simpliﬁed approach for design of raft foundations against fault rupture. part ii:...

Vol.7, No.2 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION June, 2008

Earthq Eng & Eng Vib (2008) 7:165-179 DOI: 10.1007/s11803-008-0836-5

Simplifi ed approach for design of raft foundations against fault rupture. Part II: soil–structure interaction

I. Anastasopoulos1,2†, N. Gerolymos1†, G. Gazetas1‡ and M. F. Bransby2§

1. School of Civil Engineering, National Technical University of Athens, Greece

2. Civil Engineering, University of Dundee, Scotland, UK

Abstract: This is the second paper of two, which describe the results of an integrated research effort to develop a four–step simplifi ed approach for design of raft foundations against dip-slip (normal and thrust) fault rupture. The fi rst two steps dealing with fault rupture propagation in the free-fi eld were presented in the companion paper. This paper develops an approximate analytical method to analyze soil–foundation–structure interaction (SFSI), involving two additional phenomena: (i) fault rupture diversion (Step 3); and (ii) modifi cation of the vertical displacement profi le (Step 4). For the fi rst phenomenon (Step 3), an approximate energy–based approach is developed to estimate the diversion of a fault rupture due to presence of a raft foundation. The normalized critical load for complete diversion is shown to be a function of soil strength, coeffi cient of earth pressure at rest, bedrock depth, and the horizontal position of the foundation relative to the outcropping fault rupture. For the second phenomenon (Step 4), a heuristic approach is proposed, which “scans” through possible equilibrium positions to detect the one that best satisfi es force and moment equilibrium. Thus, we account for the strong geometric nonlinearities that govern this interaction, such as uplifting and second order (P−Δ) effects. Comparisons with centrifuge–validated fi nite element analyses demonstrate the effi cacy of the method. Its simplicity makes possible its utilization for preliminary design.

Keywords: fault rupture; analytical method; raft foundation; soil-structure interaction; earthquake

Correspondence to: G. Gazetas, 36 Asimakopoulou Str., Ag. Paraskevi 15342, Athens, GreeceTel: (30) 210 600 85 78; Fax: (30) 210 772 24 05E-mail: [email protected]

†Lecturer; ‡Professor; §Senior Lecturer Supported by: OSE (the Greek Railway Organization), and the

EU Fifth Framework Programme Under Grant No. EVG1-CT-2002-00064

Received March 23, 2008; Accepted April 24, 2008

1 Introduction

Recent case histories of faulting through populated areas have shown that even modest civil engineering structures on rigid and continuous foundations can survive large surface dislocations (Youd et al., 2000; Erdik, 2001; Bray, 2001; Ural, 2001; Ulusay et al., 2002; Pamuk et al., 2005). Recent research efforts, combining fi eld studies (Anastasopoulos & Gazetas, 2007a; Faccioli et al., 2007), centrifuge model testing (Bransby et al., 2007a; 2007b), and fi nite element (FE) simulations (Anastasopoulos & Gazetas, 2007b; Anastasopoulos et al., 2007b), have shown that the rigidity and continuity of the foundation system is a crucial factor for survival. While structures on isolated footings are prone to collapse, buildings lying on rigid and continuous raft foundations can perform well even under extreme ground dislocations of the order of several meters (Faccioli et al., 2007). For such structures,

foundation and structure distress arises mainly from loss of support (Anastasopoulos et al., 2007b).

As schematically illustrated in Fig. 1, depending on the location of the structure relative to the outcropping fault rupture, loss of support may take place either under the two edges or under the middle. In the fi rst case (Fig.1(a)), the unsupported spans behave as cantilevers on “elastic” supports (producing hogging deformation); in the second (Fig.1(b)) as a single span on “elastic” supports (producing sagging deformation). The width of the unsupported span(s) has been shown to decrease as the surcharge load q increases (Anastasopoulos et al., 2007b; Faccioli et al., 2007). The latter plays a dual role: (a) it modifi es the stress fi eld underneath the foundation, facilitating fault rupture diversion (Niccum ., 1976; Youd, 1989; Kelson et al., 2001; Bray & Kelson, 2006); and (b) it tends to “fl atten” any tectonically-imposed soil anomalies, increasing the effective width (seating) of the foundation, and thus reducing the stressing. With a high bearing pressure q, detachment may even be avoided completely (e.g. Anastasopoulos et al., 2007b). Foundation rotation Δθ is also a function of q, as is the position of the foundation relative to the fault outcrop.

This paper, along with its companion (Anastasopoulos et al., 2008), develops a four-step semi-analytical approach for analysis of dip-slip fault rupture propagation through sand and its interaction with a

166 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.7

raft foundation. As already discussed in the companion paper, we consider a uniform soil deposit of thickness H, at the base of which a dip-slip fault, dipping at angle α, ruptures and produces bedrock offset of vertical amplitude (throw) h. A raft foundation of width B, carrying a uniformly distributed load q, interacts with the outcropping fault rupture and the deforming soil. The fi rst two steps, already presented in the companion paper, deal with fault rupture propagation in the free-fi eld, emphasizing estimation of the location of fault outcropping (Step 1), and of the vertical displacement profi le at the ground surface (Step 2). This paper deals with the last two steps, emphasizing soil-foundation structure interaction (SFSI).

2 Fault rupture diversion

To the best of our knowledge, the Central Bank of Nicaragua constituted one of the earliest case histories of fault rupture diversion (Niccum et al., 1976). In the 1972 Ms 6.3 Managua earthquake, the strike-slip fault rupture crossed the Bank. The building sustained almost no damage, thanks to the existence of a rigid reinforced-concrete underground vault, which successfully diverted

the rupture. Duncan & Lefebvre (1973) conducted a series of small-scale tests to investigate the interplay between a strike-slip fault and a rigid embedded cylindrical vault. As in reality, the rupture was observed to divert as it approached the rigid structure, following the “easiest” path, which certainly was not through the rigid body.

