Important Relationships Based on Soil Phase Diagram

A number of useful relationships may be derived based on the foregoing definitions and the soil-phase diagram.

1) Relationships Involving Porosity, Void Ratio, Degree of Saturation, Water Content, Percent Air Voids and Air Content

This may provide a practical approach to the determination of n.

This may provide a practical approach to the determination of e.

                                 

From the above equation, we can derive

These interrelationships between n and e facilitate computation of one if the other is known.

              

By multiplying both of these equations

By definition,

                      

By multiplying the equation of S and e

This equation is valid even if both w and S are expressed as percentages.

For saturated condition,

S = 1.

We know that,

But

2) Relationships Involving Unit Weights, Grain Specific Gravity, Void Ratio and Degree of Saturation

But

                 

              w G = S e

This is a general equation from which the unit weights corresponding to the saturated and dry states of soil may be got by substituting S = 1 and S = 0 respectively.

The submerged unit weight γ′ may be written as

                   

On solving

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Composition of Soil – Three Phase Diagram

Soil is a complex physical system. A mass of soil includes accumulated solid particles or soil grains and the void spaces that exist between the particles. The void spaces may be partially or completely filled with water or some other liquid. Void spaces not occupied by water or any other liquid are filled with air or some other gas. ‘Phase’ means any homogeneous part of the system different from other parts of the system and separated from them by abrupt transition. In other words, each physically or chemically different, homogeneous and mechanically separable part of a system constitutes a distinct phase. A system consisting of more than one phase is said to be heterogeneous.

Since the volume occupied by a soil mass may generally be expected to include material in all the three states of matter - solid, liquid and gas. Soil is referred to as a “three-phase system”. A soil mass as it exists in nature is a more or less random accumulation of soil particles, water and air-filled spaces as shown in Fig. 1 (a). For purposes of analysis it is convenient to represent this soil mass by a block diagram, called ‘Phase-diagram’, as shown in Fig. 1 (b). It may be noted that the separation of solids from voids can only be imagined. The phase-diagram provides a convenient means of developing the weight-volume relationship for a soil.

Fig. 1 (a) Actual Soil Mass (b) Representation of Soil Mass by Phase Diagram

When the soil voids are completely filled with water, the gaseous phase being absent, it is said to be ‘fully saturated’ or merely ‘saturated’. When there is no water at all in the voids, the voids will be full of air, the liquid phase being absent; the soil is said to be dry. (It may be noted that the dry condition is rare in nature and may be achieved in the laboratory through oven drying). In both these cases, the soil system reduces to a ‘two-phase’ one as shown in Fig. 2 (a) and (b). These are merely special cases of the three-phase system.

Fig. 2 (a) Saturated Soil (b) Dry Soil Represented as Two-Phase System

Basic Terminology – Weight Volume Relationship

The general three-phase diagram for soil will help in understanding the terminology and also in the development of more useful relationships between the various quantities. Conventionally, the volumes of the phases are represented on the left side of the phase diagram, while weights are represented on the right side as shown in Fig. 3.

Fig. 3 Soil-Phase Diagram

Va = Volume of air                                Wa = Weight of air (negligible or zero)

Vw = Volume of water                        Ww = Weight of water

Vv = Volume of voids                           Wv = Weight of material occupying void space

Vs = Volume of solids                           Ws = Weight of solids

V = Total volume of soil mass            W = Total weight of solid mass

        Wv = Ww

1) Porosity (n)

Porosity of a soil mass is the ratio of the volume of voids to the total volume of the soil mass. It is denoted by the letter symbol ‘n’ and is commonly expressed as a percentage. Porosity is also known as percentage voids. 

Here       Vv = Va + Vw

               V = Va + Vw + Vs

2) Void Ratio (e)

Void ratio of a soil mass is defined as the ratio of the volume of voids to the volume of solids in the soil mass. It is denoted by the letter symbol ‘e’ and is generally expressed as a decimal fraction.

Both porosity and void ratio are measures of the denseness (or looseness) of soils. As the soil becomes more and more dense, their values decrease. The term porosity is more commonly used in other disciplines such as agricultural engineering. In soil engineering, the term void ratio is more popular. It is more convenient to use void ratio than porosity. When the volume of a soil mass changes, only the numerator (i.e. Vv) in the void ratio changes and the denominator (i.e. Vs) remains constant. However, if the term porosity is used, both the numeration and the denominator change and it will become inconvenient.

