Question
What is the difference between member forces of a plate element and those of a beam element?
Answer
The member forces of a plate element and a beam element are as follows:
- Component of member forces of beam element
- Component of member forces of plate element
- Comparison member forces of beam / plate element
While there is no dispute in calculating the member force of a beam member since its cross-section is predetermined, for a plate member, it is necessary to establish a criterion for determining which area is considered as a cross-section and how to calculate the member force (section force). Therefore, the member force (section force) of the plate member displayed in Contour is based on a unit width of 1m. This can be more clearly understood by comparing it with the member force value of a beam member with a width of 1m.
The contour output of the member forces is based on a unit width (1m) of the plate element, so the same results are obtained regardless of the size of the plate element.
As mentioned earlier, the plate element's member forces are calculated per unit width (1m). However, some people may misunderstand and think that the member forces result is calculated based on the element size.
The numerical analysis process involves errors due to the discretization interval. The denser the interval, the closer the result is to the actual solution, while the error increases as the interval becomes larger. In the figure below, if the gray curve represents the actual solution, the numerical analysis result is shown by the orange line. In case (a), the result does not sufficiently reflect the distribution of the actual solution, while in case (c), the result is almost identical to the actual solution.
The accuracy of numerical analysis depends on the level of stress variation with respect to the partition interval. When the variation is not significant, as in case (A), accurate results can be obtained even with a wide partition interval. However, when the slope of the result curve changes steeply, as in case (B), the difference in the analysis results due to the partition interval becomes significant. This is a problem related to the accuracy of the analysis and not to the calculation of the structural force considering the element size. In the actual process of calculating the structural force, the stress is integrated in the thickness direction to output the force in units of kN/m or kN·m/m.
From the perspective of analysis accuracy, it can be said that more accurate results can be obtained by dividing the element finely in areas with large stress variations (such as locations of loads, points, or areas where the structure has a large deformation or changes in shape). However, it is important to understand the assumptions included in the finite element model.
The following figure shows the stress distribution in a bridge bearing. In the finite element model, the support is idealized as a point, and the stress distribution would be similar to the left side of the right image. However, in reality, the support is provided through an area, as shown on the right side of the image, and the stress distribution is expected to have a smoother shape compared to the left side.