9.3 Bearing capacity, stress and slope stability

Achievable FE Civil

9. Soil mechanics

Our FE Civil course is currently in development and is a work-in-progress.

Bearing capacity, stress and slope stability

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This chapter covers the following topics:

Bearing capacity
Vertical stress profiles
Vertical stress profiles with surcharge
Horizontal stress profiles and forces
Retaining walls
Slope stability

Bearing capacity

Definitions

Bearing capacity is the maximum pressure that soil can safely support from a foundation without experiencing shear failure or excessive settlement.

A common expression for the ultimate bearing capacity of a strip footing is:

$q_{u lt} = c N_{c} + γ^{'} D_{f} N_{q} + \frac{1}{2} γ^{'} B N_{γ}$

Where:

$N_{c}$ = bearing capacity factor for cohesion
$N_{q}$ = bearing capacity factor for depth
$N_{γ}$ = bearing capacity factor for unit weight
$D_{f}$ = depth of footing below ground surface
$B$ = width of strip footing

Vertical stress profiles

Definitions

Vertical stress is the stress acting perpendicular to a horizontal plane in soil due to self-weight and applied loads.

Vertical stress generally increases with depth because more soil (and any applied loads) lies above the point of interest.

Vertical stress profiles with surcharge

Definitions

Surcharge is an additional surface load applied to the ground, such as traffic load or structural loading, that increases vertical stress in soil.

A surcharge adds to the vertical stress in the soil mass below the loaded area, shifting the vertical stress profile upward compared with the no-surcharge case.

Horizontal stress profiles and forces

Definitions

Earth pressure refers to the lateral pressure exerted by soil on retaining structures.

Earth pressure depends on how the soil mass is allowed to deform:

Active conditions occur when the wall moves away from the soil enough for the soil to expand laterally.
Passive conditions occur when the wall moves into the soil enough to compress the soil laterally.
At-rest conditions occur when the wall does not move enough to mobilize active or passive states.

Active earth pressure coefficient (Rankine):

$K_{a} = tan^{2} (4 5^{\circ} - \frac{ϕ}{2})$

Passive earth pressure coefficient (Rankine):

$K_{p} = tan^{2} (4 5^{\circ} + \frac{ϕ}{2})$

At-rest earth pressure coefficient:

For normally consolidated soil:

$K_{0} = 1 - sin ϕ$

For overconsolidated soil:

$K_{0} = (1 - sin ϕ) \cdot OCR^{s i n ϕ}$

Retaining walls

Definitions

Retaining walls are structures designed to resist lateral earth pressure and retain soil at different elevations.

Retaining wall design checks typically focus on overturning, sliding, and bearing capacity.

Overturning:

$F S_{overturning} = \frac{\sum M _{R}}{M _{O}}$

Sliding:

$F S_{sliding} = \frac{\sum F _{R}}{\sum F _{D}}$

$F S_{sliding} = \frac{( \sum V ) tan δ + B C _{a} + P _{p}}{P _{a} cos α}$

Bearing Capacity:

$F S_{bearing capacity} = \frac{q _{UL T}}{q _{toe}}$

Toe stress:

$q_{toe} = \frac{\sum V}{B} (1 + \frac{6 e}{B})$

Eccentricity:

$e = \frac{B}{2} - (\frac{\sum M _{R} - M _{O}}{\sum V})$

Where:

$e$ = eccentricity
$B$ = width of base
$M_{R}$ = resisting moment
$M_{O}$ = overturning moment
$F_{R}$ = resisting forces
$F_{D}$ = driving forces
$V$ = vertical forces
$4 δ = k_{1} ϕ_{2}$
$C_{a} = k_{2} C_{2}$
$k_{1}$ , $k_{2}$ : given, range from 1/2 to 2/3

Slope stability

Definitions

Slope stability evaluates the ability of a soil slope to resist failure along a potential slip surface.

A common approach is to compare the available shear resistance along an assumed slip surface to the shear force required for equilibrium.

