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equilibrium and compatibility

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Tags: equilibriumandcompatibility
Design codes talk about collapse and instability but it actually means the same thing. What it should really be called is loss of equilibrium. Similarly codes talk about primary and secondary stresses which are described as load-controlled or displacement controlled. Primary stresses are also equilibrium stresses. Secondary stresses are called self-limiting or self-equilibrating because they don't cause collapse by themselves, they just thin the part enough so that the primary (equilibrium) loads cause failure. However you may be better remembering secondary stresses as discontinuity or compatibility stresses because they actually arise due to need to match displacements and slopes at geometry or load discontinuities. Unlike the equilibrium-derived stresses; eg p.r/t for pressure vessels or M.y/I for beams, these compatibility stresses dissipate with distance; typical 2.5(rt)^0.5 from the discontinuity. In the old days before FEA the design engineer would do a discontinuity analysis to find these 'secondary stresses' and hence it was easy to separate out primary and secondary stresses. The design code elastic assessment methods are based on the assumption that the designer has been able to do this discontinuity analysis and hence can separate out primary and secondary stresses easily. Alas with FEA this separation is not possible. Moreover stress concetrations, bearing stresses etc are also present in the FEA solution. The usual solution is to linearize these extra stresses away. In a future post this will be discussed.

However, now that you can understand equilibrium and compatibility you might be able to see that the area replacement technique for pressure vessel design codes is actually an equilibrium solution to a compatibility problem. The only pressure vessel code which correctly treats nozzles on shells as a compatibility problem is the now deprecated code PD5500. Area replacement is used because for most of the time it seems to work. However it may in reality just be the material ductility or the safety factors which saved the design. But let's be honest, using the wrong theory is not a great long term design tool. For ever more complex and cost-saving designs you will need a backup. Another future post will discuss this.

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