2.6:
Two-Dimensional Force System
Consider a structural element in a two-dimensional space where a force acts at an angle theta with the x-axis.
Considering that the line of action of the force passes through the origin, its components can be expressed in the Cartesian form. The direction of the force vector is always given by tan inverse of the ratio of its components.
Now, even if the line of action of the force vector does not pass through the origin, its vector components can still be expressed in Cartesian form. The sign convention of these vector components can be chosen depending on their direction.
Here, the force is directed at an angle pi minus theta, measured counterclockwise from the positive y-axis.
Now, consider a structure where the line of action of the force makes an arbitrary angle alpha minus beta to the positive x-axis of the chosen coordinate system. The components of the force can be resolved using a similar analysis.
2.6:
Two-Dimensional Force System
A two-dimensional system in mechanical engineering involves the analysis of motion and forces in a plane. A two-dimensional force vector can be resolved into its components as:
where Fx and Fy are the vector components of F in the x and y directions, respectively. Each of these vector components can be represented as a scalar (Fx and Fy) times the appropriate unit vector.
To determine the components of the force vector in a Cartesian coordinate system, one must first determine the angle θ that the force makes with the positive x-axis. Assuming that the line of action of the force passes through the origin, its components can be expressed in the Cartesian form using basic trigonometry.
where F denotes the magnitude of the force vector. The direction of the force vector is given by the inverse of the tangent of the ratio of its components.
However, in cases where the line of action of the force vector does not pass through the origin, its vector components can still be expressed in Cartesian form using the same approach. We can choose the sign of each component based on the direction of the force vector. By resolving the force vector into its components, we can determine the force's net effect on the structure in question.
Understanding the two-dimensional force system is critical for engineers to analyze and design structures that are safe and structurally sound. This knowledge provides the foundational understanding of how a structure will react to various forces and helps pinpoint the design weaknesses, if any.
Suggested Reading
- Meriam, J.L.; Kraige, L.G. and Bolton, J.N. (2020). Engineering Mechanics ‒ Statics. Hoboken, New Jersey: John Wiley. pp 33