STRENGTH OF MATERIALS- Stress & Strain

Engineering Mechanics or Solid Mechanics (also known as Strength of Materials in some institutions) is one of the most basic subject of Engineering which deals with the engineering properties of materials like Modulus of Elasticity, Stresses, Strain etc. 

In this article we will discuss information regarding Stresses (meaning, types, need) and Strain (types). First of all let us recall the Newton’s Third Law which states that “To every action there is equal and opposite reaction”. Based on the same law, Stress is induced in the materials. Whenever any force is applied on the material, it tries to deform the material but due to internal resistance of the material an equal force is induced in the opposite direction of the applied force. This induced force is the result of internal resistance of material to resist the deformation. Just like pressure is the force applied on any material per unit area, stress is the internal resistive force induced per unit area as the result of applied force. Now each material has certain strength i.e. it can take up load upto particular limit before failure. Upto that limit the material can yield or in other words its dimensions can change due to application of load. Strain is the ratio of change in the dimension due to applied force to the original dimension of the body and that change in dimension can be laterally or longitudinally.

There are different types of forces that act on any body like Tensile force, Compressive force, Shear force, Bending force and Torsion.

 Stress depends upon the applied force as already discussed and depending upon the type of force applied different types of stresses are induced as shown in the figure below.

Since stress is induced due to reaction and pressure is applied due to action, therefore stress and pressure are numerically equal only direction is different. Strain is another term related to body under stress. Whenever force is applied on any body the body deforms both longitudinally and laterally as shown in figure below. The ratio of Lateral Strain to Longitudinal Strain is termed as Poisson’s Ratio denoted by ‘μ’.

 Longitudinal Strain = 2 δL/L ;         Lateral Strain = (D-d)/D

The ratio of Direct Stress to Longitudinal Strain is known as Young’s Modulus of Elasticity denoted by ‘E’.

The ratio of Shear Stress to Shear Strain is known as Modulus of Rigidity denoted by ‘G’ or ‘C’.

The ratio of Direct Stress to Volumetric Strain is known as Bulk Modulus denoted by ‘K’.

There is a relationship among E, G, K and μ as mentioned below

E = 3K (1-2 μ) and E = 2G (1+ μ)

FOUNDATION ENGINEERING- Standard Penetration Test of Soil (SPT)

In Geotechnical Engineering one of the most preferred field test to carry out Soil Investigations is Standard Penetration Test commonly known as SPT. In this test, the strength of soil beneath ground surface is determined in terms of number of blows of hammer (63.5 kg) required to penetrate the split spoon sampler into the ground by 30 cm at regular intervals of 0.75 m or 1.5 m upto the significant depth. The apparatus used for SPT is:

• ·        Tripod Stand
• ·        Hammer of 63.5 kg
• ·        Rope
• ·        Split Spoon Sampler
• ·        Auger
• ·        Boring apparatus
• ·        Knife
• ·        Air Tight Containers
• ·        Chalk
• ·        Measuring Tape
• ·        A-Rods

The SPT is performed as per IS:2131 in following steps:

   Step 1.        Make a borehole upto 1.5 m

   Step 2.        Place the Tripod Stand over the borehole such that the hammer should be directly above the borehole

   Step 3.        Insert the Split Spoon Sampler in the borehole with the help of attaching A-Rods.

   Step 4.        Mark three readings on the top A-Rod with the help of chalk  at intervals of 15 cm above the ground surface

   Step 5.        Allow the 63.5 kg to fall freely from 75 cm height.

   Step 6.        The number of blows of hammer required for first 15 cm penetration is known as “Seating Drive” and is not counted.

   Step 7.        The number of blows of hammer required for the next 30 cm penetration is noted and is termed as ‘N’value.

(Note: If number of blows for any 15 cm penetration exceeds 50 or for 30 cm penetration exceeds 100 then SPT is stopped and termed as refusal that means hard surface is encountered and Sampler is not advanced further to protect it from damage) 

   Step 8.        Take out the sampler by striking the hammer in upward direction.

   Step 9.        From the sampler take out the 20 cm entrapped soil sample carefully from center of tube and preserve it in air tight container mentioning the depth and borehole from where sample is taken.

   Step 10.  Repeat Steps 1 to 9 at regular interval of 1.5 m below ground surface or 0.75 m if soil strata changes abruptly and complete the observation table given below.

S.No.

Depth (m)

‘N’ value

Density (kN/m3)

Total Stress ‘σ’ (kN/m2)

Effective Stress ‘σ`’ (kN/m2)

Overburden Correction ‘Cn’

Corrected ‘N’ Value N`

Nc after Dilatency Correction

·        Density is calculated from Mass of soil sample in air tight container and determining internal diameter of sampler tube to know the volume of soil sample.

·        Overburden Correction is determined by the formula: 
  Cn=0.77 log10(2000⁄(σ`)where σ` is in kN/m²        

·        Corrected N value ‘N`’ = C_n × N

·        Dilateny Correction is applied only if following conditions are satisfied:

o   Soil is Cohesionless and fully saturated

o   N value is more than 15

·        N_c after dilatency correction is given by 15+0.5(N`-15) 

   Step 11.  Once table is completed the average value of N­_c is calculated by taking weighted average i.e. N_av

·        For determining N_av count number of N values in table

·        Multiply the numbers to each N_c in decreasing order i.e. Multiply 1 with last N_c, 2 with second last N_c, 3 with third last N_c and so on.

·        Take weighted average by adding all the values calculated above and dividing the result with the sum of all the numbers that are multiplied with each N_c.

Precautions:

Ø Make sure the A-Rods are vertical while the hammer is impacted.

Ø Do not advance the sampler in the ground without making a borehole because it can damage the cutting edge of sampler.

Ø Legs of tripod stand should be anchored in the ground.

Ø Note down the readings carefully by observing the markings on the top A rod.

Result:

The results of SPT are reported in the format provided in Appendix of IS:2131 which includes the Borehole log i.e. drawing of a borehole showing SPT N_c Values at different levels as well as type of soil encountered at different levels below ground surface along with the name of site and borehole at the top.

Video:

TRANSPORTATION ENGINEERING - Superelevation on Highways

Dear readers, today I am going to discuss about the term 'Superelevation' that is frequently used in Transportation Engineering.
Transportation Engineering is the branch of Civil Engineering that mainly deals with design, construction and analysis of highways. That is why it is sometimes also known as Highway Engineering.
While designing the highways the safety of passengers is of utmost importance followed by comfort of passengers while travelling at high speeds. For designing the highways many different studies are carried out like speed study, traffic volume study, accident study etc. The main problem lies in designing the highway on curves when it should be designed such that no skidding of vehicle should take place. Skid is the term that is used when longitudinal movement of vehicles is more than circumferential movement of the tires. Slip is the term that refers to the condition when circumferential movement is more than longitudinal movement of vehicle. So skidding is the main problem that can occur on the curves when vehicle is moving in circular motion at high speed due to centrifugal force. To clear your doubts regarding circular motion there is a video below that will explain the circular motion and basic terms related to it.

(Clip showing basics of Circular Motion)

So now we will continue to the technical part that is involved in providing superelevation on highways. Superelevation is the rise provided to the outer part of the road on curves to prevent skidding of vehicle due to centrifugal force. The figure shown below represents various forces that are involved while vehicle is moving along the curve on highway.

Figure 1.1 : Showing direction of frictional force along the tires

Figure 1.2 : Showing various forces and their components

As shown in the figure 1.1 "e" is the superelevation provided. e multiplied by total width of road will give you total rise "E" required on highways. W=m×g and P=mv²/r 

where W - Weight, m - Mass, g - Acceleration due to gravity, P - Centrifugal force, v - Velocity of vehicle, r - Radius of curve.

Now as depicted from figure Pcosθ=Wsinθ+F_b, where F_b is frictional force and is equal to 

f(Wcosθ+Psinθ).

Therefore Pcosθ=Wsinθ+f(Wcosθ+Psinθ), where f is frictional coefficient. Putting values of W and P we get,

(mv²/r)cosθ-(mg×sinθ)=f(mg×cosθ)+f(mv²/r)sinθ. 

Reducing the equation after dividing by gcosθ  we get,

v²/gr - tanθ = f + f(v²/gr)tanθ, from figure 1.2 tanθ= E/B= e

v²/gr(1-fe)= f+e. Since f i.e frictional factor's maximum value is 0.14 and maximum value of e i.e. superelevation 0.07 so f×e being very small can be neglected. Hence we get

v²/gr=f+e.

Where v is design velocity

g is acceleration due to gravity

r is radius of curve

f is frictional coefficient subject to maximum of 0.14

e is superelevation provided subject to maximum of 0.07

From the above equation we can decide superelevation or we can determine new design velocity if value of superelevation is exceeding its maximum value i.e 0.07.

Therefore while designing highway on curves we should provide designed superelevation so that vehicles can take turn at design speed without skidding. The rise so provided is introduced by different methods that will be discussed in other post. Hope the topic was helpful to you.

Please do share your valuable views in comments...!!!

ENVIRONMENTAL ENGINEERING - Finding Eco-friendly ingenious ways of disposing wastewater

Dear readers today i am going to discuss some basic aspects of waste water treatment plant as to how and why it is carried out.
Environmental Engineering is the branch of Engineering that deals with the contemporary issues of society related to water resources.
Infrastructure development ensues huge harm to the environment due to increasing levels of Air Pollution, Noise Pollution and Water Pollution that is affecting the health of living organisms in one way or another. To antagonize this effect environmentalists, scientists, scholars, engineers are contemplating on this subject so as to coin various methods that can reduce the harm caused by pollution. As an initiative step towards reducing water pollution environmental engineers have designed various techniques to treat the waste water generated from houses and industries so that it can be disposed off effectively in water bodies or can be used for fulfilling other purposes like watering the gardens, medians etc.
To accomplish the above cited task water treatment is divided into three main categories namely:

1. Primary Treatment
2. Secondary Treatment
3. Tertiary Treatment

In the first step waste water is passed through primary treatment units where any solid matter present in it is filtered out by making use of screenings, filters etc. In the next step water is allowed to pass through various units like sedimentation tanks, grit chambers etc. where heavy suspended solids are settled down during detention period. Then finally primary treated water is passed through various units involving chemical and biological treatment of water that is known as tertiary treatment. After water is passed through all the three stages of treatment, it is disposed off in the water bodies without altering oxygen demand.
Oxygen Demand is the amount of oxygen required for the aquatic life to survive. As the pollution in water body increases, the living organisms in the form of bacteria, microorganisms etc. As a result those bacteria and various flora and fauna present in the water body will consume oxygen already present in the water body for their survival resulting in increased oxygen demand. This amount of oxygen required for the survival of living organisms is known as Biochemical Oxygen Demand (BOD). The high values of BOD in water sample indicates high degree of pollution due to larger number of bacteria present in water.
So 5-day BOD value of the sample is determined before disposing off the water in water body because if BOD value is higher it will pose danger to aquatic life. That is why it is necessary that before disposing off the waste water in the streams or rivers it should be properly treated in Waste Water Treatment Plant.
Following video will help in understanding the process of water treatment more effectively.

Please do leave your valuable remarks in the comments...!!!!