notes_33_matrix_grill

# notes_33_matrix_grill - Matrix Analysis Grillage intro to...

This preview shows pages 1–4. Sign up to view the full content.

Matrix Analysis, Grillage, intro to Finite Element Modeling suppose we were to analyze this pin-jointed structure what are some of the analysis tools we would use? is this statically determinant? when we write down the model, what equations result single multiple? equilibrium compatibility of displacements laws of material behavior results in set of simultaneous equations in terms of structure forces and displacements form is F = K ⋅δ Px Py it would be nice to develop an organized approach to similar problems: Matrix Analysis of Structures start with pin jointed frame: section 5.2 we want the law of material behavior: in this case a relation between force and displacement we will refer to this as a stiffness matrix and a relationship between an element and the structure it is a part of we will address the compatibility of displacements only on a single element at this stage Y,V node 1 node 2 x,u y,v φ defines an element and a structure coordinate system 1 notes_33_matrix_grillage_fem_intro.mcd X,U

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
fy2 element x,y u,v x2,u2 y1,v1 fy1 y2,v2 fx2 x1,u1 fx1 node 1 node 2 the element stiffness matrix: f = k e ⋅δ u 1 fx 1 note: even though v and fy = 0, will carry δ = v 1 f = fy 1 due to compatibility with structure u 2 fx 2 v 2 fy 2 laws of material behavior (Hooke), for details including relationship of "internal" stress/force see collapsed area f 1 = E L = E u 1 u 2 or . .. AE A L L f 1 = L ( u 1 u 2 ) fx 1 1 0 1 0 u 1 1 0 1 0 0 0 0 0 f = fy 1 = 0 0 0 0 v 1 k e = L 1 0 1 0 fx 2 L 1 0 1 0 u 2 0 0 0 0 fy 2 0 0 0 0 v 2 this is the element stiffness matrix f = k e in the equation the element stress matrix is related to the internal force = -fx1 or = fx2 u 1 f x1 E v 1 σ = A = L ( 10 1 0) u 2 => Se = E ( 1 01 L v 2 2 notes_33_matrix_grillage_fem_intro.mcd
µ φ φ µ now let's connect to the structure coordinate system: structure forces Y,V node 1 node 2 x,u y,v φ Fx 1 Fy 1 fx 1 = Fx 1 cos φ () + Fy 1 sin φ F = + Fy 1 cos φ Fx 2 fy 1 = Fx 1 sin φ Fy 2 X,U same for node 2 N.B. φ is the angle measured CCW from the structure X to the element x coordinate direction if substitute λ = cos φ µ = sin φ fx 1 λ µ 0 0 Fx 1 λ 0 0 f = fy 1 −µ λ 0 0 Fy 1 define T f = T F T := −µ λ 0 0 fx 2 = 0 0 λ µ Fx 2 0 0 λ µ fy 2 0 0 −µ λ Fy 2 0 0 −µ λ λ −µ 0 0 λ −µ 0 0 T T

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

notes_33_matrix_grill - Matrix Analysis Grillage intro to...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online