retaining walls design
TRANSCRIPT
•1
Subgrade reaction calculation method
Retaining walls design
Page 1V. Bernhardt
• Introduction
Summary
• Principles of the calculation method
• Required data
• Definition of the construction stages
• Typical output
Page 2June 2008
•2
The subgrade reaction method allows for the analysis of flexible retaining walls such as diaphragm walls, soldier-pile walls, or h t il ll
Introduction
sheet-pile walls.
It enables to calculate the horizontal displacements and bendingmoments of the retaining wall through its various construction stages: • The initial stage consists in building the retaining wall itself. • The following stages correspond to various actions such as
earthworks (excavations, fills, …), installation of anchors or
Page 3June 2008
earthworks (excavations, fills, …), installation of anchors or struts, change of the water level, or load application.
App
licat
ions
Plane and circular diaphragm walls
Page 4June 2008
•3
App
licat
ions
Soldier-pile walls / Sheet-pile walls
Page 5June 2008
• The wall is assumed to extend to infinity in the out-of-plane direction, so that the problem is plane strain (except in the case of a circular retaning wall).
• The wall inertia be defined according to depth. The wall can be subjected to:Earth and water pressures
The calculation method
Cal
cula
tion
met
hod
Earth and water pressuresHorizontal loadsForces applied by struts or anchorsImposed external momentsRotation springs (embedment of external structures).
• The earth and water pressures are modeled by horizontal pressures applied on both sides of the wall. Earth pressures are related to the wall displacements by
Page 6June 2008
an elasto-plastic soil behaviour law. The parameters for this law are calculated at each depth: they depend on the soil properties of the corresponding layer, and on the vertical stress in the soil (depending on the excavation level, the water level and the possible loads).
•4
• Soil layers are modeled as springs reacting linearly until they reach a
l tifi ti t ( ith
Reactions applied by the soilonto the beam = springsReactions applied by the soilonto the beam = springs
• The retaining wall is assumed to be a flexible beam, laying on elasto-plastic supports.
Cal
cula
tion
met
hod
The calculation method
plastification stress (either on active or passive pressure side).
• In construction stages, various actions can be defined, resulting in forces acting on the beam.
• The calculation consists in
Reaction applied by the soil onto thebeam in a given pointReaction applied by the soil onto thebeam in a given point
Page 7June 2008
finding the equilibrium state between the beam displacements and the stresses in the soil layers: iterative calculation.
Pa: pressure applied by the soil at limit equilibrium(active pressure)Pp: pressure applied by the soil at limit equilibrium(passive pressure)Kh: soil reaction modulus
Pa: pressure applied by the soil at limit equilibrium(active pressure)Pp: pressure applied by the soil at limit equilibrium(passive pressure)Kh: soil reaction modulus
• At-rest pressurepi = p0 = k0 σ’v0for the first calculation stage with σ’v0: vertical
Cal
cula
tion
met
hod
Elasto-plastic soil behaviour
stage with σ v0: vertical effective stress at rest
• Active pressurepa = ka σ’v – ca c
• Passive pressurepp = kp σ’v + cp c
• Modulus of subgrade reaction
Page 8June 2008
UphillDisplacements towards uphillUphill
Displacements towards uphillUphill
Displacements towards uphillUphill
Displacements towards uphill
reactiongradient = kh + dkh . z with kh: modulus (i.e. coefficient) of subgrade reaction
•5
Soil behaviour changes after soil plastificationC
alcu
latio
n m
etho
dElasto-plastic soil behaviour
Soil behaviour changes h h ll i
UphillDisplacements towards uphill
UphillDisplacements towards uphill
UphillDisplacements towards uphill
UphillDisplacements towards uphill
UphillDisplacements towards uphill
UphillDisplacements towards uphill
Page 9June 2008
when the wall is « separated » from the soil(no traction allowed)
UphillDisplacements towards uphill
UphillUphillDisplacements towards uphill
UphillDisplacements towards uphill
UphillUphillDisplacements towards uphill
Soil behaviour varies depending on loading conditions: consolidation phenomenon is taken into account with unloading and reloading coefficients (for soft clays for example).
Cal
cula
tion
met
hod
Unloading/reloading coefficients
Reloading conditions
UphillDisplacements towards uphillUphill
Displacements towards uphillUphill
Displacements towards uphillUphill
Displacements towards uphill
Page 10June 2008
• Δpi = kr Δσ’v if Δσ’v > 0 with kr: reloading coefficient• Δpi = kd Δσ’v if Δσ’v < 0 avec kd: unloading coefficient
As the initial state is modified, the displacement required to reach plastification as changes, especially in soft soils.
•6
Required data include:
The project dataR
equi
red
data
q
Project general settings
Soil properties
Retaining wall properties
Page 11June 2008
• Units
Req
uire
d da
ta
General settings
• Water unit weight
• Number of iterations allowed for the calculation of each stage
• Calculation step along the wall (maximum value)
Page 12June 2008
•7
Soil properties:• Zl and Zwater: top level of the layer
and water level
• PVh and PVd: moist unit weight and buoyant unit weight
Specific calculation properties:• k0, ka, kp: coefficients of at-rest,
active and passive earth pressures
• kd, kr: unloading and reloading coefficients
Req
uire
d da
taSoil properties
• c, Δc/m and ϕ • ca, cp: active and passive earth pressure coefficients for cohesion
• kh, Δkh/m: modulus of subgrade reaction and its variation with depth
K-REA includes useful wizards.
Page 13June 2008
3 wizards:
Req
uire
d da
ta
Active and passive earth pressure coefficients
Kérisel and Absi (tables)
Coulomb method (formulae)
( )2
2cos −=
ϕλaK
Page 14June 2008
( ) ( ) ( )( ) ( )
2
coscossinsin1cos ⎟⎟
⎠
⎞⎜⎜⎝
⎛−+−+
++βλδλβϕδϕδλ
a
aa
a
( )
( ) ( ) ( )( ) ( )
2
2
coscossinsin
1cos
cos
⎟⎟⎠
⎞⎜⎜⎝
⎛
−++−
−+
+=
βλδλβϕδϕ
δλ
ϕλ
p
pp
pK
•8
Rankine formulae
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−+
−−=
ϕββ
ϕβββ
22
22
coscoscoscoscoscos
cosaK
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−
−+=
ϕββ
ϕβββ
22
22
coscoscoscoscoscos
cospK
Req
uire
d da
taActive and passive earth pressure coefficients
Caquot formulae for ca and cp:
⎟⎠⎞
⎜⎝⎛ −=
24tan2 ϕπ
aK ⎟⎠⎞
⎜⎝⎛ +=
24tan2 ϕπ
pK
Note:
• If no slope (β = 0):
• The Rankine formulae do not take into accountfriction between soil and wall
Page 15June 2008
( )⎥⎦
⎤⎢⎣
⎡−
+−
= −− 1cosexpsin1
cossincostan
1 tan δϕ
γϕδϕ
ϕϕγac
( )⎥⎦
⎤⎢⎣
⎡−
−+
= + 1cosexpsin1
cossincostan
1 tan δϕ
γϕδϕ
ϕϕγpc
Balay method
3 wizards:
( )ααα *91330* += a
Ek mh
Req
uire
d da
ta
Subgrade reaction modulus
Schmitt method
( )αα 9133,02+
( )31
34
*1,2
EI
E
k
m
h
⎟⎠⎞
⎜⎝⎛
= α
Page 16June 2008
Chadeisson curves
•9
The retaining wall should be defined either by its:• Total product of inertia,
• Thickness and Young’s modulus.
Advanced properties:
Req
uire
d da
taRetaining wall properties
Advanced properties:• Working length out-of-plane,
• Circular retaining wall.
Page 17June 2008Required data
Various action types are used to define the construction stages. They are divided into 6 categories:
Definition of construction stages
Con
stru
ctio
n st
ages
Initial conditions
Loading / Forces / Couples
Earthworks
Anchors / Wall
Soil properties
Page 18June 2008
Hydraulic conditions
•10
Con
stru
ctio
n st
ages
Definition of construction stages
Page 19June 2008
• « Caquot » load (uniform and distributed. It is removed when
th k li d th
These actions can be applied only once, in the initial conditions.
UphillUphillDownhill UphillUphillDownhill UphillUphillDownhill UphillUphillDownhill
Con
stru
ctio
n st
ages
Initial conditions
earthworks are applied on the same side).
• Reduced pressures for soldier-pile walls. Pressures are applied again at 100 % (i.e. without reduction) after sheeting installation
Between z1 and z2:Active pressure multiplied by RPassive pressure multiplied by R*CWater pr. of both sides multiplied by RKh multiplied by R
Page 20June 2008
after sheeting installation.
• Maximum pressure (in the case of precast walls).
UphillDownhill UphillDownhill UphillDownhill UphillDownhill
•11
• Boussinesq load(localised, limited extent)
Con
stru
ctio
n st
ages
Loads - forces - couples
• Graux load
Page 21June 2008
(localised, limited extentand diffused) Layer 1
Layer 2 Diffusion
Layer 1
Layer 2 Diffusion
• External moments (additional moment,due to an embedded
Con
stru
ctio
n st
ages
Loads - forces - couples
floor for example)
Page 22June 2008
• Horizontal loads(trapezoidal) UphillDownhill UphillDownhill UphillDownhill UphillDownhill
•12
• Simple (possibility to excavate, change water level and apply a Caquot load on excavation side at the same time),
• With berm,
• With sheeting installation (if the « reduced pressures » option was
3 different excavation types:
Con
stru
ctio
n st
ages
Earthworks
• With sheeting installation (if the « reduced pressures » option was activated in the initial stage).
Uphill
Downhill
Uphill
Downhill
Uphill
Downhill
Uphill
Downhill
Page 23June 2008
• Fill (with the option to define a separation at formation level, and/or to apply a Caquot load on top of the fill).
Con
stru
ctio
n st
ages
Earthworks
UphillDownhill UphillDownhill UphillDownhill UphillDownhill
Page 24June 2008
•13
• Struts
3 types of anchors can be applied and superposed:
Con
stru
ctio
n st
ages
Anchors – Retaining wall
• Anchors (its prestress can be used as a linear load)
Page 25June 2008
• Embedments (allow for definition of a rotation stiffness)
These elements can be deactivated in later stages.
• Modification of the wall stiffness (the wall stiffness can only be decreased)C
onst
ruct
ion
stag
es
Anchors – Retaining wall
Page 26June 2008
• Wall upraising (additional wall element on top)
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• Modification of the soil properties (separate modification of each soil parameter, either on one side only, or for both sides at the same time).
Con
stru
ctio
n st
ages
Soil properties
Page 27June 2008
• Hydraulic gradient
Example:To apply at depth z = 50 m the hydraulic
Con
stru
ctio
n st
ages
Hydraulic conditions
pp y p ypressure of a 20 m water column,
• Z=50
• Z=30
• 20m
Page 28June 2008
•15
Main results for each stage
OutputR
esul
ts o
utpu
t
gare:
horizontal displacements of the retaining wall
bending moments
shear forces
Page 29June 2008
Additional detailed results: display of tables/graphs
For both wall sides:
Res
ults
out
put
Output
• Soil state for each cell• Earth pressures• Water pressures• Vertical pressures• Limit pressure on active
and passive sides• Annular pressure for a
circular retaining wall
Page 30June 2008
circular retaining wall
Limiting/mobilised earth resistance ratio
•16
Envelopes:
• Displacements;
• Bending moments;
• Shear forces.
Res
ults
out
put
Output
Page 31June 2008
Summary of the maximum values reached for each stageR
esul
ts o
utpu
t
Output
Page 32June 2008
Summary of the staged construction
•17
Thank you for your attention
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Page 33February 2008
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