141
S E I S M I C R E S P O N S E OF S T R U C T U R E S F R E E T O
R O C K ON T H E I R F O U N D A T I O N S
A.J.Carr***
M . J . N . P r i e s t l e y * , R.J. E v i s o n * * ,
SUMMARY
The p o s s i b i l i t y of f o u n d a t i o n r o c k i n g o f shear w a l l s t r u c t u r e s d e s i g n e d
t o N Z S 4 2 0 3 is d i s c u s s e d .
T h e o r y d e v e l o p e d by H o u s n e r for the free r o c k i n g
o f a r i g i d b l o c k is c o m p a r e d w i t h e x p e r i m e n t a l r e s u l t s f r o m a s i m p l e
structural m o d e l with a number of d i f f e r e n t foundation c o n d i t i o n s .
A simple
d e s i g n m e t h o d for assessing m a x i m u m r o c k i n g d i s p l a c e m e n t s , u s i n g e q u i v a l e n t
e l a s t i c c h a r a c t e r i s t i c s a n d a r e s p o n s e - s p e c t r a a p p r o a c h is p r o p o s e d , a n d
c o m p a r e d w i t h r e s u l t s from s i m u l a t e d s e i s m i c e x c i t a t i o n of the m o d e l u s i n g an
electro-hydraulic shake-table.
A t y p i c a l d e s i g n e x a m p l e is i n c l u d e d .
1.
INTRODUCTION
Post-mortems on structural response
subsequent to seismic attack have revealed
u n e x p e c t e d b e h a v i o u r on some types of structures
that can only be attributed to rocking of the
entire s t r u c t u r e o n its f o u n d a t i o n s .
For
example, several seemingly unstable golf-ballon-a-tee types of elevated r e i n f o r c e d - c o n c r e t e
water tanks incurred minimal damage during the
Chilean earthquake of May 1960, while more
stable ground-supported reinforced concrete
tanks were severely d a m a g e d ( 1 ) .
I n s p e c t i o n of the foundations of the
apparently unstable tanks revealed clear
evidence that the structures had rocked on
their foundations.
Investigations after the
Arvin-Tehachapi earthquake (California, July
1952) i n d i c a t e d t h a t a n u m b e r of t a l l p e t r o l e u m cracking t o w e r s had escaped serious damage by
stretching their anchor bolts, and rocking on
their foundation pads * ' ,
It w a s a l s o n o t e d
that free-standing stone pillars supporting
h e a v y s t a t u e s in I n d i a n c i t i e s r e m a i n e d i n t a c t
w h i l e m o r e s t a b l e s t r u c t u r e s in t h e v i c i n i t y
were reduced to rubble.
Seismic r e s p o n s e involving uplift of the
m a j o r p o r t i o n o f t h e b a s e (rocking) is n o t ,
h o w e v e r , l i m i t e d to s u c h u n s t a b l e s t r u c t u r e s
as d e s c r i b e d in t h e p r e v i o u s p a r a g r a p h .
Design
to t h e p r o v i s i o n s o f N Z S 4 2 0 3 : 1 9 7 6 * 3 )
result in s i t u a t i o n s where rocking of p a r t or
a l l o f t h e s t r u c t u r e u n d e r s e i s m i c a t t a c k is
probable.
Consider the simplified structural
system i l l u s t r a t e d in Fig. la, c o n s i s t i n g of
end shear walls providing the required N-S
seismic resistance, with vertical load mainly
supported by flexible interior columns w h i c h
do n o t c o n t r i b u t e s i g n i f i c a n t l y to l a t e r a l
resistance.
Despite the comparatively squat
nature of the structure, the high ratio of
tributary floor m a s s for transverse seismic
r e s p o n s e to t r i b u t a r y f l o o r m a s s for v e r t i c a l
loads c a n r e s u l t in the b a s e seismic o v e r t u r n ing m o m e n t e x c e e d i n g the r e s t o r i n g g r a v i t y
moment.
c
In
*
some
c a s e s , the
designer
is
a
required to inhibit the r o c k i n g that w o u l d
r e s u l t , by i n c o r p o r a t i n g a m a s s i v e f o u n d a t i o n
b e a m (Fig. 1 6 ) . If t h e s t r u c t u r e i n F i g . la
is d e s i g n e d t o N Z S 4 2 0 3 f o r
Z o n e A in
Reinforced Concrete using Grade 380 steel,
a S t r u c t u r a l T y p e f a c t o r S = 1.6 a n d M a t e r i a l s
F a c t o r M = 1.0 a r e a p p r o p r i a t e .
Assuming
the w a l l to be designed on the b a s i s of a
f l e x u r a l u n d e r c a p a c i t y f a c t o r o f <j)f = 0 . 9 0 ,
the f o u n d a t i o n s w o u l d n e e d to b e d e s i g n e d
for an effective S x M v a l u e of
reinforced
concrete:
S x M x - ^ = 1 . 6 x l . 0 x
cj>
^r^0.9
f
=
2.4
,
where $
= 1.3 5 is t h e a p p r o p r i a t e o v e r c a p a c i t y f a c t o r to e n s u r e no r o c k i n g o r
foundation yield.
If t h e w a l l s w e r e
c o n s t r u c t e d o f r e i n f o r c e d m a s o n r y , <J)f = 0.65
a n d M - 1.2 a r e r e q u i r e d , a n d t h u s
Q
reinforced
^
S
x
™
M
x
^°
cp^
- —
masonry:
i
r
=
1.6
=
4.0
However, NZS
4203
x
n o
1.2
x
1.35
0 . ob
7r—^-7=-
,
states
n
not
R e a d e r in C i v i l E n g i n e e r i n g , U n i v e r s i t y
of C a n t e r b u r y .
