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Seismic overturning moments in single-storey structures with ground
compliance
Rainer, J. H.
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SEISMIC OVERTURNING MOMENTS IN SINGLE -S TOREY S T R U C T U R E S WITH GROUND C O M P L I A N C E J . H. R a i n e r (1) SYNOPSIS A m e t h o d i s p r e s e n t e d f o r d e t e r m i n i n g o v e r t u r n i n g m o m e n t s in s i n g l e - s t o r e y s t r u c t u r e s with f l e x i b l e f o u n d a t i o n s s u b j e c t e d to e a r t h q u a k e s . By m e a n s of a t r a n s f o r m a t i o n of f r e q u e n c y r e s p o n s e c u r v e s , a n e q u i v a l e n t s i n g l e - d e g r e e - o f - f r e e d o m m o d e l i s d e r i v e d to r e p r e s e n t t h e o v e r t u r n i n g m o m e n t of t h e s t r u c t u r e . T h i s m o d e l i s c h a r a c t e r i z e d by t h e f u n d a m e n t a l r e s o n a n c e f r e q u e n c y of t h e i n t e r a c t i o n s t r u c t u r e and a n e q u i v a l e n t d a m p i n g r a t i o . R e - s p o n s e c a l c u l a t i o n s a r e p r e s e n t e d and r e s p o n s e s p e c t r u m t e c h n i q u e s a r e d e s c r i b e d f o r finding m a x i m u m o v e r t u r n i n g m o m e n t s .
The r e s u l t s f r o m a p a r a m e t e r s t u d y show t h a t s t r u c t u r e s with a h e i g h t - t o - b a s e width r a t i o g r e a t e r t h a n 1 . 0 g e n e r a l l y h a v e l a r g e r o v e r t u r n i n g m o - m e n t s t h a n f i x e d - b a s e d s t r u c t u r e s with t h e s a m e r e s o n a n c e f r e q u e n c y and i n - t e r s t o r e y d a m p i n g .
I
Research O f f i c e r , Building Physics S e c t i o n , Division of Building Research, National Research Council of Canada, Ottawa.
SEISMIC OVERTURNING MOMENTS IN SINGLE-STOREY STRUCTURES WITH GROUND COMPLLANCE
b Y J. H. Rainer
The subject of s t r u c t u r e-ground interaction in buildings under s e i s m i c d i s t u r b a n c e s h a s lately received considerable attention (1, 2, 3, 4, 5, 6). These studies have concerned themselves exclusively with relative s t o r e y displace- m e n t s in s t r u c t u r e s that r e s t on deformable foundations. Although the d e t e r - mination of i n t e r s t o r e y displacements and the resulting f o r c e s a r e of g r e a t significance in the a s s e s s m e n t of s t r u c t u r a l adequacy, the overturning m o - m e n t s a s s o c i a t e d with the dynamic f o r c e s a r e a l s o of importance. The over- turning moments a f f e c t the axial loads i n the v e r t i c a l supporting m e m b e r s of a building, which a r e then t r a n s m i t t e d to the foundation where finally they have to b e r e s i s t e d by the soil.
If the a x i a l load in the columns i s exceeded, c o m p r e s s i o n f a i l u r e s a r e possible, a s was observed i n s o m e buildings a f t e r the C a r a c a s 1967 earth- quake (7). When the load capacity of the soil i s exceeded, s h e a r f a i l u r e s in the foundation m a t e r i a l s may r e s u l t , e. g. Niigata Earthquake 1964 (8). The behaviour of the foundation m a t e r i a l i s f u r t h e r complicated by tendency of
s o m e soils to liquefy under dynamic loading. In either c a s e , a c l e a r defini- tion of the loads imposed by overturning moments i s e s s e n t i a l to be able to a s s e s s the safety and p e r f o r m a n c e of s t r u c t u r e and foundation under e a r t h - quake motions.
With the consideration of g r o u n d - s t r u c t u r e interaction under dynamic loads, the determination of overturning moments becomes a m o r e complex p r o b l e m than f o r a comparable fixed-based s t r u c t u r e . To aid in the under-
standing of the phenomenon involved, a relatively simple c a s e i s investigated f i r s t , namely that of the single- s t o r e y s t r u c t u r e . The r e s u l t s presented here, however, will have d i r e c t applicability to s t r u c t u r e s such a s elevated s t o r a g e tanks and to s t r u c t u r e s that c a n be idealized r e a l i s t i c a l l y by the dynamic model of a s i m p l e oscillator founded on a flexible b a s e .
The method of analysis employed h e r e i s to d e r i v e a n equivalent single-degree-of -freedom (S. D. F . ) model f o r the overturning moment of the i n t e r a c t i o n s t r u c t u r e . This approach has been developed previously f o r studies of r e l a t i v e s t o r e y displacements in s i n g l e - s t o r e y interaction s t r u c - t u r e s under earthquake loadings (9). Among the advantages of using an equiv- a l e n t S. D. F. model i s the d i r e c t applicability of techniques that have been developed f o r S. D. F. o s c i l l a t o r s , such a s n u m e r i c a l integration and the r e - sponse s p e c t r u m . F u r t h e r m o r e , extensive p a r a m e t e r studies can be p e r - f o r m e d without having to r e l y on the r e s u l t s of lengthy r e s p o n s e calculations.
h
the following pages a r e described the interaction model that was employed i n the study, the derivation of a n equivalent single-degree-of- f r e e d o m model to r e p r e s e n t the overturning moment of the interaction s t r u c - ture, and n u m e r i c a l r e s u l t s f r o m a p a r a m e t e r study. The u s e of thee q u i v a l e n t S . D. F. m o d e l i s i l l u s t r a t e d with s p e c i f i c r e s p o n s e c a l c u l a t i o n s a n d with e a r t h q u a k e r e s p o n s e s p e c t r a . The r e s u l t s a r e d i s c u s s e d and g e n e r a l c o n c l u s i o n s a r e p r e s e n t e d .
