User:Eml4500.f08.ramrod.c/HW

=Trusses, Beams, and Frames=

Diagram of Truss
Below is a diagram of a truss with 2 elastic, or deformable, bars. There is a known load P applied where the two bars join. The two bases of the truss are both fixed, or constrained, in both the x and y axes. There will be no displacement at these points. The goal is to find all of the unknown forces and displacements.



Global Free Body Diagram
Below is the Global (the whole structure) Free Body Diagram of the same 2-bar truss. Notice that the constraints have been replaced with reaction forces.



Currently, there are 4 unknown reaction forces (R1x, R1y, R3x, and R3y). However, there are only three equations that can be used (ΣFx=0, ΣFy=0, and ΣM=0). In other words, we have 4 unknowns but only 3 equations to solve for them, therefore this system is statically indeterminant.

Free Body Diagrams of the 2 bar elements
To continue, we will separate the two bars and look at them independently. A new local free body diagram is drawn for each bar element. Forces and the resulting displacements are again drawn and the nodes are renumbered accordingly.

Bar Element 1 Bar Element 2

Notes
 * fdegree of freedomundefined


 * fi(e) = ith eternal force of element e with:


 * i=1,2,3,4


 * e=1,2


 * for this example

There are two possible cases in this situation:


 * Case 1: The observer sits on node 1


 * In this case:


 * f2 = k(d2 -d1)


 * Case 2: The observer sits on node 2 (or equivalently f1+f2=0)


 * In this case


 * f1= -f2= -k(d2 -d1)


 * f1=k(d1 -d2)

Alternate Node Numbering Method
Below is another way to number the local nodes.



Force Displacement Relation
An elastic bar can be considered a spring with a spring constant of EI/L (E is Young's Modulus, I is the moment of inertia, and L is the length of the bar or spring). Recall the two following spring systems:


 * Below is a 1-dimensional spring with two free ends. A force f is applied and a displacement d results. The force-displacement relation is f=kd.




 * Another system is a 1-D (one dimensional) spring with 2 free ends. A force f is applied and a displacement d results. The matrix method (on the right) is used to find the force-displacement relationship. (The 2x2 matrix with the k's in it is known as the stiffness matrix).