# Introduction to Simple Stresses

### Contents

**Define Strength of Material**1**Explain various types of assumptions in Strength of Material**2**Differentiate between Mechanics and Strength of Materials**3**Define load and explain various types of load**4**Explain the basic concept of Stress**5**Classify the Stress and explain each of them**6**Calculate the Stress of the bar**7**State Principal of St. Venant’s**7**Define Strain**7**List and explain various types of strain**7**Calculate the Strain of the bar**7**State Hooke’s Law**7**Explain the behaviour of the Stress-strain curve**7**Define Proof Stress**7**Describe and calculate the deformation due to Axial Load-in Prismatic Body**7**Describe and calculate the deformations due to Axial Load in Tapered Cylinder**7**Describe and calculate the deformation due to Axial Load in Tapered Rectangular**7**Explain the concept of the principle of superposition**7**Describe and calculate the deformation due to the Self-Weight of the body**7**Explain the Bar in Parallel**7**Explain the Volumetric Strain due to three Mutually Perpendicular Stresses**7**Explain the following term: Modulus of Elasticity, Modulus of Rigidity, Bulk Modulus, and Poisson’s Ratio**7**Describe the relation between the elastic constants**7**Explain the concept of thermal stresses and strains when the body is free to move**7**Describe the thermal stresses when the body is constrained to move**7**Describe the thermal stresses in a tapered cylindrical bar**7**Describe the thermal stresses in a composite bar**7**Explain the concept of Strain Energy**7**Define Resilience, Proof Resilience, and, Modulus of Resilience**7**Define Toughness and Modulus of Toughness**7**Explain the relation between Impact load and Gradual load**7**List and explain Mechanical properties of materials**7**Define True Stress and True Strain**7**Explain the Factor of Safety and Permissible Stress**7**Explain the Stress-Strain curve of ideal materials**7

Define Strength of Material

Strength of Materials (SOM) is a branch of mechanical engineering that deals with the behaviour of solid materials and structures under various types of loading conditions.

Strength of Materials is the study of how solid materials, such as metals, composites, and materials like concrete, respond to external forces. The goal of Strength of Materials is to understand the behaviour of structures and materials under different loads, such as tension, compression, shear, bending, and torsion. In order to design safe and reliable structures, engineers must have a good understanding of the strength of materials that they are using, including their strengths and weaknesses under different types of loads.

In the context of Strength of Materials, “strength” refers to the ability of a material to resist deformation or failure when subjected to an external force. This strength can be characterized in terms of a material’s yield strength, tensile strength, compressive strength, shear strength, and other properties. Strength of Materials also takes into account how these properties change with different environmental conditions, such as temperature, humidity, and exposure to corrosive materials.

It is important to note that Strength of Materials is not just concerned with the strength of individual materials, but also with the behaviour of entire structures. In this context, engineers must consider not only the strength of individual components, but also how those components interact with each other and with the overall structure to withstand loads and resist failure. This requires a deep understanding of the principles of mechanics, including the concepts of stress, strain, and deformation, as well as the ability to perform complex calculations and simulations to model the behaviour of structures under different loading conditions.

In conclusion, Strength of Materials is an important branch of mechanical engineering that is concerned with the behaviour of solid materials and structures under various types of loading conditions. It is essential for engineers to understand the strength of materials in order to design safe and reliable structures.

Explain various types of assumptions in Strength of Material

Strength of Materials (SOM) is a complex subject that requires the use of various assumptions in order to simplify real-world scenarios and perform mathematical analysis.

Idealisation of material properties: In order to perform mathematical analysis, the behaviour of materials is often idealised. This means that certain assumptions are made about the material properties, such as linear elasticity, isotropy, and homogeneity. These assumptions are used to simplify the analysis, but may not always accurately reflect the real-world behaviour of the material.

- Section plane analysis: In many cases, the behaviour of a structure can be analysed by considering only a cross-sectional plane of the structure, rather than considering the entire structure. This section plane analysis assumes that the behaviour of the material is homogeneous and isotropic, and that the cross-sectional properties of the material remain constant throughout the length of the structure.
- Load application: The loading conditions applied to a structure are often idealised for the purpose of analysis. For example, a structure may be modelled as being subjected to a simple load, such as a uniform load or a concentrated load, rather than a more complex load pattern. This idealisation makes the analysis simpler, but may not accurately reflect the real-world behaviour of the structure under the given loading conditions.
- Elastic behaviour: In many cases, the behaviour of a structure is analysed under the assumption that the material is perfectly elastic, meaning that it will return to its original shape after the load is removed. This assumption is used to simplify the analysis, but may not accurately reflect the real-world behaviour of the material, particularly if the material has undergone plastic deformation.
- Infinitesimal deformations: The behaviour of a structure is often analysed by considering small deformations of the material. This infinitesimal deformation assumption is used to simplify the analysis, but may not accurately reflect the real-world behaviour of the material, particularly if the material has undergone large deformations.

In conclusion, Strength of Materials relies on a number of assumptions in order to simplify real-world scenarios and perform mathematical analysis. These assumptions include idealisation of material properties, section plane analysis, load application, elastic behaviour, and infinitesimal deformations. It is important for engineers to understand the limitations of these assumptions and to consider them when interpreting the results of Strength of Materials analysis.

Differentiate between Mechanics and Strength of Materials

Mechanics and Strength of Materials (SOM) are two branches of mechanical engineering that have a close relationship but are distinct in their focus and objectives.

Mechanics is a branch of physics that deals with the study of motion and its causes. It is concerned with the study of forces and the interactions between objects and their environment. Mechanics can be divided into several branches, including classical mechanics, which deals with the motion of bodies under the influence of forces, and quantum mechanics, which deals with the behaviour of matter and energy at the atomic and subatomic level.

Strength of Materials, on the other hand, is a branch of mechanics that specifically deals with the behaviour of solid materials and structures under various types of loading conditions. The focus of Strength of Materials is on understanding how materials and structures respond to external forces, including tension, compression, shear, bending, and torsion. Strength of Materials also takes into account the material properties, such as yield strength, tensile strength, compressive strength, and shear strength, as well as environmental conditions that can affect the behaviour of materials and structures.

While Mechanics and Strength of Materials are closely related, they differ in their focus and objectives. Mechanics is a broader subject that encompasses a wide range of topics related to motion and forces, while Strength of Materials is more focused on the behaviour of solid materials and structures under specific loading conditions.

In conclusion, Mechanics and Strength of Materials are two distinct branches of mechanical engineering that have a close relationship. Mechanics is a broad subject that deals with the study of motion and its causes, while Strength of Materials is focused on the behaviour of solid materials and structures under various types of loading conditions. It is important for engineers to understand the differences between these two subjects in order to effectively apply their knowledge to real-world problems.

Define load and explain various types of load

In the field of Strength of Materials (SOM), load refers to any external force that acts on a material or structure, causing it to deform or change shape. Load is an important factor that must be considered when analyzing the behaviour of materials and structures, as it determines how they will respond to external forces.

Dead load: Dead load is a constant load that acts on a material or structure. This type of load is often due to the weight of the structure itself and any permanent fixtures or equipment that are attached to it. Dead load is a self-weight load and is constant, meaning that it does not change over time.

- Live load: Live load is a variable load that acts on a material or structure. This type of load is due to the weight of people, vehicles, or other moving objects that may be present on the structure. Live load is variable, meaning that it can change over time, and is often influenced by external factors such as weather conditions or the number of people using the structure.
- Wind load: Wind load is a variable load that acts on a material or structure due to the movement of air. Wind load can be significant, particularly for tall structures such as buildings or bridges, and must be taken into consideration when designing these structures.
- Earthquake load: Earthquake load is a variable load that acts on a material or structure due to the ground motion caused by an earthquake. This type of load can be significant, particularly in areas that are prone to earthquakes, and must be taken into consideration when designing structures that are located in these areas.
- Hydrostatic load: Hydrostatic load is a constant load that acts on a material or structure due to the pressure exerted by a fluid. This type of load is often encountered in the design of pipelines, tanks, and other structures that are used to store or transport fluids.
- Thermal load: Thermal load is a variable load that acts on a material or structure due to changes in temperature. Thermal load can cause materials and structures to expand or contract, which can lead to deformations and changes in shape.

In conclusion, load refers to any external force that acts on a material or structure, causing it to deform or change shape. There are several types of load, including dead load, live load, wind load, earthquake load, hydrostatic load, and thermal load. It is important for engineers to understand the different types of load and how they can impact the behaviour of materials and structures, in order to effectively design and analyse these structures.

Explain the basic concept of Stress

Stress is a fundamental concept in the field of Strength of Materials (SOM) that describes the internal forces that act within a material or structure due to external loads.

Stress is defined as the force per unit area acting on a material or structure. It is a measure of the intensity of the internal forces that are generated within a material or structure as a result of an external load. The units of stress are usually given in units of force per unit area, such as pounds per square inch (psi) or Newtons per square metre (N/m^{2}).

There are two main types of stress that can act within a material or structure: tensile stress and compressive stress. Tensile stress is the stress that acts in a direction that tends to pull a material or structure apart, while compressive stress is the stress that acts in a direction that tends to squeeze a material or structure together.

Stress can be calculated by dividing the applied force by the cross-sectional area of the material or structure. For example, if a material is subjected to a force of 1000 N, and its cross-sectional area is 100 square centimetres, the stress would be calculated as 1000 N / (100 cm^{2}) = 10 N/cm^{2}.

Stress is an important factor in the design and analysis of materials and structures, as it can help to determine the strength and stability of these structures under various loading conditions. By understanding the basic concept of stress, engineers can better predict the behaviour of materials and structures and design them to meet specific performance requirements.

In conclusion, stress is a measure of the intensity of the internal forces that are generated within a material or structure as a result of an external load. Stress is calculated by dividing the applied force by the cross-sectional area of the material or structure. Understanding the basic concept of stress is essential for engineers who work in the field of Strength of Materials, as it can help to determine the strength and stability of materials and structures under various loading conditions.

Classify the Stress and explain each of them

It requires the student to classify the stress and explain each of them. There are several types of stress that can act within a material or structure, and these can be classified into three main categories: normal stress, shear stress, and torsional stress.

- Normal stress: Normal stress is the stress that acts perpendicular to the cross-sectional area of a material or structure. It is further classified into two types: tensile stress and compressive stress.

- Tensile stress: Tensile stress is the stress that acts in a direction that tends to pull a material or structure apart. It occurs when a material is subjected to a tensile force, and it increases with increasing applied force. Tensile stress is usually expressed in units of force per unit area, such as pounds per square inch (psi) or Newtons per square meter (N/m
^{2}). - Compressive stress: Compressive stress is the stress that acts in a direction that tends to squeeze a material or structure together. It occurs when a material is subjected to a compressive force, and it increases with increasing applied force. Compressive stress is also expressed in units of force per unit area, such as psi or N/m
^{2}.

- Shear stress: Shear stress is the stress that acts parallel to the cross-sectional area of a material or structure, and it is caused by a shear force. It occurs when a material is subjected to a force that acts in a direction that is perpendicular to the longitudinal axis of the material. Shear stress is expressed in units of force per unit area, such as psi or N/m
^{2}. - Torsional stress: Torsional stress is the stress that acts on a material or structure due to twisting forces. It occurs when a material is subjected to a torque, and it increases with increasing applied torque. Torsional stress is expressed in units of force per unit area, such as psi or N/m
^{2}.

In conclusion, stress is a measure of the intensity of the internal forces that are generated within a material or structure as a result of an external load. There are three main types of stress: normal stress, shear stress, and torsional stress. Normal stress is further classified into tensile stress and compressive stress, and it acts perpendicular to the cross-sectional area of a material or structure. Shear stress acts parallel to the cross-sectional area of a material or structure, and it is caused by a shear force. Torsional stress acts on a material or structure due to twisting forces. Understanding the various types of stress is essential for engineers who work in the field of Strength of Materials, as it can help to determine the strength and stability of materials and structures under various loading conditions.

Calculate the Stress of the bar

Calculate the Stress of the bar” is part of a course in Statics and Mechanics of Materials (SOM). In this learning outcome, students are expected to be able to calculate the stress in a bar.

Stress is a measure of the force per unit area that is acting on a material. In mechanics of materials, stress is a crucial concept in understanding the behaviour of materials under loads. The stress in a bar can be calculated using the formula:

Stress = Force / Area

where Force is the external force acting on the bar, and Area is the cross-sectional area of the bar. This formula is only applicable to uniaxial loads, meaning that the load is applied along a single axis.

To apply this formula, it is necessary to know the cross-sectional area of the bar, which can be found by using the formula:

Area = Ï€ * (d/2)^{2}

where d is the diameter of the bar.

It is also important to know the force that is acting on the bar. This can be found by summing up all the forces acting on the bar, including the weight of the bar itself and any external loads applied to the bar.

Once the Force and Area are known, the stress in the bar can be calculated using the formula:

Stress = Force / Area

The units of stress are usually given in Pascals (Pa) or MegaPascals (MPa).

In conclusion, the ability to calculate the stress in a bar is a fundamental skill in the study of mechanics of materials. This knowledge is important in understanding the behaviour of materials under load, which is crucial for design and analysis of structures and mechanical components.

State Principal of St. Venant’s

State the principle of Saint-Venant” is part of a course in Statics and Mechanics of Materials (SOM). In this learning outcome, students are expected to be able to state the principle of Saint-Venant, which is a fundamental concept in the analysis of the behaviour of materials under load.

The principle of Saint-Venant is named after the French mathematician and engineer Augustin-Louis Cauchy, who was also known as the Baron de Saint-Venant. The principle states that the behaviour of a material under load can be understood by considering the small changes in the shape and size of the material rather than the entire deformation. In other words, the principle of Saint-Venant suggests that the behaviour of a material can be understood by considering the effects of the loads locally, rather than globally.

This principle is particularly useful in the analysis of beams, where it is often difficult to consider the entire deformation of the material. By considering the small changes in the shape and size of the material, it becomes possible to make predictions about the behaviour of the beam under load, including the distribution of stresses and strains along the length of the beam.

The principle of Saint-Venant is based on the idea that the stresses and strains in a material are proportional to the loads applied to it. This means that the behaviour of the material can be predicted by considering the loads locally, rather than globally. This makes it possible to simplify the analysis of complex structures by breaking them down into smaller, more manageable components.

In conclusion, the principle of Saint-Venant is a fundamental concept in the analysis of the behaviour of materials under load. By considering the small changes in the shape and size of a material, it becomes possible to simplify the analysis of complex structures and make predictions about the behaviour of the material under load. This knowledge is important for the design and analysis of structures and mechanical components.

Define Strain

Define Strain” is part of a course in Statics and Mechanics of Materials (SOM). In this learning outcome, students are expected to understand the concept of strain.

Strain is a measure of the deformation of a material caused by an applied load. It is defined as the ratio of the change in length of a material to its original length. Mathematically, strain can be represented as:

Strain = Î”L / L_{0}

where Î”L is the change in length of the material, and L_{0} is the original length of the material.

Strain is an important concept in mechanics of materials because it allows us to understand how a material will deform when subjected to an external load. By knowing the strain in a material, we can make predictions about how the material will behave under different loads, including the distribution of stresses and strains along the length of the material.

There are two types of strain: longitudinal strain and shear strain. Longitudinal strain is the strain caused by the stretching or compression of a material along its length, while shear strain is the strain caused by the sliding of one layer of a material relative to another layer.

Strain is a unitless quantity, but it is commonly expressed as a percentage. For example, a strain of 0.01 represents a 1% change in length.

In conclusion, strain is a crucial concept in the analysis of the behaviour of materials under load. By understanding strain, we can make predictions about the deformation of a material when subjected to an external load, which is important for the design and analysis of structures and mechanical components.

List and explain various types of strain

List and explain various types of strain” is part of a course in Statics and Mechanics of Materials (SOM). In this learning outcome, students are expected to understand the different types of strain that can occur in a material.

There are two main types of strain: longitudinal strain and shear strain.

- Longitudinal strain: Longitudinal strain is the strain caused by the stretching or compression of a material along its length. It is the result of the change in length of the material in the direction of the applied load. Longitudinal strain is also known as axial strain.
- Shear strain: Shear strain is the strain caused by the sliding of one layer of a material relative to another layer. It occurs when a material is subjected to a load that is perpendicular to its length, causing the layers of the material to slide past each other. Shear strain is also known as transverse strain.

In addition to these two main types of strain, there are also other types of strain that can occur in a material. For example, torsional strain occurs when a material is subjected to a twisting load, and volumetric strain occurs when a material is subjected to a load that changes its volume.

It is important to understand the different types of strain because they can have different effects on the behaviour of a material. For example, longitudinal strain can result in the stretching or compression of a material, while shear strain can result in the sliding of the layers of a material. This information is important for the design and analysis of structures and mechanical components, as it allows engineers to predict the behaviour of a material under different loads.

In conclusion, the various types of strain, including longitudinal strain, shear strain, torsional strain, and volumetric strain, can have different effects on the behaviour of a material. By understanding these different types of strain, engineers can make accurate predictions about the behaviour of a material under load, which is important for the design and analysis of structures and mechanical components.

