Material and Dimensional Considerations

The selection of material and geometry in structural design depends on the magnitude and direction of maximum and minimum stresses. These stresses are often multi-directional and require tensor representation for accurate analysis.

Tensor Representation

A tensor is a multi-dimensional quantity that generalizes scalars and vectors. For a material point under three-dimensional stress, the stress and strain tensors are:

Stress Tensor:

$$ \boldsymbol{\sigma} = \begin{bmatrix} \sigma_x & \tau_{xy} & \tau_{xz} \\ \tau_{xy} & \sigma_y & \tau_{yz} \\ \tau_{xz} & \tau_{yz} & \sigma_z \end{bmatrix} $$

Strain Tensor:

$$ \boldsymbol{\varepsilon} = \begin{bmatrix} \varepsilon_{xx} & \frac{\gamma_{xy}}{2} & \frac{\gamma_{xz}}{2} \\ \frac{\gamma_{yx}}{2} & \varepsilon_{yy} & \frac{\gamma_{yz}}{2} \\ \frac{\gamma_{zx}}{2} & \frac{\gamma_{zy}}{2} & \varepsilon_{zz} \end{bmatrix} $$

Engineering shear strain \( \gamma \) is related to tensor shear strain \( \varepsilon \) by:

$$ \gamma_{xy} = 2\varepsilon_{xy}, \quad \gamma_{yz} = 2\varepsilon_{yz}, \quad \gamma_{zx} = 2\varepsilon_{zx} $$

Constitutive Relation

The material behavior under stress is described by the constitutive equation:

$$ \boldsymbol{\sigma} = \mathbf{C} \boldsymbol{\varepsilon} $$

where \( \mathbf{C} \) is the matrix of elastic constants, dependent on material properties such as Young’s modulus and Poisson’s ratio.

Cauchy’s Stress Theorem

The state of stress at a point is defined by stress vectors on all planes passing through that point. According to Cauchy’s stress theorem, knowing the stress vectors on three mutually perpendicular planes is sufficient to determine the stress vector on any other plane via coordinate transformation.

Bi-Axial System Simplification

For practical analysis, structures are often simplified to one-dimensional or two-dimensional models based on geometry and loading. In biaxial systems, two key conditions are considered:

  • Plane Stress: Thin structures where stress in the thickness direction is negligible.
  • Plane Strain: Long structures where strain in one direction is negligible.