Subgroup A - Appendix A

Appendix A of the PIANC PTCII WG28 Subgroup A report Static bearing capacity limit state equations for monolithic caissons and rubble mound breakwater parapet walls. J. Dalsgaard Sørensen and H. F. Burcharth Aalborg University, Denmark (1) Introduction This annex provides the limit state equations for 10 important foundation rupture constellations related to monolithic caissons and rubble mound breakwater parapet walls. The limit state equations are formulated as

g = Ws − Wd ≥ 0

(A1)

where Ws is the work of the stabilizing forces and Wd is the work of the destabilizing forces. In general for a specific design the minimum value of g must be larger than 0 in order to obtain stability. If this is not the case, the design must be changed. The limit state equations are representing static load situations and do not cover earthquake loadings involving inertia forces. The dynamic effect – if any – of wave loadings might be taken into account by applying a dynamic load factor to the maximum load to obtain an equivalent static load. Possible soil strength degradation due to cyclic loadings has to be taken into account. The wave induced loadings on the front and the base plate of the structures are determined either from formulae or from model tests. The waves generate pore pressure gradients in the rubble foundation and in the subsoil. The resulting horizontal component of the pore pressure acting on the rupture boundary has to be taken into account and included in the limit state equations. An approximate model for the resulting horizontal pore pressure FHU is shown in Fig. A1.

  1 2 B z − hII / tan θ FHU = min.  pu hII  B  2

(A2)

where B Bz pu θ

is the width of caisson is the effective width of the caisson base plate is the wave induced uplift pressure at the edge of the caisson is the angle between the caisson base and the rupture plane

In the model for estimation of the resulting horizontal pore pressure FHU it is assumed that the uplift pressure on the base plate is varying linearly from pu at the front to zero at the back of the A1

structure. Moreover, it is assumed that the pressure is identical at all levels in the rubble foundation vertically beneath the base plate. Further, it is assumed that the horizontal pore pressure gradients are negligible in subsoils consisting of sand or clay due to the very significant pore pressure attenuation with depth related to windgenerated waves.

Figure A1: Illustration of simple model for estimation of wave induced horizontal pore pressure along rupture boundaries. The same model eq A2, might be used also in the case of breakwater parapet wall superstructures if the wave induced pore pressure acts on the base plate. This will be the case if high water and/or wave induced internal set-up raise the phreatic surface in the rubble to levels higher than the underside of the base plate. The limit state equations are restricted to the two-dimensional case and is based on the upper bound theorem of classical plasticity theory where an associated flow rule is assumed. However, this rule is not satisfied for friction materials like sand and quarry stones for which the friction angle and the dilation angle are different. In order to overcome the problem, the following reduced effective friction angle φd is used, see Hansen (1979),

tan ϕ d =

sin ϕ' cos ψ 1 − sin ϕ' sin ψ

(A3)

where A2

φ’ is the effective friction angle ψ is the dilation angle In the bearing capacity calculations the rubble mound quarry rock can be regarded fully drained due to the large permeability. The soil strength is then characterized by φd, defined in eq A3 . The sea bed soil is normally either clay, silt or sand. In the case of clay, silt and fine sand the soil should be considered undrained during wave loadings. Coarse sand might either be drained, partially drained or non-drained dependent on the actual soil and loading conditions. For undrained conditions the soil strength is characterized by the undrained shear strength, cu. For drained subsoils φd is used as strength parameter. In the following distingtion is made between undrained and drained subsoil conditions as one set of limit state equation is given for each condition.

2) Failure modes Fig. A2 provides an overview of the failure modes for which limit state equations are presented.

A3

Fig. A2: Important geotechnical failure modes for monolithic caissons and rubble mound breakwater parapet walls. (3) Limit state equations The limit state equations presented in the following are partly reproduced from Dalsgaard Sørensen and Burcharth (2000).

