2.1 Statics Fundamentals
        Vectors and Forces | Free-Body Diagrams | Equilibrium

» Vectors and Forces
A solid understanding of VECTORS is a must in the topics of Statics and Strength of Materials. Vectors are used to describe the magnitude and direction of Forces and Torques/Moments. In many cases, it is necessary to transform or reduce vectors into a convenient coordinate system. For instance, the force at right, F, can be broken into its x and y components:

Fx = Fcos(q )   and   Fy = Fsin(q )

» Free-Body Diagrams
The proper 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 FBDs:

  • Clearly ISOLATE the part(s) of the system that you are interested in analyzing from its surroundings; visualize or actually draw a boundary between the body of interest and its surroundings.
  • IDENTIFY and REPRESENT ALL EXTERNAL FORCES (and moments) acting on the body - the forces that act across the boundary; include weight when comparable to the applied forces;
  • Include a coordinate axis and the 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 dimensions are the primary information required.
To help illustrate the construction of a FBD, take for example the rear suspension of the mountain bike below:

Full Suspension Bicycle

Free-body diagram of rear suspension

» Equilibrium
Once a FBD of a system is constructed, to identify the magnitude and direction of the forces, it is necessary to apply the concepts and conditions of STATIC EQUILIBRIUM. A body in static equilibrium is not accelerating, so the forces and moments acting on it must be in balance (sum to zero in all directions). The equations of static equilibrium require that the following conditions be satisfied:

  • The sum of the forces in any given direction is zero (3-dimensions):

    S Fx = 0 ;     S FY = 0 ;     S Fz = 0

  • The sum of the moments about any given point in a plane (about any axis perpendicular to that plane) must also be zero (3-dimensions):

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

  • In 2-dimensions, three equilibrium equations are required. When the object is in the x-y plane, one of the following three forms may be used:
S Fx = 0
S FY = 0 
S Mz = 0
S Fx = 0
S Mz,A = 0
S Mz,B = 0
S Mz,A = 0
S Mz,B = 0
S Mz,C = 0
Sum of forces in x, and in y, and the sum of moments about any point equals zero where Points A and B are two different points. where A, B and C are three different points, not all on the same line.

Many Strength of Materials problems can be reduced to 2-dimensions.
Updated: 7/12/06