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Contents:
Contents:
Concept of stress and strain, Normal stress and strain, Shear stress and strain, Bearing stress
Hooke’s Law, Stress-strain diagrams: Elastic Limit, yield strength and ultimate tensile strength
Poisson’s Ratio, Elastic constants: Modulus of elasticity, Bulk Modulus, Modulus of Rigidity
Composite Bars, Thermal Stresses, Stresses in thin-walled pressure vessel
Lecture 09: ( 06 Sep 2023)
Concept of Stress and Strain, Normal Stress and Strain, Shear Stress and Shear Strain, Bearing Stress
Stress and strain are two important concepts in engineering and physics.
They are used to describe the response of a material to an applied force.
Stress is the force per unit area that is applied to a material. It is a vector quantity, and it can be either tensile (stretching) or compressive (compressing).
Strain is the deformation of a material caused by an applied stress. It is a dimensionless quantity, and it is typically expressed as a percentage of the original length or area.
There are three main types of stress: normal stress, shear stress, and bearing stress.
Normal stress is a stress that acts perpendicular to the surface of a material. It can be tensile or compressive.
Shear stress is a stress that acts parallel to the surface of a material. It causes the material to slide in one direction relative to another.
Bearing stress is a stress that acts perpendicular to the surface of a material and is caused by a force that is trying to push the material apart.
The relationship between stress and strain is described by Hooke's law.
Hooke's law states that within the elastic limit of a material, the stress is proportional to the strain.
The elastic limit is the maximum stress that a material can withstand without permanent deformation. Once the elastic limit is exceeded, the material will undergo plastic deformation, which is permanent.
Lecture 10: ( 08 Sep 2023)
Hooke’s Law, Stress-strain diagrams: Elastic Limit, yield strength and ultimate tensile strength
Hooke's Law:
Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance.
i.e., Fs = kx,
where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small distance compared to the total possible deformation of the spring.
The modern theory of elasticity generalizes Hooke's law to say that the strain (deformation) of an elastic object or material is proportional to the stress applied to it.
However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a tensor that can be represented by a matrix of real numbers.
A stress-strain diagram is a graph that shows the relationship between stress and strain for a material. It is typically plotted with stress on the vertical axis and strain on the horizontal axis.
The stress-strain diagram for a material will typically have four distinct regions:
The elastic region is the region where Hooke's law applies. In this region, the strain is proportional to the stress.
The plastic region is the region beyond the elastic limit. In this region, the strain is no longer proportional to the stress. The material will continue to deform even after the load is removed.
The necking region is the region where the material begins to neck down. This is a localized region of deformation that occurs when the material is under a lot of stress.
The breaking point is the point at which the material breaks.
The elastic limit is the point on the stress-strain diagram where Hooke's law no longer applies. This is the maximum stress that a material can withstand without permanent deformation.
The yield strength is the stress at which the material begins to deform plastically. This is the point where the material starts to neck down.
The ultimate tensile strength is the maximum stress that a material can withstand before it breaks.
ASTM E8 Metal Tensile Testing
ASTM E8 Metal Tensile Testing
Lecture 11: ( 11 Sep 2023)
Tutorial on Problem Solving on Hooke’s Law, Stress-strain diagrams
Lecture 12: (13 Sep 2023)
Poisson’s Ratio, Elastic constants: Modulus of elasticity, Bulk Modulus, Modulus of Rigidity
Poisson's Ratio:
Poisson's Ratio:
The ratio of the transverse contraction of a material to the longitudinal extension strain in the direction of the stretching force is the Poisson's Ratio for a material.
The Poisson's ratio is negative for the compressive deformation whereas for the tensile deformation the Poisson's Ratio is Positive.
The negative Poisson's ratio suggests that the positive strain is in the transverse direction.
The Poisson's Ratio for most of the materials is in the range of 0 to 0.5.
