WEEK 6- Retaining WALL Notes PDF

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RETAINING WALLS

What is the retaining walls ?

A retaining wall is a structure designed and constructed to resist the lateral pressure of soil when there is a desired change in ground elevation that exceeds the angle of repose of the soil .

What is the Function of retaining wall ? The basic function of a retaining wall is to retain soil at a slope which is greater than it would normally assume , The natural slope taken up by any soil is called its angle of repose .

Gravity-dependent structures Gravity retaining walls depend on their own weight and any soil resting on the concrete in resisting lateral earth forces. They are generally economical up to 10 feet in height for cast concrete structures. Usually are sufficiently massive to be unreinforced.

Failure criteria Before any calculations can be carried out to establish the stability of a structure the failure modes must be known. There are five Modes of failure : 1. Sliding Failure 2.

Overturning Failure

3.

Bearing capacity Failure

4. Rotational/ Shallow shear Failure 5. Deep shear Failure/tension in wall 1. Sliding Failure Sliding failure is nothing but sliding of wall away from backfill when there is shearing failure at the base of wall.

The Factor of safety (FOS) against sliding must be at least 1,5. This means the that the friction force between the base and the wall must be 1.5 times more than the sum of the lateral forces.

the normal force=weight of the wall

F µ =µV Where:

V is the resultant ground reaction μ = coefficient of friction = tan δ Fµ is the friction force

FOS for sliding=

¿

Friction force ∑ of lateral forces

Fµ ∑ F forces

2. Overturning failure Overturning failure is rotation of wall about its toe due to exceeding of moment caused due to overturning forces to resisting forces

FOS for overturning=

∑W moments about toe ∑W moments

It must be at least 2. If its less than 2 then the wall is not safe from overturning

3. Bearing capacity failure

The pressure exerted by resultant vertical force at toe of wall must no exceed the allowable bearing capacity of the soil , the pressure distribution is assumed to be linear

FOS for ground bearing pressure=

Ultimate ground bearing pressure Max . ground bearing pressure

It must be at least 3. 4. Shallow shear/rotational failure This type of failure occurs along a cylindrical surface ABC passing through the heel of retaining wall , the failure takes place because of excessive shear stresses along the cylindrical surface within the soil mass , and the FOS against horizontal sliding is lower than that for shallow shear failure however, FOS against sliding is greater than 1.5, shallow shear failure is not likely to occur

5. Deep shear failure This type of slope failure occurs along a cylindrical surface ABC as shown in the photo below when there is a weak layer of soil under the wall at a depth of about 1.5 times height of wall .

Stability calculations

To determine the stability of a wall for the limits set out above and not to exceed them the following need to be calculated: 1. The lateral forces (F) 2. The weight of the wall (V) 3. Position of the ground reaction (Resultant position of weight) 4. Ground bearing pressure at toe and heel of wall. 5. Factor of safety (FOS)

Lateral forces (F) The lateral forces to be put under consideration are: 1. Wind (Fwi) 2. Water (Fw) 3. Soil (Fg) 4. Surcharge (Fp) NB:All calculations are to be based on a wall of width one metre.

Lateral force due to wind (Fwi) The intensity of wind is considered to be constant.

If the intensity of the wind is P N/m2 and it is acting on a rectangular area of one metre width and H metres high then:

F wi=σA ¿px Hx1

¿ pH N /m width The resultant position of the force acts at a height of H/2 from the base.

Lateral force due to water (Fw) The calculations involved follows the principle of hydrostatics which states: 1. Pressure at any point in a liquid is the same in all directions.

Therefore:

Pressure= p= ρgh( Pa ) The pressure at the water surface will be zero. Pressure varies from zero to maximum at height H. 1

Forcedue ¿ water =F w = 2σA

¿ 12ρg H 2 N /m width The line of action for resultant water force acts at H/3 from the base.

Lateral force due to soil (Fg) Rankine’s theory will be used to calculate the lateral thrust for soil.

If soil is thrown out on flat surface it will form a conical heap as shown above. The angle ɸ that the side of the cone makes with the horizontal is known as the angle of repose.

Angle of repose=angle of shear resistance

In material such as sand the lateral pressure is less than the vertical pressure. The coefficient of active lateral earth pressure Cµ will determine how much the lateral pressure will be less than the vertical.

At a given depth in the soil the lateral pressure will be given by:

Lateral pressure=Pa =C µ ρgH The vertical pressure will be given by:

Vertical pressure=ρgH

The coefficient of active lateral earth pressure is the ratio between the lateral pressure and the vertical pressure

C µ=

Pa ρgH

In terms of the angle of repose ɸ or angle of shear resistance

C µ=

1−sinɸ 1+sinɸ

For the figure above the pressure at the surface of the soil is zero and maximum at full depth. 1 2 Lateral ground force=F g = 2C µ ρg H

The line of action for this force is H/3 from the base of the wall.

Lateral Force Fp due to surcharge Surcharge is a superimposed pressure that is applied on the surface of soil for example a building. It increases the lateral force on the wall. Lateral force due to soil and lateral force due to surcharge both act on the back of the wall.

Lateral force due ¿ surchage=C µ pH N / m width The line of action of the surcharge force will be at H/2 from the base.

Step faces

A stepped face on the side of the water or soil will not influence any force acting on the wall. Its because the vertical area on which these forces are acting has not changed.

Weight of the wall W 1.weight of the wall material.

The must be divided into different sections so as to calculate the volume of each section.

Weight =W =Vρg

2.Weight of water or soil on step

Water or soil on step will add to weight of the wall

Weight=W g∨W w =V ρ g where ρ is density of water∨soil 3.Surchage acting on step When surcharge is only in front of the step it will have a lateral force only. When its spread over the step it will have: 1. Lateral force on the back of the wall 2. It will add weight to the wall It will cause a vertical force Wp acting downwards.

W p=vertival pressure x surface areaof step

¿ p x b x1 ¿ pb N /m width

Position of ground reaction Resultant ground reaction (RGR)=V=resultant weight of wall. RGR=V=W1 + W2 +….+Wg + Wp

Position of RGR from toe Take moments of the weight (clockwise) and the RGR and lateral forces (anticlockwise) about the toe.

Vx+ ∑ of lateral force moments=∑ of weight moments

Vx+∑ F moments=∑ w moments

Vx+F g x

H H + F p x =W 1 x 1 +W 2 x 2 … .. 2 3

The distance of the RGR from the toe will be x metres. The distance from the RGR from the centre of the base it’s the eccentric distance e

B e= −x 2 B is the base length between the toe and the heel.

Ground bearing pressure GBP The value of the GBP and its distribution must be determined so as to assess the factor of safety against failure of the foundation soil. The bottom view of the wall is rectangular section, 1m wide with depth of B meters.

Resultant pressure at toe=σ D + σ B

σ max =

V 6 Ve + B B2

Resultant pressure at heel=σ D −σ B σ min =

V 6 Ve − B B2

The middle third rule

a) The best position for the resultant weight would be when it is at the centre of the wall e=0.There will only be direct stress beneath the base which is evenly spread from the to the heel. b) The RGR is within the middle third and eB/6.Tension will develop in the wall.

Reinforced concrete wall The reinforcements will take up the tension developed in the wall. The tension and maximum pressure at the toe has to be calculated differently if the RGR is outside the middle third. The pressure distribution will be as shown below:

1 V = σ max x 3 x 2

σ max =

2V 3x

This only applies to steel reinforced walls where the RGR is outside the middle third. If the RGR is inside the middle third to determine maximum and minimum pressure beneath the wall we use: σ D ± σ B

WORKED EXAMPLE A retaining wall supports soil as shown below with a surcharge of 20KPa.The density of the wall material and the soil is respectively 2 400kg/m3 and 1 600kg/m3.The angle of repose is 300 with a wall friction angle of 240

Calculate: 1. The position of the RGR 2. Whether there will be tension in the wall 3. The maximum and minimum ground bearing pressure 4. The factor of safety against sliding 5. The factor of safety against overturning

Solution NB:The surcharge is not over the step therefore it will not add to the weight of the wall. 1.To calculate the position of the RGR from the toe, take moments about the toe.

Vx+ ∑ of lateral force moments=∑ of weight moments

Vx+∑ F moments=∑ w moments

The coefficient of active lateral earth pressure:

C µ=

1−sinɸ 1+sinɸ

C µ=

1−sin 30 1+sin30

C µ=

1 3

Calculate the lateral forces.

1 1 1 2 2 F g=2C µ ρg H = x x 1600 x 9.81 x 8 =167 424 3 2

1 3 ❑ F p =C µ ρ H = x 20 x 10 x 8=53 333 3 ∑ F Forces=220 757 N Take moments about the TOE

F g=

H 8 =167 424 x =446 464 3 3

Fp=

8 H =53 333 x =213 333 2 2

∑ F Moments=659 797 Nm The weight of the wall = RGR=V

W 1=Vρg=0,8 x 1 x 8 x 2400 x 9,81=150 682 W 2=Vρg=5 x 1 x 1 x 2400 x 9,81=117 720

W 1=Vρg=7 x 2.2 x 1 x 1600 x 9,81=241 718 RGR =V =510 120 N Take moments about the TOE

W 1 x x 1=150 682 x 2.4= 361636

W 2 x x 2=117 720 x 2.5=294 300 W g x x g=241 718 x 3,9= 942700

∑W Moments=1598 636 Nm

Position of RGR from TOE take moments about the toe

Vx+∑ F moments=∑ w moments

510 120 x +659 797=1 598 636 x=1,84 m RGR is 1.84m from the toe Distance from the centre=e

B e= −x 2 5 e= −1.84 2 e=660 mm

2.Tension in the wall To determine whether there is tension in the wall, compare the eccentric distance e with B/6

e=660 mm B= 5 =833 mm 6 6 e<

B 6

Therefore e...