Civil engineering - QUICK REVISION

Important Notes for Civil Engineering exam

As per IS 456:2000 Clause 17.4:

• The points from which cores are to be taken and the number of cores required shall be at the discretion of the engineer - in - charge and shall be representative of the whole of concrete concerned.

• In no case, however, shall fewer than three cores be tested.

• Cores shall be prepared and tested as described in IS 516.

• IS 456 : 2000 states that the concrete in the member represented by a core test shall be considered acceptable, if the average equivalent cube strength of core is equal to at least 85% of the cube strength of the grade of concrete specified, but no individual core has a strength less than 75%.

The effective length of groove weld:

As per IS: 800-2007, Clause 10.5.4.2 

• The effective length of butt weld or groove weld shall be taken as the length of the continuous full-size butt weld, but not less than four times the size of the weld or minimum of 40 mm.

• Maximum clear spacing between the effective length of the weld:

• For the welds in compression zone = Minimum of 12 × Thickness of the weld or 200 mm

• For the welds in tension zone = Minimum of 16 × Thickness of the weld or 200 mm

Classification of sections

• IS 800:2007, classified the cross sections into four classes depending upon the material yield strength and width to thickness ratio of the individual components within the cross section and loading arrangement.

The basic characteristics of these four classes are given below:

Plastic or Class I section:   

• 1. It has the capacity to develop plastic hinge and collapse mechanism.
• 2. They are fully effective in pure compression and capable to reaching the full plastic moment in bending and hence, used in plastic design.

Compact or Class 2 Section:

• 1. It has the capacity to form plastic hinge but does not have capacity to develop collapse mechanism because of local buckling.
• 2. They have lower deformation capacity but also fully effective in pure compression and are capable of reaching their full plastic moment in bending.

Semi- Compact or Class 3 Section:

• 1. It has capacity to develop yield moment only and local buckling is liable to prevent the development of the plastic moment resistance.
• 2. They are also fully effective in pure compression but local buckling prevents the attainment of the full plastic moment in bending.  Bending moment resistance in these cross-sections is limited to yield moment only.

Slender or Class 4 Section:

• 1. These section fails before reaching the yield stress i.e. local buckling will occur even before the attainment of the yield stress in extreme fiber.
• 2. An effective cross-section is defined based on the width to thickness ratio of the individual plate elements and this is used to determine the resistance of cross-section.

Among all these sections Plastic, Compact and semi-compact can be used as compression member but slender section cannot be used as compression member because member fails by buckling before reaching its yield strength.

Split tensile strength test of stone:

Three cylindrical test pieces of diameter not less than 50 mm and the ratio of diameter to height 1 : 2 are used to determine the tensile strength of the stone in each saturated and dry condition. Each test piece to be tested is sandwiched in between 2 steel plates of width 25 mm, thickness 10 mm and length equal to the length of test piece. The load is applied without shock and increase continuously at a uniform rate until the specimen splits and no greater load is sustained. The maximum load applied to the specimen is recorded. The average of 3 results separately for each condition should be reported as split tensile strength of the sample. In case any test piece gives a value of as much as 15% below the average.

In tensile strength test of stone,

•  diameter to height ratio is 1 : 2
• lets take
• diameter of sample is 50 mm
• diameter to height ratio = 1 : 2
• than height of cylindrical sample is 100 mm

Which of the following is NOT a coagulant?

• Ferrous sulphate
• Sodium aluminate
• Sodium sulphate - answer
• Ferric chloride

Type of milestone color code for Highways

The various type of milestone color code being used in India for highways are specified in below tabulated form:

• Type of Highway         Milestone Color Code

• National Highway         Yellow & White Strip

• State Highway         Green & White Strips

• City or Main District Roads Blue or Black & White strips

• Village Roads                 Orange & White strips

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QUICK REVISION of Important points before exam - Civil Engineering

Quick Revision

Test 1 - Most important points 
Civil Enigneering

Witney’s theory is the ultimate load theory.

Witney replaced the actual parabolic stress diagram with a rectangular stress diagram such that the C.G of both the diagrams lies at the same point and their areas are also equal.

