Download FEA 3 - Boundary Conditions

Theory of Elasticity
• Theory of elasticity governs response
– Symmetric stress & strain components
• Governing equations
– Equilibrium equations (3)
– Strain-displacement equations (6)
– Constitutive equations (6)
• Unknowns
– Stress (6)
– Strain (6)
– Displacement (3)
Boundary & Initial Conditions
• Linear elastic material
– Three partial differential equations in
displacements
– Second order in each coordinate
– Second order in time
• On a free surface in each direction
– Specify stress or displacement but not both
• Initial conditions for each direction specify
– Displacement and velocity
Surface Forces
• Specify pressure (szz), shears (szx, szy) or
• Specify displacements (u,v,w)
Rigid Body Motion – 2D
Rigid Body Displacements – 2D
• Strains vanish
 xx 
u
v
u v
 0,  yy   0,  xy 
  0,
x
y
y x
• Integrating normal strains
u  f  y , v  g  x 
• Integrating shear strain
f  y   g  x   0 or g  x   0 & g  x     constant
• Hence (U, V are constants)
f  y   , & f  y   U  y, g  x   V  x
• Displacement solution
u  x, y   U  y, v x, y   V  x
Reactions with Excessive Constraints
Extra constraint in horizontal direction will add excessive stress
Vertical constraints and loads produce point load infinite stresses
Rigid Body Displacements – 3D
• All strains vanish
 xx   yy   zz 0,  xy   yz   zx  0
• In terms of displacements

 

u  ui  vj  wk
• Integrating yields displacements
   
u U   r
• Where


 
U  Ui  Vj  Wk  constant vector
• And




  xi   y j  z k  constant vector
Self-Equilibrating Forces
• Examples include:
– Uniform pressure (submarine or bathysphere)
– Thermal expansion
• BCs remove rigid body translations & rotations
– Constrain six degrees of
freedom (3 dofs at one
point, 2 dofs at a second
and 1 dof at a third)
Plate & Beam Dofs at Each Node
• Beam – 3 translations
3 rotations
• Plate – 3 translations
2 rotations
Nastran FE Code – Plate Elements
• All nodes for all elements types have six dofs
– 3 for translation
– 3 for rotation
• Flat plate models need dofs perpendicular to
plane of model constrained (set to zero)
• Shells made of plate elements do not
• Solid elements need all three rotations at each
node set to zero
Simply Supported Beam Example
Fix six dofs – 5 translation and 1 rotation
Simply Supported Beam Example
Fix six dofs – 5 translation and 1 rotation
Cantilever Beam
Fix six dofs – 3 translation and 3 rotation
Internal Surfaces & Cracks
• Cracks
• Internal Surfaces
Hertz Contact - Gaps & Friction
• Hertz Contact
• Gap & Friction Elements
Transformations
X '  X cos  Y sin 
Y '   X sin   Y cos
Z ' Z
Use of Symmetry
•
•
•
•
Makes a large problem smaller
Axisymmetry reduces a 3D problem to 2D
Recall stress & strain symmetric
Examples:
Periodic Boundary Conditions
• Stress & Strain are periodic
• Mean displacements can vary linearly with
coordinates due to expansion and rotation
• For
u1  C1  u0
v1  v0
Multi-Point Constraints - Tying
CX  D
 c11 c12
c
c22
21

 ... ...

cn1 cn 2
or
... c1m   x1   d1 





... c2 m  x2   d 2 
    
... ...   ...   ... 
   
... cnm   xm  d m 
where xi are specified degrees of freedom,
cij dj are known constants.
Distant Boundary Conditions
• Build a sufficiently large model
– At least 20 times length of largest dimension of
interest
• Substructure a large coarse model
– Use output from large model as input to a refined
local model
– Use super-elements or substructuring
– Use infinite elements (when available)