Notes : Applications of Elastic Behaviour of Materials Class 11 Physics Notes | NCERT Chapter 8

Learn Applications of Elastic Behaviour of Materials Class 11 Physics with crane ropes, beam bending, I-girders, pillars, mountains, MCQs and FAQs. - Physics kund

Elastic behaviour of materials plays an important role in everyday life. All engineering designs require precise knowledge of the elastic behaviour of materials. Engineers use the principles of elasticity while designing bridges, cranes, buildings, pillars, and other structures to ensure safety, stability, and durability.

What is Elastic Behaviour?

Elastic behaviour is the property of a material by virtue of which it regains its original shape and size after the removal of the deforming force, provided the elastic limit is not exceeded.

Important Formulae

Stress:

$ \sigma = \frac{F}{A} $

Young's Modulus:

$ Y = \frac{\text{Stress}}{\text{Strain}} $

Sagging of Beam:

$ \delta = \frac{Wl^3}{4bd^3Y} $

Maximum Height of Mountain:

$ h\rho g = \sigma_{shear} $

1. Determination of Thickness of Crane Rope

Mechanical cranes are used for lifting and transporting heavy loads in industries, ports, and construction sites. The steel rope used in a crane must be designed carefully so that it does not undergo permanent deformation while lifting the load.

The stress produced in the rope should always remain below the yield strength of the material.

Condition for Safe Design

$ \text{Stress} = \frac{W}{A} \leq \sigma_y $

Therefore,

$ A \geq \frac{W}{\sigma_y} = \frac{Mg}{\sigma_y} $

Where:

$A$ = Cross-sectional area of rope
$W$ = Weight lifted
$M$ = Mass of load
$g$ = Acceleration due to gravity
$\sigma_y$ = Yield strength of the material

Numerical Example

Consider a crane having a lifting capacity of 10 tonnes.

Given:

$ M = 10^4\ kg $

$ g = 9.8\ ms^{-2} $

$ \sigma_y = 300 \times 10^6\ Nm^{-2} $

Weight lifted:

$ W = Mg $

$ W = 10^4 \times 9.8 $

$ W = 9.8 \times 10^4\ N $

Applying the safety condition:

$ A \geq \frac{W}{\sigma_y} $

$ A \geq \frac{9.8 \times 10^4}{300 \times 10^6} $

$ A \approx 3.3 \times 10^{-4}\ m^2 $

For a circular rope:

$ A = \pi r^2 $

$ r = \sqrt{\frac{A}{\pi}} $

$ r = \sqrt{\frac{3.3 \times 10^{-4}}{3.1416}} $

$ r \approx 0.01\ m $

$ r \approx 1\ cm $

Therefore, the minimum radius of the rope should be approximately 1 cm.

Factor of Safety

In actual engineering practice, sudden jerks, vibrations, fatigue, corrosion, and dynamic loading increase the stress acting on the rope.

Therefore, engineers use a large factor of safety, generally about 10.

Hence, a rope with a practical radius of nearly 3 cm is preferred instead of the theoretical value of 1 cm.

Why Are Crane Ropes Made of Thin Wires?

A single steel wire of large radius would behave like a rigid rod and would not be flexible enough for practical use.

Therefore, crane ropes are made of many thin steel wires braided together.

Advantages of Braided Steel Ropes

Greater flexibility
Uniform stress distribution
Better fatigue resistance
Greater durability
Reduced chances of sudden failure
Easier handling during operation

Key Concept

The thickness of a crane rope is determined using the concept of stress and yield strength so that the rope remains within its elastic limit during operation.

2. Bending of Beams

A bridge must withstand the load of moving traffic, wind forces, and its own weight. Similarly, beams used in buildings and other structures must support heavy loads safely.

When a load acts on a beam, it bends slightly downward. This downward displacement is known as sagging.

Consider a rectangular beam of:

Length = $l$
Breadth = $b$
Depth = $d$

supported at both ends and loaded at the centre by a weight $W$.

The sagging produced in the beam is given by:

$ \delta = \frac{Wl^3}{4bd^3Y} $

Where:

$\delta$ = Sagging
$W$ = Load applied
$l$ = Length of beam
$b$ = Breadth of beam
$d$ = Depth of beam
$Y$ = Young's Modulus of the material

Importance in Bridge Design

A bridge must withstand the load of moving vehicles, wind forces, and its own weight. Therefore, engineers design beams in such a way that sagging remains as small as possible.

Methods to Reduce Sagging

1. Use Material with High Young's Modulus

Since:

$ \delta \propto \frac{1}{Y} $

A material with larger Young's modulus undergoes less bending.

2. Increase Breadth

Since:

$ \delta \propto \frac{1}{b} $

If breadth is doubled:

$ b' = 2b $

Then:

$ \delta' = \frac{\delta}{2} $

Therefore, sagging becomes half.

3. Increase Depth

Since:

$ \delta \propto \frac{1}{d^3} $

If depth is doubled:

$ d' = 2d $

Then:

$ \delta' = \frac{\delta}{8} $

Thus, doubling the depth reduces sagging to one-eighth of its original value.

4. Reduce Span Length

Since:

$ \delta \propto l^3 $

Sagging increases rapidly with beam length.

Therefore, engineers try to keep the span length as small as possible.

Important Conclusion

Increasing depth is much more effective than increasing breadth in reducing sagging. Therefore, beams are generally designed with larger depth rather than larger width.

3. Buckling

Increasing the depth of a beam reduces sagging, but if the beam becomes excessively deep and narrow, another problem arises.

The beam may suddenly bend sideways under load. This phenomenon is called buckling.

Definition of Buckling

Buckling is the sudden lateral or sideways bending of a structural member when subjected to compressive forces.

Buckling may occur even before the material actually breaks.

Why Does Buckling Occur?

A very deep but narrow beam lacks sufficient lateral support. As a result, under heavy loading it becomes unstable and bends sideways.

Therefore, engineers must design beams to resist both sagging and buckling.

4. I-Shaped Girders

To overcome the problems of excessive sagging and buckling, engineers use I-shaped girders.

These girders are commonly used in:

Bridges
Railway tracks
Flyovers
Buildings
Industrial structures

Why Is the I-Section Preferred?

The I-section provides:

Large load-bearing surface
Sufficient depth
High bending strength
Resistance to buckling

The shape removes unnecessary material from regions where stresses are small, thereby reducing weight without reducing strength.

Thus, the cost of construction is also reduced.

Structure of an I-Girder

Upper Flange

The upper flange experiences compressive stress and helps resist compression.

Lower Flange

The lower flange experiences tensile stress and helps resist tension.

Web

The vertical portion connecting the two flanges is called the web.

The web provides large depth and resists shear forces acting on the girder.

Advantages of I-Girders

High load-carrying capacity
Reduced sagging
Resistance to buckling
Reduced weight
Economical construction
Efficient use of material

NCERT Concept

The I-section provides a large load-bearing surface and enough depth to prevent bending while reducing weight and cost.

5. Pillars and Columns

Pillars and columns are important structural members used to support heavy loads in buildings, bridges, and towers.

The shape of their ends significantly affects their load-bearing capacity.

Pillars with Rounded Ends

Small contact area
Higher pressure
Less stability
Lower load-bearing capacity

Because the load acts over a smaller area, the pressure becomes larger and the pillar can support less load.

Pillars with Distributed Ends

Large contact area
Lower pressure
Greater stability
Higher load-bearing capacity

The larger area distributes the load more effectively and reduces pressure on the supporting surface.

Engineering Preference

Therefore, pillars and columns are generally designed with broad or distributed ends rather than rounded ends.

Key Concept

Broad-based pillars provide greater stability and can support larger loads because pressure decreases when the area of contact increases.

6. Maximum Height of Mountains

The concept of elasticity can also explain why mountains cannot grow indefinitely in height.

A mountain base is not under uniform compression because the sides of the mountain are free. Therefore, the rocks at the base experience large shearing stresses.

If the shearing stress exceeds the elastic limit of rocks, the rocks begin to deform and flow slowly. As a result, there exists a maximum possible height for mountains on Earth.

Mathematical Explanation

Consider a mountain of height h and density ρ.

The stress produced at its base is approximately:

$ \sigma = h\rho g $

Where:

$h$ = Height of mountain
$\rho$ = Density of rock
$g$ = Acceleration due to gravity

For maximum stable height:

$ h\rho g = \sigma_{shear} $

For typical rocks:

$ \sigma_{shear} = 30 \times 10^7 \ Nm^{-2} $

$ \rho = 3 \times 10^3 \ kgm^{-3} $

$ g = 10 \ ms^{-2} $

Substituting:

$ h(3 \times 10^3)(10) = 30 \times 10^7 $

$ h(3 \times 10^4) = 30 \times 10^7 $

$ h = \frac{30 \times 10^7}{3 \times 10^4} $

$ h = 10^4 \ m $

$ h = 10 \ km $

Thus, the maximum possible height of mountains on Earth is approximately 10 km.