Elastic Constants:
Elastic Constants:
Young's modulus, Bulk Modulus and Rigidity Modulus of an elastic solid are together called Elastic constants
Elastic constants are the physical quantities that describe the elastic behavior of a material. They are used to relate the stress and strain in a material.
There are three main elastic constants:
Young's modulus (E) is the ratio of stress to strain in a tensile or compressive stress. It is a measure of the stiffness of a material.
Shear modulus (G) is the ratio of shear stress to shear strain. It is a measure of the resistance of a material to shearing.
Bulk modulus (K) is the ratio of volumetric stress to volumetric strain. It is a measure of the resistance of a material to compression.
In addition to these three main elastic constants, there are also other elastic constants that can be used to describe the elastic behavior of materials. These include:
Poisson's ratio (ν) is the ratio of the lateral strain to the axial strain in a tensile or compressive stress. It is a measure of the compressibility of a material.
The Lame constants (λ and μ) are related to the Young's modulus, shear modulus, and Poisson's ratio.
Lecture 13: ( 15 Sep 2023)
Tutorial on Poisson’s Ratio, Elastic constants: Modulus of elasticity, Bulk Modulus, Modulus of Rigidity
A material has a Young's modulus of 210 GPa and a Poisson's ratio of 0.3. Calculate the shear modulus and bulk modulus of the material.
2. The Young's modulus of steel is 200 GPa. What is the shear modulus of steel?
3. The bulk modulus of water is 2.2 GPa. What is the pressure that is required to reduce the volume of water by 10%?
4. The Young's modulus of aluminum is 70 GPa and the Poisson's ratio for aluminum is 0.3. What is the bulk modulus of aluminum?
Lecture 14: ( 18 Sep 2023)
Composite Bars and Thermal Stresses
Tutorials on Problem Solving based on Composite Bars and Thermal Stresses
A composite bar is a bar made up of two or more different materials, joined together so that the system extends or contracts as a single unit, equally when subjected to tension or compression. The different materials in a composite bar can have different properties, such as Young's modulus, yield strength, and ultimate tensile strength. This allows the composite bar to be designed to have specific properties that are not possible with a single material.
Composite bars are used in a variety of applications, including:
Structural engineering: Composite bars are used in beams, columns, and other structural members to improve their strength and stiffness.
Mechanical engineering: Composite bars are used in machine components, such as gears and shafts, to improve their strength and wear resistance.
Civil engineering: Composite bars are used in bridges, roads, and other civil engineering structures to improve their durability and resistance to corrosion.
The design of a composite bar is a complex process that takes into account the properties of the different materials, the loading conditions, and the desired properties of the composite bar. The analysis of composite bars can be done using the theory of elasticity.
Here are some of the advantages of using composite bars:
Increased strength and stiffness: Composite bars can be made to have much higher strength and stiffness than single materials. This makes them ideal for applications where high loads or stresses are present.
Reduced weight: Composite bars can be made to be much lighter than single materials. This makes them ideal for applications where weight is a critical factor, such as in aerospace engineering.
Improved durability: Composite bars can be made to be more durable than single materials. This is because the different materials in a composite bar can protect each other from damage.
Customized properties: The properties of a composite bar can be customized to meet the specific requirements of an application. This makes them ideal for applications where high performance is required.
Some of the disadvantages of using composite bars:
Cost: Composite bars can be more expensive than single materials.
Complex design: The design of a composite bar is more complex than the design of a single material.
Manufacturing: Composite bars can be more difficult to manufacture than single materials.
Availability: Composite bars may not be as readily available as single material
Thermal Stresses
Thermal Stresses
Thermal stress is the mechanical stress created by any change in the temperature of a material. These stresses can lead to fracturing or plastic deformation depending on the other variables of heating, which include material types and constraints. Temperature gradients, thermal expansion or contraction and thermal shocks are things that can lead to thermal stress. This type of stress is highly dependent on the thermal expansion coefficient which varies from material to material.
