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Forced asymmetric vibrations of an axisymmetric body in contact with an elastic half-space—A Global-Local Finite Element Approach
Journal of Sound and Vibration (1987) 114(1), 45-56
FORCED ASYMMETRIC VIBRATIONS OF AN AXISYMMETRIC BODY IN C O N T A C T WITH AN ELASTIC HALF-SPACE-A GLOBAL-LOCAL FINITE ELEMENT APPROACH V. AVANESSIAN
Ohsaki Research Institute, Shimizu Construction Co., Ltd., Fukoku Seimei Building, 2-2-2 Uchisaiwaicho, Chiyoda-ku, Tokyo, 100, Japan AND S. B. DONG AND R, MUKI
Department of Civil Engineering, School of Engineering and Applied Science, University of California, Los Angeles, California 90024, U.S.A. (Received 30 January 1986) Forced asymmetric vibrations of an axisymmetric structure in contact with an elastic half-space are investigated by means of an extended version of the Global-Local Finite Element Method (GLFEM). In GLFEM, the structure and part of the surrounding medium are modeled by conventional finite elements. The behavior beyond this finite element model, i.e., in the remainder of the half-space region, is represented by analytical functions (called global functions) in the form of spherical harmonics. The solution process requires that both displacement and traction continuity at the finite element interface with the outer field be satisfied. Also, to satisfy traction-free surface conditions, the non-vanishing spherical harmonic tractions on the free surface beyond the finite element mesh are constrained in the form of weighted-average integrals that express zero net work due to these stress components. The results for rocking vibration of a rigid circular plate in frictionless contact with the half-space compare well with existing analytical data. Response results that illustrate the effects of various embedment conditions of a rigid circular plate completely bonded to the half-space are presented. Lastly, an example on flexible transducers fully bonded to a half-space is considered. Data describing the transducers' contact displacement distribution near resonance are given.
1. INTRODUCTION D y n a m i c excitation o f a b o d y in contact with a semi-infinite m e d i u m finds considerable application in m a n y fields. This p r o b l e m ' s importance and its relation to various technical fields are obvious a n d further c o m m e n t s would be redundant. In reference [1], the authors presented a m e t h o d o f analysis for forced axisymmetric vibrations of rotationally symmetric bodies in contact with a semi-infinite elastic m e d i u m based on a version o f the global-local finite element method ( G L F E M ) . The objective herein is to extend this m e t h o d to asymmetric vibrations. A review of the literature on this class of problems may be f o u n d in reference [1], so that there is no need for an extensive s u m m a r y of previous contributions here. Only s o m e relevant studies will be m e n t i o n e d to underscore the discussion. Most contributions to date have dealt with rigid circular plates in frictionless contact with an elastic half-space. U n d e r these assumptions in references [2, 3], results for both axisymmetric and a s y m m e t r i c vibrations were presented, obtained by using an integral equation formulation a n d 45 0022-460X/87/010045+ 12 $03.00/'0
9 1987 Academic Press Inc. (London) Limited
46
V. A V A N E S S I A N , S. B. D O N G A N D
R. M U K I
numerical evaluation. Recently, Krenk and Schmidt [4, 5] and Higashihara [6] provided more refined solutions ('or this problem. Wong and Luco [7, 8] considered rectangular plates with frictionless and bonded contact with the half-space by discretizing their integral equation formulation to a system of algebraic equations. Finite element methods have also been applied. The major difficulty with finite elements is a suitable representation of the outer field. To date, most attempts to meet mesh interface continuity have relied on ad hoc methods, such as viscous dampers [9], hybrid modeling [10] or infinite elements [11-13] where an explicit outer field behavior is specified at the outset. In general, these methods do not provide sufficient kinematic freedom for capturing the true radiated field. This shortcoming in the finite element method can be overcome with a GLFEM version of the analysis described in reference [1] for axisymmetric vibration. In this GLFEM version, the rotationally symmetric body and some portion of the semi-infinite medium are modeled by conventional finite elements. The outer field in which the scattered waves occur is modeled by a complete set of spherical harmonics with undetermined amplitudes (called global coefficients). A least-square enforcement of displacement and traction continuity at the finite element mesh interface and weighted-average traction-free surface conditions are used as part of the solution procedure. This technique yields the finite element displacements and an appropriate set of global coefficients. The task in this paper is the straight-forward extension of this GLFEM method to asymmetric vibrations. Examples will be given to show the versatility of this approach.
2. SOLUTION TECHNIQUE In this GLFEM version, the axisymmetric structure and a portion of the elastic isotropic half-space that it is in contact with are modeled by conventional finite elements (see Figure 1). Let cylindrical co-ordinates (R, ~b, z) be used. For analysis of asymmetric vibrations, all mechanical variables and the forcing function are resolved in terms of trigonometric series in the ~-co-ordinate. The analysis may then be carried out by considering one harmonic at a time with the complete solution given by the summation of all harmonics considered. This method is common for structures possessing rotational symmetry; see, for example, the book by Zienkiewicz [14, Chapter 15],
~ Finiteelements (interiorregion).~ ~
X ."
jTroction-free surface
phericol
/ Y~v
k,,u.,p, ~..~
I
~
ttered
field (exteriorrer i~
Z,w Figure 1. Problem illustration.
BODY/HALF-SPACE
ASYMMETRIC
47
VIBRATIONS
The displacement components {u, v, w} expressed in terms of trigonometric series have the form
U.,(R, z, t) cos rn49+ ~ Uj(R, z, t) sin j49,
u(R, 49, z, t)=
j~0
m~O
Vm(R, z, t) sin m49+ ~. Vj(R, z, t) cos j49,
v ( R , 49, z, t) =
j=0
m=O
co o~ w(R, 49, z, t)= ~. WIn(R, z, t) cos m49+ ~ Wj(R, z, t)sin j49. m ~0
(1)
j-O
The second series does not contain any new information except for the index j = O, which is for the case of axisymmetric torsional motions. In what follows, only the first series with index m will be considered. Differentiating equation (1) according to the strain-displacement relations gives the trigonometric representation of the strain field. The corresponding stress field may be obtained by means of the stress-strain relation. In the finite element program that was developed, cylindrical orthotropy was taken. Following the finite element methodology described in reference [1], one can write the equations of motion for the discretized region for the circumferential mode number m as D.(m)
Dhi(m)
D,b(m)lfu,,.,)t:fF,(.)t Dhb(m)JlUh(,,,l~
'
(2)
~Fb(m)j
where
[Db,(m)]
to,
)ll
[Dt, b(m)]J=L[Kbi(m)
[Kbb(m)]J-~
rt ,,l rM,J l [Mhb]J
and { U~(m)}, { Ub(.,)} and {Fi(.,)}, {Fb(,n)} are nodal displacements and nodal forces, with subscripts i and b representing the degrees of freedom in the interior and on the boundary of the finite element mesh, respectively. In the half-space portion, i.e., the exterior region, the waves that are radiated from the finite element mesh will be represented by outgoing spherical harmonics that satisfy the equations of motion for the entire space. The displacements for these waves may be expressed in terms of three potential functions (see the book by Pao and Mow [15], for example). In spherical co-ordinates (r, O, 49), these potential functions have the form 9 (r, 0, 49, t ) = ~" ~ A~'(t)hl,,')(kpr)P~.(cos O) cos m49, m~O
x(r,O, 49, t) = ~
n~O
~ B.m(t)h.(1)(k,r)Pnm(cos 0) cos m49,
m~O n=0
~b(r, 0, 49, t) = ~
~ D~m(t)h.Cl)(k~r)P.,, (cos 0) sin m49,
(3)
rtl ~ 0 n = 0
where h~~) is the spherical Hankel function of the first kind, P~' the associated Legendre function of the first kind of degree n and order m, and kp and ks represent longitudinal and shear wave numbers. The constants A~', B~ and D~ are the undetermined amplitudes
48
v. A V A N E S S I A N , S. B. D O N G A N D R. M U K I
of the P, S V and S H waves, respectively, and they act as the global function coefficients. The displacement field is obtained from the above potentials by u = V ~ + ksV x (e,rx) + V x V x (errS)
(4)
and the corresponding stress field by
cr~ = X&j(Uk.k) + ~ (U,.~+ U~.,),
(5)
where V is the "del" operator, er is the unit vector in the radial direction and h and are the Lam6 parameters of the exterior region. Since the general analysis is performed for one harmonic m at a time, in what follows the solution procedure is presented for a given circumferential mode m. Let {u~,'} and n,m {crsb } represent the spatial distribution of the displacements and the tractions at the boundary nodes which can be evaluated from equation (4) and equation (5), respectively. n,ra Let (F~b~} be the effective nodal forces due to {o'gb }. Traction continuity along the boundary is enforced by using {F~,"} in the right-hand side of equation (2) resulting in finite element displacements q
" { UFEb} = Z
C ,. . .{. UFEb}. ..