A4

1 Sliding between caisson and bedding layer/rubble foundation – failure mode 1

Figure A3. Sliding failure between caisson and bedding layer/rubble foundation. Failure mechanism: horizontal sliding on the bedding layer

Limit state function: g = ( FG − FU ) tan µ − FH

(A4)

where: FG weight of caisson reduced for buoyancy FU wave induced uplift FH horizontal wave force tan µ = friction coefficient f if sliding occurs between concrete base plate and the bedding layer tan µ = ω 1V = tan ϕ d1 if sliding occurs entirely in the rubble mound, see figure 3. ϕd1 reduced effective angle of friction of the rubble mound

Figure A4 Displacement field if sliding occurs entirely in the rubble mound. 2 Failure in rubble mound – failure mode 2

A5

Figure A5. Failure in rubble mound. The effective width Bz of the caisson is determined such that the resultant force R acts on the base at a distance B z / 2 from the heel of the caisson, see figure. Failure mechanism: unit displacement along the line AB. Area of zone 1: A1 =

1 π ( Bz + a ) 2 (cosθ sin θ + sin 2 θ tan( + θ − tan −1 (hII / b))) 2 2

(A5)

Figure A6. Displacement field. Displacements: The displacement field for zone 1 is shown in figure 5. With a unit displacement δ = 1 along AB the displacements become: ω1 =

1 cos(ϕ d1 )

ω 1H = ω 1V =

(A6)

cos(ϕ d1 − θ )

(A7)

cos(ϕ d1 ) sin(ϕ d1 − θ )

(A8)

cos(ϕ d1 )

Optimization problem: Optimization variable: θ angle of rupture line Constraint: A6

  hII 0 ≤ θ ≤ tan −1   : rupture line should be in the rubble mound  Bz + a + b  The horizontal pore pressure force is taken as: FHU

1 B z2 = p u tanθ 2 B

(A9)

where B width of caisson Pu pressure at seaward edge of base plate Limit state function: g = (γ s − γ w ) A1ω 1V + ( FG − FU )ω 1V − ( FH + FHU )ω 1H

(A10)

where: FG weight of caisson reduced for buoyancy FU wave induced uplift FH horizontal wave force FHU horizontal pore pressure force γ s specific weight of rubble material γ w specific weight of water 3 Failure in rubble mound and sliding along top of subsoil (clay / sand) – failure mode 3

Figure A7. Failure in rubble mound and rubble mound subsoil interface. Failure mechanism: unit displacement δ = 1 along top of subsoil, line BC . Geometrical quantities: hII l BC = Bz + a + b − tanϕ d1

length of BC

A7

Area of zone 1: hII 1 1 hII2 + ( Bz + a − A1 = bhII + )hII 2 2 tan ϕ d1 tan ϕ d1 The horizontal pore pressure force becomes (it is assumed that point B is below the caisson) : FHU =

1  2 Bz − hII / tan ϕ d1 pu  2  B

 hII 

(A11)

Limit state function for sand subsoil: g = (γ s − γ w ) A1 tan ϕ d 2 + ( FG − FU ) tan ϕ d 2 − ( FH + FHU )

(A12)

where ϕ d 2 friction angle of sand subsoil Limit state function for clay subsoil: g = l BC cu − ( FH + FHU )

(A13)

where cu undrained shear strength of clay 4 Failure in rubble mound – failure mode 4

Figure A8. Failure in rubble mound.

A8

Figure A9. Detailed geometry of zone 3. Failure mechanism: unit displacement δ = 1 along line AB. Geometrical quantities: sin θ rBD = Bz cosϕ d1 rCD = rBD e l AB = Bz

ξ=

α tan ϕd1

length of BD

length of CD

cos(θ − ϕ d1 ) cosϕ d1

π − ϕd1 − α + θ 2

length of AB

angle CDF

 hII  β = tan −1    c  rCD sin(ξ − β ) sin( β ) (a + ∆a ) sin β lCE = sin( β + α − θ ) a sin(θ − α ) ∆rCD = cosϕ d1 ∆a =

l EF =

length of CE

(

1 α tan ϕd1 + a sin(θ − α ) B sin θe sin( β + α − θ ) z

A9

)

length of EF

Figure A10. Displacement diagram. Displacements: The displacement field for zone 1,2 and 3 are shown in figure 9. With a unit displacement δ = 1 along AB the displacements become: ω1 =

