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Two-dimensional equivalent stiffness analysis of soil-structure interaction problems Nogami, Toyoaki 1972

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TWO-DIMENSIONAL EQUIVALENT STIFFNESS ANALYSIS OF SOIL-STRUCTURE INTERACTION PROBLEMS by TOYOAKI NOGAMI B.En., Nihon Univers i ty, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of C i v i l Engineering We accept th i s thesis as conforming to the required standard: THE UNIVERSITY OF BRITISH COLUMBIA Apr i l 1972 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers ity of B r i t i s h Columbia, I agree that the Library sha l l make i t f ree ly avai lable for reference and study. I further agree that permission for extensive copying of th is thesis for scholar ly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publ icat ion of th i s thesis for f i nanc ia l gain sha l l not be allowed without my written permission. Department of C i v i l Engineering The Univers ity of B r i t i s h Columbia Vancouver 8, Canada Apr i l 24, 1972 i i ABSTRACT The f i n i t e element technique is a powerful method to study the dynamic response of a structure taking into account the effects of ground condi-t ions. However, l im i ta t ions of computer storage capacity and cost present-l y prevent i t s general appl icat ion to three-dimensional problems. In th i s thesis i t i s shown that three-dimensional problems can be analyzed by applying appropriate modif ication factors to two-dimensional (plane stra in) analyses. Modif icat ion factors are f i r s t determined ana l y t i c a l l y by comparing the dynamic response of both s t r i p and rectangular footings (uniform shear stress) fo r a range of input frequencies. I t i s found that for input f r e -quencies which are less than.the fundamental period of the s o i l d layer the modif ication factor i s e s sent ia l l y independent of the input frequency. This suggests that the modification factors could be obtained from s t a t i c analy-ses. Modif icat ion factors based on s t a t i c s t i f fnes s analyses for both un i -form shear stress and uniform shear displacement ( r i g i d foundation) conditions were obtained and were found to be in close agreement with those obtained from the dynamic analyses. Var iat ion of the modification factor with both the depth of the layer and the ra t i o of the sides of the rect -angular base are given in graphical form. These factors may be applied to f i n i t e element place s t ra in analysis to predict the dynamic response of three-dimensional structures. I l l TABLE OF CONTENTS Pac^ e Chapter 1 - INTRODUCTION Chapter 2 - STEADY STATE RESPONSE OF A STRUCTURE OVER A SOIL LAYER FOR A HORIZONTAL EXCITATION AT THE BASE OF THE SOIL LAYER 5 Chapter 3 - DYNAMIC GROUND COMPLIANCE FOR UNIFORM TANGENTIAL LOADING OVER A SOIL LAYER 10 3:1 - Ana ly t i ca l So lu t ion , Plane St ra in 10 3:2 - Solut ion by the F i n i t e Element Method, Plane St ra in 17 3:3 - Ana ly t i ca l Solut ion for Three-Dimensional Case 19 Chapter 4 - COMPARISON OF TWO-DIMENSIONAL AND THREE-DIMENSIONAL RESPONSE OF SOIL-STRUCTURES 23 Chapter 5 - AN APPROXIMATE TWO-DIMENSIONAL SHEAR STIFFNESS ANALYSIS BASED ON THE STATIC SHEAR STIFFNESS OF THE SOIL LAYER 32 5:1 - Shear S t i f fness of a Rigid Rectangular Foundation and So i l Layer System 32 5:2 - The Shear S t i f fnes s of the Rig id S t r i p Foundation So i l Layer System 39 5:3 - Ca lcu lat ion of Two and Three-Dimensional S t i f fnes s fo r the Rig id Foundation 42 5:4 - Comparison of Two and Three-Dimensional S t i f fnes s 42 5:5 - Modi f icat ion Factor 45 Chapter 6 - APPLICATION OF THE EQUIVALENT SHEAR STIFFNESS ANALYSIS TO THE DYNAMIC RESPONSE OF A RECTANGULAR FOUNDATION OVER A SOIL LAYER 53 i v Page 6:1 - Transfer Function 53 6:2 - Fundamental Frequency of the System 55 6:3 - Discussion of the Results 58 Chapter 7 - CONCLUSIONS AND SUMMARY 61 BIBLIOGRAPHY 63 APPENDIX 1 64 APPENDIX 2 - A COMPACT FORM FOR THE LINEAR SYSTEM OF EQUATIONS ARISING IN CHAPTER 5 67 APPENDIX 3 - TWO-DIMENSIONAL STATICAL GROUND COMPLIANCE 71 LIST OF FIGURES Fi gure Page 1 Accelerat ion Response Spectrum at the Base of a Bu i ld ing 2 Structure and So i l Layer System Subjected to a Harmonic Exc i tat ion at the Bedrock 3 Structure and So i l Layer Interact ion Process for a Harmonic Exc i ta t ion at the Bedrock So i l Layer Subjected to a Harmonic Uniform Horizontal S t r i p Load on the Surface 11 So i l Layer Subjected to a Harmonic Uni-form Horizontal Rectangular Load on the Surface 20 6 Dynamic Ground Compliance of a So i l Layer 26 7 Modif icat ion Factor, a 27 8 F i n i t e Element Mesh 29 9 Program for Computation of Ground Com-pliance by F i n i t e Element Method 30 10 Comparison of the Ground Compliance Factor, f i , for Uniform Stress and Uni-form Displacement 31 11 Displacement in So i l Layer and Half Space 33 12 Half Space Subjected to a Concentrated Load 34 vi Fi gure Page 13 Uniform Shear Load Over a Rectangular Area on a Half Space 34 14 Rectangular Area (2a x 2c) Divided into (2n x 2m) Elements 37 15 Rectangular Area (2a x 2c) Divided into Four Elements 37 16 S t r i p Area Divided into 2m Elements 40 17 S t r i p Area Divided into Two Sections 40 18 Effects of Number of Divis ions of Rect-angular Area on Shear S t i f fnes s 43 19 Shear S t i f fnes s of the System per Unit Length of Foundation 46 20 Shear S t i f fnes s of the System per Unit Length of Rectangular Foundation 47 21 Shear S t i f fnes s Ratio 48 22 Shear S t i f fnes s Ratio 49 23 Shear S t i f fnes s Ratio 50 24 Rigid Body on a So i l Layer 54 25 Fundamental Resonant Frequency of the System 56 26 F in i te Element Mesh for Rig id Body and So i l Layer System 59 Program for Computation of Ground Compliance by Ana ly t i ca l Method Program for Computation of Ground Compliance by F i n i te Element Method Program for Computation of Shear S t i f fness of So i l Layer and Rectangula Rigid Foundation Program for Computation of Shear S t i f fness of So i l Layer and Rigid S t r i Foundation vi i i ACKNOWLEDGEMENTS The w r i t e r wishes to thank most s incere ly Dr. W.D. Li am Finn, Dean of the Faculty of Applied Science for his advice and encouragement during th i s study. He would also l i k e to thank his colleagues i n the So i l Mechanics Lab-oratory fo r t he i r valuable suggestions. E spec ia l l y , he owes his ana l y t i ca l so lut ion of plane s t ra i n dynamic ground compliance to the invaluable advice and assistance of Dr. E. Varoglu. The w r i t e r also owes much to those who helped him with his English and his p a r t i c u l a r thanks go to Dr. P.M. Byrne. 1 CHAPTER 1 INTRODUCTION In the design of some structures i t may be important to consider the safety of the structure for dynamic loading as well as s t a t i c loading. Dynamic loading i s e spec ia l l y important for structures in seismic zones, machine foundations, mi s s i le f a c i l i t i e s and some structures which must be re s i s tant to explosions. These structures may be b u i l t on or under the ground surface. An energy exchange takes place between the structure and the ground when the structure-ground system i s subject to a dynamic loading. This energy exchange i s ca l l ed the s o i l - s t r uc tu re in teract ion phenomenon and i t i s important that i t be properly taken into cons iderat ion. Unt i l recent ly , i t was believed that a structure founded on so f t ground suffered more earthquake damage than a s im i l a r s t ructure founded on f i rm ground. However, during the Kanto earthquake more damage occurred to r i g i d structures on a f i rm ground. Now, i t i s recognized that i t i s not only the properties of the structure or the ground alone that governs the earthquake damage, but i t i s the properties of the s o i l - s t r u c t u r e system that are important. As an example to show the importance of the s o i l - s t r uc tu re in teract ion during earthquakes, the recorded accelerat ion response spectrum (5) at the base of a bu i ld ing is shown in F i g . 1. The maximum response does not occur at the resonant period of the bu i l d ing ; instead, the response at that period is close to a minimum. The same tendencies may be found in other records (5, 6) . These tendencies may be a t t r ibuted to the feedback of the motion of the structure into the ground and the response depends on the dynamical FIG. I A C C E L E R A T I O N R E S P O N S E S P E C T R U M AT T H E B A S E OF A B U I L D I N G . 3 properties of the s o i l as well as those of the s t ructure. In the analysis of s o i l - s t r u c tu re systems, in teract ion between the ground and the structure may be taken into account by: 1. replacing the ground with some mass, spr ing, dash-pot combinations, 2. taking the ground as a continuous medium and applying wave propagation theory, 3. assuming that the ground i s an assembly of a f i n i t e number of small elements and applying f i n i t e element technique. The f i r s t approach involves important approximations in choosing a mass, spr ing, dash-pot system equivalent to a continuous medium. The second approach gives an exact so lut ion under the idea l i zed conditions of s o i l . However, mathematical d i f f i c u l t i e s make i t almost impossible to con-s ider complicated ground forms and unusual geometries. In the t h i r d approach, these complexities are treated without any mathematical d i f f i c u l t y . In the invest igat ion of s o i l s t ructure systems, the f i n i t e element technique based on plane s t r a i n analysis has been used by several authors. However, when the ra t i o between the length and width of the foundation slab i s sma l l , the so lut ion obtained using the plane s t ra i n analysis does not adequately represent the response of the system. A rigorous so lut ion of th is problem requires a three-dimensional ana lys i s . However, storage l i m i -tat ions and economical reasons prevent the use of the three-dimensional f i n i t e element technique for such dynamic problems. In th i s thes i s , the problem of a structure with a rectangular r i g i d base res t ing upon a homogeneous e l a s t i c s o i l l aye r , which in turn rests upon a r i g i d base subject to a hor izontal harmonic e x c i t a t i on , i s i n v e s t i -4 gated. A method is proposed whereby th i s problem can be reduced to an equivalent p lane-stra in problem by modifying the s o i l properties to account for the three-dimensional e f fec t of the foundation. 5 CHAPTER 2 STEADY STATE RESPONSE OF A STRUCTURE OVER A SOIL LAYER FOR A  HORIZONTAL EXCITATION AT THE BASE OF THE SOIL LAYER A structure with a r i g i d rectangular base (2a x 2c) over ly ing a s o i l layer and subjected to a harmonic exc i ta t i on at the base of the s o i l layer (bedrock) i s shown in F ig. 2, where: u 0 : Input harmonic displacement at the bedrock. u : : Steady-state displacement response at the free surface of the s o i l layer alone. u 2 : Steady-state displacement response at the base of the s t ructure. The processes involved in the computation of u 2 fo r a given u Q are as fo l lows: 1. Exc i tat ion at the bedrock gives the response ux at the free surface of the s o i l l ayer . 2. Displacement at the base of the structure i s inf luenced by the motion of the s t ructure , which gives the d i sp lace-ment response u 2 at the base. This response is trans-mitted into the structure and creates the force Q at the base of the s t ructure. 3. The created force Q at the base of the structure gives the addit ional displacement u 2 at the contact surface between the structure and the s o i l layer . 4. F i n a l l y , th i s addit ional displacement u 2 and the d i s -placement at the free surface of the s o i l layer u : are combined to give the displacement at the base of the structure u 2 . 2 a STRUCTURE u 2 S O I L L A Y E R FIG.2 S T R U C T U R E A N D SOIL L A Y E R S Y S T E M S U B J E C T E D TO A HARMONIC E X C I T A T I O N A T T H E B E D R O C K . H , ( i C O ) © U 2 ® H 2 _ ( i O ) ) _ H , ( i C O ) CD FIG. 3 S T R U C T U R E A N D SOIL L A Y E R I N T E R A C T I O N P R O C E S S FOR A H A R M O N I C E X C I T A T I O N AT T H E B E D R O C K . 7 The above process to compute the displacement response at the base for a given harmonic exc i ta t ion i s d iagramatical ly i l l u s t r a t e d in F ig. 3. I f the response of the structure and s o i l layer system i s l i nea r then the processes from 1 to 4 can be expressed mathematically using Fourier transformation and transfer functions as fo l lows: 1. = H ^ i a ) ) ^ (2:1) 2. Q = H 2 ( i u ) u 2 (2:2) 3. u'z = H 3(iu))Q (2:3) 4. U2 = ul + LT2 (2:4) where: Ug> u~15 u"2, IT2, Q : Fourier transformations of u 0 , U j , u 2 , u 2 and Q respect ive ly . H 1 ( ia) ) , H 2 ( i o j ) , H 3(ia)) : Transfer functions in the processes 1, 2 and 3 respect ive ly . Subst i tut ion of the righthand side of Equation 2:2 into Equation 2:3 gives: u 2 = H 2(ia))H3(io))u 2 (2:5) Subst i tut ion of the righthand side of Equations 2:1 and 2:5 into Equation 2:4 gives: U 2 = H^icojug + H 2 ( i to)H 3 ( ico)u 2 (2:6) S imp l i f y ing , u 2 = K i w ) 3 D u 0 (2:7) where: , x HiO'«>) ^ U I -H 2 ( IOJ)H3( IOJ) The steady-state displacement response at the base of the structure can therefore be obtained by solv ing Equation 2:7. I f the structure is assumed to be i n f i n i t e l y long, then for the two-g dimensional case Equations 2:1 to 2:4 become: u x = Hjd 'co)^ (2:9) Q 2 D = H 2 ( i u ) ( u 2 ) 2 [ ) (2:10) ( u 2 ) 2 D = H 3( ico)Q 2 D (2:11) Subst i tut ing in the same manner as for the three-dimensional case, where: ( u 2 ) 2 D = I 2 D ( i m ) u 0 (2:13) H,(iu) W 1 ' " ) = T 7 — r ~ r , — r (2:14) 2DV i-H 2(io))H 3(i io) I f the three-dimensional response i s to be obtained from a two-dimensional analys i s , then the two-dimensional analysis must be modified such that the displacement and shear force at the base of the structure w i l l be the same as for a three-dimensional ana lys i s , i . e . , <"2>2D = <"2>3D ( 2 : 1 5 ) and Q 2 D = Q/2C (2:16) From Equations 2:7 and 2:13: I 2 D( ico) = I 3 D ( i " ) (2:17) hence: Miu) Mlw) i-H 2(io))H 3(ito) i-H 2(ia))H 3(ito) From Equations 2:16, 2:2 and 2:10: (2:18) H2(ico) = 2 ^H 2 ( i c o ) (2:19) I f a plane s t ra in analysis i s applied for the so i l - s t r u c tu re system with the rectangular base ( 2 a x 2 c ) , H^ iu ) w i l l not change but H2(ico) and I I 9 \ 1 w J H.(iaj) w i l l be replaced by and H'(iw) respect ive ly. 6 2C 3 The term H 3(iai) i s the ground compliance when a s o i l layer i s subject to a hor izontal load Q over a rectangular area on the surface. S i m i l a r l y , the term H 3(ito) i s the ground compliance when a s o i l layer is subject to a hor izontal load Q/2C over the per unit length of a s t r i p . These terms w i l l be evaluated in the fo l lowing chapters. 10 CHAPTER 3 DYNAMIC GROUND COMPLIANCE FOR UNIFORM TANGENTIAL  LOADING OVER A SOIL LAYER . In Chapter 2 i t i s mentioned that the response of a structure on a s o i l t ransfer functions are known. The transfer function H^iw) for the s t r i p base of a structure is the same as that for the rectangular base since i t i s not re lated to the structure but only re lated to the dynamical character of the s o i l layer. The t ransfer function H2(io>) changes simply into H 2(iu)/2c when plane s t r a i n analysis is applied to a structure with rectangular base ( 2 a x 2 c ) instead of three-dimensional analys i s . However, the change of the transfer function H 3(ito) i s not ea s rede te rm ined . This t ransfer funct ion, termed the dynamic ground compliance, w i l l be considered in th i s chapter. The fo l lowing assumptions w i l l be made: 1. Stress d i s t r i bu t i on under the base i s assumed uniform. 