INTERACTION MODEL
T h e i n t e r a c t i o n m o d e l u n d e r study i s t h a t shown i n F i g . 1, which i s i d e n t i c a l t o t h e one u s e d i n R e f e r e n c e s 3 and 9 . S i n c e d e t a i l e d d e r i v a t i o n s a r e a v a i l a b l e i n t h e s e r e f e r e n c e s , only the m a i n r e s u l t s a r e p r e s e n t e d h e r e . With t h e i n t r o d u c t i o n of r e l a t i v e h o r i z o n t a l b a s e d i s p l a c e m e n t and rocking, t h e S. D. F. s y s t e m h a s b e c o m e a t h r e e - d e g r e e - o f - f r e e d o m s y s t e m . F o r t h e t h r e e g e n e r a l i z e d c o o r d i n a t e s , r e l a t i v e i n t e r s t o r e y d i s p l a c e m e n t , h o r i z o n t a l d i s p l a c e m e n t of s t r u c t u r e , and r o c k i n g a b o u t t h e b a s e , t h e e q u a - tions of m o t i o n a r e , r e s p e c t i v e l y ,
. .
I , ,
t I 0 @ t n q h u H t M = 0 w h e r e r 2 = m - r 2 I0 0 4+
m14'
-
&
= m l h 2 , UH-
u t UB t h @ t U m , g a n d d o t s above a v a r i a b l e r e p r e s e n t d i f f e r e n t i a t i o n with r e s p e c t to t i m e . Of t h e s e , E q . 3 i s of p r i m a r y i n t e r e s t f o r t h e d e t e r m i n a t i o n of o v e r t u r n i n g m o - m e n t s . Under a s i n u s o i d a l b a s e d i s t u r b a n c e , ug = welpt, the amplification f a c t o r s X, Y and Z f o r r e l a t i v e s t o r e y d i s p l a c e m e n t Um, b a s e d i s p l a c e - m e n t UB and t h e a n g u l a r r o c k i n g component @ a r e given byi p t UB = We X = u
( 5
+
i & ) g ( 4 i p tm
= We Y = u (Y1+
i Y 2 ) g ( 6 )F o r a c i r c u l a r b a s e
w h e r e
t h e
K's
c a n b e i n t e r p r e t e d a s S. D. F. s t i f f n e s s and the C1s a s damping t e r m s f o r t h e foundation (10) and a r e plotted i n Fig. 2 . T h e s e i n t u r n a r e a function of t h e Bycroft coefficients fl and f, (11). The s u b s c r i p t s H andR
r e f e r to h o r i z o n t a l and rocking d i s p l a c e m e n t s , r e s p e c t i v e l y . G = s h e a r modulus of t h e ground, r = r a d i u s of b a s e , i =f l
a = p r / V s = non-dimensional frequency, Vs = s h e a r wave velocity of ground, p = frequency. Substitution i n t h e equation of motion, simplification and r e - a r r a n g e m e n t gives t h e f ollow- ing r e l a t i o n s h i p f o r the s t e a d y s t a t e d i s p l a c e m e n t amplification factor:which c a n b e e x p r e s s e d m o r e compactly a s
In Eq. (11)
and
A = r e l a t i v e i n t e r - s t o r e y damping r a t i o ,
p = d e n s i t y of ground
The t r a n s f e r function f o r the s y s t e m , T ~ due to ground d i s p l a c e m e n t i s ~ ,
obtained by r e a r r a n g i n g Eq. 1 1. g
The t r a n s f e r function f o r ground a c c e l e r a t i o n m a y then b e obtained a s follows:
w h e r e d d e n o t e s the g e n e r a l i z e d d i s p l a c e m e n t v e c t o r f o r t h e s t r u c t u r e . OVERTURNING MOMENTS
The o v e r t u r n i n g m o m e n t M acting on the e l a s t i c h a l f - s p a c e a s a r e s u l t of t h e dynamic r e s p o n s e of the s t r u c t u r e i s obtained f r o m Eq. 3 :
Upon s u b s t i t u t i o n of s t e a d y - s t a t e amplification f a c t o r s , Eqs. 4 to 6,
and a p p r o p r i a t e s t r u c t u r a l c o n s t a n t s , the t r a n s f e r function f o r overturning m o m e n t r e l a t i v e t o ground a c c e l e r a t i o n , M, i s given by
M l t a
Tii
-1
h Y t ~ t ZF o r a r i g i d l y b a s e d s t r u c t u r e , Y = 0, X = 1 ; thus
X, Y , and Z m a y b e computed f r o m Eq. 12, and the addition in Eq. 15 has to be c a r r i e d out with due r e g a r d to s i g n s and r e a l and i m a g i n a r y components. T h e f r e q u e n c y r e s p o n s e c u r v e f o r t h e o v e r t u r n i n g m o m e n t of a typical s t r u c - t u r e ( S t r u c t u r e No. 1, T a b l e 1 ) i s p r e s e n t e d in Fig. 3 ( a ) .
EQUIVALENT S. D. F. FOR OVERTURNING MOMENTS
The r e s p o n s e of a l i n e a r d y n a m i c a l s y s t e m i s c o m p l e t e l y d e t e r - mined by i t s f r e q u e n c y r e s p o n s e c u r v e , which r e p r e s e n t s a plot of the t r a n s - f e r function of the s y s t e m v e r s u s f r e q u e n c y . The t r a n s f e r function of o v e r - turning m o m e n t d u e to ground a c c e l e r a t i o n i s given by E q . 15, which, when plotted a s a function of f r e q u e n c y , r e s u l t s in the f r e q u e n c y r e s p o n s e c u r v e f o r o v e r t u r n i n g m o m e n t s shown in F i g . 3 ( a ) . Also shown in dotted l i n e s , i s the f r e q u e n c y r e s p o n s e c u r v e f o r o v e r t u r n i n g m o m e n t s of a n S . D. F. s y s t e m with the s a m e r e s o n a n c e f r e q u e n c y
R
when subjected to ground a c c e l e r a t i o n ug.DERIVATION O F EQUIVALENT S . D. F. MODEL
To s e t up a n equivalent S . D. F. m o d e l f o r o v e r t u r n i n g m o m e n t s , i t i s n e c e s s a r y t o c o n v e r t the f r e q u e n c y r e s p o n s e c u r v e f o r o v e r t u r n i n g m o - m e n t s shown in F i g . 3 into that of a n S . D. F. s y s t e m . With r e f e r e n c e to F i g . 3 ( b ) , t h r e e conditions a r e to b e s a t i s f i e d by the equivalent S . D. F . m o d e l and the o r i g i n a l i n t e r a c t i o n s y s t e m : ( 1 ) i d e n t i c a l r e s o n a n c e f r e q u e n c y ;
( 2 ) a g r e e m e n t of o r d i n a t e s of the f r e q u e n c y r e s p o n s e c u r v e s away f r o m r e s - onance; and ( 3 ) a g r e e m e n t of o r d i n a t e s of the f r e q u e n c y r e s p o n s e c u r v e s a t the r e s o n a n c e f r e q u e n c y .
1. I d e n t i c a l R e s o n a n c e F r e q u e n c y
S i ~ i c e the r e s o n a n c e f r e q u e n c y of the equivalent S . D. F. m o d e l iz the s a m e a s the f u n d a m e n t a l f r e q u e n c y of the i n t e r a c t i o n s y s t e m , s t a n d a r d r i g e n - v a l u e m e t h o d s applied to the m a t h e m a t i c a l m o d e l of the i n t e r a c t i o n s y s t e m will yield the r e q u i r e d r e s o n a n c e f r e q u e n c y . By e l i m i n a t i n g the r o w s and c o l u m n s c o r r e s p o n d i n g to the i m a g i n a r y t e r m s of the m a t r i x i n E q . 11, the r o o t s of the r e s u l t i n g d e t e r m i n a n t a r e the eigenvalues of the i n t e r a c t i o n s y s t e m . The l o w e s t f r e q u e n c y c o r r e s p o n d s to the fundamental, and c o n s e - quently to the r e s o n a n c e f r e q u e n c y of the equivalent S . D. F. m o d e l . A l t e r - natively, a n u m e r i c a l s e a r c h f o r the l o w e s t r e s o n a n c e peak in the f r e q u e n c y r e s p o n s e c u r v e s , using Eq. 11, will a l s o give the r e s o n a n c e f r e q u e n c y of t h e equivalent S. D. F. m o d e l .
2. A g r e e m e n t of O r d i n a t e s Away F r o m R e s o n a n c e
F o r a n S . D. F. o s c i l l a t o r the quantity t h a t i s usually computed i s t h e r e l a t i v e d i s p l a c e m e n t b e t w e e n the b a s e and t h e s p r i n g - s u p p o r t e d m a s s . In
o r d e r t h a t s p e c t r a l techniques and S . D.
F.
n u m e r i c a l i n t e g r a t i o n m e t h o d s m a y b e employed f o r r e s p o n s e c a l c u l a t i o n of i n t e r a c t i o n s t r u c t u r e s , the f r e q u e n c y r e s p o n s e c u r v e f o r o v e r t u r n i n g m o m e n t s of the i n t e r a c t i o n s y s t e m m u s t b e c o n v e r t e d i n t o the f r e q u e n c y r e s p o n s e c u r v e f o r r e l a t i v e d i s p l a c e m e n t of the equivalent S. D. F. s y s t e m . The z e r o f r e q u e n c y i n t e r c e p t of the f r e q u e n c y r e s p o n s e c u r v e f o r r e l a t i v e d i s p l a c e m e n t Um,
obtained by evaluating the t r a n s f e r function T;;Vm
f r o m E q s . 11, 12 and 13 a t p = 0, c o r r e s p o n d s t o the d i s p l a c e m e n t r e s u l t - ing f r o m a b a s e input of z e r o f r e q u e n c y , i. e . a c o n s t a n t a c c e l e r a t i o n ii = 1. 0 .The r e s u l t i n g r e l a t i v e d i s p l a c e m e n t of the . . S. D. F. s y s t e m i s then g
On the o t h e r hand, the z e r o f r e q u e n c y i n t e r c e p t f o r o v e r t u r n i n g m o m e n t s i s obtained f r o m Eq. 15, by s e t t i n g p = 0, Y = Z = 0 and X = 1, which then b e c o m e s
The o r d i n a t e s of the f r e q u e n c y r e s p o n s e c u r v e s f o r o v e r t u r n i n g m o m e n t a t f r e q u e n c i e s below r e s o n a n c e a r e brought t o c o i n c i d e with the o r d i n a t e s of a n S. D. F. s y s t e m if t h e f o r m e r a r e multiplied by 1 / R 2 . As will b e ex- plained l a t e r , a g r e e m e n t of o r d i n a t e s above r e s o n a n c e i s not c o m p l e t e l y
s a t i s f i e d by t h i s m u l t i p l i c a t i o n f a c t o r d u e t o d i f f e r e n c e s i n p h a s e . 3 . A g r e e m e n t of O r d i n a t e s a t R e s o n a n c e
T h e a m p l i t u d e of the r e s o n a n c e peak of the equivalent S. D. F. f r e - quency r e s p o n s e c u r v e obtained by m e a n s of t h e above multiplication i s now c h a r a c t e r i z e d b y a n equivalent damping r a t i o .
The damping r a t i o of a n S. D.
F.
s y s t e r n c a n b e computed f r o m the r e l a t i o n s h i p 1 ), = - 25 (18) w h e r e5
= nondimensional a m p l i f i c a t i o n f a c t o r of a n S. D. F. o s c i l l a t o r a t t h e r e s o n a n c e f r e q u e n c y R.
T h e f r e q u e n c y r e s p o n s e c u r v e of Tli of ti;e equivalent S. D.F.
m o d e l h a s to b e c o n v e r t e d t o the nondimension%l f o r mi n o r d e r t h a t E q . 1 8 m a y b e a p p l i c a b l e . A s a c o r o l l a r y to E q . 13, T..
