Element stiffness matrix

Note that the definite integrals in the members of the elemental stiffness matrix in Equation (2.77) are given, uniformly, between the limits of -1 and +1. This provides an important facility for the evaluation of the members of the elemental matrices in finite element computations by a systematic numerical integration procedure (see Section 1.8). 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

FLOW. Calculates members of the elemental stiffness matrix corresponding to the flow model. 

Step 1 To solve a Stokes flow problem by this program the inertia term in the elemental stiffness matrix should be eliminated. Multiplication of the density variable by zero enforces this conversion (this variable is identified in the program listing). 

The next step in the numerical implementation is to find a relation between each individual element stiffness matrix, element displacement vector and element force vector and the global system. Hence, we must generate a global stiffness matrix, global displacement vector and global force vector. This step in the implementation is actually only data management or book keeping. 

The element stiffness matrix, Ke, for the two elements in this problem are given by... 

Algorithm 10 Computing the element stiffness matrix and force vector for a four-noded isoparametric element by numerical integration... 

The element stiffness matrix for the /th story with respect to the Oj coordinates is ... 

The element stiffness matrix for the Zth story with respect to the DOFs (1), (2) and (3) in Figure 5.3 of the upper and lower floors is given by ... 

The distinct feature of elastic-plastic finite element computations is the presence of two iteration levels. In a standard displacement based finite element implementation, constitutive driver at each integration (Gauss) point iterates in stress and internal variable space, computes the updated stress state, constitutive stiffness tensor and delivers them to the finite element functions. Finite element functions then use the updated stresses and stiffness tensors to integrate new (internal) nodal forces and element stiffness matrix. Then, on global level, nonlinear equations are iterated on until equilibrium between internal and external forces is satisfied within some tolerance. 

Raphson method. The element stiffness matrix K, = becomes... 

The reader is referred to Thai and Kim (2011) for details on how to derive the element stiffness matrix. 

Another area of current development is in damage and failure modelling. It is impossible for a linear finite element analysis to predict failure in a structure. However, in nonlinear analysis it is possible to implement a failure model and increasingly complex failure models are now supplied as standard features of commercial FEA software. It should be noted, however, that even though the failure model may look complex, the method of implementation within FEA is usually fairly straightforward. In most cases this involves utilization of the results from the various increments of a nonlinear analysis. These are processed via some failure model to determine whether failure or damage has occurred in any of the elements. The properties of the failed or damaged elements are then modified, usually by control of the element stiffness matrix. 

In the case of quadratic or higher interpolations this has the advantage that the distribution of the primary variable is represented more closely and also that the element sides can be curved. The element stiffness matrix requires the strain-displacement relationship to be known, e.g., for the one-dimensional case ... 

It is convenient to adopt a local numbering system when evaluating the element stiffness matrix and load vector, in which the nodes are numbered in a counter clockwise direction, starting with node 1. Once the element stiffiiess matrix and load vector have been evaluated, the components can then simply be plaeed in the correct rows and columns of the global matrix and vector by recalling the global node numbers of the element. The process whereby and are computed for each element, and then added to the global matrix, is known as assembly. 

Ihe total element stiffness matrix is then written as... 

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