(6)
n~0
The effective nodal forces due to direct excitation of the plate are denoted by {FL}. The solution of equation (2) for {F L} results in {U~Eb}. Displacement continuity at the interface is expressed as L nm UvEb}+ Cnm {U~ab} = C ~ { U , bn,m }.
(7)
Rearrangement of the above equation yields
(( u ~. . } .- {.u ~. }.) c.~
=
(u~:~},
(8)
which in matrix form is [ M , ] { C " } = {B,},
(9)
where [ Mr] is a p x q matrix with p as the number of degrees of freedom on the interface, and q as the number of global functions used to represent the radiated field. The traction-free requirement on the surface of the half-space (z = 0) is met via the integral constraints imposed on the weighted-average of the tractions at 0 = 7r/2 as
fo
r
~
th) cos m~b r dr d~b = 0,
o
Iv
tl,m
r-tC~croo (r, ~ / 2 , th) cos ruth r dr d~ = 0 ,
ao
fa
r
--I~rn
n,ml
cno'o~tr,~r/2, c~)sinmdprdrddp=O,
1=0,1,2,...,
(10)
o
with ao as the finite element mesh radius, and r_,~"cos m~b~ (sin m~bJ
(11)
B O D Y / H A L F - S P A C E A S Y M M E T R I C VIBRATIONS
49
representing the weighting function. These constraints represent a weaker condition than the point-wise traction-free requirement. However, they are consistent, in formulation, with the process of enforcing traction continuity along the interface, since the consistent load vectors acting on the boundary nodes are also integrals o f weighted tractions along the mesh boundary. Equation (10) could be cast in matrix form as [M2]{C ~} ={0},
(12)
where [M2] is a ( f + 3 ) • q matrix with f as the number of integral constraints used. The procedure to determine [M2] is given in the Appendix. Combining equations (9) and (12) yields
[M]{C"} ={B}
with
and {B}={{~I} }.
(13)
(14)
The solution of equation (13) is obtained by a least square scheme yielding {C"} =
[[M]T[M]]-I[M]T{B}.
(15)
The uniqueness of the results has been discussed in reference [1]. After {C'} is determined, the displacements and the stresses in the interior and the exterior regions could be evaluated for the particular harmonic under consideration. The scattered field given in equation (3) along with the constraints given by equation (10) are capable of representing any outgoing wave in a 'dented linear elastic homogeneous half-space. The redundancy of an explicit Rayleigh wave global function was shown in reference [1]. 3, NUMERICAL EXAMPLES 3.1. FINITE E L E M E N T M O D E L I N G / G L O B A L F U N C T I O N R E P R E S E N T A T I O N In all examples presented in this paper, the finite element model used for the interior region has eight-node solid toroidal isoparametric elements based on second order Serendipity interpolations. For the outer field, the number of global functions needed to capture the scattered waves was determined by sequentially acquiring more terms in the series representation until the changes in the dominant global coefficients fell below a given tolerance. From the point of view of finite element modeling, the number of global functions should not exceed the number of degrees of freedom on the mesh boundary that interfaces with the outer field. For traction-free surface conditions, eighty (80) integral constraint equations, i.e., equation (10), were used in all examples, and this was more than sufficient. 3.2. ACCURACY OF T H E M E T H O D - - C O M P A R I S O N WITH S C H M I D T A N D K R E N K 15] The accuracy of the method for axisymmetric vibrations was demonstrated in considerable detail in reference [1], where the results were compared with those of Ka'enk and Schmidt [4]. Since the only difference is due to asymmetric motions, we anticipated a comparable level of accuracy in this case. Therefore, only a limited comparison was conducted. For comparison of G L F E M asymmetric results with those o f Schmidt and Krenk [5], two finite element models of the interior region, as shown in Figure 2, were used. The required number of global function pairs (GF) and the number of mesh degrees of freedom (D.O.F.) are indicated in this figure. To simulate the frictionless contact condition between the disk and the half-space used by Schmidt and Krenk, special thin elements
50
V. AVANESSIAN. S. B. DONG AND R. MUKI
i:ili!ifl' : !i:!!I !:i]
mmm m e m m m l mmmmmmm
I/ 111 II I
_1
// / I1 /
l/ 11 I1
I
1/ / I1 /
(a)
(b)
Figure 2. Finite element mesh layouts. (a) Mesh A, 807 D.O.F., 14GF. (b) Mesh B, 999 D.O.F., ITGF.
with virtually no shear rigidity were used to model the vanishing shear stresses o f the contact region. N o r m a l displacement amplitude at the tip of the plate due to h a r m o n i c rocking excitation, DM = [~za2~(a)/M~l, versus dimensionless wave n u m b e r Ks = aoo (p/tz)1/2 for 4~ = 0 is plotted in Figure 3. Depicted here are the results of reference [5] along with those o f the present m e t h o d obtained by employing the two mesh layouts o f Figure 2, for two inertia ratios B ' = lo/pa 5 of 1.36 and 3.26. Io is the mass m o m e n t o f inertia o f the plate a b o u t a diameter, and p and /z are the density and shear modulus o f the half-space, respectively. Poisson's ratio o f the half-space is v = 0.0. M.v is the a m p l i t u d e o f the a p p l i e d m o m e n t along the y-axis. It is evident from Figure 3 that the G L F E M results a p p r o a c h to the analytical solution by employing a more refined mesh and m o r e global functions. In fact the largest difference between the present method, with m e s h B, and the analytical solution is about 7% at the resonance frequency. Some o f the differences may be attributed to the fact that Schmidt and K_renk used a refined plate theory, w h i c h includes the effects o f transverse shear and rotatory inertia, to model the disk while in the G L F E M the plate is modeled by linear elastic elements. 3.3. RESPONSE OF CIRCULAR EMBEDDED PLATE TO investigate the effects of plate e m b e d m e n t on its mechanical response, three e m b e d ment conditions were considered as depicted in Figure 4. Mesh layout B was utilized for 4.80 r
,cop , • 3.20 E
h. . , , ~ ~
,o
/(a)
Z.40
E 1-60
Z5
0.80 0.00 0'00
I
I
0.20
0.40
I
L
I
I
I
0.60 0-80 1.00 1'20 1.40 DimensionlesswovenumberKs
~-"~
1'60
i
-
1.80 2.00
Figure 3. Normal displacement amplitude of a rigid surface frietionless circular plate subjected to rocking excitation, m = 1, D M = Iw.a2w(a)/ M,[, K, = aoo(p/o)v=; B ' = l o / p a 5, M ( t) = M.,.e -i~'. For B'= 3.26, ~ = 0-0; (a) Schmidt and Krenk; (b) GLFEM mesh B; (c) GLFEM mesh A. For B'=1.36, v=0.0: (d) Schmidt and Krenk; (e) GLFEM mesh B; (f) GLFEM mesh A.