Factor of Safety:

$FS = \frac{T _{FF}}{T _{MOB}}$

Shearing Resistance (along slip surface):

$T_{FF} = c L_{s} + W_{M} cos α_{s} tan ϕ$

Mobilized Shear Force:

$T_{MOB} = W_{M} sin α_{s}$

Where:

$c$ = cohesion
$ϕ$ = angle of internal friction
$L_{s}$ = length of assumed planar slip surface
$W_{M}$ = weight of soil above slip surface
$α_{s}$ = angle of assumed slip surface w.r.t. horizontal

Bearing capacity

Maximum soil pressure foundation can support without failure or excessive settlement
Ultimate bearing capacity formula:
- $q_{u lt} = c N_{c} + γ^{'} D_{f} N_{q} + \frac{1}{2} γ^{'} B N_{γ}$
Key factors: $N_{c}$ , $N_{q}$ , $N_{γ}$ , $D_{f}$ , $B$

Vertical stress profiles

Vertical stress increases with depth due to overlying soil and loads
Defined as stress perpendicular to horizontal soil plane

Vertical stress profiles with surcharge

Surcharge: additional surface load (e.g., traffic, structures)
Surcharge increases vertical stress and shifts stress profile upward

Horizontal stress profiles and forces

Earth pressure: lateral pressure soil exerts on structures
Types of earth pressure:
- Active: wall moves away, soil expands
- Passive: wall moves into soil, soil compresses
- At-rest: no significant wall movement
Key coefficients:
- $K_{a} = tan^{2} (4 5^{\circ} - \frac{ϕ}{2})$ (active)
- $K_{p} = tan^{2} (4 5^{\circ} + \frac{ϕ}{2})$ (passive)
- $K_{0} = 1 - sin ϕ$ (at-rest, normally consolidated)
- $K_{0} = (1 - sin ϕ) \cdot OCR^{s i n ϕ}$ (at-rest, overconsolidated)

Retaining walls

Structures to resist lateral earth pressure and retain soil
Design checks:
- Overturning: $F S_{overturning} = \frac{\sum M _{R}}{M _{O}}$
- Sliding: $F S_{sliding} = \frac{\sum F _{R}}{\sum F _{D}}$ or alternate formula
- Bearing capacity: $F S_{bearing capacity} = \frac{q _{UL T}}{q _{toe}}$
Toe stress and eccentricity formulas:
- $q_{toe} = \frac{\sum V}{B} (1 + \frac{6 e}{B})$
- $e = \frac{B}{2} - (\frac{\sum M _{R} - M _{O}}{\sum V})$

Slope stability

Assesses soil slope’s resistance to failure along slip surface
Factor of Safety: $FS = \frac{T _{FF}}{T _{MOB}}$
- $T_{FF} = c L_{s} + W_{M} cos α_{s} tan ϕ$ (shearing resistance)
- $T_{MOB} = W_{M} sin α_{s}$ (mobilized shear force)
Key terms: cohesion ( $c$ ), friction angle ( $ϕ$ ), slip surface length ( $L_{s}$ ), soil weight ( $W_{M}$ ), slip angle ( $α_{s}$ )

Next | 9.4 Soil classification

Bearing capacity, stress and slope stability

This chapter covers the following topics:

Bearing capacity
Vertical stress profiles
Vertical stress profiles with surcharge
Horizontal stress profiles and forces
Retaining walls
Slope stability

Bearing capacity

Definitions

Bearing capacity is the maximum pressure that soil can safely support from a foundation without experiencing shear failure or excessive settlement.

A common expression for the ultimate bearing capacity of a strip footing is:

$q_{u lt} = c N_{c} + γ^{'} D_{f} N_{q} + \frac{1}{2} γ^{'} B N_{γ}$

Where:

$N_{c}$ = bearing capacity factor for cohesion
$N_{q}$ = bearing capacity factor for depth
$N_{γ}$ = bearing capacity factor for unit weight
$D_{f}$ = depth of footing below ground surface
$B$ = width of strip footing

Vertical stress profiles

Definitions

Vertical stress is the stress acting perpendicular to a horizontal plane in soil due to self-weight and applied loads.

Vertical stress generally increases with depth because more soil (and any applied loads) lies above the point of interest.

Vertical stress profiles with surcharge

Definitions

Surcharge is an additional surface load applied to the ground, such as traffic load or structural loading, that increases vertical stress in soil.

A surcharge adds to the vertical stress in the soil mass below the loaded area, shifting the vertical stress profile upward compared with the no-surcharge case.

Horizontal stress profiles and forces

Definitions

Earth pressure refers to the lateral pressure exerted by soil on retaining structures.

Earth pressure depends on how the soil mass is allowed to deform:

Active conditions occur when the wall moves away from the soil enough for the soil to expand laterally.
Passive conditions occur when the wall moves into the soil enough to compress the soil laterally.
At-rest conditions occur when the wall does not move enough to mobilize active or passive states.

Active earth pressure coefficient (Rankine):

$K_{a} = tan^{2} (4 5^{\circ} - \frac{ϕ}{2})$

Passive earth pressure coefficient (Rankine):

$K_{p} = tan^{2} (4 5^{\circ} + \frac{ϕ}{2})$

At-rest earth pressure coefficient:

For normally consolidated soil:

$K_{0} = 1 - sin ϕ$

For overconsolidated soil:

$K_{0} = (1 - sin ϕ) \cdot OCR^{s i n ϕ}$

Retaining walls

Definitions

Retaining walls are structures designed to resist lateral earth pressure and retain soil at different elevations.

Retaining wall design checks typically focus on overturning, sliding, and bearing capacity.

Overturning:

$F S_{overturning} = \frac{\sum M _{R}}{M _{O}}$

Sliding:

$F S_{sliding} = \frac{\sum F _{R}}{\sum F _{D}}$

$F S_{sliding} = \frac{( \sum V ) tan δ + B C _{a} + P _{p}}{P _{a} cos α}$

Bearing Capacity:

$F S_{bearing capacity} = \frac{q _{UL T}}{q _{toe}}$

Toe stress:

$q_{toe} = \frac{\sum V}{B} (1 + \frac{6 e}{B})$

Eccentricity:

$e = \frac{B}{2} - (\frac{\sum M _{R} - M _{O}}{\sum V})$

Where:

$e$ = eccentricity
$B$ = width of base
$M_{R}$ = resisting moment
$M_{O}$ = overturning moment
$F_{R}$ = resisting forces
$F_{D}$ = driving forces
$V$ = vertical forces
$4 δ = k_{1} ϕ_{2}$
$C_{a} = k_{2} C_{2}$
$k_{1}$ , $k_{2}$ : given, range from 1/2 to 2/3

Slope stability

Definitions

Slope stability evaluates the ability of a soil slope to resist failure along a potential slip surface.

A common approach is to compare the available shear resistance along an assumed slip surface to the shear force required for equilibrium.

Factor of Safety:

$FS = \frac{T _{FF}}{T _{MOB}}$

Shearing Resistance (along slip surface):

$T_{FF} = c L_{s} + W_{M} cos α_{s} tan ϕ$

Mobilized Shear Force:

$T_{MOB} = W_{M} sin α_{s}$

Where:

$c$ = cohesion
$ϕ$ = angle of internal friction
$L_{s}$ = length of assumed planar slip surface
$W_{M}$ = weight of soil above slip surface
$α_{s}$ = angle of assumed slip surface w.r.t. horizontal

Achievable FE Civil

9. Soil mechanics

Our FE Civil course is currently in development and is a work-in-progress.

Bearing capacity, stress and slope stability

4 min read

Font

Discuss

Feedback

This chapter covers the following topics:

Bearing capacity
Vertical stress profiles
Vertical stress profiles with surcharge
Horizontal stress profiles and forces
Retaining walls
Slope stability

Bearing capacity

Definitions

Bearing capacity is the maximum pressure that soil can safely support from a foundation without experiencing shear failure or excessive settlement.

A common expression for the ultimate bearing capacity of a strip footing is:

$q_{u lt} = c N_{c} + γ^{'} D_{f} N_{q} + \frac{1}{2} γ^{'} B N_{γ}$

Where:

$N_{c}$ = bearing capacity factor for cohesion
$N_{q}$ = bearing capacity factor for depth
$N_{γ}$ = bearing capacity factor for unit weight
$D_{f}$ = depth of footing below ground surface
$B$ = width of strip footing

Vertical stress profiles

Definitions

Vertical stress is the stress acting perpendicular to a horizontal plane in soil due to self-weight and applied loads.

Vertical stress generally increases with depth because more soil (and any applied loads) lies above the point of interest.

Vertical stress profiles with surcharge

Definitions

Surcharge is an additional surface load applied to the ground, such as traffic load or structural loading, that increases vertical stress in soil.

A surcharge adds to the vertical stress in the soil mass below the loaded area, shifting the vertical stress profile upward compared with the no-surcharge case.

Horizontal stress profiles and forces

Definitions

Earth pressure refers to the lateral pressure exerted by soil on retaining structures.

Earth pressure depends on how the soil mass is allowed to deform:

Active conditions occur when the wall moves away from the soil enough for the soil to expand laterally.
Passive conditions occur when the wall moves into the soil enough to compress the soil laterally.
At-rest conditions occur when the wall does not move enough to mobilize active or passive states.

Active earth pressure coefficient (Rankine):

$K_{a} = tan^{2} (4 5^{\circ} - \frac{ϕ}{2})$

Passive earth pressure coefficient (Rankine):

$K_{p} = tan^{2} (4 5^{\circ} + \frac{ϕ}{2})$

At-rest earth pressure coefficient:

For normally consolidated soil:

$K_{0} = 1 - sin ϕ$

For overconsolidated soil:

$K_{0} = (1 - sin ϕ) \cdot OCR^{s i n ϕ}$

Retaining walls

Definitions

Retaining walls are structures designed to resist lateral earth pressure and retain soil at different elevations.