rr,
26
b e m u l t i p l i e d by t h e v a r i a b l e p2 t o give T~:". But a t the r e s o n a n c e ug f r e q u e n c y , p e q u a l s R . T h e r e f o r eI t should a l s o b e r e c a l l e d f r o m p r e c e g p a r a g r a p h s t h a t the c o n s t a n t
p l i c a t i o n f a c t o r 1 /R2 i s a p p l i e d to Tii
wm
t o a c h i e v e a g r e e m e n t with Tii?2lt
- of the equivalent S. D. I?. m o d e l a t t h g z e r o f r e q u e n c y inte;cept. c o n s e q 6 e n t l ythe a m p l i t u d e
5,
of the nondimensional amplification f a c t o r of the equivalent S. D. F. model a t r e s o n a n c e b e c o m e sThis m e a n s t h a t f o r the calculation of the equivalent damping r a t i o
X e
,
the a c t u a l a m p l i t u d e s of the overturning m o m e n t t r a n s f e r function a t r e s o n a n c e , a s given by Eq. 15, c a n be u s e d i n Eq. 18 ( L I ) .With r e s o n a n c e f r e q u e n c y R
,
the multiplication f a c t o r 1 /R,
and the equivalent damping r a t i o Xe, the f r e q u e n c y r e s p o n s e c u r v e of theequivalent S. D. F. s y s t e m i s completely d e s c r i b e d . S i n c e a l l a m p l i t u d e s of the f r e q u e n c y r e s p o n s e c u r v e f o r overturning m o m e n t s have b e e n multiplied by 1 /R any r e s p o n s e computed with t h i s equivalent S. D. F. m o d e l h a s to be multiplied by
n 2
i n o r d e r to obtain the o v e r t u r n i n g m o m e n t M / q h.RESPONSE CALCULATIONS FOR OVERTURNING MOMENTS
F i g u r e 4 shows r e s p o n s e c a l c u l a t i o n s f o r o v e r t u r n i n g m o m e n t s f o r S t r u c t u r e s No. 1 and 2 of Table 1. F o r both i n t e r a c t i o n s y s t e m s the o v e r - turning m o m e n t , shown by solid l i n e s , i s l a r g e r than f o r a r i g i d l y b a s e d s t r u c t u r e of the s a m e n a t u r a l frequency, a s r e p r e s e n t e d by the dotted c u r v e . The r e s p o n s e s obtained f o r S t r u c t u r e No. 1 f r o m the equivalent S. D. F. m o d e l differ slightly f r o m the "exact1' one obtained by F a s t F o u r i e r t r a n s - f o r m (12) with t r a n s f e r function of Eq. 24,for the following r e a s o n .
The t r a n s f e r function f o r overturning m o m e n t , Eq. 15, i s s e e n to be a c o m p o s i t e quantity of a l l t h r e e d e g r e e s of f r e e d o m . Above the r e s o n a n c e f r e q u e n c y the a m p l i t u d e s of the f r e q u e n c y r e s p o n s e c u r v e f o r overturning m o m e n t will b e diminished by the r e l a t i v e b a s e d i s p l a c e m e n t , s i n c e i t h a s p h a s e opposite to t h a t of rocking and r e l a t i v e displacement(Fig.5). Consequently, f o r f r e q u e n c i e s above r e s o n a n c e the f r e q u e n c y r e s p o n s e c u r v e of the equivalent S. D. F. m o d e l will not have the s a m e p r o p o r t i o n of a m p l i t u d e s a s the overturning m o m e n t f r e q u e n c y r e s p o n s e c u r v e . S o m e d i f f e r e n c e of r e s p o n s e can thus be expected f r o m the equivalent S. D. F. m o d e l a s c o m p a r e d to the l ' e x a c t ' l method using the t r a n s f e r function and F o u r i e r t r a n s f o r m method. Since f o r m o s t s t r u c t u r e s the r e l a t i v e b a s e d i s p l a c e m e n t i s s m a l l c o m p a r e d to the i n t e r s t o r e y and rocking d i s p l a c e - m e n t s , the e r r o r will b e s m a l l . F o r S t r u c t u r e No. 1, the d i s c r e p a n c y was i n the o r d e r of 6% of the peak r e s p o n s e amplitude.
II
In r e f e r e n c e 9 i t h a s been d e m o n s t r a t e d t h a t u n d e r c e r t a i n simplifying a s s u m p t i o n s the equivalent dampingX
f o r r e l a t i v e d i s p l a c e m e n t and over-
turning m o m e n t a r e the s a m e .MAXIMUM OVERTURNING MOMENT FROM RESPONSE S P E C T R A
D e t e r m i n a t i o n of the m a x i m u m overturning m o m e n t under a p a r t i c
-
u l a r b a s e motion m a y b e a c h i e v e d d i r e c t l y f r o m the r e s p o n s e s p e c t r u m f o r that p a r t i c u l a r e a r t h q u a k e . In a c c o r d a n c e with the d e r i v a t i o n of the equiv- a l e n t S. D. F. s y s t e m d e s c r i b e d above, the following p r o c e d u r e i s indicated: (1) The fundamental r e s o n a n t frequency, wl, i s d e t e r m i n e d , and thea m p l i t u d e s of the overturning m o m e n t r e s p o n s e c u r v e a r e multiplied by l / w t .
( 2 ) T h e equivalent damping X e i s d e t e r m i n e d a f t e r obtaining M/- h f r o m a n evaluation of Eq. 15 a t r e s o n a n c e , o r a p a r a m e t e r study s u c h a s i s p r e s e n t e d l a t e r .
( 3 ) Corresponding to the f r e q u e n c y wl, the value of m a x i m u m r e l a t i v e
d i s p l a c e m e n t i s r e a d f r o m the r e s p o n s e s p e c t r u m and multiplied by
w12 to obtain the v a l u e of the r a t i o f o r overturning moment, M / q h .
An a p p r o x i m a t i o n to t h e m a x i m u m overturning m o m e n t s of t a l l s t r u c t u r e s i s obtained by taking the undamped s p e c t r a l d i s p l a c e m e n t S D and multiplying i t by
w t
to g e t M/-h ; t h i s gives a c o n s e r v a t i v e e s t i m a t e .PARAMETER STUDY
B e s i d e s the g r e a t g e n e r a l i t y i n h e r e n t in the method of d e r i v a t i o n of the equivalent S. D. F. model, a p a r a m e t e r study c a n b e conducted f o r a wide r a n g e of p a r a m e t e r s without having to p e r f o r m lengthy r e s p o n s e c a l - c u l a t i o n s . The r e s u l t s obtained a r e p r e s e n t e d s o t h a t r e s o n a n c e f r e q u e n c i e s and equivalent damping c a n b e found f o r s t r u c t u r e s having the specific p a r a - m e t e r s c o n s i d e r e d . In addition i t i s p o s s i b l e t o a r r i v e a t s o m e g e n e r a l con- c l u s i o n s r e g a r d i n g the behaviour of s t r u c t u r e s founded on flexible founda- tions.