BODY/HALF-SPACE
ASYMMETRIC
h/2~_
+
(o)
51
VIBRATIONS
{
(SP)
(b)
(PE)
(c)
(FE)
Figure 4. Foundation embedment configurations. (a) Surface plate; (b) partially embedded; (c) fully embedded.
all the cases herein, with perfect bonding between the plate and the half-space. In all cases the plate to half-space density ratio is p J p = 4 . 0 , resulting in an inertia ratio B'= Io/pa 5 of 0.63 and a mass ratio B = M/pa 3 of 2.51. Here M is the mass of the plate. Poisson's ratio of the half-space is set to 1/3 in these examples. Figure 5 shows the effects of various embedment conditions on the normal displacement amplitude, DM, due to harmonic rocking excitation, M~,, while Figure 6 depicts the same effects on the horizontal displacement amplitude, DH---I~au/H.I, due to harmonic horizontal excitation Hx. Also presented here are the coupling terms between the rocking and the horizontal excitations DM, and DHM = r a2u/ M,I, DM., being the normal displacement amplitude due to harmonic horizontal excitation, is shown along with DM, while DHM, the horizontal displacement amplitude due to rocking excitation, is shown in Figure 6. For the case or a massless foundation these two quantities are identical by virtue of a reciprocity relation. However, since the inertia term for the rocking excitation is represented by the inertia ratio B' and for horizontal excitation by the mass ratio B, their general unequalness results in coupling terms for a foundation with mass which are markedly different from each other. For both excitations the response decreases as the penetration depth increases. However, as the frequency increases the curves tend to coalesce. It is also interesting to note that the normal displacement induced by horizontal excitation is a fraction of the normal displacement produced by rocking action. However, the horizontal displacement caused by a rocking moment is comparable in magnitude to those caused by horizontal excitation in the intermediate frequency range.
~
0.48
0"40
D,w,SP
0.32 ~ 0.24 C
I
~ 0'16 0'08 0.00 0"00
DMH~, J
0"40
D~PE~ I
0"80
1'20
D~H'~p ~
1"60
f
2'00
- -
Oimenslonles5wover~l~3erKs
2"40
2"80
.20
Figure 5. Effects of embedment configurations on normal displacement amplitudes DM, DMH of a bonded rigid circular plate. B' = 0.63, u = 3t-, DM = [I.~aZw(a)/M,.I, DMH ~ I~aw(a)/H~I.
52
V. AVANESSIAN, S. B. D O N G A N D R. MUKI 0"50
0.25 :~:~0-20 ~" 0-1 5 o.,o
.
_
_
0"05
000 0.00
I
0'40
l
0-80
I ~ t --~ ' - - - - - ~ 1'20 1.60 2.00 2.40 2.80 DimensionlesswovenumberKs
3"20
Figure 6. Effects o r e m b e d m e n t configurations on horizontal displacement amplitudes rigid circular plate. B = 2.51, u = ], D n = I~au/HJ, D,,,, = I~a2u/MI.
DH, DHM of a b o n d e d
3,4. M E C H A N I C A L R E S P O N S E O F FLEXIBLE T R A N S D U C E R For calibration
of flexible transducers
mounted
on the surface of a semi-infinite
body,
general knowledge o f their mechanical response to harmonic forced oscillation is required. In particular, the contact displacement distribution near a resonance frequency is of some interest. To shed some light on this interaction problem, two ring-shaped disks, :Jne solid and one with a circular hole, were selected to represent the transducers which are perfectly bonded to the half-space. Geometry and mesh configuration for each element are s h o w n in Figure 7. T o insure adequate representation of contact stress distributions more elements are used to discretize the flexible plate than in the case of the rigid disk o f Figure 2. Mechanical properties o f both the plate and the half-space are given in Table 1. The plate properties correspond to a PZT-5A ceramic transducer, and the half-space's to steel. These geometrical dimensions and mechanical properties are those from Ohtsu and O n o [16]. Figure 8 shows the vertical displacement amplitude distribution Do = I~w/qoal. H e r e qo is the amplitude o f the vertical uniform pressure applied to the surface o f the transducer.
,4
0
,
11
~3 (o)
NiHiii
i
~-) (b)
Figure 7. Transducer geometries and mesh layouts. (a) Solid disk, 1119 D.O.F., 16 GF, a = 4,76 ram, h = 1.45 ram; (b) hollow disk, 1107 D.O.F., 16 GF, a = 4.76 ram, b = 2-38 mm, h = 1.45 ram.
53
BODY/HALF-SPACE ASYMMETRIC VIBRATIONS TABLE 1
Mechanical properties of the plate and the half-space
Plate Half-space
Shear modulus ( N / m 2)
Poisson's ratio
Density (kg/m ~)
/zp = 2.29 x 10 I~ /z = 7.76 x 10 I~
up = 1/3 ~, = 1/3
pp = 7.75 x 10 3 p = 7.80 x 103
KS,,1.2 i 0"671"~ 0.53
aisk rrs,O. 5
0.39 0'25 0.00
KS= 0 ~ 3 ~ ~ , ~
Hollowdi I 0'20
Ks'I"Z. I ~ 0:60 0 8 0
I 0'40
N I 1.00
x/e
Figure 8, Vertical displacement amplitude of flexible disks induced by vertical uniform pressure q.(x, t)= qo e - i ' , Do= Ijzw/ qoal.
The overall largest d i s p l a c e m e n t for the solid disk occurs at its center at K, = 1.2. H o w e v e r , the tip d i s p l a c e m e n t reaches its largest value at m u c h lower frequency o f K, = 0.3, and decreases as the frequency increases. For the h o l l o w disk the overall largest d i s p l a c e m e n t is at x / a = 0.65 at K, = 0.3, although the displacement at the inner edge reaches its p e a k at K., = 1.2. As is evident f r o m Figure 8, the magnitudes of these two displacements are almost identical.
2 50 2.20
So~iddlsk
~5 1 50
1.0{7
0"0
r
0.20
1.
0"40
I
x/O
0 60
1 ,
0 80
1.00
Figure 9. Horizontal displacement amplitude of fl exible disks at ~ = 0 induced by horizontal uniform pressure
p, (X, l) "~"pIJe-i~t. .OH = [~j.lt/p(iaL
54
v. A V A N E S S 1 A N ,
S. B. D O N G
AND
R. M L I K I
The h o r i z o n t a l d i s p l a c e m e n t a m p l i t u d e d i s t r i b u t i o n D H = [#u/poa[, i n d u c e d by a harm o n i c u n i f o r m h o r i z o n t a l pressure o f a m p l i t u d e Po, is shown in Figure 9 for q5 = 0. The solid disk has a p e a k d i s p l a c e m e n t at the center at K~ = 1.8, while the hollow disk reaches its m a x i m u m d i s p l a c e m e n t a r o u n d x / a = 0 . 6 2 at K, = 1 . 9 . In the case of h o r i z o n t a l excitation, u n l i k e its vertical c o u n t e r p a r t , the m a x i m u m d i s p l a c e m e n t at different radii occurs a r o u n d the s a m e frequency.
