Document Text Contents
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Optimal design of reinforced concrete
retaining avails ,
Shravya Donkada and Devdas Menon
This paper aims at developing an understanding of
optimal design solutions for three types of reinforced
concrete retaining walls, namely, cantilever retaining
walls, counterfort retaining walls and retaining walls
with relieving platforms. Using genetic algorithms,
parametric studies were carried out to establish
heuristic rules for proportioning the wall dimensions
corresponding to the minimum cost points. Optimal
cost-estimates of the retaining walls types were compared
to establish the best design alternative for a given
height. Also, the advantages of retaining walls with
relieving platforms, which are relatively new in India,
are discussed.
Keywords: Reinforced concrete retaining walls; optimal
design; relieving platforms; cantilever walls; counterfort walls;
genetic algorithms.
Introduction
The design of retaining w a l l almost always involves
decision making with a choice or set of choices along
with their associated uncertainties and outcomes. While
designing such structures, a designer may propose a
large number of feasible designs; however, professional
considerations require that only the most opt imal
one, w i t h the least cost be chosen for construction.
For del ivering an acceptable design, today's design
practitioners increasingly rely on P C based programs
that require parameters, such as toe or heel lengths
and stem widths. The process invariably involves a
tr ial and error procedure. Obta in ing a satisfactory
design per se, does reveal its cost position against the
optimal design. The present study therefore aims at
developing an optimal design solution for reinforced
concrete retaining walls, namely, cantilever retaining
walls, counterfort retaining walls and retaining walls wi th
relieving platforms, in terms of m i n i m u m cost as per the
IS456:2000 code. 1 In this connection, this paper discusses
the heuristic rules for the required w a l l dimensions.
Incidentally, it may be noted that one of the w a l l types
studied is the retaining walls wi th relieving platforms.
This wal l type provides an innovative design alternative
and is common in Europe, but relatively new to India.
The scope of this study was confined to retaining walls
ranging from 5 m to 23 m height. A n y surcharge was
converted to an equivalent height and included i n the
heuristic rules. The study assumed that proper drainage
conditions. However, the effect of earthquake loading
was excluded, as the scope was limited to incorporating
the effects of gravity loading. The reason for doing so
was to insulate the design outcome from the complexities
that arise from the seismic zoning of sites, for example,
moderate or high seismic zone.
APRIL 2012 THE INDIAN CONCRETE JOURNAL I 9
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Types of retaining walls
It is wel l k n o w n that retaining walls
are structures that h o l d back soil
or rock f rom a bui ld ing , structure
or area. 2 They prevent down-slope
movement or erosion and provide
support for vertical or near-vertical
grade changes. The lateral earth
pressure behind the w a l l depends
on the angle of internal friction and
the cohesive strength of the retained
material, as wel l as the direction and
magnitude of movement of the stems Figure 1. Types of concrete retaining walls
of the retaining walls. Its distribution
—V\
) j t i
•1
hf! x2\
*2 "i_r
* 2 *r
Counterfort
wall \Relieving
/platforms
* 3 * 6
(a) Cantilever retaining (b) Counterfort retaining
wall wall
(c) Retaining wall with
relieving platforms
is typically triangular, least at the top
of the wal l and increasing towards
the bottom. The earth pressure could push the w a l l
forward or overturn it if not properly addressed. Also ,
the groundwater behind the w a l l should be dissipated
by a suitable drainage system; otherwise, this could
lead to an additional horizontal pressure on the wal l .
A l t h o u g h the effect of surcharge l o a d i n g was not
explicitly considered here, it can be approximated as
an equivalent height of retained earth.
A s stated earlier, this study deals wi th the fol lowing
types of retaining walls:
• Cantilever retaining wal l : Such walls transmit
loads f rom the vert ical port ion, through the
cantilever action, to a large structural footing,
converting horizontal pressures from behind the
wal l to vertical pressures on the ground below.
This wal l type is believed to be economical up
to a height of about 7 m (Figure la) . Since the
backfill acts on the base, providing most of the
dead weight, the requirement of construction
materials for this w a l l type is much less than a
traditional gravity w a l l 2 .
• Counterfort retaining wal l : Cantilever retaining
walls , sometimes, include short w i n g walls at
right angles to the main trend of the w a l l on
their back, to improve their resistance to lateral
loads. Introducing transverse supports reduces
bending moments, when the heights are large.
Such supports, called counterforts, connect the
stem w i t h the heel slab. This w a l l type is believed
to be economical for heights greater than 7 m
(Figure l b ) 2 .
• Retaining w a l l w i t h relieving platforms: When
the depth of soil to be retained is excessive, soil
pressures can be reduced by the use of a relieving
platform . Retaining wal l with relieving platforms
is relatively new to Indian construction industry.
Such walls are k n o w n to provide an economical
l ightweight design solution for relatively tall
w a l l s . 4 , 0 The retaining w a l l is shielded f r o m
active earth pressure by means of one or more
relieving platforms (Figure lc) which make the
pressure diagram discontinuous at the level of
the platform. Also , the relieving platform carries
the weight of the soil above it and any surcharge
loading, transferring them as a 'relieving' moment
to the vertical stem. The re l ieving platforms
are designed such that they intersect the plane
of rupture from the soil above and behind the
platforms preventing any load from the soil to
act on the wal l . This aspect is the key to designing
such walls.
Typically, a retaining wal l design includes :
• Performing stability checks for the retaining wal l
against sl iding and overturning.
• C o m p u t i n g the m a x i m u m a n d m i n i m u m
bearing pressures present under the toe and heel
and comparing them w i t h the allowable soi l
pressure.
• D e s i g n i n g the re in forc ing steel for the toe,
heel , stem and other parts c o n s i d e r i n g the
corresponding bending and shear forces.
Estimation of earth pressure
Two classical theories are used for estimating the lateral
earth-pressures:
• Rankine's theory
• Coulomb's theory
10 1 THE INDIAN CONCRETE JOURNAL A P R I L 2012
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Table 1. Comparison of results obtained from designs
following Rankine's and Coulomb's theories
Parameter Rankine's theory Coulomb 's theory
Cact 0.373 0.240
Cpas 2.502 5.789
Footing length, m 4.4 3.6
Concrete weight, k N / m 154.2 140.1
Reinforcement, m 3 / m 0.037 0.046
Cost estimate, R s / m 53,531 44,480
(For a cantilever retaining w a l l of height 7 m above ground level
w i t h SBC=200 k N / m 2 ; th=25°; fi=37°; thw=15°; mu=0.5; fck=25MPa;
fy = 415MPa; Cact and Cpas are the active and passive coefficients)
W h i l e R a n k i n e ' s theory cons iders the back of the w a l l
to be per fec t ly s m o o t h , C o u l o m b ' s theory cons iders the
existence of f r i c t i o n b e t w e e n the w a l l a n d the b a c k f i l l . 5 , 6
A d e s i g n e r m a y f i n d the R a n k i n e ' s d e s i g n a p p r o a c h
s i m p l e r a n d the one that g i v e s a m o r e c o n s e r v a t i v e
d e s i g n , b u t C o u l o m b ' s d e s i g n is seen as m o r e prac t i ca l
one s ince it i n v o l v e s r e a l l i fe scenar io - the f r i c t i o n
b e t w e e n the w a l l a n d the b a c k f i l l . The C o u l o m b ' s d e s i g n
a p p r o a c h g ives a cost-effective d e s i g n as c o m p a r e d to
R a n k i n e ' s d e s i g n a p p r o a c h , a n d the extent of s a v i n g s
c o u l d be as h i g h as 20 percent i n s o m e instances. Tab le
1 c o m p a r e s the results o b t a i n e d f r o m these t w o d e s i g n
approaches .
I n v i e w of the above , this p a p e r f o l l o w s the C o u l o m b ' s
d e s i g n a p p r o a c h for o p t i m i s i n g the genetic a l g o r i t h m .
Formulation for optimal design
Since the p u r p o s e of o p t i m i z a t i o n i n this s t u d y w a s to
m i n i m i z e the cost, the object ive f u n c t i o n i n c l u d e d i n
the f o r m u l a t i o n w e r e the m a t e r i a l costs of concrete a n d
steel, the carr iage cost of steel, the cost of center ing a n d
s h u t t e r i n g a n d the cost of excavat ion .
