User:Eml4500.f08.team.guan/2bar

In order to analyze a truss, first a truss must be defined. A truss is a structural object in which its members or bars carry only axial loading. Members that carry loads normal to their longitudinal axis which are loads that cause normal bending stresses are called beams. The global free body diagram of an unconstrained system is shown below. The term global means pertaining to the entire structure. The unconstrained condition means that the system is a "free,free" system in which the determinant of the global sfiffness matrix $$\displaystyle K$$is zero.

Before any analysis can be done, a labeling convention must be established. The global nodes of the system are labeled with numbers circumscribed in circles, the elements are labeled with numbers circumscribed by triangles, and local nodes (elemental nodes) are labeled with numbers circumscribed by squares. Upon drawing the global, unconstrained free body diagram of the truss system, the global displacement degrees of freedom and the global forces must be indicated. In the two-bar truss system above, there are six degrees of freedom, two at each node. Actually, the displacement dofs are partitioned into 1) a known part(i.e. fixed dofs, constraints) 2) an unknown part (these are the dofs solved by the finite element method). Similarly with the global forces, they are partitioned into a 1) known part (applied forces) and 2) an unknown part (reaction forces). Here, all the forces are drawn in accordance with the global coordinate system chosen to be a convenient system in solving the problem. The global coordinate system may or may not be different than local coordinate systems. It all depends on the convenience of calculation. Draw the local elements. Label the elemental displacement dofs and the local forces. The directions of these can be in either local or global coordinate systems. Again, the choice of the coordinate system is for convenience of calculation. A picture of element 1 of the previous two-bar truss is shown below. Now, the global force-displacement relation can be established. The element stiffness matrices, and the element force matrices must be assembled in global coordinates. $$\displaystyle F=Kd$$ Elimination of known dofs to reduce the global force-displacement relation will transition the system into a constrained system where the determinant of the stiffness matrix will not be zero. The stiffness matrix will be non singular and the system can be solved with a system of linear equations. Also, the k matrix is invertible. "n" designates the total number of displace dofs (unknown and known) where "m" designates the number of unknown displacement dofs. Solve for the d matrix by multiplying the F matrix with the inverted sitffness matrix. Finally, compute element forces form the now known d matrix and info about the stresses can be found. Compute the reaction (unknown) forces.