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r=K*a

r = nodal force vector

K = stiffness matrix

a = nodal displacement

Implicit Solver

r(i+1) = K(i+1) * a(i+1)

-uses iterative method and needs convergence

-enforces equilibium of internal and external forces

-static or low frequency dynamic analysis

-unconditionally stable, not sensitive to small elements

-more computational expensive

Explicit Solver

r(i+1) = K(i) * a(i+1)

-doesnt enforce equilibium of internal and external forces

-especially dynamic analysis

-conditionally stable, sensitive to small elements

-less computational expensive

Discretization

-domain divided into elements

-connected at nodes

-no empty space

-no overlaps

-quality depends on size and type of elements

Order of interpolation

-interpolation used between mesh nodes

-linear, quadratic, cubic

-higher order needs more nodes

-higher order is more computational expensive

Linear FEM

-linear relation between forces and displacement

-only in elastic range

-K is constant

Nonlinear FEM

-nonlinear relation between forces and displacement

-also large deforrmations

-K is not constant

Reasons for Nonlinearity

-Geometric Nonlinearity (large deformations)

-Material Nonlinearity (material doesnt follow Hooke's law)

-Contact Nonlinearity

FEM (finite element method)

-numerical technique for finding approximate solutions

-divides large problem into smaller parts

-Preprocessor

-Solver

-Postprocessor

Prepocessor

-Definition of geometry

-Discretization

-Definition of material

-Application of boundary conditions

-Setup of simulation

Postprocessor

-view results

-create animations

-plot quantities of time/elements

-create result tables

K (stiffness matrix)

-information about material and geometry

-(#unknowns) = (#DOF per node) * (#nodes)

-global K consists of element Ks of all elements

BEAM188

-standard element for line bodies

-1D

-based on Timeshenko beam theory

-DOF: UX, UY, UZ, ROTX, ROTY, ROTZ

-KEYPOT(3): 0 = linear

1 = quadratic

2 = cubic

When do you use line bodies and why?

  • beams, trusses, cables, ...
  • length is much greater than the cross-sectional dimensions
  • more simple and efficient than 2D or 3D elements


Timoshenko beam theory

-extends Euler-Bernoulli beam theory

-accounts for shear deformation and rotational inertia

-especially important for short, thick beams