Research on earthquake resistant structures
by Constantine A Markellis
A THESIS Submitted to the Graduate Committee in partial fulfillment of the requirements for the
degree of Master of Science in Civil Engineering
Montana State University
© Copyright by Constantine A Markellis (1949)
Abstract:
no abstract found in this volume EESEABCH OMEARTHQUAKE RESISTANT STBtJCTUEES
by
CONSTANTINE A. MAHtBLLIS
A THESIS
Submitted to the Graduate Committee
In
p a rtia l fulfillm ent of the requirements
fo r the degree of
Master of Science In C ivil Engineering
at
Montana State College
Approved:
In Charge of Major Work
/\\i
2-
ACKHOWIEDGEMEHT
I am gratefu lly indebted to Associate Professor B. C. DeStort
of the Department of C ir il Engineering of Montana State College
fo r h is guidance throughout my Graduate Work and fo r h is help­
fu l information and assistance with ny Thesis.
I am also indebted to my ty p is t, Valencia Kabalin, and to my
friends in Hudson House, Montana State College, fo r th e ir cooperation
in correcting the English of my paper.
C. A. M.
9 2 6 1 1
-3 -
TABLE QP CONTENTS
Patte
g
ACKSOWLEDCTfflSHT------ -------- --------------------------- -- ----------------- -IBTRODOCTICai_______________________________________________________
5
CAUSES CF EABTa^UAKES ----------------- ----------------------------------------------------
Q
EABTBQUAKE WAVES------------------- --------------------------------------------- 12
DERIVATION OF THE HORIZONTAL SEISMIC FORCE----------------------------
15
Horizontal forces produced by eona of the
disastrous earthquakes of the world - - - - - - - - - - - -
17
INVESTIGATION OF EABTHQIUAKE STRESSES IN A BUILDING FRANE - - - -
19
ELANS OF THE BUILDING FRAME INVESTIGATED-------------------- -- _ 20, 21
COMPUTATIONS OF THE MOMENTS--------------------------------------------- -- -
22
Momenta due to the v e rtic a l loads on the r o o f --------- -- —
22
"
" "
"
"
" " 5th f l o o r ------------ 25
DESIGN OF THE CEILING BEAM K-K3 ------ ---------- --------------------------- 30
HORIZONTAL WIND SHEARS------ -- ------------------------------
36
HORIZONTAL SEISMIC FORCES BY THE LOS ANGELES CITY
BUILDING CODE ----------------- -- - 37
HORIZONTAL SEISMIC FORCES BY THE UNIFORM BUILDING CQDR------------ 39
DIAGRAMS FOR WIND MOMENTS
----------------
kl
DIAGRAMS OF EARTHQUAKE MOMENTS BY THE LOG ANGELES CITY
BUILDING CODEREQUIREMENTS------------- 46
DIAGRAMS OF EARTHQUAKE MOMENTS BY THE UNIFORM BUILDING
CODEREQUIREMENTS------------------------ 51
-4Piage
DESIGN OF BEAM HS
Uniform Building Code requirem ents------ -- - --------- -- - —
56
Los Angeles City Building Code requirements ----------------- -
57
DESIGN OF COLUMN GS-HS
Uniform Building Code requirements- - - - - - - - - - - - -
58
Los Angeles City Building Code requirements --------- -- - —
60
DESIGN OF FOOTINGS---------------------------------
---------------------- - 6 l
EESION OF TIES BETWEEN FOOTINGS BI THE UNIFORM BUILDING CODE
REQUIREMENTS -----------------
69
DESIGN OF TIES BETWEEN FOOTINGS BI THE LOS AIKffiLES CITT
CONCLUSION-------------
73
LITERATURE CITED-------------------- -- -------------------------------------- -
75
LITERATURE CONSULTED ~ ----------------------------------------------------------- 76
-5lUTRODUCTION
Earthquakes have a ttra c te d universal atten tio n from the e a rlie s t
times, and cm account of th e ir destructive power i t Is not surprising
th a t they used to be regarded as supernatural phenomena.
Among the e a rlie s t existing records of earthquakes are those
mentioned in the Bible. Also the w ritings of Serodotua, Pliny, Livy,
e t c ., show the In terest which earthquakes a ttra cte d In early ages.
Speaking generally, i t may be said th a t the w ritings of the
ancients, and those of the Middle Ages, down to the commencement of the
Nineteenth Century, tended to the propagation of superstition and to the
theories based on speculations with few and imperfect facts fo r th e ir
foundation.
The development of the study of earthquakes on a ratio n al basis
may be regarded as dating from the Nineteenth Century.
The science of seismology named from the Greek celsmos, an earth­
quake, and logos, a discourse. In i t s simplest form means the study of
earthquakes, but i t now embraces the study of other earth movements due
to a variety of causes.
The English word earthquake, the German erdbeden, the French
tremble meat de te rr a , the Spanish terremoto, the Japanese jish ln , e tc .,
a l l mean "earth shaking" when lite r a lly tran slated .
The source from which the earthquake originates Is called the
"focus," and the region of the surface !m ediately above the focus is
termed the "epicentre." The time a t which a shock occurs a t the focus
Is referred to as the "time of origin" of the earthquake.
-6 -
llaers are enormous varlatioaoe in the areas cnrer which earthquakes
are perceptible and in the destructive effects which acecepany the shocks.
At one end of the scale are the great earthquakes f e l t over hundreds of
thousands of square miles# and accompanied by great loss of lif e and
extensive damage to property; a t the other are the small shocks f e l t
only by a few people and over a very lim ited region. With increasing
distance from the epicentre the disturbance gets less and le ss and
eventually la imperceptible. The movements recorded by Instruments a t
great distances beyond the area in which the earthquake can be f e l t are
said to be teleseism ic.
The earth is very nearly spherical and so the distance from an
earthquake to any place may be measured as an arc of a great c irc le .
The
study of earthquakes is a branch of the modem science of geophysics,
which, as i t s name implies, covers a l l the phenomena of the physics of
the earth .
The seismologist is interested chiefly in the development of
instruments fo r recording earthquakes, and in obtaining from the study
of the records a l l the information he can about the earthquakes and about
the m aterials inside the earth .
Seismology in v ites the cooperation of workers and thinkers in
almost every department of science. Mathematicians are faced with many
problems relatin g to the e la s tic ity of so lid s, the wave motion propa­
gated from a disturbance, and the response Cf the instruments to applied
o scillatio n s of prescribed forms.
-7 -
Turoing to the more p ractica l side of seismology, the greatest need
has been to increase our knowledge of earthquakes so th a t we can lessen
the destruction caused by them. For th is purpose i t is necessary to
examine the effects of earthquakes upon buildings and the methods which
should be adopted to avoid damage to the structures in earthquake-shaken
countries.
Here we face problems which demand the a tten tio n of engi­
neers, a rc h ite c ts, and b u ild ers,
(hi the engineering side there are
problems connected with the location and best methods of construction
to be adopted fo r houses, facto rie s, bridges, vaterm.ins, e tc .
The re su lts obtained from observations of earthquake damage are
u tiliz e d by the engineer in designing the buildings to be erected in
seismic regions.
Seiaaologlcal information is necessary when insurance companies are
requested to cover property against earthquake damage.
Earthquakes are of more common occurrence than most people realize,
and no one place in the world is much more susceptible to them than
others. According to Captain N. H. Beck, the world experiences one
major earthquake every six and one-half days, and Japan has experienced
as many as ten thousand tremors in one year.
Eecorda of the United States Coast and Geodetic Department shew
th at in the year 1933, twenty-two sta te s had a t le ast one quake that was
of moderate or greater in ten sity .
Some of the states reported several,
notably C alifornia, which in addition to the severe Long Beach earth­
quake reported almost f i f t y other fa irly strong tremors.
Washingtcxx follow with seven each.
Montana and
-8 -
CAUSES OF EARTBQUAKES
The originating or ultimate causes of earthquakes have been the
subjects of controversy fo r more than a century.
During the f i r s t h alf of the Nineteenth century these might be
divided into three groups:
(a) Those which attrib u ted earthquakes to sudden
downthrows or collapses of the ground.
(b) Those which a ttrib u ted them to volcanic action.
(c)
Those which attrib u ted them to the action of a
liquid in te rio r of the earth upon an external
rocky crust under the disturbing influence of
tid a l forces.
The b e lie f th a t earthquakes were associated with volcanic action,
th a t they were caused by i t , or were in earns way dependent upon i t , is
as old as A risto tle .
I t appears in the works of Pliny, Strabo, and
#1
Baueaniae, and was universal throughout the Middle Ages. The repeated
destruction of c itie s a t the base of Mt. Aetna, the te rrib le quakes in
Hawaii which immediately preceded the eruption of 1868, the calamity of
Casamlcciola on the island Pachia in 1883» were a l l associated with
volcanos in such a way as to leave no doubt th a t the kinetic cause of
them was volcanic in i t s nature.
Tbe downthrow, or downfall, or Einsturztheorie has had fo r i t s
advocates J . J . Scheuchzer, Swiss geologist; Bousaingault, n a tu ra list;
( numbers re fe r to lis te d lite ra tu re cited.')
-9 -
AIbert Neckser, and G. H. Otto Volger. These authors were disposed to
give the larg est possible extension to the downfall theory. Such down­
throws, Instantly followed by great and far-reaching earthquakes, had
been witnessed and v erified in the calam ities of Port Boyal In Jamaica,
1692, In the te rrib le Calabrian quake of 1783# In the Hew Madrid disaster,
1811-12, in the Bern of dutch near the mouth of the Indus, 1819, and in
Murcia in 1829.
I t remains to glance a t the tid a l theory b rie fly .
That the forces
exerted upon the earth by the moon, and th e ir variations through the
differen t parts of her o rb it, rai^ht, among other re su lts, be effective
in promoting earthquakes Is an old idea.
I t was f i r s t supported by Alexis Pterrey of Dijon.
Perrey' s view
was the prevalent b e lie f th a t the e a rth 's In terio r is In a sta te of
fusion by reason of aboriginal heat. Upon th is liquid mass, enclosed by
a th in , rocky cru st, or skin, the tid a l action of the moon continually
exercised a disturbance which reacted upon the crust so as to produce
cracks, fissu re s, and displacements, with earthquakes as accompaniments.
As the re su lt of th is study, Perrey announced three lavs which are s t i l l
known as Perrey's laws.
I.
Earthquakes are more frequent a t the syzygles, e ith e r of
the points a t which the moon Is most nearly in a lin e with
the earth and sun, and lass frequent a t the quadratures.
H.
They become more frequent as the moon approaches perigee and
lass frequent as I t approaches apogee, th a t point in the
-1 0 -
o rb it of the moon which is fa rth e st from the earth,
approximately 253,000 milea.
III.
They are more frequent when the moon is near the meridian
than when i t is 90 degrees from I t .
Two other modes of earth stre ss have been discussed which are of
in te re st in th is connection. The f i r s t is the secular cooling of the
e a rth ’s in te rio r and the slow, continuous readjustment of the cold,
outer crust to the shrinking nucleus. This view has received a remark­
able development a t the hands of Dr. Robert Mallet in 1871. The other
hypothesis originating from Babbage and HerscheI , which takes account
of the ultim ate effects of th a t process which has been going cm through­
out the whole range of geological time, in which the m aterials derived
from the disintegration of the rocks on the land are carried down by
riv ers to the sea or into the valleys and deposited there.
IM s
involves a sh iftin g of loads from one p art of the e a rth ’s surface to
others, and as i t is cumulative through the geological ages i t must
generate cumulative stra in s.
Professor George Darwin has discussed the subject of earth strain s
arisin g from unequal d istrib u tio n of loads upon the surface in a pro­
found and remarkable analysis published in the Proceedings of the Royal
Society in 1881.
The tran sfer of sediments is the only obvious and p lain ly v isib le
cause which has thus fa r been suggested as the source of those cumulative
stresses which ultim ately become re s is tle s s and lead to the collapse
-1 1 -
vhlch generates the earthquake.
A very new theory also sta te s th a t water leaking through, the leaks
around the ocean bottoms cam In contact with hot rock, change to steam
which released with explosive impact causes a trembler in the earth .
Forces added to the causes of earthquakes may be:
(a)
The lin e up of planets.
(b) Heavy rain s.
(c)
Changes In a i r pressure.
(d)
The melting of polar ice.
(e) The sun spots.