4. CONCLUDING REMARKS A v e r s i o n of the global-local finite e l e m e n t m e t h o d was a p p l i e d to the p r o b l e m o f forced a s y m m e t r i c v i b r a t i o n s of a n a x i s y m m e t r i c structure in c o n t a c t with an elastic half-space. The m e t h o d is a n extension of that used previously for a x i s y m m e t r i c vibrations I1]. H e r e i n , the details for the analysis o f a s y m m e t r i c v i b r a t i o n s are provided. With the set of global f u n c t i o n s for the radiated field, it s h o u l d be possible to represent a n y outgoing wave exiting from the finite element mesh to a n y a c c u r a c y c o m m e n s u r a t e with the n u m b e r o f terms used. E x a m p l e s are presented that d e m o n s t r a t e the capability a n d accuracy of this a p p r o a c h a n d illustrate the physical b e h a v i o r in some p r o b l e m s .
REFERENCES 1. V. AVANESSIAN, R. MUKI and S. B. DONG 1986 Journal of Sound and Vibrt~tion 104, 449-463. Forced oscillations of an axisymmetric structure in contact with an elastic half-space by a version of global local finite elements. 2. J. E. L o c o and R. A. WESTMANN 1971 Journal of the Engineering Mechanics Dioision, American Society of Civil Engineers 97, 1381-1395. Dynamic response of circular footings. 3. A. S. VELETSOS and Y. T. WEt 1971 Journal of Soil Mechanics and Foundations Division, American Society of Civil Engineers 97, 1227-1248. Lateral and rocking vibrations of footings. 4. S. KRENK and H. SCHMIDT 1981 Journal of Applied Mechanics, American Society of Mechanical Engineers 48, 161-168. Vibration of an elastic circular plate on an elastic half space--a direct approach. 5. H. SCHMIDT and S. KRENK 1982 International Journal of Solids and Structures 18, 91-105. Asymmetric Vibrations of a circular elastic plate on an elastic half space. 6. H. HIGASHIHARA 1984 Journal of the Engineering Mechanics Division, American Society of Civil Engineers 110, 1510-1523. Explicit Green's function approach to forced vertical vibrations of circular disk on semi-infinite elastic space. 7. H. L. WONG and J. E. LUCO 1976 International Journal of Earthquake Engineering and Structural Dynamics 4, 579-587. Dynamic response of rigid foundations of arbitrary shape. 8. H. L. WONG and J. E. L u c o 1978 International Journal of Earthquake Engineering and Structural Dynamics 6, 3-16. Dynamic response of rectangular foundations to obliquely incident seismic waves. 9. C. M. URLICH and R. L. KUHLEMEYER 1973 Canadian Geotechnical Journal 10, 145-160. Coupled rocking and lateral vibrations of embedded footings. 10. S. GUPTA, J. PENZIEN, T. W. LIN and C. S. YEH 1982 International Journal of Earthquake Engineering and Structural Dynamics 10, 69-87. Three-dimensional hybrid modeling of soilstructure interaction. 11. Y. K. CHOW and I. M. SMITH 1981 International Journal for Numerical Methods in Engineering 17, 503-526. Static and periodic infinite solid elements. 12. F. MEDtNA and R. L. TAYLOR 1983 International Journal for NumericaI Methods in Engineering 19, 1209-1226. Finite element techniques for problems of unbounded domains. 13. F. M ED I NA and J. PE Nm EN 1982 International Journal of Earthquake Engineering and Structural Dynamics 10, 699-709. Infinite elements for elastodynamics. 14. O. C. ZIENKIEW1CZ 1977 /'he Finite Element Method. London: McGraw-Hill, third edition. 15. Y. H. PAO and C. C. M o w 1973 Diffraction of Elastic Waves and Dynamic Stress Concentration. New York: Crane and Russak. 16. M. OHrSU and K. ONO 1983 Journal of Acoustical Emission 2, 247-260. Resonance analysis of piezoelectric transducer elements.
BODY/HALF-SPACE
ASYMMETRIC
55
VIBRATIONS
APPENDIX The expressions for tractions in equation (10) have been given in reference [15] and will not be repeated here. Substitution of these expressions reduces equation (10) into linear combinations of the integrals
where ki assumes either of k, or k~, n = 0, 1, 2 . . . . and N = - 1 , 0, 1 , . . . . The circumferential dependence could be eliminated through direct integration of equations (AI). All the integrals with radial dependence exist e• for N = - t . These correspond to I = 0 in equation (10), which represent the total forces due to tractions on the 0 = ~ / 2 plane. I~ I could be made convergent by extracting the leading term of the asymptotic expression of the spherical Hankel function
h~')(z)=(-i)"+'ei"/z[l+O(1/z)],
z--,oo
(A2)
from the integrand of I~-~. This introduces an additional set of constraints given by -1). ,+, A,,,, P,,,. (0) = O,
( _ i ) , + , B , , dP,7(O)/d 0 = O,
- ( - , ) " "+' D . ~ P,,"(O) = O,
(A3) which should be enforced along with constraints given in equation (10). Through suitable change of variables, I~ 1 and lff become
Y2'(Kfl =
L,, (Kj) =
fo
[zh(,,'l(z) - (~i) ~§ e ~] dz,
(A4)
Ki
J,,N(Kj) =
Kj
z-Nh~)(z)dz,
n,N=O, 1,2,...,
(A5)
with Ks = kjao. These integrals could be evaluated by using the following recursion formulas. No~e that J o~
=
~/2-Si
(K~) + i Ci (Kj),
(A6)
where Si and Ci are the sine and cosine integrals, respectively. Then
JY(Kj)=(i/N)[J~'-~(Kj)-KfN eiK,],
J~(Kf)=(n-N-])JN,_-tI(Kj)+ KfNh('I(Kj). (A7)
The equivalent expressions for L,(Kj) are
Lo(K~)=O,
L~(K~)=nJ~ ~(K~)+Kjh~'(Kj)-(-i)"e~G
(A8)
The constraints presented in equation (10) written in terms of Jff(Kfl, N ---0, 1, 2 . . . . . for a longitudinal wave, are given by q 1
Y~ [ ( n tl~(J
N - ,,, d P ,m, ( 0 ) / d 0 = 0 , )J,,N § ( K p)-d,,+t(Ko)]A,
q
~. [ - n 2 J f f + ' ( K p ) + ( 1 - O ' 5 K ~ /
" ,~. '(Kp)-JN+,(K.)+m2J;7*'(Kp)]A.,p.~,(O)=O, 9 K'p)J.
n=O
q
.=o[-m
d~ (Kp)]A,,
dP.~(0)/d9=0,
(A9)
56
V. A V A N E S S I A N ,
S. B. D O N G
AND
R. M U K I
for an in-plane shear wave by q
E [-(n 2- I>J~+'(K~)- 0.5J~-t(K.,)-J,,N+t(K~)]B,7 dP~(O)/dO =0, rl=0 q
Y. [ n ( n 2 + n ) S y + ' ( K . , . ) - ( n 2 + n ) J L , ( K s ) tl=0
- m2(n+l)jN~,(K,)+
m 2J ,N, + , ( K ~ ) ] B , m P,, ( 0 ) = 0 ,
q
E
m[(.+ I)Sy*,(K,)_
~ ., dP:(0)/d0 =0, Jn+l(Ks)]Bn
(AIO)
rJ = 0
and for an antiplane shear wave by q
y m[ (n - 1)J,N(K~) - JN,.~'( K.,.) ]D,TPT( O) = O, ricO
q
y. m[J~ (K,)]D,, dP, (0)/d0 =0, -
N
m
m
rl~0 q
[0"5(n2 + n ) J ~ ( K ~ ) - m z J ~ ( K . , ) ] D , ~ , P T ( O ) = 0.
