Structural response of existing spatial truss roof construction based on Cosserat rod theory

Paper presents the application of the Cosserat rod theory and newly developed associated finite elements code as the tools that support in the expert-designing engineering practice. Mechanical principles of the 3D spatially curved rods, dynamics (statics) laws, principle of virtual work are discussed. Corresponding FEM approach with interpolation and accumulation techniques of state variables are shown that enable the formulation of the $$C^{0}$$C0 Lagrangian rod elements with 6-degrees of freedom per node. Two test examples are shown proving the correctness and suitability of the proposed formulation. Next, the developed FEM code is applied to assess the structural response of the spatial truss roof of the “Olivia” Sports Arena Gdansk, Poland. The numerical results are compared with load test results. It is shown that the proposed FEM approach yields correct results.

[1]  Luca Placidi,et al.  A second gradient formulation for a 2D fabric sheet with inextensible fibres , 2016 .

[2]  Victor A. Eremeyev,et al.  Cosserat-Type Rods , 2013 .

[3]  Piotr Klikowicz,et al.  Structural Health Monitoring of Urban Structures , 2016 .

[4]  Carlos A. Felippa,et al.  A three‐dimensional non‐linear Timoshenko beam based on the core‐congruential formulation , 1993 .

[5]  J. C. Simo,et al.  On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach , 1988 .

[6]  W. Pietraszkiewicz,et al.  Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells , 2007 .

[7]  M. Crisfield,et al.  Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  Jakub Szulwic,et al.  SYSTEM OF MONITORING OF THE FOREST OPERA IN SOPOT STRUCTURE AND ROOFING , 2011 .

[9]  Francesco dell’Isola,et al.  King post truss as a motif for internal structure of (meta)material with controlled elastic properties , 2017, Royal Society Open Science.

[10]  J. Chróścielewski,et al.  Genuinely resultant shell finite elements accounting for geometric and material non-linearity , 1992 .

[11]  Wojciech Witkowski,et al.  Discrepancies of energy values in dynamics of three intersecting plates , 2010 .

[12]  Ugo Andreaus,et al.  At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola , 2013, 1310.5599.

[13]  J. C. Simo,et al.  On the Dynamics of Flexible Beams Under Large Overall Motions—The Plane Case: Part II , 1986 .

[14]  Łukasz Jankowski,et al.  An online substructure identification method for local structural health monitoring , 2013 .

[15]  J. C. Simo,et al.  On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory , 1990 .

[16]  A. Ibrahimbegovic.  On the choice of finite rotation parameters , 1997 .

[17]  P. Wriggers,et al.  An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 2: shells , 2011 .

[18]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[19]  K. Bathe,et al.  Large displacement analysis of three‐dimensional beam structures , 1979 .

[20]  M. Géradin,et al.  A beam finite element non‐linear theory with finite rotations , 1988 .

[21]  Francesco dell’Isola,et al.  Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models , 2016 .

[22]  Victor A. Eremeyev,et al.  On natural strain measures of the non-linear micropolar continuum , 2009 .

[23]  M. Rubin.  Cosserat Theories: Shells, Rods and Points , 2000 .

[24]  M. Crisfield.  A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements , 1990 .

[25]  A. Schiela,et al.  Variational analysis of the coupling between a geometrically exact Cosserat rod and an elastic continuum , 2014 .

[26]  W. Witkowski.  4-Node combined shell element with semi-EAS-ANS strain interpolations in 6-parameter shell theories with drilling degrees of freedom , 2009 .

[27]  Janusz Badur,et al.  Finite rotations in the description of continuum deformation , 1983 .

[28]  Leopoldo Greco,et al.  An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods , 2017 .

[29]  Alessandro Della Corte,et al.  Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof , 2015 .

[30]  Mikołaj Miśkiewicz,et al.  Preliminary Field Tests and Long-Term Monitoring as a Method of Design Risk Mitigation: A Case Study of Gdańsk Deepwater Container Terminal , 2017 .

[31]  W. Pietraszkiewicz,et al.  Local Symmetry Group in the General Theory of Elastic Shells , 2006 .

[32]  Krzysztof Wilde,et al.  Structural Health Monitoring of Composite Shell Footbridge for Its Design Validation , 2016, 2016 Baltic Geodetic Congress (BGC Geomatics).

[33]  Francesco dell’Isola,et al.  Buckling modes in pantographic lattices , 2016 .

[34]  The Nonlinear Thermodynamical Theory of Shells: Descent from 3-Dimensions without Thickness Expansions , 1984 .

[35]  J. Chróścielewski,et al.  Modeling of Composite Shells in 6–Parameter Nonlinear Theory with Drilling Degree of Freedom , 2011 .

[36]  Slow Motion and Metastability for a Nonlocal Evolution Equation , 2003 .

[37]  N. Laws,et al.  A general theory of rods , 1966, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[38]  Werner Wagner,et al.  A robust non‐linear mixed hybrid quadrilateral shell element , 2005 .

[39]  Krzysztof Wilde,et al.  Technical Monitoring System for a New Part of Gdańsk Deepwater Container Terminal , 2017 .

[40]  M. Pulvirenti,et al.  Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials , 2015, 1504.08015.

[41]  Ivan Giorgio,et al.  Pantographic lattices with non-orthogonal fibres: Experiments and their numerical simulations , 2017 .

[42]  F. dell’Isola,et al.  Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[43]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[44]  Victor A. Eremeyev,et al.  Deformation analysis of functionally graded beams by the direct approach , 2012 .

[45]  P. Ferrari,et al.  Flux Fluctuations in the One Dimensional Nearest Neighbors Symmetric Simple Exclusion Process , 2001, math/0103233.

[46]  Francesco dell’Isola,et al.  Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence , 2015 .

[47]  M. Géradin,et al.  Flexible Multibody Dynamics: A Finite Element Approach , 2001 .

[48]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

[49]  V. Eremeyev,et al.  The Rayleigh and Courant variational principles in the six-parameter shell theory , 2015 .

[50]  Krzysztof Wilde,et al.  Reference FEM model for SHM system of cable-stayed bridge in Rzeszów , 2016 .

[51]  Francesco dell’Isola,et al.  Linear pantographic sheets: Asymptotic micro-macro models identification , 2017 .

[52]  Holm Altenbach,et al.  On a thermodynamic theory of rods with two temperature fields , 2012 .

[53]  Krzysztof Wilde,et al.  Structural Health Monitoring System for Suspension Footbridge , 2017, 2017 Baltic Geodetic Congress (BGC Geomatics).

[54]  J. G. Simmonds,et al.  The Nonlinear Theory of Elastic Shells , 1998 .

[55]  W. Smoleński.  Statically and kinematically exact nonlinear theory of rods and its numerical verification , 1999 .

[56]  J. Chróścielewski,et al.  Four‐node semi‐EAS element in six‐field nonlinear theory of shells , 2006 .

[57]  Pierre Seppecher,et al.  Truss Modular Beams with Deformation Energy Depending on Higher Displacement Gradients , 2003 .

[58]  Francesco dell’Isola,et al.  Dynamics of 1D nonlinear pantographic continua , 2017 .