The Helena quakes occurred in a very dry season, sun spots, low
barometric pressure and water level unusually Iow .^
-1 2
#3
EAETa^UAKE WAVES
Hot a l l descriptions of tiie phenomena experienced during an
earthquake agree, but the general sensation seems to he th a t a major
shock consists of three phases. %e f i r s t phase is a trembling, or a
series of l l g i t shocks, lastin g momentarily, and building up to the
second phase which seems to be a violent wave motion, sometimes up and
down, sometimes more of a swaying, and often a combination of both.
This la s ts a few seconds and merges in to the fin a l phase, which is a
period of lesw ning trembling th a t gradually dies out a f te r a few more
seconds. Usually separate p light tremors follow a t short Intervals.
A roaring sound always accompanies th is type of earthquakes. The
f i r s t waves are fa s te r than the destructive waves, and are f e l t before
the severe motion begins unless the observer is near the epicenter.
The creaking of buildings, breaking of window panes, fa llin g of chimneys
and brick veneer, crashing of dishea fe llin g from shelves, and other
sim ilar noises often drown out much of the actual rush lin g of the
treatolor I ts e l f .
Quite generally the major quake corns without any
warning. Usually the v e rtic a l motion is only an inch or two in the very
severest quakes, and the horizontal motion a l i t t l e more. I t is the
suddenness of a very slig h t motion which causes chimneys to topple and
walls to crack.
These waves, traveling thrombi the immovable matter of the earth,
become damaging when they strik e the movable objects on the surface. The
v e rtic a l motion is the strongest over the epicenter, since the rock
movemsnt is occurring d irec tly below th is p oint. Farther from the
13-
epicenter, as a general ru le , the horizontal movement is more marked
than the v e rtic a l motion.
The wave motion trav els a t d ifferen t speeds through d ifferen t types
of rocks. Passing of waves from one type of rock to another resu lts in
a reflec tio n of the wave. Beflection of waves also occurs a t fa u lts .
Tremor vibrations tra v e l through the e a rth 's cru st a t a speed of
almost three hundred miles a minute. This refers to the ground wave of
the quakes the surface waves tra v e l considerably slower. Their velocity
has been calculated to be about 1.85 miles per second.
Fourier has shown th a t whatever the nature of the i n i t i a l motion,
i t can be expressed as the resu ltan t of a number of simple harmonic
motions. Again, th a t every wave tr a in , of whatever form, can be
expressed, in general, as the resu ltan t of a number of simple harmonic
wave tra in s .
Destructive earthquakes may la s t fo r several minutes. They may
cease and recur. The most destructive p a rt of a major earthquake,
however, occurs early and la s ts only a short time, so th a t the fate of
any structure is usually decided within the f i r s t minute from the
beginning.
The in ten sity of shocks of an earthquake varies with the distance
of the point of observation f rom the origin of the disturbance,
character of the intervening te rra in , e tc .
When earthquakes occur, the
e ffe c t on a l l structures th a t ris e above the ground is as i f they had
been gripped a t th e ir bases by gigantic farces which whip them forward
and back, to and fro , end to a certain extent up and down.
I f the period and movement of the earthquake waves have been
measured, the equivalent force may be determined as follows:
-1 5 -
3C
J)
Frs : I
S i IrVi^Ie HarmonTc Ho+Ton
Lef , T - fenod
Jl= Wave LengfK
9 = ReiahelAfion
j=
ITs Wave Veloc.i4y
;
"
j=
j=
W=
r Sin ^ f
r e f a r d a f io n * 2 tt
iL di s f mi ce i-s j6 - - - fhe, Xace
T
r'ivn a)t 3
ahda^TO^ * 2TT -Ss-
J
h* 5 in L n)L - © J s r s 1y) ^ 2 TT i . - 2 tt
^
r sir, 2 i r ( t - | )
f>vd* ,
Cre^ot-C.
JJ= If-T
=
r s in &ZT ^ t - * j
Tak i'tij partial cle.rivo-4ive5 o-f
wlfh T^jjveci* t a ,t
%
% V •
r &Z COj
T
* * *
-r is >
i n
T
«m
f t - 21 \
^
ir /
iff C i - i )
-^Oh K n*.*. a c c c /e r e t t r o K i ^
Son
(^"t ~ ^
b e etjAal t o
:.
a » ^ r 4ff2,
Tl
^vkfche.
r - <1m ^ If4vd-e, • ol
Ftoyy
"T - ^ t t mToel
NZevuf-Oh^ ZaM :
Zr =
a lso
«- 8
mo/
M <L
4 TT**
Tl
iVav*.
i
Tn*
ACC^, lehatToh
Tkete^t
CT .
4 T V ___ W
|
-17'
An earthquake producing a dynamic e ffe c t in a horizontal direction
equal to about l/lO of the weight of structures is a severe earthquake,
and m y be assuaad as the ty p ical case fa r which provision should be
made in design.
Since the v e rtic a l movements are rela tiv e ly small, and since
structures are designed to r e s is t v e rtic a l loads, the v e rtic a l earth­
quake movement produces l i t t l e . I f any, damage. P ractically a l l damage
is produced by the horizontal action.
The movements of the ground, recorded by seismographs, are traveling
in various directions when they reach the Instruments. The motion may 1»
regarded as composed of the component displacements along three axis a t
rig h t angles to each other; the directions usually chosen fo r these axis
are to the north, to the e a st, and v e rtic a lly upwards. The equipment fo r
completely recording the earth displacements is therefore two seismo­
graphs fo r the horizontal components and one fo r the v e rtic a l.
I t is
much more d if fic u lt to obtain satisfacto ry records of the motion in the
v e rtic a l than in the horizontal directions and many of the seisaological
observatories have no v e rtic a l seismograph.
Ths horizontal forces produced by some of the disastrous earth­
quakes^ per Mp of the weight of the structure is:
Sen Francisco, 1906:
F = W X a . 1000 X 6 . 18? lb . per kip
g
3 2 .2
Tokyo# 1923:
F • 1000 X 10 . 310 lb . per kip
3 2 .2
ToiMbsaR, 1923:
T = IOOO X IU = 438 lb . per Mp
"32.2
Kamkura, 1923:
F . 1000 X IS ■ 563 lb . per kip
32.2
19-
IRTESTICaTIOH OF EAHFlEiUAKE Si WtdnftW IR A BUILDING IBAIE
As an exastple, a ty p ical reinforced concrete building frame Is to
be ln rsstig sted fo r the seismic forces specified by the Los Angeles City
Building Code and the Gbiforo R iilding Code. A typical beam and a column
otf th a t frame are to be designed according to the Los Angeles City
Building Code, and the Uniform Building Code Specifications, and a com­
parison made between the two designs.