The maximum depth of concrete stress block in a balanced RCC beam is 0.53 d.

k = critical depth factor

d = effective depth of the beam

fy = characteristics strength of steel

Grade	k	Xulim
Fe 250	0.53	0.53d
Fe 415	0.48	0.48d
Fe 500	0.46	0.46d
Fe 550	0.44	0.44d
Fe 600	0.43	0.43d

For the separation of dry dust of 10 to 100 μm size, the cyclonic separator is used.

.

Method/Device	Suitability	Efficiency
Gravity settling chamber	D > 50 μm	50 – 55 %
Cyclonic separator	D > 10 μm	60 – 65 %
Electrostatic Separator	D > 1 μm	95 - 99 %
Cotton baghouse filter	Suitable for all sizes	90 – 95 %

1. Compensating errors 

These are those which remain after mistakes and systematic errors have been eliminated and are caused by the combination of errors beyond the ability of the observer to control.

They are proportional to the square root of the length of the line.

2. Accidental errors

They represent the limit of precision in the determination of a value.

They obey the law of chance and must be handled according to the mathematical law of probability.

These errors are proportional to the square root of the length of the line (L−−√L).

3. Cumulative Errors/Systematic errors  

The errors that occur in the same direction and which finally tend to accumulate are said to be Cumulative errors.

• They are cumulative in nature. Examples of systematic errors are Collimation in a level, Expansion of steel tape, etc.
• They are proportional to the length of the line.

4. Random errors 

• These are all those discrepancies remaining after the mistakes and systematic errors are removed.
• It is mainly caused by the limitations of observer and instruments and is random in nature.

CHAIN SURVEY

• Chain surveying is that type of surveying in which only linear measurements are made in the field.
• This type of surveying is suitable for surveys of small extent on open ground to secure data for exact description of the boundaries of a piece of land or to take simple details.

A chain survey may be done in following steps:

i) Reconnaissance: 

• During Reconnaissance, a reference sketch of the ground should be prepared and general arrangement of lines, principal features such as buildings, roads etc should be shown.

ii) Marking and Fixing survey stations: 

• Before selecting the stations, the surveyor should examine the intervisibility of stations and should note the positions of the buildings, roads, streams etc.
• After having selecting the survey stations, they should be marked to enable them to be easily discovered during the progress of the survey.

iii) Running survey lines:

•  After having completed the preliminary work, the chaining may be started from the base line.

FLOW TYPE

• Gradually varied flow is steady non-uniform because the velocity of water remains constant at a specified point, but it changes from one point to another point.

• The terms steady and uniform are used frequently in engineering, and thus it is important to have a clear understanding of their meanings.
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• The term steady implies no change with time. The opposite of steady is unsteady, or transient.

• The term uniform, however, implies no change with location over a specified region.

• Any devices such as turbines, compressors, boilers, condensers, and heat exchangers operate for long periods of time under the same conditions, and they are classified as steady-flow devices. 
• During steady flow, the fluid properties can change from point to point within a device, but at any fixed point, they remain constant.

Note:

• In the case of a nozzle, the velocity of water remains constant at a specified point, but it changes from the inlet to the exit (water accelerates along with the nozzle which represents the case of steady and non-uniform flow).
• You may be tempted to think that acceleration is zero in steady flow since acceleration is the rate of change of velocity with time, and in a steady flow, there is no change with time. 
• Well, a garden hose nozzle will tell us that this understanding is not correct.
• Even in a steady flow and thus constant mass flow rate, water will accelerate through the nozzle. Steady simply means no change with time at a specified location, but the value of a quantity may change from one location to another.

Flow separation 

• It occurs when the boundary layer travels far enough against an adverse pressure gradient that the speed of the boundary layer relative to the object falls almost to zero
• It has been observed that the flow is reversed in the vicinity of the wall under certain conditions

A one-dimensional flow is one which involves zero transverse components of flow.

Uniform flow

• The flow is defined as uniform flow when in the flow field the velocity and other hydrodynamic parameters do not change from point to point at any instant of time. For a uniform flow, there will be no spatial distribution of hydrodynamic and other parameters.

Non-uniform flow

• When the velocity and other hydrodynamic parameters changes from one point to another the flow is defined as non-uniform.

