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Coordinate System (Sign Convention)

Internal forces

Internal forces are defined as:

The positive direction of internal forces is evident from the following figure:

The principal moments and directions of principal axes are provided by:

The meaning of individual variables is the following: internal forces can be transformed from the (x, y) coordinate system to the (x', y') coordinate system by rotating the (x, y) plane through a certain angle about the z-axis. The angle α, in particular, corresponds to a rotation angle for which the transformed m_x'y' moment attains a zero value whereas the m_x' and m_y' moments attain their maximum and minimum values m₁ and m₂, respectively.

The maximum shear force is obtained similarly:

and the angle between v_max and the x-axis:

Load

The sign convention of the applied force and moment load is evident from the following figure:

It is worth to point out a different sign convention applied to the load moment M (at a point or along a line) and to internal moment m. While the M_x moment rotates about the x-axis (as usual for beams), the internal moment rotating about the x-axis is denoted as m_y.