M i n i m i z e cost,
C,. = 1.1 ( V A + W W VSRCC + LCSRCS) (1)
w h e r e
Vc, Vs,V=i v o l u m e s of concrete , steel a n d e x c a v a t i o n
respec t ive ly
Lc = l e n g t h of center ing a n d s h u t t e r i n g p r o v i d e d
Rc, Rs, Re, RCC,RCS = u n i t c o s t s o f c o n c r e t e , s t e e l ,
excavat ion , steel carr iage a n d center ing a n d s h u t t e r i n g
respec t ive ly
T o a r r i v e at the total , a 10 percent a d d i t i o n to the cost
•was m a d e to account for the v a r i o u s uncerta int ies i n the
' a s s u m p t i o n s . T h e costs c o n s i d e r e d w e r e based o n the
D e l h i S c h e d u l e of Rates 2007.
Design inputs
1. Site c o n d i t i o n s : h, hp thw
2. S o i l proper t ies : S B C , mu,fi, th
3. M a t e r i a l proper t ies :f&,L, dc, ds
W h e r e h, hf a n d ihw are respec t ive ly the he ight of the
re ta ined so i l o n the heel s ide of the r e t a i n i n g w a l l , he ight
of the so i l o n the toe s ide of the re ta in ing w a l l a n d b a c k f i l l
s lope; S B C , mu,fi a n d th are the safe b e a r i n g capac i ty of
the so i l , coefficient of f r i c t io n at the base of the w a l l , angle
of f r i c t i o n of the b a c k f i l l a n d angle of f r i c t i o n b e t w e e n
the w a l l a n d b a c k f i l l respect ively ;/^ andfy are the grades
of concrete a n d steel; dc a n d ds the densi t ies of concrete
a n d steel respec t ive ly .
Design variables
F i g u r e 1 s h o w s the d e s i g n v a r i a b l e s c o n s i d e r e d f o r
v a r i o u s types of r e t a i n i n g w a l l s , the same are l i s t e d
b e l o w :
1. Cantilever retaining wall
F o o t i n g th ickness (x 2 ) ; s t e m thickness at the b o t t o m (x 2 ) ;
toe slab length (x3); bar diameters i n the toe slab, heel slab
a n d s tem respec t ive ly (x 4 , x5 a n d x6) ( N o t i n F i g u r e 1)
2. Counterfort retaining wall
H e e l s lab th ickness (x 2 ) ; toe s lab th ickness (x 2 ) ; s t e m
t h i c k n e s s at the b o t t o m (x 3 ) ; c o u n t e r f o r t t h i c k n e s s
(x 4 ) ; counter for t s p a c i n g (x 5 ) ; toe s lab l e n g t h (x6); bar
d i a m e t e r s of the m a i n r e i n f o r c e m e n t i n the toe s lab,
hee l s lab a n d s tem respec t ive ly (x7, x8, x9 a n d x10) ( n o t
m a r k e d i n F i g u r e l b )
3. Retaining wall with relieving platforms
F o o t i n g th ickness (x 2 ) ; s t em thickness at the b o t t o m (x 2 ) ;
toe s lab l e n g t h (x 3 ); bar d iameters i n the toe s lab, hee l
slab, s t em a n d r e l i e v i n g p l a t f o r m respec t ive ly (x 4 , x5, x6
a n d x 7 ) ; r e l i e v i n g p l a t f o r m thickness (x8).
If
Design constraints
T h e f o l l o w i n g d e s i g n constra ints w e r e i m p o s e d o n the
var iab les :
1. Factor of safety against o v e r t u r n i n g > 1.4
2. Factor of safety against s l i d i n g > 1.4
A P R I L 2012 THE INDIAN C O N C R E T E JOURNAL I 11
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Table 2 Cantilever retaining walls - optimal solutions for various heights
h,
m m m
%2f
m m
c,
mm 2
B,
2
mm
A,
2
mm
1,
m
c „ ,
Rs. Rs.
Savings,
%
5 1.25 0.28 0.45 0.52 499 1244 1748 2.32 25316 33182 23.7
6 1.25 0.33 0.53 0.73 928 1293 2215 2.83 32574 42631 23.6
7 1.25 0.38 0.62 0.96 1525 1340 2682 3.3 41140 53531 23.1
8 1.25 0.45 0.71 1.21 2012 1305 3186 3.83 51147 66166 22.7
9 1.25 0.52 0.79 1.48 2623 1311 3821 4.38 62853 82828 24.1
10 1.25 0.6 0.89 1.75 3192 1289 4365 4.92 76295 100791 24.3
11 1.25 0.72 1.00 2.04 3289 1182 4813 5.48 91497 123585 25.9
12 1.25 0.78 1.10 2.35 4397 1224 5471 6.06 108617 147208 26.2
13 1.25 0.91 1.22 2.67 4594 1163 5945 6.64 128518 182064 29.4
14 1.25 1.02 1.33 3.00 5167 1172 6539 7.24 149741 214338 30.1
C , B, A = areas of steel i n the toe slab, heel slab and stem respectively i n m m 2 / m , as shown i n Figure 2; 1 = length of the base slab i n m
C t= tradit ional cost of construction of the wa l l per unit length i n R s / m ; C 0 = opt imal cost obtained f rom G A coding per uni t length in R s / m
Table 3. Counterfort retaining walls - optimal solutions for varies heights
h , hf , x 2 , *3' *5/ 1,
m m m m m m m m m
5 1.25 0.25 0.26 0.33 0.2 2.47 0.46 2.32
6 1.25 0.28 0.33 0.35 0.21 2.52 0.66 2.82
7 1.25 0.29 0.4 0.37 0.24 2.56 0.93 3.38
8 1.25 0.3 0.47 0.39 0.27 2.6 1.12 3.87
9 1.25 0.31 0.56 0.41 0.3 2.63 1.39 4.44
10 1.25 0.32 0.65 0.43 0.34 2.65 1.66 5
11 1.25 0.33 0.74 0.45 0.4 2.68 1.95 5.59
12 1.25 0.34 0.84 0.46 0.48 2.7 2.26 6.19
13 1.25 0.36 0.94 0.47 0.54 2.71 2.58 6.81
14 1.25 0.38 1.04 0.49 0.62 2.73 2.91 7.43
h ,
m
A ,
2
mm
B ,
2
mm
c ,
2
mm
D ,
2
mm
E ,
2
mm
F ,
2
mm
G ,
2
mm
H ,
2
mm
Rs. Rs.