**
Construction Engineer, N.Z. Structures Ltd.
* * * S e n i o r L e c t u r e r in C i v i l E n g i n e e r i n g ,
University of Canterbury.
"... no f o u n d a t i o n s y s t e m n e e d be d e s i g n e d
to r e s i s t forces and m o m e n t s g r e a t e r t h a n
those resulting from a h o r i z o n t a l force
c o r r e s p o n d i n g t o S x M = 2"
clause
3.3.6.3.1
The consequences of this loadlimitation clause are that some structural
systems will be allowed to rock on their
foundations under seismic attack.
It i s t h e w r i t e r s c o n t e n t i o n t h a t t h i s
anomoly rather than representing an unsafe
condition, may result in considerable
advantages by allowing r o c k i n g in some c a s e s .
I n F i g . lc a c o m p a r a t i v e l y l i g h t f o u n d a t i o n
is p r o v i d e d , d e s i g n e d o n the b a s i s of S x M
= 2.0 (for e x a m p l e ) .
If t h e w a l l r o c k s ,
the lateral forces developed w i l l be limited
to those inducing o v e r t u r n i n g a b o u t the t o e .
T h e w a l l could be s p e c i f i c a l l y d e s i g n e d to
remain elastic under these forces, thus
eliminating structural damage.
Similar
advantages to those resulting from b a s e isolation ^
will result.
The extent of
B U L L E T I N O F T H E N E W Z E A L A N D N A T I O N A L S O C I E T Y F O R E A R T H Q U A K E E N G I N E E R I N G , V O L . 1 1 , N O . 3, S E P T E M B E R , 1 9 7 8 .
142
r o c k i n g is l i k e l y t o b e s m a l l , a n d d a m a g e
can be expected to be confined to p l a s t i c
deformation of soil under each end of the
foundation beam.
This damage would be
easily repairable.
©
The viability of this approach would
depend on the extent of the rocking d i s placements induced.
Although instability
is i n c o n c e i v a b l e f o r t h e t y p e o f s t r u c t u r e s
r e p r e s e n t e d b y F i g . 1, e x c e p t b y s o i l
liquefaction, the incompatibility of vertical
deformation b e t w e e n the rocking shear walls
and t h e c o l u m n s h e l d in g r o u n d - c o n t a c t by
the gravity loads would result in t w i s t i n g
of floor s l a b s , and possibly some secondary
structural damage.
This paper describes
the m e c h a n i s m s involved in the rocking
response, and proposes a simple design
m e t h o d for p r e d i c t i n g rocking d i s p l a c e m e n t s
under seismic attack.
The research has
been limited to structural aspects of rocking
r e s p o n s e , and i t is a s s u m e d t h a t t h e soil
properties are such that under rocking a
c o m p a r a t i v e l y small c o n t a c t at t h e t o e is
sufficient to transmit the gravity loads
carried by the w a l l .
This assumption i s .
e x a m i n e d f u r t h e r i n S e c t i o n 4.
Bartlett
'
has recently produced analytical methods
describing the rocking r e s p o n s e of systems
where soil compliance forms a more significant
part of the total deformation.
where
2.
THEORY
D e s p i t e the importance of the rocking
p h e n o m e n o n as a seismic r e s p o n s e m e c h a n i s m ,
it h a s r e c e i v e d very l i t t l e a t t e n t i o n f r o m
researchers.
Meek(°) developed theory for
predicting the response of single degree of
freedom systems with no-tension capacity
rigid foundations, using a time-history
dynamic analysis approach.
His interest was
p r i m a r i l y in slender buildings w i t h high
fundamental p e r i o d s , for w h i c h he found
r e d u c t i o n to m a x i m u m lateral d i s p l a c e m e n t
and b a s e shear f o r c e , if r o c k i n g w a s
permitted.
S e x t o n ( ?' i n a d i s c u s s i o n t o
M e e k s p a p e r ( 6 ) d r e w a t t e n t i o n to t h e
fundamental differences between the rocking
response of squat stiff structures, and
tall flexible o n e s , and suggested that
b e n e f i t s m a y also be e x p e c t e d for s h o r t p e r i o d s t r u c t u r e s , t h o u g h r e d u c t i o n in
displacements will not necessarily result.
G
4
2.1
Housner's
Rocking
Block
A m o r e f u n d a m e n t a l study of the p r o b l e m
has been reported by H o u s n e r ^ ^ who derived
relationships governing the free vibration of
a
rigid rocking block.
(Fig. 2 ) .
He
s h o w e d that t h e f r e q u e n c y of t h e r o c k i n g
response decreased with increasing amplitude
2) a c c o r d i n g
to the
expression
. (1)
cosh
v
l-0 /a'
Q
WR
(2)
and
2b(2h)
12
m(
J
+
(2b)
- ^ 2
12
MR
+
(3)
2
is the m a s s m o m e n t of i n e r t i a of the block
about the point of rotation 0, where M =
total m a s s of block and m = m a s s / u n i t v o l u m e .
(9)
Housner
recognised that m u c h of the
a d v a n t a g e to b e g a i n e d from a l l o w i n g a
structure to rock is d e p e n d e n t o n the
e f f i c i e n c y of the r o c k i n g p h e n o m e n o n as an
energy dissipating mechanism.
He assumed
that impacts would produce no
bouncing'
and hence constitute purely inelastic
collisions, radiating energy from the rocking
system to the foundation h a l f - s p a c e .
