Benchmarks for Membrane and Bending Analysis of Laminated Shells
Part 2: Strength Analysis
Carbon fibre reinforced plastic (CFRP) is now frequently used in a large variety of applications including items of primary structure in automotive and aerospace industries, body shell structures, launch vehicle adaptors and central structure of satellite structures.
Structural components, 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.
Although confirmation of the structural integrity of CFRP components is generally achieved by qualification testing of the hardware 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. Reference (1) provides example benchmark problems for the membrane and bending stiffness characteristics and thermal characteristics of laminated shells from Classical Lamination Theory.
To assess the structural integrity of a laminated shell the engineer needs a definition of how the stresses are distributed through the individual plies and a method of assessing whether a ply has failed that may lead to subsequent failure of the laminate. The laminate stiffness matrix is used to determine the mid-plane strain and curvature in response to the applied loads. From the laminate ply stacking sequence, ply orientation and unidirectional stiffness properties of the individual plies the ply stresses can be determined. The ply stresses can be used with the ply unidirectional strengths in a number of failure criteria to assess whether ply failure has occurred. There are a number of strength failure criteria varying in complexity. They range from the simple comparison of components of ply-stress or strain with the appropriate allowable to the application of polynomial equations, similar to the Von Mise failure criteria for isotropic materials, such as Hill, Hoffman or Tsai Wu. The polynomial failure criteria uses the ply stresses in conjunction with the ply unidirectional strengths to compute a failure index taking account of the interaction of the components of stress. Some failure criteria are more detailed and endeavour to take account of the mode of failure in the ply, matrix failure or fibre failure. Background to the various failure criteria can be found in a number of text books, ESDU data sheets and articles published in journals, references (2), (3), (4) and (5) for example.
Even though the validity of the various failure criteria continues to be discussed the engineer still requires a method of quantifying the structural integrity of a proposed design. Therefore it is necessary to provide verification examples that demonstrate correct implementation of selected failure criteria in analysis software. This document defines benchmark problems for the strength analysis of laminates subject to membrane and bending loads, mechanical and thermal. The effect on structural integrity of design features, holes, notches and cutouts for example, require special attention and are dealt with by other technical papers.
The benchmark problems are not intended to recommend any particular failure criteria or analysis program. They can illustrate correct implementation of the more commonly used failure criteria analysis programs.
2. Theory of Selected Failure Criteria
2.1 HILL Failure Criteria
2.2 HOFFMAN Failure Criteria
2.3 TSAI-WU Failure Criteria
3. Failure Criteria Benchmark Problems for Mechanical Loads
3.1 Laminated Shell Strength Analysis Mechanical Load 1 (LSSAM1)
3.2 Laminated Shell Strength Analysis Mechanical Load 2 (LSSAM2)
3 Laminated Shell Strength Analysis Mechanical Load 3 (LSSAM3)
4. Failure Criteria Benchmarks for Thermal Loads
4.1 Laminated Shell Strength Analysis Thermal Load 1 (LSSAT1)
4.2 Laminated Shell Strength Analysis Thermal Load 1 (LSSAT2)
£32 | $49 | €46
£65 | $98 | €93
Order Ref: R0093
First Published - December 2005
Softback Report - 22 Pages