paraphrasing essay

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This set of studies investigated a case of stress and displacement singularity. A cantilever beam, fixed at two points to represent spot welds and receiving a 1000 N load on one face was analyzed. Using 3-D solid elements in h and p adaptive static studies, the stress and displacement results failed to converge and showed unrealistically high values. This indicates that modeling the spot welds as single points created a false restraint condition which allowed singularities to develop. Six additional static studies were performed on 2-D shell elements with various element sizes and orders. In these six studies the stress and displacement results were shown to converge, despite the faulty restraints.


The cantilever beam like the one in this study is a common sight in engineering applications. In the design of such a beam, it would be critical to know the stresses and displacements that would be experienced under the expected loading conditions. By failing to properly restrain the model and choosing the wrong type of study, results were attained that offered no insight into how the beam would perform in real life.


The CAD model is a simple flat plate with dimensions of 120 x 20 x 1 mm. Given the simplicity of the model, solid or shell elements are both a reasonable choice.

Results and Analysis

Table 2 compares the results of the eight studies. Both adaptive studies show unrealistically high stress and displacement results, indicating an error in modeling or in the FEA setup. By modeling the spot welds as single points, an unrealistic restraint condition has been created that has allowed stress and displacement singularities to develop. The mesh convergence graphs in figures 1 and 2 show the data is growing exponentially rather than leveling off as you would expect if the model was properly restrained. Convergence will not take place under these modeling conditions.

Figures 3 and 4 show convergence in the cases with 2-D shell elements. This indicates convergence can still be achieved in a 2-D simplified model when the 3-D element model has singularities.


This set of studies offered insight into different types of convergence studies and problems that can arise from improper restraints and lack of convergence. While it is essential to simplify a model compared to reality, oversimplification can lead to unrealistic and problematic results in the form of stress and displacement singularities. In this case, a flat plate with a constant cross- section, performing the FEA on 2-D shell elements resolved the singularity condition and allowed the results to converge.