3) Degree of Saturation (S)

Degree of saturation of a soil mass is defined as the ratio of the volume of water in the voids to the volume of voids. It is designated by the letter symbol ‘S’ and is commonly expressed as a percentage.

For a fully saturated soil mass, Vw = Vv

Therefore, for a saturated soil mass S = 100%.

For a dry soil mass, Vw is zero.

Therefore, for a perfectly dry soil sample S is zero.

In both these conditions, the soil is considered to be a two-phase system. The degree of saturation is between zero and 100%, the soil mass being said to be ‘partially saturated’ and is the most common condition in nature.

4) Percent Air Voids (na)

Percent air voids of a soil mass is defined as the ratio of the volume of air voids to the total volume of the soil mass. It is denoted by the letter symbol ‘na’ and is commonly expressed in percentage.

5) Air Content (ac)

Air content of a soil mass is defined as the ratio of the volume of air voids to the total volume of voids. It is designated by the letter symbol ‘ac’ and is commonly expressed as a percentage.

6) Water Content/Moisture Content (w)

Water content or Moisture content of a soil mass is defined as the ratio of the weight of water to the weight of solids (dry weight) of the soil mass. It is denoted by the letter symbol 'w' and is commonly expressed as a percentage. 

7) Bulk Unit Weight/Mass Unit Weight (𝛾)

Bulk unit weight or Mass unit weight of a soil mass is defined as the weight per unit volume of the soil mass. It is denoted by the letter symbol 'γ'. Hence, 

Here,         W = Ww + Ws

         and    V = Va + Vw + Vs

The term ‘density’ is used for ‘unit weight’ in soil mechanics, although density means the mass per unit volume and not weight.

8) Unit Weight of Solids (γs)

Unit weight of solids is the weight of soil solids per unit volume of solids alone. It is also sometimes called the ‘absolute unit weight’ of a soil. It is denoted by the letter symbol 'γs'.

9) Unit Weight of Water (γw)

Unit weight of water is the weight per unit volume of water. It is denoted by the letter symbol 'γw'.

It should be noted that the unit weight of water varies in a small range with temperature. It has a convenient value at 4°C, which is the standard temperature for this purpose. γo is the symbol used to denote the unit weight of water at 4°C. The value of γo is 1g/cm³ or 1000 kg/m³ or 9.81 kN/m³.

10) Saturated Unit Weight (γsat)

The saturated unit weight is defined as the bulk unit weight of the soil mass in the saturated condition. This is denoted by the letter symbol γsat.

11) Submerged Unit Weight/Buoyant Unit Weight (γ′)

The submerged unit weight or buoyant unit weight of a soil is its unit weight in the submerged condition. In other words, it is the submerged weight of soil solids (Ws)sub per unit of total volume, V of the soil. It is denoted by the letter symbol γ′.

(Ws)sub is equal to the weight of solids in air minus the weight of water displaced by the solids. Hence

                                                (Ws)sub = Ws – (Vs . γw)

Since the soil is submerged, the voids must be full of water.

The total volume V must be equal to (Vs + Vw) . (Ws)sub may now be written as,

                         (Ws)sub = W – Ww – Vs . γw

                                           = W – Vw . γw – Vs . γw

                                           = W – γw (Vw + Vs)

                                           = W – V . γw

Dividing throughout by V, the total volume,

Or

γ′ = γsat – γw

It may be noted that a submerged soil is invariably saturated, while a saturated soil need not be submerged. This equation may be written as a direct consequence of Archimedes’ Principle which states that the apparent loss of weight of a substance when weighed in water is equal to the weight of water displaced by it. Thus,

γ ′ = γsat – γw

12) Dry Unit Weight (γd)

The dry unit weight is defined as the weight of soil solids per unit of total volume, the former is obtained by drying the soil, while the latter would be got prior to drying. The dry unit weight is denoted by the letter symbol 'γd' and is given by

13) Mass Specific Gravity (Gm)

The mass specific gravity of a soil may be defined as the ratio of mass or bulk unit weight of soil to the unit weight of water at the standard temperature (4°C). This is denoted by the letter symbol Gm and is given by

This is also referred to as ‘bulk specific gravity’ or ‘apparent specific gravity’.

