A SEMINAR REPORT ON STONES

Report submitted to:

Department of CIVIL ENGINEERING, GURU RAMDAS KHALSA

INSTITUTE OF SCIENCE & TECHNOLOGY

JABALPUR(M.P.)

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Stones---------

SUBMITTED BY:-

ISHTDEEP SINGH HORA

0202CS131006

SESSION

2014-2015

GURU RAMDAS KHALSA INSTITUTE OF SCIENCE & TECHNOLOGY

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Acknowledgement

Apart from the efforts of me, the success of any seminar report depends largely on the

encouragement and guidelines of many others. Firstly my special thanks to Mrs. Satish Soni

Sir(Head of Department for Civil Engg.) I take this opportunity to express my gratitude to the

people who have been instrumental in the successful completion of this report.

I would like to show my greatest appreciation to Mrs. Garima Mishra (Asst. Prof.) I can’t say

thank you enough for his tremendous support and help. I feel motivated and encouraged every

time I attend his meeting. Without his encouragement and guidance this report would not have

materialized.

The guidance and support received from all the members who contributed and who are

contributing to this report, was vital for the success of the report. I am grateful for their

constant support and help.

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Contents

Introduction

Stone is the one of the major building materials. It is a versatile material and hence it can be

used from the foundation to the parapet in a building and hence the scope comprises the study

of use of different stones at these places.

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S.No. Topic discussed Pages(from – to)

1. Introduction to Stones/Rocks 4

2. India’s Glorious History 5

3. Geological Classification 5-9

4. Different types of Stones 9-16

5. Physical Properties of Stones 16-18

6. Conditions effecting disintegration 19-20

7. Conclusion 21

8. References 22

Introduction to Rocks

Rock (mineral), naturally occurring solid material consisting of one or more minerals.

Minerals are solid, naturally occurring chemical elements or compounds that are

homogenous, meaning they have a definite chemical composition and a very regular

arrangement of atoms. Rocks are everywhere, in the ground, forming mountains, and at

the bottom of the oceans. Earth’s outer layer, or crust, is made mostly of rock. Some

common rocks include granite and basalt.

Natural stone is used in building as a facing, veneer, and decoration. The major factors

affecting the suitability and use of stone fall under two broad, but overlapping

categories: physical and structural properties and aesthetic qualities. The three factors of

building stone that most influence their selection by architects for aesthetic reasons are

pattern, texture, and colour. Consideration also should be given to costs, availability,

weathering characteristics, physical properties, and size and thickness limitations.

Stone patterns are highly varied, and they provide special features that make building

stone a unique material. Texture is varied, ranging from coarse fragments to fine grains

and crystalline structures. Texture also varies with the hardness of minerals composing

the stone.

Pattern, texture, and colour all are affected by how the stone is fabricated and finished.

Granites tend to hold their colour and pattern, while limestone colour and pattern

changes with exposure. Textures may range from rough and flamed finishes to honed or

polished surfaces. The harder the stone, the better it takes and holds a polish.

India’s Glorious Tradition

India's long history, dating back to 3200 B.C. has been influenced considerably by the

disposition, development and use of stones and other construction materials. Dimension

stones have also left deep imprints on the architectural heritage of the country.

Innumerable temples, forts and palaces of Ancient Indian Civilization have been carved

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out of locally available stones. The Taj Mahal at Agra stands testimony to the age

defying beauty of Indian marble. Some of the ancient rocks cut wonders are Khajuraho

Temple, Elephanta Caves, Konark Temple, etc. Besides, all major archaeological

excavations have revealed exquisitely carved statuettes and carvings in Stone. Ancient

Buddhist monuments like the Sanchi Stupa of 3rd century BC have also been carved out

of stone.

This tradition of Stone Architecture has continued to the present era with most of the

important modern buildings in India like the Presidential House, Parliament House and

Supreme Court made from high quality sandstone of Rajasthan. The Lotus Temple of

New Delhi stands testimony to the relevance of marble in modern Indian architecture.

Stones are still the mainstays of civil construction in India, with stones being used

extensively in public buildings, hotels, temples etc. It is increasingly being used in

homes, with the use of stones now penetrating amongst the burgeoning middle class of

India.

GEOLOGICAL CLASSIFICATION OF ROCKS

1. Igneous Rocks

2. Sedimentary Rocks

3. Metamorphic Rocks

Igneous Rocks

Igneous rocks are rocks formed from a molten or partly molten material called magma.

Magma forms deep underground when rock that was once solid melts. Overlying rock

presses down on the magma, and the less dense magma rises through cracks in the

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rock. As magma moves upward, it cools and solidifies. Magma that solidifies underground

usually cools slowly, allowing large crystals to form. Magma that reaches Earth’s surface is

called lava. Lava loses heat to the atmosphere or ocean very quickly and therefore

solidifies very rapidly, forming very small crystals or glass. When lava erupts at the

surface again and again, it can form mountains called volcanoes.

Igneous rocks commonly contain the minerals feldspar, quartz, mica, pyroxene,

amphibole, and olivine. Igneous rocks are named according to which minerals they contain.

Rocks rich in feldspar and quartz are called felsic; rocks rich in pyroxene, amphibole,

and olivine, which all contain magnesium and iron, are called mafic. Common and

important igneous rocks are granite, rhyolite, gabbro, and basalt. Granite and rhyolite are

felsic; gabbro and basalt are mafic. Granite has large crystals of quartz and feldspar. Rhyolite is

the small-grained equivalent of granite. Gabbro has large crystals of pyroxene and olivine.

Basalt is the most common volcanic rock.

SedimentaryRocks

The most common types of clastic rocks are sandstone and shale (also known as

mudrock). Sandstone is made from sand, and shale is made from mud. Sand particles

have diameters in the range 2.0 to 0.06 mm (0.08 to 0.002 in), while mud particles are

smaller than 0.06 mm (0.002 in). Sand and mud form when physical or chemical

processes break down and destroy existing rocks. The sand and mud are carried by

wind, rivers, ocean currents, and glaciers, which deposit the sediment when the wind or

water slows down or where the glacier ends. Sand usually forms dunes in deserts, or

sandbars, riverbeds, beaches, and near-shore marine deposits. Mud particles are smaller

than sand particles, so they tend to stay in the wind or water longer and are deposited

only in very still environments, such as lake beds and the ocean floor.

Sedimentary rock forms when layers of sand and mud accumulate. As the sediment

accumulates, the weight of the layers of sediment presses down and compacts the

layers underneath. The sediments become cemented together into a hard rock when

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minerals (most commonly quartz or calcite) precipitate, or harden, from water in the

spaces between grains of sediment, binding the grains together. Sediment is usually

deposited in layers, and compaction and cementation preserve these layers, called beds,

in the resulting sedimentary rock.

The most common types of chemical rocks are called evaporates because they form by

evaporation of seawater or lake water. The elements dissolved in the water crystallize

to form minerals such as gypsum and halite. Gypsum is used to manufacture plaster

and wallboard; halite is used as table salt.

The most common organic rock is limestone. Many marine animals, such as corals and

shellfish, have skeletons or shells made of calcium carbonate (CaCO3). When these

animals die, their skeletons sink to the seafloor and accumulate to form large beds of

calcium carbonate. As more and more layers form, their weight compresses and

cements the layers at the bottom, forming limestone. Details of the skeletons and shells

are often preserved in the limestone as fossils.

Coal is another common organic rock. Coal comes from the carbon compounds of plants

growing in swampy environments. Plant material falling into the muck at the bottom of

the swamp is protected from decay. Burial and compaction of the accumulating plant

material can produce coal, an important fuel in many parts of the world. Coal deposits

frequently contain plant fossils.

Metamorphic Rocks

Metamorphic rock forms when pre-existing rock undergoes mineralogical and structural

changes resulting from high temperatures and pressures. These changes occur in the

rock while it remains solid (without melting).

The changes can occur while the rock is still solid because each mineral is stable only

over a specific range of temperature and pressure. If a mineral is heated or compressed

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beyond its stability range, it breaks down and forms another mineral. For example,

quartz is stable at room temperature and at pressures up to 2 Gigapascals

(corresponding to the pressure found about 65 km [about 40 mi] underground). At

pressures above 2 Gigapascals, quartz breaks down and forms the mineral coesite, in

which the silicon and oxygen atoms are packed more closely together.

In the same way, combinations of minerals are stable over specific ranges of

temperature and pressure. At temperatures and pressures outside the specific ranges,

the minerals react to form different combinations of minerals. Such combinations of

minerals are called mineral assemblages.