Berill (1983) studied the behaviour of a rigid shallow foundation of width 2b and surcharge load q, subjected to strike-slip faulting at the base of a soil deposit of thickness H. Assuming plane–strain conditions and planar fault rupture propagation, he identifi ed four possible failure mechanisms: Type 1 – no diversion of the rupture and foundation slippage; Type 1a – no diversion of the rupture and foundation structural failure; Type 2 – fault rupture diversion starting from bedrock and emerging at the edge of the foundation; and Type 2a – diversion of the rupture starting at a smaller depth and also emerging at the edge of the foundation. Assuming that the thickness and reinforcement of the foundation is adequate to avoid structural failure (i.e., ignoring the second mechanism), in order to fi nd the conditions under which the rupture may be diverted he considered the work done on the soil−foundation system per unit value of the offset of the strike-slip fault for each failure type.

Fig. 1 Schematic representation of the prevailing interaction mechanisms. Notice the qualitative difference among the two extremes: light structure on stiff soil versus heavy structure on soft soil

(a) Fault rupture close to the hanging-wall side edge of the foundation

(b) Fault rupture close to the footwall side edge of the foundation

Light structure Heavy structure

Stiff soil Soft soil Free-fi eld fault rupture

Fault rupture diversion

FaultFault

Light structure Light structure Heavy structure

Soft soil

Free-fi eld fault rupture

Fault rupture diversion

FaultFault

Stiff soil

No.2 I.Anastasopoulos et al.: Simplifi ed approach for design of raft foundations against fault rupture. Part II: SSI 167

The work for Type 1 failure was taken to be equal to the shearing resistance of the soil plus the work for foundation slippage:

W b q s y

H

1 0= + ∫tan( )δ d (1)

where s is the shear strength of the soil along the rupture path, δ is the angle of friction at the soil-foundation interface, and y is the depth below the soil surface. In the same manner, the work for Type 2 failure was taken to be equal to the shearing resistance of the soil, but along a rupture path inclined at angle ω = arctan (H/b) to the horizontal:

W s yh

2 0= ∫ sin( )ω

d (2)

Integrating along the failure paths, Berill produced closed form expressions for W1 and W2. Then, equating W1 with FS W× 2 , he estimated the minimum required surcharge load qcr for the rupture to be diverted with a factor of safety FS. To verify his analytical approach, he conducted a series of small-scale tests. However, qcr in the experiments was found to be 1.5 to 2.0 times larger than the theoretical values. This difference was attributed to the assumption of plane strain conditions.

2.1 Simplifi ed analytical method

The approximate analytical method developed here for dip-slip faulting can be seen as an extension of Berill’s work−based approach. We consider a continuous rigid strip foundation of breadth B = 2b subjected to uniform surcharge load q, lying on a soil deposit of depth H, subjected to dip-slip faulting at its base (Fig. 2).

As illustrated in Fig. 3, we consider a reference point

Fig. 3 Simplifi ed analytical method for estimation of fault rupture diversion

Fig. 2 Simplifi ed analytical method for estimation of fault rupture diversion : problem defi nition.

(a) Problem geometry and related defi nitions

(b) Stress-related defi nitions.

b

B

b bq

Strip foundation

yo

xo

Soil: φ, ψ, cH

Fault

X

yo

xoβ

βcr

r

b

y

θ1

X

x(x, y)

O

θ2

θ

Y Y

O

Stresses acting on a soil element Stresses along the rupture path

σy

σx X

Yτ

τ

σx

σy

τσ p

σ n

τ

σ n

σ pτ

X

Y

τ

τ


“O” of a fault, at depth yo and horizontal distance xo from the centreline of the foundation. In the absence of foundation (i.e., in the free fi eld), the rupture is assumed to propagate vertically. This simplifying assumption can be reasonable in the case of normal faulting; in thrust faulting it is only a practical approximation. Due to the presence of the foundation and the associated stress changes, the rupture may be diverted and “forced” to follow an inclined propagation path. We consider all possible rupture paths, inclined at angle β to the horizontal, and defi ne as critical, βcr, the angle for which the rupture is diverted to the edge of the footing:

βcr =

−⎛

⎝⎜

⎞

⎠⎟arctan

b xy

o

o

(3)

Given the previously mentioned assumption of vertical propagation in the free fi eld, the horizontal deformation component is also ignored. This means that the foundation is not directly subjected to shearing, and therefore slippage at the soil-foundation interface is not accounted for.

Using polar coordinates (r, β) with the origin at O, for a given inclination angle β (to the horizontal) the work done by internal forces can be derived through integration along the rupture path:

W s r

yo= ∫ d0

sin( )β (4)

Making use of the Mohr-Coulomb failure criterion, the shear strength along the rupture path can be taken equal to: s c= +σ φn tan( ) (5)

where c is the cohesion, φ is the friction angle, and σn is the normal effective stress along the rupture path. The latter stems from the initial overburden pressure, plus the pressure increase Δp arising from q. Following Berill’s methodology, the latter can be estimated through the increase in mean pressure Δp x y= +( ) /σ σ 2 for a uniform strip load on the surface of a linearly elastic homogeneous half-space (Carothers, 1920):

Δ

πp q= θ (6)

where θ is the angle that is subtended by the strip at the point of interest, as shown in Fig. 3a. Thus, the shear strength of the soil along the rupture path is:

s c y q≈ + +⎛

⎝⎜⎞⎠⎟

ρ θ ϕgπ

tan( ) (7)

In Cartesian and polar coordinates, the angle θ is expressed as:

θ = +⎛

⎝⎜

⎞

⎠⎟ + −⎛

⎝⎜

⎞

⎠⎟arctan arctanb x

yb x

y (8)

θβ

ββ

=+ +

−⎛

⎝⎜

⎞

⎠⎟+

− −−

arctancos( )

sin( )arctan

cos( )b x ry r

b x ryo

o

o

o

rr sin( )β⎛

⎝⎜

⎞

⎠⎟

(9)

and hence, the work done along the rupture path can be written as:

W c g y r q b x ry ro

o

o

= +⎧⎨⎩

− ++ +

−⎛

⎝⎜

⎞

⎠⎟ρ β

πβ

β[ sin( )] arctan

cos( )sin( )

++

− −−

⎛

⎝⎜

⎞

⎠⎟⎤

⎦⎥⎥

⎫⎬

∫0

y

o

o

o

b x ry r

sin( )

arctancos( )

sin( )tan( )

β

ββ

ϕ ⎪⎪

⎭⎪dr (10)

Integration of this expression is feasible only numerically. W is computed for a fi nite number (a value of 90 was used in this analysis, i.e. a discretisation of 1o) of possible inclination angles β: 0 < β ≤ π/2. The “optimum” angle βmin is defi ned as the angle for which W is minimized.

While the developed analytical method is similar to the approach of Berill, it has certain differences that should be made clear: (a) while Berill computed and compared the work for two limiting cases (i.e., no diversion, versus complete diversion to the edge of the footing), our approach detects the optimum rupture path in terms of work minimization (i.e., the output is the extent of diversion, from no diversion at all to complete diversion); (b) while Berill assumed that the rupture path in the free-fi eld is at the centerline of the foundation, our approach also takes into account the horizontal location xo of the fault rupture at depth yo.

Figure 4 illustrates the normalized work W/ρgbH with respect to β, for a foundation of width B = 2b = 10 m, for different values of surcharge load q. The fault is assumed to originate at depth yo = 2b and at horizontal distance xo = b/4. The soil is assumed to be cohesionless with φ = 30ο. As expected, for q = 0 kPa the normalized work is minimized for βmin = 90ο: i.e., the optimum rupture path is vertical. The increase of q leads to slight shifting of βmin , but not to complete diversion. For q = 60 kPa, the optimum rupture path lies at βmin = 88o. A sudden “jump” in the curve can be observed for q = 76 kPa , and βmin is reduced to 67o = βcr; i.e., the rupture is diverted to the edge of the footing. This value of the surcharge load, required for complete diversion, is defi ned as the critical surcharge load for complete diversion, denoted qcr .

Figure 5 refers to the distance D = yo tan(β) of the diversion of the rupture at the ground surface. D is plotted as a function of the surcharge load q. Up to q = 20 kPa there is absolutely no diversion: D = 0. Then, the increase of q is accompanied by a slight diversion of the rupture path. For q = 75 kPa, the diversion is still insignifi cant: D = 0.4 m. However, when q reaches


qcr = 76 kPa the rupture is diverted completely: D = 4.3 m. Interestingly, the rupture is not diverted exactly to the edge of the foundation, but about 0.5 m away(Dcr = b – xo = 3.75 m). Further increase in q leads to further increase of the diversion: D ≈ 4.6 m for q = 100 kPa as the fault avoids the stressed area around the foundation. Fig. 6 shows the FE and centrifuge model test results for an approximately equivalent case (B = 10 m, H = 25 m, and q = 91 kPa). Due to the presence of the foundation, the rupture is divided in two ruptures (bifurcation). The fi rst is diverted a few meters away from the hanging-wall side (right) edge of the foundation, in agreement with the simplifi ed analytical method. The second is diverted to the footwall side (left) edge of the foundation.

2.2 Taking account of the true stress fi eld

As discussed previously, the shear strength of the soil along the rupture path was computed using the mean pressure increase Δp in place of σn (Eq. 6). For a more rigorous approach, we take into account the actual elastic stress fi eld, which includes σy, σx, and τ (see also Fig. 3(b)):

σ ρ α α α δy y q= + + +[ ]gπ

sin( )cos( )2 (11)

σ ρ α α α δx oK y q= + − +[ ]g

πsin( )cos( )2 (12)

1.4

1.2

1.0

0.8

0.6

0.4

W/ρ

gbH

30 40 50 60 70 80 90β (deg)

Fig. 4 Normalised work W/ρgbH with respect to angle β and surcharge load q (yo = 2 b, xo = b/4, ρ = 2 Mg/m3, φ = 30ο). The rupture will propagate at the angle for which the work W is minimized, denoted βmin. If βmin= 90ο, the dislocation is not diverted, otherwise βmin represents the angle of the diverted rupture path. In this example, the critical surcharge load qcr for complete diversion (βmin = βcr , i.e. diversion to the edge of the footing), is equal to 76 kPa (thick solid line)

q=qcrq=0

2b β min=β cr β min=90˚2b

b/4 b/4

q=100q=76

q=60q=40

q=20q=0(kPa)

1.4

1.2

1.0

0.8

0.6

0.4

D (m

)

0 20 40 60 80 100q (kPa)

Partial diversionqcr=76kPa

Complete diversion

Dcr=3b/4=3.75m

Fig. 5 Diversion D with respect to the surcharge load q (yo= 2b, xo= b/4, ρ = 2 Mg/m3, φ = 30°). In this example, up to q = 20 kPa the rupture is not diverted. Then, some partial diversion is observed (clearly not enough for the rupture to "avoid" the foundation); for q > qcr= 76 kPa (for D > Dcr, = 3b/4 = 3.75 m in this case) complete diversion takes place

Fig. 6 Example of FE and centrifuge model test results illustrating the fault rupture diversion due to interaction with a strip foundation: rigid B = 10 m strip foundation, with a surcharge load q = 91 kPa. Notice that due to the presence of the foundation, the rupture is divided in two. The fi rst of the two ruptures is diverted a few meters away from the hanging-wall side (right) edge of the foundation, in accord with the simplifi ed analytical approach. The second is diverted to the footwall side (left) edge of the foundation

Centrifuge model test Finite element analysis

Free-fi eld

h=2.0m (%)

0 20 40 60h≈1.98m

Diversion due to interactionBifurcation due

to interaction

Free-fi eld


τ α α δ= +qπ

sin( )sin( )2 (13)

where:

αβ

ββ

=+ +

−⎛

⎝⎜

⎞

⎠⎟ −

− − −arctan

cos( )sin( )

arctan[ cos( )]b x r

y rb x ro

o

o

yy ro −⎛

⎝⎜ sin( )β

(14)

δ

ββ

=− − −

−⎛

⎝⎜

⎞

⎠⎟arctan

[ cos( )]sin( )

b x ry r

o

o

(15)

and Ko the coeffi cient of horizontal soil pressure at rest. Along the inclined rupture path, a normal stress

σ n and a shear stress τ will apply. These stresses are derived through a coordinate rotation from the orthogonal Cartesian system (x, y) to the coordinate system that is normal and parallel to the rupture path, denoted ( x y, ):

σ σ β σ β τ β βn x y= + −cos ( ) sin ( ) sin( )cos( )2 2 2 (16)

τ σ σ β β τ β β= − + −[ ]cos( )sin( ) [cos ( ) sin ( )]x y

2 2 (17)

Finally, the work along the inclined rupture path will be:

W c rn

yo= + ⋅ −∫ [ tan( ) ]sin( )

σ ϕ τβ

d0

(18)

The optimum angle βmin is computed as discussed previously.

2.3 Results and discussion

Figure 7 illustrates the normalized diversion, D/b, with respect to yo /b and q/ρgb for a cohesionless soil of φ = 30ο. The normalized critical surcharge load for complete fault rupture diversion, qcr /ρgb , can be seen to decrease substantially with increasing yo / b: the deeper the point of origin, the easier it is for the rupture to be diverted to the edge of the footing: qcr/ρgb = 0.52 for yo / b = 1, decreasing to 0.12 for yo / b = 8. The dotted line refers to yo/b = 2, but using the simplifi ed pressure increase assumption (see previous section) instead of the elastic stress fi eld. Observe that the critical load for complete diversion, qcr /ρgb, predicted with the simplifi ed assumption is about 2 times larger than the one predicted with the more rigorous approach (0.76 instead of 0.33). As shown in Fig. 8, FE and centrifuge model test results confi rm the validity of the elastic stress fi eld assumption. The results refer to a rigid

Fig. 8 Example of FE and centrifuge model test results illustrating fault rupture diversion due to interaction with a strip foundation: rigid B = 10 m strip foundation, with a surcharge load q = 37 kPa. The rupture is diverted to the footwall side edge of the foundation, in agreement with the prediction of the simplifi ed analytical approach with the elastic stress fi eld assumption

1.25

1.00

0.75

0.50

0.25

0

D/b

0 0.2 0.4 0.6 0.8 1.0qcr /ρgb

Fig. 7 Normalised diversion D/b (b = B/2) as a function of the normalised surcharge load q/ρgb, for four values of bedrock depth ratio yo /b, and distance of the rupture from the centerline xo /b = 0.25. The dotted line refers to the same analysis, for yo /b = 2, but using the pressure increase assumption instead of the true elastic stress fi eld

yo/b=1yo/b=2yo/b=4yo/b=8

Complete diversion

Assumption of mean pressure increase for yo/b=2

Centrifuge model test Finite element analysis

Free-fi eld

h=2.0m

0 20 40 60 80h=2.03m

Diversion dueto interaction

Free-fi eld

(%)


B = 10 m footing with surcharge load q = 37 kPa, subjected to normal faulting not far from its centerline (xo/b ≈ 0.40). Using the mean pressure increase assumption, no diversion would be predicted: qcr ≈ 60 kPa > 37 kPa (qcr /ρgb = 0.76, ρ = 1.6 Mg/m3, and b = 5 m). In stark contrast, with the elastic stress fi eld assumption fault rupture diversion is predicted for this load: qcr ≈ 26 kPa (qcr /ρgb = 0.33).

Figure 9 plots the qcr/ρgb ratio as a function of yo/b, for two values of xo/b and three different sets of soil properties, assuming Ko = 0.5 (a reasonable value for many soil types). For all soil types, qcr/ρgb is substantially larger when the origin of the rupture is in the horizontal sense close to the centerline of the foundation (xo/b = 0.25). Observe that there is a critical normalized depth, yo,cr /b, for which the required load for complete fault rupture diversion, qcr/ρgb, is minimized. This critical depth is a function of φ, c, and xo/b. As it would be expected, yo,cr /b is substantially larger (by a factor of 2 to 3) when the origin of the rupture is closer to the centerline of the foundation (xo/b = 0.25). While larger c implies larger yo,cr , the decrease of φ leads to the opposite result. For inferior depths (yo < yo,cr), the critical

load for complete diversion qcr/ρgb decreases non-linearly with the increase of yo /b, and a threshold value of yo /b exists, below which the rupture cannot be diverted.

3 Modifi cation of the settlement profi le

A continuous and rigid raft foundation subjected to faulting-induced deformation may experience loss of support. The latter is mainly responsible for its stressing. As schematically illustrated in Fig. 10, depending on the location of the foundation relative to the emerging fault scarp, loss of support may take place either at the two edges or at the middle. In the fi rst case, the unsupported spans behave as cantilevers on a central elastic support, giving hogging deformation of the foundation; in the latter case, as a single span on elastic end supports, giving sagging deformation.

An example of rigorous FE results illustrating foundation stressing due to loss of support is given in Fig. 11. It refers to a rigid B = 10 m raft foundation with surcharge load q = 20 kPa or 80 kPa, subjected to h = 2 m normal fault rupture with a 60o dip angle, through

Fig. 9 Normalised critical load qcr /ρgb , assuming Ko= 0.5, with respect to bedrock depth yo/b , soil type, and horizontal distance of the fault rupture from the centerline of the footing. (ρ = 2 Mg/m3 )

2.0

1.5

1.0

0.5

0

q cr /ρ

gb

0 2 4 6 8yo /b

1.0

0.8

0.6

0.4

0.2

0

q cr /ρ

gb

0 2 4 6 8yo /b

yo,cr/b=2.5

yo,cr/b=2

yo,cr/b=1.65

(a) xo/b=0.25 (b) xo/b=0.75

φ=30˚φ=30˚, c=40kPaφ=45˚

φ=30˚φ=30˚, c=40kPaφ=45˚

Cantilever

Cantilever CantileverHogging

deformation

Simplifi ed equivalent static system

Simplifi ed equivalent static system

Cantilever

q

Simply supportedsingle span

Simply supportedsingle span

Sagging deformation

Fig. 10. Schematic illustration of foundation distress arising from loss of support. Depending on the position of the outcropping fault rupture, loss of support may take place. More complicated equilibrium modes are also possible

