RCC Beam

A beam is a structural member which has one dimension greater than the other two and placed in a horizontal plane. It's a flexural member which can take bending moment and shear force. The cross-section of a beam can either be rectangular or T-shaped or circular etc.

To resist the bending moment and shear force, an adequate section is required which is neither too bulky nor too small.

The geometry of the section of the beam depends on section modulus and section modulus itself depends on the moment of Inertia of that section.

IS 456 has given the criteria for the selection of the section of a beam which is based on the deflection of the beam.

As per IS 456:2000 the ratio of the span and effective depth should not be greater than 20, 7 and 26 for the simply supported beam, cantilever beam and continuous beam respectively. These values are valid for span up to 10 m above which these values must be modified by multiplying them with (10/span in meter).

Effective depth of a beam is the depth of the beam from the top fibre to the CG of the reinforcement provided in the tension zone. 

The difference between the overall depth and effective depth is called effective cover.

Effective cover = nominal cover + diameter of stirrups + half the diameter of main reinforcement steel bar.

And, the Nominal cover is provided so that the steel bars are fully embedded, and it is not exposed to exterior conditions like rain. It also helps in maintaining the required connection between concrete and steel bars.
IS 456:2000 has provided the values of Nominal Cover on the bases of exposure condition. Click here

Sections of beams

There are 3 types of sections on the bases of stress condition of concrete and that of steel.

1. Under Reinforced Section: When the permissible stress in steel is reached in a beam prior to that of concrete.
2. Balanced Section: When the permissible stress in steel and that of concrete reaches at the same time.
3. Over Reinforced Section: When the permissible stress in concrete reaches prior to that of steel.

IS 456:2000 does not recommend a over reinforced section because of its brittle failure nature. In case of under reinforced section the failure is ductile which gives sufficient warning in terms of excessive cracks and deflection to the inmates before failure. 

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Methods of Design

There are 3 methods to design a structural component.

1. Working Stress Method
2. Ultimate Load Method
3. Limit State Method 

Working Stress Method

This is a classical method used to design RCC structures. In this method, the material is assumed to behave elastically. The relationship between stress and load is linear and stresses within the material is not allowed to exceed the permissible stress.

               $\text{Permissible Stress}=\frac{\text{Strength of Material}}{\text{Factor of Safety}}$

        Permissible Stress in Tension steel: $0.55{{f}_{y}}={{\sigma }_{st}}$
        FOS for Steel = 1.8

        Permissible Stress in Concrete in bending: ${{\sigma }_{cbc}}=0.33{{f}_{ck}}$
        FOS for Concrete = 3

Assumptions in Working Stress Method

1. A section which is plane before bending remains plane after bending.
2. The bond between concrete and steel is perfect within the elastic limit of steel.
3. Tension is borne entirely by steel.
4. The Modulus of Elasticity of concrete is same for all stresses.
5. There are no initial stresses in steel when it is embedded in concrete.

Deficiencies in Working Stress Method

1. Due to the long-term effect of creep and shrinkage and stress concentration, it may not be possible to keep the stresses within permissible limit.
2. Actual margin of safety is not equal to the factor of safety used in WSM because the stress-strain curve is not linear up to collapse. 
3. WSM does not discriminate between different types of loads that act simultaneously.

Although WSM has deficiencies still it is used to design structures like Bridges, Water tanks, chimneys etc because of its simplified approach.

Ultimate Load Method

This method was introduced in the 1960s. In this method stress condition at the state of impending failure is analysed and non-linear stress-strain curve of steel and concrete are made use of.
A safety measure is introduced by an appropriate choice of load factor.

               $\text{Load Factor =}\frac{\text{Ultimate Load}}{\text{Working Load}}$

In this method distribution of stress resultants at ultimate load is taken as distribution at service load magnified by load factor. This is clearly an error because significant inelastic behaviour and redistribution of stress resultant take place as loading is increased from service loads to ultimate loads.

