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ANALYSIS
AND DESIGN OF A RETAINING WALL
TABLE CONTENTS
CHAPTER ONE
Introduction
Retaining walls
Cantilever rw
Rankin theory
Forces on cantilever rw
Counterfort rw
Forces on counterfort rw
The gravity rw
Forces on gravity rw
Semi gravity rw
CHAPTER TWO
Stability of retaining wall
Soil properties of rws
Drainage in rw
Allowable bearing capacity
CHAPTER THREE
Procedure for the design of rw
Cantilever rw
Analysis example
CHAPTER FOUR
Design of retaining wall
Conclusion
NOTATION USED IN THIS PROJECT
1.
Angle of internal friction of
soil
-
O
2.
Unit weight of
sol
-&
3.
Sat unit
wt
- &sat
4.
sauntered unit
wt
- &sub
5.
Overturning moment
- Mo
6.
Base width of container wall
- B
7.
Overall height of wall
- H
8.
Effective height of wall
- He
9.
Height of
stem
- h
10.
Effective depth
- d
11.
Overall depth
- d
12.
Weight
stem
- we
13-
Weight
soil
- ws
14.
Surcharge
- S
15.
Active pressure on
wall Pa
16.
Passive pressure on wall
Pp
17.
Breadth for design
-
18.
Sum of vertical
weight
- Ew
19.
Point of application of the resultant X
20.
Eccentricity
- e
21.
Sum of
moments
- Em
22.
Characteristic strength steel
- fy
23.
“
“
‘
concrete
- fcn
24.
Lever are
factor
- z
25.
Lever
are
Z
26.
Ultimate shear
stress
- Vc
27.
Shear
force
- V
28.
Shear
stress.
- v
CHAPTER ONE
RETAINING WALLS
Retaining walls are structures used in
providing stability for earth or materials where conditions do not give room
for the material to assume its natural slope and are mostly used to hold back
soil banks, coal or over, piles and water.
Retaining walls are distinguished from
a\one another based on the method of achieving stability.
There are six type of retaining walls
and are:
i.
The gravity wall
ii.
The cantilever wall
iii.
The counterfort retaining wall
iv.
Buttressed retaining wall
v.
The crib walls
vi.
Semi-gravity wall
Bridge abutments are
often retaining walls with using wall extension to retain the approach fill and
provide protection against erosion. They differ in two major respect from the
usual retaining walls.
i.
They carry and reactions from the bridge span.
ii.
They are restrained at the top so that on active earth pressure is unlikely to
develop.
Foundation walls of
building mduding residential construction and retaining walls, whose function
is to contain the earth of the basement.
Retaining walls are
required to be of adequate proportion to resist over turning and sliding as
wall as being structurally a proportion to resist over turning and
sliding as wall as being structurally adequate.
Terms used in
retaining wall design are shown below (f.g 1-2)
The toe is both the
front base perfection and forward edge, similarly for the heel the backward perfection.
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a) gravity walls of stone masony,
brick, or plain concrete
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The retaining wall as a whole must
satisfy two basic conditions. They are
i.
The base pressure as the toe of the wall must not exceed the allowable bearing
capacity of the soil.
ii.
The factor of safety against sliding between the are and the underlying
soil must be adequate a value of at least usually being specified. (Rit
Craig 2nd edition soil mechanical published by ven nosytrand rain
hold co.ltd) 1-1 common proportion of R-10. retaining wall design goes on with
the choosing of tentative dimensions, which are then analyzed for stability and
structural requirements. Since this s a trail process overall solution to
the problem may be obtained, all of which are satisfactory.
CANTILEVER RETAINING WALL
The cantilever wall is a reinforced
concrete wall that utilizes cantilever action to retain step. The mass behind
the wall from assuming natural slope. Stability of this wall mainly depends on
the mass of the soil on the heel behind the wall. Dimensions for a retaining
wall should be adequate for structural stability and to satisfy local building
code requirement.
The date shown below may be used where
no other data is available but may result in a overly conservative design. The
200mm from a liberal interpretation. of act (1966) and preferably not
less than 600mm so that the proper placement concrete off or is broke off a
sufficient amount will remain to satisfy structural and aesthetic require
(Bowles I . E)
The base slab dimension should be such
that the resultant of the vertical loads fall within the middle one third
otherwise. The toe pressure will be too much such that only a part (Bowles J-E
and (Hansen and Peck) of the footing will be effective.
.A better is recessing in order to save
materials. A front better is more acceptable so that the forward wall movement
to develop active pressure is not noticeable.
A slight increase in wall stability is
usually obtain when the battler is on the Bach face (Ref Bowles J.E)
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RANKING THEORY
Ranking theory deals with pressure
within a soil mass under the following condition (ie assumptions)
i.
The wall is vertical
ii.
