9. Seismic Design of
RETAINING STRUCTURES
RIGID WALLS
9. Seismic Design of
RETAINING STRUCTURES
RIGID WALLS
October 2009
Part B:
G. BOUCKOVALAS & G. KOURETZIS
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.1
Solutions for “perfectly rigid” or “semi-rigid” walls
Problem outline …..
The Mononobe-Okabe method requires that the retaining wall can move freely (slide or rotate) so that active earth pressures
develop behind the wall.
Nevertheless, there are cases where the free movement of the wall is totally or partially restrained (e.g. basement walls, braced
walls, massive walls embeded in rock like formations) .
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.2
CONTENTSCONTENTS
9.6 PERFECTLY RIGID WALLS (Wood, 1973)
9.7 WALLS WITH LIMITED DISPLACEMENT (Veletsos & Yunan, 1996)
9.8 SEISMIC CODES
Sggested ReadingSggested Reading
Steven Kramer: Chapter 11
Assumptions
1. Pseudo static conditions (Τδιεγερ>>4Η/Vs) – quite usual case (why?)2. plain strain3. Elastic soil4. Smooth & rigid walls
Εwall>>Esoil
Elastic soil between two rigid walls
9.6 PERFECTLY RIGID WALLS (Wood 1973)
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.3
Analytical Solutions for ……..
dynamic earth pressures
h g
αγια = 1
heq PF F
g
αΔ = γΗ2
=F
p
h
g
α= 1
Analytical Solutions for ……..
Overturning moment and base shear
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.4
heq PF F
g
αΔ = γΗ2
heq mM F
g
αΔ = γΗ3
=F
p
=F
m
h
g
α= 1
h
g
α= 1
Overturning moment and base shear
Analytical Solutions for ……..
0 1 2 3 4 5 6 7 8 9 10
L/H
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
h/H
Application point of the resultant seismic thrust
h L. για
H H⎛ ⎞≈ >⎜ ⎟⎝ ⎠
0 55 4
eq
eq
hP
ΔΜ=
Δ
Analytical Solutions for ……..
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.5
Smooth vs. bonded (rough) wall side
Analytical Solutions for ……..
Extension to harmonic base excitation – Base shear
"στατική"λύση
συντονισμός υψίσυχνες διεγέρσεις
excit soil
soil excit
T.
T
ωΩ = = <<
ω1 0
Analytical Solutions for ……..
Static Resonance High frequency excitation
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.6
excit soil
soil excit
T.
T
ωΩ = = <<
ω1 0
Extension to harmonic base excitation – Overturning Moment
Analytical Solutions for ……..
H
ξηρή άμμοςφ, γν=0.3
This is the main reason why the elastic solutions of Wood (1973) were put aside for more than 30 yaers… (in connection with the fact that very limited wall failures were observed during strong earthquakes)
x3!!
28 32 36 40 44φ (deg)
0
0.2
0.4
0.6
0.8
1
ΔF
eq/[γΗ
2 (ah/
g)]
Wood
Mononobe-Okabe
0 0.1 0.2 0.3 0.4 0.5ah (g)
0
0.2
0.4
0.6
0.8
1
ΔF
eq/[γΗ
2 (ah/
g)]
Wood
Mononobe-Okabe
(ah=0.15g)(φ=36ο)
απλοποίησηSeed & Whitman (kv=0)
απλοποίησηSeed & Whitman (kv=0)
Comparison with Mononobe - Okabe
Dry sand
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.7
ww
GHd
D=
3
GHd
Rθθ
=2
( )w w
w
w
E tD
v
⎛ ⎞⎜ ⎟=⎜ ⎟−⎝ ⎠
3
212 1
bonded wall-soil mass-less wall5% soil damping2% wall damping
Relative translational rigidity of the wall-fill system
9.7 WALLS WITH LIMITED DISPLACEMENT (displacement & rotation, Veletsos & Yunan, 1996)
Relative rotational rigidity of the wall-fill system
Assumptions
Pseudo-static earth pressures
h
Hη =
(normal. height)
Analytical solutions for …….