Kexfc, the footings are to be investigated fo r the seismic forces
specified by the Loe Angeles City Building Code and the Uniform
Building Code, arranged and designed so th a t they can r e s is t any
horizontal displacement caused by the calculated horizontal forces.
20-
Fig. 2
Front Elevation of the building frame investigated
-2 1 -
S 'd
-JLO-
Flg. 3
Plan of building Investigated.
22
Coaputatlons of the momenta produced by the v e rtic a l loads on the
roof of the frame lmreatigatedg
D.L. 4 L.L.
Zo'
cZ
2/
7
D.L. i L«L« /3
T
/O
D.L» 4 L.L.
Dead load from slab: W1 = 2 Q x l x £ x 150 = 1500
"
"
"
snow:
"
”
" girder:
W2 = 20 x 25
=
500
2b . per f t .
W, - 10 x 14 x 150
=
146
lb . per f t .
Total * 2,146
F. S. M. of roof girders:
FEMc_2 > 2.146 x 20 x 20
« 72 k - ft.
12
FEMo_-s .
3
2.146 x 10 x 10
12
2b . per f t .
= 17.9 k - f t.
Mps per f t .
23-
Pob. Maolfl
-62.8
- vL2 - 2.146
1/3___
W
18
- 9
9
- 4.5
2.9
- 1.9
.9
- .k
•3
1/3/1/3
-l8
72
—
18
-18
9
18
- 9
- 9
4.9
2.3
- 2.9 - 2.9
1.9
1.2
- .9 - .9
.4
.4
- *3 - *3
__i/
-72
36
- 9
4.9
- 4.5
2.3
- 1.9
.8
- .4
.2
/33.7
-33.7 /62.8
-43.6
8
8
A 3.6
_______ 1 /3 /
-72
18
-18
9
—2.3
2.9
- 1.2
.9
- .4
•3
U
/h
72
-36
9
- 4.5
4.5
- 2.3
1.9
- .8
.4
• *2
H
S
M
Fm
Piw. Maa.fl fl = vL2 - 2.146 x 10 x 10 r
G-3 - 5- ’
-----------
107 k - f t.
26.8 k - ft.
+3+
-2 4 -
F ig. U
Moment diagram fo r roof girders
and columns of the frame in v esti­
gated along column lin e . (Moments
expressed in K -ft.)
-2 5 -
Cceputationa for oomante d istrib u tio n fo r the f i f t h flo o r of the
from lm restlgated:
Live Load
= 40 x 20
z 800 Xb/ft.
Slab Dead Load
= 150 x 20 x I
Z
1500
Girder Dead Load
m 10 x 14 x 150
Z
146
e
600
144
(#50 of) Brick P artitio n s
10 x 60
X
Total 3.046 k/ffc.
™ D .L.
= ^
esbW
=
ll
Pos- mooDlM
»S*L
=
= 2.246 x 20 x 20
s
75 k-ffc.
3.01*6 x 20 x 20
r
102 k-ffc.
3.046 x 20 x 20
S
-
152 k - f t.
2.24 x 20 x 20
s
112 k - f t .
12
12
=
B
FQ^ 1 Middle Span = 2.246 x 10 x 10
12
=
I S .5 k - f t .
Pos. Moante
-
28 k - f t .
= 2.246 x 10 x 10
8
LL s 0.8 k/ffc; DL = 2.246 k /ft
/ ________DL/LL _______ /
DL______ /
102
- 34
10
- 3
3
- I
- 17 -10
7
7
- 2 - 3
I
I
/7 7
-92/35
-102
21
19
21
-19 102
-21 - 21
10 I?
- 7 - 7
3
2
______
-102
34
- 10
1 - 1
MWW
m
SL/LL
-35 / 92
77
-
/
-2 6 —
Pig. 9
coliairos and girders of the frame
investigated along column lin e .
-
/
FKM 102
- 34
10
- 3
3
- I
DL/LL
TTi
L_
FEM 75
-25
6
- 2
3
- I
/56
DL
27 -
/
DL/LL
-102 25
-25 75
-12 -12
19 19
10 13
- 17 - 6
6
6
- 6 - 6
- I - 3
I
3
I
I
- I - I
T l * /42
/70
DL
/
DL/LL
1_
-7 5 25
-25 75
13 13
-13 -13
6 13
-13 - 6
5
5
- 5 - 5
- I - 3
I
3
I
I
- I - I
-70 /35
-35 /70
DL
-75
25
- 6
2
- 3
I
^56
-75
25
- 6
2
3
- I
-56
./
/
-2 8 -
Q
K)
f ig . 6
Mcment diagram fo r the 5th floor
columns and girders of the frame
Investigated along column line
fo r max. neg. mom.
(Moments expressed In K -ft.)
-2 9 -
Fig. 7
Moment diagram of the 5th floor
columns and girders of the
frame Investigated along
column lin e fo r max. pos. am .
(Moments expressed In K-f t . )
-3 0 -
Dealgn of the F ir s t Floor Celling Beam - K-K3
KU
2. 0 '
Load P la a d ire c t load from the colam above.
D etondnstlon of Pt
Boof Leads
” U girders
35 x 150 z IU % 10
IUU
5 ,1 0 0
enow
25 x 15 x 20
=
slab
I x 20 x 12 x 150
= 22.500
35,100 lb .
10 x 12 10 x 150
12 x 12 *
=
1 ,2 5 0
four girders
=
5 ,io o
slab
s
22,500
liv e load
s
12,000
p artitio n s
*
21,000
Fcwnrth Floor Loads
column above
7,500
61,850 lb .
For the r e s t three flo o rs the
flo o r, therefore
weight is added fo r each individual
61,850 x 3 ■ 185,550 lb .
So to ta l load P equals to .
35,100 / 61,850 / 185,550 . 282,500 lb s.
-31-
^262k
Dead Load plus Live Load
!
DL
7
7
A
I
B
C
Fig. 7<*Loading fo r the second flo o r.
vDL . 2.246 k /f t.
= 0.8
k /f t.
r a ^AB =
Due to the dead load
= 2.246 x 30 x 30 -
168 k - ft
12
Due to the liv e load
= 0.8 x 30 z 30
12
Due to d irect load.
z P x 20 ac IO2
30 x 30
^BA =
z 630 k -ft
282 x 10 x 20* z 1200 k -ft
30 x 30
FEM due to the dead load in span B C:
FEH = 2.246 x 20 x 20 - 75 k -ft
12
therefore.
Total FEHto
"
= 168 / 60 / 630
* / 858 k -ft
FEMba = 168 / 60 / 1200 = -1428 k -ft
r
60 k -ft
-3 2 -
D irect L. / L.L. / D.L.
858
-290
-170
56
17
- 6
0
- m
-1428
340
- 145
33
28
0
- 3
ril7 5
D.L.
T
m
75
340
13
33
- 28
0
- 3
7530
-75
25
170
-56
17
- 6
0
"775
DlfGGt L* / D> I**____________D*L» / L»L*
Z
EBH
798
-266
158
- 52
15
- 5
6
- 2
/652
-1368
316
- 133
29
- 26
13
3
___ I
-1171
102
316
17
29
- 26
13
- 3
I
M9
-102
34
158
- 52
15
- 5
6
- 2
/ 52
7
-33-
7
Io'
Z
A
Poe. Mon. AB
*
B
a o '
BL / LL / Cone. Load
KU™ = Pab = 282 x 10 x 20 = i860 k -ft
L
35
%j, V DL r 2£$ (I-*)
2
z 3«Q^6 x 10 ( 30-10) = 304.0
2
Total Poa. Mon. s 2184 k - ft
Pos. Mon. In span BC
Mpoa.
= 3.046 x^20 x 20 -
152 k -ft
Pos. Mon. In span BC when loaded only with dead load a
Mpoe
- 2.246 x 20 x 20 = 112 k -ft
8
I
C
-34-
Plg. 8
Moaent diagram of the 2nd flo o r columns end girders
along the column lin e of the frame investigated, by
loading the l e f t span AS. (Moments expressed In K-f t . )
-35-
Pig. 9
Mooant diagram of the 2nd flo o r columns and girders along the
column lin e of the frame investigated, by loading the rig h t
span BC. (Moments expressed in K -ft.)
-36-
Deteralnation of the horizontal wind shears fo r every story of the
fle a s investigated.
Wind pressure i s taken as 15 lb s. per sq. f t . fo r every portion of
the structure.
(L. A. Building Code).
Siear in the 5th floor
Siear in the 4th flo o r
-
------------ ----- -------— 1.5 / 3 s 4.5 kips
Shear in the 3rd flo o r
z 7-5 kips
Shear in the 2nd flo o r
-
—_______— - 3 y 1.5 / 3 / 3
Shear in the 1st flo o r
=
-------- 3 / 1.5 / 3 / 3 / 3.75 = 14.3 kips
s 10.5 kips
Computation of the wind forces in every individual flo o r.
5th flo o r:
20 x 5 x 15
= 1.5 kips
4th flo o r:
20 x 10 x 15
r
3
kips
3rd flo o r:
"
r
3
kips
2nd flo o r:
*
-
3
kips
1s t flo o r:
20 x 12.5 x 15 s
3.75 kips
-37-
Determination of the horizontal seismic forces produced on the
frame investigated by the horizontal force formula given by the
Lot Angeles City Building Code.
T
2 Ha
a =
N/
60
4.5
where.
T
2 Horizontal seismic force
a 2 Acceleration
N
2 Hunber of sto rie s above the story under consideration
5th Floor
J =
E
g
*
2.146 x 50 x 1000
2
(slab / snow / girders)
=
107,500
W2 * I x I x 10 x 150 x 4
»
(columns)
8
6,000
V3 « 600 x (50
2
(p artitio n s)
-
78,000
2
191, 500#
S
191,500#
8
123,000
S
78,000
*
6,000
2
398,500
*1
=
/
80)
Wt O t
Fh
=
191,500
60
32.2 * 1 / 4 . 5
2
65,000#
4th Floor
wI (from 5th floor)
W2 (L. L. / D. L.) a 2,446 x 50
w3 (p artitio n s)
W*
(columns)
-38 -
K. = 398,500 _
60
32.2
4.5 / 2
S
115, 000#
W1 (from 5th floor)
S
191, 500#
W2 (from 4th floor)
S
207,000
Wg (from 3rd floor)
Z
207,000
3rd Floor
605,500
Fh :
605,500
60
32.2 x 4 . 5 / 3
z
150, 000#
2nd Floor
W1 (from floo rs above)
Z
605,000
W2 (from second floor)
Z
207,000
812,000
F1, r
812,000 ,
32.2
, 60
4.5 / 4
=
180, 000#
1st Floor
W1 (veight of floors above)
:
K = 812,000 _
60
32.2 X 4.5 / 5
=
812, 000#
159,000#
39-
Determlntttioa of the horizontal seismic forces produced on the
frame investigated by the horizontal force formula given by the TMfcnta
Building Code.
Horizontal force formula:
F
=
CW
W z Total dead load plus one-half liv e load
C = Horizontal force facto r
Montana belongs to section 2 of the earthquake sections and the C
constant equals to 0.02 x 2 -
0.04
Confutation of Horizontal Seismic Force a t 5th Floor
F = CW
(slab and snow)
=
I x I x 10 x 150 x 4
(columns)
S
6,000
600 x 130
(p artitio n s)
S
78,000
* to t a
191,000
*1 = 2,146 x 50
it
Wg a
*h = 191,000 x 0.04
S
107,000
7,600#
4th Floor
191,000
W1
(from 5th floor)
V2
(£ live load)
e 800 x 50
2
=
w3
(dead load)
= 1646 x 50
Z
82,000
»4
(p artitio n s)
Z
78,000
W5
(4 columns)
=
6,000
=
377,000
4 ft
Wto t
= 37’,000 x 0.04
=
20,000
15, 000#
3rd Floor
M1 ( v e t f r o m flo o rs above)
=
377,000#
Wg (weIgJit from 3rd floor)
=
186,000
Wto t
= 563,000
z
Fh = 563,000 x 0.04
22,500#
2nd Floor
(from floors above)
Wg (from 2nd floor)
wt c t
=
563,000
s
186,000
«
749,000
=
Fh % 749,000 x 0.04
30, 000#
1st Floor
Wi
(from floors above)
=
749,000 x 0.04 s
749,000
30,000#
-41-
<?SO—-*■
Fig* 10
Wind Bcoants produced on the 5th flo o r.
