User:Eml4500.f08.delta 6.guzman/Lecture Notes 09/05

= Trusses = Trusses are commonly known as structural members that resist axial forces. In order to facilitate the analysis of simple truss systems the matrix method should be utilized.

= Matrix Method =

The matrix method provides basic steps that are use as a recipe while solving truss systems helping solve problems more efficiently. This is achieved because a common labeling system for the diagrams is used.

Steps Solving Truss Systems:

 * 1) Global picture.
 * 2) Element picture.
 * 3) Global Free Diagram.
 * 4) Elimination of known degrees of freedom.
 * 5) Computation of element forces.
 * 6) Computation of reactions.

Global Picture
The global picture is the description of the system as one entity. This diagram includes known reactions such as applied forces, fixed degrees of freedom (dofs), and constrains. The global picture also illustrates the unknown part of the system, the reactions.


 * Redraw the truss system.
 * Label the global nodes.
 * Label the global forces.





Element Picture
The element picture is the description of each element forming the whole structure. In this diagram the known internal forces and reactions that are acting on that element are illustrated.


 * Redraw the truss system by parts.
 * Label the element degrees of freedom (dofs).
 * Label the element forces.

Note: The element dofs and forces can be either placed in the global coordinate systems or in the local coordinate system.





Global Free Diagram Relation
In order to apply the matrix method it is necessary to develop a relationship between the applied forces and the resulting displacement. The relationship can be formed by treating the trusses as deforming springs and using the mass-spring principle. The mass-spring principle is defined by the equation F=Kd, where F is the force matrix (n x 1), d  is the displacement matrix (n x 1), and K is the spring constant matrix (n x n). Where n is the number of unknown and known displacement degrees of freedom (dofs). These matrices define the linear relationship between the element forces and the element stiffness matrix.

Elimination of known degrees of freedom
It is necessary to reduce the global free body diagram relation in order to solve the stiffness matrix. Due to the non-singularity of the stiffness matrix the mass-spring principle equation is invertible. Resulting in the following new relationship: d = K-1F, where d  is the displacement matrix (m x 1), K-1 is the spring constant matrix (m x m), and F  is the force matrix (m x 1). Where m is the number of unknown displacement degrees of freedom (dofs).

Computation of element forces
As a result of now knowing the displacement matrix d , the element forces can be calculated.

Computation of reactions
The last step is the computation of reactions of the unknown forces in the truss system. Since d and F have the same matrix size the matrix K is symmetric. By counting the support’s constrains the unknowns can be solved by using a system of linear equations.