Estimation of plastic loads

Using an elastic-plastic analysis including strain hardening and large deflections or equivalently considering experimental analysis of an actual vessel, one is confronted with the problem of defining a realistic measure of plastic loads. A number of estimations have been used. These are reviewed next. The discussion refers to pressure loading, but the same definitions can be applied to other types of loadings.

The Limit Pressure p0

Characteristic for the limit pressure definition according to the rigid perfectly-plastic theory is (with p = pressure and δ = deflection)

dp/dδ = ∞ or dδ/dp = 0 for p < p0,

and

dp/dδ = 0 or dδ/dp = ∞ for p = p0.

Characteristic for the limit pressure definition according to the elastic perfectly-plastic theory is dp/dδ > 0 for p < p0,

Design by Analysis

2.38 Design by Analysis

and

dp/dδ = 0 or dδ/dp = ∞ for p = p0.

These definitions only apply for small-deflection analyses. The Tangent-Intersection Pressure pti (Fig 2.29)

Pti = 1.35 bar 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 2.5 deflection (mm) load (bar)

Fig 2.29 : Tangent-Intersection method

The tangent-intersection pressure is the pressure at the intersection of the two tangents, drawn to the elastic and plastic parts of the pressure-deflection curves. The value of the pressure obtained by this method is sensitive to the localisation of the tangent-point in the plastic range

The 1% Plastic Strain Pressure p1

The plastic pressure is defined as the pressure with an equivalent plastic strain of 1%. Methods based upon an absolute maximum strain not only will depend on the material assumed, but more significantly on the geometry:

• Material: e.g. a 1% plastic strain is ten times the yield point strain if the yield point stress is 150 MPa, but five times the yield point strain if the yield point stress is 300 MPa. Consequently, the relative size of the elastic and plastic zones will differ and the shape of the pressure-deflection response curves will differ.

• Geometry: Ellipsoidal heads have been found to deform less than torispherical or toriconical heads. Whereas a torispherical vessel may reach a 1% strain at a certain pressure, the ellipsoidal vessel may reach the same pressure at a lower strain.

At a yield hinge location, strains will be larger than at other locations. Consequently, the selection of a strain gauge location on an experimental vessel presents a variable when yield hinge locations are not known precisely or a priori.

Thus, in summary, a strain basis for defining a plastic pressure may be subject to error in locating the exact location of maximum strain. Also, strain is a local phenomenon that is not indicative of plastic work.

The Twice-Elastic-Deformation Pressure p2y

A plastic pressure is defined to be the pressure at which the deflection or strain reaches twice the value of the elastic deflection or elastic strain at the first yield pressure py. Thus, p2y depends upon

py. Exact determination of py using a computer analysis should not be a problem. In experiments

however, determining the elastic limit on the load deflection curve may be subject to error. The Twice-Elastic-Slope Pressure pϕ

A plastic pressure is defined to be the value at the intercept of a line drawn from the origin of a pressure-deformation curve at a slope of twice the slope of the elastic portion of the curve (see Fig 2.30). y = 2.2x y = 1.1x Pφ = 1.39 bar 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 2.5 deflection (mm) load (bar)

Design by Analysis

2.40 Design by Analysis

The 0.2% Offset Strain Pressure p.2

The 0.2% offset strain pressure is a test pressure that causes a permanent strain of 0.2%. The Proportional Limit Definition ppl

Ppl is a test pressure defined as the pressure causing the displacement versus pressure curve to

deviate from linearity. The displacement of the vessel is to be measured at the weakest point, the most highly stressed point, giving the lowest value of ppl.

Analytical calculations can determine this pressure correctly. It will not necessarily be equal to the first yield pressure py. Experimental measures are subject to error in determining the point of

deviation from linearity. Values of ppl up to 30% greater than py can be estimated from an

experimental curve.

This method of determining a plastic pressure will generally give a lower bound to the plastic pressure found by most other methods.

The Plastic-Instability Pressure ppi

This is an actual plastic collapse pressure and not just an estimate of a plastic pressure. It may be identical to the limit pressure if large deflection effects are small, e.g. when the vessel is relatively thick. However, the plastic-instability pressure may be less than the small-deflection limit pressure as in the case of a large-deflection elastic-plastic solution, with geometrical weakening (see Fig 2.27, curve c). The plastic instability is defined by a zero slope on the pressure-deflection curve. A large-deflection elastic-plastic analysis is required to detect ppi. It will also be detected in

experiments on actual vessels and it is possible to have plastic instability pressures less than lower- bounds to the limit pressure where the latter are based on small-deflection analyses. It may occur that some of the above estimations of the limit pressures will be non-conservative estimates of the real plastic collapse pressure, an instability pressure, if the estimates were applied to small- deflection theoretical results.