1 cos ϕ d1

ω1V =

ω1H =

(A14)

sin(ϕd1 − θ )

(A15)

cosϕd1 cos(ϕd1 − θ )

ω2V (r , τ ) = ω1e ω 3 = ω1e

(A16)

cosϕd1 τ tan ϕd1

sin(ϕd1 − θ + τ )

0≤τ ≤α ;

α tan ϕd1

ω 3V = ω 3 sin(ϕ d1 + α − θ ) =

0 ≤ r ≤ rCD

(A17) (A18)

sin(ϕ d1 + α − θ ) cosϕ d1

e

α tan ϕd1

Area of zone 1: 1 π A1 = rBD l AB sin( − ϕ d1 ) 2 2

(A19)

(A20)

Work from selfweight in zone 1: W1 = (γ s − γ w ) A1ω1V

(A21)

Work from selfweight in zone 2: A10

rCD α

W2 = (γ s − γ w ) ∫ ∫ ω 2V (r , τ )rdrdτ 0 0

= (γ s − γ w )ω 1

2 rCD 2

2 tan ϕ d1 + 2

[e

α tan ϕ d1

( tan ϕ d

1

( tan ϕ d

1

)

sin(ϕ d1 − θ + α ) − cos(ϕ d1 − θ + α ) −

)

sin(ϕ d1 − θ ) − cos(ϕ d1 − θ ) ] (A22)

Area of zone 3: 1 1 A3 = arCD sin ξ + lCE l EF sin( β − θ + α ) 2 2

(A23)

Work from selfweight in zone 3: W3 = (γ s − γ w ) A3ω3V

(A24)

Optimization problem: Optimization variables: θ angle of rupture line α angle of zone 2 Constraints: 0≤θ 0≤α ≤θ β +α −θ > 0

The horizontal pore pressure force becomes: FHU

2 1 Bz = p u tanθ 2 B

(A25)

Limit state function : g = W1 + W2 + W3 + ( FG − FU )ω 1V − ( FH + FHU )ω 1H

A11

(A26)

5 Failure in rubble mound and drained subsoil – failure mode 5

Figure (A11). Failure in rubble mound and drained subsoil.

Figure (A12). Detailed geometry of zone 4. Failure mechanism: sliding along line AB. Geometrical quantities: h l AB = II sinθ   hII θ0 = tan −1   2 2  Bz − l AB −hII  θ1 = π − θ − θ0 θ2 = π − (θ1 + ϕ d1 − ϕ d2 ) θ3 =

π ϕd1 − 4 2

π − ϕd2 ) 2 θ5 = π − θ3 − θ4 − θ0  h  θ 6 = tan −1  II   a + b θ4 = π − (θ2 +

length of AB angle AFB angle ABF angle CBF angle DFG angle BFC angle CFD angle GFH

A12

l BF = Bz

sin θ sin θ1

rCF = l BF

sin θ2 π sin( − ϕ d2 ) 2

rDF = rCF e

θ5 tan ϕd1

hII

l D' F = l DD'

π ϕd cos( + 2 ) 4 2 = rDF − l D' F

l D' E = l DD'

π + ϕ d2 ) 2 π ϕd sin( − 2 ) 4 2

length of BF length of CF length of DF length of D’F length of DD’

sin(

l FH = (l FG + sH II ) 2 + H II2

length of D’E

length of FH

Figure (A13). Displacement diagram for zone 1.

