Construction Of Concrete And Steel Engineering Essay

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This thesis presents a lucid model to obtain the optimum cost of a cantilever retaining wall having different cases of backfill (straight and inclined) and surcharge. A code written in Java, finds out all the sections of the cantilever retaining wall possible according to stability criteria that applies to all retaining walls and gives the optimum cost of a retaining wall of a given height and the required material properties to be used, while following the provisions of the Indian Standard Code, IS 456:2000 for the sections. The freedom given for the person who uses the program to specify material properties and their costs add to the versatility of the code.

Retaining walls are structures that are used to retain earth (or any other material) in a position where the ground level changes abruptly. They can be of many types such as gravity wall, cantilever wall, counterfort wall and buttress wall among others. The 'cantilever wall' is the most common type of retaining wall and is economical heights up to about 8 m. The lateral force due to earth pressure is the main force that acts on the retaining wall which has the tendency to bend, slide and overturn it.

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The present thesis focuses on designing the cantilever type of wall giving the most economic section. The main considerations are the external stability of the section and the adherence to the recommendations of IS 456:2000. Satisfying the external stability criteria is primarily based on the section giving the required factor of safety. The ratio of resisting forces to the disturbing forces is the factor of safety, and this factor of safety should always be greater than unity for the structure to be safe against failure with respect to that particular criteria. Different modes of failure have different factors of safety.

In this thesis the most economic section for a cantilever wall is obtained using a computer program that calculates various sections satisfying the stability criteria, according to the height and properties of earth that the wall is required to support, and gives the most economical section as the output after minimizing the cost for sections adhering to provisions of IS 456:2000.

Chapter 2

2.1 Cantilever Retaining Wall:

The cantilever wall generally consists of a vertical stem, and a base slab, made up of two distinct regions, viz. a heel slab and a toe slab. All three components behave like one-way cantilever slabs: the 'stem' acts as a vertical cantilever under the lateral earth pressure; the 'heel slab' and the 'toe slab' acts as a horizontal cantilever under the action of the resulting soil pressure. The reinforcement detailing is as given in Fig. 2.1. The weight of the earth retained helps in maintaining the stability of the wall.

Fig. 2.1 Cantilever Retaining Wall

2.2 Lateral Earth Pressure on Retaining wall:

The main force acting on the retaining wall is constituted by lateral earth pressure which tends to bend, slide and overturn it. The basis for determining the magnitude and direction of the earth pressure are the principles of soil mechanics. The behavior of lateral earth pressure is similar to that of a fluid, with its magnitude p increasing nearly linearly with increasing depth z for moderate depths below the surface:

p= Kγez (2.1)

where γe is the unit weight of the earth and K is a coefficient that depends on its physical properties, and on whether the pressure is active or passive. The coefficient to be used in Eq. 2.1 is the active pressure coefficient Ka, in case of active pressure, and the passive pressure coefficient Kp, in case of passive pressure. Rankine's theory is applied for cohesionless soils and level backfills and the following expressions for Ka and Kp may be used:

Ka = (2.2 a)

Kp= (2.2 b)

where is the angle of shearing resistance.

When the backfill is sloped, the expression for Ka should be modified as follows:

Ka = cos (2.3)

where is the angle of inclination of the backfill, i.e., the angle of its surface with respect to the horizontal.

2.3 Stability of a cantilever retaining wall:

Fig. 2.1 shows a cantilever retaining wall subjected to the following forces:

Weight W1 of the stem AB.

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Weight W2 of the base slab DC

Weight W3 of the column of soil supported on the heel slab BC

Horizontal force Pa, equal to active earth pressure acting at H/3 above the base.

Fig. 2.1

2.4 Analyzing the wall for its different possible modes of failure,

Overturning:

In Fig. 2.1, the overturning moment , due to active earth pressure, at toe is

M0 = Pa H/3 = Ka γ/2. H/3 (2.4)

= Ka γ/6

The resisting moment is due to the weights W1, W2 and W3, neglecting the passive earth pressure and weight of soil above the toe slab.

Hence, MR = W1 x1 + W2 x2 + W3 x3 (2.5)

Hence the factor of safety due to overturning (F1) is given by

F1 = (2.6)

A minimum factor of safety due to 2 is adopted.

Sliding:

The horizontal force Pa tends to slide the wall away from the fill. The tendency to resist this is achieved by the friction at the base (Fig. 2.2(b)).

Shear key

∑W

Pa

F = µ∑W

∑W

Pa

Passive pressure (a

(b)

Fig.2.2 Sliding Of Retaining Wall

The force of resistance, F is given by

F = µ∑W (2.7)

where µ is the coefficient of friction between soil and concrete, and ∑W is the sum of vertical forces.

The factor of safety F2 due to sliding is given by

F2 = (2.8)

where H = Pa.

If the wall is found to be unsafe against sliding, shear key below the base is provided. Such a key develops passive pressure which resists completely the sliding tendency of the wall. A factor of safety of 1.5 is needed against sliding.

Soil pressure distribution:

Fig. 2.1 shows the various forces acting on the wall. If ∑W is the sum of all vertical forces and Pa is the horizontal active earth pressure, the resultant R will strike the base slab at a distance e (say) from the middle point of the base.

Let ∑M = W1 x1 + W2 x2 + W3 x3 - Pa . H/3 = net moment at the toe.

Then x = distance of point of application of resultant =

Hence eccentricity e = b/2 - x

The pressure distribution below the base is shown in Fig. 2.1. The intensity of soil pressure at the toe and heel is given by

p1 = at toe (2.9)

p2 = at heel (2.10)

p1 at toe should not exceed the safe bearing capacity of the soil otherwise soil will fail. Similarly, p2 at hell should be compressive. If p2 becomes tensile, the heel will be lifted above the soil, which is not permissible. In an extreme case, p2 may be zero, where e = b/6. Hence in order that tension is not developed, the resultant should strike the base within the middle third.

Bending failure:

There are three distinct parts of T-shaped cantilever retaining wall : the stem AB, the heel slab BC and the toe slab DE. The stem AB will bend as cantilever, so that tensile face will be towards the backfill. The heel slab will have net pressure acting downwards, and will bend as a cantilever, having tensile face upwards. The pressure distribution will be as shown in Fig. 2.3. The critical section will be at B, where cracks may occur if it is not reinforced properly at the upper face. The net pressure on toe slab will act upwards, and hence it must be reinforced at the bottom face. The thickness of stem, hell slab and toe slab must be sufficient to withstand compressive stresses due to bending.

Fig. 2.3 Bending failure

2.5 Basic design considerations:

Design of stem:

The stem AB is designed as a cantilever, for triangular loading. At any section hbelow the top point A, the force is equal to Ka γ/2and its bending moment about the section is Ka γ/6. The thickness at B is maximum. The minimum thickness at A should vary from 20 to 30 cm depending upon the height of the wall. Reinforcement is provided towards the inner face of stem, i.e. towards side of fill. The requirement towards the top of stem can be curtailed, since B.M. varies as h3. Distribution reinforcement is provided @ 0.15% of the area of cross-section along the length of retaining wall at inner face. Similarly, at the outer face of the stem, temperature reinforcement is provided both in horizontal as well as in vertical direction, at the rate of 0.15% of the area of cross-section.

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Design of heel slab:

The heel is also to be designed as a cantilever. It has both downward pressure (due to weight of soil and self-weight) as well as upward pressure due to soil reaction. However, the net pressure is found to act downward and hence reinforcement is provided at the upper face BC.

Design of toe slab:

Neglecting the weight of the soil above it, the toe slab will bend upwards as a cantilever due to upward soil reaction. Hence reinforcement is placed at the bottom face. Normally, the thickness of both toe slab and heel slab is kept the same, determined on the basis of greater of the cantilever bending moments.

Depth of foundation:

As shown in Fig. 2.4, the height H2 of the retaining wall, above ground level is fixed on the basis of height of the backfill to be retained. The depth of foundation yshould be such that good quality of soil to bear the induced pressure is available. However, a minimum depth of foundation given below by Rankine's formula should be provided:

ymin = (2.11)

where is the safe bearing capacity of the soil, or equal to the maximum pressure likely to occur on soil.

Specific codal provisions followed while optimizing:

The program developed in this thesis for economic design of a cantilever retaining wall is guided by certain provisions of the code IS 456:2000, which are given below:

Spacing of reinforcement:

For the purpose of this clause, the diameter of a round bar shall be its nominal diameter, and in the case of bars which are not round or in the case of deformed bars or crimped bars, the diameter shall be taken as the diameter of a circle giving an equivalent effective area. Where spacing limitations and minimum concrete cover are based on bar diameter, a group of bars bundled in contact shall be treated as a single bar of diameter derived from the total equivalent area.

Minimum distance between individual bars:

The following shall apply for spacing of bars:

The horizontal distance between two parallel main reinforcing bars shall usually be not less that ha greatest of the following:

The diameter of the bar if the diameters are equal,

The diameter of the larger bar if the diameters are unequal, and

5 mm more than the nominal maximum size of coarse aggregate.

Greater horizontal distance than the minimum specified in (a) should be provided wherever possible. However when needle vibrators are used the horizontal distance between bars of a group may be reduced to two-thirds the nominal maximum size of the coarse aggregate, provided that sufficient space is left between groups of bars to enable the vibrator to be immersed.

Where there are two or more rows of bars, the bars shall be vertically in line and the minimum vertical distance between the bars shall be 15mm, two-thirds the nominal maximum size of aggregate or the maximum size of bars, whichever is greater.

Nominal cover to reinforcement:

Nominal cover:

Nominal cover is the design depth of concrete cover to all steel reinforcements, including links. It is the dimension used in design and indicated in the drawings. It shall be not less than the diameter of the bar.

Nominal cover to meet durability requirement:

Minimum values for the nominal cover of normal-weight aggregate concrete which should be provided to all reinforcement, including links depending on the condition of exposure.

For footings minimum cover shall be 50mm.

Requirements of reinforcement for structural members:

Beams

a.1) Tension reinforcement:

i) Minimum reinforcement

The minimum area of tension reinforcement shall be not less than that given by the following:

= (3.8)

where As = minimum area of tension reinforcement

b = breadth of beam or the breadth of the web of T-beam

d = effective depth

fy = characteristic strength of reinforcement in N/mm2

ii) Maximum reinforcement - The maximum area of tensile reinforcement shall not exceed 0.04 bD.

Maximum spacing of shear reinforcement:

The maximum spacing of shear reinforcement measured along the axis of the member shall not exceed 0.75 d for vertical stirrups and d for inclined stirrups at 45Ëš, where d is the effective depth of the section under consideration. In no case shall the spacing exceed 300 mm.

Minimum shear reinforcement:

Minimum shear reinforcement in the form of stirrups shall be provided such that:

≥ (3.9)

where = total cross-sectional area of stirrup legs effective inshear,

sv = stirrups spacing along the length of the member,

b = breadth of the beam or breadth of the web of flanged beam, and

fy = characteristic strength of the stirrup reinforcement in N/mm2 which

shall not be taken greater than 415 N/mm2.

Chapter 3

3.1 Problem overview:

In this project, as we have aimed to obtain the cantilever retaining wall section costing minimum for the three cases of backfill and related input parameters, that we have considered, namely:

Horizontal backfill with static surcharge

Inclined backfill, with shear key and without surcharge load

Inclined backfill with surcharge load

Inclined backfill withoutsurcharge load and provision to vary cost of construction of different grades of concrete

Inclined backfill without traffic load and provision to vary cost of construction of different grades of steel

Inclined backfill without traffic load and provision to vary cost of construction of different grades of concrete and steel.

Detailed discussion of the problem:

Now, let us look at each of the above cases in a detailed manner:

Horizontal backfill with uniform surcharge:

If the backfill is horizontal with static surcharge, of uniform intensity w per unit area, the vertical pressure increment at any depth h increases by w. This leads to an increase in lateral pressure by Kaw. Therefore, the lateral pressure at any depth h will be given by

pa= Ka.γ.h + Ka. w (3.1)

this implies that the pressure at the base of the wall is given by

pa= Ka.γ.H + Ka. w (3.2)

w

Fig.3.1. Backfill with uniform surcharge

Fig 3.1(a) and (b) show two alternative methods of plotting the lateral pressure diagram for this case. The increase in lateral pressure due to surcharge is the same at every point of the back of the wall, and it is invariant with h. The equivalent height of the fill he is given by:

Ka.γ.he= Ka.w

or he = (3.3)

which means that the effect of surcharge is the same as that of earth retained to a height of he above the ground.

Inclined backfill with shear key and without surcharge load:

Fig.3.2. Lateral Pressure Distribution for Sloping Surcharge

Let the backfill be inclined at an angle β to the horizontal as shown in Fig. 3.2; β is called the angle of surcharge. An assumption that vertical and lateral stresses are conjugate is made while calculating the active earth pressure for this case by Rankine's theory. Fig. 3.2 shows the retaining wall with a sloping backfill. The intensity of lateral earth pressure at the base of wall is given by:

pa = cos

or pa = KaγH (3.4)

where Ka = cos.

The pressure acts parallel to the sloping surface, i.e., at β with the horizontal. The total active pressure Pa for the wall of height H is given by

Pa = KaγH2 (3.5)

The resultant acts at H/3 above the base, in direction parallel to the surcharge.

Provision of shear key:

Fig.3.3. Passive Earth Pressure Distribution and Shear Key

When the reinforcing wall fails against sliding, then an arrangement called shear key is provided. As the retaining wall pushes against the soil at the zone where shear key, as shown in Fig. 3.3 is provided, the shear key has to be designed for passive earth pressure. This can be explained by the fact that due to active earth pressure from the right side, the wall moves to the left. The soil to the left is thus compressed and in turn exerts passive earth pressure, resisting that movement.

If h is the height of fill, the intensity of passive pressure at height h is given by

pp = Kp.γh (3.6)

where Kp is the coefficient of passive earth pressure and

Kp= =

The passive pressure distribution will thus be a triangle, much like the one for active pressure distribution. The total pressure is given by

Pp= Kp. (3.7)

acting at h/3 above base.

Inclined backfill withsurcharge load:

This case is as given in Fig. 3.4 and may be treated as a combination of the first two cases wherein the part for surcharge load is done as Case 1 and the shear key part is done as Case 2.

Fig.3.4. Lateral Pressure Distribution on Inclined Backfill with Surcharge

Surcharge load and provision to change grade of concrete:

This case incorporates freedom in specifying the different cost of construction of different grades of concrete in the case considered in Case 1.

Surcharge load and provision to change grade of steel

This case incorporates freedom in specifying the different cost of construction of different grades of steel in the case considered in Case 1.

Surcharge load and provision to change grade of concrete and steel:

This case incorporates freedom in specifying the different cost of construction of different grades of both concrete and steel in the case considered in Case 1.

Chapter 4

Model formulation:

The basic purpose of the model developed here is to obtain the minimum cost of a cantilever retaining wall supporting backfill of a particular height. It is kept in mind that when the grade of the concrete becomes higher, then, the section dimensions are reduced while the cost of construction goes up significantly. So, the model is specifically formulated to give freedom to specify the different costs of construction for different grades of concrete and steel separately and in a combined way at a later stage of the program. The model is formulated in two major steps: (1) finalizing the design variables to be given importance; (2) the technique that is to be adopted to find the minimum cost of the wall.

4.2 Programming to get minimum cost:

The central idea in building the program is to provide a number of parameters that can be varied at the input level giving the program great flexibility to design the retaining wall according to various considerations like cost, aesthetics, varying site conditions, availability of materials and workmanship, requirements of the client, etc. The decision variables on the basis of which the cost optimization is done are grade of concrete (fck) and grade of steel (fy). The function that the program performs is to reduce the dimensions of the base slab, stem and the toe slab so that we may get the section where further reduction of dimensions is not possible.

Case 1:

In the case of horizontal backfill with traffic load, the input parameters given in the program are

Total Height of Retaining wall = H

Yield strength of steel = fy

Characteristic compressive strength of concrete = fck

Coefficient of friction between base slab and the ground = µ

Traffic load intensity = q

Internal friction angle of backfill soil = φ

Unit weight of backfill soil = U1

Unit weight of concrete = U2

Unit weight of foundation soil = U3

Permissible shear stress for the concrete = T

Internal friction angle of foundation soil = θ

Case 2:

The case of inclined surcharge with shear key and without traffic load has the following input parameters:

Height of soil to be retained(m) = h

Yield strength of steel for main reinforcement (N/mm2) = fy1

Yield strength of steel for distribution reinforcement (N/mm2) = fy2

Characteristic compressive strength of concrete = fck

Angle of backfill = A1

Internal friction angle of backfill soil = A2

Internal friction angle of backfill soil = A3

Unit weight of backfill soil = U1

Unit weight of concrete = U2

Unit weight of foundation soil = U3

Safe bearing capacity of soil = Pb

Coefficient of friction between base slab and the ground = ff

Cost of per m3 of concrete = Cc

Cost of per kg of reinforcement = Cs

Case 3:

The case of inclined surcharge with shear key and traffic load has the following input parameters:

Height of soil to be retained = h

Yield strength of steel for main reinforcement = fy1

Yield strength of steel for distribution reinforcement = fy2

Characteristic compressive strength of concrete = fck

Angle of backfill = A1

Internal friction angle of backfill soil = A2

Internal friction angle of backfill soil = A3

Surcharge load = q

Unit weight of backfill soil = U1

Unit weight of concrete = U2

Unit weight of foundation soil = U3

Safe bearing capacity of soil = Pb

Coefficient of friction between base slab and the ground = ff

Cost of per m3 of concrete = Cc

Cost of per kg of reinforcement = Cs

Case 4:

In addition to the input parameters given under Case 2, the cost of construction of different grades of concrete is also brought under input parameter list.

Case 5:

In addition to the input parameters given under Case 2, the cost of construction of different grades of steel is also brought under input parameter list.

Case 6:

In addition to the input parameters given under Case 2, the cost of construction of different grades of both concrete and steel is also brought under input parameter list.

Results: