An Efficient Numerical Framework for Capturing Localised 3D Stress Fields in Laminated Composites

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


Composite materials offer considerable advantages in the impetus towards lightweighting and therefore the exploitation of fibre-reinforced composites in engineering structures has been steadily diversifying from e.g. sports equipment and racing cars, to helicopters and commercial aeroplanes. Furthermore, recent advancements in composite manufacturing technology have facilitated the use of complex configurations inindustrial design applications. For reliable design of multilayered structures, accurate stress analysis tools are required. However, with an increasing structural complexity, predicting structure’s response can be non-intuitive and often can not be modelled adequately using classical lamination theory. High-fidelity finite element methods (FEM) are often employed to obtain reliable three-dimensional (3D) stress analyses with the desired level of accuracy. However, these models are computationally expensive and are prohibitive for iterative design studies. Consequently, over the years, several one-dimensional (1D) and two-dimensional (2D) models based on higher-order theories have been proposed for the analysis of multilayered composite structures with the aim of predicting accurate 3D stress fields in a computationally efficient manner. The majority of these numerical models either lack kinematic fidelity or accuracy in capturing localised regions of the structure, or have limited capabilities to model complex structures.

The work presented here uses the Unified Formulation (UF) that supersedes classical theories by exploiting a compact, hierarchical notation that allows most classic and recent theories to be retrieved from one, hence unified, model. Importantly, and unlike many classic theories, the UF applies to the partial differential equations governing three-dimensional elasticity. Full stress and strain fields are, therefore, recovered by its implementation. Although current implementations are found wanting in this respect, in a UF setting, complex geometries could easily be analysed. This is because the displacement field is expressed by means of classic 1D (beam-like case) and 2D (plate- and shell-like cases) finite element elements that need not be prismatic. Additional expansion functions are employed to approximate 3D kinematics over cross-sections (beam-like case) and through-thickness (plate- and shell-like cases). In the present work, the 1D UF is adopted and developed further by introducing a hierarchical, Serendipity Lagrange polynomialsbased, cross-sectional expansion model. The 3D stress predicting capabilities of the proposed model is verified against high-fidelity finite element models and other numerical and experimental results available in the literature by means of static analyses of isotropic, constant- and variable stiffness laminated composite, beam-like and stiffened structures. Special attention is given to the accuracy of the model in capturing 3D stress response in the localised regions, such as near geometric or constitutive discontinuities, constraints and point of load application, and using these further for predicting the structure’s failure response. Finally, to showcase a possible application, the model is applied to analyse non-prismatic and curved structures. The general formulation presented herein is well-suited for accurate and computationally efficient stress analysis for industrial design applications.
Date of Award28 Nov 2019
Original languageEnglish
Awarding Institution
  • The University of Bristol
SponsorsEC Marie Curie ITN
SupervisorAlberto Pirrera (Supervisor) & Paul M Weaver (Supervisor)


  • laminated composites
  • stress fields
  • finite element analysis
  • unified formulation

Cite this

An Efficient Numerical Framework for Capturing Localised 3D Stress Fields in Laminated Composites
Patni, M. (Author). 28 Nov 2019

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)