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This form reveals how to generalize the element stiffness to 3-D space trusses by simply extending the pattern that is evident in this formulation. After developing the element stiffness matrix in the global coordinate system, they must be merged into a single “master” or “global” stiffness matrix.
The full stiffness matrix A is the sum of the element stiffness matrices. In particular, for basis functions that are only supported locally, the stiffness matrix is sparse. For many standard choices of basis functions, i.e. piecewise linear basis functions on triangles, there are simple formulas for the element stiffness matrices.
This type of element is suitable for modeling cables, braces, trusses, beams, stiffeners, grids and frames. Straight elements usually have two nodes, one at each end, while curved elements will need at least three nodes including the end-nodes. The elements are positioned at the centroidal axis of the actual members.
Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter ...
K is the symmetric bearing or seal stiffness matrix; N is the gyroscopic matrix of deflection for inclusion of e.g., centrifugal elements; q(t) is the generalized coordinates of the rotor in inertial coordinates; f(t) is a forcing function, usually including the unbalance. The gyroscopic matrix G is proportional to spin speed Ω.
The stiffness of a structural element of a given material is the product of the material's Young's modulus and the element's second moment of area. Stiffness is measured in force per unit length (newtons per millimetre or N/mm), and is equivalent to the 'force constant' in Hooke's Law .
The conventional engineer's sign convention is used here, i.e. positive moments cause elongation at the bottom part of a beam member. For comparison purposes, the following are the results generated using a matrix method. Note that in the analysis above, the iterative process was carried to >0.01 precision.
The finite element method has been the tool of choice since civil engineer Ray W. Clough in 1940 derived the stiffness matrix of a 3-node triangular finite element (and coined the name). The precursors of FEM were elements built-up from bars (Hrennikoff, Argyris, Turner) and a conceptual variation approach suggested by R. Courant.