Part 1: Stiffness Matrix and Thermal Characteristics

Carbon fibre reinforced plastic (CFRP) is now frequently used in a large variety of applications; sports goods such as golf clubs and tennis rackets, primary structure of sports cars, from specialist suppliers such as Lotus to Formula 1 Racing Cars, components in commercial and fighter aircraft and in the space industry for primary structure of satellites, antennae systems and the launch vehicles themselves.

Structural components in the form of plates, shells, sandwich panels, sandwich shells, beams and struts, are manufactured from CFRP materials to provide a design with high strength-to-weight and stiffness-to-weight ratios. This is achieved by using the minimum number of plies of unidirectional material or woven fabric orientated at various angles to provide a laminate that satisfies the stiffness, strength, thermal distortion and functional requirements. The thickness, orientation and lay-up of the plies can produce laminates that exhibit coupling in membrane and bending behaviour. Such characteristics can have significant effect on the response of the laminate to mechanical and thermal loads.

Confirmation of the structural integrity of CFRP components is generally achieved by qualification testing of the hardware. Although verification of composite analysis against test results is desirable this is often not achievable at the start of the design process. The early design trade off studies are generally assessed by use of analysis techniques, Finite Element Analysis and other purpose-written programs, internal or proprietary, applying Classical Lamination Theory, (CLT). Experience shows that the availability of an analysis capability in a program does not guarantee correct implementation or application by engineers. There is a need to provide verification examples that demonstrate correct implementation and assist the engineer in effective application. This is evident particularly with respect to transverse shear properties of laminates where a number of different sources provided a number of different answers.

Benchmarking basic laminate analysis can be considered in three primary categories; membrane and bending stiffness characteristics; membrane and bending strength analysis; and transverse shear properties, stiffness and strength. A number of benchmark problems to address these aspects of composite analysis have been collected from a variety of sources to supplement those published some years ago by NAFEMS, (ref 12). This document discusses classical laminate analysis and defines benchmark problems for the membrane and bending stiffness analysis of laminates. The benchmarking of the more complex and specialist subjects such as fracture and crack propagation are the subject of a more recent NAFEMS report, (ref 13).

The use of benchmark problems to investigate the implementation of CLT in the available analysis tools provide the engineer with information on the significance of ply thickness, ply orientation and stacking sequence with respect to laminate stiffness characteristics, thermal characteristics and moisture absorption characteristics. They also verify the implementation of Classical Lamination Theory in proprietary software and how the membrane, bending and coupling material properties are determined in the programs.

Contents

1. Introduction

2. Classical Lamination Theory

2.1 Laminate Membrane and Bending Stiffness Characteristics

2.2 Laminate Membrane and Bending Hygrothermal Characteristics

3. Membrane and Bending Characteristics of Laminates

4. Benchmark Examples For Laminated Shell Membrane and Bending Material Characteristics

5. Finite Element Program Output for Laminated Shell Benchmark Problems

5.1 Benchmark Problem 1 LSMC1 [0/0/0/0] (Table 5.1)

5.2 Benchmark Problem 2 LSMC2 [0/0/0/0] (Table 5.1)

5.3 Benchmark Problem 3 LSMC3 [0/60/-60/0/0/-60/60/0] (Table 5.1)

5.4 Benchmark Problem 4 LSMC4 [0/90] (Table 5.2)

5.5 Benchmark Problem 5 LSMC5 [45/90/0] (Table 5.2)

5.6 Benchmark Problem 6 LSMC6 [90/45/-45/0] (Table 5.2)

6. Benchmark Examples for Material Characteristics From Unit Cases

6.1 Laminated Shell Material Characteristics Benchmark Problem LSMC1

6.2 Laminated Shell Material Characteristics Benchmark Problem LSMC2

6.3 Laminated Shell Material Characteristics Benchmark Problem LSMC3

6.4 Laminated Shell Material Characteristics Benchmark Problem LSMC4

6.5 Laminated Shell Material Characteristics Benchmark Problem LSMC5

6.6 Laminated Shell Material Characteristics Benchmark Problem LSMC6

7. References