Lecture Notes: Mechanics of Solids - Stress Transformations

Introduction

• Presenter: Manas
• Topic: Stress transformations, principal stress, and strain (compound stress and strain).

Key Concepts

Tensor Basics

• Components of Stress: A point inside a rigid body is characterized by six normal and shear stress components.
• Effective Stress Components: Though initially counted as nine, effectively there are six components:
  • 3 normal stress components
  • 3 shear stress components

Equilibrium of Forces

• Force Equilibrium: For an object in equilibrium, the summation of forces along any axis (x, y, z) is zero.
• Moment Equilibrium: The summation of moments about any axis is also zero.

Stress in Rigid Bodies

• State of Stress: Illustrated with a small element showing different stress orientations (e.g., tau xy and tau yx).
• Shear Stresses: Tau xy equals tau yx, demonstrating symmetrical stress behavior.

Plane Stress

• Definition: Planes subjected to stress that can be analyzed on a single plane are referred to as plane stress.
• Effective Components: Total six components of stress act at a point inside a rigid body.

Stress on Oblique Planes

• Normal Stress on Oblique Plane: Normal stress calculation involves resolving components along the plane.
• Shear Stress on Oblique Plane: Uses the angle theta to determine stress on inclined planes.

Calculations

• Normal Stress (σn): Calculated as sigma x * cos²(theta), representing stress on an oblique plane.
• Shear Stress (τt): Calculated as (σx / 2) * sin(2θ), maximal when 2θ equals 90 or 270 degrees.

Cases Discussed

Case 1: Direct Stress in One Direction

• Assumption: Thickness is assumed as unity.
• Stress Distribution: Normal and tangential stress on an oblique plane.
• Maximal Conditions:
  • Maximum normal stress occurs when theta = 0 degrees.
  • Maximum shear stress occurs at theta = 45 degrees or 135 degrees.

Upcoming Topics

• Case 2: Two mutually perpendicular stresses.
• Case 3: Pure shear stress.
• Case 4: Combination of two stresses and shear stress.

Conclusion

• A summary of case 1 was discussed, emphasizing understanding the stress transformations in solid mechanics.
• Next lecture will cover the mutually perpendicular stresses and complex stress cases.