### International Journal of Mathematical, Engineering and Management Sciences

#### ISSN: 2455-7749

On The Optimal Forms Finding of Shallow Foundations Made Up of Four Foothills with Explicit Considerations of Structural Perturbations

#### On The Optimal Forms Finding of Shallow Foundations Made Up of Four Foothills with Explicit Considerations of Structural Perturbations

Koumbe Mbock
Department of Mathematics and Physics, National Advanced School of Engineering, Yaounde, P.O. Box 8390, Cameroon.

Etoua Remy Magloire
Department of Mathematics and Physics, National Advanced School of Engineering, Yaounde, P.O. Box 8390, Cameroon.

Ayissi Raoul Domingo
Department of Mathematics, Faculty of Sciences, University of Yaounde, P.O Box812, Cameroon.

Mamba Mpele
Department of Civil Engineering, National Advanced School of Engineering, Yaounde, P.O. Box 8390, Cameroon.

Okpwe Mbarga Richard
Department of Civil Engineering, National Advanced School of Engineering, Yaounde, P.O. Box 8390, Cameroon.

;
Accepted on February 26, 2019

Abstract

In the absence of the exact footing form of shallow foundations, we develop a procedure to determine the optimal footing form made up of four foothills from the knowledge of the inexact footing forms. The structural perturbations that are the major cause of the inexact forms are approximated in linear elastic model whose the solution is used to formulate the evolutionary structural optimization problem. To stabilize the solution, a serial of decisions is made to minimize structural perturbations in finite element modeling, initial volume and design constraints. By using the evolutionary structural optimization technique, we examine if the material of efficient and inefficient perturbations is needed or not on the points of inexact forms. Our analysis shows that the loading forces can be transferred to structural perturbations when they are efficient and used to reinforce the design material. This transfer can modify geometric elements of footing in finite element analysis and the optimal solution. The results of the numerical experiment provide the optimal footing form of shallow foundation, the sizes of associated foothills and the form of inefficient perturbations. This approach allows to redesign the structures from the inexact forms and detects the errors of dimensioning.

Keywords- Perturbations, Ill-posed problems, Linear elastic model, Eurocode 7, Topology optimization, Foundations.

Citation

Mbock, K., Magloire, E. R., Domingo, A. R., Mpele, M., & Richard, O. M. (2019). On The Optimal Forms Finding of Shallow Foundations Made Up of Four Foothills with Explicit Considerations of Structural Perturbations. International Journal of Mathematical, Engineering and Management Sciences, 4(3), 601-618. https://dx.doi.org/10.33889/IJMEMS.2019.4.3-048.

Conflict of Interest

The authors confirm that there is no conflict of interest to declare for this publication.

Acknowledgements

We thank the Department of Mathematics and Civil Engineering for their cooperation at the National Advanced School of Engineering (NASE/ UYI) in Yaounde, Cameroon. We would also like to show our gratitude to Ayissi Raoul Domingo, Professor of Mathematics at University of Yaoundé I for assistance on the manuscript at the department of mathematics. We thank the African Center of Excellence in Information and Communication technologies at the University of Yaounde I for their collaboration and support.

References

Ayissi, R.D., & Etoua, R.M. (2017). Optimal control problem and viscosity solutions for the Vlasov equation in Yang–Mills charged Bianchi models. Advances in Pure and Applied Mathematics, 8(2), 129–140.

Bhunia, A.K., Duary, A., & Sahoo, L. (2017). A genetic algorithm based hybrid approach for reliability-redundancy optimization problem of a series system with multiple-choice. International Journal of Mathematical, Engineering and Management Sciences, 2(3), 185–212.

Bowles, J.E. (1996). Foundation analysis and design. 5th ed. New York: McGraw-Hill.

Das, B.M. (2010). Principles of foundation engineering. 7th edn. USA: Cengage Learning.

EN 1997(2004). Eurocode 7: Geotechnical design, part 1: general rules. European committee for Standardization, CEN, Brussels.

Engl, H.W., & Groetsch, C.W. (Eds.). (2014). Inverse and ill-posed problems (Vol. 4). Elsevier.

Eskin, G., & Ralston, J. (2002). On the inverse boundary value problem for linear isotropic elasticity. Inverse Problems, 18(3), 907-921.

Fellenius, B.H. (2018). Basics of foundation design. Pile Buck International, Inc.

Gajo, A., & Smith, C.C. (2018). Combined rupture mechanisms in Shallow foundation. Canadian Geotechninical Journal, 55(6), 829-838.

Green, A.E., & Zerna, W. (1968). Theoretical Elasticity, 2nd edition. Oxford University Press.

Hillyard, R.C., & Braid, I.C. (1978). Analysis of dimensions and tolerances in computer-aided mechanical design. Computer-Aided Design, 10(3), 161-166.

Huang, X., & Xie, Y.M. (2008). A new look at ESO and BESO optimization methods. Structural and Multidisciplinary Optimization, 35(1), 89-92.

Huang, X., & Xie, Y.M. (2010). A further review of ESO type methods for topology optimization. Structural and Multidisciplinary Optimization, 41(5), 671-683.

Ikpe, A.E., Orhorhoro, E.K., & Gobir, A. (2017). Design and reinforcement of a B-pillar for occupants safety in conventional vehicle applications. International Journal of Mathematical, Engineering and Management Sciences, 2(1), 37–52.

Jouve, F. (2014). Structural shape and topology optimization. In: Rozvany G.I.N., Lewiński T. (eds) Topology Optimization in Structural and Continuum Mechanics. CISM International Centre for Mechanical Sciences, (vol 549). Springer, Vienna.

Kumar, A., & Ram, M. (2018). System reliability analysis based on Weibull distribution and hesitant fuzzy set. International Journal of Mathematical, Engineering and Management Sciences, 3(4), 513–521.

Larson, M.G., & Bengzon, F. (2013). The finite element method: theory, implementation, and applications (Vol. 10). Springer Science & Business Media.

Light, R. A., & Gossard, D. C. (1983). Variational geometry: a new method for modifying part geometry for finite element analysis. Computers & Structures, 17(5-6), 903-909.

Link, T.A. (1949). Interpretation of foothills structures, Alberta, Canada. AAPG Bulletin, 33(9), 1475-1501.

Mbock, K. (2009). A Novel algorithm for motion estimation with explicit considération of perturbations. Master Thesis, University of Heidelberg, Germany.

Mbock, K., Magloire, E.R., Minsili, L.S., & Richard, O.M. (2019). Optimal mass design of 25 bar truss with loading conditions on five nodes elements. International Journal of Mathematical, Engineering and Management Sciences, 4(1), 1–16.

Miao, L., & Bernitsas, M.M. (2006). Topology redesign for performance by large admissible perturbations. Structural and Mutidsciplinary Optimization, 31(2), 117-133.

Nakamura, G., & Uhlmann, G. (2003). Global uniqueness for an inverse boundary value problem arising in elasticity. Inventiones Mathematicae, 152(1), 205-207.

Olchawa, A., & Zawalski, A. (2014). Comparison of shallow foundation design using Eurocode 7 and Polish Standard. Journal of Water and Land Development, 20(1), 57-62.

Stetson, K.A., & Palma, G.E. (1976). Inversion of first-order perturbation theory and its applications to structural design. AIAA Journal, 14(4), 454-460.

Suryatama, D., & Bernitsas, M.M. (2000). Topology and performance redesign of complex structures by large admissible perturbations. Structural and Multidisciplinary Optimization, 20(2), 138-153.

Terzaghi, K., Peck, R.B., & Mesri, G. (1996). Soil mechanics in engineering practice. Third Edition. John Wiley & Sons, Inc.

Orr, T.L., & Farrell, E.R. (2012). Geotechnical design to Eurocode 7. Springer Science & Business Media.

Verpeaux, P., Charras, T., & Millard, A. (1988). Castem 2000: une approche moderne du calcul des structures. Calcul des structures et intelligence artificielle, 2, 261-271.

Xie, Y.M, Feliceti, P., Tang, J.W. & Burry, M.C. (2005). Form finding for complex structures using evolutionary structural optimization method. Design Studies, 26(1), 55-72.

Zong, Y., & Mao, J. (2015). Tolerance mathematical modeling and analysis method based on control points of geometric element. Computer Aided Design and Applications,12(6), 723-732.