LECCION 4 - EJERCICIO 5 (15.5) v.2005 - ELEMENTO TRIANGULAR LINEAL - TENSION PLANA - DEFINICION MATRIZ RIGIDEZ

INICIO

Off[General::"spell1"]
Off[General::"spell"]

1 - DEFINICION MATRIZ RIGIDEZ

StiffnessOf3NodePlaneStressTriangle[{{x1_,y1_},{x2_,y2_},{x3_,y3_}},Emat_,{h_}]:=Module[{x21,x13,x32,y12,y31,y23,A,Be,Ke},A=Simplify[((x2*y3-x3*y2)+(x3*y1-x1*y3)+(x1*y2-x2*y1))/2];
{x21,x13,x32,y12,y31,y23}={x2-x1,x1-x3,x3-x2,y1-y2,y3-y1,y2-y3};
Be={{y23,0,y31,0,y12,0},{0,x32,0,x13,0,x21},{x32,y23,x13,y31,x21,y12}}/(2*A);
Ke=(A*h)*Transpose[Be].Emat.Be;
Return[Ke];];

DEFINICION DEL AREA A - EC. (15.3)

DaT=;

Show[DaT, ImageSize->500]

A[x1_,y1_,x2_,y2_,x3_,X]=Simplify[((x2*y3-x3*y2)+(x3*y1-x1*y3)+(x1*y2-x2*y1))/2]

DEFINICION DIFERENCIA COORDENADAS NODOS - EC. (15.11)

DdcN=;

Show[DdcN, ImageSize->450]

{x21,x13,x32,y12,y31,y23}={x2-x1,x1-x3,x3-x2,y1-y2,X,y2-y3}

x21

DEFINICION MATRIX Be - EC. (15.18)

DmbE=;

Show[DmbE, ImageSize->450]

B[x1_,y1_,x2_,y2_,x3_,y3_]={{y23,0,y31,0,y12,0},{0,x32,0,x13,0,x21},{x32,y23,x13,y31,x21,y12}}

%//MatrixForm

Be[x1_,y1_,x2_,y2_,x3_,y3_]=B[x1,y1,x2,y2,x3,y3]/(2*A[x1,y1,x2,y2,x3,y3])

DEFINICION MATRIX E - EC. (15.19)

DmE=;

Show[DmE, ImageSize->450]

Emat={{E11,E12,E13},{E12,E22,E23},{X}};

DEFINICION MATRIX DE RIGIDEZ DEL ELEMENTO TRIANGULAR LINEAL

Ke[x1_,y1_,x2_,y2_,x3_,y3_,E11_,E12_,E13_,E22_,E23_,E33_,h_]=(A[x1,y1,x2,y2,x3,y3]*h)*Transpose[Be[x1,y1,x2,y2,x3,y3]].Emat.Be[x1,y1,x2,y2,x3,y3];

EJEMPLO UTILIZACION

Ke[x1,y1,x2,y2,x3,y3,E11,E12,E13,E22,E23,E33,h]/.{x1->0,y1->0,x2->3,y2->1,x3->2,y3->2,E11->100,E12->25,E13->0,E22->100,E23->0,E33->50,h->1}//MatrixForm

Ke[x1,y1,x2,y2,x3,y3,E11,E12,E13,E22,E23,E33,h]/.{x1->0,y1->0,x2->3,y2->1,x3->2,y3->2,E11->100,E12->25,E13->0,E21->25,E22->100,E23->0,E31->0,E32->0,E33->50,h->1.0}//MatrixForm

Show[Graphics[Line[{{0,0},{3,1},{2,2},{0,0}}]],Axes->True]