By
equating momentum before and after impact,
he showed that the kinetic energy reduction
f a c t o r r, ( r a t i o o f k i n e t i c e n e r g y a f t e r
impact to kinetic energy immediately before
impact) c o u l d b e r e l a t e d to the b l o c k
dimensions by the expression
M R.2
1 c o s 2a)
(4)
(1
I
1
T h i s c o e f f i c i e n t e n a b l e d h i m t o pp ir e d i c t t h e
peak displacement
after the n
impact
during n a t u r a l decay of rocking i n terms of
t h e i n i t i a l d i s p l a c e m e n t <j> , a s
0
=
1
- /
(1- (1-
where
2.2
1
(8)
Beck and Skinner
describe similar
preliminary dynamic analyses of the South
Rangitikei Rail Bridge, a continuous
prestressed concrete box-girder with 70 m
h i g h t w i n - l e g g e d p i e r s w h i c h are free to rock
b y ' s t e p p i n g ' f r o m leg to leg in t h e t r a n s v e r s e d i r e c t i o n to l i m i t s e i s m i c f o r c e s .
H o w e v e r , in t h e i r a n a l y s e s , it w a s a s s u m e d
that structural damping would occur only
w i t h b o t h legs of a pier in contact with
the foundation, and the energy loss from
t h e s y s t e m o n i m p a c t of t h e l e g s w i t h t h e
foundation was ignored.
Consequently the
analyses of Beck and Skinner predicted
v e r y s l o w decay of rocking.
(see F i g .
and
d>
o
=
o
a
Equivalent Viscous
System
. (5)
(see F i g .
Damping
of
2)
a
Rocking
N o w the fraction of critical damping,
X , of a single degree of freedom oscillator
w i t h viscous damping under free decay of
vibration can be assessed from the relative
amplitudes of the displacements peaks :
2-rrm
n
(~
e
(6)
A
W h e r e m is the* n u m b e r o f c o m p l e t e c y c l e s
separating peak displacements A
and A .
S i n c e A a n d 4> a r e d i r e c t l y p r o p o r t i o n a l ,
and since there are two impacts per cycle.
Q
m
(7)
eqn
(6) c a n b e
irn
rewritten
(8)
e d>
n
Y
T h u s , b y d i v i d i n g e q u a t i o n (5) b y <J> , i n v e r t i n g ,
a n d s u b s t i t u t i n g i n e q u a t i o n (8) i t i s
possible to define an equivalent v i s c o u s
d a m p i n g of the rocking system.
Substituting
of trial v a l u e s r e v e a l s t h a t for 4> ^ 0.5,
t h e v a l u e o f d a m p i n g o b t a i n e d f r o m e q n s (5)
a n d (8) i s c o m p a r a t i v e l y i n s e n s i t i v e t o
initial amplitude $
a n d n u m b e r o f i m p a c t s n.
Q
0
0
143
and a r e l a t i o n s h i p b e t w e e n X and r can b e
d e v e l o p e d , w h i c h is shown g r a p h e d in f i g .
3.
V a l u e s of X will vary from this
relationship by less than ± 1 0 % for values
o f <j> < 0.5 a n d n < 1 6 .
o
2.3
Response
Spectra
Design
Approach
It is t h u s p o s s i b l e to r e p r e s e n t a
r o c k i n g b l o c k as a s i n g l e d e g r e e o f f r e e d o m
oscillator with constant damping, whose
period d e p e n d s on the amplitude of rocking.
If i t i s a s s u m e d t h a t t h e p e a k r e s p o n s e t o
seismic excitation depends only on the
e q u i v a l e n t e l a s t i c c h a r a c t e r i s t i c s (T, X )
at peak r e s p o n s e , then a t r i a l - a n d - e r r o r
response spectrum approach may be used to
determine the peak displacement.
Such
an analysis
proceeds
as
follows.
(1)
Using the no-rocking natural period and
damping of the structure, and the acceleration
response spectra of the design earthquake,
calculate the elastic response acceleration
and c h e c k t h a t t h i s w i l l in fact i n d u c e
rocking.
(2)
U s i n g e q n (4) a n d t h e r e l a t i o n s h i p
b e t w e e n r a n d X o f F i g . 3, c a l c u l a t e t h e
equivalent viscous damping X
of the r o c k i n g
system.
e
(3)
F r o m e q n (1) d e t e r m i n e t h e r e l a t i o n s h i p
between p e r i o d T and amplitude of rocking
at the c e n t r e of m a s s .
This is shown in F i g .
4a.
(4)
A n estimate of maximum rocking d i s p l a c e ment
is initially 'guessed' and the
c o r r e s p o n d i n g p e r i o d T ^ is r e a d o f f F i g . 4 a .
(5)
The maximum displacement response A2
of this equivalent elastic system
(T^,X )
is f o u n d f r o m t h e d i s p l a c e m e n t r e s p o n s e
s p e c t r a , as indicated in F i g . 4 b , w h i c h
represents the tripartite response spectra.
T h e d i s p l a c e m e n t A 2 is a r e f i n e d e s t i m a t e
o f t h e r e s p o n s e , a n d a n e w p e r i o d T 2 is r e a d
off F i g . 4 a , and u s e d in F i g . 4b to p r o d u c e
a new estimate A 3 of peak displacement.
e
(6)
The trial~and-error approach proceeds
w i t h a l t e r n a t e r e c o u r s e to F i g s . 4a and 4b
until convergence results.
This generally
takes three or four cycles.
3.
MODEL
Shake-Table
The
3.2
Rocking
Model
Dimensions of the m o d e l w e r e dictated
by t h e s i z e a n d c a p a c i t y of t h e s h a k e - t a b l e .
A simple one-sixth scale simulation of a
concrete masonry shear-wall structure with
prototype pre-rocking fundamental period of
0.4 s e c w a s a d o p t e d .
Fig. 5 shows relevant
dimensions of the m o d e l , w h i c h w a s designed
to have a model natural period of about
0.067 s e c , s a t i s f y i n g C a u c h y s i m i l i t u d e and
w a s p r o p o r t i o n e d so t h a t r o c k i n g w o u l d
initiate at a response acceleration of
0.34 g.
Although this would imply prototype
r o c k i n g at an u n r e a l i s t i c a l l y l o w r e s p o n s e
a c c e l e r a t i o n o f 0 . 0 5 7 g, i t e n a b l e d e x t e n s i v e
examination of the rocking phenomenon at
comparatively low levels of e a r t h q u a k e
excitation, and resulted in a m o d e l of
convenient dimensions.
Structural form of
the m o d e l w a s chosen to g i v e t h e r e q u i r e d
characteristics, rather than geometric
similitude.
To ensure even reaction of the model
dead w e i g h t , it w a s s u p p o r t e d o n four
circular discs of adjustable h e i g h t , with
thin insertion-rubber pads glued to the
b o t t o m of the d i s c s to p r e v e n t d a m a g e to
the table surface on impact during rocking.
Three main foundation conditions were
investigated :
(a)
Model supported directly on table,
(b)
m o d e l s u p p o r t e d o n 25 m m l a y e r o f h a r d ness IRHD rubber, simulating a deformable
foundation material.
The design of this pad
w a s such t h a t a t h r e e - f o l d i n c r e a s e in n a t u r a l
no-rocking period resulted.
(c)
A s f o r (b) b u t w i t h t h e f o u r l e v e l l i n g
feet r e m o v e d to p r o v i d e c o n t i n u o u s c o n t a c t
between the m o d e l base and rubber pad.
STUDIES
The authors are not aware of any published
reports o n simulated seismic testing of models
free to rock on their f o u n d a t i o n s , t h o u g h a
recent p a p e r by Kelly and Tsztoo^
' describes
dynamic testing of a frame where exterior
columns w e r e allowed to lift under seismic
excitation.
Because of this paucity of
experimental data, a simple structural model
was constructed, and tested on the S h a k e T a b l e a t t h e D e p a r t m e n t of C i v i l E n g i n e e r i n g ,
University of Canterbury.
Aims of the
e x p e r i m e n t a l s t u d y w e r e t o (a) c h e c k t h e
theoretical equations developed by Housner
a n d (b) c o m p a r e r e s p o n s e o f t h e m o d e l t o
simulated seismic excitation, with values
predicted using the equivalent response
spectra approach developed in Section 2.3.
3.1
a 1.5 m s q u a r e A l u m i n i u m t a b l e r u n n i n g o n
100 m m d i a m e t e r s t e e l g u i d e s t h r o u g h
Glacier DU-PTFE bearings.
D r i v e is p r o v i d e d
by a h o r i z o n t a l d o u b l e - a c t i n g 50 k N h y d r a u l i c
jack powered by a 7 5 litre/min p u m p , and
c o n t r o l l e d by an M T S s e r v o - h y d r a u l i c
electronic controller.
This enables the
t a b l e to b e d r i v e n h o r i z o n t a l l y in a n u m b e r
of m o d e s ranging from simple s i n u s o i d a l
m o t i o n , using a function g e n e r a t o r as
signal source, to simulated seismic ground
motion using an FM tape recorder with a
s u i t a b l e s c a l e d e a r t h q u a k e r e c o r d as s i g n a l
source.
In a d d i t i o n , l i m i t e d ' f r e e - r o c k i n g '
tests were carried o u t in the field,
supporting the m o d e l on w e l l - c o m p a c t e d clay
soil.
These different foundation conditions
w e r e not designed to m o d e l specific realistic
values, but rather were used to investigate
the general significance of foundation
compliance on results.
3.3
Tests
Performed
Four categories of
investigated, namely
dynamic
test
were
1.
Natural decay of free v i b r a t i o n at
amplitudes less than t h a t r e q u i r e d in
initiate rocking.
This enabled natural
n o - r o c k i n g f r e q u e n c i e s a n d d a m p i n g to be
measured.
Facility
shake-table
used
for the tests
is
2.
Natural decay of rocking m o t i o n .
The
m o d e l was displaced to an a m p l i t u d e exceeding
144
that required to cause u p l i f t , and released.
Measurements of amplitude of rocking, natural
period and rate of decay provided experimental
data for comparison w i t h H o u s n e r ' s e q u a t i o n s .
3.
Sinusoidal Excitation.
The model was
s u b j e c t e d to f o r c e d s i n u s o i d a l b a s e a c c e l e r a t i o n to f a c i l i t a t e the study o f r e s p o n s e to
different frequencies.
4.
Simulated seismic excitation.
A suitably
scaled record o f the N-S c o m p o n e n t of the
1940 El Centro earthquake was used as table
excitation.
To provide similitude, model/
prototype frequency and acceleration scales
o f 6.0 w e r e r e q u i r e d .
The earthquake
a c c e l e r a t i o n r e c o r d , so s c a l e d , w a s i n t e g r a t e d
t w i c e by c o m p u t e r and r e c o r d e d o n a n a l o g u e
magnetic tape as a displacement trace.
Scaling of magnitude was provided by
attenuation through the MTS Controller/ and
the m o d e l w a s s u b j e c t e d to v a r i o u s levels of
e x c i t a t i o n c o r r e s p o n d i n g to the r a n g e 2 0 % to
100% of El Centro 1940 N - S .
Because of space limitations only
results of the free-rocking and seismic
testing a r e included in this p a p e r .
