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Unformatted text preview: Advanced Structural Analysis Page 1 of 8 Notes by Professor Panos D. Kiousis Last Update:3/9/2007 Colorado School of Mines Division of Engineering Advanced Structural Analysis Notes (continued) MATRIX STRUCTURAL ANALYSIS – A SYNOPSIS Introduction. Structures can be thought of as springs. Generically, that means that their load-deformation relation can be expresses as: K Δ = R ( 1 ) where K is the stiffness of the structure, Δ is the generalized displacement , i.e. it can be real displacement, or rotation, etc., and R is the generalized load , i.e. it can be force, or moment. Equation (1) can be also written as: S R = Δ ( 2 ) where S = K-1 is the flexibility of the structure. Equations (1) and (2) may represent scalars (i.e. one spring) or matrices and vectors (i.e. more complex structures). Both the stiffness and flexibility approaches have been used in developing matrix solutions for structures. In the stiffness method (more common for programming complex structures), the primary unknowns are the displacements and rotations Δ , while in the flexibility method, the primary unknowns are the forces and moments R . For continuous beams, both methods are equally convenient. We shall examine both. Advanced Structural Analysis Page 2 of 8 Notes by Professor Panos D. Kiousis Last Update:3/9/2007 STIFFNESS METHOD Let us consider again a continuous beam as shown in Figure 1A. The beam deflects due to its load as demonstrated in Figure 1B. Note that the general span i can be viewed as an isolated beam which is fixed at both ends, loaded by the general load q i . To make the span i of Figure 1B and the beam of Figure 5C equivalent, we must impose the same boundary conditions at the beam of Fig. 1C, which are observed at the span i of Figure 1B. That is, we must impose rotations θ i and θ i+1 at the fixed ends that match the rotations of nodes i and i+1 . We can now write the end moment expressions for the beam of Fig. 1C. As opposed to the three moment method , where the moment expressions were written with the typical moment diagram sign convention, the moment signs here will be vectorial. This is necessary, because we shall need to examine equilibrium of moments. We shall follow the standard sign convention that is recommended in the literature and is also used in most commercial software. That, is positive forces are up and to the right, and positive...
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