(A11)
n=0
In the above expressions L , , ( K s) should replace J ,N (Ks), corresponding to 1 = 0 in equation (10), with the additional constraints given by equation (A3).
FORCED ASYMMETRIC VIBRATIONS OF AN AXISYMMETRIC BODY IN C O N T A C T WITH AN ELASTIC HALF-SPACE-A GLOBAL-LOCAL FINITE ELEMENT APPROACH V. AVANESSIAN
Ohsaki Research Institute, Shimizu Construction Co., Ltd., Fukoku Seimei Building, 2-2-2 Uchisaiwaicho, Chiyoda-ku, Tokyo, 100, Japan AND S. B. DONG AND R, MUKI
Department of Civil Engineering, School of Engineering and Applied Science, University of California, Los Angeles, California 90024, U.S.A. (Received 30 January 1986) Forced asymmetric vibrations of an axisymmetric structure in contact with an elastic half-space are investigated by means of an extended version of the Global-Local Finite Element Method (GLFEM). In GLFEM, the structure and part of the surrounding medium are modeled by conventional finite elements. The behavior beyond this finite element model, i.e., in the remainder of the half-space region, is represented by analytical functions (called global functions) in the form of spherical harmonics. The solution process requires that both displacement and traction continuity at the finite element interface with the outer field be satisfied. Also, to satisfy traction-free surface conditions, the non-vanishing spherical harmonic tractions on the free surface beyond the finite element mesh are constrained in the form of weighted-average integrals that express zero net work due to these stress components. The results for rocking vibration of a rigid circular plate in frictionless contact with the half-space compare well with existing analytical data. Response results that illustrate the effects of various embedment conditions of a rigid circular plate completely bonded to the half-space are presented. Lastly, an example on flexible transducers fully bonded to a half-space is considered. Data describing the transducers' contact displacement distribution near resonance are given.
1. INTRODUCTION D y n a m i c excitation o f a b o d y in contact with a semi-infinite m e d i u m finds considerable application in m a n y fields. This p r o b l e m ' s importance and its relation to various technical fields are obvious a n d further c o m m e n t s would be redundant. In reference [1], the authors presented a m e t h o d o f analysis for forced axisymmetric vibrations of rotationally symmetric bodies in contact with a semi-infinite elastic m e d i u m based on a version o f the global-local finite element method ( G L F E M ) . The objective herein is to extend this m e t h o d to asymmetric vibrations. A review of the literature on this class of problems may be f o u n d in reference [1], so that there is no need for an extensive s u m m a r y of previous contributions here. Only s o m e relevant studies will be m e n t i o n e d to underscore the discussion. Most contributions to date have dealt with rigid circular plates in frictionless contact with an elastic half-space. U n d e r these assumptions in references [2, 3], results for both axisymmetric and a s y m m e t r i c vibrations were presented, obtained by using an integral equation formulation a n d 45 0022-460X/87/010045+ 12 $03.00/'0
9 1987 Academic Press Inc. (London) Limited
46
V. A V A N E S S I A N , S. B. D O N G A N D
R. M U K I
numerical evaluation. Recently, Krenk and Schmidt [4, 5] and Higashihara [6] provided more refined solutions ('or this problem. Wong and Luco [7, 8] considered rectangular plates with frictionless and bonded contact with the half-space by discretizing their integral equation formulation to a system of algebraic equations. Finite element methods have also been applied. The major difficulty with finite elements is a suitable representation of the outer field. To date, most attempts to meet mesh interface continuity have relied on ad hoc methods, such as viscous dampers [9], hybrid modeling [10] or infinite elements [11-13] where an explicit outer field behavior is specified at the outset. In general, these methods do not provide sufficient kinematic freedom for capturing the true radiated field. This shortcoming in the finite element method can be overcome with a GLFEM version of the analysis described in reference [1] for axisymmetric vibration. In this GLFEM version, the rotationally symmetric body and some portion of the semi-infinite medium are modeled by conventional finite elements. The outer field in which the scattered waves occur is modeled by a complete set of spherical harmonics with undetermined amplitudes (called global coefficients). A least-square enforcement of displacement and traction continuity at the finite element mesh interface and weighted-average traction-free surface conditions are used as part of the solution procedure. This technique yields the finite element displacements and an appropriate set of global coefficients. The task in this paper is the straight-forward extension of this GLFEM method to asymmetric vibrations. Examples will be given to show the versatility of this approach.
2. SOLUTION TECHNIQUE In this GLFEM version, the axisymmetric structure and a portion of the elastic isotropic half-space that it is in contact with are modeled by conventional finite elements (see Figure 1). Let cylindrical co-ordinates (R, ~b, z) be used. For analysis of asymmetric vibrations, all mechanical variables and the forcing function are resolved in terms of trigonometric series in the ~-co-ordinate. The analysis may then be carried out by considering one harmonic at a time with the complete solution given by the summation of all harmonics considered. This method is common for structures possessing rotational symmetry; see, for example, the book by Zienkiewicz [14, Chapter 15],
~ Finiteelements (interiorregion).~ ~
X ."
jTroction-free surface
phericol
/ Y~v
k,,u.,p, ~..~
I
~
ttered
field (exteriorrer i~
Z,w Figure 1. Problem illustration.
BODY/HALF-SPACE
ASYMMETRIC
47
VIBRATIONS
The displacement components {u, v, w} expressed in terms of trigonometric series have the form
U.,(R, z, t) cos rn49+ ~ Uj(R, z, t) sin j49,
u(R, 49, z, t)=
j~0
m~O
Vm(R, z, t) sin m49+ ~. Vj(R, z, t) cos j49,
v ( R , 49, z, t) =
j=0
m=O
co o~ w(R, 49, z, t)= ~. WIn(R, z, t) cos m49+ ~ Wj(R, z, t)sin j49. m ~0
(1)
j-O
The second series does not contain any new information except for the index j = O, which is for the case of axisymmetric torsional motions. In what follows, only the first series with index m will be considered. Differentiating equation (1) according to the strain-displacement relations gives the trigonometric representation of the strain field. The corresponding stress field may be obtained by means of the stress-strain relation. In the finite element program that was developed, cylindrical orthotropy was taken. Following the finite element methodology described in reference [1], one can write the equations of motion for the discretized region for the circumferential mode number m as D.(m)
Dhi(m)
D,b(m)lfu,,.,)t:fF,(.)t Dhb(m)JlUh(,,,l~
'
(2)
~Fb(m)j
where
[Db,(m)]
to,
)ll
[Dt, b(m)]J=L[Kbi(m)
[Kbb(m)]J-~
rt ,,l rM,J l [Mhb]J
and { U~(m)}, { Ub(.,)} and {Fi(.,)}, {Fb(,n)} are nodal displacements and nodal forces, with subscripts i and b representing the degrees of freedom in the interior and on the boundary of the finite element mesh, respectively. In the half-space portion, i.e., the exterior region, the waves that are radiated from the finite element mesh will be represented by outgoing spherical harmonics that satisfy the equations of motion for the entire space. The displacements for these waves may be expressed in terms of three potential functions (see the book by Pao and Mow [15], for example). In spherical co-ordinates (r, O, 49), these potential functions have the form 9 (r, 0, 49, t ) = ~" ~ A~'(t)hl,,')(kpr)P~.(cos O) cos m49, m~O
x(r,O, 49, t) = ~
n~O
~ B.m(t)h.(1)(k,r)Pnm(cos 0) cos m49,
m~O n=0
~b(r, 0, 49, t) = ~
~ D~m(t)h.Cl)(k~r)P.,, (cos 0) sin m49,
(3)
rtl ~ 0 n = 0
where h~~) is the spherical Hankel function of the first kind, P~' the associated Legendre function of the first kind of degree n and order m, and kp and ks represent longitudinal and shear wave numbers. The constants A~', B~ and D~ are the undetermined amplitudes
48
v. A V A N E S S I A N , S. B. D O N G A N D R. M U K I
of the P, S V and S H waves, respectively, and they act as the global function coefficients. The displacement field is obtained from the above potentials by u = V ~ + ksV x (e,rx) + V x V x (errS)
(4)
and the corresponding stress field by
cr~ = X&j(Uk.k) + ~ (U,.~+ U~.,),
(5)
where V is the "del" operator, er is the unit vector in the radial direction and h and are the Lam6 parameters of the exterior region. Since the general analysis is performed for one harmonic m at a time, in what follows the solution procedure is presented for a given circumferential mode m. Let {u~,'} and n,m {crsb } represent the spatial distribution of the displacements and the tractions at the boundary nodes which can be evaluated from equation (4) and equation (5), respectively. n,ra Let (F~b~} be the effective nodal forces due to {o'gb }. Traction continuity along the boundary is enforced by using {F~,"} in the right-hand side of equation (2) resulting in finite element displacements q
" { UFEb} = Z
C ,. . .{. UFEb}. ..
