6.8.3. The Effects of Elements...
Transcript of 6.8.3. The Effects of Elements...
![Page 1: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/1.jpg)
Plan• 6.7. Contacts Conditions• 6.7.1. Continuum Mechanics Equations• 6.7.2. A Solution Approach for Contact Problems : The Constraints
Function Method
• 6.8. Some practical considerations• 6.8.1. The general Approach to Nonlinear Analysis• 6.8.2. Collapse and Buckling Analyses• 6.8.3. The Effects of Elements Distortions• 6.8.4. The Effects of Order of Numerical Integration
• Exercice 6.24• Exercice 6.27
![Page 2: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/2.jpg)
6.7. Contacts Conditions
• Difficult non linear behavior = contact between two or more bodies
• Contacts = From frictionless in small displacement to friction in general strain
• Nonlinearity of the analysis is not only geometric and material but also contact conditions
![Page 3: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/3.jpg)
Objective
• Briefly state the contact conditions in the context of finite element analysis and present a general approach for solution
![Page 4: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/4.jpg)
6.7.1. Continuum Mechanics EquationsWe consider N bodies that are in contact at time tSc is the complete area of contacttfiC : Components of contact tractionstfiS : components of the known externally applied traction
Concept of virtual work for the N bodies at time t
Usual term Contribution of contact forces
![Page 5: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/5.jpg)
We denote 2 body I and JEach body is supported such that without contact no rigid motion is possibletfIJ : vector of contact surface traction on body I due to contact with body J then tfIJ = - tfJI
The virtual work due to the contact traction can be written :
![Page 6: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/6.jpg)
We call pair of surface SIJ and SJI
• SIJ : Contractor surface• SJI : Target surface
Virtual work that the contact tractions produceOver the virtual relative displacementOn the contact pair
![Page 7: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/7.jpg)
Analyzes of the right-hand side
n : Unit outward normal to SIJ
s : Unit vector Decomposition of the contact traction tFIJ
![Page 8: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/8.jpg)
x : generic point on SIJ
y*(x,t) is the point on SJI satisfying
The distance from x to SJI is given by :
Where n* is the unit “normal vector” that we use at y*(x,t)The function g is the gap function for the contact surface pair
The conditions for normal contact can be written as :
Express the fact that if g >0 we must haveλ = 0
![Page 9: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/9.jpg)
Frictional resistanceMagnitude of the relative tangential velocity corresponding to the unit tangential vector s at y*(x,t)
Tangential velocity at time t of the material point at y* relative toThe material point at x
![Page 10: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/10.jpg)
Definitions Coulomb’s law of friction states :
![Page 11: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/11.jpg)
• Previous example consider pseudo-staic contact condition
• In dynamic analysis:Body forces mean Inertial force effect + kinematic
interface conditions must be satisfied at all instances of time required displacement, velocity and acceleration compatibility between the contacting bodies.
Various algorithms have been proposed to solve contact problems in Finite Element analysis.
![Page 12: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/12.jpg)
6.7.2. A Solution Approach for Contact Problems : The Constraints Function Method
- w is a function of g and λ that satisfiedand-Then contact conditions are given by
![Page 13: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/13.jpg)
![Page 14: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/14.jpg)
Example on the formulation governing finite element equation
We consider two-dimensional case of contractor and target bodies:Node k1 and k2 define straight boundary but not necessary corner node of an
element
![Page 15: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/15.jpg)
• Discretization of the continuum equations 6.301 and 6.314 at time t+Δt gives :
![Page 16: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/16.jpg)
• The nodal force vector is
![Page 17: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/17.jpg)
![Page 18: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/18.jpg)
Conclusion • Continuum mechanics formulation in section
6.7.1 has been given for very general conditions deformation and constitutive law.
• Formulation is applicable to frictionless contact
• 6.231 correspond to a full linearization.If too large time step Convergence difficulties
because predicted intermediate state are too far from the solution.
![Page 19: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/19.jpg)
6.8. Some practical considerations
![Page 20: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/20.jpg)
6.8.1. The general Approach to Nonlinear Analysis
• Good practice in engineering analysis :Nonlinear analysis always preceded by linear analysis so that the non
linear analysis is an extension of the complete analysis beyond the assumptions of linear analysis.
