User:Eml4500.f08.ateam.shah/Week 2

Lecture 3

Continuation from Lecture 2: Global Force Displacement relationship

The FD Relation:


\begin{bmatrix} k_{11} & k_{12} & k_{13} & k_{14} & k_{15} & k_{16}\\ k_{21} & k_{22} & k_{23} & k_{24} & k_{25} & k_{26}\\ k_{31} & k_{32} & k_{33} & k_{34} & k_{35} & k_{36}\\ k_{41} & k_{42} & k_{43} & k_{44} & k_{45} & k_{46}\\ k_{51} & k_{52} & k_{53} & k_{54} & k_{55} & k_{56}\\ k_{61} & k_{62} & k_{63} & k_{64} & k_{65} & k_{66}\\ \end{bmatrix}

\begin{bmatrix} d_{1}\\ d_{2}\\ d_{3}\\ d_{4}\\ d_{5}\\ d_{6}\\ \end{bmatrix} =

\begin{bmatrix} F_{1}\\ F_{2}\\ F_{3}\\ F_{4}\\ F_{5}\\ F_{6}\\ \end{bmatrix}

$$

In compact notation the above listed FD relationship equation simplifies to:

$$[ K_{ij} ] \cdot { d_{j} } = { F_{i} }$$

The K-spring factor is a 6x6 matrix, where as the d and F matrix are composed of a 6x1 matrix.


 * $$\sum_{j=1}^6 F_{i} = K_{ij} \cdot d_j $$


 * K = global stiffness matrix.
 * d = global displacement matrix.
 * F = global force matrix.
 * k = elemental stiffness matrix.
 * f = elemental force matrix.

How to go from an elemental matrix to a global matrix? The matrix has to go through an assembly process to go form an elemental matrix to a global matrix.

Identify the correspondence between elemental displacement dof's and global displacement dof's.

Global level:

{d1, d2, ............, d6}

Elemental level:

* Element 1:{d(1)1, d(1)2, d(1)3, d(1)4} * Element 2:{d(2)1, d(2)2, d(2)3, d(2)4}

Identification global-local dof:



Node 2 d2 = d2(1) d3 = d3(1) = d1(2) d4 = d4(1) = d2(3) Node 3 d5 = d3(2) d6 = d4(2)

Conceptual step of assembly: (topology of K)



The colored portion in the figure above represents the overlap between the two elemental matrices of K.

Matlab HW


% Two bar truss example clear all; e = [3 5]; A = [1 2]; P = 7; L=[4 2]; alpha = pi/3; beta = pi/4;

nodes = [0, 0; L(1)*cos(pi/2-alpha), L(1)*sin(pi/2-alpha); L(1)*cos(pi/2-alpha)+L(2)*sin(beta),L(1)*sin(pi/2-alpha)-L(2)*cos(beta)];

dof=2*length(nodes);

conn=[1,2; 2,3]; lmm = [1, 2, 3, 4; 3, 4, 5, 6]; elems=size(lmm,1); K=zeros(dof); R = zeros(dof,1); debc = [1, 2, 5, 6]; ebcVals = zeros(length(debc),1);

%load vector R = zeros(dof,1); R(4) = P;

% Assemble global stiffness matrix K=zeros(dof); for i=1:elems lm=lmm(i,:); con=conn(i,:); k_local=e(i)*A(i)/L(i)*[1 -1; -1 1] k=PlaneTrussElement(e(i), A(i), nodes(con,:)) K(lm, lm) = K(lm, lm) + k; end K R % Nodal solution and reactions [d, reactions] = NodalSoln(K, R, debc, ebcVals) results=[]; for i=1:elems results = [results; PlaneTrussResults(e, A, ... nodes(conn(i,:),:), d(lmm(i,:)))]; end format short g results

k_local =

0.75       -0.75        -0.75         0.75

k =

0.5625     0.32476      -0.5625     -0.32476      0.32476       0.1875     -0.32476      -0.1875      -0.5625     -0.32476       0.5625      0.32476     -0.32476      -0.1875      0.32476       0.1875

k_local =

5   -5    -5     5

k =

2.5        -2.5         -2.5          2.5         -2.5          2.5          2.5         -2.5         -2.5          2.5          2.5         -2.5          2.5         -2.5         -2.5          2.5

K =

0.5625     0.32476      -0.5625     -0.32476            0            0      0.32476       0.1875     -0.32476      -0.1875            0            0      -0.5625     -0.32476       3.0625      -2.1752         -2.5          2.5     -0.32476      -0.1875      -2.1752       2.6875          2.5         -2.5            0            0         -2.5          2.5          2.5         -2.5            0            0          2.5         -2.5         -2.5          2.5

R =

0    0     0     7     0     0

d =

0           0        4.352       6.1271            0            0

reactions =

-4.4378     -2.5622       4.4378      -4.4378

results =

1.7081      5.1244       8.5406       5.1244       17.081       0.6276       1.8828        3.138       1.8828        6.276