The r a n g e of s t r u c t u r a l and foundation p a r a m e t e r s c o n s i d e r e d i s shown in Table 2. P a r a m e t e r S e t A includes elevated s t r u c t u r e s s u c h a s w a t e r t a n k s ; P a r a m e t e r S e t B would include n u c l e a r r e a c t o r containment v e s s e l s and o t h e r m a s s i v e s t r u c t u r e s . I t should b e pointed out, however, t h a t not a l l combinations of p a r a m e t e r s p r e s e n t e d m a y r e p r e s e n t physically r e a l i z a b l e s y s t e m s .
PARAMETER S E T A
F o r P a r a m e t e r S e t A of Table 2 , when the r a t i o of f i x e d - b a s e d f r e q u e n c y t o rocking frequency, %/w + , i s plotted a g a i n s t the r a t i o of f i x e d - b a s e d f r e q u e n c y to the fundamental r e s o n a n c e frequency, wl /wo
,
the v a l u e s f a l l between the bounds shown i n F i g .6.
The lower c u r v e c o r r e s p o n d s to the t h e o r e t i c a l r e l a t i o n s h i p d e r i v e d by M e r r i t t and Housner ( 1 ) f o r a f l e x i b l e s t r u c t u r e on a b a s e with a r o t a t i o n a l d e g r e e of f r e e d o m only:F o r P a r a m e t e r S e t A of Table 2, the v a r i a t i o n of the r a t i o MS/M1 of peak m a g n i t u d e s of the i n t e r a c t i o n s y s t e m to t h a t of a f i x e d - b a s e d S . D. F. s y s t e m with t h e s a m e n a t u r a l f r e q u e n c y i s shown i n F i g s . 7 and 8. I t m a y be o b s e r v e d t h a t f o r the t a l l n a r r o w - b a s e d s t r u c t u r e s t h e p e a k f o r t h e i n t e r - a c t i o n s y s t e m i s c o n s i d e r a b l y l a r g e r than t h a t of a f i x e d - b a s e d s t r u c t u r e with t h e s a m e n a t u r a l f r e q u e n c y . Only f o r high v a l u e s of a, and r e l a t i v e l y low s t r u c t u r e s a r e the o v e r t u r n i n g m o m e n t r e s o n a n c e p e a k s f o r i n t e r a c t i o n s t r u c t u r e s s m a l l e r than t h e S . D. F. c a s e .
PARAMETER S E T B
S i i n i l a r l y a s f o r P a r a m e t e r S e t A, a plot of t h e r a t i o s
w o / w +
v e r s u sw
/w, f o r P a r a m e t e r S e t B of Table 2 g i v e s the c u r v e s shown i n F i g .9 .
Again t g e r e l a t i o n s h i p f o r a rocking b a s e i s i n d i c a t e d by b r o k e n l i n e s . I t m a y be o b s e r v e d that a s t h e s t r u c t u r e s b e c o m e t a l l e r , t h e f r e q u e n c y r e d u c t i o n a p - p r o a c h e s t h a t of the f l e x i b l e s t r u c t u r e on a rocking b a s e , E q . 21.I n Lhe study of r e s o n a n c e peak a m p l i t u d e s , f o r P a r a m e t e r S e t B, T a b l e 2, a s i m i l a r phenomenon a s f o r P a r a m e t e r S e t A i s d i s p l a y e d . In F i g u r e s 10 a n d 1 1 f o r the s t r u c t u r e s with t h e l a r g e r r a t i o of height to b a s e r a d i u s h / r , t h e r e s o n a n c e p e a k s of the o v e r t u r n i n g m o m e n t e x c e e d t h o s e of the S. D. F. (i. e . the f i x e d - b a s e d S. D. F. ), w h e r e a s f o r t h e low s t r u c t u r e the p e a k s a r e s m a l l e r than f o r t h e S . D, F. F i g u r e 10 s h o w s t h a t t h e p e a k o v e r - t u r n i n g m o m e n t a m p l i t u d e s f o r t h e s t r u c t u r e with b a s e m a s s m o = 100 000 lb s e c 2 i n . , shown by b r o k e n l i n e s , a r e only s l i g h t l y l e s s than t h o s e f o r m o = 400 000 l b s e c 2 / i n . , both top m a s s e s rrq e q u a l to 400 000 l b s e c 2 / i n . and a l l o t h e r p a r a m e t e r s being k e p t c o n s t a n t .
DISCUSSION O F RESULTS
F r o m t h e r e s u l t s p r e s e n t e d in t h e p a r a m e t e r s t u d y i t m a y b e ob- s e r v e d t h a t the r e s o n a n c e p e a k s of o v e r t u r n i n g m o m e n t s f o r s t r u c t u r e s whose h e i g h t s exceed the b a s e d i a m e t e r g e n e r a l l y a r e l a r g e r than t h e r e s o - n a n c e p e a k s of t h e f i x e d - b a s e d , S. D. F. s y s t e m . -4s the f r e q u e n c y r e s p o n s e c u r v e s f o r t h e i n t e r a c t i o n s t r u c t u r e and t h e S . D. F. s y s t e m both h a v e t h e s a m e z e r o f r e q u e n c y i n t e r c e p t , and a r e a l m o s t i d e n t i c a l e v e r y w h e r e e x c e p t n e a r t h e r e s o n a n c e f r e q u e n c y , a c o m p a r i s o n of r e s o n a n c e p e a k s then en- a b l e s one t o m a k e q u a l i t a t i v e g e n e r a l i z a t i o n s r e g a r d i n g t h e r e s p o n s e to a n a r b i t r a r y input. That i s , when t h e r e s o n a n c e p e a k s e x c e e d t h o s e of t h e S . D. F. s y s t e m , the r e s p o n s e of t h e i n t e r a c t i o n s t r u c t u r e will b e l a r g e r o r a t m o s t e q u a l to that of t h e S. D. F. s y s t e m . A n o t h e r way of i n t e r p r e t i n g t h e s e r e s u l t s i s by way of the damping r a t i o . A s y s t e m with a l a r g e r r e s o - n a n c e p e a k h a s a s m a l l e r d a m p i n g r a t i o , and c o n s e q u e n t l y a l a r g e r r e s p o n s e
u n d e r r a n d o m type or s t e a d y - s t a t e input
(LII).