More
c o m p l e t e i n f o r m a t i o n is i n c u d e d i n r e f . N o .
11.
3.4
Instrumentation
M o d e l d i s p l a c e m e n t s r e l a t i v e to the
shake-table w e r e m o n i t o r e d by H e w l e t t
Packard DCDT's , and a c c e l e r a t i o n s w e r e
measured by Kyowa AS-10B accelerometers.
Instrumentation locations are included in
F i g . 5.
In addition, table displacement
and a c c e l e r a t i o n w e r e m o n i t o r e d to p r o v i d e
comparison with specified values.
Analogue
traces of r e s p o n s e w e r e r e c o r d e d on a 25
channel Bryan Southern U-V recorder.
Fig.
4.
4.1
6 shows
EXPERIMENTAL
Natural
the model
under
of
The natural decay displacement traces
also p r o v i d e d d a t a for a s s e s s i n g H o u s n e r ' s
relationship between frequency and amplitude
of r o c k i n g , g i v e n in eqn ( 1 ) . F i g . 8b
shows the comparison between theory and
experimental results for rigid base and
rubber-pad base conditions.
A g r e e m e n t is
excellent for both base c o n d i t i o n s .
It
will be noted that for the rubber-pad base,
m a x i m u m r o c k i n g f r e q u e n c y i s 3.7 H z , c o r r e s ponding to the natural no-rocking frequency.
Above this level, H o u s n e r s theory does not
apply.
1
4.2
Response
to El Centro
1940 N-S
Excitation
A t y p i c a l m o d e l response to simulated
s e i s m i c e x c i t a t i o n s i s s h o w n i n F i g . 9,
w h i c h traces t i m e - h i s t o r i e s of m o d e l a c c e l erations and displacement resulting from
0.6 x E l C e n t r o e x c i t a t i o n f o r t h e r u b b e r - p a d
b a s e condit.ion.
It w i l l b e n o t e d t h a t
rocking initiates almost immediately in the
response record, and continues to the end
of the b a s e e x c i t a t i o n .
Peak response disp l a c e m e n t o c c u r s a t a b o u t 1.5 s e c (9.0 s e c
in p r o t o t y p e t i m e - s c a l e ) a n d d a m p s o u t
comparatively rapidly.
Vertical accelerations
r e s u l t i n g from i m p a c t of the feet o n the
rubber pad are clearly apparent and are
r e f l e c t e d in the h o r i z o n t a l a c c e l e r a t i o n
record.
test.
RESULTS
Decay
a best fit w i t h the experimental d a t a , and
is s u b s t a n t i a l l y d i f f e r e n t from the v a l u e
o f r = 0.70 p r e d i c t e d b y e q n (4) f r o m t h e
model parameters.
I t is a p p a r e n t t h a t t h e
i m p a c t s w e r e n o t t o t a l l y i n e l a s t i c , as
a s s u m e d by H o u s n e r , and t h a t a s i g n i f i c a n t
proportion of the energy of impact was
being returned to the rocking system.
It
is of i n t e r e s t t h a t t h e e x p e r i m e n t a l v a l u e
of r w a s e f f e c t i v e l y i n d e p e n d e n t of the
foundation condition.
Rocking
A typical composite record of free
rocking decay is shown for the rigid base
c o n d i t i o n i n F i g . 7.
It will be noted that
though the vertical displacements at front
and b a c k feet do not indicate s i g n i f i c a n t
* bounce', high vertical accelerations were
induced on i m p a c t , w h i c h are reflected in
significant superimposed accelerations
on the horizontal acceleration record.
These p e r t u r b a t i o n s at the natural structural
frequency do not appear to affect the basic
rocking response.
It was noticeable that
the t r a c e s r e c o r d e d for the flexible b a s e
conditions (rubber-pad foundations and
soil-base) contained much lower superimposed
a c c e l e r a t i o n s from the i m p a c t s , as w o u l d b e
expected ( H ) .
From the horizontal displacement records
for t h e f o u r b a s e c o n d i t i o n s (e.g. c e n t r a l
t r a c e i n F i g . 7) t h e e x p e r i m e n t a l r a t e o f
rocking decay, ^ n , could be determined as a
function of
^o
number of i m p a c t s , n.
The r e s u l t s for all b a s e c o n d i t i o n s fell on
t h e s a m e c u r v e , a n d a r e c o m p a r e d i n F i g . 8a
with predictions based on equation (5),
using a value of r = 0.87.
It will be seen
that the t h e o r e t i c a l decay v a l u e is in good
agreement with average experimental behaviour.
T h e v a l u e o f r = 0.87 w a s a d o p t e d t o p r o v i d e
The design approach developed in section
2.3 w a s u s e d t o e s t i m a t e t h e p e a k m o d e l
response.
First model characteristics were
converted to equivalent p r o t o t y p e values
using the m o d e l / p r o t o t y p e relationships :
accelerations
:
a
displacements
:
A
p
= L
= L
p
periods
:
T
r
p
^ a
a
r
= L
(9)
m
r
(10)
m
T
(11)
m
where L
= 6 is t h e p r o t o t y p e / m o d e l l e n g t h
r a t i o , and p a n d m r e f e r to p r o t o t y p e and
model values respectively.
r
T h e e x p e r i m e n t a l v a l u e o f r - 0.87 w a s
a d o p t e d , r e s u l t i n g , from F i g . 3, in a n
equivalent viscous damping of 4.8%.
