STRUCTURES VECTOR ANALYSIS

First used by Leonardo da Vinci, graphic vector analysis is a powerful method to analyze  and visualize the flow of forces through a structure.  However, the use of this method is restricted to statically determinate systems.  In addition to forces, vectors may represent  displacement, velocity, etc.  Though only two-dimensional forces are described here, vectors may represent forces in three-dimensional space as well.  Vectors are defined by  magnitude, line of action, and direction, represented by a straight line with an arrow and  defined as follows:

Magnitude is the vector length in a force scale, like 1” =10 k or 1 cm=50 kN
Line of Action is the vector slope and location in space
Direction is defined by an arrow pointing in the direction of action

1  Two force vectors P1 and P2 acting on a body pull in a certain direction.  The resultant R is a force with the same results as P1 and P2 combined, pulling in the  same general direction.  The resultant is found by drawing a force parallelogram [A]  or a force triangle [B].  Lines in the vector triangle must be parallel to corresponding  lines in the  vector plan [A].  The line of action of the resultant is at the intersection  of P1 / P2 in the vector plan [A].  Since most structures must be at rest it is more  useful to find the  equilibriant E that puts a set of forces in equilibrium [C].  The  equilibriant is equal in magnitude but opposite in direction to the resultant.  The  equilibriant closes a force triangle with all vectors connected head-to-tail.  The line  of action of the equilibriant is also at the intersection of P1/P2 in the vector plan [A].

2  The equilibriant of three forces [D] is found, combining interim resultant R1-2 of  forces P1 and P2 with P3 [E].  This process may be repeated for any number of  forces.  The interim resultants help to clarify the process but are not required [F].  The line of action of the equilibriant  is located at the intersection of all forces in the  vector plan [D].  Finding the equilibriant for any number of forces may be stated as  follows:

The equilibriant closes a force polygon with all forces connected head-to-tail,  and puts them in equilibrium in the force plan.

3  The equilibriant of forces without a common cross-point [G] is found in stages:   First the interim resultant R1-2 of P1 and P2 is found [H] and located at the  intersection of P1/P2 in the vector plan [G].  P3 is then combined with R1-2 to find  the equilibriant for all three forces, located at the intersection of  R1-2 with P3 in the  vector plan.  The process is repeated for any number of forces.