2. So i l layer i s assumed to be homogeneous, i s o t rop i c and 3. So i l layer i s assumed to be f i xed at the r i g i d bedrock. 3.1 Ana ly t i ca l So lut ion, Plane S t ra in A s o i l layer which i s subject to a uniformly d i s t r ibuted dynamical shear loading over a s t r i p i s shown in Figure 4. The equations of motion in terms of displacements are: layer , subjected to harmonic exc i ta t ion at bedrock, can be obtained i f the l i nea r e l a s t i c mater ia l . = (X + p) | | + uV*U (3:1) = (A + U ) — + u V 2 V ay (3:2) 11 Q = Q0e i U J t \ / SOIL L A Y E R • (x,z) \ R z F I G . 4 SOIL L A Y E R S U B J E C T E D TO A HARMONIC UNIFORM H O R I Z O N T A L S T R I P L O A D ON T H E S U R F A C E . 32W where A v 2 = (x + v) az + ™ + m ax ay az a 2 - + 9 2 — + -12 (3:3) ax 2 ay 2 az 2 Here X and y denote Lame's constants, p i s the mass density of the s o i l layer. u,v,w are displacement components. Since displacement components are independent of the long i tudina l co-ordinate y , we have: ax 2 az 2 Defining two potent ia ls $ x ( x , z , t ) and xvl ( x ,z , t ) as fo l lows: 3*, 3V. u (x ,z , t ) = — + — - (3:6) 3X 3Z a$, av, w • i f - 1 7 ( 3 : 7 ) Equations 3:1 and 3:3 reduce t o : p 32<J> f = v 2 $ 1 (3:8) X + 2y 3 t 2 = ( 3 : 9 ) Since the applied boundary stress i s harmonic, the potent ia ls should have the fo l lowing form: M x . z . t ) = * (x,z) e i t o t (3:10) ^ ( x . z . t ) = Y(X,Z) e l c o t In view of Equation 3:10, Equations 3:8 and 3:9 reduce t o : (v 2 + p 2) + (x,z) = 0 (3:11) ( v 2 + q 2 ) (x,z) = 0 where: pur put' V X+2U ' M p The s t re s s - s t ra in re lat ions (Hooke's Law) can be expressed as: 13 (3:12) (3:13) a z z - X A + 2 y az • a x x , . , 3U _ /3U , 3Wx X A + P 37 ' T X Z " P ( 3Y + ax' The boundary conditions of the problem are: u(x,h,t) = w(x,h,t) = 0 a z z ( x , o , t ) = 0 ro T x z ( x f O , t ) = Sina^ Cosx^dc = • •x0e i i o t |x|>a |xi<a (3:14) (3:15) (3:16) Taking Sine Fourier transform and Cosine Fourier transform in x of Equations 3:11 and 3:12 respect ive ly , we get: U 2 - p2) • = 0 (3:17) d z z d 2 * d z 2 U 2 - q 2 ) ^ = 0 (3:18) where: J U,z) = f-^. J 4. (x,z) Sinxcdx * U.z) OO z) Cosx^dx From the general so lut ion of Equations 3:17 and 3:18, the two potent ia l s become: • (x,z) [A Coshaz + B Sinhaz] Sinx^dc; (3:19) 14 * (x,z) = where: [C CoshBz + D SinhBz] Cosx^d^ (3 :20) « = (C 2 - P 2 ) l / 2 . 3 = U 2 - P 2 ) l / 2 (3 :21) and A,B,C,D are a rb i t ra ry functions of Employing boundary conditions of Equations 3 : 1 4 , 3:15 and 3 : 1 6 , we get: (cCoshah)A + USinhah)B + (gSinheh)C + (BCoshBh)D = 0 (aSinhah)A + (aCoshah)B + UCosheh)C + (cSinhBh)D = 0 ( 8 2 + ? 2 ) A + 2£BD = 0 2?aB + (B 2 + e 2)C = - ^ p « ( a ? ) (3 :22) Here: 6(a?) = at From these equations, A(t;) and D(?) are obtained as: Try 2?B(aBSinhBhCoshah-?2CoshBhSinhah) ^ 2aB(c 2+B 2)+aB[^ l t+(c 2+B 2) 2]CoshBhCoshah-? 2[(^ 2+B 2) 2+ l ta 2B 2]SinhBhSinhah (3 :23) Try 2?B(aBSinhBhCoshah-c 2CoshBhSinhah)(B 2+? 2) < +C 2aB(? 2+B 2)+aB[i +^+(c 2+B 2) 2]CoshBhCoshah-? 2[(^ 2+B 2) 2+' +a 2B 2]SinhBhSinhah (3 :24) From Equations 3 : 6 , 3 : 1 0 , 3:19 and 3 : 2 0 , displacement component u on the surface of the s o i l layer can be expressed as: [cAU) + BDU)]Cosx ?dc (3 :25) u(x,o,t) = e i a ) t Subst i tut ing the righthand side of Equation 3:23 and Equation 3:24 for A(s) 15 D(r), into the righthand side of Equation 3:25, f i n a l l y we get: 2 T n a e i u t u(x,o,t) = — a Try 0 3(3 2-s 2)(a3Sinh3hCoshah-s 2Cosh3hSinhah)6(3s) • Cosxcdc i t? 2a6(e 2+c 2)+ae[i t? t |+(c 2+6 2) 2]CoshBhCoshah-c 2[(t 2+3 2) 2+i ta 28 2]SinhBhSinhah (3:26) The dynamical ground compliance of a s o i l l ayer fo r a tangential s t r i p load-ing can be expressed by the r a t i o : qe where q i s the amplitude of the to ta l dynamical force act ing per un i t length of the s o i l l ayer , i . e . q = 2 T Q a . In view of Equation 3:26, the dynamical ground compliance becomes: u(o,o,t) iwt ^ o 3(6 2-c 2)( aBSinhBhCoshah-^ 2CoshShSinhah)6(Bc)dc 4^ 2aB(e 2+f: 2)+a8[(3 2+c 2) 2+4? ' t]CoshBhCoshah-? 2[(c 2+e 2) 2+i ta 2B 2]SinhshSinhah (3:27) Employing change of var iable ? jr- 5 in Equation 3:27, we get L 2 ^ - = ^ [ - ^ ^ ^ ^ 0 ^ ( 3 : 2 8 , where: D(?) = {C 2Coth(/cM aj ) - A 2 - n 2 Co th (A 2 -n 2 a j } Coth(/£M 3 l ) E ( 0 = tanh ( . ^ T a j ) F(€) = H 2 ( 2 S 2 - O A 2 - n 2 /cM Cosech(/c 2-n 2a 1)cosech(/i 2^T a j - {H^+U^- i ) 2 } A 2 - n 2 / i^T Coth(/c 2-n 2 a : ) C o t h ( / ^ T ax) 16 + C 2 K U 2 - n 2 ) U 2 - 0 + (2C 2 - i ) 2 > n 2 - 1-2V 2 ( l - v ) ' " 2 P CJ C The functions /s; 2-i and / ^ 2 - n 2 have branch points % = 1, 5 = n, for K > o» respect ive ly. To stay on one branch of these funct ions, 1 = f i v ^ i 2 f i / r r ^ i 2 " A 2 - n 2 = • 0 1 5 1 1 5 > l 0 < 5 < n / c 2 - n 2 c > n Also, C = o , C = 1 and £ = f^, k = i,2 , . . . , N are poles of the integrand where S k» k = 1,2,...N are the f i n i t e number real zeros of F(s). Since the integrand in Equation 3:28 have some s i n gu l a r i t i e s such as branch points and poles, the integral on the righthand side of Equation 3:28 must be numerically evaluated in the complex plane. From the integrat ion in the complex plane: 00 ,2D a n f £SD(5)E(5) S1n( j dK 0 C2D 11 c i O T d T D ( 5 ) E ( { ) (3:29) (3:30) where: u(o,o,t) qe I c o t ya f2 D+ i f 2 D (3:31) and P denotes Cauchy p r inc ipa l value of the i n teg ra l . The numerical study 17 2D of F(c) =0 shows that f x has three types of s i n gu l a r i t i e s as fo l lows: ( i ) a j = 2 ™ 1 TT , m = i , 2 , . . . ( i i ) a x = 2.7012, 7.455, . . . ( i i i ) aj = 2.7207, 8.162, . . . The second and th i r d type s i n gu l a r i t i e s correspond to double real roots of F(c) = 0. The function F(?) does not have any real root fo r a.l < -n/z, 2D 2D hence f 2 = 0 for al < -n/i. The flow chart for the computation of fx i s given in F ig. A l . (Appendix 1). 3.2 Solut ion by the F i n i t e Element Method, Plane St ra in In the f i n i t e element method the mass of each element i s concentrated at the nodal points of each element. The equations of motion of nodal points in a s o i l layer subjected to forced exc i tat ions are: where: [M] [C] [K] {Q} [M]{u} + [C]{Ci> + [K]{u> = {Q} (3:32) mass matrix damping matrix s t i f f ne s s matrix exc i t i ng force matrix {u}, (u>, {u} : acce le rat ion , ve loc i t y and displacement matrices respect ive ly . Equation 3:32 represents a system of coupled equations. These coupled equa-tions can be uncoupled by an orthogonal matrix [$] to y i e l d : CO + 2C [a) n ] {|} + [co2]{ C> = [M*] _ 1{Q*> (3:33) where: {£} , {|}, {?} : acce le ra t ion , ve loc i ty and displacement in normal coordinates 18 con : the natural frequencies r : % of the c r i t i c a l damping CM*] = [ * ] T [ M ] [ < J > ] {Q*} = 0] T {Q} where the column vectors of [4.] are the mode shapes. I f the exc i ta t ion i s harmonic, the displacement in the Sth normal co-ordinate can be obtained from Equation 3:33 as: where: d. = - — ( 3 :35 ) {1 - ( ^ r ) 2 } 2 + (2C - ^ - } 2 M * S {1 - (JiL) 2} - i { 2 c JiL} {1 - (^) 2 } 2 + {ir - ^ - } 2 M * S u n u n 4 + 1 5 ? (3:34) ( J ^ ) 2 "n Q* s w n Q* S 5,- = - — (3:36) 1 (1 - i^) 2) 2 + " { 2 C - T } 2 M * S  w n u n The hor izontal displacement in the nodal coordinates at the center of the loaded area i s : N s=i N = I (4 5 s + i * | 5 f ) (3:37) s=i where: u £ : horizontal displacement in the nodal coordinates at the center of the loaded area : horizontal displacement at the center of the loaded 19 area in the Sth normal coordinate N calculated number of modes From the d e f i n i t i o n , the ground compliance can be expressed as: (3:38) I t should be noted here that the imaginary part does not appear when the damping matrix i s zero. To obtain mode shapes and natural frequencies, the ex i s t i ng computer program DYNAMIC was used. This program may be obtained from the Department of C i v i l Engineering Computer Program L ibrary. The Flow chart fo r the above computations i s shown in F ig. A2. (Appendix 1) 3.3 Ana l y t i ca l Solution fo r Three-Dimensional Case Kobori (7, 8) obtained the dynamical ground compliance for th i s case and extended i t for v i s coe l a s t i c multi layers (9). According to his analy-s i s , the displacement at the center of the load as shown in F i g . 5 i s : EU)S(a 0£,9)ded£ (3:39) where: FU) = H 2(2C 2-i)/s 2-n 2 c o s e ch ( a 1 ^ 2 - n 2 ) c o s e ch ( a 1 / ? 2 - i ) - i H k + (2C2-i)2} /s 2 -n 2 /^T coth (a 1 /c 2 -n 2 )coth (a 1 v / c 2 ^) + C 2 { - U 2 - n 2 ) U 2 - i ) + (2c 2-i) 2} EU) = t a n h U j / c M ) FIG. 5 SOIL L A Y E R S U B J E C T E D TO A HARMONIC U N I F O R M H O R I Z O N T A L R E C T A N G U L A R L O A D ON T H E S U R F A C E . 21 DU) = {c 2 coth(a 1 */^i ") - A2-n2 y / ^ i c o t h t a / ^ - n 2 } c o t h ^ / f ^ i ) s in(a n ?cose) s i n ( - a n £sine) S(a 0C,8) = cose sine S i ngu l a r i t i e s in the integrand in Equation 3:39 are: Branch points 5=n, 5=1 5=0, 5=1, 5=5. , 5=5^ K (k= i , 2 , . .N) K ( k= i , 2 , . . ,N ' ) where: : real roots for F(5) = 0 5^ : real roots for (EU))" 1 = 0 Integrating Equation 3:39 i n a complex plane, the ground compliance can be obtained as: Q where: 3D u a , = f f + i f f (3 :40) 3D 1 f r = — = • P 1 =, c Tra„-oa o r - T 4 = S 'U ) - ^ 1 D(5) S ' ( ? ) [ E(5)d? 5v^7 5 F ( 5 ) 2 ™ 1 N ' f 2 = — I D (5 )E(E) S'(5)| i r a ^ k=i 5 d F ( ? ) / d 5 2 C 5 k i N ' s ; ( 0 I 1 w a oa k = 1 a i c 2 'C=5 S(c,e) s in 2 ede s 2U) 22 Tf S(5,e) cos 2ede o P : Cauchy's p r inc ipa l value of the i n t e g r a l . The real part is the Cauchy's p r inc ipa l value of the i n t eg r a l . The imaginary part i s the radius about the poles and 5^. Resonant frequencies are those where the equation F(c) = 0 has double roots. The equation ( E ( c ) ) - 1 = 0 does not govern resonant frequencies since th i s equation does not have double roots. These two equations do not have a real root fo r frequencies less than a1 = 1.57. 23 CHAPTER 4 COMPARISON OF TWO-DIMENSIONAL AND THREE-DIMENSIONAL  RESPONSE OF SOIL-STRUCTURES The dynamic response of a structure located on a s o i l layer can be ob-tained for plane s t r a i n (two-dimensional) conditions using the f i n i t e element method of analys i s . However, fo r the three-dimensional case the number of unknowns becomes very large and computer storage space and cost l i m i t the s i ze of the problem that can be solved. For a harmonic base exc i ta t ion i t has been shown (Chapter 2) that both the two and three-dimensional responses depend e s sent i a l l y on the transfer function termed the ground compliance. The ground compliance fo r both the two and three-dimensional cases have been derived in Chapter 3. In th i s chapter these ground compliances w i l l be compared and i t w i l l be shown that the three-dimensional response may be estimated from a two-dimensional plane s t r a i n analys i s . A comparison of the dynamic response of s t r i p (two-dimensional) and rectangular (three-dimensional) loaded areas shows the fo l lowing: 1. Resonant frequencies for a s t r i p load and a rectangular load are i dent i ca l s ince the functions F(c) governing the resonant frequencies for them are the same (Equations 3:28 and 3:39). 2. Radiational dampings fo r a s t r i p load and rectangular load do not appear for frequencies less than a : = 1.57, since the equations F(?) = 0 and (E (c ) ) " 1 = 0 governing the rad iat ional damping do not have any real roots for these frequencies (Equations 3:31 and 3:40). 3. Radiational damping does not appear in a f i n i t e element 24 method as shown in Section 2, Chapter 3. However, material damping may be used to simulate the rad iat iona l damping. 3D Subst i tut ing u /Q from Equation 3:40 into Equation 2:3, the t ransfer function H3(ico) i s given by: M i " ) = ± ( f f + i f f ) (4:1) and the t ransfer function I ( l c o ) ^ D from Equation 2:8 i s given by: H^co) I ( 1 u , 3 D = " 1 , f3D . , f 3D> H i s ( 4 : 2 ) 1 ' "aTT ( f i + l f 2 ' H lu> I f a plane s t ra i n analysis i s used fo r the same system, then subs t i tu t ing u/q from Equatioi be expressed as: i n 3:31 into Equation 2:11, the t ransfer funct ion H3(ico) can H3(ico) = ± ( f f + i f f ) (4:3) and the t ransfer function I ( i co ) 2 D from Equation 2:11 i s given by: H » K i c o ) 2 D = - g j -gg (4:4) i - + i — ) H2(co) ay 2C 2C ^ The d i f ference between the t ransfer functions I ( i co ) 3 D in Equation 4:2 and I ( i co ) 2 D in Equation 4:4 i s the error which would a r i se by the use of a plane s t r a i n analysis for the structure with a rectangular base. I f the system has no rad iat iona l damping the t ransfer functions I(ico) 3D and I ( i co ) 2 D w i l l be: Mco) T ^ 3 D = 1 i f3D „ M H^co) I ( o ) ) 2 D = ~ I2D ( 4 : 6 > ay 2C 2 25 A modif icat ion factor a w i l l be defined as: Mu l t ip ly ing liu)^ by ^ in Equation 4:6 gives: i H^co) ^ w ) 2 D = ~2T. <4:8> l - — — H2(to) ayct c This i s i den t i ca l with a three-dimensional t ransfer function I f w ) ^ . There-fore, i f a plane s t ra in f i n i t e element analysis i s modified to y i e l d the transfer function I ( w ) 2 D > the three-dimensional e f f ec t i s obtained for a harmonic motion at the bedrock. Although i t may be possible to give the response with three-dimensional rad iat iona l damping ef fects by choosing a su i tab le material damping, th i s thesis w i l l not discuss such damping but only the modif icat ion factor a. Since rad iat iona l damping does not appear at frequencies less than the funda-mental frequency in ground compliance (a x = 1.57), the precise considerations w i l l be given for th i s range. 2D 3D The ground compliances fx and f x fo r frequencies of a harmonic e x c i -2D tat ion less than a x = 1.57 are shown in F ig. 6, where f i s computed as 3D shown in Appendix 3 and f^ i s c i ted from Reference 8. The modif icat ion fac tor a ca lculated from these ground compliances is shown in F ig . 7. The results show that the modif icat ion factor a for a square foot ing (2a = 2c) i s almost constant with frequency (except near the resonant frequency). For a depth of s o f t , h, equal to the width of the foot ing, 2a, i . e . h/a = 2, a = 1.25 and for h/a = 4, a s 1.6. The fact that the modif ication factor a is e s sen t i a l l y independent of frequency for a given geometry i s important because i t indicates that a 26 FIG.6 DYNAMIC G R O U N D C O M P L I A N C E OF A S O I L L A Y E R . 27 2a i « POISSON'S RATIO = 0.25 c/a = 1.0 ( for f,3 D) 1 1 1 1 0 0.2 0.4 0.6 0.8 1.0 FREQUENCY OF EXCITATION/FUNDAMENTAL RESONANT FREQUENCY (a) h /a = 2 . 0 • 1 • • 0 0.2 0.4 0.6 0.8 1.0 FREQUENCY OF EXCITATION/FUNDAMENTAL RESONANT FREQUENCY ( b ) h /a =4.0 F IG.7 MODIFICATION F A C T O R , a . 28 could be obtained from a s t a t i c analysis and this w i l l be considered in Chapter 5. The analyses performed so far have been based on the assumption that the shear stress d i s t r i bu t i on at the contact between the structure and the s o i l layer is uniform. However, i t i s more l i k e l y that the displacements rather than the shear stresses would be uniform at the contact. The ground compliances (plane s t ra in) fo r these two cases were obtained using a f i n i t e element dynamic analys is . The f i n i t e element mesh used is shown in F ig . 8. A flow chart fo r the computer program used i s shown in F ig . 