Limit State Method

There is uncertainty in loading, properties of material and dimensions of a member and to account for these uncertainties FOS and Load Factor was introduced in WSM and Ultimate load Method respectively. But there was no theoretical justification for use of FOS and load factors.

To overcome this, a reliability-based analysis was performed and factors of safety for both loading and material properties were established and these factors were called Partial factors of safety. Selection of partial factors of safety was done on the probabilistic basis.

This analysis was called Limit State Method. Limit state is a state in which the structure becomes unfit for use.

There are 2 types of Limit States:

1. Limit state of serviceability:  Satisfactory performance under service load. Such as discomfort caused by excessive deflection, crack width, vibration, leakage, loss of durability etc.
2. Limit state of collapse: Adequate margin of safety for normal overloads. These include limit state of strength, overturning, sliding, buckling, fatigue etc.

Assumptions in Limit State of Collapse

1. A section which is plane before bending remains plane after bending.
2. The maximum strain in concrete at the outermost compression fibre is taken as 0.0035 in bending.
3. The relationship between compressive stress distribution in concrete and strain in concrete may be assumed to be rectangular, trapezoidal, parabolic or any other shape which results in the prediction of strength in substantial agreement with the result of the test.
   
   
                                                  Source: IS 456
   
   
4. The Tensile strength of concrete is ignored.
5. Partial Factor of Safety of Steel is 1.15 and that for concrete is 1.5.
6. Maximum strain in Tension reinforcement in the section at failure shall not be less than
   
   

                  ${{\varepsilon }_{st}}=\frac{{{f}_{y}}}{1.15{{E}_{s}}}+0.002$
   Where,
                 ${{\varepsilon }_{st}}=$ Strain in Tension Steel
                 ${{f}_{y}}=$ Characterstic Strength of Steel
                 ${{E}_{s}}=$ Modulus of Elasticity of Steel

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Roads in India

The roads in India have been in existence for at least 3000 years. In the 4th century the prime minister of Emperor Chandra Gupta Maurya, B.C. Kautilya has mentioned rules about the depth of road for different kinds of traffic in his book "Arthasastra". Later in the 5th century Asoka, the Great had improved the roads.

During the Mughal periods, the roads of India were greatly improved. Some of the highways either built or maintained by Mughals received great appreciation from the foreign visitors who visited India during that period.

Jayakar Committee

After the first world war, the use of motor vehicles increased and the demanded a better road network which can carry both bullock cart traffic and motor vehicles. Hence in 1927, the govenment appointed a Road development Committe headed by M.R. Jayakar.

Recommendations made by the committe:

1. Central Government should take the complete charge of roads as it is a matter of national interest.
2. It suggested to found "Central Road Fund" to create road development fund with an extra tax levied on petrol from road users.
3. It also suggested to found a semi official technical body to pool know how from various parts of the country and to act as an advisory body on various aspects of roads.
4. A research institute should be instituted to carry out research and development work and to be available for consultations.

Most of the recommendations were accepted by the government. The Central Road Fund was formed by the year 1929 and the semi official technical body called Indian Road Congress was formed in 1934 and Central Road Research Institute was formed in 1950.

Motor Vehicle Act

The Motor Vehicle Act was brought into force by the government in 1939 to regulate the road traffic in the form of traffic laws, ordinance and regulations.

The Motor Vehicle Act has been revised in the year 1988.

20 Year Road Development Plan

The Nagpur Plan (1943-63)

 A conference of the Chief Engineers of all the states and provinces was convened in 1943 by the Government of India at Nagpur.

Features of Nagpur Plan

1. National Highways which would connect several states and provinces and would be of national importance for strategic, administrative and other purpose.
2. State highways which would be connect imporant places within a state.
3. District Roads which would take traffic from main roads to the interior of each district.
   District Roads were further divided into Major District Roads and Other District Roads.
4. Village Roads which would connect a village or a group of villages to the road system.

Targets of Nagpur Plan:

1. 2 Lakh kms of road across the country within 20 years.
2. Construction of star and grid pattern.
3. Road density of 16 kms per 100 sq km.

 The Bombay Plan (1961-81)

 The second 20 year road development plan was initiated by the IRC and finalised in 1959 for the period 1961-81. This road development plan is known as Bombay Road Plan.

Targets of Bombay Plan: 

1. Total road length to be construct was about 10 Lakh kms.
2. Road density of 32 kms per 100 sq km.
3. 1600 KM of expressway
4. Traffic engineering cells in each state. 

The Lucknow Plan (1981-2001)

The 3rd 20 year development road plan was prepared by Road Wing of the Ministry of Shipping and Transport. 

Targets of Lucknow Plan:

1.  Road length to be construct by the year 2001 was 27 Lakh.
2. Road density of 82 km per 100 sq km.
3. National Highway of a square grid of 100 KM X 100 KM.
4. All weather road to connect villages or group of villages with a population of 500 and above. For villages with population less than 500, an all weather road shall be available at a distance of less than 3 km in plain areas and 5 km in hilly terrain.
5. Expressway to be constructed on major traffic highways.
6. The Major District Roads to connect all towns and villages with a population of 1500 and above.
7. The other District Roads to connect  villages with a population of 1000 to 1500.

 

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Early Roads

Roads have a very long history, Brick-paved streets were used in India as early as 3000 BC. Roads in the towns were straight and long, intersecting one another at right angles.

Only during the period of the Roman empire, roads were constructed in large scale. Many of the Roman roads were of elaborate construction. Some of these still exist after over 2000 years. The Appian Way was built in 312 BC extending over 580 KM which demonstrates the road building technique used by Romans.

Roman roads were built straight and they neglected the natural gradient. It was built after the soft soil was removed and hard strata were reached and that's why the depth of the pavement was as high as 0.75 m to 1.2 m although the load was very low. The wearing course was made of dressed large stone-blocks set in lime mortar.

Tresaguet Construction

In the18th century Pierre Tresaguet developed an improved method of construction of roads in France. 

The main feature of his method was that the thickness of construction need be only in the order of 30 cm. He considered the fact that drainage of water along the pavement surface and subsurface moisture content control the life of the pavement.

                                             Tresaguet Construction  (Source: Kullabs)

Telford Construction

Thomas Telford was the founder of the Institution of Civil Engineers in London. To keep the soil subgrade firm he used heavy foundation stones. He provided level subgrade of width 9m. He provided a definite cross slope for top surface by varying the thickness of the foundation stones.

                                                    Telford Construction (Source: Kullabs)

Macadam Construction

John Macadam was a Scottish road builder. He was the first one to notice that the use of large stone foundation is baseless since the stress below the wheels decreases as the depth increases. He also gave importance to the subgrade drainage and compaction. 

                                               Macadam Construction (Source: Kullabs)

Macadam's method of construction was recognised as a scientific method of construction and hence adopted by various countries with slight modifications.

One of the most popular methods which is even now prevalent in many countries is the Water Bound Macadam (WBM) construction. 

In this method broken stones of the base course and surface course, if any, are bound by the stone dust in presence of moisture. WBM roads are in use in India both as a finished pavement surface for minor roads and as a good base course for superior pavement carrying heavy traffic.

There are also bituminous bound macadam and penetration macadam which are adopted by our country.

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Highway Engineering

Transportation is a very important area for a country as it contributes to the development of the economy, infrastructure, defence etc. It is vital for the economic development since each and every commodity produced needs transport from production to distribution.

Underdeveloped colonies and tribes are improving their living conditions since the distances have apparently been reduced with the reduction in travel time. Due to the better connectivity, the urban area attracts the population from other areas resulting in enhanced economic activities.

It plays a vital role in maintaining law and order and defending the country.

Here I won't go into details, so let's take a look at what is highway engineering.

Highway Engineering is a branch of civil engineering which deals with the study of traffic and design, construction and maintenance of roads and traffic control devices etc.

Click on the topics below to see the brief details about them.

Stresses and Strains - 2

Hooke's Law

According to this law, Stress in a material is directly proportional to Strain within elastic limit.

$\sigma =E\times\varepsilon $

           $\sigma=$ Stress
           $\varepsilon=$ Strain
           $E=$ Modulus of Elasticity

Hooke's law is valid for homogeneous isotropic and linearly elastic material.

Deformation of Load under Axial Load

Case 1. Bar of uniform section

$\delta =\frac{PL}{AE}$

Case 2. Stepped Bars

$\delta ={{\delta }_{AB}}+{{\delta }_{BC}}+{{\delta }_{CD}}$

$=\frac{{{N}_{1}}{{l}_{1}}}{{{A}_{1}}{{E}_{1}}}+\frac{{{N}_{2}}{{l}_{2}}}{{{A}_{2}}{{E}_{2}}}+\frac{{{N}_{3}}{{l}_{3}}}{{{A}_{3}}{{E}_{3}}}$

${{N}_{1}},{{N}_{2}},{{N}_{3}}=$ Resultant forces in the member 1, 2 and 3 respectively

Case 3. Tapered circular bar

$\delta =\frac{PL}{\frac{\pi }{4}E\times {{D}_{1}}{{D}_{2}}}$

Case 4. Tapered Rectangular bar

$\delta =\frac{PL}{E\times t(a-b)}{{\log }_{e}}\frac{a}{b}$

Case 5. Deformation due to self weight

$\delta =\frac{\gamma {{L}^{2}}}{2E}=\frac{WL}{2AE}$

$\gamma=$ unit weight of bar
$W=$ Total Weight of bar

Case 6. Deformation in a composite bar

${{\delta }_{1}}=\frac{{{P}_{1}}L}{{{A}_{1}}{{E}_{1}}}=\frac{{{P}_{2}}L}{{{A}_{2}}{{E}_{2}}}={{\delta }_{2}}$

Case 7. Deformation due to change in temperature

$\delta =L\times \alpha \times \Delta T$

       When both ends are fixed

${{\delta }_{R}}=\frac{RL}{AE}$

$Stress=E\times \alpha \times \Delta T$

Case 8. Deformation due to change in temperature in a fixed bar with one yielding support

$\delta =L\alpha \Delta T-\frac{RL}{AE}$

$Stress=\frac{E(L\alpha \Delta T-\delta )}{L}$

Poisson's Ratio

Whenever a homogeneous and isotropic material is under stress, the longitudinal side will elongate or contract and all the transverse sides will contract or elongate.
The ratio of lateral strain and longitudinal strain is known as Poisson's ratio.

${{\varepsilon }_{x}}=\frac{{{\sigma }_{x}}}{E}$        ${{\varepsilon }_{y}}=-\mu \frac{{{\sigma }_{x}}}{E}$        ${{\varepsilon }_{z}}=-\mu \frac{{{\sigma }_{x}}}{E}$

Possion's Ratio of Engineering Materials

Cork	$0$	Concrete	$0.1-0.2$
Aluminium	$0.33$	Steel	$0.27-0.3$
Perfectly Elastic Rubber	$0.5$	Metal	$0.25-0.4$
Cast Iron	$0.2-0.3$	Rubber	$0.45-0.5$

Modulus of Rigidity (G)

Modulus of rigidity or shearing modulus is the ratio of shear stress and shear strain.

Bulk Modulus (K)

Bulk Modulus is the ratio of Direct stress and Volumetric Strain.

Relation between E,G,K and $\mu$

$E=2G(1+\mu )$       $E=3K(1-2\mu )$

$\mu =\frac{3K-2G}{6K+2G}$       $E=\frac{9KG}{G+3K}$

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Stresses and Strains - 1

Whenever an external load is applied to a solid body, it deforms if the body is restrained from motion either fully or partially.
If the body is not restrained against the motion it may undergo displacements without a change in shape or size and these displacements are called rigid body displacements.

Stress is a resistance generated by the body to counteract the external forces applied to it. It has the same unit as pressure i.e., force/area.

Types of Stresses

1. Normal Stress
2. Shear Stress

Normal Stress can be further classified as

1. Axial Stress
2. Bearing Stress
3. Bending Stress

Axial Stress -  Load is applied along the axis of a body i.e., normal to the section.
                       It can be either tensile or compressive in nature.