(All momenta are expressed In Klp-f t .)
VI
x 25 / V2 x 5 - V3 x 5 - V4 X 25 vI
Z
z
1,500 x 5 = 7,500
5 V2 z -5 V3
3*2 x 25 / V2 x 5 / 5V2 / 5V2 x 25 = 7,500
V2 (5 x 25 / 5 / 5 / 5 x 25) = 7,500
V2 s 7^0 0 s 29 lb s.
V3 B -29 lb s.
V1 :
V4 s
145 x X
145 Ibe.
1,250
I
-145 lbs
2 1,250 s 8.7 f t . (distance from le f t
3$5
corner of neutral p t.)
-42-
015*
F ig . H
Hind Momentc produced m the 4th flo o r.
(AU moments are expressed in K ip -ft.)
750 x 5 s
3 ,7 5 0 l b . - f t .
V1 = 87 x 5 = 435 Ibe.
Vg = 87 lb s.
5,ooo :
435%
X -
5,000 = 11.5 f t . (p t. of
435
contraflaxure)
-43-
Fig. 12
Wind Momenta on the 3rd flo o r.
(All momenta are expressed in K -ft.)
1,250 x 5 r
6,250 f t . lb s.
V2 » 37^500 s 144 lb s.
720 x X ;
V1 « 144 x 5 = 720 lb s.
10,000
X = 10,000 z
720
13*9 f t . (d ie. of p t. of ccxatraflexure
from 1f t . corner)
-44-
Fig. 13
Wind Moaente on the 2nd flo o r.
(All BMBante are expressed In K -ft.)
1,750 x 5 r
8,750 f t . -Ib e.
V5, = 52,500 = 202 lb .
260
*
V1 = 1,010 lb
-40
Pig. 14
Wind Momenta on the l e t flo o r,
(Momenta are expressed In K -ft.)
-46-
+JOB
Ti g. 15
Earthquake am ents on the 5th flo o r with the L. A. Building
Code requirements. (Moments expressed in K -ft.)
-U t -
Flg. 16
EartbqpiakB Boaeats an the Uth flo o r with the L. A. BulMlng
Codte requirements. (Noeeats expressed la K -ft.)
-48-220
* 2 SO
-220
»250
T ie- 17
Earthquake oonente on the 3rd flo o r with the L. A. Building
Code requirements. (Hoaents are expressed in K -ft.)
—
+ ISO
*
-Z75-
3oo
Fig. 18
Earthquake moments on the 2nd flo o r with the L. A. Building
Code requirements. (Moments expressed in K-ffc.)
+/S0
1
-5 4 7
f ig . 19
EarttiquakB xaooBirfcB on the 1 st flo o r with the L. A. Building
Code requirements. (Moments expressed in K -ft.)
-51-
Fig. 20
EarthqualBe moments oa the 5th flo o r with the Uhlfona Building
Code requirements. (Moments expressed in K -ft.)
-52-
Fig. 21
EarthquakB aoments on the 4th flo o r with the Untfora Building
Code requirements. (Momente expressed in K- f t .)
-53Ad to
♦ 36
4 38
Fig. 22
Earthquake moments an the 3rd flo o r with the Uniform Building
Code requirements. (Moments expressed in K -ft.)
tVu
-54-
+*2
5
*
5**
h
Fig. 23
Earthquake acsaente cm tibe 2nd flo o r with the Hnifora Building
Code requirements. (Moments expressed in K -ft.)
f ig . 2k
Earthciuake moments on the l e t flo o r with the Uniform BuiMing
Code requirements. (Mmaente expressed in K-f t . )
-56-
As an esmsple, the beam H2 of the frame Investigated is to be
designed according to the requirements of the Uniform Building Code
and Los Angeles City Building Code, respectively.
Design of the beam H-2 by the Uniform Building Code requirements.
D.L.
- L.L.
( le f t end)
=
- T l k - ft
Wind forces
*
- 10 k - ft
-
Earthquake forces
-
- 32 k -ft
- 11.0 k -ft
-109.4 k -ft
Bight end = - 94.0 k - ft
-119 k -ft
Max. Left End Moment = -119 k -ft
Max. Bight m
Max. Poe. Mom.
n
z
-109.4 k - ft
-
- 68 k -ft
M z 119 x 12 x 1000 Z 1,430,000 lb . in .
I 1,430.000
_ 23 in .
d z
1 220 x 12 y
As = 1.430,000 x 8
- 4.4 sq. in .
16,000 x 23 x?
Section z 12" x 25"
As « 4.4 sq. in
4.4 k - ft
-57’
Design of the bean H-2 by the Los Angeles City Building Code requlrea sats.
D.L. / L.L.
( le f t end)
z
rig h t end -
- 77 k -ft
- 94 k -ft
Wind Momenta
z - 10 k - ft
ft
"
-
- 4.4 k -ft
Earthquake Momenta
-
M
n
w
- 70
-168.4 k -ft
Max. Left End Momenta
Z
-307 k -ft
Max. Bight End Momenta
Z
-168.4 k -ft
Max.
-
/ 68 k -ft
Fo b .
Mom.
-220
-307 k - ft
M s 307 x 12 x 1000 = 3,700,000 lb . in .
d Ag :
(
3.700,000 V7* z
220 x
/
3,700,000 x 8
15,000 x 7 x 32
32 in .
8.3 eq. in .
Section * 16H x 34"
Aa = 8.3 in .2
-58-
Dealgn of the column G2-S2
D irect Load P on Column:
Eoof Loads
=
5th Floor
= 61.8
4th Floor
s
Total P
35*1
6 l,8
= 158.7 kips
E ccentricity produced by D.L. / L.L.
= 94-42 z - 52 k -ft
Wind Moments
z
/ 12.5
Earthquake Moments (L.A. Building Cods)
z /250.0
Earthquake Moments (Uniform Building Code)
z / 38.0
Design of the Column G2-H2 with the Unlfwm Building Code requirements.