Using
a n a p p r o p r i a t e l y s c a l e d v e r s i o n o f F i g . 8b
and the El Centro N-S tripartite response
s p e c t r a (see F i g . 1 2 ) , r e s u l t s i n a p r o t o t y p e
d i s p l a c e m e n t of 300 m m a f t e r f o u r t r i a l - a n d error cycles b a s e d o n an initial g u e s s
of
120 m m d i s p l a c e m e n t .
This scales to a
p r e d i c t e d m a x i m u m m o d e l d i s p l a c e m e n t of 50 m m
(eqn 1 0 ) w h i c h c o m p a r e s w i t h t h e m e a s u r e d
v a l u e of 45 m m .
T h i s a g r e e m e n t is felt to
be acceptable considering the approximations
inherant in the a n a l y s i s .
1
5.
DESIGN
1
EXAMPLE
The experimental
Housner's analysis of
results
rocking
confirm
behaviour
and
145
p r o v i d e some support for the p r o p o s e d s i m p l i fied a n a l y s i s t e c h n i q u e described in section
2.
It m a y b e o f i n t e r e s t , h o w e v e r , t o
consider a typical design example and
investigate the likely response using the
response-spectra analysis.
Consider the N-S response of the fivestorey masonry structure shown in F i g . 1 0 .
The equivalent weight of beams and slabs
i s t a k e n a s 5.0 k P a , a n d s e i s m i c l i v e - l o a d
a s 1.67 k P a .
Half the total Dead plus Live
Load contributes to the seismic shear of each
end w a l l , b u t vertical load for each wall
comes only from the wall dead weight plus a
5.0 m w i d e t r i b u t a r y s t r i p f r o m e a c h f l o o r .
Including roof and ground floor, total vertical
l o a d i s e s t i m a t e d a t 4.0 M N / w a l l .
For
normal seismic analysis, the mass distribution
may be replaced by the equivalent No-Rocking
1
F m o d e l in Fig. 1 0 , which results in a
n a t u r a l p e r i o d o f T = 0.4 3 s e c , b a s e d o n
E
- 5.0 G P
.
1
m
analysis
Will
(1)
steps
(2)
Structure
Rock
9.81 - x 1 2 . 9 > 4.0 M N x 7 . 0 ,
g
w h e r e a is t h e r e s p o n s e a c c e l e r a t i o n
required to induce rocking, and the length
of toe i n c o n t a c t w i t h soil is a s s u m e d to
b e 2.0 m , g i v i n g a l e v e r a r m o f t h e r e s t o r i n g
m o m e n t o f 7.0 m .
.*. a > 0. 2 4 5 g
(12)
for
For simplicity, assume the total mass
contributing to seismic shear to the wall
is u n i f o r m l y d i s t r i b u t e d o v e r t h e w a l l a r e a ,
rather than concentrated at floor heights.
T h e a p p r o x i m a t i o n i n v o l v e d is s m a l l .
D i s t r i b u t e d m a s s p e r u n i t a r e a is t h e n
1200
17.5 x 15.0
2
-
0.15 x
thus
1.6
Vd =
ie
a =
x
1.2 x
requirements
1.0
x
1.0
0.288
0.443
g.
(13)
g
(14)
If o v e r s t r e n g t h d u e to s t r a i n - h a r d e n i n g e t c .
d e v e l o p s , a n o v e r c a p a c i t y f a c t o r (J> = 1.35
could be appropriate for grade 380 steel.
Thus
0
1.35 x
0.443 g =
0.598
(3)
7) +
= M
1R
1.4 m
Io = m ( I x x + +
Iyy
Now
I o = 4.57
R = y/9
(17.5
g
(15)
C o m p a r i n g e q u a t i o n s (14) a n d (15) w i t h e q n
(12) i t i s c l e a r t h a t r e s p o n s e a t d e s i g n
level loading w i l l result in rocking.
For further confirmation of rocking,
check t h e e l a s t i c r e s p o n s e s p e c t r a to
ensure elastic response exceeds code requirements.
From Skinner's response spectra*
'
u s i n g T = 0.43 s e c a n d A = 5% c r i t i c a l ,
a c c e l e r a t i o n r e s p o n s e = 0.91 g .
Thus rocking
x
3
15 +
12
2
130
Io
209
107
tm
Now
W = 4.0
MN
15
3
x
17.5)
t.m
4
From Eqn (2),
P
4.0 x 1 0
=
x
2.091 x
11.4
10
8
0.463
Period
of
Rocking
Inverting
between period
_
4 cosh
-1
eqn ( 1 ) , the relationship
and amplitude is
1
(l-9o/a)
Now
a = tan ^ ~ =
and
0 o = AR /
where
0 . 2 8 8 W,
T h i s e x c e e d s t h e v a l u e g i v e n i n e q n (12)
to i n d u c e r o c k i n g .
Note also that the code
value is based on d e p e n d a b l e flexural strength,
u s i n g a n u n d e r c a p a c i t y f a c t o r <j)f = 0 . 6 5 .
Based on ideal strength, response acceleration
will be
0.288
0.65
eqn
tonnes/m
2
rocking.
Compare with NZS 4 2 0 3 ^
Zone A, Class III
CSMIR
4.57
from
T
for
Response
1200 x
?
0.900 x
the r e q u i r e m e n t
Rocking
are
Compare the wall overturning shear
with code^ shear.
Rocking will occur when
the overturning moment exceeds the restoring
moment.
That is, when
is
assured.
2
a
The
is
Thus
T
=
This
(16)
0.661 rad.
17.5
A R is t h e r o o f
eqn
8.64
displacement.
(16) r e d u c e s
to
1
-1 ,
cosh
(1-0.0864 A )
relationship
is p l o t t e d
(17)
in F i g . 1 1 .