Steady flow

• A steady flow is defined as a flow in which the various hydrodynamic parameters and fluid properties at any point do not change with time.

One-dimensional flow 

• It is the flow where all the flow parameters may be expressed as functions of time and one space coordinate only. The single space coordinate is usually the distance measured along the centre-line (not necessarily straight) in which the fluid is flowing. Example: the flow in a pipe is considered one - dimensional when variations of pressure and velocity occur along the length of the pipe, but any variation over the cross-section is assumed negligible.

Turbulent fluid motion 

• It can be considered as an irregular condition of flow in which various quantities (such as velocity components and pressure) show a random variation with time and space.

Types of equilibrium

Stable Equilibrium:

• If the body returns to its original position by retaining the originally vertical axis as vertical.

Unstable Equilibrium:

• If the body does not return to its original position but moves further from it.

Neutral Equilibrium:

• If the body neither returns to its original position nor increases its displacement further, it will simply adapt its new position.

 

Stability of Floating Bodies in Fluid:

• When the body undergoes an angular displacement about a horizontal axis, the shape of the immersed volume changes and so the centre of buoyancy moves relative to the body.

Metacentre:

• Meta Centre is defined as the point about which a body starts oscillating when the body is tilted by a small angle.

• The meta-centre may also be defined as the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given a small angular displacement.

• For the body, M is above G, and the couple acting on the body in its displaced position is a restoring couple which tends to turn the body to its original position.

• If M were below G, the couple would be an overturning couple and the original equilibrium would have been unstable.

• When M coincides with G, the body will assume its new position without any further movement and thus will be in neutral equilibrium.

•  

• Hence the condition of stable equilibrium for a floating body can be expressed in terms of metacentric height as follows:

• GM > 0 (M is above G) ⇒ Stable equilibrium

• GM = 0 (M coinciding with G) ⇒ Neutral equilibrium

• GM < 0 (M is below G) ⇒ Unstable equilibrium

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Inundation or Diversion irrigation:

• It is a type of irrigation system in which a large quantity of water flowing in a river during the flood is allowed to flood or inundate the land to be cultivated.

• Inundation canals usually flow only during the summer months and bring in a large quantity of silt beneficial to crops.

• Inundation irrigation is done by a canal taking off from a river in flood without any diversion work.

• When the water available at higher level is supplied to lower level by the action of gravity only, then the type of irrigation is called flow irrigation.

• When the dam is constructed across a river to store water during monsoons, so as to supply water in the off-taking channels during period of low flow, then it is termed as storage irrigation.

• In Combined scheme, Dam is constructed across a river to form a reservoir, the stored water is used to produce electricity apart from irrigation purpose.

Intensity of Irrigation

• The percentage of CCA (culturable commanded area) proposed to be irrigated in a given season is called intensity of irrigation.

• CCA is the portion of the gross commanded area that is cultivable.

• CCA = Gross Command Area – Uncultivable area (pasture lands, ponds, townships, waste land)

• The annual irrigation intensity is usually found to be in the range of 40 to 60%.

• But needs to be raised in the range of 100 to 180% by cultivating larger parts of CCA with more than one crop in a year.

• 1. By adding intensities of irrigation for all the crop seasons we obtain the annual intensity of irrigation.

• 2. Annual intensity of irrigation can be more than 100%.

For Closed coil fibre:

• The major stresses in a helical spring are of two types, shear stress due to torsion and direct shear due to applied load. It is observed that for both tensile load as well as compressive load on the spring, maximum shear stress always occurs at the inner side of the spring. Hence, failure of the spring, in the form of crake, is always initiated from the inner radius of the spring.

The strength of concrete is its resistance to rupture.

• It may be measured in number of ways, such as strength in compression, in tension, in shear, or in flexure.
• It is generally understood that the measured compressive strength of concrete increases with increasing loading rate as the stiffness increases with increased strain rates.
• Compressive strength decreases with decrease in rate of load because of the creep effect.
• The concrete gets less strained when subjected to load at a faster pace.
• We generally take 14 MPa per minute as load rate for cubes and,
• 12 MPa per minute for cylinder according to IS 516 which gives standard test condition while performing compressive strength test.
• The volumetric changes of concrete structures due to the loss of moisture by evaporation is known as concrete shrinkage.
• It is a time-dependent deformation which reduces the volume of concrete without the impact of external forces.

Note:

• The total shrinkage of concrete depends upon the constituents of concrete, size of member and environmental conditions.

• For a given humidity and temperature, the total shrinkage of concrete is most influenced by the total amount of water present in the concrete at the time of mixing and to a lesser extent by the cement content.

As per Clause 6.2.4.1 of IS 456:2000,

The approximate value of the total shrinkage strain for design may be taken as 0.0003.

STEAM CURING

Steam curing is advantageous where early strength gain in concrete is important or where additional heat is required to accomplish hydration, as in cold weather.

So for high alumina cement where the heat of hydration is very high, steam curing can not be applied.

Curing of concrete by steam under pressure:

• 1. Increases the rate of gain of compressive strength of concrete
• 2. Reduces the shear strength of concrete
• 3. Increases the speed of chemical reaction

Curing:

 It is the process of hardening the concrete mixes by keeping its surface moist for a certain period, in order to enable the concrete to gain more strength.

Properly cured concrete has an adequate amount of moisture for continued hydration which leads to:

• Development of strength
• volume stability.
• resistance to freezing and thawing.
• abrasion and scaling resistance.

RCC

• Limit state of serviceability of prestressed concrete should satisfy cracking, deflection, and maximum compression also.
• The crack width & deflection should not exceed the permissible limit and the maximum compressive force also should not exceed the strength of concrete.

Note: See article 19.2 & 19.3 in IS code 1343:1980

• Minimum grade of concrete to be used in the design of prestressed concrete structure as per IS 1343 is as below:
• 1. For Post-tensioning minimum grade of concrete used is M-30.
• 2. For Pre-tensioning minimum grade of concrete used is M-40.
• Hence it can be seen that grade of concrete used for prestressed member lies in the range of M30 to M60

Cover to be used in the design of prestressed concrete structure as per IS 1343 is as below:

• 1. For Posttensioning minimum cover to be used is 30 mm.
• 2. For Pre-tensioning minimum cover to be used is 20 mm.

Shear span

• It is the span between the points of application of concentrated load to its adjacent Reaction force in a beam.

Note:

• Throughout Shear Span the Shear Force is constant.
• There might be multiple shear spans for a single beam depending upon the number and position of applied force to the numbers of supports

• 1. The final deflection due to all loads including the effect of temperature, creep, and shrinkage and measured from as cast level of the support of floors, roofs, and other horizontal members should not normally exceed span/250.

• 2. The deflection including the effect of temperature, creeps, and shrinkage occurring after the erection of partition and application of finishes should not normally exceed span/350 or 20 mm whichever is less.

Purlin:

• The purlins are horizontal beams spanning between the two adjacent trusses. These are the structural members subjected to transverse loads and rest on the top chords of roof trusses. The purlins are meant to carry loads of the roofing material and to transfer it to the panel points. 

• If the slope of roof truss is not greater than 30' and steel is conforming to grades Fe 410-0 or Fe 410-W, then, the angle purlins may be designed as an alternate to the general design procedure, as recommended by IS : 800-1984. 

• Maximum allowable outstand is 16t for an unstiffened flange.

• The velocity of exit, Ve should be the minimum of 2.5 of wind speed to prevent down draught.
• The exit velocity is an important factor to prevent the spread the pollution around the surrounding. The minimum exit velocity ensures that the gases are not spread in longer vicinity.
• Exit velocity varies from region to region depending upon the topographical, geographical, temporal, etc., and many other factors.

• for the stone masonry, the direction of pressure in stone masonry is normal to the natural bedding plane.

BUILDING MATERIALS

• Rock is a hard mass of stone, or a broken-off piece of a boulder, or is slang for a piece of crack cocaine. An example of a rock is a piece of stone found at the bottom of a cliff.

Metamorphic Rock

•  iT is one of the major groups of rock that make up the crust of the Earth; it consists of pre-existing rock mass in which new minerals or textures are formed at higher temperatures and greater pressures than those present on the Earth's surface.
• Marble is a metamorphic rock that forms when limestone is subjected to the heat and pressure of metamorphism.
• A marble that contains impurities such as clay minerals, iron oxides, or bituminous material can be bluish, gray, pink, yellow, or black in colour.
• As metamorphism progresses, the crystals grow larger and become easily recognizable as interlocking crystals of calcite.
• Some other examples of metamorphic rocks are gneiss, slate, marble, schist, and quartzite.

Plutonic Rock:

•  When magma never reaches the surface and cools to form intrusions (dikes, sills, etc) the resulting rocks are called plutonic. Examples of Plutonic rocks include aplite, greisen, and syenite.
• Sedimentary Rock. : Rock formed of mechanical, chemical, or organic sediment: such as. a: clastic rock (as conglomerate, sandstone, or shale) formed of fragments of other rock transported from its source and deposited in water. Examples of sedimentary rocks include limestone, sandstone, mudstone, greywacke, chalk, coal, claystone, and flint.

Igneous Rock.

•  Igneous rock is a term used for a rock formed when molten rock cools and hardens. Extrusive, or volcanic, igneous rock cools on the surface as lava. Some examples of igneous rocks are granite, gabbro, basalt.
• Therefore, from the above facts, it is clear that Marble & Slate is an example of Metamorphic Rock.

Dressing

• The operation of removal of impurities of clay adhering to iron ore is known as dressing. By using rock crushers this operation can be carried out.
• Due to crushing ore particles of uniform size are obtained and the reducing gases penetrate the ores in a better way.
• If ore contains clay, loam and other earthy matter they are washed in a stream to remove such impurities.

Calcination

• It is the process in which the ore is heated below its melting point, either in the absence or in a limited supply of air, in an aim to drive off volatile expunges, moisture, water of hydrates and organic matter from the ore.

Purification

• In most cases, metals and their ores occur in the ground as part of complex mixtures that also contain rocks, sand, clay, silt, and other impurities.
• The first step in producing the metal for commercial use, therefore, is to separate the ore from waste materials with which it occurs. This is known as purification.

Refining

• It is a process by which the purity of the metal extracted from their ores can be improved.
• Usually, the pure metal obtained from the metal extraction is nearly 90 - 95 % only.

Schedule of rates:

A document containing a detailed description of all the items of work (but their quantities are not mentioned) together with their current rates is called schedule of rates.

These usually include general conditions, general specifications, items of different works, data for transportation, materials, and labor, method of rate analysis, plant rate analysis, and basic unit rate analysis.

​Tender:

A tender is an offer to execute some specified work or to supply some specified article at certain rates.

While inviting tender the bill of quantities, detailed specifications, conditions of the contract, and plans and drawings are supplied.

Abstract estimate

The main function of an abstract of the estimate are as follows:

The total estimated cost and the different items of works are required to complete a project can be known.

This is the basis on which percentage rate tenders are called after excluding the amounts for contingency and work-charged establishment.

This is a part of the tender document so a contractor can arrive at his own rates from the schedule of work described in the description column.

This is the basis on which bills are prepared for payment.

Analysis of rate

For the analysis of rates, knowledge of following items is necessary:

Specifications of works and materials about their quality, proportion, and construction methods.

Quantity of materials and their costs.

Cost of labors and their wages.

Location of work

Conveyance charges.

Overhead charges

Profits of contractor, consultant, and other parties involved.

Quantities so measured shall be increased by the following percentages and the results shall be included in general areas:

Corrugated steel sheets ⇒ 14%

Corrugated asbestos cement sheets ⇒ 20%

Semi-corrugated asbestos cement sheets ⇒ 10%

When color washing on asbestos cement sheets is done then the plane area of the sheets increased by 20 %.

After colour washing, area = Area + (20/100) × Area.

Corrugated surfaces shall be measured flat as fixed and not girthed.

Types of Beams	Span/effective depth
Cantilever	7
Simply supported	20
Continuous	26

The above basic ℓ/d ratio is applicable for spans up to 10 m. 

Beyond the span of 10 m the ℓ/d ratio should be multiplied by 10/span (m)

The quantity of work which can be done by an artisan for trade working of 8 hours is known as out - turn work.

The outturn for different types of work are as follows:

1. Sawing of softwood = 5.5 m2 per mason per day

2. Earthwork in the excavation in foundation trenches = 2.10 m3 per mason per day.

3. Cement concrete work = 4 m3 per mason per day.

4. Earthwork in the excavation in foundation = 2.75 m3 per mason per day.

Drip course, string course, and water coping are measured in running meter or m.

The units of measurements for civil engineering works are mainly categorized for their nature, shape, and size and for making payments to the contractor. The principle of units of measurements normally consists the following:

a) Single units work like doors, windows, trusses, etc., are expressed in numbers.

b) Works consist of linear measurements involve length like cornice, fencing, handrail, bands of specified width, etc., are expressed in running meters (m).

c) Works consist of areal surface measurements involve area like plastering, whitewashing, partitions of specified thickness etc., and are expressed in square meters (m2)

d) Works consist of cubical contents which involve volume like earthwork, cement concrete, Masonry, etc. are expressed in Cubic meters (m3).

DPC is measured in m2.

Material statement: The total quantities of all the items of materials required for the completion of the construction is shown in Material statement.

Bar bending schedule: Bar Bending Schedule, commonly referred to as “BBS” is a comprehensive list that describes the location, mark, type, size, length and number, and bending details of each bar or fabric in a Reinforcement Drawing of a Structure.

Work charged establishment: The work changed establishment will include the temporary establishment as are employed for the execution or the immediate technical supervision or departmental stores in connection with the specific work.

Sundries: Sundries is the column used to add prices for miscellaneous items which are not listed in the bow. For example, binding wire used to tie rebar, cover blocks etc.

• a) For Sandy tracks – Lead × 1.4
• b) For metal tracks – Lead × 1.0
• c) For cartze tracks – Lead × 1.1

Standard size of brick = 19 cm × 9 cm × 9 cm

Nominal size of a brick with mortar = 20 cm × 10 cm × 10 cm

Non Modular Bricks:

Conventional size of brick = 22.4 cm × 11.4 cm × 7.6 cm

SURVEYING

Cadastral survey:

•  Cadastral survey are made incident to the fixing of property lines, the calculation of land area, or the transfer of land property from one owner to another. They are also made to fix the boundaries of municipalities and of state and federal jurisdictions.

Topographical survey: 

• This consist of horizontal and vertical location of certain points by linear and angular measurements and is made to determine the natural features of a country such as rivers, streams, lakes, woods, hills etc and such artificial features like roads, railways etc.

City survey:

•  They are made in connection with the construction of streets, water supply systems, sewers and other works.

Topographical, cadastral and city survey are the parts of Land surveying

Alidade:

•  It is a straight edge ruler having some form of sighting device. One edge of the ruler is bevelled and is graduated. Always this edge is used for drawing line of sight.

Clinometers:

•  The clinometer is an optical device for measuring elevation angles above horizontal. The most common instruments of this type currently used are compass-clinometers from Suunto or Silva.

Cross staff:

•  The cross staff is used to set out the perpendicular directions for offsets

Prism square: 

• An optical square is a hand instrument used by surveyor's to lay off right angles that are multiples of 90° or of 45°.
• They normally comprise of two optical glass penta prisms in a sturdy housing. Used for placing points on a line, offset measurements, setting our curves or determining horizontal plans.

ERRORS

• For no error in the magnitude, the difference (degree) in the magnitude of the fore bearing and back bearing of any line is 180° or if the difference is exactly 180°, the two stations may be considered as not affected by local attraction
• Whenever there is any station affected by local attraction, then the difference between the fore bearing and back bearing is not equal to 180°
• A freely suspended and properly balanced magnetic needle is expected to show magnetic meridian. However, local objects like electric wires and objects of steel attract magnetic needle towards themselves
• Thus, the needle is forced to show a slightly different direction, this disturbance is called local attraction
• For detecting local attraction it is necessary to take both fore bearing and back bearing for each line.
• If difference is not 180°, better to go back to the previous station and check the fore bearing
• If that reading is same as earlier, it may be concluded that there is local attraction at one or both stations

For an open traverse with 'n' numbers of station:

• Total no of fore bearing = n - 1, and total no back bearing = n - 1.

• The stations can be primary classified as first station, intermediate station, and last station.

• For every intermediate stations: One fore bearing and back bearing is required each.

• For first station: Only one fore bearing required, No back bearing required

• For end station: Only one back bearing required, No fore bearing required

Corrections due to refraction

• CR = + 0.0112 d2

Correction due to curvature

• Cc = - 0.0785 d2

Composite correction

• C = - 0.0785 d2 + 0.0112 d2
• C = - 0.0673 d2 

where d is the distance between the staff and instrument in km, and C is in m.

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Civil Engineering - Strength of Material- quick revision

Quick Revision

Strength of Material Civil engineering

Maximum principal stress theory (Rankine’s theory)

According to this theory, permanent set takes place under a state of complex stress, when the value of maximum principal stress is equal to that of yield point stress as found in a simple tensile test.

For design criterion, the maximum principal stress (σ1) must not exceed the working stress ‘σy’ for the material.

• σ1,2 ≤ σy for no failure
• σ1,2 ≤ σ/FOS for design
• Note: For no shear failure τ ≤ 0.57 σy

Graphical representation

For brittle material, which do not fail by yielding but fail by brittle fracture, this theory gives satisfactory result.

The graph is always square even for different values of σ1 and σ2.

Maximum principal strain theory (ST. Venant’s theory)

According to this theory, a ductile material begins to yield when the maximum principal strain reaches the strain at which yielding occurs in simple tension.

• ϵ1,2≤σy/E1 For no failure in uni – axial loading.
• σ1/E−μσ2/E−μσ3/E≤σy/E For no failure in tri – axial loading.
• σ1−μσ2−μσ3≤(σy/FOS)For design, Here, ϵ = Principal strain
• σ1, σ2 and σ3 = Principal stresses   

Graphical Representation

This story over estimate the elastic strength of ductile material.

Maximum shear stress theory (Guest & Tresca’s Theory)

According to this theory, failure of specimen subjected to any combination of load when the maximum shearing stress at any point reaches the failure value equal to that developed at the yielding in an axial tensile or compressive test of the same material.

Graphical Representation

• τmax  ≤ σy/2 For no failure
• σ1−σ2 ≤ (σy/FOS)) For design

σ1 and σ2 are maximum and minimum principal stress respectively.

Here, τmax = Maximum shear stress

σy = permissible stress

This theory gives satisfactory result for ductile material.

Maximum strain energy theory (Haigh’s theory)

According to this theory, a body complex stress fails when the total strain energy at elastic limit in simple tension.

Graphical Representation.

• {σ1^2+σ2^2+σ3^2−2μ(σ1σ2+σ2σ3+σ3σ1)}≤σy^2 for no failure
• {σ1^2+σ2^2+σ3^−2μ(σ1σ2+σ2σ3+σ3σ1)}≤(σy/FOS)^2for design

This theory does not apply to brittle material for which elastic limit stress in tension and in compression are quite different.

Maximum shear strain energy / Distortion energy theory / Mises – Henky theory.

It states that inelastic action at any point in body, under any combination of stress begging, when the strain energy of distortion per unit volume absorbed at the point is equal to the strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under the state of uniaxial stress as occurs in a simple tension / compression test.

• 1/2[(σ1−σ2)^2+(σ2−σ3)^2+(σ3−σ1)^2]≤σy^2 for no failure
• 1/2[(σ1−σ2)^2+(σ2−σ3)^2+(σ3−σ1)^2]≤(σy/FOS)^2 For design

It gives very good result in ductile material.  

It cannot be applied for material under hydrostatic pressure.

All theories will give same results if loading is uniaxial​

The Strain Energy of Distortion or Distortion energy per unit volume is given by:

Distortion energy per unit volume = (Total Strain energy) - (Energy of dilation)

Total Strain energy is given by

U=12E{σ21+σ22+σ23−2μ(σ1σ2+σ2σ3+σ3σ1)}

Energy of dilation:

Uv=(1−2μ)6E×(σ1+σ2+σ3)23

Distortion energy per unit volume = (Total Strain energy) - (Energy of dilation) = Ud 

1 + μ6E[(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2

In simple tension test:

(1 + μ6E)2σ2ym

Maximum distortion energy theory (Von mises theory)

According to this theory, the failure or yielding occurs at a point in a member when the distortion strain energy per unit volume reaches the limiting distortion energy (i.e. distortion energy at yield point) per unit volume as determined from a simple tension test.

von misses stress under triaxial condition is given by:

σvm=12{(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2}−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√

Now if we compare von misses stress and distortion energy per unit volume equation then,

Ud =1 + μ3E×σ2vm$\frac{1+\mu }{3E}\times {\sigma }_{vm}^2$

Ud ∝ σ2vm

σvm ∝  Ud−−√

The von Mises stress at a point in a body subjected to forces is proportional to the square root of the distortional strain energy per unit volume.