Causes of thermal stress: Temperature gradients, thermal expansion or contraction and thermal shocks
Temperature gradients: When a material is rapidly heated or cooled, the surface and internal temperature will have a difference in temperature. Quick heating or cooling causes thermal expansion or contraction respectively, this localized movement of material causes thermal stresses.
Thermal expansion or contraction: As materials heat up, they expand, and as they cool, they contract. This is due to the increased or decreased kinetic energy of the atoms in the material. If a material is prevented from expanding or contracting freely, thermal stress will be induced.
Thermal shocks: Thermal shock is the sudden and extreme change in temperature of a material. This can be caused by quenching a hot material in cold water, or by exposing a cold material to a high temperature. Thermal shock can cause severe thermal stress, leading to cracking or fracturing of the material.
Effects of thermal stress
Cracking: Thermal stress can cause cracks to form in materials. This is especially common in brittle materials, such as glass and ceramics.
Plastic deformation: Thermal stress can also cause plastic deformation in materials. This means that the material will permanently change its shape.
Failure: In severe cases, thermal stress can cause materials to fail completely. This can be a major safety hazard in engineering applications.
A composite bar is made of steel and copper, with a steel core of radius 1 cm and a copper sheath of radius 2 cm. The bar is heated by 100°C. Find the stresses in the steel and copper.
2. A composite bar is made of two materials, steel and copper. The steel bar has a length of 1 meter and a cross-sectional area of 1 square meter. The copper bar has a length of 0.5 meters and a cross-sectional area of 0.5 square meters. The coefficient of thermal expansion of steel is 12 × 10-6 per degree Celsius, and the coefficient of thermal expansion of copper is 17 × 10-6 per degree Celsius. The bar is heated by 10 degrees Celsius.
Lecture 15: ( 20 Sep 2023)
Tutorials on Problem Solving based on Stresses in thin-walled pressure vessel
Thin-walled pressure vessels are containers that are designed to hold pressurized fluids or gases. The walls of the vessel are thin relative to the diameter of the vessel, which makes them lightweight and efficient. However, thin-walled pressure vessels are also more susceptible to stresses than thick-walled pressure vessels.
There are two main types of stresses that occur in thin-walled pressure vessels: hoop stress and longitudinal stress
Hoop stress is the stress that acts in the circumferential direction of the vessel. It is caused by the pressure of the fluid or gas pushing against the inside of the vessel.
Hoop stress is calculated using the following equation,
where:
σ_h is the hoop stress (MPa)
p is the internal pressure (MPa)
r is the inner radius of the vessel (m)
t is the thickness of the vessel wall (m)
Longitudinal stress is the stress that acts in the axial direction of the vessel. It is caused by the ends of the vessel trying to push apart due to the internal pressure.
Longitudinal stress is calculated using the following equation:
where:
σ_l is the longitudinal stress (MPa)
p is the internal pressure (MPa)
r is the inner radius of the vessel (m)
t is the thickness of the vessel wall (m)
In general, hoop stress is greater than longitudinal stress in thin-walled pressure vessels. This is because the hoop stress is proportional to the radius of the vessel, while the longitudinal stress is proportional to the radius squared. Therefore, the hoop stress increases more rapidly as the vessel diameter increase
A thin-walled cylindrical pressure vessel has an inner radius of 10 cm and an outer radius of 12 cm. The vessel is subjected to an internal pressure of 100 MPa. What are the stresses in the vessel?
In-Semester Evaluation : Activity Based Assessment
In-Semester Evaluation : Activity Based Assessment
Activity 4: "Experimental Study of Hooke's Law"
Due Date: 20 Sep 2023
Carry out the experimental study of Hooks Law using the reference material provided, prepare a neat report either written/typed
Assessment Parameters:
Explanation of Hooks Law
Craftsmanship of Experimental Setup
Data Presentation, Analysis and Inference
Activity 5: Determination of Young's Modulus and Poisson's Ratio using Ultrasonic Testing
Due Date: 25 Sep 2023
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