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modeling.ppt

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Praveen Kumar

The document discusses three dimensional problems in finite element analysis. It describes the deformation of a point in a body under forces using displacement components u, v and w. It presents the strain-displacement relations and equations for stresses, strains, body forces and tractions. It also discusses the four noded tetrahedral element, shape functions, element stiffness matrix, and dynamic analysis using the Lagrangian and equations of motion. Finally, it briefly covers guidelines for finite element modeling and debugging models.

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1 Three dimensional problems , , T u v w u , , , , , , , , , , T x y z yz xz xy T x y z yz xz xy u=displacement in x-direction v=displacement in y-direction w=displacement in z-direction The stresses and strains are given by Prof .N. Siva Prasad, Indian Institute of Technology Madras The deformation of a point in a body under forces is given by where
2 The strain-displacement relations are given by , , , , , T u v w v w u w u v x y z z y z x y x , , , , T x y z T x y z f f f T T T f T The body forces and traction vectors are given by Prof .N. Siva Prasad, Indian Institute of Technology Madras 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0.5 0 0 (1 )(1 2 ) 0 E D 0 0 0 0.5 0 0 0 0 0 0 0.5 D The stress-strain relations are given by where Three dimensional problems contd..
3 Four noded tetrahedral element Thus, the element local and global displacements 1 2 3 12 1 2 3 , , ,......, , , ,......, T T N q q q q Q Q Q Q q Q N is the total number of degrees of freedom for the structure, 3 per node Prof .N. Siva Prasad, Indian Institute of Technology Madras x y z o Three dimensional problems contd..
4 Using the master element shown in figure below, we can define the shape functions as 1 2 3 4 1 N N N N 1 2 3 4 1 2 3 4 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N N N N N N N N N N N N 4 N u Nq The displacements u, v and w at x can be written in terms of the unknown nodal values as 4 (0,0,0) 1 (1,0,0) 2 (0,1,0) 3 (0,0,1) Prof .N. Siva Prasad, Indian Institute of Technology Madras where Three dimensional problems contd..
5 The isoparametric transformation is given by 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 x N x N x N x N x y N y N y N y N y z N z N z N z N z u u x u u y u u z J Prof .N. Siva Prasad, Indian Institute of Technology Madras Using chain rule for partial derivatives, say of u, we have Three dimensional problems contd..
6 volume of the element given by 1 1 1 0 0 0 1 det det 6 e V d d d J J u u x u u y u u z A Prof .N. Siva Prasad, Indian Institute of Technology Madras We can write the following relation 14 14 14 24 24 24 34 34 34 x y z x y z x y z x y z x y z x y z J The Jacobian of the transformation is given by 14 1 4 x x x here Three dimensional problems contd..
7 Ve is the volume of the element e T e k V B DB here element stiffness matrix, ke is given by 1 1 2 2 1 1 2 2 T T T e e e T T T e e U dV dV V k D q B DBq q B DBq q q Element Stiffness Prof .N. Siva Prasad, Indian Institute of Technology Madras The element strain energy in the total potential is given by Three dimensional problems contd..
8 Force Terms The potential term associated with body force is det T T T e T e dV d d d 駕駕 u f q N f J q f , , , , , ,......., 4 T e e x y z x y z z V f f f f f f f f Consider uniformly distributed traction on the boundary surface Prof .N. Siva Prasad, Indian Institute of Technology Madras T T T T e A A e e dA dA u T q N T q T Three dimensional problems contd..
9 Dynamic analysis When loads are suddenly applied, The mass and acceleration effects are come in to picture Lagrangian L T T Kinetic energy Potential energy Hamilton Princible For an arbitrary time interval from t1 to t2,the state of motion of a body extremizes functional 2 1 t t I Ldt If L can be expressed in terms of the generalized variables 1 2 3 4 , , , ,..... , n q q q q q where i i q q t Then the equations of motions are given by 0 i i d L L dt q q i=1 to n Prof .N. Siva Prasad, Indian Institute of Technology Madras
10 1 1 , x x 2 2 , x x m2 m1 The kinetic energy and potentially energy are given by 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 T m x m x k x k x Using and L T 0 i i d L L dt q q i=1 to n The equations of motions will be 1 1 1 1 2 2 1 1 1 2 2 2 2 1 2 2 ( ) 0 ( ) 0 d L L m x k x k x x dt x x d L L m x k x x dt x x 件 件 In matrix form 1 1 1 2 2 1 2 2 2 2 2 0 ( ) 0 0 m x k k k x m k k x x 件 件 Spring mass system Stiffness matrix Mass matrix Dynamic analysis contd..
