Chapter 23: Transformations of Stress and Strain

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23.1:

Transformation of Plane Stress

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Mechanical Engineering

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JoVE Core Mechanical Engineering

Transformation of Plane Stress

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23.2: Principal Stresses

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Consider a point on a cube experiencing plane stress, defined by its stress components. If the element rotates by an angle, its stress components change. This change is determined by considering a prismatic element with faces perpendicular to its axes. By considering the area of the oblique face, the areas of the vertical and horizontal faces can be calculated using trigonometric functions of the rotation angle. The forces exerted on these faces are represented, and it is assumed that no forces act on the triangular faces of the element. The equilibrium equations are then written using components along the rotated axes. Upon solving these equations, the normal and shearing stresses are derived. The normal and shearing stresses are then re-expressed using trigonometric relations. An expression for the normal stress on the rotated vertical axis is obtained by substituting the rotation angle in one of the previous expressions with a new angle. It is observed that the sum of the normal stresses exerted on a cubic element remains independent of the element's orientation.

23.1:

Transformation of Plane Stress

Studying stress transformation is essential in understanding how stress components within a material, like a cube under plane stress, change with rotation. This change is analyzed by considering a prismatic element within the cube. As the element rotates, the stress components acting on it—both normal and shearing stresses—change in magnitude and orientation. This change is quantified using trigonometric functions of the rotation angle, relating the forces acting on the rotated element's faces to those on its original perpendicular faces.

The equilibrium equations, formulated by considering only the forces on faces perpendicular to the principal axes (excluding any forces on the triangular faces due to rotation), enable the derivation of new stress components. The normal and shearing stresses are re-expressed in terms of the original stresses.

A new expression for normal stress on the rotated vertical axis is obtained by replacing the rotation angle in a previous expression with a new one.

A notable outcome of this analysis is that the sum of normal stresses does not change regardless of the orientation of the cubic element. This invariance highlights the material's isotropic response to external stresses and is crucial for predicting material behavior under different loading conditions. Understanding how stress components transform with element orientation is vital for predicting material failure modes and designing materials and structures that are more resilient to applied loads.