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University of Nigeria Research Publications
OLUGU, Uma Agbai
Aut
hor
PG/M. Engr/82/1393
Title
Computer-Aided Optimization of
Reinforced Concrete Floors
Facu
lty
Engineering
Dep
artm
ent Civil Engineering
Dat
e September, 1984
Sign
atur
e
COMPUTER-AIDED OPTIMIZI tTION OF REINFORCED CONCRETE FLOORS
OLUGU, U m a kgbai PG/M,ENGR/82/1393
DEPIIRTI-iENT OF C I V I L ENGINEERING UNIVERSITY O F N I G E R I A
N S U K K h
SEPTEMBER 1984
COMPUTER-AIDED O P T I M I Z A T I O N O F REINFORCED CONCRETE FLOORS
OLUGU, Uma kgbai P G / M o E n g r / 8 2 / 1 3 9 3
SUBMITTED TO THE DEPi'iRTMENT OF C I V I L E N G I N E E R I N G , I N THC FiiCULTY OF E N G I N E E R I N G ,
AS PIlRT O F T H E REQUIREMENTS FOR T H L AWARD O F MASTER O F ENGINEERING DEGREE O F T H E
U N I V E R S I T Y O F N I G E R I A 4
S U P E K V I S I O R S :
D r . A. O l e i s n e w i c z Dr, lie S t r z e l c z y k Dept, of C i v i l E n g i n e e r i n g Dcpt, of 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 N i g e r i a U n i v e r s i t y of Nigeria Nsukka, N s u k k a .
DEDICATION
Dedicated to my P a r e n t s :
and
Mrs. C. Olugu
PREFACE
The t e x t a ims a t f u l f i l l i n g a need w i t h i t s systematic
a p p r o a c h t o c l a s s i f i c a t i o n o f t h e ( i t e r a t i v e ) o p t i m i z a t i o n
p r o c e s s i n t h e d e s i g n o f f l o o r s .
he p a p e r i s d i v i d e d i n t o f i v e c h a p t e r s .
C h a p t e r 1 on I n t r o d u c t i o n d i s c u s s e s w i t h i n t h e framework of
S t r u t t u r a l Design t h k o r i e s , t h e method of approach and
j u s t i f i c a t i o n o f t h e p r o j e c t . &
C h a p t e r 2 d i s c u s s e s d e s i g n and i n c o n j u n c t i o n w i t h
c h a p t e r 1 e x p l o i t s t h e d e s i g n p h i l o s o p h y t o p r o v i d e a
u n i f i c a t i o n and g e n e r a l i z a t i o n of t h e d e s i g n s t a g e s of t h e
S t r u c t u r a l d e s i g n p r o c e s s , C h a p t e r 3 d e l v e s i n t o s p e c i f i c
d e s i g n methods f o r e a c h of t h e s l a b s t h r o u g h f l o w c h a r t s
and computer programms, C h a p t e r 4 d i s c u s s e s t h e r e s u l t s
o b h a i n e d t h r o u g h g raphs . C h a p t e r 5 p r o v i d e s a c l o s u r e f o r
t h e t e x t ,
The Au tho r h a s t r i e d t o assemble and u n i f y much
s c a t t e r e d knowledge on S t r u c t u r a l Design of S l a b s which
h i s t o r i c a l l y h a s become c o m p a r t m e n t a l i s e d and t h e r e f o r e
p r o v i d e s an iterative method f o r o p t i m i z a t i o n o f R e i n f o r c e d
C o n c r e t e f l o o r s .
ACKNOWLEDGEMENTS
I wish t o thank my Prbjoct Advisers ( E n g r ) D r .
A. 0 1 e i s n i e G f a a n d Ur. n Strezlczyk; Tho accomplishmen9
of t h i s work i s l a r g e l y due to t h e i r help en
a c c o u n t o f t h e i r a d v i c e , corrections, k i n d crif;ici~ms
and i n v a l u a b l e s u g g e s t i o n s .
T h e Head of bcpa r tmen t , ( E n g r ) Dr. N . Egbuniwe
was a l w a y s ready t o c o o p e r a t e and assist dur /ng
moments of need , and I must e x p r e s s g r a t i t u d e t o
him.
t My t h a n k s go to Messrs K,Oo Uma, O.K. / c h i
A . U . Odizia, 0.0, Nwosu and O.U, Emele f o r t h e i r
dnnumerable h e l p ,
F i n a l l y , I wish to thank Engr. ( C h i e f )
F.C.N. f t g b a s i w h o gave mc t h e p r a c t i c a l background
and M r . Ukpai who typed t h e final c o p i e s of t h e
m a n u s c r i p t .
A - Area of Re in fo rcemen t
b a Width o f S e c t i o n
d 5 Effective d e p t h o f tenslbn r e i n f o r c e m e n h
F = U l t i m a t e l o a d
FCC = ~ o m ~ r c ~ s i v e f o r c e i n L t h e coneketc a c t i n g t h r o u g h t h e c e n t r o i d of t h e sti-css block
Pst E T e n s i l e f o r c e i n t h e r e i n f o r c e m e n t acting a t t h e c d n t r o i d of t h e t e n s i l e steel.
Cc = ~ o t a l C o s t o f r e i n f o r c e m e n t
Cs = T o t a l cost o f C o n c r e t e
f c u s C h a r a c t e r i s t i c concrete cube s t r e n g t h
f y = C h a r a c t e r i s t i c s t r e n g t h of r c i n f o r c c r m ~ l l t
h = O v e r a l l d e p t h o f S e c t i o n i n p l a n e o f b e n d i n g b
kl = n v e r a g e Compress ive stress i n c o n c r e t e f o r a
r e c t a n g l u s r - p a r a b o l i c stress baock
K2 = w f a c t o r t h z t re la tes t h c d e p t h to t h e c e n t r o i d o f t h e r e c t a n g u l a r - p a r a b o l i c stress b l o c k and t h c d e p t h of t h e n e u t r a l a x i s .
K2X = Depth Lo t h e c e n t r o i d o f t h e stress b lock
Lx,Ly = Span o f S l a b
M = Bcnding Moment
M u = Ultimate r e s i s i s t a n c e moment
S = S p a c i n g o f r e i n f o r c e m e n t a l o n g t h e member
V = S h e a r f o r c e , volume
X = N e u t r a l a x i s d e p t h
Z r L e v e r arm
E o = Tho c o n c r e t e s t r a i n a t t h e end of t h e p a r a b o l i c S e c t i o n
E s t a T c n s i l c s t r a i n i n t h e r e i n f o r c e m e n t
V - S h e a r S t r . ? s s
W E Thc d i s t a n c e f rom t h c n e u t r a l a x i s t o S t r a i n Eo
' i = W e i g h t o f a l e n g t h o f steel in k i l o g r a m s
WT t Summation of d i f f e r e n t w e i g h t s o f s t ee l i n k i l o g r a m s
CT = T o t a l "ost o f S l a b
0 I Diameter o f r e i n f o r c c m c n t r o d
LIST OF ,FIGURES
S o l i d Slab/Beam Ar rangemen t d o . 6
F l o o r Ar r angemen t f o r bJaff l e C o n s t r u c t i o n 7
F r e e Body s k e t c h o f a Square F l a b - S l ~ b h u a d r a n t 0 0 0 a O m 17
I n f l u e n c e o f kly on C o s t o f S o l i d S l a b 69
C o s t Against T h i c k n e s s o f S l ab .., 70
I n f l u e n c e o f FCU on C o s t o f S l a b 71
'. . I n f l u e n c e o f Rod D i a m e t e r on Co;;t o f . ; .i : ,. S o l i d S l a b - O m o o O 72
I n f l u e n c e o f Rod Diameter on Co;r F l o o r ( S o l i d S l a b + Beam) O n 5 73
I n f l u e n c e o f C o n c r e t e 5 t r i : n ~ t h 0;; Czsi o f F l o o r ( S o l i d S l a b + ~ e a m ) ,, 0 ,, 74
I n f l u e n c e o f S l a b T h i c k n e s s on C o a t o f F l o o r ( S l a b + Beam) a a 0 75
I n f l u e n c e o f S t e e l S t r e n g t h on C o s t o f F l o o r ( S o l i d S l a b + B e a m ) . o m 76
I n f l u e n c e o f R i b T h i c k n e s s on C o s t o f W a f f l e S l a b -.. . O m 77
I n f l u e n c e o f Mould S i z e on C o s t of w a f f l e F l o o r . . . . O D 78
I n f l u e n c e o f C o n c r e t e S t r e n g t h on C o s t of W a f f l e S l a b ... ... 79
I n f l u e n c e o f FY on C o s t o f W a f f l e F l o o r 80
I n f l u e n c e o f Rod Diameter on C o s t o f Waffle F l o o r . . . O O . 81
17. I n f l u e n c e o f B e a m Uepth on C o s t o f B e a m 8 2
Nomina l Cover t o Piain R e i n f o r c e m e n t M o d e r a t e E x p o s u r e a a a O o O
Nomina l C o v c r t o Main R e i n f o r c e m e n t M i l d E x p o s u r e C o n d i t i o n e a 0
S i m p l i f i e d R u l e s f o r C:!.:.tsilment-. of B a r s i n S l a b > ' p a n n i n g i n o n e Gli-c:c!:ion
M o d i f i c a t i o n F a c t o r Lor .ki:s ioirl
R e i n f o r c e m e n t ( FV - 41.G \ - . - M o d i f i c a t i o n F a c t o r I'0.r 'T't?ns:.:,r R e i n f o r c e m e n t ( F Y = [1:%3> , ,
M o d i f i c a t i o n F a c t o r F(,r ;'c\r:s."-~,; R e i n f o r c e m e n t (FY = 4 6 0 ) . L
M o d i f i c a t i o n F a c t o r For 9 'ens io ; l R e i n f o r c e m e n t (FY = 5001 O C .'