Mount Everest, with a height of about 8.85 km, lies close to this theoretical limit.

Engineering Applications of Elastic Behaviour

Design of crane ropes
Construction of bridges
Design of beams and girders
Construction of buildings
Design of pillars and columns
Railway tracks
Aircraft structures
Machine components
Structural safety analysis
Determining the maximum height of mountains

Key Points for Revision

Elastic behaviour is important in engineering design.
Crane ropes are designed using the concept of yield strength.
A factor of safety is always used in practical designs.
Sagging of a beam depends on load, length, breadth, depth, and Young's modulus.
Sagging is directly proportional to $l^3$.
Sagging is inversely proportional to $b$.
Sagging is inversely proportional to $d^3$.
Doubling depth reduces sagging to one-eighth.
Buckling is the sideways bending of a beam.
I-shaped girders provide high strength with less material.
Broad-ended pillars support larger loads.
The maximum height of mountains on Earth is about 10 km.

Frequently Asked Questions (FAQs)

1. What is elastic behaviour?

Elastic behaviour is the property by which a body regains its original shape and size after the removal of the deforming force.

2. Why are crane ropes made of braided wires?

Braided wires provide flexibility, durability, and uniform stress distribution.

3. What is sagging?

Sagging is the downward bending of a beam under load.

4. What is buckling?

Buckling is the sudden sideways bending of a structural member under compression.

5. Why are girders I-shaped?

Because they provide high strength, sufficient depth, reduced weight, and resistance to buckling.

6. Why do pillars have broad ends?

Broad ends increase the area of contact, reduce pressure, and improve stability.

7. Why can't mountains become infinitely high?

Because the rocks at the base cannot withstand unlimited shearing stress.

Multiple Choice Questions (MCQs)

1. The yield strength of mild steel is approximately:

A) $30 \times 10^6 \ Nm^{-2}$
B) $300 \times 10^6 \ Nm^{-2}$ ✅
C) $3 \times 10^6 \ Nm^{-2}$
D) $3000 \times 10^6 \ Nm^{-2}$

2. Sagging is inversely proportional to:

A) $d$
B) $d^2$
C) $d^3$ ✅
D) $d^4$

3. Sideways bending of a beam is called:

A) Stretching
B) Compression
C) Buckling ✅
D) Twisting

4. I-shaped girders are preferred because they:

A) Increase weight
B) Reduce strength
C) Reduce cost without sacrificing strength ✅
D) Increase bending

5. Maximum mountain height depends upon:

A) Wind speed
B) Temperature
C) Humidity
D) Shearing strength of rocks ✅

True or False

1. Elastic behaviour is important in engineering design. True

2. Crane ropes are made of a single thick steel rod. False

3. Sagging is proportional to the cube of beam length. True

4. Buckling is a lateral bending phenomenon. True

5. Mountains can grow without limit. False

Fill in the Blanks

1. The stress in a crane rope must remain below the __________ strength. Yield

2. Downward bending of a beam is called __________. Sagging

3. Sideways bending of a beam is called __________. Buckling

4. The most commonly used girder section is __________ shaped. I

5. The maximum height of mountains depends on the __________ strength of rocks. Shearing

Very Short Answer Questions

Q1. What is sagging?

Ans: Sagging is the downward bending of a beam under load.

Q2. What is buckling?

Ans: Buckling is the sudden sideways bending of a structural member under compression.

Q3. Why are crane ropes braided?

Ans: To increase flexibility and distribute stress uniformly.

Q4. Why are girders made I-shaped?

Ans: To obtain high strength with less material and reduced weight.

Short Answer Questions

Q1. Why is a factor of safety used in crane rope design?

Ans: A factor of safety is used because actual loads may increase due to jerks, vibrations, fatigue, corrosion, and other practical conditions.

Q2. Why are pillars designed with broad ends?

Ans: Broad ends increase the area of contact, reduce pressure, improve stability, and increase load-bearing capacity.

Long Answer Question

Q1. Explain the applications of elastic behaviour of materials.

Ans: Elastic behaviour is widely used in engineering and construction. It helps in determining the thickness of crane ropes, designing beams and girders, constructing pillars and columns, ensuring the safety of bridges and buildings, and explaining the maximum possible height of mountains. Knowledge of elasticity helps engineers build safe, durable, and economical structures.