Partial-mixed formulation and re®ned models for the analysis of composite laminates within an FSDT Ferdinando Auricchio a , Elio Sacco b, * a Dipartimento di Ingegneria Civile, Universit  a di Roma ``Tor Vergata'', Via di Tor Vergata, 00133 Roma, Italy b Dipartimento di Ingegneria Industriale, Universit  a di Cassino, Via Di Biasio 43, 03043 Cassino, Italy Abstract The present paper proposes a partial-mixed variational formulation for the analysis of composite laminates within the First-order Shear Deformation Theory (FSDT). The considered functional is recovered from the Hellinger±Reissner mixed principle and it appears to be particularly suitable for the determination of the FSDT governing equations since the transverse shear stresses are treated as independent variables. Accordingly, it is possible to obtain an accurate description of the shear stress pro®les. Herein, the attention is concentrated on two dierent re®ned FSDT models, both having piecewise parabolic shear stress pro®les. Furthermore, within one of the two re®ned models, the partial-mixed formulation is used to derive a performing ®nite element. Finally, analytical solutions from the classical and the re®ned FSDT models are compared to three-dimensional (3D) analytical solutions as well as to results obtained form the proposed laminate ®nite element. Ó 1999 Elsevier Science Ltd. All rights reserved. Keywords: Composite laminates; First-order Shear Deformation Theory (FSDT); Partial-mixed formulation 1. Introduction Laminate theories reduce the study of a three-di- mensional (3D) layered body to a two-dimensional (2D) problem. This 3D±2D reduction is in general performed assuming speci®c laminate structural behaviors, i.e. in- troducing hypotheses on the strain ®eld or on the stress ®eld or on both the strain and the stress ®elds. Ac- cording to these dierent possibilities, several laminate models have been proposed in the literature [1,2]. In general, laminate theories based on both strain and stress hypotheses lead to models characterized by good performances. Within this approach, the most used formulation is the so-called First-order Shear Defor- mation Theory (FSDT). Originally developed by Yang et al. [3] and by Whitney and Pagano [4] as an extension of the plate theory proposed by Reissner [5] and Mindlin [6], the FSDT allows the determination of accurate so- lutions for a wide class of laminate problems. Although the most common FSDT variational for- mulations are based on displacement approaches, mixed formulations have been proposed in the literature, mainly for the de®nition of innovative ®nite elements, e.g. see Refs. [7±10]. However, it is fundamental to observe that the correct use of the FSDT requires the de®nition of through-the-thickness shear stress pro®les, leading to the determination of the so-called shear correction fac- tors. Unfortunately, these pro®les are known a priori only for homogeneous plates or for cross-ply laminates under cylindrical bending and not for general lamina- tion sequences [11±13]; this aspect represents a clear limitation to a correct use of the FSDT. To overcome this diculty, two dierent approaches can be found in the literature. The ®rst approach is based on the re®nement of the model by the use of ad- ditive shear warping functions, e.g. see Refs. [14,15], leading to an increment in the kinematic unknowns. This re®nement clearly makes the method more expen- sive from a computational viewpoint. The second ap- proach consists in the development of iterative procedures, as the one proposed by Noor and coworkers and based on the analytical solution of rectangular laminates subjected to sinusoidal loading [16±18]. However, these iterative procedures are not extendable to generic laminates geometries. The present work starts with a review of the FSDT together with a detailed discussion of the basic hy- potheses introduced to construct the model. Then, from the Hellinger±Reissner functional, a partial-mixed functional for the FSDT is recovered. The obtained Composite Structures 46 (1999) 103±113 * Corresponding author. E-mail: sacco@ing.unicas.it 0263-8223/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 0 3 5 - 5