M o d i f i c a t i o n F a c t c r F o r T e n s i o n R e i n f o r c e m e n t (FY = 2 5 0 ) 0 0 0
Maximum p e r m i s s i b l e v a l u e o f N g r n i n a l U l t i m a t e S h e a r S t r e s s Uu (N/mm '1 4
U l t i m a t e S h e a r S t r e e s U c i t g a i n s t P e r c e n t a g e R e i n f o r c e m e n t (FCC = 3 0 )
U l t i m a t e S h e a r SBkess U c A g a i n s t P e r c e n t a g e R e i n f o r c e m e n t (FCU = 40 or More.) 0 0 0 0 0 0
U l t i m a t e S h e a r S t r e s s U c A g a i n s t P e r c e n t a g e R e i n f o ~ c c m e n t (FCU = 2 0 )
Ul t imate S h c a r S t r e s s U c b ~ g a i n s t P e r c e n t a g e R e i n f o r c e m e n t (FCU = 2 5 )
LIST OF TABLES P,aqe :
I n f l u t m c e o f S t e e l T e n s i l e S t r e n g t h an C o s t o f S o l i d S l a b . . D o * 52
I n f l u c n c e o f S l a b T h i c k n e s s on C o s t o f S o l i d S l a b .-. o m 5 2
I n f l u e n c e o f C o n c r e t e Compres s ive S t r e n g t h on C o s t o f S o l i d S l a b - I 53
I n f l u e n c e o f Rod D iame te r on C o s t o f S o l i d S l a b a * o O O e 53
I n f l u e n c e o f Rod Diameter on C o s t o f F l o o r ( S o l i d S l a b + B e a m ) m e -
b 54
I n f l u c n c e o f C o n c r e t e S t r e n g t h on C o s t o f F l o o r ( S o l i d S l a b + Beam) ... 54
I n f l u e n c e o f S l ' l b T h i c k n e s s on C o s t o f F l o o r ( S o l i d S l a b + B e a m ) a " > 55
I n f l u c n c e o f S t e e l Tensi1.c St rcnqc;2 on C o s t o f F l o o r ( S o l i d S l a b + Beam) 55
I n f l u c n c e o f R i b T h i c k n e s s , B , cr. C o s t o f W a f f l e F l o o r . a o D o n 56
I n f l u e n c e o f Mould S i z e on C o s t o f W a f f l e F l o o r .- .-. 56
I n f l u e n c e o f C o n c r e t e Compres s ive S t r e n g t h on C o s t o f W a f f l e F l o o r ,., .. a 5 7
I n f l u e n c e of S t e e l S t r e n g t h on C o s t of W a f f l e F l o o r .OO .om 57
I n f l u e n c e o f Rod Diameter on C o s t o f W a f f l e F l o o r ... - 58
14, I n f l u e n c e o f Beam Depth on C o s t of Beam 58
APPENDIX:
Ale Cover to Reinforcement in all Reinforced Concrete Structures (CPIIO: 3,1:2) 89
h6, Modification Factor for Tension Reinforcement (CPIIO: Clause 3.3,8,1) 94
A12 Maximum Permissible Value of Norninal Ultimate Shear Stress Vu t ~ / m m * ) ~ (CPIIO: Clause 3,3.6.1) 0 0 0 900
A14 Ultimatl Shear Stress in Beams Vc (N/rnm2lh (CPIIO: 3.30G01) U O I. 102
TABLE OF CONTENTS: Page:
D e d i c a t i o n
P r e f a c e
Acknowledgments
Notat i o n s
CHAPTER ONE:
lei I n t r o d u c t i o n i-, s o ,, O #,
1.2 O b j e c t i v e s and Sc~ptr-t c f '.V'srk ,,
1.3 F o r m u l a t i o n o f O p timixa',-.i.cr: Prob ' l Prn
CHAPTER T s
2 , 1 S e l e c t i o n o f F l o o r S y s t e m S e e
2.3 Method o f S o l u t i o n a‘ .c 0 0 . 2,4 Des ign P h i l o s o p h y . a > e n *
CHAPTER THREE :
3.1 S t r u c t u r e o f t h e Flo-d Chizi;; ,. o .
3.2 Program S p e c i f i c a t i o n O cr O
3 . 3 I n p u t Des ign I n f o r m a t i o n and l%s(:r-lpt-ion o f Flow C h a r t . O n . ) * a
3.4 Flow C h a r t f o r S o l i d S l a b s o . .
3.5 Flow C h a r t f o r Beams . O D ... 3.6 Flow C h a r t f o r W a f f l e S l a b s ...
Progranu .*a * e r n
e.. . . 4.2 Program P l o w Cot Solid Slab+ ... 4.3 Pragtam Plow for Beams .. 4.4 Program Flow for Waffle Slabs ... 4,6 Influence of V U P ~ Q U S Parameters on Cost
of the Floor ... 0 0 .
0.7 C9nelulLon and ~ e c w m n b a t ~ o n s CHAPTER FIVE:
5.1 F i n a l Remarks a . o rn o ., 5.2 Suggestions for F u t u r e Work son
Aomndix ... P a -
CHAPTER ONE:
1 1 INTRODUCTION :
This paper describes a slapie -tho4 for d d l ~ e h t
add optimum selection of reinforcod mctete f toe~r4 . A
graphical approach is employed to determine the r ~ h t
economical system, section properties followhg d saCk# of
trial analyses based on CPXIO, part 2, 1972.
CPIIO states that the purpose of design is the
achievement of acceptable probabilities t h a t t h e &ttuCtut4
being designed will not become u n f l t for use for whicd i t
is required; that is, i t will not reach a limit state.
Thus the design of reinforced concrete floors would involve
selecting section properties and floor types, Such that
an acceptable probability is provided against floors reaching
any of the limit states and being unecenomical. It is
possible to select more than one set of section properties
and floor types to satisfy these ~endikiond~
In present day design practece, d~ikable stab ~ d t t i 6 n ~
and types are usually obtained by dnelysing t r ia l des igns ,
the experience of the Engineer being used to interprete
the results, Unfortunately, the need for the Engineer to
redesign the slab considering other floor systems can lead
to an excessively long design time. This results is
o n l y a f e w t r i a l d e s i g n s b e i n g c o n s i d e r e d f o r a p a r t i c u l a r wb:c\> 2 "
f l o o r t y p a a n d thotj_rinal%'L$ a c c e p t e d may be f a r f r o m
t h e op t imum,
T h i s p r o j e c t t h e r e f o r e b r i n g s o u t t h c op t imum
s e c t i o n a n d mater ia l p r o p e r t i d s a n d c o m p a r e s t h e c o s t o f
v a r i o u s f l o o r t y p e s , Th? h s k u n d e r t d k e n a l s o s h o w s t h e
i n f l u e n c e o f t h e v a r i o u s p a r a m c t z r s o n t h e f i n a l c o s t
o f t h e g i v e n f l o o r t y p e . A l s o t h e s e t o f p a r a m e t e r s
a r e ~ v a l u e t e d t o show tht:: m c : s t e c o n o m i c s t r u c t u r a l ,
s o l u t i o n f o r thc: g i v e n s p a n .
1-2 0 3 J L C T I V i i J .i!!iD SCUPE O F ' K I R K : - -.- -.-. - .-.-. ~ . . - --..-- ...---
T h e ma in o b j e c t i v e s o f h i s p s ; ? r a r e , f i r s t t o
p r e s e n t a b r i c f d e s c r i p t i o n 0:' t h e cpt i rnum d e s i g n p r o c e d u r e
t h a t h a s b e e n c l e v c l o p c d and t r . 2n t o illustrate it$
a p p l i c a t i o n . Thi. p r o c e d u r e wc :; d e v ( : l o p e d s p e c i f i c a l l y f o r
one--way s o l i d ( ~ t ~ t t m n a l ) s l a b s , d a f f L -2 s l a b s iind s i m p l y
s u p p o r t e d Beams,
I t e m p l o y s a c o m p u t e r - a i d $ 1 i t e . ' ; l t i v e t e c h n i q u e i n
s t e p s t h , . . : t a r t : g r o u p e d i n t o s p r e l i m i n a r y d e s i g n a n d a
f i n a l d d s i g n p h a s z ,
T h i s 1 ) a p c r S o c u s p s o n t h e i l - . s c r i p t i o n o f p r e l i m i n a r y
d e s i g n p h a s z , c m p h a s i z f n g t h e o p t i m i z a t i o n t e c h n i q u e t h a t
h a s b e e n u s e d a n d it:; a p p l i c a t l c w , P r e l i m i n a r y d e s i g n s
3.
o b t a i n e d u s i n g t h e p r o p o s e d design p r o c e d u r e And a design
o b t a i n e d on t h e basis of CPIIO, ~ t r u ~ k k i ~ a l use o& Cbmerete,
1972, are compared w i t h t h e c!ooYkkuction m&l& k4.8hulted
a n d a l so w i t h r e s p e c t t o b e h a v i o u r a l characterisfitso
erdm t h i s c o m p a r i s o n , t h e ~ p t h u h dea&n and eifdei 05
m a t e r i a l p r o p e r t i e s on C o s t of kioar 16 kdbwn.
1.3 FORMULATION OF OPTIMIZATION PROBLEM:
T h e o p t i m i z a t i o n p r o b l e m i s to f i n d t h e c o n t r o l s
for a R e i n f o r c e d C o n c r e t e f l oo r which m i n i m i z e s t h e
c r i t e r i o n "Cos tw fo r a d e s i g n p r o c e s s t h a t s a t i s f i e s 'and
f o l l o w s l i m i t s t a te b e h a v i o u r and r e c o m m e n d a t i o n s i n "The
S t r u c t u r a l Use of C o n c r e t e C P I I O : 19729t.
C o n s i d e r i n g a U n i t w i d t h of s lab , t h e v a r i a t i o n of
b e n d i n g moment i s c a l c u l a t e d for t h e s l ab s u b j e c t t o t h e
u l t i m a t e l o a d s u s i n g any of t h e a n a l y t i c a l me thods
s p e c i f i e d i n CPIIO.
