User:Eml4500c.f08.gravy.mmm/Lecture 1

Notes

Online sources for free college resources.

Wikipeida, wikversity, MIT. Open course ware.

Labeling system
1.      Number in a triangle is the global label for an element

2.      Number in a circle is the global label for a node

3.      Number in a square is the local label for a node









Next big step: Force-Displacement relation (FD)





$$\begin{Bmatrix} f_1 \\ f_2 \end{Bmatrix}_{2X1}=\begin{bmatrix} k  &- k  \\-k &k    \end{bmatrix}_{2X2}*    \begin{Bmatrix} d_1 \\ d_2 \end{Bmatrix}_{2X1}$$

Case 1; observer sits on node 1

$$\ f_2=k(d_2-d_1)\,$$

Case 2: observer sits on node 2

$$\ f_1=-f_2=-k(d_2-d_1)=k(d_1-d_2) \,$$

Ch1. : Big Picture
1.1 Discretization

1.1.1         Plane Truss element

1.2      Assembly of element

Ex. 1.4 Fire-bar truss

1.4  Element solution of model validity

1.4.1         Plane Truss element

Steps to solve simple truss systems

1.      Global picture (description) At structures level

·        Global degrees of freedoms (dofs) and displacement degrees of freedoms (disp. dofs)

·        Global forces

Actually the disp dofs are partitioned into:

·        A known part, e.g fixed dofs, constrains

·        An unknown part: Solved using FEM

Similarly for the global forces:

·        A known part: Applied forces

·        An unknown part : Reactions

2.      Element Picture:

·        Element  dofs

·        Element forces

·        Either  of the above in global coordinates systems or in local coordinates systems

3.      Global FD relation

·        Element stiffness matrices in global coordinates

·        Element force matrices in global coordinates

·        Assembly of element stiffness matrix and element force matrix into global FD relation

$$\ K_{nXn} * d_{nX1}=F _{nX1} \,$$

4.      Elimination of known dofs. To reduce the global FD relation (stiffness matrix non-singular) invertible

$$\ \overline{ K_{mXm}} * \overline{ d_{mX1} }= \overline{ F _{mX1}} \,$$

M= number of unknown disp dofs

N= number of both known and unknown disp dofs

k non singular so k-1 exists (k invertible)

$$\ \overline{ d_{mX1} }= \overline{ K^{-1}_{mXm}} * \overline{ F _{mX1}} \,$$

5.      Compute element forces from  now known d= element stresses

6.      Compute reactions. (unknown forces)

Inclination angle: