Finite Elements in Analysis and Design 39 (2003) 1023–1037 www.elsevier.com/locate/nel Three-dimensional modelling of the vertical–horizontal rolling process Xiong Shangwu, J.M.C. Rodrigues, P.A.F. Martins ∗ Departamento de Engenharia Mecˆ anica, Instituto Superior T ecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Received 21 January 2002; accepted 15 April 2002 Abstract This paper discusses the utilisation of the slightly compressible nite element formulation to the numerical analysis of the vertical–horizontal rolling process. The rst part of the paper details the basis of the plasticity theory for slightly compressible materials and presents an overview of its numerical implementation. Of particular interest in vertical–horizontal rolling are topics related to the prediction of the dog-bone shape when using a Eulerian formulation, the treatment of the frictional boundary conditions along the three-dimensional roll–workpiece contact interfaces and the modelling of the material streamlines within the plastically deforming region. The second part of the paper presents a comparison between theoretical predictions obtained by means of a special purpose computer program developed under the proposed approach and experimental results taken from the literature. Assessment is made in terms of the cross-sectional shapes, roll separating force and torque. ? 2002 Elsevier B.V. All rights reserved. Keywords: Vertical–horizontal rolling; Slightly compressible material model; Finite element method 1. Introduction The vertical–horizontal rolling process is widely used to perform large width reductions of slabs in hot strip mills. The vertical rolling stand, also referred as edge rolling or slab edging, deforms the slabs in the width direction giving rise to the formation of a cross-sectional “dog-bone” shape. Vertical rolling can be performed either by means of at cylindrical rolls or calibre rolls. The latter are usually introduced to improve stability and edging eciency. The horizontal rolling stand is adopted to atten the dog-bone cross-sectional shape and to reduce the thickness of the slab (Fig. 1). * Corresponding author. Tel.: +351-21-841-7561; fax: +351-21-841-9058. E-mail address: pmartins@ist.utl.pt (P.A.F. Martins). 0168-874X/03/$-see front matter ? 2002 Elsevier B.V. All rights reserved. PII:S0168-874X(02)00154-3