Stiffness Matrix

3 Key excerpts on "Stiffness Matrix"

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  eBook - ePub

  Engineering Textiles

  Integrating the Design and Manufacture of Textile Products

  • Y El Mogahzy(Author)
  • 2008(Publication Date)
  • Woodhead Publishing

    (Publisher)

  One can also think of the elements of a mesh grid as a group of springs each of which deflects in accordance with the external loads applied until all forces are in equilibrium. It is important therefore to use matrix algebra to deal with this complex system of simultaneous equations. It is also important to consider the Stiffness Matrix for each element. By analogy with a spring system, the deflection of nodes under a system of applied forces can be described by the matrix notation, { f } = [ k ]{ δ }, where { f } is the column matrix (vector) of the forces acting on the element, [ k ] is the Stiffness Matrix for the element and { δ } is the column matrix of the deflections. The Stiffness Matrix is constructed from the coordinate locations of the nodes and the matrix of elastic constants of the materials. When all the elements of the systems are assembled, the basic matrix equation is { F } = [ k ]{ δ }, where { F } is the matrix of external forces at each node, [ K ] is the master Stiffness Matrix, assembled from the [ k ] for all the elements and { δ } is the column matrix of the displacements at each node. The force matrix is determined from the numerical values of loads and reactions computed prior to the start of the finite element analysis. The unknown displacements are determined by transposing the Stiffness Matrix. With today’s computing power, the mathematical details of finite element analysis are automated so that the analysis can be used for a wide range of applications. In practice, there are generally two types of analysis: 2D modeling and 3D modeling. The former is obviously simpler and it tends to yield less accurate results. The latter produces more accurate results but requires high computing capabilities. Within each of these modeling schemes, numerous algorithms (functions) can be utilized, some yielding linear system behavior and others yielding non-linear behavior

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  eBook - ePub

  Structural Analysis Fundamentals

  • Ramez Gayed, Amin Ghali(Authors)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  Maxwell’s reciprocal relation. In other words, the flexibility matrix is a symmetrical matrix. This property is useful in forming the flexibility matrix because some of the coefficients need not be calculated or, if they are, a check is obtained. The property of symmetry can also be used to save a part of the computational effort required for matrix inversion or for a solution of equations.

  The Stiffness Matrix [S] is a symmetrical matrix; thus, for a general stiffness coefficient

  S i j = S j i

  (5.15)

  This property can be used in the same way as in the case of the flexibility matrix.

  Another important property of the flexibility and stiffness matrices is that the elements on the main diagonal, f ii or S ii , must be positive as demonstrated below. The element f ii is the deflection at coordinate i due to a unit force at i. Obviously, the force and the displacement must be in the same direction: f ii is therefore positive. The element S ii is the force required at coordinate i to cause a unit displacement at i. Here again, the force and the displacement must be in the same direction so that the stiffness coefficient S ii is positive.

  We should note, however, that in unstable structures – for example, a strut subjected to an axial force reaching the buckling load – the stiffness coefficient S ii can be negative. This is discussed further in Chapter 10 .

  Let us now revert to Equation 5.11, which expresses the external work in terms of the force vector and the flexibility matrix. We recall that a Stiffness Matrix relates displacements {D} at a number of coordinates to the forces {F} applied at the same coordinates by the equation: [S] {D} = {F}; substituting the force vector in Equation 5.10, the work can also be expressed in terms of the displacement vector and the Stiffness Matrix, thus:

  W = 1 2   F T   f   F

  (5.16)

  or

  W = 1 2   D T   S   D

  (5.17)

  The quantity on the right-hand side of these equations is referred to as the quadratic form in variable F or D. A quadratic form is said to be positive definite if it assumes positive values for any non-zero vector of the variable and, moreover, is zero only when the vector of the variables is zero ({F} or {D

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  eBook - ePub

  Finite Element Analysis

  • M Moatamedi, Hassan Khawaja(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 2 Matrix Stiffness Methods The displacement-based finite element method is closely related to the matrix stiffness methods that were developed in the late 1950’s in order to exploit the newly-arrived digital computers. Many of the procedures that are important in finite element analysis can be illustrated clearly and simply using these methods. In this chapter, the simple bar element is, developed and it is shown how a matrix equation can be constructed to represent the behaviour of an assembly of bars. Aspects of the numerical solution of this equation are also discussed. 2.1    The Simple Bar Element 2.1.1    Stiffness in co-ordinate system parallel to element axes Figure 2.1 One-dimensional element. Consider a bar with end points (nodes) i and j. Equations are required that relate the forces at the ends (X i, X j) to the displacements at the ends (u i, u j). Throughout this chapter, the bar on the variables indicates that they are specified in terms of the local co-ordinate system of the element. By definition for a one-dimensional body, the strain (ε x) is given by: ε x = d u ¯ d x. By Hooke’s Law in one dimension: ε x = σ x E. For the bar illustrated above, with no body forces acting, the equilibrium requirement dictates that the force, F, at every cross-section must be constant. If positive F indicates a tensile force on the cross-section: F = − X ¯ i = X ¯ j. Eliminating strain from these equations, and substituting the basic definition for stress: d u ¯ d x = σ x E = F A • 1 E = − X ¯ i A E. Separating the variables: ∫ u ¯ i u ¯ j d u ¯ = ∫ x i x j − X ¯ i A E d x. Assuming that the cross-sectional area of the bar is constant along its. length: u ¯ j − u i ¯ = − X ¯ i A E (x j − x i). If the length of the bar is L: X ¯ i = A E L (u ¯ i − u ¯ j), and similarly : X ¯ j = A E L (− u ¯ i + u ¯ j). Writing the last two equations in matrix

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