Chapter – 2 : JSSC JE Mechanical Notes

(Chapter – 2) Engineering Mechanics and Strength of Materials

• Equilibrium of Forces
• Law of motion
• Friction
• Concepts of stress and strain
• Elastic limit and elastic constants
• Bending moments and shear force diagram
• Stress in composite bars
• Torsion of circular shafts
• Buckling of columns
• Euler’s and Rankin’s theories
• Thin walled pressure vessels.

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stress (σ)

• जब किसी वस्तु पर कोई बाहरी बल लगाया जाता है, तो वस्तु के अंदर से एक आंतरिक प्रतिक्रिया बल (Internal restoring force) उत्पन्न होता है जो वस्तु के आकार में होने वाले परिवर्तन का विरोध करता है।
• एकांक क्षेत्रफल पर लगने वाले इसी आंतरिक बल को प्रतिबल कहते हैं।
• Stress is the internal resistance offered by a body to an external force.
• It is defined as the force applied per unit cross-sectional area. 
  • 
• Units:
  • SI मात्रक: N/m ² या Pascal (Pa)
  • CGS मात्रक: dyne/cm ²
  • विमीय सूत्र (Dimensional Formula): [ML ⁻¹ T ⁻² ]
• Tensile Stress: Occurs when a force pulls an object, tending to elongate it.
• Compressive Stress: Occurs when a force pushes an object, tending to shorten it.
• Shear Stress: Occurs when forces act parallel to the surface, causing layers of the material to slide past one another.

विकृति | Strain (ϵ)

• जब किसी वस्तु पर बाहरी बल लगाया जाता है, तो उसके मूल आकार (लंबाई, आयतन या रूप) में परिवर्तन होता है।
• वस्तु के आकार में होने वाले इस भिन्नात्मक परिवर्तन (Fractional change) को विकृति कहते हैं।
  • विकृति का कोई मात्रक नहीं होता (Unitless), क्योंकि यह दो समान राशियों का अनुपात है।
  • यह एक विमाहीन (Dimensionless) राशि है।
• Strain is the measure of the deformation of a material in response to stress.
• It represents the ratio of the change in dimension to the original dimension.
  • 

प्रकार (Type)	प्रतिबल (Stress)	विकृति (Strain)	विवरण
अनुदैर्ध्य (Longitudinal)	F / A	∇ L / L	लंबाई में परिवर्तन होने पर।
आयतन (Volume)	F / A (दाब)	∇ V / V$	आयतन में परिवर्तन होने पर।
अपरूपण (Shearing)	F / A	θ (कोण)	आकार (Shape) में परिवर्तन होने पर।

 

Hooke’s Law

• प्रत्यास्थता की सीमा (Elastic limit) के भीतर, stress हमेशा strain के समानुपाती होता है।
  • Stress ∝ Strain
  • Stress = E × Strain
    • E =  प्रत्यास्थता गुणांक (Modulus of Elasticity)
      •  मात्रक = N/m²

Stress-Strain Relationship

• For most metals, as stress increases, strain increases proportionally up to a certain point.
  • Proportional Limit: The stage where stress is directly proportional to strain (Hooke’s Law applies).
  • Elastic Limit: The maximum stress a material can withstand and still return to its original shape once the load is removed.
  • Yield Point: The point beyond which the material starts to deform plastically (permanently).
  • Ultimate Tensile Strength (UTS): The maximum stress the material can handle before “necking” begins.
  • Fracture Point: The point where the material finally breaks.

Young’s Modulus (E)

• Within the elastic region, the ratio of stress to strain is constant. This constant is known as Young’s Modulus (or the Modulus of Elasticity).
• It measures the stiffness of a material.
  • Stiffness (कठोरता या दृढ़ता) – कोई वस्तु बाहरी बल (Force) लगाने पर अपने आकार में होने वाले परिवर्तन का कितना विरोध करती है।
  • सरल शब्दों में, जिस वस्तु को मोड़ना, खींचना या दबाना जितना कठिन होता है, उसकी Stiffness उतनी ही अधिक होती है।
  • 
• High E: The material is very stiff (e.g., Steel).
• Low E: The material is flexible (e.g., Rubber).

Shear Modulus (Modulus of Rigidity)

• The Shear Modulus (G) measures a material’s resistance to shearing strain.
• It is defined as the ratio of shear stress to shear strain within the elastic limit.
• Unlike Young’s Modulus, which deals with pulling or pushing, the Shear Modulus deals with “sliding” forces that change the shape of an object without changing its volume.
• 

Poisson’s Ratio

• Poisson’s Ratio (ν) describes the phenomenon where a material tends to expand in directions perpendicular to the direction of compression, or contract in directions perpendicular to the direction of stretching.
• It is the ratio of lateral strain to longitudinal strain.
  • Stable Materials: Most common materials have a Poisson’s ratio between 0.0 and 0.5.
  • Rubber: Approaches 0.5 (nearly incompressible).
  • Cork: Near 0.0 (shows little lateral expansion when compressed).
  • Auxetic Materials: Have a negative Poisson’s ratio (they get thicker when stretched).
• 

Relationships Between Moduli

• In isotropic materials (materials that have the same properties in all directions), these constants are mathematically linked.
• If you know two, you can find the third using Young’s Modulus (E):
  • 

Property	Symbol	Measures Resistance To…	Primary Deformation
Young’s Modulus	E	Tensile/Compressive Stress	Length change
Shear Modulus	G	Shear Stress	Shape change (angle)
Bulk Modulus	K	Uniform Pressure	Volume change
Poisson’s Ratio	ν	Lateral Expansion	Width vs. Length ratio