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Geometrically and Materially Nonlinear Analysis with Imperfections Included

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Geometrically and Materially Nonlinear Analysis with Imperfections Included, is a method of testing the structural stability of a structure via mathematical formulas and simulations. Accounting for both buckling and plasticity in structures "GMNIA" is considered to be one of the most important and beneficial methods to test buckling strength via the application of practical computational procedures.[1]

Using numerical simulations as well as applying various strength computing methods, (Ex. MNA, DDM, LDA). "GMNIA", uses hypothetical situations (EX. typhoons, earthquakes, fires) to account for a building's structural stability. Currently, the application of GMNIA has begun to decline with the increase in the use of "DDM" (Direct Design Method). An alternative design method apart from "GMNIA", "DDM" focuses primarily on reliable sourcing to complete structural testing making the method favorable to engineers.[2] "GMNIA" focuses on trial and error testing, requiring a fundamental understanding of several concepts regarding the effects of external and internal forces placed upon a structure or material.

Applications

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“GMNIA '' serves various functions applying to several different types of mechanisms, as a tool to account for all causes of deviation and keep them contained.  By using the imperfections resulting from deviations not accounted for, "GMNIA " can maintain the patterns of imperfections to solve issues relating to "buckling".[3]

Cylindrical shells

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"Buckling" is, (the assessment of stability in a structure while under the effects of constant load). These problems are common in “Cylindrical shells”(unstiffened or ring stiffened) which suffer from “geometric imperfections in their walls and welding”, resulting in buckling being affected by these geometric imperfections. “Silos”, and “Tanks” examples of cylindrical shells, are formed by rolling steel plates out and welding the ends together. This process causes deformations in the cylindrical shells due to:

  • First, the welding process which binds the steel plates together.
  • Second the thickness of the material used in the structure.
  • Third the cooling of the shells causes shrinkage and “radial deformations''(curvature towards the ends of the shell).
Image of a silo and common application of a steel cylindrical shell in the real world

Using the information regarding plate stiffness and thickness, shell stress resultant is found

  • shell resultant is defined as the effect of stress as a membrane force calculated as variables N

This often causes Cylindrical shells to be subject to “Axial forces” (forces acting upon a body in an axial direction or on the longitude of the body), which makes the structure weak to "Axial stress" (compression of the material in a perpendicular direction) not allowing it to withstand weight. Axial stress causes buckling to fail which in hand causes the structure to be more sensitive to geometric imperfections compared to other strength methods. These imperfection patterns are often used by “GMNIA” as ways to develop stronger materials fortified by tension able to withstand stress to maximum outputs without failing. Additionally using the data from testing the fortitude of cylindrical shells, "GMNIA" produces numerical estimations/data on structural integrity.[1][4][3]

Steel Beams

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Steel beams symbolize the most common application of "GMNIA" allowing for the visualization of changes in load affecting metal. By combining both information over compromised loads and load-displacement relationships, GMNIA can gather information about the behavior of metal. Behavior patterns include bending in brief moments, and changes in membrane forces (deformations in an element by forces in their local respective directions). By analyzing these behaviors "GMNIA" demonstrates the forces behind the strength of steel beams. By deflecting tensile forces (stretching force acted upon a material) in the membrane, the stronger the material will result. This is as deformations calculated by the "GMNIA" effect tensile forces acted upon steel beams. Small deformations allow for more bending and little to no tensile strength to be present, while large deformations create small bending moments and tensile forces will be greater.[5][6]

Image showing Steel beams

Limitations

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Having "GMNIA" as a go-to method for analyzing a "complex" structural system makes it difficult to determine the results of your system. As downsides require a deep understanding of the various components of which "GMNIA" is comprised.[7]

GMNIA foundations

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  • Materially nonlinear analysis (MNA)
    • Geometric nonlinearity: The structure experiences large deformations making it nonlinear
    • Material nonlinearity: structure goes beyond its yield limit causing the material to experience permanent deformations
    • Contact: the effects of two components touching causing sudden changes in stiffness leaving material deformation locally
  • Elastic bifurcation analysis (LBA): distortion or wrinkling in the plasticity of an object ( F = ∫ s { Ṁij + Ḱij + Ṅij έˢij + Nij ẇ, iẇ,j } ) a common mathematical function used to predict deformations in plasticity using elastic bifurcation. presented by "Hills theory". [8]
    • Ṁij : represents moments of bending
    • Ḱij : bending strain tensor
    • έˢij : stretching strain tensor
    • ẇ : buckling displacement
  • Buckling
  • Plasticity
  • General method: creating new methods of engineering using previous methods called method fragments.

Understanding each fundamental characteristic of "GMNIA" involves a deep understanding of each component that builds the method. This complexity in understanding makes it difficult to use in complex systems of structures. Other methods of strength computing such as "DDM" become favorable to engineers due to a decrease in limitations presented by "GMNIA" which is the series of tests and simulations performed on structures such as steel beams, while being applicable to a wider set of systems. [9]

Challenges

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Challenges resulting from "GMNIA" include the level of complexity of performing, using, and replicating the method. This makes other methods of strength testing more lucrative in the long term creating a decrease in the usage of "GMNIA" in strength computing. As a result "GMNIA" only serves functions when computing changes in structure through forces via physical repeated procedures returning data. To which more up-to-date procedures such as "DDM" are not required.

  1. ^ a b Schneider, W. (2006-05-01). "Stimulating Equivalent Geometric Imperfections for the Numerical Buckling Strength Verification of Axially Compressed Cylindrical Steel Shells". Computational Mechanics. 37 (6): 530–536. doi:10.1007/s00466-005-0728-8. ISSN 1432-0924.
  2. ^ Jaamala, Lauri; Mela, Kristo (2023-09). "Towards a practicable GMNIA procedure in the footsteps of the Direct Design Method". ce/papers. 6 (3–4): 2659–2664. doi:10.1002/cepa.2502. ISSN 2509-7075. {{cite journal}}: Check date values in: |date= (help)
  3. ^ a b Ultimate Strength of Cylindrical Shells. Butterworth-Heinemann. 2016-01-01. ISBN 978-0-08-099997-5.
  4. ^ Limit-State Design of Offshore Structures. Butterworth-Heinemann. 2016-01-01. ISBN 978-0-08-099997-5.
  5. ^ Ph.D, Łukasz Skotny (2017-03-26). "Geometrically nonlinear analysis explained". Enterfea. Retrieved 2023-11-08.
  6. ^ "ScienceDirect.com | Science, health and medical journals, full text articles and books". www.sciencedirect.com. Retrieved 2023-11-08.
  7. ^ Younis, Wasim (2009-01-01), Younis, Wasim (ed.), "Chapter 9 - An Overview of Part and Assembly Stress Analysis", Up and Running with Autodesk Inventor Simulation 2010, Oxford: Butterworth-Heinemann, pp. 233–259, ISBN 978-1-85617-694-1, retrieved 2023-11-09
  8. ^ "Bifurcation Analysis - an overview | ScienceDirect Topics". www.sciencedirect.com. Retrieved 2023-11-09.
  9. ^ Jaamala, Lauri; Mela, Kristo (2023-09). "Towards a practicable GMNIA procedure in the footsteps of the Direct Design Method". ce/papers. 6 (3–4): 2659–2664. doi:10.1002/cepa.2502. ISSN 2509-7075. {{cite journal}}: Check date values in: |date= (help)

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References

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