… In addition to general recommendations, practical aspects can be important :
![Page 21: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/21.jpg)
6.8.2. Collapse and Buckling Analyses
![Page 22: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/22.jpg)
Illustration of response structure when collapse or buckling
Arch After A :Post buckling (deformation) behavior, iduce by load increase = dynamic response
![Page 23: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/23.jpg)
The response of the column depend of βWhen β is very small we are closed to a perfectly straigth column with only compressiveload.
![Page 24: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/24.jpg)
We consider the calculation of the linearized buckling load
• t-ΔtK and tK stiffness matrix at time t-Δt and t
• t-ΔtR and tR Vectors of externally appliedload
We assume at any time
λ : Scaling factor
![Page 25: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/25.jpg)
At collapse or buckling, the stiffness matrix is singular and the condition to calculateλ is
Ф is a non zero vector
![Page 26: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/26.jpg)
We are only interested in the only smallest positive eigenvalues, so;
![Page 27: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/27.jpg)
The assumption used in the linearizedbuckling analysis are displayed in 6.323 and 6.324
We assume that the elements in the stiffness matrix vary lineary from time t-Δt onwardThe slope of the change are being given by the difference from time t-Δt to time t.The linearized buckling analysis gives a reasonable estimate of the collapse load onlyIf the the precollapse displacement are relatively small
![Page 28: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/28.jpg)
![Page 29: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/29.jpg)
Analysis of arch
• Precollapse displacement are large.• Linearized buckling analysis will give good
result if the structure displays a column type of buckling behavior.
![Page 30: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/30.jpg)
• Arch • Ten two-node isoparametricbeam element
Objectif:Predict the collapse and postcollapse response
![Page 31: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/31.jpg)
• Response calculated using a load-displacement-constraint method
• A linearized buckling analysis was performed using the state t-Δt theunstressed configuration and state Δt the configuration corresponding to a pressure of 10.
![Page 32: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/32.jpg)
Calculate response using a load displacement method with this geometric imperfection.Pressure collapse load predicted is significantly smaller than the previousThe reduction is the collapse load is associated with a nonsymmetric behavior of the Structural model
![Page 33: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/33.jpg)
Conclusion
• Structural imperfection can have a major effect on the predicted load-carrying capacity of a structure.
• Imperfection should be introduced in the structural model.
• Dynamic solution need to be considered, a dynamic buckling or collapse analysis that complete dynamic incremental analyses be performed for given different load levels.
![Page 34: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/34.jpg)
6.8.3. The Effects of Elements Distortions
• In practice, element must largely be of a general straight-sided shapes with angular distortions in order to provide
mesh grading and mesh complex geometries effectively.
![Page 35: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/35.jpg)
• In geometric nonlinear analysis significant angular and curved edge distortions due to non corner node element may arise a s consequence of deformation
• Using large displacement formulations, the principle of virtual displacement is applied to each element corresponding to the current configuration instead of the initial configuration in linear analysis.
![Page 36: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/36.jpg)
6.8.4. The Effects of Order of Numerical Integration
• Nonlinear analysis : select appropriate numerical integration and order of integration.
• In geometric nonlinear analysis, at least the same integration order should be employed as linear analysis.
• An higher integration order than in the linear analysis may be required. This consideration is very important for beam, plate and shell.
![Page 37: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/37.jpg)
Result using different order of Gauss integration for an eight-node plane stress representing the section of the beam.Analyse illustrate that predict the nonlinear response accurately a higher Integration Order in the thickness direction beam is required than linear analysis.
![Page 38: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/38.jpg)
EXAMPLE 6.24
![Page 39: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/39.jpg)
![Page 40: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/40.jpg)
![Page 41: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/41.jpg)
From table 4.2
![Page 42: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/42.jpg)
![Page 43: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/43.jpg)
![Page 44: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/44.jpg)
![Page 45: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/45.jpg)
![Page 46: 6.8.3. The Effects of Elements Distortionsarchiv.ibk.ethz.ch/emeritus/fa/education/FE_II/Lecture_7_Vuillaume.pdf · Frictional resistance Magnitude of the relative tangential velocity](https://reader030.fdocuments.net/reader030/viewer/2022040906/5e7c27c8e7d761031f2469a7/html5/thumbnails/46.jpg)