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Critique of SolidWorks Simulation on Mixing Shells with Solids

MECH 417 Rice University, J.E. Akin, 3/3/20

Introduction: In terms of using engineering concepts and expressing correct results this is one of the worst simulation tutorials; even after stating that most of the simulation tutorials are simply intended to locate new task icons. Topics not clearly addressed here are what a shell is, and when to use a thin shell, or a thick shell instead of the default solid element. A single solid element essentially cannot transmit a moment and certainly cannot give an accurate estimate of bending (flexural) normal stress or shear stress, unless there are at least five solid elements through the thickness of the bending region. The normal (flexural) stress in a thin beam or shell caused by the bending moment varies linearly through the depth, is zero at the centroid of the section and maximum at the top and bottom surfaces. The bending shear stress is maximum at the centroid of the section (3V/8A for a rectangular cross-section) and varies parabolically to zero on the surfaces. A transverse load on the top surface varies through the thickness to zero on the bottom surface. How such a through thickness stress occurs is neglected in thin beams and shells, but is included in the theory of thick beams and thick shells.

Bending normal stress Bending shear stress 𝝈 = 𝑴 𝒚 𝑰⁄ 𝝉 = 𝑽 𝑸(𝑦) 𝑰 𝒕⁄ A thin shell is the 3D combined extension of a long thin beam that is intended to transmit transverse loads primarily by bending moments and shear forces, and an extension of the axial bar which only carries uniform axial forces. The elementary ‘membrane theory of shells’ shows that some curved surfaces can carry loads by means of only membrane surface forces. However, that is not true where Dirichlet displacement boundary conditions are applied, or where there is a discontinuity in the curvature of the surface. Such regions develop very large and very localized bending moments that require the full theory of shell structures that include bending moments and transverse shear forces. Ideally, a thin shell should have a thickness to span length ratio of less than 1 to 10. A thick shell is like a short deep beam that is to transmit transverse loads by bending and a more accurate distribution of shear through its thickness. The thick shell thickness to span ratio should be in the range between 1 to 3 and 1 to 2.

Thickness to span ratios for shell and solid theories

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The next figure (taken from the SW simulation online help file) shows the distribution of the individual and combined normal stresses (acting parallel to the curved middle surface) due to a bending moment, and the mid-surface tangential force. The total normal stress (left image) differs between the top, middle, and bottom surfaces of the shell (or beam). Thus, the user must know which surface is the top of a shell element, and should know how to ‘flip’ that definition when creating the initial surface mesh.

This help section also has the following TWO IMPORTANT alerts when utilizing shell elements:

However, the SW simulation tutorial on “Mixing Shells and Solids” FAILS TO OBSERVE THESE WARNINGS and as a result compares the wrong stresses in the assembly! It also had two poor sliver surfaces near the junction. The fixtures probably over estimate the available support surfaces. If bolts with washers were used then it is more likely that the top outer circles and the lower back circles would provide the displacement restraints. A solid element has only translational displacements, while a shell element also has three (small) rotations. Thus, any connection between a shell and a solid requires the introduction of constraint equations

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to connect the shell rotations into the solid element’s translational displacements. That is best done when a split line is used on the solid face, where it meets the edge of the shell, so that a compatible mesh can assure exact displacement connections between the two element types. (That was not done in this tutorial.)

Note that the loaded side of the shell mesh is orange. That means that is the “bottom” side of the shell and that fact must be considered when selecting stresses to be displayed, say on the ‘upper’ surface. Also, the mesh was relatively uniform instead of being refined at the shell-solid interface, and at the ends of the rigid links. Most of the solid had only one element through the thickness and could not give good stress results if the supports were changed to allow significant bending of the solid. This could be a good starting mesh to be revised after the error estimate (initially 100% in the energy norm) was plotted. For important parts the error estimate should be reduced to about 1% or 0.5% by refining some regions and coarsing others. SW, and most modern FEA systems, can automatically adapt the mesh to reach a specified level of error.

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Once the solution was obtained the normal stress in the x-direction (SX) was selected and displayed on the default Top surface, and was viewed from the upper direction. The resulting figure was somewhat confusing because it looks like the shell is in compression (blue) until it connects to the solid where it suddenly jumpts to tension (red). The correct upper view of SX would be obtained by editing the plot definition and changing Top to Bottom. That would show that the upper surfaces are in tension at the shell-solid junction. A similar view from below would show the lower faces to be in compression at the junction. (My preference would have been to make the upper surfaces of the part to be the shell ‘Top’. To do that after generating the mesh right click oh the orange Botton, right click on Mesh, and select Flip Shell Elements.)

Misleading compression-jump-tension shell-surface junction (wrong view angle)

Usually, the displacements should be viewed and accepted before examining any stress results. In this case they look fine along the free edge of the shell.

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Revisions:

My approach to this problem begins by noting that the geometry, material, supports, and loading possess half symmetry. Thus, I cut the original parts in half. Since there usually is a stress concentration at a shell-solid interface I also added split lines to the curved part of the solid to have a compatible mesh with more accurate stresses in those regions.

For the necessary Dirichlet boundary conditions the cut center face of the solid had a symmetry condition (no normal displacement). That graphic may look like it imposed symmetry on the shell surface also, but it was not sufficient. That is because any shell also has rotational degrees of freedom which can only be specified on the edge of a shell surface. The symmetry condition for a shell also requires a zero rotation about an edge lying on the symmetry plane. (See the middle point of the above edge displacement plot for the full edge.) Had this requirement been missed the shell results would be wrong.

Shell edges have a zero normal displacement and

a zero tangent rotation on the symmetry plane

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The loading moment produced by the pressure must be resisted by the couple formed by the bolts. If bolts with washers were used then it is more likely that the top outer circles and the lower back circles would provide the displacement restraints. Those were applied as seen in the top half symmetry model. The mesh was created with refinement at the shell-solid interface, and at the rigid-link interface with the shell. The orange color shows that the upper pressure surface is actually the ‘Bottom” surface for showing stress results. That information must be used, or reversed by ‘flipping’ the elements, when selecting the Top, Middle, or Bottom surface as the plot option.

A mesh with selective refinements

Before examining the stresses (and material failure criterion) the deformed shape is displayed to verify that

the shell has a zero slope as it approached the center plane of symmetry. That information is most accurately

checked by a graph of the vertical displacement (UY) along the front edge from the rigid link to the symmetry

plane. That shape is similar to that of half of a simply supported beam with a uniform load.

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The deflection of the shell edge from the solid to the front free edge should start with an approximately zero

slope and then look similar to that of a cantilever beam with a uniform load.

The deflections along the edge from the solid to the lower end of the rigid link is about ten times smaller than

along the front edge. Its shape is similar to that of a propped cantilever with a uniform load.

Believing the displacement results, selected stress components are now examined. The normal stress

parallel to the x-axis (SX) was displayed in an upper view (by picking the ‘Bottom’ shell surface), and from a

lower view by displaying the ‘Top’ surface. By using the same color bar range it is clear that the upper surface

is mostly in tension (green to red), while the lower face is mainly in compression (green to blue). Their

graphed values along the center edge are basically equal and opposite (except for very small compressive

membrane stress induced by the rigid links).

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Top Center Edge SX

Bottom Center Edge SX

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Validation:

A thin solid can yield good displacement estimates, but its stress estimates are very much less accurate

than those obtained from a shell element. Nevertheless, the solid is further extended and modeled, again

with half symmetry. The shell-solid model required 46 thousand equations; while this solid required 1.45

million equations (The full solid would wastefully require solving about 3 million equations, if they would fit in

your computer.)

XXXXX

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