fem modelling of a bellows and a bellows- based micromanipulator

FEM modelling of a bellows and a bellows-based micromanipulatorIII. Modellingstrain concentration at those nodes. Even substantial strain-gaps would appear between somenodes, so that the result would then be less accurate, or maybe wrong. Hence, to avoid theseproblems and to make the aspect ratio of the elements close to 1, meshing has to be very fine.As shown in next figure, the number of elements will become enormous.(a)(b)Figure 13Mesh (a) is too coarse and couldinduce errors. So the mathematicalmodel should be more accurate,corresponding to the mesh (b).3.1.4 CHOOSING RIGHT SHELL 3D-ELEMENTBecause of the thin walls, this bellows should properly be classed as a shell of revolution forstress and displacement analyse purpose. The geometry of a shell is defined by its thicknessand its midsurface, which is a curved surface in space. Load is carried by a combination ofmembrane action and bending action. A thin shell can be very strong if membrane actiondominates, in the same way that a wire can carry great load in tension but only small load inbending. A shell of a given shape can carry a variety of distributed loadings by membraneaction alone.However, no shell is completely free of bending stresses. They appear at or near point loads,line loads, reinforcements, junctures, changes of curvature and supports. In short, anyconcentration of load or geometric discontinuity can be expected to produce bending stresses,often much larger than membrane stresses, but usually quite localised in a “boundary layer”near the load or discontinuity.Flexure stress and bending moment in a shell are related in the same way as for a plate:t / 2Mx= ∫σ x⋅ z ⋅ dz ,y= ∫−t/ 2t / 2M σy⋅ z ⋅ dz , Mxy= ∫τ xy⋅ z ⋅ dz(3.3)−t/ 2where t is the given thickness of the shell element.A quadrilateral shell element can be produced by combining quadrilateral plane and plateelement. A four-node “flat” quadrilateral is in general a warped element because its nodes arenot all coplanar. A modest amount of warping can seriously degrade the performance of anelement. Commercial software may allow only a very small amount of deformations.Curved elements based on shell theory avoid some shortcomings of flat elements, butintroduce other difficulties. More data are needed to describe the geometry of a curvedelement. Formulation is complicated, as it invokes a shell theory – of which there are many.Membrane and bending actions are coupled within the element, so it is harder to avoidmembrane locking, that is, harder to avoid great overstiffness in bending because details ofthe element formulation cause membrane strains to appear in association with bending action,and membrane stiffness is far greater than bending stiffness if the shell is thin.t / 2−t/ 2±I®L−¬°JXK¯ZYFigure 14Explanation picture of the SHELL43 element. It ismade of four nodes, I, J, K and L, and six differentfaces. Each of these surfaces can admit loads. At eachnode a different thickness t can be defined. Thiselement has plasticity, creep, stress stiffening, largedeflection and strain capabilities.15