It m a y t h e r e f o r e b e deduced f r o m F i g s . 7 , 8, 10 a n d 11 that f o r s o m e s t r u c t u r e s with foundation i n t e r - a c t i o n and a height exceeding the b a s e d i a m e t e r t h e o v e r t u r n i n g m o m e n t under e a r t h q u a k e m o t i o n s will be l a r g e r than f o r a r i g i d l y b a s e d S. D. F. s t r u c t u r e with the s a m e n a t u r a l f r e q u e n c y and i n t e r s t o r e y damping.F r o m r e s u l t s p r e s e n t e d and t h e underlying d e r i v a t i o n of t h e equiv- a l e n t S. D. F. m o d e l , s o m e s i g n i f i c a n t p a r a m e t e r s c a n b e identified. T h e s e a r e : ( 1 ) t h e f r e q u e n c y r e d u c t i o n f r o m
w o
t o wl,
( 2 ) t h e n o n - d i m e n s i o n a l f r e - quency a. = wor/Vs, and ( 3 ) t h e height and top m a s s of t h e s t r u c t u r e . T h e s e t h r e e g r o u p s of p a r a m e t e r s a r e not independent, but they p r o v i d e a conve- i e n t s e t f o r the p r e s e n t a t i o n a n d i n t e r p r e t a t i o n of r e s u l t s , a s w e l l a s f o r t h e i r s u b s e q u e n t u s e i n t h e r e s p o n s e c a l c u l a t i o n s using the equivalent S. D. F. s y s - t e m . F r o m F i g s . 9 and 10 i t m a y b e s e e n t h a t the a m p l i t u d e of r e s o n a n c e p e a k s a s w e l l a s the f r e q u e n c y r e d u c t i o n s a r e a l m o s t independent of t h e m a g n i t u d e of t h e b a s e m a s s . The f r e q u e n c y r e d u c t i o n of a l l i n t e r a c t i o n s y s t e m s i n v e s t i g a t e d i s p r i m a r i l y dependent on t h e r a t i o of rocking f r e q u e n c y t o t h e f i x e d - b a s e d n a t u r a l f r e q u e n c y of t h e s t r u c t u r e , w+/wo, a s m a y be s e e n f r o m F i g s . 6 and9. But t h e rocking f r e q u e n c y i n t u r n i s d i r e c t l y p r o p o r t i o n a l t o t h e s h e a r wave v e l o c i t y of t h e ground, Vs.
F r o m t h i s o b s e r v a t i o n and the r e s u l t s of t h e p a r a m e t e r study i n F i g s . 7, 8, 10 and 11, i t m a y be concluded t h a t f o r r e a s o n - a b l y t a l l s t r u c t u r e s t h e s a m e amplification of o v e r t u r n i n g m o m e n t s f o r a s t r u c t u r e - f o u n d a t i o n s y s t e m i s obtained a s long a s U J , / V ~ i s c o n s t a n t . Be- c a u s e of t h i s p r o p e r t y , r e s u l t s p r e s e n t e d a r e valid beyond t h e r a n g e of s p e - cific v a l u e s ofw 0
and Vs used, provided wo/Vs d o e s not exceed i t s r a n g e c o n s i d e r e d h e r e .SUMMARY AND CONCLUSIONS
The m e t h o d of a n a l y s i s of the equivalent S. D. F. m o d e l h a s b e e n a p p l i e d to a n i n v e s t i g a t i o n of o v e r t u r n i n g m o m e n t s i n s i n g l e - s t o r e y s t r u c t u r e - ground i n t e r a c t i o n s y s t e m s . The equivalent
s.
D. F. m o d e l i s obtained by m a t c h i n g t h e f r e q u e n c y r e s p o n s e c u r v e f o r o v e r t u r n i n g m o m e n t s with t h a t of a n S. D. F. s y s t e m and r e q u i r e s t h e d e t e r m i n a t i o n of t h e r e s o n a n c e f r e - quency, a m u l t i p l i c a t i o n f a c t o r a n d a n equivalent damping r a t i o . This method p e r m i t s a n e x t e n s i v e p a r a m e t e r study and t h e i s o l a t i o n of the signif- i c a n t p a r a m e t e r s f o r s i n g l e - s t o r e y s t r u c t u r e s with foundation c o m p l i a n c e . F r o m the r e s u l t s p r e s e n t e d , m a x i m u m o v e r t u r n i n g m o m e n t s under a r b i-
t r a r y b a s e m o t i o n s c a n then b e found by n u m e r i c a l i n t e g r a t i o n o r r e s p o n s e s p e c t r u m t e c h n i q u e s .LU.
I t should b e noted t h a t t h i s g e n e r a l i z a t i o n i s n o t valid f o r the c o m p a r i - s o n of r e s o n a n c e p e a k s f o r r e l a t i v e d i s p l a c e m e n t , s i n c e t h e o r d i n a t e s of t h e f r e q u e n c y r e s p o n s e c u r v e s of t h e i n t e r a c t i o n s y s t e m and the S. D. F. away f r o m r e s o n a n c e a r e not of c o m p a r a b l e m a g n i t u d e s .F r o m a p a r a m e t e r s t u d y i t i s found t h a t o v e r t u r n i n g m o m e n t s i n s i n g l e - s t o r e y s t r u c t u r e s with g r o u n d c o m p l i a n c e a n d w i t h a h e i g h t g r e a t e r t h a n b a s e d i a m e t e r g e n e r a l l y e x c e e d t h e c o r r e s p o n d i n g v a l u e f o r a f i x e d - b a s e d S. D. F. s y s t e m with t h e s a m e n a t u r a l f r e q u e n c y a n d i n t e r - s t o r e y d a m p i n g . T h i s a m p l i f i c a t i o n e f f e c t i s g r e a t e r w i t h l a r g e r h e i g h t t o b a s e - width r a t i o s . T h e p a r a m e t e r s t u d y a n d t h e m e t h o d of p r e s e n t a t i o n of t h e r e s u l t s h a v e i n d i c a t e d t h a t t h e r a t i o of s h e a r wave v e l o c i t y t o t h e n a t u r a l f r e q u e n c y of t h e f i x e d - b a s e d s t r u c t u r e i s o n e of t h e i m p o r t a n t p a r a m e t e r s i n t h e s t r u c t u r e - g r o u n d i n t e r a c t i o n p h e n o m e n o n . Although t h e r e s u l t s p r e s e n t e d a r e l i m i t e d t o a c e r t a i n c l a s s of r e l a t i v e l y s i m p l e s t r u c t u r e s , t h e y i n d i c a t e t h a t o v e r t u r n i n g m o m e n t s s h o u l d b e g i v e n c a r e f u l c o n s i d e r a t i o n i n s l e n d e r s t r u c t u r e s founded on e l a s t i c b a s e s . R E F E R E N C E S M e r r i t t , R. G. a n d H o u s n e r , G. W . , " E f f e c t of F o u n d a t i o n C o m p l i a n c e on E a r t h q u a k e S t r e s s e s i n M u l t i s t o r y Buildings, I t B u l l e t i n of S e i s . S o c .
