fem slab culvert.pdf

A SIMPLIFIED SEISMIC DESIGN METHOD FOR UNDERGROUND STRUCTURES BASED ON THE SHEAR STRAIN TRANSMITTING

CHARACTERISTICS

Tsutomu NISHIOKA1 and Shigeki UNJOH2 1Public Works Research Institute, 1-6, Minamihara, Tsukuba, 305-8516, JAPAN E-mail: [email protected] 2Public Works Research Institute, 1-6, Minamihara, Tsukuba, 305-8516, JAPAN E-mail: [email protected] ABSTRACT: This paper presents a simplified seismic design method for underground structures based on the shear strain transmitting characteristics from surrounding ground to the structure. Since seismic deformation in the cross section of underground structures is mainly shear deformation, seismic performance is estimated by the shear deformation in simplified seismic design methods. This paper clarifies that structure-ground shear strain ratio is the hyperbolic function of ground-structure stiffness ratio and proposes an analytical method to estimate the seismic shear deformation using the shear strain transmitting characteristics. 1. Introduction Underground structures were thought to be relatively safe during earthquakes until some of the subway tunnels in Kobe suffered serious damage from the 1995 Hyogoken-nanbu earthquake [1]. Seismic deformation method (SDM) is commonly applied to practical seismic design of the cross section of underground structures [2-3]. In this method, free-field ground displacement is loaded on the structure through Winkler-type soil springs. One of the problems in the SDM, however, is evaluating the soil spring that is supposed to simulate the soil-structure kinematic interaction. The adequacy of the soil spring and its evaluation method are still under discussion. Although straightforward approaches such as finite element methods (FEM) can solve the problem of the soil spring model, there still remains an important place for simple approaches for the benefit of practical design activities. Instead of the SDM, simplified seismic design methods to estimate seismic deformation of underground structures are proposed these days [3-4]. The seismic deformation in the cross section of underground structures is mainly shear deformation. Hence, seismic performance is estimated by the shear deformation based on the ground-structure shear stiffness ratio in these methods. These current methods, however, consider only horizontal deformation to evaluate the seismic shear deformation. The seismic deformation of underground structures generally includes horizontal and vertical deformation, and the rotation of the whole structure exists simultaneously. In this discussion, considering the horizontal and vertical deformation and the rotation of the structure, the seismic shear deformation is accurately evaluated. It is clarified that structure-ground shear strain ratio depends only on ground-structure shear stiffness ratio and that the hyperbolic relation between the two parameters exists. Finally, a simplified analytical method to estimate the seismic shear deformation of underground structures is proposed based on the shear strain transmitting characteristics.

2. Analysis of Shear Strain Transmitting Characteristics 2.1 Analysis Cases Common utility boxes with rectangular cross section are adopted for the analysis of the shear strain transmitting characteristics. Seven cases are analyzed, as shown in Table 1. Parameters are aspect ratio of the rectangular cross section and structure-ground weight ratio in order to investigate the effects of the kinematic and the inertia interaction on the shear strain transmitting characteristics. Case 0 is the basic case that corresponds to RC common utility box with two hollows shown in Fig. 1. The aspect ratios b/h vary from 0.22 to 2.79 in Case 0~4. The structure-ground weight ratios Ws/Wg in Case 5,6 are adjusted to be 0.0, 1.0, respectively, by changing RC weight per unit volume.

Table 1 Analysis cases

1.00.0

0.840.660.601.04

0.73

Weight ratio

Ws/Wg

1.2925.26.22.883.7051.29

0.220.472.790.61

1.29

Aspect ratio b/h

25.26.22.883.706

25.23.77.801.75425.23.77.803.65325.26.22.857.95225.26.22.851.751

25.26.22.883.700

Surface layer thickness

H (m)

Over-burdenD (m)

Heighth (m)

Widthb (m)

Case

1.00.0

0.840.660.601.04

0.73

Weight ratio

Ws/Wg

1.2925.26.22.883.7051.29

0.220.472.790.61

1.29

Aspect ratio b/h

25.26.22.883.706

25.23.77.801.75425.23.77.803.65325.26.22.857.95225.26.22.851.751

25.26.22.883.700


H (m)

Over-burdenD (m)

Heighth (m)

Widthb (m)

Case

1.00.0

0.840.660.601.04

0.73

Weight ratio

Ws/Wg

1.2925.26.22.883.7051.29

0.220.472.790.61

1.29

Aspect ratio b/h

25.26.22.883.706

25.23.77.801.75425.23.77.803.65325.26.22.857.95225.26.22.851.751

25.26.22.883.700


H (m)

Over-burdenD (m)

Heighth (m)

Widthb (m)

Case

1.00.0

0.840.660.601.04

0.73

Weight ratio

Ws/Wg

1.2925.26.22.883.7051.29

0.220.472.790.61

1.29

Aspect ratio b/h

25.26.22.883.706

25.23.77.801.75425.23.77.803.65325.26.22.857.95225.26.22.851.751

25.26.22.883.700


H (m)

Over-burdenD (m)

Heighth (m)

Widthb (m)

Case

350 1800 300 1250 350

150

150

3700

2875

2500

400

350

Rigid zone

nodeBeam element

unit:mm

350 1800 300 1250 350

150

150

3700

2875

2500

400

350

Rigid zone

nodeBeam element

unit:mm

Fig. 1 Cross section of the common utility box and the frame

model in Case 0

Ground displacement distribution

Underground structureGround level

Base layer (Fix)Quarter cosine curve

Lateral boundary (Free in the horizontal direction )

Surface layer


Underground structureGround level

Base layer (Fix)Quarter cosine curve

Lateral boundary (Free in the horizontal direction )

Surface layer

Fig. 2 Schematic diagram of the FEM analysis 155.7m

25.2

m

76.0m

155.7m

25.2

m

76.0m Fig. 3 Finite element mesh of Case 0

2.2 FEM Analysis Structure-ground shear strain ratios are analyzed by two-dimensional FEM. The structure is modeled as the beam elements illustrated in Fig. 1. The surface layer is modeled as the plain strain elements shown in Fig. 2. Static horizontal inertia forces are applied to both the ground and the structure elements so that ground displacement distribution would be the quarter cosine curve. Fig. 2 is the schematic diagram of the FEM analysis. FEM boundary conditions are fixed on the base layer and free in the horizontal direction at the lateral boundaries of the surface layer. Fig. 3 shows finite element mesh of Case 0. Horizontal distance between the structure and the lateral boundaries is set to be approximately three times as longer as the thickness of the surface layer. The surface layer is assumed to be elastic and homogeneous in order to simplify the ground shear stiffness. Two cases of shear wave velocity Vs = 50, 100 (m/s) are taken into account. Weight per unit volume and Poisson's ratio of the soil are 18 (kN/m3) and 0.45, respectively. The structure is modeled as two types of frame to investigate the effects of structural linearity and non-linearity on the shear strain transmitting characteristics. One is an equivalent linear frame model and the other is a non-linear frame model. The non-linear frame model has the tri-linear moment-curvature relationships that have the concrete crack point, the reinforcement yield point, and the ultimate point. The ultimate point is defined as the point where concrete compression strain reaches 0.0035 [5]. Weight per unit volume, modulus of elasticity, and Poisson's ratio of the RC are assumed as 24.5 (kN/m3), 2.35104 (N/mm2), and 1/6, respectively. 2.3 Computation of Shear Stiffness and Shear Strain of Ground and Structure Ground shear stiffness Gg is given by

gVG sg /2= (1)

where Vs = shear wave velocity of the surface layer, = weight per unit volume of the soil, g = acceleration of gravity. The equivalent shear stiffness Gs of the whole structure, simple-supported at the lower slab shown in Fig. 4, is estimated by

)//()/( hbPG sss = (2)

Ps = bGs(s/h)s

Ps = bGs(s/h)s

Fig. 4 Estimation of the equivalent shear stiffness Gs of the whole structure

b

s

d1d2

r

sh

g

g

b

s

d1d2

r

sh

g

g

gg

g

(a) In case of Gs > Gg

s

s

rb

h

g

g

s

s

rb

h

g

g

gg

g

(b) In case of Gs < Gg Fig. 5 Ground shear strain g and the equivalent shear strain s of the whole

structure

where Ps = horizontal force applied to the upper slab, s = horizontal displacement of the upper slab, h = height of the cross section, b = width of the cross section. The ground shear strain g on the structure's underground level is obtained by

hgg / = (3) where g = relative horizontal displacement of free-field ground on the structure's underground level shown in Fig. 5. The equivalent shear strain s of the whole structure is computed, based on the geometric property of the parallelogram in Fig. 5, by

)4/()(/ 2122 bhddhss == (4)

where s = displacement of the upper slab due to the shear deformation of the structure, d1, d2 = shorter and longer diagonals of the cross section, respectively. It should be noted that the gyrostatic rotation of the whole structure exists when Gs > Gg and that the counterclockwise rotation of the lower slab exists because of the larger shear deformation of the structure than that of the ground when Gs < Gg. 2.4 Effects of Linearity and Non-linearity of the Structure Frame Model Fig. 6 shows the comparison of structure-ground shear strain ratios s/g between the equivalent linear and the nonlinear structure frame models in Case 0. In the equivalent linear model, 5, 20, and 50 % of the initial flexural rigidity of the beam elements are analyzed. In the nonlinear model, Gs decrease as s increase. The more ground-structure shear stiffness ratios Gg/Gs increase, The more s/g increase. s/g of the equivalent linear model are very close to those of the nonlinear frame model. It is also found that, in the nonlinear model, s/g of Vs = 50, 100 (m/s) form a nearly identical curved line. 2.5 Effects of Aspect Ratio of the Cross Section s/g are analyzed in Case 0~4 where aspect ratios b/h of the rectangular cross section vary from 0.22 to 2.79. It is noted that the equivalent linear structure frame models are applied to Case 1~4 in order to simplify the analytical procedures. Fig. 7 shows the relationships between Gg/Gs and (s+r)/g, where r is the horizontal displacement due to rotation of the lower slab, obtained by subtracting s from the relative horizontal displacement (s+r) of the upper slab. Where Gg/Gs < 1, the smaller b/h are, the larger (s+r)/g are. Where Gg/Gs > 1, on the other hand, the adverse tendency is observed.

20% of Flexural Rigidity

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

s/

g



Nonlinear, Vs=50m/sNonlinear, Vs=100m/sEquivalent Linear, Vs100m/s


-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

s/

g





Fig. 6 Comparison of the non-linear and the equivalent linear structure frame models

(Case 0)

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

(s+

r)/

g

b/h=0.22b/h=0.47b/h=0.61b/h=1.29b/h=2.79

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

(s+

r)/

g

b/h=0.22b/h=0.47b/h=0.61b/h=1.29b/h=2.79

Fig. 7 Relationships between Gg /Gs and (s+r)/g (Case 0~4)

Fig. 8 shows the relationships between Gg/Gs and r/g. It is found that r/g > 0 where Gg/Gs < 1 and that r/g < 0 where Gg/Gs > 1. The slants of r/g with smaller b/h are steeper. This can be explained by the gyrostatic rotation of the whole structure in Gg/Gs < 1 and the counterclockwise rotation of the lower slab in Gg/Gs>1 as described above. If Gg/Gs=1, the shear deformation of the structure is as much as that of the ground. Fig. 9 shows the relationships between Gg/Gs and the structural deformation. Fig. 10 shows the relationships between Gg/Gs and s/g. s/g are turned to be s/g by multiplying the numerator and the denominator by the height of the cross section. s/g in Case 0~4 have the similar curved line regardless of b/h. It is made clear that b/h have little effects on s /g. 2.6 Effects of Structure-ground Weight Ratio s /g are analyzed in Case 5, 6 in which structure-ground weight ratios Ws/Wg are 0.0, 1.0, respectively. Fig. 11 shows the relationships between Gg/Gs and s/g. Ws is the equivalent weight per unit volume of the whole structure that is computed by dividing all the RC members' weight by the volume of the whole section including hollows. No inertia forces are applied to the structure when Ws/Wg=0.0, whereas the whole inertia forces of the structure are equal to those of the ground when Ws/Wg=1.0. s/g in Case 5, 6 come into the approximately same curved line. It is found that the inertia forces of the structure have little effects on s/g. 3. Estimation of Seismic Shear Deformation 3.1 Formulation of the Shear Strain Transmitting Characteristics The physical basis of the SDM is explained by the static substructure

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

r/

g

b/h=0.22b/h=0.47b/h=0.61b/h=1.29b/h=2.79

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

r/

g

b/h=0.22b/h=0.47b/h=0.61b/h=1.29b/h=2.79

Fig. 8 Relationships between Gg/Gs and r/g (Case 0~4)

(a) Gg/Gs1(a) Gg/Gs1

Fig. 9 Relationships between Gg/Gs and the structural deformation

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

s/

g

(s/

g)

b/h=0.22b/h=0.47b/h=0.61b/h=1.29b/h=2.79

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

s/

g

(s/

g)

b/h=0.22b/h=0.47b/h=0.61b/h=1.29b/h=2.79

Fig. 10 Relationships between Gg/Gs and s/g (Case 0~4)

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

s/

g

Ws/Wg=0.0Ws/Wg=1.0

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

s/

g

Ws/Wg=0.0Ws/Wg=1.0

Fig. 11 Relationships between Gg/Gs and s/g (Case 5, 6)

method derived from the dynamic substructure method [6]. Equations of motion of the SDM are given by

+

+

=

+

FI

FI

GI

I

S

II

SS

I

SGIIIIS

SISS

qrKr

rM

Mrr

KKKKK 00

00

00

(5)

where K = stiffness matrix of the structure, M = mass matrix of the structure, r = displacement vector. The subscripts I and S denote the nodes on the soil-structure interface, and the remaining nodes of the structure, respectively. The superscript dots denote time derivation. KI0G = ground impedance matrix, rIF = free-field ground displacement vector, qIF = free-field ground internal force on the soil-structure interface. The second row of Equation (5), the equilibrium on the soil-structure interface, is expressed as

=+ )( IIISIS rKrK FIIFIGIIII qrrKrM ++

)(0 (6) Since only the shear deformation of the structure and the ground are discussed herein, the equilibrium of one-dimensional shear stress in Fig. 12 is applied to Equation (6). The equilibrium is given by

ggsggss GGG += )( (7) where Gss = structure reaction shear stress, Gg(g-s) = ground shear stress due to the relative shear strain between the free-field ground and the structure, Ggg = free-field ground shear stress on the structure's underground level. The first term of the right side of Equation (6), the inertia forces of the structure are ignored because the effects of the inertia interaction are very small, shown in Fig. 11. The structure-ground shear strain ratio s/g is expressed as

)1//()/2(/ += sgsggs GGGG (8) Fig. 13 shows the comparison between s/g in Case 0~4 and Equation (8). It is found that Equation (8) approximately coincides with s /g in Case 0~4.

g s

Gg(g-s) GggGss

Base layer

Ground level


g s

Gg(g-s) GggGss

Base layer

Ground level


Fig. 12 Equilibrium of one-dimensional shear stress

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

s/

g

Case04Equation (8)

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8Gg/Gs

s/

g

Case04Equation (8)

Fig. 13 Comparison of Equation (8) and s/g

(Case 0~4)

3.2 Estimation of the Seismic Shear Deformation of Underground Structures Seismic response analysis of free-field ground gives the ground shear strain g on the structure's underground level and the ground shear stiffness Gg. Substituting g and Gg into Equation (8) yields one relation of the equivalent shear strain s and the equivalent shear stiffness Gs of the whole structure. Pushover analysis of the structure, modeled by the simple-supported non-linear frame in Fig. 4, provides another relation of s and Gs. During earthquakes, s is determined by the intersecting point of the two relations. Fig. 14 shows the flow diagram of estimation of the seismic shear deformation based on the shear strain transmitting characteristics. 3.3 Practical Example A practical example of estimation of the seismic shear deformation is provided as follows. Fig. 15 shows the multi-layered ground conditions and the common utility box located in Kobe. The one-dimensional seismic response analysis (SHAKE) of the surface layer is conducted. The seismic motion observed underground (G.L.-83m) in the Kobe reclaimed island during the 1995 Hyogoken-nanbu earthquake is applied to SHAKE. From the results of SHAKE, the peak ground shear strain g,max on the structure's underground level is 1.9810-3. In case that the thickness of the surface layer is one fourth of the shear wavelength, the equivalent shear wave velocity Vs,eq of the ground is given by

seqs THV /4, = (9) where H = total thickness of the surface layer, Ts = fundamental natural period of the surface layer analyzed by SHAKE. The equivalent ground shear stiffness Gg,eq is evaluated by Vs,eq and ave that is the average weight per unit volume weighted by each layer thickness. Fig. 16 shows the two relations of s and Gs. The intersecting point of the two relations gives s = 0.0021. At the same time, the FEM static

Evaluating g and Gg by

seismic response analysis of free-field ground

Relation of s and Gs based on Equation (8)

Equation (8), Shear strain transmitting characteristics

Pushover analysis of the structure

Structural non-linear relation of

sand Gs

Estimation of s

Solution of the two relations

Structural non-linear relation of s and Ps (Fig. 4)

Evaluating g and Gg by seismic response analysis of

free-field ground

Relation of s and Gs based on Equation (8)

Equation (8), Shear strain transmitting characteristics

Pushover analysis of the structure

Structural non-linear relation of

sand Gs

Estimation of s

Solution of the two relations

Structural non-linear relation of s and Ps (Fig. 4)

Fig. 14 Flow diagram of estimation of the seismic

shear deformation based on the shear strain transmitting characteristics

Ground level Vs(m/s)

165

147226

236226236

186

295

257

Tota

l thi

ckne

ss H

= 3

8.0

(m)

D = 2.95 (m)

6.8

Each layer thickness (m)

2.92.153.4

1.452.35

10.2

3.0

5.75

(kN/m3)

15.719.1

19.619.119.6

17.7

19.6

19.1

17.7g,max

h =

5.4

(m)

= 1.9810-3

b = 4.1 (m)

Ground level Vs(m/s)

165

147226

236226236

186

295

257

Tota

l thi

ckne

ss H

= 3

8.0

(m)

D = 2.95 (m)

6.8


2.92.153.4

1.452.35

10.2

3.0

5.75

6.8


2.92.153.4

1.452.35

10.2

3.0

5.75

(kN/m3)

15.719.1

19.619.119.6

17.7

19.6

19.1

17.7g,max

h =

5.4

(m)

= 1.9810-3

b = 4.1 (m)

Fig. 15 The multi-layered ground conditions and the

common utility box in Kobe

analysis, in which the horizontal inertia forces at the time of g,max = 1.9810-3 in SHAKE are applied to both the structure and the ground, is conducted on the identical conditions. The straightforward approach gives the more precise solution of s = 0.0014. If the capacity shear deformation su is assumed as the minimum shear deformation for the non-linear response of one structural member to reach the ultimate curvature, su results in 0.014 by the pushover analysis of the structure. It is found that s is much smaller than su, which can explain the small damage of the common utility boxes in Kobe during the Hyogoken-nanbu earthquake. The estimation of s based on the shear strain transmitting characteristics proposed herein is applicable enough for the practical seismic design method of underground structures. 4. Conclusions The shear strain transmitting characteristics from the surrounding ground to the underground structure have been intensively discussed in this paper. It is clarified that the aspect ratio of the cross section of the structure and the structure-ground weight ratio have little influence on the structure-ground shear strain ratio. The hyperbolic relationship between the ground-structure shear stiffness ratio and the structure-ground shear strain ratio is developed by the equilibrium of one-dimensional shear stress between the structure and the ground. Finally, a simplified seismic design method for underground structures based on the shear strain transmitting characteristics is proposed. References [1] SAMATA S., "Underground Subway Damage during the Hyogoken-nanbu Earthquake

and Reconstruction Technology", JSCE Journal of Construction Management and Engineering, No. 534, VI-30, 1996.3, pp. 1-17.

[2] JAPAN ROAD ASSOCIATION, "Specifications of Design and Construction for Parking Structures", Maruzen Inc., Tokyo, 1992.11, pp. 155-186.

[3] RAILWAY TECHNICAL RESEARCH INSTUTUTE, "Specifications of Seismic Design for Railway Structures", Maruzen Inc., Tokyo, 1999.10, pp. 331-341.

[4] WANG J. N., "Seismic Design of Tunnels, A State-of-the-Art Approach", Monograph 7, Parsons Brickerhoff Quade & Douglas, Inc., New York, 1993.

[5] JAPAN SOCIETY OF CIVIL ENGINEERS, "Standard Specifications for Concrete Structures, Design Edition", Maruzen Inc., Tokyo, 1996.3, pp. 23-24.

[6] TATEISHI A., "A Study on Loading Method of Seismic Deformation Method", JSCE Journal of Structural Mechanics and Earthquake Engineering, No. 441, I-18, 1992.1, pp. 157-166.

0.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

1.0E+05

0 0.005 0.01 0.015s

Gs(

kN/m

2 )

PushoveranalysisEquation (8)

s=0.0021

0.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

1.0E+05

0 0.005 0.01 0.015s

Gs(

kN/m

2 )

PushoveranalysisEquation (8)

s=0.0021

Fig. 16 Estimation of the seismic shear deformation based on the shear strain transmitting

characteristics

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