Nodal displacements

The modeling of complex solids has greatly advanced since the advent, around 1960, of the finite element method [196], Here the material is divided into a number of subdomains, termed elements, with associated nodes. The elements are considered to consist of materials, the constitutive equations of which are well known, and, upon change of the system, the nodes suffer nodal displacements and concomitant generalized nodal forces. The method involves construction of a global stiffness matrix that comprises the contributions from all elements, the relevant boundary conditions and body and thermal forces a typical problem is then to compute the nodal displacements (i. e., the local strains) by solving the system K u = F, where K is the stiffness matrix, u the... 

F is the body force vector per unit volume, [7 is the surface force vector per unit area, V is the volume and S is the area. With a nodal displacement vector d the displacement vector U is written as... 

Having constructed the microscopic mesh, we specify the microscopic problem based on the macroscopic nodal displacements. The displacements of the elemental boundaries are given by the macroscopic solution (although the internal microscopic scale displacements are not necessarily affine). The microscopic problem is to find node positions and segment lengths such that the boundary nodes are as specified by the macroscopic displacements and the internal nodes experience no net force. The boundary nodes have displacement specified and are subjected to a non-zero net force. The next step in the solution process is to convert those forces into the macroscopic stress tensor. 

In order to obtain the boundary finite element formulation, on the one hand the force-displacement relation of the discretized element layer between nodal forces P and nodal displacements u is considered, which can be written in the decomposed form ... 

Fig. 2.15. Replacement of a continuous body with a set of finite elements. In this case, the unknown displacement field has been replaced by a discrete set of unknown nodal displacements, u .

The result of this analysis is a series of coupled algebraic equations in the unknown nodal displacements. For the linear problem we have considered here, these equations are themselves linear (though they are not in general) and our problem reduces to little more than diagonalization of the relevant stiffness matrix which, as seen above, can be computed as products of material parameters and geometric factors. 

The unknown nodal displacements are obtained using Galerkin s weighted residual method. The inner product of the governing equation with respect to each of the interpolation functions is set to zero over the whole domain 2. However, the 4 order derivative term in governing equation requires the interpolation function to have continuity. In other words, the first derivatives of Nj with respect to x and z should be continuous along the inter-element boundary to avoid infinity in the integration of the so-called "stiffness" matrix. Hence, we introduce a new variable O such that... 

In the third run, vertical unit forces are applied to the released nodes in the direction needed to bring them back to their original vertical positions. The nodal displacements of these released nodes of the sample for a crack length of u + Au under the same constant load F are noted. Let represent the displacement of the bottom node in the... 

After linearization we can compute the nodal displacements from... 

Substituting Eqs. (3) and (4) into Eq. (1), the functional 77( ) can be rewritten in terms of the nodal displacement parameter /, with hierarchical structure. 

Step-(l) As initial values of nodal displacement parameters and stresses, zero is assumed. Step-(2) For the n-th node on the 1-st level, solve Eq. (9) to obtain dt/". ... 

The bending moments at the ends of a flexural element are related to the nodal displacements by... 

Define nodal displacements at the top of the columns as shown in the accompanying figure... 

After assembling Eq. 8.66 to form the global structure equations, the nodal displacement of the structure, and hence shrinkage and warpage, can be calculated. As boundary conditions, a suitable set of support constraints have to be imposed to the structure to prevent the body from undergoing unlimited rigid body motion. 

The following analysis is concerned with linear systems described by a discretized model in terms of the nodal displacement vector u(t). The equation of motion is of the form... 

Hence, the rigid body motion geometric relationships are applied to calculate nodal displacements at left edges as described by Equation (1). 

Three-control-point approach As discussed previously, instead of employing a rigid body motion assumption, beam shape functions are used for the calculation of the equivalent nodal displacements in the VecTorl model. In the above approach, the interface floor system is divided into two beams connected by three control points. In Figure 4(b), control points N, N2, and N3 form two beam members with lengths equal to half of the wall width. Cubic shape functions are used for the interpolation of continuum model nodal displacement loads [ y,from at these three... 

The nodal displacements at all the nodes along ith floor can then be computed as ... 

The modes of vibration in a cylindrical coordinate system can be obtained through the method of separation of variables (Kausel 1974 Tassoulas 1981). Using the modes of vibration, equivalent nodal loads and nodal displacements can be evaluated at the cylindrical boundary r= ro of the region r>ro and the dynamic stiffness of the transmitting boundary in the water-saturated transversely isotropic layered strata can be obtained. The amplitudes of the equivalent nodal loads are given as follows ... 

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