Surfactant lamellar phases are often complicated by the formation of multilamellar (onions) under shear, which can originate simply by shaking the sample. A systematic study has been performed on the C10E3-D2O system in which different bilayer structures under a steady shear flow were investigated by small-angle neutron scattering (SANS). In the present model-system, multilamellar vesicles exist at lower temperatures and planar bilayers (classical lamellar) exist at higher temperatures. Our study focuses on the role of the saddle-splay modulus of the bilayer, <(kappa)over bar>(b), in the transformation from onion to plane to multiply connected bilayer structure as a function of temperature. (C) 2000 Elsevier Science B.V. All rights reserved.