14) Specific Gravity of Solids (G)

The specific gravity of soil solids is defined as the ratio of the unit weight of solids (absolute unit weight of soil) to the unit weight of water at the standard temperature (4°C). This is denoted by the letter symbol G and is given by

This is also known as ‘Absolute specific gravity’ and ‘Grain Specific Gravity’.

15) Specific Gravity of Water (Gw)

Specific gravity of water is defined as the ratio of the unit weight of water to the unit weight of water at the standard temperature (4°C). It is denoted by the letter symbol, Gw and is given by

Since the variation of the unit weight of water with temperature is small, this value is very nearly unity and in practice is taken as such.

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Structure of Soil

The structure of a soil may be defined as the manner of arrangement and state of aggregation of soil grains. In a broader sense, consideration of mineralogical composition, electrical properties, orientation and shape of soil grains, nature and properties of soil water and the interaction of soil water and soil grains, also may be included in the study of soil structure, which is typical for transported or sediments soil. The engineering behaviour of soil is influenced by soil structure to varying degrees. Structural composition of sediment soil influences, many of their important engineering properties such as permeability, compressibility and shear strength. Hence, a study of the structure of soil is important. The following types of structure are commonly considered.

1) Single Grained Structure

Single grained structure is characteristic of coarse grained soils, with a particle size greater than 0.02 mm. Gravitational forces predominate the surface forces and hence grain to grain contact results. The deposition may occur in a loose state, with large voids or in a sense state, with less of voids. When such soils settle out of suspension in water, the particles settle independently of each other. The major force causing their deposition is gravitational and the surface forces are too small to produce any effect. There will be particle-to-particle contact in the deposit. The void ratio attained depends on the relative size of grains.

Fig. 1 Single-grained Structure

2) Honeycomb Structure

This structure can occur only in fine-grained soils, especially in silt and rock flour. Due to the relatively smaller size of grains, besides gravitational forces, inter-particle surface forces also play an important role in the process of settling down. When particles approach the lower region of suspension they will be attracted by particles already deposited as well as the neighbouring particles leading to formation of arches. Miniature arches are formed, which bridge over relatively large void spaces. This results in the formation of a honey-comb structure, each cell of a honey-comb being made up of numerous individual soil grains. The structure has a large void space and may carry high loads without a significant volume change. The structure can be broken down by external disturbances.

Fig. 2 Honey-comb Structure

3) Flocculent Structure

This structure is characteristic of fine-grained soils such as clays. In the case of flocculated structure, there will be edge-to-edge and edge-to-face contact between particles. The concentration of dissolved minerals in water leads to formation of flocculated structure with very high void ratio as in the case of marine deposits. Mutual repulsion of the particles may be eliminated by means of an appropriate chemical; this will result in grains coming closer together to form a ‘floc’. Formation of flocs is ‘flocculation’. But the flocs tend to settle in a honeycomb structure, in which in place of each grain, a floc occurs. Thus, grains grouping around void spaces larger than the grain-size are flocs and flocs grouping around void spaces larger than even the flocs result in the formation of a ‘flocculent’ structure.

Fig. 3 Flocculent Structure

Very fine particles or particles of colloidal size (< 0.001 mm) may be in a flocculated or dispersed state. This type of structure is common in fresh water deposits. In the case of dispersed or oriented structure, the particles will have face to face contact. This type for formation is due to net electrical forces between adjacent soil particles at the time of deposition being repulsive in nature. The flaky particles are oriented edge-to-edge or edge-to-face with respect to one another in the case of a flocculated structure. Flaky particles of clay minerals tend to from a card house structure, when flocculated. When inter-particle repulsive forces are brought back into play either by remoulding or by the transportation process, a more parallel arrangement or reorientation of the particles occurs. This means more face-to-face contacts occur for the flaky particles when these are in a dispersed state. In practice, mixed structures occur, especially in typical marine soils.

Fig. 4 Card House Structure

Fig. 5 Dispersed Structure

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Survey Stations

Survey stations are important points fixed on ground indicating the starting point and the end point of the survey line. These are also the basic control points of the survey. There can be two types of survey stations.

1) Main Stations

Main stations are control points at the ends of the chain lines commanding the boundaries of survey and the lines joining the main stations are called the main survey line or the chain lines (A, B, C, D and E in Fig.1).

2) Subsidiary or Tie Stations

These are stations selected on the main survey lines for running auxiliary lines drawn to locate, measure and plot interior details such as fences, hedges, building, etc. (a and b in Fig.1).

Fig. 1 Layout of Chain Survey

The survey stations are suitably selected with care so that at least main survey stations are mutually visible and survey lines run through as flat ground as possible and are as close to the boundaries as possible. The main survey lines should form well-conditioned triangles. These should be as few as possible and suitably selected so as to avoid obstacles in chaining and ranging.

Survey Lines

The lines joining survey stations are the survey lines. The survey lines between main stations are thus called main survey lines or chain lines. The different survey lines are listed below.

1) Base Line

The longest of the main survey line is normally called base line running primarily through the middle of the area to be surveyed. The framework of triangles shall have one or two base lines since the entire survey is built around base line. It shall be measured with higher care and accuracy. 

2) Chain Line (Main Survey) Line 

The lines that join main stations are termed as chain line or main survey line.

3) Tie or Subsidiary Line

The survey line joining the subsidiary or tie stations on main line is termed tie line. It helps to check the accuracy of surveying and to locate the interior details. The position of each tie line should be close to some features such as paths, buildings, etc.

4) Check Line or Proof Line

A check-line also termed as a proof-line is a line joining the apex of a triangle to some fixed points on any two sides of a triangle. A check-line is measured to check the accuracy of the framework. The length of a checking line, as measured on the ground should agree with its length on the plan. It is preferable to have at least one check line in each triangle of the framework.

Offsets

The details on ground such as fences, buildings and towers, etc. are to be located with reference to main chain lines by means of lateral measurements. These lateral measurements with reference to the chain line are referred to as offsets. Offsets are classified based on its length and inclination to the survey line.

1) Classification Based on Length of Offset

a) Short Offset

Offset whose length is less than 15m is called short offset.

a) Long Offset

Offset whose length is greater than 15m is called long offset.

2) Classification of Offset Based on the Inclination to the Survey Line

a) Perpendicular Offset

Perpendicular offsets are the lateral distances taken at right angles (normal) to the chain line.

b) Oblique Offset

If the inclination of offset line to chain line is anything other than 90^o, the offsets are termed as oblique offsets.

Fig. 2 Offsets 

Factors Affecting Survey Station Selection 

• Stations should be visible from at least two or more stations. 
• As far as possible, main lines should run on level ground. 
• All triangle will be well conditioned triangle. 
• Each triangle should have at least one check line. 
• Survey lines should be as few as possible. 
• Obstacles to ranging and chaining should be avoided. 
• Sides of the larger triangles should pass as close to the boundary lines as possible. 
• Trespassing and frequent crossing of the roads should be avoided.

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Clay Mineralogy

A ‘mineral’ is an inorganic chemical compound formed in nature. As a solid, it may occur in an amorphous state or in a crystalline state. A ‘crystal’ is a homogenous body bounded by smooth plane surfaces. Soil particles are largely composed of mineral crystals. Molecules of minerals are composed of atoms of chemical elements. The atoms in a crystal are arranged in a definite orderly manner to form a three dimensional net-work, called a “lattice.” Earth is about 12,500 km in diameter and most geotechnical engineering work is confined to the top few hundred meters of the crust, which is comprised essentially of oxygen (49.2%), silicon (25.7%) and aluminum (7.5%) present in the form of oxides, with some Fe³⁺, Ca²⁺, Na⁺, K⁺, Mg²⁺, etc. The atomic structure of a clay mineral is made of one of the two structural units: tetrahedrons containing a silicon atom at the center surrounded by four oxygen atoms at the corners and octahedrons containing aluminum or magnesium ions at the center surrounded by six hydroxyl or oxygen ions at the corners.

Formation of Clay Minerals

A soil particle may be a mineral or a rock fragment. A mineral is a chemical compound formed in nature during a geological process, whereas a rock fragment has a combination of one or more minerals. Based on the nature of atoms, minerals are classified as silicates, aluminates, oxides, carbonates and phosphates.

Out of these, silicate minerals are the most important as they influence the properties of clay soils. Different arrangements of atoms in the silicate minerals give rise to different silicate structures. Basic structural units soil minerals are formed from two basic structural units: tetrahedral and octahedral. Considering the valencies of the atoms forming the units, it is clear that the units are not electrically neutral and as such do not exist as single units. The basic units combine to form sheets in which the oxygen or hydroxyl ions are shared among adjacent units. Three types of sheets are thus formed, namely silica sheet, gibbsite sheet and brucite sheet.

Isomorphous substitution is the replacement of the central atom of the tetrahedral or octahedral unit by another atom during the formation of the sheets. The sheets then combine to form various two-layer or three-layer sheet minerals. As the basic units of clay minerals are sheet-like structures, the particle formed from stacking of the basic units is also plate-like. As a result, the surface area per unit mass becomes very large.

Atomic and Molecular Bonds

Forces which bind atoms and molecules to build up the structure of substances are primarily of electrical nature. They may be broadly classified into “primary bonds” and “secondary bonds’’. Primary bonds combine the atoms into molecules. Secondary bonds link atoms in one molecular to atoms in another. They are much weaker than the primary bonds. Primary bonds are the ionic bond and the covalent bond. Secondary bonds are the hydrogen bond and the Van der Waals bond.

1) Ionic Bond

The ionic bond is the simplest and strongest of the bonds which hold atoms together. This bond is formed between oppositely charged ions by the exchange of electrons. Atoms held together by ionic bonds form “ionic compounds”’, e.g. common salt (sodium chloride) and a majority of clay mineral crystals fall into this group. Ionic bonding causes a separation between centres of positive and negative charge in a molecule, which tends the molecule to orient in an electric field forming a “dipole”. Dipole is the arrangement of two equal electro-static charges of opposite sign. A dipolar molecule is one which is neutral but in which the centres of positive and negative charges are separated such that the molecule behaves like a short bar magnet with positive and negative poles.

2) Covalent Bond

The covalent bond is formed when one or more bonding electrons are shared by two atoms so that they serve to complete the outer shell for each atom.

Fig. 1 Covalent Bonding

3) Hydrogen bond

A hydrogen bond is the attractive interaction of a hydrogen atom with an electronegative atom, such as nitrogen, oxygen or fluorine, that comes from another molecule. Thus when water molecules are close together, their positive and negative regions are attracted to the oppositely-charged regions of nearby molecules. The force of attraction, shown in Fig. 2 as dotted line, is called a hydrogen bond. Each water molecule is hydrogen bonded to four others. Hydrogen bond can link the oxygen from a water molecule to the oxygen on the clay particles surface. Hydrogen bonding between two oxygen atoms is responsible for some of the weaker bonds between crystal layers for holding water at the clay surface and for bonding organic molecules to the clay surface.

Fig. 2 Hydrogen Bonding

4) Van der Waals Bond

It is the sum of the attractive or repulsive forces between molecules (or between parts of the same molecule) other than those due to covalent bonds or ionic bond. The covalent bonds within the molecules are very strong and rupture only under extreme conditions. The bonds between the molecules that allow siding and rupture to occur are called Van der Waals forces.

When ionic and covalent bonds are present, there is some imbalance in the electrical charge of the molecule. The angle hydrogen atoms are bonded to oxygen atom in water produces a positive polarity at the hydrogen-rich end of the molecule and a negative polarity at the other end. As a result of this charge imbalance the water molecules are attracted to each other. This is the force that holds the molecules together in a drop of water, shown in Fig. 3. Heat can be used to break the Van der Waal forces between the molecules and change the form of the material from solid to liquid gas.

Fig. 3 Van der Waals Bonding

Basic Structural Units of Clay Minerals

The clay minerals are a group of complex alumino-silicates, i.e., oxides of aluminium and silicon with smaller amounts of metal ions substituted within the crystal. The atomic structures of clay minerals are built up of two basic units such as Silica tetrahedral units and Aluminium (or magnesium) octahedral unit. These units are held together by ionic bonds.

1) Silica Unit

The silica unit consists of a silicon ion surrounded by four oxygen ions arranged in the form of a tetrahedron. The basic units combine in such a manner as to form a sheet. In the silica sheet, the bases of the tetrahedrals are all in the same plane and the tips all point in the same direction. Each of the three oxygen at the base is shared by two silicon of adjacent units.

2) Aluminium (or Magnesium) Octahedral Unit

The octahedral unit has an aluminium ion or a magnesium ion endorsed by six hydroxyl radicals or oxygen arranged in the form of an octahedron. In some cases, other cations (e.g. Fe) are present in place of Al and Mg. Combination of octahedral units forms an Octahedral sheet, which is called a ‘gibbsite” sheet if the central action of the unit is aluminum or a “brucite” sheet if the central cation is magnesium.

Types of Clay Minerals

From an engineering point of view, three clay minerals of interest are kaolinite, montmorillonite and illite.

1) Kaolinite

This is the most common of the Kaolin group. Each structural unit of Kaolinite is a combination of two layers with a silica layer joined to one of a gibbsite layer. Successive layers of structural units are held together to form kaolite particles which occur as platelets joined by strong H-bond. Kaolinite is used for making paper, paint and in pharmaceutical industry.

Fig. 4 Kaolinite

2) Montmorillonite

The montmorillonite mineral is a stacking of basic sheet like structural units, with each unit made up of gibbsite sheet sandwiched between two silica sheets joined by weak Van der Waal’s bond. It is easily separated by water. Because of the fact that bonding by Van der Waals forces between silica sheet of adjacent structural units is weak and there is a net negative charge deficiency in octahedral sheet, water and exchangeable cations can enter and separate the layers. Thus soil containing montmoriillonite mineral exhibits high swelling and shrinkage characteristics.

Fig. 5 Montmorillonite

3) Illite

The basic structural unit of illite is the same as that of montmorillonite except for the fact that there is some substitution of aluminium for silicon in the silica sheet and the resultant charge deficiency is balanced by potassium ions, which bond the layers in the stack. There is about 20% replacement of aluminium with silicon in the gibbsite sheet due to isomorphous substitution. The bond with the non-exchangeable K⁺ ions are weaker than the hydrogen bond in the Kaolite but is stronger than the water bond of montmorillonite. The illite crystal does not swell so much in the presence of water as does in montmorillonite particles.

Fig. 5 Illite

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Corrections to Linear Measurements

The following corrections are to be applied to the linear measurements with a chain or a tape where such accuracy is required.

1) Correction for Standard Length of Tape

Before using a tape, its actual length is ascertained by comparing it with a standard tape of known length. The designated nominal length of a tape is its designated length e.g. 30m or 100m. The absolute length of a tape is its actual length under specified conditions. Incorrect length of a tape can be one of the most important errors. It is systematic. An error due to incorrect length of a tape occurs each time the tape is used. If the true length, known by standardization, is not exactly equal to its nominal value of 100.00 m recorded for every full length, the correction can be determined and applied from the formula given below.

Where,

        l - Actual tape length at the field

        l’ - Nominal tape length

        L - Measured length of the line

Sometimes, the changes in length are quite small and of little importance in many types of surveys. The actual length of a working tape must be compared with a standard tape periodically. When its actual length is known, the tape is said to be standardized. A correction must be added or subtracted to a measured distance whenever its standardized length differs from its nominal or graduated length. In measuring unknown distances with a tape that is too long, a correction must be added. Conversely, if the tape is too short, the correction will be minus, resulting in decrease.

2) Correction for Temperature

This correction is necessary because the length of the tape or chain may be increased or decreased due to rise or fall of temperature during measurement. Steel tapes are standardized for 68⁰F or 20⁰C. A temperature higher than or lower than this value causes a change in length that must be considered. The coefficient of thermal expansion and contraction of steel used in ordinary tapes is approximately 1.16 x 10^-5 per length per ⁰C. For any tapes the correction for temperature can be computed and applied using the formula.

Ct = 𝛼 (Tm – T0) L

Corrected Length = L + Ct

Where,

     Ct - Correction in length of a line due to nonstandard temperature

     𝛼− Coefficient of thermal expansion and correction of the tape

    Tm - Tape temperature at the time of measurement

    To - Tape temperature when it has standard length

     L - Measured length of the line

3) Correction for Pull / Tension

During measurement the applied pull may be either more or less than the pull at which the chain or tape was standardized. Due to the elastic property of materials the strain will vary according to the variation of applied pull and hence necessary correction should be applied. When a steel tape is pulled with a tension greater than its standard, the tape will stretch and be no longer than its standard length. Conversely, if less than standard pull is used, the tape will be shorter than its standard length. The modulus of elasticity of the tape regulates the amount that it stretches. Correction pull can be computed and applied using the following formula.

Corrected Length = L + Cp

Where,

        Cp - Total elongation in tape length due to the pull, in meter.

        P1- Pull applied to the tape, in Kg.

        P - Standard pull for the tape, in Kg.

        A - Cross sectional area of the tape.

        E - Modulus of elasticity of the steel.

        L - Measured length of the line in meter.

