Two Force Members
Two force members are bodies that are static and only subjected to two forces. Two force bodies must fulfill two conditions in order to be static:
- The two forces must be equal in magnitude and opposite in direction to fulfill \sum F=0.
- The lines of action for both forces must pass through both points of application, preventing a force couple, fulfilling \sum M=0.
Trusses
A truss is a structure comprised solely of two force members. Due to the above mentioned constraints placed on two force members, trusses are relatively simple to analyze. Due to the simplicity of analysis trusses have been a popular structure. Trusses can be found in buildings, roofs, bridges, machines, and even the Eiffel Tower.
Two-Dimensional Equilibrium Equations
In two dimensions we can reduce the static equilibrium equations. Because the z axis will experience no forces in two dimensions we can omit the F_z term. The only moment present in a 2D system is M_z, since it is the only possible moment that can be made using F_x and F_y.
\sum F_x = 0
\sum F_y = 0
\sum M_z = 0
Method of Joints
The method of joints focuses on the pin connections of a truss structure. By applying the force equilibrium equations and newtons third law to each pin, the forces acting upon the whole structure can be solved. The method of joints can be distilled into the following steps:
- Solve the structures reaction forces.
- Begin at a joint with:
- At least one known load.
- Two or fewer connected members.
- Use the equilibrium equations to solve the forces due to the members connected to the joint. Because members are two-force, the direction of forces must align with connecting members.
- Balance forces on newly solved members by applying an equal and opposite force on the joint connected to the unsolved end of the member.
- Repeat until all joints are solved for.
It should always be remembered when using the method of joints, the forces solved for act upon the joints. Therefore, due to Newton’s third law, each pin will exert a force against neighboring members in an equal and opposite manner. As a consequence, arrows pointing outwards from members indicate that the member is in compression. Conversely, arrows pointing inwards towards a member indicate the member is in tension. Again this is because our free-body diagram using this method indicates the forces that act upon the pins of the structure.
Finally, we can double-check our solution if we want. Check whether the moment at any point in the structure is zero and ensure that all forces cancel at every joint.
Method of Sections
The method of sections is useful if information about one particular member is desired. Instead of solving the entire structure with the method of joints we can simply cut through the structure and apply the equilibrium equations to solve the internal member forces. Since we only have three equilibrium equations, we can not cut through more than three members when using the method of sections.
Zero Force Members
Some truss members will theoretically carry zero load. Identification of these zero force members before beginning analysis will simplify and speed-up the problem solving process. As such, it is recommended to identify zero force members at the start of a truss analysis.
Joints with three members will have a zero force member if the following are true:
- Two members are collinear.
- There is no applied force acting on the joint (support or external).
Both members in a joint that connects two members will be zero force members if the following are true:
- Members are not collinear.
- There is no applied force acting on the joint (support or external).