11 Systems with Distributed mass x z y u v w dv = density v The kinetic energy The velocity vector of point at x with components In the finite element method,we divide the body into elements,and in each element and 1 2 T v T u u dv , , u v w u Nq u N q The kinetic energy Te 1 2 1 2 T v T v T u u dv T q N Ndv q Mass matrix Prof .N. Siva Prasad, Indian Institute of Technology Madras Dynamic analysis contd..
12 Types of Elements One dimensional elements 1. Beam (axial) 2. Beam (bending) 3. Pipe Prof .N. Siva Prasad, Indian Institute of Technology Madras
13 Two dimensional elements 1. Triangular inplane bending 2. quadrilateral inplane bending Prof .N. Siva Prasad, Indian Institute of Technology Madras Types of Elements
14 Three dimensional elements 1. brick 2. Tetrahedral Prof .N. Siva Prasad, Indian Institute of Technology Madras Types of Elements
15 Finite element analysis Guidelines for the selection of FE package: Analysis Capability Adequacy of user oriented features Maintainability Adequacy of user support facility Portability Prof .N. Siva Prasad, Indian Institute of Technology Madras
16 Modeling Element type must be consistent Finer mesh near the stress gradient Extremely fine mesh when forces to be applied near the stress concentration areas such as fillets Uniform change in stress between adjacent elements Better aspect ratio Prof .N. Siva Prasad, Indian Institute of Technology Madras
17 Modeling (contd) Gross element distortion should be avoided Adjoining elements must share common nodes and common degrees of freedom 450 150 Prof .N. Siva Prasad, Indian Institute of Technology Madras
18 Debugging of FE models Geometry Material properties Applied forces Displacement constraints Prof .N. Siva Prasad, Indian Institute of Technology Madras
19 Common symptoms and their possible causes Symptoms Causes Excessive deflection, but anticipate stress Excessive deflection and excessive stress Internal discontinuity in stress and deflection Youngs modulus too low, missing nodal constraints Applied force too high, nodal coordinates incorrect, force applied at wrong nodes Force applied at wrong nodes, missing or double internal element Prof .N. Siva Prasad, Indian Institute of Technology Madras
20 Symptoms Causes Discontinuity along boundary Higher or lower frequency than anticipated -Static deflections, O.K. -Static deflection not O.K. Internal gap opening up in model under load, stress discontinuity Missing nodal constraint, force applied at wrong node Density incorrect Youngs modulus incorrect Improper nodal coupling Prof .N. Siva Prasad, Indian Institute of Technology Madras Common symptoms and their possible causes
21 Dynamic analysis Modal analysis - Natural frequency and mode shapes Harmonic analysis - Forced response of system to a sinusoidal forcing Transient analysis - Forced response for non-harmonic loads (impact, step or ramp forcing etc.) Prof .N. Siva Prasad, Indian Institute of Technology Madras
22 Substructure Rules for substructure A substructure may be generated from individual elements or other substructures Master nodes to be retained to be identified Nodal constraints will be retained in all subsequent uses of the substructure Along a substructure boundary that will be used for connection to the rest of the global model, all nodes must be retained as master nodes Cost effective Prof .N. Siva Prasad, Indian Institute of Technology Madras
23 Guidelines for selection of dynamic degree of freedom (DDOF) No. of the DOF must be 2 times highest mode of interest No. of reduced modes will be equal to the number of DDOF so that only bottom half of the calculated modes should be considered accurate DDOF should be placed in areas of large mass and rigidity DDOF should be distributed in such a way as to anticipate mode shapes DDOF be selected at each point of dynamic force application DDOF must be used with gap elements For plate type elements emphasise DDOF in out of plane direction Prof .N. Siva Prasad, Indian Institute of Technology Madras
24 Discretization of Structure Concentrated load . node P Abrupt change in load . node Abrupt change in thickness . node Prof .N. Siva Prasad, Indian Institute of Technology Madras
25 Discretization of Structure Abrupt change in material properties Re entrant corner . Finer mesh near stress concentration factor Nodal line node Prof .N. Siva Prasad, Indian Institute of Technology Madras

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modeling.ppt

  • 1. 1 Three dimensional problems , , T u v w u , , , , , , , , , , T x y z yz xz xy T x y z yz xz xy u=displacement in x-direction v=displacement in y-direction w=displacement in z-direction The stresses and strains are given by Prof .N. Siva Prasad, Indian Institute of Technology Madras The deformation of a point in a body under forces is given by where
  • 2. 2 The strain-displacement relations are given by , , , , , T u v w v w u w u v x y z z y z x y x , , , , T x y z T x y z f f f T T T f T The body forces and traction vectors are given by Prof .N. Siva Prasad, Indian Institute of Technology Madras 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0.5 0 0 (1 )(1 2 ) 0 E D 0 0 0 0.5 0 0 0 0 0 0 0.5 D The stress-strain relations are given by where Three dimensional problems contd..