Bearing Stress - It is a compressive stress generated when one body is supported on another.

Bending Stress - Bending tension and compression produces normal stress.

Shear Stress

It is the stress that acts parallel to the applied force. Normal stress acts perpendicular to the plane causing either tension or compression while shear stress acts tangentially, that's why it is also known as tangential stress.

                                          Source: Quora

It is of 2 types:

1. Direct Shear Stress
2. Indirect Shear Stress 

Direct Shear Stress - It is that shear stress which tries to cut the section.
Indirect Shear Stress - It arises due to either tension/ compression or torsion.

Shear Stress on opposite faces of a stressed element is equal in magnitude and opposite in nature.

                                                       Source: Javelin-Tech

Stress tensor - Stress is not a vector quantity because its results cannot be obtained by parallelogram law of vector addition. It is a mathematical quantity called tensor.

It is represented by,
                            $\sigma =\left( \begin{matrix} {{\sigma }_{xx}} & {{\tau }_{xy}} & {{\tau }_{xz}} \\ {{\tau }_{yx}} & {{\sigma }_{yy}} & {{\tau }_{yz}} \\ {{\tau }_{zx}} & {{\tau }_{zy}} & {{\sigma }_{zz}} \\ \end{matrix} \right)$

Only 6 components are required to define the condition of stress at a given point in a 3D system because of the complimentary shear stress concept.
And in a 2D system, only 3 stress components are required to define the condition of stress at a given point.

Normal Strain

It is the deformation per unit length.

                                                       $\varepsilon =\frac{\delta }{L}$

$\delta=$ Change in length
$L=$ Original Length

Stress-Strain Curve for Mild Steel

                                              Source: Quora

Failure Plane

Material	Tension Test	Torsion Test
Ductile	${{45}^{{}^\circ }}$	${{90}^{{}^\circ }}$
Brittle	${{90}^{{}^\circ }}$	${{45}^{{}^\circ }}$

Properties of Material

1. Elasticity: It is the property of a material by virtue of which, it returns to its original position after unloading.
   If the material is unloaded before the elastic limit has reached, it will follow the original curve otherwise it will have a residual strain and the unloading curve will be different from original curve.
   
   
   
   
   
2. Plasticity: It is a property of a material which describes the deformation of a solid material undergoing inelastic strain beyond the strain at the elastic limit.
3. Creep: It is a property by virtue of which a material undergoes additional deformation with the passage of time under sustained loading within elastic limit.
   
   
   
4. Relaxation: It is the decrease in stress under constant strain.
   
   
5. Fatigue: It is a deterioration of a material caused by repeatedly applied loads results in progressive and localised structural damage.
   
   
   
   
   Endurance limit is the stress level below which even large number of stress cycle cannot produce fatigue failure.
   For structural stress, it is half of the .ultimate strength and due to corrosion, endurance limit further reduced by 50%
6. Resilience: It is a property by which a material absorbs energy when it is deformed elastically and release that energy upon unloading.
   
   
                                       Source: NPTEL
   The area under the load-deformation curve within elastic limit is called resilience.
   
7. Toughness: It is the ability of a material to absorb mechanical energy upto failure.
   
   
   
                                       Source: NPTEL


8. Tenacity: It is a property of a metal to resist fracture under the action of tensile load.

Standard Stress-Strain Curves

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Strength of Material

Strength of Material is a branch of science which deals with the behaviour of solid bodies subjected to external forces.

Click on topics given below to see the brief details about them.

RCC Basics

IS 456:2000 is the code for Plain and Reinforced Concrete. As per this code, the concrete is used for an RCC structure should have the grade of M20 or above.

The letter "M" here stands for mix and the number is the characteristic strength of standard cube of size 150 mm which when tested under compression load test after 28 days, not more than 5% of the cubes are expected to fail.

For example, if 100 standard cubes of M20 is tested after 28 days then at least 95 cubes should be able to resist more than 20 MPa before failure. If it is less than 95 then the test will be declared invalid and the concrete will be rejected.

But at a construction site, site engineers cannot wait for 28 days to test the compressive strength of the concrete used for construction. To know about the strength of the concrete at early stages they test the concrete cubes at 3 and 7 days. The 3-day strength of a concrete cube is nearly 40% of the 28-day strength and 7-day strength is nearly 65% of the 28-day strength.
This test is conducted on at least 3 cubes and the average of these 3 cubes is taken, provided that individual variation should not be more than $\pm $ 15 % average.

Characteristic compressive strength compliance requirement (IS 456 Cl 16.1 and 16.3)

Specified Grade	Mean of the group of 4 Non-overlapping consecutive test results in $N/m{{m}^{2}}$	Individual test results in $N/m{{m}^{2}}$
M 15	$\ge {{f}_{ck}}+0.825\times $ established standard deviation (rounded off to nearest $0.5 N/m{{m}^{2}}$) or ${{f}_{ck}}+3$ $N/m{{m}^{2}}$ , whichever is greater	${{f}_{ck}}-3$
M 20 or above	$\ge {{f}_{ck}}+0.825\times $ established standard deviation (rounded off to nearest  $0.5 N/m{{m}^{2}}$) or ${{f}_{ck}}+4$ $N/m{{m}^{2}}$ , whichever is greater	${{f}_{ck}}-4$

Characteristic Strength (${{f}_{ck}}$)

                                              ${{f}_{ck}}={{f}_{m}}-1.65\sigma $

Where, ${{f}_{m}}=$ Mean Strength
             $ \sigma=$ Standard Deviation
             $=\sqrt{\frac{\sum\limits_{{}}^{{}}{(f-{{f}_{m}})}}{m}}$ when test samples are $\ge$ 30
             $=\sqrt{\frac{\sum\limits_{{}}^{{}}{(f-{{f}_{m}})}}{m-1}}$ when test samples are $<$ 30
             $m=$ number of samples

Compressive Strength of Concrete in Structures

Strength of concrete is found to decrease with increase in the size of the specimen. However, beyond 450 mm size, there is no decrease in the compressive strength of concrete.
Thus, compressive strength of concrete in structure is taken as $0.67{{f}_{ck}}$

Flexural Strength of Concrete (Modulus of Rupture)
                                              ${{f}_{cr}}=0.7\sqrt{{{f}_{ck}}}$

Tensile Strength of Concrete

Tensile strength of plain concrete is obtained by the splitting test.

Splitting tensile strength,

                                              ${{f}_{ct}}=\frac{2P}{\pi dL}=0.6{{f}_{cr}}$

                                              Source:Research Gate

Stress Strain Curve of Concrete

                                      Source: IS 456

Maximum compressive stress occurs at a strain value of 0.002 i.e., 0.2%. The value of stress at 0.002 strain is called compressive strength of concrete.

Modulus of elasticity of concrete for all practical purpose is taken as secant modulus at a stress of around $0.33{{f}_{ck}}$.

Modulus of elasticity of concrete is primarily influenced by the elastic properties of aggregate and to a lesser exent by the condtions of curing, mix proportion and type of cememnt.

As per IS 456:2000 short term modulus of elasticity,
                                            ${{E}_{C}}=5000\sqrt{{{f}_{ck}}}$
Long-term Modulus of elasticity depends on Creep,
                                            ${{E}_{ce}}=\frac{{{E}_{C}}}{1+\theta }$
$\theta=$ Creep Coefficient

Age at loading	Creep Coefficient
7 days	2.2
28 days	1.6
1 year	1.1

Exposure Conditions

Exposure	Minimum Grade	Minimum Cement Content ${KG}/{{{m}^{3}}}\;$	Maximum freewater cement ratio	Nominal Cover, mm
Mild	M20	300	0.55	20
Moderate	M25	300	0.50	30
Severe	M30	320	0.45	45
Very Severe	M35	340	0.45	50
Extreme	M40	360	0.40	75

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