The closed-form solution of the dynamic interaction of a shear wall and the isotropic homo- geneous and elastic half-space, previously studied only for ...
Bulletin of the Seismological Soe ety of America.
Vol. 62, NO. I. pp. 63-83.
February, 1972
I N T E R A C T I O N OF A SHEAR WALL W I T H T H E SOIL FOR I N C I D E N T PLANE
SH WAVES BY M. D. TRIFUNAC ABSTRACT The closed-form solution of the dynamic interaction of a shear wall and the isotropic homogeneous and elastic half-space, previously studied only for vertically-incident SH waves, is generalized to any angle of incidence. It is shown that the interaction equation is independent of the incidence angle, while the surface-ground displacements heavily depend on it. For the twodimensional model studied, it is demonstrated that disturbances generated by waves scattering and diffracting around the rigid foundation mass are not a local phenomenon but extend to large distances relative to the characteristic foundation length. NOMENCLATURE a,
an, bn,
Cx, Cy, f J, f b, H, Hp(2)(x), /,
G(x), k, kb, Mo, Mb, Ms, r,
R, /,
T, U/+ r U R z , Rbt z,
X , X ,t y, y',
r.(x), Z,
r, rio, 7, A, Ae, A,, 8,
radius of the semicircular foundation complex constants phase velocities in x and y directions force per unit length acting on the soil force per unit length acting on the shear wall height of the shear wall Hankel function of the second kind imaginary unit Bessel function of the first kind wave number in the soil, k = w/fl wave number in the shear wail, kb = W/rib mass of foundation per unit length mass of shear wall per unit length mass of soil per unit length distance in polar coordinates spectral ratio time coordinate period, T = 2~z/w free-field displacement in the z direction due to incident SH waves reflected waves from the semi-cylindrical rigid foundation relative displacement cartesian coordinate cartesian coordinate Bessel function of the second kind cartesian coordinate shear-wave velocity in the soil shear-wave velocity in the wall angle of incidence for SH waves amplitude of the displacement of the foundation envelope of A "backbone curve" for A dimensionless parameter ~ = rH/rba 63
64 r/,
0~ 2, Y, //b, Pb, CO,
BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA
ratio of a the radius of the foundation and 2/2 the half wavelength of incident S H waves azimuth in polar coordinate wavelength of incident S H waves rigidity of the soil rigidity of the wall density of the wall angular frequency 1NTRO DUCTION
The dynamic soil-structure interaction that takes place during the passage of seismic waves through the soil is of much interest to engineers (Housner, 1954; 1957). It has a bearing on the stresses in the building and the changes in the time history of the incident ground motion. To demonstrate some important aspects of soil-structure interaction, Luco (1969) studied the model shown in Figure l, which consists of an infinitely-long elastic shear
H
P'b B ' b I /-SCATTERING FROM FOUNDATION if,/?
/"r
"~,
/
" ~, v./" INCIDENT SH WAVE z , ~ _ _ ~ ~ /
~,,s
REFLECTED SH WAVE
Fro. 1. Shear wall, f o u n d a t i o n and soil,
wall erected on a rigid semi-cylindrical foundatiom He considered an excitation consisting of an infinite train of plane S H waves traveling vertically. In this paper, we generalize Luco's results to arbitrary incidence of S H waves and investigate the nature of surface ground motion near the foundation. Shallow and surface faults generate waves that consist predominantly of surface waves (Trifunac, 1971) in
INTERACTION OF A SHEAR WALL WITH SOIL FOR INCIDENT PLANE
SH
WAVES
65
the alluvial valleys and close to the source. It is important, therefore, to examine such simple models for which incident waves propagate nearly horizontally. Experimental studies of the soil-structure interaction are based on the simultaneous measurement of the "free-field" ground motion and building foundation (e.g., Housner, 1957; Duke e t al., 1970). By comparing these simultaneous motions, some information can be derived about the nature and significance of interaction. A common approach is to find the ratio of Fourier amplitude spectra of the "free-field" and foundation motions and to correlate the result with some theoretical expectations (Duke e t al., 1970). To fully understand the meaning of such a comparison and to determine the applicability of such an approach, however, we must first explore the nature of ground motions near buildings. THE MODEL
The model investigated here (Figure 1) consists of an infinitely-long elastic shear wall of height H and thickness 2a. This model is the same as the one studied by Luco (1969). The rigidity and the velocity of the shear waves in the isotropic and homogeneous wall are given by Pb and fib, respectively. The wall is erected on a rigid infinitely-long semicylindrical foundation of radius a. This assumption of rigidity leads to important simplifications of the method and results. The soil is assumed to be elastic, isotropic and homogeneous with rigidity p and velocity of shear waves ft. The contact between the soil and foundation is assumed to be welded. T H E G R O U N D MOTION
We take the excitation to consist of plane S H waves with incidence angle y (Figure I) and with amplitude equal to :.~ The resulting "free-field motion," i.e., the motion far from the shear wall, is then given by Ui+r = e i'°t-i(''xlcx) cos (~o3'/Cy)
(1)
where C~-sin7
and
C,.. = c o s ~ "
(2)
The subscript z indicates that the motion is in the z direction, perpendicular to the x, y plane (Figure 1) and independent ofz. The superscripts i and r indicate that the resulting motion is the sum of incident and reflected waves from the free surface. Equation (1) represents waves which propagate in the positive x direction with phase velocity C~.. The wave amplitude along the y coordinate is given by cos (~oy/Cy) where Cr is the phase velocity in the y direction. The resulting motion in the soil near the wall u, = u,i+r--+u: R, where u f is the wave reflected from the foundation, must satisfy the differential wave equation 3Zu_
1 Ou.
1
1
02/,/-
02/d_
Or2: +-r'-~ q - p 00 2 = fi2 012
(3)
for r > a and [0[ < ~z/2. It must also satisfy the boundary conditions given by r;o:-
p 0u~ -0 r 00
at
0=
rr +7'
r>a
and TC =
a
10] =
~. "-'~ 'If \:I ..............:,I .................
,III II
v
2
t
3
B F i e , 2. Effect of interaction on a shear wall.
1.8 Mb
16-
EFFECT = 2
Ms 1.4-
THE
INTERACTION
OF
ON
OF FOUNDATION
AMPLITUDE
M o
M, = I
VIBRATION
EQUATION (26) EQUATION (29)
1.2-
I.O-
IAI
/,bo
: '25:: . . . . . . .
° 4=4
1",---\.'~. I
"'~..
,8-
.6-
i i
F.,,,
".,
\
\:,~'b-7"-~-~
~/,
-../,. ,.....,,.,,i -'~,,~-"""---------~ - . ~
\
-
,
.4-
iI
,2-
iI !! ',I
~ / ~ / ~ I ' "
I
V
I'.,L..
_ ~
-~x,, ~..._>.~,
,;~
I! ......... ~, " k " ~ - ~ - . . _ . ~ 2~_ iI ........ "~ i "\ /"'-, i~"~'~'~ Ii ........... ~..l. \ / \~ ~l ~l ........ X;'"'/" ......... ~.f...............
I I
'I, ¢o°
2
P FIG. 3. Effect of interaction on a shear wall.
v
'i 3
SH
I N T E R A C T I O N OF A SHEAR W A L L W I T H SOIL FOR I N C I D E N T P L A N E
69
WAVES
1.8
1.4-
THE
.
ON A VIBRATION
- EQUATION (26) EQUAT I ON (29)
........
e~o
_ _
;=;
_
_
lal
/;.'"
,N-,,._
,,~ ....
.6-
,,.//
I'~~
.4-
I
I
=
" x , . - . . . ~ i . . . . . -:-,,-.
".../~
\
~
"-..~------,.~_~
Y #,'
t I
.2-
INTERACTION
OF FOUNDATION
~
s e=-/~ H
j"2\
OF
AMPLITUDE
MM° = I /"~ ,'-~,~
1.2-
I.O-
EFFECT
Mb Ms =4
1.6-
\
/
,
/
~"....~
t
\ \
'%..J
/
I i ..........
~!
\
................. ~,'
II
\,/
,
%,-/
-~
~ ,%
\
/
.... "~. . . . . . . . . . . . . . . . . .
i'
'7,
I
.,
/ -~,
i" a. The IA,] given by (29) also represents a "backbone curve" of equation (24) in that for values OfkbH away from (2n + 1)(~/2), [A[ is nearly [A,]. Other properties of the model in Figure 1 and equation (24) and their related consequences, such as artificial vibrations, free oscillations, influence of interaction on the base shear force, and the effect of damping in the shear wall, will not be discussed here. These questions were considered by Luco (1969) for vertically-incident SH waves. Luco's results become applicable to the general case as well, since the validity of equation (24) may be extended to all angles of incidence of the plane SH waves. RELATIVE RESPONSE Of interest in earthquake engineering is the displacement amplitude of the top of the shear wall relative to the foundation amplitude. From equation (16) we have
uz = A eight[cos k b x ' + t a n kbH sin kbx' ]. The relative displacement response amplitude RU~is then A
1
and if the interaction is neglected, A = 1 and
]RUz[ = c o s l b H - 1I.
(31)
It is readily seen that for kbH = (2n+l)n/2, n = 0, 1, 2, ..., corresponding to the natural fixed-base frequencies of the shear wall, ]Ru_.[ in (311) becomes infinite. If interaction is not neglected ]~u~[ is given by (30), where A is determined from (24) for the given values of Mo/M~, M b / M s, ka and e. Since A is zero for kbH = (2n+ 1)•/2, ]Ru~] becomes finite for all values of ka. In this way, the interaction introduces a certain amount of " d a m p i n g " into the relative displacement response and changes the peak frequencies. This is shown in Figures 5 and 6, where ]RU~[is plotted for Mo/M S = 1 and for e = 2 and ~ = 4. In addition to the curve given by (31) for a case of no interaction, three relative response curves are shown in each figure for Mb/M~ = 1, 2, and 4. As may be seen in these figures, when Mb/M~ ~ 0 the interaction becomes negligible, and, when Mb/Ms increases, it becomes more pronounced. In the case Mb/M s = 4 and e = 2 (Figure 5), the interaction effect is so pronounced that the peaks associated with the fixed-base natural frequencies of the shear wall are completely lost. Although the present model describes interaction for very long structures, relative response curves in Figures 5 and 6 suggest that similar modifications of the fixed-base response curves might take place for more realistic structural models. Thus, this is one possible reason for disagreement between the calculated relative response neglecting interaction and the experimentally observed response.
INTERACTION
OF A SHEAR WALL
WITH
i
5.0
coslKbH
71
PLANE SH WAVES
SOIL FOR INCIDENT
'1
T
V
