Matrix structural analysis and introduction to finite by Saouma V.E.
By Saouma V.E.
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Extra resources for Matrix structural analysis and introduction to finite elements
Thus, in deriving the element stiﬀness matrix, we will be reviewing concepts earlier seen. 2 The other approach, based on energy consideration through the use of assumed shape functions, will be examined in chapter 11. This second approach, exclusively used in the ﬁnite element method, will also be extended to two and three dimensional continuum elements. f. f. j for an individual element or a whole structure. 1. It should be recalled that inﬂuence lines are associated with the analysis of structures subjected to moving loads (such as bridges), and that the ﬂexibility and stiﬀness coeﬃcients are components of matrices used in structural analysis.
F: θ1 and θ2 . 8 9 If we were to analyse this problem by the force (or ﬂexibility) method, then 1. We make the structure statically determinate by removing arbitrarily two reactions (as long as the structure remains stable), and the beam is now statically determinate. 2. Assuming that we remove the two roller supports, then we determine the corresponding deﬂections due to the actual laod (∆B and ∆C ). 3. Apply a unit load at point B, and then C, and compute the deﬂections fij at note i due to a unit force at node j.
4 Stiﬀness Coeﬃcients In the ﬂexibility method, we have applied a unit force at a time and determined all the induced displacements in the statically determinate structure. 9 10 In the stiﬀness method, we 1. Constrain all the degrees of freedom 2. f. (while restraining all others to be zero) 3. f. 8) 11 Hence kij will correspond to the reaction at dof i due to a unit deformation (translation or rotation) at dof j, Fig. 2. 12 The actual stiﬀness coeﬃcients are shown in Fig. 3 for truss, beam, and grid elements in terms of elastic and geometric properties.