Damping
S u b s t i t u t i n g i n t o e q n (4) f o r t h e
e n e r g y r e d u c t i o n f a c t o r r, g i v e s
[7
L_
1 2 0 0 x 13
209 107
(1 - c o s
(2 x
37.9°)
0.21
T h u s e q u a t i o n (4) p r e d i c t s a v e r y l o w v a l u e
for squat structures such as t h e e x a m p l e
wall.
However, since the model studies
indicated the calculated v a l u e for r u n d e r estimates the true v a l u e , conservatively
adopt r = 0.50, giving, from F i g . e, X = 2 3 % .
The trial-and-error approach using the
r e l a t i o n s h i p of F i g . 11 and t h e t r i p a r t i t e
response spectra for El Centro N-S 1940
(Fig. 12) r e s u l t s in a c e n t r e - o f - m a s s d i s p l a c e m e n t of a b o u t 80 m m , c o r r e s p o n d i n g t o
a roof level displacement of 160 m m , occurring
f o r T = 1.6 s e c .
146
The extent of rocking indicated by this
e x a m p l e is n o t e x c e s s i v e , b u t p o s s i b l e d a m a g e
resulting from twisting of floor slabs and
soil yield w o u l d need to be b a l a n c e d against
r e d u c e d d a m a g e to t h e s h e a r w a l l s , a n d
s a v i n g s in f o u n d a t i o n d e s i g n .
It s h o u l d b e
noted that s e v e r a l r e c o r d e d e a r t h q u a k e s
(eg P a r k f i e l d 1 9 6 6 a n d P a c o i m a S 7 5 W 1 9 7 1 ) h a v e
m o r e s e v e r e s p e c t r a l r e s p o n s e in t h e l o n g p e r i o d r e g i o n , a n d c o u l d b e e x p e c t e d to
induce more extensive rocking than El Centro
N-S 1940.
6.
4.
5.
6.
CONCLUSIONS
(1)
T h e r e s p o n s e of a s t r u c t u r e free to
rock o n its f o u n d a t i o n s o f f e r s a m e a n s of
b a s e - i s o l a t i o n , in that lateral a c c e l e r a t i o n s
are limited to the level inducing r o c k i n g .
Structural d a m a g e can be reduced by designing
s t r u c t u r a l e l e m e n t s to r e m a i n elastic, at t h e
rocking acceleration.
7.
8.
9.
(2)
Housner's equations describing the
relationship between frequency and amplitude
for a r o c k i n g b l o c k h a v e b e e n v e r i f i e d f o r
a simple structural m o d e l .
Foundation
conditions did not have a significant influence
on the rocking r e s p o n s e .
However, Housner s
assumption that rocking impacts would
r e p r e s e n t i n e l a s t i c c o l l i s i o n s w a s found to
be u n c o n s e r v a t i v e .
10.
1
11.
(3)
Extension of H o u s n e r s theory resulted
in the d e v e l o p m e n t of a s i m p l e m e t h o d for
predicting m a x i m u m d i s p l a c e m e n t of rocking,
by u s e of d i s p l a c e m e n t r e s p o n s e s p e c t r a and an
e q u i v a l e n t e l a s t i c r e p r e s e n t a t i o n of the
rocking system.
Limited shake-table testing
provided reasonable v e r i f i c a t i o n of the
theory.
1
12.
13.
(4)
The approach developed was intended to
p r o v i d e a m e a n s for e s t i m a t i n g the rocking
response of b u i l d i n g s t r u c t u r e s .
Application
to o t h e r s t r u c t u r e s , s u c h a s b r i d g e p i e r s
and c h i m n e y s , e x a m p l e s of w h i c h have already
been designed to rock during earthquakes,
is o b v i o u s .
Non-structural applications ,
such as rocking of stacked c o n t a i n e r s , could
also be considered.
(5)
V e r i f i c a t i o n of t h e t h e o r y h a s so far
been limited to a single m o d e l and a single
earthquake record.
F u r t h e r r e s e a r c h is
needed to t e s t the s c o p e of a p p l i c a b l i l i t y
of the m e t h o d .
In p a r t i c u l a r , the h i g h
equivalent v i s c o u s damping predicted for
squat rocking structures needs verification.
ACKNOWLEDGEMENTS
T h e e x p e r i m e n t a l w o r k d e s c r i b e d in t h i s
p a p e r w a s p a r t of a n M . E . p r o j e c t by E v i s o n ,
supervised by Priestley and Carr.
Grateful
a c k n o w l e d g e m e n t is m a d e of f i n a n c i a l s u p p o r t
by H o l m e s W o o d P o o l e a n d J o h n s t o n e , C o n s u l t i n g
Engineers, Christchurch.
REFERENCES
1.
2.
3.
C l o u d , W . K., " P e r i o d M e a s u r e m e n t s of
S t r u c t u r e s in C h i l e " , B u l l . S e i s m o l o g i c a l
Society of America.
V o l . 53 N o . 2 ,
Feb. 1963, pp 359-379.
H o u s n e r , G. W . , "Limit D e s i g n of S t r u c t u r e s
t o R e s i s t E a r t h q u a k e s " , P r o c . 3rd W o r l d
Conf. on Earthquake Eng. 1956, pp 5.1-5.13.
S . A . N . Z . , "NZS 4 2 0 3 : C o d e of P r a c t i c e
for G e n e r a l S t r u c t u r a l D e s i g n and D e s i g n
Paper
Loadings for Buildings", P u b . S.A.N.Z.,
W e l l i n g t o n , 1 9 7 6 , 80 p p .
S k i n n e r , R. I. a n d M c V e r r y , G. H . ,
"Base Isolation for Increased Earthquake
Resistance of Buildings", B u l l . N . Z .