(a) Loss of support at either of the two edges (b) Loss of support at the middle

yo,cr/b=1.1

yo,cr/b=0.7

yo,cr/b=0.5


a 20 m deep idealized dense sand deposit, at position xs = 3 m (measured from the hanging-wall side edge of the foundation). As revealed by the normalized contact pressure p/q diagram, the foundation experiences loss of support at the two edges. The two unsupported spans behave as cantilevers, and the foundation experiences hogging deformation. Increasing q leads to a reduction of the length of unsupported spans (left and right cantilevers), resulting in a reduction of the normalized bending moment M/qB2. On the other hand, foundation rotation is increased for the heavily loaded foundation.

Figure 12 illustrates the rigorous FE response of the same foundation, but for xs = 9 m (i.e. xs/B = 0.9). As attested by the normalized contact pressure p/q diagrams, due to the different position of the fault the foundation now experiences loss of support at its center. The unsupported span behaves roughly as a simply supported beam, imposing sagging deformation onto the foundation. As for xs = 3 m, the increase of q leads to a decrease of the width of the central unsupported

span, and consequently a reduction in M/qB2. In contrast to the previous case, however, foundation rotation also decreases with increasing q.

3.1 Methodology

As discussed previously, the interaction is governed by strong geometric non-linearities, such as uplifting and second order (P−Δ) effects, prohibiting the solution of the problem using conventional beam-on-Winkler foundation approaches. To overcome these diffi culties, we simplify the problem by considering a rigid foundation–superstructure system. Such a crude approximation can nevertheless be claimed to be reasonable for two reasons: (a) since fault-induced displacements are of the order of meters, the superstructure and foundation can be considered relatively rigid compared to the soil; and (b) it has already been demonstrated that if foundation and superstructure are not adequately rigid they cannot survive such displacements (Anastasopoulos & Gazetas,

Fig. 11 Example of FE analysis results illustrating the stressing of the foundation due to loss of support : rigid B = 10 m strip foundation with surcharge load q = 20 and 80 kPa, subjected to h = 2 m normal fault rupture through an H = 20 m dense sand deposit, at distance xs = 3 m. As revealed by the normalized contact pressure p/q diagram, the increase of q leads to a decrease of the width of the unsupported spans of the foundation (left and right cantilevers), and of the normalised bending moment M/qB2. In contrast, foundation rotation is increased with q

1

0

-1

-2

-3

Δy (m

)

0

-2

-4

-6

-8

p/q

1

0

-1

-2

-3

Δy (m

)

-0.12

-0.09

-0.06

-0.03

0

0.03

M/q

B2

20 kPa80 kPa

20 kPa80 kPah=0

-15 -10 -5 0 5 10 15 20x (m)

-15 -10 -5 0 5 10 15 20x (m)

0 0.2 0.4 0.6 0.8 1.0 x / B

0 0.2 0.4 0.6 0.8 1.0 x / B

q=20 kPa

q=80 kPa

Loss ofsupport Effective

widthLoss ofsupport

Loss ofsupport

Effective width of foundation

Loss ofsupport

B=10 m

xs=3 m

h=2 m, α=60˚

Dense sand H=20 m


2007a; b). The geometry of the problem is depicted in Fig.

13. A rigid system of width B = 2b, carrying a uniform surcharge load q, is subjected to faulting-induced vertical displacement profi le Sp(x). To simplify the procedure, we ignore the quasi-elastic vertical displacement component, focusing on plastic displacements only. Using the semi-analytical expressions of the companion paper, and setting W = 1 – xs so that the rupture outcrops at distance xs from the hanging-wall side (left) edge of the foundation, Sp(x) can be written as:

S x x xh hy

p ( ) tanh ( )= − − + −{ }⎡⎣ ⎤⎦−

1 12

ω s (19)

where:

ωψ

=× +

−⎡

⎣⎢

⎤

⎦⎥

2

5 200 120050

50

π

dh h

dy sin( ) arcsin( )αout

(20)

As schematically illustrated in Fig.14, there exists a fi nite number of possible equilibrium positions. An equilibrium position can be defi ned using a set of two parameters: (i) the vertical coordinate yu of the footwall side (right) edge of the foundation, and (ii) the tilting angle Δθ of the structure (Fig. 13a). Using these two parameters, the vertical displacement along the base of the foundation will be:

f x y x( ) tan( )= +u Δθ (21)

where x measured from the footwall side (right) edge of the foundation.

The differential vertical displacement between the base of the foundation and the deformed soil surface mobilizes soil reactions p(x). To take into account the tensionless soil−foundation contact, the soil reactions are written as:

Fig. 12 Example of FE analysis results illustrating the stressing of the foundation due to loss of support : rigid B = 10 m strip foundation with surcharge load q = 20 and 80 kPa, subjected to h = 2 m normal fault rupture through an H = 20 m dense sand deposit, at distance xs = 9 m. As revealed by the normalized contact pressure p/q diagram, the increase of q leads to a decrease of the of the width of the unsupported span of the foundation (central simply supported span), and hence of the normalised bending moment M/qB2. Foundation rotation is also decreased with the increase of q

1

0

-1

-2

-3

Δy (m

)

0

-4

-8

-12

-16

ρ/q

1

0

-1

-2

-3

Δy (m

)

0

0.02

0.04

0.06

0.08

0.10

M/q

B2

20 kPa80 kPa

20 kPa80 kPaStatic

-20 -15 -10 -5 0 5 10 15x (m)

-20 -15 -10 -5 0 5 10 15x (m)