(6)
n~0
The effective nodal forces due to direct excitation of the plate are denoted by {FL}. The solution of equation (2) for {F L} results in {U~Eb}. Displacement continuity at the interface is expressed as L nm UvEb}+ Cnm {U~ab} = C ~ { U , bn,m }.
(7)
Rearrangement of the above equation yields
(( u ~. . } .- {.u ~. }.) c.~
=
(u~:~},
(8)
which in matrix form is [ M , ] { C " } = {B,},
(9)
where [ Mr] is a p x q matrix with p as the number of degrees of freedom on the interface, and q as the number of global functions used to represent the radiated field. The traction-free requirement on the surface of the half-space (z = 0) is met via the integral constraints imposed on the weighted-average of the tractions at 0 = 7r/2 as
fo
r
~
th) cos m~b r dr d~b = 0,
o
Iv
tl,m
r-tC~croo (r, ~ / 2 , th) cos ruth r dr d~ = 0 ,
ao
fa
r
--I~rn
n,ml
cno'o~tr,~r/2, c~)sinmdprdrddp=O,
1=0,1,2,...,
(10)
o
with ao as the finite element mesh radius, and r_,~"cos m~b~ (sin m~bJ
(11)
B O D Y / H A L F - S P A C E A S Y M M E T R I C VIBRATIONS
49
representing the weighting function. These constraints represent a weaker condition than the point-wise traction-free requirement. However, they are consistent, in formulation, with the process of enforcing traction continuity along the interface, since the consistent load vectors acting on the boundary nodes are also integrals o f weighted tractions along the mesh boundary. Equation (10) could be cast in matrix form as [M2]{C ~} ={0},
(12)
where [M2] is a ( f + 3 ) • q matrix with f as the number of integral constraints used. The procedure to determine [M2] is given in the Appendix. Combining equations (9) and (12) yields
[M]{C"} ={B}
with
and {B}={{~I} }.
(13)
(14)
The solution of equation (13) is obtained by a least square scheme yielding {C"} =
[[M]T[M]]-I[M]T{B}.
(15)
The uniqueness of the results has been discussed in reference [1]. After {C'} is determined, the displacements and the stresses in the interior and the exterior regions could be evaluated for the particular harmonic under consideration. The scattered field given in equation (3) along with the constraints given by equation (10) are capable of representing any outgoing wave in a 'dented linear elastic homogeneous half-space. The redundancy of an explicit Rayleigh wave global function was shown in reference [1]. 3, NUMERICAL EXAMPLES 3.1. FINITE E L E M E N T M O D E L I N G / G L O B A L F U N C T I O N R E P R E S E N T A T I O N In all examples presented in this paper, the finite element model used for the interior region has eight-node solid toroidal isoparametric elements based on second order Serendipity interpolations. For the outer field, the number of global functions needed to capture the scattered waves was determined by sequentially acquiring more terms in the series representation until the changes in the dominant global coefficients fell below a given tolerance. From the point of view of finite element modeling, the number of global functions should not exceed the number of degrees of freedom on the mesh boundary that interfaces with the outer field. For traction-free surface conditions, eighty (80) integral constraint equations, i.e., equation (10), were used in all examples, and this was more than sufficient. 3.2. ACCURACY OF T H E M E T H O D - - C O M P A R I S O N WITH S C H M I D T A N D K R E N K 15] The accuracy of the method for axisymmetric vibrations was demonstrated in considerable detail in reference [1], where the results were compared with those of Ka'enk and Schmidt [4]. Since the only difference is due to asymmetric motions, we anticipated a comparable level of accuracy in this case. Therefore, only a limited comparison was conducted. For comparison of G L F E M asymmetric results with those o f Schmidt and Krenk [5], two finite element models of the interior region, as shown in Figure 2, were used. The required number of global function pairs (GF) and the number of mesh degrees of freedom (D.O.F.) are indicated in this figure. To simulate the frictionless contact condition between the disk and the half-space used by Schmidt and Krenk, special thin elements
50
V. AVANESSIAN. S. B. DONG AND R. MUKI
i:ili!ifl' : !i:!!I !:i]
mmm m e m m m l mmmmmmm
I/ 111 II I
_1
// / I1 /
l/ 11 I1
I
1/ / I1 /
(a)
(b)
Figure 2. Finite element mesh layouts. (a) Mesh A, 807 D.O.F., 14GF. (b) Mesh B, 999 D.O.F., ITGF.
with virtually no shear rigidity were used to model the vanishing shear stresses o f the contact region. N o r m a l displacement amplitude at the tip of the plate due to h a r m o n i c rocking excitation, DM = [~za2~(a)/M~l, versus dimensionless wave n u m b e r Ks = aoo (p/tz)1/2 for 4~ = 0 is plotted in Figure 3. Depicted here are the results of reference [5] along with those o f the present m e t h o d obtained by employing the two mesh layouts o f Figure 2, for two inertia ratios B ' = lo/pa 5 of 1.36 and 3.26. Io is the mass m o m e n t o f inertia o f the plate a b o u t a diameter, and p and /z are the density and shear modulus o f the half-space, respectively. Poisson's ratio o f the half-space is v = 0.0. M.v is the a m p l i t u d e o f the a p p l i e d m o m e n t along the y-axis. It is evident from Figure 3 that the G L F E M results a p p r o a c h to the analytical solution by employing a more refined mesh and m o r e global functions. In fact the largest difference between the present method, with m e s h B, and the analytical solution is about 7% at the resonance frequency. Some o f the differences may be attributed to the fact that Schmidt and K_renk used a refined plate theory, w h i c h includes the effects o f transverse shear and rotatory inertia, to model the disk while in the G L F E M the plate is modeled by linear elastic elements. 3.3. RESPONSE OF CIRCULAR EMBEDDED PLATE TO investigate the effects of plate e m b e d m e n t on its mechanical response, three e m b e d ment conditions were considered as depicted in Figure 4. Mesh layout B was utilized for 4.80 r
,cop , • 3.20 E
h. . , , ~ ~
,o
/(a)
Z.40
E 1-60
Z5
0.80 0.00 0'00
I
I
0.20
0.40
I
L
I
I
I
0.60 0-80 1.00 1'20 1.40 DimensionlesswovenumberKs
~-"~
1'60
i
-
1.80 2.00
Figure 3. Normal displacement amplitude of a rigid surface frietionless circular plate subjected to rocking excitation, m = 1, D M = Iw.a2w(a)/ M,[, K, = aoo(p/o)v=; B ' = l o / p a 5, M ( t) = M.,.e -i~'. For B'= 3.26, ~ = 0-0; (a) Schmidt and Krenk; (b) GLFEM mesh B; (c) GLFEM mesh A. For B'=1.36, v=0.0: (d) Schmidt and Krenk; (e) GLFEM mesh B; (f) GLFEM mesh A.
