An analytical method for calculating stiffness of two-dimensional tri-axial braided composites Mahmood M. Shokrieh * , Mohammad S. Mazloomi Composites Research Laboratory, Center of Excellence for Experimental Solid Mechanics and Dynamics, Department of Mechanical Engineering, Iran University of Science and Technology, 16846-13114 Tehran, Iran article info Article history: Available online 11 May 2010 Keywords: Braided composites Stiffness calculation Analytical method Unit cell Tows undulation abstract This paper presents a new analytical method for calculation of the stiffness of two-dimensional tri-axial braided composites. A unit cell has been introduced as a representative cell of a braided composite and its components. The braided composite is considered as consisting of three layers. The first two layers rep- resent braided tows and the third layer is the axial tow. Then, using rule of mixtures, mechanical prop- erties of each layer are calculated. Next, using analytical relations, the undulation of representative layers of braided tows is calculated. Finally, using a volume averaging method, the total stiffness of the braided composite is calculated. The results are compared with those obtained from experimental methods and the effect of braided tows crimp on the stiffness of braided composites is examined. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Braided composites have received increasing attention from sci- entists during the past two decades. Due to their exceptional mechanical properties, braided composites are utilized in aero- space, automobile and marine industries [1]. The braiding process competes well with filament winding, pultrusion, and tape lay-up. Braiding compares favorably in terms of structural integrity of components, design flexibility, damage tolerance, repair ability, and low manufacturing cost. Braiding advantages are high rate of strand deposition on the mandrel, ability to produce complex shapes, low capital investment cost, and minimal labor cost. The most important braiding process disadvantage is the difficulty in producing low braid angle preforms [2]. Braided structures may be divided into two categories, namely two-dimensional and three-dimensional. Due to the complexity of braid structures, var- ious parameters including tow and matrix mechanical properties, braid angle, tow crimp level and tow volume fraction can affect their mechanical properties [3]. Extensive studies have been conducted to investigate mechani- cal properties of textile composites [4]. Among textile composites, woven composites are mostly used for structural applications. Hence, most studies carried out in this area are related to this type of composites. Ishikawa and Chou [5–7] have developed three ana- lytical models for two-dimensional woven composites based on Classical Lamination Theory (CLT). These three models are the mo- saic model, fiber undulating model and the fiber bridging model. When compared with the experimental data, they show that the mosaic model provides an appropriate estimate of elastic proper- ties of composites, the tow undulating model is useful for modeling of plain weave fabrics and the bridging model is desirable for satin weave fabrics. Extending the one-dimensional model of Ishikawa and Chou [7] into a two-dimensional model, Naik and Shembeker [8] presented a model for the analysis of plain woven fabric com- posites. A micromechanical model was introduced by Huang [9] for two-dimensional examination of mechanical properties of com- posites reinforced by braided tows or woven tows. Tsai et al. [10] developed a parallelogram spring model for analysis of two- dimensional tri-axial braided composites. In this model the effect of changes in various parameters of braid such as braid angle, tow volume fraction, etc. on mechanical properties of braid were investigated. Potluri et al. [11] presented an investigation of flex- ural and torsional properties of two-dimensional biaxial and tri-ax- ial braided composites with one or more layers, at different braid angles. Recent studies have shown that crimp angle and braid angle af- fect the strength and stiffness of the braided composites. Through experimental data, Phoenix [12] has shown that increase in crimp angle or braid angle causes decrease in the strength of braided composite. Chen et al. [13] have presented finite multiphase ele- ment method in their research. In this method, each element rep- resents more than one material, i.e., braided composites consist of three types of unit cells in different regions which are: interior, surface and corner regions. Masters et al. [14], presented an analytical model for predicting mechanical properties of two-dimensional tri-axial braided com- posites, using a simple rule-of-mixtures idea. Byun [15] developed 0263-8223/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2010.04.016 * Corresponding author. Tel./fax: +98 21 7720 8127. E-mail address: shokrieh@iust.ac.ir (M.M. Shokrieh). Composite Structures 92 (2010) 2901–2905 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct