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Analytical solution for deep rectangular structures subjected to far-field shear stresses

Analytical solution for deep rectangular structures subjected to far-field shear stresses
Analytical solution for deep rectangular structures subjected to far-field shear stresses

Analytical solution for deep rectangular structures subjected

to far-?eld shear stresses

H.Huo a ,A.Bobet

a,*

,G.Ferna

′ndez b ,J.Ram?′rez a

a

School of Civil Engineering,Purdue University,550Stadium Mall Drive,West Lafayette,IN 47907-1284,United States

b

Department of Civil and Environmental Engineering,University of Illinois,Urbana,IL,United States

Received 9May 2005;received in revised form 15August 2005;accepted 10December 2005

Available online 28February 2006

Abstract

Underground structures located in seismic areas have to support the static loads transferred from the surrounding ground under nor-mal working conditions,as well as the loads imposed by any seismic event.Typically underground structures have cross section dimen-sions much smaller than the wave length of ground peak velocities,in which case inertial forces can be neglected and the structure can be designed using a pseudo-static analysis,where the seismic-induced loads or deformations can be approximated by a far-?eld shear stress or strain.Current close-form solutions for deep rectangular structures subjected to a far-?eld shear stress are approximations that do not consider all the relevant variables.An analytical solution is presented in this paper for deep rectangular structures with a far-?eld shear https://www.doczj.com/doc/9517142212.html,plex variable theory and conformal mapping have been used to develop the solution,which is applicable to deep rectangular structures in a homogeneous,isotropic,elastic medium.The solution shows that the deformations of the structure depend on the relative sti?ness between the structure and the surrounding ground,and on the shape of the structure.The analytical solution has been veri?ed by comparing its predictions with results from a ?nite element method and from previously published data.ó2006Elsevier Ltd.All rights reserved.

Keywords:Deep rectangular structure;Relative sti?ness;Seismic design;Pseudo-static analysis;Analytical solution

1.Introduction

The ?nal support system of underground facilities in seismic zones must be designed to support static overbur-den loads as well as to accommodate additional deformations imposed by earthquake-induced motions.Seismic-induced deformations of underground structures can be produced by direct shearing displacements of active faults intersect-ing the structure,by ground failure,or by ground shaking.Most of the analytical work done so far has concentrated on the evaluation of the e?ects of ground shaking.

There are two basic approaches in present seismic design.One approach is to carry out dynamic,non-linear soil–structure interaction analysis using ?nite element or

?nite di?erence methods,where inertia forces are included.The input motions in these analyses are time histories emulating design response spectra.Input motions are applied to the boundaries of a ‘‘soil island’’to represent vertically propagating shear waves.In the second approach,the pseudo-static approach,inertia forces are neglected.The earthquake loading is simulated as a static far-?eld shear stress or strain applied at the boundaries of the ground where the structure is embed-ded.In the pseudo-static approach soil–structure interac-tion may or may not be considered.When the interaction is included in the analysis,?nite element or ?nite di?er-ence methods are typically used;analytical solutions exist which provide relationships to evaluate the magnitude of seismic-induced displacements or strains in underground structures (Penzien and Wu,1998;Penzien,2000;a com-prehensive review is provided by Hashash et al.(2001)).If soil–structure interaction is neglected,it is assumed that

0886-7798/$-see front matter ó2006Elsevier Ltd.All rights reserved.doi:10.1016/j.tust.2005.12.135

*

Corresponding author.Tel.:+17654945033;fax:+17654961364.E-mail address:bobet@https://www.doczj.com/doc/9517142212.html, (A.Bobet).

https://www.doczj.com/doc/9517142212.html,/locate/tust

Tunnelling and Underground Space Technology 21(2006)

613–625

Tunnelling and

Underground Space Technology

incorporating Trenchless Technology Research

the structure follows the deformations of the ground. This is the free?eld approach(Hendron and Ferna′ndez, 1983;Merritt et al.,1985),where relationships are obtained based on the premise that the structure must accommodate the free-?eld deformations without loss of its integrity.This may not be entirely correct since the presence of the structure,if more rigid than the ground, would decrease the deformations of the surrounding ground.Although this e?ect may appear to bring predic-tions on the safe side,this is so only when the structure is sti?er than the surrounding ground.If the structure is more?exible than the ground,predictions may be unsafe because the liner distortions are larger than the free?eld deformations.To account for this e?ect Hendron and Ferna′ndez(1983),and Merritt et al.(1985),suggested to consider ground deformations compatible with the strain concentration imposed by the presence of the opening.

Analytical relationships are presented in this paper fol-lowing the pseudo-static approach to estimate seismic-induced distortions(racking mode)in deep rectangular underground structures,and taking into account the inter-action between the soil and the structure.These relation-ships can be an e?ective tool for practitioners.They allow readily identi?cation of the variables controlling the magnitude of the distortions and thus provide an insight into the behavior of the structure.The insight gained from the analysis can be used to optimize potential numerical remodeling e?orts by identifying pertinent parameters for sensitivity analysis.Estimates obtained from analytical relationships can also be used in pre-feasibility and feasibility studies to obtain early and relatively accurate assessment of structural requirements.Finally,analytical relationships can also be used to check the validity of the results obtained from numerical modeling.

2.Methodologies for seismic-induced structure deformations

The subject of ground-support interaction has been trea-ted by numerous researchers(Savin,1961;Timoshenko and Goodier,1970;Peck et al.,1972;Einstein and Sch-wartz,1979;Hendron and Ferna′ndez,1983;Merritt et al.,1985;Penzien and Wu,1998;Penzien,2000;Hashash et al.,2001;Bobet,2003),and it has concentrated mostly on circular cross-sections.A circular cross section may be appropriate for lifelines and deep tunnels,but for shallow structures(e.g.,cut-and-cover)and some mines a rectangu-lar cross-section is more common.A close-form solution for deep rectangular tunnels has been proposed by Penzien (2000).

Current seismic design approaches for circular tunnels proposed by Hendron and Ferna′ndez(1983),and Merritt et al.(1985),and for rectangular tunnels proposed by Penz-ien and Wu(1998),and Penzien(2000),suggest the use of simpli?ed relationships to estimate both the seismic-induced longitudinal as well as the circumferential strains in tunnel liners.Three assumptions are usually made:

(a)The dynamic ampli?cation of stresses associated with

a stress wave impinging on the opening is negligible.

This assumption is correct if the wave length of peak velocities is at least eight times larger than the width of the opening(Hendron and Ferna′ndez,1983).

Under these conditions the free-?eld stress gradient across the opening is relatively small and the seismic loading can be considered a pseudo-static load.The wave length can be estimated as V s/f;where V s is the shear wave velocity and f,the frequency of vibra-tion of peak ground motions.In most underground openings pseudo-static conditions are usually satis?ed.

(b)Plane strain conditions are assumed on any section

perpendicular to the longitudinal axis of the tunnel.

(c)Linear elastic deformations of ground and structure.

In general racking of the structure(ovalization of the structure due to shear waves traveling perpendicular to the longitudinal axis of the structure)is the most critical deformation.In a pseudo-static analysis,the shear wave motions can be approximated by a far?eld constant shear stress or shear strain.The magnitude of the far?eld shear stress s?is equal to the shear modulus,G,of the soil times the shear strain of the soil,c?.That is:

s ff?G c ffe1TThe free?eld shear strain c?can be obtained as:

c ff?v s=V se2Twhere v s is the peak particle vibration velocity,an

d V s is th

e shear wave velocity o

f the soil.

Wang(1993)ran dynamic parametric analyses on rect-angular structures with di?erent dimensions and sti?nesses. He found that the?exibility ratio of the structure,F,corre-lates well with the structure’s deformations.The?exibility ratio is expressed as:F?Ga=S1b,where a and b are the dimensions of the structure(Fig.1),G is the shear modulus of the ground,and S1=1/D1with D1equal to the displace-ment produced on the structure by a unit lateral concen-trated force applied to the top of the

structure.

614H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625

Penzien(2000)proposed an approximate method to evaluate the racking deformation of deep rectangular tun-nels subjected to a far?eld shear stress.It is an approxi-mate solution because of the assumption that the load transfer between the ground and the structure takes place only through shear stresses at the interface,and the defor-mations of a rectangular opening are approximated by those of a circular opening.Penzien showed that the defor-mations of the structure depend on the relative sti?ness,or the sti?ness ratio,between the ground and the structure. The relative sti?ness is de?ned with the parameter k stru/ k soil,which is the ratio between k stru,the sti?ness of the structure and k soil the sti?ness of the soil.k stru is equal to the magnitude of a uniform shear stress applied to the perimeter of the structure that produces a unit displace-ment of the structure;and k soil=G/b,where G is the shear modulus of the soil and b is the height of the structure.The relative sti?ness is the inverse of Wang’s?exibility ratio.

The normalized deformation of the structure,or the ratio between the structure deformation(D stru)and the free-?eld ground deformation(D?),D stru/D?,can be obtained by:

D stru D ff ?

4e1àvT

1te3à4vTK stru

soil

e3T

where v is the Poisson’s ratio of the ground.When the structure is much sti?er than the surrounding ground (i.e.,k stru)k soil),k stru/k soil is very large and the deforma-tion of the structure approaches zero.This corresponds to a very rigid ring embedded in a much softer material;the ring will keep its shape no matter the deformation that the surrounding material undergoes.On the contrary,if the structure is much more?exible than the surrounding ground(i.e.,k stru(k soil),k stru/k soil is very small,and the deformation of the structure,D stru,is4(1àv)D?.This is as if the structure does not exist and the solution corre-sponds to the deformation of the opening.Note that the value4(1àv)D?is exact for circular openings subjected to a far?eld constant shear stress,but it is an approxima-tion for rectangular openings.Another important assump-tion made in developing Eq.(3)is that ground displacements are imposed to the structure only through shear stresses at the interface,thus neglecting the contribu-tion of any normal stresses.

A new theoretical solution is proposed in this paper. Two are the keynote factors:(1)the new solution considers the contribution of both normal and shear stresses at the interface;(2)it considers the actual deformations of a rect-angular opening.The work presented in this paper is part of an on-going research to develop practical solutions for seismic design of cut and cover rectangular structures. 3.Theoretical framework

As with other analytical methods,the following assump-tions are made in the solution proposed herein:1.Deep rectangular structure inside an in?nite medium.

2.Plane strain conditions in any section perpendicular to

the longitudinal axis of the structure.

3.Elastic response of the structure and surrounding

ground.

4.Pseudo-static analysis.The seismic deformations of

ground and structure can be approximated by a con-stant far-?eld shear stress or strain.

The solution of any problem in elasticity must satisfy force equilibrium,compatibility of deformations,and boundary conditions.This is equivalent to?nd an Airy stress function U such that:

r2er2UT?0e4Twhere$2is the Laplacian operator.

In2D Cartesian coordinates,the stress components can be expressed as:

r x?

o2U

o y2

;r y?

o2U

o x2

;s xy?à

o2U

o x o y

e5Tand the strain components,in plane strain,are:

e x?

o u x

o x

?

1

E

?e1àv2Tr xàve1tvTr y

e y?

o u y

o y

?

1

E

?e1àv2Tr yàve1tvTr x

e6T

where E is the Young’s modulus of the material and v is the Poisson’s ratio.

Complex variable theory and conformal mapping tech-niques have been used for the solution of problems con-cerning rectangular openings in an in?nite medium (Mindlin,1940,1948;Muskhelishvili,1954;Sokolniko?, 1956).The fundamental theories of complex variable and conformal mapping have been extensively described by Muskhelishvili(1954),and later on by Savin(1961)and Timoshenko and Goodier(1970).A number of researchers have successfully implemented this technique into di?erent disciplines of engineering:(Theocaris and Petrou,1989; Theocaris,1991;Motok,1997;Gercek,1997;Exadaktylos and Stavropoulou,2002;Exadaktylos et al.,2003).For cir-cular tunnels the method has been very e?ectively used by Verruijt and coworkers(Verruijt,1997,1998;Strack and Verruijt,2002).

According to complex function theory,any biharmonic function(for example,the Airy stress function)can be expressed as:

U?Re? z uezTtvezT e7Twhere U is the Airy stress function;z is a complex variable and z is the complex conjugate of z;u(z)and wezT?v0ezT?d vezT=d z are two analytic complex functions,also known as‘‘complex potential functions’’.The stress compo-nents can be expressed in terms of complex potentials as: r xtr y?2?u0ezTt

r yàr xt2i s xy?2? z u00ezTtw0ezT

e8T

H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625615

The displacements in plane strain are:

2Geu xti u yT?e3à4vTuezTàz u0ezTàwezTe9Twhere u x and u y are horizontal and vertical displacements, respectively;G is the shear modulus of the material,G= E/[2(1+v)].

Fig.1shows the problem to be solved:a rectangular structure in an in?nite medium subjected to a far-?eld stress s?.Complex variable theory is used to determine stresses and deformations of the ground,while structural theory is used to?nd stresses and deformations of the https://www.doczj.com/doc/9517142212.html,patibility of normal and shear stresses and displacements is invoked at the interface between the ground and the structure to solve the problem.Thus,the approach followed consists of the solution of two initially independent problems:(1)the ground with a rectangular opening;(2)the structure.

For the ground the objective is to?nd the two complex potentials,u(z)and w(z).The Airy stress function for the ground can be expressed as:

U?U0tU?e10Twhere U0is the Airy stress function corresponding to the in?nite medium without the rectangular opening,and U* is the Airy stress function that includes the presence of the rectangular opening.

The two complex potential functions are:

uezT?u0ezTtu?ezT

wezT?w

0ezTtw?ezT

e11T

where u0(z)and w0(z)are the complex potentials corre-sponding to the in?nite medium without the opening,and u*(z)and w*(z)are the complex potentials that account for the presence of the opening.The complex potential functions depend on the boundary conditions,which are gi-ven by the known stress?eld at in?nity,s?,and by the stress ?eld at the interface between the ground and the structure, which is unknown and depends on compatibility of defor-mations between the ground and the structure.This intro-duces an additional level of di?culty since the solution depends on the solution itself.However,numerical simula-tions show that the shape of the normal and shear stress dis-tribution at the interface is rather independent of the dimensions of the rectangular opening and of the elastic properties of the ground and structure.In fact the shear stress can be well approximated as uniformly distributed around the perimeter of the structure,and the normal stress as linearly distributed(see Fig.2).This stress distribution follows the symmetry of the problem;there are two anti-symmetric axes:one vertical and the other one horizontal through the center of the structure.Thus,a complete solu-tion can be obtained if the stress distribution at the interior boundary is known.Note that only the distribution of the stresses is required;the actual values are obtained by estab-lishing compatibility of stresses and deformations between the ground and the structure.The?nal loads on the struc-ture will result from the addition of the seismic loads,which are the ones discussed in this paper,and the static loads from the overburden(not shown in Fig.2).

The Airy stress function and the complex potentials for the in?nite plate(no structure)subjected to the far?eld stresses s?are:

U0?às ff xy

u0ezT?0

w

ezT?i s ff z

e12T

Conformal mapping allows the transformation of the rectangular opening to a unit circle(Churchill,1960).The transformation is performed with the holomorphic func-tion(Savin,1961):

z?xefT?

a0

f

ta1fta2f2táááe13T

where a0,a1,a2,etc.,are complex constants.

The complex potentials u*(f)and w*(f)can be found from the following equation(Muskhelishvili,1954;Savin, 1961):

2p i u?efTt

Z

c

xerT

x0erT

u?0erT

d r

ràf

?

Z

c

f0

1

ti f0

2

ràf

d r

2p i w?efTt

Z

c

x0erT

u?0erT

d r

ràf

?

Z

c

f0

1

ài f0

2

ràf

d r

8

>>>

<

>>>

:

e14Twhere u*(f)and w*(f)are the two complex potentials in terms of f;r denotes any point on the unit circle;c is the

unit circle;and f0

1

and f0

2

are two boundary functions de-?ned as(Muskhelishvili,1954;Savin,1961):

f0

1

ti f0

2

?f1ti f2àu0erTt

xerT

u0

erTtw0erT

"#

e15Twhere f1and f2are:

f1?à

I

àr y

d x

d s

ts xy

d y

d s

!

d s

f2?

I

r x

d y

d s

às xy

d x

d s

!

d s

e16T

where r x,r y,and s xy are the stresses at the perimeter of the opening,and the integrals are performed along the contour of the

opening.

616H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625

4.Analytical solution

The conformal mapping of a rectangle can be approxi-mated with the following terms (Savin,1961):

z ?x ef T

?aR 1f tea t a T2f tea à a T224f 3tea 2à a 2Tea à a T80f 5

!

e17T

where a =e 2k p i =cos(2k p )+i sin(2k p );k is a mathematical parameter related to the shape of the opening k ,k =a /b ,the ratio of the length ‘‘a ’’and the height ‘‘b ’’of the rect-angular opening.The following equation relates the size of the opening,given by a and b ,the aspect ratio k ,and the parameter k .k ?a =b ?

1tcos 2k p à16sin 22k p à1

20sin 2k p sin 4k p 1àcos 2k p à16

sin 22k p t120sin 2k p sin 4k p e18T

For k <1/4,the length is larger than the height (a >b );for k >1/4,the length is smaller than the height (a

R ?1

21tcos 2k p à1sin 22k p à1

sin 2k p sin 4k p àáe19TThe rectangle ABCD in the ‘‘z ’’plane,shown in Fig.3,is transformed into a unit circle A 0B 0C 0D 0in the ‘‘f ’’plane.Point A 0in the unit circle corresponds to point A in the rect-angle,and so on.Note that ABCD is in a clockwise direction,whereas A 0B 0C 0D 0is counter-clockwise.This is so because the selected conformal mapping function transforms in?nity in the ‘‘z ’’plane into the origin in the ‘‘f ’’plane,and vice versa.In other words,the area of interest (the ground)in the ‘‘z ’’plane is outside the rectangle,which corresponds to the area enclosed by the unit circle in the ‘‘f ’’plane.

The distributions of normal and shear stresses at the perimeter of the rectangular opening are also shown in Fig.3.These are the stresses imposed on the ground by the structure.Identical stresses,but with opposite sign are applied to the structure (see Fig.2).The magnitude

of the shear stress s i ,which is unknown,is constant along the perimeter of the rectangle.The normal stress has a linear distribution on the four sides of the rectan-gle with a maximum magnitude p i 1along the side of length ‘‘a ’’and p i 2along the height ‘‘b ’’;both magnitudes are unknown.The positive sign of the normal stress denotes compression whereas the negative sign denotes tension (decompression given the initial static loading due to the overburden).

According to the Cauchy theorem,the ?rst integral of the ?rst equation in (14)is zero because the function inside the integral is analytic along the unit circle c (Churchill,1960;Timoshenko and Goodier,1970).The integral along the unit circle on the right-hand side of (14)is composed of four integrals:one for each segment A 0B 0,B 0C 0,C 0D 0and D 0A 0on the unit circle (see Fig.3).The integrals are solved analytically using Taylor series expanion of the functions inside the integrals (Huo,2005).

The functions u (f )and w (f )depend on p i 1;p i 2and s i .Compatibility of deformations between the ground and the structure are needed to solve for p i 1;p i 2and s i .The dis-placements of any point of the ground at the interface,given by Eq.(9)must be the same as the displacements of the same point of the structure.Points at the corners and centers of the sides of the rectangle are chosen to impose compatibility of deformations.Also,the structure must be in self-equilibrium.Based on moment equilibrium,the relation between p i 1and p i 2is:p i 1a 2?p i 2b 2

e20T

The structure is assumed to deform only due to bending (deformations of structural elements due to axial forces are neglected).This is a common assumption in structural mechanics.The structure deformations,in plane strain,are given by:

D stru ?e1àm 2S Tes i D s i tp i

2D p i 2

Te21T

where m s is the Poisson’s ratio of the structure,D s i is the deformation of the structure due to a shear stress s i ,and D p i 2is the deformation of the structure due to a linear nor-mal stress distribution given by p i 1;p i 2,with the condition p i 1a 2?p i 2b 2.D stru is the structure’s distortion,and is equal to the di?erence between the displacements of the top and bottom of the structure (see Fig.4).The actual values of D s i and D p i 2,can be obtained analytically or numerically from structural analysis.For the case of a rectangular structure with width ‘‘a ’’and height ‘‘b ’’with an interior central column,with members’sti?ness:lateral walls,(EI)w ;top and bottom slabs,(EI)s ;central column,(EI)c ,the deformations are:

D s i ?124k b 4

2eEI Tc k eEI Ts t1eEI Tw

h i tk 2eEI Ts 3k 2eEI Ts t2

eEI T

w h i 2eEI Tc

t1eEI Tw t3k

eEI T

s D p 2i ?14b 41c k s t1w h i tk s 7k s t5

w

h i 1c

t1w

tk s

e22Tz plane

p 1i

p 2i

a

i

i

A B

C

D

A‘

B‘

C‘D‘

1

(a)(b)

Fig.3.Conformal mapping H.Huo et al./Tunnelling and Underground Space Technology 21(2006)613–625617

There is a mismatch between the dimensions of the struc-ture and the opening,with a di?erence equal to the thick-ness of the structure members.This introduces errors in the formulation,which may be important because the thick-ness of structural members may not be small compared to the size of the opening.This is generally the case for rectan-gular structures because its resisting elements must with-stand signi?cant bending moments.This issue is discussed later with the veri?cation of the analytical solution.

The D s i=D p i

2,ratio depends on the shape of the structure

and not on its absolute dimensions.Taking as reference the sti?ness of one of the structure members(E s I s),and express-ing all the other element sti?nesses as a function of the refer-ence member,Eq.(22)can be written,in a general form,as:

D s i?K sekT

E s I s

k b4

D p i

2?

K pekT

E s I s

b4

D s i=D p i

2?

K sekT

K pekT

k

e23T

where the parameters K s(k)and K p(k)depend on the shape of the structure and on the Young’s modulus and dimen-sions of the members of the structure.For a rectangular structure with no central column and members of equal sti?ness(E s I s),the above equations reduce to:

D s i?e1tkT

24E s I s

k b4

D p i

2?

e1tkT

60E s I s

b4

D s i=D p i

2?

5

2

k

e24T

Imposing compatibility of deformations between the structure and the ground at the interface,one gets:p i

1

?

p i

2

k

p i

2

?M s itN s ff

s i?L s ff

e25Twhere

where j=(3à4m)for plane strain.The functions f1(k), f2(k),etc.are included in Appendix1.Note that the coe?-cients M and N depend only on the shape of the structure, k,and on the Poisson’s ratio of the ground,m.The param-eter L also depends on the sti?ness of structure and https://www.doczj.com/doc/9517142212.html,bining Eqs.(23)and(26),one gets:

Hence the parameters M,N,and L depend only on:the Poisson’s ratios of the structure,m s,and ground,m;on the shape of the structure,k;and on the factor X?E s I s=Gb3, which is a measure of the relative sti?ness between the ground and the structure.Both factors,k and X are non-dimensional.

Fig.5is a plot of the parameters M and N as a function of the aspect ratio of the structure,k and Fig.6is a plot of L with k,and with the relative sti?ness,X,computed for a rect-angular structure with uniform thickness.The values of the Poisson’s ratios are taken as m=0.35and m s=0.15.As it will be discussed later the e?ect of the Poisson’s ratios is small.

The free?eld ground distortion,D?,is de?ned as the dis-placement di?erence between a point on the top slab of the structure and a point on the bottom slab,due to the free ?eld ground deformation c?.Thus,

D ff?c ff b?s ff

b

G

e28TNormalization of the structure’s deformation with respect to the free?eld ground deformation results in:

D stru

D ff

?

G D stru

s ff b

?e1àm2

s

T?N D p i

2

teM D p i

2

tD s iTL

G

b

e29T

M?j f1ekTtf2ekTj f3ekTtf4ekT

N?

f5ekTà0:076j ?j f3ekTtf4ekT f6ekT

L?à

e1àm2

s

TN D p i

2

àa

G

1t1

k

àá

j?Nf

7

ekTtf8ekT tN1à1

k

àá

f9ekTtf10ekT

èé

e1àm2

s

TM D p i

2

tD s i

àa1t1

àá

j?Mf

7

ekTtf11ekT tM1à1

àá

f9ekTtf12ekT

èé

e26T

M?j f1ekTtf2ekTj f3ekTtf4ekT

N?

f5ekTà0:076j ?j f3ekTtf4ekT f6ekT

L?à

e1àm2

s

TNàE s I s

Gb3

k

p

1t1

àá

j?Nf

7

ekTtf8ekT tN1à1

àá

f9ekTtf10ekT

èé

e1àm2

s

TMtK sekT

p

k

àE s I s

Gb3

k

p

1t1

àá

j?Mf

7

ekTtf11ekT tM1à1

àá

f9ekTtf12ekT

èé

e27T

618H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625

Combination of Eqs.(29)and (23)results in:D stru D ff ?e1àm 2

s Tf NK p ek Tt?MK p ek Ttk K s ek T L g Gb 3E s I s

e30T

This is a general equation that relates the geometry of the structure and the properties of the ground and the structure.

Eqs.(29)and (30)show that:(1)the normalized struc-ture displacement is independent of the absolute size of the structure;in other words the normalized displacement depends on k =a /b ,the ratio of length to height of the structure,which is a measure of the shape of structure;(2)the normalized displacement depends on the relative sti?ness between the structure and the surrounding ground,and not on the absolute sti?ness of the structure or the ground.The relative sti?ness ratio is given by X ?E s I s =Gb 3.If the structure is much more rigid than the surround-ing ground,X approaches in?nity and the deformation of the structure approaches zero.If the structure is much more ?exible than the ground,X approaches zero and the normalized deformation of the structure becomes equal to the deformation of a rectangular opening.5.Veri?cation of the analytical solution

A series of ?nite element analyses has been carried out with ABAQUS (ABAQUS,2002)to verify the accuracy of the analytical solution.The numerical tests simulate the problem of an in?nite isotropic,homogeneous,elastic medium (the ground)containing an elastic rectangular structure subjected to the far ?eld shear stress s ?(Fig.1).The veri?cation is performed assuming that the structure has no interior columns and the sti?ness of lateral walls,top and bottom slabs is the same.It has been found that the analytical solution provides best results,compared to those of the numerical model,when the dimensions a and b used in the equations correspond to those of the opening.Thus,in the following discussion a and b are measured from the interior of the structure (i.e.,from wall to wall,rather than between the axes of horizontal and vertical members).The reason for this is the constrain that the thickness of the structure slabs imposes on the ground

H.Huo et al./Tunnelling and Underground Space Technology 21(2006)613–625619

behind,and thus for practical purposes the‘‘e?ective’’structure dimensions are those of the opening.

In the analyses,the following parameters are assumed: length of the structure,a,varies from4to8m,and12m; the height of the structure,b,remains constant at4m(this corresponds to structure shapes given by k=1,2and3, respectively;note that a and b are dimensions of the open-ing);Poisson’s ratios of the ground and the structure are 0.35and0.15,respectively.The results are obtained for dif-ferent sti?ness ratios X.

Fig.7provides a comparison between analytical predic-tions and numerical results for a wide range of relative sti?-nesses ratios,ranging from X=0to10.The range of sti?ness ratios used in the comparison extends beyond those typical in normal engineering practice.However, the sti?ness range encompasses values encountered in min-ing industry and/or temporary supports for civil structures, and the extreme values are used to provide the full range of results from the analytical solution.The predictions are acceptable,with maximum errors smaller than about 12%,and typical errors in the range2–8%.The errors may be due to some of the approximations made for the

development of the analytical solution,which include:(1) the linear normal stress distribution,or the constant shear stress distribution at the interface;(2)the number of terms used for the conformal mapping function z=x(f);(3) computation of integrals with Taylor series expansion;(4) e?ects of non-negligible thickness of structural elements given the size of the opening.The di?erences obtained with the analytical solution are,however,small enough such that it can be used for practical purposes.

Fig.7shows that the normalized structure deformation is very small when X is larger than5.It also shows that the normalized deformations increase as the shape factor,k, increases.As the sti?ness ratio decreases,the structure deformations increase very quickly towards those of a rect-angular opening without a structure.Fig.8plots some of the data shown in Fig.7,but for relative sti?nesses smaller than one,and highlights the deformations of structures softer than the surrounding ground.If the structure has the same sti?ness as the ground,the structure deformations will be those of the free?eld.This occurs for square struc-tures(k=1)with a sti?ness ratio of about0.14.As the shape of the structure is more elongated,the sti?ness ratio increases;for k=3,the structure has the same deforma-tions as the free?eld for a sti?ness ratio of0.5.As the sti?-ness ratio decreases below0.1,the normalized deformations increase very quickly.At the limit,when X=0(i.e.,there is no structure),the normalized deforma-tions are:3.774,3.469and3.844for aspect ratios k=1,2 and3,respectively.These results compare very well with the numerical results,which are:3.744,3.428and3.654. The errors are smaller than5%.

The e?ect of the soil’s Poisson’s ratio is small for square structures,as shown in Fig.9,but it may be signi?cant as the aspect ratio increases,particularly for structures with small sti?ness ratios(see Fig.10).For soils,the range of the Poisson’s ratio is smaller than that shown in Figs.9 and10,with values typically in the range of0.2–0.4;thus its in?uence is even smaller.

As already mentioned,the normalized structure defor-mation depends only on the sti?ness ratio X and on the aspect ratio k.In other words,the results do not change with the sti?ness of the structure or the ground,or with the actual dimensions,length and height,of the structure as long as X and k,remain the same.Two series of numer-ical analyses have been carried out in ABAQUS to check this conclusion.

In the?rst series,the Young’s modulus of the structure elements E s and/or the shear modulus of the ground G are

620H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625

changed,keeping the sti?ness ratio X ,and the shape ratio k ,constant.The thickness of the slabs and lateral walls of the structure is always 1m,and the Poisson’s ratios of the structure and the ground are 0.15and 0.35,respec-tively.Table 1contains the results of the normalized struc-ture displacement obtained both with the analytical solution and with ABAQUS.The Young’s modulus E s of the structure material ranges from 3000to 30,000MPa,and the shear modulus of the ground G ,from 10to 100MPa.Some of these values are not realistic but are included here for theoretical purposes to investigate a wide range of values where the analytical solution should still hold.The sti?ness ratio is always X =0.391.Table 1(a)contains the results for a square structure (k =1),and

Tables 1(b)and (c)the results for rectangular structures with k =2,and 3,respectively.The numerical results,D stru /D ?(normalized structure distortion),are insensitive to the imposed changes of sti?ness of soil and structure,which indicates that the relative sti?ness is indeed a nor-malizing factor.

In the second series the relative sti?ness is kept constant while the aspect ratio is changed.The results are tabulated in Tables 2(a)–(c),which correspond to k =1,2,and 3,respectively.In each case the dimensions of the structure are changed while their ratio is the same.The Poisson’s ratios of the structure and the ground,and the moment of inertia of the structure,are the same as in the ?rst series.Table 2show that the normalized structure distortions,D stru /D ?are indeed insensitive to the actual dimensions of the structure.

Further validation of the proposed analytical solution has been done by comparing its predictions with those obtained from Wang (1993)using the ?exibility ratio and Penzien (2000)using the relative sti?ness ratio.Two cases are considered for the comparisons.In the ?rst case the structure has internal dimensions a =4,8,and 12m,and b =4m,and thickness of all members 1m.In the second case the internal dimensions are a =13m,and b =3m,with member’s thickness 1m.The two cases are used to illustrate the e?ects of changes on k ,and X while the changes on Wang (1993)?exibility ratio are small.In all cases the Poisson’s ratio of the soil is m =0.35,and of the structure,m s =0.15.Results have been obtained for di?er-ent sti?nesses of the ground,G ,and of the structure ele-ments,E s .A comparison of the results obtained with the various solutions is shown in Fig.11.Note that for Wang’s and Penzien’s output the dimensions of the structure are measured from axis to axis (i.e.,the size of the opening plus half the thickness of the slab on each side),while for the analytical solution the internal dimensions of the opening are used.The results of the analytical solution bracket Wang’s and Penzien’s results but the di?erences illustrate the combined e?ects of k and X ,which are not incorpo-rated in Wang’s or Penzien’s solutions.For about the same ?exibility ratio,the analytical solution provides di?erent results because structures with di?erent aspect ratio,k ,and sti?ness ratio,X ,can have the same ?exibility ratio,F .In general,however,the analytical solution compares well with previous results for ?exibility ratios smaller than about 1.0–1.5,which correspond to structures sti?er than the surrounding ground or slightly softer.For softer struc-tures,with ?exibility ratios larger than 2,distortions pre-dicted from the analytical solution increase much faster than those from Wang (1993)and Penzien (2000).For very large ?exibility ratios (i.e.,no structure)the analytical solu-tion predicts normalized distortions equal to those of an unsupported opening (3.774,3.469and 3.844for aspect ratios k =1,2and 3,respectively;note that those agree with theoretical and numerical solutions,as indicated before)while Wang’s and Penzien’s are 2.4–2.8and 2.6,respectively.In Wang’s analysis the type of elements used,

H.Huo et al./Tunnelling and Underground Space Technology 21(2006)613–625

621

which were beam elements with rigid connections,could have introduced constrains in the deformation of the ground.This issue could explain why the results are close

to those of a circular opening.Additional di?erences may be due to inertia e?ects in Wang’s dynamic simulations,which would not be found in a static analysis.In Penzien’s

Table 2

Normalized structure deformation with constant aspect ratios Structure length a (m)

Structure height b (m)

Aspect ratio k Sti?ness ratio X D stru /D ?ABAQUS D stru /D ?Analytical Error (%)(a)k =1,m =0.35,m s =0.154410.3910.4030.416 3.26610.3910.3910.416 6.48810.3910.3790.4169.810101

0.391

0.392

0.416

6.1

(b)k =2,m =0.35,m s =0.15842

0.3910.9320.972 4.312620.391 1.1030.97211.916820.3910.9210.972 5.5201020.391 1.0680.9729.0

(c)k =3,m =0.35,m s =0.1512430.391 1.098 1.2019.418630.391 1.231 1.201 2.424830.391 1.196 1.2010.43010

30.391 1.304 1.2017.9

Table 1

Normalized structure deformation with constant sti?ness ratio Structure Young’s modulus E s (MPa)

Structure moment of inertia I s (m 3)

Structure length a (m)Structure height b (m)Aspect ratio k Ground shear modulus G (MPa)Sti?ness ratio X D stru /D ?ABAQUS D stru /D ?Analytical Error (%)

(a)k =1,m =0.35,m s =0.1530000.083441100.3910.42010.416 1.0

60000.083441200.3910.43320.416 4.090000.083441300.3910.37950.4169.612,0000.083441400.3910.39010.416 6.615,0000.083441500.3910.39780.416 4.618,0000.083441600.3910.41340.4160.621,0000.083441700.3910.40980.416 1.524,0000.083441800.3910.37780.41610.127,0000.083441900.3910.37930.4169.730,0000.0834

4

1

100

0.391

0.3801

0.416

9.4

(b)k =2,m =0.35,m s =0.1530000.083842100.3910.9210.972 5.560000.083842200.391 1.1230.97213.490000.083842300.391 1.0560.9728.012,0000.083842400.391 1.0020.972 3.015,0000.083842500.391 1.1340.97214.318,0000.083842600.391 1.0110.972 3.921,0000.083842700.391 1.0560.9728.024,0000.083842800.391 1.1010.97211.727,0000.083842900.391 1.0340.972 6.030,0000.0838421000.391 1.0990.97211.6

(c)k =3,m =0.35,m s =0.1530000.0831243100.391 1.196 1.2010.460000.0831243200.391 1.273 1.201 5.790000.0831243300.391 1.243 1.201 3.412,0000.0831243400.391 1.145 1.201 4.915,0000.0831243500.391 1.304 1.2017.918,0000.0831243600.391 1.329 1.2019.621,0000.0831243700.391 1.321 1.2019.124,0000.0831243800.391 1.401 1.20114.327,0000.0831243900.391 1.301 1.2017.730,0000.083

12431000.391 1.297 1.2017.4

622H.Huo et al./Tunnelling and Underground Space Technology 21(2006)613–625

analysis the di?erences may be due to the approximation of

a rectangular opening by a circular opening.

6.Summary and conclusions

The seismically induced deformations imposed to an underground structure can be approximated with a pseudo-static analysis where the seismic motions are applied far from the structure as a shear stress or shear strain.

Available close-form solutions for rectangular structures are approximate and are based on assumptions that may not re?ect how stresses are transferred from the ground to the structure through the interface.A new analytical solution has been presented in this paper that addresses some of the shortcomings of previous solutions.The new solution is based on complex variable theory and confor-mal mapping.The new solution has been obtained with the following assumptions:(1)deep rectangular structure inside an in?nite medium;(2)plane strain conditions in any section perpendicular to the longitudinal axis of the structure;(3)homogeneous and isotropic ground;(4)elas-tic response of structure and surrounding ground;(5) pseudo-static analysis.

The stress distribution at the interface between the ground and the structure(interior boundary values)is decided after a series of numerical simulations,which show that the distributions of the normal and the shear stress at the interface are quite independent of the dimensions of the structure and of the elastic properties of the ground and the structure.The shear stress can be accurately approximated by a uniform distribution around the perimeter of the structure,and the normal stress by a linear distribution.These distributions are consistent with the symmetry of the problem.The actual magnitudes of normal and shear stresses at the interface are determined by imposing compatibility of deforma-tions between the structure and the ground and by equi-librium of the structure.

The results obtained from the analytical solution are veri?ed with numerical simulations,which show that the errors obtained with the solution are typically smaller than 2–8%.The analytical solution shows that the ratio between the structure displacements and the free?eld ground dis-placements,D stru/D?,only depends on the relative sti?ness between the structure and the surrounding ground and on the shape of the structure.These parameters are de?ned as follows:relative sti?ness,X?E s I s=Gb3,aspect ratio,k= a/b.The sti?ness ratio is a dimensionless factor which relates the sti?ness of the structure(E s I s)with the sti?ness of the ground G and the height of the structure‘‘b’’.The aspect ratio k is the ratio between the length‘‘a’’and height ‘‘b’’.The dimensions that should be used in the analytical solution are those of the opening,measured from internal wall to internal wall.A structure with a shape ratio k will have the same deformations as a structure with a shape factor1/k,since one structure is the90°rotation of the other one.The normalized structure deformation decreases with the increase of X.For X=0,the deformation of a rectangular opening is recovered.When X is very large, the normalized structure deformation approaches zero. This is the solution of a rigid ring embedded in a much softer medium.For the same X,the normalized structure deformation increases with the aspect ratio k.The normal-ized structure deformation increases with the decrease of the ground Poisson’s ratio,even though the in?uence of the Poisson’s ratio is small.

The analytical solution has been veri?ed by comparing its predictions with those obtained from a numerical method and those from Wang(1993)and Penzien(2000). The analytical solution predicts larger deformations than those from Wang and Penzien for?exibility ratios larger than two,which may be due to some of the limitations imposed in their analysis.