9. The ground 2D compliance factors f t f o r the two assumptions are compared in F ig. 10 where i t may be seen that the assumption of a uniform shear stress at the contact gives a higher (approximately 20%) ground compliance for the range of f r e -quencies o < a 0 < l . I t would appear reasonable to assume that the ground compliance factors 3D fl f o r the cases of uniform stress and uniform displacement would be s im i l a r to those for the two-dimensional case and hence the modif icat ion factor a obtained for the case of uniform stress would also apply for the case of a r i g i d foundation. POISSON'S RATIO = 0.3 YOUNG'S MODULUS = I 0 6 l b s . / f t 2 ro FIG.8 F IN ITE E L E M E N T M E S H 30 ^ S T A R ^ C A L C U L A T E I BY EXISTING PROGRAM "DYNAMIC READflfr], Q ) I ) C A L C U L A T E [ * ] T I C A L C U L A T E {Q'l.OO , {M - | FIG.9 P R O G R A M FOR COMPUTATION OF GROUND C O M P L I A N C E BY FINITE E L E M E N T M E T H O D . 31 l O O r 8 0 h Q M UJ o 60 h O o ZD O or o 4 0 h 2 0 0 POISSON'S RATIO = 0.3 YOUNG'S MODULUS I0 6 lbs./ft. 2 UNIFORM STRESS DISTRIBUTION -UNIFORM DISPLACEMENT DISTRIBUTION J 0 0.2 0.4 0.6 0.8 1.0 F R E Q U E N C Y OF EXCITATION/ FUNDAMENTAL FREQUENCY FIG. 10 C O M P A R I S O N OF T H E G R O U N D COMPLIANCE F A C T O R , f, FOR UNIFORM S T R E S S A N D U N I F O R M D I S P L A C E M E N T . 32 CHAPTER 5 AN APPROXIMATE TWO-DIMENSIONAL SHEAR STIFFNESS ANALYSIS BASED  ON THE STATIC SHEAR STIFFNESS OF THE SOIL LAYER Chapter 4 indicates that the dynamic response of a three-dimensional structure involv ing s o i l - s t r u c tu re in teract ion may be estimated from a two-dimensional analysis i f the appropriate modif icat ion factor i s used. Since the modif icat ion factor a i s almost constant for frequencies less than &i = 1.57, the constant value of a based on a s t a t i c analysis may be used. In th is chapter, the modif icat ion factor a based on a s t a t i c shear s t i f f ne s s w i l l be evaluated. 5.1 Shear S t i f fness of a R ig id Rectangular Foundation and So i l Layer  Sy s tern As shown in Fig. 11, the displacement at the surface of a s o i l layer of depth h subjected to a surface load i s approximately given as (10, 11): u(x,y,o) = ui(x,y,o) - U2(x,y,h) (5:1) where u ui U2 displacement at the surface of a s o i l layer of depth h. displacement at the surface of a ha l f space, displacement at the depth h in the ha l f space. The displacements ui and U2 at any point i n a horizontal plane fo r a uniform rectangular load can be ca lcu lated as fo l lows. I f a horizontal point load is applied on the surface of a s em i - i n f i n i t e e l a s t i c body (F ig . 12), the displacement at the point (x,y,z) w i l l be given by Ce r r u t i ' s formula: u(x,y,z) I+TTG X ^ 1-2U , X Z , + 1 + R( 1 ) R2 R+z R(R+z) (5:2) where applied horizontal point load. I u ( x , y , 0 ) - 0 m ui ( x , y , 0 ) -o y^//^//V/^//^/^//^//^//$y//csy/. u ( x , y , h ) = 0 a ) S O I L L A Y E R b ) H A L F S P A C E FIG.II D I S P L A C E M E N T IN SOIL L A Y E R A N D H A L F S P A C E CO CO 34 ' Z FIG.I2 H A L F SPACE S U B J E C T E D TO A C O N C E N T R A T E D L O A D FIG. 13 UNIFORM S H E A R L O A D O V E R A R E C T A N G U L A R A R E A ON A H A L F S H E A R . 35 G , v : shear modulus and poisson's r a t i o of s o i l . R = /x 2 + y 2 + z 2 Integration of Equation 5:2 over the rectangular area, as shown in Fig. 13, with z = 0 w i l l lead to the displacement at the corner of the uniform rec t -angular shear load as: ui corner 2XG - l tan c ' / a 1 f a '/cose ( l - vs in 2 e)de dr TT/2 t a n _ 1 G 7 a ' /•c'/sine ( l - vs in 2 e)de dr T a ' 2TTG F(X) (5:3) where F (A ) = ( i - v ) l o g f / m 2 + x) + xlog / x x * * 1 x = c ' / a ' This displacement at any point (x,y,o) wi th in the loaded area may be deter-mined by considering the loaded area to be comprised of four rectangular areas with the point (x,y) a common corner to the four areas. The d i sp lace-ment ui(x,y,o) i s then obtained by adding the contr ibutions from each of the four areas: ~2~Q Cx(F(J) - F ( £ £ - ) } - ( x - a ' ) { F ( ^ i r ) - F (^fr - ) } ] x>a', y<c' [x{F(£) + F ( ^ - ) } - ( x - a ' ) { F ( - f - ) - F ( ^ ) > ] x>a', y>c' u i(x,y,o)= a -x a ' - x 2TT ^C x { F ( ^ ) - r{^f-)} - ( x - a ' ) { F ( ^ ) - F ( ^ ) } ] o<x<a', y>c' (5:4) T3' (~T) — ^ 2^G" The integrat ion of Equation 5:2 over the rectangular area i s d i f f i c u l t 36 fo r z = h. I f the sides of a uniform rectangular load are small compared with the depth of a s o i l l ayer , the hor izontal displacement at the depth h can be obtained from Equation 5:2 under the coordinates shown in F ig. 13 replacing an area load with a concentrated load at the center of the area as: - T^kli^tl + ^ Rlll-^M-n (5:5) where Rx = / ( x -a ' /a ) 2 + (y -c ' /z ) + h 2 Now the displacements ui and U2 are obtained from Equation 5:4 and Equation 5:5 respect ive ly. Subst i tut ing these values into Equation 5:1, the hor izontal displacement on the surface of a s o i l layer subjected to a un i -form rectangular surface shear load can be ca lcu lated. The above so lut ion i s for a uniform shear stress over the loaded area. I t was mentioned in e a r l i e r chapters that i t i s much more l i k e l y that the displacement rather than the shear stress w i l l be uniform over the area. The so lut ion for a uniform displacement w i l l now be considered. The shear s t i f f ne s s of a rectangular r i g i d foundation on a s o i l layer i s the tota l shear force on a r i g i d foundation to cause a un i t hor izontal displacement. Div iding a r i g i d foundation into 2m x 2n number of small e l e -ments with uniform load d i s t r i bu t ions as shown in Figure 14, th i s force w i l l be ca lcu lated as fo l lows: The un i t horizontal displacement of the element ( i , j ) in a r i g i d found-at ion subjected to a hor izontal force can be expressed by superposing the influences due to each element as: 2m 2n . I I u i x k = 1 (5:6) k=i 4=1 1 , 0 ' where T. : uniform shear stress applied on the element (k,i) 37 2n n J i m - x 2m y FIG.14 R E C T A N G U L A R A R E A ( 2 a x 2 c ) D IV IDED I N T 0 ( 2 n x 2 m ) E L E M E N T S . ] V i i i I I FIG.I5 R E C T A N G U L A R A R E A ( 2 a x 2 c ) D IV IDED INTO FOUR E L E M E N T S . 38 u . ' . : horizontal displacement of the element ( i , j ) due I * to a unit uniform shear stress on the element ( k , 4 ) . Each quarter of the foundation area w i l l be denoted as I, I I , III and IV as shown in F ig. 15. Separating the influences of the element stress by the sect ion I, I I , III and IV, Equation 5:6 can be rewritten as: I II III m n = i i=i 1 , J K ' * k=m+i A=I 1 , J K , x - k=i 4=n+i 1 ' J K , ) t 2m n k ,4 m 2n ,k ,4 IV 2m 2 n I I k=m+i 4=n+i + J * U i ' j T k 4 = 1 k +i o = n + i 1 ' J ^'^ or m n v v t,,k,*> 4 . ,,m+k,4 , ,,m,n+4 , ,,m+k,n+4 > I, J t ( u i , j T k , 4 + u i , j T m + k , 4 + u i ; j T k , n + 4 + u i , j Tm+k,n+* ) = 1 (5:7) Since the contact shear stress d i s t r i bu t i on under the r i g i d foundation i s symmetrical with respect to x and y axes, the uniform shear stresses in the sections I I , III and IV can be replaced with those in the sect ion I. TT T T 2 m -k+ i , 4 1 1 1 T k , 2 n - 4 + i 2m-k+i ,2n-4+i = T k ,4 (5:8) Subst i tut ing these re lat ionships from Equation 5:8 into Equation 5:9 gives: m n f y ( u k > * + u m - . k + l ' 1 + u k > 2 n - * + 1 + u ? m : k + i , 2 n - i + i ) 39 (5:9) Applying Equation 5:9 to a l l elements in the section I gives: m n I ? ? (nk>1 + ,,2m-k+i ,£ k,2n-4+i . 2m-k+i ,2n-4+K c=i £=i 1 , 1 1 , 1 1 , 1 1 , 1 k ' £ = 1 y y (u k>* + u ? m - : k + l » 1 + u k ' 2 n " £ + 1 + u ? m : k + 1 ' 2 n " £ + 1 ) T | , ? ? (uki!L + u 2 m - k + 1 ' £ + u k,2n-4+i + 2m-k+i,2n-4+ix ^ £ £ x m,n m,n m,n m,n ; k,J = 1 = 1 • m x n equations (5:10) where the displacements can be obtained by Equations 5:1, 5:4 and 5:5 and the stresses can be obtained by solv ing Equation 5:10. Equation 5:10 has been put in a more compact form i n Appendix 2. The shear s t i f f ne s s of the rectangular r i g i d foundation and s o i l layer system K^Q can be calculated as: K 3D m 'tac r r m ' n k=l 4=1 T k > * (5:11) 5.2 The Shear S t i f fness of the Rig id S t r i p Foundation So i l Layer System The r i g i d s t r i p foundation area i s divided into 2m number of elements as shown in F ig. 16 and a shear stress d i s t r i bu t i on on each element i s assumed uniform. Then a unit hor izontal displacement of the element ( i ) in a r i g i d foundation subjected to a hor izontal force can be expressed by super-posing the influences due to each element as: 2 r m k E u i T J k=i 1 * 1 (5:12) 40 m 2m FIG.I6 S T R I P A R E A DIVIDED INTO 2m E L E M E N T S II FIG.17 S T R I P A R E A DIVIDED INTO TWO S E C T I O N S . 