A material may fail if

According to theories of failure, a material may fail for any of the following conditions and it depends upon the inherent properties of the material.

a) Maximum principal stress theory (Rankine’s Theory):

If the maximum principal stress (σ1) exceeds direct stress (σ).

b) Maximum principal strain theory (St. Venant’s Theory):

If the maximum principal strain (ε1) exceeds maximum strain (σ/E).

c) Maximum shear stress theory (Guest & Tresca’s Theory):

If maximum shear stress (τmax) exceeds half of direct stress (σ/2).

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Theories of failure	Other Name	Shape
Maximum Principal Stress Theory	RANKINE’S THEORY	Square
Maximum Principal Strain Theory	St. VENANT’S THEORY	Rhombus
Total Strain Energy Theory	HAIGH’S THEORY	Ellipse
Maximum Shear Stress Theory	GUEST AND TRESCA’S THEORY	Hexagon
Maximum Distortion Energy Theory	VON MISES AND HENCKY’S THEORY	Ellipse

Following are the assumptions made in the theory of Simple Bending:

• The material of the beam is homogenous and isotropic.
• The beam is initially straight, and all the longitudinal fibres bend in circular arcs with a common centre of curvature.
• Members have symmetric cross-sections and are subjected to bending in the plane of symmetry.
• The beam is subjected to pure bending and the effect of shear is neglected.
• Plane sections through a beam, taken normal to the axis of the beam remain plane after the beam is subjected to bending.
• The radius of curvature is large as compared to the dimensions of the beam.
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To obtain beams of uniform strength the sections of the beam may be varied by 

• keeping the width constant throughout and varying the depth, 
• keeping the depth constant throughout the length and varying the width, 
• by varying the width and depth in a suitable way and 
• a circular beam of uniform strength can be made by varying diameter in such a way that M/Z is a constant. 

Equation of pure bending:

Pure bending or bending is that in which bending moment M is constant along the length i.e. dM/dx=0, or shear force is zero.

Its empirical relationship is given by –

M/I=σ/y=E/R

where, M = Bending moment, I = MOI of cross-section about the neutral axis (NA), E = Young’s Modulus of Elasticity, σ = Bending stress at a distance y from NA 

• When a beam is suitably designed such that the extreme fibres are loaded to the maximum permissible stress σ max by varying the Cross-section it will be known as a beam of uniform strength.

Section Modulus (Z):

• The ratio of Moment of Inertia I of beam cross-section about NA to the distance of extreme fibre ymax from the neutral axis is known as section modulus.
• It also represents the strength of the section. It is given by

• Z=I/ymax
• σ=M/Z
• For pure bending, when M = Constant
•   σ=M/Z = Constant

Different assumptions made in torsion theory are as follows:

• Shaft must be straight and should have uniform cross-section.
• The shear stress induced in shaft should not exceed the elastic limit.
• Twist along the shaft is uniform.
• Twisting has no effect on circularity of shaft.

Deflection 

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Elasticity

• It is the property of a material to regain its original shape after deformation when the external forces are removed.

Plasticity

• It is the property of a material that retains the deformation produced under load permanently. 
• Thus, it is a property of material which allows it to deform without fracture

Ductility

• The property of the material that allows it to be drawn into wires or elongated before failure is known as ductility.

Malleability 

• The property of a material to deform under compression.
• The metals having malleable property can be rolled or beaten into sheets.
• An example is aluminium foil.

Toughness

• The ability of the material to withstand stress (resist fracture due to high impact loads) without fracture is known as toughness.
• It is defined as the ability to absorb energy in the plastic state.

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