Calculate the Strain of the bar

Calculate the Strain of the bar” is part of a course in Statics and Mechanics of Materials (SOM). In this learning outcome, students are expected to be able to calculate the strain of a bar.

Strain is a measure of the deformation of a material caused by an applied load, and it is defined as the ratio of the change in length of a material to its original length. To calculate the strain of a bar, the following formula can be used:

Strain = Î”L / L_{0}

where Î”L is the change in length of the bar, and L_{0} is the original length of the bar.

To calculate the strain of a bar, the following steps can be followed:

- Determine the original length of the bar: This can be done by measuring the length of the bar before it is subjected to any load.
- Determine the change in length of the bar: This can be done by measuring the length of the bar after it is subjected to a load and subtracting the original length.
- Calculate the strain: Using the formula, substitute the values of Î”L and L
_{0}into the equation and solve for the strain.

It is important to note that strain is a unitless quantity, but it is commonly expressed as a percentage. For example, a strain of 0.01 represents a 1% change in length.

In conclusion, calculating the strain of a bar is an important step in understanding the deformation of a material under load. By knowing the strain, engineers can make predictions about the behaviour of the bar under different loads, including the distribution of stresses and strains along the length of the bar. This information is important for the design and analysis of structures and mechanical components, as it allows engineers to predict the behaviour of a material under different loads.

State Hooke’s Law

State Hooke’s Law” is part of a course in Statics and Mechanics of Materials (SOM). In this learning outcome, students are expected to be familiar with Hooke’s Law.

Hooke’s Law is a fundamental principle in the field of mechanics that states that the strain experienced by a material is proportional to the applied load. It is expressed mathematically as:

Strain = K * Load

where K is the proportionality constant, also known as the modulus of elasticity, and Load is the force applied to the material.

Hooke’s Law is an important principle in the field of mechanics because it provides a simple and straightforward relationship between the load applied to a material and the resulting strain. It allows engineers to predict the behaviour of a material under different loads, and it is used as a basis for the design and analysis of structures and mechanical components.

It is important to note that Hooke’s Law is only applicable to materials that exhibit linear elastic behaviour. This means that the material will return to its original shape and size once the load is removed. Materials that do not exhibit linear elastic behaviour, such as metals that have been subjected to high loads for an extended period of time, may not obey Hooke’s Law.

In conclusion, Hooke’s Law is a fundamental principle in the field of mechanics that states that the strain experienced by a material is proportional to the applied load. It provides a simple and straightforward relationship between the load and the strain, and it is used as a basis for the design and analysis of structures and mechanical components. By understanding Hooke’s Law, engineers can make accurate predictions about the behaviour of a material under different loads.

Explain the behaviour of the Stress-strain curve

Explain the behaviour of the Stress-strain curve” is part of a course in Statics and Mechanics of Materials (SOM). In this learning outcome, students are expected to understand the behaviour of the stress-strain curve.

The stress-strain curve is a graphical representation of the relationship between stress and strain in a material. It is a plot of stress (usually on the y-axis) versus strain (usually on the x-axis) for a material subjected to increasing loads. The curve is typically obtained through testing, where the stress and strain are measured and plotted for a series of increasing loads.

The behaviour of the stress-strain curve is dependent on the material properties of the material being tested. For materials that exhibit linear elastic behaviour, the stress-strain curve is linear and the material returns to its original shape and size once the load is removed. For these materials, the modulus of elasticity (Young’s modulus) can be determined from the slope of the stress-strain curve in the elastic region.

For materials that do not exhibit linear elastic behaviour, the stress-strain curve is non-linear. The curve may show a linear elastic region, followed by a region of plastic deformation, and eventually a region of failure. In the plastic deformation region, the material will not return to its original shape and size once the load is removed, and the stress-strain curve will continue to increase, until the material eventually fails. The point at which the material fails is known as the ultimate strength, and it is usually the highest point on the stress-strain curve.

In conclusion, the stress-strain curve is a graphical representation of the relationship between stress and strain in a material. The behaviour of the curve is dependent on the material properties of the material being tested, and it provides valuable information about the behaviour of the material under different loads. By understanding the behaviour of the stress-strain curve, engineers can make accurate predictions about the behaviour of a material under different loads, and they can use this information to design and analyze structures and mechanical components.