Retaining wall design checks typically focus on overturning, sliding, and bearing capacity.

Overturning:

$F S_{overturning} = \frac{\sum M _{R}}{M _{O}}$

Sliding:

$F S_{sliding} = \frac{\sum F _{R}}{\sum F _{D}}$

$F S_{sliding} = \frac{( \sum V ) tan δ + B C _{a} + P _{p}}{P _{a} cos α}$

Bearing Capacity:

$F S_{bearing capacity} = \frac{q _{UL T}}{q _{toe}}$

Toe stress:

$q_{toe} = \frac{\sum V}{B} (1 + \frac{6 e}{B})$

Eccentricity:

$e = \frac{B}{2} - (\frac{\sum M _{R} - M _{O}}{\sum V})$

Where:

$e$ = eccentricity
$B$ = width of base
$M_{R}$ = resisting moment
$M_{O}$ = overturning moment
$F_{R}$ = resisting forces
$F_{D}$ = driving forces
$V$ = vertical forces
$4 δ = k_{1} ϕ_{2}$
$C_{a} = k_{2} C_{2}$
$k_{1}$ , $k_{2}$ : given, range from 1/2 to 2/3

Slope stability

Definitions

Slope stability evaluates the ability of a soil slope to resist failure along a potential slip surface.

A common approach is to compare the available shear resistance along an assumed slip surface to the shear force required for equilibrium.

Factor of Safety:

$FS = \frac{T _{FF}}{T _{MOB}}$

Shearing Resistance (along slip surface):

$T_{FF} = c L_{s} + W_{M} cos α_{s} tan ϕ$

Mobilized Shear Force:

$T_{MOB} = W_{M} sin α_{s}$

Where:

$c$ = cohesion
$ϕ$ = angle of internal friction
$L_{s}$ = length of assumed planar slip surface
$W_{M}$ = weight of soil above slip surface
$α_{s}$ = angle of assumed slip surface w.r.t. horizontal

Bearing capacity

Maximum soil pressure foundation can support without failure or excessive settlement
Ultimate bearing capacity formula:
- $q_{u lt} = c N_{c} + γ^{'} D_{f} N_{q} + \frac{1}{2} γ^{'} B N_{γ}$
Key factors: $N_{c}$ , $N_{q}$ , $N_{γ}$ , $D_{f}$ , $B$

Vertical stress profiles

Vertical stress increases with depth due to overlying soil and loads
Defined as stress perpendicular to horizontal soil plane

Vertical stress profiles with surcharge

Surcharge: additional surface load (e.g., traffic, structures)
Surcharge increases vertical stress and shifts stress profile upward

Horizontal stress profiles and forces

Earth pressure: lateral pressure soil exerts on structures
Types of earth pressure:
- Active: wall moves away, soil expands
- Passive: wall moves into soil, soil compresses
- At-rest: no significant wall movement
Key coefficients:
- $K_{a} = tan^{2} (4 5^{\circ} - \frac{ϕ}{2})$ (active)
- $K_{p} = tan^{2} (4 5^{\circ} + \frac{ϕ}{2})$ (passive)
- $K_{0} = 1 - sin ϕ$ (at-rest, normally consolidated)
- $K_{0} = (1 - sin ϕ) \cdot OCR^{s i n ϕ}$ (at-rest, overconsolidated)

Retaining walls

Structures to resist lateral earth pressure and retain soil
Design checks:
- Overturning: $F S_{overturning} = \frac{\sum M _{R}}{M _{O}}$
- Sliding: $F S_{sliding} = \frac{\sum F _{R}}{\sum F _{D}}$ or alternate formula
- Bearing capacity: $F S_{bearing capacity} = \frac{q _{UL T}}{q _{toe}}$
Toe stress and eccentricity formulas:
- $q_{toe} = \frac{\sum V}{B} (1 + \frac{6 e}{B})$
- $e = \frac{B}{2} - (\frac{\sum M _{R} - M _{O}}{\sum V})$

Slope stability

Assesses soil slope’s resistance to failure along slip surface
Factor of Safety: $FS = \frac{T _{FF}}{T _{MOB}}$
- $T_{FF} = c L_{s} + W_{M} cos α_{s} tan ϕ$ (shearing resistance)
- $T_{MOB} = W_{M} sin α_{s}$ (mobilized shear force)
Key terms: cohesion ( $c$ ), friction angle ( $ϕ$ ), slip surface length ( $L_{s}$ ), soil weight ( $W_{M}$ ), slip angle ( $α_{s}$ )

Next | 9.4 Soil classification

Bearing capacity, stress and slope stability

This chapter covers the following topics:

Bearing capacity
Vertical stress profiles
Vertical stress profiles with surcharge
Horizontal stress profiles and forces
Retaining walls
Slope stability

Bearing capacity

Definitions

Bearing capacity is the maximum pressure that soil can safely support from a foundation without experiencing shear failure or excessive settlement.