L e t Mmax be t h e maximum b e n d i n g moment i n the member,
The C o n s t f u n c t i o n , C p e r u n i t Area of s lab a t Maximum
moment p o i n t c a n t h e n be e x p r e s s e d a s
C = AS CS + 1 x HCC
where
As = Area of Steel
Cs = Cost of Steel per hp -
H = Depth of Concrete floor
Cc = Cost of Cubic metre af Concrete
For any grade of concrete and type of reinforcement,
Cs, Cc are constants,
CONSTRAINTS :
A) The behaviour Constraint defined by the ultimaie limit
state can be expressed as
Mmax 6 mu
where Mu is the ultimate moment of resistance
of section.
8 ) The serviceability limit state requirement appears as
a limit on the deflection. CPIIO specifies that efgher
the deflection may be calculated or certain limits on span
effective depth ratio be observed. The limits for this
ratio depends on support conditions, steel stress etc.
and given in the appendix of this paper by figures A6 to
All.
With this, the optimization problem reduces to
finding design variables which minimizes the objective
function,
D,',., .',:, ; ,ii OF COST OF SLAB:
Weight o f steel i n k i l o g r a m s i s found by u s i n g t h e
r e l a t i o n s h i p t h a t
2 'i
= 00006265.* D e L
From R e i n f o r c e d C o n c r e t e D e s i g n e r s Handbook by C h a s 2 , ReynoldS .
where Wi = "de igh t i n k i l o g r a m s
D = Diame te r o f S t e e l i n mm
L = Leng th o f s t ee l i n m e t r ~ s
T o t a l w e i g h t of s tee l f o r a s l a b is
F o r a p r i c c i n d e x o f N1 ,5 p e r 1: i logram of s tee l
T o t a l c o s t o f s tee l = C,: = A1.5 WT
Volume o f c o n c r e t e = Ares o f s l a b ~ s l a b t h i c k n e s s
T o t a l c o s t o f c o n c r e t e u s i n g a p r i c e i n d e x of
3 ,4180 p e r m o f c o n c r e t e , havl-: t h a t
T o t a l c o s t o f c o n c r e t e = Cc = t 1 8 0 X Area o f s1abYr:H
0 T o t a l c o s t o f s l ab = CT = Ss + dc 0 0
CHAPTER TWO
2 - 1 SELECTION OF FLOOR SYSTEM:
F i g u r e s 1 a n d 2 show t h e d i m e n s i o n s a n d w o r k i n g
s y s t e m o f t h e f l o o r s e l e c t e d f o r s t u d y ,
T h c s e l e c t i o n t-es.t h e a v i l y upon t h e e x p e r i e n c e a n d
j u d g 2 m c n t o f t h e w r i t e r o n conmon f l o o r s y s t e m s m e t i n
p r a c t i c e ,
T h e m a i n c o n s i d e r a t i o n i s t o a p p l y a number o f d i f f e r e n t
t y p e s o f s l a b s t h a t w o u l d rc 2 r e s e n t a much l a r g e r c l a s s 6
of f l o o r s n o r m a l l y e n c o u n t e r d i n tl - d e s i g n o f f i c e ,
T h e op t imum s l e c t i o n s h l a l ~ l d be : a r r i e d o u t f o r t h e
f o l l o w i n g a r r a n g e m e n t s :
a) S o l i d S l a b s
b ) H o l l o w S l a b s ( P r e c a s t
c ) 3 i b b e d S l a b s
d ) F l < ~ t S l a b s ( H a f f l e )
B u t b e c a u s e o f t i m e l i m i t c n d t - 3 d i f f i c u l t y i n
p r o g r a m m i n g , o n l y t h e s o l i d a n d d a f f : - s l a b s w i l l be
c o n s i d e r e d .
T h e p a r a m e t e r s c o n s i d e r e d . . g n i f i c m t i n t h e
scl:!ct i o n were
i) Span r a t i o
i i ) T h i c k n e s s of s l ab
iii) S l a b / b e a m a r r a n g e m e n t
i v ) M a t e r i a l c h a r a c t e r i s t i c s
v ) F u n c t i o n a l c o n s i d e r a t i o n
S c h e m a t i c r e p r e s e n t a t i o n s o f t h e t w o s l a b s are shown
i n f i g u r e s 1 a n d 2 r e s p e c t i v e l y , I n t h e case o f o n e way
s l a b s , t h e s e a r c h w a s c a r r i e d o u t f o r a 6 x 4 m ' i n t e r n a l
s l a b w h i l e t h a t o f t h e waff le f l o o r was f o r a n 8 x 8m' B
i n t * r n a l f l a t s l a b a n d d e s i g l e d as- two way.
2.2 P L ~ ~ A M E T E ~ ~ S $ A S S U M P T I O N $ - W i t h r c s p e c t t o t h e d e s i r j n o f f h e s l a b s a n d beams ;
a ) T h e l o a d i n g was a s s u r c.d t o be m o s t l y u n i f o r m l y
d i s t r i b u t e d ;
b) T h e s p e c i f i c w e i g h t of c o n c r e t e was t a k e n a s
2360kg/m3;
C ) Maximum s i z e o f a g g r e r l l t e h . 1 ~ a s s u m e d t o be
2 Omit1 ;
d l From e x t e n s i v e i n q u i r i : ~ , i t was d i s c o v e r e d t h a t
t h c p r i c e s of mater ia ls w e n as follows:
d 1 , S p c r k i l o g r a m f o r ct~l d i a m e t e r sizes o f s t ee l
,$I80 p e r c u b i c meter fc .- c o n c r e t e o f a n y c u b i c
str 2ng th .
P s e a r c h _were. T h e va r i ab le p a r a m e t e r s used i n t h -
A ) F o r S o l i d S l a b s / B e a m
i ) T h i c k n e s s , H , o f s l a b
i i ) C o m p r e s s i v e s t r e n g t h o f c o n c r e t e (FCU)
iii) T e n s i l e s t r e n g t h o f s teel (FY)
i v ) Diameter D, of r e i n f o r c e m e n t bars a n d
v ) B e a m d e p t h ( H I )
8 ) F o r W a f f l ? F l o o r s
i ) T h i c k n e s s o f R i b (E) t
i i) S i z e o f mou ld ( S )
i i i ) C o m p r ~ s s i v c s t r c n g t t o f corcrete (FCU)
i v ) T e n s i l e s t r e n g t h o f ,tee1 (?Y)
v ) Uiameter O , o f r e i n f o r c e m e n t ba r s
2 - 3 METI~OD OF SOLUTIOiV:
E s t a b l i s h e d S t r u c t u r a l d e : ; i y n p r o c e d u r e s are s e e n t o
be i t e r a t i v e i n n a t u r e , T h e i t 2 r a t i - m s arise f r o m t h e
a n a l y s i s - b a s e d mode o f a t t a c k c. t h e d e s i g n p r o b l e m ,
I n s h o r t , c o n v e n t i o n a l d e s i g n i: a p r o c e s s o f t r i a l
a n d error o p t i m i z a t i o n . O e s i g n s y s t e r , l s a r c commonly
d i v i d e d i n t o S u b s y s t c r , i s t o prod^ c:e a t r a c t a b l e d e s i g n
s u b p r o b l e m b u t c e r t a i n i n c o n ~ i s t e n ~ f e s make t h i s
p r o c e d u r e d i f f i c u l t , T h e r e a l s o e i x s t s a n i n t e r d e p e n d e n c e
o f e a c h l e v e l , a : e q u i r i + . * t k n o w l e f l g e a t a h i g h e r l e v e l
when d e s i g n i n g a lower l e v e l s u b s y s t e m . T h i s t h e r e f o r e
means t h a t n o i s o l a t e d s y s t e m s e x i s t t h e r e b y j u s t i f y i n g
t h e u s e of i t e r a t i o n i n t h i s p r o j e c t .
An i t e e a t i v e a p p r o a c h h a s b e e n employed f o r t h e
s o l u t i o n , The method i s b r i e f l y e x p l a i n e d . The
s e c t i o n and material p r o p e r t i e s a r e c h o s e n a s t h e known
v a r i a b l e s ,
The moments, s h e a r forces and t h e a x i a l l oad ,
d i s t r i b u t i o n on t h e f l oo r p a n e l d u e t o t h e a p p l i c a t i o n
of t h e l o a d i s c l a c u l a t e d ~ n d t h e s l a b i s d e s i g n e d f o r
a se t o f t h e s e v a l u e s .
h s i n g l e s o l u t i o n t o t h e p r o k l e m i s t r i v i a l a n d
may e a s i l y be c a r r i e d o u t b i hand c a l c u l a t i o n s a s shown i n
t h e s e c t i o n f a r p rogram f low; .
A c o m p u t e r s o l u t i o n is l o w e v e r , n e c e s s a r y s i n c e t h e
object of t h e e x e r c i s e i s t o seek m optimum
c o m b i n a t i o n o f t h e s e c t i o n an3 m a t : ? r i a l p r o p e r t i e s t o
m i n i m i z e cost , t h e r e f o r e r e q u - r i n g many s o l u t i o n s
f o r d i f f e r e n t c o n d i t i o n s .
H c o m p u t e r programme was d r i t t e n f o r t h i s p u r p o s e
and c o m p u t a t ~ o n s w e r e c a r r i e d c u t f o r t h e d i f f e r e n t
s l a b s m e n t i o n e d wf,&e.
The impu t d a t a fo r t h e programme c o n s i s t s of t h e
d e s c r i p t i o n o f t h e s e c t i o n and m a t e r i a l p r o p e r t i e s o f
t h e s l a b , The programme c a l c u l a t e d t h e moments, s h e a r
stresses a t t h e c r i t i c a l s e c t i o n s , c h e c k s d e f l e c t i o n s a n d
c r a c k i n g , d e s i g n s t h e s l a b s and t h e n c o s t s it.
The o u t p u t c o n s i s t s o f an e c h o o f t h e impu t and
t h e o u t p u t r e s u l t s o f t h e a n a l y s i s . The r e s u l t s w & e
t h e n p l o t t e d m a n u a l l y i n a g r a p h as shown.