The following provides a step-by-step procedure on how to use the solution,which can be done on a spreadsheet:

(1)Determine the internal dimensions of the opening a

and https://www.doczj.com/doc/9517142212.html,e a>b.

(2)Find the aspect ratio k=a/b,and factor k from

Eq.(18).

(3)Find the sti?ness ratio X.

(4)Use structural analysis to?nd D s i and D p i

2

,the defor-mation of the structure due to shear and normal stresses distributions,as shown in Fig.2.For simple structures use Eqs.(22)or(24).

(5)Find parameters M,N,and L,from Figs.5and6or

from Eqs.(26)or(27).Supporting equations can be found in Appendix1.

(6)Find normalized structure distortions,from Eq.(29)

or(30).

H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625623

Acknowledgements

The research has been supported by the National Sci-ence Foundation,Structural Systems and Hazards Mitiga-tion Program,under Grant CMS-0000136.This support is gratefully acknowledged.

Appendix1

The functions needed to obtain parameters M,N,and L in Eqs.(27)and(28)are:

f1ekT?A d0tB l0àA dàB l

f2ekT?àA d0tB l0tA dàB l

f3ekT?RA2b

k

tB2c R kà

RA2b0

k

àB2c0R k

f4ekT?RA2b

k

àB2c R kà

RA2b0

k

tB2c0R k

f6ekT?

X3

i?1

a i

f7ekT?R2A2b

k2

tR2B2ck

f8ekT?

R P

i?1

a i

f9ekT?R2A2b

k2

àR2B2ck

f10ekT?

R

P

i?1

a i

à3ta1à2a2t2a3

1ta1à6a2t10a3

t

1

k

R

P3

i?1

a i

1ta1t2a2t2a3

à3ta1t6a2t10a3

t1t

1

k

R1t

13

12

P3

i?1

a ie1à3a2T

"#

f11ekT?RA dtRB l

f12ekT?1t1 k

eàRA dtRB lTwhere,

k?a

b

?

1tcos2k pà1

6

sin22k pà1

20

sin2k p sin4k p

1àcos2k pà1

6

sin22k pt1

20

sin2k p sin4k p

R?

1

2e1tcos2k pà1

6

sin22k pà1

20

sin2k p sin4k pT

a1?1tcose2k pT

a2?à

1

6

sin2e2k pT

a3?à

1

20

sine4k pTsine2k pT

X3

i?1

a i?a1ta2ta3?1tcose2k pTà

1

6

sin2e2k pT

à

1

20

sine4k pTsine2k pT

A?1tcose2k pTt

1

2

sin2e2k pTà

1

4

sine4k pTsine2k pT

B?à1tcose2k pTà

1

2

sin2e2k pTà

1

4

sine4k pTsine2k pT

b?

???

2

p

p

à1t2k

c?

???

2

p

p

à2k

d?à

???

2

p

2p

t1à2kt

1

2p

ln

1àcos k p

1tcos k p

l?à

???

2

p

2p

t2k

b0?0:416à0:117ln

1tcosek pà3p=8T

1àcosek pt3p=8T

à0:009ln

1àcosek pà3p=8T

1tcosek pt3p=8T

à0:556arctan

sinek pTtsine3p=8T

cosek pTtcose3p=8T

tarctan

sinek pTtsine3p=8T

cosek pTàcose3p=8T

àp

!

t0:022arctan

sinek pTàsine3p=8T

cosek pTàcose3p=8T

tarctan

sinek pTàsine3p=8T

cosek pTtcose3p=8T

tp

!

c0?0:416à0:038ln

1àcosek pt3p=8T

1àcosek pà3p=8T

à0:084ln

1tcosek pt3p=8T

1tcosek pà3p=8T

à0:182arctan

sinek pTtsine3p=8T

cosek pTàcose3p=8T

tarctan

sinek pTàsine3p=8T

cosek pTàcose3p=8T

!

f5ekT??e0:707t2:121a2à5a3Te0:19t0:19a1à0:92a2t0:92a3Tàe0:207t0:5a1à2:121a2Te1:38à0:46a1t0:38a2t0:38a3T

e0:207t0:5a1à2:121a2Tte0:707t2:121a2à5a3T

t

13e0:38à2:76a2T

12e1t9a

2

T

à

à3ta1à2a2t2a3

1ta1à6a2t10a3

à0:08

X3

i?1

a ià

13

12e1à3a2T

624H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625

t0:407arctan àsinek pTtsine3p=8Tcosek pTtcose3p=8T

àarctan sinek pTtsine3p=8Tcosek pTtcose3p=8T

!

d0?0:208à0:089ln 1tcosek pà3p=8T1àcosek pt3p=8T

à0:032ln 1àcosek pà3p=8T1tcosek pt3p=8T

t0:589arctan sinek pTtsine3p=8Tcosek pTtcose3p=8T

tarctan sinek pTtsine3p=8T

cosek pTàcose3p=8T

àp

!

l0?à0:208à0:152ln 1tcosek pt3p=8T1tcosek pà3p=8T

t0:030ln 1àcosek pt3p=8T1àcosek pà3p=8T

à0:238arctan sinek pTtsine3p=8Tcosek pTàcose3p=8T

tarctan sinek pTàsine3p=8Tcosek pTàcose3p=8T

!

t0:147arctan àsinek pTtsine3p=8Tcosek pTtcose3p=8T

àarctan sinek pTtsine3p=8Tcosek pTtcose3p=8T

!

t0:203arctan sinek pTàsine3p=8Tcosek pTàcose3p=8T

àarctan sinek pTtsine3p=8Tcosek pTtcose3p=8T

!

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H.Huo et al./Tunnelling and Underground Space Technology21(2006)613–625625

安全风险管理大家谈

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