In a metamorphic rock, one mineral assemblage changes to another when its atoms

move about in the solid state and recombine to form new minerals. This change from

one mineral assemblage to another is called metamorphism. As temperature and

pressure increase, the rock gains energy, which fuels the chemical reactions that cause

metamorphism. As temperature and pressure decrease, the rock cools; often, it does

not have enough energy to change back to a low-temperature and low-pressure mineral

assemblage. In a sense, the rock is stuck in a state that is characteristic of its earlier

high-temperature and high-pressure environment. Thus, metamorphic rocks carry with

them information about the history of temperatures and pressures to which they were

subjected.

The size, shape, and distribution of mineral grains in a rock are called the texture of

the rock. Many metamorphic rocks are named for their main texture. Textures give important

clues as to how the rock formed. As the pressure and temperature that form a

metamorphic rock increase, the size of the mineral grains usually increases. When the

pressure is equal in all directions, mineral grains form in random orientations and point

in all directions. When the pressure is stronger in one direction than another, minerals

tend to align themselves in particular directions. In particular, thin plate-shaped

minerals, such as mica, align perpendicular to the direction of maximum pressure, giving

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rise to a layering in the rock that is known as foliation. Compositional layering, or

bands of different minerals, can also occur and cause foliation. At low pressure, foliation

forms fine, thin layers, as in the rock slate. At medium pressure, foliation becomes coarser,

forming schist. At high pressure, foliation is very coarse, forming gneiss. Commonly, the

layering is folded in complex, wavy patterns from the pressure.

DIFFERENT TYPES OF STONES

Igneous Rocks

Granite -

Granite is an igneous rock, ordinarily composed of feldspar, mica, and silica or quartz. It

is formed by the cooling and crystallization of matter below the earth's surface under

conditions of heat and pressure which do not obtain in the case of lava ejected on the

surface in a molten state. It is found in the eastern part of the United States, in

Canada, in many sections of the Rocky Mountains and as a rule, wherever the later rock

formations have been worn away by the weather, and the igneous rock has been

exposed.

Planes of Fracture -

The structure of granite is quite uniform, but there are often planes of cleavage caused

by stresses produced while the molten material was cooling. The plane along which the

rock can be split most easily is known as rift; it is often nearly horizontal. Rock can

also be split along a plane, known as the grain, which is perpendicular to the rift, but

this cleavage is not so easy as that along the rift. Sometimes, the stresses are sufficient

to cause fractures, called joints, running parallel to the surface.

Qualities of Granite -

Granite is one of the most valuable stones for construction purposes. Although the

quality of granite varies according to the proportions of the constituents and to their

method of aggregation, this kind of stone is generally durable, strong, and hard. The

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hardest and most durable granites contain a greater proportion of quartz and a smaller

proportion of feldspar and mica. Feldspar makes granite more susceptible to

decomposition by the solution potash contained in it, potash feldspar being less durable

than lime or soda feldspar. Mica, being easily decomposed, is an element of weakness

in granite. An excess of lime or soda in the mica or feldspar hastens disintegration, as

does also an excess of iron. Therefore, stones showing large and dark iron stains should

be rejected for outside work. Fine-grained granite weathers better than does granite of

coarser grain.

Granite has a pearly lustre. The colour of common granite varies from white through

yellow to deep red, and the stone is generally classified as gray and red. Feldspar

renders the stone lighter in colour.

Because of its uniform structure, granite can be quarried in large blocks. The rift, the

grain, and the joint planes are advantageous in quarrying, as it is very difficult to cut

granite in other places. The uses for which granite is suitable depend on the texture of

the stone. Medium-grained stone is best fitted for building construction. Fine-grained stone

can be carved and polished, but, on account of its extreme hardness, it cannot be

worked readily. Such stone is, therefore, costly when it has to be cut, Coarse-grained

granite should be used only for concrete aggregate.

TrapRocks -

The term trap is generally applied to a large variety of dark-coloured, igneous,

unstratified rock~ that occur in large tabular masses rising one above another in

successive steps like stairs. These rocks consist chiefly of hornblende, lime, feldspar, and

augite, with some magnetic and titanic iron. The predominance of one or the other of

these minerals gives rise to many distinctive names, as greenstone, olivine, etc. The

colour varies, being dark gray, dark green, or nearly black, according to the proportions

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of the different constituents. The texture is usually so fine and close-grained that the

character of the structure cannot be determined by the naked eye.

Trap rocks are exceedingly dense, hard, and durable. However, they are not much used for

structural purposes because of their sombre and unattractive appearance, the great cost

of working, and the difficulty of securing large blocks on account of the numerous joint

planes. As they split and break easily, trap rocks are extensively used for paving blocks,

for the aggregate in making concrete, and for the construction of macadamized roads,

for which purpose their fine texture especially fits them. They are also used for railroad

ballast.

Sedimentary Rocks

Sandstone - 

Sandstone consists of fragments of other rocks cemented together. It is a stratified rock and

belongs to the later geological periods. Most of the grains are quartz, but often feldspar

is also present in sandstone. The cementing material may be silica, oxide of iron, clay,

or carbonate of lime.

If the cementing material is silica, the rock is very durable, but difficult to work. Iron

oxide is a good cementing material and gives the stone a reddish or brownish colour.

Clay is a satisfactory binder, but it readily absorbs water, which may cause destruction

of the stone by freezing. Lime renders the stone particularly liable to disintegration

when exposed to an atmosphere containing gases, or when used for foundations in a

soil that contains acid.

Sandstones - 

Are variable in character, some being nearly as valuable as granite and others being

practically useless for permanent construction. The best stone is characterized by small

grains with a small proportion of cementing material. When broken, it has a bright, clear,

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sharp fracture. It is usually found in thick beds and shows slight evidences of

stratification.

When quarried, sandstones are usually saturated with quarry water and are very soft;

but on exposure to the air, they dry out and become hard. Water can readily penetrate

between the layers of this stone; therefore, in foundations it should be laid on its

natural bed, that is, in the same position that it occupied in the quarry, so that the

penetration of moisture and possible disintegration by freezing may be prevented as

much as possible.

The colours of sandstone are white, cream, yellow, dark brown, blue, and red. Fine-

grained blue sandstone is known as bluestone. This variety is widely used for trimmings

and for stone sidewalks, as it readily splits into slabs.

Limestone -

All limestones are of sedimentary origin and have for their principal ingredient

carbonate of lime. The presence of other minerals gives rise to the division of the

limestones into five classes, each of which is designated by the name of the

predominating mineral. When clay is present, the stone is called argillaceous limestone;

when silica predominates, siliceous limestone; when iron is prevalent, ferruginous

limestone; when magnesia is present to the extent of 15 per cent, magnesium

limestone; and when the carbonate of lime and the carbonate of magnesia are

combined in equal proportions, dolomite limestone. Limestones are either granular or

compact.

Granular limestone -

Consists of grains of carbonate of lime, cemented together by some compound of lime,

silica, and alumina. The grains are generally sea shells or fragments of shells and are

often mixed with sand. This kind of stone is always porous. It is found in various colours,

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especially white and yellowish brown. In many cases, it is so soft when first quarried

that it can be cut with a knife; it hardens, however, on exposure to the air.

The variety of granular limestone called oolitic limestone is composed of egg-shaped

grains cemented together. It is one of the most important of the limestone group and

is extensively quarried and widely used for building purposes. Each grain is usually of

concentric structure, the carbonate of lime enclosing a particle of sand or of some

substance of either animal or vegetable origin.

Compact limestone - 

Consists of carbonate of lime, either pure or mixed with sand or clay. This kind of

limestone is generally devoid of crystalline structure, and has a dull, earthy appearance

and a dark-blue, gray, black, or mottled colour. In some cases, however, it is crystalline

and full of organic remains; it is then known as crystalline limestone.

The compact limestones are easily worked with the saw and hammer. They resemble

light granite in appearance, and are extensively used for building purposes. The variety

called shelly limestone, which consists of fossil shells that are cemented together, is

sufficiently hard to take a polish; it is much used for interior ornamentation. The

condition of the minerals combined with the lime also furnishes a basis for

distinguishing names. Thus, the stone is called hornstone when very fine grained silica is

present; cherty limestone, when the silica is in the form of rounded masses or nodules;

ironstone, when the amount of iron and clay is greater than the amount of lime;

rottenstone, when the ironstone is decomposed; and hydraulic limestone, when the rock

contains silica and clay in nearly equal proportions.

Shale - 

Shale is a typical clay rock that splits readily in lines parallel to the bedding. Sand and

lime carbonate are always present in this stone and, with increase of either, the rock

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grades into shaly sandstone or shaly limestone. Shale is used for light traffic roads and

in the manufacture of brick, tile, and other burned clay products, but it is not suitable

for concrete aggregate.

Conglomerate - 

Stratified rock composed of rounded pebbles of any material, such as limestone, quartz,

shale, granite grains, feldspar, etc., cemented together is known as conglomerate. When

the pebbles are quartz with siliceous binding the rock is strong and hard to quarry or

dress. When the interstices between the pebbles are not filled by the binder, the rock

is very porous, and may hold great amounts of ground water. This stone is seldom used in

building construction.

Metamorphic Rocks 

Marble -

Metamorphosed limestone gives the masonry material known as marble. It is easily dressed

to a smooth surface and polished, and is considered one of the most valuable building

materials. It resists frost and moisture well, but like all limestones it does not withstand

fire.

Marble can be obtained in many colours, some of which are white, gray, red, blue,

green, and black. One of the most important characteristics of marble is that it is easy

to carve; the finer the grains of the stone, the more suitable it is for this purpose. The

fine white-grained varieties that are especially prized for sculpture are called saccharoid

marbles.

Some of the finest varieties of white American marble are found at Lee, Massachusetts,

and in the vicinity of Rutland, Vermont. The dark-blue marble from the Vermont

quarries is very durable and has a close grain. A fine black marble is quarried at Glens

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Falls, New York. Coloured marbles, including gray, light and dark pink, buff, chocolate,

etc., are found in Tennessee, Georgia, and other states.

Slate - 

Slate is a laminated rock of great hardness and density. It splits readily along planes called

planes of slaty cleavage. This facility of cleavage is one of the most valuable

characteristics of slate, as masses can be split into slabs and plates of small thickness

and great area.

The most common colours of slat are dark blue, bluish black, purplish gray, bluish gray,

and green; occasionally, red and cream-colored slates are also found. Some slates are

marked with bands or patches whose colour is different from that of the rest of the

stone. These marks do not affect the durability of the slate, but they spoil its

appearance.

Although slate is not strictly a building stone, it is used extensively for covering steps

and the roofs of buildings, for wall linings, and for sanitary purposes. Slate is sometimes

used to make light traffic macadam, but, although it packs well, it ultimately yields

much mud and dust, which are objectionable.

Schist -

Schist has a more crystalline structure than slate, and the crystals are easily seen. It is

composed chiefly of minerals that cleave readily, such as hornblende, mica, etc., mixed

with a variable amount of granular quartz and feldspar. The presence of the cleavage

minerals produces a fine cleavage or foliation, called schistosity.

Schist is sometimes used in building construction but it disintegrates very rapidly and is

not durable. It should always be set with the planes of schistosity horizontal.

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Gneiss - 

Gneiss is a coarse-grained, laminated rock, formed by metamorphism of either

sedimentary or igneous rock. It is often used as structural material and as concrete

aggregate.

PHYSICAL PROPERTIES OF STONE

The physical characteristics of a particular stone must be suitable for its intended use.

It is important to determine the physical properties of the actual stone being used

rather than using values from a generic table, which can be very misleading.

Considerations of the physical properties of the stone being selected include modulus of

rupture, shear strength, coefficient of expansion, permanent irreversible growth and

change in shape, creep deflection, compressive strength, modulus of elasticity, moisture

resistance, and weatherability. Epoxy adhesives, often used with stone, are affected by

cleanliness of surfaces to be bonded and ambient temperature. Curing time increases

with cold temperatures and decreases with warmer temperatures.

Fabrication and Installation

With the introduction of new systems of fabrication and installation and recent

developments in the design and detailing of stone cutting, support, and anchorage, costs

are better controlled. Correct design of joints, selection of mortars, and use of sealants

affect the quality and durability of installation. Adequate design and detailing of the

anchorage of each piece of stone are required. The size and thickness of the stone

should be established based on physical properties of the stone, its method of

anchorage, and the loads it must resist. Appropriate safety factors should be developed

based on the variability of the stone properties as well as other considerations such as

imperfect workmanship, method of support and anchorage, and degree of exposure of

the cladding installation. Relieving angles for stone support and anchorage may be

necessary to preclude unacceptable loading of the stone. The stone should be protected

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from staining and breakage during shipment, delivery, and installation.

Since stone cladding design and detailing vary with type of stone and installation, the

designer should consult stone suppliers, stone-setting specialty contractors, industry

standards (such as ASTM), and other publications to help select and implement a stone

cladding system.

Physical Properties of Representative Stones

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Weight of different materials

 

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CONDITIONS AFFECTING DISINTEGRATION

Quarrying

Disintegration of stone is hastened or retarded by the methods employed in quarrying,

seasoning, finishing, and setting the stone.

The excessive use of explosives in quarrying shatters the cohesion of the particles

composing the stone and causes cracks and flaws that make the stone more permeable

to moisture. Small charges of powder, uniformly distributed over the area to be blasted,

have a lesser weakening effect on the stone. Stone cut out by quarrying machinery is

preferable to that blasted or wedged out, because the stone is not jarred and cracked

by this method and because denser faces are produced which render the stone less

permeable to moisture.

The position of the stone in the quarry also affects its durability. Stone taken from the

exposed faces and the top ledges of the quarry is likely to be less durable than

unexposed stone.

Seasoning

Before a stone is placed in a structure, the interstitial moisture, called quarry water or

sap, must be removed by evaporation. This process is termed seasoning, and should be

effected by exposing the stone to the drying action of the atmosphere for some

months; the stone should be stored under cover for protection against rain. If the stone

is not seasoned, the quarry water will be alternately frozen and thawed during a series

of years, and the stone will be broken up.

Finishing

The life of a stone is dependent on the style of finish given to its exposed faces. A

smooth or polished surface aids in prolonging the life by facilitating the rapid discharge

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of rainwater. The methods employed in dressing the stone also affect its life. Minute

fissures that render the stone more susceptible to atmospheric influences are produced

by impact; hence, stones sawed to the required dimensions are more durable than

those hammered and broken to size.

Setting

The position in which the stone is set in the structure affects its ability to resist

disintegration. When stratified stones are placed on edge, and the mortar joints are not

properly filled, water enters between the layers and in freezing causes the stone to

scale off; therefore, laminated stones should be set with their layers horizontal.

The portions of a structure most liable to early decay are those under cornices, belt

courses, window sills, etc., on which the rainwater slowly falls or drips. As a protection

from this source of decay, the under surface of a projecting stone should have a

narrow groove, called a drip, extending its whole length. The water that collects on the

upper surface of the projection flows over the upper edge and down the face to the

under side, where its further progress is interrupted by the drip; it then falls to the

ground.

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CONCLUSION

Stones are versatile material. In order to be able to decide what kind of stone to used

under given conditions, knowledge of the different kinds employed in the various types

of construction is essential. It is not necessary to determine the exact composition of a

stone to be used in a structure, but knowledge should be sufficient to help in selecting

or specifying the stone best adapted to the type of structure.

The properties of a stone that determine its fitness for construction purposes are

durability, strength, hardness, density, and appearance. The quality of a stone is

ascertained approximately from a study of its origin and chemical composition and from

the results of tests and experiments.

Inferences

Stones are used as versatile material irrespective of the properties of it, still its use remain as same, varying the techniques of implementation and limitations.

SELECTION OF BUILDING STONES For selection of the stone should be done on the basis of:

Importance of Preliminary Investigation.-

When an important masonry structure is to be built, it is essential to select a stone

that is strong and durable. Probably nothing in engineering construction is so neglected

as the preliminary inspection of building stone.

If it is necessary to employ great quantities of building stone at points where, the

stability of the structure depends on the strength of the stone, an inspection of the

quarry from which the stone is to be obtained should be made. The engineer should

also inspect some building or structure which contains the same material and has been

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standing for a long time. It is well, however, not to depend wholly on inspection either

at the quarry or at a building, but to subject the stone also to laboratory investigation.

References –

1. Whitney, William Dwight, and Benjamin E. Smith. The Century dictionary and

cyclopedia, vol 6. New York: Century Co., 1901. 5,221. Print.

2. C.M.Harris,Dictionary of Architecture & Construction

3. World Floor Covering Association

4. Ribenboim, p.180 says that "Despite the nil practical value of the formulas ... [they] may

have some relevance to logicians who wish to understand clearly how various parts of

arithmetic may be deduced from different axiomatzations ... "

5. Hardy & Wright, pp.344—345 "Any one of these formulas (or any similar one) would

attain a different status if the exact value of the number α ... could be expressed

independently of the primes. There seems no likelihood of this, but it cannot be ruled

out as entirely impossible."

6. Ramanujan, Question 723, Papers p. 332

7. Hardy & Wright, p. 337

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