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Fundamentals

Equilibrium Equations

If a load is applied on an object, it moves in the direction of the load unless there is another load acting in the opposite direction such that the sum of forces is zero. Therefore, to prevent the object (or structure) from moving along the three orthogonal (perpendicular) directions, the sum of forces along these directions has to be equal to zero. Satisfying these conditions, ensures that the structure does not move. However, if the resultant of the forces acting on the structure are not along the same line the structure may rotate. Therefore, to prevent the rotation of the structure about any axis, the sum of moments about the three orthogonal axes has to also be equal to zero.

Concept of Equilibrium

 

In general, to prevent the structure from movements and rotations, there are six equilibrium equations that need to be satisfied (Considering the Cartesian Coordinate System):  [ Note: Σ is a Greek letter (sigma) which represent the sum. ]

efx=0 (Sum of forces along the x-axis=0)
efy=0 (Sum of forces along the y-axis=0)
efz=0 (Sum of forces along the z-axis=0)
emx=0 (Sum of moments about the x-axis=0)
emy=0 (Sum of moments about the y-axis=0)
emz=0 (Sum of moments about the z-axis=0)


If the structure can be represented in two-dimensions (within the x-y plane), only three equilibrium equations will exist:
 
 
    
     

 

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