Displacement finite element model

The displacement finite element model is based on the principle of virtual displacements. The principle requires that the sum of the external virtual work done on a body and the internal virtual work stored in the body should be equal to zero (see Reddy(46))... 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

The joint stiffness was measured between 2 and 7 kN, for which the experimental load-displacement curves were essentially linear. Table 11.3 hsts the change in stiffness as the clearance is varied from neat-fit (Cl) to 240 pm (C4), for both the experiments and the finite element models. It is evident that as clearance increases, joint stiffness decreases, and it is also evident that the finite element models provide an accurate prediction of this loss in stiffness. 

Load-displacement graphs from DIANA were plotted to compare the results from the test and the finite element model. Results are shown in Figs. 13.22 and 13.23. For numerical simulation, the envelope of the load displacement curve is... 

Finite element models similar to the experimental fabric samples are used to determine the fabric bending stiffness. Equivalent tip displacements are prescribed in the models and a range of bending-stiffness values are used to replicate the experimental profile shape of the sample. While several values... 

In the present work, these experiments have been simulated numerically. The simulation was performed with the finite element model of Figure 6. Zero displacements in x-directions were imposed on both the right and left hand side of the model. The bottom layer of the model is fully constrained. The water pressure is applied to the model via equivalent boundary forces. A comparison of the experimental results and those obtained from numerical simulation is given in Figure 4. 

During this stage the finite element model for soil (only, no structure) is loaded with soil self-weight. The finite element model for this stage excludes any structural elements, the opening (hole) where the pile will be placed is full of soil. Displacement boundary conditions on the sides of the three soil blocks are such that they allow vertical movements, and allow horizontal in boundary plane movement, while they prohibit out of boundary plane movement of soils. All the displacements are suppressed at the bottom of all three soil blocks. The soil self weight is applied in 10 incremental steps. 

The equations of the Lagrangian incremental description of motion can be derived from the principles of virtual work (i.e., virtual displacements, virtual forces, or mixed virtual displacements and forces). Since our ultimate objective is to develop the finite-element model of the equations governing a body, we will not actually derive the differential equations of motion but utilize the virtual work statements to develop the finite element models. 

Here we construct the finite-element model of Eq. (13) for the two-dimensional case (see Reddy<" 7)). Let each displacement increment be approximated as... 

The truncated expressions for the displacements also enable the application of finite element modelling formulation to investigate the lateral buckling behavior of variable cross-sectional beams. 

For the generalized displacements for each element in the finite element model, nine generalised displacements are at each end of the element, while two displacements associated with y and z axes are at the middle note of each element. The generalized displacements are depicted in Figure 1. 

The methods discussed earlier are applied to the seat-occupant-restraint system of an aircraft. A description of a computer-aided analysis environment, including a multibody model of the occupant and a nonlinear finite element model of the seat, is provided, which can be used to re-construct variety of crash scenarios. These detailed models are useful in studies of the potential human injuries in a crash environment, injuries to the head, the upper spinal column, and the lumbar area, and also structural behavior of the seat. The problem of reducing head injuries to an occupant in case of a head contact with the surroundings (bulkhead, interior walls, or instrument panels), is then considered. The head impact scenario is re-constructed using a nonlinear visco-elastic type contact force model. A measure of the optimal values for the bulkhead compliance and displacement requirements is obtained in order to keep the possibility of a head injury as little as possible. This information could in turn be used in the selection of suitable materials for the bulkhead, instrument panels, or interior walls of an aircraft. The developed analysis tool also allows aircraft designers/engineers to simulate a variety of crash events in order to obtain information on mechanisms of crash protection, designs of seats and safety features, and biodynamic responses of the occupants as related to possible injuries. 

As an application of the theory discussed earlier, the crash responses of aircraft occupant/stnicture will be presented. To improve aircraft crash safety, conditions critical to occupants survival during a crash must be known. In view of the importance of this problem, studies of post-crash dynamic behavior of victims are necessary in order to reduce severe injuries. In this study, crash dynamics program SOM-LA/TA (Seat Occupant Model - Light Aircraft / Transport Aircraft) was used (13,14]. Modifications were performed in the program for reconstruction of an occupant s head impact with the interior walls or bulkhead. A viscoelastic-type contact force model of exponential form was used to represent the compliance characteristics of the bulkhead. Correlated studies of analytical simulations with impact sled test results were accomplished. A parametric study of the coefficients in the contact force model was then performed in order to obtain the correlations between the coefficients and the Head Injury Criteria. A measure of optimal values for the bulkhead compliance and displacement requirements was thus achieved in order to keep the possibility of a head injury as little as possible. This information could in turn be usm in the selection of suitable materials for the bulkhead, instrument panel, or interior walls of an aircraft. Before introducing the contact force model representing the occupant head impacting the interior walls, descriptions of impact sled test facilities, multibody dynamics and finite element models of the occupant/seat/restraint system, duplication of experiments, and measure of head injury are provided. 

In this work, 2D finite element models of normal vessel and atherosclerotic vessels with 50% and 90% plaque deposition were developed using Comsol 3.5a. Boundary conditions were applied to the developed models and a distributed load was applied on the inner wall of the vessel. Further, the developed models were meshed using Delaunay triangulation method. The developed vessels were subjected to a transient analysis and the parameters such as total displacement, Von Mises stress and strain energy density were analyzed for normal and atherosclerotic vessels. 

The maximum radial stress as a function of the mechanical properties of the components and the geometry are calculated by finite-element modeling. The critical load must be determined experimentally. It may be observed by a kink in the force-displacement curve, by direct optical observation, or by acoustic emission analysis. The calculated stress concentration factors for different curvatures of the interface are given in Table 1. 

MSWs are complex mechanical systems and this complexity is compounded when the structures are subjected to transient dynamic loading due to earthquake. Pseudo-static and displacement methods may be sufficient for simple structures, for stable ground conditimis and MSWs in low seismic risk areas, and/or for preliminary design. Otherwise, more sophisticated analyses may be warranted using advanced dynamic finite element model or finite difference computer programs. Today commercially available computer programs offer the user a suite of constitutive models for the component materials and in some cases allow the user to implement their own constitutive models. In order to accurately model the soil in a MSW, it may be necessary to use nonlinear cyclic stress-strain models. The equivalent-linear... 

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