0 0.2 0.4 0.6 0.8 1.0 x / B

0 0.2 0.4 0.6 0.8 1.0 x / B

q=20 kPa

q=80 kPa

Loss ofsupport

Effectivewidth

B=10 m

Xs=9 m

h=2 m, α=60˚

Dense sand H=20 m

Effectivewidth

Loss ofsupport

Effectivewidth

Effectivewidth


Fig. 13 Simplifi ed analytical method to determine the modifi cation (suppression) of the vertical displacement profi le

Fig. 14 Simplifi ed analytical method for estimation of the modifi cation of the vertical displacement profi le: trial equilibrium positions of the foundation

B=2bB=2b

Center of mass h cm

yu

X

Y

Δθ

Sp

qB hcmcos(Δθ) b cos (Δθ)

ΔθX

Y

yu

Soil reactions

(a) Problem geometry (b) Forces acting on the structure for a possible equilibrium position, taking account of second order (P-Δ) effects

(a) Loss of support at the left edge (b) Loss of support at both edges (c) Loss of support at the middle

Sp Sp Sp

X

Y

X

Y

X

Y

Sp Sp Sp

Fault Fault Fault

Soil Soil Soil


p xf x S x k if f x S x

if f x S x( )

( ) ( ) , ( ) ( )

, ( ) ( )=

−⎡⎣ ⎤⎦ ≥

<

⎧⎨⎪

⎩⎪

p p

p

s

0 (22)

where ks is the coeffi cient of subgrade reaction (or Winkler spring modulus). For a given equilibrium position, the total vertical external force, Fext = qB, is assumed to act at the center of mass of the foundation–superstructure system, i.e. at height hcm (Fig. 13b). Taking account of second order (P-Δ) effects, the total external moment with respect to the footwall side (right) edge of the structure (x = 0) will be:

M qB B hext cm= +⎡

⎣⎢⎤⎦⎥2

cos( ) sin( )Δ Δθ θ (23)

Then, force and moment equilibrium,

p x x F

B( )

cos( )d ext0

0Δθ

∫ − = (24)

p x x x M

B( )

cos( )d ext0

0Δθ

∫ − = (25)

are computed for all possible pairs of yu and Δθ. The optimum equilibrium position is defi ned as the one that minimizes the root of the mean square error (rms):

e yRM F M( , )u Δθ σ σ= +2 2 (26)

where:

σ

θ

F =−∫ p x x F

F

B( )

cos( )d ext

ext

0

Δ

(27)

B=10m

1

0

-1

-2

-310 8 6 4 2 0

x (m)

Δy (m

)

0

20

40

60

80

100

p

(kPa

)

Analytical method FE analysis

Analytical method FE analysis

10 8 6 4 2 0x (m)

(a) Vertical displacement Δy at the ground surface (b) Soil reactions p

Fig. 16 Comparison of the simplifi ed analytical method with FE analysis results ; B = 10 m, q = 35 kPa, hcm = 3 m (2-storey) structure, subjected to h = 2 m normal faulting at xs = 2 m through dense sand

xs=2 m

hcm=3 m

Dense sand

h=2 m

Fig. 15 Example comparison of centrifuge model test results to FE Class “A” prediction: rigid B = 10 m foundation with surcharge load q = 91 kPa, subjected to normal faulting at xs = 1.9 m. Vertical displacement Δy at the ground surface for bedrock offset h ranging from 0.2 m to 2 m

0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

Δy (m

)

-30 -20 -10 0 10x (m)

B=10 mq=91 kPaxs=1.9 m

Free-fi eldfault outcrop

h≈0.20 mh≈0.59 mh≈0.99 mh≈1.49 mh≈1.98 m

h=0.2 mh=0.6 mh=1.0 mh=1.5 mh=2 m

B=10 m

σ

θ

M =−∫ p x x x M

M

B( )

cos( )d ext

ext

0

Δ

(28)

The necessary integrations to compute σF, σM are performed numerically.

The following section provides a comparison of the simplifi ed method with continuum analysis results. Although (due to space limitations) emphasis is given to normal faulting, the conclusions regarding the performance of the method also hold for the case of thrust faulting.

3.2 Results and discussion

Characteristic results by using the simplifi ed analytical approach are compared to the results of rigorous FE analyses. The latter have been extensively validated through successful genuine Class “A” predictions of centrifuge model tests (Anastasopoulos et al., 2007a and 2007b). Fig. 15 reproduces one such comparison, in terms of vertical displacement Δy at the ground surface (note that the footwall is now in the opposite direction). It refers to a B = 10 m rigid foundation carrying a surcharge load q = 91 kPa, subjected to normal faulting at distance xs = 1.9 m (the same example with Fig. 6). Evidently, the comparison is quite satisfactory for all levels of imposed bedrock offset h. The analysis is successful in predicting not only the bifurcation and diversion of the fault rupture (see also

Fig. 6), but also the rotation Δθ of the foundation. Figure 16 compares the results of the simplifi ed

method to FE analysis results in terms of vertical displacement Δy at the ground surface, and soil contact pressures p. The analysis refers to a foundation-structure system of width B = 10 m, with its center of mass at hcm = 3 m and q = 35 kPa (a typical 2-storey reinforced-concrete building), subjected to h = 2 m normal faulting at α = 60ο through an H = 40 m idealized dense sand deposit (see companion paper), outcropping at xs = 2 m (close to the hanging-wall edge of the foundation). The spring coeffi cient ks is estimated from the expressions of Gazetas (1991) and verifi ed against FE analyses. The analytical method compares reasonably well with FE analysis results, both in terms of Δy and contact pressures p. The analytical method correctly predicts a loss of support at both edges of the foundations, slightly underestimating the footwall side (right-hand) area of loss of support (1.4 m of cantilever, instead of 2.0 m of the FE analysis), and overestimating the left-hand (hanging-wall side) unsupported span (2.7 m of cantilever, instead of 2.3 m).

Figure 17 illustrates another comparison for a wider, taller, and thus heavier structure: B = 20 m, hcm = 7.5 m, and q = 80 kPa (a typical 5-storey building). Fault rupture outcrops at xs = 16 m (close to its footwall side edge). The analytical method correctly predicts a loss of support at the middle of the foundation. The width of the unsupported span is also predicted correctly (roughly 10 m), but its exact location is not captured with accuracy: the analytical method overestimates the hanging-wall

B=20 m

1

0

-1

-2

-3 20 15 10 5 0

x (m)

Δy (m

)

0

20

40

60

80

100

p (k

Pa)

Analytical method FE analysis Analytical method

FE analysis

20 15 10 5 0

x (m)

(a) Vertical displacement Δy at the ground surface (b) Soil reactions p

Fig. 17 Comparison of the simplifi ed method with FE analysis results; B = 20 m, q = 80 kPa, hcm = 7.5 m (5-storey) structure, subjected to h = 2 m normal faulting at xs = 16 m through dense sand

xs=16 m

hcm=7.5 m

Dense sand

h=2 m



Fig. 18 Comparison of the simplifi ed analytical method with rigorous FE analysis results. Foundation rotation Δθ as a function of the surcharge load q (B = 20 m structure subjected to h = 2 m normal faulting through a soil deposit of depth H = 40 m)

side (left-hand) effective width of the foundation (5.7 m instead of 3.0 m of the FE analysis), and underestimates the footwall side (right-hand) effective width (4.0 m instead of 6.9 m of the FE analysis). The differential settlement of the foundation is underestimated: 1.5 m instead of 2.3 m of the FE analysis (i.e. Δθ = 4.3o versus 6.6o).

To further evaluate the effectiveness of the simplifi ed analytical method, the analytically computed foundation rotation Δθ versus surcharge load q and sand type is compared with FE analysis results in Fig. 18. A rigid structure of width B = 20 m is subjected to normal faulting at xs = 4 m (i.e. the rupture outcrops close to its hanging-wall side edge), xs = 10 m (at the middle of the

6

5

4

3

2

1

00 20 40 60 80 100

q (kPa)(a) xs =4 m

Δθ (%

)

16

12

8

4

0

Δθ (%

)

FE Analysis Analytical method Dense Dense Loose Loose

0 20 40 60 80 100q (kPa)

(b) xs =10 m

16

12

8

4

0

Δθ (%

)

0 20 40 60 80 100q (kPa)

(c) xs =16 m

foundation), and xs = 16 m (close to the footwall side edge of the foundation). The overall performance of the simplifi ed method is satisfactory, because:

• it correctly predicts the magnitude of Δθ in most cases; the only exception: with the rupture emerging near the footwall side (right-hand) edge of the foundation in loose sand, the method overestimates Δθ by a factor of 2 (Phenomena such as bifurcation and diffusion, not accounted for in the simplifi ed method, may be responsible for this discrepancy.),

• it correctly predicts the trends in the variation of Δθ with increasing the applied load q: the linear increase of Δθ versus q in the case of rupture emerging near the hanging-wall side (left) edge (Fig. 18a), and a Δθ nearly independent of q in the other two cases (Fig. 18b and c), and

• it correctly predicts the varying role of increasing soil density, ranging from being benefi cial in the case of rupture near the hanging-wall side (left) edge to being detrimental in case of rupture emergence near the footwall-side (right) edge.

4 Conclusions and limitations

This paper along with its companion, present a four–step simplifi ed approach for design of raft foundations against normal and thrust faulting. By realistically modelling two important soil–structure interaction mechanisms, diversion of the rupture path and modifi cation of the displacement profi le, both arising from the presence of the foundation, the method provides a deeper insight into the nature of the problem. The main conclusions are as follows:

(1) The presence of a structure on top of the soil deposit diverts the rupture path, as the latter propagates from the base rock to the ground surface. The normalized minimum (“critical”) load (qcr /ρgb) beyond which complete fault rupture diversion occurs is a function of soil strength (φ, c), the coeffi cient of earth pressure at rest (Ko), the normalized bedrock depth (yo/b), and the horizontal position of the foundation relative to the outcropping fault rupture (xo/b). Complete diversion is easier when the rupture is close to the edge of the foundation. qcr /ρgb is shown to decrease nonlinearly with increasing yo/b. The apparent cohesion c is found to play a substantial role only at shallow depths. For large soil depths, the frictional component of shear strength prevails.

(2) A structure on top of the soil deposit will modify the faulting-induced displacement profi le. A rigid raft foundation on top of an emerging fault will experience loss of support, which is mainly responsible for its stressing. Depending on the location of the foundation relative to the emerging fault scarp, the unsupported spans behave either as cantilevers on a central elastic support, producing hogging deformation, or as a single span on elastic end supports, giving sagging deformation. Since the interaction is governed by strong geometric

non-linearities, such as uplifting and second order (P−Δ) effects, the problem cannot be solved directly using a conventional beam-on-Winkler–foundation approach. A heuristic approach is employed herein, which “scans” through possible equilibrium positions to detect the one that best satisfi es force and moment equilibrium. The simplicity of the proposed method makes its use possible for preliminary design purposes.

(3) The method gives results in general accord with rigorous fi nite element solutions, predicting the important trends and phenomena with engineering accuracy, and tending to overestimate the rotation of rigid structures.

Based on several simplifying assumptions including completely ignoring the horizontal displacement component (only a practical approximation, especially in the case of thrust faulting), the Winkler-type “springs” that do not take account of soil non-linearity and continuity, and without considering complex phenomena such as bifurcation and diffusion, the simplifi ed analytical method presented herein should only be applied with caution, and always combined with engineering judgement. It may be suitable for preliminary assessment and design purposes, and for developing a qualitative understanding of the complex interaction of a foundation with an emerging fault rupture, but not for the fi nal design of important structures, where site-specifi c FE analysis is recommended (e.g., Anastasopoulos et al., 2007b).

Acknowledgements

This work was funded by OSE (the Greek Railway Organization), as part of the research project “Railway Bridges on Active Seismic Faults”. Centrifuge testing and fi nite element analyses formed part of the EU research project “QUAKER”, funded through the EU Fifth Framework Programme, under contract number: EVG1-CT-2002-00064.

References

Anastasopoulos I and Gazetas G (2007a), “Foundation-Structure Systems over a Rupturing Normal Fault: Part I. Observations After the Kocaeli 1999 Earthquake,” Bulletin of Earthquake Engineering, 5(3): 253–275.Anastasopoulos I and Gazetas G (2007b), “Behaviour of Structure–Foundation Systems over a Rupturing Normal Fault: Part II. Analysis of the Kocaeli Case Histories,” Bulletin of Earthquake Engineering, 5(3): 277–301.Anastasopoulos I, Gazetas G, Bransby MF, Davies MCR and El Nahas A (2007a), “Fault Rupture Propagation Through Sand: Finite Element Analysis and Validation Through Centrifuge Experiments,” Journal of Geotechnical and Geoenvironmental Engineering, 133 (8): 943–958.

Anastasopoulos I, Gazetas G, Bransby MF, Davies MCR, and El Nahas A (2007b), “Normal Fault Rupture Interaction with Strip Foundations,” Journal of Geotechnical and Geoenvironmental Engineering (submitted for possible publication).Anastasopoulos I, Gerolymos N, Gazetas G and Bransby MF (2008), “Simplifi ed Approach for Design of Raft Foundations Against Fault Rupture. Part I: Free-fi eld,” Earthquake Engineering and Engineering Vibration, 7(2): - .Berill JB (1983), “Two-dimensional Analysis of the Effect of Fault Rupture on Buildings with Shallow Foundations,” Soil Dynamics and Earthquake Engineering, 2(3): 156-160.Bransby MF, Davies MCR and El Nahas A (2007a), “Centrifuge Modelling of Normal Fault-Foundation Interaction,” Bulletin of Earthquake Engineering, Special Issue: Integrated Approach to Fault Rupture- and Soil-foundation Interaction, (submitted for possible publication).Bransby MF, Davies MCR and El Nahas A (2007b), “Centrifuge Modelling of Reverse Fault-foundation Interaction,” Bulletin of Earthquake Engineering, Special Issue: Integrated Approach to Fault Rupture- and Soil-foundation Interaction, (submitted for possible publication).Bray JD (2001), “Developing Mitigation Measures for the Hazards Associated with Earthquake Surface Fault Rupture,” Proceedings, Workshop on Seismic Fault-Induced Failures–Possible Remedies for Damage to Urban Facilities, Tokyo University Press, Tokyo, pp: 55-79.Bray JD and Kelson KI (2006), “Observations of Surface Fault Rupture from the 1906 Earthquake in the Context of Current Practice,” Earthquake Spectra, 22(52): 569-589.Carothers SD (1920), “Plane Strain: the Direct Determination of Stress,” Proceedings Royal Society, Serial A, 97: 110–123.Duncan JM and Lefebvre G (1973), “Earth Pressure on Structures Due to Fault Movement,” Journal of Soil Mechanics and Foundations Division, ASCE, 99 (SM12): 1153−1163.Erdik M (2001), “Report on 1999 Kocaeli and Düzce (Turkey) Earthquakes,” Structural Control for Civil and Infrastructure Engineering, Ed. by, F. Casciati, G. Magonette, World Scientifi c.Faccioli E, Anastasopoulos I, Callerio A and Gazetas G (2007), “Case Histories of Fault–foundation Interaction,” Bulletin of Earthquake Engineering, Special Issue: Integrated Approach to Fault Rupture- and Soil-foundation Interaction, Companion paper (submitted for possible publication).Gaudin C (2002), “Experimental and Theoretical Study of the Behaviour of Supporting Walls: Validation of


Design Methods,” Ph.D Dissertation, Laboratoire Central des Ponts et Chaussées, Nantes, France.Gazetas G (1991), Foundation Vibrations. Foundation Engineering Handbook, 2nd edition, ed. HY Fang, Cluwer Publishers, pp. 553–593.Kelson KI, Kang KH, Page WD, Lee CT and Cluff LS (2001), “Representative Styles of Deformation Along the Chelungpu Fault from the 1999 Chi-Chi (Taiwan) Earthquake: Geomorphic Characteristics and Responses of Man-made Structures,” Bulletin of the Seismological Society of America, 91(5): 930-952.Niccum MR, Cluff LS, Chamoro F and Wylie L (1976), “Banco Central de Nicaragua: A Case History of a High-rise Building That Survived Surface Fault Rupture,” in Humphrey, CB, ed.,” Engineering Geology and Soils Engineering Symposium, No. 14, Idaho Transportation Department, Division of Highways, pp. 133-144.Pamuk A, Kalkanb E and Linga HI (2005), “Structural and Geotechnical Impacts of Surface Rupture on

Highway Structures During Recent Earthquakes in Turkey,” Soil Dynamics and Earthquake Engineering, 25: 581-589.Ulusay R, Aydan O and Hamada M (2002), “The Behaviour of Structures Built on Active Fault Zones: Examples from the Recent Earthquakes of Turkey,” Structural Engineering & Earthquake Engineering, JSCE, 19(2): 149–167.Ural D (2001), “The 1999 Kocaeli and Duzce Earthquakes: Lessons Learned and Possible Remedies to Minimize Future Losses,” Proc. Workshop on Seismic Fault Induced Failures, ed. Konagai, Tokyo, Japan.Youd TL (1989), “Ground Failure Damage to Buildings During Earthquakes,” Foundation Engineering — Current Principles and Practices, 1: 758-770. New York: ASCE, pp.758-770Youd TL, Bardet JP and Bray JD (2000), “Kocaeli, Turkey, Earthquake of August 17, 1999 Reconnaissance Report,” Earthquake Spectra, 16 (Suppl. A): 456.


simpliﬁed approach for design of raft foundations against fault rupture. part ii:...

Documents

fault rupture diversion

fault rupture propagation

thrust fault rupture

outcropping fault rupture

design of raft foundations

continuous raft foundations

continuous foundations

b anastasopoulos