2,4 DES1G.N PHILOSOPHY
A n a l y s i s and Des iqn o f - S l a b S e c t i o n :
The f i g u r e below r e p r e s e n t s t h e C r o s s S e c t i o n
o f a s i n g l y r e i n f o r c e d c o n c r e t e s e c t i o n and t h e r e l e v a n t
s t r a i n and stress d i s t r i b u t i o n s .
L e t r Z p c c - be t h e Compressivef~r-e i n t h e c o n c r e t e
a c t i n g t h r o u g h t h e c e n t r ~ k f o f t h e stress b l o c k ,
F . be t h e t e n s i l e f o r c e i n t h e r e i n f o r c e m e n t a c t i n g s i a t t h e c e n t r o i d o f t h e t e n s i l e s teel ,
y-.. be t h e t e n s i l e s t r a i n i n t h e r e i n f o r c e m e n t s t
f s i be t h e t e n s i l e stress i n t h e r e i n f o r c e m e n t
a n d Z be t h e l e v e r arm w h i c h i s t h e d i s t a n c e be tween
t h e p o i n t s o f a c t i o n o f FCC a n d F s t 0
T h e s e t w o f o r c e s i n t h e c o n c r e t e f a w m a c o u p l e
t o d e v e l o p t h e u l t i m a t e moment o f r e s i s t a n c e o f t h e
S e c t i o n . (Mu) v . MU = F c c . 2 = F s t ,L
ahris; . ' id ' : ' . : t h e f u n d a m e n t a l e q u a t i o n s o f e q u i l i b r u m
f o r t h e cross s e c t i o n and are a p p l i c a b l e i r r e s p e c t i v e o f * t h e n a t u r e of t h e d i s t r i b u t i o n o f t h e s t r a i n s or t h e
stresses,
Fir.:. = a v e r a g e c o n c r e t e stress x area o f c o n c r e t e \". ... i n c o m p r e s s i o n
T h e r e f o r e FCC = k b x 1
F . = S t e e l s t r e s s s a r e a o f steel .; t'
From stress b l o c k d i a g r a m ,
f r o m e q u a t i o n s ( 1) a n d ( 2 1 , w e h a v e t h a t
f r o m e q u a t i o n s (2) a n d ( 3 1 , we a l so h a v e t h a t
Mu = fst A s . Z
z) A8 = Mu/fst (d-k2X)
Using that fst = 0,87x FY, weRave that
As - MU/O.$?FY ( D - ~ ~ x ) 6
X is the depth of the neutral axis and is dot w h M the
moment of resistance of the concrete q ~ c t i o n is weaker
than the loading moment from M - 0.4 r B x pcO f X(d-k2x)
Determination of
From Strain diagram
/--'
A t maximum stress, Eo = 2.4 x ,I/( fcubm)
+ W = x w i t h = 1.5 17,86
For the stress block,
Ki = area of stress block/x
= (area pqrs - area r s t ) / x
From properties of a parabola
K 1 = (0.45fcux - O . ~ S ~ C U . W / ~ ) / X
16.
Determination of the Depth of the Controid K2X
K2 u s determined by tak ing area moments of the stress
block about the n e u t r a l a x i s
S u b s t i t u t i n g f c r W , w e have t h a t
hence
For Doubly Reinforced Sections
Compression Steel
As = M - 0.15fcu bd 2
1 0.72fy (d-d )
Tensile S t e e l
A n a l y s i s o f F l a t S l a b S e c t i o n :
A s t a t i c A n a l y s i s for a square panel of f l a b
s l a b is shown,
h c i s t h e d i a m e t e r of t h e column Capf tax, The
p o r t i o n of t h e dead and l i v e load which lies directly
o v e r t h e Column C a p i t a l is carried d i r e c t l y by t h e
columns and was p u r p o s e l y e x c l u d e d from the l o a d , W ,
which i s c o n s i d e r e d t o be u n i f o r m l y d i s t r i b u t e d .
The C e n t r c Q d s o f t h e r e a c t i o n f o r c e s a c t i n g upward
a round the q u a r t e r - c i r c l e s a t t h e b o u n d a r i e s o f t h e
c a p i t a l s are a t some d i s t a n c e , a , from t h e column c e n t r e s .
A q u a d r a n t o f t h e s l a b is t a k e n o u t for t h e study ~f
bend ing moments i n t h e east-west d i r e c t i o n o n l y .
The l o a d W/4 acts a t $o%&'i d i s t a n c e from l i ne of
Column C e n t r e s .
Mp and Mn a r e p o s i t i v e and n e g a t i v e moments t h a t
e x i s t i n t h e p a n e l , t h e h a l f v a l u e s shown a r e f o r t h e
q u a d r a n t o n l y ,
Mn/Z i n c l u d e s t h e component, i n t h e c o - o r d i n a t e
d i r e c t i o n s , of w h a t e v e r n e g a t i v e bending moment t h e r e
i s upon t h e a d j a c e n t c u r v e d boundary. From Symmetry,
t h e r e i s no v e r t i c a l shear on any of t h e f o u r plane
v e r t i c a l s u r f aces shown. T a k i n g ncrments about t h e
d i a g o n a l AC:
w (X -a) s=c4s0 = (2m - 2 ~ ~ ) COS 45' 4 P + - 2 2
L e t sum of t h e p o s i t i v e and n e g a t i v e bend ing moments
p e r p a n e l , i n e a c h d i r e c t i o n , be Plds, and t a k i n g a h a l f
p a n e l f o r s t u d y ;
From Mechanics:
W = w(L2 - 5 h g ) 4
where W i s t h e l o a d p e r U n i t square t t?
By t a k i n g moments o f a r e a s a b o u t AB:
S i n c e a = hc/X from m a t h e m a t i c s , T h e r e f o r e :
Mds = + 1 1 3 ( k c ) 8 hL L
From which by a p p r o x i m a t i o n s ,
Mds = & (1 - 3 h c l L - 8 3L
CHAJJTCR , , THREE
3.1 STRUCTUKE OF THE FLOW ,CHAFfTG
Each of t h e f low c h a r t s starts W i t h t h e d e f i n i t i o n
o f t h e i n p u t p a r a m e t e r s ,
A n a l y s i s of l o a d i n g f o l l o w s w i t h t h e C a s e of
r e s t r a i n t a s s p e c i f i e d i n C P I I O S e l e c t e d . T h i s
t h u s h e l p s i n t h e c a l c u l a t i o n of t h e b e n d i n g momen t s
a n d w h e n c e areas o f r e i n f o r c e m e n t . b
T h e f l o w c h a r t s e n d w i t h t h e c h e c k s r e q u i r e d b y
t h e l i m i t s t a t e d e s i g n v i z : C r a c k i n g C o n t r o l , D e f l e c t i o n
a n d S h e a r .
T h e m a i n f l o w c h a r t s a n d d e s i r j n s are m e a n t t o
show t h e i n f l u e n c e of S t e e l t y p e (FYI d e p t h of s l a b
( H ) , C o n c r e t e g r a d e (Fa) a n d Diameter of S t e e l
r e s p e c t i v e l y o n t h e C o s t of t h e s l a b .
3.2 PROGRAMME SPECIFICATION:
T h e p u r p o s e of t h i s p rog ramme i s t o d e m o n s t r a t e
t h e c o m p u t e r - a i d e d d e s i g n of one-way a n d w a f f l e s labs.
T h e a n a l y s i s i s r e s t r i c t e d t o a n i n t e r i o r p a n e l t y p e
i n a c o n t i n o u s f loor .
The single loading condition hnsiderecf i s tha t
of a uniformingly distributed loat) co\ictincj U;i? A l e
area of the panel,
The slabs are designed for the ultimate limit
state of bendinq and the serviceability limit states
of deflection and cracking as defined in CPIIO: 1972.
3.3 INPUT DESIGN IlIFORIVV\TION AND DESCRIPTION OF FLOh CHART:
STAGE I: - 4
The design infori~iation is read into the computer
in this stage.
The material properties which are now made the
variables are punched in,
Concrete strength (Pcu) has the values
Steel tensile strength (FYI are 250, 420, 425,
Steel Diameter (D) are 10, 12, 16, 20, 25,
Slab Thickness PHI are 100, 125, 150, 175, 200mm
Beam depth (HI) are 250, 300, 350, 400, 450,
Rib Thickness (B) are 100, 120, 130, 140, 150,
STAGE I V :
T h i s c o m p r i s e s of t h e f irst sectlor\ check. I f
t h e a p p l i e d moment i s g r e a t e r t h a n t h e m o m e n t ef
r e s i s t a n c e of t h e c o n c r e t e ( o n t h e basis of %alanced
d e s i g n ) , t h e n t h e s e c t i o n d e p t h must be i n c r e a s e d .
I f t h e check shows t h a t t h e c o n c r e t e h a s a sufficiently
h i g h moment of r e s i s t a n c e , t h e n n e x t s t a g e i n t h e
c a l c u l a t i o n i s t o d e t e r m i n e t h e a r e a o f steel r e q u i r e d .
STAGE V: 6
The maximum s p a c i n g of b a r s i s c o n t r o J 3 6 d b y t h e
c h a r a c t e r i s t i c s t r e n g t h o f s tee l , t h e steel p e r c e n t a g e
and t h e o v e r a l l s l ab dep th .
The pu rpose of t h i s , s t a g e i s t o d e t e r m i n e t h e
s p a c i n g o f t h e d e s i g n e r s p e c i f i e d b a r d i a m e t e r which
g i v e s an a r e a p e r meter wid th .
