|
Notation |
Explanation |
|
U = N. d |
- U: column matrix of unknown displacements within discrete elements (translations & rotations)
- d
: column matrix of known displacements at nodes
- N: matrix of shape functions relating known
d to unknown U
- N: derived by Pascal's triangle depending on degree of polynomial
- N: =1 at node I; =0 at others
|
|
e = B.d |
- e
: column matrix of strains
- d
: column matrix of known displacements at nodes
- B: strain-displacement relation; derivative of N w.r.t.
d (x,y,z,xy,yz,zx)
- B: =derivative
|
|
s = D.e |
- Stress-strain relation
- s
: column stress matrix
- e
: column strain matrix
- D: stress-strain relation matrix
- D: depends on material - linearly elastic; non-linearly elastic; inelastic
- D: derivation is assumed elastic or discretised into linear regions for non-elastic
- Plane strain:
e z=0
- Plane stress:
s z=0
- Axisymmetric:
s q =0
|
|
F = A. s
= K. d |
- F: force excitation column matrix
- A: area matrix
- s
: column stress matrix
- d
: column matrix of known displacements at nodes
- K: stiffness matrix, load-displacement relation
- K: 2 types - global stiffness matrix for all nodes of structure; local or element stiffness matrix for element nodes only
- K: thus transformation required
- K =
ò BT.D.B dV (or dA, dl)
|
|
Solution |
- Derivation of K matrices
- Apply excitation loads
- Apply B.C.
- Using F=K.
d , solve for d (degrees of freedom at nodes)
- Backward solution process:
- Using
e = B.d , solve for e
- Using
s = D.e , solve for s
- For both dynamic & static conditions, & total effect
|
|
Solution techniques |
- Since all variables are conveniently formulated into matrices, use matrix operations
- Inverse matrix: direct solution, but high computation needed
- Cramer's rule: limitations
- Gaussian elimination: frequently used, less computation
- Numerical techniques: finite difference method, Jacobi, Gauss-seidel, Runge-Kutta, etc.
|