Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC NUMERICAL SIMULATION OF A LAYERED ELASTIC SYSTEM USING THE BOUNDARY ELEMENT METHOD S TUDENT A. F ORNI D OCTORAL ASSISTANT S. P HAM -B A P ROFESSOR P ROF . J. F. M OLINARI
Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele I. T OPIC i. C ONTEXT ii. L AST TIME II. M ODEL IMPLEMENTATION i. P HASE 3 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED FORCE ii. E RROR IV. R ESULTS V. C ONCLUSION
Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele I. T OPIC i. C ONTEXT ii. L AST TIME II. M ODEL IMPLEMENTATION i. P HASE 3 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED FORCE ii. E RROR IV. R ESULTS V. C ONCLUSION Surface contact pressure
Phase 1 Ecole polytechnique fédérale de Lausanne q = P −1 p p = Pq Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele u = UP −1 p u = Uq I. T OPIC i. C ONTEXT K = PU −1 ii. L AST TIME 𝐯 = 𝐋 −𝟐 𝐪 II. M ODEL IMPLEMENTATION i. P HASE 3 Phase 2 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED FORCE ii. E RROR IV. R ESULTS V. C ONCLUSION
Line of concentrated force Hertzian stress (constant over line segment) Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele I. T OPIC i. C ONTEXT ii. L AST TIME II. M ODEL IMPLEMENTATION i. P HASE 3 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED Imposed displacement FORCE ii. E RROR IV. R ESULTS V. C ONCLUSION
Phase 3 Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele I. T OPIC i. C ONTEXT ii. L AST TIME II. M ODEL IMPLEMENTATION i. P HASE 3 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED p a u a K aa K ab 0 FORCE p 1 K 2 × u 1 p 2 = K 1 0 p b u b K ba K bb 0 ii. E RROR u 2 = = × 0 u c p c 0 0 K cc IV. R ESULTS V. C ONCLUSION p a = K aa + K ab −K cc − K bb −1 K ba u a = K a ′u a
Line of concentrated force (constant over line segment) Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Imposed vertical pressure on second surface Imposed vertical displacement on second surface Forni Ariele I. T OPIC i. C ONTEXT ii. L AST TIME II. M ODEL IMPLEMENTATION i. P HASE 3 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED FORCE ii. E RROR IV. R ESULTS V. C ONCLUSION
Imposed vertical displacement on second surface Ecole polytechnique fédérale de 𝑣 𝑧2 = 0 𝑣 𝑧2 = 1 Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele I. T OPIC i. C ONTEXT ii. L AST TIME II. M ODEL IMPLEMENTATION i. P HASE 3 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED FORCE ii. E RROR IV. R ESULTS V. C ONCLUSION • Relative displacements • Sum of pressures of the system is not imposed to be zero
Line of concentrated force Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele I. T OPIC i. C ONTEXT ii. L AST TIME II. M ODEL IMPLEMENTATION i. P HASE 3 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED FORCE ii. E RROR Undetermined error in assembling the stress matrices.. Results a little different IV. R ESULTS than with 2 surfaces. V. C ONCLUSION
Hertzian stress Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele I. T OPIC i. C ONTEXT ii. L AST TIME II. M ODEL IMPLEMENTATION i. P HASE 3 III. V ALIDATION WITH TWO SURFACES i. L INE OF CONCENTRATED FORCE ii. E RROR IV. R ESULTS V. C ONCLUSION Sum of pressures of the top and bottom surfaces different from zero
Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele I. T OPIC • For an exterior problem of this type, the boundary element method is i. C ONTEXT ii. L AST TIME relatively simple to develop and have a great efficiency • Relative displacements may be a problem when we simulate mixed II. M ODEL IMPLEMENTATION i. P HASE 3 boundary value systems • It is necessary to impose the sum of the pressures of two surfaces III. V ALIDATION WITH TWO SURFACES equal to zero i. L INE OF CONCENTRATED FORCE • Important to define a point of reference ii. E RROR IV. R ESULTS V. C ONCLUSION
Ecole polytechnique fédérale de Lausanne Master Cours fall 2020 CIVIL-492 Laboratoy GC Forni Ariele References Thank you Aliabadi, M. (202). Boundary Element Method. London: Wiley. Costabel, M. (1986). Principles of Boundary Element Methods. Lausanne. Crouch, S. L., & Starfield, A. M. (1983). Boundary element method for your attention in soil mechanics. London: George Allen & Unwin. Johnason, K. (1985). Contact mechanics. London: Cambridge University Press. Shield, R. (1953). Mixed boundary value problems in soil mechanics. In B. University, Quartterly of Applied Mathematics Vol. 11, No. 1 (pp. 61-75). Zhu, X. (2012). Tutorial on Hertz Contact Stress.
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