S6 l04 analytical and numerical methods of structural analysis
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This document provides an overview of analytical and numerical methods for structural analysis. It begins by explaining the process of structural analysis from the real object to the design model. It then discusses analytical methods like mechanics of materials and numerical methods like the finite element method. The document provides examples comparing analytical and numerical solutions. In summary, it outlines the appropriate uses of both methods and emphasizes the importance of understanding the underlying mechanics rather than solely relying on software tools.
S6 l04 analytical and numerical methods of structural analysis
- 1. Lecture #4 Analytical and Numerical Methods of Structural Analysis
- 2. FLOWCHART OF STRUCTURAL ANALYSIS 2 Real object Structural layout Design model Only load-carrying structure is kept Assumptions and simplifications are applied, loads are calculated according to the problem Results of analysis Implementation on real object This step is beyond the scope of structural analysis Structural analysis
- 3. FLOWCHART OF STRUCTURAL ANALYSIS 3 Real object Structural layout Design model Results of analysis Implementation on real object
- 4. FLOWCHART OF STRUCTURAL ANALYSIS 4 Real object Structural layout Design model Results of analysis Implementation on real object
- 5. FLOWCHART OF STRUCTURAL ANALYSIS 5 Real object Structural layout Design model Results of analysis Implementation on real object Depending on the kind of problem which is solved, the design model could be either as detailed as structural layout, or as generalized as below:
- 6. METHODS OF STRUCTURAL ANALYSIS Analytical methods Numerical methods Best for designing calculations, suit for checking calculations with certain limitations Best for checking calculations, practically effete for designing calculations Solutions exist for partial cases (specific objects) Versatile and flexible Need much work to be developed, but only simple software for application Need expensive and complex software and hardware 6
- 7. ANALYTICAL METHODS OF STRUCTURAL ANALYSIS 7 Widely used nowadays: • methods of Mechanics of Materials; • methods for statically indeterminate structures: • method of forces; • method of displacements; • beam theory for thin-walled structures; • method of reduction coefficients. Rarely used nowadays (mostly due to the progress of numerical methods): • methods based on Calculus of Variations; • methods of Theory of Elasticity.
- 8. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 8 Aim: Design the steel bracket to minimize the self-weight displacement of the load. Given data: D2=80 mm; L2=500 mm; L1=200 mm; mload=0.5 kg. Objectives: D1, δ1, δ2. load L1, D1, δ1 L2, D2, δ2 base
- 9. BASIC EQUATIONS OF SOLID MECHANICS 9 Equilibrium equations Constitutive equations Compatibility equations This is not only the sum of forces or moments, but applies for elementary volume as well Physical law, expresses the relation between stress and strain Solid body should remain continuous while being deformed
- 10. BASIC EQUATIONS OF SOLID MECHANICS 10 Equilibrium equations Constitutive equations Compatibility equations This is not only the sum of forces or moments, but applies for elementary volume as well. 0 0 0 xyx xz xy y yz yzxz z X , x y z Y , x y z Z , x y z where X ,Y ,Z external forces. τσ τ τ σ τ ττ σ ∂∂ ∂ + + + = ∂ ∂ ∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂∂ ∂ + + + = ∂ ∂ ∂ −
- 11. BASIC EQUATIONS OF SOLID MECHANICS 11 Equilibrium equations Constitutive equations Compatibility equations Physical law, expresses the relation between stress and strain. ( )( ) ( )( ) ( )( ) 1 1 1 x x y z y y x z z z x y xy yzxz xy xz yz , E , E , E , , G G G ε σ ν σ σ ε σ ν σ σ ε σ ν σ σ τ ττ γ γ γ = − + = − + = − + = = =
- 12. BASIC EQUATIONS OF SOLID MECHANICS 12 Equilibrium equations Constitutive equations Compatibility equations Solid body should remain continuous while being deformed. 2 22 2 22 2 2 2 2 2 22 2 2 2 2 2 2 2 2 y xyx x xzz y yzz yz xyxz x yz xy yxz yz xyxz z , , y x x y z x x z , z y y z , x x y z y z , y x y z x z . z x y z x y ε γε ε γε ε γε γ γγ ε γ γ εγ γ γγ ε ∂ ∂∂ ∂ ∂∂ + = + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ − + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ − + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ + − = ∂ ∂ ∂ ∂ ∂ ∂
- 13. BASIC EQUATIONS OF SOLID MECHANICS 13 Equilibrium equations Constitutive equations Compatibility equations Boundary conditions: • geometrical features; • supports and other means of fixation; • applied displacements and forces. Design model From analytical point of view, every design model can be expressed as a set of three basic equations with corresponding boundary conditions.
- 14. WAYS TO SOLVE A SOLID MECHANICS PROBLEM 14 Displacements are set as unknowns Strains are derived Stresses are derived Compatibility equations Constitutive equations Equilibrium equations are solved Equilibrium equations Stresses are set as unknowns Strains are derived Constitutive equations Equilibrium equations Compatibility equations are solved Compatibility equations
- 15. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 15 Analytical methods used: • Methods of Mechanics of Materials; • Mohr’s integral to calculate displacements. Software used: MathCAD V14.0 load L1, D1, δ1 L2, D2, δ2 base
- 16. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 16
- 17. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 17
- 18. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 18
- 19. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 19
- 20. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 20
- 21. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 21
- 22. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 22
- 23. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 23
- 24. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 24
- 25. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 25
- 26. METHODS OF STRUCTURAL ANALYSIS Analytical methods Numerical methods Best for designing calculations, suit for checking calculations with certain limitations Best for checking calculations, practically effete for designing calculations Solutions exist for partial cases (specific objects) Versatile and flexible Need much work to be developed, but only simple software for application Need expensive and complex software and hardware 26
- 27. NUMERICAL METHODS OF STRUCTURAL ANALYSIS 27 Most widely used nowadays is Finite Element Method (FEM). The structural analysis using FEM is called finite element analysis (FEA). Generally, FEM is just a method to solve differential equations. The physical background of FEM is the same as in analytical methods.
- 28. FINITE ELEMENT METHOD 28 The smooth field of stresses is calculated using the combination of basic functions, which are constructed on a mesh.
- 29. FINITE ELEMENT METHOD 29 Software work on principle GIGO: “garbage in – garbage out”…
- 30. FINITE ELEMENT METHOD 30 Software work on principle GIGO: “garbage in – garbage out”…
- 31. FINITE ELEMENT METHOD 31 Software work on principle GIGO: “garbage in – garbage out”…
- 32. FINITE ELEMENT METHOD 32 Even if the result is realistic, you should always check it. It’s not a big problem to create a “nice picture” using FEA. The problem is to get the reliable and verified result. That’s a difference between professional and beginner. FEA is a way to find a stress state, but not the way to design or to check the strength!
- 33. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 33load base Object FEM model
- 34. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 34 Assignment of parameters
- 35. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 35 Deformed shape 35 finite elements in model
- 36. EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS 36 Optimization took 7 minutes on average office PC. 990 points were studied.
- 37. Analytical method Finite element analysis Displacement is 7.66 µm (or 8.07 µm in FEA) Displacement is 7.81 µm (or 8.22 µm in FEA) D1=67 µm, δ1=2 µm, δ2= 11.23 mm D1=58 µm, δ1=2 µm, δ2= 13 mm Need only simple software for application Need expensive and complex software 37 EXAMPLE TO COMPARE ANALYTICAL AND NUMERICAL METHODS
- 38. METHODS OF STRUCTURAL ANALYSIS Analytical methods Numerical methods Best for designing calculations, suit for checking calculations with certain limitations Best for checking calculations, practically effete for designing calculations Solutions exist for partial cases (specific objects) Versatile and flexible Need much work to be developed, but only simple software for application Need expensive and complex software and hardware 38
- 39. EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL 39 Shell model. 12K elements in model, around 5 minutes to solve on poor office PC
- 40. EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL 40 Solid model. 250K elements, around 20 minutes to solve on powerful PC (4x4.2GHz processor, 16 GB memory).
- 41. EXAMPLE ILLUSTRATING THE DESIGN OF FINITE ELEMENT MODEL 41 Difference? Certain there is a difference, but for some results it is negligible. Thus, the model should be built according to problem, as in analytical methods
- 42. FINITE ELEMENT SOFTWARE 42 Widely used nowadays: • ANSYS; • Patran/Nastran; • ABAQUS; • Solidworks Simulation (former COSMOS); And many-many others, freeware and proprietary… Any difference? Generally, no difference. FEA packages are often customized to be more convenient for some specific problems. But for most cases, differences between packages and limitations have marketing but not the technical reason.
- 43. 1940 Theoretical aspects were developed for the first time 1950, 1960 Theoretical background was developed, the rise of computer power allowed to make first specific software 1970 Large general-purpose packages appeared (ANSYS, NASTRAN) 1980 Graphical pre- and post-processing 1990 Rise of the role of automatic meshing, and thus the model scale 2000 Multi-disciplinary and large-scale problems 43 FINITE ELEMENT METHOD HISTORICAL PROGRESS
- 44. PRINCIPAL COMPONENTS OF FEA SOTWARE 44 Pre-processor Solver Post-precessor Construction of the model, setting the properties, loads, constraints etc. Solution itself. Two main types of solvers are iterative or sparse . Looking on results, processing of results, generating “nice figures”
- 45. 45 FINITE ELEMENT METHOD – OLD INTERFACE
- 46. 46 FINITE ELEMENT METHOD – MODERN INTERFACE
- 47. CONCLUSIONS • “Analytical method” does not mean “by hand” or “on paper”, modern software can give large benefits. • Numerical methods are very exacting in terms of software and hardware. • FEA is not a magic key that opens every door. It’s just a versatile and effective tool to calculate stresses. • Deep insight is required for successful structural analysis, no matter is it analytics of FEA. • Generally, analytical methods are good for designing, FEA – for checking. However, in many cases both of methods can be used for both phases of analysis. 47
- 48. CONCLUSIONS What does it mean to be a professional in structural analysis? Knowledge of complex formulas? Usage of modern software? Neither of it is an answer. Answer is a deep understanding of how the structure works, and application of proper methods according to the given problem. Analytical and numerical methods loss their disadvantages when they are combined together. 48
- 49. WHERE TO FIND MORE INFORMATION? BOOKS AND USEFUL LINKS 49 Good book about structural analysis and Ansys is E. Madenci, I.Guven. The Finite Element Method and Applications in Engineering Using ANSYS, 2006 Very useful links are: Official website resource library ansys.com Resourse for Ansys users ansys.net Perfect forum (not for beginners, though) xansys.org Youtube (it’s not a joke, search “Ansys” for amazing training videos) youtube.com … Internet is boundless …
- 50. TOPIC OF THE NEXT LECTURE 50 Statically determinate trusses All materials of our course are available at department website k102.khai.edu 1. Go to the page “Библиотека” 2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”