BODY/HALF-SPACE
ASYMMETRIC
h/2~_
+
(o)
51
VIBRATIONS
{
(SP)
(b)
(PE)
(c)
(FE)
Figure 4. Foundation embedment configurations. (a) Surface plate; (b) partially embedded; (c) fully embedded.
all the cases herein, with perfect bonding between the plate and the half-space. In all cases the plate to half-space density ratio is p J p = 4 . 0 , resulting in an inertia ratio B'= Io/pa 5 of 0.63 and a mass ratio B = M/pa 3 of 2.51. Here M is the mass of the plate. Poisson's ratio of the half-space is set to 1/3 in these examples. Figure 5 shows the effects of various embedment conditions on the normal displacement amplitude, DM, due to harmonic rocking excitation, M~,, while Figure 6 depicts the same effects on the horizontal displacement amplitude, DH---I~au/H.I, due to harmonic horizontal excitation Hx. Also presented here are the coupling terms between the rocking and the horizontal excitations DM, and DHM = r a2u/ M,I, DM., being the normal displacement amplitude due to harmonic horizontal excitation, is shown along with DM, while DHM, the horizontal displacement amplitude due to rocking excitation, is shown in Figure 6. For the case or a massless foundation these two quantities are identical by virtue of a reciprocity relation. However, since the inertia term for the rocking excitation is represented by the inertia ratio B' and for horizontal excitation by the mass ratio B, their general unequalness results in coupling terms for a foundation with mass which are markedly different from each other. For both excitations the response decreases as the penetration depth increases. However, as the frequency increases the curves tend to coalesce. It is also interesting to note that the normal displacement induced by horizontal excitation is a fraction of the normal displacement produced by rocking action. However, the horizontal displacement caused by a rocking moment is comparable in magnitude to those caused by horizontal excitation in the intermediate frequency range.
~
0.48
0"40
D,w,SP
0.32 ~ 0.24 C
I
~ 0'16 0'08 0.00 0"00
DMH~, J
0"40
D~PE~ I
0"80
1'20
D~H'~p ~
1"60
f
2'00
- -
Oimenslonles5wover~l~3erKs
2"40
2"80
.20
Figure 5. Effects of embedment configurations on normal displacement amplitudes DM, DMH of a bonded rigid circular plate. B' = 0.63, u = 3t-, DM = [I.~aZw(a)/M,.I, DMH ~ I~aw(a)/H~I.
52
V. AVANESSIAN, S. B. D O N G A N D R. MUKI 0"50
0.25 :~:~0-20 ~" 0-1 5 o.,o
.
_
_
0"05
000 0.00
I
0'40
l
0-80
I ~ t --~ ' - - - - - ~ 1'20 1.60 2.00 2.40 2.80 DimensionlesswovenumberKs
3"20
Figure 6. Effects o r e m b e d m e n t configurations on horizontal displacement amplitudes rigid circular plate. B = 2.51, u = ], D n = I~au/HJ, D,,,, = I~a2u/MI.
DH, DHM of a b o n d e d
3,4. M E C H A N I C A L R E S P O N S E O F FLEXIBLE T R A N S D U C E R For calibration
of flexible transducers
mounted
on the surface of a semi-infinite
body,
general knowledge o f their mechanical response to harmonic forced oscillation is required. In particular, the contact displacement distribution near a resonance frequency is of some interest. To shed some light on this interaction problem, two ring-shaped disks, :Jne solid and one with a circular hole, were selected to represent the transducers which are perfectly bonded to the half-space. Geometry and mesh configuration for each element are s h o w n in Figure 7. T o insure adequate representation of contact stress distributions more elements are used to discretize the flexible plate than in the case of the rigid disk o f Figure 2. Mechanical properties o f both the plate and the half-space are given in Table 1. The plate properties correspond to a PZT-5A ceramic transducer, and the half-space's to steel. These geometrical dimensions and mechanical properties are those from Ohtsu and O n o [16]. Figure 8 shows the vertical displacement amplitude distribution Do = I~w/qoal. H e r e qo is the amplitude o f the vertical uniform pressure applied to the surface o f the transducer.
,4
0
,
11
~3 (o)
NiHiii
i
~-) (b)
Figure 7. Transducer geometries and mesh layouts. (a) Solid disk, 1119 D.O.F., 16 GF, a = 4,76 ram, h = 1.45 ram; (b) hollow disk, 1107 D.O.F., 16 GF, a = 4.76 ram, b = 2-38 mm, h = 1.45 ram.
53
BODY/HALF-SPACE ASYMMETRIC VIBRATIONS TABLE 1
Mechanical properties of the plate and the half-space
Plate Half-space
Shear modulus ( N / m 2)
Poisson's ratio
Density (kg/m ~)
/zp = 2.29 x 10 I~ /z = 7.76 x 10 I~
up = 1/3 ~, = 1/3
pp = 7.75 x 10 3 p = 7.80 x 103
KS,,1.2 i 0"671"~ 0.53
aisk rrs,O. 5
0.39 0'25 0.00
KS= 0 ~ 3 ~ ~ , ~
Hollowdi I 0'20
Ks'I"Z. I ~ 0:60 0 8 0
I 0'40
N I 1.00
x/e
Figure 8, Vertical displacement amplitude of flexible disks induced by vertical uniform pressure q.(x, t)= qo e - i ' , Do= Ijzw/ qoal.
The overall largest d i s p l a c e m e n t for the solid disk occurs at its center at K, = 1.2. H o w e v e r , the tip d i s p l a c e m e n t reaches its largest value at m u c h lower frequency o f K, = 0.3, and decreases as the frequency increases. For the h o l l o w disk the overall largest d i s p l a c e m e n t is at x / a = 0.65 at K, = 0.3, although the displacement at the inner edge reaches its p e a k at K., = 1.2. As is evident f r o m Figure 8, the magnitudes of these two displacements are almost identical.
2 50 2.20
So~iddlsk
~5 1 50
1.0{7
0"0
r
0.20
1.
0"40
I
x/O
0 60
1 ,
0 80
1.00
Figure 9. Horizontal displacement amplitude of fl exible disks at ~ = 0 induced by horizontal uniform pressure
p, (X, l) "~"pIJe-i~t. .OH = [~j.lt/p(iaL
54
v. A V A N E S S 1 A N ,
S. B. D O N G
AND
R. M L I K I
The h o r i z o n t a l d i s p l a c e m e n t a m p l i t u d e d i s t r i b u t i o n D H = [#u/poa[, i n d u c e d by a harm o n i c u n i f o r m h o r i z o n t a l pressure o f a m p l i t u d e Po, is shown in Figure 9 for q5 = 0. The solid disk has a p e a k d i s p l a c e m e n t at the center at K~ = 1.8, while the hollow disk reaches its m a x i m u m d i s p l a c e m e n t a r o u n d x / a = 0 . 6 2 at K, = 1 . 9 . In the case of h o r i z o n t a l excitation, u n l i k e its vertical c o u n t e r p a r t , the m a x i m u m d i s p l a c e m e n t at different radii occurs a r o u n d the s a m e frequency.
