I wasn't trying to be rigorous, since that could take many pages to get right in general. BrunoLevy BrunoLevy 2, 9 9 silver badges 21 21 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. Related 5. Hot Network Questions. A case in which the stiffness matrix becomes non-symmetric is when the stiffness characteristic is highly nonlocal or when the nonlocal effects become significant at a reduced scale of study.
According to the nonlocal theory, the stress at any material point is a function of not only the strain at that point but also the strains at all material points in the neighborhood.
Theoretically, This could be a source of non-symmetry in the stiffness matrix when you discretize the problem for a body of finite dimensions using FEM or any other method. In practice, however, there would be a cut-off radius which limits the neighborhood of a point causing the stiffness matrix to be symmetric at the interior material points of the finite-dimension body. However, even in this case, the non-symmetry is still expected at the material points close to the boundaries.
Can't cite a reference off the cuff and will have to think hard as to precisely where I read it , but as far as I know, if a variational formulation is available for a problem, or, to put it more specifically: if a potential function can be defined for a problem, then the resulting stiffness matrix is symmetrical. It is the existence of a potential function which, I think, provides both the necessary and sufficient conditions for Maxwell's reciprocity theorem to hold.
Someone please correct me if I was wrong there. It is Maxwell's reciprocity theorem which is usually cited as the basis for the symmetry of the stiffness matrix, in the more engineering-oriented literatureand, I don't know about the more mathematically oriented literature nor do I care much!! Caution: Sometimes, people take "variational" and "weak" as synonymous terms, equivalent in all respects, which, I think, is erroneous. A weak formulation can always be had via application of the method of weighted residuals: by lowering the differential order and thereby weakening the demand being made on continuity using integration by parts to lower that order.
However, the possibility of a weak formulation via weighted residuals does not necessarily mean that a corresponding potential function can be defined. As far as my own understanding goes, the symmetry is guarunteed only if a potential function exists. Do you have any simple demonstration of how the nonsymmetric component come into the ordinary symmetric stiffness matrix?
Also do you know of any good introductory texts to nonlocal damage theory? I do get a sense of what you have been trying to get at, but no, I can't off-hand think of a demonstration that's simple enough.
Would like to know if someone can offer one. For examples of non-symmetric stiffness matrices, search the literature on plates and shells. Hope this helps. Bye for now. Update a few minutes later: Also see CFD though, here, I was trying to think of examples from mechanics of solids. I'll try. One can illustrate this for linear problems no need for nonlocal or nonlinear effects , and I will pick boundary-value problems in one dimension extension to multi-dimensions will be apparent for it'll be easier to understand.
Have used LaTeX notation below. One must distinguish between the continuous formulation weak or variational and the choices made to construct the discrete FEM equations. Here, L is a self-adjoint operator, which points to the associated matrix being symmetric Hermitian in general: see this for its many properties.
The latter non-selfadjoint PDEs is typical of what is realized in fluid mechanics Navier-Stokes equations in its entire generality. One can use the method of weighted residuals to derive the weak form in each case. In addition, for the self-adjoint diffusion problem, there also exists a variational formulation that is equivalent to the weak form more on this later. The stiffness matrix Ke in Eq. These rigid body movements are constrained by supports or displacement constraints.
Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. What is the Global stiffness method called? Explanation: Global stiffness matrix method makes use of the members stiffness relations for computing member forces and displacements in structures. Hence Global stiffness matrix or Direct stiffness matrix or Element stiffness matrix can be called as one.
The stiffness method also known as the displacement method is the primary method used in matrix analysis of structures. As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type.
Using stiffness properties of members the memberend forces are computed and hence the internal forces through out the structure. Since nodal displacements are unknowns, the method is also called displacement method. Since stiffness properties of members are used the method is also called stiffness method.
When comparing the flexibility and stiffness methods, it is seen that the flexibility method requires the solution of equations of compatibility for unknown forces whereas the stiffness method requires the solution of equations of equilibrium for unknown displacements.
Figure 5. Stiffness is the resistance of an elastic body to deflection or deformation by an applied force — and can be expressed as.
The proportional constant k is called the spring constant.
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