Simple Planar Truss

Determination of Forces in Trusses :

Eight equations with eight unknowns imply the forces in the truss is statically determinate. There are a number of way to determine forces in trusses by the method of joints.

Solving Equilibrium Equations:

In general the eight equilibrium equations can be solved mathematically. The eight equilibrium equations can be divided into two groups, i.e. with external reactions and without external reaction. Imply

Focusing on the group without external reaction, five equations with five unknowns imply the forces in the truss member can be determinated.

The external reaction can then be determined. Imply

Solving Static Equilibrium:

Or unknown forces can also be determined by making use of the mechanical static equilibrium state. When the completely constrained truss is in static equilibrium, all the unknown reactions can be determined. Imply

Since a simple truss is constructed by adding two truss members to form a new joint, in general, a joint involving two unknown forces from two truss members can always be found.  For a statically determinate truss, the unknown forces in the truss can always be determined by the static equilibrium of the joint pin connecting to two truss members. Therefore starting with a joint with two truss members, i.e. joint pin A. Imply

After two unknowns are determined, the unknowns of the two neighborhood joints will be reduced by one. Applying the same strategy to select neighborhood joints with two unknowns, i.e. joint pin D. Imply

Repeating the procedure untill all unknown are determined. All possible joints with two unknowns  i.e. joint pins B and C can be used. It can be either to continue selecting the neighborhood joint or to start with other possible joint from other ends. Use joint C, Imply

Solving Force Vector Diagram:

Sometimes, the unkonwn forces can be determined using graphical methods through the determination of force vector by constructing the force vector diagram at each joint. Starting the joint pin with known force, so that all force vector diagram can be scaled to a known value, i.e joint pin D, Although joint pin D is constained in y dimension only, joint pin D is still a good choice. Besides the equilibrium force vector diagram should always be draw in one direction for all joint pins, i.e. counter-clockwise direction  imply

Continuing the joint pin with known dimension, which is fully constrained by the known force before so that all force vector diagram can be scaled to a known value,  i.e joint pin C, imply

Continuing the joint pin with known dimension, which is fully constrained by the known force before so that all force vector diagram can be scaled to a known value,  i.e joint pin B. The vector diagram of Joint pin A is not fully constrained because the unknown reaction R is a vector of two rectangular components. imply

Since FDB is constrained, the unconstrained force vectors at point D can be fixed also. imply

Continuing the joint pin with known dimension, which is fully constrained by the known force before so that all force vector diagram can be scaled to a known value,  i.e joint pin A.  imply

 The force vector diagtram can be joined to form a single diagram, known as Maxwell's diagram. This single diagram can give a more general idea on the characteristic of the trusses, imply