STAGE V I :
T h i s s t a g e i s a check on t h e d e f l e c t i o n c r i t e r i a
i n C P I I O . The m o d i f c a t i o n f a c t o r s are whown i n t h e
APPENDIX ( F i g u r e s A 6 t o A l l ) , The S t e e l p e r c e n t a g e
(PI which i n c o n j u n c t i o n w i t h t h e s p e c i f i e d s l a b d e p t h
s a t i s f i m t h e bench ing c r i t e r i a i s u s e d to d e t e r m i n e a n
e f f e c t i v e d e p t h and span which meets d e f l e c t i o n
r e q u i r e m e n t s . The s p a n i s &np&ed w i t h t h e available 1
s p a n . I f t h e s p a n r e q u i r e d f o r defl&t$an i s g r e a t e r
t h a n t h a t a v a i l a b l e , t h e n t h e c h a r t s i n s t r u c t t h e
d e s i g n e r t o i n c r e a s e t h e s l ab d e b t h u
STAGE VII:
The A c t u a l s h e a r stress a n d t h e a l l o w a L l e v a l u e
a r e compared. If a t t h i s j u n c t u r e t h e a c t u a l s h e a r
stress i s less t h a n t h e a l l m w a b l e v a l u e , t h e n t h e
slab w i l l h a v e s a t i s f i e d a l l t h e programmed design b
cr i ter ia b u t t h e c a l c u l a t i o n w i l l move f u r t h e r i f
s h e a r r e q u i r e m e n t s dactate t h a t t h e s l a b s h o u l d be
m o d i f i e d i n some way. The c o n c e r n o f t h e programme
i s to d e t e r m i n e t h e n a t u r e o f t h e m o d i f i c a t i o n .
STAGE; VIII:
In t h i s s t a g e , c a l c u l a t i o n of t h e number,
l e n g t h and w e i g h t of steel r e q u i r e d i s done.
C u r t a i l m e n t o f steel i s c o n s i d e r e d i n t h e c a l c u l a t i o n
a n d l e n g t h o f b a r i s c o n v e r t e d i n t o w e i g h t a s shown
ear l ier . The cost i s t h e n c a l c u l a t e d , The results
for a n y s l a b which m e t C P I I O r e q u i r e m e n t s would
a u t o m a t i c a l l y be o u t p u t . The programme r u n i s
t e r m i n a t e d a t t h i s p o i n t . A n o t h e r set o f v a r i a b l e s
are t h e n s e l e c t e d f o r a n o t h e r programme r u n ,
COM~R~SSLOW mwzec- 47 = ~ 7 r -10 .-1 W/DB-YM
NUMBeRr U S < ~ * A ~ / X D ~ ) (260ol r .la)
LEF(OTH=L~-@-~Z-~~)~~NS
b TENSWN = ~ e <~T&-ao-tbpr=~~ .a- s"aL (w-(P=P~/~ -70/4103-~2~4 UIrWBER = N B C ~ ~ ~ K C B Q ) ~ : r i rE iQ3/ f - t~ LENGTH = LB r ( L ~ - r ~ 5 ) l t CoMPReaslOHI ~ ' t =[a+=cu. erElL <09-K2 R C L ) + e - * F I - ~ f e 8 7 h
NUUSER = U6 =C41~7~0~&541f1-16
36.
P r i n t "Des ign of R.C. Slabs - Ofbe WayH I
P r i n t qtH are 1$0, 125, 159, 175, 2ggN
P r i n t "F2 are 28, 25, 38, 48, 58"
P r i n t "F3 are 25p, 418, 425, 468, 5$@,W
P r i n t 'ID are 10, 12, 16, 28, 25"
I n p u t H, F2, F3, D
L1 = 4.0$
L2 = 6p10
B = 35P)
H I = 55P)
D5 = 8
S1 = ,P)236 x H K L1
F1 = 1.5 x L1
Dl = Sl + F1
I1 = 3 3 L1
D2 = 1.4 K Dl + 1-6 X I1
Z = (3 - LGT (F2))/1,2 C 1 = 19 A Z
D3 = H - Cl - D/2 Rem l g D e s i g n S p a n 2-3"
37.
M2 = D2 L1/14
K 1 = (g.45 - g.$@84 H SQR ( ~ 2 ) ) N F2
K2 = 1 - g.45 F2/KI ( g . 5 - F2/3828)
M 1 = 4@$ F2 x X -n (D3 - K 2 +t X )
I f (b12 or- 1$ A 6 & M I ) Then G d o 290 ELSE GOT0 280
X = X + 1 GOT0 26g
2 1 = D3 - K2 +P X
A2 = M2 -x l a fi 6/(ae87/F3/2l
S2 25g w PI 3+ D D D/k23
I F 92 G 75 Thcn 410
I F 92 C 100 Then 420
I F 92 C 125 Then 430
I F $2 L 250 Then 440
I F $2 4 175 T h c n 450
I F S2 1 200 T h e n 460
I F $2 C 250 T h e n 470
I F 'S2 4 300 Then 480
I F S 2 f r3OO Then 490
S3 = 5$ GOT0 5@$
5 3 = 75 GOT0 6@$
S3 = 1$g GOT0 50g
S3 = 125 GOT0 59$
5 3 = 15$ GOT0 588
S3 = 175 GOLO 5$$
S3 = 2$$ GOT0 508
S3 = 258 GOTO 5$$
S 3 = 3$$ GOT0 5$P
A 1 = 25g H P I +E D at D/S3
R e m E 'Chcrk fo; U e f l e c t i o n f 7
P = ~ 1 / 1 $ / D 3
I F ( F 3 = 25g) T h e n 58g
I F ( F 3 = 41$) T h e n 59$
I F ( F 3 = 425) T h e n 698
I F (F3 = 46g) T h e n 618
I F ( F 3 = 588) T h e n 620
E = EXP ($3.77 - g.356 * L O G ( P ) ) - 1 @ GOT0 63$
E = EXP ( g . 3 - 9.2 u LOG (P)) - 16.34 8 GOT0 63g
E = EXP ( $ , 2 7 - $ . 3 1 * LOG ( P I ) - Be35 @ GOT0 63pl
E = EXP ( $ . I 9 - g.32 * L O G f ~ ) ) - 8.4 @ GOT0 639
39.
E = EXP (PI.12 - 16.28 u LOG ( I ? ) ) - El = 26 at E w D3
IF (El 4 lPI$jJ* C1) Then 60
R e m "Chcck f o r Crack Widtht'
C = 3 i t D 3
IF (S3 3, C ) Then 68
N1 = (1088 X- L2 - ( 2 * C1) ) /S3
L4 = 9.8 * 1\41 ae L1 CJ1 = pl.g@6165 x D x D w L4
A4 = "1 a +ti
S4 = 2 5 g ht. Dl x DS a D5/A4
IF 54 @ 75 Th$n,82$
IF S4 L lg@ Then 839
IF S4 125 Then 84PI
IF S4 4 158 Then 85P
IF 9 4 < 175 Then 86P
IF 6 4 4. 2$@ Then 878
IF 54 4 25$ Then 888
IF S4 C_ 3@@ Then 89pl
IF 94 ptz 3P)P Thcn 988
87$ S5 = 175 GOTO 91g
880 S5 = 298 GOT0 91@
090 S 5 = 25fJ GOT0 918
9$$ s5 = 3fJ@
91(a N 2 =. (1$54 x Ll - P 4 e - C 1 ) / S S
92jl LS - L 2 x ? ; 2
93$ W 2 L- , J Y ~ $ 6 1 F 5 -x D5 &.I5 xL5
94B R c m " l i u p p ~ r t E ~ r n c i . t c '
95@ M3 = D3 x L1/9
99$ R e r n "Design Su7ports 2&3"
?gag XI = 1
101$ M 4 = 4@@ E F2 1 * (D3 -K2 n XI) 1020 IF (lB@j?I@Ple) +e M3 M4) T h e n lp140
1930 XI - X I + 1 GOT0 l@l@
1@4@ 22 L D3 - L2 x XI
1$5$ A3 = '~j2(3P@@Ed * M3/Be87/F3/Z2 la68 S6 = 250 x 01 K D * D/A3
41.
197511 IF S6 C 75 Then 1160
19)89 I F S 6 4 I@@ Then 117$
1@9$ I F 56 4 125 Then lied 118$ I F S6 < 15$ Then 1198
111p IF S 6 IC, 1 7 5 Thcn 1 2 8 8
112P) I F S 6 < 290 Then 1219 1139 I F S 6 < 250 Thcn 1 2 2 8
l14B I F S 6 < 3@@ Then 1238
11SB I F S 6 3 9 3 0 8 Then 1 2 4 8
1160 S 7 J 50 GOT01;1258
1 1 7 s 57 a 75 GOT0 1250
1189 S7 4 18fl GOT0 1 2 5 8
119$ 57 = 1 2 5 GOT0 1 2 5 8
1 2 0 8 S7 = 1 5 0 GOT0 1250
121s S7 = 1 7 5 GOT0 1 2 5 0
1220 57 = 289 GOT0 1 2 5 8
1 2 3 8 S7 = 250 GOT0 12Sg
1240 S 7 = 3$a
1258 A5 = 250 +e PI I D D Dl57
1268 Rem "Check Crack liJidthw
1270 C2 = 3 * D3
1 2 8 8 If ( S 7 > C2) Then 6fd
129a N 3 = ( I@@ w L2 - 2 W C I ) / S 7
138jJ W3 = 4.086165 w D K D * L6 1329 A 6 = 3 x H
1338 S 8 = 25jU n DI * D5 % D5/A6
1348 IF S 8 4 75 Then 1438
1 3 5 0 IF S 8 r 1PB Then 1448
1360 I F 58 L 125 Then 1450
1378 I F S 8 6 1 5 0 Then 1468
138g I F S 8 L175 Then 1478
1399 IF S 8 < 2$8 Then 1488
id@$ I F ~8 f. 2SP) Then 1498
1416 IF S8< 38@ Then 15$0
142p I F S83~308 Then 1518
143$ S 9 = 5@ GOT0 1520
1448 S 9 = 75 GOT0 1528
1450 S 9 o I @ $ GOT0 1528
1468 S9 = 125 GOT0 1528
1470 S 9 = 158 GOT0 1528
148g S 9 = 175 GOT0 1528
1490 S9 r 288 GOT0 1528
157g Rem "Check for Shearu
158$ V = $3.6 E D2 +t L1
1590 Vl = 8.75 iw SQR (F2) u D3
16@@ IB (V14V) Then 60
16l$ Rem "Cost Slabw
167p PRINT H, F2, F 3 , D, CS, C6, C7
1680 END
44.