P z 159 kips;
E z 1*5 k - ft
fg S 16,000 psi
f c = 1,200 psi
n z 10
B
z 220
k
z 0.4
-59-
Gross Area = 100 sq. In.
Try section IOn x MT
Aa
z
3;
100 x 0 .0 1
Transfer Ae = (n-l)Ae = (1 0 -1 )3
I
Area Total
f.
-
1 /1 2 x 10
< 3 < 0 .0 * X 100
:
x 103 / 2 x1 3 .5 x3s
s 835 / 2*3
r
1078
= 100 / 27
=
12?
-
1500 x 12 x 5
1073
159.000 i
127
1250
2? or 1 3 .5 fear each aide
£
8*
f B m 133* x 10 = 13,3*0 < 16,000
O.K.
Section :
12" x 12"
Ag s 3 eq. in .
-60Deslgn of the column Q2-E2 with the L. A. Building Code requirements
P r
159 K
E s
210 K -ft
20" x 20"
Try Section
Area gross = ^OO eq. in .
Ag = 12 eq. in .
UOO x 0.01 < 12 < UOO x 0.0U
Transfer A8 % (n-l)Aa s 9 x 12 * 108 eq. in .
Total A s
I
-
UOO / 108 = 508 aq. in .
1/12 x 20 x 203 / 2 x 5U x Q2
s 13,300 / 6,900 s 20,200
tC _
159.000_/ 210,000 x 10 x 12
*503
20,200
3lUjf 1250
fg 5 155U x 10 s 15,5UO < 16,000
O.K.
Section s 20" x 20"
A8 r
12 sq. In.
- 6 l-
Design of fo o tin g under the columns KLt KgL8 KkL of the fr*™*
investigated.
Total load on column KL Load from the roof -
21.46 K
”
" 5th flo o r = 30.46
”
" 4th
"
"
-
21.46 K
- *9 =
28.94
”
= 30.46 - %10 =
27.4?
" 3rd
"
= 30.46
25.86
" 2nd
H
= 30.46 - $20 :
Total load on column KL -
- ^15 =
24.36
128.09 Kips
129 Kipe
Plus the to ta l load carried by column 82 which is 240 Klps and is
divided between columns KL and K3L is in such proportion:
£40*
7*L
rR„
20'
240 z 10 = Bl x 30;
sL = 240 x 10
30
%
S
240 - 80 s 160 Kipe
Therefore,
Total load on column KL :
129/80
S 209 Kips
Total load on column K3L * 276 / 160 m 436 Kips
Total load on column K4L s
329
S
80 Kips
129 Kips
—
62—
I t ie suggested th a t i t Is more economical to design the column
footings separately and then connect them against earthquake displace­
ments by means of teams which are to be poured oonolithically with the
footings rath er than to build a continuous footing under the columns of
the frame investigated.
Design of a footing under the column KL
F = 209 Kipe
A -
1.6 f t . x 1.6 f t .
S oil pressure s 4,000 lb . per sq. f t .
Total load = 209,000 lb s.
Footing 566 =
12,000
Total Load = 221,000 lb s.
Area required
Btet pressure
= 221,000 = 55 sq. f t .
"4,000
• 209
* 3.8 Kips per sq. f t .
Therefore,
Footing area z 7*4 f t . x 7.4 f t .
Column area
* 1.6 f t . x 1.6 f t .
Moments about face of column:
H s
1.6 x 2.9 x 2 ^ x 38OO / 2 (2.9 x £ x 2.9 x 38OO x 2/3 x 2.9)
2
25,500 / 61,600 :
87,100 r t .- i b s .
=
-63-
Asetzme
b ■= a / d * 1.6 / 1.5 *
Absuhb
d = 1.5 f t .
d s
f t . or 37 in .
» 12 in .
(M / Sbfi =
TJee 18 in .
Check ▼
7'
« 209.000 x (18 / 18 / 19) (18 / 18 / 19)
7.4 x t .4 %IAt
=
-
V = 209,000 - 80.000
T = 32.200 x 8 = 40
7 x 55 x 18
A. m
8o' oo°
= 129.000 = 32,200
O2K.
M = 97 ,000 x 12 x 8 = 4.6 eq. in .
f 8jd
16,000 x 7 x l8
Anchor a l l s te e l fo r bond
-6U-
Dealga of the footing under column KgL
P = U36,000 lb s.
A % 1.7 f t . x 1.7 f t .
S oil pressure
= 4,000 lb s. per f t . aq.
Total load = 436,000
Footing *6 s
25,000
Total = 461,000 lb s.
Area required = 461,000 * 115 f t.- s q .
4,000
Het pressure
= 436 = 3.8 Klps per sq. f t .
115
Footing area = 10.7 f t . x 10.7 f t .
ColtaBi area
=
1 .7 f t . x 1 .7 f t .
M « 1.7 x 4.5 x 4.5 x 3800 / 2(4.5 x 4.5 x 3,800 x 2/3 x 4.5)
2
•
2
65,000 / 230,000 s 295,000 f t . -Ib e.
Asexnae b = a / d = 1.7 / 1.7 x 3.4 f t
Take d * 1.7 f t .
Use d s 24 in
*
-65-
Check v
V* « 436,000 x (19»8 j 19.8 / 17A) (19.8 4 19.8 4 17.4)
10.7 sc 10,7 at IW
-
436,000 x 57.1 x 57.1 = 86,500
10.7 x 10.7 x IW
V z 436,000 - 86,500 T™
T a 87,000 x 8
7 x 57.1 % 2%
Ae -
349^500 z 87,000
a 72
295.000 x 12 x 8 :
7 x 16,000 x 24
O.K.
10.5 eq. In.
Anchor a l l ste e l fo r bond
-66-
Design of a footing under tbs column HkL
P % 129,000 lb s.
A s 1.5 f t . x 1.5 f t .
S oil Pressure
= 4,000 lb s. per sq. f t .
D irect Load % 129,000 lb s.
$6 Footing
r
Total
8,000
= 137,000 lb s.
Area required
- 137,000 :
4,000
34.2 sq. f t .
Set Pressure
- 129
34.2
3*8 Kips per sq. f t .
z
Footing Area * 5.85 f t . x 5.85 f t .
Column Area
= 1.5 f t . x 1.5 f t .
Moments about the face of the column
M = 1.5 x 2.17 x 2 0 7 x 3800 / 2 (2.17 x 2 0 7 x 3800 x 2/3 x 20 7 )
*2
-
d z
13,500 / 25,800
2
S
39,000 f t .- l b s .
Assume
b % a / d % 1.5 / 1.0 - 2.5 f t .
Assume
d = I ft.
(M /Kbf Z
( I k v g V f-Ig f ;
( 7 ll5 Ze = 8^ 5 1 n Take d S 10 in .
129,000 x (1 0 / 10 / 18) (10 / 10 ^ 18)
5.85 z 5-05 x l U
129.000 z 38 x 38
5.85 x 5.85 x 144
37,600
129.000 jj——
- 37.600 = 91^400 = 22,900
22,900 x 8 = 60
38 x 7 x 10
,000 x 12 x 8 r
,000 x 7 X 10'
O.K.
3.4 » i . in .
Anchor a l l s t e e l fo r bond
-68-
Fig. 25
Footings under the columns of the frame investigated.
The idea is to ti e the three footings by means of a beam which
sh a ll be poured monolithically with the columns and footings.
This
beam w ill be strong enough to take care of the large compression or
tensions which might occur by the horizontal displacements of the
footings due to the horizontal seismic forces.
i
—
69"*
Computation of the horizontal seismic forces by the requirements o f the
UNIFORM BUILDmi CODE.
Procedure:
Column KL
w*
it
W z 209,000 Iba.
CW
C s 0.04
yH = 0.04 x 209,000 % 8,390 lb s.
W r
436,000 lb s.
W
N
H
Colum K3L
436,000 x 0.04 r
17,900 lb s.
Colum K4L
W = 129,000 lb s.
it
•r
129,000 x 0.04 % 9,190 lb s.
-70-
Design o f a t i e between the footings under the coluana o f the frame
investigated fo r the horizontal displacement stated by the UNIFORM
HJILDIIiG CODE.
Procedure;
Max. Tension or Compression between the footin gs;
C or T = 17,500 / 8,350
P :
Try section
= 25,850 lb s.
25,850
8" x 8"
Area = 64 sq . in .
Try Ag = 1.7 sq. in .
64 x 0.01 < 1.7 < 64 x 0.04
Transfer Area = (n-l)Ag
r
(10-1)1.7 z 9 x 1.7 = 15.3 eq. in .
Total Area
= 64 / 15.3
f
25.850 79.3
« P/A .
= 79-3 sq. in .
325 p . s . i .
f 8 = 325 x 10 = 3,250 < 16,000
0ȣ.
Check s t e e l fo r tension
f 8 z P/Aa -
25.850 z 15,250 < 16,000
1.7
Uae section
O.K.
8" x 8"
A8 % 1*7 sq. in .
-71
Computation
a t
the horizontal seism ic forces by the requirements o f the
LOS ABGELES CITi BDHDIHG CODE.
Procedure!
Coluap KL
W = 209,000 lb s.
60
= 41,000 lbs
Fg = 209,000 x
32.2
5 f h-5
Colmm K3L
V = fo36,000 lb s.
r H = fo36,000 x
60
= 85,500 lbs
5 / 4.5
Column XfoL
W = 129,000 lb s.
60
:
Fg = 129.000 x
32.2
5 / 4.5
25,400 lbs
-72-
Design of a t i e between the footings under the columns of the frame
investigated fo r the horizontal displacement stated by the LCS ANGELES
C m BtJILDIKG CQEE.
Procedure:
Max. Tension or Congiression between the footings:
CwT
= 89,500 / 41,000 = 126,500 lb s.
P -
126,500 lb s.
Try section 15" x 15"
A % 225 sq. in.
Try Ag - 8 sq. in .
225 x 0.01 < 8 < 225 x 0.04
Transfer A = (n-l)Ae
Z
Total Area
f C
f ,
9x8
= 72 sq. in .
225 / 72 = 297 sq. in .
=
P/A = 126.500 297
425 p .s .i .
-
n x f c = 425 x 10 = 4,250 p . s . i .
Check s te e l fo r tension
f 8 = P/Aa r
126^500 :
15,800 < 16,000
OzK.
Use Section 15" x 15"
A8 = 8 sq. in .
-73-
COSCLDSICffl
As B r e su lt o f the In veetlgstion o f the "building frame fo r the
seismic farces Indicated by the Los Angeles City Building Code and the
Uniform Building Code, i t i s concluded, th at the lig h te r the v e r tic a l
loads on the building, the smaller the horizontal seismic forces tending
to displace the columns of the structure.
I f the t i e s between the footin gs are designed so that they can
r e s is t the cosnyroaeion or tension which occurs due to the horizontal
displacements o f the fo o tin g s, the structure might be considered
earthquake r e sis ta n t.
However, excess reinforcement o f the columns and girders of the
structure i s o f considerable importance.
Since the t i e s fo r the footings have only been designed for
horizontal ten sion or compression, to prevent any undesirable v e r tic a l
displacement, they should be b u ilt so th at they do not come in touch
with the earth below.
A comparison of the designs made, based on the two codes, shows
the difference between the moments on the columns and gird ers, and the
horizontal shears on the columns.
This difference i s due to the determined horizontal seism ic forces
by using the formulas given by the L. A. C ity Building Code and the
Ihiiform BuiMing Code.
The Loe Angeles C ity Building Code horizontal force formula la
based on a large factor of sa fe ty , which considerably a ffe c ts the siz e
-7%-
of the beans and coluams, and increasee the amount of reinforcing
s te e l.
Table I
Comparison of Sections
Uniform Building
Codbs
Beam size
Beam ste e l
Column size
Column ste e l
Loa Angeles City
Building Code
12" x 25"
UA sq. in .
12" x 22"
3 eq. in .
16” x 34”
8.3 sq. in .
20" x 20"
12 sq. in .
d r
18"
Ag z
4.6 sq. in.
d :
24"
A8 -
10.5 eq. in .
d s
10”
Ag r
3.4 sq. in .
Footing Under Colxsan KL
7 .4 ' x 7.4*
Footing Under Column K3L
10.7' x 10.7'
Footing Under Column K4L
5.85' z 5.85 *
Uniform Building
Code
Los Angeles City
Building Code
Ties between footings
15" x 15"
1.7 sq. in .
8 sq. in .
M
CO
Tie s te e l
CO
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