin( alpha - vphi )*sin( alpha + beta )))]^2)

$ m_O = \displaystyle\frac{1}{3} H f_H $

m_O = H * f_H / 3

$ m_R = \displaystyle\frac{1}{2}\left( d + H \text{cot} \alpha \right) w + \left( d + \displaystyle\frac{1}{3} H \text{cot} \varphi \right) f_V $

m_R = ( d + H * cot( alpha ) ) * w /2 + ( d + H * cot( vphi )/3)* f_V

$ SFO = \displaystyle\frac{ m_R }{ m_O }$

SFO = m_R / m_O

$ SFS = \displaystyle\frac{ f_R }{ f_H }$

SFS = f_R / f_H

$ \sigma_W = \displaystyle\frac{1}{ d }( w + f_V )$

sigma_W = ( w + f_V )/ d

$ w = \rho_W g H d $

w = rho_W * g * H * d

ID:(755, 0)