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H . , "Method of A n a l y s i s f o r S t r u c t u r e - G r o u n d I n t e r a c t i o n i n E a r t h q u a k e s , T e c h n i c a l P a p e r No. 340,
D i v i s i o n of Building R e - s e a r c h , N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a . K o b o r i , T . , M i n a i , R . , a n d K u s a k a b e , K . , l l D y n a m i c a l G r o u n dComplianc e of R e c t a n g u l a r Foundation, P r o c e e d i n g s 16th J a p a n Na- tional C o n g r e s s of Applied M e c h a n i c s , 1966, pp. 301 -306.
11. Bycroft, G. N . , " F o r c e d Vibrations of a Rigid C i r c u l a r P l a t e on a S e m i - i n f i n i t e E l a s t i c S p a c e and on a n E l a s t i c S t r a t u m , " P h i l o s o p h i c a l T r a n s . ,
Royal Society, London, S e r i e s A, Vol. 248, No.948, Jan. 1965, pp.323- 3 68.
12. R a i n e r , J . H . , "Dynamic R e s p o n s e by F a s t F o u r i e r T r a n s f o r m , DBR C o m p u t e r P r o g r a m , No. 29, Division of Building R e s e a r c h , National R e s e a r c h Council of Canada, Ottawa, May 1970.
P A R A M E T E R S F O R S A M P L E C A L C U L A T I O N S P a r a m c t c r U n i t S t r u c t u r e No. 1 S t r u c t u r e No. 2 ( a ) S t r u c t u r a l P a r a m c t e r s ( b ) E q u i v a l e n t S. D. F . M o d e l f o r O v e r t u r n i n g M o m e n t T A B L E 2 RANGES O F P A R A M E T E R S V a r i a b l e w '=% m 0 h r A s P a r a m e t e r S e t A 5 t o 2 0 r a d / s e c 1000 t o 4 0 0 0 l b s e c a / i n . 1000 l b s e c 2 / i n . 2 0 t o 8 0 f t 15 a n d 2 0 f t 2 p e r c e n t 300, 500 a n d 800 f . p. 8. P a r a m e t e r S e t B 5 t o 20 r a d / s e c 100 000 a n d 4 0 0 0 0 0 l b s e c 2 / i n . 100 000 t o 4 0 0 0 0 0 l b s e c 2 / i n . 4 0 t o 160 f t 60 f t 2 p e r c e n t 500, 800 a n d 1 6 0 0 f . p. s.
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INTERACTION. STRUCTURE *2 TIME. SEC FIGURE 4TIME HISTORY OF OVERTURNING MOMENTS. BASE MOTION: E L CENTRO 1940, N - S COMPONENT
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DISCUSSION OF PAPER NO. 1 0
SEISMIC OVERTURNING MOMENTS I N SINGLE STOREY STRUCTURES WITH GROUND COMPLIANCE
J . H. R a i n e r D i s c u s s i o n by: M. Novak The a u t h o r p r e s e n t s a way i n w h i c h R e i s s n e r ' s ( B y c r o f t ' s ) f r e q u e n c y f u n c t i o n s f z , f2 c a n b e i n t r o d u c e d i n t o t h e a n a l y s i s o f s i n g l e s t o r e y s t r u c t u r e s t o e x p r e s s t h e e l a s t i c and damping p r o p e r t i e s o f t h e h a l f - s p a c e ( s o i l ) . T h i s i s a good a p p r o a c h b e c a u s e t h e h a l f - s p a c e i s t a k e n i n t o a c c o u n t w i t h o u t i n - c r e a s i n g t h e number o f d e g r e e s of f r e e d o m .
