Strength of Materials (SOM), SFD, BMD, Properties of Materials, Normal Stress and Strain

Strength of Materials, often abbreviated as SOM, is a foundational branch of engineering mechanics that studies how different materials respond when subjected to forces, loads, and moments. It explains how solids deform, resist, and eventually fail under different types of stresses and strains. The study of SOM forms the basis for civil, mechanical, and structural engineering because it connects theoretical physics with real-world construction, manufacturing, and design.

The behavior of a beam under a load, the deflection of a bridge under traffic, the compression of a column, or the stretching of a cable—all can be predicted using the principles of Strength of Materials. Engineers use these principles to design safe, durable, and efficient structures. Every nut, bolt, beam, and shaft must be strong enough to carry the forces acting on it, and SOM provides the scientific method to evaluate this strength.

Understanding the Core Idea of SOM

The central idea behind SOM is the relationship between the external forces applied to a body and the internal resistance developed within the body. When an external force acts on a structural member, internal forces are generated inside the material to resist deformation. These internal forces per unit area are called stresses, and the deformation per unit length is called strain.

By studying stress and strain, engineers can understand how materials behave under various conditions—whether they stretch, compress, bend, twist, or break. Strength of Materials bridges the gap between theory and application by teaching how to calculate, control, and predict these behaviors accurately.

Shear Force and Bending Moment (SFD and BMD)

In most engineering structures, such as beams, bridges, and machine parts, loads cause bending and shearing effects. To analyze these effects, engineers use two graphical tools—the Shear Force Diagram (SFD) and the Bending Moment Diagram (BMD). These diagrams help visualize how internal shear forces and bending moments vary along the length of a beam under various loading conditions.

1. Concept of Shear Force

When a beam is loaded, each section of the beam tends to move relative to its adjacent section. The internal resistance that prevents this sliding motion is known as shear force. The shear force at a particular cross-section of a beam is equal to the algebraic sum of all the vertical forces acting to the left (or right) of the section.

A positive shear force usually acts in a clockwise direction on the left side of the section and in an anticlockwise direction on the right side. The sign convention is important because it helps ensure consistency while constructing the diagram.

Shear force gives engineers valuable information about where the beam is likely to experience sliding failure or shear cracking. High shear forces often occur near supports, where vertical reactions are large.

2. Concept of Bending Moment

When external loads act on a beam, they cause it to bend. The internal resistance offered by the material to oppose this bending is known as the bending moment. The bending moment at a section of a beam is the algebraic sum of the moments of all the external forces acting to one side of the section.

A positive bending moment (also called a sagging moment) causes the beam to bend into a concave shape—tension at the bottom and compression at the top. A negative bending moment (hogging moment) causes the beam to bend in the opposite direction.

Bending moment values are critical for determining the beam’s safe dimensions, selecting materials, and ensuring that the structure can resist failure due to bending stress.

3. Relationship between Load, Shear Force, and Bending Moment

The rate of change of shear force along the beam equals the intensity of the distributed load. Similarly, the rate of change of bending moment equals the shear force at that section. These relationships help in determining the SFD and BMD from the loading diagram.

For example:

• When the load is constant (uniformly distributed load), the shear force changes linearly and the bending moment varies parabolically.
• When there is no load between two points, the shear force remains constant and the bending moment changes linearly.

These relationships are the backbone of beam analysis and help engineers plot accurate diagrams.

4. Drawing SFD and BMD

To construct a Shear Force Diagram and a Bending Moment Diagram:

1. Identify the type of beam—simply supported, cantilever, or overhanging.
2. Determine all external loads, reactions at supports, and moments.
3. Starting from one end, calculate shear force values at key points.
4. Plot these values along the beam to form the SFD.
5. Integrate the shear force values to obtain bending moments.
6. Plot bending moments at key points to obtain the BMD.

For a simply supported beam with a central load, the shear force diagram consists of two straight lines of equal magnitude but opposite sign, while the bending moment diagram is triangular, peaking at the center.

5. Importance of SFD and BMD

SFD and BMD are powerful tools in mechanical and civil engineering design. They:

• Identify maximum shear force and bending moment locations.
• Help determine beam size, material, and reinforcement requirements.
• Predict failure zones and optimize structural safety.
• Simplify analysis of complex loading patterns.

These diagrams are not just theoretical tools but essential design instruments used daily by structural engineers.

Properties of Materials

The behavior of materials under applied forces depends on their inherent physical and mechanical properties. Understanding these properties helps in choosing the right material for a particular application. In Strength of Materials, the most significant mechanical properties include elasticity, plasticity, ductility, malleability, hardness, toughness, and resilience.

Elasticity

Elasticity is the property of a material that enables it to regain its original shape and size after the removal of load. A perfectly elastic material returns to its initial dimensions completely, though no real material is perfectly elastic. Steel is considered highly elastic among engineering materials, while rubber, though flexible, is less elastic because it does not completely return to its original shape.

Plasticity

Plasticity is the property that allows a material to undergo permanent deformation without breaking when the load exceeds the elastic limit. Materials like lead, copper, and aluminum exhibit high plasticity and can be molded or shaped easily. Plasticity is vital for manufacturing processes like forging, rolling, and extrusion.

Ductility

Ductility refers to the ability of a material to be drawn into thin wires without breaking. Ductile materials undergo considerable elongation before fracture. Steel, copper, and aluminum are ductile, whereas cast iron and glass are brittle. Ductility indicates a material’s capacity to absorb tensile stress before failure.

Malleability

Malleability is the ability of a material to be hammered or rolled into thin sheets without cracking. Gold, silver, and aluminum are highly malleable. This property is essential for sheet-metal work, car body manufacturing, and decorative applications.

Toughness

Toughness is the ability of a material to absorb energy and deform plastically before fracturing. It measures resistance to shock and impact. Tough materials like steel and brass can withstand sudden loading and are used in components such as gears, shafts, and springs.

Hardness

Hardness represents resistance to wear, scratching, and indentation. It is an important property for cutting tools and machine parts. Hardness testing methods like Brinell and Rockwell help evaluate how well a material resists deformation under concentrated loads.

Resilience

Resilience is the ability of a material to absorb energy when it is elastically deformed and to release that energy upon unloading. Materials with high resilience are used in springs, shock absorbers, and suspension systems.

Creep and Fatigue

Creep is the slow, time-dependent deformation of a material under constant stress, typically at high temperatures. Fatigue, on the other hand, is failure caused by repeated cyclic loading. Both are important in mechanical design for engines, turbines, and bridges.

Understanding these properties ensures engineers select materials that will perform reliably in their intended environments.

Normal Stress and Strain

Normal stress and strain describe how materials behave when subjected to axial (direct) loads, such as tension or compression.

1. Normal Stress

Normal stress is the internal resistance developed per unit area within a material when it is subjected to a load perpendicular to its cross-section. It can be either tensile (when the material is stretched) or compressive (when it is shortened).

The concept of normal stress helps engineers determine whether a material can safely carry a given load without exceeding its safe stress limit.

2. Normal Strain

Normal strain is the ratio of change in length to the original length of a material when a load is applied. It expresses how much a material deforms relative to its size. A small strain indicates that the material is stiff, while a large strain indicates it is flexible or ductile.

3. Hooke’s Law

Hooke’s Law establishes the fundamental relationship between stress and strain within the elastic limit of a material. It states that the strain in a material is directly proportional to the stress applied to it, up to the elastic limit. This proportionality is characterized by the modulus of elasticity or Young’s modulus, which measures stiffness.

Materials with a high modulus of elasticity, like steel, are rigid and deform very little, whereas materials with low modulus, like rubber, are more flexible.

4. Stress-Strain Curve

The stress-strain curve graphically represents how a material behaves under increasing loads. It includes important points such as:

• Proportional limit – up to which stress and strain are directly proportional.
• Elastic limit – the maximum stress up to which the material returns to its original shape.
• Yield point – the point beyond which permanent deformation begins.
• Ultimate strength – the maximum stress the material can withstand.
• Fracture point – where failure occurs.

This curve helps engineers understand the full mechanical behavior of materials under load and design accordingly.

5. Importance of Stress and Strain Analysis

By studying stress and strain, engineers can:

• Predict how much a structure will deform under load.
• Ensure that stress levels remain within safe limits.
• Compare the strength and stiffness of different materials.
• Design structures that are both safe and economical.

Normal stress and strain analysis forms the foundation of structural safety and mechanical design.

Integration of SOM Concepts in Real Engineering

All the above principles—SFD, BMD, material properties, and stress-strain relationships—are interconnected. In a real engineering structure:

• SFD shows the internal shear forces.
• BMD shows internal bending moments.
• Material properties define how the structure will resist those forces.
• Stress and strain determine whether the material will deform safely or fail.

A complete understanding of these concepts allows engineers to optimize design for strength, weight, and cost. Whether designing bridges, machines, or aircraft, SOM principles ensure performance and safety.

FAQs About SOM, SFD, BMD, and Stress–Strain

1. What is the difference between stress and strain?

Stress is the internal resistance offered by a material per unit area, while strain is the deformation per unit length caused by the stress.

2. Why are SFD and BMD important in beam analysis?

They show how shear forces and bending moments vary along a beam, helping engineers locate critical sections and design safe beams.

3. What materials are most elastic?

Steel is highly elastic, followed by aluminum and brass. Rubber, although flexible, is less elastic because it does not fully recover its shape.

4. What does Hooke’s Law state?

Hooke’s Law states that stress is directly proportional to strain within the elastic limit of a material.

5. What is the significance of the stress-strain curve?

It shows how materials behave under increasing loads and indicates important properties such as yield strength, ductility, and ultimate stress.

6. What are ductile and brittle materials?

Ductile materials, like steel and copper, deform significantly before fracture. Brittle materials, like glass and cast iron, break with little deformation.

7. How do engineers use SFD and BMD diagrams?

They use these diagrams to identify points of maximum bending moment and shear force, ensuring structures can resist these forces safely.

8. What is meant by the elastic limit?

It is the maximum stress up to which a material returns to its original shape after the load is removed.

9. Why is the study of Strength of Materials important?

It enables engineers to design structures and machines that can safely withstand applied loads without failure.

10. What is the role of material properties in design?

Material properties determine how a structure will respond to stress, strain, heat, and other environmental conditions, ensuring reliability and safety.