WSLhBl
PRINT "Design of Waffle Slabs1'
PRINT ltSarc 68@, 788, 888, 900, 1888, it@@, 1200 + 13BB, 1408"
PRINT v ' ~ Are 1@PJ, 12@, l3$, 140, 150, 160, 2$gN
PRINT "F2 Are 2P, 25 , 3fl, 48, 58"
PRINT lfD A r e , 10, 12, 16, 28, 25, 32, 40"
PRINT lgF3 Are, 259, 410, 425, 469, SgI$" , INPUT S , B , F 2 , F3, D
Rem 'lLoading Per Waffle"
D l = 4 * B
45,
K 1 = (Q.45 - BoB984 e SQR f FZ) & ~2
K2 1 - (#.45 sc F2/K1) w ( 0 , 5 k 2 f 3 2 2 8 )
R e m "Design as a F l a t Slab"
M 1 = F 1 ie L2 K (L1 - 0 .93 ) u (L1 - 8,93)/8 Z ( 3 - LGT (F2))/1.2 C1 = 1 0 A Z
D3 n D4 -C1 - D/2
Rern "Middle S tr ipqa
M2 = 6.20625 n M l
N 1 n I$P)@fd#$. JC M2/B/D3/D3/F2
X = 1
M7 = 6.4 x B bt F 2 XX K (D3 - K2 K X )
IP (M7 2 lB0P)PIBB x M2) Thcn 350
X = X + 1 @ GOTO 32g
I F ( N 1 9 0 . 1 5 ) Then,) 380
A 2 = 1jl$9)$10 -X M2/a087/F3/CD3 - K2 +t X )
N 4 s 4 * k2/PI/D/D
S 2 = 1 N S N N 4 dc L1 +c ~ 2 / ( 1 . 1 M 5 ) 8 GOT0 450
A3 = ( la@@@@$ K M2 - B.15 F2 B x (03 - K2 * 3) * ( 0 3 - K 2 * x ) / g 0 7 2 / F 3 / ( D 2 - K2 * X )
N 3 = 4 k3/PI/D/D
L3 = 1000 N 3 * L1 +t ~ 2 / ( 1 . 1 at S )
A2 = (0.2 n Fa B (D3 - K 2 R X ) + 0.72 x F3
x k 7 ) / l l / 8 7 / F 1
46.
N4 = 4 M k2/PI/D/D
L4 = 1000 * N 4 r L 1 uL2/(1.1 * S )
L2 = L 3 + L4
W2 = 0.886165 sc. D K 9 3+ L2
A1 = PI ae D * D n )44/4
P = 100 n kl/B/D3
IF (F3 = 25g) Then 53fl
IF (F3 = 41@) Then 54$
IF (F3 = 425) Then 558
IF (F3 r 468) Then 56$
IF ( F 3 = 500) Then 57@
E = EXP (0.77 - 0 356 *LOG (P)) - 1 @ GOT0 580
E = EXP (0.3 - 0.2 RLOG (P)) - 0.34 8 GOT0 58b
E = EXP (0.27 - 0.31 E LOG ( P ) ) - 0.35 @ GOTO 588
E o EXP (0.19 - 0.32 +C LOG (P)) - 0.4 @ GOTO QSB
E s EXP (0.12 - 0.28 -X LOG (PI) - 0.25 El = 26 a E e D4
Hem "Column Strip (CAP)"
M3 = 0.65 w MI - M2 XI .r 1
M 4 = 1000 F2 * XI % BD3 - K 2 n XI)
IF ( M 4 3 1000000 M M3) Thcn 668
XI = X1 + 1 @ GOT0 638
1,s = 1000000 +t M3/0/87/F3/(D3 - K2 H XI)
47.
6 7 8 S 1 = 258 x P I * 0 * D/n5
688 I F ( S 1 < 75) Then 770
69111 I F ( S 1 4 I$@) Then 78515
788 I F ( s l 4 1 2 5 ) Then 79@
718 I F (S1 r IS@) Then 88515
22$ I F ( S 1 4 175) Then 81$
730 I F (31 C 2@@) Then 8 2 9
74$ I F ( S l X 258) Then 83jB
75515 I F ( S 1 K 3P0) Then 88$
76(6 I F ( S l ' 7 o 300) Then 85111
770 S2 = %O @ GOT0 860
789 S 2 = 75 @ GOT0 860
798 S2 = 1 0 0 @ GOT0 860
888 S 2 = 125 @ GOT0 860
818 S2 = 150 @ GOT0 860
820 S 2 = 175 @ GOT0 860
8 3 8 S 2 = 200 @ GOT0 860
840 S 2 = 250 @ GOT0 860
858 52 = 300
860 ii6 =t 250* P I * D +t D/S2
48.
87f3 N 6 = 5000/S2
889 L6 = 2-5 N N 6
89P) W6 = 0,006165 x D D D D L6
900 Rcm "Column S t r i p Midw
910 M7 = 0.35 * M I - M2
92$ K 2 = 1000000 w M'7/2500/D3/D3/F2
930 X2 = 1
949 M 8 = 1000 M F2 +t X 2 (D3 - K 2 * ~ 2 ) @
95P) I F ( M 8 3r 1000000 +t M7) Then 940
968 X 2 = X2 + 1 @ GOT0 940
978 I F (K7 )0 ,15 ) Then 1010
98$ A7 r 1000000 N ~ 7 / 0 . 8 7 / F 3 / ( ~ 3 - K2 +t X2)
998 N5 = 10 ( 4 w i ~ 7 / /D/D) * (2500/1.1/S)
L7 -- ( L 1 - 2,s) w N 5 @ GOT0 1080
l@l@ ti8 = (1000000 xM7 - 0.15 x F2 K B x ( D 3 - K2 *X2)
Oe72/F3/(D3 - K2 u X 2 )
1028 N 8 = (4 +t AS/ X /U/D> +e (2500/1.1/5)
1938 L8 = ( L l - 2,s ) .K N 8
1@4$ A7 = (0.2 * F2 -n B +c (03 - K2 x x 2 ) + 0.72 M F3 +t ~ 8 )
/0087/F3
la5P) N5 = ( 4 +e k 7 / X / U / D ) -#t (2500/1.1/S)
49.
1860 L5 = ( L l - 2.5) K N 5
l$7$ L7 = L 8 +L5
1P)8@ W7 = 0.006165 * D * D * L7
1$90 Rcm " P u n c h i n g Shear"
l1$@ V 1 = F1 * L i wL2 - 96.39
1110 V2 3 1.25 % V1/2388/Pl/U3
112P P 1 = 100 117/B/D3
113$ I F ( F 2 = 20) Then 1170
1143 I F ( F 2 = 25) Then1180
1150 I F (F2 = 30) T h e n 1190
1168 I F (F2 7 = 40) Then 1200
117$ E2 = 0.6 H P I A 0.415 8 GOT0 1210
118$ E2 s 0.6 x- PI A 0.42 8 GOT0 1210
1190 E2 = 0.65 + P I A 0-43 8 GOTO 1210
12$$ E 2 = 0.65 dt P I A 0 . 5 3
121P) I F (V2 7 E2) Then 7$
1220 Rcrn "Cost Slabtg
123$ W 8 = W2 + W6 +W7
124$ C 8 = 1.5 w W8
125$ V4 = V * L 1 * L2/1.1/S/1000000000
1268 C 4 = 180 +t V4
1270 C9 = C8 + C4
1289 PRINT S t 5 , F2, F3, D t C 8 ? C4, C9
A - n f n r r r r n
CHAPTER FOUR
4.1 OUTPUT:
T a b l e 1: Table Showing I n f l u e n c e of S t e e l T e n S i l c S t r e n q t h on Cost of S o l i d S l a b
INPUT
*Cos t of Beam is n o t i n c l u d e d .
FCU I FY
50 250
Table 2: T a b l e Showinq I n f l u e n c e of S l a b T h i c k n e s s H on Cost of S o l i d S lab:
D
12
INPUT COST
P4T - *Not S a t i s f i e d
N o t S a t i s f ' e d
909.828
1050.885
1158,885 I. *These d e s i g n s d i d n o t s a t i s f y t h e L i m i t s ta te
requ irement s f u l l y .
