ASME Section VIII Div.2, Part 5 (DBA). Why is the limit 2 x Sy important? Elastic Shakedown

Let’s consider, for example, a fully pressurized pressure vessel with a load applied on the nozzle as shown in the picture. This load creates a bending stress (secondary stress) at the nozzle-to-head juncture. In the left-hand side of the nozzle, this load creates a tension strain value ε₁ (see below), somewhat beyond the yield strain as shown in the picture. Since we are considering the case of a secondary stress, we shall assume that the nature of the loading is such as to cycle the strain from zero to ε₁ and back to zero (self-limiting). When the load is removed, a residual compressive stress will be present in the area of the juncture under study.

Let’s give this a closer look. As indicated, the applied load has created a tension strain value ε₁, somewhat beyond the yield strain as shown in the path 0AB (see below). The calculated elastic stress would be S₁=E x ε₁. When the load is removed, a residual compressive stress –S₁ will be present. In fact, the magnitude of the residual compressive stress is S₁ – S_y. On any subsequent loading, this residual compression must be removed before the stress goes into tension and thus the elastic range has been increased by the quantity S₁ – S_y. This is known as Elastic shakedown.

Now, let’s assume that on a subsequent cycle the load applied moves the strain up to point D (see below), this corresponds with ε₂. The calculated elastic stress that corresponds to this strain value would be S₂=E x ε₂. It can be observed that S₂ > 2S_y. Since the magnitude of the residual compressive stress will remain the same, now the unloading line arrives to point E, showing that the external fiber of the nozzle yields in compression with the corresponding plastic strain. If the cyclic load is not increased on subsequent cycles, no additional plastic strain will accumulate. This is known as plastic shakedown.

From this analysis, it can be observed that as long as the stress levels induced by the load (S₁), are lower or equal to two times S_y, the elastic range becomes 2 S_y and the calculated secondary elastic stress will “shake down” to purely elastic action. This explains why the limit imposed by building codes on secondary stresses, or rather their variation, is exactly twice the elastic limit.

For more information:

ASME VIII | Design of Pressure Vessels for Industrial Plants

ASME VIII Div. 2 | Design of Pressure Vessels by Analysis