  • 3. 3 Four noded tetrahedral element Thus, the element local and global displacements 1 2 3 12 1 2 3 , , ,......, , , ,......, T T N q q q q Q Q Q Q q Q N is the total number of degrees of freedom for the structure, 3 per node Prof .N. Siva Prasad, Indian Institute of Technology Madras x y z o Three dimensional problems contd..
  • 4. 4 Using the master element shown in figure below, we can define the shape functions as 1 2 3 4 1 N N N N 1 2 3 4 1 2 3 4 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N N N N N N N N N N N N 4 N u Nq The displacements u, v and w at x can be written in terms of the unknown nodal values as 4 (0,0,0) 1 (1,0,0) 2 (0,1,0) 3 (0,0,1) Prof .N. Siva Prasad, Indian Institute of Technology Madras where Three dimensional problems contd..
  • 5. 5 The isoparametric transformation is given by 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 x N x N x N x N x y N y N y N y N y z N z N z N z N z u u x u u y u u z J Prof .N. Siva Prasad, Indian Institute of Technology Madras Using chain rule for partial derivatives, say of u, we have Three dimensional problems contd..
  • 6. 6 volume of the element given by 1 1 1 0 0 0 1 det det 6 e V d d d J J u u x u u y u u z A Prof .N. Siva Prasad, Indian Institute of Technology Madras We can write the following relation 14 14 14 24 24 24 34 34 34 x y z x y z x y z x y z x y z x y z J The Jacobian of the transformation is given by 14 1 4 x x x here Three dimensional problems contd..
  • 7. 7 Ve is the volume of the element e T e k V B DB here element stiffness matrix, ke is given by 1 1 2 2 1 1 2 2 T T T e e e T T T e e U dV dV V k D q B DBq q B DBq q q Element Stiffness Prof .N. Siva Prasad, Indian Institute of Technology Madras The element strain energy in the total potential is given by Three dimensional problems contd..
  • 8. 8 Force Terms The potential term associated with body force is det T T T e T e dV d d d 駕駕 u f q N f J q f , , , , , ,......., 4 T e e x y z x y z z V f f f f f f f f Consider uniformly distributed traction on the boundary surface Prof .N. Siva Prasad, Indian Institute of Technology Madras T T T T e A A e e dA dA u T q N T q T Three dimensional problems contd..
  • 9. 9 Dynamic analysis When loads are suddenly applied, The mass and acceleration effects are come in to picture Lagrangian L T T Kinetic energy Potential energy Hamilton Princible For an arbitrary time interval from t1 to t2,the state of motion of a body extremizes functional 2 1 t t I Ldt If L can be expressed in terms of the generalized variables 1 2 3 4 , , , ,..... , n q q q q q where i i q q t Then the equations of motions are given by 0 i i d L L dt q q i=1 to n Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 10. 10 1 1 , x x 2 2 , x x m2 m1 The kinetic energy and potentially energy are given by 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 T m x m x k x k x Using and L T 0 i i d L L dt q q i=1 to n The equations of motions will be 1 1 1 1 2 2 1 1 1 2 2 2 2 1 2 2 ( ) 0 ( ) 0 d L L m x k x k x x dt x x d L L m x k x x dt x x 件 件 In matrix form 1 1 1 2 2 1 2 2 2 2 2 0 ( ) 0 0 m x k k k x m k k x x 件 件 Spring mass system Stiffness matrix Mass matrix Dynamic analysis contd..
  • 11. 11 Systems with Distributed mass x z y u v w dv = density v The kinetic energy The velocity vector of point at x with components In the finite element method,we divide the body into elements,and in each element and 1 2 T v T u u dv , , u v w u Nq u N q The kinetic energy Te 1 2 1 2 T v T v T u u dv T q N Ndv q Mass matrix Prof .N. Siva Prasad, Indian Institute of Technology Madras Dynamic analysis contd..
  • 12. 12 Types of Elements One dimensional elements 1. Beam (axial) 2. Beam (bending) 3. Pipe Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 13. 13 Two dimensional elements 1. Triangular inplane bending 2. quadrilateral inplane bending Prof .N. Siva Prasad, Indian Institute of Technology Madras Types of Elements
  • 14. 14 Three dimensional elements 1. brick 2. Tetrahedral Prof .N. Siva Prasad, Indian Institute of Technology Madras Types of Elements
  • 15. 15 Finite element analysis Guidelines for the selection of FE package: Analysis Capability Adequacy of user oriented features Maintainability Adequacy of user support facility Portability Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 16. 16 Modeling Element type must be consistent Finer mesh near the stress gradient Extremely fine mesh when forces to be applied near the stress concentration areas such as fillets Uniform change in stress between adjacent elements Better aspect ratio Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 17. 17 Modeling (contd) Gross element distortion should be avoided Adjoining elements must share common nodes and common degrees of freedom 450 150 Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 18. 18 Debugging of FE models Geometry Material properties Applied forces Displacement constraints Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 19. 19 Common symptoms and their possible causes Symptoms Causes Excessive deflection, but anticipate stress Excessive deflection and excessive stress Internal discontinuity in stress and deflection Youngs modulus too low, missing nodal constraints Applied force too high, nodal coordinates incorrect, force applied at wrong nodes Force applied at wrong nodes, missing or double internal element Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 20. 20 Symptoms Causes Discontinuity along boundary Higher or lower frequency than anticipated -Static deflections, O.K. -Static deflection not O.K. Internal gap opening up in model under load, stress discontinuity Missing nodal constraint, force applied at wrong node Density incorrect Youngs modulus incorrect Improper nodal coupling Prof .N. Siva Prasad, Indian Institute of Technology Madras Common symptoms and their possible causes
  • 21. 21 Dynamic analysis Modal analysis - Natural frequency and mode shapes Harmonic analysis - Forced response of system to a sinusoidal forcing Transient analysis - Forced response for non-harmonic loads (impact, step or ramp forcing etc.) Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 22. 22 Substructure Rules for substructure A substructure may be generated from individual elements or other substructures Master nodes to be retained to be identified Nodal constraints will be retained in all subsequent uses of the substructure Along a substructure boundary that will be used for connection to the rest of the global model, all nodes must be retained as master nodes Cost effective Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 23. 23 Guidelines for selection of dynamic degree of freedom (DDOF) No. of the DOF must be 2 times highest mode of interest No. of reduced modes will be equal to the number of DDOF so that only bottom half of the calculated modes should be considered accurate DDOF should be placed in areas of large mass and rigidity DDOF should be distributed in such a way as to anticipate mode shapes DDOF be selected at each point of dynamic force application DDOF must be used with gap elements For plate type elements emphasise DDOF in out of plane direction Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 24. 24 Discretization of Structure Concentrated load . node P Abrupt change in load . node Abrupt change in thickness . node Prof .N. Siva Prasad, Indian Institute of Technology Madras
  • 25. 25 Discretization of Structure Abrupt change in material properties Re entrant corner . Finer mesh near stress concentration factor Nodal line node Prof .N. Siva Prasad, Indian Institute of Technology Madras