The retaining wall face is smooth
iii.
The wall yield above the base and satisfies the condition of plastic
equilibrium
iv.
It is isotropic
Ranking theory applies when the
soil mass is in so called rankling sate.
When a soil mass is allowed to
explain (active earth pressure) or contract (passive earth pressure rapture
surface will form within the mass. if not interrupted by the back of the
retaining wall or other strictures, the mature surface will be aeries of
straight lines making an angle 1 with the horizontal.
Active earth pressure i = 45 + Ø/2
Passive earth pressure i = 45 – Ø /2
Where Pa and Pp = unit active and
passive earth pressure respectively, at a depth Z
q= vertical pressure
or load due to the weight of the soil above Z
C = cohesion strength of the soil
Ka and kp – coefficient of active and
passive earth pressure respectively
Ka = 1-sin Ø, kp =1+ sine
Ø :. Ka = 1
1x sinØ 1
sinØ
kp
When the state above
exists the soil I said to being Rankin state and the ranking theory is
applicable.
Pa = qka – 2c√ka
Pp = qp x 2c√ kp
According to ranking theory, earth
pressure increase linearly with the depth of the same manner as the lateral
pressure exerted by a fluid. For the reason engineers often refer earth
pressure as equivalent fluid pressure.
For a cohesion less soil
Pa =1/2 ka rH2 and Pp = ½ rH2
When the back of the
wall is inclined
Ka = cos B CosB-√CosB – Cos Ø2
CosB_√CosB - Cos Ø2
Ka = 1/kp
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FORCES ON A CANTILEVER RETAINING WALL
Due to the difficulty in getting or
calculating the wall frication, Rankin active pressure is normally used (ie
O=o) mainly for walls less than 6 to 7 matters in heights
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It is more economical to use the
coulombs equation for walls over 7m in height. It should also be clear
that it is not the height it used for computation if max. shear and
bending moment that is used for
Forces on cantilever wall (a) entire unit;
free bodied for (ii) stem,(ii) toe, (iv) heel. Note that
M1+M2+M3
= 0
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Cantilever retaining wall (a) stem
shear and moment (b) to and heel shear and moments. Sliding computation to get
the driving forces it is left for the designer to decide whether to use passion
pressure from the soil in front of the toe and whether the soil covering the
top portion will be available for existing over turning moment and
sliding. Sometimes this is not considered. Just for conservative purpose
The triangular pressure diagram on the
stem, will yield a shear diagram that is a third degree curve.
The use of the different equations for
shear and moment seems to be easier.
This enables the roped
computation of the cut off points for there in forcing steel since it is
uneconomical to use a constant amount of reinforcement for the entire wall
height.
From the diagram shown below. The with
of the base slab is deduced from the gross. Soil pressure diagram before
computing the shear and moments diagrams. The eccentricity of
computed by egns (iii) and (iv) shown below differential equations can also be
used to compute the shear and moments of the base slab if the safe of the cane
is desired and even when numerical values are required.
CONTERFORT RETAINING WALL
Counterfort retaining wall, are similar
to the cantilever retaining wall only that this type has counterforts
built behind to hold the wall (stem) and base together and is used where the
cantilever is long or for very high pressure behind the wall this counterforts
behinds the wall are subjected to tensile forces.
The dimension indicated in the diagram
below, only act as a guide, some walls which are about 100mm to 150mm thick
have been built in area like united Kingdom. Any thickness which satisfies
stability of the wall can be used.
Relative costs of forms, concrete,
enforcement and labour will determine to use of counterfort but it is doubtful
if a counterfor wall will provide any relative construction economy
values it is over 7m is height.
The spacing in the counterforts is
based on the trail and error in other to huiumise cost. The most
economical method is placing them 1/3 to ½ H (height) apart. by conventional
beam theory bending moments in the face slab cantilevered part of the
wall as at the interior if the length of over hang is made 0/4h and a
spacing between counterforts of L.
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The counterfort will may be constructed
with out a toe if additional front clearance is needed and sliding and
overturning stability requirements are met.
FORCES ON A COUNTERFORT WALLS
Counterfort walls are described as
indeterminate problem. this can be solved by the use of plate theory of the
expense of large amount of labour. Simplified methods are commonly used in the
solving of this problem which makes it to be over-designed the weight of the
counterfort is not considered in the design.
A simplified and conservative solution
to a counterfort wall problem is in the diagram shown below (gig 1-7). The face
slab of the wall is considered as continuous slab constituting series of
equivalent unit –width beam since the pressure distribution is triangular, the
equivalent beam should be analyzed for strips at the junction of the wall and
base and at two or three intermediate location between the top of the wall and
base so that adjunctions in reinforcing steel can be made as the pressure
decrease moment distribution can be used to find the bending moment, although
due to approximations being made continuous beam coefficients maybe used for
lower strip wh2/12 or wl2/14may be used because of the lower edge being
fastened to the base while for the upper strips, wl2/9 or wl2/10 can be used
for a conservative solution or method. The same coefficients for both positive
and negative moments maybe employed as the designer consider appropriate the
toe of the base is considered as a cantilever beam and heel as a
continuous beam, similar to the treatment of the shrinkage steel should
be satisfied in the directions not steel should be satisfied in the
directions not analysed in this manner the counterfort member may be considered
as a wedge shaped T-beam, which include the applicable portion of the wall tem as
the flange, bat these beams are so massive that the concrete stresses will be
so low that an analysis is usually not required. Tensile steel will be required
at junction of the base (heel) and the counterfort to resist the moment,
fending to tip the wall over and the quantity can be conservatively computed
treating the conuterfort along as a beam. Tensile steel will also be required
running horizontal from the counerfort into the stem to tie the wall and
counterfort together. in some case, the bound stress requirements of this
reinforcement control the slope of the counerfiort member.
Huntington (1957) presented a method in the diagram below. the also recommended
a value for one in the middle half of the wall, at the base of
0.2qH to be used as long as he ratio of counterfort spacing to the wall
height L/H > 0.5
BOWLES J.E (1982) Treat the toe as a cantilever beam loaded with the indicated
pressure diagram.
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qL2/10 for top strips for stem with an
average ‘q’ on unit strip
qL2/10 for top strips near the bottom
of stem because of fixity of stem to base.
qL2/10 for all strips in the heel
use an average net q for heel pressure. consider both rH and the upward actins
soil press.
Fig 1-7, reduction of the complex
analysis of a counterrfort RW to a system of simple beans for rapid
design.
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Computation of bending moment in
horizontal direction for the counterfort stem (Huntington 1957)
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qs = Ws + Wcb
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b
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qb = Pa sin B
b
q1b = Pa sin B
b
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q=W1_ qs
+qb+q1b
qnet =qs+q1b+W11-qf
NOTE: The increase in heel pressure due
to moment is
W 1=2.4m: W1 = W1
= 2/3W1b
6
MT == toe moments value at front face
of wall.
Note that W1 is parabolic
but may be approximated as a uniform pressure W11
W11 =w1/b
Assure pressure q1b, qb and
q are constant and uniformly distributed across b
If B-O the is only q and will W1
to consider since W11 qb and q1bare small he design will usually be
sufficiently accurately to neglect these pressure
THE GRAVITY RETAINING WALL
This type of wall depends on its weight
in other to achieve stability, just as the name implies. it also depends used
in the constriction
No reinforcement is provided except in
concrete walls where a nominal amount of steel is placed near the exposed faces
to prevent surface cracking due to temperature changes (shrinkage
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Typical dimension for gravity walls
maybe taken as shown below. Generally, gravity walls have trepezodial shape but
it can also be constructed with broken backs. The base and other dimensions are
designed and constant in such a way that the resultant falls within the middle
on third of the base. The top width of the stem should be on the order of
0,30m. if the heel perfection is only 100 to 150mm, the coulomb equation may be
used for evaluating the lateral earth pressure, with the surface of sliding
taken along the back face of the wall. The Rankin solution may face of
used on a section taken through the heel. Due to the massive proportions and
resulting low concrete stress, low strength concrete can generally be used for
the wall construction.
A critical section for analysis of tensile flexure stress will occur, through
the junction of the toe portion at the front face of the wall.
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i.
Tentative dimensions for a gravity retaining wall
ii.
Broken –back retaining wall
Bowles J.E 1982
FORCES ON GRAVITY RETAINING WALL
The active earth pressure is computed by using either the rackine or coulomb
methods if the coulombs method is used it is assured that there is
incipient sliding on the back face of the wall, and the earth pressure
acts at the angle of wall frication to a normal with the wall. The rackine
solution applies to Pa acting at the angle B on a vertical plane through the
heel.the vector can then be added to the weight vector of the edge of
magnitude of the resultant Pa and the wall. The vertical resultant R acting on
the base is equal to the sum of the forces acting downward, and will have an
eccentricity e with respect to the geometrical center of the base
Taking moments about the toe
x = sum of overturning moment
R
If the width of the base is B1, the
eccentricity of the base can then be computed as
e = B -x
Z
a.
coulomb analysis Ph =Pa cos B
b.
Rankin analysis Pv =Pa sinB
SEMI GRAVITY WALLS
Semi gravity walls
are intermediate between a true gravity and a cantilever wall. It is somewhat
more slender than a gravity wall and requires reinforcement consisting of
vertical bars along the liner face and dowels continuing into footing. H
likewise is provided with temperature steel war the exposed face to prevent
shrinkage or expansion and contraction of wall due to temperature changes
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