4. CONCLUDING REMARKS A v e r s i o n of the global-local finite e l e m e n t m e t h o d was a p p l i e d to the p r o b l e m o f forced a s y m m e t r i c v i b r a t i o n s of a n a x i s y m m e t r i c structure in c o n t a c t with an elastic half-space. The m e t h o d is a n extension of that used previously for a x i s y m m e t r i c vibrations I1]. H e r e i n , the details for the analysis o f a s y m m e t r i c v i b r a t i o n s are provided. With the set of global f u n c t i o n s for the radiated field, it s h o u l d be possible to represent a n y outgoing wave exiting from the finite element mesh to a n y a c c u r a c y c o m m e n s u r a t e with the n u m b e r o f terms used. E x a m p l e s are presented that d e m o n s t r a t e the capability a n d accuracy of this a p p r o a c h a n d illustrate the physical b e h a v i o r in some p r o b l e m s .
REFERENCES 1. V. AVANESSIAN, R. MUKI and S. B. DONG 1986 Journal of Sound and Vibrt~tion 104, 449-463. Forced oscillations of an axisymmetric structure in contact with an elastic half-space by a version of global local finite elements. 2. J. E. L o c o and R. A. WESTMANN 1971 Journal of the Engineering Mechanics Dioision, American Society of Civil Engineers 97, 1381-1395. Dynamic response of circular footings. 3. A. S. VELETSOS and Y. T. WEt 1971 Journal of Soil Mechanics and Foundations Division, American Society of Civil Engineers 97, 1227-1248. Lateral and rocking vibrations of footings. 4. S. KRENK and H. SCHMIDT 1981 Journal of Applied Mechanics, American Society of Mechanical Engineers 48, 161-168. Vibration of an elastic circular plate on an elastic half space--a direct approach. 5. H. SCHMIDT and S. KRENK 1982 International Journal of Solids and Structures 18, 91-105. Asymmetric Vibrations of a circular elastic plate on an elastic half space. 6. H. HIGASHIHARA 1984 Journal of the Engineering Mechanics Division, American Society of Civil Engineers 110, 1510-1523. Explicit Green's function approach to forced vertical vibrations of circular disk on semi-infinite elastic space. 7. H. L. WONG and J. E. LUCO 1976 International Journal of Earthquake Engineering and Structural Dynamics 4, 579-587. Dynamic response of rigid foundations of arbitrary shape. 8. H. L. WONG and J. E. L u c o 1978 International Journal of Earthquake Engineering and Structural Dynamics 6, 3-16. Dynamic response of rectangular foundations to obliquely incident seismic waves. 9. C. M. URLICH and R. L. KUHLEMEYER 1973 Canadian Geotechnical Journal 10, 145-160. Coupled rocking and lateral vibrations of embedded footings. 10. S. GUPTA, J. PENZIEN, T. W. LIN and C. S. YEH 1982 International Journal of Earthquake Engineering and Structural Dynamics 10, 69-87. Three-dimensional hybrid modeling of soilstructure interaction. 11. Y. K. CHOW and I. M. SMITH 1981 International Journal for Numerical Methods in Engineering 17, 503-526. Static and periodic infinite solid elements. 12. F. MEDtNA and R. L. TAYLOR 1983 International Journal for NumericaI Methods in Engineering 19, 1209-1226. Finite element techniques for problems of unbounded domains. 13. F. M ED I NA and J. PE Nm EN 1982 International Journal of Earthquake Engineering and Structural Dynamics 10, 699-709. Infinite elements for elastodynamics. 14. O. C. ZIENKIEW1CZ 1977 /'he Finite Element Method. London: McGraw-Hill, third edition. 15. Y. H. PAO and C. C. M o w 1973 Diffraction of Elastic Waves and Dynamic Stress Concentration. New York: Crane and Russak. 16. M. OHrSU and K. ONO 1983 Journal of Acoustical Emission 2, 247-260. Resonance analysis of piezoelectric transducer elements.
BODY/HALF-SPACE
ASYMMETRIC
55
VIBRATIONS
APPENDIX The expressions for tractions in equation (10) have been given in reference [15] and will not be repeated here. Substitution of these expressions reduces equation (10) into linear combinations of the integrals
where ki assumes either of k, or k~, n = 0, 1, 2 . . . . and N = - 1 , 0, 1 , . . . . The circumferential dependence could be eliminated through direct integration of equations (AI). All the integrals with radial dependence exist e• for N = - t . These correspond to I = 0 in equation (10), which represent the total forces due to tractions on the 0 = ~ / 2 plane. I~ I could be made convergent by extracting the leading term of the asymptotic expression of the spherical Hankel function
h~')(z)=(-i)"+'ei"/z[l+O(1/z)],
z--,oo
(A2)
from the integrand of I~-~. This introduces an additional set of constraints given by -1). ,+, A,,,, P,,,. (0) = O,
( _ i ) , + , B , , dP,7(O)/d 0 = O,
- ( - , ) " "+' D . ~ P,,"(O) = O,
(A3) which should be enforced along with constraints given in equation (10). Through suitable change of variables, I~ 1 and lff become
Y2'(Kfl =
L,, (Kj) =
fo
[zh(,,'l(z) - (~i) ~§ e ~] dz,
(A4)
Ki
J,,N(Kj) =
Kj
z-Nh~)(z)dz,
n,N=O, 1,2,...,
(A5)
with Ks = kjao. These integrals could be evaluated by using the following recursion formulas. No~e that J o~
=
~/2-Si
(K~) + i Ci (Kj),
(A6)
where Si and Ci are the sine and cosine integrals, respectively. Then
JY(Kj)=(i/N)[J~'-~(Kj)-KfN eiK,],
J~(Kf)=(n-N-])JN,_-tI(Kj)+ KfNh('I(Kj). (A7)
The equivalent expressions for L,(Kj) are
Lo(K~)=O,
L~(K~)=nJ~ ~(K~)+Kjh~'(Kj)-(-i)"e~G
(A8)
The constraints presented in equation (10) written in terms of Jff(Kfl, N ---0, 1, 2 . . . . . for a longitudinal wave, are given by q 1
Y~ [ ( n tl~(J
N - ,,, d P ,m, ( 0 ) / d 0 = 0 , )J,,N § ( K p)-d,,+t(Ko)]A,
q
~. [ - n 2 J f f + ' ( K p ) + ( 1 - O ' 5 K ~ /
" ,~. '(Kp)-JN+,(K.)+m2J;7*'(Kp)]A.,p.~,(O)=O, 9 K'p)J.
n=O
q
.=o[-m
d~ (Kp)]A,,
dP.~(0)/d9=0,
(A9)
56
V. A V A N E S S I A N ,
S. B. D O N G
AND
R. M U K I
for an in-plane shear wave by q
E [-(n 2- I>J~+'(K~)- 0.5J~-t(K.,)-J,,N+t(K~)]B,7 dP~(O)/dO =0, rl=0 q
Y. [ n ( n 2 + n ) S y + ' ( K . , . ) - ( n 2 + n ) J L , ( K s ) tl=0
- m2(n+l)jN~,(K,)+
m 2J ,N, + , ( K ~ ) ] B , m P,, ( 0 ) = 0 ,
q
E
m[(.+ I)Sy*,(K,)_
~ ., dP:(0)/d0 =0, Jn+l(Ks)]Bn
(AIO)
rJ = 0
and for an antiplane shear wave by q
y m[ (n - 1)J,N(K~) - JN,.~'( K.,.) ]D,TPT( O) = O, ricO
q
y. m[J~ (K,)]D,, dP, (0)/d0 =0, -
N
m
m
rl~0 q
[0"5(n2 + n ) J ~ ( K ~ ) - m z J ~ ( K . , ) ] D , ~ , P T ( O ) = 0.
(A11)
n=0
In the above expressions L , , ( K s) should replace J ,N (Ks), corresponding to 1 = 0 in equation (10), with the additional constraints given by equation (A3).