# Finite element eco-approach

When the analytical or mathematical solution does not exist, numerical methods such as finite element method (Jones, 1999 and Zienkiewicz et al, 2005) may be used to yield an approached solution. However, to achieve sustainable structural performance that is also based on an assessment of environmental and health performances, it is imperative to consider these constraints in more detail and integrate them into the finite element analysis. In such situation, the finite element eco-approach for dynamic analysis will first consist in resolving for one element type (e) the system of the following equations:

M]{e(t)}+ [ke] {(t)}= {fe (t)} (18)

where fe(t)} is the vector of nodal displacements and rotations, fqe(t)} is the vector of accelerations, and {f (t)} is the vector of forcing functions (external loads). Whereas [Me] and [ke] are the mass matrix and eco-stiffness matrix, respectively. These are defined by the following expressions:

[ite] =J [] C []dQ,

Qe

element; [m] is the inertia matrix, [E] is the displacement

coupling-bending eco-stiffness matrix. The Jacobian matrix [J ] is used to allow the passage from Cartesian coordinates (x, y, z) to natural coordinates (Z, n, Z).

For static analysis, the acceleration vector vanishes and Equation (18) is independent of time and becomes as follows:

[ke] fe}= {fe } (21)

Assembling eco-stiffness and mass matrices, and forcing functions vector of each element involved into the generation of the whole structure yields the global system of equations:

• For dynamic eco-analysis: [M] f (t)} + [K] f (t)}= {F(t)} (22)

• For static eco-analysis: [K] {}= {f} (23)

where {} is the vector of global displacements, {q} is the vector of global accelerations, {f} is the vector of external loads (vanishes for free vibration analysis), [M] is the global mass matrix, and [K] is the global eco-stiffness matrix.

By comparing the eco-results with the classical ones that do not take into account environmental and health considerations, we can yield an estimate difference value called "eco-deviation", which may be calculated using the following relation:

where V is the eco-result corresponding to X=Xi (0 < Xi < 1) and V is the classical result corresponding to X=1, generally.

For better comment and understanding of this eco-approach, the final results may be presented in the form of graphs with normalized axes. A dimensionless quantity named "eco-efficiency ratio", representing the ratio between eco-results and classical ones, can be

chosen for the у-axis and denoted for example by the Greek letter Л= V/V. When performing stress analysis of composite materials and structures, Л can for instance be

equal to ct/ct or т lx. The discrepancy between classical and sustainable results can help designers and analysts to evaluate easily environmental and health performances. Moreover, this discrepancy can be minimised via alternative solutions to reach the appropriate value required by ecodesign standards.

2. Conclusion

Based on mathematical formulations, scientific, industrial and technological know-how in the field of FRP composite materials and structures, this contribution aims to innovate and develop a new approach providing the integration at each stage of the designing process three balanced key criteria characterised mainly by Quality assurance, Health protection and Environmental preservation (Q, H & E). To achieve these requirements, new criteria in the form of eco-coefficients were defined and developed. However, greater depth of study is still required to establish the rating satisfaction measure that yields the appropriate values of sustainable coefficients, which are considered as an important source of reference for comparison survey. To approach sustainability values of these coefficients, probability analysis of ecodesign function and some optimisation procedures based on a new technique of additive colours were undertaken in accordance with the three balanced key criteria. When these eco-coefficients are approved by sustainability standards, they can then be integrated into formulations of design and analysis, in characterisation tests; they can also be implemented into future finite-element computer programs, etc. Designers, analysts and engineers can make better use of ecodesign aspects to assess environmental and health performances when comparing eco-results with classical ones.

This investigation could be integrated in the international standards, codes and guidelines for sustainability research actions, and contribute to new orientations in the design of eco­friendly composite materials and structures. It may also be regarded as a stimulation of eco­innovation, sustainability and research activities in the field of FRP composite products ecodesign and as an encouragement for designers and engineers to have a great motivation towards the integration of health and environmental aspects into the designing process. In addition, new ecodesign recommendations could be developed via this innovative survey. These recommendations will, however, increase the design space of future composite materials and their products. This will offer a new data to use in evaluating the different stages of a material/process/product life-cycle.

This type of "eco-action" constitutes a multidisciplinary approach that can involve specialists in mechanical/civil/structural and process engineering, mathematics, physics,
chemistry, health, environment and sociology. In addition, the impacts that may be investigated including: (1) the undesirable substances entering in the manufacturing process, (2) the amount of emissions of greenhouse gases, (3) the level of Q-H-E interaction, (4) the quantity of the waste production and expired materials, (5) the classification of the company with regard to the authorized regulations, etc. Then, alternative solutions leading to new methods of ecodesign are suggested. The approved eco-coefficients may become a source of normative coefficients used for validating the different manufacturing stages and qualifying & certifying the new developed eco-composite materials and structures.

3. References

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Attaf, B. & Hollaway, L. (1990b). Vibrational analyses of glass reinforced polyester composite plates reinforced by a minimum mass central stiffener. Composites Vol.21, No.5, (Sept.1990) pp.425-430, ISNN 0010-4361 Attaf, B. (2007). Towards the optimisation of the ecodesign function for composites. JEC Composites. No.34, (July-August 2007) pp.58-60, ISNN 1639-965X Attaf, B. (2008). Eco-characterisation of composite materials. JEC Composites. No.42, (July – August 2008) pp.58-60, ISNN 1639-965X

Attaf, B. (2009). Probability approach in ecodesign of fibre-reinforced composite structures.

Proceedings of CAM’2009 Conference, Algeria, November 2009, Biskra Hollaway, L. & Attaf, B. (1989). On the vibration of glass/polyester composite stiffened and unstiffened rectangular plates. Proceedings of the Seventh Int. Conf. on Composite Materials, pp. 435-444, China, October 1989, Chinese Society of Aeronautics, Beijing Jones, R. M. (1999). Mechanics of composite materials. Taylor & Francis, Inc., ISBN 1-56032-712- X, Philadelphia

Saarela, O. (1994). Computer programs for mechanical analysis and design of polymer matrix composites. Prog. Polym. Sci. Vol.19, pp 58-60, ISNN 0079-6700 Yang, T. Y. (1986). Finite element structural analysis. Prentice-Hall, Inc., ISBN 0-13-317116-7, New Jersey

Zienkiewicz, O. C., Taylor, R. L. & Zhu, J. Z. (2005). The finite element method: its bases and fundamentals. Elsevier Butterworth-Heinemann, ISBN 0 7506 6320 0, Oxford

Updated: September 24, 2015 — 9:11 am