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Sunday, September 10, 2023 ,

Rounding off a number is necessary so that the accuracy of the result will be the same as that of the problem data. As a general rule, any numerical figure ending in a number greater than five is rounded up and a number less than five is not rounded up. The rules for rounding off numbers are best illustrated by examples. 

Suppose the number 3.5587 is to be rounded off to three significant figures. Because the fourth digit (8) is greater than 5, the third number is rounded up to 3.56. Likewise 0.5896 becomes 0.590 and 9.3866 becomes 9.39. If we round off 1.341 to three significant figures, because the fourth digit (1) is less than 5, then we get 1.34. Likewise 0.3762 becomes 0.376 and 9.871 becomes 9.87. 

There is a special case for any number that ends in a 5. As a general rule, if the digit preceding the 5 is an even number, then this digit is not rounded up. If the digit preceding the 5 is an odd number, then it is rounded up. For example, 75.25 rounded off to three significant digits becomes 75.2, 0.1275 becomes 0.128, and 0.2555 becomes 0.256.

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Sunday, September 10, 2023 , ,

William Navidi ... 960 pages - Language: ‎ English - Publisher: ‎ McGraw-Hill; 6th Edition (January, 2023).

Statistics for Engineers and Scientists stands out for its clear presentation of applied statistics. The book takes a practical approach to methods of statistical modeling and data analysis that are most often used in scientific work. This edition features a unique approach highlighted by an engaging writing style that explains difficult concepts clearly, along with the use of contemporary real world data sets, to help motivate students and show direct connections to industry and research. While focusing on practical applications of statistics, the text makes extensive use of examples to motivate fundamental concepts and to develop intuition.

Table of Contents: Chapter 1: Sampling and Descriptive Statistics - Chapter 2: Probability - Chapter 3: Propagation of Error - Chapter 4: Commonly Used Distributions - Chapter 5: Confidence Intervals - Chapter 6: Hypothesis Testing - Chapter 7: Correlation and Simple Linear Regression - Chapter 8: Multiple Regression - Chapter 9: Factorial Experiments - Chapter 10: Statistical Quality Control.

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Sunday, September 10, 2023
The four basic quantities -length, time, mass, and force- are not all independent from one another; in fact, they are related by Newton’s second law of motion, F = ma. Because of this, the units used to measure these quantities cannot all be selected arbitrarily. The equality F = ma is maintained only if three of the four units, called base units, are defined and the fourth unit is then derived from the equation.

The International System of units, abbreviated SI after the French Système International d’Unités, is a modern version of the metric system which has received worldwide recognition. As shown in Table 1, the system defines length in meters (m), time in seconds (s), and mass in kilograms (kg). The unit of force, called a newton (N), is derived from F = ma. Thus, 1 newton is equal to a force required to give 1 kilogram of mass an acceleration of 1 m/s2 (N = kg.m/s2). Think of this force as the weight of a small apple.

If the weight of a body located at the “standard location” is to be determined in newtons, then W = mg must be applied. Here measurements give g = 9.80665 m/s2; however, for calculations, the value g = 9.81 m/s2 will be used. Therefore, a body of mass 1 kg has a weight of 9.81 N, a 2-kg body weighs 19.62 N, and so on, Fig. 1.

Table 1. International System of Units
Prefixes: When a numerical quantity is either very large or very small, the SI units used to define its size may be modified by using a prefix. Some of these prefixes used are shown in Table 2. Each represents a multiple or submultiple of a unit which, if applied successively, moves the decimal point of a numerical quantity to every third place. For example, 4 000 000 N = 4 000 kN (kilo-newton) = 4 MN (mega-newton), or 0.005 m = 5 mm (milli-meter). Notice that the SI system does not include the multiple deca (10) or the submultiple centi (0.01), which form part of the metric system. Except for some volume and area measurements, the use of these prefixes is generally avoided in science and engineering.

Table 2. Prefixies
Rules for Use: Here are a few of the important rules that describe the proper use of the various SI symbols:

• Quantities defined by several units which are multiples of one another are separated by a dot to avoid confusion with prefix notation, as indicated by N = kg.m/s2 = kg.m.s-2. Also, m.s (meter-second), whereas ms (milli-second).

• The exponential power on a unit having a prefix refers to both the unit and its prefix. For example, mN2= (mN)2 = mN.mN. Likewise, mm2 represents (mm)2 = mm.mm.

• With the exception of the base unit the kilogram, in general avoid the use of a prefix in the denominator of composite units. For example, do not write N/mm, but rather kN/m; also, m/mg should be written as Mm/kg.

• When performing calculations, represent the numbers in terms of their base or derived units by converting all prefixes to powers of 10. The final result should then be expressed using a single prefix.

Also, after calculation, it is best to keep numerical values between 0.1 and 1000; otherwise, a suitable prefix should be chosen. For example, 

(50 kN).(60 nm) = [50.(103) N] [60.(10-9) m] = 3000.(10-6) N.m = 3.(10-3) N.m = 3 mN.m

Notes:
•Historically, the meter was defined as 1/10,000,000 the distance from the Equator to the North Pole, and the kilogram is 1/1000 of a cubic meter of water.
•The kilogram is the only base unit that is defined with a prefix.
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Friday, September 08, 2023 ,
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.

Figure 1. Soil as a three phase system
‘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. Literally speaking, phase simply means appearance and is derived from Greek. 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, in general, referred to as a “three-phase system”. (Fig. 1)

Figure 2. (a) Saturated soil (b) Dry soil represented as two-phase systems
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. 2 (a). For purposes of analysis it is convenient to represent this soil mass by a block diagram, called ‘Phase-diagram’, as shown in Fig. 2 (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.

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.
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Friday, September 08, 2023 ,
In this website, SI units have been used throughout. However, prior to this, metric units were in use, not only in India, but also in several parts of the world. Most of the equipment in these laboratories have the use of metric units. These equipment use grams and kilograms. Hence, mass units of grams and kilograms (formerly called ‘weight’) will continue to be used. Also, the term ‘mass’ and ‘weight’ are commonly used interchangeably. This may cause some problem with ‘unit weight’ terms. In this text, ‘mass’ will be represented in terms of grams and kilograms while ‘weight’ will be represented in terms of Newtons and kilo Newtons. The term density is defined as mass per unit volume; hence it will be represented in terms of g/cm3 or kg/m3 units. On the contrary, the term ‘unit weight’ is the weight per unit volume. Hence, it will be represented in terms of kN/m3.

The value of gravitational constant g will be taken as 981 cm/s2 (9.81 m/s2), for computational purposes.

Thus, in order to convert the density (expressed in terms of g/cm3) into unit weight, multiply the former by 9.81. For example, if a soil mass has a mass of 216 g and volume of 120 cm3.
  • Density = (Mass/Vol.) = 216/120 = 1.8 g/cm3
  • Unit Weight = (Weight/Vol.) = (216/120) x 9.81 = 17.66 kN/m3
Pressure, which is defined to be a force per unit area, will be expressed in terms of kN/m2 or in term of kilopascals (kPa).
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Friday, September 08, 2023 ,
The field of soil mechanics is very vast. The civil engineer has many diverse and important encounters with soil. Apart from the testing and classification of various types of soils in order to determine its physical properties, the knowledge of soil mechanics is particularly helpful in the following problems in civil engineering.

1. Foundation design and construction: Foundation is an important element of all civil engineering structures. Every structure — building, bridge, highway, tunnel, canal or dam — is founded in or on the surface of the earth. It is, therefore, necessary to know the bearing capacity of the soil, the pattern of stress distribution in the soil beneath the loaded area, the probable settlement ofthe foundation, effect of groundwater and the effect of vibrations, etc. The suitability of various types of foundations — i.e., spread foundation, pile foundation, well foundation, etc. — depend upon the type of soil strata, the magnitude of loads and groundwater conditions. A knowledge of shrinkage and swelling characteristics of soil beneath the foundation is also very essential.

2. Pavement design: A pavement can either be flexible or rigid, and its performance depends upon the subsoil on which it rests. The thickness of a pavement and its component parts, depends upon some certain characteristics of the subsoil, which should be determined before the design is made. On busy pavements, where the intensity of traffic is very high, the effect of repetition of loading and the consequent fatigue failure has to be taken into account. Apart from these, other problems of pavement design are : frost, heave and thaw with their associated problems of frost damage to pavements ; frost penetration depth ; remedial measures to prevent frost damage ; problems of ‘pumping’ of clay subsoils and suitability of a soil as a construction material for building highways or railways, earth fills or cuts, etc. A knowledge of the techniques for the improvement of the soil properties such as strength and stability is very much helpful in constructing pavements on poor soils by stabilising them.

3. Design of underground structures and earth retaining structures: The design and construction of underground (subterranean) and earth retaining structures constitute an important phase of engineering. The examples of underground structures include tunnels, underground buildings, drainage structures and pipelines. The examples of earth retaining structures are : gravity retaining wall, anchored bulk heads and cofferdams. A knowledge of soil structure interaction is essential to design properly such structures subjected to soil loadings.

4. Design of embankments and excavations: When the surface of the soil structure is not horizontal, the component of gravity tends to move the soil downward, and may disturb the stability of the earth structure. A thorough knowledge of shear-strength and related properties of soil is essential to design the slope and height (or depth) of the embankment (or excavation). The possibility of the seeping groundwater reducing the soil strength while excavating must also be taken into account. It may sometimes be essential to drain the subsoil water, to increase the soil strength and to reduce the seepage forces. Deep excavations require lateral braces and sheet walls to prevent caving in.

5. Design of earth dams: The construction of an earth dam requires a very thorough knowledge of whole of the Soil Mechanics. Since soil is used as the only construction material in an earth dam, which may either be homogeneous or of composite section, its design involves the determination of the following physical properties of soil : index properties such as density, plasticity characteristics and specific gravity, particle size distribution and gradation of the soil; permeability, consolidation and compaction characteristics, and shear strength parameters under various drainage conditions. Since huge earth mass is involved in its construction, suitable soil survey to the nearby area may be essential for the borrow-pit area. The determination of the optimum water content at which maximum density will be obtained on compaction, is probably the most essential aspect of the design. Apart from the seepage, characteristics of the dam section must be thoroughly investigated since these have the greatest impact on the stability of the slopes as well as the foundations of the dam. The consolidation characteristics help in predicting the long range behaviour of the dam toward settlement and the consequent reduction in the pore pressure. Lastly, the possible effect of vibrations during an earthquake should also be taken into account while designing. The performance of the soil in the designs cited above depends upon the characteristics of soil. Therefore, the testing of soil with relation to the determination of its physical properties, and the evaluation of effects of certain other factors such as seepage conditions, etc. forms the most essential part of the development of soil engineering. It is through research only that design and construction methods are modified to give maximum safety and/or economy, and new methods are evolved. The knowledge of theoretical soil mechanics, assuming the soil to be an ideal elastic isotropic and homogeneous material, helps in predicting the behaviour of the soil in the field.
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Friday, September 08, 2023 ,
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 soils. Structural composition of sedimented soils influences, many of their important engineering properties such as permeability, compressibility and shear strength. Hence, a study of the structure of soils is important.

The following types of structure are commonly studied: (a) Single-grained structure (b) Honey-comb structure (c) Flocculent structure.

Figure 1. Single-grained structure
Single-grained Structure: Single-grained structure is characteristic of coarsegrained 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. (Fig. 1)

Figure 2. Honey-comb structure
Honey-comb 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. 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)
 
Figure 3. Flocculent structure
Flocculent Structure: This structure is characteristic of fine-grained soils such as clays. Inter-particle forces play a predominant role in the deposition. 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)

Figure 4. Card-house structure of flaky particles
Very fine particles or particles of colloidal size (< 0.001 mm) may be in a flocculated or dispersed state. 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 (Reference: Lambe, 1953), when flocculated. This is shown in Fig. 4.

Figure 5. Dispersed structure
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, as shown in Fig. 5. 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.

The term ‘Texture’ refers to the appearance of the surface of a material, such as a fabric. It is used in a similar sense with regard to soils. Texture of a soil is reflected largely by the particle size, shape, and gradation. The concept of texture of a soil has found some use in the classification of soils to be dealt with later.
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Friday, September 08, 2023 ,
Soil is formed by the process of ‘Weathering’ of rocks, that is, disintegration and decomposition of rocks and minerals at or near the earth’s surface through the actions of natural or mechanical and chemical agents into smaller and smaller grains.

The factors of weathering may be atmospheric, such as changes in temperature and pressure; erosion and transportation by wind, water and glaciers; chemical action such as crystal growth, oxidation, hydration, carbonation and leaching by water, especially rainwater, with time.

Obviously, soils formed by mechanical weathering (that is, disintegration of rocks by the action of wind, water and glaciers) bear a similarity in certain properties to the minerals in the parent rock, since chemical changes which could destroy their identity do not take place.

It is to be noted that 95% of the earth’s crust consists of igneous rocks, and only the remaining 5% consists of sedimentary and metamorphic rocks. However, sedimentary rocks are present on 80% of the earth’s surface area. Feldspars are the minerals abundantly present (60%) in igneous rocks. Amphiboles and pyroxenes, quartz and micas come next in that order.

Rocks are altered more by the process of chemical weathering than by mechanical weathering. In chemical weathering some minerals disappear partially or fully, and new compounds are formed. The intensity of weathering depends upon the presence of water and temperature and the dissolved materials in water. Carbonic acid and oxygen are the most effective dissolved materials found in water which cause the weathering of rocks. Chemical weathering has the maximum intensity in humid and tropical climates.

‘Leaching’ is the process whereby water-soluble parts in the soil such as Calcium Carbonate, are dissolved and washed out from the soil by rainfall or percolating subsurface water. ‘Laterite’ soil, in which certain areas of Kerala abound, is formed by leaching.

Harder minerals will be more resistant to weathering action, for example, Quartz present in igneous rocks. But, prolonged chemical action may affect even such relatively stable minerals, resulting in the formation of secondary products of weatheing, such as clay minerals— illite, kaolinite and montmorillonite. ‘Clay Mineralogy’ has grown into a very complicated and broad subject  (Reference: ‘Clay Mineralogy’ by R.E. Grim).
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Tuesday, September 05, 2023 ,
Soils which are formed by weathering of rocks may remain in position at the place of region. In that case these are ‘Residual Soils’. These may get transported from the place of origin by various agencies such as wind, water, ice, gravity, etc. In this case these are termed ‘‘Transported soil’’. Residual soils differ very much from transported soils in their characteristics and engineering behaviour. The degree of disintegration may vary greatly throughout a residual soil mass and hence, only a gradual transition into rock is to be expected. An important characteristic of these soils is that the sizes of grains are not definite because of the partially disintegrated condition. The grains may break into smaller grains with the application of a little pressure.

The residual soil profile may be divided into three zones:

(i) the upper zone in which there is a high degree of weathering and removal of material; 
(ii) the intermediate zone in which there is some degree of weathering in the top portion and some deposition in the bottom portion;
(iii) the partially weathered zone where there is the transition from the weathered material to the unweathered parent rock. Residual soils tend to be more abundant in humid and warm zones where conditions are favourable to chemical weathering of rocks and have sufficient vegetation to keep the products of weathering from being easily transported as sediments. Residual soils have not received much attention from geotechnical engineers because these are located primarily in undeveloped areas. In some zones, sedimentary soil deposits range from 8 to 15 m in thickness.

Transported soils may also be referred to as ‘Sedimentary’ soils since the sediments, formed by weathering of rocks, will be transported by agencies such as wind and water to places far away from the place of origin and get deposited when favourable conditions like a decrease of velocity occur. A high degree of alteration of particle shape, size, and texture as also sorting of the grains occurs during transportation and deposition. A large range of grain sizes and a high degree of smoothness and fineness of individual grains are the typical characteristics of such soils.

Transported soils may be further subdivided, depending upon the transporting agency and the place of deposition, as under: 
  • Alluvial soils: Soils transported by rivers and streams: Sedimentary clays. 
  • Aeoline soils: Soils transported by wind: loess.
  • Glacial soils: Soils transported by glaciers: Glacial till.
  • Lacustrine soils: Soils deposited in lake beds: Lacustrine silts and lacustrine clays.
  • Marine soils: Soils deposited in sea beds: Marine silts and marine clays.
Broad classification of soils may be: 
  • Coarse-grained soils, with average grain-size greater than 0.075 mm, e.g., gravels and sands. 
  • Fine-grained soils, with average grain-size less than 0.075 mm, e.g., silts and clays.
These exhibit different properties and behaviour but certain general conclusions are possible even with this categorisation. For example, fine-grained soils exhibit the property of ‘cohesion’—bonding caused by inter-molecular attraction while coarse-grained soils do not; thus, the former may be said to be cohesive and the latter non-cohesive or cohesionless.
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