Figure (A14). Displacement diagram for zone 3. Displacements: The displacement field for zone 1,2, 3 and 4 are shown in figure 12 and 13 with a unit displacement δ = 1 along AB the displacements become: ω1 =

1 cos ϕ d1

(A27)

A13

ω1V =

ω1H =

sin(ϕd1 − θ )

(A28)

cosϕd1 cos(ϕd1 − θ )

(A29)

cosϕd1

ω2V = ω1V

ω 3V (r , τ ) = ω 1e

ω 4 = ω1e

(A30)

τ tan ϕ d 2

sin(ϕ d1 − θ + τ )

0 ≤ τ ≤ θ5 ;

θ5 tan ϕd 2

ω 4V = ω 4 sin(ϕ d1 + θ 5 − θ ) =

0 ≤ r ≤ rDF

(A31)

(A32)

sin(ϕ d1 + θ 5 − θ ) cosϕ d1

Area of zone 1: 1 A1 = Bz hII 2

e

θ 5 tan ϕ d2

(A33)

(A34)

Work from selfweight in zone 1: W1 = (γ s − γ w ) A1ω1V

(A35)

Area of zone 2: 1 A2 = l BF rCF sinθ 4 2

(A36)

Work from selfweight in zone 2:

A14

W2 = (γ s − γ w ) A2ω2V

(A37)

Work from selfweight in zone 3: rDF θ 5

W3 = (γ s − γ w ) ∫ ∫ ω 3V (r , τ )rdrdτ 0 0

= (γ s − γ w )ω 1

2 rDF 2

2 tan ϕ d2 + 2

[e

( tan ϕ d

θ 5 tan ϕ d 2

2

( tan ϕ d

2

)

sin(ϕ d1 − θ + θ 5 ) − cos(ϕ d1 − θ + θ 5 ) −

)

sin(ϕ d1 − θ ) − cos(ϕ d1 − θ ) ] (A38)

Area of zone 4: 1 1 1 π ϕd A4 = l FH a sin θ 6 + l FH (rDF − l DD' ) sin(θ 3 − θ 6 ) + l D' E l DD' sin( − 2 ) (A39) 2 2 2 4 2 Work from selfweight in zone 4: W4 = (γ s − γ w ) A4ω4V

(A40)

Optimization problem: Optimization variable: θ angle of rupture line Constraints: h  tan −1  II  ≤ θ  Bz 

rupture line should enter the subsoil

The horizontal pore pressure force becomes: FHU =

1  2 Bz − hII / tan θ pu  2  B

  hII 

(A41)

Limit state function :

A15

g = W1 + W2 + W3 + W4 + ( FG − FU )ω 1V − ( FH + FHU )ω 1H 6 Failure in rubble mound and drained subsoil – failure mode 6

Figure (A15). Failure in rubble mound and drained subsoil.

Figure (A16). Detailed geometry of zone 4. Failure mechanism: sliding along line AB. Geometrical quantities: h l AB = II sinθ   hII θ0 = tan −1   2 − hII2   Bz − l AB θ1 = π − θ − θ0 θ2 = π − (θ1 + ϕ d1 − ϕ d2 ) θ3 =

π ϕd1 − 4 2

π − ϕd2 ) 2 θ5 = π − θ3 − θ4 − θ0

θ4 = π − (θ2 +

length of AB angle AIB angle ABI angle CBI angle DIG angle BIC angle CID

A16

(A42)

 h  θ 6 = tan −1  II   a + b π ϕd θ7 = − 2 4 2 sin θ sin θ1 sin θ2 = l BI π sin( − ϕ d2 ) 2

angle GIH angle DEH

l BI = Bz

length of BI

rCI

length of CI

rDI = rCI e

θ5 tan ϕd1

l D' I =

hII

length of DI

π ϕd cos( + 2 ) 4 2 l DD' = rDI − l D' I π sin( + ϕ d2 ) 2 l D' F ' = l DD' π ϕ d2 sin( − ) 4 2

length of D’I

lGI = hII2 + (a + b) 2

length of GI

sin(θ7 − θ6 ) sin(θ7 ) lGF ' = l D' F ' − l D'G sin(θ 7 ) lGE = lGF ' π sin( − ϕ d2 ) 2 l D'G = lGI

rGF = lGE e

θ7 tan ϕd 2

length of DD’ length of D’F’

length of D’G length of GF’ length of GE length of GF

Displacements: The displacement field for zone 1, 2, 3, 4 and 5 are shown in figures A17 and A18 With a unit displacement δ = 1 along AB the displacements become:

A17

Figure (A17). Displacement diagram for zone 1.