A common expression for the ultimate bearing capacity of a strip footing is:

$q_{u lt} = c N_{c} + γ^{'} D_{f} N_{q} + \frac{1}{2} γ^{'} B N_{γ}$

Where:

$N_{c}$ = bearing capacity factor for cohesion
$N_{q}$ = bearing capacity factor for depth
$N_{γ}$ = bearing capacity factor for unit weight
$D_{f}$ = depth of footing below ground surface
$B$ = width of strip footing

Vertical stress profiles

Definitions

Vertical stress is the stress acting perpendicular to a horizontal plane in soil due to self-weight and applied loads.

Vertical stress generally increases with depth because more soil (and any applied loads) lies above the point of interest.

Vertical stress profiles with surcharge

Definitions

Surcharge is an additional surface load applied to the ground, such as traffic load or structural loading, that increases vertical stress in soil.

A surcharge adds to the vertical stress in the soil mass below the loaded area, shifting the vertical stress profile upward compared with the no-surcharge case.

Horizontal stress profiles and forces

Definitions

Earth pressure refers to the lateral pressure exerted by soil on retaining structures.

Earth pressure depends on how the soil mass is allowed to deform:

Active conditions occur when the wall moves away from the soil enough for the soil to expand laterally.
Passive conditions occur when the wall moves into the soil enough to compress the soil laterally.
At-rest conditions occur when the wall does not move enough to mobilize active or passive states.

Active earth pressure coefficient (Rankine):

$K_{a} = tan^{2} (4 5^{\circ} - \frac{ϕ}{2})$

Passive earth pressure coefficient (Rankine):

$K_{p} = tan^{2} (4 5^{\circ} + \frac{ϕ}{2})$

At-rest earth pressure coefficient:

For normally consolidated soil:

$K_{0} = 1 - sin ϕ$

For overconsolidated soil:

$K_{0} = (1 - sin ϕ) \cdot OCR^{s i n ϕ}$

Retaining walls

Definitions

Retaining walls are structures designed to resist lateral earth pressure and retain soil at different elevations.

Retaining wall design checks typically focus on overturning, sliding, and bearing capacity.

Overturning:

$F S_{overturning} = \frac{\sum M _{R}}{M _{O}}$

Sliding:

$F S_{sliding} = \frac{\sum F _{R}}{\sum F _{D}}$

$F S_{sliding} = \frac{( \sum V ) tan δ + B C _{a} + P _{p}}{P _{a} cos α}$

Bearing Capacity:

$F S_{bearing capacity} = \frac{q _{UL T}}{q _{toe}}$

Toe stress:

$q_{toe} = \frac{\sum V}{B} (1 + \frac{6 e}{B})$

Eccentricity:

$e = \frac{B}{2} - (\frac{\sum M _{R} - M _{O}}{\sum V})$

Where:

$e$ = eccentricity
$B$ = width of base
$M_{R}$ = resisting moment
$M_{O}$ = overturning moment
$F_{R}$ = resisting forces
$F_{D}$ = driving forces
$V$ = vertical forces
$4 δ = k_{1} ϕ_{2}$
$C_{a} = k_{2} C_{2}$
$k_{1}$ , $k_{2}$ : given, range from 1/2 to 2/3

Slope stability

Definitions

Slope stability evaluates the ability of a soil slope to resist failure along a potential slip surface.

A common approach is to compare the available shear resistance along an assumed slip surface to the shear force required for equilibrium.

Factor of Safety:

$FS = \frac{T _{FF}}{T _{MOB}}$

Shearing Resistance (along slip surface):

$T_{FF} = c L_{s} + W_{M} cos α_{s} tan ϕ$

Mobilized Shear Force:

$T_{MOB} = W_{M} sin α_{s}$

Where:

$c$ = cohesion
$ϕ$ = angle of internal friction
$L_{s}$ = length of assumed planar slip surface
$W_{M}$ = weight of soil above slip surface
$α_{s}$ = angle of assumed slip surface w.r.t. horizontal