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 aone 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. Semigravity 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 12)
The toe is both the front base perfection and forward edge, similarly for the heel the backward perfection.
a) gravity walls of stone masony, brick, or plain concrete


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 2^{nd} edition soil mechanical published by ven nosytrand rain hold co.ltd) 11 common proportion of R10. 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 JE 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)
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 = 1sin 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 rH^{2} and Pp = ½ rH^{2}
When the back of the wall is inclined
Ka = cos B CosB√CosB – Cos Ø^{2}
CosB_√CosB  Cos Ø^{2}
Ka = 1/kp
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

Forces on cantilever wall (a) entire unit; free bodied for (ii) stem,(ii) toe, (iv) heel. Note that
M_{1}+M_{2}+M_{3} = 0

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.

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 overdesigned 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 17). 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 Tbeam, 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.



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 17, reduction of the complex analysis of a counterrfort RW to a system of simple beans for rapid design.
Computation of bending moment in horizontal direction for the counterfort stem (Huntington 1957)
qs = Ws + Wcb


b
q^{1}b = Pa sin B
b


q=W^{1}_ qs +qb+q^{1}b
qnet =qs+q^{1}b+W^{11}qf
NOTE: The increase in heel pressure due to moment is
W ^{1}=2.4m: W1 = W^{1 } = 2/3W^{1}b
6
MT == toe moments value at front face of wall.
Note that W^{1} is parabolic but may be approximated as a uniform pressure W^{11}
W^{11} =w1/b
Assure pressure q^{1}b, qb and q are constant and uniformly distributed across b
If BO the is only q and will W^{1} to consider since W11 qb and q^{1}bare 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
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.
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