Figure (A18). Displacement diagram for zone 3.

ω1 =

1 cos ϕ d1

ω1V =

ω1H =

(A43)

sin(ϕd1 − θ )

(A44)

cosϕd1 cos(ϕd1 − θ )

(A45)

cosϕd1

ω2V = ω1V

ω 3V (r , τ ) = ω 1e

ω 4 = ω1e

(A46)

τ tan ϕ d 2

0 ≤ τ ≤ θ5 ;

sin(ϕ d1 − θ + τ )

θ5 tan ϕd 2

0 ≤ r ≤ rDI

(A47)

(A48)

A18

ω 4V = ω 4 sin(ϕ d1 + θ5 − θ ) =

ω5V (r , τ ) = ω4 e

τ tan ϕd 2

sin(ϕ d1 + θ5 − θ ) cosϕ d1

sin(ϕd1 − θ + θ5 + τ )

e

θ5 tan ϕd2

0 ≤ τ ≤ θ7 ;

(A49)

0 ≤ r ≤ rGF

Area of zone 1: 1 A1 = Bz hII 2

(A50)

(A51)

Work from selfweight in zone 1: W1 = (γ s − γ w ) A1ω1V

(A52)

Area of zone 2: 1 A2 = l BI rCI sinθ4 2

(A53)

Work from selfweight in zone 2: W2 = (γ s − γ w ) A2ω2V

(A54)

Work from selfweight in zone 3: rDI θ 5

W3 = (γ s − γ w ) ∫ ∫ ω 3V (r , τ )rdrdτ 0 0

= (γ s − γ w )ω 1

2 rDI 2

2 tan ϕ d2 + 2

[e

θ 5 tan ϕ d2

( tan ϕ d

2

( tan ϕ d

2

)

sin(ϕ d1 − θ + θ 5 ) − cos(ϕ d1 − θ + θ 5 ) −

)

sin(ϕ d1 − θ ) − cos(ϕ d1 − θ ) ] (A55)

Area of zone 4: 1 1 1 1 π ϕd A4 = lGI l IH sin θ6 + l IH (rDI − l DD' ) sin(θ7 − θ6 ) + l D ' F ' l DD' sin( − 2 ) − lGF ' lGE sin θ7 2 2 2 4 2 2 (A56)

A19

Work from selfweight in zone 4: W4 = (γ s − γ w ) A4ω4V

(A57)

Work from selfweight in zone 5: rGF θ7

W5 = (γ s − γ w ) ∫ ∫ ω5V (r , τ )rdrdτ 0 0

= (γ s − γ w )ω 4

2 rGF 2

2 tan ϕ d2 + 2

[e

( tan ϕ

θ7 tan ϕd 2

d2

( tan ϕ

d2

)

sin(ϕ d1 + θ5 − θ + θ7 ) − cos(ϕ d1 + θ5 − θ + θ7 ) −

)

sin(ϕ d1 + θ5 − θ ) − cos(ϕ d1 + θ5 − θ ) ] (A58)

Optimization problem: Optimization variable: θ angle of rupture line Constraint: h  tan −1  II  ≤ θ  Bz 

rupture line should enter the subsoil

The horizontal pore pressure force becomes: FHU =

1  2 Bz − hII / tan θ  pu   hII 2  B 

(A59)

Limit state function : g = W1 + W2 + W3 + W4 + W5 + ( FG − FU )ω 1V − ( FH + FHU )ω 1H

A20

(A60)

7 Failure in rubble mound and undrained subsoil – failure mode 7

Figure (A19). Failure in rubble mound and in undrained subsoil. Failure mechanism: unit displacement δ = 1 along the line BC. Geometrical quantities: l BF = Bz + a + b −

hII tan(θ + ϕ d1 )

length of BF

l BC = l BF cosθ

length of BC

rCF = l BF sinθ

length of CF

l DE = rCF

length of DE

Internal work from rupture along BC lBC

W1 = ∫ cu ( s)ds

(A61)

