Machine Design

How-to, in-depth technical articles for machine design engineers

Intro to Injection Molding

Provides a basic understanding of plastic injection molding presses, processes, and costs.

Autonomous Vehicle Engineering

The No. 1 media source for those developing the next generation mobility solutions.

Wind Systems

Towers, turbines, gearboxes; processes for shaping and finishing component parts.

more free magazines
Coordinate Transformation is Necessary

The generalized Hooke's law of a fiber-reinforced lamina for the principal directions is not always the most convenient form for all applications. Usually, the coordinate system used to analyze a structure is based on the shape of the structure rather than the direction of the fibers of a particular lamina.

For example, to analyze a bar or a shaft, we almost always align one axis of the coordinate system with the bar's longitudinal direction. However, the directions of the primary stresses may not line up with the chosen coordinate system. For instance, the failure plane of a brittle shaft under torsion is often at a 45° angle with the shaft. To fight this failure mode, layers with fibers running at ± 45° are usually added, resulting in a structure formed by laminae with different fiber directions. In order to "bring each layer to the same table," stress and strain transformation formulae are required.

Coordinate Transformation of Stress-Strain Relations for Lamina

If we define the coordinate transformation matrix as




The coordinate transform of plane stress can be written in the following matrix form:

Similarly, the strain transform becomes

Please notice that the tensor shear strain is used in the above formula. Suppose we define the engineering-tensor interchange matrix [R]



The stress-strain relations for a lamina of an arbitry orientation can therefore be derived as detailed below.

where the stiffness matrix is defined as

The complicance matrix is therefore

The individual components of the stiffness and compliance matrices can be found here.