An ellipse is an elongated shape that is defined by the equation: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where \(a\) and \(b\) are the semi-major and sem...
An ellipse is an elongated shape that is defined by the equation:
where (a) and (b) are the semi-major and semi-minor radii, respectively. An ellipse can be considered a combination of two circles with the same center.
A rectangular bar is a bar that is wider than it is thick. The perimeter of a rectangular bar can be calculated by multiplying the length, width, and height of the bar.
Torsion is the deformation of a shape under a twisting force. The torsion of an elliptical and rectangular bar can be calculated using the following formulas:
Torsional shear stress: (\tau = \frac{M_T}{\pi a^2})
Torsional strain: (\varepsilon = \frac{d\theta}{dr})
Torsional angle: (\theta = \tan^{-1}\left(\frac{b}{a}\right))
where (M_T) is the total torque, (d\theta) is the amount of rotation, and (r) is the radial distance.
Examples:
A circular shaft will experience torsional deformation when subjected to a twisting force. The amount of deformation will depend on the length of the shaft, the applied torque, and the material of the shaft.
A rectangle will also experience torsional deformation when subjected to a twisting force. The amount of deformation will also depend on the dimensions of the rectangle, the applied torque, and the material of the rectangle