User:Eml4500.f08.team.foskey.ckf/HW1b

The Matrix Method

The Global Picture

To demonstrate using the Matrix Method to solve a problem, we will examine a truss with two elastic, or deformable, bars.

[Truss with two elastic bars]

A force P is applied at the top of the truss where the two bars connect, and each bar is constrained to the ground. Due to these constraints, the displacement at these locations will be zero, as we will show later.

A global free-body diagram is created showing all of the forces acting upon the structure as a whole. These include the force P, as well as the horizontal and vertical reaction forces on each of the fixed locations on the bars. It is also important to label the parts of the truss. Here each node is numbered and circled. Each bar represents an element of the whole, and is numbered and enclosed by a triangle. From this point forward, the bars will be referred to as elements.

[Global FBD of Truss]

The Element Picture

Each element of the structure is now looked at as an independent part. The nodes are renumbered and enclosed by a box. The triangle enclosed element number remains. The forces acting on the element are shown and labeled using the following format: fi(e) where e is the element number and i is the ith internal force of element e. Numbering of i is done in order of the nodes, beginning with the x-direction, and then the y-direction.

Displacements are also shown and labeled for each element. They are labeled similarly to the forces, except they are denoted by the letter d, as shown: di(e)

[Element 1]

[Element 2]

The Force Displacement Relation

There is a relationship between displacement and force. A simple example would be a spring. Force is proportional to the displacement of the spring. The coefficient for the relationship is k.

[Spring Example Diagram]

[Force-displacement relationship for spring example]

This concept can be applied to the elements by relating their displacements, forces, and the axial stiffness coefficient k. The matrices for the displacements, forces, and axial stiffenesses are assembled into the force-displacement relation equation: kd=F

where k is an (n x n) matrix, and d and F are (n x 1) matrices.

Elimination of knowns and computations

Once the force-displacement equation is created, knowns can be inserted to reduce the amount of unkown values than must be solved for. These knowns include the known applied forces (force P), and any constraints that have known displacements (the constrained ends of the elements is zero because they cannot move, as stated earlier). From there, the unknown displacements and the unknown forces can be calculated using matrix algebra.