2a. Statics Fundamentals  (SSES Chapt. 2.0)
Vectors and Forces Free-Body Diagrams Equilibrium
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Vectors and Forces
A solid grasp of vectors is vital to solve problems in Statics and Strength of Materials. Vectors describe the magnitude and direction of forces and torques/moments. It is often convenient to reduce vectors into their scalar components; e.g., 2D-force F can be broken into its x- and y-components:

 Fx = Fcosq   and   Fy = Fsinq 

where F is the magnitude of force F, and q is its direction measured from the x-axis.

Free-Body Diagrams
The construction of free-body diagrams (FBDs) is the first and most important step in solving a Strength of Materials problem. The following steps are involved in constructing a FBD:
  • Isolate from its surroundings the part(s) of the system that you are interested in analyzing; visualize or actually draw a boundary around the body of interest.
  • Identify and represent ALL external forces (and moments) acting on the body; i.e., forces that act across the boundary. Include weight when comparable to the applied forces.
  • Include a coordinate system and necessary dimensions on the diagram for convenience in applying equilibrium equations and communicating geometry.
  • The diagram should be free of clutter and extensive information. The forces, moments and key dimensions are the primary information required.
To help illustrate the construction of a FBD, consider the rear suspension of the mountain bike below:

Full Suspension Bicycle

Free-body diagram of rear suspension

Once a FBD system is constructed, the next step is to apply the conditions of equilibrium. A body in static equilibrium is not accelerating, so the forces and moments acting on it must sum to zero in any direction. Equilibrium requires that:
  • the sum of the forces in any direction must be zero: S F = 0.
  • the sum of the moments about any non-accelerating point must be zero: S MO= 0.

In 3-dimensions, six equations must be satisfied:

S Fx = 0 ;     S Fy = 0 ;     S Fz = 0

S Mx = 0 ;     S My = 0 ;     S Mz = 0

In 2-dimensions, only three equilibrium equations are required. When the object is in the x-y plane, one of the following three sets of equations may be used:

S Fx = 0
S Fy = 0 
Mz,A = 0
S Fx = 0
S Mz,A = 0
S Mz,B = 0
S Mz,A = 0
S Mz,B = 0
S Mz,C = 0
The sum of forces in x and in y, and the sum of moments about any Point A (about a z-axis through Pt. A), each equal zero Points A and B are two different points. A, B and C are three points, not all on the same line. The points define a plane.

Many 3D-problems can be reduced to 2-D.

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Updated: 05/24/2009 DJD