Stiffness method

Stiffness Matrix Method. The stiffness method can be expressed in matrix form as follows ... 

In contrast to the flexibiUty method, the stiffness method considers the displacements as unknown quantities in constmcting the overall stiffness matrix (K). The force vector T is first calculated for each load case, then equation 20 is solved for the displacement D. Thermal effects, deadweight, and support displacement loads are converted to an equivalent force vector in T. Internal pipe forces and stresses are then calculated by applying the displacement vector [D] to the individual element stiffness matrices. 

This method has a simple straightforward logic for even complex systems. Multinested loops are handled like ordinary branched systems, and it can be extended easily to handle dynamic analysis. However, a huge number of equations is involved. The number of unknowns to be solved is roughly equal to six times the number of node points. Therefore, in a simple three-anchor system, the number of equations to be solved in the flexibiUty method is only 12, whereas the number of equations involved in the direct stiffness method can be substantially larger, depending on the actual number of nodes. 

The most recent developments in computational stmctural analysis are almost all based on the direct stiffness matrix method. As a result, piping stress computer programs such as SIMPLEX, ADLPIPE, NUPIPE, PIPESD, and CAESAR, to name a few, use the stiffness method. 

Eigure 11 shows a reasonably complex piping system. The number of equations involved in modeling this systems is 19 for the flexibiUty method, and 144 for the stiffness method. 

M. S. Ewing, R. J. Hinger, and A. W. Leissa, On the Validity of the Reduced Bending Stiffness Method for Laminated Composite Plate Analysis, Composite Structures, Volume 9, 1988, pp. 301-317. 

Value of the independent variable at the start of the run (0). Relative accuracy for the Auto and Stiff methods (0.01). 

The algorithm consists of the judicious application of one of two integration formulas to each equation in the system and the choice of formula is based on the time constant for each equation evaluated at the beginning of each chemical time step. Species with time constants too small are treated by the stiff method and the remaining species are treated by a classical second order method. The algorithm is characterized by a high degree of stability, moderate accuracy and low overhead which are very desirable features when applied to reactive flow calculations. 

The stiffness matrix can be formulated by the direct stiffness method and condensing the rotational DOFs. Instead we use a different method. 

Stiffness Method, The stifi iess of the polymer surface is determined by force vs. displacement curves. Very stiff cantilevers are used, and Tg is defined as the tenq)erature where a change occurs in the slope of the force-displacement curve (18, 19). 

This structure shall be analyzed by a conventional frame analysis taking account of moment distribution, flexibility and stiffness methods. 

The third parameter investigated is the effect of trachea stiffness. As explained in previous section, due to difficulty of acquiring images for bronchial tube, blood vessels images are assumed to be the trachea. In this simulation, equivalent stiffness method is used to investigate the stiffness effect at the increase rate of 1 [%] in the range from 0 5[%]. The values are as shown in table 3. 

The relationship between the intercalated nanostructure and the effective nanocomposite stiffness is constructed using an equivalent stiffness method in which the clay stacks are replaced by homogeneous nanoparticles with predetermined equivalent anisotropic stiffness, see Ref [7]. The equivalent particle aspect ratio, defined as the ratio between its thickness and its length tjL, can be expressed as ... 

Based on wave propagation, the spectral finite element or spectral element method (SEM) was introduced by Beskos in 1978, organized and seemed by Doyle (1997) in the 1990s. It allows calculating relatively complex structures with different boundary conditions and discontinuities using simple theories. It combines important characteristics of FEM, dynamic stiffness method (DSM), and spectral analysis (Lee 2009). 

There are a number of variants of the finite element method depending on the formulation and solution of the problem. The most common variant is the stiffness method in which the displacement compatibility conditions are satisfied and the equations of equilibrium are solved to yield the unknown nodal displacements. The governing equation using matrix symbolism is of the following form ... 

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