Skip to main content

2023 | Buch

A Numerical Approach to the Classical Laminate Theory of Composite Materials

The Composite Laminate Analysis Tool—CLAT v2.0


Über dieses Buch

This book first provides a systematic and thorough introduction to the classical laminate theory for composite materials based on the theory for plane elasticity elements and classical (shear-rigid) plate elements. The focus is on unidirectional lamina which can be described based on orthotropic constitutive equations and their composition to layered laminates. In addition to the elastic behavior, failure is investigated based on the maximum stress, maximum strain, Tsai-Hill, and the Tsai-Wu criteria.

The solution of the fundamental equations of the classical laminate theory is connected with extensive matrix operations, and many problems require in addition iteration loops. Thus, a classical hand calculation of related problems is extremely time consuming. In order to facilitate the application of the classical laminate theory, we decided to provide a Python-based computational tool, the so-called Composite Laminate Analysis Tool (CLAT) to easily solve some standard questions from the context of fiber-reinforced composites. The tool runs in any standard web browser and offers a user-friendly interface with many post-processing options. The functionality comprises stress and strain analysis of lamina and laminates, derivation of off-axis elastic properties of lamina, and the failure analysis based on different criteria.


Chapter 1. Introduction
This chapter introduces the major concept of composite material, i.e., the combination of different components, to obtain in total much better properties than a single component for itself. The focus is on fiber-reinforced composites where a single layer has unidirectionally aligned reinforcing fibers. The chapter concludes with a brief literature review related to textbooks which cover the mechanical behavior of fiber-reinforced materials.
Andreas Öchsner, Resam Makvandi
Chapter 2. Classical Laminate Theory
This chapter first covers the mechanical modeling of a single layer with unidirectionally aligned reinforcing fibers embedded in a homogeneous matrix, a so-called lamina. It is shown that a lamina can be treated as a combination of a plane elasticity element and a classical plate element. For both classical structural elements and their combination, the continuum mechanical modeling based on the three basic equations, i.e., the kinematics relationship, the constitutive law, and the equilibrium equation is presented. Combining these three questions results in the governing partial differential equations. The second part of the chapter covers the stacking of single laminae, generally under different angles, to a so-called laminate. Based on the approach of the classical laminate theory, a simplified stress analysis, and a subsequent failure analysis is derived, without the solution of the system of coupled differential equations for the unknown displacements in the three coordinate directions. This theory provides the solution of the statically indeterminate system based on a generalized stress-strain relationship under consideration of the constitutive relationship and the definition of the so-called stress resultants.
Andreas Öchsner, Resam Makvandi
Chapter 3. Composite Laminate Analysis Tool—CLAT
This chapter describes the programming approach for the Python-based computational tool, the so-called Composite Laminate Analysis Tool (CLAT). The choices for the different tools and libraries, particularly for our user interface, are explained in detail. The second part of this chapter covers the graphical appearance, functionality, and practical handling of CLAT.
Andreas Öchsner, Resam Makvandi
Chapter 4. Application Examples
This chapter presents the solution of some standard calculation problems based on the classical laminate theory. Each solution presents the major result plots or tables obtained in CLAT. The presented problems comprise the calculation of stresses and strains in symmetric and asymmetric laminates, the application of failure criteria, and the ply-by-ply failure of laminates. In addition, the characterization of basic properties of unidirectional laminae based on pole diagrams for the elastic properties and failure envelopes is presented.
Andreas Öchsner, Resam Makvandi
Chapter 5. Source Codes
This chapter provides the commented and structured source codes of the Python files which contains all the written routines.
Andreas Öchsner, Resam Makvandi
A Numerical Approach to the Classical Laminate Theory of Composite Materials
verfasst von
Andreas Öchsner
Resam Makvandi
Electronic ISBN
Print ISBN


    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.