Savings,
%
5 450 782 578 300 396 396 396 1237 37006 40422 8.4
6 675 820 580 336 420 420 420 1684 47161 51625 8.6
7 1088 798 567 348 444 444 444 2216 58595 63908 8.3
8 1443 924 616 360 468 468 468 2834 71499 78087 8.4
9 1776 874 598 372 492 492 492 3528 85997 94258 8.7
10 2162 836 597 384 516 516 516 4312 102559 113441 9.6
11 2701 796 572 396 540 540 540 5239 120863 133719 9.6
12 3187 751 569 416 552 * 552 552 * 6220 140847 158042 10.9
13 3770 692 526 448 564 564 564 7353 164057 186047 11.8
14 4402 653 504 480 588 588 588 8652 189504 216805 12.6
1 = length of base in m; A = area of steel reinforcement i n toe slab, as shown in Figure 3(a) i n m m 2 / m ; B, C = top reinforcement near the
counterfort and bottom reinforcement at the midd le of heel slab l m f rom the end, due to continuous beam action; D = top reinforcement
in heel slab, due to cantilever action; E, F = rear and front face reinforcements in stem, due to continuous beam action; G = rear face
reinforcement i n stem, due to cantilever action; H= counterfort reinforcement, as shown i n Figure 3(b) i n m m 2 / m ; C t = tradit ional cost of
construction of the w a l l per unit length i n Rupees /m; C 0 = opt imal cost obtained f rom G A coding per unit length in Rupees /m
12 I THE INDIAN CONCRETE JOURNAL A P R I L 2 0 1 2
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Table 4. Retaining walls with relieving pfatform - optimal solutions for varies heights
h, hf, x2, "3/ x 8 ' A, B, , C, D, I, c „ , Savings,
m m m m m m mm 2 mm 2 mm 2 mm 2 m Rs. Rs. %
5 1.25 0.32 0.39 0.74 0.20 474 .1923 673 1812 2.56 25826 28893 10.7
6 1.25 0.40 0.48 0.86 0.22 565 2549 916 2146 2.96 33464 36827 9.2
7 1.25 0.49 0.56 0.98 0.24 658 3340 1203 2620 3.35 42651 46428 8.2
8 1.25 0.58 0.67 1.09 0.29 762 4228 1323 2966 3.72 53601 56436 5.1
9 1.25 0.65 0.80 1.32 0.34 1010 5100 1604 3235 4.21 66208 69326 4.5
10 1.25 0.71 0.96 1.60 0.39 1343 5830 1963 3413 4.75 80898 85825 5.8
11 1.25 0.81 1.08 1.90 0.44 1637 6365 2413 3858 5.31 97591 100095 2.6
12 1.25 0.90 1.25 2.20 0.50 2085 6592 2833 4098 5.87 116161 122096 4.9
13 1.25 0.97 1.38 2.53 0.57 2814 7495 3232 4562 6.45 137185 142397 3.7
14 1.25 1.08 1.51 2.86 0.64 3328 7962 3712 5051 7.04 159210 16^705 5.7
15 1.25 1.18 1.67 3.22 0.71 4050 8198 4274 5455 7.65 186107 201869 7.9
16 1.25 1.28 1.81 3.58 0.80 4863 8763 4732 5963 8.27 214234 236542 9.5
17 1.25 1.39 1.96 3.96 0.89 5706 9132 5271 6472 8.91 245396 282854 13.3
18 1.25 1.50 2.13 4.34 1.15 6655 9269 4528 6915 9.53 278953 321143 13.2
Here, C, B, D and A are the areas of steel i n the toe slab, heel slab, rel ieving platform and stem respectively in mm2/m, as shown i n Fig.4; 1 is
the length of the base slab i n m; C o is the optimal cost obtained f rom G A coding and Ct is the traditional cost of construction of the w a l l per
unit length in Rupees/m.
3. 0 <Eccentricity of the resultant reaction force at
the footing< footing length / 6
4. M a x i m u m reaction pressure on the footing <
SBC
5. M i n i m u m reaction pressure on the footing > 0
6. R e s t r i c t i o n s o n m a x i m u m a n d m i n i m u m
reinforcement percentage and reinforcement
spacing as per IS 456:2000 code 1
7. Restrictions on m a x i m u m shear stress i n the
footing, stem and other parts based on concrete
grade as per IS 456:2000 code 1
Optimization using genetic algorithms
This study used Genetic algorithms (GA) for carrying out
searches within the design space. G A is a heuristic search
method, which uses the process of natural selection for
finding the global opt imum 8 . These algorithms search a
given population of potential solutions to find the best
solution. They first apply the principle of survival of the
fittest to find better and better approximations. A t each
generation of values for design variables, a new set of
approximations is created by the process of selecting
individual potential solutions (individuals) according to
their level of f itness i n the problem domain and breeding
them together using G A operators. G A does not use
the gradient but uses the values of objective functions
and hence it can be used where the search space is
discontinuous. Programs were developed incorporating
the formulations described earlier using M A T L A B . A
faster convergence was achieved when the population
size, number of generations, mutation rate and crossover
rate were at 250, 50, 0.075 and 0.8 respectively.
Results of optimization
Typical optimal solutions
The programs developed were appl ied to generate
optimal solutions for the three different types of walls
of various heights. The heights ranged from 5 m to 14 m
i n the case of cantilever and counterfort walls, however
for the walls wi th relieving platforms, the range was 5 m
to 18 m. In all the cases hc= 1.25 m and a linear tapering
i n the stem w a l l thickness assumed (0.2 m - 0.3 m at the
top). Also , the soil properties assumed were :SBC = 200
kN/m 2 ,f/z = 25°,/f = 3 7 ° , % =15° and mu = 0.5. It may
be noted that when the height considered was greater
than 14 m, no feasible solutions were possible for the
cantilever and counterfort retaining walls cases, as the
computed maximum bearing pressure on the footing
exceeded the Safe Bearing Capacity of the soil , i.e.,
bearing check failed.
This was also the case w i t h the w a l l w i t h rel ieving
platforms w h e n its height exceeded 18 m. Feasible
solutions are possible only when the SBC is higher than
the computed maximum bearing pressure on the footing.
This is explored further i n the paper.
Tables 2,3 and 4 list the optimal solutions generated for
the three w a l l types, considering M25 grade of concrete
APRIL 2012 THE INDIAN CONCRETE JOURNAL I 13
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A/3
2 A/3
Figure 2. Reinforcement detailing of cantilever retaining
wall. (A, B, and C are area of reinforcement in mm2/m;
han&hf are in m)
a n d Fe415 g r a d e b a r s f o r the m a i n r e i n f o r c e m e n t
steel. Fe250 grade bars w e r e u s e d for temperature a n d
s h r i n k a g e re inforcement . N o m i n a l r e i n f o r c e m e n t w a s
p r o v i d e d w h e r e v e r necessary. F i g u r e s 2, 3 a n d 4 s h o w
the t y p i c a l re inforcement d e t a i l i n g i n the three w a l l s . I n
the case of the counter for t w a l l , o n l y the m a i n bars are
s h o w n i n F i g u r e 3; curta i lments of re inforcement at 2 / 3 r d
a n d l / 3 r he ights of the s t e m (calculated u s i n g basic
pr inciples) are not s h o w n . A l s o , the a d d i t i o n a l h o r i z o n t a l
a n d v e r t i c a l ties p r o v i d e d i n the c o u n t e r f o r t are not
s h o w n . I n the case of w a l l s w i t h the r e l i e v i n g p l a t f o r m s ,
t w o r e l i e v i n g p l a t f o r m s at l / 3 r d a n d 2 / 3 r d l o c a t i o n s of
the w a l l he ight , w e r e a s s u m e d for a l l w a l l he ights to
m a i n t a i n cons is tency i n results .
F i g u r e s 5, 6 a n d 7 s h o w the v a r i a t i o n s i n the o p t i m a l
g e o m e t r i c d i m e n s i o n s ( w a l l / s lab thickness) for the
three types of w a l l s . These c o m p r i s e : f o o t i n g th ickness
(jj), s t e m base t h i c k n e s s (x2) a n d toe s lab length(x 3 )
i n the case of the cant i lever w a l l (F igure 5); hee l s lab
thickness (x2), toe s lab thickness (x2), s t em base thickness
(x3), counter for t th ickness (x4), counter for t s p a c i n g (x5)
a n d toe slab l e n g t h (x 6)in the case of the counter for t w a l l
(Figure 6); a n d f o o t i n g thickness (x2), s t e m base thickness
(x2), toe s lab l e n g t h (x3) a n d r e l i e v i n g p l a t f o r m thickness
Figure 3(a). Reinforcement detailing of stem and footing
slab for counterfort retaining wall. (A, B, D, E, F and G are
area of reinforcement in mm2/m)
14 I THE INDIAN CONCRETE JOURNAL A P R I L 2012
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(xg) i n the case of the w a l l w i t h rel ieving platforms
(Figure 7). From the figures, it appears that the slab
thickness increases somewhat linearly wi th the increase
i n the wal l height. Using the trends in Figures 5-7 and
Tables 2-4, it is possible to arrive at heuristic rules for
optimal proportioning of the various elements.
Figure 4. Reinforcement detailing of retaining wall with
relieving platforms. (A, B, C and D are area of
reinforcement in mm2/m)
Wall height, m
Figure 5. Variation of cantilever wall dimensions with wall
height
Cost comparison between optimal
design and conventional design
The literature suggests that Genetic algorithms always
give a better optimal solution than the conventional
design methods practised in the industry 7 . The following
inferences may be drawn from Tables 2-4, which include
the conventional design costs.
• For 5 m and 14 m high cantilever walls on a soil
of SBC 200 k N / m 2 s , for the given parameters,
the savings p r o v i d e d by the bpt imal design,
compared to the convent ional design, were
between 23.7 to 30.1 percent. The savings increase
wi th the increase i n the wal l height.
• Similarly, for the optimal counterfort walls, the
savings was 8.4 percent to 12.6 percent for the
height increase from 5 to 14 m respectively.
3- A X 6
2.5 >
M * y^"' *5
Jk .. > . . H e e l slab thickness
s U Toe slab thickness
cT 2- y — * - Stem base thickness
'55 > Couterfort thickness
S i K . r
-3
M Counterfort spacing
* Toe slab length
•3 i -
s
0.5 j
_ . • x 4
0- -1 1
5 7 ~~9 11 13 15
Wall height (in m)
Figure 6. Variation of counterfort wall dimensions with wall
height
4.5 n
1 1 1 1 1
5 8 11 14 17 20
Wall height (m)
Figure 7. Variation of retaining wall with relieving platform
dimensions with wall height
APRIL 2012 THE INDIAN CONCRETE JOURNAL J 15
\
Page 8
• In the case of walls w i t h rel ieving platforms,
the o p t i m i s a t i o n of cost savings increased
f r o m 2.6 p e r c e n t ( for w a l l h e i g h t of
11 m) to 13.2 percent (for w a l l height of 18 m) by
10.6 percent. In the case of heights less than 11 m,
no definite trend i n the saving was observed.
Effect of change in soil bearing capacity
The optimization study was extended to include the soil
bearing capacities i n the range 150 k N / m 2 t o 300 k N / m 2 .
The results suggest that the linear trends observed earlier
for SBC = 200 k N / m 2 also hold good for the extended
range. 6 However , the opt imal solutions for various
w a l l / slab thicknesses were dependent on the SBC.
These were included i n the heuristic rules.
Heuristic guidelines for optimal design
Based on the optimal solutions obtained for these wal l
types and heights, several heuristic guidelines can be
arrived at 7. The fol lowing expression can be used to
arrive at a near-optimal value of the length of the heel
slab:
Heel slab length= A J — (1)
In the case of wal l s w i t h r e l i e v i n g platforms, the
fo l lowing expression for the length of the platform,
yielded near-optimal solutions: 3
Table 5. Design heuristic rules for the three types of walls with different SBC values - optimal wall / slab
thickness values (in m)
Wall / slab thickness
SBC, kN/m 2
Wall / slab thickness
150 200 300
1. Cantilever retaining wall
Footing thickness xv m 0.064h- 0.04 0.082 h - 0.13 0.091h - 0.173
Stem base thickness x 2 , m 0.090 h 0.097 h - 0.04 0.109h - 0.096
Toe slab length x 3 , m 0.284h - 0.66 0.275 h - 0.858 0.213h - 0.576
2. Counterfort retaining wall
Heel slab thickness xv m 0.017h + 0.15 0.012 h + 0.204 0.025 h + 0.193; for h< 13m 0.480; for h> 13m
Toe slab thickness x 2 , m 0.067h - 0.058 0.087 h - 0.172 0.109 h - 0.296
Stem base thickness x 3 , m 0.020h + 0.228 0.019h + 0.242 0.013 h + 0.289
Counterfort thickness x 4 / m 0.022 h +0.117 0.047 h - 0.032 0.076 h - 0.161
Counterfort spacing x 5 , m 0.053 h + 2.234 0.027 h + 2.387 0.018 h + 2.390; for h< 15m 2.650; for h> 15m
Toe slab length x^ m 0.282 h - 0.719 0.272 h - 0 . 9 0 1 0.242 h - 0.812
3. Retaining wall with two relieving platforms
Footing thickness xv m 0.082 h - 0.088 0.089 h - 0.125 0.109 h - 0.227
Stem base thickness x 2 , m 0.129 h - 0.264 0.131h - 0.263 0.165 h - 0.434
Toe slab length x 3 , m 0.330 h - 0.899 0.268 h - 0.602 0.226 h - 0.390
Relieving platform thickness x g , m 0.054 h - 0.072 0.057 h - 0 . 0 8 7 0.072 h - 0.157
h - Height of w a l l above the ground level
' Relieving platform length = 0.33 h tan — - y (2)
A l inear m o d e l was proposed for the var iat ion of
wall/slab thickness wi th w a l l height, for the sake of
simplicity i n calculation and application. Based on the
above observations, heuristic design rules proposed
for a retaining w a l l built on soil wi th th = 25°, fi = 37°,
thb = 15° and mu = 0.5 and different SBC values (150,200
and 300 k N / m 2 ) , are tabulated i n Table 5. Using these
guidelines, a designer can select wal l and slab thickness
proportions that are likely to be close to this optimal
solutions for preliminary design, without carrying out
an optimization study.
M25 concrete and Fe415 reinforcement steel were found
to give optimal design solutions i n al l cases.
Best retaining wall design option
Based on the study, the costs corresponding to the optimal
designs for various wal l heights were compared for the
soil parameters; th = 25°; fi = 37°; thb =15°; mu = 0.5).
Figures 8, 9 and 10 show the cost per meter wal l length
for SBC = 150, 200 and 300 kN/irrespec t ive ly .
The results suggest that the cantilever retaining wal l ,
always yields the most economical solution. However,
the wal l height gets restricted when the SBC is low. When
this happens the retaining wal l wi th relieving platforms,
16 I THE INDIAN CONCRETE JOURNAL APRIL 2012
Page 9
which is a relatively new concept i n
India, provides the most economical
solution. The traditional assumption
that w a l l s w i t h counterforts are
likely to be more cost-effective than
cantilever walls for heights exceeding
about 8 m, was not found to be true.
The optimally designed counterfort
retaining w a l l was f o u n d to be a
more costly solution compared to the
optimally designed cantilever w a l l
and wal l wi th relieving platforms for
nearly all wal l heights.
It may be noted that the cost shown
i n Figures 8-10 were based on Delhi
Schedule of rates 2007. The authors
believe that even if they change with
time, the relative costs of steel and
concrete are l ike ly to remain the
same.
Conclusions
The salient conclusions, based on
the study, can be summar ized as
follows:
• C o u l o m b ' s t h e o r y , w h i c h
accounts for wal l friction, gives
a better cost-effective design
alternative for a retaining wal l
than Rankine's theory, which
is currently used i n practice,
for convenience.
• The traditional belief that walls
with counterforts are likely to
be more cost-effective than
cantilever wal ls for heights
exceeding about 8 m, was not
found to be true, w h e n an
optimal design was carried
out. The optimally designed
cantilever retaining wall was
f o u n d to be i n v a r i a b l y the
most cost-effective solution for
w a l l heights, where feasible
s o l u t i o n s w e r e p o s s i b l e
(depending on safe bearing
capacity).
• The r e t a i n i n g w a l l w i t h
relieving platforms, which is
a relatively new concept i n
250000
00
g 200000
150000
c 100000
~ 50000
Opt imal Cantilever
w a l l cost
Opt imal Counterfort
w a l l cost
• • O p t i m a l w a l l w i t h
rel ieving platforms
c'ost
15 11 13
W a l l height (h + hi) i n m
(SBC = 150kN/m 2 ; th = 2 5 ° ; ^ =37°; thb= 15°; mu = 0.5)
L7
Figure 8. Optimal design cost estimates
300000 •
g 3 250000-
£ 200000
&
& 150000 J c
" 100000
50000
- Opt imal Cantilever
w a l l cost
Opt imal Counterfort
w a l l cost
Opt imal w a l l w i t h
relieving platforms
cost
9 11 13 15 17 19
W a l l height i n m
(SBC = 200kN/m 2 ; th = 75°;fi =37°; th„ = 15°; mu = 0.5)
Figure 9. Optimal design cost estimates
500000
450000 -
/
-g ' t
to
a
400000 - / /
350000 -
- Opt imal Cantilever - Opt imal Cantilever
S 300000 - / .** w a l l cost
s / V
250000 -
;t
in
R
200000 "
• Opt imal Counterfort
w a l l cost
o
u
150000
13 100000
50000
• Opt imal w a l l w i t h
100000
50000
relieving platforms
cost
U 5 7 9 11 13 15 17 19 21 23 25
W a l l height i n m
(SBC = 300kN/m 2 ; th = 25°;fi =37°; thh= 15° mu = 0.5)
Figure 10 Optimal design cost estimates
APRIL 2012 THE INDIAN CONCRETE JOURNAL I 17
Optimal design of reinforced concrete
retaining avails ,
Shravya Donkada and Devdas Menon
This paper aims at developing an understanding of
optimal design solutions for three types of reinforced
concrete retaining walls, namely, cantilever retaining
walls, counterfort retaining walls and retaining walls
with relieving platforms. Using genetic algorithms,
parametric studies were carried out to establish
heuristic rules for proportioning the wall dimensions
corresponding to the minimum cost points. Optimal
cost-estimates of the retaining walls types were compared
to establish the best design alternative for a given
height. Also, the advantages of retaining walls with
relieving platforms, which are relatively new in India,
are discussed.
Keywords: Reinforced concrete retaining walls; optimal
design; relieving platforms; cantilever walls; counterfort walls;
genetic algorithms.
Introduction
The design of retaining w a l l almost always involves
decision making with a choice or set of choices along
with their associated uncertainties and outcomes. While
designing such structures, a designer may propose a
large number of feasible designs; however, professional
considerations require that only the most opt imal
one, w i t h the least cost be chosen for construction.
For del ivering an acceptable design, today's design
practitioners increasingly rely on P C based programs
that require parameters, such as toe or heel lengths
and stem widths. The process invariably involves a
tr ial and error procedure. Obta in ing a satisfactory
design per se, does reveal its cost position against the
optimal design. The present study therefore aims at
developing an optimal design solution for reinforced
concrete retaining walls, namely, cantilever retaining
walls, counterfort retaining walls and retaining walls wi th
relieving platforms, in terms of m i n i m u m cost as per the
IS456:2000 code. 1 In this connection, this paper discusses
the heuristic rules for the required w a l l dimensions.
Incidentally, it may be noted that one of the w a l l types
studied is the retaining walls wi th relieving platforms.
This wal l type provides an innovative design alternative
and is common in Europe, but relatively new to India.
The scope of this study was confined to retaining walls
ranging from 5 m to 23 m height. A n y surcharge was
converted to an equivalent height and included i n the
heuristic rules. The study assumed that proper drainage
conditions. However, the effect of earthquake loading
was excluded, as the scope was limited to incorporating
the effects of gravity loading. The reason for doing so
was to insulate the design outcome from the complexities
that arise from the seismic zoning of sites, for example,
moderate or high seismic zone.
APRIL 2012 THE INDIAN CONCRETE JOURNAL I 9
Page 2
Types of retaining walls
It is wel l k n o w n that retaining walls
are structures that h o l d back soil
or rock f rom a bui ld ing , structure
or area. 2 They prevent down-slope
movement or erosion and provide
support for vertical or near-vertical
grade changes. The lateral earth
pressure behind the w a l l depends
on the angle of internal friction and
the cohesive strength of the retained
material, as wel l as the direction and
magnitude of movement of the stems Figure 1. Types of concrete retaining walls
of the retaining walls. Its distribution
—V\
) j t i
•1
hf! x2\
*2 "i_r
* 2 *r
Counterfort
wall \Relieving
/platforms
* 3 * 6
(a) Cantilever retaining (b) Counterfort retaining
wall wall
(c) Retaining wall with
relieving platforms
is typically triangular, least at the top
of the wal l and increasing towards
the bottom. The earth pressure could push the w a l l
forward or overturn it if not properly addressed. Also ,
the groundwater behind the w a l l should be dissipated
by a suitable drainage system; otherwise, this could
lead to an additional horizontal pressure on the wal l .
A l t h o u g h the effect of surcharge l o a d i n g was not
explicitly considered here, it can be approximated as
an equivalent height of retained earth.
A s stated earlier, this study deals wi th the fol lowing
types of retaining walls:
• Cantilever retaining wal l : Such walls transmit
loads f rom the vert ical port ion, through the
cantilever action, to a large structural footing,
converting horizontal pressures from behind the
wal l to vertical pressures on the ground below.
This wal l type is believed to be economical up
to a height of about 7 m (Figure la) . Since the
backfill acts on the base, providing most of the
dead weight, the requirement of construction
materials for this w a l l type is much less than a
traditional gravity w a l l 2 .
• Counterfort retaining wal l : Cantilever retaining
walls , sometimes, include short w i n g walls at
right angles to the main trend of the w a l l on
their back, to improve their resistance to lateral
loads. Introducing transverse supports reduces
bending moments, when the heights are large.
Such supports, called counterforts, connect the
stem w i t h the heel slab. This w a l l type is believed
to be economical for heights greater than 7 m
(Figure l b ) 2 .
• Retaining w a l l w i t h relieving platforms: When
the depth of soil to be retained is excessive, soil
pressures can be reduced by the use of a relieving
platform . Retaining wal l with relieving platforms
is relatively new to Indian construction industry.
Such walls are k n o w n to provide an economical
l ightweight design solution for relatively tall
w a l l s . 4 , 0 The retaining w a l l is shielded f r o m
active earth pressure by means of one or more
relieving platforms (Figure lc) which make the
pressure diagram discontinuous at the level of
the platform. Also , the relieving platform carries
the weight of the soil above it and any surcharge
loading, transferring them as a 'relieving' moment
to the vertical stem. The re l ieving platforms
are designed such that they intersect the plane
of rupture from the soil above and behind the
platforms preventing any load from the soil to
act on the wal l . This aspect is the key to designing
such walls.
Typically, a retaining wal l design includes :
• Performing stability checks for the retaining wal l
against sl iding and overturning.
• C o m p u t i n g the m a x i m u m a n d m i n i m u m
bearing pressures present under the toe and heel
and comparing them w i t h the allowable soi l
pressure.
• D e s i g n i n g the re in forc ing steel for the toe,
heel , stem and other parts c o n s i d e r i n g the
corresponding bending and shear forces.
Estimation of earth pressure
Two classical theories are used for estimating the lateral
earth-pressures:
• Rankine's theory
• Coulomb's theory
10 1 THE INDIAN CONCRETE JOURNAL A P R I L 2012
Page 3
Table 1. Comparison of results obtained from designs
following Rankine's and Coulomb's theories
Parameter Rankine's theory Coulomb 's theory
Cact 0.373 0.240
Cpas 2.502 5.789
Footing length, m 4.4 3.6
Concrete weight, k N / m 154.2 140.1
Reinforcement, m 3 / m 0.037 0.046
Cost estimate, R s / m 53,531 44,480
(For a cantilever retaining w a l l of height 7 m above ground level
w i t h SBC=200 k N / m 2 ; th=25°; fi=37°; thw=15°; mu=0.5; fck=25MPa;
fy = 415MPa; Cact and Cpas are the active and passive coefficients)
W h i l e R a n k i n e ' s theory cons iders the back of the w a l l
to be per fec t ly s m o o t h , C o u l o m b ' s theory cons iders the
existence of f r i c t i o n b e t w e e n the w a l l a n d the b a c k f i l l . 5 , 6
A d e s i g n e r m a y f i n d the R a n k i n e ' s d e s i g n a p p r o a c h
s i m p l e r a n d the one that g i v e s a m o r e c o n s e r v a t i v e
d e s i g n , b u t C o u l o m b ' s d e s i g n is seen as m o r e prac t i ca l
one s ince it i n v o l v e s r e a l l i fe scenar io - the f r i c t i o n
b e t w e e n the w a l l a n d the b a c k f i l l . The C o u l o m b ' s d e s i g n
a p p r o a c h g ives a cost-effective d e s i g n as c o m p a r e d to
R a n k i n e ' s d e s i g n a p p r o a c h , a n d the extent of s a v i n g s
c o u l d be as h i g h as 20 percent i n s o m e instances. Tab le
1 c o m p a r e s the results o b t a i n e d f r o m these t w o d e s i g n
approaches .
I n v i e w of the above , this p a p e r f o l l o w s the C o u l o m b ' s
d e s i g n a p p r o a c h for o p t i m i s i n g the genetic a l g o r i t h m .
Formulation for optimal design
Since the p u r p o s e of o p t i m i z a t i o n i n this s t u d y w a s to
m i n i m i z e the cost, the object ive f u n c t i o n i n c l u d e d i n
the f o r m u l a t i o n w e r e the m a t e r i a l costs of concrete a n d
steel, the carr iage cost of steel, the cost of center ing a n d
s h u t t e r i n g a n d the cost of excavat ion .
M i n i m i z e cost,
C,. = 1.1 ( V A + W W VSRCC + LCSRCS) (1)
w h e r e
Vc, Vs,V=i v o l u m e s of concrete , steel a n d e x c a v a t i o n
respec t ive ly
Lc = l e n g t h of center ing a n d s h u t t e r i n g p r o v i d e d
Rc, Rs, Re, RCC,RCS = u n i t c o s t s o f c o n c r e t e , s t e e l ,
excavat ion , steel carr iage a n d center ing a n d s h u t t e r i n g
respec t ive ly
T o a r r i v e at the total , a 10 percent a d d i t i o n to the cost
•was m a d e to account for the v a r i o u s uncerta int ies i n the
' a s s u m p t i o n s . T h e costs c o n s i d e r e d w e r e based o n the
D e l h i S c h e d u l e of Rates 2007.
Design inputs
1. Site c o n d i t i o n s : h, hp thw
2. S o i l proper t ies : S B C , mu,fi, th
3. M a t e r i a l proper t ies :f&,L, dc, ds
W h e r e h, hf a n d ihw are respec t ive ly the he ight of the
re ta ined so i l o n the heel s ide of the r e t a i n i n g w a l l , he ight
of the so i l o n the toe s ide of the re ta in ing w a l l a n d b a c k f i l l
s lope; S B C , mu,fi a n d th are the safe b e a r i n g capac i ty of
the so i l , coefficient of f r i c t io n at the base of the w a l l , angle
of f r i c t i o n of the b a c k f i l l a n d angle of f r i c t i o n b e t w e e n
the w a l l a n d b a c k f i l l respect ively ;/^ andfy are the grades
of concrete a n d steel; dc a n d ds the densi t ies of concrete
a n d steel respec t ive ly .
Design variables
F i g u r e 1 s h o w s the d e s i g n v a r i a b l e s c o n s i d e r e d f o r
v a r i o u s types of r e t a i n i n g w a l l s , the same are l i s t e d
b e l o w :
1. Cantilever retaining wall
F o o t i n g th ickness (x 2 ) ; s t e m thickness at the b o t t o m (x 2 ) ;
toe slab length (x3); bar diameters i n the toe slab, heel slab
a n d s tem respec t ive ly (x 4 , x5 a n d x6) ( N o t i n F i g u r e 1)
2. Counterfort retaining wall
H e e l s lab th ickness (x 2 ) ; toe s lab th ickness (x 2 ) ; s t e m
t h i c k n e s s at the b o t t o m (x 3 ) ; c o u n t e r f o r t t h i c k n e s s
(x 4 ) ; counter for t s p a c i n g (x 5 ) ; toe s lab l e n g t h (x6); bar
d i a m e t e r s of the m a i n r e i n f o r c e m e n t i n the toe s lab,
hee l s lab a n d s tem respec t ive ly (x7, x8, x9 a n d x10) ( n o t
m a r k e d i n F i g u r e l b )
3. Retaining wall with relieving platforms
F o o t i n g th ickness (x 2 ) ; s t em thickness at the b o t t o m (x 2 ) ;
toe s lab l e n g t h (x 3 ); bar d iameters i n the toe s lab, hee l
slab, s t em a n d r e l i e v i n g p l a t f o r m respec t ive ly (x 4 , x5, x6
a n d x 7 ) ; r e l i e v i n g p l a t f o r m thickness (x8).
If
Design constraints
T h e f o l l o w i n g d e s i g n constra ints w e r e i m p o s e d o n the
var iab les :
1. Factor of safety against o v e r t u r n i n g > 1.4
2. Factor of safety against s l i d i n g > 1.4
A P R I L 2012 THE INDIAN C O N C R E T E JOURNAL I 11
Page 4
Table 2 Cantilever retaining walls - optimal solutions for various heights
h,
m m m
%2f
m m
c,
mm 2
B,
2
mm
A,
2
mm
1,
m
c „ ,
Rs. Rs.
Savings,
%
5 1.25 0.28 0.45 0.52 499 1244 1748 2.32 25316 33182 23.7
6 1.25 0.33 0.53 0.73 928 1293 2215 2.83 32574 42631 23.6
7 1.25 0.38 0.62 0.96 1525 1340 2682 3.3 41140 53531 23.1
8 1.25 0.45 0.71 1.21 2012 1305 3186 3.83 51147 66166 22.7
9 1.25 0.52 0.79 1.48 2623 1311 3821 4.38 62853 82828 24.1
10 1.25 0.6 0.89 1.75 3192 1289 4365 4.92 76295 100791 24.3
11 1.25 0.72 1.00 2.04 3289 1182 4813 5.48 91497 123585 25.9
12 1.25 0.78 1.10 2.35 4397 1224 5471 6.06 108617 147208 26.2
13 1.25 0.91 1.22 2.67 4594 1163 5945 6.64 128518 182064 29.4
14 1.25 1.02 1.33 3.00 5167 1172 6539 7.24 149741 214338 30.1
C , B, A = areas of steel i n the toe slab, heel slab and stem respectively i n m m 2 / m , as shown i n Figure 2; 1 = length of the base slab i n m
C t= tradit ional cost of construction of the wa l l per unit length i n R s / m ; C 0 = opt imal cost obtained f rom G A coding per uni t length in R s / m
Table 3. Counterfort retaining walls - optimal solutions for varies heights
h , hf , x 2 , *3' *5/ 1,
m m m m m m m m m
5 1.25 0.25 0.26 0.33 0.2 2.47 0.46 2.32
6 1.25 0.28 0.33 0.35 0.21 2.52 0.66 2.82
7 1.25 0.29 0.4 0.37 0.24 2.56 0.93 3.38
8 1.25 0.3 0.47 0.39 0.27 2.6 1.12 3.87
9 1.25 0.31 0.56 0.41 0.3 2.63 1.39 4.44
10 1.25 0.32 0.65 0.43 0.34 2.65 1.66 5
11 1.25 0.33 0.74 0.45 0.4 2.68 1.95 5.59
12 1.25 0.34 0.84 0.46 0.48 2.7 2.26 6.19
13 1.25 0.36 0.94 0.47 0.54 2.71 2.58 6.81
14 1.25 0.38 1.04 0.49 0.62 2.73 2.91 7.43
h ,
m
A ,
2
mm
B ,
2
mm
c ,
2
mm
D ,
2
mm
E ,
2
mm
F ,
2
mm
G ,
2
mm
H ,
2
mm
Rs. Rs.
Savings,
%
5 450 782 578 300 396 396 396 1237 37006 40422 8.4
6 675 820 580 336 420 420 420 1684 47161 51625 8.6
7 1088 798 567 348 444 444 444 2216 58595 63908 8.3
8 1443 924 616 360 468 468 468 2834 71499 78087 8.4
9 1776 874 598 372 492 492 492 3528 85997 94258 8.7
10 2162 836 597 384 516 516 516 4312 102559 113441 9.6
11 2701 796 572 396 540 540 540 5239 120863 133719 9.6
12 3187 751 569 416 552 * 552 552 * 6220 140847 158042 10.9
13 3770 692 526 448 564 564 564 7353 164057 186047 11.8
14 4402 653 504 480 588 588 588 8652 189504 216805 12.6
1 = length of base in m; A = area of steel reinforcement i n toe slab, as shown in Figure 3(a) i n m m 2 / m ; B, C = top reinforcement near the
counterfort and bottom reinforcement at the midd le of heel slab l m f rom the end, due to continuous beam action; D = top reinforcement
in heel slab, due to cantilever action; E, F = rear and front face reinforcements in stem, due to continuous beam action; G = rear face
reinforcement i n stem, due to cantilever action; H= counterfort reinforcement, as shown i n Figure 3(b) i n m m 2 / m ; C t = tradit ional cost of
construction of the w a l l per unit length i n Rupees /m; C 0 = opt imal cost obtained f rom G A coding per unit length in Rupees /m
12 I THE INDIAN CONCRETE JOURNAL A P R I L 2 0 1 2
Page 5
Table 4. Retaining walls with relieving pfatform - optimal solutions for varies heights
h, hf, x2, "3/ x 8 ' A, B, , C, D, I, c „ , Savings,
m m m m m m mm 2 mm 2 mm 2 mm 2 m Rs. Rs. %
5 1.25 0.32 0.39 0.74 0.20 474 .1923 673 1812 2.56 25826 28893 10.7
6 1.25 0.40 0.48 0.86 0.22 565 2549 916 2146 2.96 33464 36827 9.2
7 1.25 0.49 0.56 0.98 0.24 658 3340 1203 2620 3.35 42651 46428 8.2
8 1.25 0.58 0.67 1.09 0.29 762 4228 1323 2966 3.72 53601 56436 5.1
9 1.25 0.65 0.80 1.32 0.34 1010 5100 1604 3235 4.21 66208 69326 4.5
10 1.25 0.71 0.96 1.60 0.39 1343 5830 1963 3413 4.75 80898 85825 5.8
11 1.25 0.81 1.08 1.90 0.44 1637 6365 2413 3858 5.31 97591 100095 2.6
12 1.25 0.90 1.25 2.20 0.50 2085 6592 2833 4098 5.87 116161 122096 4.9
13 1.25 0.97 1.38 2.53 0.57 2814 7495 3232 4562 6.45 137185 142397 3.7
14 1.25 1.08 1.51 2.86 0.64 3328 7962 3712 5051 7.04 159210 16^705 5.7
15 1.25 1.18 1.67 3.22 0.71 4050 8198 4274 5455 7.65 186107 201869 7.9
16 1.25 1.28 1.81 3.58 0.80 4863 8763 4732 5963 8.27 214234 236542 9.5
17 1.25 1.39 1.96 3.96 0.89 5706 9132 5271 6472 8.91 245396 282854 13.3
18 1.25 1.50 2.13 4.34 1.15 6655 9269 4528 6915 9.53 278953 321143 13.2
Here, C, B, D and A are the areas of steel i n the toe slab, heel slab, rel ieving platform and stem respectively in mm2/m, as shown i n Fig.4; 1 is
the length of the base slab i n m; C o is the optimal cost obtained f rom G A coding and Ct is the traditional cost of construction of the w a l l per
unit length in Rupees/m.
3. 0 <Eccentricity of the resultant reaction force at
the footing< footing length / 6
4. M a x i m u m reaction pressure on the footing <
SBC
5. M i n i m u m reaction pressure on the footing > 0
6. R e s t r i c t i o n s o n m a x i m u m a n d m i n i m u m
reinforcement percentage and reinforcement
spacing as per IS 456:2000 code 1
7. Restrictions on m a x i m u m shear stress i n the
footing, stem and other parts based on concrete
grade as per IS 456:2000 code 1
Optimization using genetic algorithms
This study used Genetic algorithms (GA) for carrying out
searches within the design space. G A is a heuristic search
method, which uses the process of natural selection for
finding the global opt imum 8 . These algorithms search a
given population of potential solutions to find the best
solution. They first apply the principle of survival of the
fittest to find better and better approximations. A t each
generation of values for design variables, a new set of
approximations is created by the process of selecting
individual potential solutions (individuals) according to
their level of f itness i n the problem domain and breeding
them together using G A operators. G A does not use
the gradient but uses the values of objective functions
and hence it can be used where the search space is
discontinuous. Programs were developed incorporating
the formulations described earlier using M A T L A B . A
faster convergence was achieved when the population
size, number of generations, mutation rate and crossover
rate were at 250, 50, 0.075 and 0.8 respectively.
Results of optimization
Typical optimal solutions
The programs developed were appl ied to generate
optimal solutions for the three different types of walls
of various heights. The heights ranged from 5 m to 14 m
i n the case of cantilever and counterfort walls, however
for the walls wi th relieving platforms, the range was 5 m
to 18 m. In all the cases hc= 1.25 m and a linear tapering
i n the stem w a l l thickness assumed (0.2 m - 0.3 m at the
top). Also , the soil properties assumed were :SBC = 200
kN/m 2 ,f/z = 25°,/f = 3 7 ° , % =15° and mu = 0.5. It may
be noted that when the height considered was greater
than 14 m, no feasible solutions were possible for the
cantilever and counterfort retaining walls cases, as the
computed maximum bearing pressure on the footing
exceeded the Safe Bearing Capacity of the soil , i.e.,
bearing check failed.
This was also the case w i t h the w a l l w i t h rel ieving
platforms w h e n its height exceeded 18 m. Feasible
solutions are possible only when the SBC is higher than
the computed maximum bearing pressure on the footing.
This is explored further i n the paper.
Tables 2,3 and 4 list the optimal solutions generated for
the three w a l l types, considering M25 grade of concrete
APRIL 2012 THE INDIAN CONCRETE JOURNAL I 13
Page 6
A/3
2 A/3
Figure 2. Reinforcement detailing of cantilever retaining
wall. (A, B, and C are area of reinforcement in mm2/m;
han&hf are in m)
a n d Fe415 g r a d e b a r s f o r the m a i n r e i n f o r c e m e n t
steel. Fe250 grade bars w e r e u s e d for temperature a n d
s h r i n k a g e re inforcement . N o m i n a l r e i n f o r c e m e n t w a s
p r o v i d e d w h e r e v e r necessary. F i g u r e s 2, 3 a n d 4 s h o w
the t y p i c a l re inforcement d e t a i l i n g i n the three w a l l s . I n
the case of the counter for t w a l l , o n l y the m a i n bars are
s h o w n i n F i g u r e 3; curta i lments of re inforcement at 2 / 3 r d
a n d l / 3 r he ights of the s t e m (calculated u s i n g basic
pr inciples) are not s h o w n . A l s o , the a d d i t i o n a l h o r i z o n t a l
a n d v e r t i c a l ties p r o v i d e d i n the c o u n t e r f o r t are not
s h o w n . I n the case of w a l l s w i t h the r e l i e v i n g p l a t f o r m s ,
t w o r e l i e v i n g p l a t f o r m s at l / 3 r d a n d 2 / 3 r d l o c a t i o n s of
the w a l l he ight , w e r e a s s u m e d for a l l w a l l he ights to
m a i n t a i n cons is tency i n results .
F i g u r e s 5, 6 a n d 7 s h o w the v a r i a t i o n s i n the o p t i m a l
g e o m e t r i c d i m e n s i o n s ( w a l l / s lab thickness) for the
three types of w a l l s . These c o m p r i s e : f o o t i n g th ickness
(jj), s t e m base t h i c k n e s s (x2) a n d toe s lab length(x 3 )
i n the case of the cant i lever w a l l (F igure 5); hee l s lab
thickness (x2), toe s lab thickness (x2), s t em base thickness
(x3), counter for t th ickness (x4), counter for t s p a c i n g (x5)
a n d toe slab l e n g t h (x 6)in the case of the counter for t w a l l
(Figure 6); a n d f o o t i n g thickness (x2), s t e m base thickness
(x2), toe s lab l e n g t h (x3) a n d r e l i e v i n g p l a t f o r m thickness
Figure 3(a). Reinforcement detailing of stem and footing
slab for counterfort retaining wall. (A, B, D, E, F and G are
area of reinforcement in mm2/m)
14 I THE INDIAN CONCRETE JOURNAL A P R I L 2012
Page 7
(xg) i n the case of the w a l l w i t h rel ieving platforms
(Figure 7). From the figures, it appears that the slab
thickness increases somewhat linearly wi th the increase
i n the wal l height. Using the trends in Figures 5-7 and
Tables 2-4, it is possible to arrive at heuristic rules for
optimal proportioning of the various elements.
Figure 4. Reinforcement detailing of retaining wall with
relieving platforms. (A, B, C and D are area of
reinforcement in mm2/m)
Wall height, m
Figure 5. Variation of cantilever wall dimensions with wall
height
Cost comparison between optimal
design and conventional design
The literature suggests that Genetic algorithms always
give a better optimal solution than the conventional
design methods practised in the industry 7 . The following
inferences may be drawn from Tables 2-4, which include
the conventional design costs.
• For 5 m and 14 m high cantilever walls on a soil
of SBC 200 k N / m 2 s , for the given parameters,
the savings p r o v i d e d by the bpt imal design,
compared to the convent ional design, were
between 23.7 to 30.1 percent. The savings increase
wi th the increase i n the wal l height.
• Similarly, for the optimal counterfort walls, the
savings was 8.4 percent to 12.6 percent for the
height increase from 5 to 14 m respectively.
3- A X 6
2.5 >
M * y^"' *5
Jk .. > . . H e e l slab thickness
s U Toe slab thickness
cT 2- y — * - Stem base thickness
'55 > Couterfort thickness
S i K . r
-3
M Counterfort spacing
* Toe slab length
•3 i -
s
0.5 j
_ . • x 4
0- -1 1
5 7 ~~9 11 13 15
Wall height (in m)
Figure 6. Variation of counterfort wall dimensions with wall
height
4.5 n
1 1 1 1 1
5 8 11 14 17 20
Wall height (m)
Figure 7. Variation of retaining wall with relieving platform
dimensions with wall height
APRIL 2012 THE INDIAN CONCRETE JOURNAL J 15
\
Page 8
• In the case of walls w i t h rel ieving platforms,
the o p t i m i s a t i o n of cost savings increased
f r o m 2.6 p e r c e n t ( for w a l l h e i g h t of
11 m) to 13.2 percent (for w a l l height of 18 m) by
10.6 percent. In the case of heights less than 11 m,
no definite trend i n the saving was observed.
Effect of change in soil bearing capacity
The optimization study was extended to include the soil
bearing capacities i n the range 150 k N / m 2 t o 300 k N / m 2 .
The results suggest that the linear trends observed earlier
for SBC = 200 k N / m 2 also hold good for the extended
range. 6 However , the opt imal solutions for various
w a l l / slab thicknesses were dependent on the SBC.
These were included i n the heuristic rules.
Heuristic guidelines for optimal design
Based on the optimal solutions obtained for these wal l
types and heights, several heuristic guidelines can be
arrived at 7. The fol lowing expression can be used to
arrive at a near-optimal value of the length of the heel
slab:
Heel slab length= A J — (1)
In the case of wal l s w i t h r e l i e v i n g platforms, the
fo l lowing expression for the length of the platform,
yielded near-optimal solutions: 3
Table 5. Design heuristic rules for the three types of walls with different SBC values - optimal wall / slab
thickness values (in m)
Wall / slab thickness
SBC, kN/m 2
Wall / slab thickness
150 200 300
1. Cantilever retaining wall
Footing thickness xv m 0.064h- 0.04 0.082 h - 0.13 0.091h - 0.173
Stem base thickness x 2 , m 0.090 h 0.097 h - 0.04 0.109h - 0.096
Toe slab length x 3 , m 0.284h - 0.66 0.275 h - 0.858 0.213h - 0.576
2. Counterfort retaining wall
Heel slab thickness xv m 0.017h + 0.15 0.012 h + 0.204 0.025 h + 0.193; for h< 13m 0.480; for h> 13m
Toe slab thickness x 2 , m 0.067h - 0.058 0.087 h - 0.172 0.109 h - 0.296
Stem base thickness x 3 , m 0.020h + 0.228 0.019h + 0.242 0.013 h + 0.289
Counterfort thickness x 4 / m 0.022 h +0.117 0.047 h - 0.032 0.076 h - 0.161
Counterfort spacing x 5 , m 0.053 h + 2.234 0.027 h + 2.387 0.018 h + 2.390; for h< 15m 2.650; for h> 15m
Toe slab length x^ m 0.282 h - 0.719 0.272 h - 0 . 9 0 1 0.242 h - 0.812
3. Retaining wall with two relieving platforms
Footing thickness xv m 0.082 h - 0.088 0.089 h - 0.125 0.109 h - 0.227
Stem base thickness x 2 , m 0.129 h - 0.264 0.131h - 0.263 0.165 h - 0.434
Toe slab length x 3 , m 0.330 h - 0.899 0.268 h - 0.602 0.226 h - 0.390
Relieving platform thickness x g , m 0.054 h - 0.072 0.057 h - 0 . 0 8 7 0.072 h - 0.157
h - Height of w a l l above the ground level
' Relieving platform length = 0.33 h tan — - y (2)
A l inear m o d e l was proposed for the var iat ion of
wall/slab thickness wi th w a l l height, for the sake of
simplicity i n calculation and application. Based on the
above observations, heuristic design rules proposed
for a retaining w a l l built on soil wi th th = 25°, fi = 37°,
thb = 15° and mu = 0.5 and different SBC values (150,200
and 300 k N / m 2 ) , are tabulated i n Table 5. Using these
guidelines, a designer can select wal l and slab thickness
proportions that are likely to be close to this optimal
solutions for preliminary design, without carrying out
an optimization study.
M25 concrete and Fe415 reinforcement steel were found
to give optimal design solutions i n al l cases.
Best retaining wall design option
Based on the study, the costs corresponding to the optimal
designs for various wal l heights were compared for the
soil parameters; th = 25°; fi = 37°; thb =15°; mu = 0.5).
Figures 8, 9 and 10 show the cost per meter wal l length
for SBC = 150, 200 and 300 kN/irrespec t ive ly .
The results suggest that the cantilever retaining wal l ,
always yields the most economical solution. However,
the wal l height gets restricted when the SBC is low. When
this happens the retaining wal l wi th relieving platforms,
16 I THE INDIAN CONCRETE JOURNAL APRIL 2012
Page 9
which is a relatively new concept i n
India, provides the most economical
solution. The traditional assumption
that w a l l s w i t h counterforts are
likely to be more cost-effective than
cantilever walls for heights exceeding
about 8 m, was not found to be true.
The optimally designed counterfort
retaining w a l l was f o u n d to be a
more costly solution compared to the
optimally designed cantilever w a l l
and wal l wi th relieving platforms for
nearly all wal l heights.
It may be noted that the cost shown
i n Figures 8-10 were based on Delhi
Schedule of rates 2007. The authors
believe that even if they change with
time, the relative costs of steel and
concrete are l ike ly to remain the
same.
Conclusions
The salient conclusions, based on
the study, can be summar ized as
follows:
• C o u l o m b ' s t h e o r y , w h i c h
accounts for wal l friction, gives
a better cost-effective design
alternative for a retaining wal l
than Rankine's theory, which
is currently used i n practice,
for convenience.
• The traditional belief that walls
with counterforts are likely to
be more cost-effective than
cantilever wal ls for heights
exceeding about 8 m, was not
found to be true, w h e n an
optimal design was carried
out. The optimally designed
cantilever retaining wall was
f o u n d to be i n v a r i a b l y the
most cost-effective solution for
w a l l heights, where feasible
s o l u t i o n s w e r e p o s s i b l e
(depending on safe bearing
capacity).
• The r e t a i n i n g w a l l w i t h
relieving platforms, which is
a relatively new concept i n
250000
00
g 200000
150000
c 100000
~ 50000
Opt imal Cantilever
w a l l cost
Opt imal Counterfort
w a l l cost
• • O p t i m a l w a l l w i t h
rel ieving platforms
c'ost
15 11 13
W a l l height (h + hi) i n m
(SBC = 150kN/m 2 ; th = 2 5 ° ; ^ =37°; thb= 15°; mu = 0.5)
L7
Figure 8. Optimal design cost estimates
300000 •
g 3 250000-
£ 200000
&
& 150000 J c
" 100000
50000
- Opt imal Cantilever
w a l l cost
Opt imal Counterfort
w a l l cost
Opt imal w a l l w i t h
relieving platforms
cost
9 11 13 15 17 19
W a l l height i n m
(SBC = 200kN/m 2 ; th = 75°;fi =37°; th„ = 15°; mu = 0.5)
Figure 9. Optimal design cost estimates
500000
450000 -
/
-g ' t
to
a
400000 - / /
350000 -
- Opt imal Cantilever - Opt imal Cantilever
S 300000 - / .** w a l l cost
s / V
250000 -
;t
in
R
200000 "
• Opt imal Counterfort
w a l l cost
o
u
150000
13 100000
50000
• Opt imal w a l l w i t h
100000
50000
relieving platforms
cost
U 5 7 9 11 13 15 17 19 21 23 25
W a l l height i n m
(SBC = 300kN/m 2 ; th = 25°;fi =37°; thh= 15° mu = 0.5)
Figure 10 Optimal design cost estimates
APRIL 2012 THE INDIAN CONCRETE JOURNAL I 17