Table 3: Table Showinq I n f l u e n c e o f Concrete Compressive S t r e n q t h on C o s t , o f ,.Solid S l a b
T a b l e 4: Table Showinq I n f l u e n c e of Rod Diameter On Cost o f S o l i d S l a b
I BST S /N
1
2
3
4
5 I
INPUT
H
175
FCU FY I D
20 250
17 5
17 5
17 5
175 1 I I I I I
250
250
250
250
25
30
40
50
16 1187.22
16
16
16
16
1187,80
1188.20
1188.72
1189.05
T a b l e 5: T a b l e Showinq I n f l u e n c e o f Rod Diameter On C o s t o f F l o o r ( S o l i d S l a b + B e a m )
INPUT
T a b l e 6: T a b l e Showinq I n f l u e n c e o f C o n c r e t e S r e n q t h On C o s t of F l o o r ( S o l i d S l a b + ~ e a m )
H I
600
600
600
600
600
FCU
50
50
50
50
50
S/N
1
2
3
C
FY
500
500
500
500
GOO
D
10
12
16
20
25
PiT
F u Q l L i m i s t a t e C o n 2 i t i o n Not s a t i s f i e d
1849.4
1818.6
INPUT
1805.02 500
H
175
175
175
175
FCU
20
25
30
40
50
e
H I
500
500
500
500
FY
250
250
250
250
250
D
l6
16
16
16
16
T a b l e 7: - T a b l e Showinq I n f l u e n c e o f S l , a b T h i c k p e s s , Ol'd C o s t of F l o o r ( S o l i d Slab + Beam,) --
1 1 INPUT
9 / N H H1 FCU FY D
1 100 500 50 500 12
2 12 5 500 50 50 0 12
3 150 500 50 500 12 1
4 175 SO0 50 500 12
5 200 500 50 500 12
L i m i t S t a t e
C o n d i t i o n s n o t f u l l y Satisfied 136i),9
T a b l e 8: T a b l e Shwinq Influence of Steel T e n s i l e ~ t r e n ~ c h o n Cost of F l o o r (Solid S l a b + B e 9 d
S/N
1
2
3 4
5
I
3 iT * -
INPUT - D
___L_1"_Y__
R T H l I F C U -
I
FY
250
4 10
42 5
460
500
-- -
12
12
12
12
12
- - -
50
50
50
50
50
- -- - -
200
200
200
200
200
1844,O
1643.52
1635-91
1620.08
1604.69
500
500
500
500
500
-
T a b l e 9: T a b l e S h o w i n q I n f l u e n c e o f R i b T h i c k n e s s R , o n C o s t o f W a f f l e F l o o r
-*
Size of Mou 1 d s
INPUT - R i b J T h i c k n e s
FCU
Diameter
I
C o s t
HT
1196.18
1123,284
1069.52
6026.5
991.8
89S009
T a b l e 10: T a b l e Showinq I n f l u e n c e of Mould S i z e o n C o s t of W a f f l e F l o o r -
S i z e o f Mould
S 7-
600
700
800
900
1000
1100
1200
1300
1400
INPUT
T a b l e 11: - T a b l e Showinq I n f l u e n c e o f Concrete C o m p r e s i v e S t r e n q t h on C o s t o f W a f f l e F l o o r D___-
T a b l e 12: T a b l e Showinq I n f l u e n c e o f S t e e l S t r e n q t h on C o s t of Waffle F l o o r
k
S/N r
1
2
3
4
INPUT
600 120 I 30 500 I
S
600
D I WT
16
FCU
30
B
12 0 16
16
16
16
1072-83
FY
250 2145-67
1296-99
1257-18
1153.26
410
42 5
460
600
600
600
" ~ 2 0 i 30
120 1 30
1-2 0 1 30
T a b l e 13: T a b l e ShowJnq J n f l u e n c e o f Rod Diameter on C o s t o f W a f f l e F l n o r
f NPUT
FCU
T a b l e 14: T a b l e Showing I n f l u e n c e o f Deam Depth on C o s t o f Beam
INPUT
FCU
C o s t
t4T
Not S a t i s f i e d
Not S a t i s f i e d
Not S a t i s f i e d
817.56
650.44
509.834
529.486
570.011
PHOGRAMME FLON FOR DESIGN OF OIJE,&WAY, (INTERNAL) R.C. SLABS;
Print "Design of R O C e S l a b s - One Wayt'
H 100, 125, 156, 175 , 200
F2 20, 25, 30 , 40, 50
F3 25,0, 410', 425, 46$, 58,0
D 1 $ , 1 2 , 1 6 , 2 0 , 2 5
H = 175, F2 = 3,0, F3 = 250, D = 16
L 1 = 4 . p - 6
L2 = 6.@9
B = 35$
D5 = 8
S 1 = 4.236 x 175 x 4 = 16,52
F 1 = 1.5 x 4 = 6
D l = 22.52
I 1 = 3 x 4 = 1 2
D2 a 1.4 x 22.52 + 1.6 x 12 = 50.73
Z = 1.26
C1 = 18.5
D 3 = 175 - 18,s - 8 = 148.5
R e m "Design Span 2-3"
M2 = 5p.73 x 4/14 = 14.49
K 1 = 12.119
K2 = 0.45
X = i
M 1 = 1,78 x 1@6
14.49 x 1p6 7 1.78 x 1p6
x = 9
Z1 = 144.45
A1 = 461.2
S2 = 435.9
S3 = 3Qd
A 1 = 67P.2
R e m " C h e c k for D e f lect ionfl
P = 67p02/1(I/148.5 = g.45
F3 = 256
E = 1-86
E l = 26 x 1.86 x 148.5 = 7181.46
7181,4 7 40041
R e m " C h e c k for C r a c k W i d t h f 1
C = 44505
3ew L 445.5
N1 = 19.9
L4 = 63.7
W1 o 166.5
A4 E 2 lB
726 S4 = 239.3
886 S5 = 200
910 N2 = 19.8
920 L5 = ll8,8
93pI N2=46.9
94,0 Rern **Support Momentsw
950 M3 = 22.54
99a R e m *@Design S u p p o r t s 243"
1df.M Xl = 1
1glkf M4 = 1.78 x 196
1920 1.78 x 1$6 dl 22.54 x 166
103d XI = 1
1f&B 22 = 142.2
1p58 A3 = 728,77
1960 S6 = 275.89
114pI 25d L S6 L 30[J
12353 s7 = 258
1250 A5 = 8d4.2
126@ R e m "Check C r a c k width"
127$ C2 = 445,s
128pl 25g h 445.5
1290 N3 = 23.85
13fl6 L6 = 57.36
1520 N4 = 21-33
1536 L7 = 127.98
156p W 4 = 50.5
1576 Rem "Check f o r S h e a r t 1
1588 V = 121-75
1590 V1 = 6U.42
1680 121.75 L 6ld.92
1610 Rem "Cost Slabn
162.0 W5 = 288.43
163P C5 = 432.64
1640 V2 = 4.2
1650 C6 = 756
166g C7 = 1188.64
4.3: PROGRAMME FLOW FOR BEAMS
70 H = 158, H1 = 6$g, F2 5 3$, F3 = 4116, D = 2$
L1 = 4.$$
L2 = 6.a$
l$$ B1 = 250
11g P = 3 4 4 = 1 2
F = 1 . 5 ~ - 4 = 6
S1= 14,16
S2 = 2,655
S = 14.16 + 2.655 + 6 = 22,815
169 Dl = 1.4 n 22,815 + 1,6 R 12 = 51.141
17$ Rem "Design As S.S. Beamw
3 MI = 51.41 x 6 - / 8 = 234,13
Z = 1.26
C1 = 18.58
D2 - W H - 18.58 - l.0 = 571.5
K 1 = 12,119
K2 = &451
K = 8,0939
X = 1, M2 = 1.7 x 1d6
X = 148, M2 = 2135 w 10'6
X = 150, M2 = 226-7 w 106
X = 154, M2 = 231.9 w 1$6
A 1 = 1285,8/8rnrn 2
N4 = 4.169
L5 = 1.84 x- 4.P9 a 6 = 45.159
W 1 = 111.36kg
Rem "Check for D e f l e c t i o n
P 1 = K 8 9 9
E = 1 . 0 3
E l = 15.314
R e m 'lCheck for Sheart1
53 = 22.815 x 6/2 = 68.445
V l = d 4 7 9
I F ( V 1 7 4.1_8)
F2 = 3d@ GOT0 668
E 2 = #,62$9
If (6 .479 4 8'.6209) t h e n 70g
A7 = 75
N7 = 39.75
L7 = 129.267m
W2 = 79.693
W 8 = 191.0'53
W9 = 1.35m3
C3 = H286,S
C4 = 243
4.4 PROGRAMME FLOW FOR WAFFLE SLAB:
S = 6130
B = 2 d ~
F2 = 3Q/
P 3 e: 419
!D = 1 6
D l = 4 ~ d
L 1 = 8.0'
L2 = 8.6
D4 = 46,g
A = 4356,W
V = 56376D'b~
W = 1.33047 x ljh
D5 = 2.7152
F = 1.$5
D6 a , 3.7652
19pl I1 = 2.1
F 1 = 1.4 % 3,7652 + 1.6 3e2.1 = 8.6312
K 1 = 12.119
K2 = 0.451
230 Rern "Design a s A f l a t Slabw
M 1 = 8.6312 x 8 w ( 7 . 0 7 f 1 / 8 = 431.4
Rem "Middle Stripw
M2 = 88.98
N1 = $A55
X = zjd, M9 r: 83.28 K 206
X = 22g, M9 = 89.232 x d 6
A3 = 376.5
N3 = 1.87
L3 = 181.3
A2 = 879.7mm 2
N4 = 4.375
L4 = 424.2
L2 = 61D5.5
W 2 = B.Og6165 w 16 x 16 x 6g5.5 = 955,6
A1 = 879.64
D = 2.m
F3 = 41d
E = 0'.835
E l = 26 n k7.835 x- 468 = 9986.6
Rem ''Column Strip (Cap)"
M3 = 191.43
X1 = 15, M4 = 193.6 x 106
XI = 14, M 4 = 180.89 x 106
A5 = 1247.3
S1 = 161.19
72d
8W S2 = 15,0
86d A6 = 1 3 4 ~ 6 4
N6 = 33.33
L6 = 83.32
N6 = 131.498
90kf Rern V o l u m n Strip Mid1'
M7 = 62.01
K7 = B.004
X2 = 5, M8 = 65.212 x ld6
X2 = 4, M8 = 52-22 x 10'6
9881 A7 = 399.8mm 2
N5 = 7.53
L7 = 41,415
ad80 W7 = 65.36
I&$ R e m "Punching S h e a r w
V 1 = 456
V2 = fl',d0$18
1128 P I = B.914
11Sg
119$ E2 = 8,625
1220 Rem "Cost Slabv
4.6 INFLUENCE OF VARIOUS PARAMETERS ON COST OF THE FLOOR:
The design permitted the inclusion of most of
the inportant factors affecting or controdHpgt We
selection of design criteria that are in accord'with
the accepted general philosophy of slab design.
The procedures versatility permitted present
design constraints to be changed or new constraints
to be added, or both. This versatility, in conjunction
with automation of the member design, allowed several
designs to be obtained in a relatively short time.
With reference to solid skabs, the graph shows that
the higher the strength of steel, the lower the
cost of slab, There is no marked difference in the
cost of slabs with high yield bars. This might be
due to costing of the steel because the same price
unit was used far all the bar types. Moreover the
F I G 5 -
INFLUENCE . - - - . . . . . . - . . OF . ..... HOD .~ .~. DIAMETER ... . . .. . . . O N . . COST
F IG* B
REF: TABLE IG
OF W A F F L E W A L L
1000 1 I I I I I
80cf [ O W '203 _____t__
Goo 400
R E F : TABLE 1 1
INFLuEUCE OF CONCRETE 3 1 - K E N G T H
R E F : TABLE I?
F I G 16
83.
area o f bars were compensated i n t h e s e l e c t i o n of
s p a c i n g and t h e r e f o r e number o f b a t s i n t h e s l a b .
Smaller a r e a meant s m a l l e r s p a c i n g and l a r g e r number
o f b a r s and v i c e v e r s a .
Fo r t h e s p a n i n q u e s t i o n , t h e g r a p h f o r t h e
e f f e c t o f s l a b t h i c k n e s s on c o s t o f s lab shows a
r e l a t i o n s h i p o f t h e form Y = MX+C. The h i g h e r t h e
s l a b t h i c k n e s s , t h e h i g h e r t h e c o s t o f t h e s l a b .
T h i s c o u l d be a t t r i b u t e d t o t h e f a c t t h a t s m a l l plab
t h i c k n e s s e n t a i l s h i g h e r q u a n t i t y of steel and h i g h e r
s l a b t h i c k n e s s e n t a i l s lower steel q u a n t i t y .
The e x t r a cost f c r . t h e s l a b is u s e d f o r improvement
of r i g i d i t y a n d l i m i t a t i o n o f # f l e c t i o n .
The i n f l u e c e o f c o n c r e t e s t r e n g t h on t h e cost
of t h e s l a b i s d e p i c t e d i n f i g u r e 6. The g r a p h shows
t h a t economy i s a c h e i v e d w i t h l o w c o n c r e t e s t r e n g t h ,
But b e c a u s e o f s h e a r and other l i m i t s t a t e r e q u i r e m e n t s ,
h i g h e r c o n c r e t e s t r e n g t h s i n t h e o r d e r o f FCU = 30U/mm 2
are p r e f e r a b l e , I t w i l l be r e c a l l e d t h a t c o s t i n g
o f c o n c r e t e d i d n o t t a k e c o g n i s a n c e of c o n c r e t e
s t r e n g t h b u t volume of c o n c r e t e , The i n f l u e n c e o f
c o n c r e t e s t r e n g t h on cost o f t h e f l o o r ( S l a b + ~ e a m ) c a n
be s e e n t o f a v o u r t h e u s e o f h i g h e r v a l u e s f o r t h e
ach ievemen t o f lo^ cost slabs. T h i s i s shown i n
84;
For solid slabs, reinforcement should be limited
to samll diameter rods as they make for effective
control of crack widths. Smaller sized rods mean less
spacing of bars while large sized rods entail more
spacing, large area and more distribution bars and
therefore uneconomical.
For waffle floors, economy is achieved when the
rib thickness is made thick. This, firstly, improves
economy as shown in figure 12. Secondly, it makes for b
a more rigid floor espeically when it is intended to
carry large imposed loads, From the graphs, designs should
therefore use a rib thickness in the neighbourhood of
120mm.
The size of mould infleuencesthe cost of the
slab. Availability, though a factor was not taken
into consideration, The graph shows that the size of
mould should be reduced to the barest workable size.
600mm is recommended as ceiling boards come in that
size and for easy fixing of these boards.
For waffle floors, the effect of concrete strength
on its cost is shown in figure 14, For economy,
high concrete strengths are favoured to low ones,
This is so because for effective resistance to shear
8s.
S t r e s s , w e need a c l o s e l y packed and h i g h s t r e n g t h
c o n c r e t e .
High y i e l d s teel i s recommended f o r d e s i g n and
c o n s t r u c t i o n o f w a f f l e f loors. T h i s i s shown i n
f i g u r e 15, I t is n e c e s s a r y b e c a u s e t h e l a r g e c l e a r
s p a n t h a t is i n v o l v e d means h igh t e n s i l e stress on
t h e c o n c r e t e and t h e r e f o r e e n t a i l s t h e u s e o f h i g h
y i e l d steel to c o u n t e r a c t t h e a c t i o n , The h i g h e r
t h e s t r e n g t h o f s t ee l , t h e less t h e r e q u i r e d area
and t h e r e f o r e t h e less t h e number o f b a r s r e q u i r e d
p e r r ib .
With small r o d d i a m e t e r , t h e cost o f t h e f loor
i s s k y r o c k e t e d . B igge r bars b r i n g down t h e cost u n t i l
a d i a m e t e r of 25mm i s r e a c h e d , which i s t h e t ~ k n i n g
p o i n t , W i t h bars g r e a t e r t h a n 25mm, t h e w a f f l e f l o o r
also becomes uneconomica l a g a i n . F i g u r e 1 6 therefore
shows t h a t o p t i r n a l i t y i n d e s i g n i s a c h i e v e d f o r
w a f f l e f l o o r s when t h e b a r s i z e i s 25mm.
I n v e s t i g a t i o n i n t o t h e i n f l u e n c e o f beam d e p t h
on t h e c o s t o f a beam shows t h a t t h e r e i s an optimum
beam d e p t h f o r a n y span . T h i s is i l l u s t r a t e d i n
f i g u r e 1 4 where t h e minimum cost i s a c h i e v e d a t a
d e p t h o f 500mm f o r a 6m span beam.
CHAPTER F I V E 8 6 ,
eONCLUSION, SUGGESTIONS AND RECOMMENDATIONS FOR FUTUKE WORK
T h e optimum d e s i g n o f r e i n f o r c e d c o n c r e t e f l o o r s
h a s b e e n examined a n d a d e s i g n method u s i n g i t e r a t i o n
s t e p h a s b e e n d e v e l o p e d and t e s t e d , T h e r e s u l t i n g
d e s i g n method h a s b e e n shown t o b e f a s t a n d a c c u r a t e
upon t w o e x a m p l e s c h o s e n and i s c o m p e t i t i v e w i t h
a l l r e s p e c t s w i t h some o p t i m a l i t y c r i t e r i a m e t h o d s ,
One criticism o f thr . o p t i m a l i t y c r i t e r i a )
a p p r o a c h i s t h a t i t i s a p r o b l e m - o r i e n t e d t e c h n i q u e
s o l v i n g o n l y a s p e c i f i c c l a s s o f p r o b l e m s ,
I t e r a t i v e m e t h o d s a r t , f a r more g e n e r a l i n
t h e r a n g e o f a p p l i c a t i o n s , T h i s work h a s shown t h a t
t h e method employed i s a v ~ r y g o c d method f o r s o l v i n g
n o n - l i n e a r o p t i m i z a t i o n p r c 5 l e m s o f t h e t y p e s which
a r i s e f r e q u e n t l y i n s t r u c t u r a l dc - s ign ,
5 .2 SUGGKLTIONS FOR FUTUKE WORK: .- T h e r e s e a r c h was r e s t r i c t e d t o o n e way s l a b s
a n d w a f f l e f l o o r t y p e . F 'utul-e work s h o u l d c o n s i d e r
t h e e f f e c t o f t h e o u t l i n e d p a r a m e t e r s on t h e
r e m a i n i n g t y p e s of f l o o r s so t h a t a g l o b a l s o l u t i o n
c o u l d be worked o u t f o r a p a r t i c u l a r c l a s s o f f l o o r
s y s t e m .
T h e p r o c e d u r e d e v e l o p e d s h o u l d a l s o be e x t e n d e d
t o o t h e r s t r u c t u r a l e l e m e n t s l i k e c o l u m n s a n d
f o u n d a t i o n s .
HECOMMLNUATIONS r
B a s e d upon t h e d i s c u s s i o n s p r e s e n t e d i n t h i s
w o r k , i t w o u l d a p p e a r t ' a t t h c inethod e m p l o y e d
f u r n i s h e s a n a p p r o a c y ti t h e sc u t i o n o f some o p t i m a l
d e s i g n s t r u c t u r a l p r o b l t 11s w h i c I a r e o f p r a c t i c a l
i m p o r t a n c e . T h e p r o b l e m ~ f o b t , i n i n g t h e o r c t l c a l l y
e x a c t s o l u t o n s t o t h e OF_ -imurn t i e s i y n p r o b l e m i s
e x t r e m e l y c o m p l e x . T h i s , e a n s - . h a t t h c d e s i g n s
o b t a i n e d a r e n o t o p t i m a l i n any r e g o r o u s s e n s e , b u t
t h r o u g h t h e g r a p h s , ! t h e t r e n d ; n d e f f e c t of t h e
v a r i o u s p a r a m e t e r s o n t h e i ,os t c - t h e s l a b c a n be
s e e n a n d t h i s i s of r e a l p r a c t i c a l c o n s e q u e n c e ,
I t h a s a l s o b e e n shown t h a t t h e cost o f
m a t e r i a l c a n be e a s i l y i n c o r p o r a t e d i n t o a d e s i g n
p r o c e s s based o n CPIIO t o o b t a i n t h e op t imum d e s i g n
s o l u t i o n ,
N C I M I ~ ~ A L COVER TO MAIN RElpWRceMEN1
MILD EXPOSURE CONDITION
FIG. 43
S I M P L F E D R U L E S FOR CURTA\RMENT Of
B A R S IN B E A M S
M A X I M U M --.- - - PERtvlISS1BLB 'VALUE OF HOMW.I.~L --
ULTIMAE SHEAR S W C S S 0, (&,$rlrn2 ) e c@llO:3:3:G: 1 3 -- --
B o n a s i a . J . J . t tDes ign of P r e s t r e s s e d C o n c r e t e B c a m s b y Compu te r t t , J o u r n a l o f t i rnerican S o c i e t y o f E n g i n ~ e r s ( S t r u c t u f a l D i v i s i o n ) A p r i l 1960.
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J o n ~ s , G., "Minimum S t e e l Des ign o f D o u b l y R e i n f o r c e d R e c t a n ~ u p ..- BcamsW Cement and C o n c r e t e A s s o c i a t i o n J o u r n a l , S e p t . 1979 , pp. 22-24,
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