N a t . S o c f o r E . E . , V o l . 8, N o . 2 , J u n e ,
1975, pp 93-101.
B a r t l e t t , P. E . , " F o u n d a t i o n R o c k i n g
on a Clay Soil", M . E . Thesis, Rept.
N o . 1 5 4 , S c h o o l of Eng., U n i v . of
A u c k l a n d , N o v . 1976, 144 p p .
M e e k , J. W . , " E f f e c t s o f F o u n d a t i o n
Tipping on Dynamic Response", Journal
Struct. Div. ASCE Vol. 101, No. ST7
July 1975, pp 1297-1311.
S e x t o n , H . J., D i s c u s s i o n o f r e f 6.
J o u r n . S t r u c t . Div. A S C E V o l . 102 No
ST 6 June 1976, p 1262.
B e c k , J. L . a n d S k i n n e r , R . I . , " T h e
S e i s m i c R e s p o n s e of a P r o p o s e d R e i n forced Concrete Railway Viaduct", DSIR
R e p t N o . 3 6 9 , M a y 1 9 7 2 , 17 p p .
H o u s n e r , G. W . , " T h e B e h a v i o u r o f
Inverted Pendulum Structures During
E a r t h q u a k e s " , B u l l . S e i s . S o c . of Am.,
V o l . 5 3 , N o . 2, F e b . 1 9 6 3 , pp 4 0 3 - 4 1 7 .
K e l l y , J. M . a n d T s z t o o , D . F.,
"Earthquake Simulation Testing of a
Stepping Frame with Energy-Absorbing
Devices".
Bull. N . Z . Nat. Soc. for
E . E . , V o l . 1 0 , N o . 4, D e c . 1 9 7 7 p p
196-207.
E v i s o n , R. J. , " R o c k i n g F o u n d a t i o n s " ,
M . E . Thesis, Dept. Civil Eng., Res.
Rept. No. 77-8, University of Canterbury,
F e b . 1 9 7 7 , 96 p p .
Priestley, M.J.N., "Seismic Resistance
of Reinforced Concrete Masonry Shear
Walls with High Steel Percentages",
Bull. N . Z . Nat. Soc. E.E., Vol. 10,
N o . 1, M a r c h , 1 9 7 7 .
pp 1-16.
S k i n n e r , R. I . , " H a n d b o o k f o r E a r t h q u a k e G e n e r a t e d F o r c e s and M o v e m e n t s in T a l l
Buildings",N.Z. DSIR Bull. No. 166,
1964.
received
7 August,
1978.
147
N
t
f l o o r area
lateral
(a)
Floor
contributing
load ,
Plan:
to
seismic
N-S
Distribution
of
Floor
Loads
FIGURE 2:
HOUSNER'S
ROCKING
(b) Heavy
Foundation
- Ductile
Beam
(c)
Response
Light
Foundation
-Rocking
BLOCKS
Beam
Response
FIGURE 1: A L T E R N A T I V E P H I L O S O P H I E S FOR S E I S M I C
R E S P O N S E OF C A N T I L E V E R S H E A R W A L L S
T, \
Period
Energy
Reduction
Factor,
r.
FIGURE 3: A P P R O X I M A T E R E L A T I O N S H I P BETWEEN
E Q U I V A L E N T V I S C O U S D A M P I N G A N D ENERGY
REDUCTION FACTOR
{b)
TRIPARTITE
RESPONSE
SPECTRA
FIGURE 4 : E S T I M A T E O F M A X I M U M
DISPLACEMENT FROM RESPONSE
SPECTRA
148
FIGURE 5: SCHEMATIC OF MODEL ON SHAKE-TABLE
FIGURE 6: M O D E L UNDER TEST
FIGURE 7: M O D E L N A T U R A L ROCKING RESPONSE DECAY ( r i g i d base)
149
Theory
4
(a)
8
No.
DECAY
of
OF
r = O. 8 7
Exp.
rigid
Exp.
rubber
Exp.
soil
12
Impacts
16
(n)
base
pad
base
20
ROCKING
10
Theory
\
a
cr
(f-lousner)
Exp.
rigid
Exp.
rubber
base
pad
4
0)
No-rocking
for
rubber
freq.
pad
16
24
Horizontal
(b)
FREQUENCY
vs
Displacement
32
(mm)
AMPLITUDE
FIGURE 8 : THEORY vs. EXPERIMENT FOR MODEL
FIGURE 9 : M O D E L R E S P O N S E TO 0.6 x EL CENTRO 1 9 4 0 N-S (rubber pad + feet)
150
Shear
Waifs
I
"~
m
—•
N
m m m m
A
„190mm
11
•
s
30 m
E l e v a t ion
Typical
Floor
Plan
»100t
>200t
900 t
>200t
>200t
>200t
*.4sy300t
Wall
Elevation
Wall
Masses
N-S
Exci t a t i o n
for
No-Rocking
1*F
Model
FIGURE 10: SIMPLIFIED M A S O N R Y BUILDING FOR DESIGN E X A M P L E
D A M P I N G V A L U E S A R E 0. 2. 5. 10 A N D 2 0 P E R C E N T O F C R I T I C A L
cn
c
o
or
o
0
250
Roof
500
Level
750
Displacement
1000
(mm)
FIGURE 11: ROCKING C H A R A C T E R I S T I C S OF
EXAMPLE
STRUCTURE
.04
. 0 6 .08 .1
.4
£ 3 1
PERIOD
(sees)
4
6
8
10
20
FIGURE 12: TRIPARTITE R E S P O N S E SPECTRA,
EL CENTRO 1 9 4 0
N-S