4) Correction for Sag

In case of suspended measurement across a span L the chain or tape sag to take the form of curve known as catenary. Sag shortens the horizontal distance between end graduations, because the tape length remains the same. Sag can be diminished but not eliminated unless the tape is supported throughout. The following formulas are used to compute the sag correction.

Where,

        Cs - Correction for sag in meter.

        Ls - Unsupported length of the tape in meter.

        w - Weight of the tape per meter of length.

        W - Total weight of the tape between the supports in Kg.

        P1 - Pull on the tape in Kg.

In measuring lines of unknown length, the sag correction is always negative. After a line has been measured in several segments and a sag correction has been calculated for each segment, the corrected length is given by

Corrected Length = L + ΣCs

Where,

        L - Recorded length of the line

        ΣCs - Sum of individual sag corrections.

5) Normal Tension

By equating equations Cs = Cp,

i.e. the elongation due to increase in tension is made equal to the shortening due to sag; thus, the effect of the sag can be eliminated. The pull that will produce this condition, called Normal Tension Pn is given by the formula.

Where,

        Pn - Normal tension

        P - Standard pull for the tape, Kg

       W - Total weight of the tape between the support, Kg

        A - Cross sectional area of tape

        E - Modulus of elasticity of steel

6) Correction for Alignment

Generally, a survey line is set out in a continuous straight line. Sometimes, it becomes necessary due to obstruction to follow a bent line which may be composed of two or more straight portions subtending an angle other than 180º as shown in Fig. 1.

Fig.1 Correction for Alignment

Let        AC = L1, CB= L2

            < BAC = 𝜃1, BAC = 𝜃2

                                               Length AB = L1 cos 𝜃1 + L2 cos 𝜃2

                   The required correction = (L1+ L2) - (L1 cos 𝜃1 + L2 cos 𝜃2)

7) Correction for Slope

The distance measured along the slope between two stations is always greater than the horizontal distance between them. The difference in slope distance and horizontal distance is known as slope correction which is always subtractive.

Fig. 2 Slope Correction

Let,

      L - Slope distance AB

     D - Horizontal distance AC

     H - Difference in reduced levels of A and B

Question 1

The length of a survey line measured with a 30m chain was found to be 631.5m. When the chain was compared with a standard chain, it was found to be 0.1m too long. Find the true length of the survey line.

Solution

                   

                           L’ = 30.1m. L = 30m

                Measured length of the survey line = 631.5m

               Thus, true length of the survey line = 30.130 x 631.5

                                                                                         = 633.603 m.

Question 2

A 20m chain was found to be 4 cm too long after chaining 1400m. It was 8 cm too long at the end of day’s work after chaining a total distance of 2420m. If the chain was correct before commencement of the work, find the true distance.

Solution

The correct length of the chain at commencement = 20m

The length of the chain after chaining 1400m = 20.04 m

The mean length of the chain while measuring = (20+20.04)/2 

                                                                                                  = 20.02m

The true distance for the wrong chainage of 1400m = (20.02/20)x1400 

                                                                                                           = 1401.4 m

The remaining distance = 2420-1400 

                                                  = 1020m

The mean length of chain while measuring the remaining distance = (20.08+20.04)/2 

                                                                                                                                           = 20.06m

The true length of remaining 1020m = (20.06/20) x 1020 

                                                                            =1023.06m

Hence, the total true distance = 1401.4 + 1023. 06 

                                                              = 2424.46 m

Question 3

A line was measured with a steel tape which was exactly 30 meters at 20℃ at a pull of 100N, the measured length being 1650.00 meters. The temperature during measurement was 30°C and the pull applied was 150N. Find the length of the line, if the cross-sectional area of the tape was 0.025 sq.cm. The co-efficient of expansion of the material of the tape per 1 ºC = 3.5x10^-6  and the modulus of elasticity of the material of the tape = 2.1x10⁵ N/mm².

Solution

i) Correction of temperature per tape length

                            Ct = 𝛼 (Tm – T0) L

                                 = 0.0000035 (30 – 20) x 30

                                 = 0.00105m (+ve)

ii) Correction for pull per tape length

                           CP = ((P-P0) x L)/(A E)

                                 = ((150-100) x 30)/(2.5x2.1x105)

                                 = 0.00286m (+ve)

Combined correction = 0.00105+0.00286

                                             = 0.00391m

True length of the tape = 30+0.0039

                                                 =30.0039m

True length of the line = (30.0039x1650.00)/30

                                               =1650.21m

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