SUMMARYThis paper describes a method for numerically modelling the incremental plastic deformation of shells and applies the method to incremental sheet forming (ISF). An upper bound finite element shell model is developed based on sequential limit analysis under the rigid plastic assumption, which is solved by manipulating the problem into the form of a second-order cone program (SOCP). Initially, the static upper bound plate problem is investigated and the results are compared with the existing literature. The approach is then extended to a shell formulation using a linearized form of the Ilyushin yield condition and two methods for treating the Ilyushin condition are presented. The model is solved efficiently using SOCP software. The resulting model shows good geometric agreement when validated against an elasto-plastic model produced using existing commercial software and with measurements from a real product produced using ISF.

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We believe that the comments raised by Makrodimopoulos and Martin [2] are mainly based upon a misinterpretation of what we were trying to say in our paper. They raise five main points and we address them in detail below.First point: We are certainly not claiming that the method described in our article achieves a minimum upper bound nor an exact upper bound. There are several instances in the article where this is made clear, for instance from the passage on page 957 'the deformation field, or collapse mechanism, of the material is of a greater interest here than the accurate computation of the value of the load multiplier'. Had we actually derived an expression for the exact upper bound, then we would have been making this claim explicitly. What our paper does claim is much more conservative in that, considering the results in Figure 2, the solution appears to give a good approximation to the upper bound solution when compared to Andersen et al.'s work. Furthermore, unlike the method of Andersen et al., 'it approaches the minimum upper bound limit load from above with increasing mesh density.' We do accept that the terms upper bound, true upper bound and strict upper bound are used more loosely than they perhaps should be in our article, although we note that other authors are often similarly loose in their terminology.Provided that Makrodimopoulos and Martin acknowledge that our paper does not claim to derive an exact upper bound, then we believe that their commentary does give a description of some issues that need to be addressed in order to derive the exact upper bound.Second point: We do not accept this point. The statement 'restricting it to vary linearly over the element' is certainly not used to imply that a non-linear function can be linear and we feel that it is clear from (21) in the manuscript what is meant by this statement. The value of Ck is taken at the three nodes and linear interpolation is used to obtain an approximate overestimate of the value for Ck at other regions over the element, and this is done so that the integral can be calculated analytically. This is no different from the term r (x) that Makrodimopoulos and Martin