Notes on Stress Concepts and Analysis

Introduction to Stress Concepts

• Objective: Develop concepts related to stress for this course and future studies.
• Focus on slender members, but begin with three-dimensional bodies.
• Elastic Continuum: The only material aspect considered is that the material behaves as an elastic continuum.

Historical Context

• Hooke's Law: Introduced the concept of deformation and its linearity in a restricted range.
• Bernoulli's Contribution: Developed the idea of stress as force per unit area (P/A), significant for solving problems involving internal pressure on thin rings.

Stress Analysis of Materials

• Hoop Stress: For a thin ring under internal pressure, stress is given by the formula:
  [ \text{Hoop Stress} = \frac{P \cdot r}{t} ]
• Extension in Tension: Extension given by:
  [ \Delta = \frac{P \cdot L}{A imes E} ]
• Radial Elongation: Expressed as (\Delta T / (2 \pi)).

Observations from Experiments

• Chalk Failure under Axial Tension: Failure occurs in line with the cross-section.
• Torsion Effects: Failure angle is at 45 degrees, highlighting the need to analyze stress beyond simple cross-sectional views.
• Photoelasticity: Utilized to visualize stress variation in materials.

Stress at a Point

• Consider a three-dimensional body in equilibrium, subjected to body and surface forces.
• Generic Point (P): An imaginary plane is cut through point P to analyze stress.
• Stress vectors are defined through limits as the area surrounding point P tends to zero.
• Stress Vector (T): Defined as ( \frac{\Delta T}{\Delta A} ) in the limit as (\Delta A) approaches zero.

State of Stress at a Point

• Totality of Stress Vectors: To understand stress at a point, consider the stress vectors on all infinite planes through that point.
• Resolution of Stress Vectors: Stress vectors can be resolved into normal and tangential components.
  • Normal Stress (σ): (\sigma_n)
  • Tangential Stress (τ): (\tau)

Components of Stress Vectors

• Identification of Planes: Use Cartesian coordinates to define stress on mutually perpendicular planes (X, Y, Z).
• Notation for Stress Components:
  • (\sigma_{xx}, \sigma_{yy}, \sigma_{zz}) for normal stresses.
  • (\tau_{xy}, \tau_{yx}, \tau_{xz}, \tau_{yz}, \tau_{zx}, \tau_{zy}) for shear stresses.

Mathematical Representation of Stress

• Stress vectors can be represented as a matrix, which is a tensor of rank 2:
  • [ \begin{bmatrix} \sigma_{xx} & \tau_{xy} & \tau_{xz} \ \tau_{yx} & \sigma_{yy} & \tau_{yz} \ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{bmatrix} ]
• Stress vectors on any arbitrary plane can be derived from the stress vectors on three mutually perpendicular planes.

Conclusion

• Developed the concept of stress at a point, emphasizing the transition from cross-sectional analysis to point-wise stress analysis.
• Established the significance of understanding stress vectors and their resolution to predict material failure.
• Introduced the tensor representation of stress, enhancing the mathematical treatment of stress analysis.