41 where x^ : uniform horizontal stress applied on the element (k). u- : hor izontal displacement of the element ( i ) due to a unit uniform shear stress on the element (k). Separating the influences of the element stress by the sect ion I and II as shown i n F ig . 17, Equation 5:8 can be rewritten as: I II m . 2m . I U k x + I U k x = 1 k=i 1 K k=m+i 1 K (5:13) m or J (u* x, + u^k T ^ , ) = 1 (5:14) Since contact shear stress d i s t r i bu t i on under the r i g i d foundation i s symm-e t r i c a l with respect to y ax i s , the uniform shear stresses x in sect ion II can be replaced with those in sect ion I. (5:15) T 2m-k+i = X, Subst i tut ing Equation 5:15 into Equation 5:14 gives m k=i k . ,2m-k+i * _ , T '- + u - T 2 m - k+ i ) " 1 i k i (5:16) Applying Equation 5:16 to a l l elements in sect ion I gives: m J . m k=i m I ( u k + u 2 m " k + 1 ) x, ^ m m • m equations (5:17) 42 where the displacements can be obtained by Equation Al :5 (Appendix 3) and the stresses can be obtained by solv ing Equation 5:17. The shear s t i f f ne s s of a r i g i d s t r i p foundation and s o i l layer system I<2Q can be calculated as: o a m K 2 D " T £ \ ( 5 : 1 8 > 5.3 Calcu lat ion of Two and Three-Dimensional S t i f fnes s fo r the  R ig id Foundation The s t a t i c shear s t i f fnesses fo r the two and three-dimensional cases are given by Equations 5:18 and 5:11. The width of the footing i s 2a in both cases but a unit length of foot ing i s considered fo r the two-dimensional case while a length, 2c, i s considered fo r the three-dimensional case. To compare s t i f fnesses then, the s t i f f ne s s for the three-dimensional case must be d i v i -ded by the length of the foot ing , 2c. A l so , i f both s t i f fnesses are divided by Young's modulus, E, a dimensionless s t i f f ne s s i s obtained. The s t i f f ne s s values obtained w i l l depend on the number of area elements used. This i s shown for the three-dimensional case i n F ig. 18. I t may be seen that i f 30 x 30 elements are used, l i t t l e error w i l l occur. The s t i f f -ness values computed and shown in the next section were obtained using 30 area elements for the two-dimensional case and 30 x 30 area elements for the three-dimensional case. 5.4 Comparison of Two and Three-Dimensional S t i f fnes s Dimensionless s t i f fnesses for the three-dimensional case are shown in Table 1 for various rat ios of c/h, h/a and Poisson's r a t i o , v. Values shown range from 1.31 to 0.48. Dimensionless s t i f fnesses for the two-dimensional case are shown in Table 2 for various rat ios of h/a and Poisson's 43 UJ »-00 >-00 UJ X r-00 00 UJ U_ 00 or < UJ x Q I ro O oo |.4 u. O CO Q O O o >-1.0 i 8 < Q Z zo o .6 u_ UJ x O .4 x H O z UJ -» .2 0 2a >///////////////: POISSON'S RATIO = 0.0 h/a = 8 c/a = 1.0 c/a = 10.0 0 10 20 NUMBER OF E L E M E N T S 30 FIG.I8 E F F E C T S OF N U M B E R OF DIVISIONS OF R E C T A N G U L A R A R E A ON S H E A R S T I F F N E S S . T A B L E 1 3 - D SHEAR ST I FFNESS OF T H E SOIL L A Y E R YOUNG M O D U L U S x L E N G T H OF THE FOUNDATION h V - Poisson's ratio ///////////. | C h/a = 4 . h/a = 8. h/a = 16. a v - o . U--.2 7J = .3 7J--0. TJ=-2 u = o . U = -2 1. 3. 5. 10. 20. 1.314 .898 . 798 . 712 . 690 I. 21 1 .850 .763 . 691 .670 1.180 .842 .761 .695 .665 1.222 .796 .690 .598 .550 1.128 .754 .661 .581 .538 1.100 .748 .660 .586 .542 1. 179 .746 .634 .535 .489 1.090 .708 .609 .520 .478 1.065 .702 . 608 .524 .483 T A B L E 2 2 - D SHEAR S T I F F N E S S OF THE SOIL L A Y E R Y O U N G MODULUS > 1 I? = Poisson's ratio h •/////////; t h/o - 4 . h/a = 8 . h/a = 16. v - o . L?=0.3 V-O. L>=0.3 V- o. V - 0.3 . 6 9 7 . 683 . 536 .538 . 434 . 4 4 3 45 r a t i o , v, equal to 0.0 and 0.3. Values range from 0.70 to 0.44. As expected the three-dimensional s t i f f ne s s is greater than the two-dimensional s t i f f -ness. However, as the length of the foot ing , 2c, becomes large compared with the width, 2a, the three-dimensional s t i f f ne s s approaches the two-dimensional s t i f f n e s s . The dimensionless s t i f f ne s s fo r a square and s t r i p foundation are shown in F i g . 19. I t may be seen that the square foot ing i s considerably s t i f f e r than the s t r i p foot ing and that both decrease as the depth of the s o i l layer increases. The dimensionless s t i f f ne s s for the three-dimensional case i s shown as a function of c/a and h/a i n F i g . 20. I t may be seen that the s t i f f ne s s decreases with increasing values of both c/a and h/a. 5.5 Modif icat ion Factor The modif icat ion f ac to r , a, was defined in Chapter 4, Equation 4:7 as -p 2D T i /2c a = — 2 D — a n a " ">s the r a t " i o of the s t i f f ne s s e s , namely: f i a - ^ (5:19) *2D The var ia t ion of a with the geometry for Poisson's r a t i o equal to 0.3 are shown in Figs. 21, 22 and 23. It may be seen that the e f f e c t of the depth of soi1 i s as fo l lows: 1. The modif icat ion factor increases with the depth of a s o i l j a y e r and w i l l be i n f i n i t e for an i n f i n i t e depth since the displacement for a s t r i p load on the surface of a s em i - i n f i n i t e body is i n f i n i t e but not for a rec t -angular load. (F i g . 21) 2. The modif icat ion factor for a square foundation on a P r o o CO X m > CO H ~ n ~ n ZD m m CO o CO TAN OF o H <z X r ~ m > CO -< -rj CO O H (Z m a > TJ H m O c • H LE CD —1 X o s " n 3 -D S H E A R ST IFFNESS OF T H E S Y S T E M  L E N G T H OF T H E FOUNDATION x YOUNG'S MODULUS OF SOIL CD "m O ro J> °^ FIG." 21 S H E A R S T I F F N E S S R A T I O . 3-D SHEAR STIFFNESS PER UNIT LENGTH OF THE FOUNDATION 6fr z o LENGTH OF THE FOUNDATION/ THICKNESS OF T H E SOIL LAYER ( 2 c /h ) F I 6 . 2 3 S H E A R S T I F F N E S S RATIO s o i l layer with the ra t io h/a = 1 is about one (F ig . 21). The er ror caused by the use of a plane s t r a i n f i n i t e e l e -ment method fo r th is case w i l l be neg l i g ib le i f r ad ia -t iona l damping i s ignored. The e f f e c t of the r a t i o of the sides (c/a) of a rectangular foundation are as fo l lows: 1. The modif ication factor decreases with the increase in sides r a t i o c/a and w i l l be one for i n f i n i t e sides r a t i o c/a. (F ig. 22, F ig. 23) 2. The modif icat ion factor i s almost one at the r a t i o 2c/h = 5. The error caused by the use of a plane s t r a i n f i n i t e element method fo r the ra t i o 2c/h > 5 w i l l be neg-l i g i b l e i f rad iat iona l damping i s ignored. (F ig . 23) I t should be noted here that the above-mentioned results are based on the fo l lowing assumptions. 1. The r i g i d foundation is divided into small equal area elements and the stress d i s t r i bu t i on on each element i s assumed uniform. 2. The displacement of an element i s assumed to be uniform and i s equal to that at the centre of the element, although the displacement with in an element i s not uniform because of assumption 1. 3. The displacement at the surface of a s o i l layer is assumed to be-the dif ference between the surface displacement and the displacement at a depth equal to the depth of s o i l layer ( semi - i n f i n i te ana ly s i s ) . 4. The displacement in a s em i - i n f i n i t e body at the same depth as that of a s o i l layer i s ca lculated by replacing an area load on the element with a concentrated load at the centre of the element. These assumptions lead to the fo l lowing er ror s : 1. The error due to assumption 4 increases as the r a t i o of c/a increases. However, i t reduces as the r a t i o h/a increases. 2. The error due to assumption 3 increases as h/a reduces. 3. The error due to assumption 2 gives a smaller s t i f f -ness under any condit ions. 53 CHAPTER 6 APPLICATION OF THE EQUIVALENT SHEAR STIFFNESS ANALYSIS TO THE DYNAMIC  RESPONSE OF A RECTANGULAR FOUNDATION OVER A SOIL LAYER In th is chapter a plane s t r a i n f i n i t e element analysis i s modified to model the case of r i g i d rectangular base on a s o i l layer (F ig . 2 4 ) , The resu lts are compared with the ana l y t i ca l so lu t ion . 6.1 Transfer Function The force Q created by a harmonic hor izontal exc i ta t i on u 2 under the rectangular r i g i d body i s : d 2 u d t 2 where M : tota l mass of the r i g i d body on a s o i l layer. Fourier transformation of Equation 6:1 gives: Q = Mco2u2 (6 :2 ) where Q, u 2 : Fourier transformations of Q and u 2 respect ive ly . to : angular frequency of the e x c i t a t i on . Therefore, the transfer function H 2(OJ) i s : H2(co) = Mto2 (6 :3 ) Subst i tut ion of the transfer function H2(co) from Equation 6 :3 into Equation 4 :2 gives: Moo) Q = - M (6 :1 ) Kico) 3D 1 _ _ L - ( f 3 D + 1 f 3 D , ^ 2 ap i 2 T - ^ n =rr ( 6 : 4 ) i - ba 2 ( f f + i f 3D) where b = P a 3 54 h/a = 4.0 c/a =1 .0 B E D R O C K FIG. 2 4 RIGID B O D Y ON A SOIL L A Y E R ao 55 co a A P p : mass of a unit volume of s o i l . The absolute value of the complex t ransfer function in Equation 6:4 i s : H.(co) I(co) = (6:5) /( i - ba 2 f ? D ) 2 + (bag f 2 3°)2 Ignoring f | D gives: H.(co) I ( w ) , D = o 3 D (6:6) 3 0 l - ba 2 f 3 D o 1 Also, the absolute transfer function I(oo)2D by a plane s t ra i n analysis in Equation 4:4 i s : H^co) I ( t o ) 2 D = 7W~ ( 6 : 7 ) 1 - ba2 (£?E) Modif icat ion of the transfer function I (CD) 2 D to get 1(00)30 9 i v e s : Hjco) I ( U ) 2 D = ; ~2Q- ( 6 : 8 ) The transfer function H^co) can be obtained by the methods discussed in reference 12. For this special case considered here, i t can be given as: H1(co) = sec al (6:9) u coh where a = 4 For frequencies- corresponding to a x 5 -rr/2, no rad iat iona l damping w i l l occur (Chapter 3). 56 6.2 Fundamental Frequency of the System The fundamental frequency i s the lowest frequency at which the term I((D) goes to i n f i n i t y . This may occur when: H j U ) = Sec a x = » (6:10) or when l - ba 2 f = 0 (three-dimensional analysis) or l - ba2, f j • = 0 (plane s t r a i n analys is) f2D or l - ba 2 -r}-r = 0 (modified plane s t r a i n analys is) From Equation 6:10, H1(co) = °° when ax = TT / 2 . Now a 0 = t o a / / ^ (Equation 6:4) and al = coh/v^ -, hence the r a t i o a!/a 0 = h/a. For h/a = 4, then Hi (to) = °° when ao = ^ a i . Therefore resonance occurs when arj = \ a i = ir/8 = 0.391. Resonance may also occur for frequencies corresponding to a 0 less than 0.391 (a x < ir/2) provided Equation 6:11 i s s a t i s f i e d and, since a x w i l l be less than TT/2 f o r these cases, no rad iat iona l damping w i l l occur. The fundamental frequencies fo r a x < -n/z or ao < 0.391 are calculated as fo l lows: 2D 3D 1. f1 and fl are obtained from F ig. 6b fo r values of ao from zero to 0.39. 2. The mass r a t i o , b, required to s a t i s f y Equation 11 i s then ca lcu lated. The ana ly t i ca l re lat ionsh ips between ao and b are shown in F ig. 25 fo r both the plane s t ra in and square foot ing cases. Modif icat ion factors based on s t a t i c shear s t i f f ne s s were calculated i n Chapter 5. For a square foot ing with h/a = 4 the modif icat ion factor i s 1.73 (F ig . 23). The plane s t r a i n re su l t when modified for a square foot ing i s also shown in F i g . 25 and i t may be seen to be in close agreement with the ana l y t i ca l so lu t ion . (6:11) 57 .4 >-z LU ZJ o 111 or u_ r— AN .3 z o CO UJ or _i < i— z UJ .2 < Q Z ID U_ CO CO UJ _l z o i CO . i z UJ a h/a = 4 , c /a = I POISSON'S RATIO = 0.25 MODIFICATION FACTOR a 1.73 ANALYTICAL SOLUTION R E C T A N G U L A R F O U N D A T I O N S T R I P FOUNDAT ION S T R I P FOUNDAT ION (MODIF IED) FINITE E L E M E N T METHOD • S T R I P F O U N D A T I O N o STR IP F O U N D A T I O N ( MODIFIED) 2— 10 20 MASS 50 100 RATIO lb) 2 0 0 5 0 0 0 0 0 FIG.25 F U N D A M E N T A L R E S O N A N T F R E Q U E N C Y OF T H E S Y S T E M . 58 The fundamental frequencies were also obtained from a f i n i t e element analys i s . The f i n i t e element mesh used i s shown in F ig. 26. The results fo r the plane s t ra i n case and for the square foot ing ( a = 1.73) are also shown in F ig. 25 and i t i s seen that they are very s im i l a r to the ana ly t i ca l re su l t s . 6.3 Discussion of the Results The modif ication of a plane s t ra i n analysis based on the s t a t i c modi-f i c a t i o n factor gives the fundamental resonant frequency in shear close to, or the same as, that from a three-dimensional analysis as shown in F ig . 25. I t i s i n te re s t ing that the results show no s i g n i f i c a n t di f ference between the plane s t ra i n analysis and three-dimensional analysis fo r the mass r a t i o less than f i v e . The smaller ground compliance f j fo r the same frequency a 0 gives a larger resonant frequency of the system and the larger one gives a smaller resonant frequency under the constant mass r a t i o . The larger and smaller resonant frequencies obtained by a modified plane s t r a i n transfer function analysis than those by a three-dimensional one may be explained by f2D f2D — ~ r - > f ? D and — --}— < f ? D respect ive ly . When the mass r a t i o i s about 54 a 2C 1 a 2C 1 J these two analyses give the same results for this pa r t i cu l a r case, where f2D 3 D — - ^ r - and f i w i l l be the same values. The resonant frequencies from a a 2c 1 ^ f i n i t e element method are larger than those from a transfer function because the ground compliance fi fo r the uniform displacement d i s t r i bu t i on over the contact area under the foundation i s smaller than that for the uniform stress d i s t r i b u t i o n . Since the constant modif icat ion factor a i s used fo r the modif icat ion on a f i n i t e element method, the same tendencies as mentioned above for the transfer function occur. F IG .26 F INITE E L E M E N T M E S H FOR RIGID BODY AND SOIL L A Y E R S Y S T E M . cn 60 I t w i l l be possible to get exact fundamental resonant frequencies for th i s system i f a correct modif icat ion factor i s used for each frequency a 0 . Even a constant modif ication factor based on s t a t i c a l case w i l l give a sa t i s fac tory re su l t . 61 CHAPTER 7 CONCLUSIONS AND SUMMARY I f a structure with rectangular r i g i d base ( 2 a x 2c) i s attached on the surface of a homogeneous, i so t rop i c and l i nea r e l a s t i c s o i l l ayer , the trans-fe r function between the input harmonic exc i ta t ion on the bedrock and response at the base of the structure can be expressed with the ground compliance ignoring damping as: Kco) 3D . 1 -3D u / \ 1 ' ITT f i M u ' ay '1 2' I f a plane s t ra in analysis i s used for the same system the transfer function i s : M t o ) 2D f2D ! . J_^_ H ? (co) ay 2C 2 ' I f a plane s t ra i n analysis with the modif icat ion factor a i s app l ied, the transfer function Uco)g D becomes: Mco) Kco) 2D f2D ' - a i r i F M " ) The fol lowing procedures w i l l give this modif icat ion in a plane s t r a i n f i n i t e element method: f2D/ 2 C 1. Find the modif icat ion factor a from a = — ^ — . 2. Mult ip ly the mass and Young's modulus of s o i l by a. For the frequencies of a harmonic exc i ta t ion lower than &i = 1.57, the constant modif ication factor a based on the s t a t i c shear s t i f f ne s s can be used because no rad iat ional damping occurs. For these frequencies of harm-62 onic e xc i t a t i on , there w i l l be no s i gn i f i c an t difference in the response of a structure between a plane s t r a i n and three-dimensional analysis for the fol lowing condit ions: 1. Ratio 2c/h i s larger than 5. 2. Ratio h/a is about one. 3. Mass r a t i o i s smaller than certa in r a t i o ( i . e . = 5 fo r the r a t i o h/a = 4 and square foundation). The modif icat ion factor based on the s t a t i c a l shear s t i f f ne s s has the fo l low-ing tendencies: 1. Modif icat ion factor a i s larger fo r larger r a t i o h/a. 2. Modif icat ion factor i s smaller for larger r a t i o c/a. 3. Var iat ion of Poisson's r a t i o of s o i l from 0 to 0.3 has neg l i g ib le inf luence. The ca lcu lated modif icat ion factor in th i s thes is can be applied to any shape of structure as long as the structure has a rectangular base and the same rat ios h/a and c/a as those of ca lcu lated ones. Although th i s thesis considered the appl icat ion of a plane s t r a i n f i n i t e element method to the structure with rectangular base on a s o i l layer only, i t may be possible to apply this method for any case i f the transfer functions H 3 ( ico )2 D and H 3 ( i co ) - D are known. 63 BIBLIOGRAPHY 1. dough, R.W., "The F i n i t e Element Method in Plane Stra in Ana ly s i s " , Proc, 2nd ASCE Conf. on E l e c t r o n i c Computation, September 1960. 2. Argys is, J.M., "Continua and Discontinua", Conf. on Matrix Methods in Structural Mechanics, A i r Force In s t i tu te of Technology, Wright-Patterson A i r Force Base, October 1965. 3. Clough, R.W. and Chopra, "Earthquake Stress Analysis in Earth Damps", Structures and Material Research, Dept. of C i v i l Engineering, Univers ity of C a l i f o r n i a , Berkeley, Report No. 65-8, Ju ly 1965. 4. Lamb, H., "On the Propagation of Tremors over the Surface of an E l a s t i c S o l i d " , Philosophical Transactions of the Royal Society, London, Series A, Vol . 203, 1904. 5. Hisada, J . , Nakagawa, K. and Izumi, M., "Accelerat ion Spectrum and Ground Condit ion" , Architecture Research Institute, Japan, Report No. 46, 1965. 6. Hudoson, D.E., "A Comparison of Theoretical and Experimental Determination of Bui ld ing Response to Earthquakes", Proc, 2nd World Conference on Earth-quake Engineering, 1960. 7. Kobori, T., Minai, R. and Suzuki, T., "Dynamical Ground Compliance of Rectangular Foundation on an E l a s t i c Stratum over a Semi- Inf in i te Rig id Media (Part 2 ) " , I n s t i t u t e of the Prevention of Natural Disaster, Kyoto Univers i ty, Vol. 10, 1967. 8. Kobori, T., Minai, R. and Suzuki, T., "Dynamical Ground Compliance of Rectanqular Foundation on an E l a s t i c Stratum over a Semi- Inf in i te Rig id Media (Part 3 ) " , I n s t i t u t e of the Prevention of Natural Disaster, Kyoto Univers i ty , Vol. 11, 1968. 9. Kobori, T. and Suzuki, T., "Foundation Vibrations on a V i scoe las t i c M u l t i -Layered Medium", Proc. Japan Earthquake Engineering Symposium, 1970. 10. Steinbrenner, W., "Tafeln zur Setzungsberechnung", Die Strasse, Vol. 1, 1934. 11. Davis, E.H. and Taylor, H., "The Surface Displacement of E l a s t i c Layer due to Horizontal and Vert ica l Surface Loading", Proc. 5th International Conference on Soil Mechanics and Foundation Engineering, Vol. 1 , 1961. 12. Donovan, N.C. and Matthiesen, R.B., "E f fects of S i te Conditions on Ground Motions during Earthquakes", State-of-the-Art Symposium, Earthquake Engineering of Buildings, 1968. APPENDIX 1 Figure Al Program for Computation of Ground Compliance by Ana ly t i ca l Method Figure A2 Program for Computation of Ground Compliance by F i n i t e Element Method C A L C U L A T E S T A T I C f. Y E S 3! C A L C U L A T E D Y N A M I C f , FIG.AI P R O G R A M FOR COMPUTATION OF G R O U N D C O M P L I A N C E BY A N A L Y T I C A L M E T H O D . 66 CALCULATE ( I READ ([<£]» I CALCULATE l<t>]T I CALCULATE {Q"\, (K"}, {M-} BY EXISTING PROGRAM " DYNAMIC" FIG.A2 P R O G R A M FOR COMPUTAT ION OF GROUND COMPLIANCE BY FINITE E L E M E N T METHOD. 67 APPENDIX 2 A COMPACT FORM FOR THE LINEAR SYSTEM  OF EQUATIONS ARISING IN CHAPTER 5 The coe f f i c i en t s of unknowns in Equation 5:10 can be wr i t ten in terms of 2m x 2n displacements u* 3 '^ p = i , * " , 2 m , q = i , « « « , 2 n , using the r e l a t i o n : l » i up»q = u|p-i| + i»|q-j|+i i , j i , i p = i , • • • , 2 m q = I,--- , 2 n i = l , • • •, m j = l , - - - , n (A2:l) as u u k,* i ,j k ,2n+i-£ Jk - i|+ i ,U- j|+ i 1,1 |k-i|+i,2(n+i)-U+j) 1,1 2m+i-k,x, 2m+i-k ,2n+i-Ji i ,J = u = u 2(m+i)-(k+i),|ji-j +i 1,1 2(m+i)-(k+i),2(n+i)-(ji+j) i ,k = l ,• • • ,m j,z = i,-«-,n (A2:2) In view of Equation A2:2, Equation 5:10 becomes m n I I k=i i=i Jk - i |+ i ,U - j [+ i + Jk- i|+i ,2(n+i)-U+j) 1,1 1,1 2(m+i)-(k+i),U-j|+i 2(m+i)-(k+i),2(n+i)-(4+j) 1,1 i , i T k , = 1 i = l , • • • ,m j = i , - - - ,n (A2:3) This system of equations can be rewritten in matrix notation as: [a]{x} = {d} (A2:4) where a m(j-i)+i,m(i-i)+k Jk- i|+i,|i-j|+ + |k-i|+i,2(n+i)-(£+j) 1,1 i , i 2(m+i)-(k+i),|£-j|+i + 2(m+i)-(k+i),2(n+i)-(£+j) 1,1 1,1 68 k,i = i , - - - ,m ( A 2 5 ) T m ( i - 0 + j = T i , j j ( A 2 : 6 ) d r = 1 r = ,mn (A2:7) S i m i l a r l y , the displacements (k = 1 , 2 , • • • , 2 m ) in Equation 5:16 can be k expressed with the displacements (k = 1 , 2 , • • • , 2 m ) as: u k = J"" 1'* 1 (A2 :8 ) Using this re la t ionsh ip , Equation 5:17 can be wr i t ten as: I ( u l ^ l * 1 + u ? ^ ) - ( k + i ) } = T k=i K m I (u\k'^+1 + u 2 (m + i ) - ( k+ i ) ) = ] ( A 2 : 9 ) k=i 1 1 K ? ( u , k ~ m , + 1 + u^t^iJ-Ck+m)),. = T m I k=i k Therefore, i f the displacements u x (k = 1 , 2 , • • • , 2 m ) are known, the stresses T K (k = i , 2 , ' " , m ) can be obtained by so lv ing Equation A2:9. The displacement u\ can be calculated by Equation A l : 5 . F i na l l y the shear s t i f f ne s s i s obtained by Equation 5:18. Computer pro-grams for the computation of two-dimensional and three-dimensional shear s t i f f -ness of a s o i l layer is described in F ig. A3 and F ig. A4 respect ive ly. 69 MAIN P R O G R A M S U B P R O G R A M " D I S P " READ (a,c,m,n,h,E,I/) I CALL DISP z r : C A L C U L A T E z n CALCULATE K/2C CALCULATE ui.;'. ' U2V/. CALCULATE uH:i z n CALCULATE U*;$ FROM U I'.'I F IG .A3 P R O G R A M FOR C O M P U T A T I O N OF S H E A R S T I F F N E S S OF SOIL L A Y E R AND RECTANGULAR RIGID F O U N D A T I O N . 70 R E A D ( a , m , n , E , V ) C A L C U L A T E Z L Z C A L C U L A T E U ? F R O M U * ~ l ~ C A L C U L A T E C A L C U L A T E K F I G . A 4 P R O G R A M FOR C O M P U T A T I O N OF S H E A R S T I F F N E S S OF SOIL L A Y E R A N D RIGID S T R I P F O U N D A T I O N .' 71 APPENDIX 3 TWO-DIMENSIONAL STATICAL GROUND COMPLIANCE The s t a t i c compliance can be obtained from the dynamic compliance (Chapter 3.1) when New var iable n w i l l be denoted as: n = ch ( A l : l ) Expressing a, g, coshah, coshBh, sinhah and sinhsh with the new var i ab le , Maclaurins expansion for these gives: a = r ( i - n 2 E - | n 4 e 2 + ) 3 = 5 (1 - e - \e2 + ) cosh h = coshn - enn 2sinhn + ^e 2n 2n t fcoshn -2 coshgh = coshn - ensinhn + |e 2n 2coshn -(Al:2) 2 sinheh = sinhn - encoshn + ^e 2 n 2 s inhn sinhah = sinhn - £nn 2coshn + -e 2 n 2 n l | s inhn where e = \ * Q 2 (Al :3) Subst i tut ing into the appropriate equations in Chapter 3.1 and ignoring e terms of higher order when u + 0 or E ->• 0 gives: u(x,z = o,t) = 2 T h 77 )J " {(i+n 2)sinhncoshn + ( i-n 2 )n}s in(n a /h)cos(n x /h) . 2[i + ( i-n 2){( i+n 2)sinh 2n + ( i - n 2 ) n 2 } ] n 2 ° o (Al:5) Therefore, the ground compliance for a s t a t i c case with x = 0 is on oo u _ h f {(i+n2)sinhncoshn + ( i-n 2)n>sin(n a/h) • ( A I . 6 \ q ir J 2 [ ! + ( i-n 2){( i+n 2)sinh 2n + ( i - n 2 ) n 2 } ] n 2 o where q = 2at 

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