Trusses, Matrix Method

Trusses are structural systems consisting of long slender members. The members are arranged so that they are loaded axially and do not experience bending or shear forces. For more information on trusses, please refer to the Truss article on Wikipedia.

Before trusses are analyzed with the Matrix Method or any other method, it is important to develop a standardized system for labeling and drawing the systems. First, the truss system is drawn and the components are labeled. Global nodes are labeled with a number circumscribed in a circle and elements are labeled with a number circumscribed in a triangle.

The next step in the process is to draw the global free body diagram (FBD). The global FBD is a diagram of the whole structure that shows the applied forces, as well as the reaction forces. The FBD includes known and reactionary forces represented by arrows. The reaction forces are labeled Rna, where a denotes the direction of the force in reference to the global coordinate system and n represents the global node number. For example, R1X represents the reaction force R applied along the X axis at node 1.



After the global FBD is created, an element FBD must be created. The element FBD is a diagram of each element of the whole structure that includes known, reaction, and internal forces applied to that element. The elemental diagram is labeled with the element number as stated above. Each node of the element is labeled with its global node number as stated above, as well as a local node number that is a number circumscribed in a square. Each internal force of the element is labeled fi(e), where i is the force number and e is the element number. Force numbers are assigned first by local node number and then orientation related to the coordinate system. For example, a force f1(1) is the first degree of freedom (DOF) in element one. SEE ILLUSTRATION

The next "big step" in applying the matrix method to a truss problem is to develop a relation between the applied force and the resulting displacement. The deformation of an elastic truss member can be modeled as the deformation of a spring. Therefore, mass-spring relations will be examined.

First, consider a one dimensional spring with one end fixed. The force-displacement relation for this system is represented by the equation $$f=kd$$, where f is the force, d is the displacement, and k is the spring constant that it a property of the spring. This relation shows that there is linear relationship between the applied load and the resulting displacement. SEE ILLUSTRATION

Another spring system to consider is a one dimensional spring with both ends free. This system has two degrees of freedom so there is a system of two equations that govern the relationship between the applied force and the resulting displacement. Consider the spring system shown in the figure. Force 1, F1, acts at node 1, causing a displacement, d1 and force 2 acts at node 2, resulting in a displacement d2. The equations of motion for this system are developed for two cases. For the first case, consider an observer sitting on node 1. The resulting equation of motion is $$f_2 = k(d_2-d_1)$$. The second case is when an observer sits on node 2, resulting in the equation $$f_1 = k(d_1-d_2)$$. Therefore, the resulting force-displacement relation for this system is represented by the system of equations below.

$$\begin{bmatrix} F_1 \\ F_2 \\ \end{bmatrix} $$ = $$\begin{bmatrix} k & -k \\ -k & k \\ \end{bmatrix} $$ $$\begin{bmatrix} d_1 \\ d_2 \\ \end{bmatrix} $$