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.8
h
Hη =
Pseudo-static shear forces & bending moments
Analytical solutions for …….
Pseudo static base shearbase shear & & overturning momentoverturning moment
WoodWood
Seed & Whitman
Μ-O
Analytical solutions for …….
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.9
Pseudo static (?) displacements…
For a flexible (compared to the fill) concrete wall (dw=20) and a seismic excitation with amax=0.3g, the resulting displacement is U/H=0.03%...
[ U=U=0.1%0.1%÷÷0.4%0.4%··HH for active for active ““failurefailure”” →Μ-Ο]
Analytical solutions for …….
"στατική"λύση
συντονισμός υψίσυχνες διεγέρσεις
soil
soil
V
H
ππω = =
Τ12
2
the effect of harmonic excitation frequency on base shear((AF coefficientAF coefficient))
Analytical solutions for …….
High frequency excitation
StaticResonance
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.10
resonance… (Max AF)
the effect of harmonic excitation frequency on displacements
Analytical solutions for …….
Variation of amplification factor AF for base shear versus the fVariation of amplification factor AF for base shear versus the fundamental soil periodundamental soil period))
(does this remind something to you?)
Numerical solution - El Centro (1940) earthquake
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.11
Average values of the amplification factor AF for base shear andAverage values of the amplification factor AF for base shear and relative relative displacement displacement
Numerical solution - El Centro (1940) earthquake
1. Tensile cracks, at the top of the wal, are not taken into account(→shear forces and bending moments are over-estimated)
2. Uniform soil is assumed(e.g. a parabolic distribution of G with depth yields zero earth pressures at the top of the wall)
Note: (1) and (2) above have a counteracting effect for walls with rotational flexibility
Limitations . . . .
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.12
Earth press. at rest
Dynamic earth press.
H
KoγH 0.5αhγH
1.5αhγH
heqM .
g
αΔ = γΗ30 58
heq g
αΔΡ = γΗ3
Wood
EAK
Wood
9.8 SEISMIC CODES
EAK 2002 – Rigid walls
Rigid walls0.05% > U/H
ΥΠΕΧΩ∆Ε-εγκ.39/99 «Guidelines for the design of bridges»
Walls with limited displacement 0.1%>U/H>0.05%
heqM .
g
αΔ = γΗ30 58
H
0.58.H
σE=0.5.α.γ.Η
ΔPE=α.γ.H2
σE=1.5.α.γ.Η
H
H/2
σE=0.7.α.γ.Η
ΔPE=0.75.α.γ.H2
heqM .
g
αΔ = γΗ30 375
reminder:M-O
(U/H>0.1%)
H
0.60H
Δ[email protected].α.γ.H2
heqM .
g
αΔ = γΗ30 225
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.13
COMPARISON OF DIFFERENT METHODS ……….…
DIFFERENCES CAN BECOME SIGNIFICANT !
0
0.2
0.4
0.6
0.8
1
ΔF
eq/[γΗ
2 (ah/
g)]
Wood
M-O
Veletsos & Younan
EAK 2002-εγκ.39/99
U/H
HWK 9.1:Compute the total base shear force and overturning moment which develops at the base of a 5m high retaining wall during seismic excitation with αmax=0.15g. The wall is vertical and smooth, while the fill consists of sandy gravel with c=0, φ=36ο, γΞ=17kN/m3 and VS=100m/s. The computations will be performed:(α) for rigid wall, (β) for a wall with limited deformation (dw=10, dθ=1), using the V&Y methodology, (γ) for a wall with limited deformation (dw=10, dθ=1), using the seismic code provisions, Note: assume pseudo static conditions and neglect the wall mass.
HWK 9.2Repeat HWK 9.1 for the extreme case of resonance between soil and excitation.
GEORGE BOUCKOVALAS, National Technical University of Athes, Greece, 2011 9B.14