The same p r o c e d u r e c a n b e a p p l i e d t o s y s t e m s w i t h many d e g r e e s o f f r e e d o m , e . g . Korenev, Baranov, 1964 a n d t h e e f f e c t o f embedment c a n b e a l s o c o n s i d e r e d
( p a p e r n o . 7 ) . As f o r q u a n t i t a t i v e a p p l i c a t i o n s t o s o i l s , t h e damping d e r i v e d f r o m t h e h a l f - s p a c e s h o u l d b e c o n s i d e r e d w i t h c e r t a i n c a r e . E x p e r i m e n t s i n d i c a t e t h a t t h e t h e o r y t e n d s t o o v e r e s t i m a t e t h e damping i n v e r t i c a l d i r e c t i o n and u n d e r - e s t i m a t e i t w i t h r o c k i n g i f i n t e r n a l f r i c t i o n i s n e g l e c t e d ; l a y e r i n g c a n con- s i d e r a b l y r e d u c e t h e g e o m e t r i c damping. F i n a l l y , t h e t h e o r y was d e v e l o p e d f o r s t e a d y - s t a t e h a r m o n i c v i b r a t i o n n o t f o r n o n s t a t i o n a r y random m o t i o n a s e a r t h - q u a k e s a r e . Reply by: J. H. R a i n e r
The b a s i c s t r u c t u r a l model employed i n t h i s s t u d y h a s been u s e d p r e v i o u s l y by P a r m e l l e and K o b o r i , among o t h e r s . The u s e o f a n e q u i v a l e n t s i n g l e - d e g r e e - o f - f r e e d o m model f o r o v e r t u r n i n g moment i s t h o u g h t t o b e n o v e l , however. By d e t e r m i n i n g t h e r e s o n a n c e f r e q u e n c y of t h e i n t e r a c t i o n s y s t e m and a n e q u i v a l e n t damping r a t i o f r o m t h e m a g n i t u d e s o f r e s o n a n c e p e a k s , t h e r e s p o n s e t o n o n - s t a t i o n a r y random m o t i o n s s u c h a s a n e a r t h q u a k e c a n t h e n b e f o u n d d i r e c t l y f r o m a r e s p o n s e s p e c t r u m . The t e c h n i q u e p r e s e n t e d , t h e r e f o r e , b r i d g e s t h e gap b e t w e e n t h e r e s p o n s e f o r p u r e l y s i n u s o i d a l e x c i t a t i o n s and t h o s e f o r n o n - s t a t i o n a r y m o t i o n s s u c h a s e a r t h q u a k e s . Numerous s t u d i e s o f m u l t i - d e g r e e - o f - f r e e d o m s t r u c t u r e s h a v e b e e n p r e s e n t e d t h a t i n c o r p o r a t e t h e e f f e c t s o f f l e x i b l e f o u n d a t i o n s . T h e s e h a v e e i t h e r b e e n f o r s i n u s o i d a l e x c i t a t i o n s o r n u m e r i c a l r e s p o n s e c a l c u l a t i o n s u n d e r random- t y p e l o a d s . As f a r a s t h e a u t h o r i s a w a r e , t h e a b i l i t y t o draw g e n e r a l con- c l u s i o n s f r o m r e s u l t s o b t a i n e d f o r s i n u s o i d a l d i s t u r b a n c e s w h i c h would b e ap- p l i c a b l e t o random-type i n p u t s h a s n o t y e t b e e n a c h i e v e d . . I t i s i n d e e d p o s s i b l e t o i n t r o d u c e t h e a p p r o p r i a t e f r e q u e n c y f u n c t i o n s f l and f 2 f o r any t y p e of f o u n d a t i o n b e h a v i o u r , i n c l u d i n g t h e o n e s p r e s e n t e d b y t h e d i s c u s s e r i n p a p e r n o . 7. One c o u l d a l s o add a d d i t i o n a l amounts o f
damping t o t h e e l a s t i c h a l f - s p a c e r e s u l t s t o a c c o u n t f o r s m a l l h y s t e r e t i c e n e r g y l o s s i n t h e s o i l . T h i s i s a c h i e v e d by a p p r o p r i a t e m o d i f i c a t i o n s of t h e C c u r v e s s u c h a s t h o s e p r e s e n t e d i n F i g u r e 2 . A l i m i t e d s t u d y of t h i s n a t u r e h a s been p r e s e n t e d i n R e f e r e n c e 9.
Q u e s t i o n by: E . Varoglu
Could you comment on t h e r e a s o n f o r c h o o s i n g a model w i t h c i r c u l a r f o u n d a t i o n ?
I n ground c o m p l i a n c e a n a l y s i s , t h e r e l a t i o n between t h e t o t a l f o r c e and t h e d i s p l a c e m e n t depends upon t h e a s s u m p t i o n a b o u t t h e s t r e s s d i s t r i b u t i o n o v e r t h e f o u n d a t i o n a r e a . To u s e a u n i f o r m s t r e s s d i s t r i b u t i o n f o r a r i g i d f o u n d a t i o n may n o t b e a d e q u a t e i n o b t a i n i n g t o t a l f o r c e , d i s p l a c e m e n t r e l a t i o n f o r a r i g i d f o u n d a t i o n .
Reply by: J . H . R a i n e r
The c i r c u l a r f o u n d a t i o n was chosen a s m e r e l y o n e example of a f r e q u e n c y d e p e n d e n t f o u n d a t i o n . One c o u l d e q u a l l y w e l l c h o o s e a r e c t a n g u l a r , s q u a r e o r o t h e r g e o m e t r i c s h a p e a s l o n g a s t h e e x p e r i m e n t a l o r t h e o r e t i c a l complex s t i f f n e s s c h a r a c t e r i s t i c s a r e a v a i l a b l e . I t s h o u l d b e p o i n t e d o u t t h a t t h e r e s u l t s p r e s e n t e d a r e v a l i d f o r o t h e r f o u n d a t i o n s h a p e s p r o v i d e d t h e i r f r e q u e n c y - d e p e n d e n t c h a r a c t e r i s t i c s r e s e m b l e t h o s e g i v e n i n F i g u r e 2 . T h i s i s t h e c a s e f o r t h e s q u a r e and r e c t a n g u l a r f o u n d a t i o n s t r e a t e d i n R e f e r e n c e 1 0 . The w r i t e r a g r e e s w i t h t h e d i s c u s s e r t h a t t h e s t r e s s d i s t r i b u t i o n under t h e f o o t i n g a f f e c t s t h e s t i f f n e s s c h a r a c t e r i s t i c s . The work p r e s e n t e d h e r e , however, d o e s n o t c o n c e r n i t s e l f d i r e c t l y w i t h t h i s a s p e c t . R a t h e r , a s t a t e - ment of t h e problem t r e a t e d might b e a s f o l l o w s : g i v e n a p p r o p r i a t e f l e x i b l e f o u n d a t i o n p r o p e r t i e s , what i s t h e s t r u c t u r a l r e s p o n s e of t h e i n t e r a c t i o n s y s t e m u n d e r e a r t h q u a k e - t y p e l o a d i n g s ? The s u i t a b i l i t y and a p p l i c a b i l i t y of a t h e o r e t i c a l model t o r e p r e s e n t a p a r t i c u l a r f o u n d a t i o n and i t s a s s o c i a t e d s o i l must t h e n b e c o n s i d e r e d f o r any s p e c i f i c a p p l i c a t i o n . I n view of t h e r e s u l t s p r e s e n t e d i n F i g s . 6 and 9 , s m a l l v a r i a t i o n s i n f o u n d a t i o n s t i f f n e s s s h o u l d n o t g r e a t l y a f f e c t t h e f r e q u e n c y r a t i o wl/wo a n d , a s a consequence, t h e s t r u c t u r a l r e s p o n s e of t h e i n t e r a c t i o n s y s t e m .