0

where cu ( s)

undrained shear strength of subsoil as function of distance s

Internal work from rupture along CD

W2 =

π rCF ( +θ ) 4 ∫ cu ( s)ds 0

(A62)

Internal work from rupture along DE A21

lDE

W3 = ∫ cu ( s)ds

(A63)

0

Internal work from Prandl rupture zone 2 π rCF ( 4 +θ )

W4 = ∫

0

∫ cu ( s, τ )dsdτ

(A64)

0

Area of zone 4: A4 =

1 1 ( Bz + a )hII + l BF hII 2 2

(A65)

Work from selfweight in zone 4: W5 = (γ s − γ w ) A4 sinθ

(A66)

Optimization problem: Optimization variable: θ angle of rupture line Constraint: 0≤θ The horizontal pore pressure force becomes: FHU =

1  2 Bz − hII / tan(ϕ d1 + θ )   hII pu  B 2  

(A67)

Limit state function: g = W1 + W2 + W3 + W4 + W5 − ( FG − FU ) sin θ − ( FH + FHU ) cosθ

8 Failure in rubble mound and undrained subsoil – failure mode 8

A22

(A68)

Figure (A20). Failure in rubble mound and in undrained subsoil. Failure mechanism: unit rotation β = 1 about point D. Geometrical quantities: rAD =

2 2 + yD xD

length of AD

 B + a + b − xD  α = tan −1  z  hII + y D  

angle

 yD   ξ = tan −1   xD 

angle

θ=

π −α −ξ 2

rBD = rAD e

θ tan ϕ d1

angle length of BD

l BC = 2( y D + hII ) tanα

length of BC

l AE = Bz + a + b − l BC

length of AE

Area of zone 1 (approximately): 1 A1 = hII l AE 2

(A69)

Work from selfweight in zone 1:

A23

2 W1 = (γ s − γ w ) A1 ( x D − l AE ) 3

(A70)

Area of zone 2: A2 = hII (l BC − b)

(A71)

Work from selfweight in zone 2: W2 = (γ s − γ w ) A2 b

(A72)

Area of zone 3: A3 =

1 bhII 2

(A73)

Work from selfweight in zone 3: 1 2 W3 = (γ s − γ w ) A3 ( l BC − b) 2 3

(A74)

Internal work from rupture along circle BC 2α

W4 = rBD ∫ cu (τ rBD )dτ

(A75)

0

Optimization problem: Optimization variables: x D x-coordinate of point D y D y-coordinate of point D Constraints: yD ≥ 0 Bz 2

≤ x D ≤ Bz + a + b rBD cosα = y D + hII α ≥0 θ≥0

The horizontal pore pressure force becomes: A24

FHU =

1  2 B z − l AE  pu   l AE tanθ  2  B

(A76)

Limit state function g = W1 + W2 + W3 + W4 − ( FG − FU )( x D −

1 B z ) − ( FH + FHU ) y D 2

(A77)

9 Failure in rubble mound – failure mode 9

Figure (A21). Failure in rubble mound. The failure mechanism and the limit state function correspond to failure mode (3) for caisson structures. The selfweight of the soil .zone must be adjusted if fully submerged at the limit of the wave loading. 10 Failure in rubble mound – failure mode 10

Figure (A22)-. Failure in rubble mound The failure mechanism and the limit state function correspond to failure mode (4) for caisson structures. The selfweight of the soil zones must be adjusted if the rubble is not fully submerged at the time of the wave loading. REFERENCER

A25

Dalsgaard Sørensen, J. and Burcharth, H. F. (2000). Reliability analysis of geotechnical failure modes for vertical wall breakwaters. Reliability in Geotechnics 26 (2000) 225-245. Hansen, B. (1979). Deviation and use of friction angles. Proc. Int. Conf. VII ECSMFE, Brighton, U.K.

A26